Group #15674247 — subgroups

Subgroups — #15674247

Order 2944.

312 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 97 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	#13092441	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#10625018	C₂	1	2	yes	maximal, complemented
3	C₂×#10625015	C₂	1	2	yes	maximal
4	C₄×#67267	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₂₃×#124	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₈×C₂₃	C₂²	1	1	yes	characteristic
7	C₂²×D₄×C₂₃	C₂²	1	1	yes	characteristic
8	C₂₃×#124	—	2	1	no	—
9	C₂₃×#124	—	2	1	no	—
10	C₂₃×#124	—	2	2	no	—
11	C₂×C₂²⋊C₄×C₂₃	C₄	1	4	yes	—
12	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
13	C₂×#67267	C₂²	1	2	yes	—
14	C₂×#67267	C₂²	1	2	yes	—
15	#10625018	—	2	4	no	—
16	#10625015	—	2	4	no	—
17	C₄⋊(C₂₃⋊C₈)	—	2	2	no	—
18	#10624985	—	2	1	no	—
19	C₄×C₂₃⋊C₈	—	2	1	no	—
20	C₂⁴×C₂₃	C₂×C₄	1	2	yes	—
21	C₂×Q₈×C₂₃	—	2	1	no	—
22	C₂×Q₈×C₂₃	—	2	1	no	—
23	C₂×Q₈×C₂₃	—	2	2	no	—
24	C₂×Q₈×C₂₃	—	2	2	no	—
25	C₂×D₄×C₂₃	—	2	1	no	—
26	C₂×D₄×C₂₃	—	2	1	no	—
27	C₂×D₄×C₂₃	—	2	2	no	—
28	C₂×D₄×C₂₃	—	2	2	no	—
29	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
30	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
31	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
32	C₂²⋊C₄×C₂₃	—	2	4	no	—
33	C₂²⋊C₄×C₂₃	—	2	4	no	—
34	C₄²×C₂₃	D₄	1	1	yes	characteristic
35	C₄²×C₂₃	D₄	1	1	yes	characteristic
36	C₄²×C₂₃	D₄	1	2	yes	—
37	#67267	—	2	2	no	—
38	#67267	—	2	2	no	—
39	#67267	—	2	2	no	—
40	#67267	D₄	1	4	yes	—
41	C₂×C₂₃⋊C₈	—	2	2	no	—
42	C₂×C₂₃⋊C₈	—	2	2	no	—
43	C₂³×C₂₃	—	2	2	no	—
44	C₂³×C₂₃	—	2	4	no	—
45	C₂³×C₂₃	—	4	4	no	—
46	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
47	Q₈×C₂₃	—	4	2	no	—
48	Q₈×C₂₃	—	4	1	no	—
49	Q₈×C₂₃	—	4	1	no	—
50	D₄×C₂₃	—	4	2	no	—
51	D₄×C₂₃	—	4	1	no	—
52	D₄×C₂₃	—	4	1	no	—
53	C₂×C₄×C₂₃	—	2	2	no	—
54	C₂×C₄×C₂₃	—	2	4	no	—
55	C₂×C₄×C₂₃	—	4	2	no	—
56	C₂×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
57	C₂×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
58	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
59	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
60	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
61	C₂×C₄×C₂₃	#41	1	2	yes	—
62	C₂₃⋊C₈	—	2	4	no	—
63	C₂₃⋊C₈	—	4	2	no	—
64	#1967	—	23	1	no	maximal, Sylow
65	C₂²×C₂₃	—	2	2	no	—
66	C₂²×C₂₃	—	4	2	no	—
67	C₂²×C₂₃	—	4	4	no	—
68	C₂²×C₂₃	—	8	4	no	—
69	C₂²×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×C₂₃	C₄×D₄	1	2	yes	—
71	C₄×C₂₃	—	2	2	no	—
72	C₄×C₂₃	—	2	2	no	—
73	C₄×C₂₃	—	4	2	no	—
74	C₄×C₂₃	#121	1	2	yes	—
75	C₄×C₂₃	#124	1	1	yes	characteristic
76	C₄×C₂₃	#127	1	1	yes	characteristic
77	C₂×#124	D₂₃	1	1	yes	characteristic
78	#423	—	23	1	no	—
79	C₂×#101	—	23	2	no	—
80	C₂×#100	—	23	2	no	—
81	C₄×M₁₆	—	23	1	no	—
82	C₂×C₂₃	—	2	2	no	—
83	C₂×C₂₃	—	8	2	no	—
84	C₂×C₂₃	#390	1	1	yes	characteristic
85	C₂×C₂₃	#413	1	2	yes	—
86	C₂²×Q₈	C₂×D₂₃	1	1	yes	characteristic
87	C₂²×D₄	C₂×D₂₃	1	1	yes	characteristic
88	C₂×M₁₆	—	23	2	no	—
89	C₂×M₁₆	—	23	2	no	—
90	#124	—	2	1	no	—
91	#124	—	2	1	no	—
92	#124	—	2	2	no	—
93	C₂×C₂²⋊C₄	Dic₂₃	1	4	yes	—
94	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
95	C₄⋊C₈	—	46	2	no	—
96	#101	—	46	4	no	—
97	#100	—	46	4	no	—
98	#97	—	46	1	no	—
99	C₄×C₈	—	46	1	no	—
100	C₂₃	#1967	1	1	yes	characteristic, complemented, Sylow
101	C₂⁴	C₂×Dic₂₃	1	2	yes	—
102	C₂×Q₈	—	2	2	no	—
103	C₂×Q₈	—	2	2	no	—
104	C₂×Q₈	—	2	1	no	—
105	C₂×Q₈	—	2	1	no	—
106	C₂×D₄	—	2	2	no	—
107	C₂×D₄	—	2	2	no	—
108	C₂×D₄	—	2	1	no	—
109	C₂×D₄	—	2	1	no	—
110	C₂²×C₄	C₂×Dic₂₃	1	2	yes	—
111	C₂²×C₄	C₂×Dic₂₃	1	2	yes	—
112	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
113	M₁₆	—	23	4	no	—
114	M₁₆	—	46	2	no	—
115	M₁₆	—	46	2	no	—
116	M₁₆	—	46	2	no	—
117	C₂×C₈	—	46	2	no	—
118	C₂×C₈	—	46	2	no	—
119	C₂²⋊C₄	—	2	4	no	—
120	C₂²⋊C₄	—	2	4	no	—
121	C₄²	C₂²⋊D₂₃	1	1	yes	characteristic
122	C₄²	C₂²⋊D₂₃	1	1	yes	characteristic
123	C₄²	C₂²⋊D₂₃	1	2	yes	—
124	C₂³	—	2	2	no	—
125	C₂³	—	2	4	no	—
126	C₂³	—	4	4	no	—
127	C₂³	C₂²×Dic₂₃	1	1	yes	characteristic
128	Q₈	—	4	1	no	—
129	Q₈	—	4	2	no	—
130	Q₈	—	4	1	no	—
131	D₄	—	4	1	no	—
132	D₄	—	4	2	no	—
133	D₄	—	4	1	no	—
134	C₂×C₄	—	2	4	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	4	2	no	—
137	C₂×C₄	C₂²⋊Dic₂₃	1	2	yes	—
138	C₂×C₄	C₂²⋊Dic₂₃	1	2	yes	—
139	C₂×C₄	D₄×D₂₃	1	2	yes	—
140	C₂×C₄	#67290	1	2	yes	—
141	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
142	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
143	C₈	—	46	4	no	—
144	C₈	—	92	2	no	—
145	C₂²	—	2	2	no	—
146	C₂²	—	4	4	no	—
147	C₂²	—	4	2	no	—
148	C₂²	—	8	4	no	—
149	C₂²	D₄×Dic₂₃	1	2	yes	—
150	C₂²	C₂×C₂²⋊Dic₂₃	1	1	yes	characteristic
151	C₄	—	2	2	no	—
152	C₄	—	2	2	no	—
153	C₄	—	4	2	no	—
154	C₄	#10625107	1	1	yes	characteristic
155	C₄	#10625109	1	2	yes	—
156	C₄	#10625111	1	1	yes	characteristic
157	C₂	—	2	2	no	—
158	C₂	—	8	2	no	—
159	C₂	#13092320	1	1	yes	characteristic
160	C₂	#13092401	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13092441

C₂

yes

maximal, characteristic, complemented

C₂×#10625018

C₂

yes

maximal, complemented

C₂×#10625015

C₂

yes

maximal

C₄×#67267

C₂

yes

maximal, characteristic, complemented

C₂×C₂₃×#124

C₂

yes

maximal, characteristic

C₂²×Q₈×C₂₃

C₂²

yes

characteristic

C₂²×D₄×C₂₃

C₂²

yes

characteristic

C₂₃×#124

—

C₂₃×#124

—

C₂₃×#124

—

C₂×C₂²⋊C₄×C₂₃

C₄

yes

—

C₂×C₄²×C₂₃

C₂²

yes

characteristic

C₂×#67267

C₂²

yes

—

C₂×#67267

C₂²

yes

—

#10625018

—

#10625015

—

C₄⋊(C₂₃⋊C₈)

—

#10624985

—

C₄×C₂₃⋊C₈

—

C₂⁴×C₂₃

C₂×C₄

yes

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

C₂²×C₄×C₂₃

C₂×C₄

yes

—

C₂²×C₄×C₂₃

C₂×C₄

yes

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂²⋊C₄×C₂₃

—

C₂²⋊C₄×C₂₃

—

C₄²×C₂₃

D₄

yes

characteristic

C₄²×C₂₃

D₄

yes

characteristic

C₄²×C₂₃

D₄

yes

—

—

—

—

D₄

yes

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

C₂²×C₄

yes

characteristic

Q₈×C₂₃

—

Q₈×C₂₃

—

Q₈×C₂₃

—

D₄×C₂₃

—

D₄×C₂₃

—

D₄×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

—

#1967

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₃

C₄×D₄

yes

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

#121

yes

—

C₄×C₂₃

yes

C₄×C₂₃

yes

D₂₃

yes

—

—

—

C₄×M₁₆

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

#390

yes

characteristic

C₂×C₂₃

#413

yes

—

C₂²×Q₈

C₂×D₂₃

yes

characteristic

C₂²×D₄

C₂×D₂₃

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

#124

—

#124

—

#124

—

C₂×C₂²⋊C₄

Dic₂₃

yes

—

C₂×C₄²

C₂×D₂₃

yes

characteristic

C₄⋊C₈

—

#101

—

#100

—

#97

—

C₄×C₈

—

100

C₂₃

#1967

yes

characteristic, complemented, Sylow

101

C₂⁴

C₂×Dic₂₃

yes

—

102

C₂×Q₈

—

103

C₂×Q₈

—

104

C₂×Q₈

—

105

C₂×Q₈

—

106

C₂×D₄

—

107

C₂×D₄

—

108

C₂×D₄

—

109

C₂×D₄

—

110

C₂²×C₄

C₂×Dic₂₃

yes

—

111

C₂²×C₄

C₂×Dic₂₃

yes

—

112

C₂²×C₄

C₂²×D₂₃

yes

characteristic

113

M₁₆

—

114

M₁₆

—

115

M₁₆

—

116

M₁₆

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂²⋊C₄

—

120

C₂²⋊C₄

—

121

C₄²

C₂²⋊D₂₃

yes

characteristic

122

C₄²

C₂²⋊D₂₃

yes

characteristic

123

C₄²

C₂²⋊D₂₃

yes

—

124

C₂³

—

125

C₂³

—

126

C₂³

—

127

C₂³

C₂²×Dic₂₃

yes

characteristic

128

Q₈

—

129

Q₈

—

130

Q₈

—

131

D₄

—

132

D₄

—

133

D₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

C₂²⋊Dic₂₃

yes

—

138

C₂×C₄

C₂²⋊Dic₂₃

yes

—

139

C₂×C₄

D₄×D₂₃

yes

—

140

C₂×C₄

#67290

yes

—

141

C₂×C₄

C₂×C₂²⋊D₂₃

yes

characteristic

142

C₂×C₄

C₂×C₂²⋊D₂₃

yes

characteristic

143

C₈

—

144

C₈

—

145

C₂²

—

146

C₂²

—

147

C₂²

—

148

C₂²

—

149

C₂²

D₄×Dic₂₃

yes

—

150

C₂²

C₂×C₂²⋊Dic₂₃

yes

characteristic

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

#10625107

yes

characteristic

155

C₄

#10625109

yes

—

156

C₄

#10625111

yes

characteristic

157

C₂

—

158

C₂

—

159

C₂

#13092320

yes

characteristic

160

C₂

#13092401

yes

—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₂₃
commutator	C₂³×C₂₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₃
Fitting	C₂×C₂₃×#124	C₂
socle	C₂²×C₂₃	C₂×C₂²⋊C₄
prime radical	C₂²×D₄×C₂₃	C₂²
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	C₂×#124	D₂₃
hypocenter	C₂₃	#1967
LCS	G ▷ C₂³×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#124

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₂₃

commutator

C₂³×C₂₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₃

Fitting

C₂×C₂₃×#124

C₂

socle

C₂²×C₂₃

C₂×C₂²⋊C₄

prime radical

C₂²×D₄×C₂₃

C₂²

cosocle

C₂²×C₄×C₂₃

C₂³

D₂₃

C₂₃

G ▷ C₂³×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#124

N	H ≅ G/N	type
1. #13092441	C₂	semidirect
2. C₂×#10625018	C₂	semidirect
4. C₄×#67267	C₂	semidirect
100. C₂₃	#1967	semidirect

H ≅ G/N

type

1. #13092441

C₂

semidirect

2. C₂×#10625018

C₂

semidirect

4. C₄×#67267

C₂

semidirect

100. C₂₃

#1967

semidirect

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

subgroup	index
#13092441	2
C₂×#10625018	2
C₂×#10625015	2
C₄×#67267	2
C₂×C₂₃×#124	2
#1967	23

subgroup

index

C₂×C₂₃×#124

Subgroups — #15674247

Conjugacy classes of subgroups

Complements

Maximal subgroups