Group #27270736 — subgroups

Subgroups — #27270736

Order 13568.

176 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 51 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	#20272655	C₂	1	2	yes	maximal
2	C₅₃×#1843	C₂	1	1	yes	maximal, characteristic
3	C₂²⋊#11719908	—	2	2	no	—
4	C₂×C₄⋊C₈×C₅₃	—	2	2	no	—
5	C₂×C₄⋊C₈×C₅₃	C₄	1	1	yes	characteristic
6	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
7	C₅₃×#376	C₄	1	1	yes	characteristic
8	C₂×#11719908	—	4	2	no	—
9	C₂²×C₈×C₅₃	—	2	1	no	—
10	C₂²×C₈×C₅₃	D₄	1	2	yes	—
11	C₂×C₄²×C₅₃	—	2	1	no	—
12	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
13	C₄⋊C₈×C₅₃	—	4	1	no	—
14	C₄⋊C₈×C₅₃	—	4	1	no	—
15	C₄⋊C₈×C₅₃	—	4	4	no	—
16	#95×C₅₃	—	2	1	no	—
17	#95×C₅₃	C₈	1	2	yes	—
18	C₂²×C₄×C₅₃	—	2	2	no	—
19	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
20	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
21	C₂²×C₄×C₅₃	M₁₆	1	1	yes	characteristic
22	C₂×C₈×C₅₃	—	2	2	no	—
23	C₂×C₈×C₅₃	—	4	2	no	—
24	C₂×C₈×C₅₃	—	4	2	no	—
25	C₂×C₈×C₅₃	—	4	1	no	—
26	C₄²×C₅₃	—	2	2	no	—
27	C₄²×C₅₃	—	2	2	no	—
28	C₄²×C₅₃	—	2	1	no	—
29	C₄²×C₅₃	—	2	1	no	—
30	#11719908	—	8	2	no	—
31	C₂³×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
32	C₂×C₄×C₅₃	—	2	1	no	—
33	C₂×C₄×C₅₃	—	2	1	no	—
34	C₂×C₄×C₅₃	—	2	4	no	—
35	C₂×C₄×C₅₃	—	2	1	no	—
36	C₂×C₄×C₅₃	—	2	1	no	—
37	C₂×C₄×C₅₃	—	2	2	no	—
38	C₂×C₄×C₅₃	—	4	2	no	—
39	C₂×C₄×C₅₃	—	4	2	no	—
40	C₂×C₄×C₅₃	—	4	1	no	—
41	C₂×C₄×C₅₃	#99	1	1	yes	characteristic
42	C₂×C₄×C₅₃	#101	1	1	yes	characteristic
43	C₈×C₅₃	—	4	2	no	—
44	C₈×C₅₃	—	8	1	no	—
45	#7066	—	53	1	no	maximal, Sylow
46	C₂²×C₅₃	—	2	1	no	—
47	C₂²×C₅₃	—	2	1	no	—
48	C₂²×C₅₃	#324	1	1	yes	characteristic
49	C₂²×C₅₃	#350	1	2	yes	—
50	C₄×C₅₃	—	2	1	no	—
51	C₄×C₅₃	—	4	4	no	—
52	C₄×C₅₃	—	4	1	no	—
53	C₄×C₅₃	—	4	1	no	—
54	C₄×C₅₃	#353	1	1	yes	characteristic
55	C₄×C₅₃	#356	1	1	yes	characteristic
56	#1843	D₅₃	1	1	yes	characteristic
57	#1322	—	53	2	no	—
58	C₂×C₅₃	—	2	2	no	—
59	C₂×C₅₃	#1323	1	1	yes	characteristic
60	C₂×C₅₃	#1325	1	1	yes	characteristic
61	C₂×C₅₃	#1328	1	1	yes	characteristic
62	C₂×C₄⋊C₈	—	2	2	no	—
63	C₂×C₄⋊C₈	Dic₅₃	1	1	yes	characteristic
64	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
65	#376	Dic₅₃	1	1	yes	characteristic
66	C₂²⋊C₁₆	—	106	2	no	—
67	C₅₃	#7066	1	1	yes	characteristic, complemented, Sylow
68	C₂²×C₈	—	2	1	no	—
69	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
70	C₂×C₄²	—	2	1	no	—
71	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
72	C₂×C₁₆	—	212	2	no	—
73	C₄⋊C₈	—	4	1	no	—
74	C₄⋊C₈	—	4	1	no	—
75	C₄⋊C₈	—	4	4	no	—
76	#95	—	2	1	no	—
77	#95	#88322	1	2	yes	—
78	C₂²×C₄	—	2	2	no	—
79	C₂²×C₄	C₂×#88322	1	1	yes	characteristic
80	C₂²×C₄	#11719917	1	1	yes	characteristic
81	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
82	C₂×C₈	—	2	2	no	—
83	C₂×C₈	—	4	2	no	—
84	C₂×C₈	—	4	2	no	—
85	C₂×C₈	—	4	1	no	—
86	C₄²	—	2	2	no	—
87	C₄²	—	2	2	no	—
88	C₄²	—	2	1	no	—
89	C₄²	—	2	1	no	—
90	C₁₆	—	424	2	no	—
91	C₂³	C₂²⋊#88322	1	1	yes	characteristic
92	C₂×C₄	—	2	1	no	—
93	C₂×C₄	—	2	1	no	—
94	C₂×C₄	—	2	4	no	—
95	C₂×C₄	—	2	1	no	—
96	C₂×C₄	—	2	1	no	—
97	C₂×C₄	—	2	2	no	—
98	C₂×C₄	—	4	2	no	—
99	C₂×C₄	—	4	2	no	—
100	C₂×C₄	—	4	1	no	—
101	C₂×C₄	#13133132	1	1	yes	characteristic
102	C₂×C₄	#13133136	1	1	yes	characteristic
103	C₈	—	4	2	no	—
104	C₈	—	8	1	no	—
105	C₂²	—	2	1	no	—
106	C₂²	—	2	1	no	—
107	C₂²	#16810349	1	1	yes	characteristic
108	C₂²	#16810407	1	2	yes	—
109	C₄	—	2	1	no	—
110	C₄	—	4	4	no	—
111	C₄	—	4	1	no	—
112	C₄	—	4	1	no	—
113	C₄	#16810416	1	1	yes	characteristic
114	C₄	#16810422	1	1	yes	characteristic
115	C₂	—	2	2	no	—
116	C₂	#20272657	1	1	yes	characteristic
117	C₂	#20272661	1	1	yes	characteristic
118	C₂	#20272668	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20272655

C₂

yes

maximal

C₅₃×#1843

C₂

yes

maximal, characteristic

C₂²⋊#11719908

—

C₂×C₄⋊C₈×C₅₃

—

C₂×C₄⋊C₈×C₅₃

C₄

yes

characteristic

C₂×C₄⋊C₈×C₅₃

C₂²

yes

characteristic

C₅₃×#376

C₄

yes

characteristic

C₂×#11719908

—

C₂²×C₈×C₅₃

—

C₂²×C₈×C₅₃

D₄

yes

—

C₂×C₄²×C₅₃

—

C₂×C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄⋊C₈×C₅₃

—

C₄⋊C₈×C₅₃

—

C₄⋊C₈×C₅₃

—

#95×C₅₃

—

#95×C₅₃

C₈

yes

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₈

yes

characteristic

C₂²×C₄×C₅₃

M₁₆

yes

characteristic

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

#11719908

—

C₂³×C₅₃

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₈×C₅₃

—

C₈×C₅₃

—

#7066

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#324

yes

characteristic

C₂²×C₅₃

#350

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

D₅₃

yes

characteristic

#1322

—

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

Dic₅₃

yes

characteristic

C₂×C₄⋊C₈

C₂×D₅₃

yes

characteristic

#376

Dic₅₃

yes

characteristic

C₂²⋊C₁₆

—

106

—

C₅₃

#7066

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

C₂²⋊D₅₃

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×Dic₅₃

yes

characteristic

C₂×C₁₆

—

212

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

#95

—

#95

#88322

yes

—

C₂²×C₄

—

C₂²×C₄

yes

C₂²×C₄

yes

C₂²×C₄

C₂²⋊Dic₅₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₁₆

—

424

—

C₂³

C₂²⋊#88322

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

#13133132

yes

characteristic

102

C₂×C₄

#13133136

yes

characteristic

103

C₈

—

104

C₈

—

105

C₂²

—

106

C₂²

—

107

C₂²

#16810349

yes

characteristic

108

C₂²

#16810407

yes

—

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

#16810416

yes

characteristic

114

C₄

#16810422

yes

characteristic

115

C₂

—

116

C₂

#20272657

yes

characteristic

117

C₂

#20272661

yes

characteristic

118

C₂

#20272668

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#13133132
commutator	C₂²×C₄×C₅₃	C₂×C₈
Frattini	C₂×C₄⋊C₈	C₂×D₅₃
Fitting	C₅₃×#1843	C₂
socle	C₂²×C₅₃	#324
prime radical	C₂³×C₅₃	C₂²⋊C₈
cosocle	C₂×C₄⋊C₈×C₅₃	C₂²
hypercenter	#1843	D₅₃
hypocenter	C₅₃	#7066
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈ ◁ #1843

subgroup

G/H

center

C₂×C₄

#13133132

commutator

C₂²×C₄×C₅₃

C₂×C₈

Frattini

C₂×C₄⋊C₈

C₂×D₅₃

Fitting

C₅₃×#1843

C₂

socle

C₂²×C₅₃

#324

prime radical

C₂³×C₅₃

C₂²⋊C₈

cosocle

C₂×C₄⋊C₈×C₅₃

C₂²

D₅₃

C₅₃

G ▷ C₂²×C₄×C₅₃ ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈ ◁ #1843

N	H ≅ G/N	type
67. C₅₃	#7066	semidirect

H ≅ G/N

type

67. C₅₃

#7066

semidirect

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

subgroup	index
#20272655	2
C₅₃×#1843	2
#7066	53

subgroup

index

#20272655

C₅₃×#1843

#7066

Subgroups — #27270736

Conjugacy classes of subgroups

Complements

Maximal subgroups