Group #16847264 — subgroups

Subgroups — #16847264

Order 3968.

210 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 65 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	#14521098	C₂	1	1	yes	maximal, characteristic, complemented
2	#14521075	C₂	1	2	yes	maximal, complemented
3	#14521066	C₂	1	2	yes	maximal
4	#14520806	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#408	C₂	1	1	yes	maximal, characteristic
6	C₂×#41×C₃₁	C₂²	1	1	yes	characteristic
7	C₂×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
8	C₂²×C₈×C₃₁	C₂²	1	1	yes	characteristic
9	#11758503	C₂²	1	2	yes	—
10	C₂×#92474	C₂²	1	2	yes	—
11	C₃₁×C₂²⋊C₈	—	2	2	no	—
12	#11758425	—	2	4	no	—
13	#11758407	—	2	2	no	—
14	Dic₃₁⋊C₈	—	2	4	no	—
15	#41×C₃₁	—	2	2	no	—
16	#41×C₃₁	—	2	2	no	—
17	C₂×Q₈×C₃₁	Q₈	1	1	yes	characteristic
18	C₂×D₄×C₃₁	—	2	1	no	—
19	C₂×D₄×C₃₁	Q₈	1	1	yes	characteristic
20	C₂²×C₄×C₃₁	—	2	1	no	—
21	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
22	M₁₆×C₃₁	—	2	2	no	—
23	C₂×C₈×C₃₁	—	2	2	no	—
24	C₂×C₈×C₃₁	—	2	1	no	—
25	C₂×C₈×C₃₁	D₄	1	2	yes	—
26	C₂²⋊Dic₃₁	—	2	2	no	—
27	C₄⋊Dic₃₁	—	2	2	no	—
28	C₄×Dic₃₁	D₄	1	4	yes	—
29	#92474	—	2	2	no	—
30	#92474	—	2	2	no	—
31	C₂×C₃₁⋊C₈	—	2	2	no	—
32	C₂³×C₃₁	—	2	1	no	—
33	C₂³×C₃₁	#41	1	1	yes	characteristic
34	Q₈×C₃₁	—	2	2	no	—
35	D₄×C₃₁	—	2	2	no	—
36	D₄×C₃₁	—	4	2	no	—
37	C₂×C₄×C₃₁	—	2	1	no	—
38	C₂×C₄×C₃₁	—	4	1	no	—
39	C₂×C₄×C₃₁	—	4	1	no	—
40	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
41	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
43	C₂×C₄×C₃₁	#41	1	2	yes	—
44	C₂×Dic₃₁	—	2	4	no	—
45	C₈×C₃₁	—	2	2	no	—
46	C₈×C₃₁	—	4	1	no	—
47	C₃₁⋊C₈	—	4	2	no	—
48	#2098	—	31	1	no	maximal, Sylow
49	C₂²×C₃₁	—	4	1	no	—
50	C₂²×C₃₁	—	4	2	no	—
51	C₂²×C₃₁	—	4	1	no	—
52	C₂²×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
53	C₂²×C₃₁	#123	1	2	yes	—
54	C₄×C₃₁	—	4	1	no	—
55	C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
56	C₄×C₃₁	#123	1	1	yes	characteristic
57	C₄×C₃₁	#127	1	1	yes	characteristic
58	Dic₃₁	—	4	4	no	—
59	#430	—	31	1	no	—
60	#424	—	31	2	no	—
61	#421	—	31	2	no	—
62	#408	D₃₁	1	1	yes	characteristic
63	H(3,ℤ/4)	—	31	1	no	—
64	C₂×C₃₁	—	2	2	no	—
65	C₂×C₃₁	—	2	1	no	—
66	C₂×C₃₁	—	8	1	no	—
67	C₂×C₃₁	#399	1	1	yes	characteristic
68	C₂×#41	C₂×D₃₁	1	1	yes	characteristic
69	C₂×M₁₆	—	31	2	no	—
70	C₂×M₁₆	C₂×D₃₁	1	1	yes	characteristic
71	C₂²×C₈	C₂×D₃₁	1	1	yes	characteristic
72	#117	—	31	2	no	—
73	#108	—	62	2	no	—
74	C₄⋊C₈	—	62	4	no	—
75	#104	—	62	4	no	—
76	C₂²⋊C₈	—	2	2	no	—
77	C₃₁	#2098	1	1	yes	characteristic, complemented, Sylow
78	#41	—	2	2	no	—
79	#41	—	2	2	no	—
80	C₂×Q₈	Dic₆₂	1	1	yes	characteristic
81	C₂×D₄	—	2	1	no	—
82	C₂×D₄	Dic₆₂	1	1	yes	characteristic
83	C₂²×C₄	—	2	1	no	—
84	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
85	M₁₆	—	2	2	no	—
86	M₁₆	—	62	2	no	—
87	M₁₆	—	62	2	no	—
88	C₂×C₈	—	2	2	no	—
89	C₂×C₈	—	2	1	no	—
90	C₂×C₈	—	62	2	no	—
91	C₂×C₈	C₂²⋊D₃₁	1	2	yes	—
92	C₄⋊C₄	—	62	2	no	—
93	C₂²⋊C₄	—	62	2	no	—
94	C₄²	—	31	4	no	—
95	C₂³	—	2	1	no	—
96	C₂³	#92495	1	1	yes	characteristic
97	Q₈	—	2	2	no	—
98	D₄	—	2	2	no	—
99	D₄	—	4	2	no	—
100	C₂×C₄	—	2	1	no	—
101	C₂×C₄	—	4	1	no	—
102	C₂×C₄	—	4	1	no	—
103	C₂×C₄	—	62	4	no	—
104	C₂×C₄	C₂×Dic₆₂	1	1	yes	characteristic
105	C₂×C₄	D₄×D₃₁	1	2	yes	—
106	C₂×C₄	#92497	1	2	yes	—
107	C₂×C₄	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
108	C₈	—	2	2	no	—
109	C₈	—	4	1	no	—
110	C₈	—	124	2	no	—
111	C₂²	—	4	1	no	—
112	C₂²	—	4	2	no	—
113	C₂²	—	4	1	no	—
114	C₂²	#11758460	1	2	yes	—
115	C₂²	#11758501	1	1	yes	characteristic
116	C₄	—	4	1	no	—
117	C₄	—	124	4	no	—
118	C₄	#11758455	1	2	yes	—
119	C₄	#11758512	1	1	yes	characteristic
120	C₄	#11758517	1	1	yes	characteristic
121	C₂	—	2	2	no	—
122	C₂	—	2	1	no	—
123	C₂	—	8	1	no	—
124	C₂	#14520985	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14521098

C₂

yes

maximal, characteristic, complemented

#14521075

C₂

yes

maximal, complemented

#14521066

C₂

yes

maximal

#14520806

C₂

yes

maximal, characteristic, complemented

C₃₁×#408

C₂

yes

maximal, characteristic

C₂×#41×C₃₁

C₂²

yes

characteristic

C₂×M₁₆×C₃₁

C₂²

yes

characteristic

C₂²×C₈×C₃₁

C₂²

yes

characteristic

#11758503

C₂²

yes

—

C₂×#92474

C₂²

yes

—

C₃₁×C₂²⋊C₈

—

#11758425

—

#11758407

—

Dic₃₁⋊C₈

—

#41×C₃₁

—

#41×C₃₁

—

C₂×Q₈×C₃₁

Q₈

yes

characteristic

C₂×D₄×C₃₁

—

C₂×D₄×C₃₁

Q₈

yes

characteristic

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

C₂³

yes

characteristic

M₁₆×C₃₁

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

D₄

yes

—

C₂²⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₄×Dic₃₁

D₄

yes

—

#92474

—

#92474

—

C₂×C₃₁⋊C₈

—

C₂³×C₃₁

—

C₂³×C₃₁

#41

yes

characteristic

Q₈×C₃₁

—

D₄×C₃₁

—

D₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₂×D₄

yes

—

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₃₁

#41

yes

—

C₂×Dic₃₁

—

C₈×C₃₁

—

C₈×C₃₁

—

C₃₁⋊C₈

—

#2098

—

maximal, Sylow

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

C₂²⋊Q₈

yes

characteristic

C₂²×C₃₁

#123

yes

—

C₄×C₃₁

—

C₄×C₃₁

C₂²⋊Q₈

yes

—

C₄×C₃₁

yes

C₄×C₃₁

yes

Dic₃₁

—

—

—

—

D₃₁

yes

characteristic

H(3,ℤ/4)

—

C₂×C₃₁

—

C₂×C₃₁

—

C₂×C₃₁

—

C₂×C₃₁

#399

yes

characteristic

C₂×#41

C₂×D₃₁

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

C₂×D₃₁

yes

characteristic

C₂²×C₈

C₂×D₃₁

yes

characteristic

#117

—

#108

—

C₄⋊C₈

—

#104

—

C₂²⋊C₈

—

C₃₁

#2098

yes

characteristic, complemented, Sylow

#41

—

#41

—

C₂×Q₈

Dic₆₂

yes

characteristic

C₂×D₄

—

C₂×D₄

Dic₆₂

yes

characteristic

C₂²×C₄

—

C₂²×C₄

C₂²×D₃₁

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊D₃₁

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₂³

—

C₂³

#92495

yes

characteristic

Q₈

—

D₄

—

D₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

C₂×Dic₆₂

yes

characteristic

105

C₂×C₄

D₄×D₃₁

yes

—

106

C₂×C₄

#92497

yes

—

107

C₂×C₄

C₂×C₂²⋊D₃₁

yes

characteristic

108

C₈

—

109

C₈

—

110

C₈

—

124

—

111

C₂²

—

112

C₂²

—

113

C₂²

—

114

C₂²

#11758460

yes

—

115

C₂²

#11758501

yes

characteristic

116

C₄

—

117

C₄

—

124

—

118

C₄

#11758455

yes

—

119

C₄

#11758512

yes

characteristic

120

C₄

#11758517

yes

characteristic

121

C₂

—

122

C₂

—

123

C₂

—

124

C₂

#14520985

yes

characteristic

	subgroup	G/H
center	C₄	#11758517
commutator, cosocle	C₂²×C₄×C₃₁	C₂³
Frattini	C₂²×C₄	C₂²×D₃₁
Fitting	C₃₁×#408	C₂
socle	C₂×C₃₁	#399
prime radical	C₂×#41×C₃₁	C₂²
hypercenter	#408	D₃₁
hypocenter	C₃₁	#2098
LCS	G ▷ C₂²×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #408

subgroup

G/H

center

C₄

#11758517

commutator, cosocle

C₂²×C₄×C₃₁

C₂³

Frattini

C₂²×C₄

C₂²×D₃₁

Fitting

C₃₁×#408

C₂

socle

C₂×C₃₁

#399

prime radical

C₂×#41×C₃₁

C₂²

D₃₁

C₃₁

G ▷ C₂²×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #408

N	H ≅ G/N	type
1. #14521098	C₂	semidirect
2. #14521075	C₂	semidirect
4. #14520806	C₂	semidirect
77. C₃₁	#2098	semidirect

H ≅ G/N

type

1. #14521098

C₂

semidirect

2. #14521075

C₂

semidirect

4. #14520806

C₂

semidirect

77. C₃₁

#2098

semidirect

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

subgroup	index
#14521098	2
#14521075	2
#14521066	2
#14520806	2
C₃₁×#408	2
#2098	31

subgroup

index

C₃₁×#408

Subgroups — #16847264

Conjugacy classes of subgroups

Complements

Maximal subgroups