Group #21468161 — subgroups

Subgroups — C₃₁×#11402 #21468161

Order 7936.

322 conjugacy classes of subgroups; only the 246 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 246 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₃₁×M₃₂	C₂	1	2	yes	maximal
2	C₂×C₈²×C₃₁	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1312	C₂	1	4	yes	maximal, complemented
4	C₂×C₃₁×M₃₂	C₄	1	4	yes	—
5	C₂×C₄×C₈×C₃₁	C₄	1	2	yes	—
6	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
7	C₃₁×#346	C₄	1	4	yes	—
8	C₄×C₁₆×C₃₁	C₂²	1	4	yes	—
9	C₈²×C₃₁	C₄	1	1	yes	characteristic
10	C₈²×C₃₁	C₄	1	1	yes	characteristic
11	C₈²×C₃₁	C₂²	1	2	yes	—
12	C₂²×C₈×C₃₁	C₈	1	4	yes	—
13	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
14	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
15	C₃₁×M₃₂	—	2	8	no	—
16	C₂×C₁₆×C₃₁	C₂×C₄	1	8	yes	—
17	C₄×C₈×C₃₁	C₈	1	4	yes	—
18	C₄×C₈×C₃₁	C₂×C₄	1	2	yes	—
19	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
20	C₄×C₈×C₃₁	C₂×C₄	1	2	yes	—
21	C₄×C₈×C₃₁	C₂×C₄	1	2	yes	—
22	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
23	C₂²×C₄×C₃₁	C₄²	1	1	yes	characteristic
24	C₂²×C₄×C₃₁	C₂×C₈	1	2	yes	—
25	C₂×C₈×C₃₁	—	2	8	no	—
26	C₂×C₈×C₃₁	C₄²	1	2	yes	—
27	C₂×C₈×C₃₁	C₂×C₈	1	4	yes	—
28	C₂×C₈×C₃₁	C₂×C₈	1	4	yes	—
29	C₂×C₈×C₃₁	M₁₆	1	8	yes	—
30	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
31	C₄²×C₃₁	C₄²	1	1	yes	characteristic
32	C₄²×C₃₁	C₂×C₈	1	2	yes	—
33	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
34	C₁₆×C₃₁	—	2	8	no	—
35	#11402	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
36	C₂³×C₃₁	C₄×C₈	1	1	yes	characteristic
37	C₂×C₄×C₃₁	—	2	4	no	—
38	C₂×C₄×C₃₁	C₄×C₈	1	2	yes	—
39	C₂×C₄×C₃₁	C₄×C₈	1	1	yes	characteristic
40	C₂×C₄×C₃₁	#97	1	4	yes	—
41	C₂×C₄×C₃₁	C₂×C₄²	1	1	yes	characteristic
42	C₂×C₄×C₃₁	C₂²×C₈	1	2	yes	—
43	C₈×C₃₁	—	2	4	no	—
44	C₈×C₃₁	#97	1	4	yes	—
45	C₈×C₃₁	M₃₂	1	8	yes	complemented
46	C₈×C₃₁	C₂×M₁₆	1	4	yes	—
47	C₄×M₃₂	C₂×C₃₁	1	2	yes	—
48	C₂×C₈²	C₂×C₃₁	1	1	yes	characteristic
49	#1312	C₂×C₃₁	1	4	yes	complemented
50	C₂²×C₃₁	—	2	2	no	—
51	C₂²×C₃₁	#322	1	2	yes	—
52	C₂²×C₃₁	C₂×C₄×C₈	1	1	yes	characteristic
53	C₄×C₃₁	—	2	2	no	—
54	C₄×C₃₁	#322	1	2	yes	—
55	C₄×C₃₁	C₂×#97	1	2	yes	—
56	C₄×C₃₁	C₂×M₃₂	1	4	yes	—
57	C₂×M₃₂	C₄×C₃₁	1	4	yes	—
58	C₂×C₄×C₈	C₄×C₃₁	1	2	yes	—
59	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
60	#346	C₄×C₃₁	1	4	yes	—
61	C₄×C₁₆	C₂²×C₃₁	1	4	yes	—
62	C₈²	C₄×C₃₁	1	1	yes	characteristic
63	C₈²	C₄×C₃₁	1	1	yes	characteristic
64	C₈²	C₂²×C₃₁	1	2	yes	—
65	C₂×C₃₁	—	2	2	no	—
66	C₂×C₃₁	C₂×#322	1	1	yes	characteristic
67	C₂×C₃₁	C₄×M₃₂	1	2	yes	—
68	C₂²×C₈	C₈×C₃₁	1	4	yes	—
69	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
70	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
71	M₃₂	—	2	8	no	—
72	C₂×C₁₆	C₂×C₄×C₃₁	1	8	yes	—
73	C₄×C₈	C₈×C₃₁	1	4	yes	—
74	C₄×C₈	C₂×C₄×C₃₁	1	2	yes	—
75	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
76	C₄×C₈	C₂×C₄×C₃₁	1	2	yes	—
77	C₄×C₈	C₂×C₄×C₃₁	1	2	yes	—
78	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
79	C₃₁	#11402	1	1	yes	characteristic, direct product factor, Sylow
80	C₂²×C₄	C₄²×C₃₁	1	1	yes	characteristic
81	C₂²×C₄	C₂×C₈×C₃₁	1	2	yes	—
82	C₂×C₈	—	2	8	no	—
83	C₂×C₈	C₄²×C₃₁	1	2	yes	—
84	C₂×C₈	C₂×C₈×C₃₁	1	4	yes	—
85	C₂×C₈	C₂×C₈×C₃₁	1	4	yes	—
86	C₂×C₈	M₁₆×C₃₁	1	8	yes	—
87	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
88	C₄²	C₄²×C₃₁	1	1	yes	characteristic
89	C₄²	C₂×C₈×C₃₁	1	2	yes	—
90	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
91	C₁₆	—	2	8	no	—
92	C₂³	C₄×C₈×C₃₁	1	1	yes	characteristic
93	C₂×C₄	—	2	4	no	—
94	C₂×C₄	C₄×C₈×C₃₁	1	2	yes	—
95	C₂×C₄	C₄×C₈×C₃₁	1	1	yes	characteristic
96	C₂×C₄	C₃₁×#97	1	4	yes	—
97	C₂×C₄	C₂×C₄²×C₃₁	1	1	yes	characteristic
98	C₂×C₄	C₂²×C₈×C₃₁	1	2	yes	—
99	C₈	—	2	4	no	—
100	C₈	C₃₁×#97	1	4	yes	—
101	C₈	C₃₁×M₃₂	1	8	yes	complemented
102	C₈	C₂×M₁₆×C₃₁	1	4	yes	—
103	C₂²	—	2	2	no	—
104	C₂²	C₃₁×#322	1	2	yes	—
105	C₂²	C₂×C₄×C₈×C₃₁	1	1	yes	characteristic
106	C₄	—	2	2	no	—
107	C₄	C₃₁×#322	1	2	yes	—
108	C₄	C₂×C₃₁×#97	1	2	yes	—
109	C₄	C₂×C₃₁×M₃₂	1	4	yes	—
110	C₂	—	2	2	no	—
111	C₂	C₂×C₃₁×#322	1	1	yes	characteristic
112	C₂	C₄×C₃₁×M₃₂	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×C₃₁×M₃₂

C₂

yes

maximal

C₂×C₈²×C₃₁

C₂

yes

maximal, characteristic

C₃₁×#1312

C₂

yes

maximal, complemented

C₂×C₃₁×M₃₂

C₄

yes

—

C₂×C₄×C₈×C₃₁

C₄

yes

—

C₂×C₄×C₈×C₃₁

C₂²

yes

characteristic

C₃₁×#346

C₄

yes

—

C₄×C₁₆×C₃₁

C₂²

yes

—

C₈²×C₃₁

C₄

yes

characteristic

C₈²×C₃₁

C₄

yes

characteristic

C₈²×C₃₁

C₂²

yes

—

C₂²×C₈×C₃₁

C₈

yes

—

C₂²×C₈×C₃₁

C₂×C₄

yes

—

C₂×C₄²×C₃₁

C₂×C₄

yes

characteristic

C₃₁×M₃₂

—

C₂×C₁₆×C₃₁

C₂×C₄

yes

—

C₄×C₈×C₃₁

C₈

yes

—

C₄×C₈×C₃₁

C₂×C₄

yes

—

C₄×C₈×C₃₁

C₂×C₄

yes

characteristic

C₄×C₈×C₃₁

C₂×C₄

yes

—

C₄×C₈×C₃₁

C₂×C₄

yes

—

C₄×C₈×C₃₁

C₂³

yes

characteristic

C₂²×C₄×C₃₁

C₄²

yes

characteristic

C₂²×C₄×C₃₁

C₂×C₈

yes

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

C₄²

yes

—

C₂×C₈×C₃₁

C₂×C₈

yes

—

C₂×C₈×C₃₁

C₂×C₈

yes

—

C₂×C₈×C₃₁

M₁₆

yes

—

C₂×C₈×C₃₁

C₂²×C₄

yes

—

C₄²×C₃₁

C₄²

yes

characteristic

C₄²×C₃₁

C₂×C₈

yes

—

C₄²×C₃₁

C₂²×C₄

yes

characteristic

C₁₆×C₃₁

—

#11402

C₃₁

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₃₁

C₄×C₈

yes

characteristic

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₄×C₈

yes

—

C₂×C₄×C₃₁

C₄×C₈

yes

characteristic

C₂×C₄×C₃₁

#97

yes

—

C₂×C₄×C₃₁

C₂×C₄²

yes

characteristic

C₂×C₄×C₃₁

C₂²×C₈

yes

—

C₈×C₃₁

—

C₈×C₃₁

#97

yes

—

C₈×C₃₁

M₃₂

yes

complemented

C₈×C₃₁

C₂×M₁₆

yes

—

C₄×M₃₂

C₂×C₃₁

yes

—

C₂×C₈²

C₂×C₃₁

yes

characteristic

#1312

C₂×C₃₁

yes

complemented

C₂²×C₃₁

—

C₂²×C₃₁

#322

yes

—

C₂²×C₃₁

C₂×C₄×C₈

yes

characteristic

C₄×C₃₁

—

C₄×C₃₁

#322

yes

—

C₄×C₃₁

C₂×#97

yes

—

C₄×C₃₁

C₂×M₃₂

yes

—

C₂×M₃₂

C₄×C₃₁

yes

—

C₂×C₄×C₈

C₄×C₃₁

yes

—

C₂×C₄×C₈

C₂²×C₃₁

yes

characteristic

#346

C₄×C₃₁

yes

—

C₄×C₁₆

C₂²×C₃₁

yes

—

C₈²

C₄×C₃₁

yes

characteristic

C₈²

C₄×C₃₁

yes

characteristic

C₈²

C₂²×C₃₁

yes

—

C₂×C₃₁

—

C₂×C₃₁

C₂×#322

yes

characteristic

C₂×C₃₁

C₄×M₃₂

yes

—

C₂²×C₈

C₈×C₃₁

yes

—

C₂²×C₈

C₂×C₄×C₃₁

yes

—

C₂×C₄²

C₂×C₄×C₃₁

yes

characteristic

M₃₂

—

C₂×C₁₆

C₂×C₄×C₃₁

yes

—

C₄×C₈

C₈×C₃₁

yes

—

C₄×C₈

C₂×C₄×C₃₁

yes

—

C₄×C₈

C₂×C₄×C₃₁

yes

characteristic

C₄×C₈

C₂×C₄×C₃₁

yes

—

C₄×C₈

C₂×C₄×C₃₁

yes

—

C₄×C₈

C₂³×C₃₁

yes

characteristic

C₃₁

#11402

yes

characteristic, direct product factor, Sylow

C₂²×C₄

C₄²×C₃₁

yes

characteristic

C₂²×C₄

C₂×C₈×C₃₁

yes

—

C₂×C₈

—

C₂×C₈

C₄²×C₃₁

yes

—

C₂×C₈

C₂×C₈×C₃₁

yes

—

C₂×C₈

C₂×C₈×C₃₁

yes

—

C₂×C₈

M₁₆×C₃₁

yes

—

C₂×C₈

C₂²×C₄×C₃₁

yes

—

C₄²

C₄²×C₃₁

yes

characteristic

C₄²

C₂×C₈×C₃₁

yes

—

C₄²

C₂²×C₄×C₃₁

yes

characteristic

C₁₆

—

C₂³

C₄×C₈×C₃₁

yes

characteristic

C₂×C₄

—

C₂×C₄

C₄×C₈×C₃₁

yes

—

C₂×C₄

C₄×C₈×C₃₁

yes

characteristic

C₂×C₄

C₃₁×#97

yes

—

C₂×C₄

C₂×C₄²×C₃₁

yes

characteristic

C₂×C₄

C₂²×C₈×C₃₁

yes

—

C₈

—

100

C₈

C₃₁×#97

yes

—

101

C₈

C₃₁×M₃₂

yes

complemented

102

C₈

C₂×M₁₆×C₃₁

yes

—

103

C₂²

—

104

C₂²

C₃₁×#322

yes

—

105

C₂²

C₂×C₄×C₈×C₃₁

yes

characteristic

106

C₄

—

107

C₄

C₃₁×#322

yes

—

108

C₄

C₂×C₃₁×#97

yes

—

109

C₄

C₂×C₃₁×M₃₂

yes

—

110

C₂

—

111

C₂

C₂×C₃₁×#322

yes

characteristic

112

C₂

C₄×C₃₁×M₃₂

yes

—

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

subgroup

G/H

center

C₄×C₈×C₃₁

C₂³

commutator

C₂²

C₂×C₄×C₈×C₃₁

Frattini, cosocle

C₄×C₈

C₂³×C₃₁

socle

C₂²×C₃₁

C₂×C₄×C₈

prime radical

C₂³×C₃₁

C₄×C₈

LCS

G ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₄×C₈×C₃₁ ◁ G

N	H ≅ G/N	type
3. C₃₁×#1312	C₂	semidirect
35. #11402	C₃₁	direct (N × H)
45. C₈×C₃₁	M₃₂	semidirect
49. #1312	C₂×C₃₁	semidirect
79. C₃₁	#11402	direct (N × H)
101. C₈	C₃₁×M₃₂	semidirect

H ≅ G/N

type

3. C₃₁×#1312

C₂

semidirect

35. #11402

C₃₁

direct (N × H)

45. C₈×C₃₁

M₃₂

semidirect

49. #1312

C₂×C₃₁

semidirect

79. C₃₁

#11402

direct (N × H)

101. C₈

C₃₁×M₃₂

semidirect

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

subgroup	index
C₄×C₃₁×M₃₂	2
C₂×C₈²×C₃₁	2
C₃₁×#1312	2
#11402	31

subgroup

index

C₄×C₃₁×M₃₂

C₂×C₈²×C₃₁

C₃₁×#1312

#11402

Subgroups — C₃₁×#11402 #21468161

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C31×#11402 #21468161

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#11402 #21468161