Group #15672557 — subgroups

Subgroups — #15672557

Order 2944.

188 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 55 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	#13093145	C₂	1	1	yes	maximal, characteristic
2	#13092768	C₂	1	1	yes	maximal, characteristic
3	#13092713	C₂	1	1	yes	maximal, characteristic
4	#13092712	C₂	1	1	yes	maximal, characteristic
5	#13092162	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	#10625115	—	2	1	no	—
10	#10625060	—	2	2	no	—
11	#10625058	—	2	1	no	—
12	#10625041	—	2	1	no	—
13	#10625040	C₂²	1	1	yes	characteristic
14	#10625040	C₂²	1	1	yes	characteristic
15	C₄×Dic₄₆	—	2	1	no	—
16	C₄×Dic₄₆	C₂²	1	1	yes	characteristic
17	C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
18	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
19	#10625017	—	2	1	no	—
20	#10625000	—	2	1	no	—
21	#10624998	—	2	1	no	—
22	#10624998	—	2	2	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	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	C₄⋊Dic₂₃	—	2	1	no	—
36	C₄⋊Dic₂₃	—	2	1	no	—
37	C₄⋊Dic₂₃	—	2	1	no	—
38	C₄⋊Dic₂₃	D₄	1	1	yes	characteristic
39	C₄⋊Dic₂₃	D₄	1	2	yes	—
40	Dic₂₃⋊C₄	—	2	2	no	—
41	Dic₂₃⋊C₄	—	2	2	no	—
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	Q₈×C₂₃	—	2	1	no	—
48	Q₈×C₂₃	—	2	1	no	—
49	C₂×C₄×C₂₃	—	2	1	no	—
50	C₂×C₄×C₂₃	—	2	1	no	—
51	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
54	C₂×Dic₂₃	—	2	1	no	—
55	C₂×Dic₂₃	—	2	1	no	—
56	C₂×Dic₂₃	—	2	2	no	—
57	C₂×Dic₂₃	—	4	1	no	—
58	Dic₄₆	—	2	1	no	—
59	Dic₄₆	—	4	1	no	—
60	Dic₄₆	—	4	1	no	—
61	Dic₄₆	—	4	2	no	—
62	Dic₄₆	Q₁₆	1	2	yes	—
63	C₈×C₂₃	—	4	1	no	—
64	C₈×C₂₃	—	4	1	no	—
65	C₂₃⋊C₈	—	4	1	no	—
66	#1636	—	23	1	no	maximal, Sylow
67	C₂²×C₂₃	#120	1	1	yes	characteristic
68	C₄×C₂₃	—	2	1	no	—
69	C₄×C₂₃	—	4	1	no	—
70	C₄×C₂₃	—	4	1	no	—
71	C₄×C₂₃	C₂×Q₁₆	1	1	yes	characteristic
72	C₄×C₂₃	#135	1	1	yes	characteristic
73	C₄×C₂₃	#137	1	1	yes	characteristic
74	C₄×C₂₃	#137	1	1	yes	characteristic
75	Dic₂₃	—	2	2	no	—
76	Dic₂₃	—	4	2	no	—
77	Dic₂₃	—	4	1	no	—
78	Dic₂₃	—	8	1	no	—
79	#557	—	23	1	no	—
80	#487	—	23	1	no	—
81	#477	—	23	1	no	—
82	#477	—	23	1	no	—
83	#332	—	23	1	no	—
84	Q₈⋊C₈	—	23	1	no	—
85	Q₈⋊C₈	D₂₃	1	1	yes	characteristic
86	C₂×C₂₃	#451	1	1	yes	characteristic
87	C₂×C₂₃	#452	1	1	yes	characteristic
88	C₂×C₂₃	#454	1	1	yes	characteristic
89	C₄⋊Q₈	—	23	1	no	—
90	C₄⋊Q₈	—	23	1	no	—
91	C₄⋊Q₈	—	46	1	no	—
92	#125	—	46	1	no	—
93	#125	—	46	2	no	—
94	C₄×Q₈	—	23	1	no	—
95	C₄×Q₈	—	46	1	no	—
96	C₄×Q₈	—	46	1	no	—
97	C₄×Q₈	C₂×D₂₃	1	1	yes	characteristic
98	#107	—	46	1	no	—
99	#107	—	46	1	no	—
100	C₄⋊C₈	—	23	1	no	—
101	C₄⋊C₈	C₂×D₂₃	1	1	yes	characteristic
102	#103	—	46	2	no	—
103	#103	—	46	1	no	—
104	#103	—	46	1	no	—
105	C₄×C₈	C₂×D₂₃	1	1	yes	characteristic
106	C₂₃	#1636	1	1	yes	characteristic, complemented, Sylow
107	C₂×Q₈	—	23	1	no	—
108	C₂×Q₈	—	46	1	no	—
109	C₂×Q₈	—	46	1	no	—
110	C₂×Q₈	D₉₂	1	1	yes	characteristic
111	C₂×C₈	—	2	1	no	—
112	C₂×C₈	—	2	1	no	—
113	C₂×C₈	—	46	1	no	—
114	C₄⋊C₄	—	2	1	no	—
115	C₄⋊C₄	—	23	2	no	—
116	C₄⋊C₄	—	23	1	no	—
117	C₄⋊C₄	—	46	2	no	—
118	C₄⋊C₄	—	46	2	no	—
119	C₄⋊C₄	—	46	2	no	—
120	C₄⋊C₄	—	46	1	no	—
121	C₄⋊C₄	—	46	1	no	—
122	C₄⋊C₄	—	46	1	no	—
123	C₄⋊C₄	—	46	1	no	—
124	C₄⋊C₄	D₉₂	1	1	yes	characteristic
125	C₄²	—	2	1	no	—
126	C₄²	—	46	2	no	—
127	C₄²	—	46	1	no	—
128	C₄²	C₂²×D₂₃	1	1	yes	characteristic
129	Q₈	—	2	1	no	—
130	Q₈	—	2	1	no	—
131	Q₈	—	23	2	no	—
132	Q₈	—	46	1	no	—
133	Q₈	—	92	2	no	—
134	Q₈	—	92	1	no	—
135	Q₈	—	92	1	no	—
136	C₂×C₄	—	2	1	no	—
137	C₂×C₄	—	2	1	no	—
138	C₂×C₄	—	46	1	no	—
139	C₂×C₄	—	46	1	no	—
140	C₂×C₄	—	46	2	no	—
141	C₂×C₄	—	92	1	no	—
142	C₂×C₄	C₂×D₉₂	1	1	yes	characteristic
143	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
144	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
145	C₈	—	4	1	no	—
146	C₈	—	4	1	no	—
147	C₈	—	92	1	no	—
148	C₂²	#10625053	1	1	yes	characteristic
149	C₄	—	2	1	no	—
150	C₄	—	4	1	no	—
151	C₄	—	4	1	no	—
152	C₄	—	46	2	no	—
153	C₄	—	92	2	no	—
154	C₄	—	92	1	no	—
155	C₄	—	184	1	no	—
156	C₄	#10625075	1	1	yes	characteristic
157	C₄	#10625080	1	1	yes	characteristic
158	C₄	#10625086	1	1	yes	characteristic
159	C₄	Q₁₆×D₂₃	1	1	yes	characteristic
160	C₂	#13092574	1	1	yes	characteristic
161	C₂	#13092581	1	1	yes	characteristic
162	C₂	#13092591	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13093145

C₂

yes

maximal, characteristic

#13092768

C₂

yes

maximal, characteristic

#13092713

C₂

yes

maximal, characteristic

#13092712

C₂

yes

maximal, characteristic

#13092162

C₂

yes

maximal, characteristic

#13092149

C₂

yes

maximal, characteristic

C₂₃×Q₈⋊C₈

C₂

yes

maximal, characteristic

C₄×Q₈×C₂₃

C₂²

yes

—

—

—

—

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

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

D₄

yes

characteristic

C₄⋊Dic₂₃

D₄

yes

—

Dic₂₃⋊C₄

—

Dic₂₃⋊C₄

—

Dic₂₃⋊C₄

—

Dic₂₃⋊C₄

—

C₄×Dic₂₃

—

C₄×Dic₂₃

—

C₂×C₂₃⋊C₈

—

Q₈×C₂₃

—

Q₈×C₂₃

—

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

—

C₈×C₂₃

—

C₈×C₂₃

—

C₂₃⋊C₈

—

—

C₂²×C₂₃

yes

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

C₂×Q₁₆

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

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

#125

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

C₂×D₂₃

yes

characteristic

#107

—

#107

—

100

C₄⋊C₈

—

101

C₄⋊C₈

C₂×D₂₃

yes

characteristic

102

#103

—

103

#103

—

104

#103

—

105

C₄×C₈

C₂×D₂₃

yes

characteristic

106

C₂₃

#1636

yes

characteristic, complemented, Sylow

107

C₂×Q₈

—

108

C₂×Q₈

—

109

C₂×Q₈

—

110

C₂×Q₈

D₉₂

yes

characteristic

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₄⋊C₄

D₉₂

yes

characteristic

125

C₄²

—

126

C₄²

—

127

C₄²

—

128

C₄²

C₂²×D₂₃

yes

characteristic

129

Q₈

—

130

Q₈

—

131

Q₈

—

132

Q₈

—

133

Q₈

—

134

Q₈

—

135

Q₈

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

C₂×D₉₂

yes

characteristic

143

C₂×C₄

D₄×D₂₃

yes

characteristic

144

C₂×C₄

D₄×D₂₃

yes

characteristic

145

C₈

—

146

C₈

—

147

C₈

—

148

C₂²

#10625053

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

—

184

—

156

C₄

#10625075

yes

characteristic

157

C₄

#10625080

yes

characteristic

158

C₄

#10625086

yes

characteristic

159

C₄

Q₁₆×D₂₃

yes

characteristic

160

C₂

#13092574

yes

characteristic

161

C₂

#13092581

yes

characteristic

162

C₂

#13092591

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	#1636
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

#1636

LCS

G ▷ C₄²×C₂₃ ▷ C₂²×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ Q₈⋊C₈

N	H ≅ G/N	type
106. C₂₃	#1636	semidirect

H ≅ G/N

type

106. C₂₃

#1636

semidirect

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

subgroup	index
#13093145	2
#13092768	2
#13092713	2
#13092712	2
#13092162	2
#13092149	2
C₂₃×Q₈⋊C₈	2
#1636	23

subgroup

index

C₂₃×Q₈⋊C₈

Subgroups — #15672557

Conjugacy classes of subgroups

Complements

Maximal subgroups