Group #18005097 — subgroups

Subgroups — C₂×#14557932 #18005097

Order 4736.

360 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 165 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×#11924528	C₂	1	1	yes	maximal, characteristic
2	#14557932	C₂	1	8	yes	maximal, direct product factor
3	C₂×#11924469	C₂	1	2	yes	maximal
4	C₂×#11924467	C₂	1	2	yes	maximal
5	C₂×C₄×#64436	C₂	1	1	yes	maximal, characteristic
6	C₂×#125×C₃₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×#64436	C₂²	1	1	yes	characteristic
8	C₂²×#64436	C₂²	1	1	yes	characteristic
9	#11924528	—	2	2	no	—
10	#11924528	C₂²	1	4	yes	—
11	C₂×C₄⋊Dic₃₇	C₂²	1	2	yes	—
12	C₂×#10596936	—	2	2	no	—
13	#11924469	C₂²	1	8	yes	—
14	#11924467	C₂²	1	8	yes	—
15	C₄×#64436	C₂²	1	4	yes	—
16	#125×C₃₇	—	2	2	no	—
17	#125×C₃₇	C₂²	1	4	yes	—
18	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
19	C₂×C₄⋊C₄×C₃₇	C₂²	1	2	yes	—
20	C₂×C₄²×C₃₇	C₂²	1	1	yes	characteristic
21	C₂²×C₄×C₃₇	—	2	2	no	—
22	C₂²×C₄×C₃₇	D₄	1	2	yes	—
23	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
24	C₂²×Dic₃₇	—	2	2	no	—
25	C₄⋊C₄×C₃₇	—	2	8	no	—
26	C₄⋊C₄×C₃₇	—	2	2	no	—
27	C₄⋊C₄×C₃₇	C₂³	1	4	yes	—
28	C₄²×C₃₇	D₄	1	2	yes	—
29	C₄²×C₃₇	C₂³	1	2	yes	—
30	C₄⋊Dic₃₇	—	2	2	no	—
31	C₄⋊Dic₃₇	C₂³	1	4	yes	—
32	#10596936	—	2	8	no	—
33	C₂×#64436	Q₈	1	4	yes	—
34	C₂×#64436	Q₈	1	4	yes	—
35	C₂×#64436	C₂³	1	2	yes	—
36	C₂×#64436	C₂³	1	2	yes	—
37	C₂³×C₃₇	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₃₇	—	2	4	no	—
39	C₂×C₄×C₃₇	—	2	4	no	—
40	C₂×C₄×C₃₇	—	4	4	no	—
41	C₂×C₄×C₃₇	C₂×D₄	1	4	yes	—
42	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₃₇	C₂×Q₈	1	2	yes	—
44	C₂×C₄×C₃₇	C₂×Q₈	1	2	yes	—
45	C₂×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
46	C₂×Dic₃₇	—	2	4	no	—
47	C₂×Dic₃₇	—	4	4	no	—
48	#64436	C₂×Q₈	1	4	yes	—
49	#64436	C₂×Q₈	1	4	yes	—
50	C₂²×C₃₇	C₄⋊Q₈	1	2	yes	—
51	C₂²×C₃₇	#135	1	4	yes	—
52	C₂²×C₃₇	C₂²×D₄	1	1	yes	characteristic
53	C₄×C₃₇	—	2	4	no	—
54	C₄×C₃₇	—	4	4	no	—
55	C₄×C₃₇	C₄⋊Q₈	1	2	yes	—
56	C₄×C₃₇	C₂²×Q₈	1	1	yes	characteristic
57	C₄×C₃₇	C₂²×Q₈	1	1	yes	characteristic
58	Dic₃₇	—	4	4	no	—
59	C₂×#499	—	37	1	no	maximal, Sylow
60	C₂×C₃₇	#499	1	4	yes	complemented
61	C₂×C₃₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
62	C₂×C₃₇	C₂×#135	1	2	yes	—
63	C₂×#125	—	37	1	no	—
64	C₂×#125	D₃₇	1	1	yes	characteristic
65	#499	—	37	8	no	—
66	C₂×#107	—	37	2	no	—
67	C₂×#106	—	37	2	no	—
68	C₂×C₄×C₈	—	37	1	no	—
69	C₃₇	C₂×#499	1	1	yes	characteristic, complemented, Sylow
70	C₂²×C₈	—	37	1	no	—
71	C₂²×C₈	—	37	1	no	—
72	#125	—	2	2	no	—
73	#125	—	37	4	no	—
74	#125	—	74	2	no	—
75	#125	C₂×D₃₇	1	4	yes	—
76	C₂×C₄⋊C₄	—	2	2	no	—
77	C₂×C₄⋊C₄	—	37	2	no	—
78	C₂×C₄⋊C₄	—	74	2	no	—
79	C₂×C₄⋊C₄	C₂×D₃₇	1	2	yes	—
80	C₂×C₄²	C₂×D₃₇	1	1	yes	characteristic
81	#107	—	37	8	no	—
82	#106	—	37	8	no	—
83	C₄×C₈	—	37	4	no	—
84	C₂²×C₄	—	2	2	no	—
85	C₂²×C₄	—	74	2	no	—
86	C₂²×C₄	C₂²⋊D₃₇	1	2	yes	—
87	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
88	C₂×C₈	—	37	4	no	—
89	C₂×C₈	—	37	4	no	—
90	C₂×C₈	—	37	2	no	—
91	C₂×C₈	—	37	2	no	—
92	C₄⋊C₄	—	2	8	no	—
93	C₄⋊C₄	—	2	2	no	—
94	C₄⋊C₄	—	37	4	no	—
95	C₄⋊C₄	—	74	8	no	—
96	C₄⋊C₄	—	74	2	no	—
97	C₄⋊C₄	C₂²×D₃₇	1	4	yes	—
98	C₄²	C₂²⋊D₃₇	1	2	yes	—
99	C₄²	C₂²×D₃₇	1	2	yes	—
100	C₂³	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
101	C₂×C₄	—	2	4	no	—
102	C₂×C₄	—	2	4	no	—
103	C₂×C₄	—	4	4	no	—
104	C₂×C₄	—	74	4	no	—
105	C₂×C₄	—	148	4	no	—
106	C₂×C₄	Q₈×D₃₇	1	2	yes	—
107	C₂×C₄	Q₈×D₃₇	1	2	yes	—
108	C₂×C₄	C₂×C₂²⋊D₃₇	1	4	yes	—
109	C₂×C₄	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
110	C₂×C₄	C₂³×D₃₇	1	1	yes	characteristic
111	C₈	—	37	4	no	—
112	C₈	—	37	4	no	—
113	C₂²	#11924534	1	2	yes	—
114	C₂²	#11924556	1	4	yes	—
115	C₂²	C₂²×C₂²⋊D₃₇	1	1	yes	characteristic
116	C₄	—	2	4	no	—
117	C₄	—	4	4	no	—
118	C₄	—	148	4	no	—
119	C₄	#11924534	1	2	yes	—
120	C₄	C₂×Q₈×D₃₇	1	1	yes	characteristic
121	C₄	C₂×Q₈×D₃₇	1	1	yes	characteristic
122	C₂	#14557932	1	4	yes	direct product factor
123	C₂	C₂×#11924534	1	1	yes	characteristic
124	C₂	C₂×#11924556	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11924528

C₂

yes

maximal, characteristic

#14557932

C₂

yes

maximal, direct product factor

C₂×#11924469

C₂

yes

maximal

C₂×#11924467

C₂

yes

maximal

C₂×C₄×#64436

C₂

yes

maximal, characteristic

C₂×#125×C₃₇

C₂

yes

maximal, characteristic

C₂²×#64436

C₂²

yes

characteristic

C₂²×#64436

C₂²

yes

characteristic

#11924528

—

#11924528

C₂²

yes

—

C₂×C₄⋊Dic₃₇

C₂²

yes

—

C₂×#10596936

—

#11924469

C₂²

yes

—

#11924467

C₂²

yes

—

C₄×#64436

C₂²

yes

—

#125×C₃₇

—

#125×C₃₇

C₂²

yes

—

C₂×C₄⋊C₄×C₃₇

—

C₂×C₄⋊C₄×C₃₇

C₂²

yes

—

C₂×C₄²×C₃₇

C₂²

yes

characteristic

C₂²×C₄×C₃₇

—

C₂²×C₄×C₃₇

D₄

yes

—

C₂²×C₄×C₃₇

C₂³

yes

characteristic

C₂²×Dic₃₇

—

C₄⋊C₄×C₃₇

—

C₄⋊C₄×C₃₇

—

C₄⋊C₄×C₃₇

C₂³

yes

—

C₄²×C₃₇

D₄

yes

—

C₄²×C₃₇

C₂³

yes

—

C₄⋊Dic₃₇

—

C₄⋊Dic₃₇

C₂³

yes

—

#10596936

—

C₂×#64436

Q₈

yes

—

C₂×#64436

Q₈

yes

—

C₂×#64436

C₂³

yes

—

C₂×#64436

C₂³

yes

—

C₂³×C₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂×D₄

yes

—

C₂×C₄×C₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₃₇

C₂×Q₈

yes

—

C₂×C₄×C₃₇

C₂×Q₈

yes

—

C₂×C₄×C₃₇

C₂⁴

yes

characteristic

C₂×Dic₃₇

—

C₂×Dic₃₇

—

#64436

C₂×Q₈

yes

—

#64436

C₂×Q₈

yes

—

C₂²×C₃₇

C₄⋊Q₈

yes

—

C₂²×C₃₇

#135

yes

—

C₂²×C₃₇

C₂²×D₄

yes

characteristic

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

C₄⋊Q₈

yes

—

C₄×C₃₇

C₂²×Q₈

yes

characteristic

C₄×C₃₇

C₂²×Q₈

yes

characteristic

Dic₃₇

—

—

C₂×C₃₇

yes

C₂×C₃₇

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₃₇

C₂×#135

yes

—

C₂×#125

—

C₂×#125

D₃₇

yes

—

—

—

C₂×C₄×C₈

—

C₃₇

C₂×#499

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

—

—

—

C₂×D₃₇

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₃₇

yes

—

C₂×C₄²

C₂×D₃₇

yes

characteristic

#107

—

#106

—

C₄×C₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₃₇

yes

—

C₂²×C₄

C₂²×D₃₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂²×D₃₇

yes

—

C₄²

C₂²⋊D₃₇

yes

—

C₄²

C₂²×D₃₇

yes

—

100

C₂³

C₂×C₂²⋊D₃₇

yes

characteristic

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

148

—

106

C₂×C₄

Q₈×D₃₇

yes

—

107

C₂×C₄

Q₈×D₃₇

yes

—

108

C₂×C₄

C₂×C₂²⋊D₃₇

yes

—

109

C₂×C₄

C₂×C₂²⋊D₃₇

yes

characteristic

110

C₂×C₄

C₂³×D₃₇

yes

characteristic

111

C₈

—

112

C₈

—

113

C₂²

#11924534

yes

—

114

C₂²

#11924556

yes

—

115

C₂²

C₂²×C₂²⋊D₃₇

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

—

148

—

119

C₄

#11924534

yes

—

120

C₄

C₂×Q₈×D₃₇

yes

characteristic

121

C₄

C₂×Q₈×D₃₇

yes

characteristic

122

C₂

#14557932

yes

direct product factor

123

C₂

C₂×#11924534

yes

characteristic

124

C₂

C₂×#11924556

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₃₇
commutator, cosocle	C₂×C₄×C₃₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₃₇
Fitting	C₂×#125×C₃₇	C₂
socle, prime radical	C₂³×C₃₇	C₂×D₄
hypercenter	C₂×#125	D₃₇
hypocenter	C₃₇	C₂×#499
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₃₇

commutator, cosocle

C₂×C₄×C₃₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₃₇

Fitting

C₂×#125×C₃₇

C₂

socle, prime radical

C₂³×C₃₇

C₂×D₄

D₃₇

C₃₇

G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

N	H ≅ G/N	type
2. #14557932	C₂	direct (N × H)
60. C₂×C₃₇	#499	semidirect
69. C₃₇	C₂×#499	semidirect
122. C₂	#14557932	direct (N × H)

H ≅ G/N

type

2. #14557932

C₂

direct (N × H)

60. C₂×C₃₇

#499

semidirect

69. C₃₇

C₂×#499

semidirect

122. C₂

#14557932

direct (N × H)

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

subgroup	index
C₂×#11924528	2
#14557932	2
C₂×#11924469	2
C₂×#11924467	2
C₂×C₄×#64436	2
C₂×#125×C₃₇	2
C₂×#499	37

subgroup

index

C₂×C₄×#64436

C₂×#125×C₃₇

C₂×#499

Subgroups — C₂×#14557932 #18005097

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#14557932 #18005097

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14557932 #18005097