Subgroups — C₃₁×#8906 #21465665

Order 7936.

454 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₄×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1950	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#1567	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1486	C₂	1	1	yes	maximal, characteristic
5	C₂×C₃₁×D₄⋊C₈	C₂	1	1	yes	maximal, characteristic
6	C₃₁×#1301	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1278	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×M₁₆×C₃₁	—	2	1	no	—
9	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
10	C₃₁×#436	—	2	1	no	—
11	C₃₁×#435	—	2	2	no	—
12	C₈×D₄×C₃₁	—	2	1	no	—
13	C₃₁×#431	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×#108	—	2	1	no	—
15	C₃₁×#423	C₂²	1	1	yes	characteristic
16	C₃₁×#423	C₂²	1	1	yes	characteristic
17	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
18	C₃₁×#407	—	2	1	no	—
19	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
21	C₃₁×#341	—	2	1	no	—
22	C₃₁×#335	—	2	2	no	—
23	C₃₁×D₄⋊C₈	—	2	1	no	—
24	C₃₁×D₄⋊C₈	—	2	1	no	—
25	C₃₁×D₄⋊C₈	C₄	1	1	yes	characteristic
26	C₃₁×D₄⋊C₈	C₄	1	1	yes	characteristic
27	C₃₁×D₄⋊C₈	C₄	1	1	yes	characteristic
28	C₃₁×D₄⋊C₈	C₄	1	1	yes	characteristic
29	C₂²×D₄×C₃₁	Q₈	1	1	yes	characteristic
30	C₂³×C₄×C₃₁	—	2	1	no	—
31	C₂×M₁₆×C₃₁	—	2	2	no	—
32	C₂×M₁₆×C₃₁	—	2	1	no	—
33	C₂×M₁₆×C₃₁	—	2	1	no	—
34	C₂×M₁₆×C₃₁	—	4	2	no	—
35	C₂×M₁₆×C₃₁	—	4	2	no	—
36	C₂×M₁₆×C₃₁	D₄	1	1	yes	characteristic
37	C₂×M₁₆×C₃₁	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₃₁	—	2	2	no	—
39	C₂²×C₈×C₃₁	—	2	1	no	—
40	C₂²×C₈×C₃₁	—	2	1	no	—
41	C₄×D₄×C₃₁	—	2	1	no	—
42	C₄×D₄×C₃₁	—	2	1	no	—
43	C₄×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
44	C₄×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
45	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
46	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
47	C₂×C₄⋊C₄×C₃₁	Q₈	1	1	yes	characteristic
48	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
49	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
50	C₃₁×#108	—	4	2	no	—
51	C₃₁×C₄⋊C₈	—	2	1	no	—
52	C₃₁×C₄⋊C₈	—	2	1	no	—
53	C₃₁×C₄⋊C₈	—	2	1	no	—
54	C₃₁×C₄⋊C₈	—	2	1	no	—
55	C₃₁×C₄⋊C₈	—	2	1	no	—
56	C₃₁×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
57	C₃₁×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
58	C₃₁×C₄⋊C₈	D₄	1	2	yes	—
59	C₃₁×C₂²⋊C₈	—	2	2	no	—
60	C₃₁×C₂²⋊C₈	—	2	2	no	—
61	C₃₁×C₂²⋊C₈	—	2	1	no	—
62	C₃₁×#97	—	2	1	no	—
63	C₄×C₈×C₃₁	—	2	1	no	—
64	C₄×C₈×C₃₁	—	2	1	no	—
65	C₄×C₈×C₃₁	—	2	1	no	—
66	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
67	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
68	C₂⁴×C₃₁	—	2	1	no	—
69	C₂×D₄×C₃₁	—	2	2	no	—
70	C₂×D₄×C₃₁	—	2	1	no	—
71	C₂×D₄×C₃₁	—	4	1	no	—
72	C₂×D₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
73	C₂×D₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
74	C₂²×C₄×C₃₁	—	2	1	no	—
75	C₂²×C₄×C₃₁	—	2	1	no	—
76	C₂²×C₄×C₃₁	—	2	1	no	—
77	C₂²×C₄×C₃₁	—	2	1	no	—
78	C₂²×C₄×C₃₁	—	2	1	no	—
79	C₂²×C₄×C₃₁	—	4	1	no	—
80	C₂²×C₄×C₃₁	—	4	1	no	—
81	C₂²×C₄×C₃₁	—	4	1	no	—
82	C₂²×C₄×C₃₁	—	4	1	no	—
83	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
85	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
86	M₁₆×C₃₁	—	2	1	no	—
87	M₁₆×C₃₁	—	2	1	no	—
88	M₁₆×C₃₁	—	2	1	no	—
89	M₁₆×C₃₁	—	4	4	no	—
90	M₁₆×C₃₁	—	4	1	no	—
91	M₁₆×C₃₁	—	4	1	no	—
92	M₁₆×C₃₁	—	4	1	no	—
93	M₁₆×C₃₁	—	4	1	no	—
94	M₁₆×C₃₁	D₈	1	2	yes	—
95	C₂×C₈×C₃₁	—	2	1	no	—
96	C₂×C₈×C₃₁	—	2	1	no	—
97	C₂×C₈×C₃₁	—	2	1	no	—
98	C₂×C₈×C₃₁	—	2	1	no	—
99	C₂×C₈×C₃₁	—	2	1	no	—
100	C₂×C₈×C₃₁	—	2	1	no	—
101	C₂×C₈×C₃₁	—	2	2	no	—
102	C₂×C₈×C₃₁	—	4	2	no	—
103	C₂×C₈×C₃₁	—	4	4	no	—
104	C₂×C₈×C₃₁	—	4	1	no	—
105	C₂×C₈×C₃₁	—	4	1	no	—
106	C₂×C₈×C₃₁	—	4	1	no	—
107	C₂×C₈×C₃₁	—	4	1	no	—
108	C₄⋊C₄×C₃₁	—	2	1	no	—
109	C₄⋊C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
110	C₄⋊C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
111	C₂²⋊C₄×C₃₁	—	2	1	no	—
112	C₂²⋊C₄×C₃₁	—	2	1	no	—
113	C₂²⋊C₄×C₃₁	—	2	1	no	—
114	C₂²⋊C₄×C₃₁	—	2	1	no	—
115	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
116	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
117	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
118	C₄²×C₃₁	#41	1	1	yes	characteristic
119	#8906	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
120	C₂³×C₃₁	—	2	1	no	—
121	C₂³×C₃₁	—	2	1	no	—
122	C₂³×C₃₁	—	4	2	no	—
123	C₂³×C₃₁	—	4	1	no	—
124	C₂³×C₃₁	—	4	1	no	—
125	C₂³×C₃₁	—	8	1	no	—
126	C₂³×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
127	D₄×C₃₁	—	2	1	no	—
128	D₄×C₃₁	—	2	1	no	—
129	D₄×C₃₁	—	2	1	no	—
130	D₄×C₃₁	—	4	2	no	—
131	D₄×C₃₁	#108	1	2	yes	complemented
132	C₂×C₄×C₃₁	—	2	1	no	—
133	C₂×C₄×C₃₁	—	2	1	no	—
134	C₂×C₄×C₃₁	—	2	1	no	—
135	C₂×C₄×C₃₁	—	2	1	no	—
136	C₂×C₄×C₃₁	—	4	1	no	—
137	C₂×C₄×C₃₁	—	4	1	no	—
138	C₂×C₄×C₃₁	—	4	1	no	—
139	C₂×C₄×C₃₁	—	4	1	no	—
140	C₂×C₄×C₃₁	—	4	1	no	—
141	C₂×C₄×C₃₁	—	4	1	no	—
142	C₂×C₄×C₃₁	—	4	1	no	—
143	C₂×C₄×C₃₁	—	4	1	no	—
144	C₂×C₄×C₃₁	—	4	1	no	—
145	C₂×C₄×C₃₁	—	4	1	no	—
146	C₂×C₄×C₃₁	—	4	1	no	—
147	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
148	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
149	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
150	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
151	C₂×C₄×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
152	C₂×C₄×C₃₁	#123	1	1	yes	characteristic
153	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
154	C₂×C₄×C₃₁	#135	1	1	yes	characteristic
155	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
156	C₂×C₄×C₃₁	#137	1	1	yes	characteristic
157	C₈×C₃₁	—	2	2	no	—
158	C₈×C₃₁	—	2	2	no	—
159	C₈×C₃₁	—	4	1	no	—
160	C₈×C₃₁	—	4	1	no	—
161	C₈×C₃₁	—	4	1	no	—
162	C₈×C₃₁	—	4	2	no	—
163	C₈×C₃₁	—	4	1	no	—
164	C₈×C₃₁	—	8	1	no	—
165	C₈×C₃₁	—	8	1	no	—
166	D₄×M₁₆	C₂×C₃₁	1	1	yes	characteristic
167	#1950	C₂×C₃₁	1	1	yes	characteristic, complemented
168	#1567	C₂×C₃₁	1	1	yes	characteristic, complemented
169	#1486	C₂×C₃₁	1	1	yes	characteristic
170	C₂×D₄⋊C₈	C₂×C₃₁	1	1	yes	characteristic
171	#1301	C₂×C₃₁	1	1	yes	characteristic, complemented
172	#1278	C₂×C₃₁	1	1	yes	characteristic, complemented
173	C₂²×C₃₁	—	2	1	no	—
174	C₂²×C₃₁	—	2	1	no	—
175	C₂²×C₃₁	—	2	2	no	—
176	C₂²×C₃₁	—	4	1	no	—
177	C₂²×C₃₁	—	4	1	no	—
178	C₂²×C₃₁	—	4	1	no	—
179	C₂²×C₃₁	—	4	1	no	—
180	C₂²×C₃₁	—	4	1	no	—
181	C₂²×C₃₁	—	8	2	no	—
182	C₂²×C₃₁	—	8	2	no	—
183	C₂²×C₃₁	#385	1	1	yes	characteristic
184	C₂²×C₃₁	#474	1	1	yes	characteristic
185	C₂²×C₃₁	#478	1	1	yes	characteristic
186	C₄×C₃₁	—	2	1	no	—
187	C₄×C₃₁	—	2	1	no	—
188	C₄×C₃₁	—	2	1	no	—
189	C₄×C₃₁	—	4	1	no	—
190	C₄×C₃₁	—	4	1	no	—
191	C₄×C₃₁	C₂×#108	1	1	yes	characteristic
192	C₄×C₃₁	#430	1	1	yes	characteristic
193	C₄×C₃₁	C₄×D₈	1	1	yes	characteristic
194	C₄×C₃₁	#440	1	1	yes	characteristic
195	C₄×C₃₁	#447	1	1	yes	characteristic
196	C₄×C₃₁	#452	1	1	yes	characteristic
197	C₂²×M₁₆	—	2	1	no	—
198	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
199	#436	—	2	1	no	—
200	#435	—	2	2	no	—
201	C₈×D₄	—	2	1	no	—
202	#431	C₂²×C₃₁	1	1	yes	characteristic
203	C₂×#108	—	2	1	no	—
204	#423	C₂²×C₃₁	1	1	yes	characteristic
205	#423	C₂²×C₃₁	1	1	yes	characteristic
206	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
207	#407	—	2	1	no	—
208	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
209	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
210	#341	—	2	1	no	—
211	#335	—	2	2	no	—
212	D₄⋊C₈	—	2	1	no	—
213	D₄⋊C₈	—	2	1	no	—
214	D₄⋊C₈	C₄×C₃₁	1	1	yes	characteristic
215	D₄⋊C₈	C₄×C₃₁	1	1	yes	characteristic
216	D₄⋊C₈	C₄×C₃₁	1	1	yes	characteristic
217	D₄⋊C₈	C₄×C₃₁	1	1	yes	characteristic
218	C₂×C₃₁	—	2	1	no	—
219	C₂×C₃₁	—	2	1	no	—
220	C₂×C₃₁	—	4	2	no	—
221	C₂×C₃₁	—	8	1	no	—
222	C₂×C₃₁	#1855	1	1	yes	characteristic
223	C₂×C₃₁	#1861	1	1	yes	characteristic
224	C₂×C₃₁	#1862	1	1	yes	characteristic
225	C₂²×D₄	Q₈×C₃₁	1	1	yes	characteristic
226	C₂³×C₄	—	2	1	no	—
227	C₂×M₁₆	—	2	2	no	—
228	C₂×M₁₆	—	2	1	no	—
229	C₂×M₁₆	—	2	1	no	—
230	C₂×M₁₆	—	4	2	no	—
231	C₂×M₁₆	—	4	2	no	—
232	C₂×M₁₆	D₄×C₃₁	1	1	yes	characteristic
233	C₂×M₁₆	D₄×C₃₁	1	1	yes	characteristic
234	C₂²×C₈	—	2	2	no	—
235	C₂²×C₈	—	2	1	no	—
236	C₂²×C₈	—	2	1	no	—
237	C₄×D₄	—	2	1	no	—
238	C₄×D₄	—	2	1	no	—
239	C₄×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
240	C₄×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
241	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
242	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
243	C₂×C₄⋊C₄	Q₈×C₃₁	1	1	yes	characteristic
244	C₂×C₂²⋊C₄	—	2	1	no	—
245	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
246	#108	—	4	2	no	—
247	C₄⋊C₈	—	2	1	no	—
248	C₄⋊C₈	—	2	1	no	—
249	C₄⋊C₈	—	2	1	no	—
250	C₄⋊C₈	—	2	1	no	—
251	C₄⋊C₈	—	2	1	no	—
252	C₄⋊C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
253	C₄⋊C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
254	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
255	C₂²⋊C₈	—	2	2	no	—
256	C₂²⋊C₈	—	2	2	no	—
257	C₂²⋊C₈	—	2	1	no	—
258	#97	—	2	1	no	—
259	C₄×C₈	—	2	1	no	—
260	C₄×C₈	—	2	1	no	—
261	C₄×C₈	—	2	1	no	—
262	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
263	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
264	C₃₁	#8906	1	1	yes	characteristic, direct product factor, Sylow
265	C₂⁴	—	2	1	no	—
266	C₂×D₄	—	2	2	no	—
267	C₂×D₄	—	2	1	no	—
268	C₂×D₄	—	4	1	no	—
269	C₂×D₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
270	C₂×D₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
271	C₂²×C₄	—	2	1	no	—
272	C₂²×C₄	—	2	1	no	—
273	C₂²×C₄	—	2	1	no	—
274	C₂²×C₄	—	2	1	no	—
275	C₂²×C₄	—	2	1	no	—
276	C₂²×C₄	—	4	1	no	—
277	C₂²×C₄	—	4	1	no	—
278	C₂²×C₄	—	4	1	no	—
279	C₂²×C₄	—	4	1	no	—
280	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
281	C₂²×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
282	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
283	M₁₆	—	2	1	no	—
284	M₁₆	—	2	1	no	—
285	M₁₆	—	2	1	no	—
286	M₁₆	—	4	4	no	—
287	M₁₆	—	4	1	no	—
288	M₁₆	—	4	1	no	—
289	M₁₆	—	4	1	no	—
290	M₁₆	—	4	1	no	—
291	M₁₆	D₈×C₃₁	1	2	yes	—
292	C₂×C₈	—	2	1	no	—
293	C₂×C₈	—	2	1	no	—
294	C₂×C₈	—	2	1	no	—
295	C₂×C₈	—	2	1	no	—
296	C₂×C₈	—	2	1	no	—
297	C₂×C₈	—	2	1	no	—
298	C₂×C₈	—	2	2	no	—
299	C₂×C₈	—	4	2	no	—
300	C₂×C₈	—	4	4	no	—
301	C₂×C₈	—	4	1	no	—
302	C₂×C₈	—	4	1	no	—
303	C₂×C₈	—	4	1	no	—
304	C₂×C₈	—	4	1	no	—
305	C₄⋊C₄	—	2	1	no	—
306	C₄⋊C₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
307	C₄⋊C₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
308	C₂²⋊C₄	—	2	1	no	—
309	C₂²⋊C₄	—	2	1	no	—
310	C₂²⋊C₄	—	2	1	no	—
311	C₂²⋊C₄	—	2	1	no	—
312	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
313	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
314	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
315	C₄²	#41×C₃₁	1	1	yes	characteristic
316	C₂³	—	2	1	no	—
317	C₂³	—	2	1	no	—
318	C₂³	—	4	2	no	—
319	C₂³	—	4	1	no	—
320	C₂³	—	4	1	no	—
321	C₂³	—	8	1	no	—
322	C₂³	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
323	D₄	—	2	1	no	—
324	D₄	—	2	1	no	—
325	D₄	—	2	1	no	—
326	D₄	—	4	2	no	—
327	D₄	C₃₁×#108	1	2	yes	complemented
328	C₂×C₄	—	2	1	no	—
329	C₂×C₄	—	2	1	no	—
330	C₂×C₄	—	2	1	no	—
331	C₂×C₄	—	2	1	no	—
332	C₂×C₄	—	4	1	no	—
333	C₂×C₄	—	4	1	no	—
334	C₂×C₄	—	4	1	no	—
335	C₂×C₄	—	4	1	no	—
336	C₂×C₄	—	4	1	no	—
337	C₂×C₄	—	4	1	no	—
338	C₂×C₄	—	4	1	no	—
339	C₂×C₄	—	4	1	no	—
340	C₂×C₄	—	4	1	no	—
341	C₂×C₄	—	4	1	no	—
342	C₂×C₄	—	4	1	no	—
343	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
344	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
345	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
346	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
347	C₂×C₄	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
348	C₂×C₄	C₃₁×#123	1	1	yes	characteristic
349	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
350	C₂×C₄	C₃₁×#135	1	1	yes	characteristic
351	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
352	C₂×C₄	C₃₁×#137	1	1	yes	characteristic
353	C₈	—	2	2	no	—
354	C₈	—	2	2	no	—
355	C₈	—	4	1	no	—
356	C₈	—	4	1	no	—
357	C₈	—	4	1	no	—
358	C₈	—	4	2	no	—
359	C₈	—	4	1	no	—
360	C₈	—	8	1	no	—
361	C₈	—	8	1	no	—
362	C₂²	—	2	1	no	—
363	C₂²	—	2	1	no	—
364	C₂²	—	2	2	no	—
365	C₂²	—	4	1	no	—
366	C₂²	—	4	1	no	—
367	C₂²	—	4	1	no	—
368	C₂²	—	4	1	no	—
369	C₂²	—	4	1	no	—
370	C₂²	—	8	2	no	—
371	C₂²	—	8	2	no	—
372	C₂²	C₃₁×#385	1	1	yes	characteristic
373	C₂²	C₃₁×#474	1	1	yes	characteristic
374	C₂²	C₃₁×#478	1	1	yes	characteristic
375	C₄	—	2	1	no	—
376	C₄	—	2	1	no	—
377	C₄	—	2	1	no	—
378	C₄	—	4	1	no	—
379	C₄	—	4	1	no	—
380	C₄	C₂×C₃₁×#108	1	1	yes	characteristic
381	C₄	C₃₁×#430	1	1	yes	characteristic
382	C₄	C₄×D₈×C₃₁	1	1	yes	characteristic
383	C₄	C₃₁×#440	1	1	yes	characteristic
384	C₄	C₃₁×#447	1	1	yes	characteristic
385	C₄	C₃₁×#452	1	1	yes	characteristic
386	C₂	—	2	1	no	—
387	C₂	—	2	1	no	—
388	C₂	—	4	2	no	—
389	C₂	—	8	1	no	—
390	C₂	C₃₁×#1855	1	1	yes	characteristic
391	C₂	C₃₁×#1861	1	1	yes	characteristic
392	C₂	C₃₁×#1862	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	#120
commutator	C₄²	C₂²×C₄×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
socle	C₂²×C₃₁	#385
prime radical	C₂²×D₄×C₃₁	Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#1950	C₂	semidirect
3. C₃₁×#1567	C₂	semidirect
6. C₃₁×#1301	C₂	semidirect
7. C₃₁×#1278	C₂	semidirect
119. #8906	C₃₁	direct (N × H)
131. D₄×C₃₁	#108	semidirect
167. #1950	C₂×C₃₁	semidirect
168. #1567	C₂×C₃₁	semidirect
171. #1301	C₂×C₃₁	semidirect
172. #1278	C₂×C₃₁	semidirect
264. C₃₁	#8906	direct (N × H)
327. D₄	C₃₁×#108	semidirect

Maximal subgroups

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

subgroup	index
D₄×M₁₆×C₃₁	2
C₃₁×#1950	2
C₃₁×#1567	2
C₃₁×#1486	2
C₂×C₃₁×D₄⋊C₈	2
C₃₁×#1301	2
C₃₁×#1278	2
#8906	31

Subgroups — C31×#8906 #21465665

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#8906 #21465665