Group #24909514 — subgroups

Subgroups — #24909514

Order 10624.

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

normal subgroups only

This group has 154 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	C₂×Dic₆₆₄	C₂	1	2	yes	maximal
2	#19128081	C₂	1	1	yes	maximal, characteristic
3	C₄×Dic₃₃₂	C₂	1	1	yes	maximal, characteristic
4	#19127506	C₂	1	2	yes	maximal
5	C₄⋊C₁₆×C₈₃	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₃₃₂	—	2	1	no	—
7	C₂×Dic₃₃₂	C₂²	1	2	yes	—
8	C₂×Dic₃₃₂	C₂²	1	1	yes	characteristic
9	#14581792	—	2	1	no	—
10	C₄×Dic₁₆₆	—	2	1	no	—
11	Dic₆₆₄	—	2	4	no	—
12	#14581726	C₂²	1	1	yes	characteristic
13	#14581717	—	2	1	no	—
14	C₂×C₁₆×C₈₃	C₂²	1	2	yes	—
15	C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
16	C₂×Dic₁₆₆	—	2	1	no	—
17	C₂×Dic₁₆₆	—	2	2	no	—
18	Dic₃₃₂	—	2	2	no	—
19	Dic₃₃₂	—	2	1	no	—
20	Dic₃₃₂	—	2	2	no	—
21	Dic₃₃₂	—	4	2	no	—
22	Dic₃₃₂	D₄	1	2	yes	—
23	C₄⋊Dic₈₃	—	2	1	no	—
24	C₄⋊Dic₈₃	—	4	1	no	—
25	C₄⋊Dic₈₃	—	4	1	no	—
26	Dic₈₃⋊C₄	—	4	1	no	—
27	C₄×Dic₈₃	—	4	1	no	—
28	C₂×C₈×C₈₃	D₄	1	1	yes	characteristic
29	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
30	C₄²×C₈₃	D₄	1	1	yes	characteristic
31	C₁₆×C₈₃	—	2	2	no	—
32	C₂×C₄×C₈₃	D₈	1	1	yes	characteristic
33	C₂×C₄×C₈₃	D₈	1	1	yes	characteristic
34	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
35	C₂×Dic₈₃	—	4	1	no	—
36	C₂×Dic₈₃	—	4	1	no	—
37	C₂×Dic₈₃	—	4	2	no	—
38	Dic₁₆₆	—	2	2	no	—
39	Dic₁₆₆	—	4	1	no	—
40	Dic₁₆₆	—	4	2	no	—
41	Dic₁₆₆	—	4	2	no	—
42	C₈×C₈₃	—	2	1	no	—
43	C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
44	C₈×C₈₃	#41	1	1	yes	characteristic
45	C₂²×C₈₃	C₂×D₈	1	1	yes	characteristic
46	C₄×C₈₃	—	2	1	no	—
47	C₄×C₈₃	Q₃₂	1	2	yes	complemented
48	C₄×C₈₃	#121	1	1	yes	characteristic
49	C₄×C₈₃	AGL(1,ℤ/8)	1	1	yes	characteristic
50	Dic₈₃	—	4	2	no	—
51	Dic₈₃	—	8	2	no	—
52	Dic₈₃	—	8	1	no	—
53	C₂×C₈₃	#459	1	1	yes	characteristic
54	C₂×C₈₃	C₂×Q₃₂	1	1	yes	characteristic
55	C₂×C₈₃	#510	1	1	yes	characteristic
56	#2209	—	83	1	no	maximal, Sylow
57	C₈₃	#2209	1	1	yes	characteristic, complemented, Sylow
58	C₂×Q₃₂	—	83	2	no	—
59	#494	—	83	1	no	—
60	C₄×Q₁₆	—	83	1	no	—
61	C₄⋊C₁₆	D₈₃	1	1	yes	characteristic
62	#358	—	83	2	no	—
63	C₂×Q₁₆	—	83	1	no	—
64	C₂×Q₁₆	—	83	2	no	—
65	C₂×Q₁₆	—	166	1	no	—
66	C₄⋊Q₈	—	166	1	no	—
67	C₄×Q₈	—	166	1	no	—
68	Q₃₂	—	166	4	no	—
69	C₂×C₁₆	C₂×D₈₃	1	2	yes	—
70	#107	—	83	1	no	—
71	#103	—	166	1	no	—
72	C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
73	C₂×Q₈	—	166	2	no	—
74	C₂×Q₈	—	166	1	no	—
75	Q₁₆	—	332	2	no	—
76	Q₁₆	—	83	2	no	—
77	Q₁₆	—	166	2	no	—
78	Q₁₆	—	166	2	no	—
79	Q₁₆	—	166	1	no	—
80	C₂×C₈	D₃₃₂	1	1	yes	characteristic
81	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
82	C₄⋊C₄	—	332	1	no	—
83	C₄⋊C₄	—	332	1	no	—
84	C₄⋊C₄	—	332	1	no	—
85	C₄⋊C₄	—	166	1	no	—
86	C₄²	—	332	1	no	—
87	C₄²	D₃₃₂	1	1	yes	characteristic
88	C₁₆	—	2	2	no	—
89	Q₈	—	332	1	no	—
90	Q₈	—	332	2	no	—
91	Q₈	—	332	2	no	—
92	Q₈	—	166	2	no	—
93	C₂×C₄	—	332	1	no	—
94	C₂×C₄	—	332	1	no	—
95	C₂×C₄	—	332	2	no	—
96	C₂×C₄	D₆₆₄	1	1	yes	characteristic
97	C₂×C₄	D₆₆₄	1	1	yes	characteristic
98	C₂×C₄	C₂×D₃₃₂	1	1	yes	characteristic
99	C₈	—	2	1	no	—
100	C₈	D₄×D₈₃	1	1	yes	characteristic
101	C₈	#13050591	1	1	yes	characteristic
102	C₂²	C₂×D₆₆₄	1	1	yes	characteristic
103	C₄	—	2	1	no	—
104	C₄	—	332	2	no	—
105	C₄	—	664	1	no	—
106	C₄	—	664	2	no	—
107	C₄	Dic₆₆₄	1	2	yes	complemented
108	C₄	#14581767	1	1	yes	characteristic
109	C₄	#14581820	1	1	yes	characteristic
110	C₂	#19127897	1	1	yes	characteristic
111	C₂	C₂×Dic₆₆₄	1	1	yes	characteristic
112	C₂	#19128144	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Dic₆₆₄

C₂

yes

maximal

#19128081

C₂

yes

maximal, characteristic

C₄×Dic₃₃₂

C₂

yes

maximal, characteristic

#19127506

C₂

yes

maximal

C₄⋊C₁₆×C₈₃

C₂

yes

maximal, characteristic

C₂×Dic₃₃₂

—

C₂×Dic₃₃₂

C₂²

yes

—

C₂×Dic₃₃₂

C₂²

yes

characteristic

#14581792

—

C₄×Dic₁₆₆

—

Dic₆₆₄

—

#14581726

C₂²

yes

characteristic

#14581717

—

C₂×C₁₆×C₈₃

C₂²

yes

—

C₄×C₈×C₈₃

C₂²

yes

characteristic

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

—

Dic₃₃₂

—

Dic₃₃₂

—

Dic₃₃₂

—

Dic₃₃₂

—

Dic₃₃₂

D₄

yes

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

Dic₈₃⋊C₄

—

C₄×Dic₈₃

—

C₂×C₈×C₈₃

D₄

yes

characteristic

C₂×C₈×C₈₃

C₂³

yes

characteristic

C₄²×C₈₃

D₄

yes

characteristic

C₁₆×C₈₃

—

C₂×C₄×C₈₃

D₈

yes

characteristic

C₂×C₄×C₈₃

D₈

yes

characteristic

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

Dic₁₆₆

—

Dic₁₆₆

—

Dic₁₆₆

—

Dic₁₆₆

—

C₈×C₈₃

—

C₈×C₈₃

C₂×D₄

yes

characteristic

C₈×C₈₃

#41

yes

characteristic

C₂²×C₈₃

C₂×D₈

yes

characteristic

C₄×C₈₃

—

C₄×C₈₃

Q₃₂

yes

C₄×C₈₃

yes

C₄×C₈₃

yes

Dic₈₃

—

Dic₈₃

—

Dic₈₃

—

C₂×C₈₃

#459

yes

characteristic

C₂×C₈₃

C₂×Q₃₂

yes

C₂×C₈₃

yes

—

C₈₃

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

#494

—

C₄×Q₁₆

—

C₄⋊C₁₆

D₈₃

yes

characteristic

#358

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

166

—

C₄⋊Q₈

—

166

—

C₄×Q₈

—

166

—

Q₃₂

—

166

—

C₂×C₁₆

C₂×D₈₃

yes

—

#107

—

#103

—

166

—

C₄×C₈

C₂×D₈₃

yes

characteristic

C₂×Q₈

—

166

—

C₂×Q₈

—

166

—

Q₁₆

—

332

—

Q₁₆

—

Q₁₆

—

166

—

Q₁₆

—

166

—

Q₁₆

—

166

—

C₂×C₈

D₃₃₂

yes

characteristic

C₂×C₈

C₂²×D₈₃

yes

characteristic

C₄⋊C₄

—

332

—

C₄⋊C₄

—

332

—

C₄⋊C₄

—

332

—

C₄⋊C₄

—

166

—

C₄²

—

332

—

C₄²

D₃₃₂

yes

characteristic

C₁₆

—

Q₈

—

332

—

Q₈

—

332

—

Q₈

—

332

—

Q₈

—

166

—

C₂×C₄

—

332

—

C₂×C₄

—

332

—

C₂×C₄

—

332

—

C₂×C₄

D₆₆₄

yes

characteristic

C₂×C₄

D₆₆₄

yes

characteristic

C₂×C₄

C₂×D₃₃₂

yes

characteristic

C₈

—

100

C₈

D₄×D₈₃

yes

characteristic

101

C₈

#13050591

yes

characteristic

102

C₂²

C₂×D₆₆₄

yes

characteristic

103

C₄

—

104

C₄

—

332

—

105

C₄

—

664

—

106

C₄

—

664

—

107

C₄

Dic₆₆₄

yes

complemented

108

C₄

#14581767

yes

characteristic

109

C₄

#14581820

yes

characteristic

110

C₂

#19127897

yes

characteristic

111

C₂

C₂×Dic₆₆₄

yes

characteristic

112

C₂

#19128144

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₆₆₄
commutator, cosocle	C₂×C₈×C₈₃	C₂³
Frattini	C₂×C₈	C₂²×D₈₃
Fitting	C₄⋊C₁₆×C₈₃	C₂
socle, prime radical	C₂²×C₈₃	C₂×D₈
hypercenter	C₄⋊C₁₆	D₈₃
hypocenter	C₈₃	#2209
LCS	G ▷ C₂×C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄⋊C₁₆

subgroup

G/H

center

C₂²

C₂×D₆₆₄

commutator, cosocle

C₂×C₈×C₈₃

C₂³

Frattini

C₂×C₈

C₂²×D₈₃

Fitting

C₄⋊C₁₆×C₈₃

C₂

socle, prime radical

C₂²×C₈₃

C₂×D₈

hypercenter

C₄⋊C₁₆

D₈₃

hypocenter

C₈₃

#2209

LCS

G ▷ C₂×C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄⋊C₁₆

N	H ≅ G/N	type
47. C₄×C₈₃	Q₃₂	semidirect
57. C₈₃	#2209	semidirect
107. C₄	Dic₆₆₄	semidirect

H ≅ G/N

type

47. C₄×C₈₃

Q₃₂

semidirect

57. C₈₃

#2209

semidirect

107. C₄

Dic₆₆₄

semidirect

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

subgroup	index
C₂×Dic₆₆₄	2
#19128081	2
C₄×Dic₃₃₂	2
#19127506	2
C₄⋊C₁₆×C₈₃	2
#2209	83

subgroup

index

C₂×Dic₆₆₄

#19128081

C₄×Dic₃₃₂

#19127506

C₄⋊C₁₆×C₈₃

#2209

Subgroups — #24909514

Conjugacy classes of subgroups

Complements

Maximal subgroups