Group #18005252 — subgroups

Subgroups — #18005252

Order 4736.

338 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 117 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	#14558144	C₂	1	2	yes	maximal
2	#14557935	C₂	1	1	yes	maximal, characteristic, complemented
3	#14557901	C₂	1	2	yes	maximal, complemented
4	#14557788	C₂	1	2	yes	maximal
5	#14557771	C₂	1	2	yes	maximal
6	#14557640	C₂	1	2	yes	maximal, complemented
7	Q₁₆×Dic₃₇	C₂	1	2	yes	maximal, complemented
8	#14557487	C₂	1	1	yes	maximal, characteristic
9	C₃₇×#494	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#10596929	C₂²	1	1	yes	characteristic
11	C₂×C₈×D₃₇	C₂²	1	1	yes	characteristic
12	#11924531	—	2	4	no	—
13	#11924528	C₂²	1	2	yes	—
14	#11924526	—	2	4	no	—
15	#11924512	C₂²	1	4	yes	—
16	Q₈×Dic₃₇	C₂²	1	4	yes	—
17	#11924493	C₂²	1	1	yes	characteristic
18	#11924470	C₂²	1	2	yes	—
19	#11924468	C₂²	1	2	yes	—
20	#11924460	C₂²	1	4	yes	—
21	#11924460	C₂²	1	4	yes	—
22	#11924447	C₂²	1	1	yes	characteristic
23	#11924446	C₂²	1	1	yes	characteristic
24	C₈×Dic₃₇	C₂²	1	1	yes	characteristic
25	C₂×Q₁₆×C₃₇	C₂²	1	2	yes	—
26	C₂×Q₁₆×C₃₇	C₂²	1	2	yes	—
27	C₄⋊Q₈×C₃₇	C₂²	1	2	yes	—
28	C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
29	C₂×Q₈×C₃₇	C₂³	1	4	yes	—
30	C₂×C₄×D₃₇	C₂³	1	1	yes	characteristic
31	Q₁₆×C₃₇	—	2	4	no	—
32	Q₁₆×C₃₇	—	2	4	no	—
33	C₂×C₈×C₃₇	C₂³	1	1	yes	characteristic
34	C₂×C₈×C₃₇	C₂³	1	1	yes	characteristic
35	C₄⋊C₄×C₃₇	—	2	4	no	—
36	C₄²×C₃₇	C₂³	1	1	yes	characteristic
37	#10596938	—	2	4	no	—
38	#10596938	D₄	1	2	yes	—
39	C₄⋊Dic₃₇	—	2	4	no	—
40	C₄⋊Dic₃₇	C₂³	1	4	yes	—
41	#10596936	—	2	4	no	—
42	#10596936	—	2	4	no	—
43	#10596936	D₄	1	2	yes	—
44	C₄×Dic₃₇	—	2	4	no	—
45	C₄×Dic₃₇	C₂³	1	1	yes	characteristic
46	C₂×#64436	C₂³	1	1	yes	characteristic
47	C₂×#64436	C₂³	1	1	yes	characteristic
48	#10596929	—	2	2	no	—
49	C₈×D₃₇	—	2	2	no	—
50	C₂²×D₃₇	C₂×D₄	1	1	yes	characteristic
51	Q₈×C₃₇	—	2	4	no	—
52	Q₈×C₃₇	—	2	4	no	—
53	C₂×C₄×C₃₇	—	2	4	no	—
54	C₂×C₄×C₃₇	C₂×D₄	1	2	yes	—
55	C₂×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
56	C₂×Dic₃₇	—	2	4	no	—
57	C₂×Dic₃₇	C₂×D₄	1	2	yes	—
58	C₂×Dic₃₇	C₂×D₄	1	1	yes	characteristic
59	C₄×D₃₇	—	2	1	no	—
60	C₄×D₃₇	—	2	1	no	—
61	C₈×C₃₇	#41	1	2	yes	—
62	C₈×C₃₇	#41	1	2	yes	—
63	#64436	—	2	1	no	—
64	#64436	—	2	1	no	—
65	C₂²×C₃₇	C₂²×D₄	1	1	yes	characteristic
66	C₂×D₃₇	—	2	1	no	—
67	C₂×D₃₇	—	2	2	no	—
68	C₄×C₃₇	—	2	2	no	—
69	C₄×C₃₇	—	4	4	no	—
70	C₄×C₃₇	C₂×#41	1	1	yes	characteristic
71	C₄×C₃₇	C₂×#41	1	1	yes	characteristic
72	Dic₃₇	—	2	1	no	—
73	Dic₃₇	—	2	2	no	—
74	Dic₃₇	—	4	4	no	—
75	#3177	—	37	1	no	maximal, Sylow
76	C₂×C₃₇	#532	1	1	yes	characteristic
77	C₂×C₃₇	#578	1	2	yes	—
78	D₃₇	—	4	1	no	—
79	#541	—	37	2	no	—
80	#499	—	37	1	no	—
81	#494	D₃₇	1	1	yes	characteristic, complemented
82	#491	—	37	2	no	—
83	#470	—	37	2	no	—
84	#467	—	37	2	no	—
85	#441	—	37	2	no	—
86	C₄×Q₁₆	—	37	2	no	—
87	#405	—	37	1	no	—
88	C₃₇	#3177	1	1	yes	characteristic, complemented, Sylow
89	C₂×Q₁₆	C₂×D₃₇	1	2	yes	—
90	C₂×Q₁₆	C₂×D₃₇	1	2	yes	—
91	C₂×M₁₆	—	37	1	no	—
92	C₂²×C₈	—	37	1	no	—
93	C₄⋊Q₈	C₂×D₃₇	1	2	yes	—
94	#126	—	74	4	no	—
95	#125	—	37	2	no	—
96	#125	—	74	4	no	—
97	C₂²⋊Q₈	—	37	4	no	—
98	C₄×Q₈	—	37	4	no	—
99	#117	—	37	1	no	—
100	#107	—	37	2	no	—
101	#106	—	37	2	no	—
102	#103	—	37	4	no	—
103	#103	—	37	4	no	—
104	#97	—	37	1	no	—
105	#97	—	37	1	no	—
106	C₄×C₈	—	37	1	no	—
107	C₄×C₈	C₂×D₃₇	1	1	yes	characteristic
108	C₂×Q₈	C₂²×D₃₇	1	4	yes	—
109	C₂²×C₄	—	37	1	no	—
110	Q₁₆	—	2	4	no	—
111	Q₁₆	—	2	4	no	—
112	M₁₆	—	74	2	no	—
113	C₂×C₈	—	37	1	no	—
114	C₂×C₈	—	37	1	no	—
115	C₂×C₈	—	74	2	no	—
116	C₂×C₈	C₂²×D₃₇	1	1	yes	characteristic
117	C₂×C₈	C₂²×D₃₇	1	1	yes	characteristic
118	C₄⋊C₄	—	2	4	no	—
119	C₄⋊C₄	—	37	2	no	—
120	C₄⋊C₄	—	37	4	no	—
121	C₄⋊C₄	—	74	4	no	—
122	C₄⋊C₄	—	74	4	no	—
123	C₄⋊C₄	—	74	4	no	—
124	C₂²⋊C₄	—	37	2	no	—
125	C₂²⋊C₄	—	74	4	no	—
126	C₄²	—	37	1	no	—
127	C₄²	—	74	4	no	—
128	C₄²	C₂²×D₃₇	1	1	yes	characteristic
129	C₂³	—	37	1	no	—
130	Q₈	—	2	4	no	—
131	Q₈	—	2	4	no	—
132	C₂×C₄	—	2	4	no	—
133	C₂×C₄	—	37	2	no	—
134	C₂×C₄	—	37	1	no	—
135	C₂×C₄	—	74	1	no	—
136	C₂×C₄	—	74	1	no	—
137	C₂×C₄	—	74	4	no	—
138	C₂×C₄	D₄×D₃₇	1	2	yes	—
139	C₂×C₄	C₂³×D₃₇	1	1	yes	characteristic
140	C₈	—	74	1	no	—
141	C₈	—	74	1	no	—
142	C₈	#10596964	1	2	yes	—
143	C₈	#10596964	1	2	yes	—
144	C₂²	—	74	1	no	—
145	C₂²	—	74	2	no	—
146	C₂²	C₂×D₄×D₃₇	1	1	yes	characteristic
147	C₄	—	2	2	no	—
148	C₄	—	4	4	no	—
149	C₄	—	74	1	no	—
150	C₄	—	74	2	no	—
151	C₄	—	148	4	no	—
152	C₄	C₂×#10596964	1	1	yes	characteristic
153	C₄	C₂×#10596964	1	1	yes	characteristic
154	C₂	—	148	1	no	—
155	C₂	#14558081	1	1	yes	characteristic
156	C₂	#14558346	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14558144

C₂

yes

maximal

#14557935

C₂

yes

maximal, characteristic, complemented

#14557901

C₂

yes

maximal, complemented

#14557788

C₂

yes

maximal

#14557771

C₂

yes

maximal

#14557640

C₂

yes

maximal, complemented

Q₁₆×Dic₃₇

C₂

yes

maximal, complemented

#14557487

C₂

yes

maximal, characteristic

C₃₇×#494

C₂

yes

maximal, characteristic, complemented

C₂×#10596929

C₂²

yes

characteristic

C₂×C₈×D₃₇

C₂²

yes

characteristic

#11924531

—

#11924528

C₂²

yes

—

#11924526

—

#11924512

C₂²

yes

—

Q₈×Dic₃₇

C₂²

yes

—

#11924493

C₂²

yes

characteristic

#11924470

C₂²

yes

—

#11924468

C₂²

yes

—

#11924460

C₂²

yes

—

#11924460

C₂²

yes

—

#11924447

C₂²

yes

characteristic

#11924446

C₂²

yes

characteristic

C₈×Dic₃₇

C₂²

yes

characteristic

C₂×Q₁₆×C₃₇

C₂²

yes

—

C₂×Q₁₆×C₃₇

C₂²

yes

—

C₄⋊Q₈×C₃₇

C₂²

yes

—

C₄×C₈×C₃₇

C₂²

yes

characteristic

C₂×Q₈×C₃₇

C₂³

yes

—

C₂×C₄×D₃₇

C₂³

yes

characteristic

Q₁₆×C₃₇

—

Q₁₆×C₃₇

—

C₂×C₈×C₃₇

C₂³

yes

characteristic

C₂×C₈×C₃₇

C₂³

yes

characteristic

C₄⋊C₄×C₃₇

—

C₄²×C₃₇

C₂³

yes

characteristic

#10596938

—

#10596938

D₄

yes

—

C₄⋊Dic₃₇

—

C₄⋊Dic₃₇

C₂³

yes

—

#10596936

—

#10596936

—

#10596936

D₄

yes

—

C₄×Dic₃₇

—

C₄×Dic₃₇

C₂³

yes

characteristic

C₂×#64436

C₂³

yes

characteristic

C₂×#64436

C₂³

yes

characteristic

#10596929

—

C₈×D₃₇

—

C₂²×D₃₇

C₂×D₄

yes

characteristic

Q₈×C₃₇

—

Q₈×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂×D₄

yes

—

C₂×C₄×C₃₇

C₂⁴

yes

characteristic

C₂×Dic₃₇

—

C₂×Dic₃₇

C₂×D₄

yes

—

C₂×Dic₃₇

C₂×D₄

yes

characteristic

C₄×D₃₇

—

C₄×D₃₇

—

C₈×C₃₇

#41

yes

—

C₈×C₃₇

#41

yes

—

#64436

—

#64436

—

C₂²×C₃₇

C₂²×D₄

yes

characteristic

C₂×D₃₇

—

C₂×D₃₇

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

C₂×#41

yes

characteristic

C₄×C₃₇

C₂×#41

yes

characteristic

Dic₃₇

—

Dic₃₇

—

Dic₃₇

—

—

C₂×C₃₇

yes

C₂×C₃₇

yes

—

D₃₇

—

#541

—

#499

—

#494

D₃₇

yes

characteristic, complemented

—

—

—

—

C₄×Q₁₆

—

#405

—

C₃₇

#3177

yes

characteristic, complemented, Sylow

C₂×Q₁₆

C₂×D₃₇

yes

—

C₂×Q₁₆

C₂×D₃₇

yes

—

C₂×M₁₆

—

C₂²×C₈

—

C₄⋊Q₈

C₂×D₃₇

yes

—

#126

—

#125

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

100

#107

—

101

#106

—

102

#103

—

103

#103

—

104

#97

—

105

#97

—

106

C₄×C₈

—

107

C₄×C₈

C₂×D₃₇

yes

characteristic

108

C₂×Q₈

C₂²×D₃₇

yes

—

109

C₂²×C₄

—

110

Q₁₆

—

111

Q₁₆

—

112

M₁₆

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

C₂²×D₃₇

yes

characteristic

117

C₂×C₈

C₂²×D₃₇

yes

characteristic

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₂²⋊C₄

—

125

C₂²⋊C₄

—

126

C₄²

—

127

C₄²

—

128

C₄²

C₂²×D₃₇

yes

characteristic

129

C₂³

—

130

Q₈

—

131

Q₈

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

D₄×D₃₇

yes

—

139

C₂×C₄

C₂³×D₃₇

yes

characteristic

140

C₈

—

141

C₈

—

142

C₈

#10596964

yes

—

143

C₈

#10596964

yes

—

144

C₂²

—

145

C₂²

—

146

C₂²

C₂×D₄×D₃₇

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

—

148

—

152

C₄

C₂×#10596964

yes

characteristic

153

C₄

C₂×#10596964

yes

characteristic

154

C₂

—

148

—

155

C₂

#14558081

yes

characteristic

156

C₂

#14558346

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₃₇
commutator, cosocle	C₂×C₄×C₃₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₃₇
Fitting	C₃₇×#494	C₂
socle	C₂²×C₃₇	C₂²×D₄
prime radical	C₂²×D₃₇	C₂×D₄
hypercenter	#494	D₃₇
hypocenter	C₃₇	#3177
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

subgroup

G/H

center

C₂²

C₂×D₄×D₃₇

commutator, cosocle

C₂×C₄×C₃₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₃₇

Fitting

C₃₇×#494

C₂

socle

C₂²×C₃₇

C₂²×D₄

prime radical

C₂²×D₃₇

C₂×D₄

D₃₇

C₃₇

G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #494

N	H ≅ G/N	type
2. #14557935	C₂	semidirect
3. #14557901	C₂	semidirect
6. #14557640	C₂	semidirect
7. Q₁₆×Dic₃₇	C₂	semidirect
9. C₃₇×#494	C₂	semidirect
81. #494	D₃₇	semidirect
88. C₃₇	#3177	semidirect

H ≅ G/N

type

2. #14557935

C₂

semidirect

3. #14557901

C₂

semidirect

6. #14557640

C₂

semidirect

7. Q₁₆×Dic₃₇

C₂

semidirect

9. C₃₇×#494

C₂

semidirect

81. #494

D₃₇

semidirect

88. C₃₇

#3177

semidirect

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

subgroup	index
#14558144	2
#14557935	2
#14557901	2
#14557788	2
#14557771	2
#14557640	2
Q₁₆×Dic₃₇	2
#14557487	2
C₃₇×#494	2
#3177	37

subgroup

index

Q₁₆×Dic₃₇

C₃₇×#494

Subgroups — #18005252

Conjugacy classes of subgroups

Complements

Maximal subgroups