Subgroups — C₃₁×#8604 #21465363

Order 7936.

462 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₃₁×#131	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1564	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#334	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1477	C₂	1	2	yes	maximal
5	C₃₁×#1283	C₂	1	2	yes	maximal, complemented
6	C₂×C₃₁×#131	C₄	1	1	yes	characteristic
7	C₂×C₃₁×#131	C₄	1	1	yes	characteristic
8	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
9	C₃₁×#431	C₂²	1	2	yes	—
10	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
11	C₃₁×#405	—	2	1	no	—
12	C₃₁×#405	—	2	1	no	—
13	C₃₁×#405	C₄	1	1	yes	characteristic
14	C₃₁×#405	C₄	1	1	yes	characteristic
15	C₄×M₁₆×C₃₁	—	2	1	no	—
16	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₃₁	—	2	1	no	—
18	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
19	C₃₁×#335	—	2	1	no	—
20	C₃₁×#334	—	2	1	no	—
21	C₃₁×#334	—	2	1	no	—
22	C₃₁×#334	—	2	1	no	—
23	C₃₁×Q₈⋊C₈	—	2	2	no	—
24	C₃₁×D₄⋊C₈	—	2	2	no	—
25	C₂×#41×C₃₁	C₂×C₄	1	1	yes	characteristic
26	C₃₁×#131	—	2	2	no	—
27	C₃₁×#131	—	2	2	no	—
28	C₃₁×#131	—	2	2	no	—
29	C₃₁×#131	C₈	1	4	yes	complemented
30	C₂×M₁₆×C₃₁	—	2	1	no	—
31	C₂×M₁₆×C₃₁	—	2	1	no	—
32	C₂×M₁₆×C₃₁	C₂×C₄	1	1	yes	characteristic
33	C₂×M₁₆×C₃₁	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₃₁	—	2	1	no	—
35	C₂²×C₈×C₃₁	—	2	1	no	—
36	C₂²×C₈×C₃₁	—	2	2	no	—
37	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
39	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
40	C₄×Q₈×C₃₁	Q₈	1	1	yes	characteristic
41	C₄×D₄×C₃₁	—	2	1	no	—
42	C₄×D₄×C₃₁	—	2	1	no	—
43	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
44	C₄×D₄×C₃₁	Q₈	1	1	yes	characteristic
45	C₃₁×#117	—	2	1	no	—
46	C₃₁×#117	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²×C₃₁	—	2	1	no	—
48	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
49	C₃₁×C₄⋊C₈	—	2	2	no	—
50	C₃₁×C₄⋊C₈	—	2	2	no	—
51	C₃₁×C₄⋊C₈	—	2	2	no	—
52	C₃₁×C₂²⋊C₈	—	2	2	no	—
53	C₃₁×#97	—	2	1	no	—
54	C₃₁×#97	—	2	1	no	—
55	C₃₁×#97	—	2	1	no	—
56	C₄×C₈×C₃₁	—	2	1	no	—
57	C₄×C₈×C₃₁	—	2	1	no	—
58	C₄×C₈×C₃₁	—	2	1	no	—
59	C₄×C₈×C₃₁	—	2	2	no	—
60	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
61	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
62	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
63	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
64	#41×C₃₁	—	2	1	no	—
65	#41×C₃₁	—	2	1	no	—
66	#41×C₃₁	C₂×C₈	1	2	yes	—
67	#41×C₃₁	M₁₆	1	2	yes	—
68	C₂×Q₈×C₃₁	C₄⋊C₄	1	1	yes	characteristic
69	C₂×D₄×C₃₁	—	2	1	no	—
70	C₂×D₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
71	C₂²×C₄×C₃₁	—	2	1	no	—
72	C₂²×C₄×C₃₁	—	2	1	no	—
73	C₂²×C₄×C₃₁	—	2	1	no	—
74	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
76	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
77	M₁₆×C₃₁	—	2	4	no	—
78	M₁₆×C₃₁	—	2	1	no	—
79	M₁₆×C₃₁	—	2	1	no	—
80	M₁₆×C₃₁	—	2	4	no	—
81	M₁₆×C₃₁	C₂×C₈	1	2	yes	—
82	M₁₆×C₃₁	M₁₆	1	2	yes	—
83	C₂×C₈×C₃₁	—	2	4	no	—
84	C₂×C₈×C₃₁	—	2	1	no	—
85	C₂×C₈×C₃₁	—	2	1	no	—
86	C₂×C₈×C₃₁	—	2	1	no	—
87	C₂×C₈×C₃₁	—	2	1	no	—
88	C₂×C₈×C₃₁	—	2	4	no	—
89	C₂×C₈×C₃₁	—	4	2	no	—
90	C₂×C₈×C₃₁	—	4	2	no	—
91	C₂×C₈×C₃₁	—	8	2	no	—
92	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₈×C₃₁	C₂×C₈	1	2	yes	—
97	C₂×C₈×C₃₁	M₁₆	1	2	yes	—
98	C₄⋊C₄×C₃₁	—	2	1	no	—
99	C₄⋊C₄×C₃₁	—	2	1	no	—
100	C₄⋊C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
101	C₄⋊C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
102	C₂²⋊C₄×C₃₁	—	2	1	no	—
103	C₂²⋊C₄×C₃₁	—	2	1	no	—
104	C₂²⋊C₄×C₃₁	—	2	1	no	—
105	C₄²×C₃₁	—	2	1	no	—
106	C₄²×C₃₁	—	2	1	no	—
107	C₄²×C₃₁	—	2	1	no	—
108	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
110	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
111	C₄²×C₃₁	C₂×Q₈	1	1	yes	characteristic
112	#8604	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
113	C₂³×C₃₁	—	2	1	no	—
114	C₂³×C₃₁	#117	1	1	yes	characteristic
115	Q₈×C₃₁	—	2	1	no	—
116	Q₈×C₃₁	C₄⋊C₈	1	2	yes	—
117	D₄×C₃₁	—	2	2	no	—
118	D₄×C₃₁	—	2	2	no	—
119	D₄×C₃₁	—	2	1	no	—
120	D₄×C₃₁	C₄⋊C₈	1	2	yes	—
121	C₂×C₄×C₃₁	—	2	1	no	—
122	C₂×C₄×C₃₁	—	2	2	no	—
123	C₂×C₄×C₃₁	—	2	2	no	—
124	C₂×C₄×C₃₁	—	2	2	no	—
125	C₂×C₄×C₃₁	—	2	1	no	—
126	C₂×C₄×C₃₁	—	2	1	no	—
127	C₂×C₄×C₃₁	—	4	1	no	—
128	C₂×C₄×C₃₁	—	4	2	no	—
129	C₂×C₄×C₃₁	—	4	2	no	—
130	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
131	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
132	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
133	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
134	C₂×C₄×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
135	C₂×C₄×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
136	C₂×C₄×C₃₁	C₂²×C₈	1	1	yes	characteristic
137	C₂×C₄×C₃₁	C₂×M₁₆	1	1	yes	characteristic
138	C₂×C₄×C₃₁	C₂×M₁₆	1	1	yes	characteristic
139	C₂×C₄×C₃₁	C₂×M₁₆	1	1	yes	characteristic
140	C₈×C₃₁	—	2	2	no	—
141	C₈×C₃₁	—	2	1	no	—
142	C₈×C₃₁	—	2	2	no	—
143	C₈×C₃₁	—	2	1	no	—
144	C₈×C₃₁	—	8	2	no	—
145	C₈×C₃₁	—	8	2	no	—
146	C₈×C₃₁	C₂²⋊C₈	1	2	yes	—
147	C₈×C₃₁	C₂²⋊C₈	1	2	yes	—
148	C₄×#131	C₂×C₃₁	1	1	yes	characteristic
149	#1564	C₂×C₃₁	1	1	yes	characteristic, complemented
150	C₂×#334	C₂×C₃₁	1	1	yes	characteristic, complemented
151	#1477	C₂×C₃₁	1	2	yes	—
152	#1283	C₂×C₃₁	1	2	yes	complemented
153	C₂²×C₃₁	—	2	1	no	—
154	C₂²×C₃₁	—	2	1	no	—
155	C₂²×C₃₁	—	2	2	no	—
156	C₂²×C₃₁	—	4	1	no	—
157	C₂²×C₃₁	—	4	1	no	—
158	C₂²×C₃₁	#380	1	1	yes	characteristic
159	C₂²×C₃₁	#431	1	1	yes	characteristic
160	C₂²×C₃₁	#432	1	1	yes	characteristic
161	C₄×C₃₁	—	2	2	no	—
162	C₄×C₃₁	—	2	1	no	—
163	C₄×C₃₁	—	2	1	no	—
164	C₄×C₃₁	—	2	1	no	—
165	C₄×C₃₁	—	4	1	no	—
166	C₄×C₃₁	—	4	1	no	—
167	C₄×C₃₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
168	C₄×C₃₁	#407	1	1	yes	characteristic
169	C₄×C₃₁	C₂×C₄⋊C₈	1	1	yes	characteristic
170	C₄×C₃₁	#423	1	1	yes	characteristic
171	C₄×C₃₁	#471	1	2	yes	—
172	C₂×#131	C₄×C₃₁	1	1	yes	characteristic
173	C₂×#131	C₄×C₃₁	1	1	yes	characteristic
174	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
175	#431	C₂²×C₃₁	1	2	yes	—
176	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
177	#405	—	2	1	no	—
178	#405	—	2	1	no	—
179	#405	C₄×C₃₁	1	1	yes	characteristic
180	#405	C₄×C₃₁	1	1	yes	characteristic
181	C₄×M₁₆	—	2	1	no	—
182	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
183	C₂×C₄×C₈	—	2	1	no	—
184	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
185	#335	—	2	1	no	—
186	#334	—	2	1	no	—
187	#334	—	2	1	no	—
188	#334	—	2	1	no	—
189	Q₈⋊C₈	—	2	2	no	—
190	D₄⋊C₈	—	2	2	no	—
191	C₂×C₃₁	—	2	1	no	—
192	C₂×C₃₁	—	2	1	no	—
193	C₂×C₃₁	—	4	2	no	—
194	C₂×C₃₁	#1797	1	1	yes	characteristic
195	C₂×C₃₁	#1814	1	1	yes	characteristic
196	C₂×C₃₁	#1815	1	1	yes	characteristic
197	C₂×#41	C₂×C₄×C₃₁	1	1	yes	characteristic
198	#131	—	2	2	no	—
199	#131	—	2	2	no	—
200	#131	—	2	2	no	—
201	#131	C₈×C₃₁	1	4	yes	complemented
202	C₂×M₁₆	—	2	1	no	—
203	C₂×M₁₆	—	2	1	no	—
204	C₂×M₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic
205	C₂×M₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic
206	C₂²×C₈	—	2	1	no	—
207	C₂²×C₈	—	2	1	no	—
208	C₂²×C₈	—	2	2	no	—
209	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
210	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
211	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
212	C₄×Q₈	Q₈×C₃₁	1	1	yes	characteristic
213	C₄×D₄	—	2	1	no	—
214	C₄×D₄	—	2	1	no	—
215	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
216	C₄×D₄	Q₈×C₃₁	1	1	yes	characteristic
217	#117	—	2	1	no	—
218	#117	C₂×C₄×C₃₁	1	1	yes	characteristic
219	C₂×C₄²	—	2	1	no	—
220	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
221	C₄⋊C₈	—	2	2	no	—
222	C₄⋊C₈	—	2	2	no	—
223	C₄⋊C₈	—	2	2	no	—
224	C₂²⋊C₈	—	2	2	no	—
225	#97	—	2	1	no	—
226	#97	—	2	1	no	—
227	#97	—	2	1	no	—
228	C₄×C₈	—	2	1	no	—
229	C₄×C₈	—	2	1	no	—
230	C₄×C₈	—	2	1	no	—
231	C₄×C₈	—	2	2	no	—
232	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
233	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
234	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
235	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
236	C₃₁	#8604	1	1	yes	characteristic, direct product factor, Sylow
237	#41	—	2	1	no	—
238	#41	—	2	1	no	—
239	#41	C₂×C₈×C₃₁	1	2	yes	—
240	#41	M₁₆×C₃₁	1	2	yes	—
241	C₂×Q₈	C₄⋊C₄×C₃₁	1	1	yes	characteristic
242	C₂×D₄	—	2	1	no	—
243	C₂×D₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
244	C₂²×C₄	—	2	1	no	—
245	C₂²×C₄	—	2	1	no	—
246	C₂²×C₄	—	2	1	no	—
247	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
248	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
249	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
250	M₁₆	—	2	4	no	—
251	M₁₆	—	2	1	no	—
252	M₁₆	—	2	1	no	—
253	M₁₆	—	2	4	no	—
254	M₁₆	C₂×C₈×C₃₁	1	2	yes	—
255	M₁₆	M₁₆×C₃₁	1	2	yes	—
256	C₂×C₈	—	2	4	no	—
257	C₂×C₈	—	2	1	no	—
258	C₂×C₈	—	2	1	no	—
259	C₂×C₈	—	2	1	no	—
260	C₂×C₈	—	2	1	no	—
261	C₂×C₈	—	2	4	no	—
262	C₂×C₈	—	4	2	no	—
263	C₂×C₈	—	4	2	no	—
264	C₂×C₈	—	8	2	no	—
265	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
266	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
267	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
268	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
269	C₂×C₈	C₂×C₈×C₃₁	1	2	yes	—
270	C₂×C₈	M₁₆×C₃₁	1	2	yes	—
271	C₄⋊C₄	—	2	1	no	—
272	C₄⋊C₄	—	2	1	no	—
273	C₄⋊C₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
274	C₄⋊C₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
275	C₂²⋊C₄	—	2	1	no	—
276	C₂²⋊C₄	—	2	1	no	—
277	C₂²⋊C₄	—	2	1	no	—
278	C₄²	—	2	1	no	—
279	C₄²	—	2	1	no	—
280	C₄²	—	2	1	no	—
281	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
282	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
283	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
284	C₄²	C₂×Q₈×C₃₁	1	1	yes	characteristic
285	C₂³	—	2	1	no	—
286	C₂³	C₃₁×#117	1	1	yes	characteristic
287	Q₈	—	2	1	no	—
288	Q₈	C₃₁×C₄⋊C₈	1	2	yes	—
289	D₄	—	2	2	no	—
290	D₄	—	2	2	no	—
291	D₄	—	2	1	no	—
292	D₄	C₃₁×C₄⋊C₈	1	2	yes	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	4	1	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
303	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
304	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
305	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
306	C₂×C₄	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
307	C₂×C₄	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
308	C₂×C₄	C₂²×C₈×C₃₁	1	1	yes	characteristic
309	C₂×C₄	C₂×M₁₆×C₃₁	1	1	yes	characteristic
310	C₂×C₄	C₂×M₁₆×C₃₁	1	1	yes	characteristic
311	C₂×C₄	C₂×M₁₆×C₃₁	1	1	yes	characteristic
312	C₈	—	2	2	no	—
313	C₈	—	2	1	no	—
314	C₈	—	2	2	no	—
315	C₈	—	2	1	no	—
316	C₈	—	8	2	no	—
317	C₈	—	8	2	no	—
318	C₈	C₃₁×C₂²⋊C₈	1	2	yes	—
319	C₈	C₃₁×C₂²⋊C₈	1	2	yes	—
320	C₂²	—	2	1	no	—
321	C₂²	—	2	1	no	—
322	C₂²	—	2	2	no	—
323	C₂²	—	4	1	no	—
324	C₂²	—	4	1	no	—
325	C₂²	C₃₁×#380	1	1	yes	characteristic
326	C₂²	C₃₁×#431	1	1	yes	characteristic
327	C₂²	C₃₁×#432	1	1	yes	characteristic
328	C₄	—	2	2	no	—
329	C₄	—	2	1	no	—
330	C₄	—	2	1	no	—
331	C₄	—	2	1	no	—
332	C₄	—	4	1	no	—
333	C₄	—	4	1	no	—
334	C₄	C₂×C₃₁×C₂²⋊C₈	1	1	yes	characteristic
335	C₄	C₃₁×#407	1	1	yes	characteristic
336	C₄	C₂×C₃₁×C₄⋊C₈	1	1	yes	characteristic
337	C₄	C₃₁×#423	1	1	yes	characteristic
338	C₄	C₃₁×#471	1	2	yes	—
339	C₂	—	2	1	no	—
340	C₂	—	2	1	no	—
341	C₂	—	4	2	no	—
342	C₂	C₃₁×#1797	1	1	yes	characteristic
343	C₂	C₃₁×#1814	1	1	yes	characteristic
344	C₂	C₃₁×#1815	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	#121
commutator	C₂×C₄	C₂²×C₈×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
socle	C₂²×C₃₁	#380
prime radical	C₂×#41×C₃₁	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ 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×#1564	C₂	semidirect
3. C₂×C₃₁×#334	C₂	semidirect
5. C₃₁×#1283	C₂	semidirect
29. C₃₁×#131	C₈	semidirect
112. #8604	C₃₁	direct (N × H)
149. #1564	C₂×C₃₁	semidirect
150. C₂×#334	C₂×C₃₁	semidirect
152. #1283	C₂×C₃₁	semidirect
201. #131	C₈×C₃₁	semidirect
236. C₃₁	#8604	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₃₁×#131	2
C₃₁×#1564	2
C₂×C₃₁×#334	2
C₃₁×#1477	2
C₃₁×#1283	2
#8604	31

Subgroups — C31×#8604 #21465363

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#8604 #21465363