Group #15672878 — subgroups

Subgroups — #15672878

Order 2944.

184 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 55 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	#13092614	C₂	1	2	yes	maximal, complemented
2	#13092547	C₂	1	1	yes	maximal, characteristic, complemented
3	#13092168	C₂	1	1	yes	maximal, characteristic, complemented
4	#13092161	C₂	1	2	yes	maximal, complemented
5	C₂₃×#493	C₂	1	1	yes	maximal, characteristic
6	C₂×D₈×C₂₃	—	2	1	no	—
7	C₂×D₈×C₂₃	—	2	1	no	—
8	C₂₃×#127	C₂²	1	2	yes	—
9	C₂×#67273	—	2	2	no	—
10	C₄×Dic₄₆	C₂²	1	1	yes	characteristic
11	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
12	#10625014	—	2	2	no	—
13	#10624997	—	2	1	no	—
14	Dic₂₃⋊C₈	—	2	1	no	—
15	C₄⋊(C₂₃⋊C₈)	C₂²	1	2	yes	—
16	C₄⋊(C₂₃⋊C₈)	C₂²	1	1	yes	characteristic
17	C₂×D₄×C₂₃	—	2	2	no	—
18	C₂×D₄×C₂₃	—	2	2	no	—
19	C₂×D₄×C₂₃	—	2	2	no	—
20	C₂×Dic₄₆	D₄	1	1	yes	characteristic
21	D₈×C₂₃	—	4	2	no	—
22	D₈×C₂₃	—	4	2	no	—
23	C₂×C₈×C₂₃	D₄	1	1	yes	characteristic
24	C₂×C₈×C₂₃	D₄	1	1	yes	characteristic
25	C₄²×C₂₃	C₂³	1	1	yes	characteristic
26	#67273	—	4	4	no	—
27	C₄⋊Dic₂₃	D₄	1	1	yes	characteristic
28	Dic₂₃⋊C₄	—	2	1	no	—
29	C₄×Dic₂₃	—	2	1	no	—
30	C₂×C₂₃⋊C₈	—	2	2	no	—
31	C₂×C₂₃⋊C₈	—	2	1	no	—
32	C₂³×C₂₃	—	4	2	no	—
33	D₄×C₂₃	—	4	2	no	—
34	D₄×C₂₃	—	4	2	no	—
35	D₄×C₂₃	—	4	2	no	—
36	D₄×C₂₃	—	4	2	no	—
37	D₄×C₂₃	—	8	2	no	—
38	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₂₃	#41	1	1	yes	characteristic
41	C₂×Dic₂₃	—	2	1	no	—
42	C₂×Dic₂₃	—	2	1	no	—
43	Dic₄₆	—	2	2	no	—
44	C₈×C₂₃	—	2	1	no	—
45	C₈×C₂₃	SD₁₆	1	2	yes	complemented
46	C₂₃⋊C₈	—	4	1	no	—
47	C₂₃⋊C₈	—	4	2	no	—
48	#1688	—	23	1	no	maximal, Sylow
49	C₂²×C₂₃	—	8	2	no	—
50	C₂²×C₂₃	—	8	2	no	—
51	C₂²×C₂₃	—	8	2	no	—
52	C₂²×C₂₃	#121	1	1	yes	characteristic
53	C₄×C₂₃	—	2	1	no	—
54	C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
55	C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
56	C₄×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
57	Dic₂₃	—	4	1	no	—
58	Dic₂₃	—	4	1	no	—
59	#493	D₂₃	1	1	yes	characteristic
60	#460	—	23	2	no	—
61	#446	—	23	1	no	—
62	#334	—	23	1	no	—
63	#331	—	23	2	no	—
64	C₂×C₂₃	—	16	2	no	—
65	C₂×C₂₃	#460	1	1	yes	characteristic
66	C₂×C₂₃	#469	1	1	yes	characteristic
67	C₂×C₂₃	#472	1	1	yes	characteristic
68	C₂×SD₁₆	—	46	2	no	—
69	C₂×D₈	—	2	1	no	—
70	C₂×D₈	—	2	1	no	—
71	#127	C₂×D₂₃	1	2	yes	—
72	C₄×Q₈	—	23	1	no	—
73	C₄⋊C₈	—	23	1	no	—
74	C₄⋊C₈	—	23	2	no	—
75	C₄⋊C₈	—	46	1	no	—
76	#102	—	46	2	no	—
77	#97	—	46	1	no	—
78	C₄×C₈	C₂×D₂₃	1	1	yes	characteristic
79	C₂₃	#1688	1	1	yes	characteristic, complemented, Sylow
80	C₂×Q₈	—	23	1	no	—
81	C₂×D₄	—	2	2	no	—
82	C₂×D₄	—	2	2	no	—
83	C₂×D₄	—	2	2	no	—
84	SD₁₆	—	92	4	no	—
85	D₈	—	4	2	no	—
86	D₈	—	4	2	no	—
87	C₂×C₈	—	46	2	no	—
88	C₂×C₈	—	46	1	no	—
89	C₂×C₈	C₂²⋊D₂₃	1	1	yes	characteristic
90	C₂×C₈	C₂²⋊D₂₃	1	1	yes	characteristic
91	C₄⋊C₄	—	23	1	no	—
92	C₄⋊C₄	—	46	1	no	—
93	C₄²	—	46	1	no	—
94	C₄²	C₂²×D₂₃	1	1	yes	characteristic
95	C₂³	—	4	2	no	—
96	Q₈	—	46	2	no	—
97	D₄	—	4	2	no	—
98	D₄	—	4	2	no	—
99	D₄	—	4	2	no	—
100	D₄	—	4	2	no	—
101	D₄	—	8	2	no	—
102	C₂×C₄	—	46	1	no	—
103	C₂×C₄	—	46	1	no	—
104	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
105	C₂×C₄	#67290	1	1	yes	characteristic
106	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
107	C₈	—	2	1	no	—
108	C₈	—	92	2	no	—
109	C₈	—	92	1	no	—
110	C₈	#67273	1	2	yes	complemented
111	C₂²	—	8	2	no	—
112	C₂²	—	8	2	no	—
113	C₂²	—	8	2	no	—
114	C₂²	#10625109	1	1	yes	characteristic
115	C₄	—	2	1	no	—
116	C₄	—	92	1	no	—
117	C₄	—	92	1	no	—
118	C₄	#10625082	1	2	yes	—
119	C₄	#10625103	1	1	yes	characteristic
120	C₄	C₂×#67273	1	1	yes	characteristic
121	C₂	—	16	2	no	—
122	C₂	#13092614	1	1	yes	characteristic
123	C₂	#13092667	1	1	yes	characteristic
124	C₂	#13092684	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13092614

C₂

yes

maximal, complemented

#13092547

C₂

yes

maximal, characteristic, complemented

#13092168

C₂

yes

maximal, characteristic, complemented

#13092161

C₂

yes

maximal, complemented

C₂₃×#493

C₂

yes

maximal, characteristic

C₂×D₈×C₂₃

—

C₂×D₈×C₂₃

—

C₂₃×#127

C₂²

yes

—

C₂×#67273

—

C₄×Dic₄₆

C₂²

yes

characteristic

C₄×C₈×C₂₃

C₂²

yes

characteristic

#10625014

—

#10624997

—

Dic₂₃⋊C₈

—

C₄⋊(C₂₃⋊C₈)

C₂²

yes

—

C₄⋊(C₂₃⋊C₈)

C₂²

yes

characteristic

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

C₂×Dic₄₆

D₄

yes

characteristic

D₈×C₂₃

—

D₈×C₂₃

—

C₂×C₈×C₂₃

D₄

yes

characteristic

C₂×C₈×C₂₃

D₄

yes

characteristic

C₄²×C₂₃

C₂³

yes

characteristic

#67273

—

C₄⋊Dic₂₃

D₄

yes

characteristic

Dic₂₃⋊C₄

—

C₄×Dic₂₃

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂³×C₂₃

—

D₄×C₂₃

—

D₄×C₂₃

—

D₄×C₂₃

—

D₄×C₂₃

—

D₄×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

—

C₈×C₂₃

—

C₈×C₂₃

SD₁₆

yes

complemented

C₂₃⋊C₈

—

C₂₃⋊C₈

—

#1688

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

#121

yes

characteristic

C₄×C₂₃

—

C₄×C₂₃

C₂×SD₁₆

yes

characteristic

C₄×C₂₃

AGL(1,ℤ/8)

yes

—

C₄×C₂₃

AGL(1,ℤ/8)

yes

characteristic

Dic₂₃

—

Dic₂₃

—

#493

D₂₃

yes

—

—

—

—

C₂×C₂₃

—

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×SD₁₆

—

C₂×D₈

—

C₂×D₈

—

#127

C₂×D₂₃

yes

—

C₄×Q₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

#102

—

#97

—

C₄×C₈

C₂×D₂₃

yes

characteristic

C₂₃

#1688

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

SD₁₆

—

D₈

—

D₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊D₂₃

yes

characteristic

C₂×C₈

C₂²⋊D₂₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

C₂²×D₂₃

yes

characteristic

C₂³

—

Q₈

—

D₄

—

D₄

—

D₄

—

100

D₄

—

101

D₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

D₄×D₂₃

yes

characteristic

105

C₂×C₄

#67290

yes

characteristic

106

C₂×C₄

C₂×C₂²⋊D₂₃

yes

characteristic

107

C₈

—

108

C₈

—

109

C₈

—

110

C₈

#67273

yes

complemented

111

C₂²

—

112

C₂²

—

113

C₂²

—

114

C₂²

#10625109

yes

characteristic

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

#10625082

yes

—

119

C₄

#10625103

yes

characteristic

120

C₄

C₂×#67273

yes

characteristic

121

C₂

—

122

C₂

#13092614

yes

characteristic

123

C₂

#13092667

yes

characteristic

124

C₂

#13092684

yes

characteristic

	subgroup	G/H
center	C₂²	#10625109
commutator, cosocle	C₄²×C₂₃	C₂³
Frattini	C₄²	C₂²×D₂₃
Fitting, prime radical	C₂₃×#493	C₂
socle	C₂²×C₂₃	#121
hypercenter	#493	D₂₃
hypocenter	C₂₃	#1688
LCS	G ▷ C₄²×C₂₃ ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #493

subgroup

G/H

center

C₂²

#10625109

commutator, cosocle

C₄²×C₂₃

C₂³

Frattini

C₄²

C₂²×D₂₃

Fitting, prime radical

C₂₃×#493

C₂

C₂²×C₂₃

D₂₃

C₂₃

G ▷ C₄²×C₂₃ ▷ C₂²×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #493

N	H ≅ G/N	type
1. #13092614	C₂	semidirect
2. #13092547	C₂	semidirect
3. #13092168	C₂	semidirect
4. #13092161	C₂	semidirect
45. C₈×C₂₃	SD₁₆	semidirect
79. C₂₃	#1688	semidirect
110. C₈	#67273	semidirect

H ≅ G/N

type

1. #13092614

C₂

semidirect

2. #13092547

C₂

semidirect

3. #13092168

C₂

semidirect

4. #13092161

C₂

semidirect

45. C₈×C₂₃

SD₁₆

semidirect

79. C₂₃

#1688

semidirect

110. C₈

#67273

semidirect

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

subgroup	index
#13092614	2
#13092547	2
#13092168	2
#13092161	2
C₂₃×#493	2
#1688	23

subgroup

index

C₂₃×#493

Subgroups — #15672878

Conjugacy classes of subgroups

Complements

Maximal subgroups