Group #15675446 — subgroups

Subgroups — #15675446

Order 2944.

154 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 154 conjugacy classes of subgroups — too many to draw. Only its 49 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×Dic₁₈₄	C₂	1	1	yes	maximal, characteristic
2	C₂×#10625011	C₂	1	1	yes	maximal, characteristic
3	#13092795	C₂	1	1	yes	maximal, characteristic
4	#13092511	C₂	1	1	yes	maximal, characteristic
5	Dic₂₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic
6	#13092220	C₂	1	1	yes	maximal, characteristic
7	C₂₃×#358	C₂	1	1	yes	maximal, characteristic
8	C₂×Q₁₆×C₂₃	C₂²	1	1	yes	characteristic
9	#10625115	—	2	1	no	—
10	C₂×#67275	—	2	1	no	—
11	C₂×Dic₉₂	C₂²	1	1	yes	characteristic
12	C₂×Dic₉₂	C₂²	1	1	yes	characteristic
13	#10625058	—	2	1	no	—
14	C₂×C₁₆×C₂₃	C₂²	1	1	yes	characteristic
15	C₂₃×#107	C₂²	1	1	yes	characteristic
16	#10625011	—	2	2	no	—
17	C₈×Dic₂₃	C₂²	1	1	yes	characteristic
18	C₂×C₂₃⋊C₁₆	C₂²	1	1	yes	characteristic
19	#10624992	—	2	1	no	—
20	Dic₁₈₄	—	2	2	no	—
21	C₂×Q₈×C₂₃	—	2	1	no	—
22	C₂×Dic₄₆	—	2	1	no	—
23	C₂×Dic₄₆	—	2	1	no	—
24	Q₁₆×C₂₃	—	2	1	no	—
25	Q₁₆×C₂₃	—	2	1	no	—
26	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
27	C₄⋊C₄×C₂₃	—	2	1	no	—
28	#67275	—	4	2	no	—
29	Dic₂₃⋊C₄	—	4	1	no	—
30	Dic₂₃⋊C₄	—	4	1	no	—
31	Dic₂₃⋊C₄	—	4	1	no	—
32	C₄×Dic₂₃	—	4	1	no	—
33	C₄×Dic₂₃	D₄	1	1	yes	characteristic
34	C₂×C₂₃⋊C₈	D₄	1	1	yes	characteristic
35	Dic₉₂	—	2	1	no	—
36	Dic₉₂	—	2	1	no	—
37	Dic₉₂	—	2	1	no	—
38	Dic₉₂	D₄	1	2	yes	—
39	C₁₆×C₂₃	—	2	1	no	—
40	C₂₃⋊C₁₆	—	2	1	no	—
41	Q₈×C₂₃	—	4	1	no	—
42	Q₈×C₂₃	—	4	1	no	—
43	C₂×C₄×C₂₃	—	4	1	no	—
44	C₂×C₄×C₂₃	—	4	1	no	—
45	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
46	C₂×Dic₂₃	—	4	1	no	—
47	C₂×Dic₂₃	—	4	1	no	—
48	C₂×Dic₂₃	D₈	1	1	yes	characteristic
49	C₂×Dic₂₃	D₈	1	1	yes	characteristic
50	Dic₄₆	—	2	2	no	—
51	Dic₄₆	—	4	1	no	—
52	Dic₄₆	—	4	1	no	—
53	Dic₄₆	—	4	1	no	—
54	C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
55	C₈×C₂₃	#41	1	1	yes	characteristic
56	C₂₃⋊C₈	—	2	1	no	—
57	#2209	—	23	1	no	maximal, Sylow
58	C₂²×C₂₃	C₂×D₈	1	1	yes	characteristic
59	C₄×C₂₃	—	8	1	no	—
60	C₄×C₂₃	—	8	1	no	—
61	C₄×C₂₃	#121	1	1	yes	characteristic
62	C₄×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
63	Dic₂₃	—	2	1	no	—
64	Dic₂₃	—	4	2	no	—
65	Dic₂₃	—	8	1	no	—
66	Dic₂₃	Q₃₂	1	2	yes	complemented
67	C₂×Q₃₂	—	23	1	no	—
68	C₂×Q₃₂	—	23	1	no	—
69	#494	—	23	1	no	—
70	C₄×Q₁₆	—	23	1	no	—
71	C₄⋊C₁₆	—	23	1	no	—
72	#358	—	23	1	no	—
73	#358	D₂₃	1	1	yes	characteristic
74	C₂×C₂₃	#459	1	1	yes	characteristic
75	C₂×C₂₃	C₂×Q₃₂	1	1	yes	characteristic
76	C₂×C₂₃	#510	1	1	yes	characteristic
77	C₂×Q₁₆	—	23	1	no	—
78	C₂×Q₁₆	—	23	1	no	—
79	C₂×Q₁₆	—	46	1	no	—
80	C₂×Q₁₆	C₂×D₂₃	1	1	yes	characteristic
81	C₄⋊Q₈	—	46	1	no	—
82	C₄×Q₈	—	46	1	no	—
83	Q₃₂	—	46	2	no	—
84	Q₃₂	—	46	2	no	—
85	C₂×C₁₆	—	23	1	no	—
86	C₂×C₁₆	C₂×D₂₃	1	1	yes	characteristic
87	#107	C₂×D₂₃	1	1	yes	characteristic
88	#103	—	46	1	no	—
89	C₄×C₈	—	23	1	no	—
90	C₂₃	#2209	1	1	yes	characteristic, complemented, Sylow
91	C₂×Q₈	—	2	1	no	—
92	C₂×Q₈	—	46	1	no	—
93	C₂×Q₈	—	46	1	no	—
94	Q₁₆	—	2	1	no	—
95	Q₁₆	—	2	1	no	—
96	Q₁₆	—	23	2	no	—
97	Q₁₆	—	46	1	no	—
98	Q₁₆	—	46	1	no	—
99	Q₁₆	—	46	1	no	—
100	Q₁₆	—	92	2	no	—
101	C₂×C₈	—	23	1	no	—
102	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
103	C₄⋊C₄	—	2	1	no	—
104	C₄⋊C₄	—	92	1	no	—
105	C₄⋊C₄	—	92	1	no	—
106	C₄⋊C₄	—	92	1	no	—
107	C₄²	—	23	1	no	—
108	C₄²	—	92	1	no	—
109	C₁₆	—	2	1	no	—
110	C₁₆	—	46	1	no	—
111	Q₈	—	4	1	no	—
112	Q₈	—	4	1	no	—
113	Q₈	—	46	2	no	—
114	Q₈	—	92	1	no	—
115	Q₈	—	92	1	no	—
116	Q₈	—	92	1	no	—
117	C₂×C₄	—	4	1	no	—
118	C₂×C₄	—	4	1	no	—
119	C₂×C₄	—	23	1	no	—
120	C₂×C₄	—	23	1	no	—
121	C₂×C₄	—	92	1	no	—
122	C₂×C₄	—	92	1	no	—
123	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
124	C₈	—	46	1	no	—
125	C₈	#67288	1	1	yes	characteristic
126	C₈	D₄×D₂₃	1	1	yes	characteristic
127	C₂²	D₈×D₂₃	1	1	yes	characteristic
128	C₄	—	8	1	no	—
129	C₄	—	8	1	no	—
130	C₄	—	23	2	no	—
131	C₄	—	46	1	no	—
132	C₄	—	92	2	no	—
133	C₄	—	184	1	no	—
134	C₄	#10625055	1	1	yes	characteristic
135	C₄	#10625085	1	1	yes	characteristic
136	C₂	#13092611	1	1	yes	characteristic
137	C₂	Q₃₂×D₂₃	1	1	yes	characteristic
138	C₂	#13092856	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Dic₁₈₄

C₂

yes

maximal, characteristic

C₂×#10625011

C₂

yes

maximal, characteristic

#13092795

C₂

yes

maximal, characteristic

#13092511

C₂

yes

maximal, characteristic

Dic₂₃⋊C₁₆

C₂

yes

maximal, characteristic

#13092220

C₂

yes

maximal, characteristic

C₂₃×#358

C₂

yes

maximal, characteristic

C₂×Q₁₆×C₂₃

C₂²

yes

characteristic

#10625115

—

C₂×#67275

—

C₂×Dic₉₂

C₂²

yes

characteristic

C₂×Dic₉₂

C₂²

yes

characteristic

#10625058

—

C₂×C₁₆×C₂₃

C₂²

yes

characteristic

C₂₃×#107

C₂²

yes

characteristic

#10625011

—

C₈×Dic₂₃

C₂²

yes

characteristic

C₂×C₂₃⋊C₁₆

C₂²

yes

characteristic

#10624992

—

Dic₁₈₄

—

C₂×Q₈×C₂₃

—

C₂×Dic₄₆

—

C₂×Dic₄₆

—

Q₁₆×C₂₃

—

Q₁₆×C₂₃

—

C₂×C₈×C₂₃

C₂³

yes

characteristic

C₄⋊C₄×C₂₃

—

#67275

—

Dic₂₃⋊C₄

—

Dic₂₃⋊C₄

—

Dic₂₃⋊C₄

—

C₄×Dic₂₃

—

C₄×Dic₂₃

D₄

yes

characteristic

C₂×C₂₃⋊C₈

D₄

yes

characteristic

Dic₉₂

—

Dic₉₂

—

Dic₉₂

—

Dic₉₂

D₄

yes

—

C₁₆×C₂₃

—

C₂₃⋊C₁₆

—

Q₈×C₂₃

—

Q₈×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

D₈

yes

characteristic

C₂×Dic₂₃

D₈

yes

characteristic

Dic₄₆

—

Dic₄₆

—

Dic₄₆

—

Dic₄₆

—

C₈×C₂₃

C₂×D₄

yes

characteristic

C₈×C₂₃

#41

yes

characteristic

C₂₃⋊C₈

—

#2209

—

maximal, Sylow

C₂²×C₂₃

C₂×D₈

yes

characteristic

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

yes

C₄×C₂₃

yes

Dic₂₃

—

Dic₂₃

—

Dic₂₃

—

Dic₂₃

Q₃₂

yes

complemented

C₂×Q₃₂

—

C₂×Q₃₂

—

#494

—

C₄×Q₁₆

—

C₄⋊C₁₆

—

#358

—

#358

D₂₃

yes

characteristic

C₂×C₂₃

#459

yes

characteristic

C₂×C₂₃

C₂×Q₃₂

yes

characteristic

C₂×C₂₃

#510

yes

characteristic

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

C₂×D₂₃

yes

characteristic

C₄⋊Q₈

—

C₄×Q₈

—

Q₃₂

—

Q₃₂

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₂₃

yes

characteristic

#107

C₂×D₂₃

yes

characteristic

#103

—

C₄×C₈

—

C₂₃

#2209

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

100

Q₁₆

—

101

C₂×C₈

—

102

C₂×C₈

C₂²×D₂₃

yes

characteristic

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄²

—

108

C₄²

—

109

C₁₆

—

110

C₁₆

—

111

Q₈

—

112

Q₈

—

113

Q₈

—

114

Q₈

—

115

Q₈

—

116

Q₈

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

D₄×D₂₃

yes

characteristic

124

C₈

—

125

C₈

#67288

yes

characteristic

126

C₈

D₄×D₂₃

yes

characteristic

127

C₂²

D₈×D₂₃

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

—

133

C₄

—

184

—

134

C₄

#10625055

yes

characteristic

135

C₄

#10625085

yes

characteristic

136

C₂

#13092611

yes

characteristic

137

C₂

Q₃₂×D₂₃

yes

characteristic

138

C₂

#13092856

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₂₃
commutator, cosocle	C₂×C₈×C₂₃	C₂³
Frattini	C₂×C₈	C₂²×D₂₃
Fitting	C₂₃×#358	C₂
socle, prime radical	C₂²×C₂₃	C₂×D₈
hypercenter	#358	D₂₃
hypocenter	C₂₃	#2209
LCS	G ▷ C₂×C₈×C₂₃ ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #358

subgroup

G/H

center

C₂²

D₈×D₂₃

commutator, cosocle

C₂×C₈×C₂₃

C₂³

Frattini

C₂×C₈

C₂²×D₂₃

Fitting

C₂₃×#358

C₂

socle, prime radical

C₂²×C₂₃

C₂×D₈

D₂₃

C₂₃

G ▷ C₂×C₈×C₂₃ ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #358

N	H ≅ G/N	type
66. Dic₂₃	Q₃₂	semidirect
90. C₂₃	#2209	semidirect

H ≅ G/N

type

66. Dic₂₃

Q₃₂

semidirect

90. C₂₃

#2209

semidirect

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

subgroup	index
C₂×Dic₁₈₄	2
C₂×#10625011	2
#13092795	2
#13092511	2
Dic₂₃⋊C₁₆	2
#13092220	2
C₂₃×#358	2
#2209	23

subgroup

index

C₂×Dic₁₈₄

Dic₂₃⋊C₁₆

C₂₃×#358

Subgroups — #15675446

Conjugacy classes of subgroups

Complements

Maximal subgroups