Subgroups — C₃₁×#7893 #21464652

Order 7936.

714 conjugacy classes of subgroups.

The subgroup lattice has 328 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	C₃₁×#2285	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1279	C₂	1	4	yes	maximal, complemented
5	C₂³×C₈×C₃₁	—	2	1	no	—
6	C₂×C₃₁×#117	—	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	complemented
10	C₂×C₃₁×C₄⋊C₈	—	2	1	no	—
11	C₂×C₃₁×C₄⋊C₈	—	2	2	no	—
12	C₂×C₃₁×C₄⋊C₈	—	2	1	no	—
13	C₂×C₃₁×C₄⋊C₈	C₄	1	4	yes	complemented
14	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
15	C₂×C₃₁×C₂²⋊C₈	—	2	1	no	—
16	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
17	C₃₁×#384	C₄	1	2	yes	—
18	C₃₁×#382	—	2	1	no	—
19	C₃₁×#380	—	2	1	no	—
20	C₃₁×#380	C₄	1	2	yes	—
21	C₄×C₄⋊C₄×C₃₁	—	2	1	no	—
22	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
23	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
24	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
25	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₃₁	—	2	4	no	—
27	C₂²×C₈×C₃₁	—	2	2	no	—
28	C₂²×C₈×C₃₁	—	2	2	no	—
29	C₂²×C₈×C₃₁	—	4	2	no	—
30	C₂²×C₈×C₃₁	—	4	2	no	—
31	C₃₁×#117	—	4	2	no	—
32	C₃₁×#117	—	4	2	no	—
33	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
34	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
35	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
36	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
37	C₂×C₄²×C₃₁	—	2	2	no	—
38	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₃₁	D₄	1	2	yes	—
44	C₂×C₄²×C₃₁	Q₈	1	2	yes	—
45	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
46	C₃₁×C₄⋊C₈	—	2	4	no	—
47	C₃₁×C₄⋊C₈	—	2	4	no	—
48	C₃₁×C₄⋊C₈	—	2	2	no	—
49	C₃₁×C₄⋊C₈	C₂×C₄	1	4	yes	—
50	C₃₁×C₂²⋊C₈	—	2	4	no	—
51	C₄×C₈×C₃₁	—	2	2	no	—
52	C₄×C₈×C₃₁	C₂×C₄	1	4	yes	—
53	C₃₁×#95	—	2	2	no	—
54	C₃₁×#95	—	2	2	no	—
55	C₂⁴×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₃₁	—	2	4	no	—
57	C₂²×C₄×C₃₁	—	2	4	no	—
58	C₂²×C₄×C₃₁	—	4	2	no	—
59	C₂²×C₄×C₃₁	C₄²	1	2	yes	—
60	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
63	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
65	C₂²×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
66	C₂²×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
67	C₂²×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
68	C₂²×C₄×C₃₁	D₈	1	2	yes	—
69	C₂²×C₄×C₃₁	SD₁₆	1	2	yes	—
70	C₂²×C₄×C₃₁	SD₁₆	1	2	yes	—
71	C₂²×C₄×C₃₁	Q₁₆	1	2	yes	—
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	4	no	—
78	C₂×C₈×C₃₁	—	4	8	no	—
79	C₂×C₈×C₃₁	—	4	2	no	—
80	C₂×C₈×C₃₁	—	4	2	no	—
81	C₄⋊C₄×C₃₁	—	2	8	no	—
82	C₄⋊C₄×C₃₁	—	4	4	no	—
83	C₄⋊C₄×C₃₁	—	4	4	no	—
84	C₂²⋊C₄×C₃₁	—	4	4	no	—
85	C₄²×C₃₁	—	4	2	no	—
86	C₄²×C₃₁	—	4	2	no	—
87	C₄²×C₃₁	C₄²	1	2	yes	—
88	C₄²×C₃₁	C₂²⋊C₄	1	2	yes	—
89	C₄²×C₃₁	C₂²⋊C₄	1	2	yes	—
90	C₄²×C₃₁	C₄⋊C₄	1	2	yes	—
91	C₄²×C₃₁	C₄⋊C₄	1	2	yes	—
92	C₄²×C₃₁	C₄⋊C₄	1	2	yes	—
93	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
94	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
95	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
96	C₄²×C₃₁	C₂×Q₈	1	1	yes	characteristic
97	#7893	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
98	C₂³×C₃₁	—	2	2	no	—
99	C₂³×C₃₁	—	2	2	no	—
100	C₂³×C₃₁	#95	1	2	yes	—
101	C₂³×C₃₁	#102	1	2	yes	—
102	C₂³×C₃₁	#103	1	2	yes	—
103	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₃₁	—	2	2	no	—
105	C₂×C₄×C₃₁	—	2	2	no	—
106	C₂×C₄×C₃₁	—	2	2	no	—
107	C₂×C₄×C₃₁	—	2	4	no	—
108	C₂×C₄×C₃₁	—	2	4	no	—
109	C₂×C₄×C₃₁	—	4	2	no	—
110	C₂×C₄×C₃₁	—	4	2	no	—
111	C₂×C₄×C₃₁	—	4	4	no	—
112	C₂×C₄×C₃₁	—	4	4	no	—
113	C₂×C₄×C₃₁	—	4	4	no	—
114	C₂×C₄×C₃₁	#95	1	2	yes	—
115	C₂×C₄×C₃₁	#95	1	2	yes	—
116	C₂×C₄×C₃₁	#95	1	2	yes	—
117	C₂×C₄×C₃₁	#102	1	2	yes	—
118	C₂×C₄×C₃₁	#102	1	4	yes	—
119	C₂×C₄×C₃₁	#103	1	2	yes	—
120	C₂×C₄×C₃₁	#103	1	4	yes	—
121	C₂×C₄×C₃₁	#106	1	4	yes	—
122	C₂×C₄×C₃₁	#107	1	4	yes	—
123	C₂×C₄×C₃₁	C₂×C₄²	1	1	yes	characteristic
124	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
126	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
127	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
128	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
129	C₂×C₄×C₃₁	#131	1	4	yes	—
130	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
131	C₂×C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
132	C₂×C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
133	C₂×C₄×C₃₁	C₂×Q₁₆	1	1	yes	characteristic
134	C₈×C₃₁	—	4	4	no	—
135	C₈×C₃₁	—	4	4	no	—
136	C₂×#431	C₂×C₃₁	1	1	yes	characteristic
137	C₂²×C₄⋊C₈	C₂×C₃₁	1	1	yes	characteristic
138	#2285	C₂×C₃₁	1	1	yes	characteristic
139	#1279	C₂×C₃₁	1	4	yes	complemented
140	C₂²×C₃₁	—	2	2	no	—
141	C₂²×C₃₁	—	2	2	no	—
142	C₂²×C₃₁	—	2	4	no	—
143	C₂²×C₃₁	—	2	4	no	—
144	C₂²×C₃₁	#340	1	4	yes	—
145	C₂²×C₃₁	C₂×#95	1	1	yes	characteristic
146	C₂²×C₃₁	#408	1	2	yes	—
147	C₂²×C₃₁	C₂×#102	1	1	yes	characteristic
148	C₂²×C₃₁	C₂×#103	1	1	yes	characteristic
149	C₂²×C₃₁	#421	1	2	yes	—
150	C₄×C₃₁	—	2	2	no	—
151	C₄×C₃₁	—	2	1	no	—
152	C₄×C₃₁	—	2	2	no	—
153	C₄×C₃₁	—	2	1	no	—
154	C₄×C₃₁	—	8	4	no	—
155	C₄×C₃₁	#340	1	4	yes	—
156	C₄×C₃₁	#405	1	2	yes	—
157	C₄×C₃₁	C₂×#102	1	1	yes	characteristic
158	C₄×C₃₁	C₂×#103	1	1	yes	characteristic
159	C₄×C₃₁	#424	1	2	yes	—
160	C₄×C₃₁	C₂×#106	1	1	yes	characteristic
161	C₄×C₃₁	C₂×#107	1	1	yes	characteristic
162	C₂³×C₈	—	2	1	no	—
163	C₂×#117	—	2	1	no	—
164	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
165	#431	—	2	2	no	—
166	#431	C₄×C₃₁	1	4	yes	complemented
167	C₂×C₄⋊C₈	—	2	1	no	—
168	C₂×C₄⋊C₈	—	2	2	no	—
169	C₂×C₄⋊C₈	—	2	1	no	—
170	C₂×C₄⋊C₈	C₄×C₃₁	1	4	yes	complemented
171	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
172	C₂×C₂²⋊C₈	—	2	1	no	—
173	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
174	#384	C₄×C₃₁	1	2	yes	—
175	#382	—	2	1	no	—
176	#380	—	2	1	no	—
177	#380	C₄×C₃₁	1	2	yes	—
178	C₄×C₄⋊C₄	—	2	1	no	—
179	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
180	C₂×C₃₁	—	2	4	no	—
181	C₂×C₃₁	#1480	1	2	yes	—
182	C₂×C₃₁	#1481	1	2	yes	—
183	C₂×C₃₁	#1729	1	1	yes	characteristic
184	C₂×C₃₁	#1733	1	1	yes	characteristic
185	C₂×C₃₁	C₂×#340	1	1	yes	characteristic
186	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
187	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
188	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
189	C₂²×C₈	—	2	4	no	—
190	C₂²×C₈	—	2	2	no	—
191	C₂²×C₈	—	2	2	no	—
192	C₂²×C₈	—	4	2	no	—
193	C₂²×C₈	—	4	2	no	—
194	#117	—	4	2	no	—
195	#117	—	4	2	no	—
196	C₂×C₄⋊C₄	—	2	2	no	—
197	C₂×C₄⋊C₄	—	2	2	no	—
198	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	4	yes	—
199	C₂×C₂²⋊C₄	—	2	2	no	—
200	C₂×C₄²	—	2	2	no	—
201	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
202	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
203	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
204	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
205	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
206	C₂×C₄²	D₄×C₃₁	1	2	yes	—
207	C₂×C₄²	Q₈×C₃₁	1	2	yes	—
208	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
209	C₄⋊C₈	—	2	4	no	—
210	C₄⋊C₈	—	2	4	no	—
211	C₄⋊C₈	—	2	2	no	—
212	C₄⋊C₈	C₂×C₄×C₃₁	1	4	yes	—
213	C₂²⋊C₈	—	2	4	no	—
214	C₄×C₈	—	2	2	no	—
215	C₄×C₈	C₂×C₄×C₃₁	1	4	yes	—
216	#95	—	2	2	no	—
217	#95	—	2	2	no	—
218	C₃₁	#7893	1	1	yes	characteristic, direct product factor, Sylow
219	C₂⁴	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
220	C₂²×C₄	—	2	4	no	—
221	C₂²×C₄	—	2	4	no	—
222	C₂²×C₄	—	4	2	no	—
223	C₂²×C₄	C₄²×C₃₁	1	2	yes	—
224	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
225	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
226	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
227	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
228	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
229	C₂²×C₄	C₄⋊C₄×C₃₁	1	2	yes	—
230	C₂²×C₄	C₄⋊C₄×C₃₁	1	2	yes	—
231	C₂²×C₄	C₄⋊C₄×C₃₁	1	2	yes	—
232	C₂²×C₄	D₈×C₃₁	1	2	yes	—
233	C₂²×C₄	SD₁₆×C₃₁	1	2	yes	—
234	C₂²×C₄	SD₁₆×C₃₁	1	2	yes	—
235	C₂²×C₄	Q₁₆×C₃₁	1	2	yes	—
236	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
237	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
238	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
239	C₂×C₈	—	2	4	no	—
240	C₂×C₈	—	2	4	no	—
241	C₂×C₈	—	4	4	no	—
242	C₂×C₈	—	4	8	no	—
243	C₂×C₈	—	4	2	no	—
244	C₂×C₈	—	4	2	no	—
245	C₄⋊C₄	—	2	8	no	—
246	C₄⋊C₄	—	4	4	no	—
247	C₄⋊C₄	—	4	4	no	—
248	C₂²⋊C₄	—	4	4	no	—
249	C₄²	—	4	2	no	—
250	C₄²	—	4	2	no	—
251	C₄²	C₄²×C₃₁	1	2	yes	—
252	C₄²	C₂²⋊C₄×C₃₁	1	2	yes	—
253	C₄²	C₂²⋊C₄×C₃₁	1	2	yes	—
254	C₄²	C₄⋊C₄×C₃₁	1	2	yes	—
255	C₄²	C₄⋊C₄×C₃₁	1	2	yes	—
256	C₄²	C₄⋊C₄×C₃₁	1	2	yes	—
257	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
258	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
259	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
260	C₄²	C₂×Q₈×C₃₁	1	1	yes	characteristic
261	C₂³	—	2	2	no	—
262	C₂³	—	2	2	no	—
263	C₂³	C₃₁×#95	1	2	yes	—
264	C₂³	C₃₁×#102	1	2	yes	—
265	C₂³	C₃₁×#103	1	2	yes	—
266	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	—	4	4	no	—
275	C₂×C₄	—	4	4	no	—
276	C₂×C₄	—	4	4	no	—
277	C₂×C₄	C₃₁×#95	1	2	yes	—
278	C₂×C₄	C₃₁×#95	1	2	yes	—
279	C₂×C₄	C₃₁×#95	1	2	yes	—
280	C₂×C₄	C₃₁×#102	1	4	yes	—
281	C₂×C₄	C₃₁×#102	1	2	yes	—
282	C₂×C₄	C₃₁×#103	1	4	yes	—
283	C₂×C₄	C₃₁×#103	1	2	yes	—
284	C₂×C₄	C₃₁×#106	1	4	yes	—
285	C₂×C₄	C₃₁×#107	1	4	yes	—
286	C₂×C₄	C₂×C₄²×C₃₁	1	1	yes	characteristic
287	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
288	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
289	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
290	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
291	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
292	C₂×C₄	C₃₁×#131	1	4	yes	—
293	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
294	C₂×C₄	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
295	C₂×C₄	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
296	C₂×C₄	C₂×Q₁₆×C₃₁	1	1	yes	characteristic
297	C₈	—	4	4	no	—
298	C₈	—	4	4	no	—
299	C₂²	—	2	2	no	—
300	C₂²	—	2	2	no	—
301	C₂²	—	2	4	no	—
302	C₂²	—	2	4	no	—
303	C₂²	C₃₁×#340	1	4	yes	—
304	C₂²	C₂×C₃₁×#95	1	1	yes	characteristic
305	C₂²	C₃₁×#408	1	2	yes	—
306	C₂²	C₂×C₃₁×#102	1	1	yes	characteristic
307	C₂²	C₂×C₃₁×#103	1	1	yes	characteristic
308	C₂²	C₃₁×#421	1	2	yes	—
309	C₄	—	2	2	no	—
310	C₄	—	2	1	no	—
311	C₄	—	2	2	no	—
312	C₄	—	2	1	no	—
313	C₄	—	8	4	no	—
314	C₄	C₃₁×#340	1	4	yes	—
315	C₄	C₃₁×#405	1	2	yes	—
316	C₄	C₂×C₃₁×#102	1	1	yes	characteristic
317	C₄	C₂×C₃₁×#103	1	1	yes	characteristic
318	C₄	C₃₁×#424	1	2	yes	—
319	C₄	C₂×C₃₁×#106	1	1	yes	characteristic
320	C₄	C₂×C₃₁×#107	1	1	yes	characteristic
321	C₂	—	2	4	no	—
322	C₂	C₃₁×#1480	1	2	yes	—
323	C₂	C₃₁×#1481	1	2	yes	—
324	C₂	C₃₁×#1729	1	1	yes	characteristic
325	C₂	C₃₁×#1733	1	1	yes	characteristic
326	C₂	C₂×C₃₁×#340	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₃₁	C₂×D₄
commutator	C₂×C₄	C₂×C₄²×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
socle	C₂³×C₃₁	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₃₁	C₂²⋊C₄
LCS	G ▷ C₂×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×#1279	C₂	semidirect
9. C₃₁×#431	C₄	semidirect
13. C₂×C₃₁×C₄⋊C₈	C₄	semidirect
97. #7893	C₃₁	direct (N × H)
139. #1279	C₂×C₃₁	semidirect
166. #431	C₄×C₃₁	semidirect
170. C₂×C₄⋊C₈	C₄×C₃₁	semidirect
218. C₃₁	#7893	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×C₃₁×#431	2
C₂²×C₃₁×C₄⋊C₈	2
C₃₁×#2285	2
C₃₁×#1279	2
#7893	31

Subgroups — C31×#7893 #21464652

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#7893 #21464652