Group #35319108 — subgroups

Subgroups — #35319108

Order 21248.

290 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 83 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	#24909687	C₂	1	4	yes	maximal, complemented
2	#24907078	C₂	1	1	yes	maximal, characteristic
3	#24907073	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈₃×#1311	C₂	1	1	yes	maximal, characteristic, complemented
5	#19128106	C₂²	1	2	yes	—
6	#19128078	—	2	2	no	—
7	#19128078	C₂²	1	2	yes	—
8	#19128052	—	2	2	no	—
9	#19127503	—	2	8	no	—
10	C₄×C₁₆×C₈₃	C₂²	1	2	yes	—
11	C₈²×C₈₃	C₂²	1	1	yes	characteristic
12	C₂×D₆₆₄	—	2	4	no	—
13	C₂×D₆₆₄	—	4	4	no	—
14	#14581792	—	2	2	no	—
15	#14581789	—	2	2	no	—
16	#14581726	—	2	4	no	—
17	#14581717	—	4	4	no	—
18	C₂×C₁₆×C₈₃	D₄	1	4	yes	—
19	C₄×C₈×C₈₃	D₄	1	2	yes	—
20	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
21	C₂×Dic₁₆₆	—	4	2	no	—
22	C₂×D₃₃₂	—	4	2	no	—
23	C₂×D₃₃₂	—	4	4	no	—
24	D₆₆₄	—	4	8	no	—
25	D₆₆₄	—	8	8	no	—
26	C₄⋊Dic₈₃	—	4	2	no	—
27	C₄⋊Dic₈₃	—	4	2	no	—
28	C₂×C₈×C₈₃	D₈	1	4	yes	—
29	C₂×C₈×C₈₃	C₂×D₄	1	2	yes	—
30	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
31	C₁₆×C₈₃	—	2	4	no	—
32	C₂²×D₈₃	—	8	2	no	—
33	C₂×C₄×C₈₃	#127	1	1	yes	characteristic
34	C₂×C₄×C₈₃	C₂×D₈	1	2	yes	—
35	C₂×Dic₈₃	—	8	2	no	—
36	D₃₃₂	—	8	4	no	—
37	D₃₃₂	—	8	8	no	—
38	Dic₁₆₆	—	8	4	no	—
39	C₈×C₈₃	—	2	4	no	—
40	C₈×C₈₃	#135	1	4	yes	—
41	C₂²×C₈₃	#493	1	1	yes	characteristic
42	C₂×D₈₃	—	16	2	no	—
43	C₂×D₈₃	—	16	4	no	—
44	C₄×C₈₃	#495	1	2	yes	—
45	C₄×C₈₃	#509	1	4	yes	—
46	Dic₈₃	—	16	2	no	—
47	#12513	—	83	1	no	maximal, Sylow
48	C₂×C₈₃	#1709	1	1	yes	characteristic
49	C₂×C₈₃	#2250	1	2	yes	—
50	D₈₃	—	32	2	no	—
51	#2240	—	83	4	no	—
52	#1712	—	83	1	no	—
53	#1710	—	83	1	no	—
54	#1311	D₈₃	1	1	yes	characteristic, complemented
55	C₈₃	#12513	1	1	yes	characteristic, complemented, Sylow
56	#500	—	83	2	no	—
57	#493	—	83	2	no	—
58	#493	—	166	2	no	—
59	#487	—	166	2	no	—
60	#357	—	166	8	no	—
61	C₄×C₁₆	C₂×D₈₃	1	2	yes	—
62	C₈²	C₂×D₈₃	1	1	yes	characteristic
63	C₂×D₈	—	332	4	no	—
64	C₂×D₈	—	166	4	no	—
65	C₄⋊Q₈	—	166	2	no	—
66	#127	—	166	2	no	—
67	C₂×C₁₆	D₃₃₂	1	4	yes	—
68	#107	—	166	4	no	—
69	#103	—	332	4	no	—
70	C₄×C₈	D₃₃₂	1	2	yes	—
71	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
72	C₂×Q₈	—	332	2	no	—
73	C₂×D₄	—	332	2	no	—
74	C₂×D₄	—	332	4	no	—
75	D₈	—	332	8	no	—
76	D₈	—	664	8	no	—
77	C₂×C₈	D₆₆₄	1	4	yes	—
78	C₂×C₈	C₂×D₃₃₂	1	2	yes	—
79	C₄⋊C₄	—	332	2	no	—
80	C₄⋊C₄	—	332	2	no	—
81	C₄²	C₂×D₃₃₂	1	1	yes	characteristic
82	C₁₆	—	2	4	no	—
83	C₂³	—	664	2	no	—
84	Q₈	—	664	4	no	—
85	D₄	—	664	4	no	—
86	D₄	—	664	8	no	—
87	C₂×C₄	—	664	2	no	—
88	C₂×C₄	#14581789	1	1	yes	characteristic
89	C₂×C₄	C₂×D₆₆₄	1	2	yes	—
90	C₈	—	2	4	no	—
91	C₈	#14581814	1	4	yes	—
92	C₂²	—	1328	2	no	—
93	C₂²	—	1328	4	no	—
94	C₂²	#19128078	1	1	yes	characteristic
95	C₄	—	1328	2	no	—
96	C₄	#19128085	1	2	yes	—
97	C₄	#19128139	1	4	yes	—
98	C₂	—	2656	2	no	—
99	C₂	#24907069	1	1	yes	characteristic
100	C₂	#24909729	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#24909687

C₂

yes

maximal, complemented

#24907078

C₂

yes

maximal, characteristic

#24907073

C₂

yes

maximal, characteristic, complemented

C₈₃×#1311

C₂

yes

maximal, characteristic, complemented

#19128106

C₂²

yes

—

#19128078

—

#19128078

C₂²

yes

—

#19128052

—

#19127503

—

C₄×C₁₆×C₈₃

C₂²

yes

—

C₈²×C₈₃

C₂²

yes

characteristic

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

—

D₆₆₄

—

D₆₆₄

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₂×C₈×C₈₃

D₈

yes

—

C₂×C₈×C₈₃

C₂×D₄

yes

—

C₄²×C₈₃

C₂×D₄

yes

characteristic

C₁₆×C₈₃

—

C₂²×D₈₃

—

C₂×C₄×C₈₃

#127

yes

characteristic

C₂×C₄×C₈₃

C₂×D₈

yes

—

C₂×Dic₈₃

—

D₃₃₂

—

D₃₃₂

—

Dic₁₆₆

—

C₈×C₈₃

—

C₈×C₈₃

#135

yes

—

C₂²×C₈₃

#493

yes

characteristic

C₂×D₈₃

—

C₂×D₈₃

—

C₄×C₈₃

#495

yes

—

C₄×C₈₃

#509

yes

—

Dic₈₃

—

—

C₂×C₈₃

yes

C₂×C₈₃

yes

—

D₈₃

—

—

—

—

D₈₃

yes

characteristic, complemented

C₈₃

#12513

yes

characteristic, complemented, Sylow

#500

—

#493

—

#493

—

166

—

#487

—

166

—

#357

—

166

—

C₄×C₁₆

C₂×D₈₃

yes

—

C₈²

C₂×D₈₃

yes

characteristic

C₂×D₈

—

332

—

C₂×D₈

—

166

—

C₄⋊Q₈

—

166

—

#127

—

166

—

C₂×C₁₆

D₃₃₂

yes

—

#107

—

166

—

#103

—

332

—

C₄×C₈

D₃₃₂

yes

—

C₄×C₈

C₂²×D₈₃

yes

characteristic

C₂×Q₈

—

332

—

C₂×D₄

—

332

—

C₂×D₄

—

332

—

D₈

—

332

—

D₈

—

664

—

C₂×C₈

D₆₆₄

yes

—

C₂×C₈

C₂×D₃₃₂

yes

—

C₄⋊C₄

—

332

—

C₄⋊C₄

—

332

—

C₄²

C₂×D₃₃₂

yes

characteristic

C₁₆

—

C₂³

—

664

—

Q₈

—

664

—

D₄

—

664

—

D₄

—

664

—

C₂×C₄

—

664

—

C₂×C₄

#14581789

yes

characteristic

C₂×C₄

C₂×D₆₆₄

yes

—

C₈

—

C₈

#14581814

yes

—

C₂²

—

1328

—

C₂²

—

1328

—

C₂²

#19128078

yes

characteristic

C₄

—

1328

—

C₄

#19128085

yes

—

C₄

#19128139

yes

—

C₂

—

2656

—

C₂

#24907069

yes

characteristic

100

C₂

#24909729

yes

—

	subgroup	G/H
center	C₂²	#19128078
commutator, cosocle	C₄×C₈×C₈₃	C₂³
Frattini	C₄×C₈	C₂²×D₈₃
Fitting	C₈₃×#1311	C₂
socle	C₂²×C₈₃	#493
prime radical	#24907078	C₂
hypercenter	#1311	D₈₃
hypocenter	C₈₃	#12513
LCS	G ▷ C₄×C₈×C₈₃ ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1311

subgroup

G/H

center

C₂²

#19128078

commutator, cosocle

C₄×C₈×C₈₃

C₂³

Frattini

C₄×C₈

C₂²×D₈₃

Fitting

C₈₃×#1311

C₂

C₂²×C₈₃

C₂

D₈₃

C₈₃

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1311

N	H ≅ G/N	type
1. #24909687	C₂	semidirect
3. #24907073	C₂	semidirect
4. C₈₃×#1311	C₂	semidirect
54. #1311	D₈₃	semidirect
55. C₈₃	#12513	semidirect

H ≅ G/N

type

1. #24909687

C₂

semidirect

3. #24907073

C₂

semidirect

4. C₈₃×#1311

C₂

semidirect

54. #1311

D₈₃

semidirect

55. C₈₃

#12513

semidirect

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

subgroup	index
#24909687	2
#24907078	2
#24907073	2
C₈₃×#1311	2
#12513	83

subgroup

index

C₈₃×#1311

Subgroups — #35319108

Conjugacy classes of subgroups

Complements

Maximal subgroups