Subgroups — C₃₁×#34391 #21491150

Order 7936.

2458 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×#525	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2375	C₂	1	4	yes	maximal, complemented
3	C₃₁×#2363	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2362	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2357	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2353	C₂	1	2	yes	maximal, complemented
7	C₃₁×#2347	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#2346	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#2341	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#2339	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#2335	C₂	1	4	yes	maximal, complemented
12	C₃₁×#2324	C₂	1	2	yes	maximal, complemented
13	C₃₁×#2323	C₂	1	2	yes	maximal, complemented
14	C₃₁×#2319	C₂	1	2	yes	maximal, complemented
15	C₃₁×#2318	C₂	1	2	yes	maximal, complemented
16	C₃₁×#2280	C₂	1	1	yes	maximal, characteristic, complemented
17	C₃₁×#2280	C₂	1	1	yes	maximal, characteristic, complemented
18	C₂²×#41×C₃₁	C₂²	1	1	yes	characteristic
19	C₂⁴×C₄×C₃₁	C₂²	1	1	yes	characteristic
20	C₃₁×#525	C₄	1	16	yes	complemented
21	C₂×C₃₁×#123	C₂²	1	2	yes	—
22	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
23	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
24	C₂×C₃₁×#121	C₂²	1	1	yes	characteristic
25	C₂×C₃₁×#121	C₂²	1	1	yes	characteristic
26	C₂×C₃₁×#120	C₂²	1	2	yes	—
27	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	complemented
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	2	yes	complemented
31	C₂×C₃₁×#117	C₂²	1	2	yes	—
32	C₂×C₃₁×#117	C₂²	1	2	yes	—
33	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
34	C₃₁×#391	C₂²	1	2	yes	complemented
35	C₃₁×#390	C₂²	1	4	yes	—
36	C₃₁×#390	C₂²	1	4	yes	—
37	C₃₁×#390	C₂²	1	4	yes	—
38	C₃₁×#390	C₂²	1	2	yes	complemented
39	C₃₁×#389	C₂²	1	4	yes	complemented
40	C₃₁×#389	C₂²	1	2	yes	complemented
41	C₃₁×#389	C₂²	1	2	yes	complemented
42	C₃₁×#388	C₂²	1	4	yes	—
43	C₃₁×#388	C₂²	1	4	yes	—
44	C₃₁×#388	C₂²	1	4	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	complemented
49	C₃₁×#388	C₂²	1	4	yes	complemented
50	C₃₁×#387	C₂²	1	4	yes	complemented
51	C₃₁×#387	C₂²	1	4	yes	complemented
52	C₃₁×#386	C₂²	1	4	yes	—
53	C₃₁×#386	C₂²	1	4	yes	—
54	C₃₁×#386	C₂²	1	4	yes	—
55	C₃₁×#385	C₂²	1	4	yes	complemented
56	C₃₁×#383	C₂²	1	2	yes	complemented
57	C₃₁×#382	C₂²	1	2	yes	complemented
58	C₃₁×#381	C₂²	1	2	yes	—
59	C₃₁×#381	C₂²	1	1	yes	characteristic
60	C₃₁×#381	C₂²	1	1	yes	characteristic
61	C₃₁×#381	C₂²	1	2	yes	complemented
62	C₃₁×#380	C₂²	1	1	yes	characteristic
63	C₃₁×#380	C₂²	1	1	yes	characteristic
64	C₃₁×#380	C₂²	1	2	yes	—
65	C₃₁×#380	C₂²	1	2	yes	—
66	C₃₁×#380	C₂²	1	1	yes	characteristic
67	C₃₁×#380	C₂²	1	1	yes	characteristic
68	C₃₁×#380	C₂²	1	2	yes	complemented
69	C₃₁×#379	C₂²	1	2	yes	—
70	C₃₁×#379	C₂²	1	2	yes	—
71	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	complemented
72	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	complemented
73	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	complemented
74	C₄×C₄⋊C₄×C₃₁	C₂²	1	4	yes	complemented
75	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
76	C₄×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
77	C₄×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
78	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
79	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
80	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	complemented
81	C₄×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	complemented
82	C₃₁×#376	C₂²	1	2	yes	complemented
83	C₂×C₃₁×#95	C₂²	1	2	yes	complemented
84	C₂⁵×C₃₁	C₂³	1	1	yes	characteristic
85	C₂×#41×C₃₁	—	4	4	no	—
86	C₂×#41×C₃₁	C₂×C₄	1	8	yes	—
87	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
88	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
89	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
90	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
91	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
92	C₂³×C₄×C₃₁	—	4	4	no	—
93	C₂³×C₄×C₃₁	C₂×C₄	1	8	yes	complemented
94	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
95	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
96	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
97	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
98	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
99	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
100	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
101	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
102	C₃₁×#123	C₂×C₄	1	16	yes	complemented
103	C₃₁×C₂²⋊Q₈	C₂×C₄	1	8	yes	complemented
104	C₃₁×C₂²⋊Q₈	C₂×C₄	1	8	yes	complemented
105	C₃₁×#121	C₂×C₄	1	8	yes	—
106	C₃₁×#121	C₂×C₄	1	8	yes	—
107	C₃₁×#120	C₂×C₄	1	16	yes	complemented
108	C₄×Q₈×C₃₁	—	4	4	no	—
109	C₄×D₄×C₃₁	—	4	4	no	—
110	C₄×D₄×C₃₁	C₂×C₄	1	16	yes	complemented
111	C₄×D₄×C₃₁	C₂×C₄	1	16	yes	complemented
112	C₃₁×#117	—	4	4	no	—
113	C₃₁×#117	—	4	4	no	—
114	C₃₁×#117	C₂×C₄	1	8	yes	complemented
115	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
116	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	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	4	yes	—
120	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
121	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
122	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
123	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
124	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
125	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
126	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
127	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
128	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
129	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
130	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
131	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
132	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
133	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
134	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
135	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
136	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
137	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
138	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	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×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
143	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
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	4	yes	—
148	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
149	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
150	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
151	C₃₁×#95	C₂³	1	1	yes	characteristic
152	C₃₁×#95	C₂³	1	2	yes	—
153	C₃₁×#95	C₂³	1	4	yes	—
154	C₃₁×#95	C₂³	1	1	yes	characteristic
155	C₃₁×#95	C₂³	1	4	yes	—
156	C₃₁×#95	C₂³	1	1	yes	characteristic
157	C₃₁×#95	C₂³	1	2	yes	—
158	C₃₁×#95	C₂³	1	2	yes	—
159	C₃₁×#95	C₂³	1	1	yes	characteristic
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	C₂⁴×C₃₁	—	4	2	no	—
167	C₂⁴×C₃₁	—	4	1	no	—
168	C₂⁴×C₃₁	—	4	2	no	—
169	C₂⁴×C₃₁	—	4	1	no	—
170	C₂⁴×C₃₁	C₂²×C₄	1	4	yes	—
171	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
172	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
173	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
174	C₂⁴×C₃₁	C₂⁴	1	2	yes	—
175	#41×C₃₁	—	4	16	no	—
176	C₂×Q₈×C₃₁	—	2	4	no	—
177	C₂×Q₈×C₃₁	C₂²×C₄	1	4	yes	—
178	C₂×D₄×C₃₁	—	2	4	no	—
179	C₂×D₄×C₃₁	—	4	2	no	—
180	C₂×D₄×C₃₁	—	4	4	no	—
181	C₂×D₄×C₃₁	—	4	2	no	—
182	C₂×D₄×C₃₁	—	4	4	no	—
183	C₂×D₄×C₃₁	C₂²×C₄	1	4	yes	—
184	C₂×D₄×C₃₁	C₂²×C₄	1	8	yes	—
185	C₂×D₄×C₃₁	C₂²×C₄	1	8	yes	—
186	C₂×D₄×C₃₁	C₂²×C₄	1	8	yes	—
187	C₂²×C₄×C₃₁	—	2	4	no	—
188	C₂²×C₄×C₃₁	—	2	4	no	—
189	C₂²×C₄×C₃₁	—	2	4	no	—
190	C₂²×C₄×C₃₁	—	4	16	no	—
191	C₂²×C₄×C₃₁	—	4	2	no	—
192	C₂²×C₄×C₃₁	—	4	2	no	—
193	C₂²×C₄×C₃₁	—	4	2	no	—
194	C₂²×C₄×C₃₁	—	4	4	no	—
195	C₂²×C₄×C₃₁	—	4	4	no	—
196	C₂²×C₄×C₃₁	—	4	4	no	—
197	C₂²×C₄×C₃₁	—	4	4	no	—
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	4	yes	—
201	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
202	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
203	C₂²×C₄×C₃₁	C₂²×C₄	1	8	yes	—
204	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
205	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
206	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
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	4	yes	—
211	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
212	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
213	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
214	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
215	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
216	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
217	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
218	C₄⋊C₄×C₃₁	—	2	4	no	—
219	C₄⋊C₄×C₃₁	—	2	4	no	—
220	C₄⋊C₄×C₃₁	—	4	4	no	—
221	C₄⋊C₄×C₃₁	—	4	4	no	—
222	C₄⋊C₄×C₃₁	—	4	2	no	—
223	C₄⋊C₄×C₃₁	—	4	2	no	—
224	C₄⋊C₄×C₃₁	—	4	4	no	—
225	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
226	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
227	C₄⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
228	C₄⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
229	C₂²⋊C₄×C₃₁	—	4	2	no	—
230	C₂²⋊C₄×C₃₁	—	4	2	no	—
231	C₂²⋊C₄×C₃₁	—	4	2	no	—
232	C₂²⋊C₄×C₃₁	—	4	4	no	—
233	C₂²⋊C₄×C₃₁	—	4	4	no	—
234	C₂²⋊C₄×C₃₁	—	4	4	no	—
235	C₂²⋊C₄×C₃₁	—	4	2	no	—
236	C₂²⋊C₄×C₃₁	—	4	2	no	—
237	C₂²⋊C₄×C₃₁	—	4	2	no	—
238	C₂²⋊C₄×C₃₁	—	4	2	no	—
239	C₂²⋊C₄×C₃₁	—	4	2	no	—
240	C₂²⋊C₄×C₃₁	—	4	4	no	—
241	C₂²⋊C₄×C₃₁	—	4	4	no	—
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	C₂²⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
247	C₄²×C₃₁	—	2	4	no	—
248	C₄²×C₃₁	—	2	4	no	—
249	C₄²×C₃₁	—	4	4	no	—
250	C₄²×C₃₁	—	4	4	no	—
251	C₄²×C₃₁	—	4	4	no	—
252	C₄²×C₃₁	—	4	4	no	—
253	C₄²×C₃₁	C₂²×C₄	1	8	yes	—
254	#34391	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
255	C₂³×C₃₁	—	2	2	no	—
256	C₂³×C₃₁	—	2	2	no	—
257	C₂³×C₃₁	—	2	4	no	—
258	C₂³×C₃₁	—	2	2	no	—
259	C₂³×C₃₁	—	2	2	no	—
260	C₂³×C₃₁	—	2	1	no	—
261	C₂³×C₃₁	—	2	1	no	—
262	C₂³×C₃₁	—	2	2	no	—
263	C₂³×C₃₁	—	2	2	no	—
264	C₂³×C₃₁	—	2	1	no	—
265	C₂³×C₃₁	—	2	1	no	—
266	C₂³×C₃₁	—	2	4	no	—
267	C₂³×C₃₁	—	2	2	no	—
268	C₂³×C₃₁	—	2	1	no	—
269	C₂³×C₃₁	—	2	2	no	—
270	C₂³×C₃₁	—	2	1	no	—
271	C₂³×C₃₁	—	4	4	no	—
272	C₂³×C₃₁	—	4	4	no	—
273	C₂³×C₃₁	—	4	4	no	—
274	C₂³×C₃₁	—	4	8	no	—
275	C₂³×C₃₁	—	4	8	no	—
276	C₂³×C₃₁	C₂³×C₄	1	2	yes	—
277	C₂³×C₃₁	C₂³×C₄	1	2	yes	—
278	C₂³×C₃₁	C₂³×C₄	1	2	yes	—
279	C₂³×C₃₁	C₂³×C₄	1	4	yes	—
280	C₂³×C₃₁	C₂⁵	1	1	yes	characteristic
281	Q₈×C₃₁	—	2	8	no	—
282	D₄×C₃₁	—	2	8	no	—
283	D₄×C₃₁	—	4	8	no	—
284	D₄×C₃₁	—	4	8	no	—
285	D₄×C₃₁	—	4	8	no	—
286	C₂×C₄×C₃₁	—	2	2	no	—
287	C₂×C₄×C₃₁	—	2	4	no	—
288	C₂×C₄×C₃₁	—	2	2	no	—
289	C₂×C₄×C₃₁	—	2	2	no	—
290	C₂×C₄×C₃₁	—	2	2	no	—
291	C₂×C₄×C₃₁	—	2	4	no	—
292	C₂×C₄×C₃₁	—	2	2	no	—
293	C₂×C₄×C₃₁	—	2	8	no	—
294	C₂×C₄×C₃₁	—	2	8	no	—
295	C₂×C₄×C₃₁	—	2	8	no	—
296	C₂×C₄×C₃₁	—	2	2	no	—
297	C₂×C₄×C₃₁	—	2	2	no	—
298	C₂×C₄×C₃₁	—	2	2	no	—
299	C₂×C₄×C₃₁	—	2	2	no	—
300	C₂×C₄×C₃₁	—	2	4	no	—
301	C₂×C₄×C₃₁	—	2	4	no	—
302	C₂×C₄×C₃₁	—	2	2	no	—
303	C₂×C₄×C₃₁	—	2	2	no	—
304	C₂×C₄×C₃₁	—	2	4	no	—
305	C₂×C₄×C₃₁	—	2	4	no	—
306	C₂×C₄×C₃₁	—	2	2	no	—
307	C₂×C₄×C₃₁	—	2	2	no	—
308	C₂×C₄×C₃₁	—	2	4	no	—
309	C₂×C₄×C₃₁	—	2	2	no	—
310	C₂×C₄×C₃₁	—	2	2	no	—
311	C₂×C₄×C₃₁	—	2	4	no	—
312	C₂×C₄×C₃₁	—	2	2	no	—
313	C₂×C₄×C₃₁	—	2	2	no	—
314	C₂×C₄×C₃₁	—	4	4	no	—
315	C₂×C₄×C₃₁	—	4	4	no	—
316	C₂×C₄×C₃₁	—	4	4	no	—
317	C₂×C₄×C₃₁	—	4	8	no	—
318	C₂×C₄×C₃₁	—	8	8	no	—
319	C₂×C₄×C₃₁	C₂³×C₄	1	2	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	C₂³×C₄	1	4	yes	—
325	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
326	C₂×C₄×C₃₁	H(5,𝔽₂)	1	4	yes	—
327	C₂×#525	C₂×C₃₁	1	1	yes	characteristic
328	#2375	C₂×C₃₁	1	4	yes	complemented
329	#2363	C₂×C₃₁	1	2	yes	complemented
330	#2362	C₂×C₃₁	1	2	yes	complemented
331	#2357	C₂×C₃₁	1	2	yes	complemented
332	#2353	C₂×C₃₁	1	2	yes	complemented
333	#2347	C₂×C₃₁	1	1	yes	characteristic, complemented
334	#2346	C₂×C₃₁	1	1	yes	characteristic, complemented
335	#2341	C₂×C₃₁	1	1	yes	characteristic, complemented
336	#2339	C₂×C₃₁	1	1	yes	characteristic, complemented
337	#2335	C₂×C₃₁	1	4	yes	complemented
338	#2324	C₂×C₃₁	1	2	yes	complemented
339	#2323	C₂×C₃₁	1	2	yes	complemented
340	#2319	C₂×C₃₁	1	2	yes	complemented
341	#2318	C₂×C₃₁	1	2	yes	complemented
342	#2280	C₂×C₃₁	1	1	yes	characteristic, complemented
343	#2280	C₂×C₃₁	1	1	yes	characteristic, complemented
344	C₂²×C₃₁	—	2	2	no	—
345	C₂²×C₃₁	—	2	2	no	—
346	C₂²×C₃₁	—	2	2	no	—
347	C₂²×C₃₁	—	2	4	no	—
348	C₂²×C₃₁	—	2	2	no	—
349	C₂²×C₃₁	—	2	2	no	—
350	C₂²×C₃₁	—	2	4	no	—
351	C₂²×C₃₁	—	2	4	no	—
352	C₂²×C₃₁	—	2	2	no	—
353	C₂²×C₃₁	—	2	2	no	—
354	C₂²×C₃₁	—	2	2	no	—
355	C₂²×C₃₁	—	2	2	no	—
356	C₂²×C₃₁	—	4	16	no	—
357	C₂²×C₃₁	—	4	4	no	—
358	C₂²×C₃₁	—	4	2	no	—
359	C₂²×C₃₁	—	4	2	no	—
360	C₂²×C₃₁	—	4	2	no	—
361	C₂²×C₃₁	—	4	4	no	—
362	C₂²×C₃₁	—	4	2	no	—
363	C₂²×C₃₁	—	4	2	no	—
364	C₂²×C₃₁	—	4	2	no	—
365	C₂²×C₃₁	C₂⁴×C₄	1	1	yes	characteristic
366	C₂²×C₃₁	C₂×H(5,𝔽₂)	1	2	yes	—
367	C₂²×C₃₁	#585	1	1	yes	characteristic
368	C₂²×C₃₁	#585	1	2	yes	—
369	C₂²×C₃₁	#585	1	1	yes	characteristic
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	—	4	2	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	—	4	2	no	—
382	C₄×C₃₁	#518	1	4	yes	—
383	C₂²×#41	C₂²×C₃₁	1	1	yes	characteristic
384	C₂⁴×C₄	C₂²×C₃₁	1	1	yes	characteristic
385	#525	C₄×C₃₁	1	16	yes	complemented
386	C₂×#123	C₂²×C₃₁	1	2	yes	—
387	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
388	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
389	C₂×#121	C₂²×C₃₁	1	1	yes	characteristic
390	C₂×#121	C₂²×C₃₁	1	1	yes	characteristic
391	C₂×#120	C₂²×C₃₁	1	2	yes	—
392	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	complemented
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	2	yes	complemented
396	C₂×#117	C₂²×C₃₁	1	2	yes	—
397	C₂×#117	C₂²×C₃₁	1	2	yes	—
398	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
399	#391	C₂²×C₃₁	1	2	yes	complemented
400	#390	C₂²×C₃₁	1	4	yes	—
401	#390	C₂²×C₃₁	1	4	yes	—
402	#390	C₂²×C₃₁	1	4	yes	—
403	#390	C₂²×C₃₁	1	2	yes	complemented
404	#389	C₂²×C₃₁	1	4	yes	complemented
405	#389	C₂²×C₃₁	1	2	yes	complemented
406	#389	C₂²×C₃₁	1	2	yes	complemented
407	#388	C₂²×C₃₁	1	4	yes	—
408	#388	C₂²×C₃₁	1	4	yes	—
409	#388	C₂²×C₃₁	1	4	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	complemented
414	#388	C₂²×C₃₁	1	4	yes	complemented
415	#387	C₂²×C₃₁	1	4	yes	complemented
416	#387	C₂²×C₃₁	1	4	yes	complemented
417	#386	C₂²×C₃₁	1	4	yes	—
418	#386	C₂²×C₃₁	1	4	yes	—
419	#386	C₂²×C₃₁	1	4	yes	—
420	#385	C₂²×C₃₁	1	4	yes	complemented
421	#383	C₂²×C₃₁	1	2	yes	complemented
422	#382	C₂²×C₃₁	1	2	yes	complemented
423	#381	C₂²×C₃₁	1	2	yes	—
424	#381	C₂²×C₃₁	1	1	yes	characteristic
425	#381	C₂²×C₃₁	1	1	yes	characteristic
426	#381	C₂²×C₃₁	1	2	yes	complemented
427	#380	C₂²×C₃₁	1	1	yes	characteristic
428	#380	C₂²×C₃₁	1	1	yes	characteristic
429	#380	C₂²×C₃₁	1	2	yes	—
430	#380	C₂²×C₃₁	1	2	yes	—
431	#380	C₂²×C₃₁	1	1	yes	characteristic
432	#380	C₂²×C₃₁	1	1	yes	characteristic
433	#380	C₂²×C₃₁	1	2	yes	complemented
434	#379	C₂²×C₃₁	1	2	yes	—
435	#379	C₂²×C₃₁	1	2	yes	—
436	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	complemented
437	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	complemented
438	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	complemented
439	C₄×C₄⋊C₄	C₂²×C₃₁	1	4	yes	complemented
440	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
441	C₄×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
442	C₄×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
443	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
444	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
445	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	complemented
446	C₄×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	complemented
447	#376	C₂²×C₃₁	1	2	yes	complemented
448	C₂×#95	C₂²×C₃₁	1	2	yes	complemented
449	C₂×C₃₁	—	4	4	no	—
450	C₂×C₃₁	—	4	2	no	—
451	C₂×C₃₁	—	4	2	no	—
452	C₂×C₃₁	—	4	2	no	—
453	C₂×C₃₁	C₂×#518	1	1	yes	characteristic
454	C₂×C₃₁	#3428	1	1	yes	characteristic
455	C₂×C₃₁	#3428	1	1	yes	characteristic
456	C₂×C₃₁	#3539	1	2	yes	—
457	C₂×C₃₁	#3540	1	2	yes	—
458	C₂⁵	C₂³×C₃₁	1	1	yes	characteristic
459	C₂×#41	—	4	4	no	—
460	C₂×#41	C₂×C₄×C₃₁	1	8	yes	—
461	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
462	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
463	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
464	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
465	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
466	C₂³×C₄	—	4	4	no	—
467	C₂³×C₄	C₂×C₄×C₃₁	1	8	yes	complemented
468	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
469	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
470	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
471	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
472	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
473	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
474	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
475	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
476	#123	C₂×C₄×C₃₁	1	16	yes	complemented
477	C₂²⋊Q₈	C₂×C₄×C₃₁	1	8	yes	complemented
478	C₂²⋊Q₈	C₂×C₄×C₃₁	1	8	yes	complemented
479	#121	C₂×C₄×C₃₁	1	8	yes	—
480	#121	C₂×C₄×C₃₁	1	8	yes	—
481	#120	C₂×C₄×C₃₁	1	16	yes	complemented
482	C₄×Q₈	—	4	4	no	—
483	C₄×D₄	—	4	4	no	—
484	C₄×D₄	C₂×C₄×C₃₁	1	16	yes	complemented
485	C₄×D₄	C₂×C₄×C₃₁	1	16	yes	complemented
486	#117	—	4	4	no	—
487	#117	—	4	4	no	—
488	#117	C₂×C₄×C₃₁	1	8	yes	complemented
489	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
490	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	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	4	yes	—
494	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
495	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
496	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
497	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
498	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
499	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
500	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
501	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
502	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
503	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
504	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
505	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
506	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
507	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
508	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
509	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
510	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
511	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
512	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	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×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
517	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
518	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
519	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
520	C₂×C₄²	C₂³×C₃₁	1	4	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	#95	C₂³×C₃₁	1	1	yes	characteristic
526	#95	C₂³×C₃₁	1	2	yes	—
527	#95	C₂³×C₃₁	1	4	yes	—
528	#95	C₂³×C₃₁	1	1	yes	characteristic
529	#95	C₂³×C₃₁	1	4	yes	—
530	#95	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	1	yes	characteristic
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	C₃₁	#34391	1	1	yes	characteristic, direct product factor, Sylow
541	C₂⁴	—	4	2	no	—
542	C₂⁴	—	4	1	no	—
543	C₂⁴	—	4	2	no	—
544	C₂⁴	—	4	1	no	—
545	C₂⁴	C₂²×C₄×C₃₁	1	4	yes	—
546	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
547	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
548	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
549	C₂⁴	C₂⁴×C₃₁	1	2	yes	—
550	#41	—	4	16	no	—
551	C₂×Q₈	—	2	4	no	—
552	C₂×Q₈	C₂²×C₄×C₃₁	1	4	yes	—
553	C₂×D₄	—	2	4	no	—
554	C₂×D₄	—	4	2	no	—
555	C₂×D₄	—	4	4	no	—
556	C₂×D₄	—	4	2	no	—
557	C₂×D₄	—	4	4	no	—
558	C₂×D₄	C₂²×C₄×C₃₁	1	4	yes	—
559	C₂×D₄	C₂²×C₄×C₃₁	1	8	yes	—
560	C₂×D₄	C₂²×C₄×C₃₁	1	8	yes	—
561	C₂×D₄	C₂²×C₄×C₃₁	1	8	yes	—
562	C₂²×C₄	—	2	4	no	—
563	C₂²×C₄	—	2	4	no	—
564	C₂²×C₄	—	2	4	no	—
565	C₂²×C₄	—	4	16	no	—
566	C₂²×C₄	—	4	2	no	—
567	C₂²×C₄	—	4	2	no	—
568	C₂²×C₄	—	4	2	no	—
569	C₂²×C₄	—	4	4	no	—
570	C₂²×C₄	—	4	4	no	—
571	C₂²×C₄	—	4	4	no	—
572	C₂²×C₄	—	4	4	no	—
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	4	yes	—
576	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
577	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
578	C₂²×C₄	C₂²×C₄×C₃₁	1	8	yes	—
579	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
580	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
581	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
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	4	yes	—
586	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
587	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
588	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
589	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
590	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
591	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
592	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
593	C₄⋊C₄	—	2	4	no	—
594	C₄⋊C₄	—	2	4	no	—
595	C₄⋊C₄	—	4	4	no	—
596	C₄⋊C₄	—	4	4	no	—
597	C₄⋊C₄	—	4	2	no	—
598	C₄⋊C₄	—	4	2	no	—
599	C₄⋊C₄	—	4	4	no	—
600	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
601	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
602	C₄⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
603	C₄⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
604	C₂²⋊C₄	—	4	2	no	—
605	C₂²⋊C₄	—	4	2	no	—
606	C₂²⋊C₄	—	4	2	no	—
607	C₂²⋊C₄	—	4	4	no	—
608	C₂²⋊C₄	—	4	4	no	—
609	C₂²⋊C₄	—	4	4	no	—
610	C₂²⋊C₄	—	4	2	no	—
611	C₂²⋊C₄	—	4	2	no	—
612	C₂²⋊C₄	—	4	2	no	—
613	C₂²⋊C₄	—	4	2	no	—
614	C₂²⋊C₄	—	4	2	no	—
615	C₂²⋊C₄	—	4	4	no	—
616	C₂²⋊C₄	—	4	4	no	—
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	C₂²⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
622	C₄²	—	2	4	no	—
623	C₄²	—	2	4	no	—
624	C₄²	—	4	4	no	—
625	C₄²	—	4	4	no	—
626	C₄²	—	4	4	no	—
627	C₄²	—	4	4	no	—
628	C₄²	C₂²×C₄×C₃₁	1	8	yes	—
629	C₂³	—	2	2	no	—
630	C₂³	—	2	2	no	—
631	C₂³	—	2	4	no	—
632	C₂³	—	2	2	no	—
633	C₂³	—	2	2	no	—
634	C₂³	—	2	1	no	—
635	C₂³	—	2	1	no	—
636	C₂³	—	2	2	no	—
637	C₂³	—	2	2	no	—
638	C₂³	—	2	1	no	—
639	C₂³	—	2	1	no	—
640	C₂³	—	2	4	no	—
641	C₂³	—	2	2	no	—
642	C₂³	—	2	1	no	—
643	C₂³	—	2	2	no	—
644	C₂³	—	2	1	no	—
645	C₂³	—	4	4	no	—
646	C₂³	—	4	4	no	—
647	C₂³	—	4	4	no	—
648	C₂³	—	4	8	no	—
649	C₂³	—	4	8	no	—
650	C₂³	C₂³×C₄×C₃₁	1	2	yes	—
651	C₂³	C₂³×C₄×C₃₁	1	2	yes	—
652	C₂³	C₂³×C₄×C₃₁	1	2	yes	—
653	C₂³	C₂³×C₄×C₃₁	1	4	yes	—
654	C₂³	C₂⁵×C₃₁	1	1	yes	characteristic
655	Q₈	—	2	8	no	—
656	D₄	—	2	8	no	—
657	D₄	—	4	8	no	—
658	D₄	—	4	8	no	—
659	D₄	—	4	8	no	—
660	C₂×C₄	—	2	2	no	—
661	C₂×C₄	—	2	4	no	—
662	C₂×C₄	—	2	2	no	—
663	C₂×C₄	—	2	2	no	—
664	C₂×C₄	—	2	2	no	—
665	C₂×C₄	—	2	4	no	—
666	C₂×C₄	—	2	2	no	—
667	C₂×C₄	—	2	8	no	—
668	C₂×C₄	—	2	8	no	—
669	C₂×C₄	—	2	8	no	—
670	C₂×C₄	—	2	2	no	—
671	C₂×C₄	—	2	2	no	—
672	C₂×C₄	—	2	2	no	—
673	C₂×C₄	—	2	2	no	—
674	C₂×C₄	—	2	4	no	—
675	C₂×C₄	—	2	4	no	—
676	C₂×C₄	—	2	2	no	—
677	C₂×C₄	—	2	2	no	—
678	C₂×C₄	—	2	4	no	—
679	C₂×C₄	—	2	4	no	—
680	C₂×C₄	—	2	2	no	—
681	C₂×C₄	—	2	2	no	—
682	C₂×C₄	—	2	4	no	—
683	C₂×C₄	—	2	2	no	—
684	C₂×C₄	—	2	2	no	—
685	C₂×C₄	—	2	4	no	—
686	C₂×C₄	—	2	2	no	—
687	C₂×C₄	—	2	2	no	—
688	C₂×C₄	—	4	4	no	—
689	C₂×C₄	—	4	4	no	—
690	C₂×C₄	—	4	4	no	—
691	C₂×C₄	—	4	8	no	—
692	C₂×C₄	—	8	8	no	—
693	C₂×C₄	C₂³×C₄×C₃₁	1	2	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	C₂³×C₄×C₃₁	1	4	yes	—
699	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
700	C₂×C₄	C₃₁×H(5,𝔽₂)	1	4	yes	—
701	C₂²	—	2	2	no	—
702	C₂²	—	2	2	no	—
703	C₂²	—	2	2	no	—
704	C₂²	—	2	4	no	—
705	C₂²	—	2	2	no	—
706	C₂²	—	2	2	no	—
707	C₂²	—	2	4	no	—
708	C₂²	—	2	4	no	—
709	C₂²	—	2	2	no	—
710	C₂²	—	2	2	no	—
711	C₂²	—	2	2	no	—
712	C₂²	—	2	2	no	—
713	C₂²	—	4	16	no	—
714	C₂²	—	4	2	no	—
715	C₂²	—	4	2	no	—
716	C₂²	—	4	2	no	—
717	C₂²	—	4	4	no	—
718	C₂²	—	4	4	no	—
719	C₂²	—	4	2	no	—
720	C₂²	—	4	2	no	—
721	C₂²	—	4	2	no	—
722	C₂²	C₂⁴×C₄×C₃₁	1	1	yes	characteristic
723	C₂²	C₂×C₃₁×H(5,𝔽₂)	1	2	yes	—
724	C₂²	C₃₁×#585	1	1	yes	characteristic
725	C₂²	C₃₁×#585	1	2	yes	—
726	C₂²	C₃₁×#585	1	1	yes	characteristic
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	—	4	2	no	—
734	C₄	—	4	2	no	—
735	C₄	—	4	4	no	—
736	C₄	—	4	2	no	—
737	C₄	—	4	4	no	—
738	C₄	—	4	2	no	—
739	C₄	C₃₁×#518	1	4	yes	—
740	C₂	—	4	2	no	—
741	C₂	—	4	2	no	—
742	C₂	—	4	2	no	—
743	C₂	—	4	4	no	—
744	C₂	C₂×C₃₁×#518	1	1	yes	characteristic
745	C₂	C₃₁×#3428	1	1	yes	characteristic
746	C₂	C₃₁×#3428	1	1	yes	characteristic
747	C₂	C₃₁×#3539	1	2	yes	—
748	C₂	C₃₁×#3540	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₃₁
Frattini, cosocle	C₂³	C₂⁵×C₃₁
prime radical	C₂×C₃₁×#525	C₂
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×#2375	C₂	semidirect
3. C₃₁×#2363	C₂	semidirect
4. C₃₁×#2362	C₂	semidirect
5. C₃₁×#2357	C₂	semidirect
6. C₃₁×#2353	C₂	semidirect
7. C₃₁×#2347	C₂	semidirect
8. C₃₁×#2346	C₂	semidirect
9. C₃₁×#2341	C₂	semidirect
10. C₃₁×#2339	C₂	semidirect
11. C₃₁×#2335	C₂	semidirect
12. C₃₁×#2324	C₂	semidirect
13. C₃₁×#2323	C₂	semidirect
14. C₃₁×#2319	C₂	semidirect
15. C₃₁×#2318	C₂	semidirect
16. C₃₁×#2280	C₂	semidirect
17. C₃₁×#2280	C₂	semidirect
20. C₃₁×#525	C₄	semidirect
27. C₂×C₄×Q₈×C₃₁	C₂²	semidirect
30. C₂×C₄×D₄×C₃₁	C₂²	semidirect
34. C₃₁×#391	C₂²	semidirect
38. C₃₁×#390	C₂²	semidirect
39. C₃₁×#389	C₂²	semidirect
40. C₃₁×#389	C₂²	semidirect
41. C₃₁×#389	C₂²	semidirect
48. C₃₁×#388	C₂²	semidirect
49. C₃₁×#388	C₂²	semidirect
50. C₃₁×#387	C₂²	semidirect
51. C₃₁×#387	C₂²	semidirect
55. C₃₁×#385	C₂²	semidirect
56. C₃₁×#383	C₂²	semidirect
57. C₃₁×#382	C₂²	semidirect
61. C₃₁×#381	C₂²	semidirect
68. C₃₁×#380	C₂²	semidirect
71. C₄×C₄⋊C₄×C₃₁	C₂²	semidirect
72. C₄×C₄⋊C₄×C₃₁	C₂²	semidirect
73. C₄×C₄⋊C₄×C₃₁	C₂²	semidirect
74. C₄×C₄⋊C₄×C₃₁	C₂²	semidirect
80. C₄×C₂²⋊C₄×C₃₁	C₂²	semidirect
81. C₄×C₂²⋊C₄×C₃₁	C₂²	semidirect
82. C₃₁×#376	C₂²	semidirect
83. C₂×C₃₁×#95	C₂²	semidirect
93. C₂³×C₄×C₃₁	C₂×C₄	semidirect
102. C₃₁×#123	C₂×C₄	semidirect
103. C₃₁×C₂²⋊Q₈	C₂×C₄	semidirect
104. C₃₁×C₂²⋊Q₈	C₂×C₄	semidirect
107. C₃₁×#120	C₂×C₄	semidirect
110. C₄×D₄×C₃₁	C₂×C₄	semidirect
111. C₄×D₄×C₃₁	C₂×C₄	semidirect
114. C₃₁×#117	C₂×C₄	semidirect
254. #34391	C₃₁	direct (N × H)
328. #2375	C₂×C₃₁	semidirect
329. #2363	C₂×C₃₁	semidirect
330. #2362	C₂×C₃₁	semidirect
331. #2357	C₂×C₃₁	semidirect
332. #2353	C₂×C₃₁	semidirect
333. #2347	C₂×C₃₁	semidirect
334. #2346	C₂×C₃₁	semidirect
335. #2341	C₂×C₃₁	semidirect
336. #2339	C₂×C₃₁	semidirect
337. #2335	C₂×C₃₁	semidirect
338. #2324	C₂×C₃₁	semidirect
339. #2323	C₂×C₃₁	semidirect
340. #2319	C₂×C₃₁	semidirect
341. #2318	C₂×C₃₁	semidirect
342. #2280	C₂×C₃₁	semidirect
343. #2280	C₂×C₃₁	semidirect
385. #525	C₄×C₃₁	semidirect
392. C₂×C₄×Q₈	C₂²×C₃₁	semidirect
395. C₂×C₄×D₄	C₂²×C₃₁	semidirect
399. #391	C₂²×C₃₁	semidirect
403. #390	C₂²×C₃₁	semidirect
404. #389	C₂²×C₃₁	semidirect
405. #389	C₂²×C₃₁	semidirect
406. #389	C₂²×C₃₁	semidirect
413. #388	C₂²×C₃₁	semidirect
414. #388	C₂²×C₃₁	semidirect
415. #387	C₂²×C₃₁	semidirect
416. #387	C₂²×C₃₁	semidirect
420. #385	C₂²×C₃₁	semidirect
421. #383	C₂²×C₃₁	semidirect
422. #382	C₂²×C₃₁	semidirect
426. #381	C₂²×C₃₁	semidirect
433. #380	C₂²×C₃₁	semidirect
436. C₄×C₄⋊C₄	C₂²×C₃₁	semidirect
437. C₄×C₄⋊C₄	C₂²×C₃₁	semidirect
438. C₄×C₄⋊C₄	C₂²×C₃₁	semidirect
439. C₄×C₄⋊C₄	C₂²×C₃₁	semidirect
445. C₄×C₂²⋊C₄	C₂²×C₃₁	semidirect
446. C₄×C₂²⋊C₄	C₂²×C₃₁	semidirect
447. #376	C₂²×C₃₁	semidirect
448. C₂×#95	C₂²×C₃₁	semidirect
467. C₂³×C₄	C₂×C₄×C₃₁	semidirect
476. #123	C₂×C₄×C₃₁	semidirect
477. C₂²⋊Q₈	C₂×C₄×C₃₁	semidirect
478. C₂²⋊Q₈	C₂×C₄×C₃₁	semidirect
481. #120	C₂×C₄×C₃₁	semidirect
484. C₄×D₄	C₂×C₄×C₃₁	semidirect
485. C₄×D₄	C₂×C₄×C₃₁	semidirect
488. #117	C₂×C₄×C₃₁	semidirect
540. C₃₁	#34391	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×#525	2
C₃₁×#2375	2
C₃₁×#2363	2
C₃₁×#2362	2
C₃₁×#2357	2
C₃₁×#2353	2
C₃₁×#2347	2
C₃₁×#2346	2
C₃₁×#2341	2
C₃₁×#2339	2
C₃₁×#2335	2
C₃₁×#2324	2
C₃₁×#2323	2
C₃₁×#2319	2
C₃₁×#2318	2
C₃₁×#2280	2
#34391	31

Subgroups — C31×#34391 #21491150

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#34391 #21491150