Group #15687545 — subgroups

Subgroups — #15687545

Order 2944.

930 conjugacy classes of subgroups; only the 451 normal subgroups are shown.

normal subgroups only

This group has 930 conjugacy classes of subgroups — too many to draw. Only its 451 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	#13093171	C₂	1	10	yes	maximal, complemented
2	#13093165	C₂	1	5	yes	maximal, complemented
3	#13092973	C₂	1	5	yes	maximal, complemented
4	#13092952	C₂	1	10	yes	maximal, complemented
5	C₂₃×#564	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#125	C₂²	1	5	yes	—
7	C₂₃×#125	C₂²	1	10	yes	—
8	#10625069	C₂²	1	20	yes	—
9	#10625068	C₂²	1	10	yes	—
10	#10625067	C₂²	1	20	yes	—
11	#10625066	C₂²	1	10	yes	—
12	#10625065	C₂²	1	20	yes	—
13	#10625064	C₂²	1	10	yes	—
14	#10625063	C₂²	1	10	yes	—
15	#10625059	C₂²	1	10	yes	—
16	#10625058	C₂²	1	10	yes	—
17	#10625046	C₂²	1	10	yes	—
18	C₄²×D₂₃	C₂²	1	5	yes	—
19	#10625041	C₂²	1	5	yes	—
20	C₂×D₉₂	C₂³	1	10	yes	—
21	C₂×C₄×D₂₃	C₂³	1	5	yes	—
22	C₂×C₄×D₂₃	C₂³	1	10	yes	—
23	C₂×Dic₄₆	C₂³	1	10	yes	—
24	C₄⋊C₄×C₂₃	C₂³	1	10	yes	—
25	C₄⋊C₄×C₂₃	C₂³	1	10	yes	—
26	C₄⋊C₄×C₂₃	C₂³	1	10	yes	—
27	C₄²×C₂₃	C₂³	1	5	yes	—
28	(C₂×D₂₃)⋊C₄	C₂³	1	20	yes	—
29	(C₂×D₂₃)⋊C₄	C₂³	1	20	yes	—
30	C₄⋊Dic₂₃	C₂³	1	10	yes	—
31	Dic₂₃⋊C₄	C₂³	1	20	yes	—
32	C₄×Dic₂₃	C₂³	1	10	yes	—
33	C₄×Dic₂₃	C₂³	1	5	yes	—
34	C₂²×D₂₃	C₂⁴	1	5	yes	—
35	C₂×C₄×C₂₃	C₂⁴	1	10	yes	—
36	C₂×C₄×C₂₃	C₂⁴	1	5	yes	—
37	C₂×Dic₂₃	C₂⁴	1	10	yes	—
38	C₂×Dic₂₃	C₂⁴	1	1	yes	characteristic
39	D₉₂	—	4	10	no	—
40	C₄×D₂₃	—	2	10	no	—
41	C₄×D₂₃	—	4	10	no	—
42	Dic₄₆	—	4	10	no	—
43	#3531	—	23	1	no	maximal, Sylow
44	C₂²×C₂₃	C₂⁵	1	1	yes	characteristic
45	C₂×D₂₃	—	2	5	no	—
46	C₂×D₂₃	—	2	10	no	—
47	C₄×C₂₃	—	2	5	no	—
48	C₄×C₂₃	—	2	10	no	—
49	Dic₂₃	—	2	10	no	—
50	Dic₂₃	2₋¹⁺⁴	1	2	yes	—
51	#564	D₂₃	1	1	yes	characteristic, complemented
52	#563	—	23	10	no	—
53	#562	—	23	5	no	—
54	#533	—	23	5	no	—
55	#529	—	23	10	no	—
56	C₂×C₂₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
57	C₂×C₂₃	#585	1	2	yes	—
58	D₂₃	—	4	5	no	—
59	C₄⋊Q₈	—	23	10	no	—
60	#126	—	23	20	no	—
61	#125	—	23	5	no	—
62	#125	C₂×D₂₃	1	10	yes	—
63	#125	C₂×D₂₃	1	5	yes	—
64	#124	—	23	10	no	—
65	#123	—	23	20	no	—
66	C₂²⋊Q₈	—	23	10	no	—
67	C₂²⋊Q₈	—	23	20	no	—
68	#121	—	23	10	no	—
69	C₄×Q₈	—	23	10	no	—
70	C₄×D₄	—	23	10	no	—
71	#117	—	23	10	no	—
72	C₂×C₄²	—	23	5	no	—
73	C₂₃	#3531	1	1	yes	characteristic, complemented, Sylow
74	C₂×Q₈	—	23	10	no	—
75	C₂×D₄	—	23	10	no	—
76	C₂²×C₄	—	23	5	no	—
77	C₂²×C₄	—	23	10	no	—
78	C₄⋊C₄	—	23	10	no	—
79	C₄⋊C₄	—	23	20	no	—
80	C₄⋊C₄	C₂²×D₂₃	1	10	yes	—
81	C₄⋊C₄	C₂²×D₂₃	1	10	yes	—
82	C₄⋊C₄	C₂²×D₂₃	1	10	yes	—
83	C₂²⋊C₄	—	23	20	no	—
84	C₂²⋊C₄	—	23	20	no	—
85	C₄²	—	23	10	no	—
86	C₄²	—	23	5	no	—
87	C₄²	C₂²×D₂₃	1	5	yes	—
88	C₂³	—	23	5	no	—
89	Q₈	—	92	10	no	—
90	D₄	—	92	10	no	—
91	C₂×C₄	—	23	1	no	—
92	C₂×C₄	—	23	10	no	—
93	C₂×C₄	—	46	10	no	—
94	C₂×C₄	—	92	10	no	—
95	C₂×C₄	C₂³×D₂₃	1	5	yes	—
96	C₂×C₄	C₂³×D₂₃	1	10	yes	—
97	C₂²	—	46	5	no	—
98	C₂²	—	46	10	no	—
99	C₂²	C₂⁴×D₂₃	1	1	yes	characteristic
100	C₄	—	2	5	no	—
101	C₄	—	2	10	no	—
102	C₄	—	23	2	no	—
103	C₄	—	46	10	no	—
104	C₂	—	92	5	no	—
105	C₂	2₋¹⁺⁴×D₂₃	1	1	yes	characteristic
106	C₂	#13093254	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13093171

C₂

yes

maximal, complemented

#13093165

C₂

yes

maximal, complemented

#13092973

C₂

yes

maximal, complemented

#13092952

C₂

yes

maximal, complemented

C₂₃×#564

C₂

yes

maximal, characteristic, complemented

C₂₃×#125

C₂²

yes

—

C₂₃×#125

C₂²

yes

—

#10625069

C₂²

yes

—

#10625068

C₂²

yes

—

#10625067

C₂²

yes

—

#10625066

C₂²

yes

—

#10625065

C₂²

yes

—

#10625064

C₂²

yes

—

#10625063

C₂²

yes

—

#10625059

C₂²

yes

—

#10625058

C₂²

yes

—

#10625046

C₂²

yes

—

C₄²×D₂₃

C₂²

yes

—

#10625041

C₂²

yes

—

C₂×D₉₂

C₂³

yes

—

C₂×C₄×D₂₃

C₂³

yes

—

C₂×C₄×D₂₃

C₂³

yes

—

C₂×Dic₄₆

C₂³

yes

—

C₄⋊C₄×C₂₃

C₂³

yes

—

C₄⋊C₄×C₂₃

C₂³

yes

—

C₄⋊C₄×C₂₃

C₂³

yes

—

C₄²×C₂₃

C₂³

yes

—

(C₂×D₂₃)⋊C₄

C₂³

yes

—

(C₂×D₂₃)⋊C₄

C₂³

yes

—

C₄⋊Dic₂₃

C₂³

yes

—

Dic₂₃⋊C₄

C₂³

yes

—

C₄×Dic₂₃

C₂³

yes

—

C₄×Dic₂₃

C₂³

yes

—

C₂²×D₂₃

C₂⁴

yes

—

C₂×C₄×C₂₃

C₂⁴

yes

—

C₂×C₄×C₂₃

C₂⁴

yes

—

C₂×Dic₂₃

C₂⁴

yes

—

C₂×Dic₂₃

C₂⁴

yes

characteristic

D₉₂

—

C₄×D₂₃

—

C₄×D₂₃

—

Dic₄₆

—

#3531

—

maximal, Sylow

C₂²×C₂₃

C₂⁵

yes

characteristic

C₂×D₂₃

—

C₂×D₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

Dic₂₃

—

Dic₂₃

2₋¹⁺⁴

yes

—

#564

D₂₃

yes

characteristic, complemented

—

—

—

—

C₂×C₂₃

C₂×2₋¹⁺⁴

yes

characteristic

C₂×C₂₃

#585

yes

—

D₂₃

—

C₄⋊Q₈

—

#126

—

#125

—

#125

C₂×D₂₃

yes

—

#125

C₂×D₂₃

yes

—

#124

—

#123

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

#121

—

C₄×Q₈

—

C₄×D₄

—

#117

—

C₂×C₄²

—

C₂₃

#3531

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂²×D₂₃

yes

—

C₄⋊C₄

C₂²×D₂₃

yes

—

C₄⋊C₄

C₂²×D₂₃

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₂²×D₂₃

yes

—

C₂³

—

Q₈

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂³×D₂₃

yes

—

C₂×C₄

C₂³×D₂₃

yes

—

C₂²

—

C₂²

—

C₂²

C₂⁴×D₂₃

yes

characteristic

100

C₄

—

101

C₄

—

102

C₄

—

103

C₄

—

104

C₂

—

105

C₂

2₋¹⁺⁴×D₂₃

yes

characteristic

106

C₂

#13093254

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₂₃
commutator, socle, cosocle	C₂²×C₂₃	C₂⁵
Fitting	C₂₃×#564	C₂
hypercenter	#564	D₂₃
hypocenter	C₂₃	#3531
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ #564

subgroup

G/H

center, Frattini

C₂²

C₂⁴×D₂₃

commutator, socle, cosocle

C₂²×C₂₃

C₂⁵

Fitting

C₂₃×#564

C₂

D₂₃

C₂₃

G ▷ C₂²×C₂₃ ▷ C₂₃

C₁ ◁ C₂² ◁ #564

N	H ≅ G/N	type
1. #13093171	C₂	semidirect
2. #13093165	C₂	semidirect
3. #13092973	C₂	semidirect
4. #13092952	C₂	semidirect
5. C₂₃×#564	C₂	semidirect
51. #564	D₂₃	semidirect
73. C₂₃	#3531	semidirect

H ≅ G/N

type

1. #13093171

C₂

semidirect

2. #13093165

C₂

semidirect

3. #13092973

C₂

semidirect

4. #13092952

C₂

semidirect

5. C₂₃×#564

C₂

semidirect

51. #564

D₂₃

semidirect

73. C₂₃

#3531

semidirect

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

subgroup	index
#13093171	2
#13093165	2
#13092973	2
#13092952	2
C₂₃×#564	2
#3531	23

subgroup

index

C₂₃×#564

Subgroups — #15687545

Conjugacy classes of subgroups

Complements

Maximal subgroups