Subgroups — C₃₁×#9674 #21466433

Order 7936.

736 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₃₁×#127	C₂	1	1	yes	maximal, characteristic
2	C₄×C₃₁×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₃₁×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃₁×D₄⋊C₈	C₂	1	4	yes	maximal, complemented
5	C₂×C₃₁×#127	C₄	1	1	yes	characteristic
6	C₂×C₄×D₄×C₃₁	—	2	1	no	—
7	C₂×C₄×D₄×C₃₁	—	2	1	no	—
8	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
9	C₂×C₃₁×C₄⋊C₈	—	2	1	no	—
10	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
11	C₂×C₄×C₈×C₃₁	—	2	1	no	—
12	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
13	C₃₁×#390	—	2	1	no	—
14	C₃₁×#390	—	2	1	no	—
15	C₃₁×#390	C₄	1	2	yes	—
16	C₃₁×#384	C₄	1	1	yes	characteristic
17	C₄³×C₃₁	C₂²	1	1	yes	characteristic
18	C₃₁×#336	—	2	1	no	—
19	C₃₁×#336	—	2	1	no	—
20	C₃₁×D₄⋊C₈	—	2	8	no	—
21	C₂²×D₄×C₃₁	—	2	1	no	—
22	C₂²×D₄×C₃₁	—	2	1	no	—
23	C₂²×D₄×C₃₁	C₂×C₄	1	2	yes	—
24	C₂³×C₄×C₃₁	—	2	2	no	—
25	C₂²×C₈×C₃₁	—	2	2	no	—
26	C₂²×C₈×C₃₁	—	2	2	no	—
27	C₃₁×#127	—	2	2	no	—
28	C₃₁×#127	C₈	1	4	yes	complemented
29	C₄×D₄×C₃₁	—	2	4	no	—
30	C₄×D₄×C₃₁	—	2	4	no	—
31	C₄×D₄×C₃₁	—	4	4	no	—
32	C₄×D₄×C₃₁	—	4	2	no	—
33	C₄×D₄×C₃₁	—	4	2	no	—
34	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
35	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
36	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
37	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
38	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
39	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
40	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
47	C₃₁×C₄⋊C₈	—	2	4	no	—
48	C₃₁×C₄⋊C₈	—	2	2	no	—
49	C₃₁×C₄⋊C₈	D₄	1	4	yes	complemented
50	C₄×C₈×C₃₁	—	2	4	no	—
51	C₄×C₈×C₃₁	—	4	2	no	—
52	C₂⁴×C₃₁	—	2	2	no	—
53	C₂×D₄×C₃₁	—	2	4	no	—
54	C₂×D₄×C₃₁	—	2	4	no	—
55	C₂×D₄×C₃₁	—	2	4	no	—
56	C₂×D₄×C₃₁	—	2	2	no	—
57	C₂×D₄×C₃₁	—	2	4	no	—
58	C₂×D₄×C₃₁	—	4	2	no	—
59	C₂×D₄×C₃₁	—	4	2	no	—
60	C₂×D₄×C₃₁	—	4	2	no	—
61	C₂×D₄×C₃₁	—	4	2	no	—
62	C₂×D₄×C₃₁	C₂×C₈	1	4	yes	—
63	C₂²×C₄×C₃₁	—	2	2	no	—
64	C₂²×C₄×C₃₁	—	4	4	no	—
65	C₂²×C₄×C₃₁	—	4	2	no	—
66	C₂²×C₄×C₃₁	—	4	2	no	—
67	C₂²×C₄×C₃₁	—	4	4	no	—
68	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
71	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
75	C₂×C₈×C₃₁	—	2	4	no	—
76	C₂×C₈×C₃₁	—	2	4	no	—
77	C₂×C₈×C₃₁	—	4	2	no	—
78	C₂×C₈×C₃₁	—	4	2	no	—
79	C₂×C₈×C₃₁	—	4	2	no	—
80	C₂×C₈×C₃₁	—	4	2	no	—
81	C₄⋊C₄×C₃₁	—	2	4	no	—
82	C₄⋊C₄×C₃₁	—	4	2	no	—
83	C₄⋊C₄×C₃₁	—	4	2	no	—
84	C₂²⋊C₄×C₃₁	—	4	4	no	—
85	C₂²⋊C₄×C₃₁	—	4	4	no	—
86	C₂²⋊C₄×C₃₁	—	4	2	no	—
87	C₂²⋊C₄×C₃₁	—	4	2	no	—
88	C₄²×C₃₁	—	2	2	no	—
89	C₄²×C₃₁	—	2	2	no	—
90	C₄²×C₃₁	—	2	2	no	—
91	C₄²×C₃₁	—	4	1	no	—
92	C₄²×C₃₁	C₂×C₈	1	2	yes	—
93	C₄²×C₃₁	M₁₆	1	2	yes	—
94	C₄²×C₃₁	D₈	1	2	yes	—
95	C₄²×C₃₁	SD₁₆	1	2	yes	—
96	C₄²×C₃₁	C₂×D₄	1	2	yes	—
97	C₄²×C₃₁	#41	1	2	yes	—
98	#9674	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
99	C₂³×C₃₁	—	2	4	no	—
100	C₂³×C₃₁	—	4	2	no	—
101	C₂³×C₃₁	—	4	2	no	—
102	C₂³×C₃₁	—	4	2	no	—
103	C₂³×C₃₁	—	4	2	no	—
104	C₂³×C₃₁	—	4	2	no	—
105	C₂³×C₃₁	—	4	2	no	—
106	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	D₄×C₃₁	—	2	8	no	—
108	D₄×C₃₁	—	4	8	no	—
109	D₄×C₃₁	—	4	8	no	—
110	D₄×C₃₁	—	4	4	no	—
111	C₂×C₄×C₃₁	—	2	1	no	—
112	C₂×C₄×C₃₁	—	2	1	no	—
113	C₂×C₄×C₃₁	—	2	1	no	—
114	C₂×C₄×C₃₁	—	2	1	no	—
115	C₂×C₄×C₃₁	—	2	1	no	—
116	C₂×C₄×C₃₁	—	2	1	no	—
117	C₂×C₄×C₃₁	—	2	1	no	—
118	C₂×C₄×C₃₁	—	2	1	no	—
119	C₂×C₄×C₃₁	—	2	1	no	—
120	C₂×C₄×C₃₁	—	4	2	no	—
121	C₂×C₄×C₃₁	—	4	2	no	—
122	C₂×C₄×C₃₁	—	4	2	no	—
123	C₂×C₄×C₃₁	—	8	4	no	—
124	C₂×C₄×C₃₁	—	8	4	no	—
125	C₂×C₄×C₃₁	C₂²⋊C₈	1	2	yes	—
126	C₂×C₄×C₃₁	C₂²⋊C₈	1	2	yes	—
127	C₂×C₄×C₃₁	#102	1	2	yes	—
128	C₂×C₄×C₃₁	#102	1	2	yes	—
129	C₂×C₄×C₃₁	#104	1	2	yes	—
130	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
131	C₂×C₄×C₃₁	#121	1	2	yes	—
132	C₂×C₄×C₃₁	#124	1	1	yes	characteristic
133	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
134	C₂×C₄×C₃₁	C₂²×C₈	1	1	yes	characteristic
135	C₂×C₄×C₃₁	C₂×M₁₆	1	1	yes	characteristic
136	C₂×C₄×C₃₁	#131	1	2	yes	—
137	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
138	C₂×C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
139	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
140	C₈×C₃₁	—	4	4	no	—
141	C₈×C₃₁	—	4	4	no	—
142	C₄×#127	C₂×C₃₁	1	1	yes	characteristic
143	C₄×C₄⋊C₈	C₂×C₃₁	1	1	yes	characteristic, complemented
144	C₄×C₄⋊C₈	C₂×C₃₁	1	1	yes	characteristic, complemented
145	C₂×D₄⋊C₈	C₂×C₃₁	1	4	yes	complemented
146	C₂²×C₃₁	—	4	4	no	—
147	C₂²×C₃₁	—	4	4	no	—
148	C₂²×C₃₁	—	4	4	no	—
149	C₂²×C₃₁	—	8	2	no	—
150	C₂²×C₃₁	—	8	2	no	—
151	C₂²×C₃₁	—	8	2	no	—
152	C₂²×C₃₁	—	8	2	no	—
153	C₂²×C₃₁	#390	1	1	yes	characteristic
154	C₂²×C₃₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
155	C₂²×C₃₁	#408	1	1	yes	characteristic
156	C₂²×C₃₁	C₂×#102	1	1	yes	characteristic
157	C₂²×C₃₁	#418	1	1	yes	characteristic
158	C₂²×C₃₁	C₂×#104	1	1	yes	characteristic
159	C₂²×C₃₁	#421	1	1	yes	characteristic
160	C₄×C₃₁	—	2	1	no	—
161	C₄×C₃₁	—	2	1	no	—
162	C₄×C₃₁	—	2	2	no	—
163	C₄×C₃₁	—	2	2	no	—
164	C₄×C₃₁	—	4	1	no	—
165	C₄×C₃₁	—	8	2	no	—
166	C₄×C₃₁	D₄⋊C₈	1	4	yes	complemented
167	C₄×C₃₁	C₈×D₄	1	2	yes	—
168	C₄×C₃₁	#436	1	2	yes	—
169	C₄×C₃₁	#459	1	2	yes	—
170	C₄×C₃₁	#460	1	2	yes	—
171	C₂×#127	C₄×C₃₁	1	1	yes	characteristic
172	C₂×C₄×D₄	—	2	1	no	—
173	C₂×C₄×D₄	—	2	1	no	—
174	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
175	C₂×C₄⋊C₈	—	2	1	no	—
176	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
177	C₂×C₄×C₈	—	2	1	no	—
178	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
179	#390	—	2	1	no	—
180	#390	—	2	1	no	—
181	#390	C₄×C₃₁	1	2	yes	—
182	#384	C₄×C₃₁	1	1	yes	characteristic
183	C₄³	C₂²×C₃₁	1	1	yes	characteristic
184	#336	—	2	1	no	—
185	#336	—	2	1	no	—
186	D₄⋊C₈	—	2	8	no	—
187	C₂×C₃₁	—	8	4	no	—
188	C₂×C₃₁	C₂×D₄⋊C₈	1	1	yes	characteristic
189	C₂×C₃₁	#1478	1	1	yes	characteristic
190	C₂×C₃₁	#1480	1	1	yes	characteristic
191	C₂×C₃₁	#1484	1	1	yes	characteristic
192	C₂×C₃₁	#1954	1	1	yes	characteristic
193	C₂×C₃₁	#1980	1	1	yes	characteristic
194	C₂×C₃₁	#1982	1	1	yes	characteristic
195	C₂²×D₄	—	2	1	no	—
196	C₂²×D₄	—	2	1	no	—
197	C₂²×D₄	C₂×C₄×C₃₁	1	2	yes	—
198	C₂³×C₄	—	2	2	no	—
199	C₂²×C₈	—	2	2	no	—
200	C₂²×C₈	—	2	2	no	—
201	#127	—	2	2	no	—
202	#127	C₈×C₃₁	1	4	yes	complemented
203	C₄×D₄	—	2	4	no	—
204	C₄×D₄	—	2	4	no	—
205	C₄×D₄	—	4	4	no	—
206	C₄×D₄	—	4	2	no	—
207	C₄×D₄	—	4	2	no	—
208	C₂×C₄⋊C₄	—	2	1	no	—
209	C₂×C₄⋊C₄	—	2	1	no	—
210	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
211	C₂×C₂²⋊C₄	—	2	2	no	—
212	C₂×C₂²⋊C₄	—	2	2	no	—
213	C₂×C₂²⋊C₄	—	2	2	no	—
214	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
215	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
216	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
217	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
218	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
219	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
220	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
221	C₄⋊C₈	—	2	4	no	—
222	C₄⋊C₈	—	2	2	no	—
223	C₄⋊C₈	D₄×C₃₁	1	4	yes	complemented
224	C₄×C₈	—	2	4	no	—
225	C₄×C₈	—	4	2	no	—
226	C₃₁	#9674	1	1	yes	characteristic, direct product factor, Sylow
227	C₂⁴	—	2	2	no	—
228	C₂×D₄	—	2	4	no	—
229	C₂×D₄	—	2	4	no	—
230	C₂×D₄	—	2	4	no	—
231	C₂×D₄	—	2	2	no	—
232	C₂×D₄	—	2	4	no	—
233	C₂×D₄	—	4	2	no	—
234	C₂×D₄	—	4	2	no	—
235	C₂×D₄	—	4	2	no	—
236	C₂×D₄	—	4	2	no	—
237	C₂×D₄	C₂×C₈×C₃₁	1	4	yes	—
238	C₂²×C₄	—	2	2	no	—
239	C₂²×C₄	—	4	4	no	—
240	C₂²×C₄	—	4	2	no	—
241	C₂²×C₄	—	4	2	no	—
242	C₂²×C₄	—	4	4	no	—
243	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
244	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
245	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
246	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
247	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
248	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
249	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
250	C₂×C₈	—	2	4	no	—
251	C₂×C₈	—	2	4	no	—
252	C₂×C₈	—	4	2	no	—
253	C₂×C₈	—	4	2	no	—
254	C₂×C₈	—	4	2	no	—
255	C₂×C₈	—	4	2	no	—
256	C₄⋊C₄	—	2	4	no	—
257	C₄⋊C₄	—	4	2	no	—
258	C₄⋊C₄	—	4	2	no	—
259	C₂²⋊C₄	—	4	4	no	—
260	C₂²⋊C₄	—	4	4	no	—
261	C₂²⋊C₄	—	4	2	no	—
262	C₂²⋊C₄	—	4	2	no	—
263	C₄²	—	2	2	no	—
264	C₄²	—	2	2	no	—
265	C₄²	—	2	2	no	—
266	C₄²	—	4	1	no	—
267	C₄²	C₂×C₈×C₃₁	1	2	yes	—
268	C₄²	M₁₆×C₃₁	1	2	yes	—
269	C₄²	D₈×C₃₁	1	2	yes	—
270	C₄²	SD₁₆×C₃₁	1	2	yes	—
271	C₄²	C₂×D₄×C₃₁	1	2	yes	—
272	C₄²	#41×C₃₁	1	2	yes	—
273	C₂³	—	2	4	no	—
274	C₂³	—	4	2	no	—
275	C₂³	—	4	2	no	—
276	C₂³	—	4	2	no	—
277	C₂³	—	4	2	no	—
278	C₂³	—	4	2	no	—
279	C₂³	—	4	2	no	—
280	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
281	D₄	—	2	8	no	—
282	D₄	—	4	8	no	—
283	D₄	—	4	4	no	—
284	D₄	—	4	8	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	8	4	no	—
298	C₂×C₄	—	8	4	no	—
299	C₂×C₄	C₃₁×C₂²⋊C₈	1	2	yes	—
300	C₂×C₄	C₃₁×C₂²⋊C₈	1	2	yes	—
301	C₂×C₄	C₃₁×#102	1	2	yes	—
302	C₂×C₄	C₃₁×#102	1	2	yes	—
303	C₂×C₄	C₃₁×#104	1	2	yes	—
304	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
305	C₂×C₄	C₃₁×#121	1	2	yes	—
306	C₂×C₄	C₃₁×#124	1	1	yes	characteristic
307	C₂×C₄	C₃₁×#127	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	C₃₁×#131	1	2	yes	—
311	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
312	C₂×C₄	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
313	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
314	C₈	—	4	4	no	—
315	C₈	—	4	4	no	—
316	C₂²	—	4	4	no	—
317	C₂²	—	4	4	no	—
318	C₂²	—	4	4	no	—
319	C₂²	—	8	2	no	—
320	C₂²	—	8	2	no	—
321	C₂²	—	8	2	no	—
322	C₂²	—	8	2	no	—
323	C₂²	C₃₁×#390	1	1	yes	characteristic
324	C₂²	C₂×C₃₁×C₂²⋊C₈	1	1	yes	characteristic
325	C₂²	C₃₁×#408	1	1	yes	characteristic
326	C₂²	C₂×C₃₁×#102	1	1	yes	characteristic
327	C₂²	C₃₁×#418	1	1	yes	characteristic
328	C₂²	C₂×C₃₁×#104	1	1	yes	characteristic
329	C₂²	C₃₁×#421	1	1	yes	characteristic
330	C₄	—	2	1	no	—
331	C₄	—	2	1	no	—
332	C₄	—	2	2	no	—
333	C₄	—	2	2	no	—
334	C₄	—	4	1	no	—
335	C₄	—	8	2	no	—
336	C₄	C₃₁×D₄⋊C₈	1	4	yes	complemented
337	C₄	C₈×D₄×C₃₁	1	2	yes	—
338	C₄	C₃₁×#436	1	2	yes	—
339	C₄	C₃₁×#459	1	2	yes	—
340	C₄	C₃₁×#460	1	2	yes	—
341	C₂	—	8	4	no	—
342	C₂	C₂×C₃₁×D₄⋊C₈	1	1	yes	characteristic
343	C₂	C₃₁×#1478	1	1	yes	characteristic
344	C₂	C₃₁×#1480	1	1	yes	characteristic
345	C₂	C₃₁×#1484	1	1	yes	characteristic
346	C₂	C₃₁×#1954	1	1	yes	characteristic
347	C₂	C₃₁×#1980	1	1	yes	characteristic
348	C₂	C₃₁×#1982	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₃₁	C₂×D₄
commutator	C₂×C₄	C₂²×C₈×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
socle	C₂³×C₃₁	C₂×C₂²⋊C₄
prime radical	C₂×C₃₁×#127	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₃₁ ◁ C₄³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄×C₃₁×C₄⋊C₈	C₂	semidirect
3. C₄×C₃₁×C₄⋊C₈	C₂	semidirect
4. C₂×C₃₁×D₄⋊C₈	C₂	semidirect
28. C₃₁×#127	C₈	semidirect
49. C₃₁×C₄⋊C₈	D₄	semidirect
98. #9674	C₃₁	direct (N × H)
143. C₄×C₄⋊C₈	C₂×C₃₁	semidirect
144. C₄×C₄⋊C₈	C₂×C₃₁	semidirect
145. C₂×D₄⋊C₈	C₂×C₃₁	semidirect
166. C₄×C₃₁	D₄⋊C₈	semidirect
202. #127	C₈×C₃₁	semidirect
223. C₄⋊C₈	D₄×C₃₁	semidirect
226. C₃₁	#9674	direct (N × H)
336. C₄	C₃₁×D₄⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₃₁×#127	2
C₄×C₃₁×C₄⋊C₈	2
C₂×C₃₁×D₄⋊C₈	2
#9674	31

Subgroups — C31×#9674 #21466433

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#9674 #21466433