Group #28303554 — subgroups

Subgroups — #28303554

Order 13568.

228 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 76 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	#20290278	C₂	1	2	yes	maximal
2	#20289572	C₂	1	4	yes	maximal
3	#20272997	C₂	1	1	yes	maximal, characteristic
4	#16811645	C₂²	1	4	yes	—
5	#16811487	—	2	2	no	—
6	#16811485	—	2	2	no	—
7	#16810674	C₂²	1	2	yes	—
8	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
9	#16810345	C₄	1	2	yes	—
10	C₈×#88322	C₄	1	1	yes	characteristic
11	C₈²×C₅₃	C₄	1	1	yes	characteristic
12	C₄⋊#88324	—	2	4	no	—
13	#13133274	—	2	4	no	—
14	#13133272	—	2	4	no	—
15	C₂×C₈×D₅₃	Q₈	1	2	yes	—
16	C₄²×D₅₃	C₂³	1	1	yes	characteristic
17	#13133146	—	2	2	no	—
18	#13133129	—	2	2	no	—
19	C₄×#88322	C₂×C₄	1	2	yes	—
20	C₄×#88322	C₂×C₄	1	1	yes	characteristic
21	C₈×Dic₅₃	D₄	1	2	yes	—
22	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
23	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×D₅₃	C₂×Q₈	1	1	yes	characteristic
25	C₂×C₄×D₅₃	#41	1	2	yes	—
26	C₂×#88324	—	4	4	no	—
27	C₂×C₈×C₅₃	—	2	2	no	—
28	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
29	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
30	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
31	C₄×Dic₅₃	#41	1	2	yes	—
32	C₂×#88322	—	2	2	no	—
33	C₂×#88322	C₄⋊C₄	1	2	yes	—
34	C₈×D₅₃	—	2	4	no	—
35	C₂²×D₅₃	#125	1	1	yes	characteristic
36	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₄×C₅₃	#117	1	2	yes	—
38	C₂×Dic₅₃	#124	1	2	yes	—
39	C₂×Dic₅₃	#125	1	1	yes	characteristic
40	C₄×D₅₃	—	2	2	no	—
41	C₄×D₅₃	—	4	2	no	—
42	#88324	—	8	4	no	—
43	C₈×C₅₃	—	2	2	no	—
44	C₈×C₅₃	—	4	2	no	—
45	#88322	—	4	2	no	—
46	#88322	#108	1	4	yes	—
47	#8797	—	53	1	no	maximal, Sylow
48	C₂²×C₅₃	#382	1	1	yes	characteristic
49	C₂×D₅₃	—	2	2	no	—
50	C₂×D₅₃	—	2	1	no	—
51	C₄×C₅₃	—	2	2	no	—
52	C₄×C₅₃	C₂×#108	1	2	yes	—
53	Dic₅₃	—	2	2	no	—
54	Dic₅₃	—	2	1	no	—
55	#1568	—	53	2	no	—
56	#1450	—	53	1	no	—
57	#1287	—	53	4	no	—
58	C₂×C₅₃	#1833	1	2	yes	—
59	C₂×C₅₃	#1837	1	1	yes	characteristic
60	D₅₃	—	4	1	no	—
61	#432	—	53	4	no	—
62	#431	—	53	2	no	—
63	C₂×C₄×C₈	—	53	1	no	—
64	#335	—	106	2	no	—
65	#334	—	106	2	no	—
66	#322	—	53	2	no	—
67	C₈²	—	53	1	no	—
68	C₈²	#6157	1	1	yes	characteristic
69	C₅₃	#8797	1	1	yes	characteristic, complemented, Sylow
70	C₂²×C₈	—	53	2	no	—
71	C₂×C₄²	—	53	1	no	—
72	C₄⋊C₈	—	106	2	no	—
73	C₄⋊C₈	—	106	4	no	—
74	C₂²⋊C₈	—	106	2	no	—
75	C₂²⋊C₈	—	106	4	no	—
76	#97	—	106	4	no	—
77	C₄×C₈	—	53	2	no	—
78	C₄×C₈	—	53	1	no	—
79	C₄×C₈	—	53	2	no	—
80	C₄×C₈	C₂×#6157	1	2	yes	—
81	C₄×C₈	C₂×#6157	1	1	yes	characteristic
82	C₂²×C₄	—	53	1	no	—
83	C₂²×C₄	—	53	2	no	—
84	C₂×C₈	—	2	2	no	—
85	C₂×C₈	—	53	2	no	—
86	C₂×C₈	—	106	4	no	—
87	C₂×C₈	—	106	2	no	—
88	C₂×C₈	—	212	4	no	—
89	C₂×C₈	C₄⋊#6157	1	2	yes	—
90	C₄²	—	53	1	no	—
91	C₄²	—	53	2	no	—
92	C₄²	C₂²×#6157	1	1	yes	characteristic
93	C₂³	—	53	1	no	—
94	C₂×C₄	—	53	2	no	—
95	C₂×C₄	—	53	1	no	—
96	C₂×C₄	—	106	2	no	—
97	C₂×C₄	—	212	2	no	—
98	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
99	C₂×C₄	#13133296	1	2	yes	—
100	C₈	—	2	2	no	—
101	C₈	—	4	2	no	—
102	C₈	—	53	4	no	—
103	C₈	—	424	4	no	—
104	C₈	—	212	2	no	—
105	C₂²	—	106	2	no	—
106	C₂²	—	106	1	no	—
107	C₂²	#16811567	1	1	yes	characteristic
108	C₄	—	2	2	no	—
109	C₄	—	106	2	no	—
110	C₄	—	106	1	no	—
111	C₄	D₅₃⋊#108	1	2	yes	—
112	C₂	—	212	1	no	—
113	C₂	#20290634	1	2	yes	—
114	C₂	#20290640	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20290278

C₂

yes

maximal

#20289572

C₂

yes

maximal

#20272997

C₂

yes

maximal, characteristic

#16811645

C₂²

yes

—

#16811487

—

#16811485

—

#16810674

C₂²

yes

—

C₄×C₈×D₅₃

C₂²

yes

characteristic

#16810345

C₄

yes

—

C₈×#88322

C₄

yes

characteristic

C₈²×C₅₃

C₄

yes

—

—

—

C₂×C₈×D₅₃

Q₈

yes

—

C₄²×D₅₃

C₂³

yes

—

—

C₂×C₄

yes

—

C₄×#88322

C₂×C₄

yes

characteristic

C₈×Dic₅₃

D₄

yes

—

C₄×C₈×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₄×D₅₃

C₂×Q₈

yes

characteristic

C₂×C₄×D₅₃

#41

yes

—

C₂×#88324

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₄⋊C₄

yes

—

C₄²×C₅₃

C₂²×C₄

yes

characteristic

C₄×Dic₅₃

C₂×D₄

yes

characteristic

C₄×Dic₅₃

#41

yes

—

C₂×#88322

—

C₂×#88322

C₄⋊C₄

yes

—

C₈×D₅₃

—

C₂²×D₅₃

#125

yes

characteristic

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

#117

yes

—

C₂×Dic₅₃

#124

yes

—

C₂×Dic₅₃

#125

yes

characteristic

C₄×D₅₃

—

C₄×D₅₃

—

#88324

—

C₈×C₅₃

—

C₈×C₅₃

—

#88322

—

#88322

#108

yes

—

—

C₂²×C₅₃

yes

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×#108

yes

—

Dic₅₃

—

Dic₅₃

—

#1568

—

#1450

—

#1287

—

C₂×C₅₃

#1833

yes

—

C₂×C₅₃

#1837

yes

characteristic

D₅₃

—

#432

—

#431

—

C₂×C₄×C₈

—

#335

—

106

—

#334

—

106

—

#322

—

C₈²

—

C₈²

#6157

yes

characteristic

C₅₃

#8797

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂×C₄²

—

C₄⋊C₈

—

106

—

C₄⋊C₈

—

106

—

C₂²⋊C₈

—

106

—

C₂²⋊C₈

—

106

—

#97

—

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

—

212

—

C₂×C₈

C₄⋊#6157

yes

—

C₄²

—

C₄²

—

C₄²

C₂²×#6157

yes

characteristic

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

106

—

C₂×C₄

—

212

—

C₂×C₄

C₂×C₄⋊#6157

yes

characteristic

C₂×C₄

#13133296

yes

—

100

C₈

—

101

C₈

—

102

C₈

—

103

C₈

—

424

—

104

C₈

—

212

—

105

C₂²

—

106

—

106

C₂²

—

106

—

107

C₂²

#16811567

yes

characteristic

108

C₄

—

109

C₄

—

106

—

110

C₄

—

106

—

111

C₄

D₅₃⋊#108

yes

—

112

C₂

—

212

—

113

C₂

#20290634

yes

—

114

C₂

#20290640

yes

characteristic

	subgroup	G/H
center	C₂²	#16811567
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₄²	C₂²×#6157
Fitting	C₈²×C₅₃	C₄
socle	C₂²×C₅₃	#382
prime radical	C₂²×D₅₃	#125
cosocle	C₄²×D₅₃	C₂³
hypercenter	C₈²	#6157
hypocenter	C₅₃	#8797
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈²

subgroup

G/H

center

C₂²

#16811567

commutator

C₄²×C₅₃

C₂²×C₄

Frattini

C₄²

C₂²×#6157

Fitting

C₈²×C₅₃

C₄

C₂²×C₅₃

C₂²×D₅₃

C₄²×D₅₃

C₂³

C₈²

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈²

N	H ≅ G/N	type
69. C₅₃	#8797	semidirect

H ≅ G/N

type

69. C₅₃

#8797

semidirect

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

subgroup	index
#20290278	2
#20289572	2
#20272997	2
#8797	53

subgroup

index

Subgroups — #28303554

Conjugacy classes of subgroups

Complements

Maximal subgroups