Group #15674354 — subgroups

Subgroups — #15674354

Order 2944.

260 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 109 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×#10625041	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₂₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	#13092179	C₂	1	2	yes	maximal
4	#13092179	C₂	1	2	yes	maximal
5	C₄×C₄⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
7	C₂×C₄²×C₂₃	—	2	1	no	—
8	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
11	C₂×Dic₂₃⋊C₄	—	2	1	no	—
12	C₂×Dic₂₃⋊C₄	—	2	1	no	—
13	C₂²×C₂₃⋊C₈	C₂²	1	2	yes	—
14	#10625041	—	2	2	no	—
15	#10625041	C₄	1	4	yes	complemented
16	C₂₃×#95	—	2	1	no	—
17	C₄⋊(C₂₃⋊C₈)	—	2	2	no	—
18	C₂²×C₄×C₂₃	—	2	2	no	—
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	C₄⋊C₄×C₂₃	—	2	2	no	—
25	C₄⋊C₄×C₂₃	Q₈	1	4	yes	—
26	C₄²×C₂₃	—	4	2	no	—
27	C₄²×C₂₃	C₂×C₄	1	2	yes	—
28	C₄²×C₂₃	D₄	1	2	yes	—
29	C₄⋊Dic₂₃	—	2	1	no	—
30	C₄⋊Dic₂₃	—	2	1	no	—
31	C₄⋊Dic₂₃	C₂×C₄	1	2	yes	—
32	C₄⋊Dic₂₃	C₂×C₄	1	2	yes	—
33	Dic₂₃⋊C₄	—	2	4	no	—
34	Dic₂₃⋊C₄	—	2	4	no	—
35	C₂×C₂₃⋊C₈	—	2	2	no	—
36	C₂×C₂₃⋊C₈	—	2	4	no	—
37	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₂₃	—	2	2	no	—
39	C₂×C₄×C₂₃	—	2	1	no	—
40	C₂×C₄×C₂₃	—	2	1	no	—
41	C₂×C₄×C₂₃	—	2	4	no	—
42	C₂×C₄×C₂₃	—	4	2	no	—
43	C₂×C₄×C₂₃	—	4	2	no	—
44	C₂×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
45	C₂×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
46	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₂₃	C₂×Q₈	1	2	yes	—
49	C₂×C₄×C₂₃	#41	1	2	yes	—
50	C₂×Dic₂₃	—	2	2	no	—
51	C₂×Dic₂₃	—	2	2	no	—
52	C₂×Dic₂₃	—	4	2	no	—
53	C₂×Dic₂₃	—	4	1	no	—
54	C₂×Dic₂₃	—	4	1	no	—
55	C₂₃⋊C₈	—	4	2	no	—
56	#1989	—	23	1	no	maximal, Sylow
57	C₂²×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₂₃	C₂²⋊Q₈	1	2	yes	—
59	C₂²×C₂₃	#135	1	2	yes	—
60	C₂²×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
61	C₂²×C₂₃	#137	1	1	yes	characteristic
62	C₄×C₂₃	—	2	2	no	—
63	C₄×C₂₃	—	2	2	no	—
64	C₄×C₂₃	—	4	4	no	—
65	C₄×C₂₃	C₄×Q₈	1	2	yes	—
66	C₄×C₂₃	#124	1	1	yes	characteristic
67	C₄×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
68	Dic₂₃	—	4	2	no	—
69	Dic₂₃	—	4	2	no	—
70	C₂×#125	—	23	1	no	—
71	C₂×C₄⋊C₈	—	23	1	no	—
72	C₄×C₄⋊C₄	D₂₃	1	1	yes	characteristic
73	#340	—	23	2	no	—
74	#340	—	23	2	no	—
75	C₂×C₂₃	#391	1	1	yes	characteristic
76	C₂×C₂₃	#416	1	1	yes	characteristic
77	C₂×C₂₃	#417	1	1	yes	characteristic
78	C₂×C₂₃	#478	1	2	yes	—
79	C₂×C₂₃	#479	1	2	yes	—
80	C₂²×C₈	—	23	2	no	—
81	#125	—	23	4	no	—
82	#125	—	46	2	no	—
83	C₂×C₄⋊C₄	—	23	1	no	—
84	C₂×C₄⋊C₄	—	23	1	no	—
85	C₂×C₄⋊C₄	—	46	1	no	—
86	C₂×C₄⋊C₄	—	46	1	no	—
87	C₂×C₄⋊C₄	C₂×D₂₃	1	2	yes	—
88	C₂×C₄²	—	2	1	no	—
89	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
90	C₄⋊C₈	—	46	2	no	—
91	#95	—	2	1	no	—
92	C₂₃	#1989	1	1	yes	characteristic, complemented, Sylow
93	C₂²×C₄	—	2	2	no	—
94	C₂²×C₄	—	46	1	no	—
95	C₂²×C₄	—	46	1	no	—
96	C₂²×C₄	C₂²⋊D₂₃	1	1	yes	characteristic
97	C₂²×C₄	C₂²⋊D₂₃	1	1	yes	characteristic
98	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
99	C₂×C₈	—	46	2	no	—
100	C₂×C₈	—	46	4	no	—
101	C₄⋊C₄	—	2	2	no	—
102	C₄⋊C₄	—	23	2	no	—
103	C₄⋊C₄	—	23	2	no	—
104	C₄⋊C₄	—	46	4	no	—
105	C₄⋊C₄	—	46	4	no	—
106	C₄⋊C₄	—	46	1	no	—
107	C₄⋊C₄	—	46	1	no	—
108	C₄⋊C₄	Dic₄₆	1	4	yes	—
109	C₄²	—	4	2	no	—
110	C₄²	C₄×D₂₃	1	2	yes	—
111	C₄²	D₉₂	1	2	yes	—
112	C₂³	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
113	C₂×C₄	—	2	1	no	—
114	C₂×C₄	—	2	1	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	4	no	—
117	C₂×C₄	—	4	2	no	—
118	C₂×C₄	—	4	2	no	—
119	C₂×C₄	—	46	2	no	—
120	C₂×C₄	—	46	2	no	—
121	C₂×C₄	—	92	1	no	—
122	C₂×C₄	—	92	1	no	—
123	C₂×C₄	—	92	2	no	—
124	C₂×C₄	(C₂×D₂₃)⋊C₄	1	2	yes	—
125	C₂×C₄	(C₂×D₂₃)⋊C₄	1	2	yes	—
126	C₂×C₄	C₂×Dic₄₆	1	2	yes	—
127	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
128	C₂×C₄	C₂×D₉₂	1	1	yes	characteristic
129	C₂×C₄	#67288	1	2	yes	—
130	C₈	—	92	2	no	—
131	C₂²	#10625095	1	2	yes	—
132	C₂²	C₂×(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
133	C₂²	#10625103	1	1	yes	characteristic
134	C₂²	#10625113	1	1	yes	characteristic
135	C₂²	#10625121	1	2	yes	—
136	C₄	—	2	2	no	—
137	C₄	—	2	2	no	—
138	C₄	—	4	4	no	—
139	C₄	—	92	2	no	—
140	C₄	—	92	2	no	—
141	C₄	C₄×Dic₄₆	1	2	yes	—
142	C₄	#10625040	1	1	yes	characteristic
143	C₄	#10625046	1	1	yes	characteristic
144	C₂	#13092323	1	1	yes	characteristic
145	C₂	#13092411	1	1	yes	characteristic
146	C₂	#13092416	1	1	yes	characteristic
147	C₂	#13092717	1	2	yes	—
148	C₂	#13092724	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10625041

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₂₃⋊C₈)

C₂

yes

maximal, characteristic

#13092179

C₂

yes

maximal

#13092179

C₂

yes

maximal

C₄×C₄⋊C₄×C₂₃

C₂

yes

maximal, characteristic

C₂×C₄⋊C₄×C₂₃

C₂²

yes

—

C₂×C₄²×C₂₃

—

C₂×C₄²×C₂₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₂₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₂₃

C₂²

yes

characteristic

C₂×Dic₂₃⋊C₄

—

C₂×Dic₂₃⋊C₄

—

C₂²×C₂₃⋊C₈

C₂²

yes

—

#10625041

—

#10625041

C₄

yes

complemented

C₂₃×#95

—

C₄⋊(C₂₃⋊C₈)

—

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

Q₈

yes

—

C₄²×C₂₃

—

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

—

Dic₂₃⋊C₄

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂³×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

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

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×Q₈

yes

—

C₂×C₄×C₂₃

#41

yes

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂₃⋊C₈

—

#1989

—

maximal, Sylow

C₂²×C₂₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₃

C₂²⋊Q₈

yes

—

C₂²×C₂₃

#135

yes

—

C₂²×C₂₃

yes

C₂²×C₂₃

yes

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

C₄×Q₈

yes

—

C₄×C₂₃

#124

yes

characteristic

C₄×C₂₃

C₄⋊Q₈

yes

characteristic

Dic₂₃

—

Dic₂₃

—

C₂×#125

—

C₂×C₄⋊C₈

—

C₄×C₄⋊C₄

D₂₃

yes

characteristic

#340

—

#340

—

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

—

C₂×C₂₃

#479

yes

—

C₂²×C₈

—

#125

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₂₃

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×D₂₃

yes

characteristic

C₄⋊C₈

—

#95

—

C₂₃

#1989

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₂₃

yes

characteristic

C₂²×C₄

C₂²⋊D₂₃

yes

characteristic

C₂²×C₄

C₂²×D₂₃

yes

characteristic

C₂×C₈

—

100

C₂×C₈

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

Dic₄₆

yes

—

109

C₄²

—

110

C₄²

C₄×D₂₃

yes

—

111

C₄²

D₉₂

yes

—

112

C₂³

C₂×C₂²⋊D₂₃

yes

characteristic

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

(C₂×D₂₃)⋊C₄

yes

—

125

C₂×C₄

(C₂×D₂₃)⋊C₄

yes

—

126

C₂×C₄

C₂×Dic₄₆

yes

—

127

C₂×C₄

C₂×C₄×D₂₃

yes

characteristic

128

C₂×C₄

C₂×D₉₂

yes

characteristic

129

C₂×C₄

#67288

yes

—

130

C₈

—

131

C₂²

#10625095

yes

—

132

C₂²

C₂×(C₂×D₂₃)⋊C₄

yes

characteristic

133

C₂²

#10625103

yes

characteristic

134

C₂²

#10625113

yes

characteristic

135

C₂²

#10625121

yes

—

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

C₄×Dic₄₆

yes

—

142

C₄

#10625040

yes

characteristic

143

C₄

#10625046

yes

characteristic

144

C₂

#13092323

yes

characteristic

145

C₂

#13092411

yes

characteristic

146

C₂

#13092416

yes

characteristic

147

C₂

#13092717

yes

—

148

C₂

#13092724

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₂₃
commutator	C₂×C₄×C₂₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₃
Fitting	C₄×C₄⋊C₄×C₂₃	C₂
socle, prime radical	C₂³×C₂₃	C₂×D₄
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	C₄×C₄⋊C₄	D₂₃
hypocenter	C₂₃	#1989
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₄×C₄⋊C₄

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₂₃

commutator

C₂×C₄×C₂₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₃

Fitting

C₄×C₄⋊C₄×C₂₃

C₂

socle, prime radical

C₂³×C₂₃

C₂×D₄

cosocle

C₂²×C₄×C₂₃

C₂³

hypercenter

C₄×C₄⋊C₄

D₂₃

hypocenter

C₂₃

#1989

LCS

G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₄×C₄⋊C₄

N	H ≅ G/N	type
15. #10625041	C₄	semidirect
92. C₂₃	#1989	semidirect

H ≅ G/N

type

15. #10625041

C₄

semidirect

92. C₂₃

#1989

semidirect

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

subgroup	index
C₂×#10625041	2
C₂×C₄⋊(C₂₃⋊C₈)	2
#13092179	2
C₄×C₄⋊C₄×C₂₃	2
#1989	23

subgroup

index

C₂×#10625041

C₂×C₄⋊(C₂₃⋊C₈)

#13092179

C₄×C₄⋊C₄×C₂₃

#1989

Subgroups — #15674354

Conjugacy classes of subgroups

Complements

Maximal subgroups