Group #24906982 — subgroups

Subgroups — #24906982

Order 10624.

152 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 55 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	#19128106	C₂	1	1	yes	maximal, characteristic
2	#19127912	C₂	1	2	yes	maximal
3	#19127831	C₂	1	1	yes	maximal, characteristic
4	#19127448	C₂	1	2	yes	maximal
5	#334×C₈₃	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₃₃₂	—	2	2	no	—
7	#14581792	C₂²	1	2	yes	—
8	C₄×Dic₁₆₆	C₂²	1	1	yes	characteristic
9	#14581726	—	2	1	no	—
10	#14581726	—	2	1	no	—
11	C₄⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
12	Dic₈₃⋊C₈	—	2	1	no	—
13	#14581717	—	2	2	no	—
14	#14581702	—	2	1	no	—
15	C₄⋊C₈×C₈₃	C₂²	1	2	yes	—
16	C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
17	C₂×Dic₁₆₆	—	2	2	no	—
18	C₂×Dic₁₆₆	D₄	1	1	yes	characteristic
19	Dic₃₃₂	—	4	4	no	—
20	C₂×C₈₃⋊C₈	—	2	1	no	—
21	C₄⋊Dic₈₃	—	2	2	no	—
22	C₄⋊Dic₈₃	—	2	2	no	—
23	C₄⋊Dic₈₃	D₄	1	1	yes	characteristic
24	Dic₈₃⋊C₄	—	2	1	no	—
25	C₄×Dic₈₃	—	2	1	no	—
26	C₂×C₈×C₈₃	—	2	2	no	—
27	C₂×C₈×C₈₃	D₄	1	1	yes	characteristic
28	C₂×C₈×C₈₃	D₄	1	1	yes	characteristic
29	C₄²×C₈₃	C₂³	1	1	yes	characteristic
30	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
31	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₈₃	#41	1	1	yes	characteristic
33	C₂×Dic₈₃	—	2	1	no	—
34	C₂×Dic₈₃	—	2	1	no	—
35	C₂×Dic₈₃	—	4	2	no	—
36	Dic₁₆₆	—	2	2	no	—
37	Dic₁₆₆	—	4	2	no	—
38	Dic₁₆₆	—	4	2	no	—
39	C₈×C₈₃	—	2	1	no	—
40	C₈×C₈₃	—	4	2	no	—
41	C₈×C₈₃	Q₁₆	1	2	yes	complemented
42	C₈₃⋊C₈	—	4	1	no	—
43	C₂²×C₈₃	#121	1	1	yes	characteristic
44	C₄×C₈₃	—	2	1	no	—
45	C₄×C₈₃	C₂×Q₁₆	1	1	yes	characteristic
46	C₄×C₈₃	AGL(1,ℤ/8)	1	2	yes	—
47	C₄×C₈₃	#137	1	1	yes	characteristic
48	Dic₈₃	—	4	1	no	—
49	Dic₈₃	—	4	1	no	—
50	Dic₈₃	—	8	2	no	—
51	C₂×C₈₃	#462	1	1	yes	characteristic
52	C₂×C₈₃	#469	1	1	yes	characteristic
53	C₂×C₈₃	#473	1	1	yes	characteristic
54	#1694	—	83	1	no	maximal, Sylow
55	C₈₃	#1694	1	1	yes	characteristic, complemented, Sylow
56	#500	—	83	1	no	—
57	#462	—	83	2	no	—
58	#446	—	83	1	no	—
59	#334	D₈₃	1	1	yes	characteristic
60	#332	—	83	2	no	—
61	C₂×Q₁₆	—	166	2	no	—
62	C₄⋊Q₈	—	83	2	no	—
63	C₄×Q₈	—	83	1	no	—
64	#107	—	166	1	no	—
65	#107	—	166	1	no	—
66	C₄⋊C₈	—	83	1	no	—
67	C₄⋊C₈	—	166	1	no	—
68	C₄⋊C₈	C₂×D₈₃	1	2	yes	—
69	#103	—	166	2	no	—
70	#97	—	166	1	no	—
71	C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
72	C₂×Q₈	—	83	1	no	—
73	C₂×Q₈	—	166	2	no	—
74	Q₁₆	—	332	4	no	—
75	C₂×C₈	—	2	2	no	—
76	C₂×C₈	—	166	1	no	—
77	C₂×C₈	D₃₃₂	1	1	yes	characteristic
78	C₂×C₈	D₃₃₂	1	1	yes	characteristic
79	C₄⋊C₄	—	83	1	no	—
80	C₄⋊C₄	—	166	2	no	—
81	C₄⋊C₄	—	166	2	no	—
82	C₄⋊C₄	—	166	1	no	—
83	C₄²	—	166	1	no	—
84	C₄²	C₂²×D₈₃	1	1	yes	characteristic
85	Q₈	—	332	2	no	—
86	Q₈	—	332	2	no	—
87	Q₈	—	166	2	no	—
88	C₂×C₄	—	332	2	no	—
89	C₂×C₄	—	166	1	no	—
90	C₂×C₄	—	166	1	no	—
91	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
92	C₂×C₄	C₂×D₃₃₂	1	1	yes	characteristic
93	C₂×C₄	#13050591	1	1	yes	characteristic
94	C₈	—	2	1	no	—
95	C₈	—	4	2	no	—
96	C₈	—	332	1	no	—
97	C₈	Dic₃₃₂	1	2	yes	complemented
98	C₂²	#14581767	1	1	yes	characteristic
99	C₄	—	2	1	no	—
100	C₄	—	332	1	no	—
101	C₄	—	332	1	no	—
102	C₄	—	664	2	no	—
103	C₄	C₂×Dic₃₃₂	1	1	yes	characteristic
104	C₄	#14581819	1	2	yes	—
105	C₄	#14581825	1	1	yes	characteristic
106	C₂	#19127912	1	1	yes	characteristic
107	C₂	#19127955	1	1	yes	characteristic
108	C₂	#19127977	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19128106

C₂

yes

maximal, characteristic

#19127912

C₂

yes

maximal

#19127831

C₂

yes

maximal, characteristic

#19127448

C₂

yes

maximal

#334×C₈₃

C₂

yes

maximal, characteristic

C₂×Dic₃₃₂

—

#14581792

C₂²

yes

—

C₄×Dic₁₆₆

C₂²

yes

characteristic

#14581726

—

#14581726

—

C₄⋊(C₈₃⋊C₈)

C₂²

yes

characteristic

Dic₈₃⋊C₈

—

#14581717

—

#14581702

—

C₄⋊C₈×C₈₃

C₂²

yes

—

C₄×C₈×C₈₃

C₂²

yes

characteristic

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

D₄

yes

characteristic

Dic₃₃₂

—

C₂×C₈₃⋊C₈

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

D₄

yes

characteristic

Dic₈₃⋊C₄

—

C₄×Dic₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

D₄

yes

characteristic

C₂×C₈×C₈₃

D₄

yes

characteristic

C₄²×C₈₃

C₂³

yes

characteristic

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₈₃

#41

yes

characteristic

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

Dic₁₆₆

—

Dic₁₆₆

—

Dic₁₆₆

—

C₈×C₈₃

—

C₈×C₈₃

—

C₈×C₈₃

Q₁₆

yes

complemented

C₈₃⋊C₈

—

C₂²×C₈₃

#121

yes

characteristic

C₄×C₈₃

—

C₄×C₈₃

C₂×Q₁₆

yes

characteristic

C₄×C₈₃

AGL(1,ℤ/8)

yes

—

C₄×C₈₃

#137

yes

characteristic

Dic₈₃

—

Dic₈₃

—

Dic₈₃

—

C₂×C₈₃

yes

C₂×C₈₃

yes

C₂×C₈₃

yes

—

C₈₃

yes

characteristic, complemented, Sylow

—

—

—

D₈₃

yes

characteristic

#332

—

C₂×Q₁₆

—

166

—

C₄⋊Q₈

—

C₄×Q₈

—

#107

—

166

—

#107

—

166

—

C₄⋊C₈

—

C₄⋊C₈

—

166

—

C₄⋊C₈

C₂×D₈₃

yes

—

#103

—

166

—

#97

—

166

—

C₄×C₈

C₂×D₈₃

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

166

—

Q₁₆

—

332

—

C₂×C₈

—

C₂×C₈

—

166

—

C₂×C₈

D₃₃₂

yes

characteristic

C₂×C₈

D₃₃₂

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

166

—

C₄⋊C₄

—

166

—

C₄⋊C₄

—

166

—

C₄²

—

166

—

C₄²

C₂²×D₈₃

yes

characteristic

Q₈

—

332

—

Q₈

—

332

—

Q₈

—

166

—

C₂×C₄

—

332

—

C₂×C₄

—

166

—

C₂×C₄

—

166

—

C₂×C₄

D₄×D₈₃

yes

characteristic

C₂×C₄

C₂×D₃₃₂

yes

characteristic

C₂×C₄

#13050591

yes

characteristic

C₈

—

C₈

—

C₈

—

332

—

C₈

Dic₃₃₂

yes

complemented

C₂²

#14581767

yes

characteristic

C₄

—

100

C₄

—

332

—

101

C₄

—

332

—

102

C₄

—

664

—

103

C₄

C₂×Dic₃₃₂

yes

characteristic

104

C₄

#14581819

yes

—

105

C₄

#14581825

yes

characteristic

106

C₂

#19127912

yes

characteristic

107

C₂

#19127955

yes

characteristic

108

C₂

#19127977

yes

characteristic

	subgroup	G/H
center	C₂²	#14581767
commutator, cosocle	C₄²×C₈₃	C₂³
Frattini	C₄²	C₂²×D₈₃
Fitting	#334×C₈₃	C₂
socle, prime radical	C₂²×C₈₃	#121
hypercenter	#334	D₈₃
hypocenter	C₈₃	#1694
LCS	G ▷ C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #334

subgroup

G/H

center

C₂²

#14581767

commutator, cosocle

C₄²×C₈₃

C₂³

Frattini

C₄²

C₂²×D₈₃

Fitting

#334×C₈₃

C₂

C₂²×C₈₃

D₈₃

C₈₃

G ▷ C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #334

N	H ≅ G/N	type
41. C₈×C₈₃	Q₁₆	semidirect
55. C₈₃	#1694	semidirect
97. C₈	Dic₃₃₂	semidirect

H ≅ G/N

type

41. C₈×C₈₃

Q₁₆

semidirect

55. C₈₃

#1694

semidirect

97. C₈

Dic₃₃₂

semidirect

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

subgroup	index
#19128106	2
#19127912	2
#19127831	2
#19127448	2
#334×C₈₃	2
#1694	83

subgroup

index

#334×C₈₃

Subgroups — #24906982

Conjugacy classes of subgroups

Complements

Maximal subgroups