Subgroups — C₃₁×#34243 #21491002

Order 7936.

2430 conjugacy classes of subgroups.

The subgroup lattice has 750 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#529	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2370	C₂	1	2	yes	maximal, complemented
3	C₃₁×#2367	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2359	C₂	1	4	yes	maximal, complemented
5	C₃₁×#2355	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#2354	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#2352	C₂	1	2	yes	maximal, complemented
8	Q₈×C₄⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
9	D₄×C₄⋊C₄×C₃₁	C₂	1	2	yes	maximal, complemented
10	D₄×C₄⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#2334	C₂	1	4	yes	maximal, complemented
12	C₃₁×#2314	C₂	1	2	yes	maximal, complemented
13	C₄×C₃₁×C₂²⋊Q₈	C₂	1	2	yes	maximal, complemented
14	C₄×C₃₁×#121	C₂	1	2	yes	maximal
15	C₃₁×#2285	C₂	1	1	yes	maximal, characteristic, complemented
16	C₃₁×#2274	C₂	1	2	yes	maximal, complemented
17	C₂×C₄×C₄⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
18	C₃₁×#529	C₄	1	16	yes	complemented
19	C₂×C₃₁×#126	C₂²	1	2	yes	—
20	C₂×C₃₁×#125	C₂²	1	1	yes	characteristic
21	C₂×C₃₁×#124	C₂²	1	1	yes	characteristic
22	C₂×C₃₁×#123	C₂²	1	2	yes	—
23	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
24	C₂×C₃₁×#121	C₂²	1	1	yes	characteristic
25	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
26	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
27	C₂×C₄×D₄×C₃₁	C₂²	1	4	yes	—
28	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
29	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
30	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
31	C₂×C₃₁×#117	C₂²	1	2	yes	—
32	C₂×C₃₁×#117	C₂²	1	2	yes	—
33	C₂×C₃₁×#117	C₂²	1	2	yes	—
34	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
35	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
36	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
37	C₂²×C₄²×C₃₁	C₂²	1	2	yes	—
38	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
39	C₃₁×#391	C₂²	1	2	yes	—
40	C₃₁×#390	C₂²	1	2	yes	—
41	C₃₁×#389	C₂²	1	4	yes	—
42	C₃₁×#389	C₂²	1	4	yes	—
43	C₃₁×#389	C₂²	1	2	yes	—
44	C₃₁×#389	C₂²	1	2	yes	—
45	C₃₁×#388	C₂²	1	4	yes	—
46	C₃₁×#388	C₂²	1	4	yes	—
47	C₃₁×#388	C₂²	1	4	yes	—
48	C₃₁×#388	C₂²	1	4	yes	—
49	C₃₁×#387	C₂²	1	4	yes	—
50	C₃₁×#387	C₂²	1	4	yes	—
51	C₃₁×#387	C₂²	1	4	yes	—
52	C₃₁×#387	C₂²	1	4	yes	—
53	C₃₁×#386	C₂²	1	4	yes	—
54	C₃₁×#385	C₂²	1	4	yes	—
55	C₃₁×#385	C₂²	1	4	yes	—
56	C₃₁×#385	C₂²	1	4	yes	—
57	C₃₁×#384	C₂²	1	2	yes	—
58	C₃₁×#384	C₂²	1	1	yes	characteristic
59	C₃₁×#384	C₂²	1	1	yes	characteristic
60	C₃₁×#382	C₂²	1	2	yes	—
61	C₃₁×#382	C₂²	1	2	yes	—
62	C₃₁×#380	C₂²	1	2	yes	—
63	C₃₁×#380	C₂²	1	2	yes	—
64	C₃₁×#380	C₂²	1	2	yes	—
65	C₃₁×#380	C₂²	1	2	yes	—
66	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
67	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
68	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
69	C₄×C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
70	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
71	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
72	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
73	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
74	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
75	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
76	C₄×C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
77	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
78	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
79	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
80	C₄×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	—
81	C₄×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	—
82	C₄×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	—
83	C₂×C₃₁×#95	C₂²	1	2	yes	—
84	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
85	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
86	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
87	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
88	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
89	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
90	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
91	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
92	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
93	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
94	C₃₁×#126	C₂×C₄	1	16	yes	complemented
95	C₃₁×#125	C₂×C₄	1	8	yes	complemented
96	C₃₁×#124	C₂×C₄	1	8	yes	complemented
97	C₃₁×#123	C₂×C₄	1	16	yes	complemented
98	C₃₁×C₂²⋊Q₈	C₂×C₄	1	8	yes	complemented
99	C₃₁×#121	C₂×C₄	1	8	yes	—
100	C₄×Q₈×C₃₁	—	4	4	no	—
101	C₄×Q₈×C₃₁	C₂×C₄	1	8	yes	complemented
102	C₄×D₄×C₃₁	—	4	4	no	—
103	C₄×D₄×C₃₁	—	4	8	no	—
104	C₄×D₄×C₃₁	C₂×C₄	1	8	yes	complemented
105	C₄×D₄×C₃₁	C₂×C₄	1	16	yes	complemented
106	C₃₁×#117	—	2	8	no	—
107	C₃₁×#117	—	4	4	no	—
108	C₃₁×#117	C₂×C₄	1	16	yes	complemented
109	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
110	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
111	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
112	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
113	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
114	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
115	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
116	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
117	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
118	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
119	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
120	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
121	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
122	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
123	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
124	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
125	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
126	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
127	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
128	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
129	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
130	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
131	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
132	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
133	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
134	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
135	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
136	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
137	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
138	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
139	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
140	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
141	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
142	C₂×C₄²×C₃₁	—	4	4	no	—
143	C₂×C₄²×C₃₁	C₂×C₄	1	8	yes	complemented
144	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
145	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
146	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
147	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
148	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
149	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
150	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
151	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
152	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
153	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
154	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
155	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
156	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
157	C₃₁×#95	C₂³	1	2	yes	—
158	C₃₁×#95	C₂³	1	2	yes	—
159	C₃₁×#95	C₂³	1	2	yes	—
160	C₃₁×#95	C₂³	1	2	yes	—
161	C₃₁×#95	C₂³	1	4	yes	—
162	C₃₁×#95	C₂³	1	4	yes	—
163	C₃₁×#95	C₂³	1	4	yes	—
164	C₃₁×#95	C₂³	1	4	yes	—
165	C₃₁×#95	C₂³	1	4	yes	—
166	C₃₁×#95	C₂³	1	4	yes	—
167	C₃₁×#95	C₂³	1	4	yes	—
168	C₃₁×#95	C₂³	1	4	yes	—
169	C₂⁴×C₃₁	C₂⁴	1	2	yes	—
170	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
171	C₂×Q₈×C₃₁	—	2	2	no	—
172	C₂×Q₈×C₃₁	—	2	2	no	—
173	C₂×Q₈×C₃₁	C₂²×C₄	1	4	yes	—
174	C₂×D₄×C₃₁	—	2	4	no	—
175	C₂×D₄×C₃₁	—	2	4	no	—
176	C₂×D₄×C₃₁	—	2	2	no	—
177	C₂×D₄×C₃₁	—	2	2	no	—
178	C₂×D₄×C₃₁	C₂²×C₄	1	8	yes	—
179	C₂×D₄×C₃₁	C₂²×C₄	1	4	yes	—
180	C₂²×C₄×C₃₁	—	2	2	no	—
181	C₂²×C₄×C₃₁	—	2	2	no	—
182	C₂²×C₄×C₃₁	—	2	2	no	—
183	C₂²×C₄×C₃₁	—	2	2	no	—
184	C₂²×C₄×C₃₁	—	2	4	no	—
185	C₂²×C₄×C₃₁	—	2	4	no	—
186	C₂²×C₄×C₃₁	—	2	2	no	—
187	C₂²×C₄×C₃₁	—	2	2	no	—
188	C₂²×C₄×C₃₁	—	2	4	no	—
189	C₂²×C₄×C₃₁	—	2	4	no	—
190	C₂²×C₄×C₃₁	—	2	4	no	—
191	C₂²×C₄×C₃₁	—	2	4	no	—
192	C₂²×C₄×C₃₁	—	2	8	no	—
193	C₂²×C₄×C₃₁	—	4	2	no	—
194	C₂²×C₄×C₃₁	—	4	2	no	—
195	C₂²×C₄×C₃₁	—	4	4	no	—
196	C₂²×C₄×C₃₁	—	4	4	no	—
197	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
198	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
199	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
200	C₂²×C₄×C₃₁	C₂²×C₄	1	8	yes	—
201	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
202	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
203	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
204	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
205	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
206	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
207	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
208	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
209	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
210	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
211	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
212	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
213	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
214	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
215	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
216	C₄⋊C₄×C₃₁	—	2	4	no	—
217	C₄⋊C₄×C₃₁	—	2	4	no	—
218	C₄⋊C₄×C₃₁	—	2	8	no	—
219	C₄⋊C₄×C₃₁	—	2	2	no	—
220	C₄⋊C₄×C₃₁	—	2	2	no	—
221	C₄⋊C₄×C₃₁	—	2	4	no	—
222	C₄⋊C₄×C₃₁	—	2	4	no	—
223	C₄⋊C₄×C₃₁	—	2	4	no	—
224	C₄⋊C₄×C₃₁	—	2	4	no	—
225	C₄⋊C₄×C₃₁	—	4	2	no	—
226	C₄⋊C₄×C₃₁	—	4	2	no	—
227	C₄⋊C₄×C₃₁	—	4	4	no	—
228	C₄⋊C₄×C₃₁	—	4	4	no	—
229	C₄⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
230	C₄⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
231	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
232	C₄⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
233	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
234	C₄⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
235	C₄⋊C₄×C₃₁	#41	1	4	yes	—
236	C₄⋊C₄×C₃₁	#41	1	4	yes	—
237	C₂²⋊C₄×C₃₁	—	2	8	no	—
238	C₂²⋊C₄×C₃₁	—	2	8	no	—
239	C₂²⋊C₄×C₃₁	—	4	4	no	—
240	C₂²⋊C₄×C₃₁	—	4	4	no	—
241	C₂²⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
242	C₂²⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
243	C₂²⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
244	C₂²⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
245	C₂²⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
246	C₄²×C₃₁	—	2	8	no	—
247	C₄²×C₃₁	—	2	4	no	—
248	C₄²×C₃₁	—	2	4	no	—
249	C₄²×C₃₁	—	4	2	no	—
250	C₄²×C₃₁	—	4	2	no	—
251	C₄²×C₃₁	—	4	4	no	—
252	C₄²×C₃₁	—	4	4	no	—
253	C₄²×C₃₁	—	4	4	no	—
254	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
255	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
256	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
257	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
258	C₄²×C₃₁	C₂²×C₄	1	8	yes	—
259	#34243	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
260	C₂³×C₃₁	—	2	1	no	—
261	C₂³×C₃₁	—	2	1	no	—
262	C₂³×C₃₁	—	2	1	no	—
263	C₂³×C₃₁	—	2	1	no	—
264	C₂³×C₃₁	—	2	2	no	—
265	C₂³×C₃₁	—	2	2	no	—
266	C₂³×C₃₁	—	2	2	no	—
267	C₂³×C₃₁	—	2	2	no	—
268	C₂³×C₃₁	—	2	2	no	—
269	C₂³×C₃₁	—	2	2	no	—
270	C₂³×C₃₁	C₂³×C₄	1	4	yes	—
271	C₂³×C₃₁	C₂³×C₄	1	2	yes	—
272	C₂³×C₃₁	H(5,𝔽₂)	1	2	yes	—
273	C₂³×C₃₁	2₋¹⁺⁴	1	2	yes	—
274	C₂³×C₃₁	C₂⁵	1	1	yes	characteristic
275	Q₈×C₃₁	—	2	8	no	—
276	D₄×C₃₁	—	2	16	no	—
277	D₄×C₃₁	—	2	8	no	—
278	C₂×C₄×C₃₁	—	2	8	no	—
279	C₂×C₄×C₃₁	—	2	2	no	—
280	C₂×C₄×C₃₁	—	2	2	no	—
281	C₂×C₄×C₃₁	—	2	1	no	—
282	C₂×C₄×C₃₁	—	2	1	no	—
283	C₂×C₄×C₃₁	—	2	2	no	—
284	C₂×C₄×C₃₁	—	2	2	no	—
285	C₂×C₄×C₃₁	—	2	1	no	—
286	C₂×C₄×C₃₁	—	2	1	no	—
287	C₂×C₄×C₃₁	—	2	2	no	—
288	C₂×C₄×C₃₁	—	2	2	no	—
289	C₂×C₄×C₃₁	—	2	2	no	—
290	C₂×C₄×C₃₁	—	2	1	no	—
291	C₂×C₄×C₃₁	—	2	1	no	—
292	C₂×C₄×C₃₁	—	2	1	no	—
293	C₂×C₄×C₃₁	—	2	1	no	—
294	C₂×C₄×C₃₁	—	2	2	no	—
295	C₂×C₄×C₃₁	—	2	2	no	—
296	C₂×C₄×C₃₁	—	2	2	no	—
297	C₂×C₄×C₃₁	—	2	8	no	—
298	C₂×C₄×C₃₁	—	2	16	no	—
299	C₂×C₄×C₃₁	—	2	8	no	—
300	C₂×C₄×C₃₁	—	2	4	no	—
301	C₂×C₄×C₃₁	—	2	2	no	—
302	C₂×C₄×C₃₁	—	2	4	no	—
303	C₂×C₄×C₃₁	—	2	2	no	—
304	C₂×C₄×C₃₁	—	2	2	no	—
305	C₂×C₄×C₃₁	—	2	2	no	—
306	C₂×C₄×C₃₁	—	2	4	no	—
307	C₂×C₄×C₃₁	—	2	4	no	—
308	C₂×C₄×C₃₁	—	2	4	no	—
309	C₂×C₄×C₃₁	—	2	2	no	—
310	C₂×C₄×C₃₁	—	2	4	no	—
311	C₂×C₄×C₃₁	—	2	2	no	—
312	C₂×C₄×C₃₁	—	4	8	no	—
313	C₂×C₄×C₃₁	—	4	8	no	—
314	C₂×C₄×C₃₁	—	8	8	no	—
315	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
316	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
317	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
318	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
319	C₂×C₄×C₃₁	C₂³×C₄	1	4	yes	—
320	C₂×C₄×C₃₁	C₂³×C₄	1	4	yes	—
321	C₂×C₄×C₃₁	C₂³×C₄	1	4	yes	—
322	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
323	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
324	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
325	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
326	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
327	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
328	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
329	C₂×C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
330	C₂×#529	C₂×C₃₁	1	1	yes	characteristic
331	#2370	C₂×C₃₁	1	2	yes	complemented
332	#2367	C₂×C₃₁	1	2	yes	complemented
333	#2359	C₂×C₃₁	1	4	yes	complemented
334	#2355	C₂×C₃₁	1	1	yes	characteristic, complemented
335	#2354	C₂×C₃₁	1	1	yes	characteristic, complemented
336	#2352	C₂×C₃₁	1	2	yes	complemented
337	Q₈×C₄⋊C₄	C₂×C₃₁	1	1	yes	characteristic, complemented
338	D₄×C₄⋊C₄	C₂×C₃₁	1	2	yes	complemented
339	D₄×C₄⋊C₄	C₂×C₃₁	1	1	yes	characteristic, complemented
340	#2334	C₂×C₃₁	1	4	yes	complemented
341	#2314	C₂×C₃₁	1	2	yes	complemented
342	C₄×C₂²⋊Q₈	C₂×C₃₁	1	2	yes	complemented
343	C₄×#121	C₂×C₃₁	1	2	yes	—
344	#2285	C₂×C₃₁	1	1	yes	characteristic, complemented
345	#2274	C₂×C₃₁	1	2	yes	complemented
346	C₂×C₄×C₄⋊C₄	C₂×C₃₁	1	1	yes	characteristic, complemented
347	C₂²×C₃₁	—	2	2	no	—
348	C₂²×C₃₁	—	2	2	no	—
349	C₂²×C₃₁	—	2	2	no	—
350	C₂²×C₃₁	—	2	2	no	—
351	C₂²×C₃₁	—	2	2	no	—
352	C₂²×C₃₁	—	2	2	no	—
353	C₂²×C₃₁	—	2	4	no	—
354	C₂²×C₃₁	—	2	4	no	—
355	C₂²×C₃₁	—	2	4	no	—
356	C₂²×C₃₁	—	4	2	no	—
357	C₂²×C₃₁	—	4	2	no	—
358	C₂²×C₃₁	—	4	2	no	—
359	C₂²×C₃₁	—	4	2	no	—
360	C₂²×C₃₁	#520	1	4	yes	—
361	C₂²×C₃₁	C₂⁴×C₄	1	1	yes	characteristic
362	C₂²×C₃₁	C₂²×#41	1	1	yes	characteristic
363	C₂²×C₃₁	C₂²×#41	1	1	yes	characteristic
364	C₂²×C₃₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
365	C₂²×C₃₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
366	C₂²×C₃₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
367	C₂²×C₃₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
368	C₄×C₃₁	—	2	4	no	—
369	C₄×C₃₁	—	2	2	no	—
370	C₄×C₃₁	—	2	2	no	—
371	C₄×C₃₁	—	2	2	no	—
372	C₄×C₃₁	—	2	2	no	—
373	C₄×C₃₁	—	2	4	no	—
374	C₄×C₃₁	—	2	4	no	—
375	C₄×C₃₁	—	2	4	no	—
376	C₄×C₃₁	—	2	4	no	—
377	C₄×C₃₁	—	4	2	no	—
378	C₄×C₃₁	—	4	4	no	—
379	C₄×C₃₁	—	4	2	no	—
380	C₄×C₃₁	—	4	4	no	—
381	C₄×C₃₁	C₄×#41	1	4	yes	—
382	C₄×C₃₁	#520	1	4	yes	—
383	#529	C₄×C₃₁	1	16	yes	complemented
384	C₂×#126	C₂²×C₃₁	1	2	yes	—
385	C₂×#125	C₂²×C₃₁	1	1	yes	characteristic
386	C₂×#124	C₂²×C₃₁	1	1	yes	characteristic
387	C₂×#123	C₂²×C₃₁	1	2	yes	—
388	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
389	C₂×#121	C₂²×C₃₁	1	1	yes	characteristic
390	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
391	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
392	C₂×C₄×D₄	C₂²×C₃₁	1	4	yes	—
393	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
394	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
395	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
396	C₂×#117	C₂²×C₃₁	1	2	yes	—
397	C₂×#117	C₂²×C₃₁	1	2	yes	—
398	C₂×#117	C₂²×C₃₁	1	2	yes	—
399	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
400	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
401	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
402	C₂²×C₄²	C₂²×C₃₁	1	2	yes	—
403	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
404	#391	C₂²×C₃₁	1	2	yes	—
405	#390	C₂²×C₃₁	1	2	yes	—
406	#389	C₂²×C₃₁	1	4	yes	—
407	#389	C₂²×C₃₁	1	4	yes	—
408	#389	C₂²×C₃₁	1	2	yes	—
409	#389	C₂²×C₃₁	1	2	yes	—
410	#388	C₂²×C₃₁	1	4	yes	—
411	#388	C₂²×C₃₁	1	4	yes	—
412	#388	C₂²×C₃₁	1	4	yes	—
413	#388	C₂²×C₃₁	1	4	yes	—
414	#387	C₂²×C₃₁	1	4	yes	—
415	#387	C₂²×C₃₁	1	4	yes	—
416	#387	C₂²×C₃₁	1	4	yes	—
417	#387	C₂²×C₃₁	1	4	yes	—
418	#386	C₂²×C₃₁	1	4	yes	—
419	#385	C₂²×C₃₁	1	4	yes	—
420	#385	C₂²×C₃₁	1	4	yes	—
421	#385	C₂²×C₃₁	1	4	yes	—
422	#384	C₂²×C₃₁	1	2	yes	—
423	#384	C₂²×C₃₁	1	1	yes	characteristic
424	#384	C₂²×C₃₁	1	1	yes	characteristic
425	#382	C₂²×C₃₁	1	2	yes	—
426	#382	C₂²×C₃₁	1	2	yes	—
427	#380	C₂²×C₃₁	1	2	yes	—
428	#380	C₂²×C₃₁	1	2	yes	—
429	#380	C₂²×C₃₁	1	2	yes	—
430	#380	C₂²×C₃₁	1	2	yes	—
431	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
432	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
433	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
434	C₄×C₄⋊C₄	C₂²×C₃₁	1	4	yes	—
435	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
436	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
437	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
438	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
439	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
440	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
441	C₄×C₄⋊C₄	C₂²×C₃₁	1	4	yes	—
442	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
443	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
444	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
445	C₄×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	—
446	C₄×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	—
447	C₄×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	—
448	C₂×#95	C₂²×C₃₁	1	2	yes	—
449	C₂×C₃₁	—	2	4	no	—
450	C₂×C₃₁	—	4	4	no	—
451	C₂×C₃₁	C₂×C₄×#41	1	1	yes	characteristic
452	C₂×C₃₁	C₂×#520	1	1	yes	characteristic
453	C₂×C₃₁	C₂×#520	1	1	yes	characteristic
454	C₂×C₃₁	#3519	1	1	yes	characteristic
455	C₂×C₃₁	#3520	1	1	yes	characteristic
456	C₂×C₃₁	#3522	1	1	yes	characteristic
457	C₂×C₃₁	#3523	1	1	yes	characteristic
458	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
459	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
460	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
461	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
462	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
463	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
464	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
465	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
466	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
467	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
468	#126	C₂×C₄×C₃₁	1	16	yes	complemented
469	#125	C₂×C₄×C₃₁	1	8	yes	complemented
470	#124	C₂×C₄×C₃₁	1	8	yes	complemented
471	#123	C₂×C₄×C₃₁	1	16	yes	complemented
472	C₂²⋊Q₈	C₂×C₄×C₃₁	1	8	yes	complemented
473	#121	C₂×C₄×C₃₁	1	8	yes	—
474	C₄×Q₈	—	4	4	no	—
475	C₄×Q₈	C₂×C₄×C₃₁	1	8	yes	complemented
476	C₄×D₄	—	4	4	no	—
477	C₄×D₄	—	4	8	no	—
478	C₄×D₄	C₂×C₄×C₃₁	1	8	yes	complemented
479	C₄×D₄	C₂×C₄×C₃₁	1	16	yes	complemented
480	#117	—	2	8	no	—
481	#117	—	4	4	no	—
482	#117	C₂×C₄×C₃₁	1	16	yes	complemented
483	C₂×C₄⋊C₄	—	2	4	no	—
484	C₂×C₄⋊C₄	—	2	8	no	—
485	C₂×C₄⋊C₄	—	2	4	no	—
486	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
487	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
488	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
489	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
490	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
491	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
492	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
493	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
494	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
495	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
496	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
497	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
498	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
499	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
500	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
501	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
502	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
503	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
504	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
505	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
506	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
507	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
508	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
509	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
510	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
511	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
512	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
513	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
514	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
515	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
516	C₂×C₄²	—	4	4	no	—
517	C₂×C₄²	C₂×C₄×C₃₁	1	8	yes	complemented
518	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
519	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
520	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
521	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
522	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
523	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
524	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
525	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
526	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
527	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
528	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
529	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
530	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
531	#95	C₂³×C₃₁	1	2	yes	—
532	#95	C₂³×C₃₁	1	2	yes	—
533	#95	C₂³×C₃₁	1	2	yes	—
534	#95	C₂³×C₃₁	1	2	yes	—
535	#95	C₂³×C₃₁	1	4	yes	—
536	#95	C₂³×C₃₁	1	4	yes	—
537	#95	C₂³×C₃₁	1	4	yes	—
538	#95	C₂³×C₃₁	1	4	yes	—
539	#95	C₂³×C₃₁	1	4	yes	—
540	#95	C₂³×C₃₁	1	4	yes	—
541	#95	C₂³×C₃₁	1	4	yes	—
542	#95	C₂³×C₃₁	1	4	yes	—
543	C₃₁	#34243	1	1	yes	characteristic, direct product factor, Sylow
544	C₂⁴	C₂⁴×C₃₁	1	2	yes	—
545	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
546	C₂×Q₈	—	2	2	no	—
547	C₂×Q₈	—	2	2	no	—
548	C₂×Q₈	C₂²×C₄×C₃₁	1	4	yes	—
549	C₂×D₄	—	2	4	no	—
550	C₂×D₄	—	2	4	no	—
551	C₂×D₄	—	2	2	no	—
552	C₂×D₄	—	2	2	no	—
553	C₂×D₄	C₂²×C₄×C₃₁	1	8	yes	—
554	C₂×D₄	C₂²×C₄×C₃₁	1	4	yes	—
555	C₂²×C₄	—	2	2	no	—
556	C₂²×C₄	—	2	2	no	—
557	C₂²×C₄	—	2	2	no	—
558	C₂²×C₄	—	2	2	no	—
559	C₂²×C₄	—	2	4	no	—
560	C₂²×C₄	—	2	4	no	—
561	C₂²×C₄	—	2	2	no	—
562	C₂²×C₄	—	2	2	no	—
563	C₂²×C₄	—	2	4	no	—
564	C₂²×C₄	—	2	4	no	—
565	C₂²×C₄	—	2	4	no	—
566	C₂²×C₄	—	2	4	no	—
567	C₂²×C₄	—	2	8	no	—
568	C₂²×C₄	—	4	2	no	—
569	C₂²×C₄	—	4	2	no	—
570	C₂²×C₄	—	4	4	no	—
571	C₂²×C₄	—	4	4	no	—
572	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
573	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
574	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
575	C₂²×C₄	C₂²×C₄×C₃₁	1	8	yes	—
576	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
577	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
578	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
579	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
580	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
581	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
582	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
583	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
584	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
585	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
586	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
587	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
588	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
589	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
590	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
591	C₄⋊C₄	—	2	4	no	—
592	C₄⋊C₄	—	2	4	no	—
593	C₄⋊C₄	—	2	8	no	—
594	C₄⋊C₄	—	2	2	no	—
595	C₄⋊C₄	—	2	2	no	—
596	C₄⋊C₄	—	2	4	no	—
597	C₄⋊C₄	—	2	4	no	—
598	C₄⋊C₄	—	2	4	no	—
599	C₄⋊C₄	—	2	4	no	—
600	C₄⋊C₄	—	4	2	no	—
601	C₄⋊C₄	—	4	2	no	—
602	C₄⋊C₄	—	4	4	no	—
603	C₄⋊C₄	—	4	4	no	—
604	C₄⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
605	C₄⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
606	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
607	C₄⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
608	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
609	C₄⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
610	C₄⋊C₄	#41×C₃₁	1	4	yes	—
611	C₄⋊C₄	#41×C₃₁	1	4	yes	—
612	C₂²⋊C₄	—	2	8	no	—
613	C₂²⋊C₄	—	2	8	no	—
614	C₂²⋊C₄	—	4	4	no	—
615	C₂²⋊C₄	—	4	4	no	—
616	C₂²⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
617	C₂²⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
618	C₂²⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
619	C₂²⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
620	C₂²⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
621	C₄²	—	2	8	no	—
622	C₄²	—	2	4	no	—
623	C₄²	—	2	4	no	—
624	C₄²	—	4	2	no	—
625	C₄²	—	4	2	no	—
626	C₄²	—	4	4	no	—
627	C₄²	—	4	4	no	—
628	C₄²	—	4	4	no	—
629	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
630	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
631	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
632	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
633	C₄²	C₂²×C₄×C₃₁	1	8	yes	—
634	C₂³	—	2	1	no	—
635	C₂³	—	2	1	no	—
636	C₂³	—	2	1	no	—
637	C₂³	—	2	1	no	—
638	C₂³	—	2	2	no	—
639	C₂³	—	2	2	no	—
640	C₂³	—	2	2	no	—
641	C₂³	—	2	2	no	—
642	C₂³	—	2	2	no	—
643	C₂³	—	2	2	no	—
644	C₂³	C₂³×C₄×C₃₁	1	4	yes	—
645	C₂³	C₂³×C₄×C₃₁	1	2	yes	—
646	C₂³	C₃₁×H(5,𝔽₂)	1	2	yes	—
647	C₂³	C₃₁×2₋¹⁺⁴	1	2	yes	—
648	C₂³	C₂⁵×C₃₁	1	1	yes	characteristic
649	Q₈	—	2	8	no	—
650	D₄	—	2	16	no	—
651	D₄	—	2	8	no	—
652	C₂×C₄	—	2	8	no	—
653	C₂×C₄	—	2	2	no	—
654	C₂×C₄	—	2	2	no	—
655	C₂×C₄	—	2	1	no	—
656	C₂×C₄	—	2	1	no	—
657	C₂×C₄	—	2	2	no	—
658	C₂×C₄	—	2	2	no	—
659	C₂×C₄	—	2	1	no	—
660	C₂×C₄	—	2	1	no	—
661	C₂×C₄	—	2	2	no	—
662	C₂×C₄	—	2	2	no	—
663	C₂×C₄	—	2	2	no	—
664	C₂×C₄	—	2	1	no	—
665	C₂×C₄	—	2	1	no	—
666	C₂×C₄	—	2	1	no	—
667	C₂×C₄	—	2	1	no	—
668	C₂×C₄	—	2	2	no	—
669	C₂×C₄	—	2	2	no	—
670	C₂×C₄	—	2	2	no	—
671	C₂×C₄	—	2	8	no	—
672	C₂×C₄	—	2	16	no	—
673	C₂×C₄	—	2	8	no	—
674	C₂×C₄	—	2	4	no	—
675	C₂×C₄	—	2	2	no	—
676	C₂×C₄	—	2	4	no	—
677	C₂×C₄	—	2	2	no	—
678	C₂×C₄	—	2	2	no	—
679	C₂×C₄	—	2	2	no	—
680	C₂×C₄	—	2	4	no	—
681	C₂×C₄	—	2	4	no	—
682	C₂×C₄	—	2	4	no	—
683	C₂×C₄	—	2	2	no	—
684	C₂×C₄	—	2	4	no	—
685	C₂×C₄	—	2	2	no	—
686	C₂×C₄	—	4	8	no	—
687	C₂×C₄	—	4	8	no	—
688	C₂×C₄	—	8	8	no	—
689	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
690	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
691	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
692	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
693	C₂×C₄	C₂³×C₄×C₃₁	1	4	yes	—
694	C₂×C₄	C₂³×C₄×C₃₁	1	4	yes	—
695	C₂×C₄	C₂³×C₄×C₃₁	1	4	yes	—
696	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
697	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
698	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
699	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
700	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
701	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
702	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
703	C₂×C₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
704	C₂²	—	2	2	no	—
705	C₂²	—	2	2	no	—
706	C₂²	—	2	2	no	—
707	C₂²	—	2	2	no	—
708	C₂²	—	2	2	no	—
709	C₂²	—	2	2	no	—
710	C₂²	—	2	4	no	—
711	C₂²	—	2	4	no	—
712	C₂²	—	2	4	no	—
713	C₂²	—	4	2	no	—
714	C₂²	—	4	2	no	—
715	C₂²	—	4	2	no	—
716	C₂²	—	4	2	no	—
717	C₂²	C₃₁×#520	1	4	yes	—
718	C₂²	C₂⁴×C₄×C₃₁	1	1	yes	characteristic
719	C₂²	C₂²×#41×C₃₁	1	1	yes	characteristic
720	C₂²	C₂²×#41×C₃₁	1	1	yes	characteristic
721	C₂²	C₂×C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
722	C₂²	C₂×C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
723	C₂²	C₂×C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
724	C₂²	C₂×C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
725	C₄	—	2	4	no	—
726	C₄	—	2	2	no	—
727	C₄	—	2	2	no	—
728	C₄	—	2	2	no	—
729	C₄	—	2	2	no	—
730	C₄	—	2	4	no	—
731	C₄	—	2	4	no	—
732	C₄	—	2	4	no	—
733	C₄	—	2	4	no	—
734	C₄	—	4	2	no	—
735	C₄	—	4	4	no	—
736	C₄	—	4	2	no	—
737	C₄	—	4	4	no	—
738	C₄	C₄×#41×C₃₁	1	4	yes	—
739	C₄	C₃₁×#520	1	4	yes	—
740	C₂	—	2	4	no	—
741	C₂	—	4	4	no	—
742	C₂	C₂×C₄×#41×C₃₁	1	1	yes	characteristic
743	C₂	C₂×C₃₁×#520	1	1	yes	characteristic
744	C₂	C₂×C₃₁×#520	1	1	yes	characteristic
745	C₂	C₃₁×#3519	1	1	yes	characteristic
746	C₂	C₃₁×#3520	1	1	yes	characteristic
747	C₂	C₃₁×#3522	1	1	yes	characteristic
748	C₂	C₃₁×#3523	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₃₁
Frattini, cosocle	C₂³	C₂⁵×C₃₁
prime radical	C₂×C₃₁×#121	C₂²
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
2. C₃₁×#2370	C₂	semidirect
3. C₃₁×#2367	C₂	semidirect
4. C₃₁×#2359	C₂	semidirect
5. C₃₁×#2355	C₂	semidirect
6. C₃₁×#2354	C₂	semidirect
7. C₃₁×#2352	C₂	semidirect
8. Q₈×C₄⋊C₄×C₃₁	C₂	semidirect
9. D₄×C₄⋊C₄×C₃₁	C₂	semidirect
10. D₄×C₄⋊C₄×C₃₁	C₂	semidirect
11. C₃₁×#2334	C₂	semidirect
12. C₃₁×#2314	C₂	semidirect
13. C₄×C₃₁×C₂²⋊Q₈	C₂	semidirect
15. C₃₁×#2285	C₂	semidirect
16. C₃₁×#2274	C₂	semidirect
17. C₂×C₄×C₄⋊C₄×C₃₁	C₂	semidirect
18. C₃₁×#529	C₄	semidirect
94. C₃₁×#126	C₂×C₄	semidirect
95. C₃₁×#125	C₂×C₄	semidirect
96. C₃₁×#124	C₂×C₄	semidirect
97. C₃₁×#123	C₂×C₄	semidirect
98. C₃₁×C₂²⋊Q₈	C₂×C₄	semidirect
101. C₄×Q₈×C₃₁	C₂×C₄	semidirect
104. C₄×D₄×C₃₁	C₂×C₄	semidirect
105. C₄×D₄×C₃₁	C₂×C₄	semidirect
108. C₃₁×#117	C₂×C₄	semidirect
143. C₂×C₄²×C₃₁	C₂×C₄	semidirect
259. #34243	C₃₁	direct (N × H)
331. #2370	C₂×C₃₁	semidirect
332. #2367	C₂×C₃₁	semidirect
333. #2359	C₂×C₃₁	semidirect
334. #2355	C₂×C₃₁	semidirect
335. #2354	C₂×C₃₁	semidirect
336. #2352	C₂×C₃₁	semidirect
337. Q₈×C₄⋊C₄	C₂×C₃₁	semidirect
338. D₄×C₄⋊C₄	C₂×C₃₁	semidirect
339. D₄×C₄⋊C₄	C₂×C₃₁	semidirect
340. #2334	C₂×C₃₁	semidirect
341. #2314	C₂×C₃₁	semidirect
342. C₄×C₂²⋊Q₈	C₂×C₃₁	semidirect
344. #2285	C₂×C₃₁	semidirect
345. #2274	C₂×C₃₁	semidirect
346. C₂×C₄×C₄⋊C₄	C₂×C₃₁	semidirect
383. #529	C₄×C₃₁	semidirect
468. #126	C₂×C₄×C₃₁	semidirect
469. #125	C₂×C₄×C₃₁	semidirect
470. #124	C₂×C₄×C₃₁	semidirect
471. #123	C₂×C₄×C₃₁	semidirect
472. C₂²⋊Q₈	C₂×C₄×C₃₁	semidirect
475. C₄×Q₈	C₂×C₄×C₃₁	semidirect
478. C₄×D₄	C₂×C₄×C₃₁	semidirect
479. C₄×D₄	C₂×C₄×C₃₁	semidirect
482. #117	C₂×C₄×C₃₁	semidirect
517. C₂×C₄²	C₂×C₄×C₃₁	semidirect
543. C₃₁	#34243	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×#529	2
C₃₁×#2370	2
C₃₁×#2367	2
C₃₁×#2359	2
C₃₁×#2355	2
C₃₁×#2354	2
C₃₁×#2352	2
Q₈×C₄⋊C₄×C₃₁	2
D₄×C₄⋊C₄×C₃₁	2
C₃₁×#2334	2
C₃₁×#2314	2
C₄×C₃₁×C₂²⋊Q₈	2
C₄×C₃₁×#121	2
C₃₁×#2285	2
C₃₁×#2274	2
C₂×C₄×C₄⋊C₄×C₃₁	2
#34243	31

Subgroups — C31×#34243 #21491002

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#34243 #21491002