Group #27218924 — subgroups

Subgroups — C₅₃×#11412 #27218924

Order 13568.

260 conjugacy classes of subgroups; only the 190 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 190 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×#2162	C₂	1	2	yes	maximal
2	C₅₃×#1450	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1312	C₂	1	2	yes	maximal, complemented
4	C₅₃×#1311	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#431	C₄	1	2	yes	—
7	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
8	C₅₃×C₄⋊C₁₆	C₄	1	4	yes	—
9	C₅₃×C₂²⋊C₁₆	C₄	1	4	yes	—
10	C₄×C₁₆×C₅₃	C₂²	1	2	yes	—
11	C₄×C₁₆×C₅₃	C₂²	1	2	yes	—
12	C₅₃×#322	C₄	1	2	yes	—
13	C₈²×C₅₃	C₂²	1	1	yes	characteristic
14	C₈²×C₅₃	C₂²	1	1	yes	characteristic
15	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
16	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
17	C₂×C₁₆×C₅₃	C₂×C₄	1	4	yes	—
18	C₂×C₁₆×C₅₃	C₂×C₄	1	4	yes	—
19	C₄⋊C₈×C₅₃	C₈	1	4	yes	—
20	C₂²⋊C₈×C₅₃	C₈	1	4	yes	—
21	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
22	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
23	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
24	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
25	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
26	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
27	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
28	C₂×C₈×C₅₃	—	2	4	no	—
29	C₂×C₈×C₅₃	C₄²	1	2	yes	—
30	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	—
31	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	—
32	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
33	C₄²×C₅₃	C₄²	1	1	yes	characteristic
34	C₄²×C₅₃	C₂×C₈	1	2	yes	—
35	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
36	C₁₆×C₅₃	—	2	4	no	—
37	C₁₆×C₅₃	—	2	4	no	—
38	C₂³×C₅₃	C₄×C₈	1	1	yes	characteristic
39	C₂×C₄×C₅₃	—	2	2	no	—
40	C₂×C₄×C₅₃	—	2	4	no	—
41	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
42	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
43	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
44	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
45	C₈×C₅₃	—	2	2	no	—
46	C₈×C₅₃	—	2	4	no	—
47	C₈×C₅₃	—	2	4	no	—
48	C₈×C₅₃	#131	1	4	yes	—
49	#11412	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
50	C₂²×C₅₃	—	2	2	no	—
51	C₂²×C₅₃	—	2	1	no	—
52	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
53	C₄×C₅₃	—	2	2	no	—
54	C₄×C₅₃	—	2	1	no	—
55	C₄×C₅₃	#405	1	2	yes	—
56	C₄×C₅₃	#504	1	4	yes	—
57	#2162	C₂×C₅₃	1	2	yes	—
58	#1450	C₂×C₅₃	1	1	yes	characteristic
59	#1312	C₂×C₅₃	1	2	yes	complemented
60	#1311	C₂×C₅₃	1	1	yes	characteristic, complemented
61	C₈×C₁₆	C₂×C₅₃	1	1	yes	characteristic, complemented
62	C₂×C₅₃	—	4	1	no	—
63	C₂×C₅₃	#1450	1	1	yes	characteristic
64	C₂×C₅₃	#2108	1	2	yes	—
65	#431	C₄×C₅₃	1	2	yes	—
66	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
67	C₄⋊C₁₆	C₄×C₅₃	1	4	yes	—
68	C₂²⋊C₁₆	C₄×C₅₃	1	4	yes	—
69	C₄×C₁₆	C₂²×C₅₃	1	2	yes	—
70	C₄×C₁₆	C₂²×C₅₃	1	2	yes	—
71	#322	C₄×C₅₃	1	2	yes	—
72	C₈²	C₂²×C₅₃	1	1	yes	characteristic
73	C₈²	C₂²×C₅₃	1	1	yes	characteristic
74	C₅₃	#11412	1	1	yes	characteristic, direct product factor, Sylow
75	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
76	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
77	C₂×C₁₆	C₂×C₄×C₅₃	1	4	yes	—
78	C₂×C₁₆	C₂×C₄×C₅₃	1	4	yes	—
79	C₄⋊C₈	C₈×C₅₃	1	4	yes	—
80	C₂²⋊C₈	C₈×C₅₃	1	4	yes	—
81	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
82	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
83	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
84	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
85	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
86	C₂²×C₄	C₄²×C₅₃	1	1	yes	characteristic
87	C₂²×C₄	C₂×C₈×C₅₃	1	2	yes	—
88	C₂×C₈	—	2	4	no	—
89	C₂×C₈	C₄²×C₅₃	1	2	yes	—
90	C₂×C₈	C₂×C₈×C₅₃	1	4	yes	—
91	C₂×C₈	C₂×C₈×C₅₃	1	4	yes	—
92	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
93	C₄²	C₄²×C₅₃	1	1	yes	characteristic
94	C₄²	C₂×C₈×C₅₃	1	2	yes	—
95	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
96	C₁₆	—	2	4	no	—
97	C₁₆	—	2	4	no	—
98	C₂³	C₄×C₈×C₅₃	1	1	yes	characteristic
99	C₂×C₄	—	2	4	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	C₄×C₈×C₅₃	1	2	yes	—
102	C₂×C₄	C₄×C₈×C₅₃	1	1	yes	characteristic
103	C₂×C₄	C₂×C₄²×C₅₃	1	1	yes	characteristic
104	C₂×C₄	C₂²×C₈×C₅₃	1	2	yes	—
105	C₈	—	2	4	no	—
106	C₈	—	2	4	no	—
107	C₈	—	2	2	no	—
108	C₈	#131×C₅₃	1	4	yes	—
109	C₂²	—	2	1	no	—
110	C₂²	—	2	2	no	—
111	C₂²	C₂×C₄×C₈×C₅₃	1	1	yes	characteristic
112	C₄	—	2	1	no	—
113	C₄	—	2	2	no	—
114	C₄	C₅₃×#405	1	2	yes	—
115	C₄	C₅₃×#504	1	4	yes	—
116	C₂	—	4	1	no	—
117	C₂	C₅₃×#1450	1	1	yes	characteristic
118	C₂	C₅₃×#2108	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₅₃×#2162

C₂

yes

maximal

C₅₃×#1450

C₂

yes

maximal, characteristic

C₅₃×#1312

C₂

yes

maximal, complemented

C₅₃×#1311

C₂

yes

maximal, characteristic, complemented

C₈×C₁₆×C₅₃

C₂

yes

maximal, characteristic, complemented

C₅₃×#431

C₄

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×C₄⋊C₁₆

C₄

yes

—

C₅₃×C₂²⋊C₁₆

C₄

yes

—

C₄×C₁₆×C₅₃

C₂²

yes

—

C₄×C₁₆×C₅₃

C₂²

yes

—

C₅₃×#322

C₄

yes

—

C₈²×C₅₃

C₂²

yes

characteristic

C₈²×C₅₃

C₂²

yes

characteristic

C₂²×C₈×C₅₃

C₂×C₄

yes

—

C₂×C₄²×C₅₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₅₃

C₂×C₄

yes

—

C₂×C₁₆×C₅₃

C₂×C₄

yes

—

C₄⋊C₈×C₅₃

C₈

yes

—

C₂²⋊C₈×C₅₃

C₈

yes

—

C₄×C₈×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄×C₈×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

C₂³

yes

characteristic

C₂²×C₄×C₅₃

C₄²

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₈

yes

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₄²

yes

—

C₂×C₈×C₅₃

C₂×C₈

yes

—

C₂×C₈×C₅₃

C₂×C₈

yes

—

C₂×C₈×C₅₃

C₂²×C₄

yes

—

C₄²×C₅₃

C₄²

yes

characteristic

C₄²×C₅₃

C₂×C₈

yes

—

C₄²×C₅₃

C₂²×C₄

yes

characteristic

C₁₆×C₅₃

—

C₁₆×C₅₃

—

C₂³×C₅₃

C₄×C₈

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₄×C₈

yes

characteristic

C₂×C₄×C₅₃

C₄×C₈

yes

—

C₂×C₄×C₅₃

C₂×C₄²

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₈

yes

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

#131

yes

—

#11412

C₅₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×C₄×C₈

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#405

yes

—

C₄×C₅₃

#504

yes

—

#2162

C₂×C₅₃

yes

—

#1450

C₂×C₅₃

yes

characteristic

#1312

C₂×C₅₃

yes

complemented

#1311

C₂×C₅₃

yes

characteristic, complemented

C₈×C₁₆

C₂×C₅₃

yes

characteristic, complemented

C₂×C₅₃

—

C₂×C₅₃

#1450

yes

characteristic

C₂×C₅₃

#2108

yes

—

#431

C₄×C₅₃

yes

—

C₂×C₄×C₈

C₂²×C₅₃

yes

characteristic

C₄⋊C₁₆

C₄×C₅₃

yes

—

C₂²⋊C₁₆

C₄×C₅₃

yes

—

C₄×C₁₆

C₂²×C₅₃

yes

—

C₄×C₁₆

C₂²×C₅₃

yes

—

#322

C₄×C₅₃

yes

—

C₈²

C₂²×C₅₃

yes

characteristic

C₈²

C₂²×C₅₃

yes

characteristic

C₅₃

#11412

yes

characteristic, direct product factor, Sylow

C₂²×C₈

C₂×C₄×C₅₃

yes

—

C₂×C₄²

C₂×C₄×C₅₃

yes

characteristic

C₂×C₁₆

C₂×C₄×C₅₃

yes

—

C₂×C₁₆

C₂×C₄×C₅₃

yes

—

C₄⋊C₈

C₈×C₅₃

yes

—

C₂²⋊C₈

C₈×C₅₃

yes

—

C₄×C₈

C₂×C₄×C₅₃

yes

characteristic

C₄×C₈

C₂×C₄×C₅₃

yes

—

C₄×C₈

C₂×C₄×C₅₃

yes

—

C₄×C₈

C₂×C₄×C₅₃

yes

—

C₄×C₈

C₂³×C₅₃

yes

characteristic

C₂²×C₄

C₄²×C₅₃

yes

characteristic

C₂²×C₄

C₂×C₈×C₅₃

yes

—

C₂×C₈

—

C₂×C₈

C₄²×C₅₃

yes

—

C₂×C₈

C₂×C₈×C₅₃

yes

—

C₂×C₈

C₂×C₈×C₅₃

yes

—

C₂×C₈

C₂²×C₄×C₅₃

yes

—

C₄²

C₄²×C₅₃

yes

characteristic

C₄²

C₂×C₈×C₅₃

yes

—

C₄²

C₂²×C₄×C₅₃

yes

characteristic

C₁₆

—

C₁₆

—

C₂³

C₄×C₈×C₅₃

yes

characteristic

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

C₄×C₈×C₅₃

yes

—

102

C₂×C₄

C₄×C₈×C₅₃

yes

characteristic

103

C₂×C₄

C₂×C₄²×C₅₃

yes

characteristic

104

C₂×C₄

C₂²×C₈×C₅₃

yes

—

105

C₈

—

106

C₈

—

107

C₈

—

108

C₈

#131×C₅₃

yes

—

109

C₂²

—

110

C₂²

—

111

C₂²

C₂×C₄×C₈×C₅₃

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

C₅₃×#405

yes

—

115

C₄

C₅₃×#504

yes

—

116

C₂

—

117

C₂

C₅₃×#1450

yes

characteristic

118

C₂

C₅₃×#2108

yes

—

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

subgroup

G/H

center

C₄×C₈×C₅₃

C₂³

commutator

C₂²

C₂×C₄×C₈×C₅₃

Frattini, cosocle

C₄×C₈

C₂³×C₅₃

socle

C₂²×C₅₃

C₂×C₄×C₈

prime radical

C₂³×C₅₃

C₄×C₈

LCS

G ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₄×C₈×C₅₃ ◁ G

N	H ≅ G/N	type
3. C₅₃×#1312	C₂	semidirect
4. C₅₃×#1311	C₂	semidirect
5. C₈×C₁₆×C₅₃	C₂	semidirect
49. #11412	C₅₃	direct (N × H)
59. #1312	C₂×C₅₃	semidirect
60. #1311	C₂×C₅₃	semidirect
61. C₈×C₁₆	C₂×C₅₃	semidirect
74. C₅₃	#11412	direct (N × H)

H ≅ G/N

type

3. C₅₃×#1312

C₂

semidirect

4. C₅₃×#1311

C₂

semidirect

5. C₈×C₁₆×C₅₃

C₂

semidirect

49. #11412

C₅₃

direct (N × H)

59. #1312

C₂×C₅₃

semidirect

60. #1311

C₂×C₅₃

semidirect

61. C₈×C₁₆

C₂×C₅₃

semidirect

74. C₅₃

#11412

direct (N × H)

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

subgroup	index
C₅₃×#2162	2
C₅₃×#1450	2
C₅₃×#1312	2
C₅₃×#1311	2
C₈×C₁₆×C₅₃	2
#11412	53

subgroup

index

C₅₃×#2162

C₅₃×#1450

C₅₃×#1312

C₅₃×#1311

C₈×C₁₆×C₅₃

#11412

Subgroups — C₅₃×#11412 #27218924

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C53×#11412 #27218924

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#11412 #27218924