Group #18011504 — subgroups

Subgroups — #18011504

Order 4736.

322 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 148 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	#14558584	C₂	1	4	yes	maximal, complemented
2	#14558583	C₂	1	2	yes	maximal, complemented
3	#14558582	C₂	1	2	yes	maximal, complemented
4	#14558558	C₂	1	4	yes	maximal
5	#14558555	C₂	1	1	yes	maximal, characteristic
6	#14558551	C₂	1	1	yes	maximal, characteristic
7	#14558088	C₂	1	1	yes	maximal, characteristic
8	#11924603	C₂²	1	2	yes	—
9	#11924603	C₂²	1	4	yes	—
10	C₄⋊#64438	C₂²	1	2	yes	—
11	C₄⋊#64438	C₂²	1	4	yes	—
12	#11924593	C₂²	1	2	yes	—
13	#11924593	C₂²	1	2	yes	—
14	#11924591	C₂²	1	2	yes	—
15	#11924591	C₂²	1	4	yes	—
16	C₄×#64438	C₂²	1	1	yes	characteristic
17	C₄×#64438	C₂²	1	1	yes	characteristic
18	C₄×#64438	C₂²	1	4	yes	—
19	#11924536	C₄	1	1	yes	characteristic
20	#11924527	C₄	1	2	yes	—
21	#11924516	C₄	1	2	yes	—
22	#11924512	C₄	1	2	yes	—
23	Q₈×Dic₃₇	C₂²	1	2	yes	—
24	#11924501	C₂²	1	2	yes	—
25	#11924494	C₂²	1	2	yes	—
26	C₄²×D₃₇	C₂²	1	1	yes	characteristic
27	C₄⋊Q₈×C₃₇	C₄	1	1	yes	characteristic
28	C₂×Q₈×C₃₇	C₂×C₄	1	2	yes	—
29	C₂×C₄×D₃₇	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄×D₃₇	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×D₃₇	C₂³	1	1	yes	characteristic
32	C₂×Dic₇₄	C₂×C₄	1	2	yes	—
33	C₂×#64438	C₂³	1	2	yes	—
34	C₂×#64438	C₂³	1	2	yes	—
35	C₂×#64438	C₂³	1	4	yes	—
36	C₄⋊C₄×C₃₇	C₂×C₄	1	2	yes	—
37	C₄⋊C₄×C₃₇	C₂×C₄	1	2	yes	—
38	C₄²×C₃₇	C₂×C₄	1	1	yes	characteristic
39	#10596938	C₂×C₄	1	4	yes	—
40	C₄⋊Dic₃₇	C₂×C₄	1	2	yes	—
41	C₄⋊Dic₃₇	C₂×C₄	1	2	yes	—
42	C₄⋊Dic₃₇	C₂×C₄	1	4	yes	—
43	#10596936	C₂×C₄	1	4	yes	—
44	C₄×Dic₃₇	C₂×C₄	1	1	yes	characteristic
45	C₄×Dic₃₇	C₂³	1	1	yes	characteristic
46	C₄×Dic₃₇	C₂³	1	1	yes	characteristic
47	C₄×Dic₃₇	C₂³	1	4	yes	—
48	C₂²×D₃₇	C₂²×C₄	1	1	yes	characteristic
49	Q₈×C₃₇	—	4	2	no	—
50	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₃₇	C₂²×C₄	1	4	yes	—
52	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
54	C₂×Dic₃₇	C₂²×C₄	1	1	yes	characteristic
55	C₂×Dic₃₇	C₂²×C₄	1	1	yes	characteristic
56	C₂×Dic₃₇	C₂²×C₄	1	4	yes	—
57	C₂×Dic₃₇	C₂⁴	1	1	yes	characteristic
58	C₄×D₃₇	—	2	2	no	—
59	C₄×D₃₇	—	2	2	no	—
60	C₄×D₃₇	—	4	1	no	—
61	Dic₇₄	—	2	4	no	—
62	#64438	—	2	2	no	—
63	#64438	—	2	2	no	—
64	#64438	—	2	4	no	—
65	C₂²×C₃₇	C₂³×C₄	1	1	yes	characteristic
66	C₂×D₃₇	—	2	1	no	—
67	C₂×D₃₇	—	2	1	no	—
68	C₂×D₃₇	—	2	1	no	—
69	C₄×C₃₇	—	2	1	no	—
70	C₄×C₃₇	—	2	1	no	—
71	C₄×C₃₇	—	2	4	no	—
72	C₄×C₃₇	#131	1	2	yes	—
73	Dic₃₇	—	2	4	no	—
74	Dic₃₇	—	2	1	no	—
75	Dic₃₇	—	2	1	no	—
76	Dic₃₇	2₋¹⁺⁴	1	2	yes	—
77	#2985	—	37	1	no	maximal, Sylow
78	C₂×C₃₇	#519	1	1	yes	characteristic
79	C₂×C₃₇	C₂×#131	1	1	yes	characteristic
80	C₂×C₃₇	#568	1	1	yes	characteristic
81	D₃₇	—	4	1	no	—
82	#533	—	37	1	no	—
83	#446	—	37	2	no	—
84	#446	—	37	4	no	—
85	C₈×Q₈	—	37	2	no	—
86	#433	—	37	4	no	—
87	#432	—	37	1	no	—
88	#431	—	37	1	no	—
89	C₃₇	#2985	1	1	yes	characteristic, complemented, Sylow
90	C₄⋊Q₈	—	37	1	no	—
91	C₄⋊Q₈	#3866	1	1	yes	characteristic
92	#125	—	37	2	no	—
93	C₂²⋊Q₈	—	37	2	no	—
94	C₂²⋊Q₈	—	37	2	no	—
95	C₄×Q₈	—	37	2	no	—
96	C₄×Q₈	—	37	2	no	—
97	#117	—	37	2	no	—
98	C₂×C₄²	—	37	1	no	—
99	C₄⋊C₈	—	37	2	no	—
100	C₄⋊C₈	—	37	4	no	—
101	C₄⋊C₈	—	37	2	no	—
102	C₄⋊C₈	—	37	4	no	—
103	C₂²⋊C₈	—	37	2	no	—
104	C₂²⋊C₈	—	37	2	no	—
105	#97	—	37	2	no	—
106	#97	—	37	4	no	—
107	C₄×C₈	—	37	1	no	—
108	C₄×C₈	—	37	1	no	—
109	C₄×C₈	—	37	4	no	—
110	C₂×Q₈	—	37	2	no	—
111	C₂×Q₈	C₂×#3866	1	2	yes	—
112	C₂²×C₄	—	37	1	no	—
113	C₂²×C₄	—	37	1	no	—
114	C₂²×C₄	—	37	1	no	—
115	C₂×C₈	—	37	2	no	—
116	C₂×C₈	—	37	2	no	—
117	C₂×C₈	—	37	4	no	—
118	C₄⋊C₄	—	37	4	no	—
119	C₄⋊C₄	—	37	2	no	—
120	C₄⋊C₄	—	37	4	no	—
121	C₄⋊C₄	—	37	2	no	—
122	C₄⋊C₄	C₂×#3866	1	2	yes	—
123	C₄⋊C₄	C₂×#3866	1	2	yes	—
124	C₂²⋊C₄	—	37	4	no	—
125	C₄²	—	37	1	no	—
126	C₄²	—	37	1	no	—
127	C₄²	—	37	4	no	—
128	C₄²	—	37	1	no	—
129	C₄²	C₂×#3866	1	1	yes	characteristic
130	C₂³	—	37	1	no	—
131	Q₈	—	4	2	no	—
132	Q₈	—	74	4	no	—
133	C₂×C₄	—	37	1	no	—
134	C₂×C₄	—	37	1	no	—
135	C₂×C₄	—	37	1	no	—
136	C₂×C₄	—	37	4	no	—
137	C₂×C₄	—	74	2	no	—
138	C₂×C₄	—	74	2	no	—
139	C₂×C₄	—	148	1	no	—
140	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
141	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
142	C₂×C₄	C₂²×#3866	1	4	yes	—
143	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
144	C₈	—	74	2	no	—
145	C₈	—	74	2	no	—
146	C₈	—	74	4	no	—
147	C₂²	—	74	1	no	—
148	C₂²	—	74	1	no	—
149	C₂²	—	74	1	no	—
150	C₂²	C₂³×#3866	1	1	yes	characteristic
151	C₄	—	2	1	no	—
152	C₄	—	2	1	no	—
153	C₄	—	2	4	no	—
154	C₄	—	37	2	no	—
155	C₄	—	74	1	no	—
156	C₄	—	74	4	no	—
157	C₄	—	74	1	no	—
158	C₄	#11924624	1	2	yes	—
159	C₂	—	148	1	no	—
160	C₂	#14558601	1	1	yes	characteristic
161	C₂	C₂×#11924624	1	1	yes	characteristic
162	C₂	#14558611	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14558584

C₂

yes

maximal, complemented

#14558583

C₂

yes

maximal, complemented

#14558582

C₂

yes

maximal, complemented

#14558558

C₂

yes

maximal

#14558555

C₂

yes

maximal, characteristic

#14558551

C₂

yes

maximal, characteristic

#14558088

C₂

yes

maximal, characteristic

#11924603

C₂²

yes

—

#11924603

C₂²

yes

—

C₄⋊#64438

C₂²

yes

—

C₄⋊#64438

C₂²

yes

—

#11924593

C₂²

yes

—

#11924593

C₂²

yes

—

#11924591

C₂²

yes

—

#11924591

C₂²

yes

—

C₄×#64438

C₂²

yes

characteristic

C₄×#64438

C₂²

yes

characteristic

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

—

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

—

#64438

—

C₂²×C₃₇

C₂³×C₄

yes

characteristic

C₂×D₃₇

—

C₂×D₃₇

—

C₂×D₃₇

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

#131

yes

—

Dic₃₇

—

Dic₃₇

—

Dic₃₇

—

Dic₃₇

2₋¹⁺⁴

yes

—

—

C₂×C₃₇

yes

C₂×C₃₇

yes

C₂×C₃₇

yes

D₃₇

—

#533

—

#446

—

#446

—

C₈×Q₈

—

#433

—

#432

—

#431

—

C₃₇

#2985

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

—

100

C₄⋊C₈

—

101

C₄⋊C₈

—

102

C₄⋊C₈

—

103

C₂²⋊C₈

—

104

C₂²⋊C₈

—

105

#97

—

106

#97

—

107

C₄×C₈

—

108

C₄×C₈

—

109

C₄×C₈

—

110

C₂×Q₈

—

111

C₂×Q₈

C₂×#3866

yes

—

112

C₂²×C₄

—

113

C₂²×C₄

—

114

C₂²×C₄

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

C₂×#3866

yes

—

123

C₄⋊C₄

C₂×#3866

yes

—

124

C₂²⋊C₄

—

125

C₄²

—

126

C₄²

—

127

C₄²

—

128

C₄²

—

129

C₄²

C₂×#3866

yes

characteristic

130

C₂³

—

131

Q₈

—

132

Q₈

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

148

—

140

C₂×C₄

C₂²×#3866

yes

characteristic

141

C₂×C₄

C₂²×#3866

yes

characteristic

142

C₂×C₄

C₂²×#3866

yes

—

143

C₂×C₄

C₂²×#3866

yes

characteristic

144

C₈

—

145

C₈

—

146

C₈

—

147

C₂²

—

148

C₂²

—

149

C₂²

—

150

C₂²

C₂³×#3866

yes

characteristic

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

—

158

C₄

#11924624

yes

—

159

C₂

—

148

—

160

C₂

#14558601

yes

characteristic

161

C₂

C₂×#11924624

yes

characteristic

162

C₂

#14558611

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	#2985
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. #14558584	C₂	semidirect
2. #14558583	C₂	semidirect
3. #14558582	C₂	semidirect
89. C₃₇	#2985	semidirect

H ≅ G/N

type

1. #14558584

C₂

semidirect

2. #14558583

C₂

semidirect

3. #14558582

C₂

semidirect

89. C₃₇

#2985

semidirect

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

subgroup	index
#14558584	2
#14558583	2
#14558582	2
#14558558	2
#14558555	2
#14558551	2
#14558088	2
#2985	37

subgroup

index

Subgroups — #18011504

Conjugacy classes of subgroups

Complements

Maximal subgroups