Group #15672539 — subgroups

Subgroups — #15672539

Order 2944.

206 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 59 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	Q₈×Dic₄₆	C₂	1	1	yes	maximal, characteristic
2	#13092797	C₂	1	1	yes	maximal, characteristic
3	#13092713	C₂	1	1	yes	maximal, characteristic
4	#13092712	C₂	1	1	yes	maximal, characteristic
5	#13092165	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×Dic₉₂	—	2	2	no	—
11	#10625059	—	2	2	no	—
12	#10625059	—	2	1	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	#10625017	—	2	1	no	—
21	#10625000	—	2	1	no	—
22	#10624998	—	2	1	no	—
23	#10624989	—	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	2	no	—
28	C₂×Dic₄₆	D₄	1	1	yes	characteristic
29	C₂×Dic₄₆	D₄	1	2	yes	—
30	C₂×C₈×C₂₃	—	2	1	no	—
31	C₂×C₈×C₂₃	—	2	1	no	—
32	C₄⋊C₄×C₂₃	—	2	1	no	—
33	C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
34	C₄²×C₂₃	—	2	1	no	—
35	C₄²×C₂₃	C₂³	1	1	yes	characteristic
36	C₄⋊Dic₂₃	—	2	2	no	—
37	C₄⋊Dic₂₃	—	2	1	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	Dic₂₃⋊C₄	—	2	2	no	—
43	Dic₂₃⋊C₄	—	2	1	no	—
44	C₄×Dic₂₃	—	2	2	no	—
45	C₄×Dic₂₃	—	2	1	no	—
46	C₂×C₂₃⋊C₈	—	2	1	no	—
47	Dic₉₂	—	4	4	no	—
48	Q₈×C₂₃	—	2	1	no	—
49	Q₈×C₂₃	Q₁₆	1	2	yes	complemented
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	2	no	—
57	C₂×Dic₂₃	—	2	1	no	—
58	C₂×Dic₂₃	—	4	1	no	—
59	Dic₄₆	—	2	1	no	—
60	Dic₄₆	—	4	2	no	—
61	Dic₄₆	—	4	2	no	—
62	Dic₄₆	—	8	2	no	—
63	Dic₄₆	Q₁₆	1	2	yes	complemented
64	C₈×C₂₃	—	2	2	no	—
65	C₈×C₂₃	—	4	1	no	—
66	C₂₃⋊C₈	—	4	1	no	—
67	#1633	—	23	1	no	maximal, Sylow
68	C₂²×C₂₃	#120	1	1	yes	characteristic
69	C₄×C₂₃	—	2	2	no	—
70	C₄×C₂₃	—	2	1	no	—
71	C₄×C₂₃	—	4	1	no	—
72	C₄×C₂₃	C₂×Q₁₆	1	1	yes	characteristic
73	C₄×C₂₃	C₂×Q₁₆	1	1	yes	characteristic
74	C₄×C₂₃	#137	1	1	yes	characteristic
75	C₄×C₂₃	#137	1	1	yes	characteristic
76	Dic₂₃	—	2	2	no	—
77	Dic₂₃	—	4	2	no	—
78	Dic₂₃	—	4	1	no	—
79	Dic₂₃	—	8	1	no	—
80	Q₈²	—	23	1	no	—
81	#494	—	23	1	no	—
82	#477	—	23	1	no	—
83	#477	—	23	1	no	—
84	#333	—	23	1	no	—
85	Q₈⋊C₈	—	23	1	no	—
86	Q₈⋊C₈	D₂₃	1	1	yes	characteristic
87	C₂×C₂₃	#451	1	1	yes	characteristic
88	C₂×C₂₃	#451	1	1	yes	characteristic
89	C₂×C₂₃	#453	1	1	yes	characteristic
90	C₂×Q₁₆	—	46	2	no	—
91	C₄⋊Q₈	—	23	1	no	—
92	C₄⋊Q₈	—	23	1	no	—
93	C₄⋊Q₈	—	46	1	no	—
94	C₄⋊Q₈	—	46	2	no	—
95	C₄⋊Q₈	—	46	1	no	—
96	C₄×Q₈	—	23	1	no	—
97	C₄×Q₈	—	46	1	no	—
98	C₄×Q₈	—	46	1	no	—
99	C₄×Q₈	C₂×D₂₃	1	1	yes	characteristic
100	#107	—	46	1	no	—
101	#107	—	46	1	no	—
102	C₄⋊C₈	—	23	1	no	—
103	C₄⋊C₈	C₂×D₂₃	1	1	yes	characteristic
104	#103	—	46	1	no	—
105	#103	—	46	1	no	—
106	C₄×C₈	C₂×D₂₃	1	1	yes	characteristic
107	C₂₃	#1633	1	1	yes	characteristic, complemented, Sylow
108	C₂×Q₈	—	23	1	no	—
109	C₂×Q₈	—	23	2	no	—
110	C₂×Q₈	—	46	2	no	—
111	C₂×Q₈	—	46	1	no	—
112	C₂×Q₈	D₉₂	1	1	yes	characteristic
113	Q₁₆	—	92	4	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	—	46	2	no	—
120	C₄⋊C₄	—	46	2	no	—
121	C₄⋊C₄	—	46	1	no	—
122	C₄⋊C₄	—	46	1	no	—
123	C₄⋊C₄	—	46	1	no	—
124	C₄⋊C₄	—	46	1	no	—
125	C₄⋊C₄	—	46	1	no	—
126	C₄⋊C₄	D₉₂	1	1	yes	characteristic
127	C₄²	—	2	1	no	—
128	C₄²	—	46	2	no	—
129	C₄²	—	46	1	no	—
130	C₄²	C₂²×D₂₃	1	1	yes	characteristic
131	Q₈	—	2	1	no	—
132	Q₈	—	23	2	no	—
133	Q₈	—	46	1	no	—
134	Q₈	—	92	2	no	—
135	Q₈	—	92	2	no	—
136	Q₈	—	184	2	no	—
137	Q₈	Dic₉₂	1	2	yes	complemented
138	C₂×C₄	—	2	1	no	—
139	C₂×C₄	—	2	1	no	—
140	C₂×C₄	—	46	1	no	—
141	C₂×C₄	—	46	1	no	—
142	C₂×C₄	—	46	2	no	—
143	C₂×C₄	—	92	1	no	—
144	C₂×C₄	C₂×D₉₂	1	1	yes	characteristic
145	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
146	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
147	C₈	—	2	2	no	—
148	C₈	—	4	1	no	—
149	C₈	—	92	1	no	—
150	C₂²	#10625053	1	1	yes	characteristic
151	C₄	—	2	2	no	—
152	C₄	—	2	1	no	—
153	C₄	—	4	1	no	—
154	C₄	—	46	2	no	—
155	C₄	—	92	2	no	—
156	C₄	—	92	1	no	—
157	C₄	—	184	1	no	—
158	C₄	C₂×Dic₉₂	1	1	yes	characteristic
159	C₄	#10625080	1	1	yes	characteristic
160	C₄	#10625086	1	1	yes	characteristic
161	C₄	Q₁₆×D₂₃	1	1	yes	characteristic
162	C₂	#13092574	1	1	yes	characteristic
163	C₂	#13092575	1	1	yes	characteristic
164	C₂	#13092586	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×Dic₄₆

C₂

yes

maximal, characteristic

#13092797

C₂

yes

maximal, characteristic

#13092713

C₂

yes

maximal, characteristic

#13092712

C₂

yes

maximal, characteristic

#13092165

C₂

yes

maximal, characteristic

#13092149

C₂

yes

maximal, characteristic

C₂₃×Q₈⋊C₈

C₂

yes

maximal, characteristic

C₄×Q₈×C₂₃

C₂²

yes

characteristic

Q₈×Dic₂₃

—

C₂×Dic₉₂

—

—

—

—

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×Dic₄₆

D₄

yes

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

D₄

yes

characteristic

C₄²×C₂₃

—

C₄²×C₂₃

C₂³

yes

characteristic

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

D₄

yes

characteristic

Dic₂₃⋊C₄

—

Dic₂₃⋊C₄

—

C₄×Dic₂₃

—

C₄×Dic₂₃

—

C₂×C₂₃⋊C₈

—

Dic₉₂

—

Q₈×C₂₃

—

Q₈×C₂₃

Q₁₆

yes

complemented

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

Q₁₆

yes

complemented

C₈×C₂₃

—

C₈×C₂₃

—

C₂₃⋊C₈

—

—

C₂²×C₂₃

yes

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

C₂×Q₁₆

yes

characteristic

C₄×C₂₃

C₂×Q₁₆

yes

C₄×C₂₃

yes

C₄×C₂₃

yes

Dic₂₃

—

Dic₂₃

—

Dic₂₃

—

Dic₂₃

—

Q₈²

—

—

—

—

—

Q₈⋊C₈

—

Q₈⋊C₈

D₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

C₂×D₂₃

yes

characteristic

100

#107

—

101

#107

—

102

C₄⋊C₈

—

103

C₄⋊C₈

C₂×D₂₃

yes

characteristic

104

#103

—

105

#103

—

106

C₄×C₈

C₂×D₂₃

yes

characteristic

107

C₂₃

#1633

yes

characteristic, complemented, Sylow

108

C₂×Q₈

—

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

D₉₂

yes

characteristic

127

C₄²

—

128

C₄²

—

129

C₄²

—

130

C₄²

C₂²×D₂₃

yes

characteristic

131

Q₈

—

132

Q₈

—

133

Q₈

—

134

Q₈

—

135

Q₈

—

136

Q₈

—

184

—

137

Q₈

Dic₉₂

yes

complemented

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

C₂×D₉₂

yes

characteristic

145

C₂×C₄

D₄×D₂₃

yes

characteristic

146

C₂×C₄

D₄×D₂₃

yes

characteristic

147

C₈

—

148

C₈

—

149

C₈

—

150

C₂²

#10625053

yes

characteristic

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

—

184

—

158

C₄

C₂×Dic₉₂

yes

characteristic

159

C₄

#10625080

yes

characteristic

160

C₄

#10625086

yes

characteristic

161

C₄

Q₁₆×D₂₃

yes

characteristic

162

C₂

#13092574

yes

characteristic

163

C₂

#13092575

yes

characteristic

164

C₂

#13092586

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	#1633
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

#1633

LCS

G ▷ C₄²×C₂₃ ▷ C₂²×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ Q₈⋊C₈

N	H ≅ G/N	type
49. Q₈×C₂₃	Q₁₆	semidirect
63. Dic₄₆	Q₁₆	semidirect
107. C₂₃	#1633	semidirect
137. Q₈	Dic₉₂	semidirect

H ≅ G/N

type

49. Q₈×C₂₃

Q₁₆

semidirect

63. Dic₄₆

Q₁₆

semidirect

107. C₂₃

#1633

semidirect

137. Q₈

Dic₉₂

semidirect

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

subgroup	index
Q₈×Dic₄₆	2
#13092797	2
#13092713	2
#13092712	2
#13092165	2
#13092149	2
C₂₃×Q₈⋊C₈	2
#1633	23

subgroup

index

Q₈×Dic₄₆

C₂₃×Q₈⋊C₈

Subgroups — #15672539

Conjugacy classes of subgroups

Complements

Maximal subgroups