Group #15672914 — subgroups

Subgroups — #15672914

Order 2944.

152 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 51 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	#13092820	C₂	1	1	yes	maximal, characteristic
2	#13092628	C₂	1	1	yes	maximal, characteristic
3	#13092624	C₂	1	1	yes	maximal, characteristic
4	#13092544	C₂	1	1	yes	maximal, characteristic
5	#13092167	C₂	1	1	yes	maximal, characteristic
6	#13092162	C₂	1	1	yes	maximal, characteristic
7	C₂₃×#332	C₂	1	1	yes	maximal, characteristic
8	C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	C₂×#67275	—	2	1	no	—
10	C₂×Dic₉₂	—	2	1	no	—
11	#10625040	C₂²	1	1	yes	characteristic
12	C₄×Dic₄₆	C₂²	1	1	yes	characteristic
13	C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
15	#10625017	—	2	1	no	—
16	#10624998	—	2	1	no	—
17	#10624997	—	2	1	no	—
18	Dic₂₃⋊C₈	—	2	1	no	—
19	#10624989	—	2	1	no	—
20	#10624989	—	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	—	2	1	no	—
24	C₂×Dic₄₆	—	2	1	no	—
25	C₂×Dic₄₆	D₄	1	1	yes	characteristic
26	C₂×C₈×C₂₃	—	2	1	no	—
27	C₂×C₈×C₂₃	—	2	1	no	—
28	C₄⋊C₄×C₂₃	—	2	1	no	—
29	C₄⋊C₄×C₂₃	—	2	1	no	—
30	C₄²×C₂₃	C₂³	1	1	yes	characteristic
31	#67275	—	4	2	no	—
32	C₄⋊Dic₂₃	—	2	1	no	—
33	C₄⋊Dic₂₃	—	2	1	no	—
34	C₄⋊Dic₂₃	D₄	1	1	yes	characteristic
35	Dic₂₃⋊C₄	—	2	1	no	—
36	C₄×Dic₂₃	—	2	1	no	—
37	C₂×C₂₃⋊C₈	—	2	1	no	—
38	C₂×C₂₃⋊C₈	D₄	1	1	yes	characteristic
39	C₂×C₂₃⋊C₈	D₄	1	1	yes	characteristic
40	Dic₉₂	—	4	2	no	—
41	Q₈×C₂₃	—	4	1	no	—
42	Q₈×C₂₃	—	4	1	no	—
43	C₂×C₄×C₂₃	—	4	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	—	2	1	no	—
48	C₂×Dic₂₃	—	2	1	no	—
49	C₂×Dic₂₃	—	4	1	no	—
50	Dic₄₆	—	2	1	no	—
51	Dic₄₆	—	2	1	no	—
52	Dic₄₆	—	4	1	no	—
53	Dic₄₆	—	4	1	no	—
54	C₈×C₂₃	—	4	1	no	—
55	C₈×C₂₃	—	4	1	no	—
56	C₂₃⋊C₈	—	2	1	no	—
57	C₂₃⋊C₈	—	4	1	no	—
58	C₂₃⋊C₈	Q₁₆	1	2	yes	complemented
59	#1694	—	23	1	no	maximal, Sylow
60	C₂²×C₂₃	#121	1	1	yes	characteristic
61	C₄×C₂₃	—	2	1	no	—
62	C₄×C₂₃	—	8	1	no	—
63	C₄×C₂₃	C₂×Q₁₆	1	1	yes	characteristic
64	C₄×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
65	C₄×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
66	C₄×C₂₃	#137	1	1	yes	characteristic
67	Dic₂₃	—	4	1	no	—
68	Dic₂₃	—	4	1	no	—
69	Dic₂₃	—	8	1	no	—
70	#500	—	23	1	no	—
71	#462	—	23	1	no	—
72	#462	—	23	1	no	—
73	#446	—	23	1	no	—
74	#334	—	23	1	no	—
75	#332	—	23	1	no	—
76	#332	D₂₃	1	1	yes	characteristic
77	C₂×C₂₃	#462	1	1	yes	characteristic
78	C₂×C₂₃	#469	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	—	23	1	no	—
85	#107	—	46	1	no	—
86	#107	—	46	1	no	—
87	C₄⋊C₈	—	23	1	no	—
88	C₄⋊C₈	—	46	1	no	—
89	C₄⋊C₈	C₂×D₂₃	1	1	yes	characteristic
90	C₄⋊C₈	C₂×D₂₃	1	1	yes	characteristic
91	#103	—	46	1	no	—
92	#103	—	46	1	no	—
93	#97	—	46	1	no	—
94	C₄×C₈	—	23	1	no	—
95	C₂₃	#1694	1	1	yes	characteristic, complemented, Sylow
96	C₂×Q₈	—	2	1	no	—
97	C₂×Q₈	—	23	1	no	—
98	C₂×Q₈	—	46	1	no	—
99	Q₁₆	—	92	2	no	—
100	Q₁₆	—	92	2	no	—
101	C₂×C₈	—	2	1	no	—
102	C₂×C₈	—	2	1	no	—
103	C₂×C₈	—	23	1	no	—
104	C₂×C₈	—	23	1	no	—
105	C₂×C₈	—	46	1	no	—
106	C₄⋊C₄	—	2	1	no	—
107	C₄⋊C₄	—	2	1	no	—
108	C₄⋊C₄	—	23	1	no	—
109	C₄⋊C₄	—	46	1	no	—
110	C₄⋊C₄	—	46	1	no	—
111	C₄⋊C₄	—	46	1	no	—
112	C₄²	—	46	1	no	—
113	C₄²	C₂²×D₂₃	1	1	yes	characteristic
114	Q₈	—	4	1	no	—
115	Q₈	—	4	1	no	—
116	Q₈	—	46	1	no	—
117	Q₈	—	46	1	no	—
118	Q₈	—	92	1	no	—
119	Q₈	—	92	1	no	—
120	C₂×C₄	—	4	1	no	—
121	C₂×C₄	—	46	1	no	—
122	C₂×C₄	—	46	1	no	—
123	C₂×C₄	—	92	1	no	—
124	C₂×C₄	#67288	1	1	yes	characteristic
125	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
126	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
127	C₈	—	4	1	no	—
128	C₈	—	4	1	no	—
129	C₈	—	23	2	no	—
130	C₈	—	46	1	no	—
131	C₈	—	92	1	no	—
132	C₂²	#10625055	1	1	yes	characteristic
133	C₄	—	2	1	no	—
134	C₄	—	8	1	no	—
135	C₄	—	92	1	no	—
136	C₄	—	92	1	no	—
137	C₄	—	184	1	no	—
138	C₄	#10625082	1	1	yes	characteristic
139	C₄	#10625085	1	1	yes	characteristic
140	C₄	#10625086	1	1	yes	characteristic
141	C₄	Q₁₆×D₂₃	1	1	yes	characteristic
142	C₂	#13092626	1	1	yes	characteristic
143	C₂	#13092669	1	1	yes	characteristic
144	C₂	#13092691	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13092820

C₂

yes

maximal, characteristic

#13092628

C₂

yes

maximal, characteristic

#13092624

C₂

yes

maximal, characteristic

#13092544

C₂

yes

maximal, characteristic

#13092167

C₂

yes

maximal, characteristic

#13092162

C₂

yes

maximal, characteristic

C₂₃×#332

C₂

yes

maximal, characteristic

C₂₃×C₄⋊Q₈

C₂²

yes

characteristic

C₂×#67275

—

C₂×Dic₉₂

—

#10625040

C₂²

yes

characteristic

C₄×Dic₄₆

C₂²

yes

characteristic

C₂₃×C₄⋊C₈

C₂²

yes

characteristic

C₂₃×C₄⋊C₈

C₂²

yes

—

—

—

Dic₂₃⋊C₈

—

#10624989

—

#10624989

—

C₄⋊(C₂₃⋊C₈)

C₂²

yes

characteristic

C₄×C₂₃⋊C₈

C₂²

yes

characteristic

C₂×Q₈×C₂₃

—

C₂×Dic₄₆

—

C₂×Dic₄₆

D₄

yes

characteristic

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄²×C₂₃

C₂³

yes

characteristic

#67275

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

D₄

yes

characteristic

Dic₂₃⋊C₄

—

C₄×Dic₂₃

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

D₄

yes

characteristic

C₂×C₂₃⋊C₈

D₄

yes

characteristic

Dic₉₂

—

Q₈×C₂₃

—

Q₈×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

—

C₂×Dic₂₃

—

Dic₄₆

—

Dic₄₆

—

Dic₄₆

—

Dic₄₆

—

C₈×C₂₃

—

C₈×C₂₃

—

C₂₃⋊C₈

—

C₂₃⋊C₈

—

C₂₃⋊C₈

Q₁₆

yes

—

C₂²×C₂₃

yes

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

C₂×Q₁₆

yes

C₄×C₂₃

yes

C₄×C₂₃

yes

C₄×C₂₃

yes

Dic₂₃

—

Dic₂₃

—

Dic₂₃

—

—

—

—

—

—

—

D₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₂₃

yes

characteristic

C₄×Q₈

—

#107

—

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₂₃

yes

characteristic

C₄⋊C₈

C₂×D₂₃

yes

—

—

—

C₄×C₈

—

C₂₃

#1694

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

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

—

112

C₄²

—

113

C₄²

C₂²×D₂₃

yes

characteristic

114

Q₈

—

115

Q₈

—

116

Q₈

—

117

Q₈

—

118

Q₈

—

119

Q₈

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

#67288

yes

characteristic

125

C₂×C₄

D₄×D₂₃

yes

characteristic

126

C₂×C₄

D₄×D₂₃

yes

characteristic

127

C₈

—

128

C₈

—

129

C₈

—

130

C₈

—

131

C₈

—

132

C₂²

#10625055

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

—

137

C₄

—

184

—

138

C₄

#10625082

yes

characteristic

139

C₄

#10625085

yes

characteristic

140

C₄

#10625086

yes

characteristic

141

C₄

Q₁₆×D₂₃

yes

characteristic

142

C₂

#13092626

yes

characteristic

143

C₂

#13092669

yes

characteristic

144

C₂

#13092691

yes

characteristic

	subgroup	G/H
center	C₂²	#10625055
commutator, cosocle	C₄²×C₂₃	C₂³
Frattini	C₄²	C₂²×D₂₃
Fitting	C₂₃×#332	C₂
socle, prime radical	C₂²×C₂₃	#121
hypercenter	#332	D₂₃
hypocenter	C₂₃	#1694
LCS	G ▷ C₄²×C₂₃ ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #332

subgroup

G/H

center

C₂²

#10625055

commutator, cosocle

C₄²×C₂₃

C₂³

Frattini

C₄²

C₂²×D₂₃

Fitting

C₂₃×#332

C₂

C₂²×C₂₃

D₂₃

C₂₃

G ▷ C₄²×C₂₃ ▷ C₂²×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #332

N	H ≅ G/N	type
58. C₂₃⋊C₈	Q₁₆	semidirect
95. C₂₃	#1694	semidirect

H ≅ G/N

type

58. C₂₃⋊C₈

Q₁₆

semidirect

95. C₂₃

#1694

semidirect

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

subgroup	index
#13092820	2
#13092628	2
#13092624	2
#13092544	2
#13092167	2
#13092162	2
C₂₃×#332	2
#1694	23

subgroup

index

C₂₃×#332

Subgroups — #15672914

Conjugacy classes of subgroups

Complements

Maximal subgroups