Group #15674196 — subgroups

Subgroups — #15674196

Order 2944.

384 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 121 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⋊(C₂₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂²⋊(C₂₃⋊C₈)	C₂	1	4	yes	maximal, complemented
3	C₂×#10624985	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₄×D₄×C₂₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×D₄×C₂₃	C₄	1	1	yes	characteristic
6	C₂³×C₄×C₂₃	C₂²	1	2	yes	—
7	C₄×D₄×C₂₃	—	2	2	no	—
8	C₄×D₄×C₂₃	—	2	2	no	—
9	C₂×C₄⋊C₄×C₂₃	C₄	1	1	yes	characteristic
10	C₂×C₂²⋊C₄×C₂₃	C₄	1	2	yes	—
11	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
12	C₂²×C₂₃⋊C₈	C₂²	1	2	yes	—
13	C₂²×C₂₃⋊C₈	C₂²	1	2	yes	—
14	C₂²⋊(C₂₃⋊C₈)	—	2	8	no	—
15	C₄⋊(C₂₃⋊C₈)	—	2	2	no	—
16	#10624985	—	2	2	no	—
17	C₂⁴×C₂₃	C₂×C₄	1	2	yes	—
18	C₂×D₄×C₂₃	—	2	4	no	—
19	C₂×D₄×C₂₃	—	2	2	no	—
20	C₂²×C₄×C₂₃	—	2	2	no	—
21	C₂²×C₄×C₂₃	—	2	2	no	—
22	C₂²×C₄×C₂₃	—	2	4	no	—
23	C₂²×C₄×C₂₃	—	2	4	no	—
24	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
25	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
28	C₄⋊C₄×C₂₃	—	2	2	no	—
29	C₂²⋊C₄×C₂₃	—	2	4	no	—
30	C₄²×C₂₃	D₄	1	2	yes	—
31	C₄²×C₂₃	D₄	1	2	yes	—
32	C₂×C₂₃⋊C₈	—	2	2	no	—
33	C₂×C₂₃⋊C₈	—	2	2	no	—
34	C₂×C₂₃⋊C₈	—	2	2	no	—
35	C₂×C₂₃⋊C₈	—	2	4	no	—
36	C₂×C₂₃⋊C₈	D₄	1	4	yes	complemented
37	C₂³×C₂₃	—	2	2	no	—
38	C₂³×C₂₃	—	2	4	no	—
39	C₂³×C₂₃	—	2	2	no	—
40	C₂³×C₂₃	—	2	4	no	—
41	C₂³×C₂₃	M₁₆	1	4	yes	—
42	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
43	D₄×C₂₃	—	2	8	no	—
44	C₂×C₄×C₂₃	—	2	2	no	—
45	C₂×C₄×C₂₃	—	2	2	no	—
46	C₂×C₄×C₂₃	—	2	2	no	—
47	C₂×C₄×C₂₃	—	2	4	no	—
48	C₂×C₄×C₂₃	—	4	4	no	—
49	C₂×C₄×C₂₃	—	4	2	no	—
50	C₂×C₄×C₂₃	—	4	2	no	—
51	C₂×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
52	C₂×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
53	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
54	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₂₃	#41	1	2	yes	—
57	C₂₃⋊C₈	—	2	4	no	—
58	C₂₃⋊C₈	—	4	2	no	—
59	#1955	—	23	1	no	maximal, Sylow
60	C₂²×C₂₃	—	2	4	no	—
61	C₂²×C₂₃	—	2	8	no	—
62	C₂²×C₂₃	—	4	2	no	—
63	C₂²×C₂₃	—	4	2	no	—
64	C₂²×C₂₃	—	4	4	no	—
65	C₂²×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₂₃	C₄×D₄	1	2	yes	—
67	C₂²×C₂₃	C₂×M₁₆	1	2	yes	—
68	C₂²×C₂₃	#131	1	2	yes	—
69	C₄×C₂₃	—	2	2	no	—
70	C₄×C₂₃	—	2	2	no	—
71	C₄×C₂₃	—	4	2	no	—
72	C₄×C₂₃	#121	1	2	yes	—
73	C₄×C₂₃	#124	1	1	yes	characteristic
74	C₄×C₂₃	#127	1	1	yes	characteristic
75	C₂×C₄×D₄	D₂₃	1	1	yes	characteristic
76	C₂×C₄⋊C₈	—	23	1	no	—
77	C₂×C₂²⋊C₈	—	23	4	no	—
78	C₂×#97	—	23	1	no	—
79	C₂×C₂₃	—	4	4	no	—
80	C₂×C₂₃	#390	1	1	yes	characteristic
81	C₂×C₂₃	#407	1	1	yes	characteristic
82	C₂×C₂₃	#408	1	1	yes	characteristic
83	C₂×C₂₃	#435	1	4	yes	—
84	C₂²×D₄	Dic₂₃	1	1	yes	characteristic
85	C₂³×C₄	C₂×D₂₃	1	2	yes	—
86	C₂²×C₈	—	23	2	no	—
87	C₂²×C₈	—	23	2	no	—
88	C₄×D₄	—	2	2	no	—
89	C₄×D₄	—	2	2	no	—
90	C₂×C₄⋊C₄	Dic₂₃	1	1	yes	characteristic
91	C₂×C₂²⋊C₄	Dic₂₃	1	2	yes	—
92	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
93	C₄⋊C₈	—	46	2	no	—
94	C₂²⋊C₈	—	46	8	no	—
95	#97	—	46	2	no	—
96	C₂₃	#1955	1	1	yes	characteristic, complemented, Sylow
97	C₂⁴	C₂×Dic₂₃	1	2	yes	—
98	C₂×D₄	—	2	4	no	—
99	C₂×D₄	—	2	2	no	—
100	C₂²×C₄	—	2	4	no	—
101	C₂²×C₄	—	2	4	no	—
102	C₂²×C₄	—	2	2	no	—
103	C₂²×C₄	—	2	2	no	—
104	C₂²×C₄	C₂×Dic₂₃	1	2	yes	—
105	C₂²×C₄	C₂×Dic₂₃	1	1	yes	characteristic
106	C₂²×C₄	C₂×Dic₂₃	1	1	yes	characteristic
107	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
108	C₂×C₈	—	23	4	no	—
109	C₂×C₈	—	46	2	no	—
110	C₂×C₈	—	46	2	no	—
111	C₂×C₈	—	46	4	no	—
112	C₂×C₈	—	46	2	no	—
113	C₄⋊C₄	—	2	2	no	—
114	C₂²⋊C₄	—	2	4	no	—
115	C₄²	C₂²⋊D₂₃	1	2	yes	—
116	C₄²	C₂²⋊D₂₃	1	2	yes	—
117	C₂³	—	2	2	no	—
118	C₂³	—	2	4	no	—
119	C₂³	—	2	4	no	—
120	C₂³	—	2	2	no	—
121	C₂³	#67267	1	4	yes	—
122	C₂³	C₂²×Dic₂₃	1	1	yes	characteristic
123	D₄	—	2	8	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	2	4	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	4	2	no	—
129	C₂×C₄	—	4	4	no	—
130	C₂×C₄	—	4	2	no	—
131	C₂×C₄	C₂²⋊Dic₂₃	1	2	yes	—
132	C₂×C₄	C₂²⋊Dic₂₃	1	2	yes	—
133	C₂×C₄	D₄×D₂₃	1	2	yes	—
134	C₂×C₄	#67290	1	2	yes	—
135	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
136	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
137	C₈	—	46	4	no	—
138	C₈	—	92	2	no	—
139	C₂²	—	2	4	no	—
140	C₂²	—	2	8	no	—
141	C₂²	—	4	2	no	—
142	C₂²	—	4	4	no	—
143	C₂²	—	4	2	no	—
144	C₂²	C₂×#67267	1	2	yes	—
145	C₂²	D₄×Dic₂₃	1	2	yes	—
146	C₂²	#10625119	1	2	yes	—
147	C₂²	C₂×C₂²⋊Dic₂₃	1	1	yes	characteristic
148	C₄	—	2	2	no	—
149	C₄	—	2	2	no	—
150	C₄	—	4	2	no	—
151	C₄	#10625107	1	1	yes	characteristic
152	C₄	#10625109	1	2	yes	—
153	C₄	#10625111	1	1	yes	characteristic
154	C₂	—	4	4	no	—
155	C₂	#13092320	1	1	yes	characteristic
156	C₂	#13092381	1	1	yes	characteristic
157	C₂	#13092384	1	1	yes	characteristic
158	C₂	#13092489	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₂₃⋊C₈)

C₂

yes

maximal, characteristic, complemented

C₂×C₂²⋊(C₂₃⋊C₈)

C₂

yes

maximal, complemented

C₂×#10624985

C₂

yes

maximal, characteristic, complemented

C₂×C₄×D₄×C₂₃

C₂

yes

maximal, characteristic

C₂²×D₄×C₂₃

C₄

yes

characteristic

C₂³×C₄×C₂₃

C₂²

yes

—

C₄×D₄×C₂₃

—

C₄×D₄×C₂₃

—

C₂×C₄⋊C₄×C₂₃

C₄

yes

characteristic

C₂×C₂²⋊C₄×C₂₃

C₄

yes

—

C₂×C₄²×C₂₃

C₂²

yes

characteristic

C₂²×C₂₃⋊C₈

C₂²

yes

—

C₂²×C₂₃⋊C₈

C₂²

yes

—

C₂²⋊(C₂₃⋊C₈)

—

C₄⋊(C₂₃⋊C₈)

—

#10624985

—

C₂⁴×C₂₃

C₂×C₄

yes

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

C₂×C₄

yes

—

C₂²×C₄×C₂₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₄⋊C₄×C₂₃

—

C₂²⋊C₄×C₂₃

—

C₄²×C₂₃

D₄

yes

—

C₄²×C₂₃

D₄

yes

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

D₄

yes

complemented

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

M₁₆

yes

—

C₂³×C₂₃

C₂²×C₄

yes

characteristic

D₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂²⋊C₄

yes

—

C₂×C₄×C₂₃

C₂²⋊C₄

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

#41

yes

—

C₂₃⋊C₈

—

C₂₃⋊C₈

—

#1955

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₃

C₄×D₄

yes

—

C₂²×C₂₃

C₂×M₁₆

yes

—

C₂²×C₂₃

#131

yes

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

#121

yes

—

C₄×C₂₃

yes

C₄×C₂₃

yes

C₂×C₄×D₄

D₂₃

yes

characteristic

C₂×C₄⋊C₈

—

C₂×C₂²⋊C₈

—

C₂×#97

—

C₂×C₂₃

—

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

—

C₂²×D₄

Dic₂₃

yes

characteristic

C₂³×C₄

C₂×D₂₃

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₄×D₄

—

C₄×D₄

—

C₂×C₄⋊C₄

Dic₂₃

yes

characteristic

C₂×C₂²⋊C₄

Dic₂₃

yes

—

C₂×C₄²

C₂×D₂₃

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

C₂₃

#1955

yes

characteristic, complemented, Sylow

C₂⁴

C₂×Dic₂₃

yes

—

C₂×D₄

—

C₂×D₄

—

100

C₂²×C₄

—

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

—

104

C₂²×C₄

C₂×Dic₂₃

yes

—

105

C₂²×C₄

C₂×Dic₂₃

yes

characteristic

106

C₂²×C₄

C₂×Dic₂₃

yes

characteristic

107

C₂²×C₄

C₂²×D₂₃

yes

characteristic

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₄⋊C₄

—

114

C₂²⋊C₄

—

115

C₄²

C₂²⋊D₂₃

yes

—

116

C₄²

C₂²⋊D₂₃

yes

—

117

C₂³

—

118

C₂³

—

119

C₂³

—

120

C₂³

—

121

C₂³

#67267

yes

—

122

C₂³

C₂²×Dic₂₃

yes

characteristic

123

D₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

C₂²⋊Dic₂₃

yes

—

132

C₂×C₄

C₂²⋊Dic₂₃

yes

—

133

C₂×C₄

D₄×D₂₃

yes

—

134

C₂×C₄

#67290

yes

—

135

C₂×C₄

C₂×C₂²⋊D₂₃

yes

characteristic

136

C₂×C₄

C₂×C₂²⋊D₂₃

yes

characteristic

137

C₈

—

138

C₈

—

139

C₂²

—

140

C₂²

—

141

C₂²

—

142

C₂²

—

143

C₂²

—

144

C₂²

C₂×#67267

yes

—

145

C₂²

D₄×Dic₂₃

yes

—

146

C₂²

#10625119

yes

—

147

C₂²

C₂×C₂²⋊Dic₂₃

yes

characteristic

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

#10625107

yes

characteristic

152

C₄

#10625109

yes

—

153

C₄

#10625111

yes

characteristic

154

C₂

—

155

C₂

#13092320

yes

characteristic

156

C₂

#13092381

yes

characteristic

157

C₂

#13092384

yes

characteristic

158

C₂

#13092489

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₂₃
commutator, socle	C₂³×C₂₃	C₂²×C₄
Fitting	C₂×C₄×D₄×C₂₃	C₂
prime radical	C₂²×D₄×C₂₃	C₄
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	C₂×C₄×D₄	D₂₃
hypocenter	C₂₃	#1955
LCS	G ▷ C₂³×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×D₄

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₂₃

commutator, socle

C₂³×C₂₃

C₂²×C₄

Fitting

C₂×C₄×D₄×C₂₃

C₂

prime radical

C₂²×D₄×C₂₃

C₄

cosocle

C₂²×C₄×C₂₃

C₂³

hypercenter

C₂×C₄×D₄

D₂₃

C₂₃

G ▷ C₂³×C₂₃ ▷ C₂₃

C₁ ◁ C₂²×C₄ ◁ C₂×C₄×D₄

N	H ≅ G/N	type
1. C₂×C₄⋊(C₂₃⋊C₈)	C₂	semidirect
2. C₂×C₂²⋊(C₂₃⋊C₈)	C₂	semidirect
3. C₂×#10624985	C₂	semidirect
36. C₂×C₂₃⋊C₈	D₄	semidirect
96. C₂₃	#1955	semidirect

H ≅ G/N

type

1. C₂×C₄⋊(C₂₃⋊C₈)

C₂

semidirect

2. C₂×C₂²⋊(C₂₃⋊C₈)

C₂

semidirect

3. C₂×#10624985

C₂

semidirect

36. C₂×C₂₃⋊C₈

D₄

semidirect

96. C₂₃

#1955

semidirect

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

subgroup	index
C₂×C₄⋊(C₂₃⋊C₈)	2
C₂×C₂²⋊(C₂₃⋊C₈)	2
C₂×#10624985	2
C₂×C₄×D₄×C₂₃	2
#1955	23

subgroup

index

C₂×C₄⋊(C₂₃⋊C₈)

C₂×C₂²⋊(C₂₃⋊C₈)

C₂×#10624985

C₂×C₄×D₄×C₂₃

#1955

Subgroups — #15674196

Conjugacy classes of subgroups

Complements

Maximal subgroups