Subgroups — C₃₁×#11934 #21468693

Order 7936.

306 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3160	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1564	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1363	C₂	1	4	yes	maximal, complemented
5	C₃₁×#533	—	2	1	no	—
6	C₂²×C₁₆×C₃₁	—	2	1	no	—
7	C₃₁×#500	C₄	1	1	yes	characteristic
8	C₃₁×#500	C₄	1	1	yes	characteristic
9	C₃₁×#499	—	2	1	no	—
10	C₃₁×#499	C₂²	1	2	yes	—
11	C₃₁×#498	—	2	1	no	—
12	C₃₁×#431	—	2	1	no	—
13	C₃₁×#427	—	2	1	no	—
14	C₃₁×#427	C₄	1	2	yes	—
15	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
16	C₃₁×C₄⋊C₁₆	—	2	1	no	—
17	C₃₁×C₄⋊C₁₆	C₂²	1	2	yes	—
18	C₃₁×#335	—	2	1	no	—
19	C₃₁×#334	C₂²	1	2	yes	—
20	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
22	C₃₁×C₄⋊Q₈	—	2	1	no	—
23	C₃₁×C₄⋊Q₈	—	2	1	no	—
24	C₃₁×#125	—	2	2	no	—
25	C₃₁×C₂²⋊Q₈	—	4	1	no	—
26	C₃₁×C₂²⋊Q₈	—	4	1	no	—
27	C₄×Q₈×C₃₁	—	4	1	no	—
28	C₄×Q₈×C₃₁	—	4	1	no	—
29	C₃₁×#117	—	2	2	no	—
30	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
31	C₂×C₁₆×C₃₁	—	2	2	no	—
32	C₂×C₁₆×C₃₁	—	4	2	no	—
33	C₃₁×#107	—	2	1	no	—
34	C₃₁×#107	—	2	1	no	—
35	C₃₁×#107	C₂×C₄	1	2	yes	—
36	C₃₁×#107	C₂×C₄	1	2	yes	—
37	C₃₁×#106	—	2	2	no	—
38	C₃₁×#106	—	2	2	no	—
39	C₃₁×C₄⋊C₈	—	2	2	no	—
40	C₃₁×C₂²⋊C₈	—	4	1	no	—
41	C₄×C₈×C₃₁	—	4	1	no	—
42	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
44	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
45	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
46	C₂×Q₈×C₃₁	—	4	1	no	—
47	C₂×Q₈×C₃₁	—	4	1	no	—
48	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
50	C₂²×C₄×C₃₁	SD₁₆	1	1	yes	characteristic
51	C₂×C₈×C₃₁	—	2	2	no	—
52	C₂×C₈×C₃₁	—	4	2	no	—
53	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₈×C₃₁	SD₁₆	1	2	yes	—
56	C₂×C₈×C₃₁	Q₁₆	1	2	yes	—
57	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
59	C₄⋊C₄×C₃₁	—	2	2	no	—
60	C₄⋊C₄×C₃₁	—	2	2	no	—
61	C₄⋊C₄×C₃₁	—	4	1	no	—
62	C₄⋊C₄×C₃₁	—	4	1	no	—
63	C₄⋊C₄×C₃₁	—	4	2	no	—
64	C₄⋊C₄×C₃₁	—	4	2	no	—
65	C₂²⋊C₄×C₃₁	—	4	2	no	—
66	C₄²×C₃₁	—	4	2	no	—
67	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
68	C₄²×C₃₁	D₈	1	1	yes	characteristic
69	C₄²×C₃₁	SD₁₆	1	1	yes	characteristic
70	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
71	C₁₆×C₃₁	—	4	2	no	—
72	#11934	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
73	C₂³×C₃₁	#102	1	1	yes	characteristic
74	Q₈×C₃₁	—	8	2	no	—
75	Q₈×C₃₁	—	8	2	no	—
76	C₂×C₄×C₃₁	—	2	2	no	—
77	C₂×C₄×C₃₁	—	4	1	no	—
78	C₂×C₄×C₃₁	—	4	2	no	—
79	C₂×C₄×C₃₁	—	4	2	no	—
80	C₂×C₄×C₃₁	#101	1	2	yes	—
81	C₂×C₄×C₃₁	#102	1	1	yes	characteristic
82	C₂×C₄×C₃₁	#102	1	1	yes	characteristic
83	C₂×C₄×C₃₁	#102	1	1	yes	characteristic
84	C₂×C₄×C₃₁	#103	1	2	yes	—
85	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
87	C₂×C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
88	C₈×C₃₁	—	2	1	no	—
89	C₈×C₃₁	—	2	1	no	—
90	C₈×C₃₁	—	8	2	no	—
91	C₈×C₃₁	#103	1	2	yes	—
92	C₈×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
93	C₈×C₃₁	C₂×Q₁₆	1	1	yes	characteristic
94	#3160	C₂×C₃₁	1	1	yes	characteristic
95	C₂×C₄⋊C₁₆	C₂×C₃₁	1	1	yes	characteristic
96	#1564	C₂×C₃₁	1	1	yes	characteristic
97	#1363	C₂×C₃₁	1	4	yes	complemented
98	C₂²×C₃₁	—	2	1	no	—
99	C₂²×C₃₁	—	2	1	no	—
100	C₂²×C₃₁	#332	1	2	yes	—
101	C₂²×C₃₁	C₂×#102	1	1	yes	characteristic
102	C₄×C₃₁	—	2	1	no	—
103	C₄×C₃₁	—	2	1	no	—
104	C₄×C₃₁	—	2	1	no	—
105	C₄×C₃₁	—	8	2	no	—
106	C₄×C₃₁	—	8	2	no	—
107	C₄×C₃₁	#332	1	2	yes	—
108	C₄×C₃₁	#359	1	2	yes	—
109	C₄×C₃₁	C₂×#101	1	1	yes	characteristic
110	C₄×C₃₁	C₂×#103	1	1	yes	characteristic
111	#533	—	2	1	no	—
112	C₂²×C₁₆	—	2	1	no	—
113	#500	C₄×C₃₁	1	1	yes	characteristic
114	#500	C₄×C₃₁	1	1	yes	characteristic
115	#499	—	2	1	no	—
116	#499	C₂²×C₃₁	1	2	yes	—
117	#498	—	2	1	no	—
118	#431	—	2	1	no	—
119	#427	—	2	1	no	—
120	#427	C₄×C₃₁	1	2	yes	—
121	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
122	C₄⋊C₁₆	—	2	1	no	—
123	C₄⋊C₁₆	C₂²×C₃₁	1	2	yes	—
124	#335	—	2	1	no	—
125	#334	C₂²×C₃₁	1	2	yes	—
126	C₂×C₃₁	—	2	2	no	—
127	C₂×C₃₁	C₂×#332	1	1	yes	characteristic
128	C₂×C₃₁	C₂×#359	1	1	yes	characteristic
129	C₂×C₃₁	#2145	1	1	yes	characteristic
130	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
131	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
132	C₄⋊Q₈	—	2	1	no	—
133	C₄⋊Q₈	—	2	1	no	—
134	#125	—	2	2	no	—
135	C₂²⋊Q₈	—	4	1	no	—
136	C₂²⋊Q₈	—	4	1	no	—
137	C₄×Q₈	—	4	1	no	—
138	C₄×Q₈	—	4	1	no	—
139	#117	—	2	2	no	—
140	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
141	C₂×C₁₆	—	2	2	no	—
142	C₂×C₁₆	—	4	2	no	—
143	#107	—	2	1	no	—
144	#107	—	2	1	no	—
145	#107	C₂×C₄×C₃₁	1	2	yes	—
146	#107	C₂×C₄×C₃₁	1	2	yes	—
147	#106	—	2	2	no	—
148	#106	—	2	2	no	—
149	C₄⋊C₈	—	2	2	no	—
150	C₂²⋊C₈	—	4	1	no	—
151	C₄×C₈	—	4	1	no	—
152	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
153	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
154	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
155	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
156	C₃₁	#11934	1	1	yes	characteristic, direct product factor, Sylow
157	C₂×Q₈	—	4	1	no	—
158	C₂×Q₈	—	4	1	no	—
159	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
160	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
161	C₂²×C₄	SD₁₆×C₃₁	1	1	yes	characteristic
162	C₂×C₈	—	2	2	no	—
163	C₂×C₈	—	4	2	no	—
164	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
165	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
166	C₂×C₈	SD₁₆×C₃₁	1	2	yes	—
167	C₂×C₈	Q₁₆×C₃₁	1	2	yes	—
168	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
169	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
170	C₄⋊C₄	—	2	2	no	—
171	C₄⋊C₄	—	2	2	no	—
172	C₄⋊C₄	—	4	1	no	—
173	C₄⋊C₄	—	4	1	no	—
174	C₄⋊C₄	—	4	2	no	—
175	C₄⋊C₄	—	4	2	no	—
176	C₂²⋊C₄	—	4	2	no	—
177	C₄²	—	4	2	no	—
178	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
179	C₄²	D₈×C₃₁	1	1	yes	characteristic
180	C₄²	SD₁₆×C₃₁	1	1	yes	characteristic
181	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
182	C₁₆	—	4	2	no	—
183	C₂³	C₃₁×#102	1	1	yes	characteristic
184	Q₈	—	8	2	no	—
185	Q₈	—	8	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	4	1	no	—
188	C₂×C₄	—	4	2	no	—
189	C₂×C₄	—	4	2	no	—
190	C₂×C₄	C₃₁×#101	1	2	yes	—
191	C₂×C₄	C₃₁×#102	1	1	yes	characteristic
192	C₂×C₄	C₃₁×#102	1	1	yes	characteristic
193	C₂×C₄	C₃₁×#102	1	1	yes	characteristic
194	C₂×C₄	C₃₁×#103	1	2	yes	—
195	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
196	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
197	C₂×C₄	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
198	C₈	—	2	1	no	—
199	C₈	—	2	1	no	—
200	C₈	—	8	2	no	—
201	C₈	C₃₁×#103	1	2	yes	—
202	C₈	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
203	C₈	C₂×Q₁₆×C₃₁	1	1	yes	characteristic
204	C₂²	—	2	1	no	—
205	C₂²	—	2	1	no	—
206	C₂²	C₃₁×#332	1	2	yes	—
207	C₂²	C₂×C₃₁×#102	1	1	yes	characteristic
208	C₄	—	2	1	no	—
209	C₄	—	2	1	no	—
210	C₄	—	2	1	no	—
211	C₄	—	8	2	no	—
212	C₄	—	8	2	no	—
213	C₄	C₃₁×#332	1	2	yes	—
214	C₄	C₃₁×#359	1	2	yes	—
215	C₄	C₂×C₃₁×#101	1	1	yes	characteristic
216	C₄	C₂×C₃₁×#103	1	1	yes	characteristic
217	C₂	—	2	2	no	—
218	C₂	C₂×C₃₁×#332	1	1	yes	characteristic
219	C₂	C₂×C₃₁×#359	1	1	yes	characteristic
220	C₂	C₃₁×#2145	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	#102
commutator	C₂×C₈	C₂²×C₄×C₃₁
Frattini, cosocle	C₄×C₈	C₂³×C₃₁
socle	C₂²×C₃₁	C₂×#102
prime radical	C₂³×C₃₁	#102
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₃₁×#1363	C₂	semidirect
72. #11934	C₃₁	direct (N × H)
97. #1363	C₂×C₃₁	semidirect
156. C₃₁	#11934	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃₁×#3160	2
C₂×C₃₁×C₄⋊C₁₆	2
C₃₁×#1564	2
C₃₁×#1363	2
#11934	31

Subgroups — C31×#11934 #21468693

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#11934 #21468693