Group #18010769 — subgroups

Subgroups — #18010769

Order 4736.

280 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 92 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×C₄⋊#64438	C₂	1	1	yes	maximal, characteristic
2	C₂×#11924591	C₂	1	1	yes	maximal, characteristic
3	#14558399	C₂	1	2	yes	maximal
4	#14558399	C₂	1	2	yes	maximal
5	C₂×#11924501	C₂	1	1	yes	maximal, characteristic
6	C₂²×#64438	C₂²	1	2	yes	—
7	C₂²×#64438	C₂²	1	2	yes	—
8	C₄⋊#64438	—	2	2	no	—
9	#11924591	—	2	2	no	—
10	C₂²×Dic₇₄	C₄	1	1	yes	characteristic
11	#11924501	—	2	2	no	—
12	#11924501	—	2	2	no	—
13	C₂×#10596936	C₄	1	1	yes	characteristic
14	C₂×#10596936	C₄	1	1	yes	characteristic
15	C₂×C₄×Dic₃₇	C₂²	1	1	yes	characteristic
16	C₂×C₄×Dic₃₇	C₂²	1	1	yes	characteristic
17	C₂×C₄×Dic₃₇	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₃₇	C₄	1	1	yes	characteristic
19	C₂²×C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
22	C₂²×Dic₃₇	C₂×C₄	1	1	yes	characteristic
23	C₂²×Dic₃₇	C₂×C₄	1	1	yes	characteristic
24	C₂²×Dic₃₇	C₂×C₄	1	1	yes	characteristic
25	C₂²×Dic₃₇	C₂³	1	1	yes	characteristic
26	C₂×Dic₇₄	—	2	2	no	—
27	C₂×Dic₇₄	—	2	2	no	—
28	C₂×Dic₇₄	—	2	2	no	—
29	C₂×#64438	—	2	2	no	—
30	C₂×#64438	—	2	2	no	—
31	C₂×#64438	—	2	2	no	—
32	C₂×#64438	—	2	4	no	—
33	C₂×#64438	Q₈	1	4	yes	complemented
34	C₄⋊C₄×C₃₇	—	2	2	no	—
35	#10596936	—	2	2	no	—
36	#10596936	—	2	2	no	—
37	C₄×Dic₃₇	—	2	1	no	—
38	C₄×Dic₃₇	—	2	1	no	—
39	C₄×Dic₃₇	—	2	1	no	—
40	C₄×Dic₃₇	—	2	1	no	—
41	C₄×Dic₃₇	D₄	1	2	yes	—
42	C₄×Dic₃₇	D₄	1	2	yes	—
43	C₂³×C₃₇	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₃₇	—	2	1	no	—
45	C₂×C₄×C₃₇	—	2	1	no	—
46	C₂×C₄×C₃₇	—	2	2	no	—
47	C₂×C₄×C₃₇	—	2	1	no	—
48	C₂×C₄×C₃₇	—	2	2	no	—
49	C₂×C₄×C₃₇	C₂²⋊C₄	1	2	yes	—
50	C₂×C₄×C₃₇	M₁₆	1	2	yes	—
51	C₂×Dic₃₇	—	2	1	no	—
52	C₂×Dic₃₇	—	2	1	no	—
53	C₂×Dic₃₇	—	2	1	no	—
54	C₂×Dic₃₇	—	2	1	no	—
55	C₂×Dic₃₇	—	2	1	no	—
56	C₂×Dic₃₇	—	2	2	no	—
57	C₂×Dic₃₇	C₂²⋊C₄	1	2	yes	—
58	C₂×Dic₃₇	M₁₆	1	2	yes	—
59	C₂×Dic₃₇	C₂×D₄	1	1	yes	characteristic
60	C₂×Dic₃₇	C₂×D₄	1	1	yes	characteristic
61	C₂×Dic₃₇	C₂×Q₈	1	2	yes	—
62	C₂×Dic₃₇	#41	1	2	yes	—
63	Dic₇₄	—	2	8	no	—
64	#64438	—	2	4	no	—
65	#64438	—	4	2	no	—
66	C₂²×C₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₃₇	C₄×Q₈	1	2	yes	—
68	C₂²×C₃₇	C₂×M₁₆	1	1	yes	characteristic
69	C₂²×C₃₇	C₂×M₁₆	1	1	yes	characteristic
70	C₂²×C₃₇	#131	1	2	yes	—
71	C₄×C₃₇	—	2	2	no	—
72	C₄×C₃₇	—	2	2	no	—
73	C₄×C₃₇	—	4	1	no	—
74	Dic₃₇	—	2	2	no	—
75	Dic₃₇	—	2	2	no	—
76	Dic₃₇	—	4	1	no	—
77	Dic₃₇	C₂²⋊Q₈	1	2	yes	—
78	Dic₃₇	#124	1	1	yes	characteristic
79	Dic₃₇	C₄⋊Q₈	1	1	yes	characteristic
80	#1985	—	37	1	no	maximal, Sylow
81	C₂×C₃₇	#391	1	1	yes	characteristic
82	C₂×C₃₇	#407	1	1	yes	characteristic
83	C₂×C₃₇	#408	1	1	yes	characteristic
84	C₂×C₃₇	#446	1	2	yes	—
85	C₂×C₃₇	#446	1	2	yes	—
86	C₂×C₄×Q₈	—	37	1	no	—
87	C₂×C₄⋊C₈	—	37	1	no	—
88	C₂×#97	—	37	1	no	—
89	#336	—	37	2	no	—
90	#336	—	37	2	no	—
91	C₃₇	#1985	1	1	yes	characteristic, complemented, Sylow
92	C₂²×Q₈	—	37	1	no	—
93	C₂²×C₈	—	37	2	no	—
94	C₂²×C₈	—	37	2	no	—
95	C₄×Q₈	—	74	2	no	—
96	C₄×Q₈	—	74	2	no	—
97	C₂×C₄⋊C₄	—	37	1	no	—
98	C₂×C₄⋊C₄	—	37	1	no	—
99	C₂×C₄⋊C₄	#3866	1	1	yes	characteristic
100	C₂×C₄²	—	37	1	no	—
101	C₂×C₄²	—	37	1	no	—
102	C₂×C₄²	—	37	1	no	—
103	C₄⋊C₈	—	74	2	no	—
104	#97	—	74	2	no	—
105	C₂×Q₈	—	74	2	no	—
106	C₂×Q₈	—	74	2	no	—
107	C₂×Q₈	—	74	2	no	—
108	C₂²×C₄	—	37	1	no	—
109	C₂²×C₄	—	37	1	no	—
110	C₂²×C₄	—	37	1	no	—
111	C₂²×C₄	—	37	1	no	—
112	C₂²×C₄	C₂×#3866	1	1	yes	characteristic
113	C₂²×C₄	C₂×#3866	1	1	yes	characteristic
114	C₂²×C₄	C₂×#3866	1	1	yes	characteristic
115	C₂×C₈	—	37	4	no	—
116	C₂×C₈	—	74	2	no	—
117	C₂×C₈	—	74	2	no	—
118	C₂×C₈	—	74	2	no	—
119	C₂×C₈	—	74	4	no	—
120	C₄⋊C₄	—	2	2	no	—
121	C₄⋊C₄	—	74	2	no	—
122	C₄⋊C₄	—	74	2	no	—
123	C₄²	—	37	2	no	—
124	C₄²	—	37	2	no	—
125	C₄²	—	74	1	no	—
126	C₄²	—	74	1	no	—
127	C₄²	—	74	1	no	—
128	C₄²	—	74	1	no	—
129	C₂³	C₂²×#3866	1	1	yes	characteristic
130	Q₈	—	74	8	no	—
131	C₂×C₄	—	2	1	no	—
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	1	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	37	1	no	—
137	C₂×C₄	—	37	1	no	—
138	C₂×C₄	—	37	2	no	—
139	C₂×C₄	—	37	2	no	—
140	C₂×C₄	—	37	2	no	—
141	C₂×C₄	—	37	2	no	—
142	C₂×C₄	—	74	1	no	—
143	C₂×C₄	—	74	1	no	—
144	C₂×C₄	—	74	1	no	—
145	C₂×C₄	—	74	1	no	—
146	C₂×C₄	—	74	1	no	—
147	C₂×C₄	—	74	2	no	—
148	C₂×C₄	#10596953	1	2	yes	—
149	C₂×C₄	C₂²⋊#3866	1	2	yes	—
150	C₈	—	74	4	no	—
151	C₈	—	148	2	no	—
152	C₂²	C₂×C₂²⋊#3866	1	1	yes	characteristic
153	C₂²	Q₈×#3866	1	2	yes	—
154	C₂²	D₃₇⋊M₁₆	1	1	yes	characteristic
155	C₂²	C₂×#10596953	1	1	yes	characteristic
156	C₂²	#11924623	1	2	yes	—
157	C₄	—	2	2	no	—
158	C₄	—	2	2	no	—
159	C₄	—	4	1	no	—
160	C₄	—	37	2	no	—
161	C₄	—	37	1	no	—
162	C₄	—	37	1	no	—
163	C₄	—	74	2	no	—
164	C₄	—	74	2	no	—
165	C₄	—	148	1	no	—
166	C₂	(C₂×#3866)⋊Q₈	1	1	yes	characteristic
167	C₂	#14558504	1	1	yes	characteristic
168	C₂	#14558506	1	1	yes	characteristic
169	C₂	#14558583	1	2	yes	—
170	C₂	#14558585	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊#64438

C₂

yes

maximal, characteristic

C₂×#11924591

C₂

yes

maximal, characteristic

#14558399

C₂

yes

maximal

#14558399

C₂

yes

maximal

C₂×#11924501

C₂

yes

maximal, characteristic

C₂²×#64438

C₂²

yes

—

C₂²×#64438

C₂²

yes

—

C₄⋊#64438

—

#11924591

—

C₂²×Dic₇₄

C₄

yes

—

—

C₄

yes

characteristic

C₂×#10596936

C₄

yes

characteristic

C₂×C₄×Dic₃₇

C₂²

yes

characteristic

C₂×C₄×Dic₃₇

C₂²

yes

characteristic

C₂×C₄×Dic₃₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₃₇

C₄

yes

characteristic

C₂²×C₄×C₃₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₃₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₃₇

C₂×C₄

yes

characteristic

C₂²×Dic₃₇

C₂×C₄

yes

characteristic

C₂²×Dic₃₇

C₂×C₄

yes

characteristic

C₂²×Dic₃₇

C₂×C₄

yes

characteristic

C₂²×Dic₃₇

C₂³

yes

characteristic

C₂×Dic₇₄

—

C₂×Dic₇₄

—

C₂×Dic₇₄

—

—

—

—

—

Q₈

yes

complemented

C₄⋊C₄×C₃₇

—

#10596936

—

#10596936

—

C₄×Dic₃₇

—

C₄×Dic₃₇

—

C₄×Dic₃₇

—

C₄×Dic₃₇

—

C₄×Dic₃₇

D₄

yes

—

C₄×Dic₃₇

D₄

yes

—

C₂³×C₃₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂²⋊C₄

yes

—

C₂×C₄×C₃₇

M₁₆

yes

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

C₂²⋊C₄

yes

—

C₂×Dic₃₇

M₁₆

yes

—

C₂×Dic₃₇

C₂×D₄

yes

characteristic

C₂×Dic₃₇

C₂×D₄

yes

characteristic

C₂×Dic₃₇

C₂×Q₈

yes

—

C₂×Dic₃₇

#41

yes

—

Dic₇₄

—

#64438

—

#64438

—

C₂²×C₃₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₃₇

C₄×Q₈

yes

—

C₂²×C₃₇

C₂×M₁₆

yes

characteristic

C₂²×C₃₇

C₂×M₁₆

yes

characteristic

C₂²×C₃₇

#131

yes

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

—

Dic₃₇

—

Dic₃₇

—

Dic₃₇

—

Dic₃₇

C₂²⋊Q₈

yes

—

Dic₃₇

#124

yes

characteristic

Dic₃₇

C₄⋊Q₈

yes

—

C₂×C₃₇

yes

C₂×C₃₇

yes

C₂×C₃₇

yes

C₂×C₃₇

yes

—

C₂×C₃₇

#446

yes

—

C₂×C₄×Q₈

—

C₂×C₄⋊C₈

—

C₂×#97

—

#336

—

#336

—

C₃₇

#1985

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×C₈

—

C₂²×C₈

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

#3866

yes

characteristic

100

C₂×C₄²

—

101

C₂×C₄²

—

102

C₂×C₄²

—

103

C₄⋊C₈

—

104

#97

—

105

C₂×Q₈

—

106

C₂×Q₈

—

107

C₂×Q₈

—

108

C₂²×C₄

—

109

C₂²×C₄

—

110

C₂²×C₄

—

111

C₂²×C₄

—

112

C₂²×C₄

C₂×#3866

yes

characteristic

113

C₂²×C₄

C₂×#3866

yes

characteristic

114

C₂²×C₄

C₂×#3866

yes

characteristic

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄²

—

124

C₄²

—

125

C₄²

—

126

C₄²

—

127

C₄²

—

128

C₄²

—

129

C₂³

C₂²×#3866

yes

characteristic

130

Q₈

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

#10596953

yes

—

149

C₂×C₄

C₂²⋊#3866

yes

—

150

C₈

—

151

C₈

—

148

—

152

C₂²

C₂×C₂²⋊#3866

yes

characteristic

153

C₂²

Q₈×#3866

yes

—

154

C₂²

D₃₇⋊M₁₆

yes

characteristic

155

C₂²

C₂×#10596953

yes

characteristic

156

C₂²

#11924623

yes

—

157

C₄

—

158

C₄

—

159

C₄

—

160

C₄

—

161

C₄

—

162

C₄

—

163

C₄

—

164

C₄

—

165

C₄

—

148

—

166

C₂

(C₂×#3866)⋊Q₈

yes

characteristic

167

C₂

#14558504

yes

characteristic

168

C₂

#14558506

yes

characteristic

169

C₂

#14558583

yes

—

170

C₂

#14558585

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂²×#3866
commutator, socle, prime radical	C₂³×C₃₇	C₂²×C₄
Fitting	C₂×C₄⋊C₄×C₃₇	C₄
cosocle	C₂²×Dic₃₇	C₂³
hypercenter	C₂×C₄⋊C₄	#3866
hypocenter	C₃₇	#1985
LCS	G ▷ C₂³×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

subgroup

G/H

center, Frattini

C₂³

C₂²×#3866

commutator, socle, prime radical

C₂³×C₃₇

C₂²×C₄

Fitting

C₂×C₄⋊C₄×C₃₇

C₄

cosocle

C₂²×Dic₃₇

C₂³

C₂×C₄⋊C₄

C₃₇

G ▷ C₂³×C₃₇ ▷ C₃₇

C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

N	H ≅ G/N	type
33. C₂×#64438	Q₈	semidirect
91. C₃₇	#1985	semidirect

H ≅ G/N

type

33. C₂×#64438

Q₈

semidirect

91. C₃₇

#1985

semidirect

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

subgroup	index
C₂×C₄⋊#64438	2
C₂×#11924591	2
#14558399	2
C₂×#11924501	2
#1985	37

subgroup

index

C₂×C₄⋊#64438

Subgroups — #18010769

Conjugacy classes of subgroups

Complements

Maximal subgroups