Group #15671247 — subgroups

Subgroups — #15671247

Order 2944.

160 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 81 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	#13092470	C₂	1	1	yes	maximal, characteristic
2	C₄×#67267	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₈×C₂₃	C₄	1	2	yes	—
5	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
6	C₂×#67267	C₄	1	2	yes	—
7	C₂₃×C₄⋊C₈	—	2	1	no	—
8	C₂₃×C₄⋊C₈	—	2	1	no	—
9	C₂²⋊(C₂₃⋊C₈)	C₄	1	2	yes	—
10	C₄⋊(C₂₃⋊C₈)	—	2	1	no	—
11	#10624985	—	2	1	no	—
12	C₄×C₂₃⋊C₈	—	2	1	no	—
13	C₄×C₂₃⋊C₈	—	2	1	no	—
14	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
17	C₂×C₈×C₂₃	—	2	2	no	—
18	C₂×C₈×C₂₃	—	4	2	no	—
19	C₄²×C₂₃	D₄	1	1	yes	characteristic
20	C₄²×C₂₃	D₄	1	1	yes	characteristic
21	C₄²×C₂₃	D₄	1	1	yes	characteristic
22	C₄²×C₂₃	Q₈	1	1	yes	characteristic
23	#67267	—	2	2	no	—
24	#67267	C₈	1	4	yes	complemented
25	C₂×C₂₃⋊C₈	—	2	2	no	—
26	C₂×C₂₃⋊C₈	—	2	2	no	—
27	C₂³×C₂₃	C₄²	1	1	yes	characteristic
28	C₂×C₄×C₂₃	—	2	1	no	—
29	C₂×C₄×C₂₃	—	2	1	no	—
30	C₂×C₄×C₂₃	—	4	1	no	—
31	C₂×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
32	C₂×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₄×C₂₃	C₂×C₈	1	2	yes	—
38	C₂×C₄×C₂₃	M₁₆	1	2	yes	—
39	C₈×C₂₃	—	4	2	no	—
40	C₂₃⋊C₈	—	2	4	no	—
41	C₂₃⋊C₈	—	4	2	no	—
42	#1283	—	23	1	no	maximal, Sylow
43	C₂²×C₂₃	—	2	1	no	—
44	C₂²×C₂₃	—	2	1	no	—
45	C₂²×C₂₃	#95	1	1	yes	characteristic
46	C₂²×C₂₃	C₄×C₈	1	1	yes	characteristic
47	C₂²×C₂₃	#97	1	1	yes	characteristic
48	C₄×C₂₃	—	2	1	no	—
49	C₄×C₂₃	—	2	1	no	—
50	C₄×C₂₃	—	2	1	no	—
51	C₄×C₂₃	C₂²⋊C₈	1	2	yes	—
52	C₄×C₂₃	#100	1	1	yes	characteristic
53	C₄×C₂₃	#101	1	1	yes	characteristic
54	C₄×C₂₃	C₄⋊C₈	1	2	yes	—
55	#431	—	23	1	no	—
56	C₂×C₄⋊C₈	D₂₃	1	1	yes	characteristic
57	C₄×M₁₆	—	23	1	no	—
58	C₂×C₂₃	—	2	1	no	—
59	C₂×C₂₃	—	2	1	no	—
60	C₂×C₂₃	#336	1	1	yes	characteristic
61	C₂×C₂₃	#343	1	1	yes	characteristic
62	C₂×C₂₃	#344	1	1	yes	characteristic
63	C₂×M₁₆	—	23	2	no	—
64	C₂²×C₈	Dic₂₃	1	2	yes	—
65	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
66	C₄⋊C₈	—	2	1	no	—
67	C₄⋊C₈	—	2	1	no	—
68	C₄⋊C₈	—	46	1	no	—
69	C₂²⋊C₈	—	23	2	no	—
70	#97	—	46	1	no	—
71	C₄×C₈	—	46	1	no	—
72	C₄×C₈	—	46	1	no	—
73	C₂₃	#1283	1	1	yes	characteristic, complemented, Sylow
74	C₂²×C₄	C₄×D₂₃	1	1	yes	characteristic
75	C₂²×C₄	C₄×D₂₃	1	1	yes	characteristic
76	C₂²×C₄	C₂×Dic₂₃	1	1	yes	characteristic
77	M₁₆	—	23	4	no	—
78	M₁₆	—	46	2	no	—
79	C₂×C₈	—	2	2	no	—
80	C₂×C₈	—	4	2	no	—
81	C₂×C₈	—	46	2	no	—
82	C₂×C₈	—	46	2	no	—
83	C₄²	Dic₄₆	1	1	yes	characteristic
84	C₄²	D₉₂	1	1	yes	characteristic
85	C₄²	C₂²⋊D₂₃	1	1	yes	characteristic
86	C₄²	C₂²⋊D₂₃	1	1	yes	characteristic
87	C₂³	C₄×Dic₂₃	1	1	yes	characteristic
88	C₂×C₄	—	2	1	no	—
89	C₂×C₄	—	2	1	no	—
90	C₂×C₄	—	4	1	no	—
91	C₂×C₄	C₈×D₂₃	1	2	yes	—
92	C₂×C₄	#67262	1	2	yes	—
93	C₂×C₄	Dic₂₃⋊C₄	1	1	yes	characteristic
94	C₂×C₄	Dic₂₃⋊C₄	1	1	yes	characteristic
95	C₂×C₄	C₄⋊Dic₂₃	1	1	yes	characteristic
96	C₂×C₄	(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
97	C₂×C₄	(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
98	C₂×C₄	C₂²⋊Dic₂₃	1	1	yes	characteristic
99	C₈	—	4	2	no	—
100	C₈	—	46	4	no	—
101	C₈	—	92	2	no	—
102	C₂²	—	2	1	no	—
103	C₂²	—	2	1	no	—
104	C₂²	C₈×Dic₂₃	1	1	yes	characteristic
105	C₂²	#10624997	1	1	yes	characteristic
106	C₂²	C₂₃⋊#95	1	1	yes	characteristic
107	C₄	—	2	1	no	—
108	C₄	—	2	1	no	—
109	C₄	—	2	1	no	—
110	C₄	Dic₂₃⋊C₈	1	2	yes	—
111	C₄	(C₂×D₂₃)⋊C₈	1	2	yes	—
112	C₄	#10625015	1	1	yes	characteristic
113	C₄	#10625018	1	1	yes	characteristic
114	C₂	—	2	1	no	—
115	C₂	—	2	1	no	—
116	C₂	#13092171	1	1	yes	characteristic
117	C₂	#13092186	1	1	yes	characteristic
118	C₂	#13092188	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13092470

C₂

yes

maximal, characteristic

C₄×#67267

C₂

yes

maximal, characteristic

C₂×C₂₃×C₄⋊C₈

C₂

yes

maximal, characteristic

C₂²×C₈×C₂₃

C₄

yes

—

C₂×C₄²×C₂₃

C₂²

yes

characteristic

C₂×#67267

C₄

yes

—

C₂₃×C₄⋊C₈

—

C₂₃×C₄⋊C₈

—

C₂²⋊(C₂₃⋊C₈)

C₄

yes

—

C₄⋊(C₂₃⋊C₈)

—

#10624985

—

C₄×C₂₃⋊C₈

—

C₄×C₂₃⋊C₈

—

C₂²×C₄×C₂₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₃

C₂×C₄

yes

characteristic

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₄²×C₂₃

D₄

yes

characteristic

C₄²×C₂₃

D₄

yes

characteristic

C₄²×C₂₃

D₄

yes

characteristic

C₄²×C₂₃

Q₈

yes

characteristic

#67267

—

#67267

C₈

yes

complemented

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂³×C₂₃

C₄²

yes

characteristic

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₂₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₂₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₂₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₂₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₂₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₂₃

C₂×C₈

yes

—

C₂×C₄×C₂₃

M₁₆

yes

—

C₈×C₂₃

—

C₂₃⋊C₈

—

C₂₃⋊C₈

—

#1283

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

#95

yes

characteristic

C₂²×C₂₃

C₄×C₈

yes

characteristic

C₂²×C₂₃

#97

yes

characteristic

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

C₂²⋊C₈

yes

—

C₄×C₂₃

yes

C₄×C₂₃

yes

C₄×C₂₃

C₄⋊C₈

yes

—

#431

—

C₂×C₄⋊C₈

D₂₃

yes

characteristic

C₄×M₁₆

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×M₁₆

—

C₂²×C₈

Dic₂₃

yes

—

C₂×C₄²

C₂×D₂₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₂₃

#1283

yes

characteristic, complemented, Sylow

C₂²×C₄

C₄×D₂₃

yes

characteristic

C₂²×C₄

C₄×D₂₃

yes

characteristic

C₂²×C₄

C₂×Dic₂₃

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄²

Dic₄₆

yes

characteristic

C₄²

D₉₂

yes

characteristic

C₄²

C₂²⋊D₂₃

yes

characteristic

C₄²

C₂²⋊D₂₃

yes

characteristic

C₂³

C₄×Dic₂₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₈×D₂₃

yes

—

C₂×C₄

#67262

yes

—

C₂×C₄

Dic₂₃⋊C₄

yes

characteristic

C₂×C₄

Dic₂₃⋊C₄

yes

characteristic

C₂×C₄

C₄⋊Dic₂₃

yes

characteristic

C₂×C₄

(C₂×D₂₃)⋊C₄

yes

characteristic

C₂×C₄

(C₂×D₂₃)⋊C₄

yes

characteristic

C₂×C₄

C₂²⋊Dic₂₃

yes

characteristic

C₈

—

100

C₈

—

101

C₈

—

102

C₂²

—

103

C₂²

—

104

C₂²

C₈×Dic₂₃

yes

characteristic

105

C₂²

#10624997

yes

characteristic

106

C₂²

C₂₃⋊#95

yes

characteristic

107

C₄

—

108

C₄

—

109

C₄

—

110

C₄

Dic₂₃⋊C₈

yes

—

111

C₄

(C₂×D₂₃)⋊C₈

yes

—

112

C₄

#10625015

yes

characteristic

113

C₄

#10625018

yes

characteristic

114

C₂

—

115

C₂

—

116

C₂

#13092171

yes

characteristic

117

C₂

#13092186

yes

characteristic

118

C₂

#13092188

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊Dic₂₃
commutator	C₂²×C₂₃	C₄×C₈
Frattini	C₂×C₄²	C₂×D₂₃
Fitting	C₂×C₂₃×C₄⋊C₈	C₂
socle	C₂²×C₂₃	#95
prime radical	C₂³×C₂₃	C₄²
cosocle	C₂×C₄²×C₂₃	C₂²
hypercenter	C₂×C₄⋊C₈	D₂₃
hypocenter	C₂₃	#1283
LCS	G ▷ C₂²×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂×C₄

C₂²⋊Dic₂₃

commutator

C₂²×C₂₃

C₄×C₈

Frattini

C₂×C₄²

C₂×D₂₃

Fitting

C₂×C₂₃×C₄⋊C₈

C₂

socle

C₂²×C₂₃

#95

prime radical

C₂³×C₂₃

C₄²

cosocle

C₂×C₄²×C₂₃

C₂²

hypercenter

C₂×C₄⋊C₈

D₂₃

hypocenter

C₂₃

#1283

LCS

G ▷ C₂²×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
24. #67267	C₈	semidirect
73. C₂₃	#1283	semidirect

H ≅ G/N

type

24. #67267

C₈

semidirect

73. C₂₃

#1283

semidirect

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

subgroup	index
#13092470	2
C₄×#67267	2
C₂×C₂₃×C₄⋊C₈	2
#1283	23

subgroup

index

#13092470

C₄×#67267

C₂×C₂₃×C₄⋊C₈

#1283

Subgroups — #15671247

Conjugacy classes of subgroups

Complements

Maximal subgroups