Group #27214651 — subgroups

Subgroups — C₅₃×#7139 #27214651

Order 13568.

262 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 166 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	C₄×M₃₂×C₅₃	C₂	1	2	yes	maximal
2	C₂×C₅₃×#322	C₂	1	1	yes	maximal, characteristic
3	C₂×M₃₂×C₅₃	C₄	1	4	yes	—
4	C₂×C₄×C₈×C₅₃	C₄	1	2	yes	—
5	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
6	C₅₃×#346	—	2	2	no	—
7	C₄×C₁₆×C₅₃	—	2	2	no	—
8	C₅₃×#322	—	2	2	no	—
9	C₂²×C₈×C₅₃	C₈	1	4	yes	—
10	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
11	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
12	M₃₂×C₅₃	—	2	4	no	—
13	M₃₂×C₅₃	C₈	1	8	yes	complemented
14	C₂×C₁₆×C₅₃	—	2	4	no	—
15	C₄×C₈×C₅₃	—	2	2	no	—
16	C₄×C₈×C₅₃	—	2	2	no	—
17	C₄×C₈×C₅₃	D₄	1	2	yes	—
18	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
19	C₄×C₈×C₅₃	Q₈	1	1	yes	characteristic
20	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
21	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
22	C₂×C₈×C₅₃	—	2	4	no	—
23	C₂×C₈×C₅₃	—	4	4	no	—
24	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
25	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
26	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	—
27	C₂×C₈×C₅₃	M₁₆	1	4	yes	—
28	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
29	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
30	C₄²×C₅₃	M₁₆	1	2	yes	—
31	C₁₆×C₅₃	—	2	8	no	—
32	C₂³×C₅₃	C₄×C₈	1	1	yes	characteristic
33	C₂×C₄×C₅₃	—	2	4	no	—
34	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
35	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
36	C₂×C₄×C₅₃	#97	1	1	yes	characteristic
37	C₂×C₄×C₅₃	#97	1	2	yes	—
38	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
39	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
40	C₈×C₅₃	—	4	4	no	—
41	C₈×C₅₃	C₂²⋊C₈	1	4	yes	—
42	C₈×C₅₃	C₄⋊C₈	1	4	yes	—
43	#7139	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
44	C₂²×C₅₃	—	2	2	no	—
45	C₂²×C₅₃	C₈²	1	1	yes	characteristic
46	C₂²×C₅₃	#322	1	1	yes	characteristic
47	C₂²×C₅₃	#336	1	1	yes	characteristic
48	C₄×C₅₃	—	2	2	no	—
49	C₄×C₅₃	#322	1	2	yes	—
50	C₄×C₅₃	#336	1	2	yes	—
51	C₄×C₅₃	#349	1	4	yes	—
52	C₄×M₃₂	C₂×C₅₃	1	2	yes	—
53	C₂×#322	C₂×C₅₃	1	1	yes	characteristic
54	C₂×C₅₃	—	2	1	no	—
55	C₂×C₅₃	—	2	1	no	—
56	C₂×C₅₃	#1273	1	1	yes	characteristic
57	C₂×C₅₃	#1377	1	2	yes	—
58	C₂×M₃₂	C₄×C₅₃	1	4	yes	—
59	C₂×C₄×C₈	C₄×C₅₃	1	2	yes	—
60	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
61	#346	—	2	2	no	—
62	C₄×C₁₆	—	2	2	no	—
63	#322	—	2	2	no	—
64	C₅₃	#7139	1	1	yes	characteristic, direct product factor, Sylow
65	C₂²×C₈	C₈×C₅₃	1	4	yes	—
66	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
67	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
68	M₃₂	—	2	4	no	—
69	M₃₂	C₈×C₅₃	1	8	yes	complemented
70	C₂×C₁₆	—	2	4	no	—
71	C₄×C₈	—	2	2	no	—
72	C₄×C₈	—	2	2	no	—
73	C₄×C₈	D₄×C₅₃	1	2	yes	—
74	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
75	C₄×C₈	Q₈×C₅₃	1	1	yes	characteristic
76	C₂²×C₄	C₄²×C₅₃	1	1	yes	characteristic
77	C₂²×C₄	C₂×C₈×C₅₃	1	2	yes	—
78	C₂×C₈	—	2	4	no	—
79	C₂×C₈	—	4	4	no	—
80	C₂×C₈	C₂²⋊C₄×C₅₃	1	2	yes	—
81	C₂×C₈	C₄⋊C₄×C₅₃	1	2	yes	—
82	C₂×C₈	C₂×C₈×C₅₃	1	4	yes	—
83	C₂×C₈	M₁₆×C₅₃	1	4	yes	—
84	C₄²	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
85	C₄²	C₄⋊C₄×C₅₃	1	1	yes	characteristic
86	C₄²	M₁₆×C₅₃	1	2	yes	—
87	C₁₆	—	2	8	no	—
88	C₂³	C₄×C₈×C₅₃	1	1	yes	characteristic
89	C₂×C₄	—	2	4	no	—
90	C₂×C₄	#95×C₅₃	1	1	yes	characteristic
91	C₂×C₄	C₄×C₈×C₅₃	1	2	yes	—
92	C₂×C₄	#97×C₅₃	1	1	yes	characteristic
93	C₂×C₄	#97×C₅₃	1	2	yes	—
94	C₂×C₄	C₂²⋊C₈×C₅₃	1	2	yes	—
95	C₂×C₄	C₄⋊C₈×C₅₃	1	2	yes	—
96	C₈	—	4	4	no	—
97	C₈	C₂²⋊C₈×C₅₃	1	4	yes	—
98	C₈	C₄⋊C₈×C₅₃	1	4	yes	—
99	C₂²	—	2	2	no	—
100	C₂²	C₈²×C₅₃	1	1	yes	characteristic
101	C₂²	C₅₃×#322	1	1	yes	characteristic
102	C₂²	C₅₃×#336	1	1	yes	characteristic
103	C₄	—	2	2	no	—
104	C₄	C₅₃×#322	1	2	yes	—
105	C₄	C₅₃×#336	1	2	yes	—
106	C₄	C₅₃×#349	1	4	yes	—
107	C₂	—	2	1	no	—
108	C₂	—	2	1	no	—
109	C₂	C₅₃×#1273	1	1	yes	characteristic
110	C₂	C₅₃×#1377	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×M₃₂×C₅₃

C₂

yes

maximal

C₂×C₅₃×#322

C₂

yes

maximal, characteristic

C₂×M₃₂×C₅₃

C₄

yes

—

C₂×C₄×C₈×C₅₃

C₄

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×#346

—

C₄×C₁₆×C₅₃

—

C₅₃×#322

—

C₂²×C₈×C₅₃

C₈

yes

—

C₂²×C₈×C₅₃

C₂×C₄

yes

—

C₂×C₄²×C₅₃

C₂×C₄

yes

characteristic

M₃₂×C₅₃

—

M₃₂×C₅₃

C₈

yes

complemented

C₂×C₁₆×C₅₃

—

C₄×C₈×C₅₃

—

C₄×C₈×C₅₃

—

C₄×C₈×C₅₃

D₄

yes

—

C₄×C₈×C₅₃

D₄

yes

characteristic

C₄×C₈×C₅₃

Q₈

yes

characteristic

C₂²×C₄×C₅₃

C₄²

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₈

yes

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂²⋊C₄

yes

—

C₂×C₈×C₅₃

C₄⋊C₄

yes

—

C₂×C₈×C₅₃

C₂×C₈

yes

—

C₂×C₈×C₅₃

M₁₆

yes

—

C₄²×C₅₃

C₂²⋊C₄

yes

characteristic

C₄²×C₅₃

C₄⋊C₄

yes

characteristic

C₄²×C₅₃

M₁₆

yes

—

C₁₆×C₅₃

—

C₂³×C₅₃

C₄×C₈

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#95

yes

characteristic

C₂×C₄×C₅₃

C₄×C₈

yes

—

C₂×C₄×C₅₃

#97

yes

characteristic

C₂×C₄×C₅₃

#97

yes

—

C₂×C₄×C₅₃

C₂²⋊C₈

yes

—

C₂×C₄×C₅₃

C₄⋊C₈

yes

—

C₈×C₅₃

—

C₈×C₅₃

C₂²⋊C₈

yes

—

C₈×C₅₃

C₄⋊C₈

yes

—

#7139

C₅₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

C₈²

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

#322

yes

—

C₄×C₅₃

#336

yes

—

C₄×C₅₃

#349

yes

—

C₄×M₃₂

C₂×C₅₃

yes

—

C₂×#322

C₂×C₅₃

yes

characteristic

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#1273

yes

characteristic

C₂×C₅₃

#1377

yes

—

C₂×M₃₂

C₄×C₅₃

yes

—

C₂×C₄×C₈

C₄×C₅₃

yes

—

C₂×C₄×C₈

C₂²×C₅₃

yes

characteristic

#346

—

C₄×C₁₆

—

#322

—

C₅₃

#7139

yes

characteristic, direct product factor, Sylow

C₂²×C₈

C₈×C₅₃

yes

—

C₂²×C₈

C₂×C₄×C₅₃

yes

—

C₂×C₄²

C₂×C₄×C₅₃

yes

characteristic

M₃₂

—

M₃₂

C₈×C₅₃

yes

complemented

C₂×C₁₆

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

D₄×C₅₃

yes

—

C₄×C₈

D₄×C₅₃

yes

characteristic

C₄×C₈

Q₈×C₅₃

yes

characteristic

C₂²×C₄

C₄²×C₅₃

yes

characteristic

C₂²×C₄

C₂×C₈×C₅₃

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊C₄×C₅₃

yes

—

C₂×C₈

C₄⋊C₄×C₅₃

yes

—

C₂×C₈

C₂×C₈×C₅₃

yes

—

C₂×C₈

M₁₆×C₅₃

yes

—

C₄²

C₂²⋊C₄×C₅₃

yes

characteristic

C₄²

C₄⋊C₄×C₅₃

yes

characteristic

C₄²

M₁₆×C₅₃

yes

—

C₁₆

—

C₂³

C₄×C₈×C₅₃

yes

characteristic

C₂×C₄

—

C₂×C₄

#95×C₅₃

yes

characteristic

C₂×C₄

C₄×C₈×C₅₃

yes

—

C₂×C₄

#97×C₅₃

yes

characteristic

C₂×C₄

#97×C₅₃

yes

—

C₂×C₄

C₂²⋊C₈×C₅₃

yes

—

C₂×C₄

C₄⋊C₈×C₅₃

yes

—

C₈

—

C₈

C₂²⋊C₈×C₅₃

yes

—

C₈

C₄⋊C₈×C₅₃

yes

—

C₂²

—

100

C₂²

C₈²×C₅₃

yes

characteristic

101

C₂²

C₅₃×#322

yes

characteristic

102

C₂²

C₅₃×#336

yes

characteristic

103

C₄

—

104

C₄

C₅₃×#322

yes

—

105

C₄

C₅₃×#336

yes

—

106

C₄

C₅₃×#349

yes

—

107

C₂

—

108

C₂

—

109

C₂

C₅₃×#1273

yes

characteristic

110

C₂

C₅₃×#1377

yes

—

	subgroup	G/H
center	C₄²×C₅₃	C₂²⋊C₄
commutator	C₂²	C₈²×C₅₃
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₅₃
socle	C₂²×C₅₃	#336
prime radical	C₂³×C₅₃	C₄×C₈
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄²×C₅₃ ◁ C₂×C₄×C₈×C₅₃ ◁ G

subgroup

G/H

center

C₄²×C₅₃

C₂²⋊C₄

commutator

C₂²

C₈²×C₅₃

Frattini, cosocle

C₂×C₄×C₈

C₂²×C₅₃

socle

C₂²×C₅₃

#336

prime radical

C₂³×C₅₃

C₄×C₈

LCS

G ▷ C₂² ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₄²×C₅₃ ◁ C₂×C₄×C₈×C₅₃ ◁ G

N	H ≅ G/N	type
13. M₃₂×C₅₃	C₈	semidirect
43. #7139	C₅₃	direct (N × H)
64. C₅₃	#7139	direct (N × H)
69. M₃₂	C₈×C₅₃	semidirect

H ≅ G/N

type

13. M₃₂×C₅₃

C₈

semidirect

43. #7139

C₅₃

direct (N × H)

64. C₅₃

#7139

direct (N × H)

69. M₃₂

C₈×C₅₃

semidirect

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

subgroup	index
C₄×M₃₂×C₅₃	2
C₂×C₅₃×#322	2
#7139	53

subgroup

index

C₄×M₃₂×C₅₃

C₂×C₅₃×#322

#7139

Subgroups — C₅₃×#7139 #27214651

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C53×#7139 #27214651

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#7139 #27214651