Group #15683053 — subgroups

Subgroups — #15683053

Order 2944.

460 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 141 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	C₂²×Dic₉₂	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#10625072	C₂	1	1	yes	maximal, characteristic
3	#13092993	C₂	1	2	yes	maximal
4	#13092676	C₂	1	4	yes	maximal, complemented
5	#13092659	C₂	1	4	yes	maximal, complemented
6	#13092425	C₂	1	2	yes	maximal, complemented
7	C₂₃×#427	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#67288	C₂²	1	1	yes	characteristic
9	C₂²×Dic₄₆	C₂²	1	2	yes	—
10	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
11	C₂₃×#117	C₂²	1	2	yes	—
12	C₂×#67267	C₂²	1	1	yes	characteristic
13	C₂×Dic₉₂	—	2	4	no	—
14	C₂×Dic₉₂	C₂²	1	2	yes	—
15	C₂×Dic₉₂	C₂²	1	2	yes	—
16	#10625072	—	2	4	no	—
17	C₂×#67262	C₂²	1	2	yes	—
18	C₂×C₈×D₂₃	C₂²	1	2	yes	—
19	#10625068	C₂²	1	8	yes	—
20	#10625057	—	2	4	no	—
21	#10625046	—	2	2	no	—
22	#10625040	—	2	2	no	—
23	C₂₃×#107	C₂²	1	2	yes	—
24	C₂₃×#106	C₂²	1	2	yes	—
25	#10624992	C₂²	1	8	yes	—
26	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
27	C₂×C₂²⋊D₂₃	D₄	1	2	yes	—
28	C₂²×Dic₂₃	—	2	2	no	—
29	#67288	—	2	2	no	—
30	#67288	—	2	2	no	—
31	C₂×D₉₂	D₄	1	1	yes	characteristic
32	C₂×C₄×D₂₃	C₂³	1	2	yes	—
33	C₂×Dic₄₆	—	2	4	no	—
34	C₂×Dic₄₆	—	2	4	no	—
35	C₂×Dic₄₆	—	2	2	no	—
36	C₂×Dic₄₆	D₄	1	1	yes	characteristic
37	C₂×Dic₄₆	C₂³	1	4	yes	—
38	C₂×C₈×C₂₃	D₄	1	4	yes	—
39	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
40	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
41	C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
42	C₂²⋊C₄×C₂₃	—	2	2	no	—
43	C₄²×C₂₃	—	2	2	no	—
44	(C₂×D₂₃)⋊C₄	—	2	8	no	—
45	C₄⋊Dic₂₃	—	2	8	no	—
46	#67267	—	2	2	no	—
47	C₂×C₂₃⋊C₈	C₂³	1	2	yes	—
48	Dic₉₂	—	2	4	no	—
49	Dic₉₂	—	2	4	no	—
50	#67262	—	2	4	no	—
51	C₈×D₂₃	—	2	4	no	—
52	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
53	C₂²×D₂₃	C₂×D₄	1	2	yes	—
54	C₂×C₄×C₂₃	—	2	4	no	—
55	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
56	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₂₃	#41	1	2	yes	—
58	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
59	C₂²⋊D₂₃	—	2	4	no	—
60	C₂×Dic₂₃	—	2	4	no	—
61	C₂×Dic₂₃	—	4	4	no	—
62	C₂×Dic₂₃	C₂×D₄	1	2	yes	—
63	D₉₂	—	2	2	no	—
64	C₄×D₂₃	—	2	2	no	—
65	C₄×D₂₃	—	2	2	no	—
66	Dic₄₆	—	2	2	no	—
67	Dic₄₆	—	2	4	no	—
68	Dic₄₆	—	2	4	no	—
69	Dic₄₆	—	4	4	no	—
70	C₈×C₂₃	C₂×D₄	1	2	yes	—
71	C₈×C₂₃	C₂×D₄	1	2	yes	—
72	C₂₃⋊C₈	—	2	2	no	—
73	#3059	—	23	1	no	maximal, Sylow
74	C₂²×C₂₃	—	2	2	no	—
75	C₂²×C₂₃	#121	1	2	yes	—
76	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
77	C₂×D₂₃	—	2	2	no	—
78	C₂×D₂₃	—	2	4	no	—
79	C₄×C₂₃	—	4	4	no	—
80	C₄×C₂₃	#121	1	2	yes	—
81	C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
82	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
83	Dic₂₃	—	2	2	no	—
84	Dic₂₃	—	4	4	no	—
85	C₂²×Q₁₆	—	23	1	no	—
86	C₂×#131	—	23	1	no	—
87	#536	—	23	2	no	—
88	#470	—	23	4	no	—
89	#467	—	23	4	no	—
90	#427	D₂₃	1	1	yes	characteristic, complemented
91	#419	—	23	2	no	—
92	C₂×C₂₃	—	2	2	no	—
93	C₂×C₂₃	C₂×#121	1	1	yes	characteristic
94	C₂×C₂₃	#578	1	2	yes	—
95	D₂₃	—	4	2	no	—
96	C₂×#41	—	23	1	no	—
97	C₂²×Q₈	—	23	2	no	—
98	C₂×Q₁₆	—	23	2	no	—
99	C₂×Q₁₆	—	23	2	no	—
100	C₂×Q₁₆	—	46	4	no	—
101	#131	—	46	4	no	—
102	C₂×M₁₆	—	23	1	no	—
103	C₂×M₁₆	—	23	2	no	—
104	C₂²×C₈	—	23	2	no	—
105	C₂²×C₈	C₂×D₂₃	1	1	yes	characteristic
106	C₄⋊Q₈	—	46	2	no	—
107	#124	—	46	2	no	—
108	#123	—	46	4	no	—
109	C₂²⋊Q₈	—	23	8	no	—
110	#117	C₂×D₂₃	1	2	yes	—
111	#107	C₂×D₂₃	1	2	yes	—
112	#106	C₂×D₂₃	1	2	yes	—
113	#103	—	23	8	no	—
114	C₂₃	#3059	1	1	yes	characteristic, complemented, Sylow
115	#41	—	46	2	no	—
116	#41	—	46	2	no	—
117	C₂×Q₈	—	23	1	no	—
118	C₂×Q₈	—	23	4	no	—
119	C₂×Q₈	—	46	2	no	—
120	C₂×Q₈	—	46	4	no	—
121	C₂×Q₈	—	46	4	no	—
122	C₂×D₄	—	23	2	no	—
123	C₂×D₄	—	23	1	no	—
124	C₂²×C₄	—	23	2	no	—
125	C₂²×C₄	—	46	2	no	—
126	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
127	Q₁₆	—	46	4	no	—
128	Q₁₆	—	46	4	no	—
129	M₁₆	—	46	2	no	—
130	M₁₆	—	46	4	no	—
131	C₂×C₈	—	23	2	no	—
132	C₂×C₈	—	46	4	no	—
133	C₂×C₈	D₉₂	1	4	yes	—
134	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
135	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
136	C₄⋊C₄	—	46	8	no	—
137	C₄⋊C₄	C₂²×D₂₃	1	4	yes	—
138	C₂²⋊C₄	—	2	2	no	—
139	C₂²⋊C₄	—	46	8	no	—
140	C₄²	—	2	2	no	—
141	C₂³	—	23	2	no	—
142	C₂³	D₄×D₂₃	1	1	yes	characteristic
143	Q₈	—	46	2	no	—
144	Q₈	—	46	4	no	—
145	Q₈	—	46	4	no	—
146	Q₈	—	92	4	no	—
147	D₄	—	46	4	no	—
148	D₄	—	46	2	no	—
149	C₂×C₄	—	2	4	no	—
150	C₂×C₄	—	23	2	no	—
151	C₂×C₄	—	46	2	no	—
152	C₂×C₄	—	46	2	no	—
153	C₂×C₄	—	46	4	no	—
154	C₂×C₄	—	92	4	no	—
155	C₂×C₄	C₂×D₉₂	1	2	yes	—
156	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
157	C₂×C₄	#67292	1	2	yes	—
158	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
159	C₈	—	46	2	no	—
160	C₈	C₂×D₉₂	1	2	yes	—
161	C₈	C₂×D₉₂	1	2	yes	—
162	C₂²	—	2	2	no	—
163	C₂²	—	46	2	no	—
164	C₂²	—	46	4	no	—
165	C₂²	#10625066	1	2	yes	—
166	C₂²	C₂×D₄×D₂₃	1	1	yes	characteristic
167	C₄	—	4	4	no	—
168	C₄	—	46	2	no	—
169	C₄	—	92	4	no	—
170	C₄	#10625066	1	2	yes	—
171	C₄	C₂²×D₉₂	1	1	yes	characteristic
172	C₄	C₂×#67292	1	1	yes	characteristic
173	C₂	—	2	2	no	—
174	C₂	—	92	2	no	—
175	C₂	C₂×#10625066	1	1	yes	characteristic
176	C₂	#13093231	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₉₂

C₂

yes

maximal, characteristic, complemented

C₂×#10625072

C₂

yes

maximal, characteristic

#13092993

C₂

yes

maximal

#13092676

C₂

yes

maximal, complemented

#13092659

C₂

yes

maximal, complemented

#13092425

C₂

yes

maximal, complemented

C₂₃×#427

C₂

yes

maximal, characteristic, complemented

C₂×#67288

C₂²

yes

characteristic

C₂²×Dic₄₆

C₂²

yes

—

C₂²×C₈×C₂₃

C₂²

yes

characteristic

C₂₃×#117

C₂²

yes

—

C₂×#67267

C₂²

yes

characteristic

C₂×Dic₉₂

—

C₂×Dic₉₂

C₂²

yes

—

C₂×Dic₉₂

C₂²

yes

—

#10625072

—

C₂×#67262

C₂²

yes

—

C₂×C₈×D₂₃

C₂²

yes

—

#10625068

C₂²

yes

—

#10625057

—

#10625046

—

#10625040

—

C₂₃×#107

C₂²

yes

—

C₂₃×#106

C₂²

yes

—

#10624992

C₂²

yes

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂×C₂²⋊D₂₃

D₄

yes

—

C₂²×Dic₂₃

—

#67288

—

#67288

—

C₂×D₉₂

D₄

yes

characteristic

C₂×C₄×D₂₃

C₂³

yes

—

C₂×Dic₄₆

—

C₂×Dic₄₆

—

C₂×Dic₄₆

—

C₂×Dic₄₆

D₄

yes

characteristic

C₂×Dic₄₆

C₂³

yes

—

C₂×C₈×C₂₃

D₄

yes

—

C₂×C₈×C₂₃

C₂³

yes

characteristic

C₂×C₈×C₂₃

C₂³

yes

characteristic

C₄⋊C₄×C₂₃

C₂³

yes

—

C₂²⋊C₄×C₂₃

—

C₄²×C₂₃

—

(C₂×D₂₃)⋊C₄

—

C₄⋊Dic₂₃

—

#67267

—

C₂×C₂₃⋊C₈

C₂³

yes

—

Dic₉₂

—

Dic₉₂

—

#67262

—

C₈×D₂₃

—

C₂³×C₂₃

C₂×D₄

yes

characteristic

C₂²×D₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

#41

yes

—

C₂×C₄×C₂₃

C₂⁴

yes

characteristic

C₂²⋊D₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

C₂×D₄

yes

—

D₉₂

—

C₄×D₂₃

—

C₄×D₂₃

—

Dic₄₆

—

Dic₄₆

—

Dic₄₆

—

Dic₄₆

—

C₈×C₂₃

C₂×D₄

yes

—

C₈×C₂₃

C₂×D₄

yes

—

C₂₃⋊C₈

—

#3059

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

#121

yes

—

C₂²×C₂₃

C₂²×D₄

yes

characteristic

C₂×D₂₃

—

C₂×D₂₃

—

C₄×C₂₃

—

C₄×C₂₃

#121

yes

—

C₄×C₂₃

C₂²×D₄

yes

characteristic

C₄×C₂₃

C₂×#41

yes

characteristic

Dic₂₃

—

Dic₂₃

—

C₂²×Q₁₆

—

—

—

—

—

D₂₃

yes

characteristic, complemented

#419

—

C₂×C₂₃

—

C₂×C₂₃

C₂×#121

yes

characteristic

C₂×C₂₃

#578

yes

—

D₂₃

—

C₂×#41

—

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

100

C₂×Q₁₆

—

101

#131

—

102

C₂×M₁₆

—

103

C₂×M₁₆

—

104

C₂²×C₈

—

105

C₂²×C₈

C₂×D₂₃

yes

characteristic

106

C₄⋊Q₈

—

107

#124

—

108

#123

—

109

C₂²⋊Q₈

—

110

#117

C₂×D₂₃

yes

—

111

#107

C₂×D₂₃

yes

—

112

#106

C₂×D₂₃

yes

—

113

#103

—

114

C₂₃

#3059

yes

characteristic, complemented, Sylow

115

#41

—

116

#41

—

117

C₂×Q₈

—

118

C₂×Q₈

—

119

C₂×Q₈

—

120

C₂×Q₈

—

121

C₂×Q₈

—

122

C₂×D₄

—

123

C₂×D₄

—

124

C₂²×C₄

—

125

C₂²×C₄

—

126

C₂²×C₄

C₂²×D₂₃

yes

characteristic

127

Q₁₆

—

128

Q₁₆

—

129

M₁₆

—

130

M₁₆

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

D₉₂

yes

—

134

C₂×C₈

C₂²×D₂₃

yes

characteristic

135

C₂×C₈

C₂²×D₂₃

yes

characteristic

136

C₄⋊C₄

—

137

C₄⋊C₄

C₂²×D₂₃

yes

—

138

C₂²⋊C₄

—

139

C₂²⋊C₄

—

140

C₄²

—

141

C₂³

—

142

C₂³

D₄×D₂₃

yes

characteristic

143

Q₈

—

144

Q₈

—

145

Q₈

—

146

Q₈

—

147

D₄

—

148

D₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

C₂×D₉₂

yes

—

156

C₂×C₄

D₄×D₂₃

yes

characteristic

157

C₂×C₄

#67292

yes

—

158

C₂×C₄

C₂³×D₂₃

yes

characteristic

159

C₈

—

160

C₈

C₂×D₉₂

yes

—

161

C₈

C₂×D₉₂

yes

—

162

C₂²

—

163

C₂²

—

164

C₂²

—

165

C₂²

#10625066

yes

—

166

C₂²

C₂×D₄×D₂₃

yes

characteristic

167

C₄

—

168

C₄

—

169

C₄

—

170

C₄

#10625066

yes

—

171

C₄

C₂²×D₉₂

yes

characteristic

172

C₄

C₂×#67292

yes

characteristic

173

C₂

—

174

C₂

—

175

C₂

C₂×#10625066

yes

characteristic

176

C₂

#13093231

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₂₃
commutator, cosocle	C₂×C₄×C₂₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₃
Fitting	C₂₃×#427	C₂
socle	C₂²×C₂₃	C₂²×D₄
prime radical	C₂×#67288	C₂²
hypercenter	#427	D₂₃
hypocenter	C₂₃	#3059
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #427

subgroup

G/H

center

C₂²

C₂×D₄×D₂₃

commutator, cosocle

C₂×C₄×C₂₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₂₃

Fitting

C₂₃×#427

C₂

socle

C₂²×C₂₃

C₂²×D₄

C₂²

D₂₃

C₂₃

G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #427

N	H ≅ G/N	type
1. C₂²×Dic₉₂	C₂	semidirect
4. #13092676	C₂	semidirect
5. #13092659	C₂	semidirect
6. #13092425	C₂	semidirect
7. C₂₃×#427	C₂	semidirect
90. #427	D₂₃	semidirect
114. C₂₃	#3059	semidirect

H ≅ G/N

type

1. C₂²×Dic₉₂

C₂

semidirect

4. #13092676

C₂

semidirect

5. #13092659

C₂

semidirect

6. #13092425

C₂

semidirect

7. C₂₃×#427

C₂

semidirect

90. #427

D₂₃

semidirect

114. C₂₃

#3059

semidirect

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

subgroup	index
C₂²×Dic₉₂	2
C₂×#10625072	2
#13092993	2
#13092676	2
#13092659	2
#13092425	2
C₂₃×#427	2
#3059	23

subgroup

index

C₂²×Dic₉₂

C₂₃×#427

Subgroups — #15683053

Conjugacy classes of subgroups

Complements

Maximal subgroups