Group #14558132 — subgroups

Subgroups — #14558132

Order 2368.

218 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 95 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	#11924531	C₂	1	2	yes	maximal, complemented
2	#11924527	C₂	1	2	yes	maximal, complemented
3	#11924521	C₂	1	2	yes	maximal, complemented
4	#11924520	C₂	1	2	yes	maximal, complemented
5	#11924512	C₂	1	2	yes	maximal, complemented
6	#11924510	C₂	1	1	yes	maximal, characteristic
7	#11924496	C₂	1	2	yes	maximal, complemented
8	C₂×#10596936	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²⋊Q₈×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×Q₈×C₃₇	C₂²	1	1	yes	characteristic
11	C₂²×C₄×C₃₇	C₂²	1	1	yes	characteristic
12	C₂×C₂²⋊D₃₇	C₂²	1	2	yes	—
13	C₂²×Dic₃₇	C₂²	1	1	yes	characteristic
14	C₂²×Dic₃₇	C₂²	1	1	yes	characteristic
15	C₂×D₁₄₈	C₂²	1	1	yes	characteristic
16	C₂×C₄×D₃₇	C₂²	1	2	yes	—
17	C₄⋊C₄×C₃₇	C₂²	1	2	yes	—
18	C₄⋊C₄×C₃₇	C₂²	1	1	yes	characteristic
19	C₂²⋊C₄×C₃₇	C₂²	1	2	yes	—
20	#10596938	C₂²	1	2	yes	—
21	#10596938	C₂²	1	2	yes	—
22	#10596938	C₂²	1	2	yes	—
23	#10596938	C₂²	1	2	yes	—
24	C₄⋊Dic₃₇	C₂²	1	1	yes	characteristic
25	C₄⋊Dic₃₇	C₂²	1	2	yes	—
26	C₄⋊Dic₃₇	C₂²	1	2	yes	—
27	#10596936	C₂²	1	2	yes	—
28	#10596936	C₂²	1	2	yes	—
29	#10596936	C₂²	1	2	yes	—
30	C₄×Dic₃₇	C₂²	1	2	yes	—
31	C₂³×C₃₇	C₂³	1	1	yes	characteristic
32	C₂²×D₃₇	C₂³	1	2	yes	—
33	Q₈×C₃₇	—	4	1	no	—
34	C₂×C₄×C₃₇	—	4	1	no	—
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×C₄×C₃₇	C₂³	1	2	yes	—
39	C₂²⋊D₃₇	—	2	4	no	—
40	C₂×Dic₃₇	—	2	2	no	—
41	C₂×Dic₃₇	—	4	1	no	—
42	C₂×Dic₃₇	C₂³	1	2	yes	—
43	C₂×Dic₃₇	C₂³	1	2	yes	—
44	C₂×Dic₃₇	C₂³	1	2	yes	—
45	D₁₄₈	—	4	1	no	—
46	C₄×D₃₇	—	4	2	no	—
47	C₂²×C₃₇	—	2	1	no	—
48	C₂²×C₃₇	—	2	1	no	—
49	C₂²×C₃₇	#41	1	2	yes	—
50	C₂²×C₃₇	C₂⁴	1	1	yes	characteristic
51	C₂×D₃₇	—	2	2	no	—
52	C₂×D₃₇	—	2	2	no	—
53	C₂×D₃₇	—	2	2	no	—
54	C₄×C₃₇	—	2	2	no	—
55	C₄×C₃₇	—	2	1	no	—
56	C₄×C₃₇	—	2	1	no	—
57	C₄×C₃₇	—	2	2	no	—
58	Dic₃₇	—	2	2	no	—
59	Dic₃₇	—	2	2	no	—
60	Dic₃₇	—	2	2	no	—
61	C₂×C₃₇	—	2	2	no	—
62	C₂×C₃₇	C₂×#41	1	1	yes	characteristic
63	C₂×C₃₇	H(5,𝔽₂)	1	1	yes	characteristic
64	C₂×C₃₇	2₋¹⁺⁴	1	1	yes	characteristic
65	D₃₇	—	4	2	no	—
66	#539	—	37	1	no	maximal, Sylow
67	C₃₇	#539	1	1	yes	characteristic, complemented, Sylow
68	#126	—	37	2	no	—
69	#125	—	37	2	no	—
70	#123	—	37	2	no	—
71	#123	—	37	2	no	—
72	C₂²⋊Q₈	—	37	2	no	—
73	C₂²⋊Q₈	D₃₇	1	1	yes	characteristic, complemented
74	#121	—	37	1	no	—
75	C₄×D₄	—	37	2	no	—
76	C₂×C₄⋊C₄	—	37	1	no	—
77	C₂×Q₈	C₂×D₃₇	1	1	yes	characteristic
78	C₂×D₄	—	37	2	no	—
79	C₂×D₄	—	37	1	no	—
80	C₂²×C₄	—	37	1	no	—
81	C₂²×C₄	—	37	1	no	—
82	C₂²×C₄	—	37	2	no	—
83	C₂²×C₄	C₂×D₃₇	1	1	yes	characteristic
84	C₄⋊C₄	—	37	2	no	—
85	C₄⋊C₄	—	37	2	no	—
86	C₄⋊C₄	—	37	2	no	—
87	C₄⋊C₄	—	37	1	no	—
88	C₄⋊C₄	—	37	2	no	—
89	C₄⋊C₄	—	37	2	no	—
90	C₄⋊C₄	C₂×D₃₇	1	2	yes	—
91	C₄⋊C₄	C₂×D₃₇	1	1	yes	characteristic
92	C₂²⋊C₄	—	37	2	no	—
93	C₂²⋊C₄	—	37	2	no	—
94	C₂²⋊C₄	—	37	2	no	—
95	C₂²⋊C₄	—	37	2	no	—
96	C₂²⋊C₄	C₂×D₃₇	1	2	yes	—
97	C₄²	—	37	2	no	—
98	C₂³	—	37	2	no	—
99	C₂³	C₂²×D₃₇	1	1	yes	characteristic
100	Q₈	—	4	1	no	—
101	D₄	—	74	4	no	—
102	D₄	—	148	1	no	—
103	C₂×C₄	—	4	1	no	—
104	C₂×C₄	—	37	2	no	—
105	C₂×C₄	—	37	2	no	—
106	C₂×C₄	—	37	2	no	—
107	C₂×C₄	—	74	2	no	—
108	C₂×C₄	—	148	2	no	—
109	C₂×C₄	—	148	1	no	—
110	C₂×C₄	C₂²×D₃₇	1	2	yes	—
111	C₂×C₄	C₂²×D₃₇	1	1	yes	characteristic
112	C₂×C₄	C₂²×D₃₇	1	2	yes	—
113	C₂×C₄	C₂²×D₃₇	1	1	yes	characteristic
114	C₂²	—	2	1	no	—
115	C₂²	—	2	1	no	—
116	C₂²	—	74	2	no	—
117	C₂²	—	74	2	no	—
118	C₂²	—	74	2	no	—
119	C₂²	#10596966	1	2	yes	—
120	C₂²	C₂³×D₃₇	1	1	yes	characteristic
121	C₄	—	2	1	no	—
122	C₄	—	2	1	no	—
123	C₄	—	2	2	no	—
124	C₄	—	2	2	no	—
125	C₄	—	74	2	no	—
126	C₄	—	74	2	no	—
127	C₄	—	74	2	no	—
128	C₂	—	2	2	no	—
129	C₂	—	148	2	no	—
130	C₂	C₂×#10596966	1	1	yes	characteristic
131	C₂	#11924581	1	1	yes	characteristic
132	C₂	#11924583	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11924531

C₂

yes

maximal, complemented

#11924527

C₂

yes

maximal, complemented

#11924521

C₂

yes

maximal, complemented

#11924520

C₂

yes

maximal, complemented

#11924512

C₂

yes

maximal, complemented

#11924510

C₂

yes

maximal, characteristic

#11924496

C₂

yes

maximal, complemented

C₂×#10596936

C₂

yes

maximal, characteristic, complemented

C₂²⋊Q₈×C₃₇

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×C₃₇

C₂²

yes

characteristic

C₂²×C₄×C₃₇

C₂²

yes

characteristic

C₂×C₂²⋊D₃₇

C₂²

yes

—

C₂²×Dic₃₇

C₂²

yes

characteristic

C₂²×Dic₃₇

C₂²

yes

characteristic

C₂×D₁₄₈

C₂²

yes

characteristic

C₂×C₄×D₃₇

C₂²

yes

—

C₄⋊C₄×C₃₇

C₂²

yes

—

C₄⋊C₄×C₃₇

C₂²

yes

characteristic

C₂²⋊C₄×C₃₇

C₂²

yes

—

#10596938

C₂²

yes

—

#10596938

C₂²

yes

—

#10596938

C₂²

yes

—

#10596938

C₂²

yes

—

C₄⋊Dic₃₇

C₂²

yes

characteristic

C₄⋊Dic₃₇

C₂²

yes

—

C₄⋊Dic₃₇

C₂²

yes

—

#10596936

C₂²

yes

—

#10596936

C₂²

yes

—

#10596936

C₂²

yes

—

C₄×Dic₃₇

C₂²

yes

—

C₂³×C₃₇

C₂³

yes

characteristic

C₂²×D₃₇

C₂³

yes

—

Q₈×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂³

yes

—

C₂×C₄×C₃₇

C₂³

yes

characteristic

C₂×C₄×C₃₇

C₂³

yes

characteristic

C₂×C₄×C₃₇

C₂³

yes

—

C₂²⋊D₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

C₂³

yes

—

C₂×Dic₃₇

C₂³

yes

—

C₂×Dic₃₇

C₂³

yes

—

D₁₄₈

—

C₄×D₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

#41

yes

—

C₂²×C₃₇

C₂⁴

yes

characteristic

C₂×D₃₇

—

C₂×D₃₇

—

C₂×D₃₇

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

—

Dic₃₇

—

Dic₃₇

—

Dic₃₇

—

C₂×C₃₇

—

C₂×C₃₇

C₂×#41

yes

characteristic

C₂×C₃₇

H(5,𝔽₂)

yes

characteristic

C₂×C₃₇

2₋¹⁺⁴

yes

characteristic

D₃₇

—

#539

—

maximal, Sylow

C₃₇

#539

yes

characteristic, complemented, Sylow

—

—

—

—

C₂²⋊Q₈

—

C₂²⋊Q₈

D₃₇

yes

characteristic, complemented

#121

—

C₄×D₄

—

C₂×C₄⋊C₄

—

C₂×Q₈

C₂×D₃₇

yes

characteristic

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₃₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₃₇

yes

—

C₄⋊C₄

C₂×D₃₇

yes

characteristic

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

C₂×D₃₇

yes

—

C₄²

—

C₂³

—

C₂³

C₂²×D₃₇

yes

characteristic

100

Q₈

—

101

D₄

—

102

D₄

—

148

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

148

—

109

C₂×C₄

—

148

—

110

C₂×C₄

C₂²×D₃₇

yes

—

111

C₂×C₄

C₂²×D₃₇

yes

characteristic

112

C₂×C₄

C₂²×D₃₇

yes

—

113

C₂×C₄

C₂²×D₃₇

yes

characteristic

114

C₂²

—

115

C₂²

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

#10596966

yes

—

120

C₂²

C₂³×D₃₇

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

—

128

C₂

—

129

C₂

—

148

—

130

C₂

C₂×#10596966

yes

characteristic

131

C₂

#11924581

yes

characteristic

132

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	#11924510	C₂
hypercenter	C₂²⋊Q₈	D₃₇
hypocenter	C₃₇	#539
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

#11924510

C₂

hypercenter

C₂²⋊Q₈

D₃₇

C₃₇

G ▷ C₂²×C₃₇ ▷ C₃₇

C₁ ◁ C₂² ◁ C₂²⋊Q₈

N	H ≅ G/N	type
1. #11924531	C₂	semidirect
2. #11924527	C₂	semidirect
3. #11924521	C₂	semidirect
4. #11924520	C₂	semidirect
5. #11924512	C₂	semidirect
7. #11924496	C₂	semidirect
8. C₂×#10596936	C₂	semidirect
9. C₂²⋊Q₈×C₃₇	C₂	semidirect
67. C₃₇	#539	semidirect
73. C₂²⋊Q₈	D₃₇	semidirect

H ≅ G/N

type

1. #11924531

C₂

semidirect

2. #11924527

C₂

semidirect

3. #11924521

C₂

semidirect

4. #11924520

C₂

semidirect

5. #11924512

C₂

semidirect

7. #11924496

C₂

semidirect

8. C₂×#10596936

C₂

semidirect

9. C₂²⋊Q₈×C₃₇

C₂

semidirect

67. C₃₇

#539

semidirect

73. C₂²⋊Q₈

D₃₇

semidirect

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

subgroup	index
#11924531	2
#11924527	2
#11924521	2
#11924520	2
#11924512	2
#11924510	2
#11924496	2
C₂×#10596936	2
C₂²⋊Q₈×C₃₇	2
#539	37

subgroup

index

C₂²⋊Q₈×C₃₇

Subgroups — #14558132

Conjugacy classes of subgroups

Complements

Maximal subgroups