Group #18011510 — subgroups

Subgroups — #18011510

Order 4736.

314 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 144 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	#14558585	C₂	1	4	yes	maximal, complemented
2	#14558584	C₂	1	4	yes	maximal, complemented
3	#14558558	C₂	1	4	yes	maximal
4	#14558556	C₂	1	1	yes	maximal, characteristic
5	#14558556	C₂	1	1	yes	maximal, characteristic
6	#14558088	C₂	1	1	yes	maximal, characteristic
7	#11924603	C₂²	1	2	yes	—
8	#11924603	C₂²	1	2	yes	—
9	#11924603	C₂²	1	4	yes	—
10	C₄⋊#64438	C₂²	1	4	yes	—
11	#11924593	C₂²	1	2	yes	—
12	#11924593	C₂²	1	2	yes	—
13	#11924591	C₂²	1	2	yes	—
14	#11924591	C₂²	1	2	yes	—
15	#11924591	C₂²	1	4	yes	—
16	C₄×#64438	C₂²	1	4	yes	—
17	#11924536	C₄	1	1	yes	characteristic
18	#11924527	C₄	1	2	yes	—
19	#11924516	C₄	1	2	yes	—
20	#11924512	C₄	1	2	yes	—
21	Q₈×Dic₃₇	C₂²	1	2	yes	—
22	#11924501	C₂²	1	2	yes	—
23	#11924494	C₂²	1	2	yes	—
24	C₄²×D₃₇	C₂²	1	1	yes	characteristic
25	C₄⋊Q₈×C₃₇	C₄	1	1	yes	characteristic
26	C₂×Q₈×C₃₇	C₂×C₄	1	2	yes	—
27	C₂×C₄×D₃₇	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄×D₃₇	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×D₃₇	C₂³	1	1	yes	characteristic
30	C₂×Dic₇₄	C₂×C₄	1	2	yes	—
31	C₂×#64438	C₂³	1	4	yes	—
32	C₂×#64438	C₂³	1	4	yes	—
33	C₄⋊C₄×C₃₇	C₂×C₄	1	2	yes	—
34	C₄⋊C₄×C₃₇	C₂×C₄	1	2	yes	—
35	C₄²×C₃₇	C₂×C₄	1	1	yes	characteristic
36	#10596938	C₂×C₄	1	4	yes	—
37	C₄⋊Dic₃₇	C₂×C₄	1	2	yes	—
38	C₄⋊Dic₃₇	C₂×C₄	1	2	yes	—
39	C₄⋊Dic₃₇	C₂×C₄	1	4	yes	—
40	#10596936	C₂×C₄	1	4	yes	—
41	C₄×Dic₃₇	C₂×C₄	1	1	yes	characteristic
42	C₄×Dic₃₇	C₂³	1	1	yes	characteristic
43	C₄×Dic₃₇	C₂³	1	1	yes	characteristic
44	C₄×Dic₃₇	C₂³	1	4	yes	—
45	C₂²×D₃₇	C₂²×C₄	1	1	yes	characteristic
46	Q₈×C₃₇	—	4	2	no	—
47	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₄×C₃₇	C₂²×C₄	1	4	yes	—
49	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
51	C₂×Dic₃₇	C₂²×C₄	1	1	yes	characteristic
52	C₂×Dic₃₇	C₂²×C₄	1	1	yes	characteristic
53	C₂×Dic₃₇	C₂²×C₄	1	4	yes	—
54	C₂×Dic₃₇	C₂⁴	1	1	yes	characteristic
55	C₄×D₃₇	—	2	2	no	—
56	C₄×D₃₇	—	4	1	no	—
57	C₄×D₃₇	—	4	1	no	—
58	Dic₇₄	—	4	2	no	—
59	#64438	—	2	4	no	—
60	#64438	—	2	4	no	—
61	C₂²×C₃₇	C₂³×C₄	1	1	yes	characteristic
62	C₂×D₃₇	—	2	1	no	—
63	C₂×D₃₇	—	2	1	no	—
64	C₂×D₃₇	—	2	1	no	—
65	C₄×C₃₇	—	2	1	no	—
66	C₄×C₃₇	—	2	1	no	—
67	C₄×C₃₇	—	2	1	no	—
68	C₄×C₃₇	—	2	4	no	—
69	Dic₃₇	—	2	4	no	—
70	Dic₃₇	—	2	1	no	—
71	Dic₃₇	—	2	1	no	—
72	Dic₃₇	2₋¹⁺⁴	1	2	yes	—
73	#2987	—	37	1	no	maximal, Sylow
74	C₂×C₃₇	#519	1	1	yes	characteristic
75	C₂×C₃₇	#568	1	1	yes	characteristic
76	C₂×C₃₇	#568	1	1	yes	characteristic
77	D₃₇	—	4	1	no	—
78	#533	—	37	1	no	—
79	#446	—	37	4	no	—
80	#446	—	37	4	no	—
81	#433	—	37	4	no	—
82	#432	—	37	1	no	—
83	#432	—	37	1	no	—
84	C₃₇	#2987	1	1	yes	characteristic, complemented, Sylow
85	C₄⋊Q₈	—	37	1	no	—
86	C₄⋊Q₈	#3866	1	1	yes	characteristic
87	#125	—	37	2	no	—
88	C₂²⋊Q₈	—	37	2	no	—
89	C₂²⋊Q₈	—	37	2	no	—
90	C₄×Q₈	—	37	2	no	—
91	C₄×Q₈	—	37	2	no	—
92	#117	—	37	2	no	—
93	C₂×C₄²	—	37	1	no	—
94	C₄⋊C₈	—	37	4	no	—
95	C₄⋊C₈	—	37	2	no	—
96	C₄⋊C₈	—	37	2	no	—
97	C₄⋊C₈	—	37	4	no	—
98	C₂²⋊C₈	—	37	2	no	—
99	C₂²⋊C₈	—	37	2	no	—
100	#97	—	37	2	no	—
101	#97	—	37	2	no	—
102	#97	—	37	4	no	—
103	C₄×C₈	—	37	4	no	—
104	C₂×Q₈	—	37	2	no	—
105	C₂×Q₈	C₂×#3866	1	2	yes	—
106	C₂²×C₄	—	37	1	no	—
107	C₂²×C₄	—	37	1	no	—
108	C₂²×C₄	—	37	1	no	—
109	C₂×C₈	—	37	4	no	—
110	C₂×C₈	—	37	4	no	—
111	C₄⋊C₄	—	37	4	no	—
112	C₄⋊C₄	—	37	2	no	—
113	C₄⋊C₄	—	37	4	no	—
114	C₄⋊C₄	—	37	2	no	—
115	C₄⋊C₄	C₂×#3866	1	2	yes	—
116	C₄⋊C₄	C₂×#3866	1	2	yes	—
117	C₂²⋊C₄	—	37	4	no	—
118	C₄²	—	37	1	no	—
119	C₄²	—	37	4	no	—
120	C₄²	—	37	1	no	—
121	C₄²	—	37	1	no	—
122	C₄²	C₂×#3866	1	1	yes	characteristic
123	C₂³	—	37	1	no	—
124	Q₈	—	4	2	no	—
125	Q₈	—	148	2	no	—
126	C₂×C₄	—	37	1	no	—
127	C₂×C₄	—	37	1	no	—
128	C₂×C₄	—	37	1	no	—
129	C₂×C₄	—	37	4	no	—
130	C₂×C₄	—	74	2	no	—
131	C₂×C₄	—	148	1	no	—
132	C₂×C₄	—	148	1	no	—
133	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
134	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
135	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
136	C₂×C₄	C₂²×#3866	1	4	yes	—
137	C₈	—	74	4	no	—
138	C₈	—	74	4	no	—
139	C₂²	—	74	1	no	—
140	C₂²	—	74	1	no	—
141	C₂²	—	74	1	no	—
142	C₂²	C₂³×#3866	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	2	1	no	—
145	C₄	—	2	4	no	—
146	C₄	—	2	1	no	—
147	C₄	—	37	2	no	—
148	C₄	—	74	1	no	—
149	C₄	—	74	1	no	—
150	C₄	—	74	4	no	—
151	C₂	—	148	1	no	—
152	C₂	#14558601	1	1	yes	characteristic
153	C₂	#14558611	1	1	yes	characteristic
154	C₂	#14558613	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14558585

C₂

yes

maximal, complemented

#14558584

C₂

yes

maximal, complemented

#14558558

C₂

yes

maximal

#14558556

C₂

yes

maximal, characteristic

#14558556

C₂

yes

maximal, characteristic

#14558088

C₂

yes

maximal, characteristic

#11924603

C₂²

yes

—

#11924603

C₂²

yes

—

#11924603

C₂²

yes

—

C₄⋊#64438

C₂²

yes

—

#11924593

C₂²

yes

—

#11924593

C₂²

yes

—

#11924591

C₂²

yes

—

#11924591

C₂²

yes

—

#11924591

C₂²

yes

—

C₄×#64438

C₂²

yes

—

#11924536

C₄

yes

characteristic

#11924527

C₄

yes

—

#11924516

C₄

yes

—

#11924512

C₄

yes

—

Q₈×Dic₃₇

C₂²

yes

—

#11924501

C₂²

yes

—

#11924494

C₂²

yes

—

C₄²×D₃₇

C₂²

yes

characteristic

C₄⋊Q₈×C₃₇

C₄

yes

characteristic

C₂×Q₈×C₃₇

C₂×C₄

yes

—

C₂×C₄×D₃₇

C₂×C₄

yes

characteristic

C₂×C₄×D₃₇

C₂×C₄

yes

characteristic

C₂×C₄×D₃₇

C₂³

yes

characteristic

C₂×Dic₇₄

C₂×C₄

yes

—

C₂×#64438

C₂³

yes

—

C₂×#64438

C₂³

yes

—

C₄⋊C₄×C₃₇

C₂×C₄

yes

—

C₄⋊C₄×C₃₇

C₂×C₄

yes

—

C₄²×C₃₇

C₂×C₄

yes

characteristic

#10596938

C₂×C₄

yes

—

C₄⋊Dic₃₇

C₂×C₄

yes

—

C₄⋊Dic₃₇

C₂×C₄

yes

—

C₄⋊Dic₃₇

C₂×C₄

yes

—

#10596936

C₂×C₄

yes

—

C₄×Dic₃₇

C₂×C₄

yes

characteristic

C₄×Dic₃₇

C₂³

yes

characteristic

C₄×Dic₃₇

C₂³

yes

characteristic

C₄×Dic₃₇

C₂³

yes

—

C₂²×D₃₇

C₂²×C₄

yes

characteristic

Q₈×C₃₇

—

C₂×C₄×C₃₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₇

C₂²×C₄

yes

—

C₂×C₄×C₃₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₇

C₂²×C₄

yes

characteristic

C₂×Dic₃₇

C₂²×C₄

yes

characteristic

C₂×Dic₃₇

C₂²×C₄

yes

characteristic

C₂×Dic₃₇

C₂²×C₄

yes

—

C₂×Dic₃₇

C₂⁴

yes

characteristic

C₄×D₃₇

—

C₄×D₃₇

—

C₄×D₃₇

—

Dic₇₄

—

#64438

—

#64438

—

C₂²×C₃₇

C₂³×C₄

yes

characteristic

C₂×D₃₇

—

C₂×D₃₇

—

C₂×D₃₇

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

—

Dic₃₇

—

Dic₃₇

—

Dic₃₇

—

Dic₃₇

2₋¹⁺⁴

yes

—

—

C₂×C₃₇

yes

C₂×C₃₇

yes

C₂×C₃₇

yes

D₃₇

—

—

—

—

—

—

—

C₃₇

#2987

yes

characteristic, complemented, Sylow

C₄⋊Q₈

—

C₄⋊Q₈

#3866

yes

characteristic

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

100

#97

—

101

#97

—

102

#97

—

103

C₄×C₈

—

104

C₂×Q₈

—

105

C₂×Q₈

C₂×#3866

yes

—

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₂²×C₄

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

C₂×#3866

yes

—

116

C₄⋊C₄

C₂×#3866

yes

—

117

C₂²⋊C₄

—

118

C₄²

—

119

C₄²

—

120

C₄²

—

121

C₄²

—

122

C₄²

C₂×#3866

yes

characteristic

123

C₂³

—

124

Q₈

—

125

Q₈

—

148

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

148

—

132

C₂×C₄

—

148

—

133

C₂×C₄

C₂²×#3866

yes

characteristic

134

C₂×C₄

C₂²×#3866

yes

characteristic

135

C₂×C₄

C₂²×#3866

yes

characteristic

136

C₂×C₄

C₂²×#3866

yes

—

137

C₈

—

138

C₈

—

139

C₂²

—

140

C₂²

—

141

C₂²

—

142

C₂²

C₂³×#3866

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

—

151

C₂

—

148

—

152

C₂

#14558601

yes

characteristic

153

C₂

#14558611

yes

characteristic

154

C₂

#14558613

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#3866
commutator, socle	C₂²×C₃₇	C₂³×C₄
Fitting	C₄⋊Q₈×C₃₇	C₄
prime radical	C₂²×D₃₇	C₂²×C₄
cosocle	C₂×Dic₃₇	C₂⁴
hypercenter	C₄⋊Q₈	#3866
hypocenter	C₃₇	#2987
LCS	G ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×#3866

commutator, socle

C₂²×C₃₇

C₂³×C₄

Fitting

C₄⋊Q₈×C₃₇

C₄

prime radical

C₂²×D₃₇

C₂²×C₄

cosocle

C₂×Dic₃₇

C₂⁴

C₄⋊Q₈

C₃₇

G ▷ C₂²×C₃₇ ▷ C₃₇

C₁ ◁ C₂² ◁ C₄⋊Q₈

N	H ≅ G/N	type
1. #14558585	C₂	semidirect
2. #14558584	C₂	semidirect
84. C₃₇	#2987	semidirect

H ≅ G/N

type

1. #14558585

C₂

semidirect

2. #14558584

C₂

semidirect

84. C₃₇

#2987

semidirect

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

subgroup	index
#14558585	2
#14558584	2
#14558558	2
#14558556	2
#14558088	2
#2987	37

subgroup

index

Subgroups — #18011510

Conjugacy classes of subgroups

Complements

Maximal subgroups