Group #15682115 — subgroups

Subgroups — #15682115

Order 2944.

484 conjugacy classes of subgroups; only the 221 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 221 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×#10625119	C₂	1	2	yes	maximal, complemented
2	#13092489	C₂	1	8	yes	maximal, complemented
3	#13092441	C₂	1	1	yes	maximal, characteristic, complemented
4	#13092384	C₂	1	2	yes	maximal, complemented
5	C₂×#10624985	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#532	C₂	1	1	yes	maximal, characteristic
7	C₂×#41×C₂₃	C₂²	1	2	yes	—
8	C₂₃×C₄⋊Q₈	C₄	1	1	yes	characteristic
9	C₂₃×#127	C₄	1	1	yes	characteristic
10	C₂₃×#124	C₄	1	2	yes	—
11	C₂₃×#121	C₄	1	4	yes	—
12	C₄×D₄×C₂₃	C₂²	1	4	yes	—
13	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
14	#10625119	—	2	8	no	—
15	C₂×#67267	C₂²	1	2	yes	—
16	C₂×#67267	C₂²	1	4	yes	—
17	C₂²×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
18	C₂²×C₂₃⋊C₈	C₂²	1	4	yes	—
19	C₂²×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
20	C₂²⋊(C₂₃⋊C₈)	C₂²	1	8	yes	—
21	C₄⋊(C₂₃⋊C₈)	C₂²	1	4	yes	—
22	#10624985	C₂²	1	4	yes	—
23	#41×C₂₃	—	2	4	no	—
24	#41×C₂₃	—	2	4	no	—
25	C₂×Q₈×C₂₃	C₂×C₄	1	2	yes	—
26	C₂×D₄×C₂₃	C₂×C₄	1	4	yes	—
27	C₂×D₄×C₂₃	C₂×C₄	1	2	yes	—
28	C₂×D₄×C₂₃	C₂×C₄	1	4	yes	—
29	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
30	C₂²×C₄×C₂₃	C₂³	1	4	yes	—
31	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
32	C₄⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
33	C₂²⋊C₄×C₂₃	C₂×C₄	1	8	yes	—
34	C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
35	C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
36	C₄²×C₂₃	C₂³	1	2	yes	—
37	#67267	—	2	4	no	—
38	#67267	—	2	8	no	—
39	C₂×C₂₃⋊C₈	—	2	2	no	—
40	C₂×C₂₃⋊C₈	—	2	8	no	—
41	C₂×C₂₃⋊C₈	D₄	1	4	yes	complemented
42	C₂×C₂₃⋊C₈	C₂³	1	4	yes	—
43	C₂×C₂₃⋊C₈	C₂³	1	2	yes	—
44	C₂×C₂₃⋊C₈	C₂³	1	2	yes	—
45	C₂³×C₂₃	C₂²×C₄	1	4	yes	—
46	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
47	Q₈×C₂₃	—	2	4	no	—
48	D₄×C₂₃	—	2	8	no	—
49	D₄×C₂₃	—	2	4	no	—
50	D₄×C₂₃	—	4	4	no	—
51	C₂×C₄×C₂₃	—	2	4	no	—
52	C₂×C₄×C₂₃	—	2	4	no	—
53	C₂×C₄×C₂₃	—	4	2	no	—
54	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
55	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
56	C₂×C₄×C₂₃	C₂²×C₄	1	4	yes	—
57	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
59	C₂×C₄×C₂₃	#41	1	2	yes	—
60	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
61	C₂₃⋊C₈	—	2	4	no	—
62	C₂₃⋊C₈	—	2	2	no	—
63	C₂₃⋊C₈	C₂×D₄	1	4	yes	—
64	#2932	—	23	1	no	maximal, Sylow
65	C₂²×C₂₃	—	2	2	no	—
66	C₂²×C₂₃	—	2	4	no	—
67	C₂²×C₂₃	—	2	8	no	—
68	C₂²×C₂₃	C₄×D₄	1	2	yes	—
69	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
70	C₄×C₂₃	—	2	4	no	—
71	C₄×C₂₃	—	2	2	no	—
72	C₄×C₂₃	—	2	2	no	—
73	C₄×C₂₃	C₄×D₄	1	2	yes	—
74	C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
75	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
76	C₂×#131	—	23	2	no	—
77	#532	D₂₃	1	1	yes	characteristic
78	#435	—	23	8	no	—
79	#423	—	23	1	no	—
80	#408	—	23	2	no	—
81	C₂×#97	—	23	1	no	—
82	C₂×C₂₃	—	2	2	no	—
83	C₂×C₂₃	—	4	4	no	—
84	C₂×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
85	C₂×C₂₃	#568	1	2	yes	—
86	C₂×#41	C₂×D₂₃	1	2	yes	—
87	#131	—	46	8	no	—
88	C₂×M₁₆	—	23	2	no	—
89	C₂×M₁₆	—	23	4	no	—
90	C₂²×C₈	—	23	1	no	—
91	C₂²×C₈	—	23	4	no	—
92	C₂²×C₈	—	23	1	no	—
93	C₄⋊Q₈	Dic₂₃	1	1	yes	characteristic
94	#127	Dic₂₃	1	1	yes	characteristic
95	#124	Dic₂₃	1	2	yes	—
96	#121	Dic₂₃	1	4	yes	—
97	C₄×D₄	C₂×D₂₃	1	4	yes	—
98	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
99	C₄⋊C₈	—	23	4	no	—
100	C₂²⋊C₈	—	23	8	no	—
101	#97	—	23	4	no	—
102	C₂₃	#2932	1	1	yes	characteristic, complemented, Sylow
103	#41	—	2	4	no	—
104	#41	—	2	4	no	—
105	C₂×Q₈	C₂×Dic₂₃	1	2	yes	—
106	C₂×D₄	C₂×Dic₂₃	1	4	yes	—
107	C₂×D₄	C₂×Dic₂₃	1	2	yes	—
108	C₂×D₄	C₂×Dic₂₃	1	4	yes	—
109	C₂²×C₄	C₂×Dic₂₃	1	2	yes	—
110	C₂²×C₄	C₂²×D₂₃	1	4	yes	—
111	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
112	M₁₆	—	46	8	no	—
113	M₁₆	—	46	4	no	—
114	C₂×C₈	—	23	4	no	—
115	C₂×C₈	—	23	2	no	—
116	C₂×C₈	—	23	2	no	—
117	C₂×C₈	—	23	4	no	—
118	C₂×C₈	—	46	8	no	—
119	C₂×C₈	—	46	2	no	—
120	C₄⋊C₄	C₂×Dic₂₃	1	4	yes	—
121	C₂²⋊C₄	C₂×Dic₂₃	1	8	yes	—
122	C₄²	C₂×Dic₂₃	1	1	yes	characteristic
123	C₄²	C₂×Dic₂₃	1	1	yes	characteristic
124	C₄²	C₂²×D₂₃	1	2	yes	—
125	C₂³	C₂²×Dic₂₃	1	4	yes	—
126	C₂³	C₂²×Dic₂₃	1	1	yes	characteristic
127	Q₈	—	2	4	no	—
128	D₄	—	2	8	no	—
129	D₄	—	2	4	no	—
130	D₄	—	4	4	no	—
131	C₂×C₄	—	2	4	no	—
132	C₂×C₄	—	2	4	no	—
133	C₂×C₄	—	4	2	no	—
134	C₂×C₄	D₄×D₂₃	1	2	yes	—
135	C₂×C₄	#67290	1	2	yes	—
136	C₂×C₄	C₂²×Dic₂₃	1	2	yes	—
137	C₂×C₄	C₂²×Dic₂₃	1	2	yes	—
138	C₂×C₄	C₂²×Dic₂₃	1	4	yes	—
139	C₂×C₄	C₂²×Dic₂₃	1	1	yes	characteristic
140	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
141	C₈	—	23	4	no	—
142	C₈	—	46	4	no	—
143	C₈	—	46	2	no	—
144	C₂²	—	2	2	no	—
145	C₂²	—	2	4	no	—
146	C₂²	—	2	8	no	—
147	C₂²	D₄×Dic₂₃	1	2	yes	—
148	C₂²	C₂³×Dic₂₃	1	1	yes	characteristic
149	C₄	—	2	4	no	—
150	C₄	—	2	2	no	—
151	C₄	—	2	2	no	—
152	C₄	D₄×Dic₂₃	1	2	yes	—
153	C₄	C₂×D₄×D₂₃	1	1	yes	characteristic
154	C₄	C₂×#67290	1	1	yes	characteristic
155	C₂	—	2	2	no	—
156	C₂	—	4	4	no	—
157	C₂	C₂×D₄×Dic₂₃	1	1	yes	characteristic
158	C₂	#13093184	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10625119

C₂

yes

maximal, complemented

#13092489

C₂

yes

maximal, complemented

#13092441

C₂

yes

maximal, characteristic, complemented

#13092384

C₂

yes

maximal, complemented

C₂×#10624985

C₂

yes

maximal, characteristic, complemented

C₂₃×#532

C₂

yes

maximal, characteristic

C₂×#41×C₂₃

C₂²

yes

—

C₂₃×C₄⋊Q₈

C₄

yes

characteristic

C₂₃×#127

C₄

yes

characteristic

C₂₃×#124

C₄

yes

—

C₂₃×#121

C₄

yes

—

C₄×D₄×C₂₃

C₂²

yes

—

C₂×C₄²×C₂₃

C₂²

yes

characteristic

#10625119

—

C₂×#67267

C₂²

yes

—

C₂×#67267

C₂²

yes

—

C₂²×C₂₃⋊C₈

C₂²

yes

characteristic

C₂²×C₂₃⋊C₈

C₂²

yes

—

C₂²×C₂₃⋊C₈

C₂²

yes

characteristic

C₂²⋊(C₂₃⋊C₈)

C₂²

yes

—

C₄⋊(C₂₃⋊C₈)

C₂²

yes

—

#10624985

C₂²

yes

—

#41×C₂₃

—

#41×C₂₃

—

C₂×Q₈×C₂₃

C₂×C₄

yes

—

C₂×D₄×C₂₃

C₂×C₄

yes

—

C₂×D₄×C₂₃

C₂×C₄

yes

—

C₂×D₄×C₂₃

C₂×C₄

yes

—

C₂²×C₄×C₂₃

C₂×C₄

yes

—

C₂²×C₄×C₂₃

C₂³

yes

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₄⋊C₄×C₂₃

C₂×C₄

yes

—

C₂²⋊C₄×C₂₃

C₂×C₄

yes

—

C₄²×C₂₃

C₂×C₄

yes

characteristic

C₄²×C₂₃

C₂×C₄

yes

characteristic

C₄²×C₂₃

C₂³

yes

—

#67267

—

#67267

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

D₄

yes

complemented

C₂×C₂₃⋊C₈

C₂³

yes

—

C₂×C₂₃⋊C₈

C₂³

yes

—

C₂×C₂₃⋊C₈

C₂³

yes

—

C₂³×C₂₃

C₂²×C₄

yes

—

C₂³×C₂₃

C₂²×C₄

yes

characteristic

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

C₂²×C₄

yes

—

C₂×C₄×C₂₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

#41

yes

—

C₂×C₄×C₂₃

C₂⁴

yes

characteristic

C₂₃⋊C₈

—

C₂₃⋊C₈

—

C₂₃⋊C₈

C₂×D₄

yes

—

#2932

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

C₄×D₄

yes

—

C₂²×C₂₃

C₂³×C₄

yes

characteristic

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

C₄×D₄

yes

—

C₄×C₂₃

C₂²×D₄

yes

characteristic

C₄×C₂₃

C₂×#41

yes

characteristic

C₂×#131

—

#532

D₂₃

yes

—

—

—

C₂×#97

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

C₂×C₄×D₄

yes

characteristic

C₂×C₂₃

#568

yes

—

C₂×#41

C₂×D₂₃

yes

—

#131

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₄⋊Q₈

Dic₂₃

yes

characteristic

#127

Dic₂₃

yes

characteristic

#124

Dic₂₃

yes

—

#121

Dic₂₃

yes

—

C₄×D₄

C₂×D₂₃

yes

—

C₂×C₄²

C₂×D₂₃

yes

characteristic

C₄⋊C₈

—

100

C₂²⋊C₈

—

101

#97

—

102

C₂₃

#2932

yes

characteristic, complemented, Sylow

103

#41

—

104

#41

—

105

C₂×Q₈

C₂×Dic₂₃

yes

—

106

C₂×D₄

C₂×Dic₂₃

yes

—

107

C₂×D₄

C₂×Dic₂₃

yes

—

108

C₂×D₄

C₂×Dic₂₃

yes

—

109

C₂²×C₄

C₂×Dic₂₃

yes

—

110

C₂²×C₄

C₂²×D₂₃

yes

—

111

C₂²×C₄

C₂²×D₂₃

yes

characteristic

112

M₁₆

—

113

M₁₆

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₄⋊C₄

C₂×Dic₂₃

yes

—

121

C₂²⋊C₄

C₂×Dic₂₃

yes

—

122

C₄²

C₂×Dic₂₃

yes

characteristic

123

C₄²

C₂×Dic₂₃

yes

characteristic

124

C₄²

C₂²×D₂₃

yes

—

125

C₂³

C₂²×Dic₂₃

yes

—

126

C₂³

C₂²×Dic₂₃

yes

characteristic

127

Q₈

—

128

D₄

—

129

D₄

—

130

D₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

D₄×D₂₃

yes

—

135

C₂×C₄

#67290

yes

—

136

C₂×C₄

C₂²×Dic₂₃

yes

—

137

C₂×C₄

C₂²×Dic₂₃

yes

—

138

C₂×C₄

C₂²×Dic₂₃

yes

—

139

C₂×C₄

C₂²×Dic₂₃

yes

characteristic

140

C₂×C₄

C₂³×D₂₃

yes

characteristic

141

C₈

—

142

C₈

—

143

C₈

—

144

C₂²

—

145

C₂²

—

146

C₂²

—

147

C₂²

D₄×Dic₂₃

yes

—

148

C₂²

C₂³×Dic₂₃

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

D₄×Dic₂₃

yes

—

153

C₄

C₂×D₄×D₂₃

yes

characteristic

154

C₄

C₂×#67290

yes

characteristic

155

C₂

—

156

C₂

—

157

C₂

C₂×D₄×Dic₂₃

yes

characteristic

158

C₂

#13093184

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₂₃
commutator, socle	C₂²×C₂₃	C₂³×C₄
Fitting, prime radical	C₂₃×#532	C₂
cosocle	C₂×C₄×C₂₃	C₂⁴
hypercenter	#532	D₂₃
hypocenter	C₂₃	#2932
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ #532

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₂₃

commutator, socle

C₂²×C₂₃

C₂³×C₄

Fitting, prime radical

C₂₃×#532

C₂

cosocle

C₂×C₄×C₂₃

C₂⁴

D₂₃

C₂₃

G ▷ C₂²×C₂₃ ▷ C₂₃

C₁ ◁ C₂×C₄ ◁ #532

N	H ≅ G/N	type
1. C₂×#10625119	C₂	semidirect
2. #13092489	C₂	semidirect
3. #13092441	C₂	semidirect
4. #13092384	C₂	semidirect
5. C₂×#10624985	C₂	semidirect
41. C₂×C₂₃⋊C₈	D₄	semidirect
102. C₂₃	#2932	semidirect

H ≅ G/N

type

1. C₂×#10625119

C₂

semidirect

2. #13092489

C₂

semidirect

3. #13092441

C₂

semidirect

4. #13092384

C₂

semidirect

5. C₂×#10624985

C₂

semidirect

41. C₂×C₂₃⋊C₈

D₄

semidirect

102. C₂₃

#2932

semidirect

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

subgroup	index
C₂×#10625119	2
#13092489	2
#13092441	2
#13092384	2
C₂×#10624985	2
C₂₃×#532	2
#2932	23

subgroup

index

C₂₃×#532

Subgroups — #15682115

Conjugacy classes of subgroups

Complements

Maximal subgroups