Group #15672111 — subgroups

Subgroups — #15672111

Order 2944.

332 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 85 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	#13092474	C₂	1	2	yes	maximal, complemented
2	#13092161	C₂	1	4	yes	maximal, complemented
3	C₂×C₂₃×#127	C₂	1	1	yes	maximal, characteristic
4	C₂²×D₄×C₂₃	—	2	2	no	—
5	C₂²×D₄×C₂₃	C₄	1	2	yes	—
6	C₂₃×#127	—	2	2	no	—
7	C₂₃×#127	C₄	1	2	yes	—
8	C₂₃×#127	C₂²	1	2	yes	—
9	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
10	C₂²⋊(C₂₃⋊C₈)	—	2	2	no	—
11	C₄⋊(C₂₃⋊C₈)	C₂²	1	4	yes	complemented
12	#10624985	—	2	2	no	—
13	C₂⁴×C₂₃	—	2	2	no	—
14	C₂×D₄×C₂₃	—	2	2	no	—
15	C₂×D₄×C₂₃	—	2	4	no	—
16	C₂×D₄×C₂₃	—	2	4	no	—
17	C₂×D₄×C₂₃	—	4	4	no	—
18	C₂×D₄×C₂₃	—	4	8	no	—
19	C₂×D₄×C₂₃	C₂×C₄	1	4	yes	—
20	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₂₃	D₄	1	2	yes	—
22	C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
23	C₄²×C₂₃	D₄	1	2	yes	—
24	C₄²×C₂₃	C₂³	1	1	yes	characteristic
25	C₂×C₂₃⋊C₈	—	2	4	no	—
26	C₂³×C₂₃	—	2	4	no	—
27	C₂³×C₂₃	—	4	8	no	—
28	C₂³×C₂₃	—	4	4	no	—
29	C₂³×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
30	D₄×C₂₃	—	4	4	no	—
31	D₄×C₂₃	—	4	4	no	—
32	D₄×C₂₃	—	4	8	no	—
33	D₄×C₂₃	—	4	8	no	—
34	C₂×C₄×C₂₃	—	2	2	no	—
35	C₂×C₄×C₂₃	—	2	4	no	—
36	C₂×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
37	C₂×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
38	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
40	C₂₃⋊C₈	—	4	4	no	—
41	#1555	—	23	1	no	maximal, Sylow
42	C₂²×C₂₃	—	2	2	no	—
43	C₂²×C₂₃	—	4	4	no	—
44	C₂²×C₂₃	—	4	8	no	—
45	C₂²×C₂₃	—	8	8	no	—
46	C₂²×C₂₃	#100	1	2	yes	—
47	C₂²×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
48	C₄×C₂₃	—	2	1	no	—
49	C₄×C₂₃	—	2	1	no	—
50	C₄×C₂₃	—	2	2	no	—
51	C₄×C₂₃	AGL(1,ℤ/8)	1	4	yes	—
52	C₂×#127	D₂₃	1	1	yes	characteristic
53	#432	—	23	2	no	—
54	#331	—	23	4	no	—
55	C₂×C₂₃	—	2	2	no	—
56	C₂×C₂₃	—	8	4	no	—
57	C₂×C₂₃	C₂×#100	1	1	yes	characteristic
58	C₂×C₂₃	#418	1	2	yes	—
59	C₂²×D₄	—	2	2	no	—
60	C₂²×D₄	Dic₂₃	1	2	yes	—
61	#127	—	2	2	no	—
62	#127	Dic₂₃	1	2	yes	—
63	#127	C₂×D₂₃	1	2	yes	—
64	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
65	C₄⋊C₈	—	23	4	no	—
66	C₂²⋊C₈	—	46	2	no	—
67	#97	—	46	2	no	—
68	C₂₃	#1555	1	1	yes	characteristic, complemented, Sylow
69	C₂⁴	—	2	2	no	—
70	C₂×D₄	—	2	2	no	—
71	C₂×D₄	—	2	4	no	—
72	C₂×D₄	—	2	4	no	—
73	C₂×D₄	—	4	4	no	—
74	C₂×D₄	—	4	8	no	—
75	C₂×D₄	C₂×Dic₂₃	1	4	yes	—
76	C₂²×C₄	C₂×Dic₂₃	1	1	yes	characteristic
77	C₂²×C₄	C₂²⋊D₂₃	1	2	yes	—
78	C₂×C₈	—	46	4	no	—
79	C₄²	C₂×Dic₂₃	1	1	yes	characteristic
80	C₄²	C₂²⋊D₂₃	1	2	yes	—
81	C₄²	C₂²×D₂₃	1	1	yes	characteristic
82	C₂³	—	2	4	no	—
83	C₂³	—	4	8	no	—
84	C₂³	—	4	4	no	—
85	C₂³	C₂²⋊Dic₂₃	1	1	yes	characteristic
86	D₄	—	4	4	no	—
87	D₄	—	4	4	no	—
88	D₄	—	4	8	no	—
89	D₄	—	4	8	no	—
90	C₂×C₄	—	2	2	no	—
91	C₂×C₄	—	2	4	no	—
92	C₂×C₄	C₂²⋊Dic₂₃	1	2	yes	—
93	C₂×C₄	C₂²⋊Dic₂₃	1	1	yes	characteristic
94	C₂×C₄	C₂²×Dic₂₃	1	1	yes	characteristic
95	C₂×C₄	C₂×C₂²⋊D₂₃	1	2	yes	—
96	C₈	—	92	4	no	—
97	C₂²	—	2	2	no	—
98	C₂²	—	4	4	no	—
99	C₂²	—	4	8	no	—
100	C₂²	—	8	8	no	—
101	C₂²	#10625015	1	2	yes	—
102	C₂²	C₂×C₂²⋊Dic₂₃	1	1	yes	characteristic
103	C₄	—	2	1	no	—
104	C₄	—	2	1	no	—
105	C₄	—	2	2	no	—
106	C₄	#10625103	1	4	yes	—
107	C₂	—	2	2	no	—
108	C₂	—	8	4	no	—
109	C₂	C₂×#10625015	1	1	yes	characteristic
110	C₂	#13092422	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13092474

C₂

yes

maximal, complemented

#13092161

C₂

yes

maximal, complemented

C₂×C₂₃×#127

C₂

yes

maximal, characteristic

C₂²×D₄×C₂₃

—

C₂²×D₄×C₂₃

C₄

yes

—

C₂₃×#127

—

C₂₃×#127

C₄

yes

—

C₂₃×#127

C₂²

yes

—

C₂×C₄²×C₂₃

C₂²

yes

characteristic

C₂²⋊(C₂₃⋊C₈)

—

C₄⋊(C₂₃⋊C₈)

C₂²

yes

complemented

#10624985

—

C₂⁴×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

C₂×C₄

yes

—

C₂²×C₄×C₂₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₃

D₄

yes

—

C₄²×C₂₃

C₂×C₄

yes

characteristic

C₄²×C₂₃

D₄

yes

—

C₄²×C₂₃

C₂³

yes

characteristic

C₂×C₂₃⋊C₈

—

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

C₂²⋊C₄

yes

characteristic

D₄×C₂₃

—

D₄×C₂₃

—

D₄×C₂₃

—

D₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂²⋊C₄

yes

—

C₂×C₄×C₂₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₂₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂₃⋊C₈

—

#1555

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

#100

yes

—

C₂²×C₂₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

AGL(1,ℤ/8)

yes

—

C₂×#127

D₂₃

yes

characteristic

#432

—

#331

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

C₂×#100

yes

characteristic

C₂×C₂₃

#418

yes

—

C₂²×D₄

—

C₂²×D₄

Dic₂₃

yes

—

#127

—

#127

Dic₂₃

yes

—

#127

C₂×D₂₃

yes

—

C₂×C₄²

C₂×D₂₃

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

C₂₃

#1555

yes

characteristic, complemented, Sylow

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×Dic₂₃

yes

—

C₂²×C₄

C₂×Dic₂₃

yes

characteristic

C₂²×C₄

C₂²⋊D₂₃

yes

—

C₂×C₈

—

C₄²

C₂×Dic₂₃

yes

characteristic

C₄²

C₂²⋊D₂₃

yes

—

C₄²

C₂²×D₂₃

yes

characteristic

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂²⋊Dic₂₃

yes

characteristic

D₄

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊Dic₂₃

yes

—

C₂×C₄

C₂²⋊Dic₂₃

yes

characteristic

C₂×C₄

C₂²×Dic₂₃

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₂₃

yes

—

C₈

—

C₂²

—

C₂²

—

C₂²

—

100

C₂²

—

101

C₂²

#10625015

yes

—

102

C₂²

C₂×C₂²⋊Dic₂₃

yes

characteristic

103

C₄

—

104

C₄

—

105

C₄

—

106

C₄

#10625103

yes

—

107

C₂

—

108

C₂

—

109

C₂

C₂×#10625015

yes

characteristic

110

C₂

#13092422

yes

—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₂₃
commutator	C₂×C₄×C₂₃	C₂²×C₄
Frattini	C₄²	C₂²×D₂₃
Fitting, prime radical	C₂×C₂₃×#127	C₂
socle	C₂²×C₂₃	C₂×C₂²⋊C₄
cosocle	C₄²×C₂₃	C₂³
hypercenter	C₂×#127	D₂₃
hypocenter	C₂₃	#1555
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#127

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₂₃

commutator

C₂×C₄×C₂₃

C₂²×C₄

Frattini

C₄²

C₂²×D₂₃

Fitting, prime radical

C₂×C₂₃×#127

C₂

socle

C₂²×C₂₃

C₂×C₂²⋊C₄

cosocle

C₄²×C₂₃

C₂³

D₂₃

C₂₃

G ▷ C₂×C₄×C₂₃ ▷ C₂²×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#127

N	H ≅ G/N	type
1. #13092474	C₂	semidirect
2. #13092161	C₂	semidirect
11. C₄⋊(C₂₃⋊C₈)	C₂²	semidirect
68. C₂₃	#1555	semidirect

H ≅ G/N

type

1. #13092474

C₂

semidirect

2. #13092161

C₂

semidirect

11. C₄⋊(C₂₃⋊C₈)

C₂²

semidirect

68. C₂₃

#1555

semidirect

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

subgroup	index
#13092474	2
#13092161	2
C₂×C₂₃×#127	2
#1555	23

subgroup

index

#13092474

#13092161

C₂×C₂₃×#127

#1555

Subgroups — #15672111

Conjugacy classes of subgroups

Complements

Maximal subgroups