Subgroups — C₃₁×#34300 #21491059

Order 7936.

2128 conjugacy classes of subgroups.

The subgroup lattice has 968 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#552	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2374	C₂	1	2	yes	maximal
3	C₃₁×#2373	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2371	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2370	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#2362	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#2362	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#2355	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#2354	C₂	1	1	yes	maximal, characteristic
10	C₃₁×#2346	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#2346	C₂	1	2	yes	maximal, complemented
12	C₃₁×#2345	C₂	1	1	yes	maximal, characteristic, complemented
13	C₃₁×#2335	C₂	1	2	yes	maximal
14	C₃₁×#2334	C₂	1	2	yes	maximal, complemented
15	C₃₁×#2330	C₂	1	1	yes	maximal, characteristic, complemented
16	C₃₁×#2323	C₂	1	1	yes	maximal, characteristic, complemented
17	C₄×C₃₁×#125	C₂	1	2	yes	maximal, complemented
18	C₄×C₃₁×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
19	C₃₁×#2298	C₂	1	2	yes	maximal, complemented
20	C₃₁×#2290	C₂	1	1	yes	maximal, characteristic
21	C₃₁×#2276	C₂	1	1	yes	maximal, characteristic, complemented
22	C₃₁×#2274	C₂	1	1	yes	maximal, characteristic, complemented
23	C₄×C₃₁×#117	C₂	1	1	yes	maximal, characteristic, complemented
24	C₃₁×#552	C₄	1	16	yes	complemented
25	C₂×C₃₁×#126	C₂²	1	2	yes	—
26	C₂×C₃₁×#125	C₂²	1	1	yes	characteristic
27	C₂×C₃₁×#125	C₂²	1	2	yes	—
28	C₂×C₃₁×#123	C₂²	1	2	yes	—
29	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
30	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
31	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
32	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
33	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
34	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
35	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
36	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
37	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
38	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
39	C₂×C₃₁×#117	C₂²	1	2	yes	—
40	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
41	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
42	C₃₁×#391	C₂²	1	1	yes	characteristic
43	C₃₁×#391	C₂²	1	1	yes	characteristic
44	C₃₁×#390	C₂²	1	2	yes	—
45	C₃₁×#389	C₂²	1	2	yes	—
46	C₃₁×#389	C₂²	1	1	yes	characteristic
47	C₃₁×#389	C₂²	1	2	yes	—
48	C₃₁×#389	C₂²	1	1	yes	characteristic
49	C₃₁×#389	C₂²	1	1	yes	characteristic
50	C₃₁×#389	C₂²	1	2	yes	—
51	C₃₁×#389	C₂²	1	2	yes	—
52	C₃₁×#389	C₂²	1	1	yes	characteristic
53	C₃₁×#388	C₂²	1	2	yes	—
54	C₃₁×#388	C₂²	1	2	yes	—
55	C₃₁×#388	C₂²	1	2	yes	—
56	C₃₁×#388	C₂²	1	2	yes	—
57	C₃₁×#388	C₂²	1	2	yes	—
58	C₃₁×#388	C₂²	1	2	yes	—
59	C₃₁×#388	C₂²	1	2	yes	—
60	C₃₁×#388	C₂²	1	2	yes	—
61	C₃₁×#388	C₂²	1	1	yes	characteristic
62	C₃₁×#388	C₂²	1	2	yes	—
63	C₃₁×#388	C₂²	1	2	yes	—
64	C₃₁×#388	C₂²	1	1	yes	characteristic
65	C₃₁×#387	C₂²	1	2	yes	—
66	C₃₁×#387	C₂²	1	2	yes	—
67	C₃₁×#387	C₂²	1	2	yes	—
68	C₃₁×#387	C₂²	1	2	yes	—
69	C₃₁×#387	C₂²	1	2	yes	—
70	C₃₁×#387	C₂²	1	2	yes	—
71	C₃₁×#387	C₂²	1	2	yes	—
72	C₃₁×#387	C₂²	1	2	yes	—
73	C₃₁×#387	C₂²	1	2	yes	—
74	C₃₁×#387	C₂²	1	2	yes	—
75	C₃₁×#387	C₂²	1	2	yes	—
76	C₃₁×#387	C₂²	1	1	yes	characteristic
77	C₃₁×#387	C₂²	1	1	yes	characteristic
78	C₃₁×#387	C₂²	1	2	yes	—
79	C₃₁×#386	C₂²	1	1	yes	characteristic
80	C₃₁×#386	C₂²	1	1	yes	characteristic
81	C₃₁×#385	C₂²	1	1	yes	characteristic
82	C₃₁×#385	C₂²	1	2	yes	—
83	C₃₁×#385	C₂²	1	1	yes	characteristic
84	C₃₁×#384	C₂²	1	1	yes	characteristic
85	C₃₁×#382	C₂²	1	2	yes	—
86	C₃₁×#382	C₂²	1	2	yes	—
87	C₃₁×#380	C₂²	1	1	yes	characteristic
88	C₃₁×#380	C₂²	1	2	yes	—
89	C₃₁×#380	C₂²	1	1	yes	characteristic
90	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
91	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
92	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
93	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
94	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
95	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
96	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
97	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
98	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
99	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
100	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
101	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
102	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
103	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
104	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
105	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
106	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
107	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
108	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
109	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
110	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
111	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
112	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
113	C₄×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
114	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
115	C₄×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
116	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
117	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
118	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
119	C₃₁×#376	C₂²	1	1	yes	characteristic
120	C₃₁×#376	C₂²	1	2	yes	—
121	C₃₁×#376	C₂²	1	2	yes	—
122	C₃₁×#376	C₂²	1	2	yes	—
123	C₄³×C₃₁	C₂²	1	2	yes	—
124	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
125	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
126	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
127	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
128	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
129	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
130	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
131	C₃₁×#126	C₂×C₄	1	16	yes	—
132	C₃₁×#125	C₂×C₄	1	16	yes	complemented
133	C₃₁×#125	C₂×C₄	1	8	yes	complemented
134	C₃₁×#123	C₂×C₄	1	16	yes	—
135	C₃₁×C₂²⋊Q₈	C₂×C₄	1	8	yes	—
136	C₃₁×C₂²⋊Q₈	C₂×C₄	1	8	yes	complemented
137	C₄×Q₈×C₃₁	—	2	4	no	—
138	C₄×Q₈×C₃₁	—	2	4	no	—
139	C₄×Q₈×C₃₁	C₂×C₄	1	8	yes	complemented
140	C₄×D₄×C₃₁	C₂×C₄	1	8	yes	—
141	C₃₁×#117	—	2	4	no	—
142	C₃₁×#117	—	2	4	no	—
143	C₃₁×#117	—	2	4	no	—
144	C₃₁×#117	C₂×C₄	1	8	yes	—
145	C₃₁×#117	C₂×C₄	1	16	yes	complemented
146	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	8	yes	complemented
147	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
148	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
149	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
150	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
151	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
152	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
153	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
154	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
155	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
156	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
157	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
158	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
159	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
160	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
161	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
162	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
163	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
164	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
165	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
166	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
167	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
168	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
169	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
170	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
171	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
172	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
173	C₂×C₂²⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
174	C₂×C₂²⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
175	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
176	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
177	C₂×C₂²⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
178	C₂×C₂²⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
179	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
180	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
181	C₂×C₂²⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
182	C₂×C₂²⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
183	C₂×C₂²⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
184	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
185	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
186	C₂×C₂²⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
187	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
188	C₂×C₄²×C₃₁	—	2	4	no	—
189	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
190	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
191	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
192	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
193	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
194	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
195	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
196	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
197	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
198	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
199	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
200	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
201	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
202	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
203	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
204	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
205	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
206	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
207	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
208	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
209	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
210	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
211	C₃₁×#95	C₂³	1	1	yes	characteristic
212	C₃₁×#95	C₂³	1	2	yes	—
213	C₃₁×#95	C₂³	1	1	yes	characteristic
214	C₃₁×#95	C₂³	1	2	yes	—
215	C₃₁×#95	C₂³	1	2	yes	—
216	C₃₁×#95	C₂³	1	2	yes	—
217	C₃₁×#95	C₂³	1	2	yes	—
218	C₃₁×#95	C₂³	1	2	yes	—
219	C₃₁×#95	C₂³	1	1	yes	characteristic
220	C₃₁×#95	C₂³	1	1	yes	characteristic
221	C₃₁×#95	C₂³	1	1	yes	characteristic
222	C₃₁×#95	C₂³	1	2	yes	—
223	C₃₁×#95	C₂³	1	2	yes	—
224	C₃₁×#95	C₂³	1	2	yes	—
225	C₃₁×#95	C₂³	1	2	yes	—
226	C₃₁×#95	C₂³	1	1	yes	characteristic
227	C₃₁×#95	C₂³	1	2	yes	—
228	C₃₁×#95	C₂³	1	1	yes	characteristic
229	C₃₁×#95	C₂³	1	2	yes	—
230	C₃₁×#95	C₂³	1	2	yes	—
231	C₃₁×#95	C₂³	1	2	yes	—
232	C₃₁×#95	C₂³	1	1	yes	characteristic
233	C₃₁×#95	C₂³	1	2	yes	—
234	C₃₁×#95	C₂³	1	1	yes	characteristic
235	C₃₁×#95	C₂³	1	2	yes	—
236	C₃₁×#95	C₂³	1	2	yes	—
237	C₃₁×#95	C₂³	1	2	yes	—
238	C₃₁×#95	C₂³	1	1	yes	characteristic
239	C₃₁×#95	C₂³	1	2	yes	—
240	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
241	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
242	C₂×Q₈×C₃₁	—	2	2	no	—
243	C₂×Q₈×C₃₁	—	2	2	no	—
244	C₂×Q₈×C₃₁	C₂²×C₄	1	4	yes	—
245	C₂×D₄×C₃₁	—	4	1	no	—
246	C₂×D₄×C₃₁	—	4	1	no	—
247	C₂×D₄×C₃₁	C₂²×C₄	1	4	yes	—
248	C₂²×C₄×C₃₁	—	2	2	no	—
249	C₂²×C₄×C₃₁	—	2	2	no	—
250	C₂²×C₄×C₃₁	—	2	2	no	—
251	C₂²×C₄×C₃₁	—	2	2	no	—
252	C₂²×C₄×C₃₁	—	2	2	no	—
253	C₂²×C₄×C₃₁	—	2	2	no	—
254	C₂²×C₄×C₃₁	—	2	2	no	—
255	C₂²×C₄×C₃₁	—	4	1	no	—
256	C₂²×C₄×C₃₁	—	4	2	no	—
257	C₂²×C₄×C₃₁	—	4	2	no	—
258	C₂²×C₄×C₃₁	—	4	1	no	—
259	C₂²×C₄×C₃₁	—	4	1	no	—
260	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
261	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
262	C₂²×C₄×C₃₁	C₂²×C₄	1	8	yes	—
263	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
264	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
265	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
266	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
267	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
268	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
269	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
270	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
271	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
272	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
273	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
274	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
275	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
276	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
277	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
278	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
279	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
280	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
281	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
282	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
283	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
284	C₄⋊C₄×C₃₁	—	2	2	no	—
285	C₄⋊C₄×C₃₁	—	2	2	no	—
286	C₄⋊C₄×C₃₁	—	2	2	no	—
287	C₄⋊C₄×C₃₁	—	2	2	no	—
288	C₄⋊C₄×C₃₁	—	2	4	no	—
289	C₄⋊C₄×C₃₁	—	2	4	no	—
290	C₄⋊C₄×C₃₁	—	2	2	no	—
291	C₄⋊C₄×C₃₁	—	2	2	no	—
292	C₄⋊C₄×C₃₁	—	4	2	no	—
293	C₄⋊C₄×C₃₁	—	4	2	no	—
294	C₄⋊C₄×C₃₁	—	4	2	no	—
295	C₄⋊C₄×C₃₁	—	4	2	no	—
296	C₄⋊C₄×C₃₁	—	4	2	no	—
297	C₄⋊C₄×C₃₁	—	4	1	no	—
298	C₄⋊C₄×C₃₁	—	4	2	no	—
299	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
300	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
301	C₄⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
302	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
303	C₄⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
304	C₄⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
305	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
306	C₄⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
307	C₄⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
308	C₄⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
309	C₄⋊C₄×C₃₁	#41	1	4	yes	—
310	C₂²⋊C₄×C₃₁	—	2	2	no	—
311	C₂²⋊C₄×C₃₁	—	2	4	no	—
312	C₂²⋊C₄×C₃₁	—	2	4	no	—
313	C₂²⋊C₄×C₃₁	—	2	2	no	—
314	C₂²⋊C₄×C₃₁	—	2	2	no	—
315	C₂²⋊C₄×C₃₁	—	4	1	no	—
316	C₂²⋊C₄×C₃₁	—	4	1	no	—
317	C₂²⋊C₄×C₃₁	—	4	2	no	—
318	C₂²⋊C₄×C₃₁	—	4	2	no	—
319	C₂²⋊C₄×C₃₁	—	4	1	no	—
320	C₂²⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
321	C₂²⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
322	C₂²⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
323	C₂²⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
324	C₂²⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
325	C₄²×C₃₁	—	2	2	no	—
326	C₄²×C₃₁	—	2	2	no	—
327	C₄²×C₃₁	—	2	2	no	—
328	C₄²×C₃₁	—	2	4	no	—
329	C₄²×C₃₁	—	2	4	no	—
330	C₄²×C₃₁	—	2	2	no	—
331	C₄²×C₃₁	—	2	4	no	—
332	C₄²×C₃₁	—	2	4	no	—
333	C₄²×C₃₁	—	2	2	no	—
334	C₄²×C₃₁	—	2	4	no	—
335	C₄²×C₃₁	—	4	2	no	—
336	C₄²×C₃₁	—	4	2	no	—
337	C₄²×C₃₁	—	4	2	no	—
338	C₄²×C₃₁	—	4	2	no	—
339	C₄²×C₃₁	—	4	2	no	—
340	C₄²×C₃₁	—	4	2	no	—
341	C₄²×C₃₁	—	4	2	no	—
342	C₄²×C₃₁	—	4	2	no	—
343	C₄²×C₃₁	C₂²×C₄	1	8	yes	—
344	C₄²×C₃₁	C₂²×C₄	1	8	yes	—
345	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
346	C₄²×C₃₁	#41	1	4	yes	—
347	#34300	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
348	C₂³×C₃₁	—	2	1	no	—
349	C₂³×C₃₁	—	2	1	no	—
350	C₂³×C₃₁	—	2	1	no	—
351	C₂³×C₃₁	—	2	1	no	—
352	C₂³×C₃₁	—	2	1	no	—
353	C₂³×C₃₁	—	2	1	no	—
354	C₂³×C₃₁	—	2	1	no	—
355	C₂³×C₃₁	—	2	1	no	—
356	C₂³×C₃₁	—	2	1	no	—
357	C₂³×C₃₁	—	2	1	no	—
358	C₂³×C₃₁	—	2	1	no	—
359	C₂³×C₃₁	—	2	1	no	—
360	C₂³×C₃₁	C₂³×C₄	1	2	yes	—
361	C₂³×C₃₁	C₂³×C₄	1	2	yes	—
362	C₂³×C₃₁	C₂⁵	1	1	yes	characteristic
363	Q₈×C₃₁	—	2	8	no	—
364	D₄×C₃₁	—	4	4	no	—
365	C₂×C₄×C₃₁	—	2	1	no	—
366	C₂×C₄×C₃₁	—	2	1	no	—
367	C₂×C₄×C₃₁	—	2	1	no	—
368	C₂×C₄×C₃₁	—	2	1	no	—
369	C₂×C₄×C₃₁	—	2	2	no	—
370	C₂×C₄×C₃₁	—	2	2	no	—
371	C₂×C₄×C₃₁	—	2	1	no	—
372	C₂×C₄×C₃₁	—	2	2	no	—
373	C₂×C₄×C₃₁	—	2	2	no	—
374	C₂×C₄×C₃₁	—	2	1	no	—
375	C₂×C₄×C₃₁	—	2	1	no	—
376	C₂×C₄×C₃₁	—	2	2	no	—
377	C₂×C₄×C₃₁	—	2	2	no	—
378	C₂×C₄×C₃₁	—	2	2	no	—
379	C₂×C₄×C₃₁	—	2	1	no	—
380	C₂×C₄×C₃₁	—	2	1	no	—
381	C₂×C₄×C₃₁	—	2	1	no	—
382	C₂×C₄×C₃₁	—	2	1	no	—
383	C₂×C₄×C₃₁	—	2	1	no	—
384	C₂×C₄×C₃₁	—	2	2	no	—
385	C₂×C₄×C₃₁	—	2	2	no	—
386	C₂×C₄×C₃₁	—	2	2	no	—
387	C₂×C₄×C₃₁	—	2	1	no	—
388	C₂×C₄×C₃₁	—	2	1	no	—
389	C₂×C₄×C₃₁	—	2	8	no	—
390	C₂×C₄×C₃₁	—	2	8	no	—
391	C₂×C₄×C₃₁	—	2	1	no	—
392	C₂×C₄×C₃₁	—	2	1	no	—
393	C₂×C₄×C₃₁	—	2	1	no	—
394	C₂×C₄×C₃₁	—	2	1	no	—
395	C₂×C₄×C₃₁	—	2	2	no	—
396	C₂×C₄×C₃₁	—	2	2	no	—
397	C₂×C₄×C₃₁	—	2	2	no	—
398	C₂×C₄×C₃₁	—	2	1	no	—
399	C₂×C₄×C₃₁	—	2	1	no	—
400	C₂×C₄×C₃₁	—	2	1	no	—
401	C₂×C₄×C₃₁	—	2	1	no	—
402	C₂×C₄×C₃₁	—	2	1	no	—
403	C₂×C₄×C₃₁	—	2	1	no	—
404	C₂×C₄×C₃₁	—	2	2	no	—
405	C₂×C₄×C₃₁	—	2	2	no	—
406	C₂×C₄×C₃₁	—	2	2	no	—
407	C₂×C₄×C₃₁	—	2	2	no	—
408	C₂×C₄×C₃₁	—	2	2	no	—
409	C₂×C₄×C₃₁	—	2	1	no	—
410	C₂×C₄×C₃₁	—	2	1	no	—
411	C₂×C₄×C₃₁	—	2	2	no	—
412	C₂×C₄×C₃₁	—	2	1	no	—
413	C₂×C₄×C₃₁	—	2	1	no	—
414	C₂×C₄×C₃₁	—	2	1	no	—
415	C₂×C₄×C₃₁	—	4	8	no	—
416	C₂×C₄×C₃₁	—	4	4	no	—
417	C₂×C₄×C₃₁	—	4	4	no	—
418	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
419	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
420	C₂×C₄×C₃₁	C₂³×C₄	1	4	yes	—
421	C₂×C₄×C₃₁	C₂³×C₄	1	4	yes	—
422	C₂×C₄×C₃₁	C₂³×C₄	1	4	yes	—
423	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
424	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
425	C₂×C₄×C₃₁	C₂³×C₄	1	4	yes	—
426	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
427	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
428	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
429	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
430	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
431	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
432	C₂×C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
433	C₂×#552	C₂×C₃₁	1	1	yes	characteristic
434	#2374	C₂×C₃₁	1	2	yes	—
435	#2373	C₂×C₃₁	1	2	yes	complemented
436	#2371	C₂×C₃₁	1	2	yes	complemented
437	#2370	C₂×C₃₁	1	1	yes	characteristic, complemented
438	#2362	C₂×C₃₁	1	1	yes	characteristic, complemented
439	#2362	C₂×C₃₁	1	1	yes	characteristic, complemented
440	#2355	C₂×C₃₁	1	1	yes	characteristic, complemented
441	#2354	C₂×C₃₁	1	1	yes	characteristic
442	#2346	C₂×C₃₁	1	1	yes	characteristic, complemented
443	#2346	C₂×C₃₁	1	2	yes	complemented
444	#2345	C₂×C₃₁	1	1	yes	characteristic, complemented
445	#2335	C₂×C₃₁	1	2	yes	—
446	#2334	C₂×C₃₁	1	2	yes	complemented
447	#2330	C₂×C₃₁	1	1	yes	characteristic, complemented
448	#2323	C₂×C₃₁	1	1	yes	characteristic, complemented
449	C₄×#125	C₂×C₃₁	1	2	yes	complemented
450	C₄×C₂²⋊Q₈	C₂×C₃₁	1	1	yes	characteristic, complemented
451	#2298	C₂×C₃₁	1	2	yes	complemented
452	#2290	C₂×C₃₁	1	1	yes	characteristic
453	#2276	C₂×C₃₁	1	1	yes	characteristic, complemented
454	#2274	C₂×C₃₁	1	1	yes	characteristic, complemented
455	C₄×#117	C₂×C₃₁	1	1	yes	characteristic, complemented
456	C₂²×C₃₁	—	2	2	no	—
457	C₂²×C₃₁	—	2	2	no	—
458	C₂²×C₃₁	—	2	2	no	—
459	C₂²×C₃₁	—	2	2	no	—
460	C₂²×C₃₁	—	2	2	no	—
461	C₂²×C₃₁	—	2	2	no	—
462	C₂²×C₃₁	—	4	1	no	—
463	C₂²×C₃₁	—	4	1	no	—
464	C₂²×C₃₁	—	4	1	no	—
465	C₂²×C₃₁	—	4	1	no	—
466	C₂²×C₃₁	—	4	1	no	—
467	C₂²×C₃₁	—	4	1	no	—
468	C₂²×C₃₁	—	4	1	no	—
469	C₂²×C₃₁	—	4	1	no	—
470	C₂²×C₃₁	C₂⁴×C₄	1	1	yes	characteristic
471	C₂²×C₃₁	C₂²×#41	1	1	yes	characteristic
472	C₂²×C₃₁	C₂²×#41	1	1	yes	characteristic
473	C₂²×C₃₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
474	C₂²×C₃₁	#585	1	1	yes	characteristic
475	C₂²×C₃₁	#585	1	1	yes	characteristic
476	C₂²×C₃₁	#585	1	1	yes	characteristic
477	C₄×C₃₁	—	2	2	no	—
478	C₄×C₃₁	—	2	2	no	—
479	C₄×C₃₁	—	2	4	no	—
480	C₄×C₃₁	—	2	4	no	—
481	C₄×C₃₁	—	2	4	no	—
482	C₄×C₃₁	—	2	2	no	—
483	C₄×C₃₁	—	2	4	no	—
484	C₄×C₃₁	—	2	2	no	—
485	C₄×C₃₁	—	2	2	no	—
486	C₄×C₃₁	—	2	2	no	—
487	C₄×C₃₁	—	2	2	no	—
488	C₄×C₃₁	—	2	4	no	—
489	C₄×C₃₁	—	4	1	no	—
490	C₄×C₃₁	—	4	2	no	—
491	C₄×C₃₁	—	4	1	no	—
492	C₄×C₃₁	—	4	1	no	—
493	C₄×C₃₁	—	4	1	no	—
494	C₄×C₃₁	—	4	1	no	—
495	C₄×C₃₁	—	4	2	no	—
496	C₄×C₃₁	—	4	2	no	—
497	C₄×C₃₁	C₄×#41	1	4	yes	—
498	#552	C₄×C₃₁	1	16	yes	complemented
499	C₂×#126	C₂²×C₃₁	1	2	yes	—
500	C₂×#125	C₂²×C₃₁	1	2	yes	—
501	C₂×#125	C₂²×C₃₁	1	1	yes	characteristic
502	C₂×#123	C₂²×C₃₁	1	2	yes	—
503	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
504	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
505	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
506	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
507	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
508	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
509	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
510	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
511	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
512	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
513	C₂×#117	C₂²×C₃₁	1	2	yes	—
514	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
515	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
516	#391	C₂²×C₃₁	1	1	yes	characteristic
517	#391	C₂²×C₃₁	1	1	yes	characteristic
518	#390	C₂²×C₃₁	1	2	yes	—
519	#389	C₂²×C₃₁	1	2	yes	—
520	#389	C₂²×C₃₁	1	2	yes	—
521	#389	C₂²×C₃₁	1	1	yes	characteristic
522	#389	C₂²×C₃₁	1	1	yes	characteristic
523	#389	C₂²×C₃₁	1	1	yes	characteristic
524	#389	C₂²×C₃₁	1	2	yes	—
525	#389	C₂²×C₃₁	1	2	yes	—
526	#389	C₂²×C₃₁	1	1	yes	characteristic
527	#388	C₂²×C₃₁	1	2	yes	—
528	#388	C₂²×C₃₁	1	2	yes	—
529	#388	C₂²×C₃₁	1	2	yes	—
530	#388	C₂²×C₃₁	1	2	yes	—
531	#388	C₂²×C₃₁	1	2	yes	—
532	#388	C₂²×C₃₁	1	2	yes	—
533	#388	C₂²×C₃₁	1	2	yes	—
534	#388	C₂²×C₃₁	1	2	yes	—
535	#388	C₂²×C₃₁	1	1	yes	characteristic
536	#388	C₂²×C₃₁	1	2	yes	—
537	#388	C₂²×C₃₁	1	2	yes	—
538	#388	C₂²×C₃₁	1	1	yes	characteristic
539	#387	C₂²×C₃₁	1	1	yes	characteristic
540	#387	C₂²×C₃₁	1	2	yes	—
541	#387	C₂²×C₃₁	1	2	yes	—
542	#387	C₂²×C₃₁	1	2	yes	—
543	#387	C₂²×C₃₁	1	2	yes	—
544	#387	C₂²×C₃₁	1	2	yes	—
545	#387	C₂²×C₃₁	1	2	yes	—
546	#387	C₂²×C₃₁	1	2	yes	—
547	#387	C₂²×C₃₁	1	2	yes	—
548	#387	C₂²×C₃₁	1	2	yes	—
549	#387	C₂²×C₃₁	1	2	yes	—
550	#387	C₂²×C₃₁	1	2	yes	—
551	#387	C₂²×C₃₁	1	2	yes	—
552	#387	C₂²×C₃₁	1	1	yes	characteristic
553	#386	C₂²×C₃₁	1	1	yes	characteristic
554	#386	C₂²×C₃₁	1	1	yes	characteristic
555	#385	C₂²×C₃₁	1	1	yes	characteristic
556	#385	C₂²×C₃₁	1	2	yes	—
557	#385	C₂²×C₃₁	1	1	yes	characteristic
558	#384	C₂²×C₃₁	1	1	yes	characteristic
559	#382	C₂²×C₃₁	1	2	yes	—
560	#382	C₂²×C₃₁	1	2	yes	—
561	#380	C₂²×C₃₁	1	1	yes	characteristic
562	#380	C₂²×C₃₁	1	2	yes	—
563	#380	C₂²×C₃₁	1	1	yes	characteristic
564	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
565	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
566	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
567	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
568	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
569	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
570	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
571	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
572	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
573	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
574	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
575	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
576	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
577	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
578	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
579	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
580	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
581	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
582	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
583	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
584	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
585	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
586	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
587	C₄×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
588	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
589	C₄×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
590	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
591	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
592	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
593	#376	C₂²×C₃₁	1	1	yes	characteristic
594	#376	C₂²×C₃₁	1	2	yes	—
595	#376	C₂²×C₃₁	1	2	yes	—
596	#376	C₂²×C₃₁	1	2	yes	—
597	C₄³	C₂²×C₃₁	1	2	yes	—
598	C₂×C₃₁	—	4	2	no	—
599	C₂×C₃₁	—	4	2	no	—
600	C₂×C₃₁	C₂×C₄×#41	1	1	yes	characteristic
601	C₂×C₃₁	#3428	1	1	yes	characteristic
602	C₂×C₃₁	C₄×2₋¹⁺⁴	1	1	yes	characteristic
603	C₂×C₃₁	#3493	1	1	yes	characteristic
604	C₂×C₃₁	#3495	1	1	yes	characteristic
605	C₂×C₃₁	#3515	1	1	yes	characteristic
606	C₂×C₃₁	#3524	1	1	yes	characteristic
607	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
608	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
609	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
610	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
611	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
612	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
613	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
614	#126	C₂×C₄×C₃₁	1	16	yes	—
615	#125	C₂×C₄×C₃₁	1	16	yes	complemented
616	#125	C₂×C₄×C₃₁	1	8	yes	complemented
617	#123	C₂×C₄×C₃₁	1	16	yes	—
618	C₂²⋊Q₈	C₂×C₄×C₃₁	1	8	yes	—
619	C₂²⋊Q₈	C₂×C₄×C₃₁	1	8	yes	complemented
620	C₄×Q₈	—	2	4	no	—
621	C₄×Q₈	—	2	4	no	—
622	C₄×Q₈	C₂×C₄×C₃₁	1	8	yes	complemented
623	C₄×D₄	C₂×C₄×C₃₁	1	8	yes	—
624	#117	—	2	4	no	—
625	#117	—	2	4	no	—
626	#117	—	2	4	no	—
627	#117	C₂×C₄×C₃₁	1	8	yes	—
628	#117	C₂×C₄×C₃₁	1	16	yes	complemented
629	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	8	yes	complemented
630	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
631	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
632	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
633	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
634	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
635	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
636	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
637	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
638	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
639	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
640	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
641	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
642	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
643	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
644	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
645	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
646	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
647	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
648	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
649	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
650	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
651	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
652	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
653	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
654	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
655	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
656	C₂×C₂²⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
657	C₂×C₂²⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
658	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
659	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
660	C₂×C₂²⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
661	C₂×C₂²⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
662	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
663	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
664	C₂×C₂²⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
665	C₂×C₂²⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
666	C₂×C₂²⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
667	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
668	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
669	C₂×C₂²⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
670	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
671	C₂×C₄²	—	2	4	no	—
672	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
673	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
674	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
675	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
676	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
677	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
678	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
679	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
680	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
681	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
682	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
683	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
684	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
685	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
686	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
687	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
688	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
689	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
690	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
691	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
692	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
693	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
694	#95	C₂³×C₃₁	1	1	yes	characteristic
695	#95	C₂³×C₃₁	1	2	yes	—
696	#95	C₂³×C₃₁	1	1	yes	characteristic
697	#95	C₂³×C₃₁	1	2	yes	—
698	#95	C₂³×C₃₁	1	2	yes	—
699	#95	C₂³×C₃₁	1	2	yes	—
700	#95	C₂³×C₃₁	1	2	yes	—
701	#95	C₂³×C₃₁	1	2	yes	—
702	#95	C₂³×C₃₁	1	1	yes	characteristic
703	#95	C₂³×C₃₁	1	1	yes	characteristic
704	#95	C₂³×C₃₁	1	1	yes	characteristic
705	#95	C₂³×C₃₁	1	2	yes	—
706	#95	C₂³×C₃₁	1	2	yes	—
707	#95	C₂³×C₃₁	1	2	yes	—
708	#95	C₂³×C₃₁	1	2	yes	—
709	#95	C₂³×C₃₁	1	1	yes	characteristic
710	#95	C₂³×C₃₁	1	2	yes	—
711	#95	C₂³×C₃₁	1	1	yes	characteristic
712	#95	C₂³×C₃₁	1	2	yes	—
713	#95	C₂³×C₃₁	1	2	yes	—
714	#95	C₂³×C₃₁	1	2	yes	—
715	#95	C₂³×C₃₁	1	1	yes	characteristic
716	#95	C₂³×C₃₁	1	2	yes	—
717	#95	C₂³×C₃₁	1	1	yes	characteristic
718	#95	C₂³×C₃₁	1	2	yes	—
719	#95	C₂³×C₃₁	1	2	yes	—
720	#95	C₂³×C₃₁	1	2	yes	—
721	#95	C₂³×C₃₁	1	1	yes	characteristic
722	#95	C₂³×C₃₁	1	2	yes	—
723	C₃₁	#34300	1	1	yes	characteristic, direct product factor, Sylow
724	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
725	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
726	C₂×Q₈	—	2	2	no	—
727	C₂×Q₈	—	2	2	no	—
728	C₂×Q₈	C₂²×C₄×C₃₁	1	4	yes	—
729	C₂×D₄	—	4	1	no	—
730	C₂×D₄	—	4	1	no	—
731	C₂×D₄	C₂²×C₄×C₃₁	1	4	yes	—
732	C₂²×C₄	—	2	2	no	—
733	C₂²×C₄	—	2	2	no	—
734	C₂²×C₄	—	2	2	no	—
735	C₂²×C₄	—	2	2	no	—
736	C₂²×C₄	—	2	2	no	—
737	C₂²×C₄	—	2	2	no	—
738	C₂²×C₄	—	2	2	no	—
739	C₂²×C₄	—	4	1	no	—
740	C₂²×C₄	—	4	2	no	—
741	C₂²×C₄	—	4	2	no	—
742	C₂²×C₄	—	4	1	no	—
743	C₂²×C₄	—	4	1	no	—
744	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
745	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
746	C₂²×C₄	C₂²×C₄×C₃₁	1	8	yes	—
747	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
748	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
749	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
750	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
751	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
752	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
753	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
754	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
755	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
756	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
757	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
758	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
759	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
760	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
761	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
762	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
763	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
764	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
765	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
766	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
767	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
768	C₄⋊C₄	—	2	2	no	—
769	C₄⋊C₄	—	2	2	no	—
770	C₄⋊C₄	—	2	2	no	—
771	C₄⋊C₄	—	2	2	no	—
772	C₄⋊C₄	—	2	4	no	—
773	C₄⋊C₄	—	2	4	no	—
774	C₄⋊C₄	—	2	2	no	—
775	C₄⋊C₄	—	2	2	no	—
776	C₄⋊C₄	—	4	2	no	—
777	C₄⋊C₄	—	4	2	no	—
778	C₄⋊C₄	—	4	2	no	—
779	C₄⋊C₄	—	4	2	no	—
780	C₄⋊C₄	—	4	2	no	—
781	C₄⋊C₄	—	4	1	no	—
782	C₄⋊C₄	—	4	2	no	—
783	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
784	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
785	C₄⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
786	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
787	C₄⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
788	C₄⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
789	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
790	C₄⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
791	C₄⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
792	C₄⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
793	C₄⋊C₄	#41×C₃₁	1	4	yes	—
794	C₂²⋊C₄	—	2	2	no	—
795	C₂²⋊C₄	—	2	4	no	—
796	C₂²⋊C₄	—	2	4	no	—
797	C₂²⋊C₄	—	2	2	no	—
798	C₂²⋊C₄	—	2	2	no	—
799	C₂²⋊C₄	—	4	1	no	—
800	C₂²⋊C₄	—	4	1	no	—
801	C₂²⋊C₄	—	4	2	no	—
802	C₂²⋊C₄	—	4	2	no	—
803	C₂²⋊C₄	—	4	1	no	—
804	C₂²⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
805	C₂²⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
806	C₂²⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
807	C₂²⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
808	C₂²⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
809	C₄²	—	2	2	no	—
810	C₄²	—	2	2	no	—
811	C₄²	—	2	2	no	—
812	C₄²	—	2	4	no	—
813	C₄²	—	2	4	no	—
814	C₄²	—	2	2	no	—
815	C₄²	—	2	4	no	—
816	C₄²	—	2	4	no	—
817	C₄²	—	2	2	no	—
818	C₄²	—	2	4	no	—
819	C₄²	—	4	2	no	—
820	C₄²	—	4	2	no	—
821	C₄²	—	4	2	no	—
822	C₄²	—	4	2	no	—
823	C₄²	—	4	2	no	—
824	C₄²	—	4	2	no	—
825	C₄²	—	4	2	no	—
826	C₄²	—	4	2	no	—
827	C₄²	C₂²×C₄×C₃₁	1	8	yes	—
828	C₄²	C₂²×C₄×C₃₁	1	8	yes	—
829	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
830	C₄²	#41×C₃₁	1	4	yes	—
831	C₂³	—	2	1	no	—
832	C₂³	—	2	1	no	—
833	C₂³	—	2	1	no	—
834	C₂³	—	2	1	no	—
835	C₂³	—	2	1	no	—
836	C₂³	—	2	1	no	—
837	C₂³	—	2	1	no	—
838	C₂³	—	2	1	no	—
839	C₂³	—	2	1	no	—
840	C₂³	—	2	1	no	—
841	C₂³	—	2	1	no	—
842	C₂³	—	2	1	no	—
843	C₂³	C₂³×C₄×C₃₁	1	2	yes	—
844	C₂³	C₂³×C₄×C₃₁	1	2	yes	—
845	C₂³	C₂⁵×C₃₁	1	1	yes	characteristic
846	Q₈	—	2	8	no	—
847	D₄	—	4	4	no	—
848	C₂×C₄	—	2	1	no	—
849	C₂×C₄	—	2	1	no	—
850	C₂×C₄	—	2	1	no	—
851	C₂×C₄	—	2	1	no	—
852	C₂×C₄	—	2	2	no	—
853	C₂×C₄	—	2	2	no	—
854	C₂×C₄	—	2	1	no	—
855	C₂×C₄	—	2	2	no	—
856	C₂×C₄	—	2	2	no	—
857	C₂×C₄	—	2	1	no	—
858	C₂×C₄	—	2	1	no	—
859	C₂×C₄	—	2	2	no	—
860	C₂×C₄	—	2	2	no	—
861	C₂×C₄	—	2	2	no	—
862	C₂×C₄	—	2	1	no	—
863	C₂×C₄	—	2	1	no	—
864	C₂×C₄	—	2	1	no	—
865	C₂×C₄	—	2	1	no	—
866	C₂×C₄	—	2	1	no	—
867	C₂×C₄	—	2	2	no	—
868	C₂×C₄	—	2	2	no	—
869	C₂×C₄	—	2	2	no	—
870	C₂×C₄	—	2	1	no	—
871	C₂×C₄	—	2	1	no	—
872	C₂×C₄	—	2	8	no	—
873	C₂×C₄	—	2	8	no	—
874	C₂×C₄	—	2	1	no	—
875	C₂×C₄	—	2	1	no	—
876	C₂×C₄	—	2	1	no	—
877	C₂×C₄	—	2	1	no	—
878	C₂×C₄	—	2	2	no	—
879	C₂×C₄	—	2	2	no	—
880	C₂×C₄	—	2	2	no	—
881	C₂×C₄	—	2	1	no	—
882	C₂×C₄	—	2	1	no	—
883	C₂×C₄	—	2	1	no	—
884	C₂×C₄	—	2	1	no	—
885	C₂×C₄	—	2	1	no	—
886	C₂×C₄	—	2	1	no	—
887	C₂×C₄	—	2	2	no	—
888	C₂×C₄	—	2	2	no	—
889	C₂×C₄	—	2	2	no	—
890	C₂×C₄	—	2	2	no	—
891	C₂×C₄	—	2	2	no	—
892	C₂×C₄	—	2	1	no	—
893	C₂×C₄	—	2	1	no	—
894	C₂×C₄	—	2	2	no	—
895	C₂×C₄	—	2	1	no	—
896	C₂×C₄	—	2	1	no	—
897	C₂×C₄	—	2	1	no	—
898	C₂×C₄	—	4	8	no	—
899	C₂×C₄	—	4	4	no	—
900	C₂×C₄	—	4	4	no	—
901	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
902	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
903	C₂×C₄	C₂³×C₄×C₃₁	1	4	yes	—
904	C₂×C₄	C₂³×C₄×C₃₁	1	4	yes	—
905	C₂×C₄	C₂³×C₄×C₃₁	1	4	yes	—
906	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
907	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
908	C₂×C₄	C₂³×C₄×C₃₁	1	4	yes	—
909	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
910	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
911	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
912	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
913	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
914	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
915	C₂×C₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
916	C₂²	—	2	2	no	—
917	C₂²	—	2	2	no	—
918	C₂²	—	2	2	no	—
919	C₂²	—	2	2	no	—
920	C₂²	—	2	2	no	—
921	C₂²	—	2	2	no	—
922	C₂²	—	4	1	no	—
923	C₂²	—	4	1	no	—
924	C₂²	—	4	1	no	—
925	C₂²	—	4	1	no	—
926	C₂²	—	4	1	no	—
927	C₂²	—	4	1	no	—
928	C₂²	—	4	1	no	—
929	C₂²	—	4	1	no	—
930	C₂²	C₂⁴×C₄×C₃₁	1	1	yes	characteristic
931	C₂²	C₂²×#41×C₃₁	1	1	yes	characteristic
932	C₂²	C₂²×#41×C₃₁	1	1	yes	characteristic
933	C₂²	C₂×C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
934	C₂²	C₃₁×#585	1	1	yes	characteristic
935	C₂²	C₃₁×#585	1	1	yes	characteristic
936	C₂²	C₃₁×#585	1	1	yes	characteristic
937	C₄	—	2	2	no	—
938	C₄	—	2	2	no	—
939	C₄	—	2	4	no	—
940	C₄	—	2	4	no	—
941	C₄	—	2	4	no	—
942	C₄	—	2	2	no	—
943	C₄	—	2	4	no	—
944	C₄	—	2	2	no	—
945	C₄	—	2	2	no	—
946	C₄	—	2	2	no	—
947	C₄	—	2	2	no	—
948	C₄	—	2	4	no	—
949	C₄	—	4	1	no	—
950	C₄	—	4	1	no	—
951	C₄	—	4	2	no	—
952	C₄	—	4	1	no	—
953	C₄	—	4	1	no	—
954	C₄	—	4	1	no	—
955	C₄	—	4	2	no	—
956	C₄	—	4	2	no	—
957	C₄	C₄×#41×C₃₁	1	4	yes	—
958	C₂	—	4	2	no	—
959	C₂	—	4	2	no	—
960	C₂	C₂×C₄×#41×C₃₁	1	1	yes	characteristic
961	C₂	C₃₁×#3428	1	1	yes	characteristic
962	C₂	C₄×C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
963	C₂	C₃₁×#3493	1	1	yes	characteristic
964	C₂	C₃₁×#3495	1	1	yes	characteristic
965	C₂	C₃₁×#3515	1	1	yes	characteristic
966	C₂	C₃₁×#3524	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₃₁
Frattini, cosocle	C₂³	C₂⁵×C₃₁
prime radical	C₂²×D₄×C₃₁	C₂³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
3. C₃₁×#2373	C₂	semidirect
4. C₃₁×#2371	C₂	semidirect
5. C₃₁×#2370	C₂	semidirect
6. C₃₁×#2362	C₂	semidirect
7. C₃₁×#2362	C₂	semidirect
8. C₃₁×#2355	C₂	semidirect
10. C₃₁×#2346	C₂	semidirect
11. C₃₁×#2346	C₂	semidirect
12. C₃₁×#2345	C₂	semidirect
14. C₃₁×#2334	C₂	semidirect
15. C₃₁×#2330	C₂	semidirect
16. C₃₁×#2323	C₂	semidirect
17. C₄×C₃₁×#125	C₂	semidirect
18. C₄×C₃₁×C₂²⋊Q₈	C₂	semidirect
19. C₃₁×#2298	C₂	semidirect
21. C₃₁×#2276	C₂	semidirect
22. C₃₁×#2274	C₂	semidirect
23. C₄×C₃₁×#117	C₂	semidirect
24. C₃₁×#552	C₄	semidirect
132. C₃₁×#125	C₂×C₄	semidirect
133. C₃₁×#125	C₂×C₄	semidirect
136. C₃₁×C₂²⋊Q₈	C₂×C₄	semidirect
139. C₄×Q₈×C₃₁	C₂×C₄	semidirect
145. C₃₁×#117	C₂×C₄	semidirect
146. C₂×C₄⋊C₄×C₃₁	C₂×C₄	semidirect
347. #34300	C₃₁	direct (N × H)
435. #2373	C₂×C₃₁	semidirect
436. #2371	C₂×C₃₁	semidirect
437. #2370	C₂×C₃₁	semidirect
438. #2362	C₂×C₃₁	semidirect
439. #2362	C₂×C₃₁	semidirect
440. #2355	C₂×C₃₁	semidirect
442. #2346	C₂×C₃₁	semidirect
443. #2346	C₂×C₃₁	semidirect
444. #2345	C₂×C₃₁	semidirect
446. #2334	C₂×C₃₁	semidirect
447. #2330	C₂×C₃₁	semidirect
448. #2323	C₂×C₃₁	semidirect
449. C₄×#125	C₂×C₃₁	semidirect
450. C₄×C₂²⋊Q₈	C₂×C₃₁	semidirect
451. #2298	C₂×C₃₁	semidirect
453. #2276	C₂×C₃₁	semidirect
454. #2274	C₂×C₃₁	semidirect
455. C₄×#117	C₂×C₃₁	semidirect
498. #552	C₄×C₃₁	semidirect
615. #125	C₂×C₄×C₃₁	semidirect
616. #125	C₂×C₄×C₃₁	semidirect
619. C₂²⋊Q₈	C₂×C₄×C₃₁	semidirect
622. C₄×Q₈	C₂×C₄×C₃₁	semidirect
628. #117	C₂×C₄×C₃₁	semidirect
629. C₂×C₄⋊C₄	C₂×C₄×C₃₁	semidirect
723. C₃₁	#34300	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₃₁×#552	2
C₃₁×#2374	2
C₃₁×#2373	2
C₃₁×#2371	2
C₃₁×#2370	2
C₃₁×#2362	2
C₃₁×#2355	2
C₃₁×#2354	2
C₃₁×#2346	2
C₃₁×#2345	2
C₃₁×#2335	2
C₃₁×#2334	2
C₃₁×#2330	2
C₃₁×#2323	2
C₄×C₃₁×#125	2
C₄×C₃₁×C₂²⋊Q₈	2
C₃₁×#2298	2
C₃₁×#2290	2
C₃₁×#2276	2
C₃₁×#2274	2
C₄×C₃₁×#117	2
#34300	31

Subgroups — C31×#34300 #21491059

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#34300 #21491059