Group #15675008 — subgroups

Subgroups — #15675008

Order 2944.

260 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 117 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	C₂×#10624994	C₂	1	2	yes	maximal, complemented
2	C₂²⋊(C₂₃⋊C₁₆)	C₂	1	4	yes	maximal, complemented
3	C₂³×C₈×C₂₃	C₂	1	1	yes	maximal, characteristic
4	C₂³×C₄×C₂₃	C₄	1	1	yes	characteristic
5	C₂²×C₈×C₂₃	—	2	4	no	—
6	C₂²×C₈×C₂₃	C₄	1	1	yes	characteristic
7	C₂²×C₈×C₂₃	C₄	1	2	yes	—
8	C₂²×C₈×C₂₃	C₂²	1	2	yes	—
9	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
10	#10624994	—	2	4	no	—
11	C₂×C₂₃⋊C₁₆	C₂²	1	4	yes	complemented
12	C₂⁴×C₂₃	C₈	1	1	yes	characteristic
13	C₂²×C₄×C₂₃	—	2	2	no	—
14	C₂²×C₄×C₂₃	—	2	2	no	—
15	C₂²×C₄×C₂₃	C₈	1	1	yes	characteristic
16	C₂²×C₄×C₂₃	C₈	1	2	yes	—
17	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
18	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₈×C₂₃	—	2	2	no	—
20	C₂×C₈×C₂₃	—	2	4	no	—
21	C₂×C₈×C₂₃	—	2	4	no	—
22	C₂×C₈×C₂₃	C₂×C₄	1	2	yes	—
23	C₂×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈×C₂₃	D₄	1	4	yes	—
25	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
26	C₂₃⋊C₁₆	—	2	4	no	—
27	C₂³×C₂₃	—	2	2	no	—
28	C₂³×C₂₃	—	2	4	no	—
29	C₂³×C₂₃	C₂×C₈	1	1	yes	characteristic
30	C₂³×C₂₃	C₂×C₈	1	2	yes	—
31	C₂×C₄×C₂₃	—	2	2	no	—
32	C₂×C₄×C₂₃	—	2	4	no	—
33	C₂×C₄×C₂₃	—	2	2	no	—
34	C₂×C₄×C₂₃	—	2	2	no	—
35	C₂×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
36	C₂×C₄×C₂₃	C₂×C₈	1	2	yes	—
37	C₂×C₄×C₂₃	C₂×C₈	1	1	yes	characteristic
38	C₂×C₄×C₂₃	M₁₆	1	2	yes	—
39	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
40	C₈×C₂₃	—	2	2	no	—
41	C₈×C₂₃	C₂²⋊C₄	1	2	yes	—
42	C₈×C₂₃	C₂×D₄	1	2	yes	—
43	#2112	—	23	1	no	maximal, Sylow
44	C₂²×C₂₃	—	2	8	no	—
45	C₂²×C₂₃	—	2	2	no	—
46	C₂²×C₂₃	—	2	2	no	—
47	C₂²×C₂₃	—	2	2	no	—
48	C₂²×C₂₃	C₂²⋊C₈	1	2	yes	—
49	C₂²×C₂₃	M₃₂	1	4	yes	complemented
50	C₂²×C₂₃	C₂²×C₈	1	1	yes	characteristic
51	C₄×C₂₃	—	2	2	no	—
52	C₄×C₂₃	C₂²⋊C₈	1	2	yes	—
53	C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
54	C₄×C₂₃	C₂×M₁₆	1	1	yes	characteristic
55	C₂³×C₈	D₂₃	1	1	yes	characteristic
56	C₂×M₃₂	—	23	2	no	—
57	C₂²⋊C₁₆	—	23	4	no	—
58	C₂×C₂₃	—	2	2	no	—
59	C₂×C₂₃	—	2	4	no	—
60	C₂×C₂₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
61	C₂×C₂₃	C₂×M₃₂	1	2	yes	—
62	C₂³×C₄	Dic₂₃	1	1	yes	characteristic
63	C₂²×C₈	—	2	4	no	—
64	C₂²×C₈	Dic₂₃	1	1	yes	characteristic
65	C₂²×C₈	Dic₂₃	1	2	yes	—
66	C₂²×C₈	C₂×D₂₃	1	2	yes	—
67	C₂²×C₈	C₂×D₂₃	1	1	yes	characteristic
68	M₃₂	—	46	4	no	—
69	C₂×C₁₆	—	23	4	no	—
70	C₂₃	#2112	1	1	yes	characteristic, complemented, Sylow
71	C₂⁴	C₂₃⋊C₈	1	1	yes	characteristic
72	C₂²×C₄	—	2	2	no	—
73	C₂²×C₄	—	2	2	no	—
74	C₂²×C₄	C₂₃⋊C₈	1	1	yes	characteristic
75	C₂²×C₄	C₂₃⋊C₈	1	2	yes	—
76	C₂²×C₄	C₂×Dic₂₃	1	2	yes	—
77	C₂²×C₄	C₂×Dic₂₃	1	1	yes	characteristic
78	C₂×C₈	—	2	2	no	—
79	C₂×C₈	—	2	4	no	—
80	C₂×C₈	—	2	4	no	—
81	C₂×C₈	C₂×Dic₂₃	1	2	yes	—
82	C₂×C₈	C₂×Dic₂₃	1	1	yes	characteristic
83	C₂×C₈	C₂²⋊D₂₃	1	4	yes	—
84	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
85	C₁₆	—	46	4	no	—
86	C₂³	—	2	4	no	—
87	C₂³	—	2	2	no	—
88	C₂³	C₂×C₂₃⋊C₈	1	2	yes	—
89	C₂³	C₂×C₂₃⋊C₈	1	1	yes	characteristic
90	C₂×C₄	—	2	2	no	—
91	C₂×C₄	—	2	2	no	—
92	C₂×C₄	—	2	2	no	—
93	C₂×C₄	—	2	4	no	—
94	C₂×C₄	C₂×C₂₃⋊C₈	1	2	yes	—
95	C₂×C₄	C₂×C₂₃⋊C₈	1	1	yes	characteristic
96	C₂×C₄	#67267	1	2	yes	—
97	C₂×C₄	C₂²⋊Dic₂₃	1	2	yes	—
98	C₂×C₄	C₂²×Dic₂₃	1	1	yes	characteristic
99	C₈	—	2	2	no	—
100	C₈	C₂²⋊Dic₂₃	1	2	yes	—
101	C₈	C₂×C₂²⋊D₂₃	1	2	yes	—
102	C₂²	—	2	8	no	—
103	C₂²	—	2	2	no	—
104	C₂²	—	2	2	no	—
105	C₂²	—	2	2	no	—
106	C₂²	#10624994	1	4	yes	complemented
107	C₂²	C₂²⋊(C₂₃⋊C₈)	1	2	yes	—
108	C₂²	C₂²×C₂₃⋊C₈	1	1	yes	characteristic
109	C₄	—	2	2	no	—
110	C₄	C₂²⋊(C₂₃⋊C₈)	1	2	yes	—
111	C₄	C₂×#67267	1	1	yes	characteristic
112	C₄	C₂×C₂²⋊Dic₂₃	1	1	yes	characteristic
113	C₂	—	2	4	no	—
114	C₂	—	2	2	no	—
115	C₂	C₂×C₂²⋊(C₂₃⋊C₈)	1	1	yes	characteristic
116	C₂	C₂×#10624994	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10624994

C₂

yes

maximal, complemented

C₂²⋊(C₂₃⋊C₁₆)

C₂

yes

maximal, complemented

C₂³×C₈×C₂₃

C₂

yes

maximal, characteristic

C₂³×C₄×C₂₃

C₄

yes

characteristic

C₂²×C₈×C₂₃

—

C₂²×C₈×C₂₃

C₄

yes

characteristic

C₂²×C₈×C₂₃

C₄

yes

—

C₂²×C₈×C₂₃

C₂²

yes

—

C₂²×C₈×C₂₃

C₂²

yes

characteristic

#10624994

—

C₂×C₂₃⋊C₁₆

C₂²

yes

complemented

C₂⁴×C₂₃

C₈

yes

characteristic

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

C₈

yes

characteristic

C₂²×C₄×C₂₃

C₈

yes

—

C₂²×C₄×C₂₃

C₂×C₄

yes

—

C₂²×C₄×C₂₃

C₂×C₄

yes

characteristic

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

C₂×C₄

yes

—

C₂×C₈×C₂₃

C₂×C₄

yes

characteristic

C₂×C₈×C₂₃

D₄

yes

—

C₂×C₈×C₂₃

C₂³

yes

characteristic

C₂₃⋊C₁₆

—

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

C₂×C₈

yes

characteristic

C₂³×C₂₃

C₂×C₈

yes

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂²⋊C₄

yes

—

C₂×C₄×C₂₃

C₂×C₈

yes

—

C₂×C₄×C₂₃

C₂×C₈

yes

characteristic

C₂×C₄×C₂₃

M₁₆

yes

—

C₂×C₄×C₂₃

C₂²×C₄

yes

characteristic

C₈×C₂₃

—

C₈×C₂₃

C₂²⋊C₄

yes

—

C₈×C₂₃

C₂×D₄

yes

—

#2112

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

C₂²⋊C₈

yes

—

C₂²×C₂₃

M₃₂

yes

complemented

C₂²×C₂₃

C₂²×C₈

yes

characteristic

C₄×C₂₃

—

C₄×C₂₃

C₂²⋊C₈

yes

—

C₄×C₂₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₂₃

C₂×M₁₆

yes

characteristic

C₂³×C₈

D₂₃

yes

characteristic

C₂×M₃₂

—

C₂²⋊C₁₆

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

C₂×C₂²⋊C₈

yes

characteristic

C₂×C₂₃

C₂×M₃₂

yes

—

C₂³×C₄

Dic₂₃

yes

characteristic

C₂²×C₈

—

C₂²×C₈

Dic₂₃

yes

characteristic

C₂²×C₈

Dic₂₃

yes

—

C₂²×C₈

C₂×D₂₃

yes

—

C₂²×C₈

C₂×D₂₃

yes

characteristic

M₃₂

—

C₂×C₁₆

—

C₂₃

#2112

yes

characteristic, complemented, Sylow

C₂⁴

C₂₃⋊C₈

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂₃⋊C₈

yes

characteristic

C₂²×C₄

C₂₃⋊C₈

yes

—

C₂²×C₄

C₂×Dic₂₃

yes

—

C₂²×C₄

C₂×Dic₂₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×Dic₂₃

yes

—

C₂×C₈

C₂×Dic₂₃

yes

characteristic

C₂×C₈

C₂²⋊D₂₃

yes

—

C₂×C₈

C₂²×D₂₃

yes

characteristic

C₁₆

—

C₂³

—

C₂³

—

C₂³

C₂×C₂₃⋊C₈

yes

—

C₂³

C₂×C₂₃⋊C₈

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₂₃⋊C₈

yes

—

C₂×C₄

C₂×C₂₃⋊C₈

yes

characteristic

C₂×C₄

#67267

yes

—

C₂×C₄

C₂²⋊Dic₂₃

yes

—

C₂×C₄

C₂²×Dic₂₃

yes

characteristic

C₈

—

100

C₈

C₂²⋊Dic₂₃

yes

—

101

C₈

C₂×C₂²⋊D₂₃

yes

—

102

C₂²

—

103

C₂²

—

104

C₂²

—

105

C₂²

—

106

C₂²

#10624994

yes

complemented

107

C₂²

C₂²⋊(C₂₃⋊C₈)

yes

—

108

C₂²

C₂²×C₂₃⋊C₈

yes

characteristic

109

C₄

—

110

C₄

C₂²⋊(C₂₃⋊C₈)

yes

—

111

C₄

C₂×#67267

yes

characteristic

112

C₄

C₂×C₂²⋊Dic₂₃

yes

characteristic

113

C₂

—

114

C₂

—

115

C₂

C₂×C₂²⋊(C₂₃⋊C₈)

yes

characteristic

116

C₂

C₂×#10624994

yes

—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₂₃
commutator, socle	C₂²×C₂₃	C₂²×C₈
Fitting	C₂³×C₈×C₂₃	C₂
prime radical	C₂⁴×C₂₃	C₈
cosocle	C₂×C₈×C₂₃	C₂³
hypercenter	C₂³×C₈	D₂₃
hypocenter	C₂₃	#2112
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×D₂₃

commutator, socle

C₂²×C₂₃

C₂²×C₈

Fitting

C₂³×C₈×C₂₃

C₂

prime radical

C₂⁴×C₂₃

C₈

cosocle

C₂×C₈×C₂₃

C₂³

hypercenter

C₂³×C₈

D₂₃

C₂₃

G ▷ C₂²×C₂₃ ▷ C₂₃

C₁ ◁ C₂×C₈ ◁ C₂³×C₈

N	H ≅ G/N	type
1. C₂×#10624994	C₂	semidirect
2. C₂²⋊(C₂₃⋊C₁₆)	C₂	semidirect
11. C₂×C₂₃⋊C₁₆	C₂²	semidirect
49. C₂²×C₂₃	M₃₂	semidirect
70. C₂₃	#2112	semidirect
106. C₂²	#10624994	semidirect

H ≅ G/N

type

1. C₂×#10624994

C₂

semidirect

2. C₂²⋊(C₂₃⋊C₁₆)

C₂

semidirect

11. C₂×C₂₃⋊C₁₆

C₂²

semidirect

49. C₂²×C₂₃

M₃₂

semidirect

70. C₂₃

#2112

semidirect

106. C₂²

#10624994

semidirect

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

subgroup	index
C₂×#10624994	2
C₂²⋊(C₂₃⋊C₁₆)	2
C₂³×C₈×C₂₃	2
#2112	23

subgroup

index

C₂×#10624994

C₂²⋊(C₂₃⋊C₁₆)

C₂³×C₈×C₂₃

#2112

Subgroups — #15675008

Conjugacy classes of subgroups

Complements

Maximal subgroups