Group #18005099 — subgroups

Subgroups — C₂×#14557934 #18005099

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×#11924527	C₂	1	2	yes	maximal
2	#14557934	C₂	1	8	yes	maximal, direct product factor
3	C₂×#11924470	C₂	1	1	yes	maximal, characteristic
4	C₂×#11924467	C₂	1	2	yes	maximal
5	C₂×C₈×Dic₃₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#107×C₃₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×#64436	C₂²	1	1	yes	characteristic
8	#11924527	—	2	4	no	—
9	#11924527	C₂²	1	8	yes	—
10	C₂×C₄⋊Dic₃₇	—	2	2	no	—
11	C₂×C₄⋊Dic₃₇	C₂²	1	2	yes	—
12	C₂×#10596936	—	2	2	no	—
13	C₂×C₄×Dic₃₇	C₂²	1	1	yes	characteristic
14	#11924470	C₂²	1	4	yes	—
15	#11924467	C₂²	1	8	yes	—
16	C₈×Dic₃₇	C₂²	1	4	yes	—
17	C₂²×C₈×C₃₇	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₃₇	C₂²	1	2	yes	—
19	#107×C₃₇	C₂²	1	4	yes	—
20	C₂²×C₄×C₃₇	—	2	2	no	—
21	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
22	C₂²×Dic₃₇	—	2	2	no	—
23	C₂²×Dic₃₇	D₄	1	1	yes	characteristic
24	C₂²×Dic₃₇	D₄	1	1	yes	characteristic
25	C₂×C₈×C₃₇	Q₈	1	4	yes	—
26	C₂×C₈×C₃₇	C₂³	1	2	yes	—
27	C₄⋊C₄×C₃₇	—	2	2	no	—
28	C₄⋊C₄×C₃₇	C₂³	1	4	yes	—
29	C₄⋊Dic₃₇	—	2	2	no	—
30	C₄⋊Dic₃₇	—	2	8	no	—
31	C₄⋊Dic₃₇	C₂³	1	4	yes	—
32	#10596936	—	2	8	no	—
33	C₄×Dic₃₇	D₄	1	2	yes	—
34	C₄×Dic₃₇	C₂³	1	2	yes	—
35	C₂×#64436	Q₈	1	4	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	—	4	4	no	—
40	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₃₇	C₂×Q₈	1	2	yes	—
42	C₂×C₄×C₃₇	C₂×Q₈	1	2	yes	—
43	C₂×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
44	C₂×Dic₃₇	—	2	2	no	—
45	C₂×Dic₃₇	—	2	2	no	—
46	C₂×Dic₃₇	—	2	4	no	—
47	C₂×Dic₃₇	—	4	4	no	—
48	C₂×Dic₃₇	C₂×D₄	1	2	yes	—
49	C₂×Dic₃₇	C₂×D₄	1	2	yes	—
50	C₈×C₃₇	C₂×Q₈	1	4	yes	—
51	#64436	C₂×Q₈	1	4	yes	—
52	C₂²×C₃₇	C₄⋊Q₈	1	2	yes	—
53	C₂²×C₃₇	#135	1	2	yes	—
54	C₂²×C₃₇	#135	1	2	yes	—
55	C₂²×C₃₇	C₂²×D₄	1	1	yes	characteristic
56	C₄×C₃₇	—	4	4	no	—
57	C₄×C₃₇	C₄⋊Q₈	1	2	yes	—
58	C₄×C₃₇	C₂²×Q₈	1	1	yes	characteristic
59	C₄×C₃₇	C₂²×Q₈	1	1	yes	characteristic
60	Dic₃₇	—	2	2	no	—
61	Dic₃₇	—	2	2	no	—
62	Dic₃₇	—	4	4	no	—
63	C₂×#499	—	37	1	no	maximal, Sylow
64	C₂×C₃₇	#499	1	4	yes	complemented
65	C₂×C₃₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
66	C₂×C₃₇	C₂×#135	1	1	yes	characteristic
67	C₂×C₃₇	C₂×#135	1	1	yes	characteristic
68	C₂×#125	—	37	2	no	—
69	#499	—	37	8	no	—
70	C₂×#107	—	37	1	no	—
71	C₂×#107	D₃₇	1	1	yes	characteristic
72	C₂×#106	—	37	2	no	—
73	C₂×C₄×C₈	—	37	1	no	—
74	C₃₇	C₂×#499	1	1	yes	characteristic, complemented, Sylow
75	C₂²×C₈	—	37	1	no	—
76	C₂²×C₈	C₂×D₃₇	1	1	yes	characteristic
77	#125	—	37	8	no	—
78	#125	—	74	4	no	—
79	C₂×C₄⋊C₄	—	37	2	no	—
80	C₂×C₄⋊C₄	—	74	2	no	—
81	C₂×C₄⋊C₄	—	74	2	no	—
82	C₂×C₄⋊C₄	C₂×D₃₇	1	2	yes	—
83	C₂×C₄²	—	37	1	no	—
84	#107	—	37	4	no	—
85	#107	C₂×D₃₇	1	4	yes	—
86	#106	—	37	8	no	—
87	C₄×C₈	—	37	4	no	—
88	C₂²×C₄	—	2	2	no	—
89	C₂²×C₄	—	37	1	no	—
90	C₂²×C₄	—	37	1	no	—
91	C₂²×C₄	—	74	2	no	—
92	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
93	C₂×C₈	—	37	4	no	—
94	C₂×C₈	—	37	2	no	—
95	C₂×C₈	Dic₇₄	1	4	yes	—
96	C₂×C₈	C₂²×D₃₇	1	2	yes	—
97	C₄⋊C₄	—	2	2	no	—
98	C₄⋊C₄	—	37	4	no	—
99	C₄⋊C₄	—	74	8	no	—
100	C₄⋊C₄	—	74	8	no	—
101	C₄⋊C₄	—	74	2	no	—
102	C₄⋊C₄	C₂²×D₃₇	1	4	yes	—
103	C₄²	—	37	2	no	—
104	C₄²	—	37	2	no	—
105	C₂³	D₄×D₃₇	1	1	yes	characteristic
106	C₂×C₄	—	2	4	no	—
107	C₂×C₄	—	4	4	no	—
108	C₂×C₄	—	37	2	no	—
109	C₂×C₄	—	37	2	no	—
110	C₂×C₄	—	74	2	no	—
111	C₂×C₄	—	74	2	no	—
112	C₂×C₄	—	74	4	no	—
113	C₂×C₄	—	148	4	no	—
114	C₂×C₄	C₂×Dic₇₄	1	2	yes	—
115	C₂×C₄	D₄×D₃₇	1	1	yes	characteristic
116	C₂×C₄	Q₈×D₃₇	1	2	yes	—
117	C₂×C₄	C₂³×D₃₇	1	1	yes	characteristic
118	C₈	—	37	4	no	—
119	C₈	C₂×Dic₇₄	1	4	yes	—
120	C₂²	#11924535	1	2	yes	—
121	C₂²	#11924555	1	2	yes	—
122	C₂²	#11924559	1	2	yes	—
123	C₂²	C₂×D₄×D₃₇	1	1	yes	characteristic
124	C₄	—	4	4	no	—
125	C₄	—	74	2	no	—
126	C₄	—	74	2	no	—
127	C₄	—	148	4	no	—
128	C₄	#11924535	1	2	yes	—
129	C₄	C₂×Q₈×D₃₇	1	1	yes	characteristic
130	C₄	C₂²×Dic₇₄	1	1	yes	characteristic
131	C₂	#14557934	1	4	yes	direct product factor
132	C₂	C₂×#11924535	1	1	yes	characteristic
133	C₂	C₂×#11924555	1	1	yes	characteristic
134	C₂	C₂×#11924559	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11924527

C₂

yes

maximal

#14557934

C₂

yes

maximal, direct product factor

C₂×#11924470

C₂

yes

maximal, characteristic

C₂×#11924467

C₂

yes

maximal

C₂×C₈×Dic₃₇

C₂

yes

maximal, characteristic

C₂×#107×C₃₇

C₂

yes

maximal, characteristic

C₂²×#64436

C₂²

yes

characteristic

#11924527

—

#11924527

C₂²

yes

—

C₂×C₄⋊Dic₃₇

—

C₂×C₄⋊Dic₃₇

C₂²

yes

—

C₂×#10596936

—

C₂×C₄×Dic₃₇

C₂²

yes

characteristic

#11924470

C₂²

yes

—

#11924467

C₂²

yes

—

C₈×Dic₃₇

C₂²

yes

—

C₂²×C₈×C₃₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₃₇

C₂²

yes

—

#107×C₃₇

C₂²

yes

—

C₂²×C₄×C₃₇

—

C₂²×C₄×C₃₇

C₂³

yes

characteristic

C₂²×Dic₃₇

—

C₂²×Dic₃₇

D₄

yes

characteristic

C₂²×Dic₃₇

D₄

yes

characteristic

C₂×C₈×C₃₇

Q₈

yes

—

C₂×C₈×C₃₇

C₂³

yes

—

C₄⋊C₄×C₃₇

—

C₄⋊C₄×C₃₇

C₂³

yes

—

C₄⋊Dic₃₇

—

C₄⋊Dic₃₇

—

C₄⋊Dic₃₇

C₂³

yes

—

#10596936

—

C₄×Dic₃₇

D₄

yes

—

C₄×Dic₃₇

C₂³

yes

—

C₂×#64436

Q₈

yes

—

C₂×#64436

C₂³

yes

—

C₂³×C₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

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

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

C₂×D₄

yes

—

C₂×Dic₃₇

C₂×D₄

yes

—

C₈×C₃₇

C₂×Q₈

yes

—

#64436

C₂×Q₈

yes

—

C₂²×C₃₇

C₄⋊Q₈

yes

—

C₂²×C₃₇

#135

yes

—

C₂²×C₃₇

#135

yes

—

C₂²×C₃₇

C₂²×D₄

yes

characteristic

C₄×C₃₇

—

C₄×C₃₇

C₄⋊Q₈

yes

—

C₄×C₃₇

C₂²×Q₈

yes

characteristic

C₄×C₃₇

C₂²×Q₈

yes

characteristic

Dic₃₇

—

Dic₃₇

—

Dic₃₇

—

—

C₂×C₃₇

yes

C₂×C₃₇

C₂×C₄⋊Q₈

yes

C₂×C₃₇

yes

C₂×C₃₇

yes

—

—

—

D₃₇

yes

characteristic

C₂×#106

—

C₂×C₄×C₈

—

C₃₇

C₂×#499

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

C₂×D₃₇

yes

characteristic

#125

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₃₇

yes

—

C₂×C₄²

—

#107

—

#107

C₂×D₃₇

yes

—

#106

—

C₄×C₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₃₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Dic₇₄

yes

—

C₂×C₈

C₂²×D₃₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

C₂²×D₃₇

yes

—

103

C₄²

—

104

C₄²

—

105

C₂³

D₄×D₃₇

yes

characteristic

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

148

—

114

C₂×C₄

C₂×Dic₇₄

yes

—

115

C₂×C₄

D₄×D₃₇

yes

characteristic

116

C₂×C₄

Q₈×D₃₇

yes

—

117

C₂×C₄

C₂³×D₃₇

yes

characteristic

118

C₈

—

119

C₈

C₂×Dic₇₄

yes

—

120

C₂²

#11924535

yes

—

121

C₂²

#11924555

yes

—

122

C₂²

#11924559

yes

—

123

C₂²

C₂×D₄×D₃₇

yes

characteristic

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

—

148

—

128

C₄

#11924535

yes

—

129

C₄

C₂×Q₈×D₃₇

yes

characteristic

130

C₄

C₂²×Dic₇₄

yes

characteristic

131

C₂

#14557934

yes

direct product factor

132

C₂

C₂×#11924535

yes

characteristic

133

C₂

C₂×#11924555

yes

characteristic

134

C₂

C₂×#11924559

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₃₇
commutator, cosocle	C₂×C₄×C₃₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₃₇
Fitting	C₂×#107×C₃₇	C₂
socle, prime radical	C₂³×C₃₇	C₂×D₄
hypercenter	C₂×#107	D₃₇
hypocenter	C₃₇	C₂×#499
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

subgroup

G/H

center

C₂³

D₄×D₃₇

commutator, cosocle

C₂×C₄×C₃₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₃₇

Fitting

C₂×#107×C₃₇

C₂

socle, prime radical

C₂³×C₃₇

C₂×D₄

D₃₇

C₃₇

G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

N	H ≅ G/N	type
2. #14557934	C₂	direct (N × H)
64. C₂×C₃₇	#499	semidirect
74. C₃₇	C₂×#499	semidirect
131. C₂	#14557934	direct (N × H)

H ≅ G/N

type

2. #14557934

C₂

direct (N × H)

64. C₂×C₃₇

#499

semidirect

74. C₃₇

C₂×#499

semidirect

131. C₂

#14557934

direct (N × H)

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

subgroup	index
C₂×#11924527	2
#14557934	2
C₂×#11924470	2
C₂×#11924467	2
C₂×C₈×Dic₃₇	2
C₂×#107×C₃₇	2
C₂×#499	37

subgroup

index

C₂×C₈×Dic₃₇

C₂×#107×C₃₇

C₂×#499

Subgroups — C₂×#14557934 #18005099

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#14557934 #18005099

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14557934 #18005099