Group #27276660 — subgroups

Subgroups — #27276660

Order 13568.

656 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 139 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	#20275693	C₂	1	3	yes	maximal
2	#20273453	C₂	1	3	yes	maximal
3	C₅₃×#3556	C₂	1	1	yes	maximal, characteristic
4	#16810647	C₂²	1	3	yes	—
5	C₄²×Dic₅₃	C₂²	1	1	yes	characteristic
6	#16810382	—	2	6	no	—
7	#16810369	—	2	6	no	—
8	C₅₃×#564	C₄	1	4	yes	complemented
9	C₅₃×#557	—	2	6	no	—
10	C₅₃×#541	—	2	6	no	—
11	C₅₃×#533	C₂²	1	3	yes	—
12	C₂×#11719917	—	2	3	no	—
13	C₂×C₄×Dic₅₃	—	2	3	no	—
14	C₄⋊#88322	—	2	6	no	—
15	C₄⋊Q₈×C₅₃	—	2	6	no	—
16	C₄⋊Q₈×C₅₃	D₄	1	6	yes	—
17	#126×C₅₃	—	2	8	no	—
18	#125×C₅₃	—	2	6	no	—
19	#125×C₅₃	—	2	24	no	—
20	#125×C₅₃	C₂×C₄	1	6	yes	—
21	C₂²⋊Q₈×C₅₃	—	2	6	no	—
22	C₄×Q₈×C₅₃	—	2	12	no	—
23	C₄×Q₈×C₅₃	—	2	6	no	—
24	#117×C₅₃	D₄	1	6	yes	—
25	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
26	C₂×Q₈×C₅₃	—	2	6	no	—
27	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
28	C₂²×Dic₅₃	—	4	1	no	—
29	C₄⋊C₄×C₅₃	—	2	24	no	—
30	C₄⋊C₄×C₅₃	—	2	12	no	—
31	C₄⋊C₄×C₅₃	—	2	6	no	—
32	C₄⋊C₄×C₅₃	C₂²⋊C₄	1	12	yes	—
33	C₂²⋊C₄×C₅₃	—	2	6	no	—
34	C₄²×C₅₃	—	2	8	no	—
35	C₄²×C₅₃	—	2	6	no	—
36	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
37	C₄²×C₅₃	C₂×D₄	1	3	yes	—
38	C₄×Dic₅₃	—	2	6	no	—
39	C₄×Dic₅₃	—	4	3	no	—
40	#11719917	—	4	6	no	—
41	C₂×#88322	—	4	3	no	—
42	C₂³×C₅₃	#120	1	1	yes	characteristic
43	Q₈×C₅₃	—	4	6	no	—
44	Q₈×C₅₃	—	4	6	no	—
45	C₂×C₄×C₅₃	—	2	3	no	—
46	C₂×C₄×C₅₃	—	2	3	no	—
47	C₂×C₄×C₅₃	—	2	12	no	—
48	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	3	yes	—
49	C₂×C₄×C₅₃	#120	1	3	yes	—
50	C₂×Dic₅₃	—	4	6	no	—
51	#88322	—	8	3	no	—
52	#8533	—	53	1	no	maximal, Sylow
53	C₂²×C₅₃	—	2	3	no	—
54	C₂²×C₅₃	#379	1	1	yes	characteristic
55	C₄×C₅₃	—	2	3	no	—
56	C₄×C₅₃	—	4	12	no	—
57	C₄×C₅₃	#454	1	3	yes	—
58	C₄×C₅₃	#456	1	3	yes	—
59	Dic₅₃	—	4	4	no	—
60	#3556	D₅₃	1	1	yes	characteristic
61	#1994	—	53	3	no	—
62	#1561	—	53	3	no	—
63	C₂×C₅₃	—	4	1	no	—
64	C₂×C₅₃	#1793	1	3	yes	—
65	#564	Dic₅₃	1	4	yes	complemented
66	#557	—	2	6	no	—
67	#541	—	2	6	no	—
68	#533	C₂×D₅₃	1	3	yes	—
69	#423	—	53	3	no	—
70	C₄³	—	53	1	no	—
71	#339	—	106	6	no	—
72	#332	—	106	6	no	—
73	C₅₃	#8533	1	1	yes	characteristic, complemented, Sylow
74	C₂×M₁₆	—	106	3	no	—
75	C₄⋊Q₈	—	2	6	no	—
76	C₄⋊Q₈	C₂²⋊D₅₃	1	6	yes	—
77	#126	—	2	8	no	—
78	#125	—	2	6	no	—
79	#125	—	2	24	no	—
80	#125	C₂×Dic₅₃	1	6	yes	—
81	C₂²⋊Q₈	—	2	6	no	—
82	C₄×Q₈	—	2	12	no	—
83	C₄×Q₈	—	2	6	no	—
84	#117	C₂²⋊D₅₃	1	6	yes	—
85	C₂×C₄²	—	106	3	no	—
86	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
87	C₄⋊C₈	—	106	6	no	—
88	C₂×Q₈	—	2	6	no	—
89	C₂²×C₄	—	212	1	no	—
90	C₂²×C₄	C₂×C₂²⋊D₅₃	1	3	yes	—
91	M₁₆	—	212	6	no	—
92	C₂×C₈	—	212	3	no	—
93	C₄⋊C₄	—	2	24	no	—
94	C₄⋊C₄	—	2	12	no	—
95	C₄⋊C₄	—	2	6	no	—
96	C₄⋊C₄	C₂²⋊Dic₅₃	1	12	yes	—
97	C₂²⋊C₄	—	2	6	no	—
98	C₄²	—	2	8	no	—
99	C₄²	—	2	6	no	—
100	C₄²	—	106	6	no	—
101	C₄²	—	212	3	no	—
102	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
103	C₄²	C₂×C₂²⋊D₅₃	1	3	yes	—
104	C₂³	#13133186	1	1	yes	characteristic
105	Q₈	—	4	6	no	—
106	Q₈	—	4	6	no	—
107	C₂×C₄	—	2	3	no	—
108	C₂×C₄	—	2	3	no	—
109	C₂×C₄	—	2	12	no	—
110	C₂×C₄	—	212	6	no	—
111	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	3	yes	—
112	C₂×C₄	#13133186	1	3	yes	—
113	C₈	—	424	3	no	—
114	C₂²	—	2	3	no	—
115	C₂²	#16810479	1	1	yes	characteristic
116	C₄	—	2	3	no	—
117	C₄	—	4	12	no	—
118	C₄	—	212	4	no	—
119	C₄	#16810796	1	3	yes	—
120	C₄	#16810806	1	3	yes	—
121	C₂	—	4	1	no	—
122	C₂	#20274674	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20275693

C₂

yes

maximal

#20273453

C₂

yes

maximal

C₅₃×#3556

C₂

yes

maximal, characteristic

#16810647

C₂²

yes

—

C₄²×Dic₅₃

C₂²

yes

characteristic

#16810382

—

#16810369

—

C₅₃×#564

C₄

yes

complemented

C₅₃×#557

—

C₅₃×#541

—

C₅₃×#533

C₂²

yes

—

C₂×#11719917

—

C₂×C₄×Dic₅₃

—

C₄⋊#88322

—

C₄⋊Q₈×C₅₃

—

C₄⋊Q₈×C₅₃

D₄

yes

—

#126×C₅₃

—

#125×C₅₃

—

#125×C₅₃

—

#125×C₅₃

C₂×C₄

yes

—

C₂²⋊Q₈×C₅₃

—

C₄×Q₈×C₅₃

—

C₄×Q₈×C₅₃

—

#117×C₅₃

D₄

yes

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₂×Q₈×C₅₃

—

C₂²×C₄×C₅₃

C₂×D₄

yes

—

C₂²×Dic₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

C₂²⋊C₄

yes

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂²×C₄

yes

characteristic

C₄²×C₅₃

C₂×D₄

yes

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

#11719917

—

C₂×#88322

—

C₂³×C₅₃

#120

yes

characteristic

Q₈×C₅₃

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×C₂²⋊C₄

yes

—

C₂×C₄×C₅₃

#120

yes

—

C₂×Dic₅₃

—

#88322

—

#8533

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

#379

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#454

yes

—

C₄×C₅₃

#456

yes

—

Dic₅₃

—

#3556

D₅₃

yes

characteristic

#1994

—

#1561

—

C₂×C₅₃

—

C₂×C₅₃

#1793

yes

—

#564

Dic₅₃

yes

—

—

C₂×D₅₃

yes

—

#423

—

C₄³

—

#339

—

106

—

#332

—

106

—

C₅₃

#8533

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

106

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂²⋊D₅₃

yes

—

—

—

—

C₂×Dic₅₃

yes

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

C₂²⋊D₅₃

yes

—

C₂×C₄²

—

106

—

C₂×C₄²

C₂²×D₅₃

yes

characteristic

C₄⋊C₈

—

106

—

C₂×Q₈

—

C₂²×C₄

—

212

—

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

—

M₁₆

—

212

—

C₂×C₈

—

212

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂²⋊Dic₅₃

yes

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

100

C₄²

—

106

—

101

C₄²

—

212

—

102

C₄²

C₂²×Dic₅₃

yes

characteristic

103

C₄²

C₂×C₂²⋊D₅₃

yes

—

104

C₂³

#13133186

yes

characteristic

105

Q₈

—

106

Q₈

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

212

—

111

C₂×C₄

C₂×C₂²⋊Dic₅₃

yes

—

112

C₂×C₄

#13133186

yes

—

113

C₈

—

424

—

114

C₂²

—

115

C₂²

#16810479

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

—

212

—

119

C₄

#16810796

yes

—

120

C₄

#16810806

yes

—

121

C₂

—

122

C₂

#20274674

yes

—

	subgroup	G/H
center	C₂²	#16810479
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#3556	C₂
socle	C₂²×C₅₃	#379
prime radical	C₂³×C₅₃	#120
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#3556	D₅₃
hypocenter	C₅₃	#8533
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3556

subgroup

G/H

center

C₂²

#16810479

commutator

C₄²×C₅₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₅₃×#3556

C₂

C₂²×C₅₃

C₂³×C₅₃

C₂×C₄²×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3556

N	H ≅ G/N	type
8. C₅₃×#564	C₄	semidirect
65. #564	Dic₅₃	semidirect
73. C₅₃	#8533	semidirect

H ≅ G/N

type

8. C₅₃×#564

C₄

semidirect

65. #564

Dic₅₃

semidirect

73. C₅₃

#8533

semidirect

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

subgroup	index
#20275693	2
#20273453	2
C₅₃×#3556	2
#8533	53

subgroup

index

#20275693

#20273453

C₅₃×#3556

#8533

Subgroups — #27276660

Conjugacy classes of subgroups

Complements

Maximal subgroups