Subgroups — C₃₁×#7981 #21464740

Order 7936.

714 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	C₃₁×#2285	C₂	1	2	yes	maximal
3	C₃₁×#1289	C₂	1	4	yes	maximal
4	C₂×C₃₁×#117	—	2	2	no	—
5	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
6	C₃₁×#431	—	2	2	no	—
7	C₃₁×#431	—	2	2	no	—
8	C₂×C₃₁×C₄⋊C₈	C₄	1	2	yes	complemented
9	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
10	C₂×C₄×C₈×C₃₁	C₄	1	2	yes	complemented
11	C₃₁×#384	—	2	2	no	—
12	C₃₁×#382	C₄	1	4	yes	complemented
13	C₃₁×#380	—	2	2	no	—
14	C₃₁×#380	C₂²	1	4	yes	—
15	C₄×C₄⋊C₄×C₃₁	—	2	2	no	—
16	C₄×C₄⋊C₄×C₃₁	C₄	1	4	yes	complemented
17	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
18	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
19	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
20	C₂²×C₈×C₃₁	C₂×C₄	1	4	yes	—
21	C₃₁×#117	—	4	4	no	—
22	C₃₁×#117	—	4	4	no	—
23	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
24	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
25	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
26	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
27	C₂×C₄²×C₃₁	—	2	4	no	—
28	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₃₁	D₄	1	2	yes	—
33	C₂×C₄²×C₃₁	D₄	1	2	yes	—
34	C₂×C₄²×C₃₁	Q₈	1	1	yes	characteristic
35	C₂×C₄²×C₃₁	Q₈	1	1	yes	characteristic
36	C₃₁×C₄⋊C₈	—	2	4	no	—
37	C₃₁×C₂²⋊C₈	—	2	4	no	—
38	C₄×C₈×C₃₁	—	2	2	no	—
39	C₄×C₈×C₃₁	—	2	2	no	—
40	C₃₁×#95	—	2	4	no	—
41	C₃₁×#95	C₂×C₄	1	8	yes	—
42	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
43	C₂²×C₄×C₃₁	—	2	8	no	—
44	C₂²×C₄×C₃₁	—	4	2	no	—
45	C₂²×C₄×C₃₁	—	4	2	no	—
46	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
47	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
48	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
49	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
50	C₂²×C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₃₁	D₈	1	2	yes	—
55	C₂²×C₄×C₃₁	SD₁₆	1	4	yes	—
56	C₂²×C₄×C₃₁	Q₁₆	1	2	yes	—
57	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
59	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
60	C₂²×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
61	C₂×C₈×C₃₁	—	2	4	no	—
62	C₂×C₈×C₃₁	—	4	4	no	—
63	C₄⋊C₄×C₃₁	—	4	8	no	—
64	C₄⋊C₄×C₃₁	—	4	8	no	—
65	C₄⋊C₄×C₃₁	—	4	8	no	—
66	C₂²⋊C₄×C₃₁	—	4	4	no	—
67	C₂²⋊C₄×C₃₁	—	4	4	no	—
68	C₄²×C₃₁	—	4	8	no	—
69	C₄²×C₃₁	C₄²	1	1	yes	characteristic
70	C₄²×C₃₁	C₄²	1	2	yes	—
71	C₄²×C₃₁	C₄²	1	1	yes	characteristic
72	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
73	C₄²×C₃₁	C₂²⋊C₄	1	2	yes	—
74	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
75	C₄²×C₃₁	C₄⋊C₄	1	2	yes	—
76	C₄²×C₃₁	C₄⋊C₄	1	2	yes	—
77	C₄²×C₃₁	C₄⋊C₄	1	2	yes	—
78	C₄²×C₃₁	C₄⋊C₄	1	2	yes	—
79	#7981	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
80	C₂³×C₃₁	—	4	1	no	—
81	C₂³×C₃₁	—	4	1	no	—
82	C₂³×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
83	C₂³×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
84	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
85	C₂³×C₃₁	C₂×SD₁₆	1	2	yes	—
86	C₂³×C₃₁	C₂×Q₁₆	1	1	yes	characteristic
87	C₂×C₄×C₃₁	—	2	4	no	—
88	C₂×C₄×C₃₁	—	2	2	no	—
89	C₂×C₄×C₃₁	—	2	4	no	—
90	C₂×C₄×C₃₁	—	2	2	no	—
91	C₂×C₄×C₃₁	—	4	2	no	—
92	C₂×C₄×C₃₁	—	4	1	no	—
93	C₂×C₄×C₃₁	—	4	2	no	—
94	C₂×C₄×C₃₁	—	4	1	no	—
95	C₂×C₄×C₃₁	—	4	8	no	—
96	C₂×C₄×C₃₁	—	4	8	no	—
97	C₂×C₄×C₃₁	—	4	8	no	—
98	C₂×C₄×C₃₁	#95	1	2	yes	—
99	C₂×C₄×C₃₁	#95	1	1	yes	characteristic
100	C₂×C₄×C₃₁	#95	1	1	yes	characteristic
101	C₂×C₄×C₃₁	#95	1	1	yes	characteristic
102	C₂×C₄×C₃₁	#95	1	2	yes	—
103	C₂×C₄×C₃₁	#95	1	1	yes	characteristic
104	C₂×C₄×C₃₁	#102	1	4	yes	—
105	C₂×C₄×C₃₁	#102	1	4	yes	—
106	C₂×C₄×C₃₁	#103	1	4	yes	—
107	C₂×C₄×C₃₁	#103	1	4	yes	—
108	C₂×C₄×C₃₁	#106	1	2	yes	—
109	C₂×C₄×C₃₁	#106	1	2	yes	—
110	C₂×C₄×C₃₁	#107	1	2	yes	—
111	C₂×C₄×C₃₁	#107	1	2	yes	—
112	C₂×C₄×C₃₁	C₂×C₄²	1	1	yes	characteristic
113	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
115	C₈×C₃₁	—	4	4	no	—
116	C₂×#431	C₂×C₃₁	1	1	yes	characteristic
117	#2285	C₂×C₃₁	1	2	yes	—
118	#1289	C₂×C₃₁	1	4	yes	—
119	C₂²×C₃₁	—	2	2	no	—
120	C₂²×C₃₁	—	2	1	no	—
121	C₂²×C₃₁	—	2	2	no	—
122	C₂²×C₃₁	—	2	1	no	—
123	C₂²×C₃₁	—	4	1	no	—
124	C₂²×C₃₁	—	4	2	no	—
125	C₂²×C₃₁	—	4	1	no	—
126	C₂²×C₃₁	C₂×#95	1	1	yes	characteristic
127	C₂²×C₃₁	C₂×#102	1	2	yes	—
128	C₂²×C₃₁	C₂×#103	1	2	yes	—
129	C₂²×C₃₁	C₂×#106	1	1	yes	characteristic
130	C₂²×C₃₁	C₂×#107	1	1	yes	characteristic
131	C₄×C₃₁	—	2	2	no	—
132	C₄×C₃₁	—	2	1	no	—
133	C₄×C₃₁	—	2	1	no	—
134	C₄×C₃₁	—	2	2	no	—
135	C₄×C₃₁	—	8	8	no	—
136	C₄×C₃₁	H(3,ℤ/4)	1	2	yes	—
137	C₄×C₃₁	#340	1	4	yes	—
138	C₄×C₃₁	#340	1	2	yes	—
139	C₄×C₃₁	#340	1	2	yes	—
140	C₄×C₃₁	#410	1	2	yes	—
141	C₂×#117	—	2	2	no	—
142	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
143	#431	—	2	2	no	—
144	#431	—	2	2	no	—
145	C₂×C₄⋊C₈	C₄×C₃₁	1	2	yes	complemented
146	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
147	C₂×C₄×C₈	C₄×C₃₁	1	2	yes	complemented
148	#384	—	2	2	no	—
149	#382	C₄×C₃₁	1	4	yes	complemented
150	#380	—	2	2	no	—
151	#380	C₂²×C₃₁	1	4	yes	—
152	C₄×C₄⋊C₄	—	2	2	no	—
153	C₄×C₄⋊C₄	C₄×C₃₁	1	4	yes	complemented
154	C₂×C₃₁	—	2	2	no	—
155	C₂×C₃₁	—	2	2	no	—
156	C₂×C₃₁	—	2	1	no	—
157	C₂×C₃₁	—	2	1	no	—
158	C₂×C₃₁	C₂×H(3,ℤ/4)	1	1	yes	characteristic
159	C₂×C₃₁	C₂×#340	1	1	yes	characteristic
160	C₂×C₃₁	#1740	1	1	yes	characteristic
161	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
162	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
163	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
164	C₂²×C₈	C₂×C₄×C₃₁	1	4	yes	—
165	#117	—	4	4	no	—
166	#117	—	4	4	no	—
167	C₂×C₄⋊C₄	—	2	4	no	—
168	C₂×C₄⋊C₄	—	2	4	no	—
169	C₂×C₄⋊C₄	—	2	4	no	—
170	C₂×C₂²⋊C₄	—	2	4	no	—
171	C₂×C₄²	—	2	4	no	—
172	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
173	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
174	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
175	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
176	C₂×C₄²	D₄×C₃₁	1	2	yes	—
177	C₂×C₄²	D₄×C₃₁	1	2	yes	—
178	C₂×C₄²	Q₈×C₃₁	1	1	yes	characteristic
179	C₂×C₄²	Q₈×C₃₁	1	1	yes	characteristic
180	C₄⋊C₈	—	2	4	no	—
181	C₂²⋊C₈	—	2	4	no	—
182	C₄×C₈	—	2	2	no	—
183	C₄×C₈	—	2	2	no	—
184	#95	—	2	4	no	—
185	#95	C₂×C₄×C₃₁	1	8	yes	—
186	C₃₁	#7981	1	1	yes	characteristic, direct product factor, Sylow
187	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
188	C₂²×C₄	—	2	8	no	—
189	C₂²×C₄	—	4	2	no	—
190	C₂²×C₄	—	4	2	no	—
191	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
192	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
193	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
194	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
195	C₂²×C₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
196	C₂²×C₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
197	C₂²×C₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
198	C₂²×C₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
199	C₂²×C₄	D₈×C₃₁	1	2	yes	—
200	C₂²×C₄	SD₁₆×C₃₁	1	4	yes	—
201	C₂²×C₄	Q₁₆×C₃₁	1	2	yes	—
202	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
203	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
204	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
205	C₂²×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
206	C₂×C₈	—	2	4	no	—
207	C₂×C₈	—	4	4	no	—
208	C₄⋊C₄	—	4	8	no	—
209	C₄⋊C₄	—	4	8	no	—
210	C₄⋊C₄	—	4	8	no	—
211	C₂²⋊C₄	—	4	4	no	—
212	C₂²⋊C₄	—	4	4	no	—
213	C₄²	—	4	8	no	—
214	C₄²	C₄²×C₃₁	1	1	yes	characteristic
215	C₄²	C₄²×C₃₁	1	2	yes	—
216	C₄²	C₄²×C₃₁	1	1	yes	characteristic
217	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
218	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
219	C₄²	C₂²⋊C₄×C₃₁	1	2	yes	—
220	C₄²	C₄⋊C₄×C₃₁	1	2	yes	—
221	C₄²	C₄⋊C₄×C₃₁	1	2	yes	—
222	C₄²	C₄⋊C₄×C₃₁	1	2	yes	—
223	C₄²	C₄⋊C₄×C₃₁	1	2	yes	—
224	C₂³	—	4	1	no	—
225	C₂³	—	4	1	no	—
226	C₂³	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
227	C₂³	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
228	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
229	C₂³	C₂×SD₁₆×C₃₁	1	2	yes	—
230	C₂³	C₂×Q₁₆×C₃₁	1	1	yes	characteristic
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	4	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	—	4	1	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	4	1	no	—
239	C₂×C₄	—	4	8	no	—
240	C₂×C₄	—	4	8	no	—
241	C₂×C₄	—	4	8	no	—
242	C₂×C₄	C₃₁×#95	1	1	yes	characteristic
243	C₂×C₄	C₃₁×#95	1	2	yes	—
244	C₂×C₄	C₃₁×#95	1	1	yes	characteristic
245	C₂×C₄	C₃₁×#95	1	1	yes	characteristic
246	C₂×C₄	C₃₁×#95	1	2	yes	—
247	C₂×C₄	C₃₁×#95	1	1	yes	characteristic
248	C₂×C₄	C₃₁×#102	1	4	yes	—
249	C₂×C₄	C₃₁×#102	1	4	yes	—
250	C₂×C₄	C₃₁×#103	1	4	yes	—
251	C₂×C₄	C₃₁×#103	1	4	yes	—
252	C₂×C₄	C₃₁×#106	1	2	yes	—
253	C₂×C₄	C₃₁×#106	1	2	yes	—
254	C₂×C₄	C₃₁×#107	1	2	yes	—
255	C₂×C₄	C₃₁×#107	1	2	yes	—
256	C₂×C₄	C₂×C₄²×C₃₁	1	1	yes	characteristic
257	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
258	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
259	C₈	—	4	4	no	—
260	C₂²	—	2	2	no	—
261	C₂²	—	2	1	no	—
262	C₂²	—	2	2	no	—
263	C₂²	—	2	1	no	—
264	C₂²	—	4	1	no	—
265	C₂²	—	4	2	no	—
266	C₂²	—	4	1	no	—
267	C₂²	C₂×C₃₁×#95	1	1	yes	characteristic
268	C₂²	C₂×C₃₁×#102	1	2	yes	—
269	C₂²	C₂×C₃₁×#103	1	2	yes	—
270	C₂²	C₂×C₃₁×#106	1	1	yes	characteristic
271	C₂²	C₂×C₃₁×#107	1	1	yes	characteristic
272	C₄	—	2	2	no	—
273	C₄	—	2	1	no	—
274	C₄	—	2	1	no	—
275	C₄	—	2	2	no	—
276	C₄	—	8	8	no	—
277	C₄	C₃₁×H(3,ℤ/4)	1	2	yes	—
278	C₄	C₃₁×#340	1	4	yes	—
279	C₄	C₃₁×#340	1	2	yes	—
280	C₄	C₃₁×#340	1	2	yes	—
281	C₄	C₃₁×#410	1	2	yes	—
282	C₂	—	2	2	no	—
283	C₂	—	2	2	no	—
284	C₂	—	2	1	no	—
285	C₂	—	2	1	no	—
286	C₂	C₂×C₃₁×H(3,ℤ/4)	1	1	yes	characteristic
287	C₂	C₂×C₃₁×#340	1	1	yes	characteristic
288	C₂	C₃₁×#1740	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	#95
commutator	C₂×C₄	C₂×C₄²×C₃₁
Frattini, cosocle	C₂³×C₄	C₂³×C₃₁
socle	C₂²×C₃₁	C₂×#95
prime radical	C₂⁴×C₃₁	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂²×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
8. C₂×C₃₁×C₄⋊C₈	C₄	semidirect
10. C₂×C₄×C₈×C₃₁	C₄	semidirect
12. C₃₁×#382	C₄	semidirect
16. C₄×C₄⋊C₄×C₃₁	C₄	semidirect
79. #7981	C₃₁	direct (N × H)
145. C₂×C₄⋊C₈	C₄×C₃₁	semidirect
147. C₂×C₄×C₈	C₄×C₃₁	semidirect
149. #382	C₄×C₃₁	semidirect
153. C₄×C₄⋊C₄	C₄×C₃₁	semidirect
186. C₃₁	#7981	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
C₃₁×#2285	2
C₃₁×#1289	2
#7981	31

Subgroups — C31×#7981 #21464740

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#7981 #21464740