Group #18004996 — subgroups

Subgroups — #18004996

Order 4736.

392 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 133 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, complemented
2	#14557901	C₂	1	2	yes	maximal, complemented
3	#14557897	C₂	1	2	yes	maximal, complemented
4	#14557887	C₂	1	4	yes	maximal, complemented
5	#14557534	C₂	1	4	yes	maximal, complemented
6	C₄×#10596934	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₇×#532	C₂	1	1	yes	maximal, characteristic
8	C₂×#10596934	C₂²	1	2	yes	—
9	#11924528	—	2	2	no	—
10	#11924528	C₂²	1	2	yes	—
11	#11924528	C₂²	1	2	yes	—
12	C₂×C₄⋊Dic₃₇	C₂²	1	2	yes	—
13	C₂×#10596936	—	2	2	no	—
14	#11924460	C₂²	1	8	yes	—
15	#11924457	C₂²	1	8	yes	—
16	#11924447	C₂²	1	2	yes	—
17	C₄×#64436	C₂²	1	2	yes	—
18	C₂×#41×C₃₇	C₂²	1	2	yes	—
19	C₄⋊Q₈×C₃₇	C₂²	1	1	yes	characteristic
20	#127×C₃₇	C₂²	1	1	yes	characteristic
21	#124×C₃₇	C₂²	1	2	yes	—
22	#121×C₃₇	—	2	2	no	—
23	C₄×D₄×C₃₇	—	2	2	no	—
24	C₂×C₄²×C₃₇	C₂²	1	1	yes	characteristic
25	#41×C₃₇	—	2	8	no	—
26	C₂×Q₈×C₃₇	C₂³	1	2	yes	—
27	C₂×D₄×C₃₇	—	2	2	no	—
28	C₂×D₄×C₃₇	—	2	2	no	—
29	C₂×D₄×C₃₇	C₂³	1	2	yes	—
30	C₂²×C₄×C₃₇	—	2	2	no	—
31	C₂²×C₄×C₃₇	D₄	1	2	yes	—
32	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
33	C₂²×Dic₃₇	—	2	2	no	—
34	C₄⋊C₄×C₃₇	—	2	2	no	—
35	C₂²⋊C₄×C₃₇	—	2	4	no	—
36	C₄²×C₃₇	D₄	1	2	yes	—
37	C₄²×C₃₇	C₂³	1	1	yes	characteristic
38	C₄²×C₃₇	C₂³	1	1	yes	characteristic
39	C₄⋊Dic₃₇	—	2	2	no	—
40	C₄⋊Dic₃₇	C₂³	1	4	yes	—
41	#10596936	—	2	4	no	—
42	#10596936	—	2	4	no	—
43	#10596934	—	2	4	no	—
44	C₂×#64436	C₂³	1	4	yes	—
45	C₂³×C₃₇	—	2	2	no	—
46	C₂³×C₃₇	C₂×D₄	1	1	yes	characteristic
47	Q₈×C₃₇	—	2	4	no	—
48	D₄×C₃₇	—	2	4	no	—
49	D₄×C₃₇	—	4	4	no	—
50	D₄×C₃₇	—	4	4	no	—
51	C₂×C₄×C₃₇	—	2	2	no	—
52	C₂×C₄×C₃₇	—	4	2	no	—
53	C₂×C₄×C₃₇	—	4	4	no	—
54	C₂×C₄×C₃₇	C₂×D₄	1	2	yes	—
55	C₂×C₄×C₃₇	C₂×D₄	1	2	yes	—
56	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₃₇	#41	1	4	yes	—
58	C₂×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
59	C₂×Dic₃₇	—	2	4	no	—
60	C₂×Dic₃₇	—	4	4	no	—
61	#64436	—	2	4	no	—
62	C₂²×C₃₇	—	2	2	no	—
63	C₂²×C₃₇	—	4	2	no	—
64	C₂²×C₃₇	—	4	4	no	—
65	C₂²×C₃₇	#124	1	2	yes	—
66	C₂²×C₃₇	C₂²×D₄	1	1	yes	characteristic
67	C₄×C₃₇	—	2	2	no	—
68	C₄×C₃₇	—	2	2	no	—
69	C₄×C₃₇	—	4	2	no	—
70	C₄×C₃₇	#124	1	2	yes	—
71	C₄×C₃₇	C₂×#41	1	2	yes	—
72	Dic₃₇	—	4	4	no	—
73	#3142	—	37	1	no	maximal, Sylow
74	C₂×C₃₇	—	2	2	no	—
75	C₂×C₃₇	—	8	2	no	—
76	C₂×C₃₇	C₂×#124	1	1	yes	characteristic
77	C₂×C₃₇	#575	1	2	yes	—
78	#532	D₃₇	1	1	yes	characteristic
79	C₂×#125	—	37	1	no	—
80	#491	—	37	2	no	—
81	#490	—	37	2	no	—
82	#488	—	37	4	no	—
83	#417	—	37	4	no	—
84	C₄×M₁₆	—	37	1	no	—
85	C₃₇	#3142	1	1	yes	characteristic, complemented, Sylow
86	C₂×#41	C₂×D₃₇	1	2	yes	—
87	C₂×M₁₆	—	37	2	no	—
88	C₄⋊Q₈	C₂×D₃₇	1	1	yes	characteristic
89	#127	C₂×D₃₇	1	1	yes	characteristic
90	#125	—	37	2	no	—
91	#125	—	37	2	no	—
92	#125	—	74	2	no	—
93	#124	C₂×D₃₇	1	2	yes	—
94	#121	—	2	2	no	—
95	C₄×D₄	—	2	2	no	—
96	C₂×C₄⋊C₄	—	37	2	no	—
97	C₂×C₄⋊C₄	—	74	2	no	—
98	C₂×C₄²	C₂×D₃₇	1	1	yes	characteristic
99	#103	—	37	8	no	—
100	#102	—	37	8	no	—
101	#97	—	37	2	no	—
102	C₄×C₈	—	37	2	no	—
103	#41	—	2	8	no	—
104	C₂×Q₈	C₂²×D₃₇	1	2	yes	—
105	C₂×D₄	—	2	2	no	—
106	C₂×D₄	—	2	2	no	—
107	C₂×D₄	C₂²×D₃₇	1	2	yes	—
108	C₂²×C₄	—	2	2	no	—
109	C₂²×C₄	—	74	2	no	—
110	C₂²×C₄	C₂²⋊D₃₇	1	2	yes	—
111	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
112	M₁₆	—	74	4	no	—
113	C₂×C₈	—	37	4	no	—
114	C₄⋊C₄	—	2	2	no	—
115	C₄⋊C₄	—	37	4	no	—
116	C₄⋊C₄	—	74	4	no	—
117	C₄⋊C₄	—	74	4	no	—
118	C₄⋊C₄	—	74	2	no	—
119	C₂²⋊C₄	—	2	4	no	—
120	C₄²	C₂²⋊D₃₇	1	2	yes	—
121	C₄²	C₂²×D₃₇	1	1	yes	characteristic
122	C₄²	C₂²×D₃₇	1	1	yes	characteristic
123	C₂³	—	2	2	no	—
124	C₂³	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
125	Q₈	—	2	4	no	—
126	D₄	—	2	4	no	—
127	D₄	—	4	4	no	—
128	D₄	—	4	4	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	4	2	no	—
131	C₂×C₄	—	4	4	no	—
132	C₂×C₄	—	74	4	no	—
133	C₂×C₄	—	148	4	no	—
134	C₂×C₄	#10596964	1	4	yes	—
135	C₂×C₄	C₂×C₂²⋊D₃₇	1	2	yes	—
136	C₂×C₄	C₂×C₂²⋊D₃₇	1	2	yes	—
137	C₂×C₄	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
138	C₂×C₄	C₂³×D₃₇	1	1	yes	characteristic
139	C₈	—	74	4	no	—
140	C₂²	—	2	2	no	—
141	C₂²	—	4	2	no	—
142	C₂²	—	4	4	no	—
143	C₂²	#11924523	1	2	yes	—
144	C₂²	C₂²×C₂²⋊D₃₇	1	1	yes	characteristic
145	C₄	—	2	2	no	—
146	C₄	—	2	2	no	—
147	C₄	—	4	2	no	—
148	C₄	—	148	4	no	—
149	C₄	#11924523	1	2	yes	—
150	C₄	C₂×#10596964	1	2	yes	—
151	C₂	—	2	2	no	—
152	C₂	—	8	2	no	—
153	C₂	C₂×#11924523	1	1	yes	characteristic
154	C₂	#14558331	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11924528

C₂

yes

maximal, characteristic, complemented

#14557901

C₂

yes

maximal, complemented

#14557897

C₂

yes

maximal, complemented

#14557887

C₂

yes

maximal, complemented

#14557534

C₂

yes

maximal, complemented

C₄×#10596934

C₂

yes

maximal, characteristic, complemented

C₃₇×#532

C₂

yes

maximal, characteristic

C₂×#10596934

C₂²

yes

—

#11924528

—

#11924528

C₂²

yes

—

#11924528

C₂²

yes

—

C₂×C₄⋊Dic₃₇

C₂²

yes

—

C₂×#10596936

—

#11924460

C₂²

yes

—

#11924457

C₂²

yes

—

#11924447

C₂²

yes

—

C₄×#64436

C₂²

yes

—

C₂×#41×C₃₇

C₂²

yes

—

C₄⋊Q₈×C₃₇

C₂²

yes

characteristic

#127×C₃₇

C₂²

yes

characteristic

#124×C₃₇

C₂²

yes

—

#121×C₃₇

—

C₄×D₄×C₃₇

—

C₂×C₄²×C₃₇

C₂²

yes

characteristic

#41×C₃₇

—

C₂×Q₈×C₃₇

C₂³

yes

—

C₂×D₄×C₃₇

—

C₂×D₄×C₃₇

—

C₂×D₄×C₃₇

C₂³

yes

—

C₂²×C₄×C₃₇

—

C₂²×C₄×C₃₇

D₄

yes

—

C₂²×C₄×C₃₇

C₂³

yes

characteristic

C₂²×Dic₃₇

—

C₄⋊C₄×C₃₇

—

C₂²⋊C₄×C₃₇

—

C₄²×C₃₇

D₄

yes

—

C₄²×C₃₇

C₂³

yes

characteristic

C₄²×C₃₇

C₂³

yes

characteristic

C₄⋊Dic₃₇

—

C₄⋊Dic₃₇

C₂³

yes

—

—

—

—

C₂³

yes

—

C₂³×C₃₇

—

C₂³×C₃₇

C₂×D₄

yes

characteristic

Q₈×C₃₇

—

D₄×C₃₇

—

D₄×C₃₇

—

D₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂×D₄

yes

—

C₂×C₄×C₃₇

C₂×D₄

yes

—

C₂×C₄×C₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₃₇

#41

yes

—

C₂×C₄×C₃₇

C₂⁴

yes

characteristic

C₂×Dic₃₇

—

C₂×Dic₃₇

—

#64436

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

#124

yes

—

C₂²×C₃₇

C₂²×D₄

yes

characteristic

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

#124

yes

—

C₄×C₃₇

C₂×#41

yes

—

Dic₃₇

—

#3142

—

maximal, Sylow

C₂×C₃₇

—

C₂×C₃₇

—

C₂×C₃₇

C₂×#124

yes

characteristic

C₂×C₃₇

#575

yes

—

#532

D₃₇

yes

—

—

—

—

—

C₄×M₁₆

—

C₃₇

#3142

yes

characteristic, complemented, Sylow

C₂×#41

C₂×D₃₇

yes

—

C₂×M₁₆

—

C₄⋊Q₈

C₂×D₃₇

yes

characteristic

#127

C₂×D₃₇

yes

—

—

—

C₂×D₃₇

yes

—

#121

—

C₄×D₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₃₇

yes

characteristic

#103

—

100

#102

—

101

#97

—

102

C₄×C₈

—

103

#41

—

104

C₂×Q₈

C₂²×D₃₇

yes

—

105

C₂×D₄

—

106

C₂×D₄

—

107

C₂×D₄

C₂²×D₃₇

yes

—

108

C₂²×C₄

—

109

C₂²×C₄

—

110

C₂²×C₄

C₂²⋊D₃₇

yes

—

111

C₂²×C₄

C₂²×D₃₇

yes

characteristic

112

M₁₆

—

113

C₂×C₈

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₂²⋊C₄

—

120

C₄²

C₂²⋊D₃₇

yes

—

121

C₄²

C₂²×D₃₇

yes

characteristic

122

C₄²

C₂²×D₃₇

yes

characteristic

123

C₂³

—

124

C₂³

C₂×C₂²⋊D₃₇

yes

characteristic

125

Q₈

—

126

D₄

—

127

D₄

—

128

D₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

148

—

134

C₂×C₄

#10596964

yes

—

135

C₂×C₄

C₂×C₂²⋊D₃₇

yes

—

136

C₂×C₄

C₂×C₂²⋊D₃₇

yes

—

137

C₂×C₄

C₂×C₂²⋊D₃₇

yes

characteristic

138

C₂×C₄

C₂³×D₃₇

yes

characteristic

139

C₈

—

140

C₂²

—

141

C₂²

—

142

C₂²

—

143

C₂²

#11924523

yes

—

144

C₂²

C₂²×C₂²⋊D₃₇

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

148

—

149

C₄

#11924523

yes

—

150

C₄

C₂×#10596964

yes

—

151

C₂

—

152

C₂

—

153

C₂

C₂×#11924523

yes

characteristic

154

C₂

#14558331

yes

—

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₃₇
commutator, cosocle	C₂×C₄×C₃₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₃₇
Fitting, prime radical	C₃₇×#532	C₂
socle	C₂²×C₃₇	C₂²×D₄
hypercenter	#532	D₃₇
hypocenter	C₃₇	#3142
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #532

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₃₇

commutator, cosocle

C₂×C₄×C₃₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₃₇

Fitting, prime radical

C₃₇×#532

C₂

socle

C₂²×C₃₇

C₂²×D₄

D₃₇

C₃₇

G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #532

N	H ≅ G/N	type
1. C₂×#11924528	C₂	semidirect
2. #14557901	C₂	semidirect
3. #14557897	C₂	semidirect
4. #14557887	C₂	semidirect
5. #14557534	C₂	semidirect
6. C₄×#10596934	C₂	semidirect
85. C₃₇	#3142	semidirect

H ≅ G/N

type

1. C₂×#11924528

C₂

semidirect

2. #14557901

C₂

semidirect

3. #14557897

C₂

semidirect

4. #14557887

C₂

semidirect

5. #14557534

C₂

semidirect

6. C₄×#10596934

C₂

semidirect

85. C₃₇

#3142

semidirect

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

subgroup	index
C₂×#11924528	2
#14557901	2
#14557897	2
#14557887	2
#14557534	2
C₄×#10596934	2
C₃₇×#532	2
#3142	37

subgroup

index

C₃₇×#532

Subgroups — #18004996

Conjugacy classes of subgroups

Complements

Maximal subgroups