Group #24907129 — subgroups

Subgroups — #24907129

Order 10624.

242 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 69 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	#19128078	C₂	1	2	yes	maximal, complemented
2	#19128078	C₂	1	1	yes	maximal, characteristic, complemented
3	#19128074	C₂	1	1	yes	maximal, characteristic, complemented
4	#19128048	C₂	1	2	yes	maximal, complemented
5	#322×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13050579	—	2	2	no	—
7	C₂×D₆₆₄	—	2	4	no	—
8	C₂×D₆₆₄	—	2	2	no	—
9	#14581792	C₂²	1	1	yes	characteristic
10	#14581789	C₂²	1	1	yes	characteristic
11	#14581789	C₂²	1	2	yes	—
12	#14581714	—	2	4	no	—
13	C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₈₃	C₂²	1	2	yes	—
15	C₂×Dic₁₆₆	—	2	1	no	—
16	C₂×D₃₃₂	—	2	2	no	—
17	C₂×D₃₃₂	—	2	2	no	—
18	C₂×D₃₃₂	—	2	2	no	—
19	C₂×D₃₃₂	—	2	1	no	—
20	C₂×D₃₃₂	—	2	2	no	—
21	#13050579	—	4	4	no	—
22	D₆₆₄	—	4	8	no	—
23	D₆₆₄	—	4	4	no	—
24	C₄⋊Dic₈₃	—	2	2	no	—
25	C₂×C₈×C₈₃	D₄	1	4	yes	—
26	C₂×C₈×C₈₃	D₄	1	2	yes	—
27	C₄²×C₈₃	C₂³	1	1	yes	characteristic
28	C₂²×D₈₃	—	4	1	no	—
29	C₂²×D₈₃	—	4	2	no	—
30	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
31	C₂×C₄×C₈₃	C₂×D₄	1	2	yes	—
32	C₂×Dic₈₃	—	4	1	no	—
33	D₃₃₂	—	4	4	no	—
34	D₃₃₂	—	4	2	no	—
35	D₃₃₂	—	4	4	no	—
36	D₃₃₂	—	4	4	no	—
37	D₃₃₂	—	4	4	no	—
38	Dic₁₆₆	—	4	2	no	—
39	C₈×C₈₃	—	2	4	no	—
40	C₈×C₈₃	D₈	1	4	yes	complemented
41	C₂²×C₈₃	#127	1	1	yes	characteristic
42	C₂×D₈₃	—	8	2	no	—
43	C₂×D₈₃	—	8	1	no	—
44	C₂×D₈₃	—	8	2	no	—
45	C₂×D₈₃	—	8	2	no	—
46	C₂×D₈₃	—	8	2	no	—
47	C₄×C₈₃	C₂×D₈	1	2	yes	—
48	C₄×C₈₃	AGL(1,ℤ/8)	1	4	yes	—
49	Dic₈₃	—	8	1	no	—
50	C₂×C₈₃	#493	1	1	yes	characteristic
51	C₂×C₈₃	#496	1	2	yes	—
52	D₈₃	—	16	2	no	—
53	D₈₃	—	16	1	no	—
54	#1721	—	83	1	no	maximal, Sylow
55	C₈₃	#1721	1	1	yes	characteristic, complemented, Sylow
56	#493	—	83	2	no	—
57	#493	—	83	1	no	—
58	#492	—	83	1	no	—
59	#486	—	83	2	no	—
60	#322	D₈₃	1	1	yes	characteristic, complemented
61	C₂×SD₁₆	—	166	2	no	—
62	C₂×D₈	—	166	4	no	—
63	C₂×D₈	—	166	2	no	—
64	C₄⋊Q₈	—	83	1	no	—
65	#127	—	83	2	no	—
66	#127	—	83	1	no	—
67	#102	—	166	4	no	—
68	C₄×C₈	C₂×D₈₃	1	2	yes	—
69	C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
70	C₂×Q₈	—	166	1	no	—
71	C₂×D₄	—	166	2	no	—
72	C₂×D₄	—	166	2	no	—
73	C₂×D₄	—	166	2	no	—
74	C₂×D₄	—	166	2	no	—
75	C₂×D₄	—	166	1	no	—
76	SD₁₆	—	332	4	no	—
77	D₈	—	332	8	no	—
78	D₈	—	332	4	no	—
79	C₂×C₈	D₃₃₂	1	4	yes	—
80	C₂×C₈	D₃₃₂	1	2	yes	—
81	C₄⋊C₄	—	166	2	no	—
82	C₄²	C₂²×D₈₃	1	1	yes	characteristic
83	C₂³	—	332	2	no	—
84	C₂³	—	332	1	no	—
85	Q₈	—	332	2	no	—
86	D₄	—	332	4	no	—
87	D₄	—	332	2	no	—
88	D₄	—	332	4	no	—
89	D₄	—	332	4	no	—
90	D₄	—	332	4	no	—
91	C₂×C₄	—	332	1	no	—
92	C₂×C₄	C₂×D₃₃₂	1	1	yes	characteristic
93	C₂×C₄	C₂×D₃₃₂	1	2	yes	—
94	C₈	—	2	4	no	—
95	C₈	D₆₆₄	1	4	yes	complemented
96	C₂²	—	664	2	no	—
97	C₂²	—	664	1	no	—
98	C₂²	—	664	2	no	—
99	C₂²	—	664	2	no	—
100	C₂²	—	664	2	no	—
101	C₂²	#14581789	1	1	yes	characteristic
102	C₄	—	664	1	no	—
103	C₄	C₂×D₆₆₄	1	2	yes	—
104	C₄	#14581820	1	4	yes	—
105	C₂	—	1328	2	no	—
106	C₂	—	1328	1	no	—
107	C₂	#19128078	1	1	yes	characteristic
108	C₂	#19128090	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19128078

C₂

yes

maximal, complemented

#19128078

C₂

yes

maximal, characteristic, complemented

#19128074

C₂

yes

maximal, characteristic, complemented

#19128048

C₂

yes

maximal, complemented

#322×C₈₃

C₂

yes

maximal, characteristic, complemented

C₂×#13050579

—

C₂×D₆₆₄

—

C₂×D₆₆₄

—

#14581792

C₂²

yes

characteristic

#14581789

C₂²

yes

characteristic

#14581789

C₂²

yes

—

#14581714

—

C₄×C₈×C₈₃

C₂²

yes

characteristic

C₄×C₈×C₈₃

C₂²

yes

—

C₂×Dic₁₆₆

—

C₂×D₃₃₂

—

C₂×D₃₃₂

—

C₂×D₃₃₂

—

C₂×D₃₃₂

—

C₂×D₃₃₂

—

#13050579

—

D₆₆₄

—

D₆₆₄

—

C₄⋊Dic₈₃

—

C₂×C₈×C₈₃

D₄

yes

—

C₂×C₈×C₈₃

D₄

yes

—

C₄²×C₈₃

C₂³

yes

characteristic

C₂²×D₈₃

—

C₂²×D₈₃

—

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₈₃

C₂×D₄

yes

—

C₂×Dic₈₃

—

D₃₃₂

—

D₃₃₂

—

D₃₃₂

—

D₃₃₂

—

D₃₃₂

—

Dic₁₆₆

—

C₈×C₈₃

—

C₈×C₈₃

D₈

yes

complemented

C₂²×C₈₃

#127

yes

characteristic

C₂×D₈₃

—

C₂×D₈₃

—

C₂×D₈₃

—

C₂×D₈₃

—

C₂×D₈₃

—

C₄×C₈₃

C₂×D₈

yes

—

C₄×C₈₃

AGL(1,ℤ/8)

yes

—

Dic₈₃

—

C₂×C₈₃

#493

yes

characteristic

C₂×C₈₃

#496

yes

—

D₈₃

—

D₈₃

—

#1721

—

maximal, Sylow

C₈₃

#1721

yes

characteristic, complemented, Sylow

—

—

—

—

D₈₃

yes

characteristic, complemented

C₂×SD₁₆

—

166

—

C₂×D₈

—

166

—

C₂×D₈

—

166

—

C₄⋊Q₈

—

#127

—

#127

—

#102

—

166

—

C₄×C₈

C₂×D₈₃

yes

—

C₄×C₈

C₂×D₈₃

yes

characteristic

C₂×Q₈

—

166

—

C₂×D₄

—

166

—

C₂×D₄

—

166

—

C₂×D₄

—

166

—

C₂×D₄

—

166

—

C₂×D₄

—

166

—

SD₁₆

—

332

—

D₈

—

332

—

D₈

—

332

—

C₂×C₈

D₃₃₂

yes

—

C₂×C₈

D₃₃₂

yes

—

C₄⋊C₄

—

166

—

C₄²

C₂²×D₈₃

yes

characteristic

C₂³

—

332

—

C₂³

—

332

—

Q₈

—

332

—

D₄

—

332

—

D₄

—

332

—

D₄

—

332

—

D₄

—

332

—

D₄

—

332

—

C₂×C₄

—

332

—

C₂×C₄

C₂×D₃₃₂

yes

characteristic

C₂×C₄

C₂×D₃₃₂

yes

—

C₈

—

C₈

D₆₆₄

yes

complemented

C₂²

—

664

—

C₂²

—

664

—

C₂²

—

664

—

C₂²

—

664

—

100

C₂²

—

664

—

101

C₂²

#14581789

yes

characteristic

102

C₄

—

664

—

103

C₄

C₂×D₆₆₄

yes

—

104

C₄

#14581820

yes

—

105

C₂

—

1328

—

106

C₂

—

1328

—

107

C₂

#19128078

yes

characteristic

108

C₂

#19128090

yes

—

	subgroup	G/H
center	C₂²	#14581789
commutator, cosocle	C₄²×C₈₃	C₂³
Frattini	C₄²	C₂²×D₈₃
Fitting	#322×C₈₃	C₂
socle	C₂²×C₈₃	#127
hypercenter	#322	D₈₃
hypocenter	C₈₃	#1721
LCS	G ▷ C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #322

subgroup

G/H

center

C₂²

#14581789

commutator, cosocle

C₄²×C₈₃

C₂³

Frattini

C₄²

C₂²×D₈₃

Fitting

#322×C₈₃

C₂

C₂²×C₈₃

D₈₃

C₈₃

G ▷ C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #322

N	H ≅ G/N	type
1. #19128078	C₂	semidirect
2. #19128078	C₂	semidirect
3. #19128074	C₂	semidirect
4. #19128048	C₂	semidirect
5. #322×C₈₃	C₂	semidirect
40. C₈×C₈₃	D₈	semidirect
55. C₈₃	#1721	semidirect
60. #322	D₈₃	semidirect
95. C₈	D₆₆₄	semidirect

H ≅ G/N

type

1. #19128078

C₂

semidirect

2. #19128078

C₂

semidirect

3. #19128074

C₂

semidirect

4. #19128048

C₂

semidirect

5. #322×C₈₃

C₂

semidirect

40. C₈×C₈₃

D₈

semidirect

55. C₈₃

#1721

semidirect

60. #322

D₈₃

semidirect

95. C₈

D₆₆₄

semidirect

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

subgroup	index
#19128078	2
#19128074	2
#19128048	2
#322×C₈₃	2
#1721	83

subgroup

index

#322×C₈₃

Subgroups — #24907129

Conjugacy classes of subgroups

Complements

Maximal subgroups