Subgroups — C₃₁×#7848 #21464607

Order 7936.

798 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#431	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄×C₈×C₃₁	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1273	C₂	1	4	yes	maximal, complemented
5	C₂²×M₁₆×C₃₁	C₄	1	2	yes	—
6	C₂³×C₈×C₃₁	C₄	1	2	yes	—
7	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
8	C₃₁×#431	—	2	2	no	—
9	C₃₁×#431	—	2	2	no	—
10	C₂×C₃₁×C₄⋊C₈	C₄	1	2	yes	—
11	C₂×C₃₁×C₂²⋊C₈	C₄	1	2	yes	—
12	C₄×M₁₆×C₃₁	—	2	2	no	—
13	C₄×M₁₆×C₃₁	—	2	2	no	—
14	C₂×C₃₁×#97	C₄	1	2	yes	—
15	C₂×C₄×C₈×C₃₁	—	2	2	no	—
16	C₂×C₄×C₈×C₃₁	—	2	2	no	—
17	C₂×C₄×C₈×C₃₁	C₄	1	2	yes	—
18	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
21	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
23	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
24	C₂×M₁₆×C₃₁	—	2	4	no	—
25	C₂×M₁₆×C₃₁	—	2	4	no	—
26	C₂×M₁₆×C₃₁	C₈	1	8	yes	complemented
27	C₂²×C₈×C₃₁	—	2	4	no	—
28	C₂²×C₈×C₃₁	—	2	8	no	—
29	C₂²×C₈×C₃₁	C₂×C₄	1	4	yes	—
30	C₂²×C₈×C₃₁	C₂×C₄	1	4	yes	—
31	C₂²×C₈×C₃₁	C₂×C₄	1	4	yes	—
32	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₃₁	D₄	1	2	yes	—
36	C₂×C₄²×C₃₁	D₄	1	2	yes	—
37	C₂×C₄²×C₃₁	D₄	1	2	yes	—
38	C₂×C₄²×C₃₁	Q₈	1	2	yes	—
39	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
40	C₃₁×C₄⋊C₈	—	2	4	no	—
41	C₃₁×C₂²⋊C₈	—	2	4	no	—
42	C₃₁×#97	—	2	4	no	—
43	C₄×C₈×C₃₁	—	2	4	no	—
44	C₄×C₈×C₃₁	—	2	2	no	—
45	C₄×C₈×C₃₁	—	2	2	no	—
46	C₄×C₈×C₃₁	—	2	2	no	—
47	C₄×C₈×C₃₁	—	2	2	no	—
48	C₄×C₈×C₃₁	—	2	2	no	—
49	C₄×C₈×C₃₁	—	2	2	no	—
50	C₂⁴×C₃₁	C₄²	1	1	yes	characteristic
51	C₂²×C₄×C₃₁	—	2	4	no	—
52	C₂²×C₄×C₃₁	—	2	4	no	—
53	C₂²×C₄×C₃₁	C₄²	1	1	yes	characteristic
54	C₂²×C₄×C₃₁	C₄²	1	1	yes	characteristic
55	C₂²×C₄×C₃₁	C₄²	1	1	yes	characteristic
56	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
57	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
58	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
59	C₂²×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
60	C₂²×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
61	C₂²×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
62	C₂²×C₄×C₃₁	C₂×C₈	1	4	yes	—
63	C₂²×C₄×C₃₁	M₁₆	1	4	yes	—
64	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
67	M₁₆×C₃₁	—	2	16	no	—
68	C₂×C₈×C₃₁	—	2	16	no	—
69	C₂×C₈×C₃₁	—	2	4	no	—
70	C₂×C₈×C₃₁	—	2	4	no	—
71	C₂×C₈×C₃₁	—	2	4	no	—
72	C₂×C₈×C₃₁	—	2	4	no	—
73	C₂×C₈×C₃₁	—	2	4	no	—
74	C₂×C₈×C₃₁	—	2	8	no	—
75	C₂×C₈×C₃₁	C₂×C₈	1	8	yes	complemented
76	C₂×C₈×C₃₁	M₁₆	1	8	yes	complemented
77	C₄²×C₃₁	C₂²⋊C₄	1	2	yes	—
78	C₄²×C₃₁	C₂²⋊C₄	1	2	yes	—
79	C₄²×C₃₁	C₂²⋊C₄	1	2	yes	—
80	C₄²×C₃₁	C₄⋊C₄	1	2	yes	—
81	C₄²×C₃₁	C₄⋊C₄	1	2	yes	—
82	C₄²×C₃₁	C₄⋊C₄	1	2	yes	—
83	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
85	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
86	C₄²×C₃₁	C₂×Q₈	1	1	yes	characteristic
87	#7848	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
88	C₂³×C₃₁	—	2	2	no	—
89	C₂³×C₃₁	—	2	2	no	—
90	C₂³×C₃₁	#95	1	2	yes	—
91	C₂³×C₃₁	C₄×C₈	1	2	yes	—
92	C₂³×C₃₁	#97	1	2	yes	—
93	C₂³×C₃₁	C₂×C₄²	1	1	yes	characteristic
94	C₂×C₄×C₃₁	—	2	2	no	—
95	C₂×C₄×C₃₁	—	2	2	no	—
96	C₂×C₄×C₃₁	—	2	4	no	—
97	C₂×C₄×C₃₁	—	2	4	no	—
98	C₂×C₄×C₃₁	—	2	4	no	—
99	C₂×C₄×C₃₁	—	2	4	no	—
100	C₂×C₄×C₃₁	#95	1	2	yes	—
101	C₂×C₄×C₃₁	#95	1	2	yes	—
102	C₂×C₄×C₃₁	#95	1	2	yes	—
103	C₂×C₄×C₃₁	C₄×C₈	1	2	yes	—
104	C₂×C₄×C₃₁	#97	1	2	yes	—
105	C₂×C₄×C₃₁	C₂²⋊C₈	1	8	yes	—
106	C₂×C₄×C₃₁	C₄⋊C₈	1	8	yes	—
107	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₃₁	C₂²×C₈	1	2	yes	—
114	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
115	C₂×C₄×C₃₁	C₂×M₁₆	1	4	yes	—
116	C₂×C₄×C₃₁	#131	1	4	yes	—
117	C₈×C₃₁	—	2	8	no	—
118	C₈×C₃₁	—	2	8	no	—
119	C₈×C₃₁	—	4	4	no	—
120	C₂×#431	C₂×C₃₁	1	1	yes	characteristic
121	C₂×C₄×M₁₆	C₂×C₃₁	1	1	yes	characteristic
122	C₂²×C₄×C₈	C₂×C₃₁	1	1	yes	characteristic
123	#1273	C₂×C₃₁	1	4	yes	complemented
124	C₂²×C₃₁	—	2	4	no	—
125	C₂²×C₃₁	—	2	4	no	—
126	C₂²×C₃₁	—	2	2	no	—
127	C₂²×C₃₁	—	2	2	no	—
128	C₂²×C₃₁	#336	1	4	yes	—
129	C₂²×C₃₁	C₂×#95	1	1	yes	characteristic
130	C₂²×C₃₁	C₂×C₄×C₈	1	1	yes	characteristic
131	C₂²×C₃₁	C₂×#97	1	1	yes	characteristic
132	C₂²×C₃₁	C₄×M₁₆	1	2	yes	—
133	C₂²×C₃₁	#405	1	2	yes	—
134	C₄×C₃₁	—	2	2	no	—
135	C₄×C₃₁	—	2	2	no	—
136	C₄×C₃₁	#336	1	4	yes	—
137	C₄×C₃₁	C₂×C₂²⋊C₈	1	2	yes	—
138	C₄×C₃₁	#407	1	2	yes	—
139	C₄×C₃₁	#408	1	2	yes	—
140	C₄×C₃₁	C₂×C₄⋊C₈	1	2	yes	—
141	C₄×C₃₁	#423	1	2	yes	—
142	C₄×C₃₁	#424	1	2	yes	—
143	C₂²×M₁₆	C₄×C₃₁	1	2	yes	—
144	C₂³×C₈	C₄×C₃₁	1	2	yes	—
145	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
146	#431	—	2	2	no	—
147	#431	—	2	2	no	—
148	C₂×C₄⋊C₈	C₄×C₃₁	1	2	yes	—
149	C₂×C₂²⋊C₈	C₄×C₃₁	1	2	yes	—
150	C₄×M₁₆	—	2	2	no	—
151	C₄×M₁₆	—	2	2	no	—
152	C₂×#97	C₄×C₃₁	1	2	yes	—
153	C₂×C₄×C₈	—	2	2	no	—
154	C₂×C₄×C₈	—	2	2	no	—
155	C₂×C₄×C₈	C₄×C₃₁	1	2	yes	—
156	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
157	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
158	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
159	C₂×C₃₁	—	2	4	no	—
160	C₂×C₃₁	C₈×M₁₆	1	2	yes	—
161	C₂×C₃₁	#1451	1	2	yes	—
162	C₂×C₃₁	C₂×#336	1	1	yes	characteristic
163	C₂×C₃₁	#1728	1	1	yes	characteristic
164	C₂×C₃₁	#1729	1	1	yes	characteristic
165	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
166	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
167	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
168	C₂×M₁₆	—	2	4	no	—
169	C₂×M₁₆	—	2	4	no	—
170	C₂×M₁₆	C₈×C₃₁	1	8	yes	complemented
171	C₂²×C₈	—	2	4	no	—
172	C₂²×C₈	—	2	8	no	—
173	C₂²×C₈	C₂×C₄×C₃₁	1	4	yes	—
174	C₂²×C₈	C₂×C₄×C₃₁	1	4	yes	—
175	C₂²×C₈	C₂×C₄×C₃₁	1	4	yes	—
176	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
177	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
178	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
179	C₂×C₄²	D₄×C₃₁	1	2	yes	—
180	C₂×C₄²	D₄×C₃₁	1	2	yes	—
181	C₂×C₄²	D₄×C₃₁	1	2	yes	—
182	C₂×C₄²	Q₈×C₃₁	1	2	yes	—
183	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
184	C₄⋊C₈	—	2	4	no	—
185	C₂²⋊C₈	—	2	4	no	—
186	#97	—	2	4	no	—
187	C₄×C₈	—	2	4	no	—
188	C₄×C₈	—	2	2	no	—
189	C₄×C₈	—	2	2	no	—
190	C₄×C₈	—	2	2	no	—
191	C₄×C₈	—	2	2	no	—
192	C₄×C₈	—	2	2	no	—
193	C₄×C₈	—	2	2	no	—
194	C₃₁	#7848	1	1	yes	characteristic, direct product factor, Sylow
195	C₂⁴	C₄²×C₃₁	1	1	yes	characteristic
196	C₂²×C₄	—	2	4	no	—
197	C₂²×C₄	—	2	4	no	—
198	C₂²×C₄	C₄²×C₃₁	1	1	yes	characteristic
199	C₂²×C₄	C₄²×C₃₁	1	1	yes	characteristic
200	C₂²×C₄	C₄²×C₃₁	1	1	yes	characteristic
201	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
202	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
203	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
204	C₂²×C₄	C₄⋊C₄×C₃₁	1	2	yes	—
205	C₂²×C₄	C₄⋊C₄×C₃₁	1	2	yes	—
206	C₂²×C₄	C₄⋊C₄×C₃₁	1	2	yes	—
207	C₂²×C₄	C₂×C₈×C₃₁	1	4	yes	—
208	C₂²×C₄	M₁₆×C₃₁	1	4	yes	—
209	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
210	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
211	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
212	M₁₆	—	2	16	no	—
213	C₂×C₈	—	2	16	no	—
214	C₂×C₈	—	2	4	no	—
215	C₂×C₈	—	2	4	no	—
216	C₂×C₈	—	2	4	no	—
217	C₂×C₈	—	2	4	no	—
218	C₂×C₈	—	2	4	no	—
219	C₂×C₈	—	2	8	no	—
220	C₂×C₈	C₂×C₈×C₃₁	1	8	yes	complemented
221	C₂×C₈	M₁₆×C₃₁	1	8	yes	complemented
222	C₄²	C₂²⋊C₄×C₃₁	1	2	yes	—
223	C₄²	C₂²⋊C₄×C₃₁	1	2	yes	—
224	C₄²	C₂²⋊C₄×C₃₁	1	2	yes	—
225	C₄²	C₄⋊C₄×C₃₁	1	2	yes	—
226	C₄²	C₄⋊C₄×C₃₁	1	2	yes	—
227	C₄²	C₄⋊C₄×C₃₁	1	2	yes	—
228	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
229	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
230	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
231	C₄²	C₂×Q₈×C₃₁	1	1	yes	characteristic
232	C₂³	—	2	2	no	—
233	C₂³	—	2	2	no	—
234	C₂³	C₃₁×#95	1	2	yes	—
235	C₂³	C₄×C₈×C₃₁	1	2	yes	—
236	C₂³	C₃₁×#97	1	2	yes	—
237	C₂³	C₂×C₄²×C₃₁	1	1	yes	characteristic
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	2	4	no	—
244	C₂×C₄	C₃₁×#95	1	2	yes	—
245	C₂×C₄	C₃₁×#95	1	2	yes	—
246	C₂×C₄	C₃₁×#95	1	2	yes	—
247	C₂×C₄	C₄×C₈×C₃₁	1	2	yes	—
248	C₂×C₄	C₃₁×#97	1	2	yes	—
249	C₂×C₄	C₃₁×C₂²⋊C₈	1	8	yes	—
250	C₂×C₄	C₃₁×C₄⋊C₈	1	8	yes	—
251	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
252	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
253	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
254	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
255	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
256	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
257	C₂×C₄	C₂²×C₈×C₃₁	1	2	yes	—
258	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
259	C₂×C₄	C₂×M₁₆×C₃₁	1	4	yes	—
260	C₂×C₄	C₃₁×#131	1	4	yes	—
261	C₈	—	2	8	no	—
262	C₈	—	2	8	no	—
263	C₈	—	4	4	no	—
264	C₂²	—	2	4	no	—
265	C₂²	—	2	4	no	—
266	C₂²	—	2	2	no	—
267	C₂²	—	2	2	no	—
268	C₂²	C₃₁×#336	1	4	yes	—
269	C₂²	C₂×C₃₁×#95	1	1	yes	characteristic
270	C₂²	C₂×C₄×C₈×C₃₁	1	1	yes	characteristic
271	C₂²	C₂×C₃₁×#97	1	1	yes	characteristic
272	C₂²	C₄×M₁₆×C₃₁	1	2	yes	—
273	C₂²	C₃₁×#405	1	2	yes	—
274	C₄	—	2	2	no	—
275	C₄	—	2	2	no	—
276	C₄	C₃₁×#336	1	4	yes	—
277	C₄	C₂×C₃₁×C₂²⋊C₈	1	2	yes	—
278	C₄	C₃₁×#407	1	2	yes	—
279	C₄	C₃₁×#408	1	2	yes	—
280	C₄	C₂×C₃₁×C₄⋊C₈	1	2	yes	—
281	C₄	C₃₁×#423	1	2	yes	—
282	C₄	C₃₁×#424	1	2	yes	—
283	C₂	—	2	4	no	—
284	C₂	C₈×M₁₆×C₃₁	1	2	yes	—
285	C₂	C₃₁×#1451	1	2	yes	—
286	C₂	C₂×C₃₁×#336	1	1	yes	characteristic
287	C₂	C₃₁×#1728	1	1	yes	characteristic
288	C₂	C₃₁×#1729	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄²×C₃₁	C₂³
commutator	C₂²	C₂×C₄×C₈×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
socle	C₂³×C₃₁	C₂×C₄²
prime radical	C₂⁴×C₃₁	C₄²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₃₁×#1273	C₂	semidirect
26. C₂×M₁₆×C₃₁	C₈	semidirect
75. C₂×C₈×C₃₁	C₂×C₈	semidirect
76. C₂×C₈×C₃₁	M₁₆	semidirect
87. #7848	C₃₁	direct (N × H)
123. #1273	C₂×C₃₁	semidirect
170. C₂×M₁₆	C₈×C₃₁	semidirect
194. C₃₁	#7848	direct (N × H)
220. C₂×C₈	C₂×C₈×C₃₁	semidirect
221. C₂×C₈	M₁₆×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#431	2
C₂×C₄×M₁₆×C₃₁	2
C₂²×C₄×C₈×C₃₁	2
C₃₁×#1273	2
#7848	31

Subgroups — C31×#7848 #21464607

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#7848 #21464607