Group #27216045 — subgroups

Subgroups — C₅₃×#8533 #27216045

Order 13568.

656 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 150 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	C₅₃×#3556	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1994	C₂	1	3	yes	maximal
3	C₅₃×#1561	C₂	1	3	yes	maximal
4	C₅₃×#564	C₄	1	4	yes	complemented
5	C₅₃×#557	—	2	6	no	—
6	C₅₃×#541	—	2	6	no	—
7	C₅₃×#533	C₂²	1	3	yes	—
8	C₅₃×#423	C₂²	1	3	yes	—
9	C₄³×C₅₃	C₂²	1	1	yes	characteristic
10	C₅₃×#339	—	2	6	no	—
11	C₅₃×#332	—	2	6	no	—
12	C₂×M₁₆×C₅₃	—	2	3	no	—
13	C₄⋊Q₈×C₅₃	—	2	6	no	—
14	C₄⋊Q₈×C₅₃	D₄	1	6	yes	—
15	#126×C₅₃	—	2	8	no	—
16	#125×C₅₃	—	2	6	no	—
17	#125×C₅₃	—	2	24	no	—
18	#125×C₅₃	C₂×C₄	1	6	yes	—
19	C₂²⋊Q₈×C₅₃	—	2	6	no	—
20	C₄×Q₈×C₅₃	—	2	12	no	—
21	C₄×Q₈×C₅₃	—	2	6	no	—
22	#117×C₅₃	D₄	1	6	yes	—
23	C₂×C₄²×C₅₃	—	2	3	no	—
24	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
25	C₄⋊C₈×C₅₃	—	2	6	no	—
26	C₂×Q₈×C₅₃	—	2	6	no	—
27	C₂²×C₄×C₅₃	—	4	1	no	—
28	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
29	M₁₆×C₅₃	—	4	6	no	—
30	C₂×C₈×C₅₃	—	4	3	no	—
31	C₄⋊C₄×C₅₃	—	2	24	no	—
32	C₄⋊C₄×C₅₃	—	2	12	no	—
33	C₄⋊C₄×C₅₃	—	2	6	no	—
34	C₄⋊C₄×C₅₃	C₂²⋊C₄	1	12	yes	—
35	C₂²⋊C₄×C₅₃	—	2	6	no	—
36	C₄²×C₅₃	—	2	8	no	—
37	C₄²×C₅₃	—	2	6	no	—
38	C₄²×C₅₃	—	2	6	no	—
39	C₄²×C₅₃	—	4	3	no	—
40	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
41	C₄²×C₅₃	C₂×D₄	1	3	yes	—
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	—	4	6	no	—
49	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	3	yes	—
50	C₂×C₄×C₅₃	#120	1	3	yes	—
51	C₈×C₅₃	—	8	3	no	—
52	#8533	C₅₃	1	1	yes	maximal, characteristic, direct product factor, 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	4	no	—
57	C₄×C₅₃	—	4	12	no	—
58	C₄×C₅₃	#454	1	3	yes	—
59	C₄×C₅₃	#456	1	3	yes	—
60	#3556	C₂×C₅₃	1	1	yes	characteristic
61	#1994	C₂×C₅₃	1	3	yes	—
62	#1561	C₂×C₅₃	1	3	yes	—
63	C₂×C₅₃	—	4	1	no	—
64	C₂×C₅₃	#1793	1	3	yes	—
65	#564	C₄×C₅₃	1	4	yes	complemented
66	#557	—	2	6	no	—
67	#541	—	2	6	no	—
68	#533	C₂²×C₅₃	1	3	yes	—
69	#423	C₂²×C₅₃	1	3	yes	—
70	C₄³	C₂²×C₅₃	1	1	yes	characteristic
71	#339	—	2	6	no	—
72	#332	—	2	6	no	—
73	C₅₃	#8533	1	1	yes	characteristic, direct product factor, Sylow
74	C₂×M₁₆	—	2	3	no	—
75	C₄⋊Q₈	—	2	6	no	—
76	C₄⋊Q₈	D₄×C₅₃	1	6	yes	—
77	#126	—	2	8	no	—
78	#125	—	2	6	no	—
79	#125	—	2	24	no	—
80	#125	C₂×C₄×C₅₃	1	6	yes	—
81	C₂²⋊Q₈	—	2	6	no	—
82	C₄×Q₈	—	2	12	no	—
83	C₄×Q₈	—	2	6	no	—
84	#117	D₄×C₅₃	1	6	yes	—
85	C₂×C₄²	—	2	3	no	—
86	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
87	C₄⋊C₈	—	2	6	no	—
88	C₂×Q₈	—	2	6	no	—
89	C₂²×C₄	—	4	1	no	—
90	C₂²×C₄	C₂×D₄×C₅₃	1	3	yes	—
91	M₁₆	—	4	6	no	—
92	C₂×C₈	—	4	3	no	—
93	C₄⋊C₄	—	2	24	no	—
94	C₄⋊C₄	—	2	12	no	—
95	C₄⋊C₄	—	2	6	no	—
96	C₄⋊C₄	C₂²⋊C₄×C₅₃	1	12	yes	—
97	C₂²⋊C₄	—	2	6	no	—
98	C₄²	—	2	8	no	—
99	C₄²	—	2	6	no	—
100	C₄²	—	2	6	no	—
101	C₄²	—	4	3	no	—
102	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
103	C₄²	C₂×D₄×C₅₃	1	3	yes	—
104	C₂³	#120×C₅₃	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	—	4	6	no	—
111	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	3	yes	—
112	C₂×C₄	#120×C₅₃	1	3	yes	—
113	C₈	—	8	3	no	—
114	C₂²	—	2	3	no	—
115	C₂²	C₅₃×#379	1	1	yes	characteristic
116	C₄	—	2	3	no	—
117	C₄	—	4	4	no	—
118	C₄	—	4	12	no	—
119	C₄	C₅₃×#454	1	3	yes	—
120	C₄	C₅₃×#456	1	3	yes	—
121	C₂	—	4	1	no	—
122	C₂	C₅₃×#1793	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₅₃×#3556

C₂

yes

maximal, characteristic

C₅₃×#1994

C₂

yes

maximal

C₅₃×#1561

C₂

yes

maximal

C₅₃×#564

C₄

yes

complemented

C₅₃×#557

—

C₅₃×#541

—

C₅₃×#533

C₂²

yes

—

C₅₃×#423

C₂²

yes

—

C₄³×C₅₃

C₂²

yes

characteristic

C₅₃×#339

—

C₅₃×#332

—

C₂×M₁₆×C₅₃

—

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

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊C₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×D₄

yes

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

C₂²⋊C₄

yes

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂²×C₄

yes

characteristic

C₄²×C₅₃

C₂×D₄

yes

—

C₂³×C₅₃

#120

yes

characteristic

Q₈×C₅₃

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×C₂²⋊C₄

yes

—

C₂×C₄×C₅₃

#120

yes

—

C₈×C₅₃

—

#8533

C₅₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

#379

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#454

yes

—

C₄×C₅₃

#456

yes

—

#3556

C₂×C₅₃

yes

characteristic

#1994

C₂×C₅₃

yes

—

#1561

C₂×C₅₃

yes

—

C₂×C₅₃

—

C₂×C₅₃

#1793

yes

—

#564

C₄×C₅₃

yes

—

—

C₂²×C₅₃

yes

—

#423

C₂²×C₅₃

yes

—

C₄³

C₂²×C₅₃

yes

characteristic

#339

—

#332

—

C₅₃

#8533

yes

characteristic, direct product factor, Sylow

C₂×M₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

D₄×C₅₃

yes

—

—

—

—

C₂×C₄×C₅₃

yes

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

D₄×C₅₃

yes

—

C₂×C₄²

—

C₂×C₄²

C₂³×C₅₃

yes

characteristic

C₄⋊C₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄×C₅₃

yes

—

M₁₆

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂²⋊C₄×C₅₃

yes

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

100

C₄²

—

101

C₄²

—

102

C₄²

C₂²×C₄×C₅₃

yes

characteristic

103

C₄²

C₂×D₄×C₅₃

yes

—

104

C₂³

#120×C₅₃

yes

characteristic

105

Q₈

—

106

Q₈

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

C₂×C₂²⋊C₄×C₅₃

yes

—

112

C₂×C₄

#120×C₅₃

yes

—

113

C₈

—

114

C₂²

—

115

C₂²

C₅₃×#379

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

C₅₃×#454

yes

—

120

C₄

C₅₃×#456

yes

—

121

C₂

—

122

C₂

C₅₃×#1793

yes

—

	subgroup	G/H
center, socle	C₂²×C₅₃	#379
commutator	C₄²	C₂²×C₄×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
prime radical	C₂³×C₅₃	#120
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

subgroup

G/H

center, socle

C₂²×C₅₃

#379

commutator

C₄²

C₂²×C₄×C₅₃

Frattini, cosocle

C₂×C₄²

C₂³×C₅₃

C₂³×C₅₃

G ▷ C₄² ▷ C₂² ▷ C₁

C₁ ◁ C₂²×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

N	H ≅ G/N	type
4. C₅₃×#564	C₄	semidirect
52. #8533	C₅₃	direct (N × H)
65. #564	C₄×C₅₃	semidirect
73. C₅₃	#8533	direct (N × H)

H ≅ G/N

type

4. C₅₃×#564

C₄

semidirect

52. #8533

C₅₃

direct (N × H)

65. #564

C₄×C₅₃

semidirect

73. C₅₃

#8533

direct (N × H)

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

subgroup	index
C₅₃×#3556	2
C₅₃×#1994	2
C₅₃×#1561	2
#8533	53

subgroup

index

C₅₃×#3556

C₅₃×#1994

C₅₃×#1561

#8533

Subgroups — C₅₃×#8533 #27216045

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C53×#8533 #27216045

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#8533 #27216045