Group #15674113 — subgroups

Subgroups — #15674113

Order 2944.

260 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 129 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×Dic₂₃⋊C₈	C₂	1	2	yes	maximal
2	C₂×C₄×C₂₃⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₄×C₄⋊Dic₂₃	C₂	1	1	yes	maximal, characteristic
4	#13092171	C₂	1	2	yes	maximal
5	C₂×C₂₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₈×C₂₃	C₂²	1	2	yes	—
7	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₂₃	C₄	1	1	yes	characteristic
9	C₂×C₄⋊Dic₂₃	C₄	1	1	yes	characteristic
10	C₂×C₄×Dic₂₃	C₂²	1	2	yes	—
11	C₂²×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
12	C₂²×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
13	C₂₃×C₄⋊C₈	—	2	2	no	—
14	C₂₃⋊#95	C₄	1	2	yes	—
15	Dic₂₃⋊C₈	—	2	4	no	—
16	C₄×C₂₃⋊C₈	—	2	2	no	—
17	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
20	C₂²×Dic₂₃	C₂×C₄	1	2	yes	—
21	C₂²×Dic₂₃	C₂×C₄	1	2	yes	—
22	C₂×C₈×C₂₃	—	2	2	no	—
23	C₂×C₈×C₂₃	—	2	2	no	—
24	C₂×C₈×C₂₃	—	2	2	no	—
25	C₄²×C₂₃	D₄	1	2	yes	—
26	C₄²×C₂₃	Q₈	1	2	yes	—
27	C₄⋊Dic₂₃	—	2	2	no	—
28	C₄⋊Dic₂₃	C₈	1	4	yes	complemented
29	C₄×Dic₂₃	—	2	2	no	—
30	C₄×Dic₂₃	—	2	2	no	—
31	C₂×C₂₃⋊C₈	—	2	1	no	—
32	C₂×C₂₃⋊C₈	—	2	1	no	—
33	C₂×C₂₃⋊C₈	—	2	1	no	—
34	C₂×C₂₃⋊C₈	—	2	1	no	—
35	C₂×C₂₃⋊C₈	D₄	1	2	yes	—
36	C₂×C₂₃⋊C₈	Q₈	1	2	yes	—
37	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₂₃	—	2	1	no	—
39	C₂×C₄×C₂₃	—	2	1	no	—
40	C₂×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
41	C₂×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
42	C₂×C₄×C₂₃	C₂×C₈	1	2	yes	—
43	C₂×C₄×C₂₃	M₁₆	1	2	yes	—
44	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₄×C₂₃	#41	1	1	yes	characteristic
49	C₂×C₄×C₂₃	#41	1	1	yes	characteristic
50	C₂×Dic₂₃	—	2	2	no	—
51	C₂×Dic₂₃	—	2	2	no	—
52	C₂×Dic₂₃	—	2	2	no	—
53	C₂×Dic₂₃	—	2	2	no	—
54	C₂×Dic₂₃	—	2	2	no	—
55	C₂×Dic₂₃	C₂×C₈	1	4	yes	—
56	C₈×C₂₃	—	4	2	no	—
57	C₂₃⋊C₈	—	2	2	no	—
58	C₂₃⋊C₈	—	2	2	no	—
59	#1939	—	23	1	no	maximal, Sylow
60	C₂²×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂²×C₂₃	C₄×D₄	1	1	yes	characteristic
62	C₂²×C₂₃	C₄×Q₈	1	1	yes	characteristic
63	C₂²×C₂₃	C₂²×C₈	1	1	yes	characteristic
64	C₂²×C₂₃	C₂×M₁₆	1	1	yes	characteristic
65	C₂²×C₂₃	#131	1	1	yes	characteristic
66	C₂²×C₂₃	#131	1	1	yes	characteristic
67	C₄×C₂₃	—	2	2	no	—
68	C₄×C₂₃	C₄⋊C₈	1	4	yes	complemented
69	C₄×C₂₃	#121	1	1	yes	characteristic
70	C₄×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
71	C₄×C₂₃	#125	1	1	yes	characteristic
72	C₄×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
73	Dic₂₃	—	2	4	no	—
74	Dic₂₃	—	4	2	no	—
75	C₂×C₄⋊C₈	—	23	2	no	—
76	C₂×C₄⋊C₈	D₂₃	1	1	yes	characteristic
77	C₂×C₄×C₈	—	23	1	no	—
78	C₄×C₄⋊C₄	—	23	1	no	—
79	#336	—	23	2	no	—
80	C₂×C₂₃	#389	1	1	yes	characteristic
81	C₂×C₂₃	C₂×C₄⋊C₈	1	1	yes	characteristic
82	C₂×C₂₃	#424	1	1	yes	characteristic
83	C₂×C₂₃	C₈×D₄	1	1	yes	characteristic
84	C₂×C₂₃	#436	1	1	yes	characteristic
85	C₂×C₂₃	C₈×Q₈	1	1	yes	characteristic
86	C₂×C₂₃	#446	1	1	yes	characteristic
87	C₂²×C₈	—	23	1	no	—
88	C₂²×C₈	—	23	1	no	—
89	C₂²×C₈	C₂×D₂₃	1	2	yes	—
90	C₂×C₄⋊C₄	—	23	1	no	—
91	C₂×C₄⋊C₄	—	23	1	no	—
92	C₂×C₄²	—	23	2	no	—
93	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
94	C₄⋊C₈	—	2	2	no	—
95	C₄⋊C₈	—	46	4	no	—
96	C₄×C₈	—	46	2	no	—
97	#95	—	23	2	no	—
98	C₂₃	#1939	1	1	yes	characteristic, complemented, Sylow
99	C₂²×C₄	—	23	2	no	—
100	C₂²×C₄	—	23	2	no	—
101	C₂²×C₄	C₄×D₂₃	1	1	yes	characteristic
102	C₂²×C₄	C₄×D₂₃	1	1	yes	characteristic
103	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
104	C₂×C₈	—	2	2	no	—
105	C₂×C₈	—	2	2	no	—
106	C₂×C₈	—	2	2	no	—
107	C₂×C₈	—	23	2	no	—
108	C₂×C₈	—	23	2	no	—
109	C₂×C₈	—	46	1	no	—
110	C₂×C₈	—	46	1	no	—
111	C₂×C₈	—	46	1	no	—
112	C₂×C₈	—	46	1	no	—
113	C₄⋊C₄	—	23	4	no	—
114	C₄⋊C₄	—	46	2	no	—
115	C₄²	—	46	2	no	—
116	C₄²	—	46	2	no	—
117	C₄²	Dic₄₆	1	2	yes	—
118	C₄²	C₂²⋊D₂₃	1	2	yes	—
119	C₂³	C₂×C₄×D₂₃	1	1	yes	characteristic
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	23	4	no	—
123	C₂×C₄	—	46	2	no	—
124	C₂×C₄	—	46	2	no	—
125	C₂×C₄	—	46	2	no	—
126	C₂×C₄	—	46	2	no	—
127	C₂×C₄	—	46	2	no	—
128	C₂×C₄	C₈×D₂₃	1	2	yes	—
129	C₂×C₄	#67262	1	2	yes	—
130	C₂×C₄	Dic₂₃⋊C₄	1	2	yes	—
131	C₂×C₄	Dic₂₃⋊C₄	1	2	yes	—
132	C₂×C₄	C₂×Dic₄₆	1	1	yes	characteristic
133	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
134	C₂×C₄	#67290	1	1	yes	characteristic
135	C₂×C₄	Q₈×D₂₃	1	1	yes	characteristic
136	C₂×C₄	#67292	1	1	yes	characteristic
137	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
138	C₈	—	4	2	no	—
139	C₈	—	46	2	no	—
140	C₈	—	46	2	no	—
141	C₂²	#10625058	1	1	yes	characteristic
142	C₂²	#10625064	1	1	yes	characteristic
143	C₂²	C₂×C₈×D₂₃	1	1	yes	characteristic
144	C₂²	C₂×#67262	1	1	yes	characteristic
145	C₂²	#10625078	1	1	yes	characteristic
146	C₂²	#10625078	1	1	yes	characteristic
147	C₂²	C₂×Dic₂₃⋊C₄	1	1	yes	characteristic
148	C₄	—	2	2	no	—
149	C₄	—	46	4	no	—
150	C₄	—	92	2	no	—
151	C₄	Dic₂₃⋊C₈	1	4	yes	complemented
152	C₄	#10625059	1	1	yes	characteristic
153	C₄	#10625061	1	1	yes	characteristic
154	C₄	#10625109	1	1	yes	characteristic
155	C₄	#10625117	1	1	yes	characteristic
156	C₂	#13092313	1	1	yes	characteristic
157	C₂	C₂×Dic₂₃⋊C₈	1	1	yes	characteristic
158	C₂	#13092445	1	1	yes	characteristic
159	C₂	#13092482	1	1	yes	characteristic
160	C₂	#13092494	1	1	yes	characteristic
161	C₂	#13092541	1	1	yes	characteristic
162	C₂	#13092544	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Dic₂₃⋊C₈

C₂

yes

maximal

C₂×C₄×C₂₃⋊C₈

C₂

yes

maximal, characteristic

C₄×C₄⋊Dic₂₃

C₂

yes

maximal, characteristic

#13092171

C₂

yes

maximal

C₂×C₂₃×C₄⋊C₈

C₂

yes

maximal, characteristic

C₂²×C₈×C₂₃

C₂²

yes

—

C₂×C₄²×C₂₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₂₃

C₄

yes

characteristic

C₂×C₄⋊Dic₂₃

C₄

yes

characteristic

C₂×C₄×Dic₂₃

C₂²

yes

—

C₂²×C₂₃⋊C₈

C₂²

yes

characteristic

C₂²×C₂₃⋊C₈

C₂²

yes

characteristic

C₂₃×C₄⋊C₈

—

C₂₃⋊#95

C₄

yes

—

Dic₂₃⋊C₈

—

C₄×C₂₃⋊C₈

—

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

—

C₄²×C₂₃

D₄

yes

—

C₄²×C₂₃

Q₈

yes

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

C₈

yes

complemented

C₄×Dic₂₃

—

C₄×Dic₂₃

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

D₄

yes

—

C₂×C₂₃⋊C₈

Q₈

yes

—

C₂³×C₂₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₄⋊C₄

yes

—

C₂×C₄×C₂₃

C₄⋊C₄

yes

—

C₂×C₄×C₂₃

C₂×C₈

yes

—

C₂×C₄×C₂₃

M₁₆

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×Q₈

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

C₂×C₈

yes

—

C₈×C₂₃

—

C₂₃⋊C₈

—

C₂₃⋊C₈

—

#1939

—

maximal, Sylow

C₂²×C₂₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₂₃

C₄×D₄

yes

characteristic

C₂²×C₂₃

C₄×Q₈

yes

characteristic

C₂²×C₂₃

C₂²×C₈

yes

characteristic

C₂²×C₂₃

C₂×M₁₆

yes

C₂²×C₂₃

yes

C₂²×C₂₃

yes

C₄×C₂₃

—

C₄×C₂₃

C₄⋊C₈

yes

complemented

C₄×C₂₃

#121

yes

characteristic

C₄×C₂₃

C₂²⋊Q₈

yes

characteristic

C₄×C₂₃

#125

yes

characteristic

C₄×C₂₃

C₄⋊Q₈

yes

characteristic

Dic₂₃

—

Dic₂₃

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

D₂₃

yes

characteristic

C₂×C₄×C₈

—

C₄×C₄⋊C₄

—

#336

—

C₂×C₂₃

#389

yes

characteristic

C₂×C₂₃

C₂×C₄⋊C₈

yes

characteristic

C₂×C₂₃

#424

yes

characteristic

C₂×C₂₃

C₈×D₄

yes

characteristic

C₂×C₂₃

#436

yes

characteristic

C₂×C₂₃

C₈×Q₈

yes

characteristic

C₂×C₂₃

#446

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×D₂₃

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₂₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

#95

—

C₂₃

#1939

yes

characteristic, complemented, Sylow

C₂²×C₄

—

100

C₂²×C₄

—

101

C₂²×C₄

C₄×D₂₃

yes

characteristic

102

C₂²×C₄

C₄×D₂₃

yes

characteristic

103

C₂²×C₄

C₂²×D₂₃

yes

characteristic

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄²

—

116

C₄²

—

117

C₄²

Dic₄₆

yes

—

118

C₄²

C₂²⋊D₂₃

yes

—

119

C₂³

C₂×C₄×D₂₃

yes

characteristic

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

C₈×D₂₃

yes

—

129

C₂×C₄

#67262

yes

—

130

C₂×C₄

Dic₂₃⋊C₄

yes

—

131

C₂×C₄

Dic₂₃⋊C₄

yes

—

132

C₂×C₄

C₂×Dic₄₆

yes

characteristic

133

C₂×C₄

D₄×D₂₃

yes

characteristic

134

C₂×C₄

#67290

yes

characteristic

135

C₂×C₄

Q₈×D₂₃

yes

characteristic

136

C₂×C₄

#67292

yes

characteristic

137

C₂×C₄

C₂×C₂²⋊D₂₃

yes

characteristic

138

C₈

—

139

C₈

—

140

C₈

—

141

C₂²

#10625058

yes

characteristic

142

C₂²

#10625064

yes

characteristic

143

C₂²

C₂×C₈×D₂₃

yes

characteristic

144

C₂²

C₂×#67262

yes

characteristic

145

C₂²

#10625078

yes

characteristic

146

C₂²

#10625078

yes

characteristic

147

C₂²

C₂×Dic₂₃⋊C₄

yes

characteristic

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

Dic₂₃⋊C₈

yes

complemented

152

C₄

#10625059

yes

characteristic

153

C₄

#10625061

yes

characteristic

154

C₄

#10625109

yes

characteristic

155

C₄

#10625117

yes

characteristic

156

C₂

#13092313

yes

characteristic

157

C₂

C₂×Dic₂₃⋊C₈

yes

characteristic

158

C₂

#13092445

yes

characteristic

159

C₂

#13092482

yes

characteristic

160

C₂

#13092494

yes

characteristic

161

C₂

#13092541

yes

characteristic

162

C₂

#13092544

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₂₃
commutator	C₂²×C₂₃	C₂²×C₈
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	#1939
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₂₃

commutator

C₂²×C₂₃

C₂²×C₈

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
28. C₄⋊Dic₂₃	C₈	semidirect
68. C₄×C₂₃	C₄⋊C₈	semidirect
98. C₂₃	#1939	semidirect
151. C₄	Dic₂₃⋊C₈	semidirect

H ≅ G/N

type

28. C₄⋊Dic₂₃

C₈

semidirect

68. C₄×C₂₃

C₄⋊C₈

semidirect

98. C₂₃

#1939

semidirect

151. C₄

Dic₂₃⋊C₈

semidirect

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

subgroup	index
C₂×Dic₂₃⋊C₈	2
C₂×C₄×C₂₃⋊C₈	2
C₄×C₄⋊Dic₂₃	2
#13092171	2
C₂×C₂₃×C₄⋊C₈	2
#1939	23

subgroup

index

C₂×Dic₂₃⋊C₈

C₂×C₄×C₂₃⋊C₈

C₄×C₄⋊Dic₂₃

#13092171

C₂×C₂₃×C₄⋊C₈

#1939

Subgroups — #15674113

Conjugacy classes of subgroups

Complements

Maximal subgroups