Group #16847322 — subgroups

Subgroups — #16847322

Order 3968.

260 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 117 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	C₂×#11758400	C₂	1	2	yes	maximal, complemented
2	C₂²⋊(C₃₁⋊C₁₆)	C₂	1	4	yes	maximal, complemented
3	C₂³×C₈×C₃₁	C₂	1	1	yes	maximal, characteristic
4	C₂³×C₄×C₃₁	C₄	1	1	yes	characteristic
5	C₂²×C₈×C₃₁	—	2	4	no	—
6	C₂²×C₈×C₃₁	C₄	1	1	yes	characteristic
7	C₂²×C₈×C₃₁	C₄	1	2	yes	—
8	C₂²×C₈×C₃₁	C₂²	1	2	yes	—
9	C₂²×C₈×C₃₁	C₂²	1	1	yes	characteristic
10	#11758400	—	2	4	no	—
11	C₂×C₃₁⋊C₁₆	C₂²	1	4	yes	complemented
12	C₂⁴×C₃₁	C₈	1	1	yes	characteristic
13	C₂²×C₄×C₃₁	—	2	2	no	—
14	C₂²×C₄×C₃₁	—	2	2	no	—
15	C₂²×C₄×C₃₁	C₈	1	1	yes	characteristic
16	C₂²×C₄×C₃₁	C₈	1	2	yes	—
17	C₂²×C₄×C₃₁	C₂×C₄	1	2	yes	—
18	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
19	C₂×C₈×C₃₁	—	2	2	no	—
20	C₂×C₈×C₃₁	—	2	4	no	—
21	C₂×C₈×C₃₁	—	2	4	no	—
22	C₂×C₈×C₃₁	C₂×C₄	1	2	yes	—
23	C₂×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈×C₃₁	D₄	1	4	yes	—
25	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
26	C₃₁⋊C₁₆	—	2	4	no	—
27	C₂³×C₃₁	—	2	2	no	—
28	C₂³×C₃₁	—	2	4	no	—
29	C₂³×C₃₁	C₂×C₈	1	1	yes	characteristic
30	C₂³×C₃₁	C₂×C₈	1	2	yes	—
31	C₂×C₄×C₃₁	—	2	2	no	—
32	C₂×C₄×C₃₁	—	2	4	no	—
33	C₂×C₄×C₃₁	—	2	2	no	—
34	C₂×C₄×C₃₁	—	2	2	no	—
35	C₂×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
36	C₂×C₄×C₃₁	C₂×C₈	1	2	yes	—
37	C₂×C₄×C₃₁	C₂×C₈	1	1	yes	characteristic
38	C₂×C₄×C₃₁	M₁₆	1	2	yes	—
39	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
40	C₈×C₃₁	—	2	2	no	—
41	C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
42	C₈×C₃₁	C₂×D₄	1	2	yes	—
43	#2112	—	31	1	no	maximal, Sylow
44	C₂²×C₃₁	—	2	8	no	—
45	C₂²×C₃₁	—	2	2	no	—
46	C₂²×C₃₁	—	2	2	no	—
47	C₂²×C₃₁	—	2	2	no	—
48	C₂²×C₃₁	C₂²⋊C₈	1	2	yes	—
49	C₂²×C₃₁	M₃₂	1	4	yes	complemented
50	C₂²×C₃₁	C₂²×C₈	1	1	yes	characteristic
51	C₄×C₃₁	—	2	2	no	—
52	C₄×C₃₁	C₂²⋊C₈	1	2	yes	—
53	C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
54	C₄×C₃₁	C₂×M₁₆	1	1	yes	characteristic
55	C₂³×C₈	D₃₁	1	1	yes	characteristic
56	C₂×M₃₂	—	31	2	no	—
57	C₂²⋊C₁₆	—	31	4	no	—
58	C₂×C₃₁	—	2	2	no	—
59	C₂×C₃₁	—	2	4	no	—
60	C₂×C₃₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
61	C₂×C₃₁	C₂×M₃₂	1	2	yes	—
62	C₂³×C₄	Dic₃₁	1	1	yes	characteristic
63	C₂²×C₈	—	2	4	no	—
64	C₂²×C₈	Dic₃₁	1	1	yes	characteristic
65	C₂²×C₈	Dic₃₁	1	2	yes	—
66	C₂²×C₈	C₂×D₃₁	1	2	yes	—
67	C₂²×C₈	C₂×D₃₁	1	1	yes	characteristic
68	M₃₂	—	62	4	no	—
69	C₂×C₁₆	—	31	4	no	—
70	C₃₁	#2112	1	1	yes	characteristic, complemented, Sylow
71	C₂⁴	C₃₁⋊C₈	1	1	yes	characteristic
72	C₂²×C₄	—	2	2	no	—
73	C₂²×C₄	—	2	2	no	—
74	C₂²×C₄	C₃₁⋊C₈	1	1	yes	characteristic
75	C₂²×C₄	C₃₁⋊C₈	1	2	yes	—
76	C₂²×C₄	C₂×Dic₃₁	1	2	yes	—
77	C₂²×C₄	C₂×Dic₃₁	1	1	yes	characteristic
78	C₂×C₈	—	2	2	no	—
79	C₂×C₈	—	2	4	no	—
80	C₂×C₈	—	2	4	no	—
81	C₂×C₈	C₂×Dic₃₁	1	2	yes	—
82	C₂×C₈	C₂×Dic₃₁	1	1	yes	characteristic
83	C₂×C₈	C₂²⋊D₃₁	1	4	yes	—
84	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
85	C₁₆	—	62	4	no	—
86	C₂³	—	2	4	no	—
87	C₂³	—	2	2	no	—
88	C₂³	C₂×C₃₁⋊C₈	1	2	yes	—
89	C₂³	C₂×C₃₁⋊C₈	1	1	yes	characteristic
90	C₂×C₄	—	2	2	no	—
91	C₂×C₄	—	2	4	no	—
92	C₂×C₄	—	2	2	no	—
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	C₂×C₃₁⋊C₈	1	2	yes	—
95	C₂×C₄	C₂×C₃₁⋊C₈	1	1	yes	characteristic
96	C₂×C₄	#92474	1	2	yes	—
97	C₂×C₄	C₂²⋊Dic₃₁	1	2	yes	—
98	C₂×C₄	C₂²×Dic₃₁	1	1	yes	characteristic
99	C₈	—	2	2	no	—
100	C₈	C₂²⋊Dic₃₁	1	2	yes	—
101	C₈	C₂×C₂²⋊D₃₁	1	2	yes	—
102	C₂²	—	2	8	no	—
103	C₂²	—	2	2	no	—
104	C₂²	—	2	2	no	—
105	C₂²	—	2	2	no	—
106	C₂²	#11758400	1	4	yes	complemented
107	C₂²	C₂²⋊(C₃₁⋊C₈)	1	2	yes	—
108	C₂²	C₂²×C₃₁⋊C₈	1	1	yes	characteristic
109	C₄	—	2	2	no	—
110	C₄	C₂²⋊(C₃₁⋊C₈)	1	2	yes	—
111	C₄	C₂×#92474	1	1	yes	characteristic
112	C₄	C₂×C₂²⋊Dic₃₁	1	1	yes	characteristic
113	C₂	—	2	4	no	—
114	C₂	—	2	2	no	—
115	C₂	C₂×C₂²⋊(C₃₁⋊C₈)	1	1	yes	characteristic
116	C₂	C₂×#11758400	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11758400

C₂

yes

maximal, complemented

C₂²⋊(C₃₁⋊C₁₆)

C₂

yes

maximal, complemented

C₂³×C₈×C₃₁

C₂

yes

maximal, characteristic

C₂³×C₄×C₃₁

C₄

yes

characteristic

C₂²×C₈×C₃₁

—

C₂²×C₈×C₃₁

C₄

yes

characteristic

C₂²×C₈×C₃₁

C₄

yes

—

C₂²×C₈×C₃₁

C₂²

yes

—

C₂²×C₈×C₃₁

C₂²

yes

characteristic

#11758400

—

C₂×C₃₁⋊C₁₆

C₂²

yes

complemented

C₂⁴×C₃₁

C₈

yes

characteristic

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

C₈

yes

characteristic

C₂²×C₄×C₃₁

C₈

yes

—

C₂²×C₄×C₃₁

C₂×C₄

yes

—

C₂²×C₄×C₃₁

C₂×C₄

yes

characteristic

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

C₂×C₄

yes

—

C₂×C₈×C₃₁

C₂×C₄

yes

characteristic

C₂×C₈×C₃₁

D₄

yes

—

C₂×C₈×C₃₁

C₂³

yes

characteristic

C₃₁⋊C₁₆

—

C₂³×C₃₁

—

C₂³×C₃₁

—

C₂³×C₃₁

C₂×C₈

yes

characteristic

C₂³×C₃₁

C₂×C₈

yes

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₂²⋊C₄

yes

—

C₂×C₄×C₃₁

C₂×C₈

yes

—

C₂×C₄×C₃₁

C₂×C₈

yes

characteristic

C₂×C₄×C₃₁

M₁₆

yes

—

C₂×C₄×C₃₁

C₂²×C₄

yes

characteristic

C₈×C₃₁

—

C₈×C₃₁

C₂²⋊C₄

yes

—

C₈×C₃₁

C₂×D₄

yes

—

#2112

—

maximal, Sylow

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

C₂²⋊C₈

yes

—

C₂²×C₃₁

M₃₂

yes

complemented

C₂²×C₃₁

C₂²×C₈

yes

characteristic

C₄×C₃₁

—

C₄×C₃₁

C₂²⋊C₈

yes

—

C₄×C₃₁

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₃₁

C₂×M₁₆

yes

characteristic

C₂³×C₈

D₃₁

yes

characteristic

C₂×M₃₂

—

C₂²⋊C₁₆

—

C₂×C₃₁

—

C₂×C₃₁

—

C₂×C₃₁

C₂×C₂²⋊C₈

yes

characteristic

C₂×C₃₁

C₂×M₃₂

yes

—

C₂³×C₄

Dic₃₁

yes

characteristic

C₂²×C₈

—

C₂²×C₈

Dic₃₁

yes

characteristic

C₂²×C₈

Dic₃₁

yes

—

C₂²×C₈

C₂×D₃₁

yes

—

C₂²×C₈

C₂×D₃₁

yes

characteristic

M₃₂

—

C₂×C₁₆

—

C₃₁

#2112

yes

characteristic, complemented, Sylow

C₂⁴

C₃₁⋊C₈

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₃₁⋊C₈

yes

characteristic

C₂²×C₄

C₃₁⋊C₈

yes

—

C₂²×C₄

C₂×Dic₃₁

yes

—

C₂²×C₄

C₂×Dic₃₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×Dic₃₁

yes

—

C₂×C₈

C₂×Dic₃₁

yes

characteristic

C₂×C₈

C₂²⋊D₃₁

yes

—

C₂×C₈

C₂²×D₃₁

yes

characteristic

C₁₆

—

C₂³

—

C₂³

—

C₂³

C₂×C₃₁⋊C₈

yes

—

C₂³

C₂×C₃₁⋊C₈

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₃₁⋊C₈

yes

—

C₂×C₄

C₂×C₃₁⋊C₈

yes

characteristic

C₂×C₄

#92474

yes

—

C₂×C₄

C₂²⋊Dic₃₁

yes

—

C₂×C₄

C₂²×Dic₃₁

yes

characteristic

C₈

—

100

C₈

C₂²⋊Dic₃₁

yes

—

101

C₈

C₂×C₂²⋊D₃₁

yes

—

102

C₂²

—

103

C₂²

—

104

C₂²

—

105

C₂²

—

106

C₂²

#11758400

yes

complemented

107

C₂²

C₂²⋊(C₃₁⋊C₈)

yes

—

108

C₂²

C₂²×C₃₁⋊C₈

yes

characteristic

109

C₄

—

110

C₄

C₂²⋊(C₃₁⋊C₈)

yes

—

111

C₄

C₂×#92474

yes

characteristic

112

C₄

C₂×C₂²⋊Dic₃₁

yes

characteristic

113

C₂

—

114

C₂

—

115

C₂

C₂×C₂²⋊(C₃₁⋊C₈)

yes

characteristic

116

C₂

C₂×#11758400

yes

—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₃₁
commutator, socle	C₂²×C₃₁	C₂²×C₈
Fitting	C₂³×C₈×C₃₁	C₂
prime radical	C₂⁴×C₃₁	C₈
cosocle	C₂×C₈×C₃₁	C₂³
hypercenter	C₂³×C₈	D₃₁
hypocenter	C₃₁	#2112
LCS	G ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×D₃₁

commutator, socle

C₂²×C₃₁

C₂²×C₈

Fitting

C₂³×C₈×C₃₁

C₂

prime radical

C₂⁴×C₃₁

C₈

cosocle

C₂×C₈×C₃₁

C₂³

hypercenter

C₂³×C₈

D₃₁

C₃₁

G ▷ C₂²×C₃₁ ▷ C₃₁

C₁ ◁ C₂×C₈ ◁ C₂³×C₈

N	H ≅ G/N	type
1. C₂×#11758400	C₂	semidirect
2. C₂²⋊(C₃₁⋊C₁₆)	C₂	semidirect
11. C₂×C₃₁⋊C₁₆	C₂²	semidirect
49. C₂²×C₃₁	M₃₂	semidirect
70. C₃₁	#2112	semidirect
106. C₂²	#11758400	semidirect

H ≅ G/N

type

1. C₂×#11758400

C₂

semidirect

2. C₂²⋊(C₃₁⋊C₁₆)

C₂

semidirect

11. C₂×C₃₁⋊C₁₆

C₂²

semidirect

49. C₂²×C₃₁

M₃₂

semidirect

70. C₃₁

#2112

semidirect

106. C₂²

#11758400

semidirect

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

subgroup	index
C₂×#11758400	2
C₂²⋊(C₃₁⋊C₁₆)	2
C₂³×C₈×C₃₁	2
#2112	31

subgroup

index

C₂×#11758400

C₂²⋊(C₃₁⋊C₁₆)

C₂³×C₈×C₃₁

#2112

Subgroups — #16847322

Conjugacy classes of subgroups

Complements

Maximal subgroups