Group #15673348 — subgroups

Subgroups — #15673348

Order 2944.

548 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 121 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	#13092401	C₂	1	3	yes	maximal, complemented
2	#13092384	C₂	1	3	yes	maximal, complemented
3	C₂₃×#585	C₂	1	1	yes	maximal, characteristic
4	C₂₃×2₋¹⁺⁴	—	2	3	no	—
5	C₂₃×H(5,𝔽₂)	—	2	3	no	—
6	C₂₃×H(5,𝔽₂)	C₄	1	4	yes	—
7	C₂×#41×C₂₃	—	2	6	no	—
8	C₂×#41×C₂₃	C₂²	1	3	yes	—
9	C₂×#67267	C₂²	1	3	yes	complemented
10	C₂²×C₂₃⋊C₈	C₂²	1	1	yes	characteristic, complemented
11	#10625018	—	2	3	no	—
12	#10625015	—	2	3	no	—
13	C₂²⋊(C₂₃⋊C₈)	—	2	6	no	—
14	#41×C₂₃	—	2	4	no	—
15	#41×C₂₃	—	2	12	no	—
16	#41×C₂₃	—	2	12	no	—
17	#41×C₂₃	—	2	6	no	—
18	#41×C₂₃	—	2	6	no	—
19	C₂×Q₈×C₂₃	—	2	6	no	—
20	C₂×Q₈×C₂₃	D₄	1	3	yes	—
21	C₂×D₄×C₂₃	—	2	6	no	—
22	C₂×D₄×C₂₃	—	2	12	no	—
23	C₂×D₄×C₂₃	C₂×C₄	1	6	yes	—
24	C₂×D₄×C₂₃	D₄	1	3	yes	—
25	C₂²×C₄×C₂₃	—	2	4	no	—
26	C₂²×C₄×C₂₃	D₄	1	6	yes	—
27	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
28	#67267	—	2	6	no	—
29	C₂×C₂₃⋊C₈	—	2	3	no	—
30	C₂×C₂₃⋊C₈	—	2	3	no	—
31	C₂³×C₂₃	—	2	4	no	—
32	C₂³×C₂₃	C₂²⋊C₄	1	6	yes	—
33	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
34	Q₈×C₂₃	—	2	4	no	—
35	Q₈×C₂₃	—	2	6	no	—
36	D₄×C₂₃	—	2	12	no	—
37	D₄×C₂₃	—	2	12	no	—
38	D₄×C₂₃	—	2	6	no	—
39	C₂×C₄×C₂₃	—	2	12	no	—
40	C₂×C₄×C₂₃	—	2	6	no	—
41	C₂×C₄×C₂₃	—	2	6	no	—
42	C₂×C₄×C₂₃	C₂²⋊C₄	1	6	yes	—
43	C₂×C₄×C₂₃	C₂×D₄	1	3	yes	—
44	C₂×C₄×C₂₃	C₂×D₄	1	3	yes	—
45	C₂₃⋊C₈	—	4	1	no	—
46	C₂₃⋊C₈	—	4	3	no	—
47	#1791	—	23	1	no	maximal, Sylow
48	C₂²×C₂₃	—	2	6	no	—
49	C₂²×C₂₃	—	4	1	no	—
50	C₂²×C₂₃	—	4	6	no	—
51	C₂²×C₂₃	—	8	4	no	—
52	C₂²×C₂₃	C₂×C₂²⋊C₄	1	3	yes	—
53	C₄×C₂₃	—	2	6	no	—
54	C₄×C₂₃	#120	1	3	yes	—
55	C₄×C₂₃	#120	1	1	yes	characteristic
56	#585	D₂₃	1	1	yes	characteristic
57	#413	—	23	3	no	—
58	#408	—	23	3	no	—
59	C₂×C₂₃	—	2	3	no	—
60	C₂×C₂₃	—	4	6	no	—
61	C₂×C₂₃	#379	1	1	yes	characteristic
62	2₋¹⁺⁴	—	2	3	no	—
63	H(5,𝔽₂)	—	2	3	no	—
64	H(5,𝔽₂)	Dic₂₃	1	4	yes	—
65	C₂×#41	—	2	6	no	—
66	C₂×#41	C₂×D₂₃	1	3	yes	—
67	C₂×M₁₆	—	23	3	no	—
68	C₂²×C₈	—	23	1	no	—
69	#101	—	46	3	no	—
70	#100	—	46	3	no	—
71	C₂²⋊C₈	—	46	6	no	—
72	C₂₃	#1791	1	1	yes	characteristic, complemented, Sylow
73	#41	—	2	4	no	—
74	#41	—	2	12	no	—
75	#41	—	2	12	no	—
76	#41	—	2	6	no	—
77	#41	—	2	6	no	—
78	C₂×Q₈	—	2	6	no	—
79	C₂×Q₈	C₂²⋊D₂₃	1	3	yes	—
80	C₂×D₄	—	2	6	no	—
81	C₂×D₄	—	2	12	no	—
82	C₂×D₄	C₂×Dic₂₃	1	6	yes	—
83	C₂×D₄	C₂²⋊D₂₃	1	3	yes	—
84	C₂²×C₄	—	2	4	no	—
85	C₂²×C₄	C₂²⋊D₂₃	1	6	yes	—
86	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
87	M₁₆	—	46	6	no	—
88	C₂×C₈	—	46	3	no	—
89	C₂×C₈	—	46	3	no	—
90	C₂³	—	2	4	no	—
91	C₂³	C₂²⋊Dic₂₃	1	6	yes	—
92	C₂³	C₂²×Dic₂₃	1	1	yes	characteristic
93	Q₈	—	2	4	no	—
94	Q₈	—	2	6	no	—
95	D₄	—	2	12	no	—
96	D₄	—	2	12	no	—
97	D₄	—	2	6	no	—
98	C₂×C₄	—	2	12	no	—
99	C₂×C₄	—	2	6	no	—
100	C₂×C₄	—	2	6	no	—
101	C₂×C₄	C₂²⋊Dic₂₃	1	6	yes	—
102	C₂×C₄	C₂×C₂²⋊D₂₃	1	3	yes	—
103	C₂×C₄	C₂×C₂²⋊D₂₃	1	3	yes	—
104	C₈	—	92	3	no	—
105	C₈	—	92	1	no	—
106	C₂²	—	2	6	no	—
107	C₂²	—	4	1	no	—
108	C₂²	—	4	6	no	—
109	C₂²	—	8	4	no	—
110	C₂²	C₂×C₂²⋊Dic₂₃	1	3	yes	—
111	C₄	—	2	6	no	—
112	C₄	#10625124	1	3	yes	—
113	C₄	#10625124	1	1	yes	characteristic
114	C₂	—	2	3	no	—
115	C₂	—	4	6	no	—
116	C₂	#13092273	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13092401

C₂

yes

maximal, complemented

#13092384

C₂

yes

maximal, complemented

C₂₃×#585

C₂

yes

maximal, characteristic

C₂₃×2₋¹⁺⁴

—

C₂₃×H(5,𝔽₂)

—

C₂₃×H(5,𝔽₂)

C₄

yes

—

C₂×#41×C₂₃

—

C₂×#41×C₂₃

C₂²

yes

—

C₂×#67267

C₂²

yes

complemented

C₂²×C₂₃⋊C₈

C₂²

yes

characteristic, complemented

#10625018

—

#10625015

—

C₂²⋊(C₂₃⋊C₈)

—

#41×C₂₃

—

#41×C₂₃

—

#41×C₂₃

—

#41×C₂₃

—

#41×C₂₃

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

D₄

yes

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

C₂×C₄

yes

—

C₂×D₄×C₂₃

D₄

yes

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

D₄

yes

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

#67267

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂³×C₂₃

—

C₂³×C₂₃

C₂²⋊C₄

yes

—

C₂³×C₂₃

C₂²×C₄

yes

characteristic

Q₈×C₂₃

—

Q₈×C₂₃

—

D₄×C₂₃

—

D₄×C₂₃

—

D₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂²⋊C₄

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂₃⋊C₈

—

C₂₃⋊C₈

—

#1791

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

C₂×C₂²⋊C₄

yes

—

C₄×C₂₃

—

C₄×C₂₃

#120

yes

—

C₄×C₂₃

#120

yes

characteristic

#585

D₂₃

yes

characteristic

#413

—

#408

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

#379

yes

characteristic

2₋¹⁺⁴

—

H(5,𝔽₂)

—

H(5,𝔽₂)

Dic₂₃

yes

—

C₂×#41

—

C₂×#41

C₂×D₂₃

yes

—

C₂×M₁₆

—

C₂²×C₈

—

#101

—

#100

—

C₂²⋊C₈

—

C₂₃

#1791

yes

characteristic, complemented, Sylow

—

—

—

—

—

C₂×Q₈

—

C₂×Q₈

C₂²⋊D₂₃

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×Dic₂₃

yes

—

C₂×D₄

C₂²⋊D₂₃

yes

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₂₃

yes

—

C₂²×C₄

C₂²×D₂₃

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂³

—

C₂³

C₂²⋊Dic₂₃

yes

—

C₂³

C₂²×Dic₂₃

yes

characteristic

Q₈

—

Q₈

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

C₂²⋊Dic₂₃

yes

—

102

C₂×C₄

C₂×C₂²⋊D₂₃

yes

—

103

C₂×C₄

C₂×C₂²⋊D₂₃

yes

—

104

C₈

—

105

C₈

—

106

C₂²

—

107

C₂²

—

108

C₂²

—

109

C₂²

—

110

C₂²

C₂×C₂²⋊Dic₂₃

yes

—

111

C₄

—

112

C₄

#10625124

yes

—

113

C₄

#10625124

yes

characteristic

114

C₂

—

115

C₂

—

116

C₂

#13092273

yes

characteristic

	subgroup	G/H
center	C₄	#10625124
commutator	C₂³×C₂₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₃
Fitting, prime radical	C₂₃×#585	C₂
socle	C₂×C₂₃	#379
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	#585	D₂₃
hypocenter	C₂₃	#1791
LCS	G ▷ C₂³×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #585

subgroup

G/H

center

C₄

#10625124

commutator

C₂³×C₂₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₃

Fitting, prime radical

C₂₃×#585

C₂

socle

C₂×C₂₃

#379

cosocle

C₂²×C₄×C₂₃

C₂³

D₂₃

C₂₃

G ▷ C₂³×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #585

N	H ≅ G/N	type
1. #13092401	C₂	semidirect
2. #13092384	C₂	semidirect
9. C₂×#67267	C₂²	semidirect
10. C₂²×C₂₃⋊C₈	C₂²	semidirect
72. C₂₃	#1791	semidirect

H ≅ G/N

type

1. #13092401

C₂

semidirect

2. #13092384

C₂

semidirect

9. C₂×#67267

C₂²

semidirect

10. C₂²×C₂₃⋊C₈

C₂²

semidirect

72. C₂₃

#1791

semidirect

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

subgroup	index
#13092401	2
#13092384	2
C₂₃×#585	2
#1791	23

subgroup

index

#13092401

#13092384

C₂₃×#585

#1791

Subgroups — #15673348

Conjugacy classes of subgroups

Complements

Maximal subgroups