Group #15675229 — subgroups

Subgroups — #15675229

Order 2944.

162 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 162 conjugacy classes of subgroups — too many to draw. Only its 87 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	C₂×#10624994	C₂	1	2	yes	maximal, complemented
2	C₄⋊(C₂₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²⋊(C₂₃⋊C₁₆)	C₂	1	2	yes	maximal, complemented
4	C₄×C₂₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈×D₄×C₂₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₈×C₂₃	C₂²	1	2	yes	—
7	C₄×D₄×C₂₃	C₄	1	1	yes	characteristic
8	C₂₃×C₄⋊C₈	C₄	1	1	yes	characteristic
9	C₂₃×C₂²⋊C₈	C₄	1	2	yes	—
10	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
11	#10624994	—	2	4	no	—
12	C₂×C₂₃⋊C₁₆	C₂²	1	1	yes	characteristic
13	C₂×C₂₃⋊C₁₆	C₂²	1	2	yes	—
14	C₂×C₂₃⋊C₁₆	C₂²	1	1	yes	characteristic
15	C₂×D₄×C₂₃	C₈	1	1	yes	characteristic
16	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
17	C₂×C₈×C₂₃	—	2	2	no	—
18	C₂×C₈×C₂₃	—	2	2	no	—
19	C₂×C₈×C₂₃	C₂×C₄	1	2	yes	—
20	C₂×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
22	C₄⋊C₄×C₂₃	C₈	1	1	yes	characteristic
23	C₂²⋊C₄×C₂₃	C₈	1	2	yes	—
24	C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
25	C₂₃⋊C₁₆	—	2	2	no	—
26	C₂₃⋊C₁₆	—	2	1	no	—
27	C₂₃⋊C₁₆	D₄	1	2	yes	complemented
28	C₂³×C₂₃	C₂×C₈	1	2	yes	—
29	D₄×C₂₃	—	2	2	no	—
30	C₂×C₄×C₂₃	—	2	2	no	—
31	C₂×C₄×C₂₃	—	2	2	no	—
32	C₂×C₄×C₂₃	C₂×C₈	1	1	yes	characteristic
33	C₂×C₄×C₂₃	C₂×C₈	1	2	yes	—
34	C₂×C₄×C₂₃	C₂×C₈	1	1	yes	characteristic
35	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
36	C₈×C₂₃	—	2	2	no	—
37	C₈×C₂₃	—	2	1	no	—
38	C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
39	C₈×C₂₃	#41	1	1	yes	characteristic
40	#2169	—	23	1	no	maximal, Sylow
41	C₂²×C₂₃	—	2	2	no	—
42	C₂²×C₂₃	—	2	2	no	—
43	C₂²×C₂₃	—	2	2	no	—
44	C₂²×C₂₃	C₂²×C₈	1	1	yes	characteristic
45	C₄×C₂₃	—	2	2	no	—
46	C₄×C₂₃	—	2	1	no	—
47	C₄×C₂₃	M₃₂	1	2	yes	complemented
48	C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
49	C₄×C₂₃	#131	1	1	yes	characteristic
50	C₂×M₃₂	—	23	2	no	—
51	C₈×D₄	D₂₃	1	1	yes	characteristic
52	C₄⋊C₁₆	—	23	1	no	—
53	C₂²⋊C₁₆	—	23	2	no	—
54	C₄×C₁₆	—	23	1	no	—
55	C₂×C₂₃	—	4	2	no	—
56	C₂×C₂₃	C₈×D₄	1	1	yes	characteristic
57	C₂×C₂₃	C₂×M₃₂	1	1	yes	characteristic
58	C₂×C₂₃	#504	1	1	yes	characteristic
59	C₂²×C₈	C₂×D₂₃	1	2	yes	—
60	C₄×D₄	Dic₂₃	1	1	yes	characteristic
61	M₃₂	—	46	4	no	—
62	C₂×C₁₆	—	23	1	no	—
63	C₂×C₁₆	—	23	1	no	—
64	C₂×C₁₆	—	23	2	no	—
65	C₄⋊C₈	Dic₂₃	1	1	yes	characteristic
66	C₂²⋊C₈	Dic₂₃	1	2	yes	—
67	C₄×C₈	C₂×D₂₃	1	1	yes	characteristic
68	C₂₃	#2169	1	1	yes	characteristic, complemented, Sylow
69	C₂×D₄	C₂₃⋊C₈	1	1	yes	characteristic
70	C₂²×C₄	C₂×Dic₂₃	1	2	yes	—
71	C₂×C₈	—	2	2	no	—
72	C₂×C₈	—	2	2	no	—
73	C₂×C₈	C₂×Dic₂₃	1	1	yes	characteristic
74	C₂×C₈	C₂×Dic₂₃	1	2	yes	—
75	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
76	C₄⋊C₄	C₂₃⋊C₈	1	1	yes	characteristic
77	C₂²⋊C₄	C₂₃⋊C₈	1	2	yes	—
78	C₄²	C₂×Dic₂₃	1	1	yes	characteristic
79	C₁₆	—	23	2	no	—
80	C₁₆	—	46	2	no	—
81	C₁₆	—	46	1	no	—
82	C₂³	C₂×C₂₃⋊C₈	1	2	yes	—
83	D₄	—	2	2	no	—
84	C₂×C₄	—	2	2	no	—
85	C₂×C₄	—	2	2	no	—
86	C₂×C₄	C₂×C₂₃⋊C₈	1	1	yes	characteristic
87	C₂×C₄	C₂×C₂₃⋊C₈	1	2	yes	—
88	C₂×C₄	C₂×C₂₃⋊C₈	1	1	yes	characteristic
89	C₂×C₄	C₂²×Dic₂₃	1	1	yes	characteristic
90	C₈	—	2	2	no	—
91	C₈	—	2	1	no	—
92	C₈	D₄×D₂₃	1	1	yes	characteristic
93	C₈	#67290	1	1	yes	characteristic
94	C₂²	—	2	2	no	—
95	C₂²	—	2	2	no	—
96	C₂²	—	2	2	no	—
97	C₂²	C₂²×C₂₃⋊C₈	1	1	yes	characteristic
98	C₄	—	2	2	no	—
99	C₄	—	2	1	no	—
100	C₄	#10624994	1	2	yes	complemented
101	C₄	D₄×Dic₂₃	1	1	yes	characteristic
102	C₄	#10625119	1	1	yes	characteristic
103	C₂	—	4	2	no	—
104	C₂	D₄×C₂₃⋊C₈	1	1	yes	characteristic
105	C₂	C₂×#10624994	1	1	yes	characteristic
106	C₂	#13092834	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10624994

C₂

yes

maximal, complemented

C₄⋊(C₂₃⋊C₁₆)

C₂

yes

maximal, characteristic, complemented

C₂²⋊(C₂₃⋊C₁₆)

C₂

yes

maximal, complemented

C₄×C₂₃⋊C₁₆

C₂

yes

maximal, characteristic, complemented

C₈×D₄×C₂₃

C₂

yes

maximal, characteristic

C₂²×C₈×C₂₃

C₂²

yes

—

C₄×D₄×C₂₃

C₄

yes

characteristic

C₂₃×C₄⋊C₈

C₄

yes

characteristic

C₂₃×C₂²⋊C₈

C₄

yes

—

C₄×C₈×C₂₃

C₂²

yes

characteristic

#10624994

—

C₂×C₂₃⋊C₁₆

C₂²

yes

characteristic

C₂×C₂₃⋊C₁₆

C₂²

yes

—

C₂×C₂₃⋊C₁₆

C₂²

yes

characteristic

C₂×D₄×C₂₃

C₈

yes

characteristic

C₂²×C₄×C₂₃

C₂×C₄

yes

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

C₂×C₄

yes

—

C₂×C₈×C₂₃

C₂×C₄

yes

characteristic

C₂×C₈×C₂₃

C₂³

yes

characteristic

C₄⋊C₄×C₂₃

C₈

yes

characteristic

C₂²⋊C₄×C₂₃

C₈

yes

—

C₄²×C₂₃

C₂×C₄

yes

characteristic

C₂₃⋊C₁₆

—

C₂₃⋊C₁₆

—

C₂₃⋊C₁₆

D₄

yes

complemented

C₂³×C₂₃

C₂×C₈

yes

—

D₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂×C₈

yes

characteristic

C₂×C₄×C₂₃

C₂×C₈

yes

—

C₂×C₄×C₂₃

C₂×C₈

yes

characteristic

C₂×C₄×C₂₃

C₂²×C₄

yes

characteristic

C₈×C₂₃

—

C₈×C₂₃

—

C₈×C₂₃

C₂×D₄

yes

C₈×C₂₃

yes

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

C₂²×C₈

yes

characteristic

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

M₃₂

yes

complemented

C₄×C₂₃

C₄×D₄

yes

characteristic

C₄×C₂₃

#131

yes

characteristic

C₂×M₃₂

—

C₈×D₄

D₂₃

yes

characteristic

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

C₄×C₁₆

—

C₂×C₂₃

—

C₂×C₂₃

C₈×D₄

yes

characteristic

C₂×C₂₃

C₂×M₃₂

yes

characteristic

C₂×C₂₃

#504

yes

characteristic

C₂²×C₈

C₂×D₂₃

yes

—

C₄×D₄

Dic₂₃

yes

characteristic

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₄⋊C₈

Dic₂₃

yes

characteristic

C₂²⋊C₈

Dic₂₃

yes

—

C₄×C₈

C₂×D₂₃

yes

characteristic

C₂₃

#2169

yes

characteristic, complemented, Sylow

C₂×D₄

C₂₃⋊C₈

yes

characteristic

C₂²×C₄

C₂×Dic₂₃

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×Dic₂₃

yes

characteristic

C₂×C₈

C₂×Dic₂₃

yes

—

C₂×C₈

C₂²×D₂₃

yes

characteristic

C₄⋊C₄

C₂₃⋊C₈

yes

characteristic

C₂²⋊C₄

C₂₃⋊C₈

yes

—

C₄²

C₂×Dic₂₃

yes

characteristic

C₁₆

—

C₁₆

—

C₁₆

—

C₂³

C₂×C₂₃⋊C₈

yes

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₂₃⋊C₈

yes

characteristic

C₂×C₄

C₂×C₂₃⋊C₈

yes

—

C₂×C₄

C₂×C₂₃⋊C₈

yes

characteristic

C₂×C₄

C₂²×Dic₂₃

yes

characteristic

C₈

—

C₈

—

C₈

D₄×D₂₃

yes

characteristic

C₈

#67290

yes

characteristic

C₂²

—

C₂²

—

C₂²

—

C₂²

C₂²×C₂₃⋊C₈

yes

characteristic

C₄

—

C₄

—

100

C₄

#10624994

yes

complemented

101

C₄

D₄×Dic₂₃

yes

characteristic

102

C₄

#10625119

yes

characteristic

103

C₂

—

104

C₂

D₄×C₂₃⋊C₈

yes

characteristic

105

C₂

C₂×#10624994

yes

characteristic

106

C₂

#13092834

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₂₃
commutator, socle	C₂²×C₂₃	C₂²×C₈
Fitting	C₈×D₄×C₂₃	C₂
prime radical	C₂×D₄×C₂₃	C₈
cosocle	C₂×C₈×C₂₃	C₂³
hypercenter	C₈×D₄	D₂₃
hypocenter	C₂₃	#2169
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₈ ◁ C₈×D₄

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×D₂₃

commutator, socle

C₂²×C₂₃

C₂²×C₈

Fitting

C₈×D₄×C₂₃

C₂

prime radical

C₂×D₄×C₂₃

C₈

cosocle

C₂×C₈×C₂₃

C₂³

hypercenter

C₈×D₄

D₂₃

C₂₃

G ▷ C₂²×C₂₃ ▷ C₂₃

C₁ ◁ C₂×C₈ ◁ C₈×D₄

N	H ≅ G/N	type
1. C₂×#10624994	C₂	semidirect
2. C₄⋊(C₂₃⋊C₁₆)	C₂	semidirect
3. C₂²⋊(C₂₃⋊C₁₆)	C₂	semidirect
4. C₄×C₂₃⋊C₁₆	C₂	semidirect
27. C₂₃⋊C₁₆	D₄	semidirect
47. C₄×C₂₃	M₃₂	semidirect
68. C₂₃	#2169	semidirect
100. C₄	#10624994	semidirect

H ≅ G/N

type

1. C₂×#10624994

C₂

semidirect

2. C₄⋊(C₂₃⋊C₁₆)

C₂

semidirect

3. C₂²⋊(C₂₃⋊C₁₆)

C₂

semidirect

4. C₄×C₂₃⋊C₁₆

C₂

semidirect

27. C₂₃⋊C₁₆

D₄

semidirect

47. C₄×C₂₃

M₃₂

semidirect

68. C₂₃

#2169

semidirect

100. C₄

#10624994

semidirect

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

subgroup	index
C₂×#10624994	2
C₄⋊(C₂₃⋊C₁₆)	2
C₂²⋊(C₂₃⋊C₁₆)	2
C₄×C₂₃⋊C₁₆	2
C₈×D₄×C₂₃	2
#2169	23

subgroup

index

C₂×#10624994

C₄⋊(C₂₃⋊C₁₆)

C₂²⋊(C₂₃⋊C₁₆)

C₄×C₂₃⋊C₁₆

C₈×D₄×C₂₃

#2169

Subgroups — #15675229

Conjugacy classes of subgroups

Complements

Maximal subgroups