Group #15672725 — subgroups

Subgroups — #15672725

Order 2944.

172 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 49 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	#13092827	C₂	1	2	yes	maximal
2	#13092787	C₂	1	1	yes	maximal, characteristic
3	#13092159	C₂	1	1	yes	maximal, characteristic
4	#13092159	C₂	1	2	yes	maximal
5	C₂₃×#564	C₂	1	1	yes	maximal, characteristic
6	C₂₃×#125	—	2	2	no	—
7	C₂₃×#125	—	2	2	no	—
8	C₂₃×#125	—	2	2	no	—
9	C₂₃×#125	C₂²	1	1	yes	characteristic
10	C₂₃×#125	C₂²	1	2	yes	—
11	#10625040	C₂²	1	1	yes	characteristic
12	#10624992	—	2	2	no	—
13	#10624990	—	2	2	no	—
14	#10624989	—	2	2	no	—
15	#10624985	C₂²	1	1	yes	characteristic
16	#10624985	C₂²	1	2	yes	—
17	C₂×Dic₄₆	—	2	1	no	—
18	C₄⋊C₄×C₂₃	—	2	2	no	—
19	C₄⋊C₄×C₂₃	—	2	2	no	—
20	C₄⋊C₄×C₂₃	—	2	2	no	—
21	C₄⋊C₄×C₂₃	—	2	2	no	—
22	C₄⋊C₄×C₂₃	—	2	2	no	—
23	C₄⋊C₄×C₂₃	—	2	2	no	—
24	C₄⋊C₄×C₂₃	D₄	1	2	yes	—
25	C₄⋊C₄×C₂₃	D₄	1	2	yes	—
26	C₄⋊C₄×C₂₃	D₄	1	2	yes	—
27	C₄²×C₂₃	—	2	2	no	—
28	C₄²×C₂₃	C₂³	1	1	yes	characteristic
29	C₄⋊Dic₂₃	—	2	2	no	—
30	C₂×C₂₃⋊C₈	—	2	2	no	—
31	C₂×C₂₃⋊C₈	—	2	1	no	—
32	C₂×C₄×C₂₃	—	2	2	no	—
33	C₂×C₄×C₂₃	—	2	2	no	—
34	C₂×C₄×C₂₃	—	2	2	no	—
35	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
36	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
37	C₂×Dic₂₃	—	4	1	no	—
38	Dic₄₆	—	4	2	no	—
39	C₂₃⋊C₈	—	2	4	no	—
40	C₂₃⋊C₈	—	4	1	no	—
41	#1663	—	23	1	no	maximal, Sylow
42	C₂²×C₂₃	#120	1	1	yes	characteristic
43	C₄×C₂₃	—	2	1	no	—
44	C₄×C₂₃	—	2	2	no	—
45	C₄×C₂₃	—	4	2	no	—
46	C₄×C₂₃	—	4	2	no	—
47	C₄×C₂₃	—	4	2	no	—
48	Dic₂₃	—	8	1	no	—
49	#564	D₂₃	1	1	yes	characteristic
50	#501	—	23	2	no	—
51	#491	—	23	1	no	—
52	#330	—	23	2	no	—
53	#330	—	23	1	no	—
54	C₂×C₂₃	#454	1	1	yes	characteristic
55	C₂×C₂₃	#456	1	2	yes	—
56	C₄⋊Q₈	—	23	1	no	—
57	#125	—	2	2	no	—
58	#125	—	2	2	no	—
59	#125	—	2	2	no	—
60	#125	C₂×D₂₃	1	1	yes	characteristic
61	#125	C₂×D₂₃	1	2	yes	—
62	#107	—	46	2	no	—
63	#106	—	46	2	no	—
64	#103	—	46	2	no	—
65	#97	—	23	1	no	—
66	#97	—	23	2	no	—
67	C₂₃	#1663	1	1	yes	characteristic, complemented, Sylow
68	C₂×Q₈	—	46	1	no	—
69	C₂×C₈	—	46	2	no	—
70	C₂×C₈	—	46	1	no	—
71	C₄⋊C₄	—	2	2	no	—
72	C₄⋊C₄	—	2	2	no	—
73	C₄⋊C₄	—	2	2	no	—
74	C₄⋊C₄	—	2	2	no	—
75	C₄⋊C₄	—	2	2	no	—
76	C₄⋊C₄	—	2	2	no	—
77	C₄⋊C₄	—	46	2	no	—
78	C₄⋊C₄	C₂²⋊D₂₃	1	2	yes	—
79	C₄⋊C₄	C₂²⋊D₂₃	1	2	yes	—
80	C₄⋊C₄	C₂²⋊D₂₃	1	2	yes	—
81	C₄²	—	2	2	no	—
82	C₄²	C₂²×D₂₃	1	1	yes	characteristic
83	Q₈	—	92	2	no	—
84	C₂×C₄	—	2	2	no	—
85	C₂×C₄	—	2	2	no	—
86	C₂×C₄	—	2	2	no	—
87	C₂×C₄	—	92	1	no	—
88	C₂×C₄	C₂×C₂²⋊D₂₃	1	2	yes	—
89	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
90	C₈	—	46	4	no	—
91	C₈	—	92	1	no	—
92	C₂²	#10625124	1	1	yes	characteristic
93	C₄	—	2	1	no	—
94	C₄	—	2	2	no	—
95	C₄	—	4	2	no	—
96	C₄	—	4	2	no	—
97	C₄	—	4	2	no	—
98	C₄	—	184	1	no	—
99	C₂	#13092590	1	1	yes	characteristic
100	C₂	#13092600	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13092827

C₂

yes

maximal

#13092787

C₂

yes

maximal, characteristic

#13092159

C₂

yes

maximal, characteristic

#13092159

C₂

yes

maximal

C₂₃×#564

C₂

yes

maximal, characteristic

C₂₃×#125

—

C₂₃×#125

—

C₂₃×#125

—

C₂₃×#125

C₂²

yes

characteristic

C₂₃×#125

C₂²

yes

—

#10625040

C₂²

yes

—

—

—

C₂²

yes

characteristic

#10624985

C₂²

yes

—

C₂×Dic₄₆

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

D₄

yes

—

C₄⋊C₄×C₂₃

D₄

yes

—

C₄⋊C₄×C₂₃

D₄

yes

—

C₄²×C₂₃

—

C₄²×C₂₃

C₂³

yes

characteristic

C₄⋊Dic₂₃

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×Dic₂₃

—

Dic₄₆

—

C₂₃⋊C₈

—

C₂₃⋊C₈

—

—

C₂²×C₂₃

yes

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

Dic₂₃

—

#564

D₂₃

yes

—

—

—

—

C₂×C₂₃

#454

yes

characteristic

C₂×C₂₃

#456

yes

—

C₄⋊Q₈

—

—

—

—

C₂×D₂₃

yes

characteristic

#125

C₂×D₂₃

yes

—

—

—

—

—

—

C₂₃

#1663

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂²⋊D₂₃

yes

—

C₄⋊C₄

C₂²⋊D₂₃

yes

—

C₄⋊C₄

C₂²⋊D₂₃

yes

—

C₄²

—

C₄²

C₂²×D₂₃

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₂²⋊D₂₃

yes

—

C₂×C₄

C₂×C₂²⋊D₂₃

yes

characteristic

C₈

—

C₈

—

C₂²

#10625124

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

—

C₄

—

C₄

—

184

—

C₂

#13092590

yes

characteristic

100

C₂

#13092600

yes

—

	subgroup	G/H
center	C₂²	#10625124
commutator, cosocle	C₄²×C₂₃	C₂³
Frattini	C₄²	C₂²×D₂₃
Fitting	C₂₃×#564	C₂
socle, prime radical	C₂²×C₂₃	#120
hypercenter	#564	D₂₃
hypocenter	C₂₃	#1663
LCS	G ▷ C₄²×C₂₃ ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #564

subgroup

G/H

center

C₂²

#10625124

commutator, cosocle

C₄²×C₂₃

C₂³

Frattini

C₄²

C₂²×D₂₃

Fitting

C₂₃×#564

C₂

C₂²×C₂₃

D₂₃

C₂₃

G ▷ C₄²×C₂₃ ▷ C₂²×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #564

N	H ≅ G/N	type
67. C₂₃	#1663	semidirect

H ≅ G/N

type

67. C₂₃

#1663

semidirect

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

subgroup	index
#13092827	2
#13092787	2
#13092159	2
C₂₃×#564	2
#1663	23

subgroup

index

C₂₃×#564

Subgroups — #15672725

Conjugacy classes of subgroups

Complements

Maximal subgroups