Group #18003315 — subgroups

Subgroups — #18003315

Order 4736.

484 conjugacy classes of subgroups; only the 189 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 189 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×#11924542	C₂	1	2	yes	maximal, complemented
2	#14558083	C₂	1	1	yes	maximal, characteristic
3	#14557606	C₂	1	8	yes	maximal, complemented
4	#14557557	C₂	1	1	yes	maximal, characteristic, complemented
5	#14557499	C₂	1	2	yes	maximal, complemented
6	C₂×#97×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#10596962	C₂²	1	2	yes	—
8	#11924542	—	2	8	no	—
9	C₂×#10596934	C₂²	1	2	yes	—
10	C₂×#10596929	C₂²	1	4	yes	—
11	C₂×C₈×D₃₇	C₂²	1	4	yes	—
12	#11924536	C₄	1	1	yes	characteristic
13	#11924533	C₄	1	1	yes	characteristic
14	#11924524	C₄	1	2	yes	—
15	#11924510	C₄	1	4	yes	—
16	C₄×D₁₄₈	C₂²	1	4	yes	—
17	C₄⋊#64436	C₂²	1	4	yes	—
18	#11924448	C₂²	1	8	yes	—
19	C₂²×C₈×C₃₇	C₂²	1	1	yes	characteristic
20	C₂²×C₈×C₃₇	C₂²	1	1	yes	characteristic
21	C₂×C₄²×C₃₇	C₂²	1	1	yes	characteristic
22	#97×C₃₇	C₂²	1	4	yes	—
23	C₂²×C₄×C₃₇	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
25	C₂×C₂²⋊D₃₇	C₂×C₄	1	4	yes	—
26	#10596962	—	2	4	no	—
27	#10596962	—	2	4	no	—
28	C₂×D₁₄₈	C₂×C₄	1	2	yes	—
29	C₂×D₁₄₈	C₂×C₄	1	4	yes	—
30	C₂×C₄×D₃₇	C₂³	1	4	yes	—
31	C₂×Dic₇₄	C₂×C₄	1	2	yes	—
32	C₂×C₈×C₃₇	—	2	2	no	—
33	C₂×C₈×C₃₇	D₄	1	4	yes	complemented
34	C₂×C₈×C₃₇	C₂³	1	2	yes	—
35	C₂×C₈×C₃₇	C₂³	1	2	yes	—
36	C₄²×C₃₇	C₂×C₄	1	1	yes	characteristic
37	C₄²×C₃₇	C₂×C₄	1	1	yes	characteristic
38	C₄²×C₃₇	C₂³	1	2	yes	—
39	#10596938	C₂×C₄	1	8	yes	—
40	C₄⋊Dic₃₇	C₂×C₄	1	4	yes	—
41	#10596934	—	2	4	no	—
42	C₂×#64436	C₂³	1	4	yes	—
43	#10596929	—	2	8	no	—
44	C₈×D₃₇	—	2	8	no	—
45	C₂³×C₃₇	C₂²×C₄	1	1	yes	characteristic
46	C₂²×D₃₇	C₂²×C₄	1	4	yes	—
47	C₂×C₄×C₃₇	—	4	2	no	—
48	C₂×C₄×C₃₇	C₂²×C₄	1	2	yes	—
49	C₂×C₄×C₃₇	C₂²×C₄	1	2	yes	—
50	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₃₇	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₃₇	#41	1	2	yes	—
53	C₂×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
54	C₂²⋊D₃₇	—	2	8	no	—
55	C₂×Dic₃₇	C₂²×C₄	1	4	yes	—
56	D₁₄₈	—	2	4	no	—
57	D₁₄₈	—	4	4	no	—
58	C₄×D₃₇	—	2	4	no	—
59	C₄×D₃₇	—	2	4	no	—
60	Dic₇₄	—	2	4	no	—
61	C₈×C₃₇	—	2	2	no	—
62	C₈×C₃₇	C₂×D₄	1	4	yes	—
63	#64436	—	2	4	no	—
64	C₂²×C₃₇	—	2	2	no	—
65	C₂²×C₃₇	C₄×D₄	1	2	yes	—
66	C₂²×C₃₇	C₂³×C₄	1	1	yes	characteristic
67	C₂×D₃₇	—	2	4	no	—
68	C₂×D₃₇	—	2	8	no	—
69	C₄×C₃₇	—	2	2	no	—
70	C₄×C₃₇	—	2	2	no	—
71	C₄×C₃₇	C₄×D₄	1	2	yes	—
72	C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
73	C₄×C₃₇	C₂×#41	1	1	yes	characteristic
74	Dic₃₇	—	2	4	no	—
75	#2932	—	37	1	no	maximal, Sylow
76	C₂×C₃₇	—	2	2	no	—
77	C₂×C₃₇	C₂×C₄×D₄	1	1	yes	characteristic
78	C₂×C₃₇	#568	1	2	yes	—
79	D₃₇	—	4	4	no	—
80	C₂×#131	—	37	2	no	—
81	#532	—	37	1	no	—
82	#435	—	37	8	no	—
83	#423	—	37	1	no	—
84	#408	—	37	2	no	—
85	C₂×#97	D₃₇	1	1	yes	characteristic, complemented
86	C₃₇	#2932	1	1	yes	characteristic, complemented, Sylow
87	C₂×#41	—	37	2	no	—
88	#131	—	74	8	no	—
89	C₂×M₁₆	—	37	2	no	—
90	C₂×M₁₆	—	37	4	no	—
91	C₂²×C₈	—	37	4	no	—
92	C₂²×C₈	C₂×D₃₇	1	1	yes	characteristic
93	C₂²×C₈	C₂×D₃₇	1	1	yes	characteristic
94	C₄⋊Q₈	—	37	1	no	—
95	#127	—	37	1	no	—
96	#124	—	37	2	no	—
97	#121	—	37	4	no	—
98	C₄×D₄	—	37	4	no	—
99	C₂×C₄²	C₂×D₃₇	1	1	yes	characteristic
100	C₄⋊C₈	—	37	4	no	—
101	C₂²⋊C₈	—	37	8	no	—
102	#97	C₂×D₃₇	1	4	yes	—
103	#41	—	74	4	no	—
104	#41	—	74	4	no	—
105	C₂×Q₈	—	37	2	no	—
106	C₂×D₄	—	37	4	no	—
107	C₂×D₄	—	37	2	no	—
108	C₂×D₄	—	37	4	no	—
109	C₂²×C₄	—	37	4	no	—
110	C₂²×C₄	C₄×D₃₇	1	2	yes	—
111	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
112	M₁₆	—	74	4	no	—
113	M₁₆	—	74	8	no	—
114	C₂×C₈	—	2	2	no	—
115	C₂×C₈	—	37	4	no	—
116	C₂×C₈	—	74	8	no	—
117	C₂×C₈	D₁₄₈	1	4	yes	complemented
118	C₂×C₈	C₂²×D₃₇	1	2	yes	—
119	C₂×C₈	C₂²×D₃₇	1	2	yes	—
120	C₄⋊C₄	—	37	4	no	—
121	C₂²⋊C₄	—	37	8	no	—
122	C₄²	C₄×D₃₇	1	1	yes	characteristic
123	C₄²	C₄×D₃₇	1	1	yes	characteristic
124	C₄²	C₂²×D₃₇	1	2	yes	—
125	C₂³	—	37	4	no	—
126	C₂³	C₂×C₄×D₃₇	1	1	yes	characteristic
127	Q₈	—	74	4	no	—
128	D₄	—	74	8	no	—
129	D₄	—	74	4	no	—
130	D₄	—	148	4	no	—
131	C₂×C₄	—	4	2	no	—
132	C₂×C₄	—	37	4	no	—
133	C₂×C₄	—	74	4	no	—
134	C₂×C₄	—	74	4	no	—
135	C₂×C₄	C₂×C₄×D₃₇	1	2	yes	—
136	C₂×C₄	C₂×C₄×D₃₇	1	2	yes	—
137	C₂×C₄	C₂×C₄×D₃₇	1	1	yes	characteristic
138	C₂×C₄	C₂×D₁₄₈	1	2	yes	—
139	C₂×C₄	#10596962	1	2	yes	—
140	C₂×C₄	C₂³×D₃₇	1	1	yes	characteristic
141	C₈	—	2	2	no	—
142	C₈	—	74	4	no	—
143	C₈	C₂×D₁₄₈	1	4	yes	—
144	C₂²	—	2	2	no	—
145	C₂²	—	74	4	no	—
146	C₂²	—	74	8	no	—
147	C₂²	C₄×D₁₄₈	1	2	yes	—
148	C₂²	C₂²×C₄×D₃₇	1	1	yes	characteristic
149	C₄	—	2	2	no	—
150	C₄	—	2	2	no	—
151	C₄	—	74	4	no	—
152	C₄	C₄×D₁₄₈	1	2	yes	—
153	C₄	C₂²×D₁₄₈	1	1	yes	characteristic
154	C₄	C₂×#10596962	1	1	yes	characteristic
155	C₂	—	2	2	no	—
156	C₂	—	148	4	no	—
157	C₂	C₂×C₄×D₁₄₈	1	1	yes	characteristic
158	C₂	#14558299	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11924542

C₂

yes

maximal, complemented

#14558083

C₂

yes

maximal, characteristic

#14557606

C₂

yes

maximal, complemented

#14557557

C₂

yes

maximal, characteristic, complemented

#14557499

C₂

yes

maximal, complemented

C₂×#97×C₃₇

C₂

yes

maximal, characteristic, complemented

C₂×#10596962

C₂²

yes

—

#11924542

—

C₂×#10596934

C₂²

yes

—

C₂×#10596929

C₂²

yes

—

C₂×C₈×D₃₇

C₂²

yes

—

#11924536

C₄

yes

characteristic

#11924533

C₄

yes

characteristic

#11924524

C₄

yes

—

#11924510

C₄

yes

—

C₄×D₁₄₈

C₂²

yes

—

C₄⋊#64436

C₂²

yes

—

#11924448

C₂²

yes

—

C₂²×C₈×C₃₇

C₂²

yes

characteristic

C₂²×C₈×C₃₇

C₂²

yes

characteristic

C₂×C₄²×C₃₇

C₂²

yes

characteristic

#97×C₃₇

C₂²

yes

—

C₂²×C₄×C₃₇

C₂×C₄

yes

—

C₂²×C₄×C₃₇

C₂³

yes

characteristic

C₂×C₂²⋊D₃₇

C₂×C₄

yes

—

#10596962

—

#10596962

—

C₂×D₁₄₈

C₂×C₄

yes

—

C₂×D₁₄₈

C₂×C₄

yes

—

C₂×C₄×D₃₇

C₂³

yes

—

C₂×Dic₇₄

C₂×C₄

yes

—

C₂×C₈×C₃₇

—

C₂×C₈×C₃₇

D₄

yes

complemented

C₂×C₈×C₃₇

C₂³

yes

—

C₂×C₈×C₃₇

C₂³

yes

—

C₄²×C₃₇

C₂×C₄

yes

characteristic

C₄²×C₃₇

C₂×C₄

yes

characteristic

C₄²×C₃₇

C₂³

yes

—

#10596938

C₂×C₄

yes

—

C₄⋊Dic₃₇

C₂×C₄

yes

—

#10596934

—

C₂×#64436

C₂³

yes

—

#10596929

—

C₈×D₃₇

—

C₂³×C₃₇

C₂²×C₄

yes

characteristic

C₂²×D₃₇

C₂²×C₄

yes

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂²×C₄

yes

—

C₂×C₄×C₃₇

C₂²×C₄

yes

—

C₂×C₄×C₃₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₇

C₂×D₄

yes

—

C₂×C₄×C₃₇

#41

yes

—

C₂×C₄×C₃₇

C₂⁴

yes

characteristic

C₂²⋊D₃₇

—

C₂×Dic₃₇

C₂²×C₄

yes

—

D₁₄₈

—

D₁₄₈

—

C₄×D₃₇

—

C₄×D₃₇

—

Dic₇₄

—

C₈×C₃₇

—

C₈×C₃₇

C₂×D₄

yes

—

#64436

—

C₂²×C₃₇

—

C₂²×C₃₇

C₄×D₄

yes

—

C₂²×C₃₇

C₂³×C₄

yes

characteristic

C₂×D₃₇

—

C₂×D₃₇

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

C₄×D₄

yes

—

C₄×C₃₇

C₂²×D₄

yes

characteristic

C₄×C₃₇

C₂×#41

yes

characteristic

Dic₃₇

—

#2932

—

maximal, Sylow

C₂×C₃₇

—

C₂×C₃₇

C₂×C₄×D₄

yes

characteristic

C₂×C₃₇

#568

yes

—

D₃₇

—

—

—

—

—

—

C₂×#97

D₃₇

yes

characteristic, complemented

C₃₇

#2932

yes

characteristic, complemented, Sylow

C₂×#41

—

#131

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂×D₃₇

yes

characteristic

C₂²×C₈

C₂×D₃₇

yes

characteristic

C₄⋊Q₈

—

#127

—

#124

—

#121

—

C₄×D₄

—

C₂×C₄²

C₂×D₃₇

yes

characteristic

100

C₄⋊C₈

—

101

C₂²⋊C₈

—

102

#97

C₂×D₃₇

yes

—

103

#41

—

104

#41

—

105

C₂×Q₈

—

106

C₂×D₄

—

107

C₂×D₄

—

108

C₂×D₄

—

109

C₂²×C₄

—

110

C₂²×C₄

C₄×D₃₇

yes

—

111

C₂²×C₄

C₂²×D₃₇

yes

characteristic

112

M₁₆

—

113

M₁₆

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

D₁₄₈

yes

complemented

118

C₂×C₈

C₂²×D₃₇

yes

—

119

C₂×C₈

C₂²×D₃₇

yes

—

120

C₄⋊C₄

—

121

C₂²⋊C₄

—

122

C₄²

C₄×D₃₇

yes

characteristic

123

C₄²

C₄×D₃₇

yes

characteristic

124

C₄²

C₂²×D₃₇

yes

—

125

C₂³

—

126

C₂³

C₂×C₄×D₃₇

yes

characteristic

127

Q₈

—

128

D₄

—

129

D₄

—

130

D₄

—

148

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

C₂×C₄×D₃₇

yes

—

136

C₂×C₄

C₂×C₄×D₃₇

yes

—

137

C₂×C₄

C₂×C₄×D₃₇

yes

characteristic

138

C₂×C₄

C₂×D₁₄₈

yes

—

139

C₂×C₄

#10596962

yes

—

140

C₂×C₄

C₂³×D₃₇

yes

characteristic

141

C₈

—

142

C₈

—

143

C₈

C₂×D₁₄₈

yes

—

144

C₂²

—

145

C₂²

—

146

C₂²

—

147

C₂²

C₄×D₁₄₈

yes

—

148

C₂²

C₂²×C₄×D₃₇

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

C₄×D₁₄₈

yes

—

153

C₄

C₂²×D₁₄₈

yes

characteristic

154

C₄

C₂×#10596962

yes

characteristic

155

C₂

—

156

C₂

—

148

—

157

C₂

C₂×C₄×D₁₄₈

yes

characteristic

158

C₂

#14558299

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₃₇
commutator, socle	C₂²×C₃₇	C₂³×C₄
Fitting	C₂×#97×C₃₇	C₂
prime radical	#14558083	C₂
cosocle	C₂×C₄×C₃₇	C₂⁴
hypercenter	C₂×#97	D₃₇
hypocenter	C₃₇	#2932
LCS	G ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₃₇

commutator, socle

C₂²×C₃₇

C₂³×C₄

Fitting

C₂×#97×C₃₇

C₂

prime radical

#14558083

C₂

cosocle

C₂×C₄×C₃₇

C₂⁴

hypercenter

C₂×#97

D₃₇

C₃₇

G ▷ C₂²×C₃₇ ▷ C₃₇

C₁ ◁ C₂×C₄ ◁ C₂×#97

N	H ≅ G/N	type
1. C₂×#11924542	C₂	semidirect
3. #14557606	C₂	semidirect
4. #14557557	C₂	semidirect
5. #14557499	C₂	semidirect
6. C₂×#97×C₃₇	C₂	semidirect
33. C₂×C₈×C₃₇	D₄	semidirect
85. C₂×#97	D₃₇	semidirect
86. C₃₇	#2932	semidirect
117. C₂×C₈	D₁₄₈	semidirect

H ≅ G/N

type

1. C₂×#11924542

C₂

semidirect

3. #14557606

C₂

semidirect

4. #14557557

C₂

semidirect

5. #14557499

C₂

semidirect

6. C₂×#97×C₃₇

C₂

semidirect

33. C₂×C₈×C₃₇

D₄

semidirect

85. C₂×#97

D₃₇

semidirect

86. C₃₇

#2932

semidirect

117. C₂×C₈

D₁₄₈

semidirect

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

subgroup	index
C₂×#11924542	2
#14558083	2
#14557606	2
#14557557	2
#14557499	2
C₂×#97×C₃₇	2
#2932	37

subgroup

index

C₂×#97×C₃₇

Subgroups — #18003315

Conjugacy classes of subgroups

Complements

Maximal subgroups