Group #28298175 — subgroups

Subgroups — #28298175

Order 13568.

186 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 46 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	#20289745	C₂	1	2	yes	maximal, complemented
2	#20274329	C₂	1	1	yes	maximal, characteristic
3	C₄⋊#11719910	—	2	2	no	—
4	#16811000	C₄	1	1	yes	characteristic
5	#16810999	—	2	2	no	—
6	C₈×#88322	C₂²	1	1	yes	characteristic
7	C₅₃×#493	C₄	1	1	yes	characteristic
8	C₂×#11719910	—	4	2	no	—
9	C₂×D₄₂₄	—	4	1	no	—
10	C₂×#11719922	—	4	2	no	—
11	C₂×#11719922	—	4	2	no	—
12	#13133214	—	2	1	no	—
13	C₄×#88322	D₄	1	2	yes	—
14	C₂×D₈×C₅₃	—	4	1	no	—
15	#127×C₅₃	—	2	1	no	—
16	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
17	C₂×D₄×C₅₃	—	4	1	no	—
18	C₂×D₄×C₅₃	—	4	2	no	—
19	C₂×D₂₁₂	—	4	1	no	—
20	C₂×D₂₁₂	—	4	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	4	no	—
25	#11719922	—	8	2	no	—
26	#11719922	—	8	2	no	—
27	C₂×#88322	D₈	1	2	yes	—
28	C₂×#88322	SD₁₆	1	2	yes	—
29	D₄₂₄	—	8	2	no	—
30	#11719910	—	8	2	no	—
31	C₂³×C₅₃	—	8	1	no	—
32	C₂²×D₅₃	—	8	1	no	—
33	D₄×C₅₃	—	8	2	no	—
34	D₄×C₅₃	—	8	2	no	—
35	D₄×C₅₃	—	8	2	no	—
36	C₂×C₄×C₅₃	#100	1	1	yes	characteristic
37	C₂×C₄×C₅₃	#102	1	2	yes	—
38	D₂₁₂	—	8	2	no	—
39	D₂₁₂	—	8	2	no	—
40	D₂₁₂	—	8	2	no	—
41	C₈×C₅₃	—	2	2	no	—
42	#88322	—	2	2	no	—
43	#88322	D₁₆	1	2	yes	—
44	#88322	SD₃₂	1	2	yes	—
45	#7096	—	53	1	no	maximal, Sylow
46	C₂²×C₅₃	—	16	1	no	—
47	C₂²×C₅₃	—	16	2	no	—
48	C₂²×C₅₃	#331	1	1	yes	characteristic
49	C₂×D₅₃	—	16	1	no	—
50	C₂×D₅₃	—	16	2	no	—
51	C₄×C₅₃	—	2	1	no	—
52	C₄×C₅₃	#357	1	2	yes	—
53	C₄×C₅₃	#361	1	2	yes	—
54	#1712	—	53	1	no	—
55	#1370	—	53	2	no	—
56	C₂×C₅₃	—	32	1	no	—
57	C₂×C₅₃	#1357	1	1	yes	characteristic
58	C₂×C₅₃	#1361	1	2	yes	—
59	D₅₃	—	32	1	no	—
60	#493	—	53	1	no	—
61	#493	—	106	2	no	—
62	#493	#6157	1	1	yes	characteristic
63	C₄⋊C₁₆	—	106	2	no	—
64	C₈²	—	53	1	no	—
65	C₅₃	#7096	1	1	yes	characteristic, complemented, Sylow
66	C₂×D₈	—	4	1	no	—
67	C₂×D₈	—	212	1	no	—
68	C₂×D₈	—	212	2	no	—
69	C₂×D₈	—	212	2	no	—
70	#127	—	2	1	no	—
71	#127	—	106	1	no	—
72	C₂×C₁₆	—	212	2	no	—
73	C₄×C₈	—	53	2	no	—
74	C₄×C₈	C₂×#6157	1	1	yes	characteristic
75	C₂×D₄	—	4	1	no	—
76	C₂×D₄	—	4	2	no	—
77	C₂×D₄	—	212	1	no	—
78	C₂×D₄	—	212	2	no	—
79	D₈	—	8	2	no	—
80	D₈	—	424	2	no	—
81	D₈	—	424	2	no	—
82	D₈	—	424	2	no	—
83	D₈	—	424	4	no	—
84	C₂×C₈	—	2	1	no	—
85	C₂×C₈	—	53	2	no	—
86	C₂×C₈	—	53	2	no	—
87	C₄²	C₂²⋊#6157	1	1	yes	characteristic
88	C₁₆	—	424	2	no	—
89	C₂³	—	8	1	no	—
90	C₂³	—	424	1	no	—
91	D₄	—	8	2	no	—
92	D₄	—	8	2	no	—
93	D₄	—	8	2	no	—
94	D₄	—	424	2	no	—
95	D₄	—	424	2	no	—
96	D₄	—	424	2	no	—
97	C₂×C₄	#13133277	1	1	yes	characteristic
98	C₂×C₄	#13133279	1	2	yes	—
99	C₈	—	2	2	no	—
100	C₈	—	53	2	no	—
101	C₈	—	53	2	no	—
102	C₈	—	106	2	no	—
103	C₂²	—	16	1	no	—
104	C₂²	—	16	2	no	—
105	C₂²	—	848	1	no	—
106	C₂²	—	848	2	no	—
107	C₂²	#16811481	1	1	yes	characteristic
108	C₄	—	2	1	no	—
109	C₄	#16811535	1	2	yes	—
110	C₄	#16811541	1	2	yes	—
111	C₂	—	32	1	no	—
112	C₂	—	1696	1	no	—
113	C₂	#20289723	1	1	yes	characteristic
114	C₂	#20289729	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20289745

C₂

yes

maximal, complemented

#20274329

C₂

yes

maximal, characteristic

C₄⋊#11719910

—

#16811000

C₄

yes

characteristic

#16810999

—

C₈×#88322

C₂²

yes

characteristic

C₅₃×#493

C₄

yes

characteristic

C₂×#11719910

—

C₂×D₄₂₄

—

—

—

—

D₄

yes

—

C₂×D₈×C₅₃

—

#127×C₅₃

—

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

D₈×C₅₃

—

C₂×C₈×C₅₃

—

C₄²×C₅₃

C₂²⋊C₄

yes

—

—

—

D₈

yes

—

C₂×#88322

SD₁₆

yes

—

D₄₂₄

—

#11719910

—

C₂³×C₅₃

—

C₂²×D₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

#100

yes

characteristic

C₂×C₄×C₅₃

#102

yes

—

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

C₈×C₅₃

—

#88322

—

#88322

D₁₆

yes

—

#88322

SD₃₂

yes

—

#7096

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#331

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#357

yes

—

C₄×C₅₃

#361

yes

—

#1712

—

#1370

—

C₂×C₅₃

—

C₂×C₅₃

#1357

yes

characteristic

C₂×C₅₃

#1361

yes

—

D₅₃

—

#493

—

#493

—

106

—

#493

#6157

yes

characteristic

C₄⋊C₁₆

—

106

—

C₈²

—

C₅₃

#7096

yes

characteristic, complemented, Sylow

C₂×D₈

—

C₂×D₈

—

212

—

C₂×D₈

—

212

—

C₂×D₈

—

212

—

#127

—

#127

—

106

—

C₂×C₁₆

—

212

—

C₄×C₈

—

C₄×C₈

C₂×#6157

yes

characteristic

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

212

—

C₂×D₄

—

212

—

D₈

—

D₈

—

424

—

D₈

—

424

—

D₈

—

424

—

D₈

—

424

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄²

C₂²⋊#6157

yes

characteristic

C₁₆

—

424

—

C₂³

—

C₂³

—

424

—

D₄

—

D₄

—

D₄

—

D₄

—

424

—

D₄

—

424

—

D₄

—

424

—

C₂×C₄

#13133277

yes

characteristic

C₂×C₄

#13133279

yes

—

C₈

—

100

C₈

—

101

C₈

—

102

C₈

—

106

—

103

C₂²

—

104

C₂²

—

105

C₂²

—

848

—

106

C₂²

—

848

—

107

C₂²

#16811481

yes

characteristic

108

C₄

—

109

C₄

#16811535

yes

—

110

C₄

#16811541

yes

—

111

C₂

—

112

C₂

—

1696

—

113

C₂

#20289723

yes

characteristic

114

C₂

#20289729

yes

—

	subgroup	G/H
center	C₂²	#16811481
commutator	C₄×C₈×C₅₃	C₂×C₄
Frattini	C₄×C₈	C₂×#6157
Fitting	C₅₃×#493	C₄
socle	C₂²×C₅₃	#331
prime radical	#20274329	C₂
cosocle	C₈×#88322	C₂²
hypercenter	#493	#6157
hypocenter	C₅₃	#7096
LCS	G ▷ C₄×C₈×C₅₃ ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #493

subgroup

G/H

center

C₂²

#16811481

commutator

C₄×C₈×C₅₃

C₂×C₄

Frattini

C₄×C₈

C₂×#6157

Fitting

C₅₃×#493

C₄

C₂²×C₅₃

C₂

C₂²

C₅₃

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #493

N	H ≅ G/N	type
1. #20289745	C₂	semidirect
65. C₅₃	#7096	semidirect

H ≅ G/N

type

1. #20289745

C₂

semidirect

65. C₅₃

#7096

semidirect

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

subgroup	index
#20289745	2
#20274329	2
#7096	53

subgroup

index

#20289745

#20274329

#7096

Subgroups — #28298175

Conjugacy classes of subgroups

Complements

Maximal subgroups