Group #15672676 — subgroups

Subgroups — #15672676

Order 2944.

194 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 194 conjugacy classes of subgroups — too many to draw. Only its 57 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	#13093146	C₂	1	1	yes	maximal, characteristic
2	#13092815	C₂	1	1	yes	maximal, characteristic
3	#13092703	C₂	1	1	yes	maximal, characteristic
4	#13092627	C₂	1	1	yes	maximal, characteristic
5	#13092164	C₂	1	1	yes	maximal, characteristic
6	#13092149	C₂	1	1	yes	maximal, characteristic
7	C₂₃×Q₈⋊C₈	C₂	1	1	yes	maximal, characteristic
8	C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
9	Q₈×Dic₂₃	—	2	1	no	—
10	C₂×#67275	—	2	1	no	—
11	#10625061	—	2	1	no	—
12	#10625061	—	2	2	no	—
13	#10625040	—	2	1	no	—
14	#10625040	C₂²	1	1	yes	characteristic
15	#10625040	C₂²	1	1	yes	characteristic
16	C₄×Dic₄₆	—	2	1	no	—
17	C₄×Dic₄₆	C₂²	1	1	yes	characteristic
18	C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
19	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
20	#10624999	—	2	1	no	—
21	#10624999	—	2	2	no	—
22	#10624998	—	2	1	no	—
23	#10624992	—	2	1	no	—
24	C₄⋊(C₂₃⋊C₈)	C₂²	1	1	yes	characteristic
25	C₂×Q₈×C₂₃	D₄	1	1	yes	characteristic
26	C₂×Dic₄₆	—	2	1	no	—
27	C₂×Dic₄₆	—	2	1	no	—
28	C₂×Dic₄₆	D₄	1	1	yes	characteristic
29	C₂×C₈×C₂₃	—	2	1	no	—
30	C₂×C₈×C₂₃	—	2	1	no	—
31	C₄⋊C₄×C₂₃	—	2	1	no	—
32	C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
33	C₄²×C₂₃	—	2	1	no	—
34	C₄²×C₂₃	C₂³	1	1	yes	characteristic
35	#67275	—	4	2	no	—
36	C₄⋊Dic₂₃	—	2	2	no	—
37	C₄⋊Dic₂₃	—	2	2	no	—
38	C₄⋊Dic₂₃	—	2	1	no	—
39	C₄⋊Dic₂₃	—	2	1	no	—
40	C₄⋊Dic₂₃	—	2	1	no	—
41	C₄⋊Dic₂₃	D₄	1	1	yes	characteristic
42	C₄⋊Dic₂₃	D₄	1	2	yes	—
43	Dic₂₃⋊C₄	—	2	2	no	—
44	Dic₂₃⋊C₄	—	2	1	no	—
45	C₄×Dic₂₃	—	2	2	no	—
46	C₄×Dic₂₃	—	2	1	no	—
47	C₂×C₂₃⋊C₈	—	2	1	no	—
48	Q₈×C₂₃	—	2	1	no	—
49	Q₈×C₂₃	SD₁₆	1	2	yes	—
50	C₂×C₄×C₂₃	—	2	1	no	—
51	C₂×C₄×C₂₃	—	2	1	no	—
52	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
55	C₂×Dic₂₃	—	2	1	no	—
56	C₂×Dic₂₃	—	2	1	no	—
57	C₂×Dic₂₃	—	2	2	no	—
58	C₂×Dic₂₃	—	4	1	no	—
59	Dic₄₆	—	2	1	no	—
60	Dic₄₆	—	2	1	no	—
61	Dic₄₆	—	4	1	no	—
62	Dic₄₆	—	4	1	no	—
63	Dic₄₆	—	4	1	no	—
64	Dic₄₆	—	4	1	no	—
65	C₈×C₂₃	—	2	2	no	—
66	C₈×C₂₃	—	4	1	no	—
67	C₂₃⋊C₈	—	4	1	no	—
68	#1653	—	23	1	no	maximal, Sylow
69	C₂²×C₂₃	#120	1	1	yes	characteristic
70	C₄×C₂₃	—	2	2	no	—
71	C₄×C₂₃	—	2	1	no	—
72	C₄×C₂₃	—	4	1	no	—
73	C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
74	C₄×C₂₃	#135	1	1	yes	characteristic
75	C₄×C₂₃	#137	1	1	yes	characteristic
76	C₄×C₂₃	#137	1	1	yes	characteristic
77	Dic₂₃	—	4	1	no	—
78	Dic₂₃	—	4	2	no	—
79	Dic₂₃	—	4	1	no	—
80	Dic₂₃	—	8	1	no	—
81	#557	—	23	1	no	—
82	#498	—	23	1	no	—
83	#475	—	23	1	no	—
84	#462	—	23	1	no	—
85	#332	—	23	1	no	—
86	Q₈⋊C₈	—	23	1	no	—
87	Q₈⋊C₈	D₂₃	1	1	yes	characteristic
88	C₂×C₂₃	#448	1	1	yes	characteristic
89	C₂×C₂₃	#452	1	1	yes	characteristic
90	C₂×C₂₃	#456	1	1	yes	characteristic
91	C₂×Q₁₆	—	46	1	no	—
92	C₄⋊Q₈	—	23	1	no	—
93	C₄⋊Q₈	—	23	1	no	—
94	C₄⋊Q₈	—	46	1	no	—
95	#125	—	46	2	no	—
96	#125	—	46	1	no	—
97	C₄×Q₈	—	23	1	no	—
98	C₄×Q₈	—	46	1	no	—
99	C₄×Q₈	—	46	1	no	—
100	C₄×Q₈	C₂×D₂₃	1	1	yes	characteristic
101	#106	—	46	1	no	—
102	#106	—	46	2	no	—
103	C₄⋊C₈	—	23	1	no	—
104	C₄⋊C₈	C₂×D₂₃	1	1	yes	characteristic
105	#103	—	46	1	no	—
106	#103	—	46	1	no	—
107	C₄×C₈	C₂×D₂₃	1	1	yes	characteristic
108	C₂₃	#1653	1	1	yes	characteristic, complemented, Sylow
109	C₂×Q₈	—	23	1	no	—
110	C₂×Q₈	—	46	1	no	—
111	C₂×Q₈	—	46	1	no	—
112	C₂×Q₈	D₉₂	1	1	yes	characteristic
113	Q₁₆	—	92	2	no	—
114	C₂×C₈	—	2	1	no	—
115	C₂×C₈	—	2	1	no	—
116	C₂×C₈	—	46	1	no	—
117	C₄⋊C₄	—	2	1	no	—
118	C₄⋊C₄	—	23	1	no	—
119	C₄⋊C₄	—	23	2	no	—
120	C₄⋊C₄	—	46	2	no	—
121	C₄⋊C₄	—	46	2	no	—
122	C₄⋊C₄	—	46	2	no	—
123	C₄⋊C₄	—	46	1	no	—
124	C₄⋊C₄	—	46	1	no	—
125	C₄⋊C₄	—	46	1	no	—
126	C₄⋊C₄	—	46	1	no	—
127	C₄⋊C₄	D₉₂	1	1	yes	characteristic
128	C₄²	—	2	1	no	—
129	C₄²	—	46	2	no	—
130	C₄²	—	46	1	no	—
131	C₄²	C₂²×D₂₃	1	1	yes	characteristic
132	Q₈	—	2	1	no	—
133	Q₈	—	46	1	no	—
134	Q₈	—	46	1	no	—
135	Q₈	—	92	1	no	—
136	Q₈	—	92	1	no	—
137	Q₈	—	92	1	no	—
138	Q₈	—	92	1	no	—
139	Q₈	#67263	1	2	yes	—
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	2	1	no	—
142	C₂×C₄	—	46	1	no	—
143	C₂×C₄	—	46	2	no	—
144	C₂×C₄	—	46	1	no	—
145	C₂×C₄	—	92	1	no	—
146	C₂×C₄	C₂×D₉₂	1	1	yes	characteristic
147	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
148	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
149	C₈	—	2	2	no	—
150	C₈	—	4	1	no	—
151	C₈	—	92	1	no	—
152	C₂²	#10625053	1	1	yes	characteristic
153	C₄	—	2	2	no	—
154	C₄	—	2	1	no	—
155	C₄	—	4	1	no	—
156	C₄	—	92	1	no	—
157	C₄	—	92	2	no	—
158	C₄	—	92	1	no	—
159	C₄	—	184	1	no	—
160	C₄	C₂×#67263	1	1	yes	characteristic
161	C₄	#10625080	1	1	yes	characteristic
162	C₄	#10625087	1	1	yes	characteristic
163	C₄	#10625089	1	1	yes	characteristic
164	C₂	#13092558	1	1	yes	characteristic
165	C₂	#13092582	1	1	yes	characteristic
166	C₂	#13092601	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13093146

C₂

yes

maximal, characteristic

#13092815

C₂

yes

maximal, characteristic

#13092703

C₂

yes

maximal, characteristic

#13092627

C₂

yes

maximal, characteristic

#13092164

C₂

yes

maximal, characteristic

#13092149

C₂

yes

maximal, characteristic

C₂₃×Q₈⋊C₈

C₂

yes

maximal, characteristic

C₄×Q₈×C₂₃

C₂²

yes

characteristic

Q₈×Dic₂₃

—

—

—

—

—

C₂²

yes

characteristic

#10625040

C₂²

yes

characteristic

C₄×Dic₄₆

—

C₄×Dic₄₆

C₂²

yes

characteristic

C₂₃×C₄⋊C₈

C₂²

yes

characteristic

C₄×C₈×C₂₃

C₂²

yes

—

—

—

—

C₄⋊(C₂₃⋊C₈)

C₂²

yes

characteristic

C₂×Q₈×C₂₃

D₄

yes

characteristic

C₂×Dic₄₆

—

C₂×Dic₄₆

—

C₂×Dic₄₆

D₄

yes

characteristic

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

D₄

yes

characteristic

C₄²×C₂₃

—

C₄²×C₂₃

C₂³

yes

characteristic

#67275

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

D₄

yes

characteristic

C₄⋊Dic₂₃

D₄

yes

—

Dic₂₃⋊C₄

—

Dic₂₃⋊C₄

—

C₄×Dic₂₃

—

C₄×Dic₂₃

—

C₂×C₂₃⋊C₈

—

Q₈×C₂₃

—

Q₈×C₂₃

SD₁₆

yes

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

Dic₄₆

—

Dic₄₆

—

Dic₄₆

—

Dic₄₆

—

Dic₄₆

—

Dic₄₆

—

C₈×C₂₃

—

C₈×C₂₃

—

C₂₃⋊C₈

—

—

C₂²×C₂₃

yes

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

C₂×SD₁₆

yes

C₄×C₂₃

yes

C₄×C₂₃

yes

C₄×C₂₃

yes

Dic₂₃

—

Dic₂₃

—

Dic₂₃

—

Dic₂₃

—

—

—

—

—

—

Q₈⋊C₈

—

Q₈⋊C₈

D₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

#125

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

100

C₄×Q₈

C₂×D₂₃

yes

characteristic

101

#106

—

102

#106

—

103

C₄⋊C₈

—

104

C₄⋊C₈

C₂×D₂₃

yes

characteristic

105

#103

—

106

#103

—

107

C₄×C₈

C₂×D₂₃

yes

characteristic

108

C₂₃

#1653

yes

characteristic, complemented, Sylow

109

C₂×Q₈

—

110

C₂×Q₈

—

111

C₂×Q₈

—

112

C₂×Q₈

D₉₂

yes

characteristic

113

Q₁₆

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₄⋊C₄

D₉₂

yes

characteristic

128

C₄²

—

129

C₄²

—

130

C₄²

—

131

C₄²

C₂²×D₂₃

yes

characteristic

132

Q₈

—

133

Q₈

—

134

Q₈

—

135

Q₈

—

136

Q₈

—

137

Q₈

—

138

Q₈

—

139

Q₈

#67263

yes

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

C₂×D₉₂

yes

characteristic

147

C₂×C₄

D₄×D₂₃

yes

characteristic

148

C₂×C₄

D₄×D₂₃

yes

characteristic

149

C₈

—

150

C₈

—

151

C₈

—

152

C₂²

#10625053

yes

characteristic

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

—

158

C₄

—

159

C₄

—

184

—

160

C₄

C₂×#67263

yes

characteristic

161

C₄

#10625080

yes

characteristic

162

C₄

#10625087

yes

characteristic

163

C₄

#10625089

yes

characteristic

164

C₂

#13092558

yes

characteristic

165

C₂

#13092582

yes

characteristic

166

C₂

#13092601

yes

characteristic

	subgroup	G/H
center	C₂²	#10625053
commutator, cosocle	C₄²×C₂₃	C₂³
Frattini	C₄²	C₂²×D₂₃
Fitting	C₂₃×Q₈⋊C₈	C₂
socle, prime radical	C₂²×C₂₃	#120
hypercenter	Q₈⋊C₈	D₂₃
hypocenter	C₂₃	#1653
LCS	G ▷ C₄²×C₂₃ ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ Q₈⋊C₈

subgroup

G/H

center

C₂²

#10625053

commutator, cosocle

C₄²×C₂₃

C₂³

Frattini

C₄²

C₂²×D₂₃

Fitting

C₂₃×Q₈⋊C₈

C₂

socle, prime radical

C₂²×C₂₃

#120

hypercenter

Q₈⋊C₈

D₂₃

hypocenter

C₂₃

#1653

LCS

G ▷ C₄²×C₂₃ ▷ C₂²×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ Q₈⋊C₈

N	H ≅ G/N	type
108. C₂₃	#1653	semidirect

H ≅ G/N

type

108. C₂₃

#1653

semidirect

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

subgroup	index
#13093146	2
#13092815	2
#13092703	2
#13092627	2
#13092164	2
#13092149	2
C₂₃×Q₈⋊C₈	2
#1653	23

subgroup

index

C₂₃×Q₈⋊C₈

Subgroups — #15672676

Conjugacy classes of subgroups

Complements

Maximal subgroups