Group #15672765 — subgroups

Subgroups — #15672765

Order 2944.

156 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 156 conjugacy classes of subgroups — too many to draw. Only its 57 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	#13092796	C₂	1	1	yes	maximal, characteristic
2	#13092702	C₂	1	1	yes	maximal, characteristic
3	Q₈×C₂₃⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#13092169	C₂	1	1	yes	maximal, characteristic
5	#13092163	C₂	1	1	yes	maximal, characteristic
6	#13092162	C₂	1	1	yes	maximal, characteristic
7	C₂₃×#475	C₂	1	1	yes	maximal, characteristic
8	C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
10	C₂×#67275	—	2	1	no	—
11	C₂×#67275	—	2	1	no	—
12	#10625040	C₂²	1	1	yes	characteristic
13	C₂₃×#106	—	2	1	no	—
14	C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
15	C₂₃×#103	—	2	1	no	—
16	#10625017	—	2	1	no	—
17	#10624990	—	2	1	no	—
18	C₄⋊(C₂₃⋊C₈)	—	2	1	no	—
19	C₄⋊(C₂₃⋊C₈)	C₂²	1	1	yes	characteristic
20	C₄⋊(C₂₃⋊C₈)	C₂²	1	1	yes	characteristic
21	C₄×C₂₃⋊C₈	—	2	1	no	—
22	C₄×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
23	C₂×Q₈×C₂₃	—	2	1	no	—
24	C₂×Q₈×C₂₃	D₄	1	1	yes	characteristic
25	C₂×Dic₄₆	—	2	1	no	—
26	C₂×C₈×C₂₃	—	2	1	no	—
27	C₄⋊C₄×C₂₃	—	2	1	no	—
28	C₄⋊C₄×C₂₃	—	2	1	no	—
29	C₄⋊C₄×C₂₃	—	2	1	no	—
30	C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
31	C₄²×C₂₃	—	2	1	no	—
32	C₄²×C₂₃	C₂³	1	1	yes	characteristic
33	#67275	—	4	2	no	—
34	#67275	—	4	2	no	—
35	C₄⋊Dic₂₃	—	2	1	no	—
36	C₄⋊Dic₂₃	—	2	1	no	—
37	C₂×C₂₃⋊C₈	—	2	1	no	—
38	C₂×C₂₃⋊C₈	—	2	1	no	—
39	C₂×C₂₃⋊C₈	D₄	1	1	yes	characteristic
40	C₂×C₂₃⋊C₈	D₄	1	1	yes	characteristic
41	Q₈×C₂₃	—	2	1	no	—
42	Q₈×C₂₃	—	4	1	no	—
43	Q₈×C₂₃	—	4	1	no	—
44	Q₈×C₂₃	Q₁₆	1	2	yes	complemented
45	C₂×C₄×C₂₃	—	2	1	no	—
46	C₂×C₄×C₂₃	—	2	1	no	—
47	C₂×C₄×C₂₃	—	4	1	no	—
48	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₂₃	#41	1	1	yes	characteristic
51	C₂×Dic₂₃	—	4	1	no	—
52	Dic₄₆	—	4	1	no	—
53	Dic₄₆	—	4	1	no	—
54	C₈×C₂₃	—	4	1	no	—
55	C₂₃⋊C₈	—	2	1	no	—
56	C₂₃⋊C₈	—	4	1	no	—
57	C₂₃⋊C₈	—	4	1	no	—
58	C₂₃⋊C₈	SD₁₆	1	2	yes	—
59	#1670	—	23	1	no	maximal, Sylow
60	C₂²×C₂₃	#121	1	1	yes	characteristic
61	C₄×C₂₃	—	2	2	no	—
62	C₄×C₂₃	—	2	1	no	—
63	C₄×C₂₃	—	4	1	no	—
64	C₄×C₂₃	—	8	1	no	—
65	C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
66	C₄×C₂₃	C₂×Q₁₆	1	1	yes	characteristic
67	C₄×C₂₃	#135	1	1	yes	characteristic
68	C₄×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
69	Dic₂₃	—	8	1	no	—
70	#494	—	23	1	no	—
71	#475	—	23	1	no	—
72	#475	D₂₃	1	1	yes	characteristic
73	C₈×Q₈	—	23	1	no	—
74	#335	—	23	1	no	—
75	#332	—	23	1	no	—
76	#332	—	23	1	no	—
77	C₂×C₂₃	#460	1	1	yes	characteristic
78	C₂×C₂₃	#467	1	1	yes	characteristic
79	C₂×C₂₃	#473	1	1	yes	characteristic
80	C₂×Q₁₆	—	46	1	no	—
81	C₂×Q₁₆	—	46	1	no	—
82	C₄⋊Q₈	—	23	1	no	—
83	C₄⋊Q₈	C₂×D₂₃	1	1	yes	characteristic
84	C₄×Q₈	C₂×D₂₃	1	1	yes	characteristic
85	#106	—	2	1	no	—
86	#106	—	46	1	no	—
87	C₄⋊C₈	—	23	1	no	—
88	C₄⋊C₈	—	23	1	no	—
89	C₄⋊C₈	—	46	1	no	—
90	C₄⋊C₈	C₂×D₂₃	1	1	yes	characteristic
91	#103	—	2	1	no	—
92	#103	—	46	1	no	—
93	C₄×C₈	—	23	1	no	—
94	C₄×C₈	—	46	1	no	—
95	C₂₃	#1670	1	1	yes	characteristic, complemented, Sylow
96	C₂×Q₈	—	2	1	no	—
97	C₂×Q₈	—	46	1	no	—
98	C₂×Q₈	C₂²⋊D₂₃	1	1	yes	characteristic
99	Q₁₆	—	92	2	no	—
100	Q₁₆	—	92	2	no	—
101	C₂×C₈	—	2	1	no	—
102	C₂×C₈	—	23	1	no	—
103	C₂×C₈	—	23	1	no	—
104	C₂×C₈	—	46	1	no	—
105	C₂×C₈	—	46	1	no	—
106	C₄⋊C₄	—	2	1	no	—
107	C₄⋊C₄	—	2	1	no	—
108	C₄⋊C₄	—	2	1	no	—
109	C₄⋊C₄	—	46	1	no	—
110	C₄⋊C₄	—	46	1	no	—
111	C₄⋊C₄	C₂²⋊D₂₃	1	1	yes	characteristic
112	C₄²	—	2	1	no	—
113	C₄²	C₂²×D₂₃	1	1	yes	characteristic
114	Q₈	—	2	1	no	—
115	Q₈	—	4	1	no	—
116	Q₈	—	4	1	no	—
117	Q₈	—	92	1	no	—
118	Q₈	—	92	1	no	—
119	Q₈	#67275	1	2	yes	complemented
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	4	1	no	—
123	C₂×C₄	—	92	1	no	—
124	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
125	C₂×C₄	#67290	1	1	yes	characteristic
126	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
127	C₈	—	4	1	no	—
128	C₈	—	23	2	no	—
129	C₈	—	46	1	no	—
130	C₈	—	92	1	no	—
131	C₈	—	92	1	no	—
132	C₂²	#10625110	1	1	yes	characteristic
133	C₄	—	2	2	no	—
134	C₄	—	2	1	no	—
135	C₄	—	4	1	no	—
136	C₄	—	8	1	no	—
137	C₄	—	184	1	no	—
138	C₄	SD₁₆×D₂₃	1	1	yes	characteristic
139	C₄	#10625085	1	1	yes	characteristic
140	C₄	C₂×#67275	1	1	yes	characteristic
141	C₄	#10625121	1	1	yes	characteristic
142	C₂	#13092615	1	1	yes	characteristic
143	C₂	#13092656	1	1	yes	characteristic
144	C₂	#13092690	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13092796

C₂

yes

maximal, characteristic

#13092702

C₂

yes

maximal, characteristic

Q₈×C₂₃⋊C₈

C₂

yes

maximal, characteristic

#13092169

C₂

yes

maximal, characteristic

#13092163

C₂

yes

maximal, characteristic

#13092162

C₂

yes

maximal, characteristic

C₂₃×#475

C₂

yes

maximal, characteristic

C₂₃×C₄⋊Q₈

C₂²

yes

characteristic

C₄×Q₈×C₂₃

C₂²

yes

—

—

C₂²

yes

characteristic

C₂₃×#106

—

C₂₃×C₄⋊C₈

C₂²

yes

characteristic

C₂₃×#103

—

#10625017

—

#10624990

—

C₄⋊(C₂₃⋊C₈)

—

C₄⋊(C₂₃⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₂₃⋊C₈)

C₂²

yes

characteristic

C₄×C₂₃⋊C₈

—

C₄×C₂₃⋊C₈

C₂²

yes

characteristic

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

D₄

yes

characteristic

C₂×Dic₄₆

—

C₂×C₈×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

D₄

yes

characteristic

C₄²×C₂₃

—

C₄²×C₂₃

C₂³

yes

characteristic

#67275

—

#67275

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

D₄

yes

characteristic

C₂×C₂₃⋊C₈

D₄

yes

characteristic

Q₈×C₂₃

—

Q₈×C₂₃

—

Q₈×C₂₃

—

Q₈×C₂₃

Q₁₆

yes

complemented

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

#41

yes

characteristic

C₂×Dic₂₃

—

Dic₄₆

—

Dic₄₆

—

C₈×C₂₃

—

C₂₃⋊C₈

—

C₂₃⋊C₈

—

C₂₃⋊C₈

—

C₂₃⋊C₈

SD₁₆

yes

—

—

C₂²×C₂₃

yes

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

C₂×SD₁₆

yes

characteristic

C₄×C₂₃

C₂×Q₁₆

yes

C₄×C₂₃

yes

C₄×C₂₃

yes

Dic₂₃

—

#494

—

#475

—

#475

D₂₃

yes

characteristic

C₈×Q₈

—

#335

—

#332

—

#332

—

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₂₃

yes

characteristic

C₄×Q₈

C₂×D₂₃

yes

characteristic

#106

—

#106

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₂₃

yes

characteristic

#103

—

#103

—

C₄×C₈

—

C₄×C₈

—

C₂₃

#1670

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂²⋊D₂₃

yes

characteristic

Q₁₆

—

100

Q₁₆

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

C₂²⋊D₂₃

yes

characteristic

112

C₄²

—

113

C₄²

C₂²×D₂₃

yes

characteristic

114

Q₈

—

115

Q₈

—

116

Q₈

—

117

Q₈

—

118

Q₈

—

119

Q₈

#67275

yes

complemented

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

D₄×D₂₃

yes

characteristic

125

C₂×C₄

#67290

yes

characteristic

126

C₂×C₄

C₂×C₂²⋊D₂₃

yes

characteristic

127

C₈

—

128

C₈

—

129

C₈

—

130

C₈

—

131

C₈

—

132

C₂²

#10625110

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

—

137

C₄

—

184

—

138

C₄

SD₁₆×D₂₃

yes

characteristic

139

C₄

#10625085

yes

characteristic

140

C₄

C₂×#67275

yes

characteristic

141

C₄

#10625121

yes

characteristic

142

C₂

#13092615

yes

characteristic

143

C₂

#13092656

yes

characteristic

144

C₂

#13092690

yes

characteristic

	subgroup	G/H
center	C₂²	#10625110
commutator, cosocle	C₄²×C₂₃	C₂³
Frattini	C₄²	C₂²×D₂₃
Fitting	C₂₃×#475	C₂
socle, prime radical	C₂²×C₂₃	#121
hypercenter	#475	D₂₃
hypocenter	C₂₃	#1670
LCS	G ▷ C₄²×C₂₃ ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #475

subgroup

G/H

center

C₂²

#10625110

commutator, cosocle

C₄²×C₂₃

C₂³

Frattini

C₄²

C₂²×D₂₃

Fitting

C₂₃×#475

C₂

C₂²×C₂₃

D₂₃

C₂₃

G ▷ C₄²×C₂₃ ▷ C₂²×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #475

N	H ≅ G/N	type
44. Q₈×C₂₃	Q₁₆	semidirect
95. C₂₃	#1670	semidirect
119. Q₈	#67275	semidirect

H ≅ G/N

type

44. Q₈×C₂₃

Q₁₆

semidirect

95. C₂₃

#1670

semidirect

119. Q₈

#67275

semidirect

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

subgroup	index
#13092796	2
#13092702	2
Q₈×C₂₃⋊C₈	2
#13092169	2
#13092163	2
#13092162	2
C₂₃×#475	2
#1670	23

subgroup

index

Q₈×C₂₃⋊C₈

C₂₃×#475

Subgroups — #15672765

Conjugacy classes of subgroups

Complements

Maximal subgroups