Group #15673228 — subgroups

Subgroups — #15673228

Order 2944.

324 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 129 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×C₄×Dic₄₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#10624998	C₂	1	2	yes	maximal
3	C₄×C₄⋊Dic₂₃	C₂	1	1	yes	maximal, characteristic
4	#13092180	C₂	1	2	yes	maximal
5	C₂×C₂₃×#97	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₄₆	C₂²	1	1	yes	characteristic
7	C₂²×C₈×C₂₃	C₂²	1	2	yes	—
8	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₂₃	C₂²	1	2	yes	—
11	C₂×Dic₂₃⋊C₄	—	2	1	no	—
12	C₂×C₄×Dic₂₃	—	2	1	no	—
13	C₂×C₄×Dic₂₃	—	2	1	no	—
14	C₄×Dic₄₆	—	2	2	no	—
15	C₄×Dic₄₆	C₄	1	4	yes	complemented
16	C₂₃×#97	—	2	2	no	—
17	C₂₃⋊#95	—	2	1	no	—
18	#10624998	C₄	1	8	yes	complemented
19	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
20	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
21	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
22	C₂²×Dic₂₃	—	2	1	no	—
23	C₂²×Dic₂₃	—	2	1	no	—
24	C₂²×Dic₂₃	—	2	2	no	—
25	C₂×Dic₄₆	—	2	1	no	—
26	C₂×Dic₄₆	—	2	2	no	—
27	C₂×Dic₄₆	C₂×C₄	1	2	yes	—
28	C₂×Dic₄₆	C₂×C₄	1	4	yes	—
29	C₂×C₈×C₂₃	—	2	2	no	—
30	C₂×C₈×C₂₃	—	2	2	no	—
31	C₂×C₈×C₂₃	C₂×C₄	1	4	yes	—
32	C₄²×C₂₃	C₂×C₄	1	2	yes	—
33	C₄²×C₂₃	D₄	1	2	yes	—
34	C₄⋊Dic₂₃	—	2	1	no	—
35	C₄⋊Dic₂₃	—	2	2	no	—
36	C₄⋊Dic₂₃	C₂×C₄	1	2	yes	—
37	C₄⋊Dic₂₃	C₂×C₄	1	4	yes	—
38	Dic₂₃⋊C₄	—	2	4	no	—
39	C₄×Dic₂₃	—	2	4	no	—
40	C₄×Dic₂₃	—	4	2	no	—
41	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₂₃	—	2	1	no	—
43	C₂×C₄×C₂₃	—	2	1	no	—
44	C₂×C₄×C₂₃	—	2	2	no	—
45	C₂×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
46	C₂×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
47	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
49	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₂₃	#41	1	2	yes	—
51	C₂×Dic₂₃	—	2	2	no	—
52	C₂×Dic₂₃	—	2	4	no	—
53	C₂×Dic₂₃	—	2	2	no	—
54	C₂×Dic₂₃	—	2	4	no	—
55	C₂×Dic₂₃	—	4	2	no	—
56	C₂×Dic₂₃	—	4	1	no	—
57	C₂×Dic₂₃	—	4	1	no	—
58	C₂×Dic₂₃	—	4	2	no	—
59	Dic₄₆	—	2	2	no	—
60	Dic₄₆	—	2	4	no	—
61	Dic₄₆	C₄²	1	4	yes	complemented
62	C₈×C₂₃	—	2	4	no	—
63	#1763	—	23	1	no	maximal, Sylow
64	C₂²×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₂₃	C₄×D₄	1	2	yes	—
66	C₂²×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
67	C₂²×C₂₃	#137	1	1	yes	characteristic
68	C₂²×C₂₃	#137	1	2	yes	—
69	C₄×C₂₃	—	2	2	no	—
70	C₄×C₂₃	—	2	2	no	—
71	C₄×C₂₃	C₂×C₄²	1	1	yes	characteristic
72	C₄×C₂₃	#117	1	1	yes	characteristic
73	C₄×C₂₃	C₄×D₄	1	2	yes	—
74	Dic₂₃	—	2	4	no	—
75	Dic₂₃	—	4	2	no	—
76	Dic₂₃	—	4	4	no	—
77	C₂×C₄×Q₈	—	23	1	no	—
78	C₂×#103	—	23	2	no	—
79	C₂×#97	D₂₃	1	1	yes	characteristic
80	C₄×C₄⋊C₄	—	23	1	no	—
81	#340	—	23	2	no	—
82	C₂×C₂₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₂₃	#417	1	1	yes	characteristic
84	C₂×C₂₃	#419	1	1	yes	characteristic
85	C₂×C₂₃	#440	1	2	yes	—
86	C₂×C₂₃	#441	1	2	yes	—
87	C₂²×Q₈	—	23	1	no	—
88	C₂²×C₈	C₂×D₂₃	1	2	yes	—
89	C₄×Q₈	—	23	4	no	—
90	C₄×Q₈	—	46	2	no	—
91	C₂×C₄⋊C₄	—	23	1	no	—
92	C₂×C₄⋊C₄	—	23	2	no	—
93	C₂×C₄⋊C₄	—	46	1	no	—
94	C₂×C₄²	—	46	1	no	—
95	C₂×C₄²	—	46	1	no	—
96	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
97	#103	—	23	8	no	—
98	#97	—	2	2	no	—
99	#95	—	46	1	no	—
100	C₂₃	#1763	1	1	yes	characteristic, complemented, Sylow
101	C₂×Q₈	—	23	4	no	—
102	C₂×Q₈	—	23	2	no	—
103	C₂×Q₈	—	46	2	no	—
104	C₂×Q₈	—	46	1	no	—
105	C₂²×C₄	—	46	1	no	—
106	C₂²×C₄	—	46	2	no	—
107	C₂²×C₄	—	46	1	no	—
108	C₂²×C₄	D₉₂	1	1	yes	characteristic
109	C₂²×C₄	D₉₂	1	1	yes	characteristic
110	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
111	C₂×C₈	—	2	2	no	—
112	C₂×C₈	—	2	2	no	—
113	C₂×C₈	C₄×D₂₃	1	4	yes	—
114	C₄⋊C₄	—	23	4	no	—
115	C₄⋊C₄	—	23	2	no	—
116	C₄⋊C₄	—	46	4	no	—
117	C₄⋊C₄	—	46	2	no	—
118	C₄⋊C₄	—	46	1	no	—
119	C₄²	—	46	4	no	—
120	C₄²	—	92	2	no	—
121	C₄²	C₄×D₂₃	1	2	yes	—
122	C₄²	C₂²⋊D₂₃	1	2	yes	—
123	C₂³	C₂×D₉₂	1	1	yes	characteristic
124	Q₈	—	23	4	no	—
125	Q₈	—	46	2	no	—
126	Q₈	—	46	4	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	2	1	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	46	4	no	—
131	C₂×C₄	—	46	2	no	—
132	C₂×C₄	—	46	4	no	—
133	C₂×C₄	—	46	2	no	—
134	C₂×C₄	—	92	2	no	—
135	C₂×C₄	—	92	1	no	—
136	C₂×C₄	—	92	1	no	—
137	C₂×C₄	—	92	2	no	—
138	C₂×C₄	(C₂×D₂₃)⋊C₄	1	2	yes	—
139	C₂×C₄	(C₂×D₂₃)⋊C₄	1	2	yes	—
140	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
141	C₂×C₄	C₂×C₄×D₂₃	1	2	yes	—
142	C₂×C₄	#67288	1	2	yes	—
143	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
144	C₈	—	2	4	no	—
145	C₂²	C₄×D₉₂	1	2	yes	—
146	C₂²	#10625079	1	1	yes	characteristic
147	C₂²	#10625080	1	1	yes	characteristic
148	C₂²	#10625080	1	2	yes	—
149	C₂²	C₂×(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
150	C₄	—	2	2	no	—
151	C₄	—	2	2	no	—
152	C₄	—	46	4	no	—
153	C₄	—	92	2	no	—
154	C₄	—	92	4	no	—
155	C₄	C₄²×D₂₃	1	1	yes	characteristic
156	C₄	#10625043	1	1	yes	characteristic
157	C₄	C₄×C₂²⋊D₂₃	1	2	yes	—
158	C₂	C₄×(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
159	C₂	#13092419	1	1	yes	characteristic
160	C₂	#13092427	1	1	yes	characteristic
161	C₂	#13092519	1	2	yes	—
162	C₂	#13092525	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Dic₄₆

C₂

yes

maximal, characteristic

C₂×#10624998

C₂

yes

maximal

C₄×C₄⋊Dic₂₃

C₂

yes

maximal, characteristic

#13092180

C₂

yes

maximal

C₂×C₂₃×#97

C₂

yes

maximal, characteristic

C₂²×Dic₄₆

C₂²

yes

characteristic

C₂²×C₈×C₂₃

C₂²

yes

—

C₂×C₄²×C₂₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₂₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₂₃

C₂²

yes

—

C₂×Dic₂₃⋊C₄

—

C₂×C₄×Dic₂₃

—

C₂×C₄×Dic₂₃

—

C₄×Dic₄₆

—

C₄×Dic₄₆

C₄

yes

complemented

C₂₃×#97

—

C₂₃⋊#95

—

#10624998

C₄

yes

complemented

C₂²×C₄×C₂₃

D₄

yes

characteristic

C₂²×C₄×C₂₃

D₄

yes

characteristic

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂²×Dic₂₃

—

C₂²×Dic₂₃

—

C₂²×Dic₂₃

—

C₂×Dic₄₆

—

C₂×Dic₄₆

—

C₂×Dic₄₆

C₂×C₄

yes

—

C₂×Dic₄₆

C₂×C₄

yes

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

C₂×C₄

yes

—

C₄²×C₂₃

C₂×C₄

yes

—

C₄²×C₂₃

D₄

yes

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

C₂×C₄

yes

—

C₄⋊Dic₂₃

C₂×C₄

yes

—

Dic₂₃⋊C₄

—

C₄×Dic₂₃

—

C₄×Dic₂₃

—

C₂³×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂²⋊C₄

yes

—

C₂×C₄×C₂₃

C₂²⋊C₄

yes

—

C₂×C₄×C₂₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃

C₂²×C₄

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

#41

yes

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

Dic₄₆

—

Dic₄₆

—

Dic₄₆

C₄²

yes

complemented

C₈×C₂₃

—

#1763

—

maximal, Sylow

C₂²×C₂₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₃

C₄×D₄

yes

—

C₂²×C₂₃

yes

C₂²×C₂₃

yes

C₂²×C₂₃

yes

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

C₂×C₄²

yes

characteristic

C₄×C₂₃

#117

yes

characteristic

C₄×C₂₃

C₄×D₄

yes

—

Dic₂₃

—

Dic₂₃

—

Dic₂₃

—

C₂×C₄×Q₈

—

C₂×#103

—

C₂×#97

D₂₃

yes

characteristic

C₄×C₄⋊C₄

—

#340

—

C₂×C₂₃

C₄×C₂²⋊C₄

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

—

C₂×C₂₃

#441

yes

—

C₂²×Q₈

—

C₂²×C₈

C₂×D₂₃

yes

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₂₃

yes

—

—

—

100

C₂₃

#1763

yes

characteristic, complemented, Sylow

101

C₂×Q₈

—

102

C₂×Q₈

—

103

C₂×Q₈

—

104

C₂×Q₈

—

105

C₂²×C₄

—

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₂²×C₄

D₉₂

yes

characteristic

109

C₂²×C₄

D₉₂

yes

characteristic

110

C₂²×C₄

C₂²×D₂₃

yes

characteristic

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

C₄×D₂₃

yes

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄²

—

120

C₄²

—

121

C₄²

C₄×D₂₃

yes

—

122

C₄²

C₂²⋊D₂₃

yes

—

123

C₂³

C₂×D₉₂

yes

characteristic

124

Q₈

—

125

Q₈

—

126

Q₈

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

(C₂×D₂₃)⋊C₄

yes

—

139

C₂×C₄

(C₂×D₂₃)⋊C₄

yes

—

140

C₂×C₄

C₂×C₄×D₂₃

yes

characteristic

141

C₂×C₄

C₂×C₄×D₂₃

yes

—

142

C₂×C₄

#67288

yes

—

143

C₂×C₄

C₂×C₂²⋊D₂₃

yes

characteristic

144

C₈

—

145

C₂²

C₄×D₉₂

yes

—

146

C₂²

#10625079

yes

characteristic

147

C₂²

#10625080

yes

characteristic

148

C₂²

#10625080

yes

—

149

C₂²

C₂×(C₂×D₂₃)⋊C₄

yes

characteristic

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

C₄²×D₂₃

yes

characteristic

156

C₄

#10625043

yes

characteristic

157

C₄

C₄×C₂²⋊D₂₃

yes

—

158

C₂

C₄×(C₂×D₂₃)⋊C₄

yes

characteristic

159

C₂

#13092419

yes

characteristic

160

C₂

#13092427

yes

characteristic

161

C₂

#13092519

yes

—

162

C₂

#13092525

yes

—

	subgroup	G/H
center	C₂³	C₂×D₉₂
commutator	C₄×C₂₃	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₂₃
Fitting	C₂×C₂₃×#97	C₂
socle, prime radical	C₂³×C₂₃	C₂×D₄
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	C₂×#97	D₂₃
hypocenter	C₂₃	#1763
LCS	G ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

subgroup

G/H

center

C₂³

C₂×D₉₂

commutator

C₄×C₂₃

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₂₃

Fitting

C₂×C₂₃×#97

C₂

socle, prime radical

C₂³×C₂₃

C₂×D₄

cosocle

C₂²×C₄×C₂₃

C₂³

hypercenter

C₂×#97

D₂₃

hypocenter

C₂₃

#1763

LCS

G ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

N	H ≅ G/N	type
15. C₄×Dic₄₆	C₄	semidirect
18. #10624998	C₄	semidirect
61. Dic₄₆	C₄²	semidirect
100. C₂₃	#1763	semidirect

H ≅ G/N

type

15. C₄×Dic₄₆

C₄

semidirect

18. #10624998

C₄

semidirect

61. Dic₄₆

C₄²

semidirect

100. C₂₃

#1763

semidirect

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

subgroup	index
C₂×C₄×Dic₄₆	2
C₂×#10624998	2
C₄×C₄⋊Dic₂₃	2
#13092180	2
C₂×C₂₃×#97	2
#1763	23

subgroup

index

C₂×C₄×Dic₄₆

C₂×#10624998

C₄×C₄⋊Dic₂₃

#13092180

C₂×C₂₃×#97

#1763

Subgroups — #15673228

Conjugacy classes of subgroups

Complements

Maximal subgroups