Group #14558162 — subgroups

Subgroups — #14558162

Order 2368.

206 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 103 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	#11924535	C₂	1	2	yes	maximal, complemented
2	#11924534	C₂	1	2	yes	maximal, complemented
3	#11924526	C₂	1	2	yes	maximal, complemented
4	#11924526	C₂	1	2	yes	maximal, complemented
5	#11924515	C₂	1	1	yes	maximal, characteristic
6	#11924513	C₂	1	2	yes	maximal
7	#11924492	C₂	1	2	yes	maximal
8	C₂×#10596936	C₂	1	1	yes	maximal, characteristic
9	C₂²⋊Q₈×C₃₇	C₂	1	1	yes	maximal, characteristic
10	C₂×Q₈×C₃₇	C₂²	1	1	yes	characteristic
11	C₂²×C₄×C₃₇	C₂²	1	1	yes	characteristic
12	C₂²×Dic₃₇	C₂²	1	1	yes	characteristic
13	C₂²×Dic₃₇	C₂²	1	1	yes	characteristic
14	C₂×Dic₇₄	C₂²	1	2	yes	—
15	C₂×Dic₇₄	C₂²	1	1	yes	characteristic
16	C₄⋊C₄×C₃₇	C₂²	1	1	yes	characteristic
17	C₄⋊C₄×C₃₇	C₂²	1	2	yes	—
18	C₂²⋊C₄×C₃₇	C₂²	1	2	yes	—
19	C₂²⋊Dic₃₇	C₂²	1	2	yes	—
20	C₄⋊Dic₃₇	C₂²	1	2	yes	—
21	C₄⋊Dic₃₇	C₂²	1	1	yes	characteristic
22	#10596936	C₂²	1	2	yes	—
23	#10596936	C₂²	1	2	yes	—
24	#10596936	C₂²	1	2	yes	—
25	#10596936	C₂²	1	2	yes	—
26	#10596936	C₂²	1	2	yes	—
27	#10596936	C₂²	1	2	yes	—
28	#10596936	C₂²	1	2	yes	—
29	C₄×Dic₃₇	C₂²	1	2	yes	—
30	C₄×Dic₃₇	C₂²	1	2	yes	—
31	C₂³×C₃₇	C₂³	1	1	yes	characteristic
32	Q₈×C₃₇	—	4	1	no	—
33	C₂×C₄×C₃₇	—	4	1	no	—
34	C₂×C₄×C₃₇	C₂³	1	2	yes	—
35	C₂×C₄×C₃₇	C₂³	1	2	yes	—
36	C₂×C₄×C₃₇	C₂³	1	1	yes	characteristic
37	C₂×C₄×C₃₇	C₂³	1	1	yes	characteristic
38	C₂×Dic₃₇	—	4	1	no	—
39	C₂×Dic₃₇	Q₈	1	4	yes	complemented
40	C₂×Dic₃₇	C₂³	1	2	yes	—
41	C₂×Dic₃₇	C₂³	1	2	yes	—
42	C₂×Dic₃₇	C₂³	1	2	yes	—
43	C₂×Dic₃₇	C₂³	1	2	yes	—
44	Dic₇₄	—	4	1	no	—
45	Dic₇₄	—	4	2	no	—
46	C₂²×C₃₇	—	2	1	no	—
47	C₂²×C₃₇	—	2	1	no	—
48	C₂²×C₃₇	C₂×Q₈	1	2	yes	—
49	C₂²×C₃₇	C₂⁴	1	1	yes	characteristic
50	C₄×C₃₇	—	2	1	no	—
51	C₄×C₃₇	—	2	2	no	—
52	C₄×C₃₇	—	2	1	no	—
53	C₄×C₃₇	—	2	2	no	—
54	Dic₃₇	—	2	2	no	—
55	Dic₃₇	—	2	2	no	—
56	Dic₃₇	—	2	2	no	—
57	Dic₃₇	C₂×Q₈	1	4	yes	—
58	C₂×C₃₇	—	2	2	no	—
59	C₂×C₃₇	C₂²×Q₈	1	1	yes	characteristic
60	C₂×C₃₇	H(5,𝔽₂)	1	1	yes	characteristic
61	C₂×C₃₇	2₋¹⁺⁴	1	1	yes	characteristic
62	#544	—	37	1	no	maximal, Sylow
63	C₃₇	#544	1	1	yes	characteristic, complemented, Sylow
64	C₄⋊Q₈	—	37	2	no	—
65	C₄⋊Q₈	—	37	2	no	—
66	#125	—	37	2	no	—
67	#125	—	37	2	no	—
68	C₂²⋊Q₈	—	37	1	no	—
69	C₂²⋊Q₈	—	37	2	no	—
70	C₂²⋊Q₈	D₃₇	1	1	yes	characteristic
71	#117	—	37	2	no	—
72	C₂×C₄⋊C₄	—	37	1	no	—
73	C₂×Q₈	—	37	1	no	—
74	C₂×Q₈	—	37	2	no	—
75	C₂×Q₈	C₂×D₃₇	1	1	yes	characteristic
76	C₂²×C₄	—	37	1	no	—
77	C₂²×C₄	—	37	1	no	—
78	C₂²×C₄	C₂×D₃₇	1	1	yes	characteristic
79	C₄⋊C₄	—	37	2	no	—
80	C₄⋊C₄	—	37	2	no	—
81	C₄⋊C₄	—	37	2	no	—
82	C₄⋊C₄	—	37	2	no	—
83	C₄⋊C₄	—	37	2	no	—
84	C₄⋊C₄	—	37	2	no	—
85	C₄⋊C₄	—	37	2	no	—
86	C₄⋊C₄	—	37	1	no	—
87	C₄⋊C₄	—	37	2	no	—
88	C₄⋊C₄	C₂×D₃₇	1	1	yes	characteristic
89	C₄⋊C₄	C₂×D₃₇	1	2	yes	—
90	C₂²⋊C₄	—	37	2	no	—
91	C₂²⋊C₄	C₂×D₃₇	1	2	yes	—
92	C₄²	—	37	2	no	—
93	C₄²	—	37	2	no	—
94	C₂³	C₂²×D₃₇	1	1	yes	characteristic
95	Q₈	—	4	1	no	—
96	Q₈	—	148	1	no	—
97	Q₈	—	148	2	no	—
98	C₂×C₄	—	4	1	no	—
99	C₂×C₄	—	37	4	no	—
100	C₂×C₄	—	37	2	no	—
101	C₂×C₄	—	37	2	no	—
102	C₂×C₄	—	37	2	no	—
103	C₂×C₄	—	37	2	no	—
104	C₂×C₄	—	148	1	no	—
105	C₂×C₄	C₂²×D₃₇	1	2	yes	—
106	C₂×C₄	C₂²×D₃₇	1	2	yes	—
107	C₂×C₄	C₂²×D₃₇	1	1	yes	characteristic
108	C₂×C₄	C₂²×D₃₇	1	1	yes	characteristic
109	C₂²	—	2	1	no	—
110	C₂²	—	2	1	no	—
111	C₂²	Q₈×D₃₇	1	2	yes	—
112	C₂²	C₂³×D₃₇	1	1	yes	characteristic
113	C₄	—	2	1	no	—
114	C₄	—	2	1	no	—
115	C₄	—	2	2	no	—
116	C₄	—	2	2	no	—
117	C₄	—	37	4	no	—
118	C₄	—	74	2	no	—
119	C₄	—	74	2	no	—
120	C₄	—	74	2	no	—
121	C₂	—	2	2	no	—
122	C₂	C₂×Q₈×D₃₇	1	1	yes	characteristic
123	C₂	#11924581	1	1	yes	characteristic
124	C₂	#11924583	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11924535

C₂

yes

maximal, complemented

#11924534

C₂

yes

maximal, complemented

#11924526

C₂

yes

maximal, complemented

#11924526

C₂

yes

maximal, complemented

#11924515

C₂

yes

maximal, characteristic

#11924513

C₂

yes

maximal

#11924492

C₂

yes

maximal

C₂×#10596936

C₂

yes

maximal, characteristic

C₂²⋊Q₈×C₃₇

C₂

yes

maximal, characteristic

C₂×Q₈×C₃₇

C₂²

yes

characteristic

C₂²×C₄×C₃₇

C₂²

yes

characteristic

C₂²×Dic₃₇

C₂²

yes

characteristic

C₂²×Dic₃₇

C₂²

yes

characteristic

C₂×Dic₇₄

C₂²

yes

—

C₂×Dic₇₄

C₂²

yes

characteristic

C₄⋊C₄×C₃₇

C₂²

yes

characteristic

C₄⋊C₄×C₃₇

C₂²

yes

—

C₂²⋊C₄×C₃₇

C₂²

yes

—

C₂²⋊Dic₃₇

C₂²

yes

—

C₄⋊Dic₃₇

C₂²

yes

—

C₄⋊Dic₃₇

C₂²

yes

characteristic

#10596936

C₂²

yes

—

#10596936

C₂²

yes

—

#10596936

C₂²

yes

—

#10596936

C₂²

yes

—

#10596936

C₂²

yes

—

#10596936

C₂²

yes

—

#10596936

C₂²

yes

—

C₄×Dic₃₇

C₂²

yes

—

C₄×Dic₃₇

C₂²

yes

—

C₂³×C₃₇

C₂³

yes

characteristic

Q₈×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂³

yes

—

C₂×C₄×C₃₇

C₂³

yes

—

C₂×C₄×C₃₇

C₂³

yes

characteristic

C₂×C₄×C₃₇

C₂³

yes

characteristic

C₂×Dic₃₇

—

C₂×Dic₃₇

Q₈

yes

complemented

C₂×Dic₃₇

C₂³

yes

—

C₂×Dic₃₇

C₂³

yes

—

C₂×Dic₃₇

C₂³

yes

—

C₂×Dic₃₇

C₂³

yes

—

Dic₇₄

—

Dic₇₄

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

C₂×Q₈

yes

—

C₂²×C₃₇

C₂⁴

yes

characteristic

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

—

Dic₃₇

—

Dic₃₇

—

Dic₃₇

—

Dic₃₇

C₂×Q₈

yes

—

C₂×C₃₇

—

C₂×C₃₇

C₂²×Q₈

yes

characteristic

C₂×C₃₇

H(5,𝔽₂)

yes

characteristic

C₂×C₃₇

2₋¹⁺⁴

yes

—

C₃₇

yes

characteristic, complemented, Sylow

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₂²⋊Q₈

D₃₇

yes

characteristic

#117

—

C₂×C₄⋊C₄

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×D₃₇

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₃₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₃₇

yes

characteristic

C₄⋊C₄

C₂×D₃₇

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

C₂×D₃₇

yes

—

C₄²

—

C₄²

—

C₂³

C₂²×D₃₇

yes

characteristic

Q₈

—

Q₈

—

148

—

Q₈

—

148

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

148

—

105

C₂×C₄

C₂²×D₃₇

yes

—

106

C₂×C₄

C₂²×D₃₇

yes

—

107

C₂×C₄

C₂²×D₃₇

yes

characteristic

108

C₂×C₄

C₂²×D₃₇

yes

characteristic

109

C₂²

—

110

C₂²

—

111

C₂²

Q₈×D₃₇

yes

—

112

C₂²

C₂³×D₃₇

yes

characteristic

113

C₄

—

114

C₄

—

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₂

—

122

C₂

C₂×Q₈×D₃₇

yes

characteristic

123

C₂

#11924581

yes

characteristic

124

C₂

#11924583

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃₇
commutator, socle, cosocle	C₂²×C₃₇	C₂⁴
Fitting	C₂²⋊Q₈×C₃₇	C₂
prime radical	C₂³×C₃₇	C₂³
hypercenter	C₂²⋊Q₈	D₃₇
hypocenter	C₃₇	#544
LCS	G ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₂²⋊Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₃₇

commutator, socle, cosocle

C₂²×C₃₇

C₂⁴

Fitting

C₂²⋊Q₈×C₃₇

C₂

prime radical

C₂³×C₃₇

C₂³

hypercenter

C₂²⋊Q₈

D₃₇

C₃₇

G ▷ C₂²×C₃₇ ▷ C₃₇

C₁ ◁ C₂² ◁ C₂²⋊Q₈

N	H ≅ G/N	type
1. #11924535	C₂	semidirect
2. #11924534	C₂	semidirect
3. #11924526	C₂	semidirect
4. #11924526	C₂	semidirect
39. C₂×Dic₃₇	Q₈	semidirect
63. C₃₇	#544	semidirect

H ≅ G/N

type

1. #11924535

C₂

semidirect

2. #11924534

C₂

semidirect

3. #11924526

C₂

semidirect

4. #11924526

C₂

semidirect

39. C₂×Dic₃₇

Q₈

semidirect

63. C₃₇

#544

semidirect

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

subgroup	index
#11924535	2
#11924534	2
#11924526	2
#11924515	2
#11924513	2
#11924492	2
C₂×#10596936	2
C₂²⋊Q₈×C₃₇	2
#544	37

subgroup

index

C₂²⋊Q₈×C₃₇

Subgroups — #14558162

Conjugacy classes of subgroups

Complements

Maximal subgroups