Subgroups — C₃₁×#22457 #21479216

Order 7936.

1294 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#578	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2950	C₂	1	2	yes	maximal, complemented
3	C₃₁×#2884	C₂	1	2	yes	maximal, complemented
4	C₃₁×#1976	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1905	C₂	1	4	yes	maximal, complemented
6	C₃₁×#1812	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1796	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1757	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₃₁×2₋¹⁺⁴	C₂²	1	2	yes	—
10	C₃₁×#578	—	2	4	no	—
11	C₃₁×#578	—	2	4	no	—
12	C₂×C₃₁×#137	C₄	1	4	yes	complemented
13	C₂×C₃₁×#137	C₂²	1	2	yes	—
14	C₂×C₃₁×#135	C₄	1	2	yes	complemented
15	C₂×C₃₁×#135	C₂²	1	1	yes	characteristic
16	C₂²×Q₁₆×C₃₁	C₄	1	2	yes	complemented
17	C₂²×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
18	C₂×C₃₁×#131	C₂²	1	1	yes	characteristic
19	C₃₁×#519	C₂²	1	4	yes	—
20	C₃₁×#441	—	2	4	no	—
21	C₄×Q₁₆×C₃₁	—	2	4	no	—
22	C₃₁×#427	C₂²	1	2	yes	—
23	C₃₁×#421	C₂²	1	4	yes	—
24	C₃₁×#419	C₂²	1	4	yes	—
25	C₃₁×#413	C₂²	1	1	yes	characteristic
26	C₃₁×#410	—	2	8	no	—
27	C₃₁×#410	C₂²	1	2	yes	—
28	C₃₁×#410	C₂²	1	4	yes	—
29	C₃₁×#410	C₂²	1	1	yes	characteristic
30	C₃₁×#405	C₂²	1	2	yes	—
31	C₃₁×#344	C₂²	1	2	yes	—
32	C₃₁×H(3,ℤ/4)	C₂²	1	2	yes	—
33	C₃₁×2₋¹⁺⁴	—	2	8	no	—
34	C₃₁×2₋¹⁺⁴	—	4	4	no	—
35	C₂×#41×C₃₁	—	2	4	no	—
36	C₂×#41×C₃₁	—	2	2	no	—
37	C₂×#41×C₃₁	C₂×C₄	1	4	yes	—
38	C₂×#41×C₃₁	C₂³	1	2	yes	—
39	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
40	C₂²×Q₈×C₃₁	—	2	2	no	—
41	C₂²×Q₈×C₃₁	C₂×C₄	1	4	yes	—
42	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
43	C₃₁×#137	—	2	8	no	—
44	C₃₁×#137	—	2	16	no	—
45	C₃₁×#135	—	2	4	no	—
46	C₃₁×#135	—	2	4	no	—
47	C₃₁×#135	—	2	2	no	—
48	C₃₁×#135	—	2	2	no	—
49	C₂×Q₁₆×C₃₁	—	2	4	no	—
50	C₂×Q₁₆×C₃₁	—	2	4	no	—
51	C₂×Q₁₆×C₃₁	C₂×C₄	1	1	yes	characteristic
52	C₂×Q₁₆×C₃₁	C₂×C₄	1	4	yes	—
53	C₂×Q₁₆×C₃₁	C₂×C₄	1	2	yes	—
54	C₂×Q₁₆×C₃₁	C₂×C₄	1	2	yes	—
55	C₂×Q₁₆×C₃₁	D₄	1	4	yes	—
56	C₂×Q₁₆×C₃₁	D₄	1	4	yes	—
57	C₂×SD₁₆×C₃₁	C₂×C₄	1	4	yes	—
58	C₂×SD₁₆×C₃₁	C₂×C₄	1	2	yes	—
59	C₂×D₈×C₃₁	C₂×C₄	1	1	yes	characteristic
60	C₃₁×#131	—	2	2	no	—
61	C₃₁×#131	—	2	2	no	—
62	C₂×M₁₆×C₃₁	C₂×C₄	1	2	yes	—
63	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
64	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
65	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
66	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
67	C₄×Q₈×C₃₁	—	2	8	no	—
68	C₄×Q₈×C₃₁	—	4	4	no	—
69	C₃₁×#117	—	2	8	no	—
70	C₃₁×#117	C₂³	1	4	yes	—
71	C₃₁×#117	C₂³	1	2	yes	—
72	C₃₁×#107	—	2	2	no	—
73	C₃₁×#106	—	2	2	no	—
74	C₃₁×#104	—	4	4	no	—
75	C₃₁×#103	—	2	8	no	—
76	C₃₁×#101	—	2	1	no	—
77	C₃₁×#100	—	2	1	no	—
78	C₃₁×#99	—	2	4	no	—
79	C₃₁×#99	—	2	8	no	—
80	C₃₁×#99	D₄	1	4	yes	—
81	C₃₁×#97	—	2	2	no	—
82	C₄×C₈×C₃₁	—	2	2	no	—
83	#41×C₃₁	—	2	2	no	—
84	#41×C₃₁	—	2	2	no	—
85	#41×C₃₁	—	2	8	no	—
86	#41×C₃₁	—	2	4	no	—
87	#41×C₃₁	—	2	4	no	—
88	#41×C₃₁	—	4	4	no	—
89	#41×C₃₁	—	4	16	no	—
90	#41×C₃₁	—	4	8	no	—
91	C₂×Q₈×C₃₁	—	2	4	no	—
92	C₂×Q₈×C₃₁	—	2	2	no	—
93	C₂×Q₈×C₃₁	—	2	4	no	—
94	C₂×Q₈×C₃₁	—	2	8	no	—
95	C₂×Q₈×C₃₁	—	4	4	no	—
96	C₂×Q₈×C₃₁	—	4	8	no	—
97	C₂×Q₈×C₃₁	C₂²×C₄	1	2	yes	—
98	C₂×Q₈×C₃₁	C₂²×C₄	1	4	yes	—
99	C₂×Q₈×C₃₁	C₂×D₄	1	8	yes	—
100	C₂×Q₈×C₃₁	#41	1	1	yes	characteristic
101	C₂×D₄×C₃₁	—	2	4	no	—
102	C₂×D₄×C₃₁	—	2	2	no	—
103	C₂×D₄×C₃₁	C₂²×C₄	1	2	yes	—
104	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
105	C₂×D₄×C₃₁	#41	1	1	yes	characteristic
106	C₂²×C₄×C₃₁	—	2	2	no	—
107	C₂²×C₄×C₃₁	—	2	4	no	—
108	C₂²×C₄×C₃₁	—	2	4	no	—
109	C₂²×C₄×C₃₁	—	2	2	no	—
110	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
111	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
112	Q₁₆×C₃₁	—	2	8	no	—
113	Q₁₆×C₃₁	—	2	1	no	—
114	Q₁₆×C₃₁	—	2	1	no	—
115	Q₁₆×C₃₁	C₂²⋊C₄	1	4	yes	—
116	Q₁₆×C₃₁	C₂²⋊C₄	1	4	yes	—
117	Q₁₆×C₃₁	C₂²⋊C₄	1	8	yes	—
118	SD₁₆×C₃₁	—	2	8	no	—
119	SD₁₆×C₃₁	—	2	4	no	—
120	D₈×C₃₁	—	2	1	no	—
121	D₈×C₃₁	—	2	1	no	—
122	M₁₆×C₃₁	—	2	2	no	—
123	M₁₆×C₃₁	—	2	4	no	—
124	M₁₆×C₃₁	—	2	2	no	—
125	M₁₆×C₃₁	—	2	2	no	—
126	C₂×C₈×C₃₁	—	2	2	no	—
127	C₂×C₈×C₃₁	—	2	2	no	—
128	C₂×C₈×C₃₁	—	2	2	no	—
129	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
130	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
131	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
132	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
133	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
134	C₄⋊C₄×C₃₁	—	2	2	no	—
135	C₄⋊C₄×C₃₁	—	2	4	no	—
136	C₄⋊C₄×C₃₁	—	4	8	no	—
137	C₂²⋊C₄×C₃₁	—	2	4	no	—
138	C₂²⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
139	C₄²×C₃₁	—	2	2	no	—
140	C₄²×C₃₁	—	2	4	no	—
141	C₄²×C₃₁	—	4	8	no	—
142	#22457	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
143	C₂³×C₃₁	—	2	2	no	—
144	C₂³×C₃₁	C₄×D₄	1	2	yes	—
145	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
146	Q₈×C₃₁	—	2	4	no	—
147	Q₈×C₃₁	—	2	8	no	—
148	Q₈×C₃₁	—	2	2	no	—
149	Q₈×C₃₁	—	2	2	no	—
150	Q₈×C₃₁	—	2	2	no	—
151	Q₈×C₃₁	—	4	8	no	—
152	Q₈×C₃₁	C₂×C₂²⋊C₄	1	8	yes	—
153	D₄×C₃₁	—	2	4	no	—
154	D₄×C₃₁	—	2	2	no	—
155	D₄×C₃₁	—	2	2	no	—
156	D₄×C₃₁	—	2	2	no	—
157	D₄×C₃₁	—	4	8	no	—
158	D₄×C₃₁	—	4	4	no	—
159	C₂×C₄×C₃₁	—	2	4	no	—
160	C₂×C₄×C₃₁	—	2	8	no	—
161	C₂×C₄×C₃₁	—	2	2	no	—
162	C₂×C₄×C₃₁	—	2	2	no	—
163	C₂×C₄×C₃₁	—	2	2	no	—
164	C₂×C₄×C₃₁	—	2	4	no	—
165	C₂×C₄×C₃₁	—	4	8	no	—
166	C₂×C₄×C₃₁	—	4	8	no	—
167	C₂×C₄×C₃₁	—	4	4	no	—
168	C₂×C₄×C₃₁	—	4	4	no	—
169	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
170	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
171	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
172	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
173	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
174	C₈×C₃₁	—	2	2	no	—
175	C₈×C₃₁	—	4	2	no	—
176	C₈×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
177	C₈×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
178	C₈×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
179	C₂×#578	C₂×C₃₁	1	1	yes	characteristic
180	#2950	C₂×C₃₁	1	2	yes	complemented
181	#2884	C₂×C₃₁	1	2	yes	complemented
182	#1976	C₂×C₃₁	1	2	yes	complemented
183	#1905	C₂×C₃₁	1	4	yes	complemented
184	#1812	C₂×C₃₁	1	1	yes	characteristic, complemented
185	#1796	C₂×C₃₁	1	2	yes	complemented
186	#1757	C₂×C₃₁	1	1	yes	characteristic, complemented
187	C₂²×C₃₁	—	2	2	no	—
188	C₂²×C₃₁	—	4	1	no	—
189	C₂²×C₃₁	—	4	2	no	—
190	C₂²×C₃₁	—	4	2	no	—
191	C₂²×C₃₁	—	8	2	no	—
192	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
193	C₂²×C₃₁	D₄²	1	2	yes	—
194	C₄×C₃₁	—	2	8	no	—
195	C₄×C₃₁	—	2	2	no	—
196	C₄×C₃₁	—	4	4	no	—
197	C₄×C₃₁	—	4	2	no	—
198	C₄×C₃₁	—	8	4	no	—
199	C₄×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
200	C₄×C₃₁	#518	1	1	yes	characteristic
201	C₄×C₃₁	#546	1	2	yes	—
202	C₂×2₋¹⁺⁴	C₂²×C₃₁	1	2	yes	—
203	#578	—	2	4	no	—
204	#578	—	2	4	no	—
205	C₂×#137	C₄×C₃₁	1	4	yes	complemented
206	C₂×#137	C₂²×C₃₁	1	2	yes	—
207	C₂×#135	C₄×C₃₁	1	2	yes	complemented
208	C₂×#135	C₂²×C₃₁	1	1	yes	characteristic
209	C₂²×Q₁₆	C₄×C₃₁	1	2	yes	complemented
210	C₂²×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
211	C₂×#131	C₂²×C₃₁	1	1	yes	characteristic
212	#519	C₂²×C₃₁	1	4	yes	—
213	#441	—	2	4	no	—
214	C₄×Q₁₆	—	2	4	no	—
215	#427	C₂²×C₃₁	1	2	yes	—
216	#421	C₂²×C₃₁	1	4	yes	—
217	#419	C₂²×C₃₁	1	4	yes	—
218	#413	C₂²×C₃₁	1	1	yes	characteristic
219	#410	—	2	8	no	—
220	#410	C₂²×C₃₁	1	2	yes	—
221	#410	C₂²×C₃₁	1	4	yes	—
222	#410	C₂²×C₃₁	1	1	yes	characteristic
223	#405	C₂²×C₃₁	1	2	yes	—
224	#344	C₂²×C₃₁	1	2	yes	—
225	H(3,ℤ/4)	C₂²×C₃₁	1	2	yes	—
226	C₂×C₃₁	—	2	2	no	—
227	C₂×C₃₁	—	2	1	no	—
228	C₂×C₃₁	—	4	2	no	—
229	C₂×C₃₁	—	8	2	no	—
230	C₂×C₃₁	D₄×C₂²⋊C₄	1	1	yes	characteristic
231	2₋¹⁺⁴	—	2	8	no	—
232	2₋¹⁺⁴	—	4	4	no	—
233	C₂×#41	—	2	4	no	—
234	C₂×#41	—	2	2	no	—
235	C₂×#41	C₂×C₄×C₃₁	1	4	yes	—
236	C₂×#41	C₂³×C₃₁	1	2	yes	—
237	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
238	C₂²×Q₈	—	2	2	no	—
239	C₂²×Q₈	C₂×C₄×C₃₁	1	4	yes	—
240	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
241	#137	—	2	8	no	—
242	#137	—	2	16	no	—
243	#135	—	2	4	no	—
244	#135	—	2	4	no	—
245	#135	—	2	2	no	—
246	#135	—	2	2	no	—
247	C₂×Q₁₆	—	2	4	no	—
248	C₂×Q₁₆	—	2	4	no	—
249	C₂×Q₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic
250	C₂×Q₁₆	C₂×C₄×C₃₁	1	4	yes	—
251	C₂×Q₁₆	C₂×C₄×C₃₁	1	2	yes	—
252	C₂×Q₁₆	C₂×C₄×C₃₁	1	2	yes	—
253	C₂×Q₁₆	D₄×C₃₁	1	4	yes	—
254	C₂×Q₁₆	D₄×C₃₁	1	4	yes	—
255	C₂×SD₁₆	C₂×C₄×C₃₁	1	4	yes	—
256	C₂×SD₁₆	C₂×C₄×C₃₁	1	2	yes	—
257	C₂×D₈	C₂×C₄×C₃₁	1	1	yes	characteristic
258	#131	—	2	2	no	—
259	#131	—	2	2	no	—
260	C₂×M₁₆	C₂×C₄×C₃₁	1	2	yes	—
261	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
262	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
263	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
264	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
265	C₄×Q₈	—	2	8	no	—
266	C₄×Q₈	—	4	4	no	—
267	#117	—	2	8	no	—
268	#117	C₂³×C₃₁	1	4	yes	—
269	#117	C₂³×C₃₁	1	2	yes	—
270	#107	—	2	2	no	—
271	#106	—	2	2	no	—
272	#104	—	4	4	no	—
273	#103	—	2	8	no	—
274	#101	—	2	1	no	—
275	#100	—	2	1	no	—
276	#99	—	2	4	no	—
277	#99	—	2	8	no	—
278	#99	D₄×C₃₁	1	4	yes	—
279	#97	—	2	2	no	—
280	C₄×C₈	—	2	2	no	—
281	C₃₁	#22457	1	1	yes	characteristic, direct product factor, Sylow
282	#41	—	2	2	no	—
283	#41	—	2	2	no	—
284	#41	—	2	8	no	—
285	#41	—	2	4	no	—
286	#41	—	2	4	no	—
287	#41	—	4	4	no	—
288	#41	—	4	16	no	—
289	#41	—	4	8	no	—
290	C₂×Q₈	—	2	4	no	—
291	C₂×Q₈	—	2	2	no	—
292	C₂×Q₈	—	2	4	no	—
293	C₂×Q₈	—	2	8	no	—
294	C₂×Q₈	—	4	4	no	—
295	C₂×Q₈	—	4	8	no	—
296	C₂×Q₈	C₂²×C₄×C₃₁	1	2	yes	—
297	C₂×Q₈	C₂²×C₄×C₃₁	1	4	yes	—
298	C₂×Q₈	C₂×D₄×C₃₁	1	8	yes	—
299	C₂×Q₈	#41×C₃₁	1	1	yes	characteristic
300	C₂×D₄	—	2	4	no	—
301	C₂×D₄	—	2	2	no	—
302	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
303	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
304	C₂×D₄	#41×C₃₁	1	1	yes	characteristic
305	C₂²×C₄	—	2	2	no	—
306	C₂²×C₄	—	2	4	no	—
307	C₂²×C₄	—	2	4	no	—
308	C₂²×C₄	—	2	2	no	—
309	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
310	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
311	Q₁₆	—	2	8	no	—
312	Q₁₆	—	2	1	no	—
313	Q₁₆	—	2	1	no	—
314	Q₁₆	C₂²⋊C₄×C₃₁	1	4	yes	—
315	Q₁₆	C₂²⋊C₄×C₃₁	1	4	yes	—
316	Q₁₆	C₂²⋊C₄×C₃₁	1	8	yes	—
317	SD₁₆	—	2	8	no	—
318	SD₁₆	—	2	4	no	—
319	D₈	—	2	1	no	—
320	D₈	—	2	1	no	—
321	M₁₆	—	2	2	no	—
322	M₁₆	—	2	4	no	—
323	M₁₆	—	2	2	no	—
324	M₁₆	—	2	2	no	—
325	C₂×C₈	—	2	2	no	—
326	C₂×C₈	—	2	2	no	—
327	C₂×C₈	—	2	2	no	—
328	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
329	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
330	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
331	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
332	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
333	C₄⋊C₄	—	2	2	no	—
334	C₄⋊C₄	—	2	4	no	—
335	C₄⋊C₄	—	4	8	no	—
336	C₂²⋊C₄	—	2	4	no	—
337	C₂²⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
338	C₄²	—	2	2	no	—
339	C₄²	—	2	4	no	—
340	C₄²	—	4	8	no	—
341	C₂³	—	2	2	no	—
342	C₂³	C₄×D₄×C₃₁	1	2	yes	—
343	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
344	Q₈	—	2	4	no	—
345	Q₈	—	2	8	no	—
346	Q₈	—	2	2	no	—
347	Q₈	—	2	2	no	—
348	Q₈	—	2	2	no	—
349	Q₈	—	4	8	no	—
350	Q₈	C₂×C₂²⋊C₄×C₃₁	1	8	yes	—
351	D₄	—	2	4	no	—
352	D₄	—	2	2	no	—
353	D₄	—	2	2	no	—
354	D₄	—	2	2	no	—
355	D₄	—	4	8	no	—
356	D₄	—	4	4	no	—
357	C₂×C₄	—	2	4	no	—
358	C₂×C₄	—	2	8	no	—
359	C₂×C₄	—	2	2	no	—
360	C₂×C₄	—	2	2	no	—
361	C₂×C₄	—	2	2	no	—
362	C₂×C₄	—	2	4	no	—
363	C₂×C₄	—	4	8	no	—
364	C₂×C₄	—	4	8	no	—
365	C₂×C₄	—	4	4	no	—
366	C₂×C₄	—	4	4	no	—
367	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
368	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
369	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
370	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
371	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
372	C₈	—	2	2	no	—
373	C₈	—	4	2	no	—
374	C₈	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
375	C₈	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
376	C₈	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
377	C₂²	—	2	2	no	—
378	C₂²	—	4	1	no	—
379	C₂²	—	4	2	no	—
380	C₂²	—	4	2	no	—
381	C₂²	—	8	2	no	—
382	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
383	C₂²	D₄²×C₃₁	1	2	yes	—
384	C₄	—	2	8	no	—
385	C₄	—	2	2	no	—
386	C₄	—	4	4	no	—
387	C₄	—	4	2	no	—
388	C₄	—	8	4	no	—
389	C₄	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
390	C₄	C₃₁×#518	1	1	yes	characteristic
391	C₄	C₃₁×#546	1	2	yes	—
392	C₂	—	2	2	no	—
393	C₂	—	2	1	no	—
394	C₂	—	4	2	no	—
395	C₂	—	8	2	no	—
396	C₂	D₄×C₂²⋊C₄×C₃₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃₁	D₄×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂×C₃₁×#578	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃₁ ◁ C₂²×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2950	C₂	semidirect
3. C₃₁×#2884	C₂	semidirect
4. C₃₁×#1976	C₂	semidirect
5. C₃₁×#1905	C₂	semidirect
6. C₃₁×#1812	C₂	semidirect
7. C₃₁×#1796	C₂	semidirect
8. C₃₁×#1757	C₂	semidirect
12. C₂×C₃₁×#137	C₄	semidirect
14. C₂×C₃₁×#135	C₄	semidirect
16. C₂²×Q₁₆×C₃₁	C₄	semidirect
142. #22457	C₃₁	direct (N × H)
180. #2950	C₂×C₃₁	semidirect
181. #2884	C₂×C₃₁	semidirect
182. #1976	C₂×C₃₁	semidirect
183. #1905	C₂×C₃₁	semidirect
184. #1812	C₂×C₃₁	semidirect
185. #1796	C₂×C₃₁	semidirect
186. #1757	C₂×C₃₁	semidirect
205. C₂×#137	C₄×C₃₁	semidirect
207. C₂×#135	C₄×C₃₁	semidirect
209. C₂²×Q₁₆	C₄×C₃₁	semidirect
281. C₃₁	#22457	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×#578	2
C₃₁×#2950	2
C₃₁×#2884	2
C₃₁×#1976	2
C₃₁×#1905	2
C₃₁×#1812	2
C₃₁×#1796	2
C₃₁×#1757	2
#22457	31

Subgroups — C31×#22457 #21479216

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22457 #21479216