Group #28307867 — subgroups

Subgroups — #28307867

Order 13568.

228 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 92 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	#20291078	C₂	1	2	yes	maximal
2	#20289633	C₂	1	4	yes	maximal, complemented
3	#20272997	C₂	1	1	yes	maximal, characteristic
4	#16811544	C₄	1	4	yes	—
5	#16811518	C₄	1	4	yes	—
6	#16811511	C₂²	1	4	yes	—
7	#16810674	C₄	1	2	yes	—
8	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
9	#16810345	C₂²	1	2	yes	—
10	C₈×#88322	C₄	1	1	yes	characteristic
11	C₈²×C₅₃	C₄	1	1	yes	characteristic
12	C₂×#11719910	C₂×C₄	1	8	yes	—
13	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
14	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
15	#13133146	—	2	2	no	—
16	#13133129	—	2	2	no	—
17	C₄×#88322	C₂×C₄	1	2	yes	—
18	C₄×#88322	C₂³	1	1	yes	characteristic
19	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
20	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
21	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×D₅₃	C₄²	1	1	yes	characteristic
23	C₂×C₄×D₅₃	M₁₆	1	2	yes	—
24	C₂×C₈×C₅₃	—	2	2	no	—
25	C₂×C₈×C₅₃	C₄²	1	2	yes	—
26	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
27	C₄×Dic₅₃	C₄²	1	1	yes	characteristic
28	C₄×Dic₅₃	M₁₆	1	2	yes	—
29	C₂×#88322	—	2	2	no	—
30	C₂×#88322	C₂²×C₄	1	2	yes	—
31	C₈×D₅₃	—	2	4	no	—
32	#11719910	—	2	8	no	—
33	C₂²×D₅₃	#97	1	1	yes	characteristic
34	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
35	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
36	C₂×Dic₅₃	#97	1	1	yes	characteristic
37	C₂×Dic₅₃	#97	1	2	yes	—
38	C₄×D₅₃	—	2	2	no	—
39	C₄×D₅₃	—	4	2	no	—
40	C₈×C₅₃	—	2	2	no	—
41	C₈×C₅₃	—	4	2	no	—
42	#88322	—	4	2	no	—
43	#88322	#131	1	4	yes	—
44	#11416	—	53	1	no	maximal, Sylow
45	C₂²×C₅₃	C₂×#97	1	1	yes	characteristic
46	C₂×D₅₃	—	2	2	no	—
47	C₂×D₅₃	—	2	1	no	—
48	C₄×C₅₃	—	2	2	no	—
49	C₄×C₅₃	#405	1	2	yes	—
50	Dic₅₃	—	2	2	no	—
51	Dic₅₃	—	2	1	no	—
52	#2163	—	53	2	no	—
53	#1450	—	53	1	no	—
54	#1313	—	53	4	no	—
55	C₂×C₅₃	#1452	1	1	yes	characteristic
56	C₂×C₅₃	#2110	1	2	yes	—
57	D₅₃	—	4	1	no	—
58	#431	—	53	2	no	—
59	C₂×C₄×C₈	—	53	1	no	—
60	C₄⋊C₁₆	—	53	4	no	—
61	C₂²⋊C₁₆	—	53	4	no	—
62	#346	—	53	4	no	—
63	#322	—	53	2	no	—
64	C₈²	—	53	1	no	—
65	C₈²	#6157	1	1	yes	characteristic
66	C₅₃	#11416	1	1	yes	characteristic, complemented, Sylow
67	C₂²×C₈	—	53	2	no	—
68	C₂×C₄²	—	53	1	no	—
69	C₂×C₁₆	—	53	8	no	—
70	C₄⋊C₈	—	106	2	no	—
71	C₂²⋊C₈	—	106	2	no	—
72	C₄×C₈	—	53	2	no	—
73	C₄×C₈	—	53	1	no	—
74	C₄×C₈	—	53	2	no	—
75	C₄×C₈	C₂×#6157	1	2	yes	—
76	C₄×C₈	C₂×#6157	1	1	yes	characteristic
77	C₂²×C₄	—	53	1	no	—
78	C₂²×C₄	—	53	2	no	—
79	C₂×C₈	—	2	2	no	—
80	C₂×C₈	—	53	2	no	—
81	C₂×C₈	—	106	4	no	—
82	C₂×C₈	—	106	2	no	—
83	C₂×C₈	C₄×#6157	1	2	yes	—
84	C₄²	—	53	1	no	—
85	C₄²	—	53	2	no	—
86	C₄²	C₂²×#6157	1	1	yes	characteristic
87	C₁₆	—	106	8	no	—
88	C₂³	—	53	1	no	—
89	C₂×C₄	—	53	2	no	—
90	C₂×C₄	—	53	1	no	—
91	C₂×C₄	—	106	2	no	—
92	C₂×C₄	—	212	2	no	—
93	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
94	C₂×C₄	D₅₃⋊M₁₆	1	2	yes	—
95	C₈	—	2	2	no	—
96	C₈	—	4	2	no	—
97	C₈	—	53	4	no	—
98	C₈	—	212	2	no	—
99	C₂²	—	106	2	no	—
100	C₂²	—	106	1	no	—
101	C₂²	#16811582	1	1	yes	characteristic
102	C₄	—	2	2	no	—
103	C₄	—	106	2	no	—
104	C₄	—	106	1	no	—
105	C₄	#16811586	1	2	yes	—
106	C₂	—	212	1	no	—
107	C₂	#20289929	1	1	yes	characteristic
108	C₂	#20290937	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20291078

C₂

yes

maximal

#20289633

C₂

yes

maximal, complemented

#20272997

C₂

yes

maximal, characteristic

#16811544

C₄

yes

—

#16811518

C₄

yes

—

#16811511

C₂²

yes

—

#16810674

C₄

yes

—

C₄×C₈×D₅₃

C₂²

yes

characteristic

#16810345

C₂²

yes

—

C₈×#88322

C₄

yes

characteristic

C₈²×C₅₃

C₄

yes

characteristic

C₂×#11719910

C₂×C₄

yes

—

C₂×C₈×D₅₃

C₂×C₄

yes

—

C₄²×D₅₃

C₂×C₄

yes

—

—

C₂×C₄

yes

—

C₄×#88322

C₂³

yes

characteristic

C₈×Dic₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₄×D₅₃

C₄²

yes

characteristic

C₂×C₄×D₅₃

M₁₆

yes

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₄²

yes

—

C₄²×C₅₃

C₂²×C₄

yes

characteristic

C₄×Dic₅₃

C₄²

yes

characteristic

C₄×Dic₅₃

M₁₆

yes

—

C₂×#88322

—

C₂×#88322

C₂²×C₄

yes

—

C₈×D₅₃

—

#11719910

—

C₂²×D₅₃

#97

yes

characteristic

C₂×C₄×C₅₃

C₂×C₄²

yes

characteristic

C₂×C₄×C₅₃

C₂×M₁₆

yes

—

C₂×Dic₅₃

#97

yes

characteristic

C₂×Dic₅₃

#97

yes

—

C₄×D₅₃

—

C₄×D₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

#88322

—

#88322

#131

yes

—

#11416

—

maximal, Sylow

C₂²×C₅₃

C₂×#97

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#405

yes

—

Dic₅₃

—

Dic₅₃

—

#2163

—

#1450

—

#1313

—

C₂×C₅₃

#1452

yes

characteristic

C₂×C₅₃

#2110

yes

—

D₅₃

—

#431

—

C₂×C₄×C₈

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

#346

—

#322

—

C₈²

—

C₈²

#6157

yes

characteristic

C₅₃

#11416

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂×C₄²

—

C₂×C₁₆

—

C₄⋊C₈

—

106

—

C₂²⋊C₈

—

106

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×#6157

yes

—

C₄×C₈

C₂×#6157

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₂×C₈

C₄×#6157

yes

—

C₄²

—

C₄²

—

C₄²

C₂²×#6157

yes

characteristic

C₁₆

—

106

—

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

106

—

C₂×C₄

—

212

—

C₂×C₄

C₂×C₄×#6157

yes

characteristic

C₂×C₄

D₅₃⋊M₁₆

yes

—

C₈

—

C₈

—

C₈

—

C₈

—

212

—

C₂²

—

106

—

100

C₂²

—

106

—

101

C₂²

#16811582

yes

characteristic

102

C₄

—

103

C₄

—

106

—

104

C₄

—

106

—

105

C₄

#16811586

yes

—

106

C₂

—

212

—

107

C₂

#20289929

yes

characteristic

108

C₂

#20290937

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₄×#6157
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₄²	C₂²×#6157
Fitting	C₈²×C₅₃	C₄
socle	C₂²×C₅₃	C₂×#97
prime radical	C₂²×D₅₃	#97
cosocle	C₄×#88322	C₂³
hypercenter	C₈²	#6157
hypocenter	C₅₃	#11416
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₈²

subgroup

G/H

center

C₂×C₄

C₂×C₄×#6157

commutator

C₂×C₄×C₅₃

C₂×C₄²

Frattini

C₄²

C₂²×#6157

Fitting

C₈²×C₅₃

C₄

socle

C₂²×C₅₃

C₂×#97

C₂²×D₅₃

C₂³

C₈²

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₅₃

C₁ ◁ C₂×C₄ ◁ C₈²

N	H ≅ G/N	type
2. #20289633	C₂	semidirect
66. C₅₃	#11416	semidirect

H ≅ G/N

type

2. #20289633

C₂

semidirect

66. C₅₃

#11416

semidirect

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

subgroup	index
#20291078	2
#20289633	2
#20272997	2
#11416	53

subgroup

index

Subgroups — #28307867

Conjugacy classes of subgroups

Complements

Maximal subgroups