Subgroups — C₃₁×#11346 #21468105

Order 7936.

878 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₄⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#336	C₂	1	4	yes	maximal
3	C₂×C₃₁×#336	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃₁×#336	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₈×C₃₁	C₂²	1	2	yes	—
6	C₂³×C₈×C₃₁	C₂²	1	2	yes	—
7	C₂²×C₄⋊C₄×C₃₁	C₄	1	1	yes	characteristic
8	C₂²×C₄⋊C₄×C₃₁	C₄	1	1	yes	characteristic
9	C₂²×C₄²×C₃₁	C₂²	1	2	yes	—
10	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
11	C₄×C₄⋊C₄×C₃₁	—	2	2	no	—
12	C₄×C₄⋊C₄×C₃₁	—	2	2	no	—
13	C₂×C₃₁×#95	C₄	1	2	yes	—
14	C₃₁×#336	—	2	8	no	—
15	C₃₁×#336	—	2	2	no	—
16	C₃₁×#336	—	2	2	no	—
17	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
18	C₂³×C₄×C₃₁	C₂×C₄	1	2	yes	—
19	C₂³×C₄×C₃₁	C₂×C₄	1	2	yes	—
20	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
21	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
22	C₂²×C₈×C₃₁	—	2	2	no	—
23	C₂²×C₈×C₃₁	—	2	4	no	—
24	C₂²×C₈×C₃₁	—	2	2	no	—
25	C₂²×C₈×C₃₁	—	2	2	no	—
26	C₂²×C₈×C₃₁	—	2	4	no	—
27	C₂²×C₈×C₃₁	—	2	2	no	—
28	C₂²×C₈×C₃₁	—	2	4	no	—
29	C₂²×C₈×C₃₁	—	2	4	no	—
30	C₂²×C₈×C₃₁	D₄	1	4	yes	—
31	C₂²×C₈×C₃₁	Q₈	1	4	yes	—
32	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
33	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
34	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
35	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
36	C₂×C₄⋊C₄×C₃₁	C₈	1	4	yes	complemented
37	C₂×C₄²×C₃₁	—	2	4	no	—
38	C₂×C₄²×C₃₁	—	2	4	no	—
39	C₂×C₄²×C₃₁	—	2	2	no	—
40	C₂×C₄²×C₃₁	—	2	2	no	—
41	C₂×C₄²×C₃₁	—	2	4	no	—
42	C₂×C₄²×C₃₁	D₄	1	2	yes	—
43	C₂×C₄²×C₃₁	D₄	1	2	yes	—
44	C₃₁×#95	—	2	4	no	—
45	C₂⁴×C₃₁	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₃₁	—	2	2	no	—
47	C₂²×C₄×C₃₁	—	2	2	no	—
48	C₂²×C₄×C₃₁	—	2	2	no	—
49	C₂²×C₄×C₃₁	—	2	2	no	—
50	C₂²×C₄×C₃₁	—	2	2	no	—
51	C₂²×C₄×C₃₁	—	2	4	no	—
52	C₂²×C₄×C₃₁	—	2	4	no	—
53	C₂²×C₄×C₃₁	—	2	2	no	—
54	C₂²×C₄×C₃₁	—	2	2	no	—
55	C₂²×C₄×C₃₁	—	2	2	no	—
56	C₂²×C₄×C₃₁	—	2	2	no	—
57	C₂²×C₄×C₃₁	—	2	4	no	—
58	C₂²×C₄×C₃₁	—	2	4	no	—
59	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
60	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
61	C₂²×C₄×C₃₁	C₂×C₈	1	4	yes	—
62	C₂²×C₄×C₃₁	C₂×C₈	1	2	yes	—
63	C₂²×C₄×C₃₁	M₁₆	1	4	yes	—
64	C₂²×C₄×C₃₁	M₁₆	1	2	yes	—
65	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
67	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₃₁	C₂×Q₈	1	2	yes	—
69	C₂²×C₄×C₃₁	#41	1	2	yes	—
70	C₂²×C₄×C₃₁	#41	1	4	yes	—
71	C₂²×C₄×C₃₁	#41	1	2	yes	—
72	C₂×C₈×C₃₁	—	2	4	no	—
73	C₂×C₈×C₃₁	—	2	8	no	—
74	C₂×C₈×C₃₁	—	2	4	no	—
75	C₂×C₈×C₃₁	—	2	8	no	—
76	C₂×C₈×C₃₁	—	4	2	no	—
77	C₂×C₈×C₃₁	—	4	4	no	—
78	C₂×C₈×C₃₁	—	4	2	no	—
79	C₂×C₈×C₃₁	—	4	2	no	—
80	C₂×C₈×C₃₁	—	4	4	no	—
81	C₂×C₈×C₃₁	—	4	2	no	—
82	C₄⋊C₄×C₃₁	—	2	8	no	—
83	C₄⋊C₄×C₃₁	—	2	8	no	—
84	C₄²×C₃₁	—	2	4	no	—
85	C₄²×C₃₁	—	2	4	no	—
86	C₄²×C₃₁	—	4	4	no	—
87	C₄²×C₃₁	—	4	2	no	—
88	C₄²×C₃₁	—	4	2	no	—
89	#11346	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
90	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	C₂³×C₃₁	#117	1	2	yes	—
92	C₂³×C₃₁	C₄×D₄	1	2	yes	—
93	C₂³×C₃₁	C₄×Q₈	1	2	yes	—
94	C₂³×C₃₁	C₂²×C₈	1	1	yes	characteristic
95	C₂³×C₃₁	C₂×M₁₆	1	1	yes	characteristic
96	C₂³×C₃₁	C₂×M₁₆	1	2	yes	—
97	C₂³×C₃₁	#131	1	2	yes	—
98	C₂³×C₃₁	#131	1	2	yes	—
99	C₂×C₄×C₃₁	—	2	2	no	—
100	C₂×C₄×C₃₁	—	2	2	no	—
101	C₂×C₄×C₃₁	—	2	4	no	—
102	C₂×C₄×C₃₁	—	2	4	no	—
103	C₂×C₄×C₃₁	—	2	2	no	—
104	C₂×C₄×C₃₁	—	2	4	no	—
105	C₂×C₄×C₃₁	—	2	4	no	—
106	C₂×C₄×C₃₁	—	2	2	no	—
107	C₂×C₄×C₃₁	—	2	2	no	—
108	C₂×C₄×C₃₁	—	2	8	no	—
109	C₂×C₄×C₃₁	—	2	8	no	—
110	C₂×C₄×C₃₁	—	4	2	no	—
111	C₂×C₄×C₃₁	—	4	2	no	—
112	C₂×C₄×C₃₁	—	4	2	no	—
113	C₂×C₄×C₃₁	—	4	2	no	—
114	C₂×C₄×C₃₁	—	4	4	no	—
115	C₂×C₄×C₃₁	—	4	4	no	—
116	C₂×C₄×C₃₁	—	4	4	no	—
117	C₂×C₄×C₃₁	C₂²⋊C₈	1	4	yes	—
118	C₂×C₄×C₃₁	#121	1	2	yes	—
119	C₂×C₄×C₃₁	C₂²⋊Q₈	1	4	yes	—
120	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
121	C₂×C₄×C₃₁	#123	1	2	yes	—
122	C₂×C₄×C₃₁	#123	1	4	yes	—
123	C₂×C₄×C₃₁	#123	1	2	yes	—
124	C₂×C₄×C₃₁	#124	1	1	yes	characteristic
125	C₂×C₄×C₃₁	#124	1	1	yes	characteristic
126	C₂×C₄×C₃₁	#125	1	4	yes	—
127	C₂×C₄×C₃₁	#126	1	4	yes	—
128	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
129	C₂×C₄×C₃₁	C₄⋊Q₈	1	1	yes	characteristic
130	C₈×C₃₁	—	4	4	no	—
131	C₈×C₃₁	—	4	4	no	—
132	C₂×C₄×C₄⋊C₄	C₂×C₃₁	1	1	yes	characteristic
133	C₂×#336	C₂×C₃₁	1	1	yes	characteristic
134	C₂×#336	C₂×C₃₁	1	4	yes	—
135	C₂×#336	C₂×C₃₁	1	1	yes	characteristic
136	C₂²×C₃₁	#381	1	1	yes	characteristic
137	C₂²×C₃₁	#387	1	4	yes	—
138	C₂²×C₃₁	#390	1	1	yes	characteristic
139	C₂²×C₃₁	#391	1	1	yes	characteristic
140	C₂²×C₃₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
141	C₂²×C₃₁	#407	1	1	yes	characteristic
142	C₂²×C₃₁	#408	1	1	yes	characteristic
143	C₂²×C₃₁	#408	1	1	yes	characteristic
144	C₂²×C₃₁	#431	1	2	yes	—
145	C₂²×C₃₁	#432	1	2	yes	—
146	C₂²×C₃₁	#433	1	4	yes	—
147	C₂²×C₃₁	C₈×D₄	1	2	yes	—
148	C₂²×C₃₁	#435	1	4	yes	—
149	C₂²×C₃₁	#436	1	2	yes	—
150	C₂²×C₃₁	C₈×Q₈	1	2	yes	—
151	C₂²×C₃₁	#446	1	4	yes	—
152	C₂²×C₃₁	#446	1	2	yes	—
153	C₄×C₃₁	—	2	4	no	—
154	C₄×C₃₁	—	4	2	no	—
155	C₄×C₃₁	—	4	4	no	—
156	C₄×C₃₁	—	4	4	no	—
157	C₄×C₃₁	#397	1	2	yes	—
158	C₄×C₃₁	#398	1	2	yes	—
159	C₄×C₃₁	#399	1	2	yes	—
160	C₄×C₃₁	#400	1	2	yes	—
161	C₂³×C₈	C₂²×C₃₁	1	2	yes	—
162	C₂³×C₈	C₂²×C₃₁	1	2	yes	—
163	C₂²×C₄⋊C₄	C₄×C₃₁	1	1	yes	characteristic
164	C₂²×C₄⋊C₄	C₄×C₃₁	1	1	yes	characteristic
165	C₂²×C₄²	C₂²×C₃₁	1	2	yes	—
166	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
167	C₄×C₄⋊C₄	—	2	2	no	—
168	C₄×C₄⋊C₄	—	2	2	no	—
169	C₂×#95	C₄×C₃₁	1	2	yes	—
170	#336	—	2	8	no	—
171	#336	—	2	2	no	—
172	#336	—	2	2	no	—
173	C₂×C₃₁	#1445	1	1	yes	characteristic
174	C₂×C₃₁	#1817	1	1	yes	characteristic
175	C₂×C₃₁	#1818	1	1	yes	characteristic
176	C₂×C₃₁	#1916	1	4	yes	—
177	C₂×C₃₁	#1917	1	4	yes	—
178	C₂×C₃₁	#1954	1	1	yes	characteristic
179	C₂×C₃₁	#1955	1	1	yes	characteristic
180	C₂×C₃₁	#1984	1	1	yes	characteristic
181	C₂×C₃₁	#1985	1	1	yes	characteristic
182	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
183	C₂³×C₄	C₂×C₄×C₃₁	1	2	yes	—
184	C₂³×C₄	C₂×C₄×C₃₁	1	2	yes	—
185	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
186	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
187	C₂²×C₈	—	2	2	no	—
188	C₂²×C₈	—	2	4	no	—
189	C₂²×C₈	—	2	2	no	—
190	C₂²×C₈	—	2	2	no	—
191	C₂²×C₈	—	2	4	no	—
192	C₂²×C₈	—	2	2	no	—
193	C₂²×C₈	—	2	4	no	—
194	C₂²×C₈	—	2	4	no	—
195	C₂²×C₈	D₄×C₃₁	1	4	yes	—
196	C₂²×C₈	Q₈×C₃₁	1	4	yes	—
197	C₂×C₄⋊C₄	—	2	2	no	—
198	C₂×C₄⋊C₄	—	2	2	no	—
199	C₂×C₄⋊C₄	—	2	4	no	—
200	C₂×C₄⋊C₄	—	2	2	no	—
201	C₂×C₄⋊C₄	C₈×C₃₁	1	4	yes	complemented
202	C₂×C₄²	—	2	4	no	—
203	C₂×C₄²	—	2	4	no	—
204	C₂×C₄²	—	2	2	no	—
205	C₂×C₄²	—	2	2	no	—
206	C₂×C₄²	—	2	4	no	—
207	C₂×C₄²	D₄×C₃₁	1	2	yes	—
208	C₂×C₄²	D₄×C₃₁	1	2	yes	—
209	#95	—	2	4	no	—
210	C₃₁	#11346	1	1	yes	characteristic, direct product factor, Sylow
211	C₂⁴	C₂²×C₄×C₃₁	1	1	yes	characteristic
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	—	2	2	no	—
216	C₂²×C₄	—	2	2	no	—
217	C₂²×C₄	—	2	4	no	—
218	C₂²×C₄	—	2	4	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	2	2	no	—
221	C₂²×C₄	—	2	2	no	—
222	C₂²×C₄	—	2	2	no	—
223	C₂²×C₄	—	2	4	no	—
224	C₂²×C₄	—	2	4	no	—
225	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
226	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
227	C₂²×C₄	C₂×C₈×C₃₁	1	4	yes	—
228	C₂²×C₄	C₂×C₈×C₃₁	1	2	yes	—
229	C₂²×C₄	M₁₆×C₃₁	1	4	yes	—
230	C₂²×C₄	M₁₆×C₃₁	1	2	yes	—
231	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
232	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
233	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
234	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
235	C₂²×C₄	#41×C₃₁	1	2	yes	—
236	C₂²×C₄	#41×C₃₁	1	4	yes	—
237	C₂²×C₄	#41×C₃₁	1	2	yes	—
238	C₂×C₈	—	2	4	no	—
239	C₂×C₈	—	2	8	no	—
240	C₂×C₈	—	2	4	no	—
241	C₂×C₈	—	2	8	no	—
242	C₂×C₈	—	4	2	no	—
243	C₂×C₈	—	4	4	no	—
244	C₂×C₈	—	4	2	no	—
245	C₂×C₈	—	4	2	no	—
246	C₂×C₈	—	4	4	no	—
247	C₂×C₈	—	4	2	no	—
248	C₄⋊C₄	—	2	8	no	—
249	C₄⋊C₄	—	2	8	no	—
250	C₄²	—	2	4	no	—
251	C₄²	—	2	4	no	—
252	C₄²	—	4	4	no	—
253	C₄²	—	4	2	no	—
254	C₄²	—	4	2	no	—
255	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
256	C₂³	C₃₁×#117	1	2	yes	—
257	C₂³	C₄×D₄×C₃₁	1	2	yes	—
258	C₂³	C₄×Q₈×C₃₁	1	2	yes	—
259	C₂³	C₂²×C₈×C₃₁	1	1	yes	characteristic
260	C₂³	C₂×M₁₆×C₃₁	1	1	yes	characteristic
261	C₂³	C₂×M₁₆×C₃₁	1	2	yes	—
262	C₂³	C₃₁×#131	1	2	yes	—
263	C₂³	C₃₁×#131	1	2	yes	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	8	no	—
273	C₂×C₄	—	2	8	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	4	no	—
278	C₂×C₄	—	4	4	no	—
279	C₂×C₄	—	4	4	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	C₃₁×C₂²⋊C₈	1	4	yes	—
283	C₂×C₄	C₃₁×#121	1	2	yes	—
284	C₂×C₄	C₃₁×C₂²⋊Q₈	1	4	yes	—
285	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
286	C₂×C₄	C₃₁×#123	1	2	yes	—
287	C₂×C₄	C₃₁×#123	1	4	yes	—
288	C₂×C₄	C₃₁×#123	1	2	yes	—
289	C₂×C₄	C₃₁×#124	1	1	yes	characteristic
290	C₂×C₄	C₃₁×#124	1	1	yes	characteristic
291	C₂×C₄	C₃₁×#125	1	4	yes	—
292	C₂×C₄	C₃₁×#126	1	4	yes	—
293	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
294	C₂×C₄	C₃₁×C₄⋊Q₈	1	1	yes	characteristic
295	C₈	—	4	4	no	—
296	C₈	—	4	4	no	—
297	C₂²	C₃₁×#381	1	1	yes	characteristic
298	C₂²	C₃₁×#387	1	4	yes	—
299	C₂²	C₃₁×#390	1	1	yes	characteristic
300	C₂²	C₃₁×#391	1	1	yes	characteristic
301	C₂²	C₂×C₃₁×C₂²⋊C₈	1	1	yes	characteristic
302	C₂²	C₃₁×#407	1	1	yes	characteristic
303	C₂²	C₃₁×#408	1	1	yes	characteristic
304	C₂²	C₃₁×#408	1	1	yes	characteristic
305	C₂²	C₃₁×#431	1	2	yes	—
306	C₂²	C₃₁×#432	1	2	yes	—
307	C₂²	C₃₁×#433	1	4	yes	—
308	C₂²	C₈×D₄×C₃₁	1	2	yes	—
309	C₂²	C₃₁×#435	1	4	yes	—
310	C₂²	C₃₁×#436	1	2	yes	—
311	C₂²	C₈×Q₈×C₃₁	1	2	yes	—
312	C₂²	C₃₁×#446	1	4	yes	—
313	C₂²	C₃₁×#446	1	2	yes	—
314	C₄	—	2	4	no	—
315	C₄	—	4	2	no	—
316	C₄	—	4	4	no	—
317	C₄	—	4	4	no	—
318	C₄	C₃₁×#397	1	2	yes	—
319	C₄	C₃₁×#398	1	2	yes	—
320	C₄	C₃₁×#399	1	2	yes	—
321	C₄	C₃₁×#400	1	2	yes	—
322	C₂	C₃₁×#1445	1	1	yes	characteristic
323	C₂	C₃₁×#1817	1	1	yes	characteristic
324	C₂	C₃₁×#1818	1	1	yes	characteristic
325	C₂	C₃₁×#1916	1	4	yes	—
326	C₂	C₃₁×#1917	1	4	yes	—
327	C₂	C₃₁×#1954	1	1	yes	characteristic
328	C₂	C₃₁×#1955	1	1	yes	characteristic
329	C₂	C₃₁×#1984	1	1	yes	characteristic
330	C₂	C₃₁×#1985	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄×C₃₁	C₂³
commutator	C₂³	C₂²×C₈×C₃₁
Frattini, cosocle	C₂³×C₄	C₂³×C₃₁
socle, prime radical	C₂⁴×C₃₁	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
36. C₂×C₄⋊C₄×C₃₁	C₈	semidirect
89. #11346	C₃₁	direct (N × H)
201. C₂×C₄⋊C₄	C₈×C₃₁	semidirect
210. C₃₁	#11346	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×C₄×C₄⋊C₄×C₃₁	2
C₂×C₃₁×#336	2
#11346	31

Subgroups — C31×#11346 #21468105

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#11346 #21468105