Group #28299360 — subgroups

Subgroups — #28299360

Order 13568.

186 conjugacy classes of subgroups; only the 22 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 22 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	#20289847	C₂	1	2	yes	maximal, complemented
2	#20274329	C₂	1	1	yes	maximal, characteristic
3	#16811528	—	2	2	no	—
4	#16811000	C₂²	1	1	yes	characteristic
5	#16810999	—	2	2	no	—
6	C₈×#88322	C₄	1	1	yes	characteristic, complemented
7	C₅₃×#493	C₄	1	1	yes	characteristic, complemented
8	#13133276	—	4	2	no	—
9	C₂×D₄₂₄	—	4	1	no	—
10	C₂×#11719922	—	4	4	no	—
11	#13133214	D₄	1	2	yes	—
12	C₄×#88322	—	2	1	no	—
13	C₂×D₈×C₅₃	—	4	1	no	—
14	#127×C₅₃	—	2	1	no	—
15	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
16	C₂×D₄×C₅₃	—	4	2	no	—
17	C₂×D₄×C₅₃	—	4	1	no	—
18	C₂×D₂₁₂	—	2	2	no	—
19	C₂×D₂₁₂	—	4	2	no	—
20	C₂²⋊#6157	—	8	2	no	—
21	D₈×C₅₃	—	8	2	no	—
22	C₂×C₈×C₅₃	—	2	1	no	—
23	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
24	#11719922	—	8	8	no	—
25	C₂×#88322	—	2	2	no	—
26	D₄₂₄	—	8	2	no	—
27	C₂³×C₅₃	—	8	1	no	—
28	C₂²×D₅₃	—	4	2	no	—
29	C₂×#6157	—	16	2	no	—
30	D₄×C₅₃	—	8	4	no	—
31	D₄×C₅₃	—	8	2	no	—
32	C₂×C₄×C₅₃	—	2	1	no	—
33	C₂×C₄×C₅₃	#99	1	1	yes	characteristic
34	D₂₁₂	—	8	2	no	—
35	D₂₁₂	—	8	4	no	—
36	C₈×C₅₃	—	2	2	no	—
37	#88322	—	2	4	no	—
38	#7529	—	53	1	no	maximal, Sylow
39	C₂²×C₅₃	—	16	2	no	—
40	C₂²×C₅₃	—	16	1	no	—
41	C₂²×C₅₃	#353	1	1	yes	characteristic
42	C₂×D₅₃	—	8	2	no	—
43	C₂×D₅₃	—	16	2	no	—
44	#6157	—	32	2	no	—
45	C₄×C₅₃	—	2	2	no	—
46	C₄×C₅₃	—	2	1	no	—
47	#1712	—	53	1	no	—
48	#1408	—	53	2	no	—
49	C₂×C₅₃	—	2	1	no	—
50	C₂×C₅₃	—	32	1	no	—
51	C₂×C₅₃	#1408	1	1	yes	characteristic
52	D₅₃	—	16	2	no	—
53	#493	—	53	1	no	—
54	#493	—	106	2	no	—
55	#493	#6157	1	1	yes	characteristic, complemented
56	#353	—	106	2	no	—
57	C₈²	—	53	1	no	—
58	C₅₃	#7529	1	1	yes	characteristic, complemented, Sylow
59	C₂×D₈	—	4	1	no	—
60	C₂×D₈	—	212	1	no	—
61	C₂×D₈	—	212	4	no	—
62	#127	—	2	1	no	—
63	#127	—	53	2	no	—
64	#99	—	212	2	no	—
65	C₄×C₈	—	106	1	no	—
66	C₄×C₈	C₂×#6157	1	1	yes	characteristic
67	C₂×D₄	—	4	2	no	—
68	C₂×D₄	—	4	1	no	—
69	C₂×D₄	—	106	2	no	—
70	C₂×D₄	—	212	2	no	—
71	D₈	—	8	2	no	—
72	D₈	—	424	2	no	—
73	D₈	—	424	8	no	—
74	C₂×C₈	—	2	1	no	—
75	C₂×C₈	—	106	2	no	—
76	C₂²⋊C₄	—	424	2	no	—
77	C₄²	C₂²⋊#6157	1	1	yes	characteristic
78	C₂³	—	8	1	no	—
79	C₂³	—	212	2	no	—
80	D₄	—	8	2	no	—
81	D₄	—	8	4	no	—
82	D₄	—	424	4	no	—
83	D₄	—	424	2	no	—
84	C₂×C₄	—	2	1	no	—
85	C₂×C₄	—	848	2	no	—
86	C₂×C₄	#13133275	1	1	yes	characteristic
87	C₈	—	2	2	no	—
88	C₈	—	106	4	no	—
89	C₂²	—	16	2	no	—
90	C₂²	—	16	1	no	—
91	C₂²	—	848	2	no	—
92	C₂²	—	424	2	no	—
93	C₂²	#16811527	1	1	yes	characteristic
94	C₄	—	2	2	no	—
95	C₄	—	2	1	no	—
96	C₄	—	1696	2	no	—
97	C₂	—	2	1	no	—
98	C₂	—	32	1	no	—
99	C₂	—	848	2	no	—
100	C₂	#20289846	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20289847

C₂

yes

maximal, complemented

#20274329

C₂

yes

maximal, characteristic

#16811528

—

#16811000

C₂²

yes

characteristic

#16810999

—

C₈×#88322

C₄

yes

characteristic, complemented

C₅₃×#493

C₄

yes

characteristic, complemented

#13133276

—

C₂×D₄₂₄

—

C₂×#11719922

—

#13133214

D₄

yes

—

C₄×#88322

—

C₂×D₈×C₅₃

—

#127×C₅₃

—

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂²⋊#6157

—

D₈×C₅₃

—

C₂×C₈×C₅₃

—

C₄²×C₅₃

C₂²⋊C₄

yes

characteristic

#11719922

—

C₂×#88322

—

D₄₂₄

—

C₂³×C₅₃

—

C₂²×D₅₃

—

C₂×#6157

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#99

yes

characteristic

D₂₁₂

—

D₂₁₂

—

C₈×C₅₃

—

#88322

—

#7529

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#353

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

#6157

—

C₄×C₅₃

—

C₄×C₅₃

—

#1712

—

#1408

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#1408

yes

characteristic

D₅₃

—

#493

—

#493

—

106

—

#493

#6157

yes

characteristic, complemented

#353

—

106

—

C₈²

—

C₅₃

#7529

yes

characteristic, complemented, Sylow

C₂×D₈

—

C₂×D₈

—

212

—

C₂×D₈

—

212

—

#127

—

#127

—

#99

—

212

—

C₄×C₈

—

106

—

C₄×C₈

C₂×#6157

yes

characteristic

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

106

—

C₂×D₄

—

212

—

D₈

—

D₈

—

424

—

D₈

—

424

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂²⋊C₄

—

424

—

C₄²

C₂²⋊#6157

yes

characteristic

C₂³

—

C₂³

—

212

—

D₄

—

D₄

—

D₄

—

424

—

D₄

—

424

—

C₂×C₄

—

C₂×C₄

—

848

—

C₂×C₄

#13133275

yes

characteristic

C₈

—

C₈

—

106

—

C₂²

—

C₂²

—

C₂²

—

848

—

C₂²

—

424

—

C₂²

#16811527

yes

characteristic

C₄

—

C₄

—

C₄

—

1696

—

C₂

—

C₂

—

C₂

—

848

—

100

C₂

#20289846

yes

characteristic

	subgroup	G/H
center	C₂	#20289846
commutator	C₄×C₈×C₅₃	C₂×C₄
Frattini	C₄×C₈	C₂×#6157
Fitting	C₅₃×#493	C₄
socle	C₂×C₅₃	#1408
prime radical	#20274329	C₂
cosocle	#16811000	C₂²
hypercenter	#493	#6157
hypocenter	C₅₃	#7529
LCS	G ▷ C₄×C₈×C₅₃ ▷ C₄²×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ #493

subgroup

G/H

center

C₂

#20289846

commutator

C₄×C₈×C₅₃

C₂×C₄

Frattini

C₄×C₈

C₂×#6157

Fitting

C₅₃×#493

C₄

C₂×C₅₃

C₂

C₂²

C₅₃

G ▷ C₄×C₈×C₅₃ ▷ C₄²×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ #493

N	H ≅ G/N	type
1. #20289847	C₂	semidirect
6. C₈×#88322	C₄	semidirect
7. C₅₃×#493	C₄	semidirect
55. #493	#6157	semidirect
58. C₅₃	#7529	semidirect

H ≅ G/N

type

1. #20289847

C₂

semidirect

6. C₈×#88322

C₄

semidirect

7. C₅₃×#493

C₄

semidirect

55. #493

#6157

semidirect

58. C₅₃

#7529

semidirect

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

subgroup	index
#20289847	2
#20274329	2
#7529	53

subgroup

index

#20289847

#20274329

#7529

Subgroups — #28299360

Conjugacy classes of subgroups

Complements

Maximal subgroups