Group #24909295 — subgroups

Subgroups — #24909295

Order 10624.

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×#14581732	C₂	1	2	yes	maximal, complemented
2	C₄⋊(C₈₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²⋊(C₈₃⋊C₁₆)	C₂	1	2	yes	maximal, complemented
4	C₄×C₈₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈×D₄×C₈₃	C₂	1	1	yes	maximal, characteristic
6	#14581732	—	2	4	no	—
7	C₂×C₈₃⋊C₁₆	C₂²	1	2	yes	—
8	C₂×C₈₃⋊C₁₆	C₂²	1	1	yes	characteristic
9	C₂×C₈₃⋊C₁₆	C₂²	1	1	yes	characteristic
10	C₂²×C₈×C₈₃	C₂²	1	2	yes	—
11	C₄×D₄×C₈₃	C₄	1	1	yes	characteristic
12	C₄⋊C₈×C₈₃	C₄	1	1	yes	characteristic
13	C₂²⋊C₈×C₈₃	C₄	1	2	yes	—
14	C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
15	C₈₃⋊C₁₆	—	2	2	no	—
16	C₈₃⋊C₁₆	—	2	1	no	—
17	C₈₃⋊C₁₆	D₄	1	2	yes	complemented
18	C₂×D₄×C₈₃	C₈	1	1	yes	characteristic
19	C₂²×C₄×C₈₃	C₂×C₄	1	2	yes	—
20	C₂×C₈×C₈₃	—	2	2	no	—
21	C₂×C₈×C₈₃	—	2	2	no	—
22	C₂×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₈×C₈₃	C₂×C₄	1	2	yes	—
24	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
25	C₄⋊C₄×C₈₃	C₈	1	1	yes	characteristic
26	C₂²⋊C₄×C₈₃	C₈	1	2	yes	—
27	C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₈₃	C₂×C₈	1	2	yes	—
29	D₄×C₈₃	—	2	2	no	—
30	C₂×C₄×C₈₃	—	2	2	no	—
31	C₂×C₄×C₈₃	—	2	2	no	—
32	C₂×C₄×C₈₃	C₂×C₈	1	1	yes	characteristic
33	C₂×C₄×C₈₃	C₂×C₈	1	2	yes	—
34	C₂×C₄×C₈₃	C₂×C₈	1	1	yes	characteristic
35	C₂×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
36	C₈×C₈₃	—	2	2	no	—
37	C₈×C₈₃	—	2	1	no	—
38	C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
39	C₈×C₈₃	#41	1	1	yes	characteristic
40	C₂²×C₈₃	—	2	2	no	—
41	C₂²×C₈₃	—	2	2	no	—
42	C₂²×C₈₃	—	2	2	no	—
43	C₂²×C₈₃	C₂²×C₈	1	1	yes	characteristic
44	C₄×C₈₃	—	2	2	no	—
45	C₄×C₈₃	—	2	1	no	—
46	C₄×C₈₃	M₃₂	1	2	yes	complemented
47	C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
48	C₄×C₈₃	#131	1	1	yes	characteristic
49	C₂×C₈₃	—	4	2	no	—
50	C₂×C₈₃	C₈×D₄	1	1	yes	characteristic
51	C₂×C₈₃	C₂×M₃₂	1	1	yes	characteristic
52	C₂×C₈₃	#504	1	1	yes	characteristic
53	#2169	—	83	1	no	maximal, Sylow
54	C₈₃	#2169	1	1	yes	characteristic, complemented, Sylow
55	C₂×M₃₂	—	83	2	no	—
56	C₈×D₄	D₈₃	1	1	yes	characteristic
57	C₄⋊C₁₆	—	83	1	no	—
58	C₂²⋊C₁₆	—	83	2	no	—
59	C₄×C₁₆	—	83	1	no	—
60	C₂²×C₈	C₂×D₈₃	1	2	yes	—
61	C₄×D₄	Dic₈₃	1	1	yes	characteristic
62	M₃₂	—	166	4	no	—
63	C₂×C₁₆	—	83	1	no	—
64	C₂×C₁₆	—	83	1	no	—
65	C₂×C₁₆	—	83	2	no	—
66	C₄⋊C₈	Dic₈₃	1	1	yes	characteristic
67	C₂²⋊C₈	Dic₈₃	1	2	yes	—
68	C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
69	C₂×D₄	C₈₃⋊C₈	1	1	yes	characteristic
70	C₂²×C₄	C₂×Dic₈₃	1	2	yes	—
71	C₂×C₈	—	2	2	no	—
72	C₂×C₈	—	2	2	no	—
73	C₂×C₈	C₂×Dic₈₃	1	1	yes	characteristic
74	C₂×C₈	C₂×Dic₈₃	1	2	yes	—
75	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
76	C₄⋊C₄	C₈₃⋊C₈	1	1	yes	characteristic
77	C₂²⋊C₄	C₈₃⋊C₈	1	2	yes	—
78	C₄²	C₂×Dic₈₃	1	1	yes	characteristic
79	C₁₆	—	83	2	no	—
80	C₁₆	—	166	2	no	—
81	C₁₆	—	166	1	no	—
82	C₂³	C₂×C₈₃⋊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×C₈₃⋊C₈	1	1	yes	characteristic
87	C₂×C₄	C₂×C₈₃⋊C₈	1	2	yes	—
88	C₂×C₄	C₂×C₈₃⋊C₈	1	1	yes	characteristic
89	C₂×C₄	C₂²×Dic₈₃	1	1	yes	characteristic
90	C₈	—	2	1	no	—
91	C₈	—	2	2	no	—
92	C₈	D₄×D₈₃	1	1	yes	characteristic
93	C₈	#13050589	1	1	yes	characteristic
94	C₂²	—	2	2	no	—
95	C₂²	—	2	2	no	—
96	C₂²	—	2	2	no	—
97	C₂²	C₂²×C₈₃⋊C₈	1	1	yes	characteristic
98	C₄	—	2	1	no	—
99	C₄	—	2	2	no	—
100	C₄	#14581732	1	2	yes	complemented
101	C₄	D₄×Dic₈₃	1	1	yes	characteristic
102	C₄	#14581799	1	1	yes	characteristic
103	C₂	—	4	2	no	—
104	C₂	D₄×C₈₃⋊C₈	1	1	yes	characteristic
105	C₂	C₂×#14581732	1	1	yes	characteristic
106	C₂	#19128118	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14581732

C₂

yes

maximal, complemented

C₄⋊(C₈₃⋊C₁₆)

C₂

yes

maximal, characteristic, complemented

C₂²⋊(C₈₃⋊C₁₆)

C₂

yes

maximal, complemented

C₄×C₈₃⋊C₁₆

C₂

yes

maximal, characteristic, complemented

C₈×D₄×C₈₃

C₂

yes

maximal, characteristic

#14581732

—

C₂×C₈₃⋊C₁₆

C₂²

yes

—

C₂×C₈₃⋊C₁₆

C₂²

yes

characteristic

C₂×C₈₃⋊C₁₆

C₂²

yes

characteristic

C₂²×C₈×C₈₃

C₂²

yes

—

C₄×D₄×C₈₃

C₄

yes

characteristic

C₄⋊C₈×C₈₃

C₄

yes

characteristic

C₂²⋊C₈×C₈₃

C₄

yes

—

C₄×C₈×C₈₃

C₂²

yes

characteristic

C₈₃⋊C₁₆

—

C₈₃⋊C₁₆

—

C₈₃⋊C₁₆

D₄

yes

complemented

C₂×D₄×C₈₃

C₈

yes

characteristic

C₂²×C₄×C₈₃

C₂×C₄

yes

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

C₂×C₄

yes

characteristic

C₂×C₈×C₈₃

C₂×C₄

yes

—

C₂×C₈×C₈₃

C₂³

yes

characteristic

C₄⋊C₄×C₈₃

C₈

yes

characteristic

C₂²⋊C₄×C₈₃

C₈

yes

—

C₄²×C₈₃

C₂×C₄

yes

characteristic

C₂³×C₈₃

C₂×C₈

yes

—

D₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂×C₈

yes

characteristic

C₂×C₄×C₈₃

C₂×C₈

yes

—

C₂×C₄×C₈₃

C₂×C₈

yes

characteristic

C₂×C₄×C₈₃

C₂²×C₄

yes

characteristic

C₈×C₈₃

—

C₈×C₈₃

—

C₈×C₈₃

C₂×D₄

yes

characteristic

C₈×C₈₃

#41

yes

characteristic

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

C₂²×C₈

yes

characteristic

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

M₃₂

yes

complemented

C₄×C₈₃

C₄×D₄

yes

characteristic

C₄×C₈₃

#131

yes

characteristic

C₂×C₈₃

—

C₂×C₈₃

C₈×D₄

yes

characteristic

C₂×C₈₃

C₂×M₃₂

yes

C₂×C₈₃

yes

—

C₈₃

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₈×D₄

D₈₃

yes

characteristic

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

C₄×C₁₆

—

C₂²×C₈

C₂×D₈₃

yes

—

C₄×D₄

Dic₈₃

yes

characteristic

M₃₂

—

166

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₄⋊C₈

Dic₈₃

yes

characteristic

C₂²⋊C₈

Dic₈₃

yes

—

C₄×C₈

C₂×D₈₃

yes

characteristic

C₂×D₄

C₈₃⋊C₈

yes

characteristic

C₂²×C₄

C₂×Dic₈₃

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×Dic₈₃

yes

characteristic

C₂×C₈

C₂×Dic₈₃

yes

—

C₂×C₈

C₂²×D₈₃

yes

characteristic

C₄⋊C₄

C₈₃⋊C₈

yes

characteristic

C₂²⋊C₄

C₈₃⋊C₈

yes

—

C₄²

C₂×Dic₈₃

yes

characteristic

C₁₆

—

C₁₆

—

166

—

C₁₆

—

166

—

C₂³

C₂×C₈₃⋊C₈

yes

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₈₃⋊C₈

yes

characteristic

C₂×C₄

C₂×C₈₃⋊C₈

yes

—

C₂×C₄

C₂×C₈₃⋊C₈

yes

characteristic

C₂×C₄

C₂²×Dic₈₃

yes

characteristic

C₈

—

C₈

—

C₈

D₄×D₈₃

yes

characteristic

C₈

#13050589

yes

characteristic

C₂²

—

C₂²

—

C₂²

—

C₂²

C₂²×C₈₃⋊C₈

yes

characteristic

C₄

—

C₄

—

100

C₄

#14581732

yes

complemented

101

C₄

D₄×Dic₈₃

yes

characteristic

102

C₄

#14581799

yes

characteristic

103

C₂

—

104

C₂

D₄×C₈₃⋊C₈

yes

characteristic

105

C₂

C₂×#14581732

yes

characteristic

106

C₂

#19128118

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₈₃
commutator, socle	C₂²×C₈₃	C₂²×C₈
Fitting	C₈×D₄×C₈₃	C₂
prime radical	C₂×D₄×C₈₃	C₈
cosocle	C₂×C₈×C₈₃	C₂³
hypercenter	C₈×D₄	D₈₃
hypocenter	C₈₃	#2169
LCS	G ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₈ ◁ C₈×D₄

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×D₈₃

commutator, socle

C₂²×C₈₃

C₂²×C₈

Fitting

C₈×D₄×C₈₃

C₂

prime radical

C₂×D₄×C₈₃

C₈

cosocle

C₂×C₈×C₈₃

C₂³

hypercenter

C₈×D₄

D₈₃

C₈₃

G ▷ C₂²×C₈₃ ▷ C₈₃

C₁ ◁ C₂×C₈ ◁ C₈×D₄

N	H ≅ G/N	type
1. C₂×#14581732	C₂	semidirect
2. C₄⋊(C₈₃⋊C₁₆)	C₂	semidirect
3. C₂²⋊(C₈₃⋊C₁₆)	C₂	semidirect
4. C₄×C₈₃⋊C₁₆	C₂	semidirect
17. C₈₃⋊C₁₆	D₄	semidirect
46. C₄×C₈₃	M₃₂	semidirect
54. C₈₃	#2169	semidirect
100. C₄	#14581732	semidirect

H ≅ G/N

type

1. C₂×#14581732

C₂

semidirect

2. C₄⋊(C₈₃⋊C₁₆)

C₂

semidirect

3. C₂²⋊(C₈₃⋊C₁₆)

C₂

semidirect

4. C₄×C₈₃⋊C₁₆

C₂

semidirect

17. C₈₃⋊C₁₆

D₄

semidirect

46. C₄×C₈₃

M₃₂

semidirect

54. C₈₃

#2169

semidirect

100. C₄

#14581732

semidirect

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

subgroup	index
C₂×#14581732	2
C₄⋊(C₈₃⋊C₁₆)	2
C₂²⋊(C₈₃⋊C₁₆)	2
C₄×C₈₃⋊C₁₆	2
C₈×D₄×C₈₃	2
#2169	83

subgroup

index

C₂×#14581732

C₄⋊(C₈₃⋊C₁₆)

C₂²⋊(C₈₃⋊C₁₆)

C₄×C₈₃⋊C₁₆

C₈×D₄×C₈₃

#2169

Subgroups — #24909295

Conjugacy classes of subgroups

Complements

Maximal subgroups