Group #24909480 — subgroups

Subgroups — #24909480

Order 10624.

248 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 45 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	#19127873	C₂	1	3	yes	maximal, complemented
2	#19127498	C₂	1	3	yes	maximal, complemented
3	H(3,𝔽₄)×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#14581840	C₂²	1	1	yes	characteristic, complemented
5	#14581818	—	2	3	no	—
6	#14581720	—	2	3	no	—
7	#14581711	C₂²	1	3	yes	complemented
8	#124×C₈₃	—	2	3	no	—
9	#124×C₈₃	C₂²	1	3	yes	—
10	#120×C₈₃	—	2	3	no	—
11	#13050591	—	4	1	no	—
12	#13050590	—	2	3	no	—
13	Q₈×D₈₃	—	4	1	no	—
14	#13050578	—	4	3	no	—
15	#13050575	—	4	3	no	—
16	#13050574	—	2	3	no	—
17	C₂⁴×C₈₃	—	2	1	no	—
18	C₂×Q₈×C₈₃	—	2	1	no	—
19	C₂×Q₈×C₈₃	C₂³	1	1	yes	characteristic
20	C₂×D₄×C₈₃	—	2	3	no	—
21	C₂×D₄×C₈₃	D₄	1	3	yes	—
22	C₂²⋊C₄×C₈₃	—	2	3	no	—
23	C₂²⋊C₄×C₈₃	—	2	6	no	—
24	C₄²×C₈₃	D₄	1	3	yes	—
25	C₂³×C₈₃	—	2	3	no	—
26	C₂³×C₈₃	—	8	2	no	—
27	C₂³×C₈₃	—	8	1	no	—
28	Q₈×C₈₃	—	4	1	no	—
29	Q₈×C₈₃	—	8	1	no	—
30	D₄×C₈₃	—	4	3	no	—
31	D₄×C₈₃	—	8	3	no	—
32	C₂×C₄×C₈₃	—	2	3	no	—
33	C₂×C₄×C₈₃	C₂×D₄	1	3	yes	—
34	C₂²⋊D₈₃	—	4	1	no	—
35	D₃₃₂	—	2	3	no	—
36	C₄×D₈₃	—	4	3	no	—
37	Dic₁₆₆	—	2	3	no	—
38	C₈₃⋊C₈	—	4	3	no	—
39	C₂²×C₈₃	—	4	6	no	—
40	C₂²×C₈₃	—	4	3	no	—
41	C₂²×C₈₃	—	8	4	no	—
42	C₂²×C₈₃	#120	1	1	yes	characteristic
43	C₂×D₈₃	—	4	1	no	—
44	C₄×C₈₃	—	2	3	no	—
45	C₄×C₈₃	—	4	3	no	—
46	Dic₈₃	—	4	1	no	—
47	C₂×C₈₃	—	2	1	no	—
48	C₂×C₈₃	—	8	3	no	—
49	C₂×C₈₃	B(4,𝔽₂)	1	1	yes	characteristic
50	D₈₃	—	8	1	no	—
51	#2202	—	83	1	no	maximal, Sylow
52	C₈₃	#2202	1	1	yes	characteristic, complemented, Sylow
53	H(3,𝔽₄)	D₈₃	1	1	yes	characteristic, complemented
54	#454	—	83	3	no	—
55	#355	—	83	3	no	—
56	2₋¹⁺⁴	—	83	1	no	—
57	AGL(1,ℤ/8)	—	166	3	no	—
58	#124	—	2	3	no	—
59	#124	C₂×D₈₃	1	3	yes	—
60	#120	—	2	3	no	—
61	#104	—	166	3	no	—
62	#101	—	83	3	no	—
63	C₂⁴	—	2	1	no	—
64	#41	—	332	1	no	—
65	#41	—	166	3	no	—
66	C₂×Q₈	—	2	1	no	—
67	C₂×Q₈	—	332	1	no	—
68	C₂×Q₈	C₂²×D₈₃	1	1	yes	characteristic
69	C₂×D₄	—	2	3	no	—
70	C₂×D₄	C₂²⋊D₈₃	1	3	yes	—
71	SD₁₆	—	332	3	no	—
72	D₈	—	332	3	no	—
73	M₁₆	—	166	3	no	—
74	C₂²⋊C₄	—	2	3	no	—
75	C₂²⋊C₄	—	2	6	no	—
76	C₄²	C₂²⋊D₈₃	1	3	yes	—
77	C₂³	—	2	3	no	—
78	C₂³	—	8	2	no	—
79	C₂³	—	8	1	no	—
80	Q₈	—	4	1	no	—
81	Q₈	—	8	1	no	—
82	Q₈	—	166	3	no	—
83	D₄	—	4	3	no	—
84	D₄	—	8	3	no	—
85	D₄	—	332	1	no	—
86	D₄	—	166	3	no	—
87	C₂×C₄	—	2	3	no	—
88	C₂×C₄	—	332	3	no	—
89	C₂×C₄	C₂×C₂²⋊D₈₃	1	3	yes	—
90	C₈	—	332	3	no	—
91	C₂²	—	4	6	no	—
92	C₂²	—	4	3	no	—
93	C₂²	—	8	4	no	—
94	C₂²	—	332	1	no	—
95	C₂²	#14581761	1	1	yes	characteristic
96	C₄	—	2	3	no	—
97	C₄	—	4	3	no	—
98	C₄	—	332	1	no	—
99	C₂	—	2	1	no	—
100	C₂	—	8	3	no	—
101	C₂	—	664	1	no	—
102	C₂	#19127888	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19127873

C₂

yes

maximal, complemented

#19127498

C₂

yes

maximal, complemented

H(3,𝔽₄)×C₈₃

C₂

yes

maximal, characteristic, complemented

#14581840

C₂²

yes

characteristic, complemented

#14581818

—

#14581720

—

#14581711

C₂²

yes

complemented

#124×C₈₃

—

#124×C₈₃

C₂²

yes

—

#120×C₈₃

—

#13050591

—

#13050590

—

Q₈×D₈₃

—

#13050578

—

#13050575

—

#13050574

—

C₂⁴×C₈₃

—

C₂×Q₈×C₈₃

—

C₂×Q₈×C₈₃

C₂³

yes

characteristic

C₂×D₄×C₈₃

—

C₂×D₄×C₈₃

D₄

yes

—

C₂²⋊C₄×C₈₃

—

C₂²⋊C₄×C₈₃

—

C₄²×C₈₃

D₄

yes

—

C₂³×C₈₃

—

C₂³×C₈₃

—

C₂³×C₈₃

—

Q₈×C₈₃

—

Q₈×C₈₃

—

D₄×C₈₃

—

D₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂×D₄

yes

—

C₂²⋊D₈₃

—

D₃₃₂

—

C₄×D₈₃

—

Dic₁₆₆

—

C₈₃⋊C₈

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

#120

yes

characteristic

C₂×D₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

Dic₈₃

—

C₂×C₈₃

—

C₂×C₈₃

—

C₂×C₈₃

B(4,𝔽₂)

yes

characteristic

D₈₃

—

#2202

—

maximal, Sylow

C₈₃

#2202

yes

characteristic, complemented, Sylow

H(3,𝔽₄)

D₈₃

yes

characteristic, complemented

#454

—

#355

—

2₋¹⁺⁴

—

AGL(1,ℤ/8)

—

166

—

#124

—

#124

C₂×D₈₃

yes

—

#120

—

#104

—

166

—

#101

—

C₂⁴

—

#41

—

332

—

#41

—

166

—

C₂×Q₈

—

C₂×Q₈

—

332

—

C₂×Q₈

C₂²×D₈₃

yes

characteristic

C₂×D₄

—

C₂×D₄

C₂²⋊D₈₃

yes

—

SD₁₆

—

332

—

D₈

—

332

—

M₁₆

—

166

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

C₂²⋊D₈₃

yes

—

C₂³

—

C₂³

—

C₂³

—

Q₈

—

Q₈

—

Q₈

—

166

—

D₄

—

D₄

—

D₄

—

332

—

D₄

—

166

—

C₂×C₄

—

C₂×C₄

—

332

—

C₂×C₄

C₂×C₂²⋊D₈₃

yes

—

C₈

—

332

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

332

—

C₂²

#14581761

yes

characteristic

C₄

—

C₄

—

C₄

—

332

—

C₂

—

100

C₂

—

101

C₂

—

664

—

102

C₂

#19127888

yes

characteristic

	subgroup	G/H
center	C₂	#19127888
commutator, cosocle	C₂×Q₈×C₈₃	C₂³
Frattini	C₂×Q₈	C₂²×D₈₃
Fitting	H(3,𝔽₄)×C₈₃	C₂
socle	C₂×C₈₃	B(4,𝔽₂)
hypercenter	H(3,𝔽₄)	D₈₃
hypocenter	C₈₃	#2202
LCS	G ▷ C₂×Q₈×C₈₃ ▷ C₂²×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×Q₈ ◁ H(3,𝔽₄)

subgroup

G/H

center

C₂

#19127888

commutator, cosocle

C₂×Q₈×C₈₃

C₂³

Frattini

C₂×Q₈

C₂²×D₈₃

Fitting

H(3,𝔽₄)×C₈₃

C₂

C₂×C₈₃

D₈₃

C₈₃

G ▷ C₂×Q₈×C₈₃ ▷ C₂²×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂ ◁ C₂² ◁ C₂×Q₈ ◁ H(3,𝔽₄)

N	H ≅ G/N	type
1. #19127873	C₂	semidirect
2. #19127498	C₂	semidirect
3. H(3,𝔽₄)×C₈₃	C₂	semidirect
4. #14581840	C₂²	semidirect
7. #14581711	C₂²	semidirect
52. C₈₃	#2202	semidirect
53. H(3,𝔽₄)	D₈₃	semidirect

H ≅ G/N

type

1. #19127873

C₂

semidirect

2. #19127498

C₂

semidirect

3. H(3,𝔽₄)×C₈₃

C₂

semidirect

4. #14581840

C₂²

semidirect

7. #14581711

C₂²

semidirect

52. C₈₃

#2202

semidirect

53. H(3,𝔽₄)

D₈₃

semidirect

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

subgroup	index
#19127873	2
#19127498	2
H(3,𝔽₄)×C₈₃	2
#2202	83

subgroup

index

#19127873

#19127498

H(3,𝔽₄)×C₈₃

#2202

Subgroups — #24909480

Conjugacy classes of subgroups

Complements

Maximal subgroups