Group #15672858 — subgroups

Subgroups — #15672858

Order 2944.

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

normal subgroups only

This group has 144 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	#13092769	C₂	1	1	yes	maximal, characteristic
2	#13092709	C₂	1	1	yes	maximal, characteristic
3	#13092703	C₂	1	1	yes	maximal, characteristic
4	#13092541	C₂	1	1	yes	maximal, characteristic
5	#13092169	C₂	1	1	yes	maximal, characteristic
6	#13092164	C₂	1	1	yes	maximal, characteristic
7	C₂₃×#333	C₂	1	1	yes	maximal, characteristic
8	C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	#10625040	C₂²	1	1	yes	characteristic
10	C₄×Dic₄₆	C₂²	1	1	yes	characteristic
11	C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
12	C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
13	#10624999	—	2	1	no	—
14	#10624998	—	2	1	no	—
15	Dic₂₃⋊C₈	—	2	1	no	—
16	C₈×Dic₂₃	—	2	1	no	—
17	#10624992	—	2	1	no	—
18	#10624992	—	2	1	no	—
19	#10624992	—	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	C₄⋊Dic₂₃	—	2	1	no	—
32	C₄⋊Dic₂₃	—	2	1	no	—
33	C₄⋊Dic₂₃	D₄	1	1	yes	characteristic
34	Dic₂₃⋊C₄	—	2	1	no	—
35	C₄×Dic₂₃	—	2	1	no	—
36	C₂×C₂₃⋊C₈	—	2	1	no	—
37	C₂×C₂₃⋊C₈	D₄	1	1	yes	characteristic
38	C₂×C₂₃⋊C₈	D₄	1	1	yes	characteristic
39	Q₈×C₂₃	—	8	1	no	—
40	C₂×C₄×C₂₃	—	4	1	no	—
41	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₂₃	#41	1	1	yes	characteristic
44	C₂×Dic₂₃	—	2	1	no	—
45	C₂×Dic₂₃	—	2	1	no	—
46	C₂×Dic₂₃	—	4	1	no	—
47	Dic₄₆	—	2	1	no	—
48	Dic₄₆	—	4	1	no	—
49	Dic₄₆	—	4	1	no	—
50	Dic₄₆	Q₁₆	1	2	yes	—
51	C₈×C₂₃	—	4	1	no	—
52	C₈×C₂₃	—	4	1	no	—
53	C₂₃⋊C₈	—	2	1	no	—
54	C₂₃⋊C₈	—	2	1	no	—
55	C₂₃⋊C₈	—	4	1	no	—
56	#1684	—	23	1	no	maximal, Sylow
57	C₂²×C₂₃	#121	1	1	yes	characteristic
58	C₄×C₂₃	—	2	1	no	—
59	C₄×C₂₃	—	8	1	no	—
60	C₄×C₂₃	C₂×Q₁₆	1	1	yes	characteristic
61	C₄×C₂₃	#135	1	1	yes	characteristic
62	C₄×C₂₃	#135	1	1	yes	characteristic
63	C₄×C₂₃	#137	1	1	yes	characteristic
64	Dic₂₃	—	2	2	no	—
65	Dic₂₃	—	4	1	no	—
66	Dic₂₃	—	8	1	no	—
67	#487	—	23	1	no	—
68	#477	—	23	1	no	—
69	#475	—	23	1	no	—
70	C₈×Q₈	—	23	1	no	—
71	#335	—	23	1	no	—
72	#333	D₂₃	1	1	yes	characteristic
73	#332	—	23	1	no	—
74	C₂×C₂₃	#464	1	1	yes	characteristic
75	C₂×C₂₃	#467	1	1	yes	characteristic
76	C₂×C₂₃	#471	1	1	yes	characteristic
77	C₄⋊Q₈	—	23	1	no	—
78	C₄⋊Q₈	C₂×D₂₃	1	1	yes	characteristic
79	C₄×Q₈	—	23	1	no	—
80	#107	—	46	1	no	—
81	#106	—	46	1	no	—
82	C₄⋊C₈	—	23	1	no	—
83	C₄⋊C₈	—	46	1	no	—
84	C₄⋊C₈	C₂×D₂₃	1	1	yes	characteristic
85	C₄⋊C₈	C₂×D₂₃	1	1	yes	characteristic
86	#103	—	46	1	no	—
87	#103	—	46	1	no	—
88	#103	—	46	1	no	—
89	#103	—	46	1	no	—
90	C₄×C₈	—	23	1	no	—
91	C₄×C₈	—	46	1	no	—
92	C₂₃	#1684	1	1	yes	characteristic, complemented, Sylow
93	C₂×Q₈	—	2	1	no	—
94	C₂×Q₈	—	23	1	no	—
95	C₂×Q₈	—	46	1	no	—
96	C₂×C₈	—	2	1	no	—
97	C₂×C₈	—	2	1	no	—
98	C₂×C₈	—	23	1	no	—
99	C₂×C₈	—	23	1	no	—
100	C₂×C₈	—	46	1	no	—
101	C₄⋊C₄	—	2	1	no	—
102	C₄⋊C₄	—	2	1	no	—
103	C₄⋊C₄	—	23	1	no	—
104	C₄⋊C₄	—	46	1	no	—
105	C₄⋊C₄	—	46	1	no	—
106	C₄⋊C₄	—	46	1	no	—
107	C₄²	—	46	1	no	—
108	C₄²	C₂²×D₂₃	1	1	yes	characteristic
109	Q₈	—	8	1	no	—
110	Q₈	—	23	2	no	—
111	Q₈	—	46	1	no	—
112	Q₈	—	92	1	no	—
113	Q₈	—	92	1	no	—
114	C₂×C₄	—	4	1	no	—
115	C₂×C₄	—	46	1	no	—
116	C₂×C₄	—	46	1	no	—
117	C₂×C₄	—	92	1	no	—
118	C₂×C₄	#67288	1	1	yes	characteristic
119	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
120	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
121	C₈	—	4	1	no	—
122	C₈	—	4	1	no	—
123	C₈	—	46	1	no	—
124	C₈	—	46	1	no	—
125	C₈	—	92	1	no	—
126	C₂²	#10625055	1	1	yes	characteristic
127	C₄	—	2	1	no	—
128	C₄	—	8	1	no	—
129	C₄	—	46	2	no	—
130	C₄	—	92	1	no	—
131	C₄	—	184	1	no	—
132	C₄	#10625087	1	1	yes	characteristic
133	C₄	#10625087	1	1	yes	characteristic
134	C₄	Q₁₆×D₂₃	1	1	yes	characteristic
135	C₄	#10625089	1	1	yes	characteristic
136	C₂	#13092642	1	1	yes	characteristic
137	C₂	#13092657	1	1	yes	characteristic
138	C₂	#13092679	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13092769

C₂

yes

maximal, characteristic

#13092709

C₂

yes

maximal, characteristic

#13092703

C₂

yes

maximal, characteristic

#13092541

C₂

yes

maximal, characteristic

#13092169

C₂

yes

maximal, characteristic

#13092164

C₂

yes

maximal, characteristic

C₂₃×#333

C₂

yes

maximal, characteristic

C₂₃×C₄⋊Q₈

C₂²

yes

characteristic

#10625040

C₂²

yes

characteristic

C₄×Dic₄₆

C₂²

yes

characteristic

C₂₃×C₄⋊C₈

C₂²

yes

characteristic

C₂₃×C₄⋊C₈

C₂²

yes

characteristic

#10624999

—

#10624998

—

Dic₂₃⋊C₈

—

C₈×Dic₂₃

—

—

—

—

—

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

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

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

Q₁₆

yes

—

C₈×C₂₃

—

C₈×C₂₃

—

C₂₃⋊C₈

—

C₂₃⋊C₈

—

C₂₃⋊C₈

—

—

C₂²×C₂₃

yes

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

C₂×Q₁₆

yes

C₄×C₂₃

yes

C₄×C₂₃

yes

C₄×C₂₃

yes

Dic₂₃

—

Dic₂₃

—

Dic₂₃

—

#487

—

#477

—

#475

—

C₈×Q₈

—

#335

—

#333

D₂₃

yes

characteristic

#332

—

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₂₃

yes

characteristic

C₄×Q₈

—

#107

—

#106

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₂₃

yes

characteristic

C₄⋊C₈

C₂×D₂₃

yes

—

—

—

—

C₄×C₈

—

C₄×C₈

—

C₂₃

#1684

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄²

—

108

C₄²

C₂²×D₂₃

yes

characteristic

109

Q₈

—

110

Q₈

—

111

Q₈

—

112

Q₈

—

113

Q₈

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

#67288

yes

characteristic

119

C₂×C₄

D₄×D₂₃

yes

characteristic

120

C₂×C₄

D₄×D₂₃

yes

characteristic

121

C₈

—

122

C₈

—

123

C₈

—

124

C₈

—

125

C₈

—

126

C₂²

#10625055

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

184

—

132

C₄

#10625087

yes

characteristic

133

C₄

#10625087

yes

characteristic

134

C₄

Q₁₆×D₂₃

yes

characteristic

135

C₄

#10625089

yes

characteristic

136

C₂

#13092642

yes

characteristic

137

C₂

#13092657

yes

characteristic

138

C₂

#13092679

yes

characteristic

	subgroup	G/H
center	C₂²	#10625055
commutator, cosocle	C₄²×C₂₃	C₂³
Frattini	C₄²	C₂²×D₂₃
Fitting	C₂₃×#333	C₂
socle, prime radical	C₂²×C₂₃	#121
hypercenter	#333	D₂₃
hypocenter	C₂₃	#1684
LCS	G ▷ C₄²×C₂₃ ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #333

subgroup

G/H

center

C₂²

#10625055

commutator, cosocle

C₄²×C₂₃

C₂³

Frattini

C₄²

C₂²×D₂₃

Fitting

C₂₃×#333

C₂

C₂²×C₂₃

D₂₃

C₂₃

G ▷ C₄²×C₂₃ ▷ C₂²×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #333

N	H ≅ G/N	type
92. C₂₃	#1684	semidirect

H ≅ G/N

type

92. C₂₃

#1684

semidirect

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

subgroup	index
#13092769	2
#13092709	2
#13092703	2
#13092541	2
#13092169	2
#13092164	2
C₂₃×#333	2
#1684	23

subgroup

index

C₂₃×#333

Subgroups — #15672858

Conjugacy classes of subgroups

Complements

Maximal subgroups