Group #15673256 — subgroups

Subgroups — #15673256

Order 2944.

260 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 133 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
2	C₂×#10624997	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×Dic₂₃	C₂	1	1	yes	maximal, characteristic
4	#13092263	C₂	1	1	yes	maximal, characteristic
5	#13092171	C₂	1	1	yes	maximal, characteristic
6	#13092171	C₂	1	1	yes	maximal, characteristic
7	C₂×C₂₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
9	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
10	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
11	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
12	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
13	C₂²×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
14	C₂²×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
15	C₂₃×C₄⋊C₈	C₄	1	4	yes	complemented
16	C₂₃⋊#95	C₄	1	2	yes	—
17	C₂₃⋊#95	C₄	1	2	yes	—
18	#10624997	—	2	2	no	—
19	C₈×Dic₂₃	—	2	2	no	—
20	C₄⋊(C₂₃⋊C₈)	C₄	1	4	yes	complemented
21	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
24	C₂²×Dic₂₃	C₂×C₄	1	2	yes	—
25	C₂²×Dic₂₃	C₂×C₄	1	2	yes	—
26	C₂×C₈×C₂₃	—	2	1	no	—
27	C₂×C₈×C₂₃	—	2	1	no	—
28	C₂×C₈×C₂₃	—	2	1	no	—
29	C₂×C₈×C₂₃	—	2	1	no	—
30	C₂×C₈×C₂₃	C₂×C₄	1	2	yes	—
31	C₂×C₈×C₂₃	C₂×C₄	1	2	yes	—
32	C₄²×C₂₃	C₂×C₄	1	2	yes	—
33	C₄²×C₂₃	C₂×C₄	1	2	yes	—
34	C₄×Dic₂₃	—	2	1	no	—
35	C₄×Dic₂₃	—	2	1	no	—
36	C₄×Dic₂₃	D₄	1	2	yes	—
37	C₄×Dic₂₃	Q₈	1	2	yes	—
38	C₂×C₂₃⋊C₈	—	2	1	no	—
39	C₂×C₂₃⋊C₈	—	2	1	no	—
40	C₂×C₂₃⋊C₈	—	2	1	no	—
41	C₂×C₂₃⋊C₈	—	2	1	no	—
42	C₂×C₂₃⋊C₈	C₂×C₄	1	2	yes	—
43	C₂×C₂₃⋊C₈	C₂×C₄	1	2	yes	—
44	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₂₃	—	2	2	no	—
46	C₂×C₄×C₂₃	—	2	2	no	—
47	C₂×C₄×C₂₃	C₄²	1	2	yes	—
48	C₂×C₄×C₂₃	C₄²	1	2	yes	—
49	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₄×C₂₃	#41	1	1	yes	characteristic
54	C₂×C₄×C₂₃	#41	1	1	yes	characteristic
55	C₂×Dic₂₃	—	2	2	no	—
56	C₂×Dic₂₃	—	2	2	no	—
57	C₂×Dic₂₃	—	2	2	no	—
58	C₂×Dic₂₃	—	2	2	no	—
59	C₂×Dic₂₃	—	2	2	no	—
60	C₂×Dic₂₃	C₄⋊C₄	1	4	yes	—
61	C₈×C₂₃	—	2	2	no	—
62	C₈×C₂₃	—	2	2	no	—
63	C₂₃⋊C₈	—	2	2	no	—
64	C₂₃⋊C₈	—	2	2	no	—
65	#1768	—	23	1	no	maximal, Sylow
66	C₂²×C₂₃	C₂×C₄²	1	1	yes	characteristic
67	C₂²×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₂²×C₂₃	#117	1	1	yes	characteristic
69	C₂²×C₂₃	#131	1	2	yes	—
70	C₂²×C₂₃	#131	1	2	yes	—
71	C₄×C₂₃	—	2	2	no	—
72	C₄×C₂₃	—	2	2	no	—
73	C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
74	C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
75	C₄×C₂₃	C₄×Q₈	1	1	yes	characteristic
76	C₄×C₂₃	C₄×Q₈	1	1	yes	characteristic
77	Dic₂₃	—	2	4	no	—
78	Dic₂₃	—	4	2	no	—
79	C₂×C₄⋊C₈	—	23	1	no	—
80	C₂×C₄⋊C₈	D₂₃	1	1	yes	characteristic
81	C₂×#97	—	23	1	no	—
82	C₂×C₄×C₈	—	23	1	no	—
83	#376	—	23	1	no	—
84	#336	—	23	1	no	—
85	#336	—	23	1	no	—
86	C₂×C₂₃	C₄×C₄⋊C₄	1	1	yes	characteristic
87	C₂×C₂₃	#405	1	1	yes	characteristic
88	C₂×C₂₃	#405	1	1	yes	characteristic
89	C₂×C₂₃	#424	1	2	yes	—
90	C₂×C₂₃	#433	1	2	yes	—
91	C₂²×C₈	—	23	1	no	—
92	C₂²×C₈	—	23	1	no	—
93	C₂²×C₈	C₂×D₂₃	1	1	yes	characteristic
94	C₂²×C₈	C₂×D₂₃	1	1	yes	characteristic
95	C₂×C₄²	—	23	1	no	—
96	C₂×C₄²	—	23	1	no	—
97	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
98	C₄⋊C₈	—	23	4	no	—
99	C₄⋊C₈	Dic₂₃	1	4	yes	complemented
100	#97	—	46	2	no	—
101	C₄×C₈	—	46	2	no	—
102	#95	—	23	2	no	—
103	#95	—	23	2	no	—
104	C₂₃	#1768	1	1	yes	characteristic, complemented, Sylow
105	C₂²×C₄	—	23	2	no	—
106	C₂²×C₄	—	23	2	no	—
107	C₂²×C₄	C₄×D₂₃	1	1	yes	characteristic
108	C₂²×C₄	C₄×D₂₃	1	1	yes	characteristic
109	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
110	C₂×C₈	—	2	1	no	—
111	C₂×C₈	—	2	1	no	—
112	C₂×C₈	—	2	1	no	—
113	C₂×C₈	—	2	1	no	—
114	C₂×C₈	—	23	2	no	—
115	C₂×C₈	—	23	2	no	—
116	C₂×C₈	—	46	1	no	—
117	C₂×C₈	—	46	1	no	—
118	C₂×C₈	—	46	1	no	—
119	C₂×C₈	—	46	1	no	—
120	C₂×C₈	C₂×Dic₂₃	1	2	yes	—
121	C₂×C₈	C₂×Dic₂₃	1	2	yes	—
122	C₄²	—	23	2	no	—
123	C₄²	—	23	2	no	—
124	C₄²	—	46	1	no	—
125	C₄²	—	46	1	no	—
126	C₄²	C₄×D₂₃	1	2	yes	—
127	C₄²	C₂×Dic₂₃	1	2	yes	—
128	C₂³	C₂×C₄×D₂₃	1	1	yes	characteristic
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	23	4	no	—
132	C₂×C₄	—	46	2	no	—
133	C₂×C₄	—	46	2	no	—
134	C₂×C₄	—	46	2	no	—
135	C₂×C₄	—	46	2	no	—
136	C₂×C₄	—	46	2	no	—
137	C₂×C₄	C₄×Dic₂₃	1	2	yes	—
138	C₂×C₄	C₄×Dic₂₃	1	2	yes	—
139	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
140	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
141	C₂×C₄	#67290	1	1	yes	characteristic
142	C₂×C₄	Q₈×D₂₃	1	1	yes	characteristic
143	C₂×C₄	#67292	1	1	yes	characteristic
144	C₂×C₄	C₂²×Dic₂₃	1	1	yes	characteristic
145	C₈	—	2	2	no	—
146	C₈	—	2	2	no	—
147	C₈	—	46	2	no	—
148	C₈	—	46	2	no	—
149	C₂²	C₄⋊C₄×D₂₃	1	1	yes	characteristic
150	C₂²	#10625063	1	1	yes	characteristic
151	C₂²	#10625072	1	2	yes	—
152	C₂²	#10625078	1	2	yes	—
153	C₂²	C₂×C₄×Dic₂₃	1	1	yes	characteristic
154	C₄	—	2	2	no	—
155	C₄	—	2	2	no	—
156	C₄	—	46	4	no	—
157	C₄	—	92	2	no	—
158	C₄	#10625058	1	1	yes	characteristic
159	C₄	#10625064	1	1	yes	characteristic
160	C₄	D₄×Dic₂₃	1	1	yes	characteristic
161	C₄	Q₈×Dic₂₃	1	1	yes	characteristic
162	C₂	C₄⋊C₄×Dic₂₃	1	1	yes	characteristic
163	C₂	#13092372	1	1	yes	characteristic
164	C₂	#13092375	1	1	yes	characteristic
165	C₂	#13092442	1	2	yes	—
166	C₂	#13092477	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₂₃⋊C₈)

C₂

yes

maximal, characteristic

C₂×#10624997

C₂

yes

maximal, characteristic

C₂×C₈×Dic₂₃

C₂

yes

maximal, characteristic

#13092263

C₂

yes

maximal, characteristic

#13092171

C₂

yes

maximal, characteristic

#13092171

C₂

yes

maximal, characteristic

C₂×C₂₃×C₄⋊C₈

C₂

yes

maximal, characteristic

C₂²×C₈×C₂₃

C₂²

yes

characteristic

C₂²×C₈×C₂₃

C₂²

yes

characteristic

C₂×C₄²×C₂₃

C₂²

yes

characteristic

C₂×C₄×Dic₂₃

C₂²

yes

characteristic

C₂×C₄×Dic₂₃

C₂²

yes

characteristic

C₂²×C₂₃⋊C₈

C₂²

yes

characteristic

C₂²×C₂₃⋊C₈

C₂²

yes

characteristic

C₂₃×C₄⋊C₈

C₄

yes

complemented

C₂₃⋊#95

C₄

yes

—

C₂₃⋊#95

C₄

yes

—

#10624997

—

C₈×Dic₂₃

—

C₄⋊(C₂₃⋊C₈)

C₄

yes

complemented

C₂²×C₄×C₂₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂²×Dic₂₃

C₂×C₄

yes

—

C₂²×Dic₂₃

C₂×C₄

yes

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

C₂×C₄

yes

—

C₂×C₈×C₂₃

C₂×C₄

yes

—

C₄²×C₂₃

C₂×C₄

yes

—

C₄²×C₂₃

C₂×C₄

yes

—

C₄×Dic₂₃

—

C₄×Dic₂₃

—

C₄×Dic₂₃

D₄

yes

—

C₄×Dic₂₃

Q₈

yes

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

C₂×C₄

yes

—

C₂×C₂₃⋊C₈

C₂×C₄

yes

—

C₂³×C₂₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₄²

yes

—

C₂×C₄×C₂₃

C₄²

yes

—

C₂×C₄×C₂₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×Q₈

yes

C₂×C₄×C₂₃

yes

C₂×C₄×C₂₃

yes

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

C₄⋊C₄

yes

—

C₈×C₂₃

—

C₈×C₂₃

—

C₂₃⋊C₈

—

C₂₃⋊C₈

—

#1768

—

maximal, Sylow

C₂²×C₂₃

C₂×C₄²

yes

characteristic

C₂²×C₂₃

C₂×C₄⋊C₄

yes

C₂²×C₂₃

yes

C₂²×C₂₃

yes

—

C₂²×C₂₃

#131

yes

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

C₄×D₄

yes

characteristic

C₄×C₂₃

C₄×D₄

yes

characteristic

C₄×C₂₃

C₄×Q₈

yes

characteristic

C₄×C₂₃

C₄×Q₈

yes

characteristic

Dic₂₃

—

Dic₂₃

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

D₂₃

yes

characteristic

C₂×#97

—

C₂×C₄×C₈

—

#376

—

#336

—

#336

—

C₂×C₂₃

C₄×C₄⋊C₄

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

—

C₂×C₂₃

#433

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×D₂₃

yes

characteristic

C₂²×C₈

C₂×D₂₃

yes

characteristic

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₂₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

Dic₂₃

yes

complemented

100

#97

—

101

C₄×C₈

—

102

#95

—

103

#95

—

104

C₂₃

#1768

yes

characteristic, complemented, Sylow

105

C₂²×C₄

—

106

C₂²×C₄

—

107

C₂²×C₄

C₄×D₂₃

yes

characteristic

108

C₂²×C₄

C₄×D₂₃

yes

characteristic

109

C₂²×C₄

C₂²×D₂₃

yes

characteristic

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

C₂×Dic₂₃

yes

—

121

C₂×C₈

C₂×Dic₂₃

yes

—

122

C₄²

—

123

C₄²

—

124

C₄²

—

125

C₄²

—

126

C₄²

C₄×D₂₃

yes

—

127

C₄²

C₂×Dic₂₃

yes

—

128

C₂³

C₂×C₄×D₂₃

yes

characteristic

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

C₄×Dic₂₃

yes

—

138

C₂×C₄

C₄×Dic₂₃

yes

—

139

C₂×C₄

C₂×C₄×D₂₃

yes

characteristic

140

C₂×C₄

D₄×D₂₃

yes

characteristic

141

C₂×C₄

#67290

yes

characteristic

142

C₂×C₄

Q₈×D₂₃

yes

characteristic

143

C₂×C₄

#67292

yes

characteristic

144

C₂×C₄

C₂²×Dic₂₃

yes

characteristic

145

C₈

—

146

C₈

—

147

C₈

—

148

C₈

—

149

C₂²

C₄⋊C₄×D₂₃

yes

characteristic

150

C₂²

#10625063

yes

characteristic

151

C₂²

#10625072

yes

—

152

C₂²

#10625078

yes

—

153

C₂²

C₂×C₄×Dic₂₃

yes

characteristic

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

—

158

C₄

#10625058

yes

characteristic

159

C₄

#10625064

yes

characteristic

160

C₄

D₄×Dic₂₃

yes

characteristic

161

C₄

Q₈×Dic₂₃

yes

characteristic

162

C₂

C₄⋊C₄×Dic₂₃

yes

characteristic

163

C₂

#13092372

yes

characteristic

164

C₂

#13092375

yes

characteristic

165

C₂

#13092442

yes

—

166

C₂

#13092477

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₂₃
commutator	C₂²×C₂₃	C₂×C₄²
Fitting	C₂×C₂₃×C₄⋊C₈	C₂
socle, prime radical	C₂³×C₂₃	C₂²×C₄
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	C₂×C₄⋊C₈	D₂₃
hypocenter	C₂₃	#1768
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₂₃

commutator

C₂²×C₂₃

C₂×C₄²

Fitting

C₂×C₂₃×C₄⋊C₈

C₂

socle, prime radical

C₂³×C₂₃

C₂²×C₄

cosocle

C₂²×C₄×C₂₃

C₂³

hypercenter

C₂×C₄⋊C₈

D₂₃

C₂₃

G ▷ C₂²×C₂₃ ▷ C₂₃

C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
15. C₂₃×C₄⋊C₈	C₄	semidirect
20. C₄⋊(C₂₃⋊C₈)	C₄	semidirect
99. C₄⋊C₈	Dic₂₃	semidirect
104. C₂₃	#1768	semidirect

H ≅ G/N

type

15. C₂₃×C₄⋊C₈

C₄

semidirect

20. C₄⋊(C₂₃⋊C₈)

C₄

semidirect

99. C₄⋊C₈

Dic₂₃

semidirect

104. C₂₃

#1768

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×#10624997	2
C₂×C₈×Dic₂₃	2
#13092263	2
#13092171	2
C₂×C₂₃×C₄⋊C₈	2
#1768	23

subgroup

index

C₂×C₄⋊(C₂₃⋊C₈)

C₂×C₈×Dic₂₃

C₂×C₂₃×C₄⋊C₈

Subgroups — #15673256

Conjugacy classes of subgroups

Complements

Maximal subgroups