Subgroups — C₃₁×#11996 #21468755

Order 7936.

270 conjugacy classes of subgroups.

The subgroup lattice has 162 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2150	C₂	1	2	yes	maximal
2	C₂×C₈²×C₃₁	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1367	C₂	1	4	yes	maximal, complemented
4	C₂×C₃₁×M₃₂	—	2	2	no	—
5	C₂×C₄×C₈×C₃₁	C₄	1	1	yes	characteristic
6	C₂×C₄×C₈×C₃₁	C₄	1	1	yes	characteristic
7	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
8	C₃₁×C₄⋊C₁₆	—	2	2	no	—
9	C₃₁×C₄⋊C₁₆	C₂²	1	4	yes	—
10	C₈²×C₃₁	C₄	1	2	yes	—
11	C₈²×C₃₁	C₂²	1	2	yes	—
12	C₂²×C₈×C₃₁	—	2	1	no	—
13	C₂²×C₈×C₃₁	C₈	1	2	yes	—
14	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
15	C₂²×C₈×C₃₁	Q₈	1	1	yes	characteristic
16	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
17	C₃₁×M₃₂	—	4	4	no	—
18	C₂×C₁₆×C₃₁	—	2	4	no	—
19	C₄×C₈×C₃₁	—	2	1	no	—
20	C₄×C₈×C₃₁	C₈	1	1	yes	characteristic
21	C₄×C₈×C₃₁	C₈	1	1	yes	characteristic
22	C₄×C₈×C₃₁	C₂×C₄	1	2	yes	—
23	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
24	C₄×C₈×C₃₁	C₂×C₄	1	2	yes	—
25	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
26	C₄×C₈×C₃₁	Q₈	1	1	yes	characteristic
27	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
28	C₂²×C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
29	C₂²×C₄×C₃₁	C₂×C₈	1	1	yes	characteristic
30	C₂²×C₄×C₃₁	M₁₆	1	1	yes	characteristic
31	C₂×C₈×C₃₁	—	2	4	no	—
32	C₂×C₈×C₃₁	—	2	2	no	—
33	C₂×C₈×C₃₁	—	2	2	no	—
34	C₂×C₈×C₃₁	—	2	4	no	—
35	C₂×C₈×C₃₁	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₈×C₃₁	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₈×C₃₁	C₂×C₈	1	2	yes	—
38	C₂×C₈×C₃₁	C₂×C₈	1	2	yes	—
39	C₂×C₈×C₃₁	SD₁₆	1	4	yes	—
40	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
41	C₂×C₈×C₃₁	C₂×Q₈	1	1	yes	characteristic
42	C₄²×C₃₁	C₄⋊C₄	1	1	yes	characteristic
43	C₄²×C₃₁	C₂×C₈	1	1	yes	characteristic
44	C₄²×C₃₁	M₁₆	1	1	yes	characteristic
45	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
46	C₁₆×C₃₁	—	4	4	no	—
47	#11996	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
48	C₂³×C₃₁	C₄⋊C₈	1	1	yes	characteristic
49	C₂×C₄×C₃₁	—	2	2	no	—
50	C₂×C₄×C₃₁	—	2	2	no	—
51	C₂×C₄×C₃₁	C₄⋊C₈	1	1	yes	characteristic
52	C₂×C₄×C₃₁	C₄⋊C₈	1	1	yes	characteristic
53	C₂×C₄×C₃₁	C₄⋊C₈	1	1	yes	characteristic
54	C₂×C₄×C₃₁	#106	1	2	yes	—
55	C₂×C₄×C₃₁	#108	1	2	yes	—
56	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₄×C₃₁	C₂²×C₈	1	1	yes	characteristic
58	C₂×C₄×C₃₁	C₂×M₁₆	1	1	yes	characteristic
59	C₈×C₃₁	—	2	4	no	—
60	C₈×C₃₁	—	2	1	no	—
61	C₈×C₃₁	—	2	1	no	—
62	C₈×C₃₁	—	2	2	no	—
63	C₈×C₃₁	#106	1	2	yes	—
64	C₈×C₃₁	M₃₂	1	4	yes	complemented
65	C₈×C₃₁	C₂×SD₁₆	1	2	yes	—
66	#2150	C₂×C₃₁	1	2	yes	—
67	C₂×C₈²	C₂×C₃₁	1	1	yes	characteristic
68	#1367	C₂×C₃₁	1	4	yes	complemented
69	C₂²×C₃₁	—	2	1	no	—
70	C₂²×C₃₁	—	2	1	no	—
71	C₂²×C₃₁	#334	1	2	yes	—
72	C₂²×C₃₁	C₂×C₄⋊C₈	1	1	yes	characteristic
73	C₄×C₃₁	—	2	1	no	—
74	C₄×C₃₁	—	2	1	no	—
75	C₄×C₃₁	#334	1	2	yes	—
76	C₄×C₃₁	#364	1	2	yes	—
77	C₄×C₃₁	C₂×#106	1	1	yes	characteristic
78	C₄×C₃₁	C₂×#108	1	1	yes	characteristic
79	C₄×C₃₁	C₂×M₃₂	1	2	yes	—
80	C₂×M₃₂	—	2	2	no	—
81	C₂×C₄×C₈	C₄×C₃₁	1	1	yes	characteristic
82	C₂×C₄×C₈	C₄×C₃₁	1	1	yes	characteristic
83	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
84	C₄⋊C₁₆	—	2	2	no	—
85	C₄⋊C₁₆	C₂²×C₃₁	1	4	yes	—
86	C₈²	C₄×C₃₁	1	2	yes	—
87	C₈²	C₂²×C₃₁	1	2	yes	—
88	C₂×C₃₁	—	2	2	no	—
89	C₂×C₃₁	C₂×#334	1	1	yes	characteristic
90	C₂×C₃₁	#2150	1	1	yes	characteristic
91	C₂×C₃₁	C₂×#364	1	1	yes	characteristic
92	C₂²×C₈	—	2	1	no	—
93	C₂²×C₈	C₈×C₃₁	1	2	yes	—
94	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
95	C₂²×C₈	Q₈×C₃₁	1	1	yes	characteristic
96	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
97	M₃₂	—	4	4	no	—
98	C₂×C₁₆	—	2	4	no	—
99	C₄×C₈	—	2	1	no	—
100	C₄×C₈	C₈×C₃₁	1	1	yes	characteristic
101	C₄×C₈	C₈×C₃₁	1	1	yes	characteristic
102	C₄×C₈	C₂×C₄×C₃₁	1	2	yes	—
103	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
104	C₄×C₈	C₂×C₄×C₃₁	1	2	yes	—
105	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
106	C₄×C₈	Q₈×C₃₁	1	1	yes	characteristic
107	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
108	C₃₁	#11996	1	1	yes	characteristic, direct product factor, Sylow
109	C₂²×C₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
110	C₂²×C₄	C₂×C₈×C₃₁	1	1	yes	characteristic
111	C₂²×C₄	M₁₆×C₃₁	1	1	yes	characteristic
112	C₂×C₈	—	2	4	no	—
113	C₂×C₈	—	2	2	no	—
114	C₂×C₈	—	2	2	no	—
115	C₂×C₈	—	2	4	no	—
116	C₂×C₈	C₄⋊C₄×C₃₁	1	1	yes	characteristic
117	C₂×C₈	C₄⋊C₄×C₃₁	1	1	yes	characteristic
118	C₂×C₈	C₂×C₈×C₃₁	1	2	yes	—
119	C₂×C₈	C₂×C₈×C₃₁	1	2	yes	—
120	C₂×C₈	SD₁₆×C₃₁	1	4	yes	—
121	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
122	C₂×C₈	C₂×Q₈×C₃₁	1	1	yes	characteristic
123	C₄²	C₄⋊C₄×C₃₁	1	1	yes	characteristic
124	C₄²	C₂×C₈×C₃₁	1	1	yes	characteristic
125	C₄²	M₁₆×C₃₁	1	1	yes	characteristic
126	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
127	C₁₆	—	4	4	no	—
128	C₂³	C₃₁×C₄⋊C₈	1	1	yes	characteristic
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	C₃₁×C₄⋊C₈	1	1	yes	characteristic
132	C₂×C₄	C₃₁×C₄⋊C₈	1	1	yes	characteristic
133	C₂×C₄	C₃₁×C₄⋊C₈	1	1	yes	characteristic
134	C₂×C₄	C₃₁×#106	1	2	yes	—
135	C₂×C₄	C₃₁×#108	1	2	yes	—
136	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
137	C₂×C₄	C₂²×C₈×C₃₁	1	1	yes	characteristic
138	C₂×C₄	C₂×M₁₆×C₃₁	1	1	yes	characteristic
139	C₈	—	2	4	no	—
140	C₈	—	2	1	no	—
141	C₈	—	2	1	no	—
142	C₈	—	2	2	no	—
143	C₈	C₃₁×#106	1	2	yes	—
144	C₈	C₃₁×M₃₂	1	4	yes	complemented
145	C₈	C₂×SD₁₆×C₃₁	1	2	yes	—
146	C₂²	—	2	1	no	—
147	C₂²	—	2	1	no	—
148	C₂²	C₃₁×#334	1	2	yes	—
149	C₂²	C₂×C₃₁×C₄⋊C₈	1	1	yes	characteristic
150	C₄	—	2	1	no	—
151	C₄	—	2	1	no	—
152	C₄	C₃₁×#334	1	2	yes	—
153	C₄	C₃₁×#364	1	2	yes	—
154	C₄	C₂×C₃₁×#106	1	1	yes	characteristic
155	C₄	C₂×C₃₁×#108	1	1	yes	characteristic
156	C₄	C₂×C₃₁×M₃₂	1	2	yes	—
157	C₂	—	2	2	no	—
158	C₂	C₂×C₃₁×#334	1	1	yes	characteristic
159	C₂	C₃₁×#2150	1	1	yes	characteristic
160	C₂	C₂×C₃₁×#364	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
3. C₃₁×#1367	C₂	semidirect
47. #11996	C₃₁	direct (N × H)
64. C₈×C₃₁	M₃₂	semidirect
68. #1367	C₂×C₃₁	semidirect
108. C₃₁	#11996	direct (N × H)
144. C₈	C₃₁×M₃₂	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#2150	2
C₂×C₈²×C₃₁	2
C₃₁×#1367	2
#11996	31

Subgroups — C31×#11996 #21468755

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#11996 #21468755