Group #17997199 — subgroups

Subgroups — #17997199

Order 4736.

512 conjugacy classes of subgroups; only the 229 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 229 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	#14557419	C₂	1	8	yes	maximal, complemented
2	#14557396	C₂	1	2	yes	maximal
3	#14557393	C₂	1	2	yes	maximal
4	C₄×C₂²⋊Dic₃₇	C₂	1	2	yes	maximal
5	C₂²×C₄⋊C₄×C₃₇	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊Dic₃₇	C₂²	1	4	yes	—
7	C₂×C₂²⋊Dic₃₇	C₂²	1	4	yes	—
8	C₂×C₄×Dic₃₇	C₂²	1	4	yes	—
9	#11924443	C₂²	1	4	yes	—
10	#11924443	C₂²	1	4	yes	—
11	#11924443	C₂²	1	8	yes	—
12	C₂³×C₄×C₃₇	C₂²	1	1	yes	characteristic
13	C₂³×C₄×C₃₇	C₂²	1	1	yes	characteristic
14	C₂³×C₄×C₃₇	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₄×C₃₇	C₄	1	8	yes	complemented
16	C₂×C₄⋊C₄×C₃₇	C₂²	1	4	yes	—
17	C₂⁴×C₃₇	C₂³	1	1	yes	characteristic
18	C₂²×C₄×C₃₇	—	2	4	no	—
19	C₂²×C₄×C₃₇	—	4	2	no	—
20	C₂²×C₄×C₃₇	—	4	2	no	—
21	C₂²×C₄×C₃₇	C₂×C₄	1	4	yes	—
22	C₂²×C₄×C₃₇	C₂×C₄	1	4	yes	—
23	C₂²×C₄×C₃₇	C₂×C₄	1	4	yes	—
24	C₂²×C₄×C₃₇	C₂³	1	2	yes	—
25	C₂²×C₄×C₃₇	C₂³	1	2	yes	—
26	C₂²×C₄×C₃₇	C₂³	1	2	yes	—
27	C₂²×Dic₃₇	C₂³	1	8	yes	—
28	C₄⋊C₄×C₃₇	C₂×C₄	1	16	yes	—
29	C₂²⋊Dic₃₇	—	2	8	no	—
30	C₄⋊Dic₃₇	—	4	4	no	—
31	C₄×Dic₃₇	—	4	4	no	—
32	C₂³×C₃₇	—	2	2	no	—
33	C₂³×C₃₇	—	2	2	no	—
34	C₂³×C₃₇	C₂²×C₄	1	2	yes	—
35	C₂³×C₃₇	#41	1	4	yes	—
36	C₂³×C₃₇	C₂⁴	1	1	yes	characteristic
37	C₂×C₄×C₃₇	—	2	8	no	—
38	C₂×C₄×C₃₇	—	2	4	no	—
39	C₂×C₄×C₃₇	—	2	4	no	—
40	C₂×C₄×C₃₇	—	2	4	no	—
41	C₂×C₄×C₃₇	—	4	4	no	—
42	C₂×C₄×C₃₇	—	4	4	no	—
43	C₂×C₄×C₃₇	C₂²×C₄	1	4	yes	—
44	C₂×C₄×C₃₇	C₂²×C₄	1	4	yes	—
45	C₂×C₄×C₃₇	C₂²×C₄	1	4	yes	—
46	C₂×Dic₃₇	—	2	8	no	—
47	C₂×Dic₃₇	—	2	8	no	—
48	C₂×Dic₃₇	—	2	8	no	—
49	C₂²×C₃₇	—	2	2	no	—
50	C₂²×C₃₇	—	2	2	no	—
51	C₂²×C₃₇	—	2	8	no	—
52	C₂²×C₃₇	#117	1	4	yes	—
53	C₂²×C₃₇	C₂³×C₄	1	1	yes	characteristic
54	C₂²×C₃₇	C₂×#41	1	2	yes	—
55	C₂²×C₃₇	H(5,𝔽₂)	1	2	yes	—
56	C₂²×C₃₇	2₋¹⁺⁴	1	2	yes	—
57	C₄×C₃₇	—	2	4	no	—
58	C₄×C₃₇	—	2	4	no	—
59	C₄×C₃₇	—	2	4	no	—
60	Dic₃₇	—	4	8	no	—
61	#2313	—	37	1	no	maximal, Sylow
62	C₂×C₃₇	—	2	4	no	—
63	C₂×C₃₇	C₂×#117	1	1	yes	characteristic
64	C₂×C₃₇	#518	1	1	yes	characteristic
65	C₂×C₃₇	#519	1	1	yes	characteristic
66	C₂×C₃₇	#539	1	4	yes	—
67	C₂²×C₄⋊C₄	D₃₇	1	1	yes	characteristic
68	#387	—	37	8	no	—
69	#381	—	37	2	no	—
70	#380	—	37	2	no	—
71	C₄×C₂²⋊C₄	—	37	2	no	—
72	C₃₇	#2313	1	1	yes	characteristic, complemented, Sylow
73	C₂³×C₄	C₂×D₃₇	1	1	yes	characteristic
74	C₂³×C₄	C₂×D₃₇	1	1	yes	characteristic
75	C₂³×C₄	C₂×D₃₇	1	1	yes	characteristic
76	C₂×C₄⋊C₄	—	37	4	no	—
77	C₂×C₄⋊C₄	Dic₃₇	1	8	yes	complemented
78	C₂×C₄⋊C₄	C₂×D₃₇	1	4	yes	—
79	C₂×C₂²⋊C₄	—	37	4	no	—
80	C₂×C₄²	—	37	4	no	—
81	#95	—	37	4	no	—
82	#95	—	37	4	no	—
83	#95	—	37	8	no	—
84	C₂⁴	C₂²×D₃₇	1	1	yes	characteristic
85	C₂²×C₄	—	2	4	no	—
86	C₂²×C₄	—	4	2	no	—
87	C₂²×C₄	—	4	2	no	—
88	C₂²×C₄	—	37	8	no	—
89	C₂²×C₄	C₂×Dic₃₇	1	4	yes	—
90	C₂²×C₄	C₂×Dic₃₇	1	4	yes	—
91	C₂²×C₄	C₂×Dic₃₇	1	4	yes	—
92	C₂²×C₄	C₂²×D₃₇	1	2	yes	—
93	C₂²×C₄	C₂²×D₃₇	1	2	yes	—
94	C₂²×C₄	C₂²×D₃₇	1	2	yes	—
95	C₄⋊C₄	—	148	4	no	—
96	C₄⋊C₄	C₂×Dic₃₇	1	16	yes	—
97	C₂²⋊C₄	—	74	8	no	—
98	C₄²	—	148	4	no	—
99	C₂³	—	2	2	no	—
100	C₂³	—	2	2	no	—
101	C₂³	#10596962	1	4	yes	—
102	C₂³	C₂²×Dic₃₇	1	2	yes	—
103	C₂³	C₂³×D₃₇	1	1	yes	characteristic
104	C₂×C₄	—	2	8	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	2	4	no	—
107	C₂×C₄	—	2	4	no	—
108	C₂×C₄	—	4	4	no	—
109	C₂×C₄	—	4	4	no	—
110	C₂×C₄	—	74	8	no	—
111	C₂×C₄	—	74	8	no	—
112	C₂×C₄	—	74	8	no	—
113	C₂×C₄	C₂²×Dic₃₇	1	4	yes	—
114	C₂×C₄	C₂²×Dic₃₇	1	4	yes	—
115	C₂×C₄	C₂²×Dic₃₇	1	4	yes	—
116	C₂²	—	2	2	no	—
117	C₂²	—	2	2	no	—
118	C₂²	—	2	8	no	—
119	C₂²	#11924495	1	4	yes	—
120	C₂²	C₂³×Dic₃₇	1	1	yes	characteristic
121	C₂²	C₂×#10596962	1	2	yes	—
122	C₂²	#11924581	1	2	yes	—
123	C₂²	#11924584	1	2	yes	—
124	C₄	—	2	4	no	—
125	C₄	—	2	4	no	—
126	C₄	—	2	4	no	—
127	C₄	—	148	8	no	—
128	C₂	—	2	4	no	—
129	C₂	C₂×#11924495	1	1	yes	characteristic
130	C₂	#14558008	1	1	yes	characteristic
131	C₂	#14558011	1	1	yes	characteristic
132	C₂	#14558130	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14557419

C₂

yes

maximal, complemented

#14557396

C₂

yes

maximal

#14557393

C₂

yes

maximal

C₄×C₂²⋊Dic₃₇

C₂

yes

maximal

C₂²×C₄⋊C₄×C₃₇

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₃₇

C₂²

yes

—

C₂×C₂²⋊Dic₃₇

C₂²

yes

—

C₂×C₄×Dic₃₇

C₂²

yes

—

#11924443

C₂²

yes

—

#11924443

C₂²

yes

—

#11924443

C₂²

yes

—

C₂³×C₄×C₃₇

C₂²

yes

characteristic

C₂³×C₄×C₃₇

C₂²

yes

characteristic

C₂³×C₄×C₃₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₃₇

C₄

yes

complemented

C₂×C₄⋊C₄×C₃₇

C₂²

yes

—

C₂⁴×C₃₇

C₂³

yes

characteristic

C₂²×C₄×C₃₇

—

C₂²×C₄×C₃₇

—

C₂²×C₄×C₃₇

—

C₂²×C₄×C₃₇

C₂×C₄

yes

—

C₂²×C₄×C₃₇

C₂×C₄

yes

—

C₂²×C₄×C₃₇

C₂×C₄

yes

—

C₂²×C₄×C₃₇

C₂³

yes

—

C₂²×C₄×C₃₇

C₂³

yes

—

C₂²×C₄×C₃₇

C₂³

yes

—

C₂²×Dic₃₇

C₂³

yes

—

C₄⋊C₄×C₃₇

C₂×C₄

yes

—

C₂²⋊Dic₃₇

—

C₄⋊Dic₃₇

—

C₄×Dic₃₇

—

C₂³×C₃₇

—

C₂³×C₃₇

—

C₂³×C₃₇

C₂²×C₄

yes

—

C₂³×C₃₇

#41

yes

—

C₂³×C₃₇

C₂⁴

yes

characteristic

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂²×C₄

yes

—

C₂×C₄×C₃₇

C₂²×C₄

yes

—

C₂×C₄×C₃₇

C₂²×C₄

yes

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

#117

yes

—

C₂²×C₃₇

C₂³×C₄

yes

characteristic

C₂²×C₃₇

C₂×#41

yes

—

C₂²×C₃₇

H(5,𝔽₂)

yes

—

C₂²×C₃₇

2₋¹⁺⁴

yes

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

—

Dic₃₇

—

#2313

—

maximal, Sylow

C₂×C₃₇

—

C₂×C₃₇

yes

C₂×C₃₇

yes

C₂×C₃₇

yes

C₂×C₃₇

yes

—

C₂²×C₄⋊C₄

D₃₇

yes

—

—

—

C₄×C₂²⋊C₄

—

C₃₇

#2313

yes

characteristic, complemented, Sylow

C₂³×C₄

C₂×D₃₇

yes

characteristic

C₂³×C₄

C₂×D₃₇

yes

characteristic

C₂³×C₄

C₂×D₃₇

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

Dic₃₇

yes

complemented

C₂×C₄⋊C₄

C₂×D₃₇

yes

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

#95

—

#95

—

#95

—

C₂⁴

C₂²×D₃₇

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×Dic₃₇

yes

—

C₂²×C₄

C₂×Dic₃₇

yes

—

C₂²×C₄

C₂×Dic₃₇

yes

—

C₂²×C₄

C₂²×D₃₇

yes

—

C₂²×C₄

C₂²×D₃₇

yes

—

C₂²×C₄

C₂²×D₃₇

yes

—

C₄⋊C₄

—

148

—

C₄⋊C₄

C₂×Dic₃₇

yes

—

C₂²⋊C₄

—

C₄²

—

148

—

C₂³

—

100

C₂³

—

101

C₂³

#10596962

yes

—

102

C₂³

C₂²×Dic₃₇

yes

—

103

C₂³

C₂³×D₃₇

yes

characteristic

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

C₂²×Dic₃₇

yes

—

114

C₂×C₄

C₂²×Dic₃₇

yes

—

115

C₂×C₄

C₂²×Dic₃₇

yes

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

#11924495

yes

—

120

C₂²

C₂³×Dic₃₇

yes

characteristic

121

C₂²

C₂×#10596962

yes

—

122

C₂²

#11924581

yes

—

123

C₂²

#11924584

yes

—

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

—

148

—

128

C₂

—

129

C₂

C₂×#11924495

yes

characteristic

130

C₂

#14558008

yes

characteristic

131

C₂

#14558011

yes

characteristic

132

C₂

#14558130

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₃₇
commutator	C₂²×C₃₇	C₂³×C₄
Fitting	C₂²×C₄⋊C₄×C₃₇	C₂
socle, cosocle	C₂³×C₃₇	C₂⁴
prime radical	C₂⁴×C₃₇	C₂³
hypercenter	C₂²×C₄⋊C₄	D₃₇
hypocenter	C₃₇	#2313
LCS	G ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄⋊C₄

subgroup

G/H

center, Frattini

C₂³

C₂³×D₃₇

commutator

C₂²×C₃₇

C₂³×C₄

Fitting

C₂²×C₄⋊C₄×C₃₇

C₂

socle, cosocle

C₂³×C₃₇

C₂⁴

prime radical

C₂⁴×C₃₇

C₂³

hypercenter

C₂²×C₄⋊C₄

D₃₇

C₃₇

G ▷ C₂²×C₃₇ ▷ C₃₇

C₁ ◁ C₂³ ◁ C₂²×C₄⋊C₄

N	H ≅ G/N	type
1. #14557419	C₂	semidirect
15. C₂×C₄⋊C₄×C₃₇	C₄	semidirect
72. C₃₇	#2313	semidirect
77. C₂×C₄⋊C₄	Dic₃₇	semidirect

H ≅ G/N

type

1. #14557419

C₂

semidirect

15. C₂×C₄⋊C₄×C₃₇

C₄

semidirect

72. C₃₇

#2313

semidirect

77. C₂×C₄⋊C₄

Dic₃₇

semidirect

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

subgroup	index
#14557419	2
#14557396	2
#14557393	2
C₄×C₂²⋊Dic₃₇	2
C₂²×C₄⋊C₄×C₃₇	2
#2313	37

subgroup

index

#14557419

#14557396

#14557393

C₄×C₂²⋊Dic₃₇

C₂²×C₄⋊C₄×C₃₇

#2313

Subgroups — #17997199

Conjugacy classes of subgroups

Complements

Maximal subgroups