Group #18009653 — subgroups

Subgroups — #18009653

Order 4736.

152 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 60 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×D₃₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#11924605	C₂	1	1	yes	maximal, characteristic
3	#107×D₃₇	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊#3866	—	2	1	no	—
5	D₃₇⋊C₁₆	—	2	1	no	—
6	D₃₇⋊C₁₆	C₄	1	2	yes	complemented
7	C₂×#10596927	—	2	1	no	—
8	#11924605	—	2	1	no	—
9	#11924605	C₄	1	2	yes	complemented
10	C₂×C₈×D₃₇	C₂²	1	1	yes	characteristic
11	C₄⋊C₄×D₃₇	—	2	1	no	—
12	#11924470	C₄	1	1	yes	characteristic, complemented
13	#11924469	—	2	1	no	—
14	#107×C₃₇	C₄	1	1	yes	characteristic, complemented
15	C₂²×#3866	—	4	1	no	—
16	C₂×C₄×D₃₇	—	4	1	no	—
17	C₂×C₄×D₃₇	D₄	1	1	yes	characteristic
18	C₄⋊#3866	—	2	2	no	—
19	C₄⋊#3866	—	4	1	no	—
20	C₂×C₈×C₃₇	C₂×C₄	1	1	yes	characteristic
21	C₄⋊C₄×C₃₇	—	2	1	no	—
22	C₄⋊Dic₃₇	—	2	1	no	—
23	#10596936	—	4	1	no	—
24	C₂×#64436	D₄	1	1	yes	characteristic
25	C₈×D₃₇	C₂×C₄	1	1	yes	characteristic
26	C₈×D₃₇	C₂×C₄	1	1	yes	characteristic
27	C₈×D₃₇	D₄	1	1	yes	characteristic
28	C₈×D₃₇	Q₈	1	1	yes	characteristic
29	#10596927	—	2	2	no	—
30	C₂²×D₃₇	D₈	1	1	yes	characteristic
31	C₂×#3866	—	4	2	no	—
32	C₂×#3866	—	8	1	no	—
33	C₂×#3866	—	8	1	no	—
34	C₂×C₄×C₃₇	—	4	1	no	—
35	C₂×C₄×C₃₇	C₂²⋊C₄	1	1	yes	characteristic
36	C₂×Dic₃₇	—	4	1	no	—
37	C₂×Dic₃₇	SD₁₆	1	1	yes	characteristic
38	C₄×D₃₇	—	8	1	no	—
39	C₄×D₃₇	—	8	1	no	—
40	C₄×D₃₇	C₂²⋊C₄	1	1	yes	characteristic
41	C₄×D₃₇	C₄⋊C₄	1	1	yes	characteristic
42	C₄×D₃₇	SD₁₆	1	1	yes	characteristic
43	C₄×D₃₇	Q₁₆	1	1	yes	characteristic
44	C₈×C₃₇	C₄²	1	1	yes	characteristic
45	C₈×C₃₇	C₄⋊C₄	1	1	yes	characteristic
46	#64436	C₂²⋊C₄	1	1	yes	characteristic
47	#64436	C₄⋊C₄	1	1	yes	characteristic
48	C₂²×C₃₇	#102	1	1	yes	characteristic
49	C₂×D₃₇	#102	1	1	yes	characteristic
50	C₂×D₃₇	#107	1	1	yes	characteristic
51	C₂×D₃₇	D₁₆	1	1	yes	characteristic
52	C₂×D₃₇	SD₃₂	1	1	yes	characteristic
53	C₂×D₃₇	SD₃₂	1	1	yes	characteristic
54	C₂×D₃₇	Q₃₂	1	1	yes	characteristic
55	#3866	—	8	2	no	—
56	C₄×C₃₇	—	8	1	no	—
57	C₄×C₃₇	#95	1	1	yes	characteristic
58	C₄×C₃₇	#103	1	1	yes	characteristic
59	Dic₃₇	—	8	1	no	—
60	Dic₃₇	#103	1	1	yes	characteristic
61	Dic₃₇	#106	1	1	yes	characteristic
62	#1380	—	37	1	no	maximal, Sylow
63	C₂×C₃₇	#340	1	1	yes	characteristic
64	C₂×C₃₇	#357	1	1	yes	characteristic
65	C₂×C₃₇	#358	1	1	yes	characteristic
66	D₃₇	#357	1	1	yes	characteristic
67	D₃₇	#358	1	1	yes	characteristic
68	D₃₇	#366	1	1	yes	characteristic
69	D₃₇	#367	1	1	yes	characteristic
70	C₂²×C₁₆	—	37	1	no	—
71	C₂×#107	—	37	1	no	—
72	C₂×#107	—	37	1	no	—
73	C₃₇	#1380	1	1	yes	characteristic, complemented, Sylow
74	C₂²×C₈	—	37	1	no	—
75	C₂×C₄⋊C₄	—	74	1	no	—
76	C₂×C₄⋊C₄	—	74	1	no	—
77	C₂×C₁₆	—	37	2	no	—
78	C₂×C₁₆	—	74	1	no	—
79	C₂×C₁₆	—	74	1	no	—
80	#107	—	37	2	no	—
81	#107	—	37	1	no	—
82	#107	—	74	1	no	—
83	#107	—	74	1	no	—
84	#107	#3866	1	1	yes	characteristic, complemented
85	C₂²×C₄	—	37	1	no	—
86	C₂²×C₄	—	148	1	no	—
87	C₂²×C₄	—	148	1	no	—
88	C₂×C₈	—	37	1	no	—
89	C₂×C₈	—	37	1	no	—
90	C₂×C₈	—	37	1	no	—
91	C₂×C₈	—	37	1	no	—
92	C₂×C₈	—	37	1	no	—
93	C₂×C₈	C₂×#3866	1	1	yes	characteristic
94	C₄⋊C₄	—	2	1	no	—
95	C₄⋊C₄	—	74	2	no	—
96	C₄⋊C₄	—	74	1	no	—
97	C₄⋊C₄	—	148	1	no	—
98	C₄⋊C₄	—	148	1	no	—
99	C₁₆	—	74	2	no	—
100	C₂³	—	37	1	no	—
101	C₂×C₄	—	4	1	no	—
102	C₂×C₄	—	37	1	no	—
103	C₂×C₄	—	37	1	no	—
104	C₂×C₄	—	37	1	no	—
105	C₂×C₄	—	37	1	no	—
106	C₂×C₄	—	37	1	no	—
107	C₂×C₄	—	296	1	no	—
108	C₂×C₄	—	296	1	no	—
109	C₂×C₄	—	296	1	no	—
110	C₂×C₄	—	296	1	no	—
111	C₂×C₄	—	148	2	no	—
112	C₂×C₄	—	148	1	no	—
113	C₂×C₄	C₂²⋊#3866	1	1	yes	characteristic
114	C₈	—	37	1	no	—
115	C₈	—	37	1	no	—
116	C₈	C₄×#3866	1	1	yes	characteristic
117	C₈	C₄⋊#3866	1	1	yes	characteristic
118	C₂²	—	37	1	no	—
119	C₂²	—	37	1	no	—
120	C₂²	—	37	1	no	—
121	C₂²	—	37	1	no	—
122	C₂²	—	37	1	no	—
123	C₂²	—	37	1	no	—
124	C₂²	#11924598	1	1	yes	characteristic
125	C₄	—	8	1	no	—
126	C₄	—	37	1	no	—
127	C₄	—	37	1	no	—
128	C₄	—	296	2	no	—
129	C₄	—	296	1	no	—
130	C₄	(C₂×#3866)⋊C₄	1	1	yes	characteristic
131	C₄	#11924599	1	1	yes	characteristic
132	C₂	—	37	1	no	—
133	C₂	—	37	1	no	—
134	C₂	—	37	1	no	—
135	C₂	—	37	1	no	—
136	C₂	#14558406	1	1	yes	characteristic
137	C₂	#14558445	1	1	yes	characteristic
138	C₂	#14558446	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×D₃₇⋊C₁₆

C₂

yes

maximal, characteristic

C₂×#11924605

C₂

yes

maximal, characteristic

#107×D₃₇

C₂

yes

maximal, characteristic

C₂×C₄⋊#3866

—

D₃₇⋊C₁₆

—

D₃₇⋊C₁₆

C₄

yes

—

—

C₄

yes

complemented

C₂×C₈×D₃₇

C₂²

yes

characteristic

C₄⋊C₄×D₃₇

—

#11924470

C₄

yes

characteristic, complemented

#11924469

—

#107×C₃₇

C₄

yes

characteristic, complemented

C₂²×#3866

—

C₂×C₄×D₃₇

—

C₂×C₄×D₃₇

D₄

yes

characteristic

C₄⋊#3866

—

C₄⋊#3866

—

C₂×C₈×C₃₇

C₂×C₄

yes

characteristic

C₄⋊C₄×C₃₇

—

C₄⋊Dic₃₇

—

#10596936

—

C₂×#64436

D₄

yes

characteristic

C₈×D₃₇

C₂×C₄

yes

characteristic

C₈×D₃₇

C₂×C₄

yes

characteristic

C₈×D₃₇

D₄

yes

characteristic

C₈×D₃₇

Q₈

yes

characteristic

#10596927

—

C₂²×D₃₇

D₈

yes

—

—

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂²⋊C₄

yes

characteristic

C₂×Dic₃₇

—

C₂×Dic₃₇

SD₁₆

yes

characteristic

C₄×D₃₇

—

C₄×D₃₇

—

C₄×D₃₇

C₂²⋊C₄

yes

characteristic

C₄×D₃₇

C₄⋊C₄

yes

characteristic

C₄×D₃₇

SD₁₆

yes

characteristic

C₄×D₃₇

Q₁₆

yes

characteristic

C₈×C₃₇

C₄²

yes

characteristic

C₈×C₃₇

C₄⋊C₄

yes

characteristic

#64436

C₂²⋊C₄

yes

characteristic

#64436

C₄⋊C₄

yes

C₂²×C₃₇

yes

C₂×D₃₇

yes

C₂×D₃₇

yes

C₂×D₃₇

D₁₆

yes

characteristic

C₂×D₃₇

SD₃₂

yes

characteristic

C₂×D₃₇

SD₃₂

yes

characteristic

C₂×D₃₇

Q₃₂

yes

characteristic

#3866

—

C₄×C₃₇

—

C₄×C₃₇

yes

C₄×C₃₇

yes

Dic₃₇

—

Dic₃₇

yes

Dic₃₇

yes

—

C₂×C₃₇

yes

C₂×C₃₇

yes

C₂×C₃₇

yes

D₃₇

yes

D₃₇

yes

D₃₇

yes

D₃₇

yes

C₂²×C₁₆

—

C₂×#107

—

C₂×#107

—

C₃₇

#1380

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

—

—

—

—

yes

characteristic, complemented

C₂²×C₄

—

C₂²×C₄

—

148

—

C₂²×C₄

—

148

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×#3866

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

148

—

C₄⋊C₄

—

148

—

C₁₆

—

100

C₂³

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

296

—

108

C₂×C₄

—

296

—

109

C₂×C₄

—

296

—

110

C₂×C₄

—

296

—

111

C₂×C₄

—

148

—

112

C₂×C₄

—

148

—

113

C₂×C₄

C₂²⋊#3866

yes

characteristic

114

C₈

—

115

C₈

—

116

C₈

C₄×#3866

yes

characteristic

117

C₈

C₄⋊#3866

yes

characteristic

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

—

122

C₂²

—

123

C₂²

—

124

C₂²

#11924598

yes

characteristic

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

—

296

—

129

C₄

—

296

—

130

C₄

(C₂×#3866)⋊C₄

yes

characteristic

131

C₄

#11924599

yes

characteristic

132

C₂

—

133

C₂

—

134

C₂

—

135

C₂

—

136

C₂

#14558406

yes

characteristic

137

C₂

#14558445

yes

characteristic

138

C₂

#14558446

yes

characteristic

	subgroup	G/H
center	C₂²	#11924598
commutator	C₈×C₃₇	C₄²
Frattini	C₂×C₈	C₂×#3866
Fitting	#107×C₃₇	C₄
socle	C₂²×C₃₇	#102
prime radical	C₂²×D₃₇	D₈
cosocle	C₂×C₈×D₃₇	C₂²
hypercenter	#107	#3866
hypocenter	C₃₇	#1380
LCS	G ▷ C₈×C₃₇ ▷ C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #107

subgroup

G/H

center

C₂²

#11924598

commutator

C₈×C₃₇

C₄²

Frattini

C₂×C₈

C₂×#3866

Fitting

#107×C₃₇

C₄

socle

C₂²×C₃₇

#102

prime radical

C₂²×D₃₇

D₈

cosocle

C₂×C₈×D₃₇

C₂²

C₃₇

G ▷ C₈×C₃₇ ▷ C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #107

N	H ≅ G/N	type
6. D₃₇⋊C₁₆	C₄	semidirect
9. #11924605	C₄	semidirect
12. #11924470	C₄	semidirect
14. #107×C₃₇	C₄	semidirect
73. C₃₇	#1380	semidirect
84. #107	#3866	semidirect

H ≅ G/N

type

6. D₃₇⋊C₁₆

C₄

semidirect

9. #11924605

C₄

semidirect

12. #11924470

C₄

semidirect

14. #107×C₃₇

C₄

semidirect

73. C₃₇

#1380

semidirect

84. #107

#3866

semidirect

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

subgroup	index
C₂×D₃₇⋊C₁₆	2
C₂×#11924605	2
#107×D₃₇	2
#1380	37

subgroup

index

C₂×D₃₇⋊C₁₆

C₂×#11924605

#107×D₃₇

#1380

Subgroups — #18009653

Conjugacy classes of subgroups

Complements

Maximal subgroups