Subgroups — C₃₁×#18665 #21475424

Order 7936.

926 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3501	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#3249	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3170	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#3077	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1649	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1644	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1637	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1632	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1550	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1493	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₃₁×D₄⋊C₈	C₂	1	1	yes	maximal, characteristic
12	C₃₁×#554	—	2	1	no	—
13	C₃₁×#554	—	2	2	no	—
14	C₃₁×#554	—	2	2	no	—
15	C₃₁×#554	C₂²	1	2	yes	—
16	D₄×Q₈×C₃₁	—	2	1	no	—
17	D₄×Q₈×C₃₁	C₂²	1	2	yes	—
18	C₃₁×#544	—	2	1	no	—
19	C₃₁×#543	—	2	2	no	—
20	C₃₁×#533	C₂²	1	1	yes	characteristic
21	C₃₁×#533	C₂²	1	1	yes	characteristic
22	C₃₁×#533	C₂²	1	1	yes	characteristic
23	C₂×C₃₁×C₂²⋊Q₈	—	2	2	no	—
24	C₃₁×#518	—	2	1	no	—
25	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
26	C₃₁×#500	C₂²	1	1	yes	characteristic, complemented
27	C₃₁×#494	C₂²	1	1	yes	characteristic, complemented
28	C₃₁×#487	C₂²	1	1	yes	characteristic, complemented
29	C₃₁×#487	C₂²	1	1	yes	characteristic, complemented
30	C₃₁×#478	—	2	2	no	—
31	C₃₁×#477	—	2	1	no	—
32	C₃₁×#477	C₂²	1	2	yes	complemented
33	C₃₁×#476	C₂²	1	2	yes	—
34	C₃₁×#474	C₂²	1	2	yes	—
35	C₃₁×#467	—	2	2	no	—
36	C₃₁×#467	—	2	2	no	—
37	C₃₁×#462	—	2	1	no	—
38	C₃₁×#462	C₂²	1	2	yes	complemented
39	C₄×Q₁₆×C₃₁	—	2	2	no	—
40	C₃₁×#428	—	2	1	no	—
41	C₃₁×#423	C₂²	1	1	yes	characteristic
42	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
43	C₃₁×#418	—	2	1	no	—
44	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
45	C₃₁×#333	C₂²	1	2	yes	complemented
46	C₃₁×#332	C₂²	1	2	yes	complemented
47	C₃₁×Q₈⋊C₈	C₂²	1	2	yes	complemented
48	C₃₁×Q₈⋊C₈	C₂²	1	2	yes	complemented
49	C₃₁×D₄⋊C₈	C₂²	1	2	yes	—
50	C₃₁×D₄⋊C₈	C₂²	1	2	yes	—
51	C₂²×Q₈×C₃₁	—	2	2	no	—
52	C₂²×D₄×C₃₁	D₄	1	1	yes	characteristic
53	C₂³×C₄×C₃₁	—	2	1	no	—
54	C₂×Q₁₆×C₃₁	—	2	2	no	—
55	C₂×Q₁₆×C₃₁	—	2	2	no	—
56	C₂×M₁₆×C₃₁	—	2	1	no	—
57	C₂²×C₈×C₃₁	—	2	1	no	—
58	C₂²×C₈×C₃₁	—	2	1	no	—
59	C₃₁×C₄⋊Q₈	—	2	2	no	—
60	C₃₁×C₄⋊Q₈	—	2	2	no	—
61	C₃₁×C₄⋊Q₈	D₄	1	1	yes	characteristic
62	C₃₁×C₄⋊Q₈	D₄	1	1	yes	characteristic
63	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
64	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
65	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
66	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
67	C₃₁×#125	—	2	2	no	—
68	C₃₁×#125	—	2	2	no	—
69	C₃₁×#125	—	2	1	no	—
70	C₃₁×#125	—	2	1	no	—
71	C₃₁×#125	—	2	1	no	—
72	C₃₁×C₂²⋊Q₈	—	2	4	no	—
73	C₃₁×C₂²⋊Q₈	—	2	4	no	—
74	C₃₁×C₂²⋊Q₈	—	2	4	no	—
75	C₃₁×C₂²⋊Q₈	—	2	4	no	—
76	C₃₁×C₂²⋊Q₈	—	2	2	no	—
77	C₃₁×C₂²⋊Q₈	—	2	4	no	—
78	C₃₁×C₂²⋊Q₈	—	2	1	no	—
79	C₃₁×C₂²⋊Q₈	—	2	1	no	—
80	C₃₁×C₂²⋊Q₈	—	2	2	no	—
81	C₃₁×C₂²⋊Q₈	—	2	4	no	—
82	C₃₁×C₂²⋊Q₈	—	2	1	no	—
83	C₃₁×C₂²⋊Q₈	—	2	1	no	—
84	C₃₁×C₂²⋊Q₈	D₄	1	2	yes	—
85	C₃₁×C₂²⋊Q₈	D₄	1	2	yes	—
86	C₄×Q₈×C₃₁	—	2	1	no	—
87	C₄×Q₈×C₃₁	—	2	1	no	—
88	C₄×Q₈×C₃₁	—	2	1	no	—
89	C₄×Q₈×C₃₁	—	2	1	no	—
90	C₄×Q₈×C₃₁	D₄	1	2	yes	—
91	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
92	C₄×D₄×C₃₁	—	2	2	no	—
93	C₄×D₄×C₃₁	—	2	2	no	—
94	C₄×D₄×C₃₁	—	2	2	no	—
95	C₄×D₄×C₃₁	—	2	2	no	—
96	C₄×D₄×C₃₁	—	2	1	no	—
97	C₄×D₄×C₃₁	—	2	1	no	—
98	C₄×D₄×C₃₁	D₄	1	2	yes	—
99	C₄×D₄×C₃₁	C₂³	1	2	yes	—
100	C₃₁×#117	—	2	1	no	—
101	C₃₁×#117	—	2	1	no	—
102	C₃₁×#117	—	2	1	no	—
103	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
104	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
105	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
106	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
107	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
108	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
109	C₃₁×#107	—	2	2	no	—
110	C₃₁×#107	—	2	2	no	—
111	C₃₁×#107	—	2	2	no	—
112	C₃₁×#106	—	2	2	no	—
113	C₃₁×C₄⋊C₈	—	2	1	no	—
114	C₃₁×C₄⋊C₈	—	2	1	no	—
115	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
116	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
117	C₃₁×#103	—	2	2	no	—
118	C₃₁×#103	—	2	2	no	—
119	C₃₁×#103	—	2	2	no	—
120	C₃₁×#103	—	2	2	no	—
121	C₃₁×#102	—	2	2	no	—
122	C₃₁×#102	—	2	2	no	—
123	C₄×C₈×C₃₁	—	2	1	no	—
124	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
125	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
126	C₂⁴×C₃₁	—	2	1	no	—
127	C₂×Q₈×C₃₁	—	2	4	no	—
128	C₂×Q₈×C₃₁	—	2	1	no	—
129	C₂×Q₈×C₃₁	—	2	1	no	—
130	C₂×Q₈×C₃₁	—	2	1	no	—
131	C₂×Q₈×C₃₁	—	2	1	no	—
132	C₂×Q₈×C₃₁	—	8	2	no	—
133	C₂×Q₈×C₃₁	—	8	2	no	—
134	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
135	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
136	C₂×D₄×C₃₁	—	2	1	no	—
137	C₂×D₄×C₃₁	—	2	2	no	—
138	C₂×D₄×C₃₁	Q₁₆	1	4	yes	complemented
139	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
140	C₂²×C₄×C₃₁	—	2	4	no	—
141	C₂²×C₄×C₃₁	—	2	4	no	—
142	C₂²×C₄×C₃₁	—	2	1	no	—
143	C₂²×C₄×C₃₁	—	2	2	no	—
144	C₂²×C₄×C₃₁	—	2	2	no	—
145	C₂²×C₄×C₃₁	—	8	1	no	—
146	C₂²×C₄×C₃₁	—	8	1	no	—
147	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
148	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
149	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
150	Q₁₆×C₃₁	—	4	4	no	—
151	Q₁₆×C₃₁	—	8	2	no	—
152	M₁₆×C₃₁	—	4	2	no	—
153	C₂×C₈×C₃₁	—	2	1	no	—
154	C₂×C₈×C₃₁	—	2	1	no	—
155	C₂×C₈×C₃₁	—	2	2	no	—
156	C₂×C₈×C₃₁	—	2	2	no	—
157	C₂×C₈×C₃₁	—	4	2	no	—
158	C₂×C₈×C₃₁	—	8	1	no	—
159	C₄⋊C₄×C₃₁	—	2	4	no	—
160	C₄⋊C₄×C₃₁	—	2	4	no	—
161	C₄⋊C₄×C₃₁	—	2	4	no	—
162	C₄⋊C₄×C₃₁	—	2	2	no	—
163	C₄⋊C₄×C₃₁	—	2	2	no	—
164	C₄⋊C₄×C₃₁	—	2	1	no	—
165	C₄⋊C₄×C₃₁	—	2	1	no	—
166	C₄⋊C₄×C₃₁	—	2	1	no	—
167	C₄⋊C₄×C₃₁	—	2	1	no	—
168	C₄⋊C₄×C₃₁	—	2	2	no	—
169	C₄⋊C₄×C₃₁	—	2	1	no	—
170	C₄⋊C₄×C₃₁	—	2	1	no	—
171	C₄⋊C₄×C₃₁	—	2	1	no	—
172	C₄⋊C₄×C₃₁	—	2	2	no	—
173	C₄⋊C₄×C₃₁	—	2	1	no	—
174	C₄⋊C₄×C₃₁	—	2	1	no	—
175	C₄⋊C₄×C₃₁	—	2	1	no	—
176	C₄⋊C₄×C₃₁	—	2	1	no	—
177	C₄⋊C₄×C₃₁	—	4	1	no	—
178	C₄⋊C₄×C₃₁	—	4	1	no	—
179	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
180	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
181	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
182	C₂²⋊C₄×C₃₁	—	2	4	no	—
183	C₂²⋊C₄×C₃₁	—	2	4	no	—
184	C₂²⋊C₄×C₃₁	—	2	2	no	—
185	C₂²⋊C₄×C₃₁	—	2	2	no	—
186	C₂²⋊C₄×C₃₁	—	2	2	no	—
187	C₂²⋊C₄×C₃₁	—	2	2	no	—
188	C₂²⋊C₄×C₃₁	—	4	1	no	—
189	C₄²×C₃₁	—	2	2	no	—
190	C₄²×C₃₁	—	2	2	no	—
191	C₄²×C₃₁	—	4	1	no	—
192	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
193	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
194	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
195	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
196	#18665	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
197	C₂³×C₃₁	—	2	2	no	—
198	C₂³×C₃₁	—	2	4	no	—
199	C₂³×C₃₁	—	8	1	no	—
200	C₂³×C₃₁	—	8	1	no	—
201	C₂³×C₃₁	#120	1	1	yes	characteristic
202	Q₈×C₃₁	—	2	2	no	—
203	Q₈×C₃₁	—	2	2	no	—
204	Q₈×C₃₁	—	4	2	no	—
205	Q₈×C₃₁	—	4	2	no	—
206	Q₈×C₃₁	—	4	1	no	—
207	Q₈×C₃₁	—	4	1	no	—
208	Q₈×C₃₁	—	4	1	no	—
209	Q₈×C₃₁	—	4	1	no	—
210	Q₈×C₃₁	—	8	1	no	—
211	Q₈×C₃₁	—	8	4	no	—
212	D₄×C₃₁	—	2	2	no	—
213	D₄×C₃₁	—	2	4	no	—
214	D₄×C₃₁	C₂×Q₁₆	1	4	yes	—
215	C₂×C₄×C₃₁	—	2	1	no	—
216	C₂×C₄×C₃₁	—	2	1	no	—
217	C₂×C₄×C₃₁	—	2	2	no	—
218	C₂×C₄×C₃₁	—	2	2	no	—
219	C₂×C₄×C₃₁	—	2	2	no	—
220	C₂×C₄×C₃₁	—	2	2	no	—
221	C₂×C₄×C₃₁	—	2	2	no	—
222	C₂×C₄×C₃₁	—	4	1	no	—
223	C₂×C₄×C₃₁	—	4	2	no	—
224	C₂×C₄×C₃₁	—	4	2	no	—
225	C₂×C₄×C₃₁	—	4	1	no	—
226	C₂×C₄×C₃₁	—	4	1	no	—
227	C₂×C₄×C₃₁	—	8	2	no	—
228	C₂×C₄×C₃₁	—	8	1	no	—
229	C₂×C₄×C₃₁	—	8	4	no	—
230	C₂×C₄×C₃₁	—	8	4	no	—
231	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
232	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
233	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
234	C₂×C₄×C₃₁	C₂×Q₁₆	1	2	yes	—
235	C₂×C₄×C₃₁	#137	1	2	yes	—
236	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
237	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
238	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
239	C₈×C₃₁	—	2	2	no	—
240	C₈×C₃₁	—	4	1	no	—
241	C₈×C₃₁	—	4	2	no	—
242	C₈×C₃₁	—	4	2	no	—
243	#3501	C₂×C₃₁	1	1	yes	characteristic
244	#3249	C₂×C₃₁	1	1	yes	characteristic, complemented
245	#3170	C₂×C₃₁	1	1	yes	characteristic, complemented
246	#3077	C₂×C₃₁	1	1	yes	characteristic
247	#1649	C₂×C₃₁	1	2	yes	complemented
248	#1644	C₂×C₃₁	1	2	yes	complemented
249	#1637	C₂×C₃₁	1	2	yes	complemented
250	#1632	C₂×C₃₁	1	2	yes	complemented
251	#1550	C₂×C₃₁	1	1	yes	characteristic, complemented
252	#1493	C₂×C₃₁	1	1	yes	characteristic, complemented
253	C₂×D₄⋊C₈	C₂×C₃₁	1	1	yes	characteristic
254	C₂²×C₃₁	—	2	1	no	—
255	C₂²×C₃₁	—	2	1	no	—
256	C₂²×C₃₁	—	2	4	no	—
257	C₂²×C₃₁	—	4	2	no	—
258	C₂²×C₃₁	—	4	2	no	—
259	C₂²×C₃₁	—	8	4	no	—
260	C₂²×C₃₁	#451	1	2	yes	—
261	C₂²×C₃₁	C₂×#120	1	1	yes	characteristic
262	C₄×C₃₁	—	2	1	no	—
263	C₄×C₃₁	—	2	1	no	—
264	C₄×C₃₁	—	2	1	no	—
265	C₄×C₃₁	—	4	2	no	—
266	C₄×C₃₁	—	4	2	no	—
267	C₄×C₃₁	—	4	2	no	—
268	C₄×C₃₁	—	4	2	no	—
269	C₄×C₃₁	—	4	2	no	—
270	C₄×C₃₁	—	8	1	no	—
271	C₄×C₃₁	—	8	1	no	—
272	C₄×C₃₁	#451	1	2	yes	—
273	C₄×C₃₁	C₂²×Q₁₆	1	1	yes	characteristic
274	C₄×C₃₁	C₂×#137	1	1	yes	characteristic
275	C₄×C₃₁	#576	1	1	yes	characteristic
276	C₄×C₃₁	#577	1	1	yes	characteristic
277	#554	—	2	1	no	—
278	#554	—	2	2	no	—
279	#554	—	2	2	no	—
280	#554	C₂²×C₃₁	1	2	yes	—
281	D₄×Q₈	—	2	1	no	—
282	D₄×Q₈	C₂²×C₃₁	1	2	yes	—
283	#544	—	2	1	no	—
284	#543	—	2	2	no	—
285	#533	C₂²×C₃₁	1	1	yes	characteristic
286	#533	C₂²×C₃₁	1	1	yes	characteristic
287	#533	C₂²×C₃₁	1	1	yes	characteristic
288	C₂×C₂²⋊Q₈	—	2	2	no	—
289	#518	—	2	1	no	—
290	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
291	#500	C₂²×C₃₁	1	1	yes	characteristic, complemented
292	#494	C₂²×C₃₁	1	1	yes	characteristic, complemented
293	#487	C₂²×C₃₁	1	1	yes	characteristic, complemented
294	#487	C₂²×C₃₁	1	1	yes	characteristic, complemented
295	#478	—	2	2	no	—
296	#477	—	2	1	no	—
297	#477	C₂²×C₃₁	1	2	yes	complemented
298	#476	C₂²×C₃₁	1	2	yes	—
299	#474	C₂²×C₃₁	1	2	yes	—
300	#467	—	2	2	no	—
301	#467	—	2	2	no	—
302	#462	—	2	1	no	—
303	#462	C₂²×C₃₁	1	2	yes	complemented
304	C₄×Q₁₆	—	2	2	no	—
305	#428	—	2	1	no	—
306	#423	C₂²×C₃₁	1	1	yes	characteristic
307	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
308	#418	—	2	1	no	—
309	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
310	#333	C₂²×C₃₁	1	2	yes	complemented
311	#332	C₂²×C₃₁	1	2	yes	complemented
312	Q₈⋊C₈	C₂²×C₃₁	1	2	yes	complemented
313	Q₈⋊C₈	C₂²×C₃₁	1	2	yes	complemented
314	D₄⋊C₈	C₂²×C₃₁	1	2	yes	—
315	D₄⋊C₈	C₂²×C₃₁	1	2	yes	—
316	C₂×C₃₁	—	2	2	no	—
317	C₂×C₃₁	—	4	4	no	—
318	C₂×C₃₁	C₂×#451	1	1	yes	characteristic
319	C₂×C₃₁	#3012	1	1	yes	characteristic
320	C₂×C₃₁	#3018	1	1	yes	characteristic
321	C₂²×Q₈	—	2	2	no	—
322	C₂²×D₄	D₄×C₃₁	1	1	yes	characteristic
323	C₂³×C₄	—	2	1	no	—
324	C₂×Q₁₆	—	2	2	no	—
325	C₂×Q₁₆	—	2	2	no	—
326	C₂×M₁₆	—	2	1	no	—
327	C₂²×C₈	—	2	1	no	—
328	C₂²×C₈	—	2	1	no	—
329	C₄⋊Q₈	—	2	2	no	—
330	C₄⋊Q₈	—	2	2	no	—
331	C₄⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
332	C₄⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
333	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
334	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
335	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
336	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
337	#125	—	2	2	no	—
338	#125	—	2	2	no	—
339	#125	—	2	1	no	—
340	#125	—	2	1	no	—
341	#125	—	2	1	no	—
342	C₂²⋊Q₈	—	2	4	no	—
343	C₂²⋊Q₈	—	2	4	no	—
344	C₂²⋊Q₈	—	2	4	no	—
345	C₂²⋊Q₈	—	2	4	no	—
346	C₂²⋊Q₈	—	2	2	no	—
347	C₂²⋊Q₈	—	2	4	no	—
348	C₂²⋊Q₈	—	2	1	no	—
349	C₂²⋊Q₈	—	2	1	no	—
350	C₂²⋊Q₈	—	2	2	no	—
351	C₂²⋊Q₈	—	2	4	no	—
352	C₂²⋊Q₈	—	2	1	no	—
353	C₂²⋊Q₈	—	2	1	no	—
354	C₂²⋊Q₈	D₄×C₃₁	1	2	yes	—
355	C₂²⋊Q₈	D₄×C₃₁	1	2	yes	—
356	C₄×Q₈	—	2	1	no	—
357	C₄×Q₈	—	2	1	no	—
358	C₄×Q₈	—	2	1	no	—
359	C₄×Q₈	—	2	1	no	—
360	C₄×Q₈	D₄×C₃₁	1	2	yes	—
361	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
362	C₄×D₄	—	2	2	no	—
363	C₄×D₄	—	2	2	no	—
364	C₄×D₄	—	2	2	no	—
365	C₄×D₄	—	2	2	no	—
366	C₄×D₄	—	2	1	no	—
367	C₄×D₄	—	2	1	no	—
368	C₄×D₄	D₄×C₃₁	1	2	yes	—
369	C₄×D₄	C₂³×C₃₁	1	2	yes	—
370	#117	—	2	1	no	—
371	#117	—	2	1	no	—
372	#117	—	2	1	no	—
373	C₂×C₄⋊C₄	—	2	4	no	—
374	C₂×C₄⋊C₄	—	2	2	no	—
375	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
376	C₂×C₂²⋊C₄	—	2	4	no	—
377	C₂×C₂²⋊C₄	—	2	1	no	—
378	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
379	#107	—	2	2	no	—
380	#107	—	2	2	no	—
381	#107	—	2	2	no	—
382	#106	—	2	2	no	—
383	C₄⋊C₈	—	2	1	no	—
384	C₄⋊C₈	—	2	1	no	—
385	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
386	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
387	#103	—	2	2	no	—
388	#103	—	2	2	no	—
389	#103	—	2	2	no	—
390	#103	—	2	2	no	—
391	#102	—	2	2	no	—
392	#102	—	2	2	no	—
393	C₄×C₈	—	2	1	no	—
394	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
395	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
396	C₃₁	#18665	1	1	yes	characteristic, direct product factor, Sylow
397	C₂⁴	—	2	1	no	—
398	C₂×Q₈	—	2	4	no	—
399	C₂×Q₈	—	2	1	no	—
400	C₂×Q₈	—	2	1	no	—
401	C₂×Q₈	—	2	1	no	—
402	C₂×Q₈	—	2	1	no	—
403	C₂×Q₈	—	8	2	no	—
404	C₂×Q₈	—	8	2	no	—
405	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
406	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
407	C₂×D₄	—	2	1	no	—
408	C₂×D₄	—	2	2	no	—
409	C₂×D₄	Q₁₆×C₃₁	1	4	yes	complemented
410	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
411	C₂²×C₄	—	2	4	no	—
412	C₂²×C₄	—	2	4	no	—
413	C₂²×C₄	—	2	1	no	—
414	C₂²×C₄	—	2	2	no	—
415	C₂²×C₄	—	2	2	no	—
416	C₂²×C₄	—	8	1	no	—
417	C₂²×C₄	—	8	1	no	—
418	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
419	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
420	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
421	Q₁₆	—	4	4	no	—
422	Q₁₆	—	8	2	no	—
423	M₁₆	—	4	2	no	—
424	C₂×C₈	—	2	1	no	—
425	C₂×C₈	—	2	1	no	—
426	C₂×C₈	—	2	2	no	—
427	C₂×C₈	—	2	2	no	—
428	C₂×C₈	—	4	2	no	—
429	C₂×C₈	—	8	1	no	—
430	C₄⋊C₄	—	2	4	no	—
431	C₄⋊C₄	—	2	4	no	—
432	C₄⋊C₄	—	2	4	no	—
433	C₄⋊C₄	—	2	2	no	—
434	C₄⋊C₄	—	2	2	no	—
435	C₄⋊C₄	—	2	1	no	—
436	C₄⋊C₄	—	2	1	no	—
437	C₄⋊C₄	—	2	1	no	—
438	C₄⋊C₄	—	2	1	no	—
439	C₄⋊C₄	—	2	2	no	—
440	C₄⋊C₄	—	2	1	no	—
441	C₄⋊C₄	—	2	1	no	—
442	C₄⋊C₄	—	2	1	no	—
443	C₄⋊C₄	—	2	2	no	—
444	C₄⋊C₄	—	2	1	no	—
445	C₄⋊C₄	—	2	1	no	—
446	C₄⋊C₄	—	2	1	no	—
447	C₄⋊C₄	—	2	1	no	—
448	C₄⋊C₄	—	4	1	no	—
449	C₄⋊C₄	—	4	1	no	—
450	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
451	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
452	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
453	C₂²⋊C₄	—	2	4	no	—
454	C₂²⋊C₄	—	2	4	no	—
455	C₂²⋊C₄	—	2	2	no	—
456	C₂²⋊C₄	—	2	2	no	—
457	C₂²⋊C₄	—	2	2	no	—
458	C₂²⋊C₄	—	2	2	no	—
459	C₂²⋊C₄	—	4	1	no	—
460	C₄²	—	2	2	no	—
461	C₄²	—	2	2	no	—
462	C₄²	—	4	1	no	—
463	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
464	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
465	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
466	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
467	C₂³	—	2	2	no	—
468	C₂³	—	2	4	no	—
469	C₂³	—	8	1	no	—
470	C₂³	—	8	1	no	—
471	C₂³	C₃₁×#120	1	1	yes	characteristic
472	Q₈	—	2	2	no	—
473	Q₈	—	2	2	no	—
474	Q₈	—	4	2	no	—
475	Q₈	—	4	2	no	—
476	Q₈	—	4	1	no	—
477	Q₈	—	4	1	no	—
478	Q₈	—	4	1	no	—
479	Q₈	—	4	1	no	—
480	Q₈	—	8	1	no	—
481	Q₈	—	8	4	no	—
482	D₄	—	2	4	no	—
483	D₄	—	2	2	no	—
484	D₄	C₂×Q₁₆×C₃₁	1	4	yes	—
485	C₂×C₄	—	2	1	no	—
486	C₂×C₄	—	2	1	no	—
487	C₂×C₄	—	2	2	no	—
488	C₂×C₄	—	2	2	no	—
489	C₂×C₄	—	2	2	no	—
490	C₂×C₄	—	2	2	no	—
491	C₂×C₄	—	2	2	no	—
492	C₂×C₄	—	4	1	no	—
493	C₂×C₄	—	4	2	no	—
494	C₂×C₄	—	4	2	no	—
495	C₂×C₄	—	4	1	no	—
496	C₂×C₄	—	4	1	no	—
497	C₂×C₄	—	8	2	no	—
498	C₂×C₄	—	8	1	no	—
499	C₂×C₄	—	8	4	no	—
500	C₂×C₄	—	8	4	no	—
501	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
502	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
503	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
504	C₂×C₄	C₂×Q₁₆×C₃₁	1	2	yes	—
505	C₂×C₄	C₃₁×#137	1	2	yes	—
506	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
507	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
508	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
509	C₈	—	2	2	no	—
510	C₈	—	4	1	no	—
511	C₈	—	4	2	no	—
512	C₈	—	4	2	no	—
513	C₂²	—	2	1	no	—
514	C₂²	—	2	1	no	—
515	C₂²	—	2	4	no	—
516	C₂²	—	4	2	no	—
517	C₂²	—	4	2	no	—
518	C₂²	—	8	4	no	—
519	C₂²	C₃₁×#451	1	2	yes	—
520	C₂²	C₂×C₃₁×#120	1	1	yes	characteristic
521	C₄	—	2	1	no	—
522	C₄	—	2	1	no	—
523	C₄	—	2	1	no	—
524	C₄	—	4	2	no	—
525	C₄	—	4	2	no	—
526	C₄	—	4	2	no	—
527	C₄	—	4	2	no	—
528	C₄	—	4	2	no	—
529	C₄	—	8	1	no	—
530	C₄	—	8	1	no	—
531	C₄	C₃₁×#451	1	2	yes	—
532	C₄	C₂²×Q₁₆×C₃₁	1	1	yes	characteristic
533	C₄	C₂×C₃₁×#137	1	1	yes	characteristic
534	C₄	C₃₁×#576	1	1	yes	characteristic
535	C₄	C₃₁×#577	1	1	yes	characteristic
536	C₂	—	2	2	no	—
537	C₂	—	4	4	no	—
538	C₂	C₂×C₃₁×#451	1	1	yes	characteristic
539	C₂	C₃₁×#3012	1	1	yes	characteristic
540	C₂	C₃₁×#3018	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	C₂²×D₄×C₃₁	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#3249	C₂	semidirect
3. C₃₁×#3170	C₂	semidirect
5. C₃₁×#1649	C₂	semidirect
6. C₃₁×#1644	C₂	semidirect
7. C₃₁×#1637	C₂	semidirect
8. C₃₁×#1632	C₂	semidirect
9. C₃₁×#1550	C₂	semidirect
10. C₃₁×#1493	C₂	semidirect
26. C₃₁×#500	C₂²	semidirect
27. C₃₁×#494	C₂²	semidirect
28. C₃₁×#487	C₂²	semidirect
29. C₃₁×#487	C₂²	semidirect
32. C₃₁×#477	C₂²	semidirect
38. C₃₁×#462	C₂²	semidirect
45. C₃₁×#333	C₂²	semidirect
46. C₃₁×#332	C₂²	semidirect
47. C₃₁×Q₈⋊C₈	C₂²	semidirect
48. C₃₁×Q₈⋊C₈	C₂²	semidirect
138. C₂×D₄×C₃₁	Q₁₆	semidirect
196. #18665	C₃₁	direct (N × H)
244. #3249	C₂×C₃₁	semidirect
245. #3170	C₂×C₃₁	semidirect
247. #1649	C₂×C₃₁	semidirect
248. #1644	C₂×C₃₁	semidirect
249. #1637	C₂×C₃₁	semidirect
250. #1632	C₂×C₃₁	semidirect
251. #1550	C₂×C₃₁	semidirect
252. #1493	C₂×C₃₁	semidirect
291. #500	C₂²×C₃₁	semidirect
292. #494	C₂²×C₃₁	semidirect
293. #487	C₂²×C₃₁	semidirect
294. #487	C₂²×C₃₁	semidirect
297. #477	C₂²×C₃₁	semidirect
303. #462	C₂²×C₃₁	semidirect
310. #333	C₂²×C₃₁	semidirect
311. #332	C₂²×C₃₁	semidirect
312. Q₈⋊C₈	C₂²×C₃₁	semidirect
313. Q₈⋊C₈	C₂²×C₃₁	semidirect
396. C₃₁	#18665	direct (N × H)
409. C₂×D₄	Q₁₆×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#3501	2
C₃₁×#3249	2
C₃₁×#3170	2
C₃₁×#3077	2
C₃₁×#1649	2
C₃₁×#1644	2
C₃₁×#1637	2
C₃₁×#1632	2
C₃₁×#1550	2
C₃₁×#1493	2
C₂×C₃₁×D₄⋊C₈	2
#18665	31

Subgroups — C31×#18665 #21475424

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#18665 #21475424