Subgroups — C₃₁×#22716 #21479475

Order 7936.

906 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#436	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2352	C₂	1	2	yes	maximal
3	C₃₁×#1949	C₂	1	2	yes	maximal, complemented
4	C₃₁×#1854	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1852	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1835	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1809	C₂	1	2	yes	maximal, complemented
8	C₄×C₃₁×#107	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1736	C₂	1	2	yes	maximal, complemented
10	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
11	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
12	C₂×C₃₁×#117	C₂²	1	2	yes	—
13	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
14	C₃₁×#436	C₄	1	8	yes	complemented
15	C₂×C₃₁×#107	C₂²	1	2	yes	—
16	C₂×C₃₁×#107	C₂²	1	2	yes	—
17	C₂×C₃₁×#106	C₂²	1	2	yes	—
18	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
19	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
21	C₃₁×#388	—	2	2	no	—
22	C₃₁×#387	—	2	2	no	—
23	C₃₁×#386	—	2	2	no	—
24	C₃₁×#382	C₂²	1	2	yes	—
25	C₃₁×#380	C₂²	1	2	yes	—
26	C₃₁×#380	C₂²	1	2	yes	—
27	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
28	C₄×C₂²⋊C₄×C₃₁	—	2	2	no	—
29	C₃₁×#340	C₂²	1	1	yes	characteristic
30	C₃₁×#340	C₂²	1	2	yes	—
31	C₃₁×#340	C₂²	1	1	yes	characteristic
32	C₃₁×#340	C₂²	1	1	yes	characteristic
33	C₃₁×#340	C₂²	1	2	yes	—
34	C₃₁×#340	C₂²	1	2	yes	—
35	C₃₁×#340	C₂²	1	1	yes	characteristic
36	C₂²×D₄×C₃₁	D₄	1	1	yes	characteristic
37	C₂³×C₄×C₃₁	—	2	2	no	—
38	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
39	C₂×M₁₆×C₃₁	—	4	2	no	—
40	C₂×M₁₆×C₃₁	—	4	2	no	—
41	C₂×M₁₆×C₃₁	C₂×C₄	1	8	yes	complemented
42	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
43	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
44	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
45	C₄×D₄×C₃₁	C₂×C₄	1	4	yes	—
46	C₄×D₄×C₃₁	Q₈	1	4	yes	—
47	C₃₁×#117	—	2	4	no	—
48	C₃₁×#117	—	4	2	no	—
49	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
50	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
51	C₂×C₄⋊C₄×C₃₁	—	4	2	no	—
52	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
53	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
54	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
55	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
56	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
57	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
58	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
59	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
60	C₂×C₂²⋊C₄×C₃₁	D₄	1	2	yes	—
61	C₂×C₄²×C₃₁	—	2	2	no	—
62	C₂×C₄²×C₃₁	—	2	2	no	—
63	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
64	C₃₁×#107	—	2	4	no	—
65	C₃₁×#107	—	4	2	no	—
66	C₃₁×#106	—	4	2	no	—
67	C₃₁×C₄⋊C₈	C₂×C₄	1	4	yes	complemented
68	C₃₁×C₂²⋊C₈	C₂×C₄	1	8	yes	complemented
69	C₄×C₈×C₃₁	C₂×C₄	1	4	yes	complemented
70	C₃₁×#95	—	2	2	no	—
71	C₃₁×#95	—	2	2	no	—
72	C₃₁×#95	—	2	2	no	—
73	C₃₁×#95	—	2	2	no	—
74	C₃₁×#95	—	2	2	no	—
75	C₂⁴×C₃₁	C₂×D₄	1	2	yes	—
76	C₂×D₄×C₃₁	—	4	2	no	—
77	C₂×D₄×C₃₁	C₄⋊C₄	1	4	yes	—
78	C₂²×C₄×C₃₁	—	2	2	no	—
79	C₂²×C₄×C₃₁	—	2	2	no	—
80	C₂²×C₄×C₃₁	—	2	2	no	—
81	C₂²×C₄×C₃₁	—	2	2	no	—
82	C₂²×C₄×C₃₁	—	2	2	no	—
83	C₂²×C₄×C₃₁	—	2	2	no	—
84	C₂²×C₄×C₃₁	—	2	2	no	—
85	C₂²×C₄×C₃₁	—	2	2	no	—
86	C₂²×C₄×C₃₁	—	4	4	no	—
87	C₂²×C₄×C₃₁	—	4	4	no	—
88	C₂²×C₄×C₃₁	—	8	2	no	—
89	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
90	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
93	C₂²×C₄×C₃₁	C₂×Q₈	1	4	yes	—
94	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
95	M₁₆×C₃₁	—	4	8	no	—
96	C₂×C₈×C₃₁	—	2	2	no	—
97	C₂×C₈×C₃₁	—	2	2	no	—
98	C₂×C₈×C₃₁	—	2	2	no	—
99	C₂×C₈×C₃₁	—	2	2	no	—
100	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
101	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
102	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
103	C₄⋊C₄×C₃₁	—	2	2	no	—
104	C₄⋊C₄×C₃₁	—	2	2	no	—
105	C₄⋊C₄×C₃₁	—	2	2	no	—
106	C₄⋊C₄×C₃₁	—	2	2	no	—
107	C₄⋊C₄×C₃₁	—	2	2	no	—
108	C₄⋊C₄×C₃₁	—	2	2	no	—
109	C₄⋊C₄×C₃₁	—	2	2	no	—
110	C₄⋊C₄×C₃₁	—	4	4	no	—
111	C₄⋊C₄×C₃₁	C₄⋊C₄	1	4	yes	—
112	C₄⋊C₄×C₃₁	#41	1	4	yes	—
113	C₂²⋊C₄×C₃₁	—	4	4	no	—
114	C₂²⋊C₄×C₃₁	—	4	4	no	—
115	C₂²⋊C₄×C₃₁	—	8	2	no	—
116	C₂²⋊C₄×C₃₁	C₄⋊C₄	1	8	yes	—
117	C₄²×C₃₁	—	4	4	no	—
118	C₄²×C₃₁	—	8	2	no	—
119	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
120	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
121	#22716	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
122	C₂³×C₃₁	—	2	2	no	—
123	C₂³×C₃₁	—	2	2	no	—
124	C₂³×C₃₁	—	2	2	no	—
125	C₂³×C₃₁	—	2	2	no	—
126	C₂³×C₃₁	—	2	2	no	—
127	C₂³×C₃₁	—	2	2	no	—
128	C₂³×C₃₁	C₂×C₄⋊C₄	1	4	yes	—
129	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
130	D₄×C₃₁	—	4	4	no	—
131	C₂×C₄×C₃₁	—	2	2	no	—
132	C₂×C₄×C₃₁	—	2	2	no	—
133	C₂×C₄×C₃₁	—	2	2	no	—
134	C₂×C₄×C₃₁	—	2	1	no	—
135	C₂×C₄×C₃₁	—	2	1	no	—
136	C₂×C₄×C₃₁	—	2	4	no	—
137	C₂×C₄×C₃₁	—	2	4	no	—
138	C₂×C₄×C₃₁	—	2	4	no	—
139	C₂×C₄×C₃₁	—	2	4	no	—
140	C₂×C₄×C₃₁	—	2	4	no	—
141	C₂×C₄×C₃₁	—	4	2	no	—
142	C₂×C₄×C₃₁	—	4	2	no	—
143	C₂×C₄×C₃₁	—	4	2	no	—
144	C₂×C₄×C₃₁	—	4	2	no	—
145	C₂×C₄×C₃₁	—	4	2	no	—
146	C₂×C₄×C₃₁	—	4	2	no	—
147	C₂×C₄×C₃₁	—	4	2	no	—
148	C₂×C₄×C₃₁	—	4	2	no	—
149	C₂×C₄×C₃₁	—	4	2	no	—
150	C₂×C₄×C₃₁	—	4	2	no	—
151	C₂×C₄×C₃₁	—	4	2	no	—
152	C₂×C₄×C₃₁	—	8	8	no	—
153	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
154	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
155	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	4	yes	—
156	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
157	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
158	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
159	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
160	C₈×C₃₁	—	2	2	no	—
161	C₈×C₃₁	—	2	4	no	—
162	C₈×C₃₁	—	2	2	no	—
163	C₂×#436	C₂×C₃₁	1	1	yes	characteristic
164	#2352	C₂×C₃₁	1	2	yes	—
165	#1949	C₂×C₃₁	1	2	yes	complemented
166	#1854	C₂×C₃₁	1	2	yes	complemented
167	#1852	C₂×C₃₁	1	2	yes	complemented
168	#1835	C₂×C₃₁	1	1	yes	characteristic, complemented
169	#1809	C₂×C₃₁	1	2	yes	complemented
170	C₄×#107	C₂×C₃₁	1	1	yes	characteristic, complemented
171	#1736	C₂×C₃₁	1	2	yes	complemented
172	C₂²×C₃₁	—	2	4	no	—
173	C₂²×C₃₁	—	2	4	no	—
174	C₂²×C₃₁	—	2	4	no	—
175	C₂²×C₃₁	—	4	2	no	—
176	C₂²×C₃₁	—	4	2	no	—
177	C₂²×C₃₁	—	4	2	no	—
178	C₂²×C₃₁	—	4	2	no	—
179	C₂²×C₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
180	C₂²×C₃₁	#546	1	2	yes	—
181	C₂²×C₃₁	#575	1	2	yes	—
182	C₂²×C₃₁	#576	1	1	yes	characteristic
183	C₂²×C₃₁	#578	1	1	yes	characteristic
184	C₄×C₃₁	—	2	4	no	—
185	C₄×C₃₁	—	2	2	no	—
186	C₄×C₃₁	—	2	2	no	—
187	C₄×C₃₁	—	4	4	no	—
188	C₄×C₃₁	—	8	2	no	—
189	C₄×C₃₁	—	8	2	no	—
190	C₄×C₃₁	—	8	2	no	—
191	C₄×C₃₁	C₄×#41	1	1	yes	characteristic
192	C₄×C₃₁	#518	1	1	yes	characteristic
193	C₄×C₃₁	#554	1	2	yes	—
194	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
195	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
196	C₂×#117	C₂²×C₃₁	1	2	yes	—
197	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
198	#436	C₄×C₃₁	1	8	yes	complemented
199	C₂×#107	C₂²×C₃₁	1	2	yes	—
200	C₂×#107	C₂²×C₃₁	1	2	yes	—
201	C₂×#106	C₂²×C₃₁	1	2	yes	—
202	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
203	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
204	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
205	#388	—	2	2	no	—
206	#387	—	2	2	no	—
207	#386	—	2	2	no	—
208	#382	C₂²×C₃₁	1	2	yes	—
209	#380	C₂²×C₃₁	1	2	yes	—
210	#380	C₂²×C₃₁	1	2	yes	—
211	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
212	C₄×C₂²⋊C₄	—	2	2	no	—
213	#340	C₂²×C₃₁	1	1	yes	characteristic
214	#340	C₂²×C₃₁	1	2	yes	—
215	#340	C₂²×C₃₁	1	1	yes	characteristic
216	#340	C₂²×C₃₁	1	1	yes	characteristic
217	#340	C₂²×C₃₁	1	2	yes	—
218	#340	C₂²×C₃₁	1	2	yes	—
219	#340	C₂²×C₃₁	1	1	yes	characteristic
220	C₂×C₃₁	—	4	4	no	—
221	C₂×C₃₁	#2349	1	1	yes	characteristic
222	C₂×C₃₁	#2911	1	1	yes	characteristic
223	C₂×C₃₁	#2913	1	1	yes	characteristic
224	C₂×C₃₁	#3322	1	2	yes	—
225	C₂×C₃₁	#3327	1	2	yes	—
226	C₂²×D₄	D₄×C₃₁	1	1	yes	characteristic
227	C₂³×C₄	—	2	2	no	—
228	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
229	C₂×M₁₆	—	4	2	no	—
230	C₂×M₁₆	—	4	2	no	—
231	C₂×M₁₆	C₂×C₄×C₃₁	1	8	yes	complemented
232	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
233	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
234	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
235	C₄×D₄	C₂×C₄×C₃₁	1	4	yes	—
236	C₄×D₄	Q₈×C₃₁	1	4	yes	—
237	#117	—	2	4	no	—
238	#117	—	4	2	no	—
239	C₂×C₄⋊C₄	—	2	4	no	—
240	C₂×C₄⋊C₄	—	2	2	no	—
241	C₂×C₄⋊C₄	—	4	2	no	—
242	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
243	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
244	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
245	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
246	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
247	C₂×C₂²⋊C₄	—	2	2	no	—
248	C₂×C₂²⋊C₄	—	2	2	no	—
249	C₂×C₂²⋊C₄	—	2	2	no	—
250	C₂×C₂²⋊C₄	D₄×C₃₁	1	2	yes	—
251	C₂×C₄²	—	2	2	no	—
252	C₂×C₄²	—	2	2	no	—
253	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
254	#107	—	2	4	no	—
255	#107	—	4	2	no	—
256	#106	—	4	2	no	—
257	C₄⋊C₈	C₂×C₄×C₃₁	1	4	yes	complemented
258	C₂²⋊C₈	C₂×C₄×C₃₁	1	8	yes	complemented
259	C₄×C₈	C₂×C₄×C₃₁	1	4	yes	complemented
260	#95	—	2	2	no	—
261	#95	—	2	2	no	—
262	#95	—	2	2	no	—
263	#95	—	2	2	no	—
264	#95	—	2	2	no	—
265	C₃₁	#22716	1	1	yes	characteristic, direct product factor, Sylow
266	C₂⁴	C₂×D₄×C₃₁	1	2	yes	—
267	C₂×D₄	—	4	2	no	—
268	C₂×D₄	C₄⋊C₄×C₃₁	1	4	yes	—
269	C₂²×C₄	—	2	2	no	—
270	C₂²×C₄	—	2	2	no	—
271	C₂²×C₄	—	2	2	no	—
272	C₂²×C₄	—	2	2	no	—
273	C₂²×C₄	—	2	2	no	—
274	C₂²×C₄	—	2	2	no	—
275	C₂²×C₄	—	2	2	no	—
276	C₂²×C₄	—	2	2	no	—
277	C₂²×C₄	—	4	4	no	—
278	C₂²×C₄	—	4	4	no	—
279	C₂²×C₄	—	8	2	no	—
280	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
281	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
282	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
283	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
284	C₂²×C₄	C₂×Q₈×C₃₁	1	4	yes	—
285	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
286	M₁₆	—	4	8	no	—
287	C₂×C₈	—	2	2	no	—
288	C₂×C₈	—	2	2	no	—
289	C₂×C₈	—	2	2	no	—
290	C₂×C₈	—	2	2	no	—
291	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
292	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
293	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
294	C₄⋊C₄	—	2	2	no	—
295	C₄⋊C₄	—	2	2	no	—
296	C₄⋊C₄	—	2	2	no	—
297	C₄⋊C₄	—	2	2	no	—
298	C₄⋊C₄	—	2	2	no	—
299	C₄⋊C₄	—	2	2	no	—
300	C₄⋊C₄	—	2	2	no	—
301	C₄⋊C₄	—	4	4	no	—
302	C₄⋊C₄	C₄⋊C₄×C₃₁	1	4	yes	—
303	C₄⋊C₄	#41×C₃₁	1	4	yes	—
304	C₂²⋊C₄	—	4	4	no	—
305	C₂²⋊C₄	—	4	4	no	—
306	C₂²⋊C₄	—	8	2	no	—
307	C₂²⋊C₄	C₄⋊C₄×C₃₁	1	8	yes	—
308	C₄²	—	4	4	no	—
309	C₄²	—	8	2	no	—
310	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
311	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
312	C₂³	—	2	2	no	—
313	C₂³	—	2	2	no	—
314	C₂³	—	2	2	no	—
315	C₂³	—	2	2	no	—
316	C₂³	—	2	2	no	—
317	C₂³	—	2	2	no	—
318	C₂³	C₂×C₄⋊C₄×C₃₁	1	4	yes	—
319	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
320	D₄	—	4	4	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	2	1	no	—
325	C₂×C₄	—	2	1	no	—
326	C₂×C₄	—	2	4	no	—
327	C₂×C₄	—	2	4	no	—
328	C₂×C₄	—	2	4	no	—
329	C₂×C₄	—	2	4	no	—
330	C₂×C₄	—	2	4	no	—
331	C₂×C₄	—	4	2	no	—
332	C₂×C₄	—	4	2	no	—
333	C₂×C₄	—	4	2	no	—
334	C₂×C₄	—	4	2	no	—
335	C₂×C₄	—	4	2	no	—
336	C₂×C₄	—	4	2	no	—
337	C₂×C₄	—	4	2	no	—
338	C₂×C₄	—	4	2	no	—
339	C₂×C₄	—	4	2	no	—
340	C₂×C₄	—	4	2	no	—
341	C₂×C₄	—	4	2	no	—
342	C₂×C₄	—	8	8	no	—
343	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
344	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
345	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	4	yes	—
346	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
347	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
348	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
349	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
350	C₈	—	2	2	no	—
351	C₈	—	2	4	no	—
352	C₈	—	2	2	no	—
353	C₂²	—	2	4	no	—
354	C₂²	—	2	4	no	—
355	C₂²	—	2	4	no	—
356	C₂²	—	4	2	no	—
357	C₂²	—	4	2	no	—
358	C₂²	—	4	2	no	—
359	C₂²	—	4	2	no	—
360	C₂²	C₂²×C₄⋊C₄×C₃₁	1	1	yes	characteristic
361	C₂²	C₃₁×#546	1	2	yes	—
362	C₂²	C₃₁×#575	1	2	yes	—
363	C₂²	C₃₁×#576	1	1	yes	characteristic
364	C₂²	C₃₁×#578	1	1	yes	characteristic
365	C₄	—	2	4	no	—
366	C₄	—	2	2	no	—
367	C₄	—	2	2	no	—
368	C₄	—	4	4	no	—
369	C₄	—	8	2	no	—
370	C₄	—	8	2	no	—
371	C₄	—	8	2	no	—
372	C₄	C₄×#41×C₃₁	1	1	yes	characteristic
373	C₄	C₃₁×#518	1	1	yes	characteristic
374	C₄	C₃₁×#554	1	2	yes	—
375	C₂	—	4	4	no	—
376	C₂	C₃₁×#2349	1	1	yes	characteristic
377	C₂	C₃₁×#2911	1	1	yes	characteristic
378	C₂	C₃₁×#2913	1	1	yes	characteristic
379	C₂	C₃₁×#3322	1	2	yes	—
380	C₂	C₃₁×#3327	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₃₁×#1949	C₂	semidirect
4. C₃₁×#1854	C₂	semidirect
5. C₃₁×#1852	C₂	semidirect
6. C₃₁×#1835	C₂	semidirect
7. C₃₁×#1809	C₂	semidirect
8. C₄×C₃₁×#107	C₂	semidirect
9. C₃₁×#1736	C₂	semidirect
14. C₃₁×#436	C₄	semidirect
41. C₂×M₁₆×C₃₁	C₂×C₄	semidirect
67. C₃₁×C₄⋊C₈	C₂×C₄	semidirect
68. C₃₁×C₂²⋊C₈	C₂×C₄	semidirect
69. C₄×C₈×C₃₁	C₂×C₄	semidirect
121. #22716	C₃₁	direct (N × H)
165. #1949	C₂×C₃₁	semidirect
166. #1854	C₂×C₃₁	semidirect
167. #1852	C₂×C₃₁	semidirect
168. #1835	C₂×C₃₁	semidirect
169. #1809	C₂×C₃₁	semidirect
170. C₄×#107	C₂×C₃₁	semidirect
171. #1736	C₂×C₃₁	semidirect
198. #436	C₄×C₃₁	semidirect
231. C₂×M₁₆	C₂×C₄×C₃₁	semidirect
257. C₄⋊C₈	C₂×C₄×C₃₁	semidirect
258. C₂²⋊C₈	C₂×C₄×C₃₁	semidirect
259. C₄×C₈	C₂×C₄×C₃₁	semidirect
265. C₃₁	#22716	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×#436	2
C₃₁×#2352	2
C₃₁×#1949	2
C₃₁×#1854	2
C₃₁×#1852	2
C₃₁×#1835	2
C₃₁×#1809	2
C₄×C₃₁×#107	2
C₃₁×#1736	2
#22716	31

Subgroups — C31×#22716 #21479475

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22716 #21479475