Group #24906830 — subgroups

Subgroups — #24906830

Order 10624.

156 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 156 conjugacy classes of subgroups — too many to draw. Only its 57 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	#19127983	C₂	1	2	yes	maximal
2	C₈×Dic₁₆₆	C₂	1	1	yes	maximal, characteristic
3	#19127454	C₂	1	1	yes	maximal, characteristic
4	#19127447	C₂	1	2	yes	maximal
5	#494×C₈₃	C₂	1	1	yes	maximal, characteristic
6	C₄×Dic₁₆₆	C₂²	1	1	yes	characteristic
7	#14581723	—	2	2	no	—
8	C₄⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
9	C₄⋊(C₈₃⋊C₈)	C₂²	1	2	yes	—
10	Dic₈₃⋊C₈	—	2	1	no	—
11	#14581715	—	2	2	no	—
12	C₈×Dic₈₃	—	2	1	no	—
13	C₂×Q₁₆×C₈₃	—	2	1	no	—
14	C₂×Q₁₆×C₈₃	—	2	1	no	—
15	C₄⋊Q₈×C₈₃	C₂²	1	2	yes	—
16	C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
17	C₂×Dic₁₆₆	D₄	1	1	yes	characteristic
18	C₂×C₈₃⋊C₈	—	2	1	no	—
19	C₂×C₈₃⋊C₈	—	2	2	no	—
20	C₄⋊Dic₈₃	D₄	1	1	yes	characteristic
21	Dic₈₃⋊C₄	—	2	1	no	—
22	C₄×Dic₈₃	—	2	1	no	—
23	C₂×Q₈×C₈₃	—	2	2	no	—
24	Q₁₆×C₈₃	—	4	2	no	—
25	Q₁₆×C₈₃	—	4	2	no	—
26	C₂×C₈×C₈₃	D₄	1	1	yes	characteristic
27	C₂×C₈×C₈₃	D₄	1	1	yes	characteristic
28	C₄⋊C₄×C₈₃	—	2	2	no	—
29	C₄⋊C₄×C₈₃	—	2	2	no	—
30	C₄²×C₈₃	C₂³	1	1	yes	characteristic
31	Q₈×C₈₃	—	4	2	no	—
32	Q₈×C₈₃	—	4	2	no	—
33	C₂×C₄×C₈₃	—	4	2	no	—
34	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₈₃	#41	1	1	yes	characteristic
37	C₂×Dic₈₃	—	2	1	no	—
38	C₂×Dic₈₃	—	2	1	no	—
39	Dic₁₆₆	—	2	1	no	—
40	Dic₁₆₆	Q₁₆	1	2	yes	complemented
41	C₈×C₈₃	—	2	1	no	—
42	C₈×C₈₃	SD₁₆	1	2	yes	—
43	C₈₃⋊C₈	—	4	2	no	—
44	C₈₃⋊C₈	—	4	1	no	—
45	C₂²×C₈₃	#121	1	1	yes	characteristic
46	C₄×C₈₃	—	2	1	no	—
47	C₄×C₈₃	—	8	2	no	—
48	C₄×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
49	C₄×C₈₃	C₂×Q₁₆	1	1	yes	characteristic
50	C₄×C₈₃	#135	1	1	yes	characteristic
51	C₄×C₈₃	AGL(1,ℤ/8)	1	1	yes	characteristic
52	Dic₈₃	—	2	2	no	—
53	Dic₈₃	—	4	1	no	—
54	C₂×C₈₃	#460	1	1	yes	characteristic
55	C₂×C₈₃	#467	1	1	yes	characteristic
56	C₂×C₈₃	#473	1	1	yes	characteristic
57	#1670	—	83	1	no	maximal, Sylow
58	C₈₃	#1670	1	1	yes	characteristic, complemented, Sylow
59	#494	D₈₃	1	1	yes	characteristic
60	#475	—	83	2	no	—
61	C₈×Q₈	—	83	1	no	—
62	#335	—	83	1	no	—
63	#332	—	83	2	no	—
64	C₂×Q₁₆	—	2	1	no	—
65	C₂×Q₁₆	—	2	1	no	—
66	C₄⋊Q₈	C₂×D₈₃	1	2	yes	—
67	C₄×Q₈	—	83	1	no	—
68	#106	—	166	2	no	—
69	C₄⋊C₈	—	83	1	no	—
70	C₄⋊C₈	—	83	2	no	—
71	C₄⋊C₈	—	166	1	no	—
72	#103	—	166	2	no	—
73	C₄×C₈	—	166	1	no	—
74	C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
75	C₂×Q₈	—	2	2	no	—
76	C₂×Q₈	—	83	1	no	—
77	Q₁₆	—	4	2	no	—
78	Q₁₆	—	4	2	no	—
79	C₂×C₈	—	166	2	no	—
80	C₂×C₈	—	166	1	no	—
81	C₂×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
82	C₂×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
83	C₄⋊C₄	—	2	2	no	—
84	C₄⋊C₄	—	2	2	no	—
85	C₄⋊C₄	—	83	1	no	—
86	C₄⋊C₄	—	166	1	no	—
87	C₄²	—	166	1	no	—
88	C₄²	C₂²×D₈₃	1	1	yes	characteristic
89	Q₈	—	4	2	no	—
90	Q₈	—	4	2	no	—
91	Q₈	—	83	2	no	—
92	Q₈	—	166	1	no	—
93	C₂×C₄	—	4	2	no	—
94	C₂×C₄	—	166	1	no	—
95	C₂×C₄	—	166	1	no	—
96	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
97	C₂×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
98	C₂×C₄	#13050589	1	1	yes	characteristic
99	C₈	—	2	1	no	—
100	C₈	—	332	2	no	—
101	C₈	—	332	1	no	—
102	C₈	#13050578	1	2	yes	—
103	C₂²	#14581763	1	1	yes	characteristic
104	C₄	—	2	1	no	—
105	C₄	—	8	2	no	—
106	C₄	—	332	1	no	—
107	C₄	—	166	2	no	—
108	C₄	C₂×#13050578	1	1	yes	characteristic
109	C₄	Q₁₆×D₈₃	1	1	yes	characteristic
110	C₄	#14581815	1	1	yes	characteristic
111	C₄	#14581818	1	1	yes	characteristic
112	C₂	#19127898	1	1	yes	characteristic
113	C₂	#19127939	1	1	yes	characteristic
114	C₂	#19127973	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19127983

C₂

yes

maximal

C₈×Dic₁₆₆

C₂

yes

maximal, characteristic

#19127454

C₂

yes

maximal, characteristic

#19127447

C₂

yes

maximal

#494×C₈₃

C₂

yes

maximal, characteristic

C₄×Dic₁₆₆

C₂²

yes

characteristic

#14581723

—

C₄⋊(C₈₃⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₈₃⋊C₈)

C₂²

yes

—

Dic₈₃⋊C₈

—

#14581715

—

C₈×Dic₈₃

—

C₂×Q₁₆×C₈₃

—

C₂×Q₁₆×C₈₃

—

C₄⋊Q₈×C₈₃

C₂²

yes

—

C₄×C₈×C₈₃

C₂²

yes

characteristic

C₂×Dic₁₆₆

D₄

yes

characteristic

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₄⋊Dic₈₃

D₄

yes

characteristic

Dic₈₃⋊C₄

—

C₄×Dic₈₃

—

C₂×Q₈×C₈₃

—

Q₁₆×C₈₃

—

Q₁₆×C₈₃

—

C₂×C₈×C₈₃

D₄

yes

characteristic

C₂×C₈×C₈₃

D₄

yes

characteristic

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

—

C₄²×C₈₃

C₂³

yes

characteristic

Q₈×C₈₃

—

Q₈×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₈₃

#41

yes

characteristic

C₂×Dic₈₃

—

C₂×Dic₈₃

—

Dic₁₆₆

—

Dic₁₆₆

Q₁₆

yes

complemented

C₈×C₈₃

—

C₈×C₈₃

SD₁₆

yes

—

C₈₃⋊C₈

—

C₈₃⋊C₈

—

C₂²×C₈₃

#121

yes

characteristic

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

C₂×SD₁₆

yes

characteristic

C₄×C₈₃

C₂×Q₁₆

yes

C₄×C₈₃

yes

C₄×C₈₃

yes

Dic₈₃

—

Dic₈₃

—

C₂×C₈₃

yes

C₂×C₈₃

yes

C₂×C₈₃

yes

—

C₈₃

yes

characteristic, complemented, Sylow

#494

D₈₃

yes

characteristic

#475

—

C₈×Q₈

—

#335

—

#332

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

C₂×D₈₃

yes

—

C₄×Q₈

—

#106

—

166

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

166

—

#103

—

166

—

C₄×C₈

—

166

—

C₄×C₈

C₂×D₈₃

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

Q₁₆

—

Q₁₆

—

C₂×C₈

—

166

—

C₂×C₈

—

166

—

C₂×C₈

C₂²⋊D₈₃

yes

characteristic

C₂×C₈

C₂²⋊D₈₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

166

—

C₄²

—

166

—

C₄²

C₂²×D₈₃

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

Q₈

—

166

—

C₂×C₄

—

C₂×C₄

—

166

—

C₂×C₄

—

166

—

C₂×C₄

D₄×D₈₃

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₈₃

yes

characteristic

C₂×C₄

#13050589

yes

characteristic

C₈

—

100

C₈

—

332

—

101

C₈

—

332

—

102

C₈

#13050578

yes

—

103

C₂²

#14581763

yes

characteristic

104

C₄

—

105

C₄

—

106

C₄

—

332

—

107

C₄

—

166

—

108

C₄

C₂×#13050578

yes

characteristic

109

C₄

Q₁₆×D₈₃

yes

characteristic

110

C₄

#14581815

yes

characteristic

111

C₄

#14581818

yes

characteristic

112

C₂

#19127898

yes

characteristic

113

C₂

#19127939

yes

characteristic

114

C₂

#19127973

yes

characteristic

	subgroup	G/H
center	C₂²	#14581763
commutator, cosocle	C₄²×C₈₃	C₂³
Frattini	C₄²	C₂²×D₈₃
Fitting	#494×C₈₃	C₂
socle, prime radical	C₂²×C₈₃	#121
hypercenter	#494	D₈₃
hypocenter	C₈₃	#1670
LCS	G ▷ C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

subgroup

G/H

center

C₂²

#14581763

commutator, cosocle

C₄²×C₈₃

C₂³

Frattini

C₄²

C₂²×D₈₃

Fitting

#494×C₈₃

C₂

C₂²×C₈₃

D₈₃

C₈₃

G ▷ C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #494

N	H ≅ G/N	type
40. Dic₁₆₆	Q₁₆	semidirect
58. C₈₃	#1670	semidirect

H ≅ G/N

type

40. Dic₁₆₆

Q₁₆

semidirect

58. C₈₃

#1670

semidirect

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

subgroup	index
#19127983	2
C₈×Dic₁₆₆	2
#19127454	2
#19127447	2
#494×C₈₃	2
#1670	83

subgroup

index

C₈×Dic₁₆₆

#494×C₈₃

Subgroups — #24906830

Conjugacy classes of subgroups

Complements

Maximal subgroups