Group #15683052 — subgroups

Subgroups — #15683052

Order 2944.

460 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 149 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	#13092988	C₂	1	2	yes	maximal
3	#13092675	C₂	1	4	yes	maximal, complemented
4	#13092658	C₂	1	4	yes	maximal, complemented
5	#13092455	C₂	1	1	yes	maximal, characteristic, complemented
6	#13092427	C₂	1	2	yes	maximal, complemented
7	C₂×C₂₃×#131	C₂	1	1	yes	maximal, characteristic
8	C₂×#41×C₂₃	C₂²	1	1	yes	characteristic
9	C₂²×Dic₄₆	C₂²	1	2	yes	—
10	C₂₃×#131	—	2	4	no	—
11	C₂×M₁₆×C₂₃	C₂²	1	2	yes	—
12	C₂×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₂₃	C₂²	1	2	yes	—
14	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
15	#10625115	—	2	2	no	—
16	#10625107	—	2	2	no	—
17	#10625106	—	2	4	no	—
18	#10625097	C₂²	1	2	yes	—
19	#10625095	C₂²	1	8	yes	—
20	C₂×Dic₉₂	—	2	4	no	—
21	C₂×Dic₉₂	C₂²	1	2	yes	—
22	C₂×Dic₉₂	C₂²	1	2	yes	—
23	#10625000	C₂²	1	2	yes	—
24	#10624999	C₂²	1	2	yes	—
25	#10624998	C₂²	1	8	yes	—
26	#41×C₂₃	—	2	2	no	—
27	#41×C₂₃	—	2	2	no	—
28	C₂×Q₈×C₂₃	D₄	1	1	yes	characteristic
29	C₂×D₄×C₂₃	D₄	1	2	yes	—
30	C₂×D₄×C₂₃	D₄	1	1	yes	characteristic
31	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
32	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
33	C₂²×Dic₂₃	—	2	2	no	—
34	C₂×Dic₄₆	—	2	4	no	—
35	C₂×Dic₄₆	—	2	4	no	—
36	C₂×Dic₄₆	—	2	2	no	—
37	C₂×Dic₄₆	C₂³	1	4	yes	—
38	M₁₆×C₂₃	—	2	2	no	—
39	M₁₆×C₂₃	—	2	4	no	—
40	C₂×C₈×C₂₃	—	2	4	no	—
41	C₂×C₈×C₂₃	D₄	1	4	yes	—
42	C₂×C₈×C₂₃	C₂³	1	2	yes	—
43	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
44	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
45	C₂²⋊Dic₂₃	—	2	8	no	—
46	C₂²⋊Dic₂₃	—	2	2	no	—
47	C₄⋊Dic₂₃	C₂³	1	4	yes	—
48	Dic₂₃⋊C₄	—	2	8	no	—
49	C₄×Dic₂₃	—	2	2	no	—
50	Dic₉₂	—	2	4	no	—
51	Dic₉₂	—	2	4	no	—
52	C₂³×C₂₃	C₂×D₄	1	2	yes	—
53	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
54	Q₈×C₂₃	—	2	2	no	—
55	D₄×C₂₃	—	2	4	no	—
56	D₄×C₂₃	—	2	2	no	—
57	C₂×C₄×C₂₃	—	2	2	no	—
58	C₂×C₄×C₂₃	—	2	2	no	—
59	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
61	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
62	C₂×C₄×C₂₃	#41	1	2	yes	—
63	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
64	C₂×Dic₂₃	—	2	4	no	—
65	C₂×Dic₂₃	—	2	4	no	—
66	C₂×Dic₂₃	—	4	4	no	—
67	Dic₄₆	—	2	4	no	—
68	Dic₄₆	—	2	4	no	—
69	Dic₄₆	—	4	4	no	—
70	C₈×C₂₃	—	2	2	no	—
71	C₈×C₂₃	C₂×D₄	1	2	yes	—
72	C₈×C₂₃	C₂×D₄	1	2	yes	—
73	#3059	—	23	1	no	maximal, Sylow
74	C₂²×C₂₃	—	2	2	no	—
75	C₂²×C₂₃	—	2	2	no	—
76	C₂²×C₂₃	—	2	4	no	—
77	C₂²×C₂₃	#121	1	2	yes	—
78	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
79	C₄×C₂₃	—	2	2	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	—	4	4	no	—
84	Dic₂₃	—	4	4	no	—
85	C₂²×Q₁₆	—	23	1	no	—
86	C₂×#131	D₂₃	1	1	yes	characteristic
87	#536	—	23	2	no	—
88	#470	—	23	4	no	—
89	#467	—	23	4	no	—
90	#427	—	23	1	no	—
91	#419	—	23	2	no	—
92	C₂×C₂₃	—	2	2	no	—
93	C₂×C₂₃	—	4	2	no	—
94	C₂×C₂₃	C₂×#121	1	1	yes	characteristic
95	C₂×C₂₃	#578	1	2	yes	—
96	C₂×#41	C₂×D₂₃	1	1	yes	characteristic
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	—	2	4	no	—
102	C₂×M₁₆	C₂×D₂₃	1	1	yes	characteristic
103	C₂×M₁₆	C₂×D₂₃	1	2	yes	—
104	C₂²×C₈	C₂×D₂₃	1	2	yes	—
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	—	23	2	no	—
111	#107	—	23	2	no	—
112	#106	—	23	2	no	—
113	#103	—	23	8	no	—
114	C₂₃	#3059	1	1	yes	characteristic, complemented, Sylow
115	#41	—	2	2	no	—
116	#41	—	2	2	no	—
117	C₂×Q₈	—	23	4	no	—
118	C₂×Q₈	—	46	2	no	—
119	C₂×Q₈	—	46	4	no	—
120	C₂×Q₈	—	46	4	no	—
121	C₂×Q₈	D₉₂	1	1	yes	characteristic
122	C₂×D₄	D₉₂	1	2	yes	—
123	C₂×D₄	D₉₂	1	1	yes	characteristic
124	C₂²×C₄	—	46	2	no	—
125	C₂²×C₄	C₂²×D₂₃	1	2	yes	—
126	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
127	Q₁₆	—	46	4	no	—
128	Q₁₆	—	46	4	no	—
129	M₁₆	—	2	2	no	—
130	M₁₆	—	2	4	no	—
131	C₂×C₈	—	2	4	no	—
132	C₂×C₈	C₂²⋊D₂₃	1	4	yes	—
133	C₂×C₈	C₂²×D₂₃	1	2	yes	—
134	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
135	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
136	C₄⋊C₄	—	23	4	no	—
137	C₄⋊C₄	—	46	8	no	—
138	C₂²⋊C₄	—	46	2	no	—
139	C₂²⋊C₄	—	46	8	no	—
140	C₄²	—	46	2	no	—
141	C₂³	C₂×D₉₂	1	2	yes	—
142	C₂³	C₂×D₉₂	1	1	yes	characteristic
143	Q₈	—	2	2	no	—
144	Q₈	—	46	4	no	—
145	Q₈	—	46	4	no	—
146	Q₈	—	92	4	no	—
147	D₄	—	2	4	no	—
148	D₄	—	2	2	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	46	4	no	—
152	C₂×C₄	—	46	4	no	—
153	C₂×C₄	—	92	4	no	—
154	C₂×C₄	C₂×D₉₂	1	2	yes	—
155	C₂×C₄	C₂×D₉₂	1	1	yes	characteristic
156	C₂×C₄	#67288	1	2	yes	—
157	C₂×C₄	C₂×C₂²⋊D₂₃	1	2	yes	—
158	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
159	C₈	—	2	2	no	—
160	C₈	C₂×C₂²⋊D₂₃	1	2	yes	—
161	C₈	C₂×C₂²⋊D₂₃	1	2	yes	—
162	C₂²	—	2	2	no	—
163	C₂²	—	2	2	no	—
164	C₂²	—	2	4	no	—
165	C₂²	#10625101	1	2	yes	—
166	C₂²	C₂²×D₉₂	1	1	yes	characteristic
167	C₄	—	2	2	no	—
168	C₄	—	92	4	no	—
169	C₄	—	92	4	no	—
170	C₄	#10625101	1	2	yes	—
171	C₄	C₂×#67288	1	1	yes	characteristic
172	C₄	C₂²×C₂²⋊D₂₃	1	1	yes	characteristic
173	C₂	—	2	2	no	—
174	C₂	—	4	2	no	—
175	C₂	C₂×#10625101	1	1	yes	characteristic
176	C₂	#13093234	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₉₂

C₂

yes

maximal, characteristic, complemented

#13092988

C₂

yes

maximal

#13092675

C₂

yes

maximal, complemented

#13092658

C₂

yes

maximal, complemented

#13092455

C₂

yes

maximal, characteristic, complemented

#13092427

C₂

yes

maximal, complemented

C₂×C₂₃×#131

C₂

yes

maximal, characteristic

C₂×#41×C₂₃

C₂²

yes

characteristic

C₂²×Dic₄₆

C₂²

yes

—

C₂₃×#131

—

C₂×M₁₆×C₂₃

C₂²

yes

—

C₂×M₁₆×C₂₃

C₂²

yes

characteristic

C₂²×C₈×C₂₃

C₂²

yes

—

C₂²×C₈×C₂₃

C₂²

yes

—

—

—

C₂²

yes

—

#10625095

C₂²

yes

—

C₂×Dic₉₂

—

C₂×Dic₉₂

C₂²

yes

—

C₂×Dic₉₂

C₂²

yes

—

#10625000

C₂²

yes

—

#10624999

C₂²

yes

—

#10624998

C₂²

yes

—

#41×C₂₃

—

#41×C₂₃

—

C₂×Q₈×C₂₃

D₄

yes

characteristic

C₂×D₄×C₂₃

D₄

yes

—

C₂×D₄×C₂₃

D₄

yes

characteristic

C₂²×C₄×C₂₃

C₂³

yes

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂²×Dic₂₃

—

C₂×Dic₄₆

—

C₂×Dic₄₆

—

C₂×Dic₄₆

—

C₂×Dic₄₆

C₂³

yes

—

M₁₆×C₂₃

—

M₁₆×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

D₄

yes

—

C₂×C₈×C₂₃

C₂³

yes

—

C₂×C₈×C₂₃

C₂³

yes

characteristic

C₂×C₈×C₂₃

C₂³

yes

characteristic

C₂²⋊Dic₂₃

—

C₂²⋊Dic₂₃

—

C₄⋊Dic₂₃

C₂³

yes

—

Dic₂₃⋊C₄

—

C₄×Dic₂₃

—

Dic₉₂

—

Dic₉₂

—

C₂³×C₂₃

C₂×D₄

yes

—

C₂³×C₂₃

C₂×D₄

yes

characteristic

Q₈×C₂₃

—

D₄×C₂₃

—

D₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

#41

yes

—

C₂×C₄×C₂₃

C₂⁴

yes

characteristic

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

Dic₄₆

—

Dic₄₆

—

Dic₄₆

—

C₈×C₂₃

—

C₈×C₂₃

C₂×D₄

yes

—

C₈×C₂₃

C₂×D₄

yes

—

#3059

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

#121

yes

—

C₂²×C₂₃

C₂²×D₄

yes

characteristic

C₄×C₂₃

—

C₄×C₂₃

#121

yes

—

C₄×C₂₃

C₂²×D₄

yes

characteristic

C₄×C₂₃

C₂×#41

yes

characteristic

Dic₂₃

—

Dic₂₃

—

C₂²×Q₁₆

—

C₂×#131

D₂₃

yes

—

—

—

—

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

C₂×#121

yes

characteristic

C₂×C₂₃

#578

yes

—

C₂×#41

C₂×D₂₃

yes

characteristic

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

100

C₂×Q₁₆

—

101

#131

—

102

C₂×M₁₆

C₂×D₂₃

yes

characteristic

103

C₂×M₁₆

C₂×D₂₃

yes

—

104

C₂²×C₈

C₂×D₂₃

yes

—

105

C₂²×C₈

C₂×D₂₃

yes

characteristic

106

C₄⋊Q₈

—

107

#124

—

108

#123

—

109

C₂²⋊Q₈

—

110

#117

—

111

#107

—

112

#106

—

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

D₉₂

yes

characteristic

122

C₂×D₄

D₉₂

yes

—

123

C₂×D₄

D₉₂

yes

characteristic

124

C₂²×C₄

—

125

C₂²×C₄

C₂²×D₂₃

yes

—

126

C₂²×C₄

C₂²×D₂₃

yes

characteristic

127

Q₁₆

—

128

Q₁₆

—

129

M₁₆

—

130

M₁₆

—

131

C₂×C₈

—

132

C₂×C₈

C₂²⋊D₂₃

yes

—

133

C₂×C₈

C₂²×D₂₃

yes

—

134

C₂×C₈

C₂²×D₂₃

yes

characteristic

135

C₂×C₈

C₂²×D₂₃

yes

characteristic

136

C₄⋊C₄

—

137

C₄⋊C₄

—

138

C₂²⋊C₄

—

139

C₂²⋊C₄

—

140

C₄²

—

141

C₂³

C₂×D₉₂

yes

—

142

C₂³

C₂×D₉₂

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

C₂×D₉₂

yes

—

155

C₂×C₄

C₂×D₉₂

yes

characteristic

156

C₂×C₄

#67288

yes

—

157

C₂×C₄

C₂×C₂²⋊D₂₃

yes

—

158

C₂×C₄

C₂³×D₂₃

yes

characteristic

159

C₈

—

160

C₈

C₂×C₂²⋊D₂₃

yes

—

161

C₈

C₂×C₂²⋊D₂₃

yes

—

162

C₂²

—

163

C₂²

—

164

C₂²

—

165

C₂²

#10625101

yes

—

166

C₂²

C₂²×D₉₂

yes

characteristic

167

C₄

—

168

C₄

—

169

C₄

—

170

C₄

#10625101

yes

—

171

C₄

C₂×#67288

yes

characteristic

172

C₄

C₂²×C₂²⋊D₂₃

yes

characteristic

173

C₂

—

174

C₂

—

175

C₂

C₂×#10625101

yes

characteristic

176

C₂

#13093234

yes

—

	subgroup	G/H
center	C₂²	C₂²×D₉₂
commutator, cosocle	C₂×C₄×C₂₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₃
Fitting	C₂×C₂₃×#131	C₂
socle	C₂²×C₂₃	C₂²×D₄
prime radical	C₂×#41×C₂₃	C₂²
hypercenter	C₂×#131	D₂₃
hypocenter	C₂₃	#3059
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ C₂×#131

subgroup

G/H

center

C₂²

C₂²×D₉₂

commutator, cosocle

C₂×C₄×C₂₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₂₃

Fitting

C₂×C₂₃×#131

C₂

socle

C₂²×C₂₃

C₂²×D₄

prime radical

C₂×#41×C₂₃

C₂²

D₂₃

C₂₃

G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂² ◁ C₂×#41 ◁ C₂×#131

N	H ≅ G/N	type
1. C₂²×Dic₉₂	C₂	semidirect
3. #13092675	C₂	semidirect
4. #13092658	C₂	semidirect
5. #13092455	C₂	semidirect
6. #13092427	C₂	semidirect
114. C₂₃	#3059	semidirect

H ≅ G/N

type

1. C₂²×Dic₉₂

C₂

semidirect

3. #13092675

C₂

semidirect

4. #13092658

C₂

semidirect

5. #13092455

C₂

semidirect

6. #13092427

C₂

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
#13092988	2
#13092675	2
#13092658	2
#13092455	2
#13092427	2
C₂×C₂₃×#131	2
#3059	23

subgroup

index

C₂²×Dic₉₂

C₂×C₂₃×#131

Subgroups — #15683052

Conjugacy classes of subgroups

Complements

Maximal subgroups