Subgroups — C₃₁×#22517 #21479276

Order 7936.

938 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#557	C₂	1	1	yes	maximal, characteristic
2	Q₈×M₁₆×C₃₁	C₂	1	2	yes	maximal
3	C₃₁×#1998	C₂	1	4	yes	maximal
4	C₃₁×#1997	C₂	1	2	yes	maximal
5	C₃₁×#1801	C₂	1	1	yes	maximal, characteristic
6	C₃₁×#1801	C₂	1	2	yes	maximal
7	C₄×C₃₁×#101	C₂	1	1	yes	maximal, characteristic
8	C₄×C₃₁×#101	C₂	1	2	yes	maximal
9	C₃₁×#557	—	2	2	no	—
10	C₃₁×#557	—	2	4	no	—
11	C₃₁×#557	—	2	2	no	—
12	C₂×C₃₁×C₄⋊Q₈	C₄	1	2	yes	—
13	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×#125	C₄	1	4	yes	—
15	C₂×C₃₁×#125	C₂²	1	2	yes	—
16	C₂×C₄×Q₈×C₃₁	C₄	1	2	yes	—
17	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
18	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
19	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
20	C₃₁×#446	—	2	4	no	—
21	C₃₁×#446	—	2	2	no	—
22	C₈×Q₈×C₃₁	—	2	2	no	—
23	C₃₁×#423	C₂²	1	2	yes	—
24	C₃₁×#423	C₂²	1	4	yes	—
25	C₂×C₃₁×#101	C₂²	1	1	yes	characteristic
26	C₂×C₃₁×#101	C₂²	1	2	yes	—
27	C₂×C₃₁×#101	C₂²	1	1	yes	characteristic
28	C₄×M₁₆×C₃₁	C₂²	1	2	yes	—
29	C₄×M₁₆×C₃₁	C₂²	1	4	yes	—
30	C₃₁×#343	C₂²	1	4	yes	—
31	C₃₁×#343	C₂²	1	2	yes	—
32	C₃₁×#343	C₂²	1	4	yes	—
33	C₃₁×#343	C₂²	1	2	yes	—
34	C₂²×Q₈×C₃₁	C₂×C₄	1	2	yes	—
35	C₂²×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
36	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
37	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
38	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
39	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
40	C₃₁×C₄⋊Q₈	—	2	4	no	—
41	C₃₁×C₄⋊Q₈	—	2	4	no	—
42	C₃₁×C₄⋊Q₈	—	2	2	no	—
43	C₃₁×C₄⋊Q₈	—	2	2	no	—
44	C₃₁×#125	—	2	8	no	—
45	C₃₁×#125	—	2	8	no	—
46	C₃₁×#125	—	2	4	no	—
47	C₃₁×#125	—	2	4	no	—
48	C₄×Q₈×C₃₁	—	2	4	no	—
49	C₄×Q₈×C₃₁	—	2	2	no	—
50	C₄×Q₈×C₃₁	—	2	2	no	—
51	C₄×Q₈×C₃₁	—	2	4	no	—
52	C₄×Q₈×C₃₁	—	2	4	no	—
53	C₄×Q₈×C₃₁	—	2	2	no	—
54	C₄×Q₈×C₃₁	—	2	2	no	—
55	C₄×Q₈×C₃₁	D₄	1	2	yes	—
56	C₄×Q₈×C₃₁	D₄	1	4	yes	—
57	C₄×Q₈×C₃₁	D₄	1	2	yes	—
58	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
59	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
60	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
61	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
62	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
63	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
65	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
66	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
67	C₂×C₄²×C₃₁	C₂×C₄	1	4	yes	—
68	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
69	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
70	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
71	C₃₁×C₄⋊C₈	—	2	4	no	—
72	C₃₁×C₄⋊C₈	—	2	2	no	—
73	C₃₁×C₄⋊C₈	—	2	4	no	—
74	C₃₁×C₄⋊C₈	—	2	2	no	—
75	C₃₁×#101	—	2	2	no	—
76	C₃₁×#101	—	2	4	no	—
77	C₃₁×#101	Q₈	1	4	yes	complemented
78	C₃₁×#97	—	2	4	no	—
79	C₃₁×#97	—	2	2	no	—
80	C₄×C₈×C₃₁	—	2	4	no	—
81	C₄×C₈×C₃₁	—	2	2	no	—
82	C₂×Q₈×C₃₁	—	2	2	no	—
83	C₂×Q₈×C₃₁	—	2	2	no	—
84	C₂×Q₈×C₃₁	—	2	2	no	—
85	C₂×Q₈×C₃₁	—	2	2	no	—
86	C₂×Q₈×C₃₁	—	4	2	no	—
87	C₂×Q₈×C₃₁	—	4	4	no	—
88	C₂×Q₈×C₃₁	C₂²⋊C₄	1	2	yes	—
89	C₂×Q₈×C₃₁	C₂²⋊C₄	1	2	yes	—
90	C₂×Q₈×C₃₁	C₂×Q₈	1	2	yes	—
91	C₂×Q₈×C₃₁	#41	1	2	yes	—
92	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
93	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
94	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
95	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
96	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
97	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
98	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
99	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
100	M₁₆×C₃₁	—	2	2	no	—
101	M₁₆×C₃₁	—	2	2	no	—
102	M₁₆×C₃₁	—	2	4	no	—
103	M₁₆×C₃₁	—	2	2	no	—
104	M₁₆×C₃₁	—	2	4	no	—
105	M₁₆×C₃₁	C₂×Q₈	1	4	yes	—
106	C₂×C₈×C₃₁	—	2	2	no	—
107	C₂×C₈×C₃₁	—	2	4	no	—
108	C₂×C₈×C₃₁	—	2	2	no	—
109	C₄⋊C₄×C₃₁	—	2	4	no	—
110	C₄⋊C₄×C₃₁	—	2	4	no	—
111	C₄⋊C₄×C₃₁	—	2	4	no	—
112	C₄⋊C₄×C₃₁	—	2	8	no	—
113	C₄⋊C₄×C₃₁	—	2	8	no	—
114	C₄⋊C₄×C₃₁	—	2	4	no	—
115	C₄⋊C₄×C₃₁	—	2	2	no	—
116	C₄⋊C₄×C₃₁	—	2	4	no	—
117	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
118	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
119	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
120	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	4	yes	—
121	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	4	yes	—
122	C₄²×C₃₁	—	2	4	no	—
123	C₄²×C₃₁	—	2	4	no	—
124	C₄²×C₃₁	—	2	2	no	—
125	C₄²×C₃₁	—	2	2	no	—
126	C₄²×C₃₁	C₂×D₄	1	4	yes	—
127	C₄²×C₃₁	C₂×D₄	1	2	yes	—
128	C₄²×C₃₁	C₂×D₄	1	4	yes	—
129	C₄²×C₃₁	C₂×D₄	1	2	yes	—
130	#22517	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
131	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
132	Q₈×C₃₁	—	2	4	no	—
133	Q₈×C₃₁	—	2	4	no	—
134	Q₈×C₃₁	—	2	4	no	—
135	Q₈×C₃₁	—	4	2	no	—
136	Q₈×C₃₁	—	4	4	no	—
137	Q₈×C₃₁	—	4	4	no	—
138	C₂×C₄×C₃₁	—	2	2	no	—
139	C₂×C₄×C₃₁	—	2	2	no	—
140	C₂×C₄×C₃₁	—	2	2	no	—
141	C₂×C₄×C₃₁	—	2	2	no	—
142	C₂×C₄×C₃₁	—	2	4	no	—
143	C₂×C₄×C₃₁	—	4	2	no	—
144	C₂×C₄×C₃₁	—	4	1	no	—
145	C₂×C₄×C₃₁	—	4	2	no	—
146	C₂×C₄×C₃₁	—	4	2	no	—
147	C₂×C₄×C₃₁	—	4	4	no	—
148	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
149	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
150	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
151	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
152	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	4	yes	—
153	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
154	C₂×C₄×C₃₁	C₄×Q₈	1	4	yes	—
155	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
156	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
157	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
158	C₂×C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
159	C₈×C₃₁	—	2	4	no	—
160	C₈×C₃₁	—	4	4	no	—
161	C₈×C₃₁	—	4	2	no	—
162	C₂×#557	C₂×C₃₁	1	1	yes	characteristic
163	Q₈×M₁₆	C₂×C₃₁	1	2	yes	—
164	#1998	C₂×C₃₁	1	4	yes	—
165	#1997	C₂×C₃₁	1	2	yes	—
166	#1801	C₂×C₃₁	1	2	yes	—
167	#1801	C₂×C₃₁	1	1	yes	characteristic
168	C₄×#101	C₂×C₃₁	1	2	yes	—
169	C₄×#101	C₂×C₃₁	1	1	yes	characteristic
170	C₂²×C₃₁	—	2	1	no	—
171	C₂²×C₃₁	—	2	1	no	—
172	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
173	C₂²×C₃₁	C₂×C₄×Q₈	1	1	yes	characteristic
174	C₂²×C₃₁	#519	1	1	yes	characteristic
175	C₄×C₃₁	—	2	4	no	—
176	C₄×C₃₁	—	2	1	no	—
177	C₄×C₃₁	—	2	4	no	—
178	C₄×C₃₁	—	2	2	no	—
179	C₄×C₃₁	—	2	2	no	—
180	C₄×C₃₁	—	2	2	no	—
181	C₄×C₃₁	—	4	4	no	—
182	C₄×C₃₁	—	4	2	no	—
183	C₄×C₃₁	#413	1	2	yes	—
184	C₄×C₃₁	#548	1	2	yes	—
185	C₄×C₃₁	D₄×Q₈	1	2	yes	—
186	#557	—	2	2	no	—
187	#557	—	2	4	no	—
188	#557	—	2	2	no	—
189	C₂×C₄⋊Q₈	C₄×C₃₁	1	2	yes	—
190	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
191	C₂×#125	C₄×C₃₁	1	4	yes	—
192	C₂×#125	C₂²×C₃₁	1	2	yes	—
193	C₂×C₄×Q₈	C₄×C₃₁	1	2	yes	—
194	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
195	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
196	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
197	#446	—	2	4	no	—
198	#446	—	2	2	no	—
199	C₈×Q₈	—	2	2	no	—
200	#423	C₂²×C₃₁	1	2	yes	—
201	#423	C₂²×C₃₁	1	4	yes	—
202	C₂×#101	C₂²×C₃₁	1	1	yes	characteristic
203	C₂×#101	C₂²×C₃₁	1	2	yes	—
204	C₂×#101	C₂²×C₃₁	1	1	yes	characteristic
205	C₄×M₁₆	C₂²×C₃₁	1	2	yes	—
206	C₄×M₁₆	C₂²×C₃₁	1	4	yes	—
207	#343	C₂²×C₃₁	1	4	yes	—
208	#343	C₂²×C₃₁	1	2	yes	—
209	#343	C₂²×C₃₁	1	4	yes	—
210	#343	C₂²×C₃₁	1	2	yes	—
211	C₂×C₃₁	—	2	1	no	—
212	C₂×C₃₁	—	2	1	no	—
213	C₂×C₃₁	Q₈×C₂²⋊C₄	1	1	yes	characteristic
214	C₂×C₃₁	C₂×#413	1	1	yes	characteristic
215	C₂×C₃₁	#2889	1	1	yes	characteristic
216	C₂²×Q₈	C₂×C₄×C₃₁	1	2	yes	—
217	C₂²×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
218	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
219	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
220	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
221	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
222	C₄⋊Q₈	—	2	4	no	—
223	C₄⋊Q₈	—	2	4	no	—
224	C₄⋊Q₈	—	2	2	no	—
225	C₄⋊Q₈	—	2	2	no	—
226	#125	—	2	8	no	—
227	#125	—	2	8	no	—
228	#125	—	2	4	no	—
229	#125	—	2	4	no	—
230	C₄×Q₈	—	2	4	no	—
231	C₄×Q₈	—	2	2	no	—
232	C₄×Q₈	—	2	2	no	—
233	C₄×Q₈	—	2	4	no	—
234	C₄×Q₈	—	2	4	no	—
235	C₄×Q₈	—	2	2	no	—
236	C₄×Q₈	—	2	2	no	—
237	C₄×Q₈	D₄×C₃₁	1	2	yes	—
238	C₄×Q₈	D₄×C₃₁	1	4	yes	—
239	C₄×Q₈	D₄×C₃₁	1	2	yes	—
240	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	4	yes	—
241	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
242	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	4	yes	—
243	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
244	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	4	yes	—
245	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
246	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
247	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
248	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
249	C₂×C₄²	C₂×C₄×C₃₁	1	4	yes	—
250	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
251	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
252	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
253	C₄⋊C₈	—	2	4	no	—
254	C₄⋊C₈	—	2	2	no	—
255	C₄⋊C₈	—	2	4	no	—
256	C₄⋊C₈	—	2	2	no	—
257	#101	—	2	2	no	—
258	#101	—	2	4	no	—
259	#101	Q₈×C₃₁	1	4	yes	complemented
260	#97	—	2	4	no	—
261	#97	—	2	2	no	—
262	C₄×C₈	—	2	4	no	—
263	C₄×C₈	—	2	2	no	—
264	C₃₁	#22517	1	1	yes	characteristic, direct product factor, Sylow
265	C₂×Q₈	—	2	2	no	—
266	C₂×Q₈	—	2	2	no	—
267	C₂×Q₈	—	2	2	no	—
268	C₂×Q₈	—	2	2	no	—
269	C₂×Q₈	—	4	2	no	—
270	C₂×Q₈	—	4	4	no	—
271	C₂×Q₈	C₂²⋊C₄×C₃₁	1	2	yes	—
272	C₂×Q₈	C₂²⋊C₄×C₃₁	1	2	yes	—
273	C₂×Q₈	C₂×Q₈×C₃₁	1	2	yes	—
274	C₂×Q₈	#41×C₃₁	1	2	yes	—
275	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
276	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
277	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
278	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
279	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
280	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
281	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
282	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
283	M₁₆	—	2	2	no	—
284	M₁₆	—	2	2	no	—
285	M₁₆	—	2	4	no	—
286	M₁₆	—	2	2	no	—
287	M₁₆	—	2	4	no	—
288	M₁₆	C₂×Q₈×C₃₁	1	4	yes	—
289	C₂×C₈	—	2	2	no	—
290	C₂×C₈	—	2	4	no	—
291	C₂×C₈	—	2	2	no	—
292	C₄⋊C₄	—	2	4	no	—
293	C₄⋊C₄	—	2	4	no	—
294	C₄⋊C₄	—	2	4	no	—
295	C₄⋊C₄	—	2	8	no	—
296	C₄⋊C₄	—	2	8	no	—
297	C₄⋊C₄	—	2	4	no	—
298	C₄⋊C₄	—	2	2	no	—
299	C₄⋊C₄	—	2	4	no	—
300	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
301	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
302	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
303	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	4	yes	—
304	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	4	yes	—
305	C₄²	—	2	4	no	—
306	C₄²	—	2	4	no	—
307	C₄²	—	2	2	no	—
308	C₄²	—	2	2	no	—
309	C₄²	C₂×D₄×C₃₁	1	4	yes	—
310	C₄²	C₂×D₄×C₃₁	1	2	yes	—
311	C₄²	C₂×D₄×C₃₁	1	4	yes	—
312	C₄²	C₂×D₄×C₃₁	1	2	yes	—
313	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
314	Q₈	—	2	4	no	—
315	Q₈	—	2	4	no	—
316	Q₈	—	2	4	no	—
317	Q₈	—	4	2	no	—
318	Q₈	—	4	4	no	—
319	Q₈	—	4	4	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	2	4	no	—
325	C₂×C₄	—	4	2	no	—
326	C₂×C₄	—	4	1	no	—
327	C₂×C₄	—	4	2	no	—
328	C₂×C₄	—	4	2	no	—
329	C₂×C₄	—	4	4	no	—
330	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
331	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
332	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
333	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
334	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	4	yes	—
335	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
336	C₂×C₄	C₄×Q₈×C₃₁	1	4	yes	—
337	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
338	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
339	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
340	C₂×C₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
341	C₈	—	2	4	no	—
342	C₈	—	4	4	no	—
343	C₈	—	4	2	no	—
344	C₂²	—	2	1	no	—
345	C₂²	—	2	1	no	—
346	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
347	C₂²	C₂×C₄×Q₈×C₃₁	1	1	yes	characteristic
348	C₂²	C₃₁×#519	1	1	yes	characteristic
349	C₄	—	2	4	no	—
350	C₄	—	2	1	no	—
351	C₄	—	2	4	no	—
352	C₄	—	2	2	no	—
353	C₄	—	2	2	no	—
354	C₄	—	2	2	no	—
355	C₄	—	4	4	no	—
356	C₄	—	4	2	no	—
357	C₄	C₃₁×#413	1	2	yes	—
358	C₄	C₃₁×#548	1	2	yes	—
359	C₄	D₄×Q₈×C₃₁	1	2	yes	—
360	C₂	—	2	1	no	—
361	C₂	—	2	1	no	—
362	C₂	Q₈×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
363	C₂	C₂×C₃₁×#413	1	1	yes	characteristic
364	C₂	C₃₁×#2889	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×C₂²⋊C₄
commutator	C₂³	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂³×C₃₁	C₂³×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄×Q₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
77. C₃₁×#101	Q₈	semidirect
130. #22517	C₃₁	direct (N × H)
259. #101	Q₈×C₃₁	semidirect
264. C₃₁	#22517	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×#557	2
Q₈×M₁₆×C₃₁	2
C₃₁×#1998	2
C₃₁×#1997	2
C₃₁×#1801	2
C₄×C₃₁×#101	2
#22517	31

Subgroups — C31×#22517 #21479276

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22517 #21479276