Group #15672827 — subgroups

Subgroups — #15672827

Order 2944.

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

normal subgroups only

This group has 150 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	#13092768	C₂	1	1	yes	maximal, characteristic
2	#13092702	C₂	1	1	yes	maximal, characteristic
3	#13092627	C₂	1	1	yes	maximal, characteristic
4	#13092168	C₂	1	1	yes	maximal, characteristic
5	#13092164	C₂	1	1	yes	maximal, characteristic
6	#13092162	C₂	1	1	yes	maximal, characteristic
7	C₈×Q₈×C₂₃	C₂	1	1	yes	maximal, characteristic
8	C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
9	C₂×#67275	—	2	1	no	—
10	#10625040	C₂²	1	1	yes	characteristic
11	#10625040	C₂²	1	1	yes	characteristic
12	C₂₃×C₄⋊C₈	—	2	1	no	—
13	C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₂₃	—	2	1	no	—
15	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
16	#10625017	—	2	1	no	—
17	#10624998	—	2	1	no	—
18	#10624998	—	2	1	no	—
19	#10624992	—	2	1	no	—
20	#10624990	—	2	1	no	—
21	C₄⋊(C₂₃⋊C₈)	C₂²	1	1	yes	characteristic
22	C₄⋊(C₂₃⋊C₈)	C₂²	1	1	yes	characteristic
23	C₂×Q₈×C₂₃	D₄	1	1	yes	characteristic
24	C₂×Dic₄₆	—	2	1	no	—
25	C₂×Dic₄₆	—	2	1	no	—
26	C₂×C₈×C₂₃	—	2	1	no	—
27	C₂×C₈×C₂₃	D₄	1	1	yes	characteristic
28	C₂×C₈×C₂₃	D₄	1	1	yes	characteristic
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	C₄⋊Dic₂₃	—	2	1	no	—
35	C₄⋊Dic₂₃	—	2	1	no	—
36	C₄⋊Dic₂₃	—	2	1	no	—
37	C₄⋊Dic₂₃	—	2	1	no	—
38	C₂×C₂₃⋊C₈	—	2	1	no	—
39	C₂×C₂₃⋊C₈	—	2	1	no	—
40	Q₈×C₂₃	—	2	1	no	—
41	Q₈×C₂₃	SD₁₆	1	2	yes	—
42	C₂×C₄×C₂₃	—	2	1	no	—
43	C₂×C₄×C₂₃	—	2	1	no	—
44	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₂₃	#41	1	1	yes	characteristic
47	C₂×Dic₂₃	—	4	1	no	—
48	C₂×Dic₂₃	—	4	1	no	—
49	Dic₄₆	—	4	1	no	—
50	Dic₄₆	—	4	1	no	—
51	Dic₄₆	—	4	1	no	—
52	Dic₄₆	—	4	1	no	—
53	C₈×C₂₃	—	2	1	no	—
54	C₈×C₂₃	—	2	1	no	—
55	C₈×C₂₃	—	4	1	no	—
56	C₂₃⋊C₈	—	4	1	no	—
57	C₂₃⋊C₈	—	4	1	no	—
58	#1680	—	23	1	no	maximal, Sylow
59	C₂²×C₂₃	#121	1	1	yes	characteristic
60	C₄×C₂₃	—	2	2	no	—
61	C₄×C₂₃	—	2	1	no	—
62	C₄×C₂₃	—	4	1	no	—
63	C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
64	C₄×C₂₃	#135	1	1	yes	characteristic
65	C₄×C₂₃	#135	1	1	yes	characteristic
66	C₄×C₂₃	#137	1	1	yes	characteristic
67	Dic₂₃	—	8	1	no	—
68	Dic₂₃	—	8	1	no	—
69	#487	—	23	1	no	—
70	#475	—	23	1	no	—
71	#462	—	23	1	no	—
72	C₈×Q₈	D₂₃	1	1	yes	characteristic
73	#334	—	23	1	no	—
74	#332	—	23	1	no	—
75	#332	—	23	1	no	—
76	C₂×C₂₃	#464	1	1	yes	characteristic
77	C₂×C₂₃	#465	1	1	yes	characteristic
78	C₂×C₂₃	#473	1	1	yes	characteristic
79	C₂×Q₁₆	—	46	1	no	—
80	C₄⋊Q₈	—	23	1	no	—
81	C₄⋊Q₈	—	23	1	no	—
82	C₄×Q₈	C₂×D₂₃	1	1	yes	characteristic
83	#106	—	46	1	no	—
84	C₄⋊C₈	—	2	1	no	—
85	C₄⋊C₈	—	23	1	no	—
86	C₄⋊C₈	—	23	1	no	—
87	C₄⋊C₈	C₂×D₂₃	1	1	yes	characteristic
88	#103	—	46	1	no	—
89	#103	—	46	1	no	—
90	#103	—	46	1	no	—
91	#103	—	46	1	no	—
92	C₄×C₈	—	2	1	no	—
93	C₄×C₈	C₂×D₂₃	1	1	yes	characteristic
94	C₂₃	#1680	1	1	yes	characteristic, complemented, Sylow
95	C₂×Q₈	—	46	1	no	—
96	C₂×Q₈	—	46	1	no	—
97	C₂×Q₈	D₉₂	1	1	yes	characteristic
98	Q₁₆	—	92	2	no	—
99	C₂×C₈	—	2	1	no	—
100	C₂×C₈	—	46	1	no	—
101	C₂×C₈	—	46	1	no	—
102	C₂×C₈	C₂²⋊D₂₃	1	1	yes	characteristic
103	C₂×C₈	C₂²⋊D₂₃	1	1	yes	characteristic
104	C₄⋊C₄	—	2	1	no	—
105	C₄⋊C₄	—	46	1	no	—
106	C₄⋊C₄	—	46	1	no	—
107	C₄⋊C₄	—	46	1	no	—
108	C₄⋊C₄	—	46	1	no	—
109	C₄⋊C₄	D₉₂	1	1	yes	characteristic
110	C₄²	—	2	1	no	—
111	C₄²	C₂²×D₂₃	1	1	yes	characteristic
112	Q₈	—	2	1	no	—
113	Q₈	—	92	1	no	—
114	Q₈	—	92	1	no	—
115	Q₈	—	92	1	no	—
116	Q₈	—	92	1	no	—
117	Q₈	#67263	1	2	yes	—
118	C₂×C₄	—	2	1	no	—
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	—	92	1	no	—
121	C₂×C₄	—	92	1	no	—
122	C₂×C₄	C₂×D₉₂	1	1	yes	characteristic
123	C₂×C₄	#67288	1	1	yes	characteristic
124	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
125	C₈	—	2	1	no	—
126	C₈	—	2	1	no	—
127	C₈	—	4	1	no	—
128	C₈	—	92	1	no	—
129	C₈	—	92	1	no	—
130	C₂²	#10625101	1	1	yes	characteristic
131	C₄	—	2	2	no	—
132	C₄	—	2	1	no	—
133	C₄	—	4	1	no	—
134	C₄	—	184	1	no	—
135	C₄	—	184	1	no	—
136	C₄	C₂×#67263	1	1	yes	characteristic
137	C₄	#10625075	1	1	yes	characteristic
138	C₄	#10625113	1	1	yes	characteristic
139	C₄	#10625121	1	1	yes	characteristic
140	C₂	#13092639	1	1	yes	characteristic
141	C₂	#13092646	1	1	yes	characteristic
142	C₂	#13092692	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13092768

C₂

yes

maximal, characteristic

#13092702

C₂

yes

maximal, characteristic

#13092627

C₂

yes

maximal, characteristic

#13092168

C₂

yes

maximal, characteristic

#13092164

C₂

yes

maximal, characteristic

#13092162

C₂

yes

maximal, characteristic

C₈×Q₈×C₂₃

C₂

yes

maximal, characteristic

C₄×Q₈×C₂₃

C₂²

yes

characteristic

C₂×#67275

—

#10625040

C₂²

yes

characteristic

#10625040

C₂²

yes

characteristic

C₂₃×C₄⋊C₈

—

C₂₃×C₄⋊C₈

C₂²

yes

characteristic

C₄×C₈×C₂₃

—

C₄×C₈×C₂₃

C₂²

yes

—

—

—

—

—

C₄⋊(C₂₃⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₂₃⋊C₈)

C₂²

yes

characteristic

C₂×Q₈×C₂₃

D₄

yes

characteristic

C₂×Dic₄₆

—

C₂×Dic₄₆

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

D₄

yes

characteristic

C₂×C₈×C₂₃

D₄

yes

characteristic

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

D₄

yes

characteristic

C₄²×C₂₃

—

C₄²×C₂₃

C₂³

yes

characteristic

#67275

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

Q₈×C₂₃

—

Q₈×C₂₃

SD₁₆

yes

—

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

—

C₂×Dic₂₃

—

Dic₄₆

—

Dic₄₆

—

Dic₄₆

—

Dic₄₆

—

C₈×C₂₃

—

C₈×C₂₃

—

C₈×C₂₃

—

C₂₃⋊C₈

—

C₂₃⋊C₈

—

—

C₂²×C₂₃

yes

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

C₂×SD₁₆

yes

C₄×C₂₃

yes

C₄×C₂₃

yes

C₄×C₂₃

yes

Dic₂₃

—

Dic₂₃

—

#487

—

#475

—

#462

—

C₈×Q₈

D₂₃

yes

—

—

—

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄×Q₈

C₂×D₂₃

yes

characteristic

#106

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₂₃

yes

—

—

—

—

C₄×C₈

—

C₄×C₈

C₂×D₂₃

yes

characteristic

C₂₃

#1680

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

D₉₂

yes

characteristic

Q₁₆

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

C₂²⋊D₂₃

yes

characteristic

103

C₂×C₈

C₂²⋊D₂₃

yes

characteristic

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄⋊C₄

D₉₂

yes

characteristic

110

C₄²

—

111

C₄²

C₂²×D₂₃

yes

characteristic

112

Q₈

—

113

Q₈

—

114

Q₈

—

115

Q₈

—

116

Q₈

—

117

Q₈

#67263

yes

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

C₂×D₉₂

yes

characteristic

123

C₂×C₄

#67288

yes

characteristic

124

C₂×C₄

C₂×C₂²⋊D₂₃

yes

characteristic

125

C₈

—

126

C₈

—

127

C₈

—

128

C₈

—

129

C₈

—

130

C₂²

#10625101

yes

characteristic

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

—

184

—

135

C₄

—

184

—

136

C₄

C₂×#67263

yes

characteristic

137

C₄

#10625075

yes

characteristic

138

C₄

#10625113

yes

characteristic

139

C₄

#10625121

yes

characteristic

140

C₂

#13092639

yes

characteristic

141

C₂

#13092646

yes

characteristic

142

C₂

#13092692

yes

characteristic

	subgroup	G/H
center	C₂²	#10625101
commutator, cosocle	C₄²×C₂₃	C₂³
Frattini	C₄²	C₂²×D₂₃
Fitting	C₈×Q₈×C₂₃	C₂
socle, prime radical	C₂²×C₂₃	#121
hypercenter	C₈×Q₈	D₂₃
hypocenter	C₂₃	#1680
LCS	G ▷ C₄²×C₂₃ ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈

subgroup

G/H

center

C₂²

#10625101

commutator, cosocle

C₄²×C₂₃

C₂³

Frattini

C₄²

C₂²×D₂₃

Fitting

C₈×Q₈×C₂₃

C₂

socle, prime radical

C₂²×C₂₃

#121

hypercenter

C₈×Q₈

D₂₃

hypocenter

C₂₃

#1680

LCS

G ▷ C₄²×C₂₃ ▷ C₂²×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈

N	H ≅ G/N	type
94. C₂₃	#1680	semidirect

H ≅ G/N

type

94. C₂₃

#1680

semidirect

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

subgroup	index
#13092768	2
#13092702	2
#13092627	2
#13092168	2
#13092164	2
#13092162	2
C₈×Q₈×C₂₃	2
#1680	23

subgroup

index

C₈×Q₈×C₂₃

Subgroups — #15672827

Conjugacy classes of subgroups

Complements

Maximal subgroups