Group #35319122 — subgroups

Subgroups — #35319122

Order 21248.

354 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 115 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	#24909719	C₂	1	4	yes	maximal, complemented
2	#24907081	C₂	1	1	yes	maximal, characteristic, complemented
3	#24907078	C₂	1	1	yes	maximal, characteristic
4	C₈×C₁₆×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#14581737	—	2	8	no	—
6	#19128106	—	2	2	no	—
7	#19128081	C₂²	1	2	yes	—
8	#19128078	—	2	2	no	—
9	#19128078	C₂²	1	2	yes	—
10	C₄×C₁₆×C₈₃	C₂²	1	2	yes	—
11	C₈²×C₈₃	C₂²	1	1	yes	characteristic
12	C₂×Dic₃₃₂	—	2	4	no	—
13	C₂×D₆₆₄	—	2	4	no	—
14	C₂×D₆₆₄	—	4	4	no	—
15	#14581792	—	2	2	no	—
16	#14581789	—	2	2	no	—
17	#14581737	—	4	16	no	—
18	#14581726	—	4	4	no	—
19	C₂×C₁₆×C₈₃	D₄	1	4	yes	—
20	C₄×C₈×C₈₃	D₄	1	2	yes	—
21	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
22	C₂×Dic₁₆₆	—	4	2	no	—
23	C₂×D₃₃₂	—	4	2	no	—
24	C₂×D₃₃₂	—	4	4	no	—
25	Dic₃₃₂	—	4	8	no	—
26	D₆₆₄	—	4	8	no	—
27	D₆₆₄	—	8	8	no	—
28	C₄⋊Dic₈₃	—	4	4	no	—
29	C₂×C₈×C₈₃	D₈	1	4	yes	—
30	C₂×C₈×C₈₃	C₂×D₄	1	2	yes	—
31	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
32	C₁₆×C₈₃	D₈	1	8	yes	complemented
33	C₂²×D₈₃	—	8	2	no	—
34	C₂×C₄×C₈₃	#127	1	1	yes	characteristic
35	C₂×C₄×C₈₃	C₂×D₈	1	2	yes	—
36	C₂×Dic₈₃	—	8	2	no	—
37	D₃₃₂	—	8	4	no	—
38	D₃₃₂	—	8	8	no	—
39	Dic₁₆₆	—	8	4	no	—
40	C₈×C₈₃	SD₃₂	1	8	yes	complemented
41	C₈×C₈₃	C₂×D₈	1	4	yes	—
42	C₂²×C₈₃	#493	1	1	yes	characteristic
43	C₂×D₈₃	—	16	2	no	—
44	C₂×D₈₃	—	16	4	no	—
45	C₄×C₈₃	#493	1	2	yes	—
46	C₄×C₈₃	C₂×SD₃₂	1	4	yes	—
47	Dic₈₃	—	16	2	no	—
48	#12517	—	83	1	no	maximal, Sylow
49	C₂×C₈₃	#1712	1	1	yes	characteristic
50	C₂×C₈₃	#2248	1	2	yes	—
51	D₈₃	—	32	2	no	—
52	#2248	—	83	4	no	—
53	#1713	—	83	1	no	—
54	#1712	—	83	1	no	—
55	C₈×C₁₆	D₈₃	1	1	yes	characteristic, complemented
56	C₈₃	#12517	1	1	yes	characteristic, complemented, Sylow
57	C₂×SD₃₂	—	166	8	no	—
58	#500	—	166	2	no	—
59	#494	—	83	2	no	—
60	#493	—	83	2	no	—
61	#493	—	166	2	no	—
62	C₄×C₁₆	C₂×D₈₃	1	2	yes	—
63	C₈²	C₂×D₈₃	1	1	yes	characteristic
64	C₂×Q₁₆	—	166	4	no	—
65	C₂×D₈	—	332	4	no	—
66	C₂×D₈	—	166	4	no	—
67	C₄⋊Q₈	—	166	2	no	—
68	#127	—	166	2	no	—
69	SD₃₂	—	332	16	no	—
70	C₂×C₁₆	D₃₃₂	1	4	yes	—
71	#107	—	332	4	no	—
72	C₄×C₈	D₃₃₂	1	2	yes	—
73	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
74	C₂×Q₈	—	332	2	no	—
75	C₂×D₄	—	332	2	no	—
76	C₂×D₄	—	332	4	no	—
77	Q₁₆	—	332	8	no	—
78	D₈	—	332	8	no	—
79	D₈	—	664	8	no	—
80	C₂×C₈	D₆₆₄	1	4	yes	—
81	C₂×C₈	C₂×D₃₃₂	1	2	yes	—
82	C₄⋊C₄	—	332	4	no	—
83	C₄²	C₂×D₃₃₂	1	1	yes	characteristic
84	C₁₆	D₆₆₄	1	8	yes	complemented
85	C₂³	—	664	2	no	—
86	Q₈	—	664	4	no	—
87	D₄	—	664	4	no	—
88	D₄	—	664	8	no	—
89	C₂×C₄	—	664	2	no	—
90	C₂×C₄	#14581789	1	1	yes	characteristic
91	C₂×C₄	C₂×D₆₆₄	1	2	yes	—
92	C₈	#14581737	1	8	yes	complemented
93	C₈	C₂×D₆₆₄	1	4	yes	—
94	C₂²	—	1328	2	no	—
95	C₂²	—	1328	4	no	—
96	C₂²	#19128078	1	1	yes	characteristic
97	C₄	—	1328	2	no	—
98	C₄	#19128078	1	2	yes	—
99	C₄	C₂×#14581737	1	4	yes	—
100	C₂	—	2656	2	no	—
101	C₂	#24907078	1	1	yes	characteristic
102	C₂	#24909719	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#24909719

C₂

yes

maximal, complemented

#24907081

C₂

yes

maximal, characteristic, complemented

#24907078

C₂

yes

maximal, characteristic

C₈×C₁₆×C₈₃

C₂

yes

maximal, characteristic, complemented

C₂×#14581737

—

#19128106

—

#19128081

C₂²

yes

—

#19128078

—

#19128078

C₂²

yes

—

C₄×C₁₆×C₈₃

C₂²

yes

—

C₈²×C₈₃

C₂²

yes

characteristic

C₂×Dic₃₃₂

—

C₂×D₆₆₄

—

C₂×D₆₆₄

—

—

—

—

—

C₂×C₁₆×C₈₃

D₄

yes

—

C₄×C₈×C₈₃

D₄

yes

—

C₄×C₈×C₈₃

C₂³

yes

characteristic

C₂×Dic₁₆₆

—

C₂×D₃₃₂

—

C₂×D₃₃₂

—

Dic₃₃₂

—

D₆₆₄

—

D₆₆₄

—

C₄⋊Dic₈₃

—

C₂×C₈×C₈₃

D₈

yes

—

C₂×C₈×C₈₃

C₂×D₄

yes

—

C₄²×C₈₃

C₂×D₄

yes

characteristic

C₁₆×C₈₃

D₈

yes

complemented

C₂²×D₈₃

—

C₂×C₄×C₈₃

#127

yes

characteristic

C₂×C₄×C₈₃

C₂×D₈

yes

—

C₂×Dic₈₃

—

D₃₃₂

—

D₃₃₂

—

Dic₁₆₆

—

C₈×C₈₃

SD₃₂

yes

complemented

C₈×C₈₃

C₂×D₈

yes

—

C₂²×C₈₃

#493

yes

characteristic

C₂×D₈₃

—

C₂×D₈₃

—

C₄×C₈₃

#493

yes

—

C₄×C₈₃

C₂×SD₃₂

yes

—

Dic₈₃

—

—

C₂×C₈₃

yes

C₂×C₈₃

yes

—

D₈₃

—

#2248

—

#1713

—

#1712

—

C₈×C₁₆

D₈₃

yes

characteristic, complemented

C₈₃

#12517

yes

characteristic, complemented, Sylow

C₂×SD₃₂

—

166

—

#500

—

166

—

#494

—

#493

—

#493

—

166

—

C₄×C₁₆

C₂×D₈₃

yes

—

C₈²

C₂×D₈₃

yes

characteristic

C₂×Q₁₆

—

166

—

C₂×D₈

—

332

—

C₂×D₈

—

166

—

C₄⋊Q₈

—

166

—

#127

—

166

—

SD₃₂

—

332

—

C₂×C₁₆

D₃₃₂

yes

—

#107

—

332

—

C₄×C₈

D₃₃₂

yes

—

C₄×C₈

C₂²×D₈₃

yes

characteristic

C₂×Q₈

—

332

—

C₂×D₄

—

332

—

C₂×D₄

—

332

—

Q₁₆

—

332

—

D₈

—

332

—

D₈

—

664

—

C₂×C₈

D₆₆₄

yes

—

C₂×C₈

C₂×D₃₃₂

yes

—

C₄⋊C₄

—

332

—

C₄²

C₂×D₃₃₂

yes

characteristic

C₁₆

D₆₆₄

yes

complemented

C₂³

—

664

—

Q₈

—

664

—

D₄

—

664

—

D₄

—

664

—

C₂×C₄

—

664

—

C₂×C₄

#14581789

yes

characteristic

C₂×C₄

C₂×D₆₆₄

yes

—

C₈

#14581737

yes

complemented

C₈

C₂×D₆₆₄

yes

—

C₂²

—

1328

—

C₂²

—

1328

—

C₂²

#19128078

yes

characteristic

C₄

—

1328

—

C₄

#19128078

yes

—

C₄

C₂×#14581737

yes

—

100

C₂

—

2656

—

101

C₂

#24907078

yes

characteristic

102

C₂

#24909719

yes

—

	subgroup	G/H
center	C₂²	#19128078
commutator, cosocle	C₄×C₈×C₈₃	C₂³
Frattini	C₄×C₈	C₂²×D₈₃
Fitting	C₈×C₁₆×C₈₃	C₂
socle	C₂²×C₈₃	#493
prime radical	#24907078	C₂
hypercenter	C₈×C₁₆	D₈₃
hypocenter	C₈₃	#12517
LCS	G ▷ C₄×C₈×C₈₃ ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆

subgroup

G/H

center

C₂²

#19128078

commutator, cosocle

C₄×C₈×C₈₃

C₂³

Frattini

C₄×C₈

C₂²×D₈₃

Fitting

C₈×C₁₆×C₈₃

C₂

C₂²×C₈₃

C₂

C₈×C₁₆

D₈₃

hypocenter

C₈₃

#12517

LCS

G ▷ C₄×C₈×C₈₃ ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆

N	H ≅ G/N	type
1. #24909719	C₂	semidirect
2. #24907081	C₂	semidirect
4. C₈×C₁₆×C₈₃	C₂	semidirect
32. C₁₆×C₈₃	D₈	semidirect
40. C₈×C₈₃	SD₃₂	semidirect
55. C₈×C₁₆	D₈₃	semidirect
56. C₈₃	#12517	semidirect
84. C₁₆	D₆₆₄	semidirect
92. C₈	#14581737	semidirect

H ≅ G/N

type

1. #24909719

C₂

semidirect

2. #24907081

C₂

semidirect

4. C₈×C₁₆×C₈₃

C₂

semidirect

32. C₁₆×C₈₃

D₈

semidirect

40. C₈×C₈₃

SD₃₂

semidirect

55. C₈×C₁₆

D₈₃

semidirect

56. C₈₃

#12517

semidirect

84. C₁₆

D₆₆₄

semidirect

92. C₈

#14581737

semidirect

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

subgroup	index
#24909719	2
#24907081	2
#24907078	2
C₈×C₁₆×C₈₃	2
#12517	83

subgroup

index

C₈×C₁₆×C₈₃

Subgroups — #35319122

Conjugacy classes of subgroups

Complements

Maximal subgroups