Group #15675234 — subgroups

Subgroups — #15675234

Order 2944.

146 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 146 conjugacy classes of subgroups — too many to draw. Only its 79 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	#13092834	C₂	1	2	yes	maximal, complemented
2	#13092238	C₂	1	1	yes	maximal, characteristic, complemented
3	#13092201	C₂	1	2	yes	maximal, complemented
4	C₄×C₂₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#443	C₂	1	1	yes	maximal, characteristic
6	C₂₃×#135	C₄	1	1	yes	characteristic
7	C₂₃×#131	C₂²	1	2	yes	—
8	C₂₃×#108	C₄	1	1	yes	characteristic
9	C₂₃×#104	C₄	1	2	yes	—
10	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
11	#10624994	—	2	2	no	—
12	#10624994	C₂²	1	2	yes	—
13	C₂×C₂₃⋊C₁₆	—	2	2	no	—
14	C₂×C₂₃⋊C₁₆	C₂²	1	1	yes	characteristic
15	C₂×C₂₃⋊C₁₆	C₂²	1	1	yes	characteristic
16	#41×C₂₃	C₂×C₄	1	2	yes	—
17	Q₁₆×C₂₃	C₈	1	1	yes	characteristic
18	SD₁₆×C₂₃	C₈	1	2	yes	—
19	D₈×C₂₃	C₈	1	1	yes	characteristic
20	M₁₆×C₂₃	—	2	2	no	—
21	M₁₆×C₂₃	C₂×C₄	1	2	yes	—
22	C₂×C₈×C₂₃	—	2	2	no	—
23	C₂×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
25	C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
26	C₂₃⋊C₁₆	—	2	2	no	—
27	C₂₃⋊C₁₆	—	2	1	no	—
28	C₂₃⋊C₁₆	D₄	1	1	yes	characteristic
29	C₂₃⋊C₁₆	D₄	1	1	yes	characteristic
30	Q₈×C₂₃	C₂×C₈	1	2	yes	—
31	D₄×C₂₃	—	2	2	no	—
32	D₄×C₂₃	C₂×C₈	1	2	yes	—
33	C₂×C₄×C₂₃	—	2	1	no	—
34	C₂×C₄×C₂₃	—	2	2	no	—
35	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
36	C₈×C₂₃	—	2	2	no	—
37	C₈×C₂₃	C₂×C₈	1	1	yes	characteristic
38	C₈×C₂₃	C₂×C₈	1	1	yes	characteristic
39	C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
40	C₈×C₂₃	#41	1	1	yes	characteristic
41	#2170	—	23	1	no	maximal, Sylow
42	C₂²×C₂₃	—	2	2	no	—
43	C₂²×C₂₃	#131	1	1	yes	characteristic
44	C₄×C₂₃	—	2	2	no	—
45	C₄×C₂₃	—	2	2	no	—
46	C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
47	C₄×C₂₃	C₂²×C₈	1	1	yes	characteristic
48	#504	—	23	2	no	—
49	#443	D₂₃	1	1	yes	characteristic
50	#364	—	23	1	no	—
51	#350	—	23	2	no	—
52	C₄×C₁₆	—	23	1	no	—
53	C₂×C₂₃	—	2	1	no	—
54	C₂×C₂₃	—	4	2	no	—
55	C₂×C₂₃	C₈×D₄	1	1	yes	characteristic
56	#135	Dic₂₃	1	1	yes	characteristic
57	#131	C₂×D₂₃	1	2	yes	—
58	M₃₂	—	23	2	no	—
59	M₃₂	—	46	2	no	—
60	C₂×C₁₆	—	23	1	no	—
61	C₂×C₁₆	—	23	1	no	—
62	C₂×C₁₆	—	46	2	no	—
63	#108	Dic₂₃	1	1	yes	characteristic
64	#104	Dic₂₃	1	2	yes	—
65	C₄×C₈	C₂×D₂₃	1	1	yes	characteristic
66	C₂₃	#2170	1	1	yes	characteristic, complemented, Sylow
67	#41	C₂×Dic₂₃	1	2	yes	—
68	Q₁₆	C₂₃⋊C₈	1	1	yes	characteristic
69	SD₁₆	C₂₃⋊C₈	1	2	yes	—
70	D₈	C₂₃⋊C₈	1	1	yes	characteristic
71	M₁₆	—	2	2	no	—
72	M₁₆	C₂×Dic₂₃	1	2	yes	—
73	C₂×C₈	—	2	2	no	—
74	C₂×C₈	C₂×Dic₂₃	1	1	yes	characteristic
75	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
76	C₄²	C₂×Dic₂₃	1	1	yes	characteristic
77	C₁₆	—	23	1	no	—
78	C₁₆	—	23	1	no	—
79	C₁₆	—	46	2	no	—
80	C₁₆	—	46	1	no	—
81	Q₈	C₂×C₂₃⋊C₈	1	2	yes	—
82	D₄	—	2	2	no	—
83	D₄	C₂×C₂₃⋊C₈	1	2	yes	—
84	C₂×C₄	—	2	1	no	—
85	C₂×C₄	—	2	2	no	—
86	C₂×C₄	C₂²×Dic₂₃	1	1	yes	characteristic
87	C₈	—	2	2	no	—
88	C₈	C₂×C₂₃⋊C₈	1	1	yes	characteristic
89	C₈	C₂×C₂₃⋊C₈	1	1	yes	characteristic
90	C₈	D₄×D₂₃	1	1	yes	characteristic
91	C₈	#67290	1	1	yes	characteristic
92	C₂²	—	2	2	no	—
93	C₂²	#10625119	1	1	yes	characteristic
94	C₄	—	2	2	no	—
95	C₄	—	2	2	no	—
96	C₄	C₂²×C₂₃⋊C₈	1	1	yes	characteristic
97	C₄	D₄×Dic₂₃	1	1	yes	characteristic
98	C₂	—	2	1	no	—
99	C₂	—	4	2	no	—
100	C₂	D₄×C₂₃⋊C₈	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13092834

C₂

yes

maximal, complemented

#13092238

C₂

yes

maximal, characteristic, complemented

#13092201

C₂

yes

maximal, complemented

C₄×C₂₃⋊C₁₆

C₂

yes

maximal, characteristic, complemented

C₂₃×#443

C₂

yes

maximal, characteristic

C₂₃×#135

C₄

yes

characteristic

C₂₃×#131

C₂²

yes

—

C₂₃×#108

C₄

yes

characteristic

C₂₃×#104

C₄

yes

—

C₄×C₈×C₂₃

C₂²

yes

characteristic

#10624994

—

#10624994

C₂²

yes

—

C₂×C₂₃⋊C₁₆

—

C₂×C₂₃⋊C₁₆

C₂²

yes

characteristic

C₂×C₂₃⋊C₁₆

C₂²

yes

characteristic

#41×C₂₃

C₂×C₄

yes

—

Q₁₆×C₂₃

C₈

yes

characteristic

SD₁₆×C₂₃

C₈

yes

—

D₈×C₂₃

C₈

yes

characteristic

M₁₆×C₂₃

—

M₁₆×C₂₃

C₂×C₄

yes

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

C₂×C₄

yes

characteristic

C₂×C₈×C₂₃

C₂³

yes

characteristic

C₄²×C₂₃

C₂×C₄

yes

characteristic

C₂₃⋊C₁₆

—

C₂₃⋊C₁₆

—

C₂₃⋊C₁₆

D₄

yes

characteristic

C₂₃⋊C₁₆

D₄

yes

characteristic

Q₈×C₂₃

C₂×C₈

yes

—

D₄×C₂₃

—

D₄×C₂₃

C₂×C₈

yes

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂²×C₄

yes

characteristic

C₈×C₂₃

—

C₈×C₂₃

C₂×C₈

yes

characteristic

C₈×C₂₃

C₂×C₈

yes

characteristic

C₈×C₂₃

C₂×D₄

yes

C₈×C₂₃

yes

—

C₂²×C₂₃

—

C₂²×C₂₃

#131

yes

characteristic

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

C₄×D₄

yes

characteristic

C₄×C₂₃

C₂²×C₈

yes

characteristic

#504

—

#443

D₂₃

yes

characteristic

#364

—

#350

—

C₄×C₁₆

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

C₈×D₄

yes

characteristic

#135

Dic₂₃

yes

characteristic

#131

C₂×D₂₃

yes

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

#108

Dic₂₃

yes

characteristic

#104

Dic₂₃

yes

—

C₄×C₈

C₂×D₂₃

yes

characteristic

C₂₃

#2170

yes

characteristic, complemented, Sylow

#41

C₂×Dic₂₃

yes

—

Q₁₆

C₂₃⋊C₈

yes

characteristic

SD₁₆

C₂₃⋊C₈

yes

—

D₈

C₂₃⋊C₈

yes

characteristic

M₁₆

—

M₁₆

C₂×Dic₂₃

yes

—

C₂×C₈

—

C₂×C₈

C₂×Dic₂₃

yes

characteristic

C₂×C₈

C₂²×D₂₃

yes

characteristic

C₄²

C₂×Dic₂₃

yes

characteristic

C₁₆

—

C₁₆

—

C₁₆

—

C₁₆

—

Q₈

C₂×C₂₃⋊C₈

yes

—

D₄

—

D₄

C₂×C₂₃⋊C₈

yes

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²×Dic₂₃

yes

characteristic

C₈

—

C₈

C₂×C₂₃⋊C₈

yes

characteristic

C₈

C₂×C₂₃⋊C₈

yes

characteristic

C₈

D₄×D₂₃

yes

characteristic

C₈

#67290

yes

characteristic

C₂²

—

C₂²

#10625119

yes

characteristic

C₄

—

C₄

—

C₄

C₂²×C₂₃⋊C₈

yes

characteristic

C₄

D₄×Dic₂₃

yes

characteristic

C₂

—

C₂

—

100

C₂

D₄×C₂₃⋊C₈

yes

characteristic

	subgroup	G/H
center	C₈	D₄×D₂₃
commutator	C₄×C₂₃	C₂²×C₈
Frattini	C₂×C₈	C₂²×D₂₃
Fitting	C₂₃×#443	C₂
socle	C₂×C₂₃	C₈×D₄
prime radical	C₂₃×#135	C₄
cosocle	C₂×C₈×C₂₃	C₂³
hypercenter	#443	D₂₃
hypocenter	C₂₃	#2170
LCS	G ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ #443

subgroup

G/H

center

C₈

D₄×D₂₃

commutator

C₄×C₂₃

C₂²×C₈

Frattini

C₂×C₈

C₂²×D₂₃

Fitting

C₂₃×#443

C₂

socle

C₂×C₂₃

C₈×D₄

prime radical

C₂₃×#135

C₄

cosocle

C₂×C₈×C₂₃

C₂³

D₂₃

C₂₃

G ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₈ ◁ C₂×C₈ ◁ #443

N	H ≅ G/N	type
1. #13092834	C₂	semidirect
2. #13092238	C₂	semidirect
3. #13092201	C₂	semidirect
4. C₄×C₂₃⋊C₁₆	C₂	semidirect
66. C₂₃	#2170	semidirect

H ≅ G/N

type

1. #13092834

C₂

semidirect

2. #13092238

C₂

semidirect

3. #13092201

C₂

semidirect

4. C₄×C₂₃⋊C₁₆

C₂

semidirect

66. C₂₃

#2170

semidirect

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

subgroup	index
#13092834	2
#13092238	2
#13092201	2
C₄×C₂₃⋊C₁₆	2
C₂₃×#443	2
#2170	23

subgroup

index

#13092834

#13092238

#13092201

C₄×C₂₃⋊C₁₆

C₂₃×#443

#2170

Subgroups — #15675234

Conjugacy classes of subgroups

Complements

Maximal subgroups