Group #24906359 — subgroups

Subgroups — #24906359

Order 10624.

152 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 65 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	#19127828	C₂	1	2	yes	maximal
2	C₄×Dic₃₃₂	C₂	1	1	yes	maximal, characteristic
3	#19127431	C₂	1	2	yes	maximal
4	C₈×C₈₃⋊C₈	C₂	1	1	yes	maximal, characteristic
5	#335×C₈₃	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₃₃₂	C₄	1	1	yes	characteristic
7	C₄×Dic₁₆₆	C₂²	1	2	yes	—
8	#14581726	C₄	1	1	yes	characteristic
9	Dic₈₃⋊C₈	—	2	2	no	—
10	#14581717	C₄	1	2	yes	—
11	#14581702	—	2	2	no	—
12	C₄×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
13	C₄×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
14	C₄⋊C₈×C₈₃	C₂²	1	2	yes	—
15	C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
16	C₂×Dic₁₆₆	C₂×C₄	1	2	yes	—
17	Dic₃₃₂	—	2	2	no	—
18	C₂×C₈₃⋊C₈	—	2	1	no	—
19	C₂×C₈₃⋊C₈	D₄	1	1	yes	characteristic
20	C₂×C₈₃⋊C₈	D₄	1	1	yes	characteristic
21	C₄⋊Dic₈₃	C₂×C₄	1	2	yes	—
22	Dic₈₃⋊C₄	—	2	2	no	—
23	C₄×Dic₈₃	—	2	2	no	—
24	C₂×C₈×C₈₃	—	2	2	no	—
25	C₂×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
26	C₂×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
27	C₄²×C₈₃	C₂³	1	1	yes	characteristic
28	C₂×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
29	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
30	C₂×C₄×C₈₃	#41	1	1	yes	characteristic
31	C₂×Dic₈₃	—	2	2	no	—
32	C₂×Dic₈₃	—	2	2	no	—
33	Dic₁₆₆	—	2	2	no	—
34	Dic₁₆₆	—	2	2	no	—
35	C₈×C₈₃	—	2	1	no	—
36	C₈×C₈₃	—	4	2	no	—
37	C₈×C₈₃	M₁₆	1	2	yes	—
38	C₈₃⋊C₈	—	2	2	no	—
39	C₈₃⋊C₈	—	2	1	no	—
40	C₈₃⋊C₈	Q₁₆	1	2	yes	complemented
41	C₂²×C₈₃	C₄×D₄	1	1	yes	characteristic
42	C₄×C₈₃	—	2	1	no	—
43	C₄×C₈₃	C₂×M₁₆	1	1	yes	characteristic
44	C₄×C₈₃	#131	1	1	yes	characteristic
45	C₄×C₈₃	C₂×Q₁₆	1	1	yes	characteristic
46	C₄×C₈₃	#135	1	1	yes	characteristic
47	Dic₈₃	—	4	2	no	—
48	Dic₈₃	—	4	2	no	—
49	C₂×C₈₃	#436	1	1	yes	characteristic
50	C₂×C₈₃	C₄×Q₁₆	1	1	yes	characteristic
51	C₂×C₈₃	#443	1	1	yes	characteristic
52	#1591	—	83	1	no	maximal, Sylow
53	C₈₃	#1591	1	1	yes	characteristic, complemented, Sylow
54	#446	—	83	2	no	—
55	C₄×Q₁₆	—	83	1	no	—
56	#335	D₈₃	1	1	yes	characteristic
57	Q₈⋊C₈	—	83	2	no	—
58	C₈²	—	83	1	no	—
59	C₂×Q₁₆	—	83	1	no	—
60	C₄×Q₈	—	83	2	no	—
61	#107	—	83	1	no	—
62	C₄⋊C₈	—	166	2	no	—
63	C₄⋊C₈	C₂×D₈₃	1	2	yes	—
64	#103	—	83	2	no	—
65	#97	—	166	2	no	—
66	C₄×C₈	—	83	1	no	—
67	C₄×C₈	—	83	1	no	—
68	C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
69	C₂×Q₈	—	83	2	no	—
70	Q₁₆	—	166	2	no	—
71	C₂×C₈	—	2	2	no	—
72	C₂×C₈	—	83	1	no	—
73	C₂×C₈	—	83	1	no	—
74	C₂×C₈	—	166	1	no	—
75	C₂×C₈	C₄×D₈₃	1	1	yes	characteristic
76	C₂×C₈	C₄×D₈₃	1	1	yes	characteristic
77	C₄⋊C₄	—	83	2	no	—
78	C₄⋊C₄	—	166	2	no	—
79	C₄²	—	166	2	no	—
80	C₄²	C₂²×D₈₃	1	1	yes	characteristic
81	Q₈	—	166	2	no	—
82	Q₈	—	166	2	no	—
83	C₂×C₄	—	166	2	no	—
84	C₂×C₄	—	166	2	no	—
85	C₂×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
86	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
87	C₂×C₄	#13050591	1	1	yes	characteristic
88	C₈	—	2	1	no	—
89	C₈	—	4	2	no	—
90	C₈	—	83	2	no	—
91	C₈	—	166	2	no	—
92	C₈	—	166	1	no	—
93	C₈	#13050573	1	2	yes	—
94	C₂²	#14581754	1	1	yes	characteristic
95	C₄	—	2	1	no	—
96	C₄	—	332	2	no	—
97	C₄	—	332	2	no	—
98	C₄	C₂×#13050573	1	1	yes	characteristic
99	C₄	#14581800	1	1	yes	characteristic
100	C₄	Q₁₆×D₈₃	1	1	yes	characteristic
101	C₄	#14581811	1	1	yes	characteristic
102	C₂	#19127778	1	1	yes	characteristic
103	C₂	#19127795	1	1	yes	characteristic
104	C₂	#19127817	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19127828

C₂

yes

maximal

C₄×Dic₃₃₂

C₂

yes

maximal, characteristic

#19127431

C₂

yes

maximal

C₈×C₈₃⋊C₈

C₂

yes

maximal, characteristic

#335×C₈₃

C₂

yes

maximal, characteristic

C₂×Dic₃₃₂

C₄

yes

characteristic

C₄×Dic₁₆₆

C₂²

yes

—

#14581726

C₄

yes

characteristic

Dic₈₃⋊C₈

—

#14581717

C₄

yes

—

#14581702

—

C₄×C₈₃⋊C₈

C₂²

yes

characteristic

C₄×C₈₃⋊C₈

C₂²

yes

characteristic

C₄⋊C₈×C₈₃

C₂²

yes

—

C₄×C₈×C₈₃

C₂²

yes

characteristic

C₂×Dic₁₆₆

C₂×C₄

yes

—

Dic₃₃₂

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

D₄

yes

characteristic

C₂×C₈₃⋊C₈

D₄

yes

characteristic

C₄⋊Dic₈₃

C₂×C₄

yes

—

Dic₈₃⋊C₄

—

C₄×Dic₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

C₂×C₄

yes

characteristic

C₂×C₈×C₈₃

C₂×C₄

yes

characteristic

C₄²×C₈₃

C₂³

yes

characteristic

C₂×C₄×C₈₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₈₃

#41

yes

characteristic

C₂×Dic₈₃

—

C₂×Dic₈₃

—

Dic₁₆₆

—

Dic₁₆₆

—

C₈×C₈₃

—

C₈×C₈₃

—

C₈×C₈₃

M₁₆

yes

—

C₈₃⋊C₈

—

C₈₃⋊C₈

—

C₈₃⋊C₈

Q₁₆

yes

complemented

C₂²×C₈₃

C₄×D₄

yes

characteristic

C₄×C₈₃

—

C₄×C₈₃

C₂×M₁₆

yes

characteristic

C₄×C₈₃

#131

yes

characteristic

C₄×C₈₃

C₂×Q₁₆

yes

characteristic

C₄×C₈₃

#135

yes

characteristic

Dic₈₃

—

Dic₈₃

—

C₂×C₈₃

#436

yes

characteristic

C₂×C₈₃

C₄×Q₁₆

yes

C₂×C₈₃

yes

—

C₈₃

yes

characteristic, complemented, Sylow

#446

—

C₄×Q₁₆

—

#335

D₈₃

yes

characteristic

Q₈⋊C₈

—

C₈²

—

C₂×Q₁₆

—

C₄×Q₈

—

#107

—

C₄⋊C₈

—

166

—

C₄⋊C₈

C₂×D₈₃

yes

—

#103

—

#97

—

166

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₈₃

yes

characteristic

C₂×Q₈

—

Q₁₆

—

166

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

166

—

C₂×C₈

C₄×D₈₃

yes

characteristic

C₂×C₈

C₄×D₈₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

166

—

C₄²

—

166

—

C₄²

C₂²×D₈₃

yes

characteristic

Q₈

—

166

—

Q₈

—

166

—

C₂×C₄

—

166

—

C₂×C₄

—

166

—

C₂×C₄

C₂×C₄×D₈₃

yes

characteristic

C₂×C₄

D₄×D₈₃

yes

characteristic

C₂×C₄

#13050591

yes

characteristic

C₈

—

C₈

—

C₈

—

C₈

—

166

—

C₈

—

166

—

C₈

#13050573

yes

—

C₂²

#14581754

yes

characteristic

C₄

—

C₄

—

332

—

C₄

—

332

—

C₄

yes

C₄

yes

100

C₄

Q₁₆×D₈₃

yes

characteristic

101

C₄

#14581811

yes

characteristic

102

C₂

#19127778

yes

characteristic

103

C₂

#19127795

yes

characteristic

104

C₂

#19127817

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₈₃
commutator	C₂×C₄×C₈₃	C₂²×C₄
Frattini	C₄²	C₂²×D₈₃
Fitting	#335×C₈₃	C₂
socle, prime radical	C₂²×C₈₃	C₄×D₄
cosocle	C₄²×C₈₃	C₂³
hypercenter	#335	D₈₃
hypocenter	C₈₃	#1591
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ #335

subgroup

G/H

center

C₂×C₄

D₄×D₈₃

commutator

C₂×C₄×C₈₃

C₂²×C₄

Frattini

C₄²

C₂²×D₈₃

Fitting

#335×C₈₃

C₂

socle, prime radical

C₂²×C₈₃

C₄×D₄

cosocle

C₄²×C₈₃

C₂³

D₈₃

C₈₃

G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ #335

N	H ≅ G/N	type
40. C₈₃⋊C₈	Q₁₆	semidirect
53. C₈₃	#1591	semidirect

H ≅ G/N

type

40. C₈₃⋊C₈

Q₁₆

semidirect

53. C₈₃

#1591

semidirect

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

subgroup	index
#19127828	2
C₄×Dic₃₃₂	2
#19127431	2
C₈×C₈₃⋊C₈	2
#335×C₈₃	2
#1591	83

subgroup

index

#19127828

C₄×Dic₃₃₂

#19127431

C₈×C₈₃⋊C₈

#335×C₈₃

#1591

Subgroups — #24906359

Conjugacy classes of subgroups

Complements

Maximal subgroups