Subgroups — C₃₁×#7873 #21464632

Order 7936.

722 conjugacy classes of subgroups.

The subgroup lattice has 292 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	C₂²×C₃₁×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1277	C₂	1	4	yes	maximal, complemented
5	C₂²×M₁₆×C₃₁	C₄	1	2	yes	—
6	C₂³×C₈×C₃₁	—	2	1	no	—
7	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
8	C₃₁×#431	—	2	2	no	—
9	C₃₁×#431	C₄	1	4	yes	—
10	C₂×C₃₁×C₄⋊C₈	—	2	1	no	—
11	C₂×C₃₁×C₄⋊C₈	—	2	2	no	—
12	C₂×C₃₁×C₄⋊C₈	C₄	1	4	yes	—
13	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
14	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
15	C₂×C₃₁×C₂²⋊C₈	—	2	1	no	—
16	C₄×M₁₆×C₃₁	—	2	2	no	—
17	C₄×M₁₆×C₃₁	—	2	2	no	—
18	C₂×C₃₁×#97	C₄	1	2	yes	—
19	C₂×C₄×C₈×C₃₁	—	2	1	no	—
20	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
21	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
23	C₂³×C₄×C₃₁	Q₈	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	2	no	—
29	C₂²×C₈×C₃₁	—	2	2	no	—
30	C₂²×C₈×C₃₁	—	4	4	no	—
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	2	yes	—
34	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
35	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₃₁	D₄	1	2	yes	—
37	C₂×C₄²×C₃₁	D₄	1	2	yes	—
38	C₂×C₄²×C₃₁	Q₈	1	1	yes	characteristic
39	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
40	C₃₁×C₄⋊C₈	—	2	4	no	—
41	C₃₁×C₄⋊C₈	—	2	2	no	—
42	C₃₁×C₄⋊C₈	—	2	2	no	—
43	C₃₁×C₄⋊C₈	C₂×C₄	1	4	yes	—
44	C₃₁×C₄⋊C₈	C₂×C₄	1	4	yes	—
45	C₃₁×C₂²⋊C₈	—	2	4	no	—
46	C₃₁×#97	—	2	4	no	—
47	C₄×C₈×C₃₁	—	2	4	no	—
48	C₄×C₈×C₃₁	—	2	2	no	—
49	C₄×C₈×C₃₁	—	2	2	no	—
50	C₂⁴×C₃₁	C₄⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₃₁	—	2	2	no	—
52	C₂²×C₄×C₃₁	—	2	2	no	—
53	C₂²×C₄×C₃₁	—	4	2	no	—
54	C₂²×C₄×C₃₁	C₄²	1	2	yes	—
55	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
56	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
57	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
58	C₂²×C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
61	C₂²×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
62	C₂²×C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₃₁	C₂×C₈	1	4	yes	—
64	C₂²×C₄×C₃₁	M₁₆	1	4	yes	—
65	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
68	M₁₆×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	—	4	8	no	—
74	C₂×C₈×C₃₁	—	4	2	no	—
75	C₂×C₈×C₃₁	—	4	4	no	—
76	C₂×C₈×C₃₁	—	4	2	no	—
77	C₂×C₈×C₃₁	C₂×C₈	1	8	yes	complemented
78	C₄²×C₃₁	C₄²	1	2	yes	—
79	C₄²×C₃₁	C₂²⋊C₄	1	2	yes	—
80	C₄²×C₃₁	C₂²⋊C₄	1	2	yes	—
81	C₄²×C₃₁	C₂²⋊C₄	1	2	yes	—
82	C₄²×C₃₁	C₄⋊C₄	1	2	yes	—
83	C₄²×C₃₁	C₄⋊C₄	1	2	yes	—
84	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
85	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
86	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
87	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
88	#7873	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
89	C₂³×C₃₁	—	2	2	no	—
90	C₂³×C₃₁	—	2	2	no	—
91	C₂³×C₃₁	#95	1	2	yes	—
92	C₂³×C₃₁	C₄⋊C₈	1	4	yes	—
93	C₂³×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₃₁	—	2	2	no	—
95	C₂×C₄×C₃₁	—	2	2	no	—
96	C₂×C₄×C₃₁	—	2	2	no	—
97	C₂×C₄×C₃₁	—	2	4	no	—
98	C₂×C₄×C₃₁	—	2	2	no	—
99	C₂×C₄×C₃₁	—	2	2	no	—
100	C₂×C₄×C₃₁	—	4	2	no	—
101	C₂×C₄×C₃₁	—	4	2	no	—
102	C₂×C₄×C₃₁	#95	1	2	yes	—
103	C₂×C₄×C₃₁	#95	1	2	yes	—
104	C₂×C₄×C₃₁	#95	1	2	yes	—
105	C₂×C₄×C₃₁	C₄×C₈	1	4	yes	—
106	C₂×C₄×C₃₁	#97	1	4	yes	—
107	C₂×C₄×C₃₁	C₂²⋊C₈	1	8	yes	—
108	C₂×C₄×C₃₁	C₄⋊C₈	1	4	yes	—
109	C₂×C₄×C₃₁	C₂×C₄²	1	1	yes	characteristic
110	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₃₁	C₂²×C₈	1	2	yes	—
116	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
117	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
118	C₂×C₄×C₃₁	#137	1	2	yes	—
119	C₈×C₃₁	—	2	8	no	—
120	C₈×C₃₁	—	4	4	no	—
121	C₈×C₃₁	—	4	4	no	—
122	C₂×#431	C₂×C₃₁	1	1	yes	characteristic
123	C₂²×C₄⋊C₈	C₂×C₃₁	1	1	yes	characteristic
124	C₂×C₄×M₁₆	C₂×C₃₁	1	1	yes	characteristic
125	#1277	C₂×C₃₁	1	4	yes	complemented
126	C₂²×C₃₁	—	2	8	no	—
127	C₂²×C₃₁	—	2	2	no	—
128	C₂²×C₃₁	—	2	2	no	—
129	C₂²×C₃₁	#336	1	4	yes	—
130	C₂²×C₃₁	C₂×#95	1	1	yes	characteristic
131	C₂²×C₃₁	C₂×C₄⋊C₈	1	2	yes	—
132	C₂²×C₃₁	#428	1	2	yes	—
133	C₂²×C₃₁	#430	1	2	yes	—
134	C₄×C₃₁	—	2	2	no	—
135	C₄×C₃₁	—	2	2	no	—
136	C₄×C₃₁	—	2	1	no	—
137	C₄×C₃₁	—	2	1	no	—
138	C₄×C₃₁	#336	1	4	yes	—
139	C₄×C₃₁	C₂×C₄×C₈	1	1	yes	characteristic
140	C₄×C₃₁	C₂×#97	1	1	yes	characteristic
141	C₄×C₃₁	C₂×C₂²⋊C₈	1	2	yes	—
142	C₄×C₃₁	#417	1	2	yes	—
143	C₄×C₃₁	#418	1	1	yes	characteristic
144	C₄×C₃₁	#419	1	1	yes	characteristic
145	C₂²×M₁₆	C₄×C₃₁	1	2	yes	—
146	C₂³×C₈	—	2	1	no	—
147	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
148	#431	—	2	2	no	—
149	#431	C₄×C₃₁	1	4	yes	—
150	C₂×C₄⋊C₈	—	2	1	no	—
151	C₂×C₄⋊C₈	—	2	2	no	—
152	C₂×C₄⋊C₈	C₄×C₃₁	1	4	yes	—
153	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
154	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
155	C₂×C₂²⋊C₈	—	2	1	no	—
156	C₄×M₁₆	—	2	2	no	—
157	C₄×M₁₆	—	2	2	no	—
158	C₂×#97	C₄×C₃₁	1	2	yes	—
159	C₂×C₄×C₈	—	2	1	no	—
160	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
161	C₂×C₃₁	—	2	4	no	—
162	C₂×C₃₁	#1565	1	4	yes	—
163	C₂×C₃₁	C₂×#336	1	1	yes	characteristic
164	C₂×C₃₁	#1736	1	1	yes	characteristic
165	C₂×C₃₁	#1738	1	1	yes	characteristic
166	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
167	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
168	C₂³×C₄	Q₈×C₃₁	1	1	yes	characteristic
169	C₂×M₁₆	—	2	4	no	—
170	C₂×M₁₆	—	2	4	no	—
171	C₂×M₁₆	C₈×C₃₁	1	8	yes	complemented
172	C₂²×C₈	—	2	4	no	—
173	C₂²×C₈	—	2	2	no	—
174	C₂²×C₈	—	2	2	no	—
175	C₂²×C₈	—	4	4	no	—
176	C₂²×C₈	C₂×C₄×C₃₁	1	4	yes	—
177	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
178	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
179	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
180	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
181	C₂×C₄²	D₄×C₃₁	1	2	yes	—
182	C₂×C₄²	D₄×C₃₁	1	2	yes	—
183	C₂×C₄²	Q₈×C₃₁	1	1	yes	characteristic
184	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
185	C₄⋊C₈	—	2	4	no	—
186	C₄⋊C₈	—	2	2	no	—
187	C₄⋊C₈	—	2	2	no	—
188	C₄⋊C₈	C₂×C₄×C₃₁	1	4	yes	—
189	C₄⋊C₈	C₂×C₄×C₃₁	1	4	yes	—
190	C₂²⋊C₈	—	2	4	no	—
191	#97	—	2	4	no	—
192	C₄×C₈	—	2	4	no	—
193	C₄×C₈	—	2	2	no	—
194	C₄×C₈	—	2	2	no	—
195	C₃₁	#7873	1	1	yes	characteristic, direct product factor, Sylow
196	C₂⁴	C₄⋊C₄×C₃₁	1	1	yes	characteristic
197	C₂²×C₄	—	2	2	no	—
198	C₂²×C₄	—	2	2	no	—
199	C₂²×C₄	—	4	2	no	—
200	C₂²×C₄	C₄²×C₃₁	1	2	yes	—
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	1	yes	characteristic
205	C₂²×C₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
206	C₂²×C₄	C₄⋊C₄×C₃₁	1	2	yes	—
207	C₂²×C₄	C₄⋊C₄×C₃₁	1	2	yes	—
208	C₂²×C₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
209	C₂²×C₄	C₂×C₈×C₃₁	1	4	yes	—
210	C₂²×C₄	M₁₆×C₃₁	1	4	yes	—
211	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
212	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
213	C₂²×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
214	M₁₆	—	2	16	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	—	4	8	no	—
220	C₂×C₈	—	4	2	no	—
221	C₂×C₈	—	4	4	no	—
222	C₂×C₈	—	4	2	no	—
223	C₂×C₈	C₂×C₈×C₃₁	1	8	yes	complemented
224	C₄²	C₄²×C₃₁	1	2	yes	—
225	C₄²	C₂²⋊C₄×C₃₁	1	2	yes	—
226	C₄²	C₂²⋊C₄×C₃₁	1	2	yes	—
227	C₄²	C₂²⋊C₄×C₃₁	1	2	yes	—
228	C₄²	C₄⋊C₄×C₃₁	1	2	yes	—
229	C₄²	C₄⋊C₄×C₃₁	1	2	yes	—
230	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
231	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
232	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
233	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
234	C₂³	—	2	2	no	—
235	C₂³	—	2	2	no	—
236	C₂³	C₃₁×#95	1	2	yes	—
237	C₂³	C₃₁×C₄⋊C₈	1	4	yes	—
238	C₂³	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	C₃₁×#95	1	2	yes	—
248	C₂×C₄	C₃₁×#95	1	2	yes	—
249	C₂×C₄	C₃₁×#95	1	2	yes	—
250	C₂×C₄	C₄×C₈×C₃₁	1	4	yes	—
251	C₂×C₄	C₃₁×#97	1	4	yes	—
252	C₂×C₄	C₃₁×C₂²⋊C₈	1	8	yes	—
253	C₂×C₄	C₃₁×C₄⋊C₈	1	4	yes	—
254	C₂×C₄	C₂×C₄²×C₃₁	1	1	yes	characteristic
255	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
256	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
257	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
258	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
259	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
260	C₂×C₄	C₂²×C₈×C₃₁	1	2	yes	—
261	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
262	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
263	C₂×C₄	C₃₁×#137	1	2	yes	—
264	C₈	—	2	8	no	—
265	C₈	—	4	4	no	—
266	C₈	—	4	4	no	—
267	C₂²	—	2	8	no	—
268	C₂²	—	2	2	no	—
269	C₂²	—	2	2	no	—
270	C₂²	C₃₁×#336	1	4	yes	—
271	C₂²	C₂×C₃₁×#95	1	1	yes	characteristic
272	C₂²	C₂×C₃₁×C₄⋊C₈	1	2	yes	—
273	C₂²	C₃₁×#428	1	2	yes	—
274	C₂²	C₃₁×#430	1	2	yes	—
275	C₄	—	2	2	no	—
276	C₄	—	2	2	no	—
277	C₄	—	2	1	no	—
278	C₄	—	2	1	no	—
279	C₄	C₃₁×#336	1	4	yes	—
280	C₄	C₂×C₄×C₈×C₃₁	1	1	yes	characteristic
281	C₄	C₂×C₃₁×#97	1	1	yes	characteristic
282	C₄	C₂×C₃₁×C₂²⋊C₈	1	2	yes	—
283	C₄	C₃₁×#417	1	2	yes	—
284	C₄	C₃₁×#418	1	1	yes	characteristic
285	C₄	C₃₁×#419	1	1	yes	characteristic
286	C₂	—	2	4	no	—
287	C₂	C₃₁×#1565	1	4	yes	—
288	C₂	C₂×C₃₁×#336	1	1	yes	characteristic
289	C₂	C₃₁×#1736	1	1	yes	characteristic
290	C₂	C₃₁×#1738	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₃₁	C₂×D₄
commutator	C₄	C₂×C₄×C₈×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
socle	C₂³×C₃₁	C₂×C₄⋊C₄
prime radical	C₂⁴×C₃₁	C₄⋊C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₃₁ ◁ C₂²×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₃₁×#1277	C₂	semidirect
26. C₂×M₁₆×C₃₁	C₈	semidirect
77. C₂×C₈×C₃₁	C₂×C₈	semidirect
88. #7873	C₃₁	direct (N × H)
125. #1277	C₂×C₃₁	semidirect
171. C₂×M₁₆	C₈×C₃₁	semidirect
195. C₃₁	#7873	direct (N × H)
223. C₂×C₈	C₂×C₈×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
C₂²×C₃₁×C₄⋊C₈	2
C₂×C₄×M₁₆×C₃₁	2
C₃₁×#1277	2
#7873	31

Subgroups — C31×#7873 #21464632

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#7873 #21464632