Group #15672135 — subgroups

Subgroups — #15672135

Order 2944.

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

normal subgroups only

This group has 210 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	#13092441	C₂	1	2	yes	maximal
2	#13092163	C₂	1	4	yes	maximal, complemented
3	C₂₃×#533	C₂	1	1	yes	maximal, characteristic
4	C₂₃×C₄⋊Q₈	C₂²	1	2	yes	—
5	C₂₃×#125	C₄	1	2	yes	—
6	C₂₃×C₂²⋊Q₈	—	2	2	no	—
7	C₄×Q₈×C₂₃	—	2	2	no	—
8	C₂₃×#117	C₄	1	2	yes	—
9	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
10	C₂×#67267	—	2	2	no	—
11	C₄⋊(C₂₃⋊C₈)	—	2	2	no	—
12	C₄⋊(C₂₃⋊C₈)	C₂²	1	4	yes	—
13	C₂×Q₈×C₂₃	—	2	2	no	—
14	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₄×C₂₃	D₄	1	2	yes	—
16	C₄⋊C₄×C₂₃	—	2	4	no	—
17	C₄⋊C₄×C₂₃	—	2	2	no	—
18	C₄⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
19	C₂²⋊C₄×C₂₃	—	2	2	no	—
20	C₄²×C₂₃	—	2	2	no	—
21	C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
22	C₄²×C₂₃	D₄	1	2	yes	—
23	C₄²×C₂₃	C₂³	1	1	yes	characteristic
24	#67267	—	4	4	no	—
25	C₂×C₂₃⋊C₈	—	2	4	no	—
26	C₂³×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
27	Q₈×C₂₃	—	4	2	no	—
28	Q₈×C₂₃	—	4	2	no	—
29	C₂×C₄×C₂₃	—	2	2	no	—
30	C₂×C₄×C₂₃	—	2	2	no	—
31	C₂×C₄×C₂₃	—	2	2	no	—
32	C₂×C₄×C₂₃	—	2	4	no	—
33	C₂×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
34	C₂×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
37	C₂₃⋊C₈	—	4	4	no	—
38	#1561	—	23	1	no	maximal, Sylow
39	C₂²×C₂₃	—	2	1	no	—
40	C₂²×C₂₃	—	2	2	no	—
41	C₂²×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
42	C₄×C₂₃	—	2	1	no	—
43	C₄×C₂₃	—	2	2	no	—
44	C₄×C₂₃	—	4	4	no	—
45	C₄×C₂₃	#101	1	2	yes	—
46	C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
47	C₄×C₂₃	#137	1	2	yes	—
48	#533	D₂₃	1	1	yes	characteristic
49	#423	—	23	2	no	—
50	#332	—	23	4	no	—
51	C₂×C₂₃	—	4	1	no	—
52	C₂×C₂₃	C₂×#101	1	1	yes	characteristic
53	C₂×C₂₃	#417	1	2	yes	—
54	C₂×M₁₆	—	46	2	no	—
55	C₄⋊Q₈	C₂×D₂₃	1	2	yes	—
56	#125	Dic₂₃	1	2	yes	—
57	C₂²⋊Q₈	—	2	2	no	—
58	C₄×Q₈	—	2	2	no	—
59	#117	Dic₂₃	1	2	yes	—
60	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
61	C₄⋊C₈	—	23	4	no	—
62	C₄⋊C₈	—	46	2	no	—
63	C₂₃	#1561	1	1	yes	characteristic, complemented, Sylow
64	C₂×Q₈	—	2	2	no	—
65	C₂²×C₄	C₂×Dic₂₃	1	1	yes	characteristic
66	C₂²×C₄	C₂²⋊D₂₃	1	2	yes	—
67	M₁₆	—	92	4	no	—
68	C₂×C₈	—	46	4	no	—
69	C₄⋊C₄	—	2	4	no	—
70	C₄⋊C₄	—	2	2	no	—
71	C₄⋊C₄	C₂×Dic₂₃	1	4	yes	—
72	C₂²⋊C₄	—	2	2	no	—
73	C₄²	—	2	2	no	—
74	C₄²	C₂×Dic₂₃	1	1	yes	characteristic
75	C₄²	C₂²⋊D₂₃	1	2	yes	—
76	C₄²	C₂²×D₂₃	1	1	yes	characteristic
77	C₂³	C₂²⋊Dic₂₃	1	1	yes	characteristic
78	Q₈	—	4	2	no	—
79	Q₈	—	4	2	no	—
80	C₂×C₄	—	2	2	no	—
81	C₂×C₄	—	2	2	no	—
82	C₂×C₄	—	2	2	no	—
83	C₂×C₄	—	2	4	no	—
84	C₂×C₄	C₂²⋊Dic₂₃	1	2	yes	—
85	C₂×C₄	C₂²⋊Dic₂₃	1	1	yes	characteristic
86	C₂×C₄	C₂²×Dic₂₃	1	1	yes	characteristic
87	C₂×C₄	C₂×C₂²⋊D₂₃	1	2	yes	—
88	C₈	—	92	4	no	—
89	C₂²	—	2	1	no	—
90	C₂²	—	2	2	no	—
91	C₂²	C₂×C₂²⋊Dic₂₃	1	1	yes	characteristic
92	C₄	—	2	1	no	—
93	C₄	—	2	2	no	—
94	C₄	—	4	4	no	—
95	C₄	#10625018	1	2	yes	—
96	C₄	#10625120	1	2	yes	—
97	C₄	#10625122	1	2	yes	—
98	C₂	—	4	1	no	—
99	C₂	C₂×#10625018	1	1	yes	characteristic
100	C₂	#13092418	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13092441

C₂

yes

maximal

#13092163

C₂

yes

maximal, complemented

C₂₃×#533

C₂

yes

maximal, characteristic

C₂₃×C₄⋊Q₈

C₂²

yes

—

C₂₃×#125

C₄

yes

—

C₂₃×C₂²⋊Q₈

—

C₄×Q₈×C₂₃

—

C₂₃×#117

C₄

yes

—

C₂×C₄²×C₂₃

C₂²

yes

characteristic

C₂×#67267

—

C₄⋊(C₂₃⋊C₈)

—

C₄⋊(C₂₃⋊C₈)

C₂²

yes

—

C₂×Q₈×C₂₃

—

C₂²×C₄×C₂₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₃

D₄

yes

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

C₂×C₄

yes

—

C₂²⋊C₄×C₂₃

—

C₄²×C₂₃

—

C₄²×C₂₃

C₂×C₄

yes

characteristic

C₄²×C₂₃

D₄

yes

—

C₄²×C₂₃

C₂³

yes

characteristic

#67267

—

C₂×C₂₃⋊C₈

—

C₂³×C₂₃

C₂²⋊C₄

yes

characteristic

Q₈×C₂₃

—

Q₈×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×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

—

#1561

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

#101

yes

—

C₄×C₂₃

AGL(1,ℤ/8)

yes

—

C₄×C₂₃

#137

yes

—

#533

D₂₃

yes

characteristic

#423

—

#332

—

C₂×C₂₃

—

C₂×C₂₃

C₂×#101

yes

characteristic

C₂×C₂₃

#417

yes

—

C₂×M₁₆

—

C₄⋊Q₈

C₂×D₂₃

yes

—

#125

Dic₂₃

yes

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

Dic₂₃

yes

—

C₂×C₄²

C₂×D₂₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₂₃

#1561

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂²×C₄

C₂×Dic₂₃

yes

characteristic

C₂²×C₄

C₂²⋊D₂₃

yes

—

M₁₆

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×Dic₂₃

yes

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂×Dic₂₃

yes

characteristic

C₄²

C₂²⋊D₂₃

yes

—

C₄²

C₂²×D₂₃

yes

characteristic

C₂³

C₂²⋊Dic₂₃

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

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

C₂×C₂²⋊Dic₂₃

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

#10625018

yes

—

C₄

#10625120

yes

—

C₄

#10625122

yes

—

C₂

—

C₂

C₂×#10625018

yes

characteristic

100

C₂

#13092418

yes

—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₂₃
commutator	C₂×C₄×C₂₃	C₂²×C₄
Frattini	C₄²	C₂²×D₂₃
Fitting	C₂₃×#533	C₂
socle	C₂²×C₂₃	C₂×C₂²⋊C₄
prime radical	C₂³×C₂₃	C₂²⋊C₄
cosocle	C₄²×C₂₃	C₂³
hypercenter	#533	D₂₃
hypocenter	C₂₃	#1561
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₂₃

commutator

C₂×C₄×C₂₃

C₂²×C₄

Frattini

C₄²

C₂²×D₂₃

Fitting

C₂₃×#533

C₂

socle

C₂²×C₂₃

C₂×C₂²⋊C₄

prime radical

C₂³×C₂₃

C₂²⋊C₄

cosocle

C₄²×C₂₃

C₂³

D₂₃

C₂₃

G ▷ C₂×C₄×C₂₃ ▷ C₂²×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
2. #13092163	C₂	semidirect
63. C₂₃	#1561	semidirect

H ≅ G/N

type

2. #13092163

C₂

semidirect

63. C₂₃

#1561

semidirect

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

subgroup	index
#13092441	2
#13092163	2
C₂₃×#533	2
#1561	23

subgroup

index

#13092441

#13092163

C₂₃×#533

#1561

Subgroups — #15672135

Conjugacy classes of subgroups

Complements

Maximal subgroups