Group #27221142 — subgroups

Subgroups — C₅₃×#13630 #27221142

Order 13568.

198 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 154 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	C₂×C₅₃×M₆₄	C₂	1	2	yes	maximal
2	C₅₃×#2108	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1422	C₂	1	4	yes	maximal, complemented
4	C₂×M₃₂×C₅₃	C₄	1	1	yes	characteristic
5	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
6	C₅₃×#405	C₄	1	1	yes	characteristic
7	C₅₃×M₆₄	—	2	2	no	—
8	C₅₃×M₆₄	C₂²	1	4	yes	—
9	C₂×C₃₂×C₅₃	—	2	2	no	—
10	C₅₃×#346	C₄	1	2	yes	—
11	C₄×C₁₆×C₅₃	C₂²	1	2	yes	—
12	C₂×M₁₆×C₅₃	C₈	1	1	yes	characteristic
13	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
14	#117×C₅₃	C₈	1	1	yes	characteristic
15	M₃₂×C₅₃	—	2	2	no	—
16	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
17	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
18	C₂×C₁₆×C₅₃	D₄	1	2	yes	—
19	C₂×C₁₆×C₅₃	Q₈	1	2	yes	—
20	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
21	#97×C₅₃	C₈	1	2	yes	—
22	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
23	C₃₂×C₅₃	—	2	4	no	—
24	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
25	M₁₆×C₅₃	C₁₆	1	4	yes	—
26	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
27	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
28	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
29	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
30	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
31	C₄⋊C₄×C₅₃	—	2	1	no	—
32	C₂²⋊C₄×C₅₃	—	2	1	no	—
33	C₄²×C₅₃	C₂×C₈	1	2	yes	—
34	C₁₆×C₅₃	—	2	2	no	—
35	C₁₆×C₅₃	C₄⋊C₄	1	2	yes	—
36	C₁₆×C₅₃	C₂×D₄	1	1	yes	characteristic
37	C₁₆×C₅₃	C₂×Q₈	1	1	yes	characteristic
38	C₂³×C₅₃	M₃₂	1	1	yes	characteristic
39	C₂×C₄×C₅₃	—	2	2	no	—
40	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
41	C₂×C₄×C₅₃	C₂×C₁₆	1	2	yes	—
42	C₂×C₄×C₅₃	M₃₂	1	1	yes	characteristic
43	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
44	C₈×C₅₃	C₄⋊C₈	1	2	yes	—
45	C₈×C₅₃	C₂×C₁₆	1	4	yes	—
46	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
47	C₈×C₅₃	C₂×M₁₆	1	1	yes	characteristic
48	#13630	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
49	C₂²×C₅₃	—	4	1	no	—
50	C₂²×C₅₃	C₄⋊C₁₆	1	2	yes	—
51	C₂²×C₅₃	C₂×M₃₂	1	1	yes	characteristic
52	C₄×C₅₃	—	4	2	no	—
53	C₄×C₅₃	C₄⋊C₁₆	1	2	yes	—
54	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
55	C₄×C₅₃	C₂²×C₁₆	1	1	yes	characteristic
56	C₂×M₆₄	C₂×C₅₃	1	2	yes	—
57	#2108	C₂×C₅₃	1	1	yes	characteristic
58	#1422	C₂×C₅₃	1	4	yes	complemented
59	C₂×C₅₃	—	2	1	no	—
60	C₂×C₅₃	—	2	2	no	—
61	C₂×C₅₃	C₂×C₄⋊C₁₆	1	1	yes	characteristic
62	C₂×M₃₂	C₄×C₅₃	1	1	yes	characteristic
63	C₂²×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
64	#405	C₄×C₅₃	1	1	yes	characteristic
65	M₆₄	—	2	2	no	—
66	M₆₄	C₂²×C₅₃	1	4	yes	—
67	C₂×C₃₂	—	2	2	no	—
68	#346	C₄×C₅₃	1	2	yes	—
69	C₄×C₁₆	C₂²×C₅₃	1	2	yes	—
70	C₅₃	#13630	1	1	yes	characteristic, direct product factor, Sylow
71	C₂×M₁₆	C₈×C₅₃	1	1	yes	characteristic
72	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
73	#117	C₈×C₅₃	1	1	yes	characteristic
74	M₃₂	—	2	2	no	—
75	C₂×C₁₆	C₂×C₄×C₅₃	1	2	yes	—
76	C₂×C₁₆	C₂×C₄×C₅₃	1	1	yes	characteristic
77	C₂×C₁₆	D₄×C₅₃	1	2	yes	—
78	C₂×C₁₆	Q₈×C₅₃	1	2	yes	—
79	C₂×C₁₆	C₂³×C₅₃	1	1	yes	characteristic
80	#97	C₈×C₅₃	1	2	yes	—
81	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
82	C₃₂	—	2	4	no	—
83	C₂²×C₄	C₂×C₈×C₅₃	1	1	yes	characteristic
84	M₁₆	C₁₆×C₅₃	1	4	yes	—
85	C₂×C₈	C₄⋊C₄×C₅₃	1	2	yes	—
86	C₂×C₈	C₂×C₈×C₅₃	1	1	yes	characteristic
87	C₂×C₈	C₂×C₈×C₅₃	1	2	yes	—
88	C₂×C₈	M₁₆×C₅₃	1	2	yes	—
89	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
90	C₄⋊C₄	—	2	1	no	—
91	C₂²⋊C₄	—	2	1	no	—
92	C₄²	C₂×C₈×C₅₃	1	2	yes	—
93	C₁₆	—	2	2	no	—
94	C₁₆	C₄⋊C₄×C₅₃	1	2	yes	—
95	C₁₆	C₂×D₄×C₅₃	1	1	yes	characteristic
96	C₁₆	C₂×Q₈×C₅₃	1	1	yes	characteristic
97	C₂³	M₃₂×C₅₃	1	1	yes	characteristic
98	C₂×C₄	—	2	2	no	—
99	C₂×C₄	C₄⋊C₈×C₅₃	1	2	yes	—
100	C₂×C₄	C₂×C₁₆×C₅₃	1	2	yes	—
101	C₂×C₄	M₃₂×C₅₃	1	1	yes	characteristic
102	C₂×C₄	C₂²×C₈×C₅₃	1	1	yes	characteristic
103	C₈	C₄⋊C₈×C₅₃	1	2	yes	—
104	C₈	C₂×C₁₆×C₅₃	1	4	yes	—
105	C₈	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
106	C₈	C₂×M₁₆×C₅₃	1	1	yes	characteristic
107	C₂²	—	4	1	no	—
108	C₂²	C₅₃×C₄⋊C₁₆	1	2	yes	—
109	C₂²	C₂×M₃₂×C₅₃	1	1	yes	characteristic
110	C₄	—	4	2	no	—
111	C₄	C₅₃×C₄⋊C₁₆	1	2	yes	—
112	C₄	C₂×C₄⋊C₈×C₅₃	1	1	yes	characteristic
113	C₄	C₂²×C₁₆×C₅₃	1	1	yes	characteristic
114	C₂	—	2	1	no	—
115	C₂	—	2	2	no	—
116	C₂	C₂×C₅₃×C₄⋊C₁₆	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₅₃×M₆₄

C₂

yes

maximal

C₅₃×#2108

C₂

yes

maximal, characteristic

C₅₃×#1422

C₂

yes

maximal, complemented

C₂×M₃₂×C₅₃

C₄

yes

characteristic

C₂²×C₁₆×C₅₃

C₂²

yes

characteristic

C₅₃×#405

C₄

yes

characteristic

C₅₃×M₆₄

—

C₅₃×M₆₄

C₂²

yes

—

C₂×C₃₂×C₅₃

—

C₅₃×#346

C₄

yes

—

C₄×C₁₆×C₅₃

C₂²

yes

—

C₂×M₁₆×C₅₃

C₈

yes

characteristic

C₂²×C₈×C₅₃

C₂×C₄

yes

characteristic

#117×C₅₃

C₈

yes

characteristic

M₃₂×C₅₃

—

C₂×C₁₆×C₅₃

C₂×C₄

yes

—

C₂×C₁₆×C₅₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₅₃

D₄

yes

—

C₂×C₁₆×C₅₃

Q₈

yes

—

C₂×C₁₆×C₅₃

C₂³

yes

characteristic

#97×C₅₃

C₈

yes

—

C₄×C₈×C₅₃

C₂×C₄

yes

—

C₃₂×C₅₃

—

C₂²×C₄×C₅₃

C₂×C₈

yes

characteristic

M₁₆×C₅₃

C₁₆

yes

—

C₂×C₈×C₅₃

C₄⋊C₄

yes

—

C₂×C₈×C₅₃

C₂×C₈

yes

characteristic

C₂×C₈×C₅₃

C₂×C₈

yes

—

C₂×C₈×C₅₃

M₁₆

yes

—

C₂×C₈×C₅₃

C₂²×C₄

yes

characteristic

C₄⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

C₂×C₈

yes

—

C₁₆×C₅₃

—

C₁₆×C₅₃

C₄⋊C₄

yes

—

C₁₆×C₅₃

C₂×D₄

yes

characteristic

C₁₆×C₅₃

C₂×Q₈

yes

characteristic

C₂³×C₅₃

M₃₂

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₄⋊C₈

yes

—

C₂×C₄×C₅₃

C₂×C₁₆

yes

—

C₂×C₄×C₅₃

M₃₂

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₈

yes

characteristic

C₈×C₅₃

C₄⋊C₈

yes

—

C₈×C₅₃

C₂×C₁₆

yes

—

C₈×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₈×C₅₃

C₂×M₁₆

yes

characteristic

#13630

C₅₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

C₄⋊C₁₆

yes

—

C₂²×C₅₃

C₂×M₃₂

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

C₄⋊C₁₆

yes

—

C₄×C₅₃

C₂×C₄⋊C₈

yes

characteristic

C₄×C₅₃

C₂²×C₁₆

yes

characteristic

C₂×M₆₄

C₂×C₅₃

yes

—

#2108

C₂×C₅₃

yes

characteristic

#1422

C₂×C₅₃

yes

complemented

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

C₂×C₄⋊C₁₆

yes

characteristic

C₂×M₃₂

C₄×C₅₃

yes

characteristic

C₂²×C₁₆

C₂²×C₅₃

yes

characteristic

#405

C₄×C₅₃

yes

characteristic

M₆₄

—

M₆₄

C₂²×C₅₃

yes

—

C₂×C₃₂

—

#346

C₄×C₅₃

yes

—

C₄×C₁₆

C₂²×C₅₃

yes

—

C₅₃

#13630

yes

characteristic, direct product factor, Sylow

C₂×M₁₆

C₈×C₅₃

yes

characteristic

C₂²×C₈

C₂×C₄×C₅₃

yes

characteristic

#117

C₈×C₅₃

yes

characteristic

M₃₂

—

C₂×C₁₆

C₂×C₄×C₅₃

yes

—

C₂×C₁₆

C₂×C₄×C₅₃

yes

characteristic

C₂×C₁₆

D₄×C₅₃

yes

—

C₂×C₁₆

Q₈×C₅₃

yes

—

C₂×C₁₆

C₂³×C₅₃

yes

characteristic

#97

C₈×C₅₃

yes

—

C₄×C₈

C₂×C₄×C₅₃

yes

—

C₃₂

—

C₂²×C₄

C₂×C₈×C₅₃

yes

characteristic

M₁₆

C₁₆×C₅₃

yes

—

C₂×C₈

C₄⋊C₄×C₅₃

yes

—

C₂×C₈

C₂×C₈×C₅₃

yes

characteristic

C₂×C₈

C₂×C₈×C₅₃

yes

—

C₂×C₈

M₁₆×C₅₃

yes

—

C₂×C₈

C₂²×C₄×C₅₃

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

C₂×C₈×C₅₃

yes

—

C₁₆

—

C₁₆

C₄⋊C₄×C₅₃

yes

—

C₁₆

C₂×D₄×C₅₃

yes

characteristic

C₁₆

C₂×Q₈×C₅₃

yes

characteristic

C₂³

M₃₂×C₅₃

yes

characteristic

C₂×C₄

—

C₂×C₄

C₄⋊C₈×C₅₃

yes

—

100

C₂×C₄

C₂×C₁₆×C₅₃

yes

—

101

C₂×C₄

M₃₂×C₅₃

yes

characteristic

102

C₂×C₄

C₂²×C₈×C₅₃

yes

characteristic

103

C₈

C₄⋊C₈×C₅₃

yes

—

104

C₈

C₂×C₁₆×C₅₃

yes

—

105

C₈

C₂×C₄⋊C₄×C₅₃

yes

characteristic

106

C₈

C₂×M₁₆×C₅₃

yes

characteristic

107

C₂²

—

108

C₂²

C₅₃×C₄⋊C₁₆

yes

—

109

C₂²

C₂×M₃₂×C₅₃

yes

characteristic

110

C₄

—

111

C₄

C₅₃×C₄⋊C₁₆

yes

—

112

C₄

C₂×C₄⋊C₈×C₅₃

yes

characteristic

113

C₄

C₂²×C₁₆×C₅₃

yes

characteristic

114

C₂

—

115

C₂

—

116

C₂

C₂×C₅₃×C₄⋊C₁₆

yes

characteristic

	subgroup	G/H
center	C₁₆×C₅₃	C₂×D₄
commutator	C₄	C₂²×C₁₆×C₅₃
Frattini, cosocle	C₂×C₁₆	C₂³×C₅₃
socle	C₂×C₅₃	C₂×C₄⋊C₁₆
prime radical	C₂³×C₅₃	M₃₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₁₆×C₅₃ ◁ C₂²×C₁₆×C₅₃ ◁ G

subgroup

G/H

center

C₁₆×C₅₃

C₂×D₄

commutator

C₄

C₂²×C₁₆×C₅₃

Frattini, cosocle

C₂×C₁₆

C₂³×C₅₃

socle

C₂×C₅₃

C₂×C₄⋊C₁₆

prime radical

C₂³×C₅₃

M₃₂

LCS

G ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₁₆×C₅₃ ◁ C₂²×C₁₆×C₅₃ ◁ G

N	H ≅ G/N	type
3. C₅₃×#1422	C₂	semidirect
48. #13630	C₅₃	direct (N × H)
58. #1422	C₂×C₅₃	semidirect
70. C₅₃	#13630	direct (N × H)

H ≅ G/N

type

3. C₅₃×#1422

C₂

semidirect

48. #13630

C₅₃

direct (N × H)

58. #1422

C₂×C₅₃

semidirect

70. C₅₃

#13630

direct (N × H)

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

subgroup	index
C₂×C₅₃×M₆₄	2
C₅₃×#2108	2
C₅₃×#1422	2
#13630	53

subgroup

index

C₂×C₅₃×M₆₄

C₅₃×#2108

C₅₃×#1422

#13630

Subgroups — C₅₃×#13630 #27221142

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C53×#13630 #27221142

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#13630 #27221142