Group #15674640 — subgroups

Subgroups — #15674640

Order 2944.

298 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 95 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×#10625017	C₂	1	2	yes	maximal
2	C₂×#10625017	C₂	1	2	yes	maximal
3	#13092315	C₂	1	1	yes	maximal, characteristic
4	#13092172	C₂	1	1	yes	maximal, characteristic
5	C₂×C₂₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₈×C₂₃	C₂²	1	2	yes	—
7	C₂₃×C₄⋊Q₈	—	2	4	no	—
8	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
9	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
10	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
13	C₂×Dic₂₃⋊C₄	—	2	1	no	—
14	C₂²×C₂₃⋊C₈	C₂²	1	2	yes	—
15	#10625017	—	2	4	no	—
16	#10625017	—	2	4	no	—
17	C₂₃⋊#95	—	2	1	no	—
18	C₂×Q₈×C₂₃	—	2	2	no	—
19	C₂×Q₈×C₂₃	—	2	2	no	—
20	C₂×Q₈×C₂₃	—	2	2	no	—
21	C₂×Q₈×C₂₃	—	2	2	no	—
22	C₂×Q₈×C₂₃	—	2	2	no	—
23	C₂×Q₈×C₂₃	D₄	1	4	yes	—
24	C₂²×C₄×C₂₃	—	2	2	no	—
25	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
26	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
27	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
28	C₂²×Dic₂₃	—	2	1	no	—
29	C₂²×Dic₂₃	—	2	1	no	—
30	C₄⋊C₄×C₂₃	—	2	4	no	—
31	C₄⋊C₄×C₂₃	—	2	4	no	—
32	C₄²×C₂₃	—	2	2	no	—
33	C₄⋊Dic₂₃	—	2	1	no	—
34	C₄⋊Dic₂₃	—	2	2	no	—
35	C₄⋊Dic₂₃	D₄	1	2	yes	—
36	Dic₂₃⋊C₄	—	4	2	no	—
37	C₂×C₂₃⋊C₈	—	2	2	no	—
38	C₂×C₂₃⋊C₈	—	2	2	no	—
39	C₂×C₂₃⋊C₈	—	2	2	no	—
40	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
41	Q₈×C₂₃	—	2	4	no	—
42	Q₈×C₂₃	—	2	4	no	—
43	Q₈×C₂₃	—	4	2	no	—
44	Q₈×C₂₃	—	4	2	no	—
45	C₂×C₄×C₂₃	—	2	1	no	—
46	C₂×C₄×C₂₃	—	2	1	no	—
47	C₂×C₄×C₂₃	—	2	1	no	—
48	C₂×C₄×C₂₃	—	2	1	no	—
49	C₂×C₄×C₂₃	—	2	4	no	—
50	C₂×C₄×C₂₃	—	4	2	no	—
51	C₂×C₄×C₂₃	—	4	2	no	—
52	C₂×C₄×C₂₃	SD₁₆	1	2	yes	—
53	C₂×C₄×C₂₃	Q₁₆	1	2	yes	—
54	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
56	C₂×C₄×C₂₃	#41	1	2	yes	—
57	C₂×C₄×C₂₃	#41	1	1	yes	characteristic
58	C₂×Dic₂₃	—	2	2	no	—
59	C₂×Dic₂₃	—	4	1	no	—
60	C₂×Dic₂₃	—	4	1	no	—
61	C₂×Dic₂₃	—	4	1	no	—
62	C₂×Dic₂₃	—	4	1	no	—
63	C₂×Dic₂₃	—	4	1	no	—
64	C₂₃⋊C₈	—	4	2	no	—
65	#2043	—	23	1	no	maximal, Sylow
66	C₂²×C₂₃	#120	1	1	yes	characteristic
67	C₂²×C₂₃	#121	1	1	yes	characteristic
68	C₂²×C₂₃	#124	1	1	yes	characteristic
69	C₂²×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
70	C₂²×C₂₃	C₂×Q₁₆	1	1	yes	characteristic
71	C₂²×C₂₃	#137	1	1	yes	characteristic
72	C₂²×C₂₃	#137	1	1	yes	characteristic
73	C₄×C₂₃	—	2	2	no	—
74	C₄×C₂₃	—	2	2	no	—
75	C₄×C₂₃	—	4	4	no	—
76	C₄×C₂₃	#121	1	2	yes	—
77	C₄×C₂₃	#123	1	2	yes	—
78	Dic₂₃	—	4	2	no	—
79	Dic₂₃	—	8	1	no	—
80	C₂×C₄⋊Q₈	D₂₃	1	1	yes	characteristic
81	C₂×#103	—	23	2	no	—
82	C₂×#103	—	23	2	no	—
83	#389	—	23	1	no	—
84	#336	—	23	1	no	—
85	C₂×C₂₃	#394	1	1	yes	characteristic
86	C₂×C₂₃	#448	1	1	yes	characteristic
87	C₂×C₂₃	#451	1	1	yes	characteristic
88	C₂×C₂₃	#461	1	1	yes	characteristic
89	C₂×C₂₃	#462	1	1	yes	characteristic
90	C₂×C₂₃	#487	1	1	yes	characteristic
91	C₂×C₂₃	#491	1	1	yes	characteristic
92	C₂²×Q₈	C₂×D₂₃	1	2	yes	—
93	C₂²×C₈	—	23	2	no	—
94	C₄⋊Q₈	—	2	4	no	—
95	C₂×C₄⋊C₄	—	2	1	no	—
96	C₂×C₄⋊C₄	—	2	1	no	—
97	C₂×C₄⋊C₄	—	23	1	no	—
98	C₂×C₄⋊C₄	—	23	1	no	—
99	C₂×C₄⋊C₄	—	46	1	no	—
100	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
101	#103	—	46	4	no	—
102	#103	—	46	4	no	—
103	#95	—	46	1	no	—
104	C₂₃	#2043	1	1	yes	characteristic, complemented, Sylow
105	C₂×Q₈	—	2	2	no	—
106	C₂×Q₈	—	2	2	no	—
107	C₂×Q₈	—	2	2	no	—
108	C₂×Q₈	—	2	2	no	—
109	C₂×Q₈	—	2	2	no	—
110	C₂×Q₈	C₂²⋊D₂₃	1	4	yes	—
111	C₂²×C₄	—	2	2	no	—
112	C₂²×C₄	—	46	1	no	—
113	C₂²×C₄	—	46	1	no	—
114	C₂²×C₄	C₂²⋊D₂₃	1	1	yes	characteristic
115	C₂²×C₄	C₂²⋊D₂₃	1	1	yes	characteristic
116	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
117	C₂×C₈	—	46	2	no	—
118	C₂×C₈	—	46	2	no	—
119	C₂×C₈	—	46	2	no	—
120	C₄⋊C₄	—	2	4	no	—
121	C₄⋊C₄	—	2	4	no	—
122	C₄⋊C₄	—	23	2	no	—
123	C₄⋊C₄	—	46	1	no	—
124	C₄⋊C₄	—	46	2	no	—
125	C₄⋊C₄	—	92	2	no	—
126	C₄²	—	2	2	no	—
127	C₂³	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
128	Q₈	—	2	4	no	—
129	Q₈	—	2	4	no	—
130	Q₈	—	4	2	no	—
131	Q₈	—	4	2	no	—
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	2	1	no	—
135	C₂×C₄	—	2	1	no	—
136	C₂×C₄	—	2	4	no	—
137	C₂×C₄	—	4	2	no	—
138	C₂×C₄	—	4	2	no	—
139	C₂×C₄	—	46	2	no	—
140	C₂×C₄	—	92	1	no	—
141	C₂×C₄	—	92	1	no	—
142	C₂×C₄	—	92	1	no	—
143	C₂×C₄	—	92	1	no	—
144	C₂×C₄	—	92	1	no	—
145	C₂×C₄	#67274	1	2	yes	—
146	C₂×C₄	#67275	1	2	yes	—
147	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
148	C₂×C₄	#67290	1	1	yes	characteristic
149	C₂×C₄	#67290	1	2	yes	—
150	C₂×C₄	C₂×C₂²⋊D₂₃	1	2	yes	—
151	C₈	—	92	2	no	—
152	C₂²	#10625107	1	1	yes	characteristic
153	C₂²	#10625109	1	1	yes	characteristic
154	C₂²	C₂×#67274	1	1	yes	characteristic
155	C₂²	#10625113	1	1	yes	characteristic
156	C₂²	#10625113	1	1	yes	characteristic
157	C₂²	C₂×#67275	1	1	yes	characteristic
158	C₂²	#10625124	1	1	yes	characteristic
159	C₄	—	2	2	no	—
160	C₄	—	2	2	no	—
161	C₄	—	4	4	no	—
162	C₄	—	92	2	no	—
163	C₄	—	184	1	no	—
164	C₄	#10625106	1	2	yes	—
165	C₄	#10625110	1	2	yes	—
166	C₂	#13092331	1	1	yes	characteristic
167	C₂	#13092556	1	1	yes	characteristic
168	C₂	#13092573	1	1	yes	characteristic
169	C₂	#13092619	1	1	yes	characteristic
170	C₂	#13092624	1	1	yes	characteristic
171	C₂	#13092767	1	1	yes	characteristic
172	C₂	#13092785	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10625017

C₂

yes

maximal

C₂×#10625017

C₂

yes

maximal

#13092315

C₂

yes

maximal, characteristic

#13092172

C₂

yes

maximal, characteristic

C₂×C₂₃×C₄⋊Q₈

C₂

yes

maximal, characteristic

C₂²×Q₈×C₂₃

C₂²

yes

—

C₂₃×C₄⋊Q₈

—

C₂×C₄⋊C₄×C₂₃

—

C₂×C₄⋊C₄×C₂₃

—

C₂×C₄²×C₂₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₂₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₂₃

C₂²

yes

characteristic

C₂×Dic₂₃⋊C₄

—

C₂²×C₂₃⋊C₈

C₂²

yes

—

#10625017

—

#10625017

—

C₂₃⋊#95

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

D₄

yes

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

D₄

yes

characteristic

C₂²×C₄×C₂₃

D₄

yes

characteristic

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂²×Dic₂₃

—

C₂²×Dic₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄²×C₂₃

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

D₄

yes

—

Dic₂₃⋊C₄

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂³×C₂₃

C₂×D₄

yes

characteristic

Q₈×C₂₃

—

Q₈×C₂₃

—

Q₈×C₂₃

—

Q₈×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

SD₁₆

yes

—

C₂×C₄×C₂₃

Q₁₆

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

#41

yes

—

C₂×C₄×C₂₃

#41

yes

characteristic

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂₃⋊C₈

—

—

C₂²×C₂₃

yes

C₂²×C₂₃

yes

C₂²×C₂₃

yes

C₂²×C₂₃

C₂×SD₁₆

yes

characteristic

C₂²×C₂₃

C₂×Q₁₆

yes

C₂²×C₂₃

yes

C₂²×C₂₃

yes

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

#121

yes

—

C₄×C₂₃

#123

yes

—

Dic₂₃

—

Dic₂₃

—

C₂×C₄⋊Q₈

D₂₃

yes

—

—

—

—

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂²×Q₈

C₂×D₂₃

yes

—

C₂²×C₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

100

C₂×C₄²

C₂×D₂₃

yes

characteristic

101

#103

—

102

#103

—

103

#95

—

104

C₂₃

#2043

yes

characteristic, complemented, Sylow

105

C₂×Q₈

—

106

C₂×Q₈

—

107

C₂×Q₈

—

108

C₂×Q₈

—

109

C₂×Q₈

—

110

C₂×Q₈

C₂²⋊D₂₃

yes

—

111

C₂²×C₄

—

112

C₂²×C₄

—

113

C₂²×C₄

—

114

C₂²×C₄

C₂²⋊D₂₃

yes

characteristic

115

C₂²×C₄

C₂²⋊D₂₃

yes

characteristic

116

C₂²×C₄

C₂²×D₂₃

yes

characteristic

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

125

C₄⋊C₄

—

126

C₄²

—

127

C₂³

C₂×C₂²⋊D₂₃

yes

characteristic

128

Q₈

—

129

Q₈

—

130

Q₈

—

131

Q₈

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

#67274

yes

—

146

C₂×C₄

#67275

yes

—

147

C₂×C₄

D₄×D₂₃

yes

characteristic

148

C₂×C₄

#67290

yes

characteristic

149

C₂×C₄

#67290

yes

—

150

C₂×C₄

C₂×C₂²⋊D₂₃

yes

—

151

C₈

—

152

C₂²

#10625107

yes

characteristic

153

C₂²

#10625109

yes

characteristic

154

C₂²

C₂×#67274

yes

characteristic

155

C₂²

#10625113

yes

characteristic

156

C₂²

#10625113

yes

characteristic

157

C₂²

C₂×#67275

yes

characteristic

158

C₂²

#10625124

yes

characteristic

159

C₄

—

160

C₄

—

161

C₄

—

162

C₄

—

163

C₄

—

184

—

164

C₄

#10625106

yes

—

165

C₄

#10625110

yes

—

166

C₂

#13092331

yes

characteristic

167

C₂

#13092556

yes

characteristic

168

C₂

#13092573

yes

characteristic

169

C₂

#13092619

yes

characteristic

170

C₂

#13092624

yes

characteristic

171

C₂

#13092767

yes

characteristic

172

C₂

#13092785

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₂₃
commutator, cosocle	C₂²×C₄×C₂₃	C₂³
Frattini	C₂²×C₄	C₂²×D₂₃
Fitting	C₂×C₂₃×C₄⋊Q₈	C₂
socle, prime radical	C₂³×C₂₃	C₂×D₄
hypercenter	C₂×C₄⋊Q₈	D₂₃
hypocenter	C₂₃	#2043
LCS	G ▷ C₂²×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₂₃

commutator, cosocle

C₂²×C₄×C₂₃

C₂³

Frattini

C₂²×C₄

C₂²×D₂₃

Fitting

C₂×C₂₃×C₄⋊Q₈

C₂

socle, prime radical

C₂³×C₂₃

C₂×D₄

hypercenter

C₂×C₄⋊Q₈

D₂₃

hypocenter

C₂₃

#2043

LCS

G ▷ C₂²×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
104. C₂₃	#2043	semidirect

H ≅ G/N

type

104. C₂₃

#2043

semidirect

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

subgroup	index
C₂×#10625017	2
#13092315	2
#13092172	2
C₂×C₂₃×C₄⋊Q₈	2
#2043	23

subgroup

index

C₂×C₂₃×C₄⋊Q₈

Subgroups — #15674640

Conjugacy classes of subgroups

Complements

Maximal subgroups