Group #27270814 — subgroups

Subgroups — #27270814

Order 13568.

130 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 130 conjugacy classes of subgroups — too many to draw. Only its 49 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	#20274325	C₂	1	1	yes	maximal, characteristic
2	#20272779	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1367	C₂	1	1	yes	maximal, characteristic
4	#16811028	C₄	1	1	yes	characteristic
5	#16811028	C₄	1	1	yes	characteristic
6	#16810974	—	2	1	no	—
7	#16810974	—	2	1	no	—
8	C₄⋊#11719908	—	2	1	no	—
9	C₅₃×C₄⋊C₁₆	—	2	1	no	—
10	C₈²×C₅₃	C₂²	1	1	yes	characteristic
11	#13133217	—	2	1	no	—
12	#13133217	—	2	1	no	—
13	C₂×#11719908	—	4	1	no	—
14	#13133151	—	4	1	no	—
15	#13133151	—	4	1	no	—
16	#13133142	—	4	1	no	—
17	#13133142	—	4	1	no	—
18	#13133142	—	4	1	no	—
19	#13133142	—	4	1	no	—
20	C₂×C₁₆×C₅₃	—	4	1	no	—
21	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
23	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
24	C₂×Dic₁₀₆	—	4	1	no	—
25	C₂×Dic₁₀₆	—	4	1	no	—
26	C₂×C₈×C₅₃	—	2	1	no	—
27	C₂×C₈×C₅₃	D₈	1	1	yes	characteristic
28	C₂×C₈×C₅₃	D₈	1	1	yes	characteristic
29	C₂×C₈×C₅₃	SD₁₆	1	1	yes	characteristic
30	C₂×C₈×C₅₃	SD₁₆	1	1	yes	characteristic
31	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
32	C₄⋊Dic₅₃	—	4	1	no	—
33	C₄⋊Dic₅₃	—	4	1	no	—
34	C₄⋊Dic₅₃	—	4	1	no	—
35	C₄⋊Dic₅₃	—	4	1	no	—
36	C₁₆×C₅₃	—	8	1	no	—
37	#11719908	—	8	1	no	—
38	C₂×C₄×C₅₃	#100	1	1	yes	characteristic
39	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
40	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
41	C₂×Dic₅₃	—	8	1	no	—
42	C₂×Dic₅₃	—	8	1	no	—
43	Dic₁₀₆	—	8	1	no	—
44	Dic₁₀₆	—	8	1	no	—
45	Dic₁₀₆	—	8	1	no	—
46	Dic₁₀₆	—	8	1	no	—
47	C₈×C₅₃	—	2	2	no	—
48	C₈×C₅₃	—	2	1	no	—
49	C₈×C₅₃	—	2	1	no	—
50	C₈×C₅₃	—	2	1	no	—
51	C₈×C₅₃	—	2	1	no	—
52	#7093	—	53	1	no	maximal, Sylow
53	C₂²×C₅₃	#331	1	1	yes	characteristic
54	C₄×C₅₃	—	2	1	no	—
55	C₄×C₅₃	#359	1	1	yes	characteristic
56	C₄×C₅₃	#359	1	1	yes	characteristic
57	C₄×C₅₃	#360	1	1	yes	characteristic
58	C₄×C₅₃	#360	1	1	yes	characteristic
59	Dic₅₃	—	16	1	no	—
60	Dic₅₃	—	16	1	no	—
61	#1710	—	53	1	no	—
62	#1367	—	53	1	no	—
63	#1367	D₅₃	1	1	yes	characteristic
64	C₂×C₅₃	#1358	1	1	yes	characteristic
65	C₂×C₅₃	#1360	1	1	yes	characteristic
66	C₂×C₅₃	#1360	1	1	yes	characteristic
67	#500	—	53	1	no	—
68	#500	—	53	1	no	—
69	#487	—	106	1	no	—
70	#487	—	106	1	no	—
71	C₄⋊C₁₆	—	2	1	no	—
72	C₄⋊C₁₆	—	106	1	no	—
73	C₈²	C₂×D₅₃	1	1	yes	characteristic
74	C₅₃	#7093	1	1	yes	characteristic, complemented, Sylow
75	C₄⋊Q₈	—	106	1	no	—
76	C₄⋊Q₈	—	106	1	no	—
77	C₂×C₁₆	—	4	1	no	—
78	C₂×C₁₆	—	212	1	no	—
79	#107	—	212	1	no	—
80	#107	—	212	1	no	—
81	#103	—	212	1	no	—
82	#103	—	212	1	no	—
83	#103	—	212	1	no	—
84	#103	—	212	1	no	—
85	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
86	C₄×C₈	D₂₁₂	1	1	yes	characteristic
87	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
88	C₂×Q₈	—	212	1	no	—
89	C₂×Q₈	—	212	1	no	—
90	C₂×C₈	—	2	1	no	—
91	C₂×C₈	#11719913	1	1	yes	characteristic
92	C₂×C₈	D₄₂₄	1	1	yes	characteristic
93	C₂×C₈	#11719922	1	1	yes	characteristic
94	C₂×C₈	#11719924	1	1	yes	characteristic
95	C₄⋊C₄	—	212	1	no	—
96	C₄⋊C₄	—	212	1	no	—
97	C₄⋊C₄	—	212	1	no	—
98	C₄⋊C₄	—	212	1	no	—
99	C₄²	#11719921	1	1	yes	characteristic
100	C₁₆	—	8	1	no	—
101	C₁₆	—	424	1	no	—
102	Q₈	—	424	1	no	—
103	Q₈	—	424	1	no	—
104	Q₈	—	424	1	no	—
105	Q₈	—	424	1	no	—
106	C₂×C₄	—	424	1	no	—
107	C₂×C₄	—	424	1	no	—
108	C₂×C₄	#13133133	1	1	yes	characteristic
109	C₂×C₄	#13133137	1	1	yes	characteristic
110	C₂×C₄	#13133139	1	1	yes	characteristic
111	C₈	—	2	2	no	—
112	C₈	—	2	1	no	—
113	C₈	—	2	1	no	—
114	C₈	—	2	1	no	—
115	C₈	—	2	1	no	—
116	C₂²	#16810366	1	1	yes	characteristic
117	C₄	—	2	1	no	—
118	C₄	—	848	1	no	—
119	C₄	—	848	1	no	—
120	C₄	#16810429	1	1	yes	characteristic
121	C₄	#16810431	1	1	yes	characteristic
122	C₄	#16810432	1	1	yes	characteristic
123	C₄	#16810434	1	1	yes	characteristic
124	C₂	#20272753	1	1	yes	characteristic
125	C₂	#20272757	1	1	yes	characteristic
126	C₂	#20272759	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20274325

C₂

yes

maximal, characteristic

#20272779

C₂

yes

maximal, characteristic

C₅₃×#1367

C₂

yes

maximal, characteristic

#16811028

C₄

yes

characteristic

#16811028

C₄

yes

—

—

—

C₅₃×C₄⋊C₁₆

—

C₈²×C₅₃

C₂²

yes

—

—

—

—

—

—

—

—

—

C₂×C₁₆×C₅₃

—

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄×C₈×C₅₃

D₄

yes

characteristic

C₄×C₈×C₅₃

D₄

yes

characteristic

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

D₈

yes

characteristic

C₂×C₈×C₅₃

D₈

yes

characteristic

C₂×C₈×C₅₃

SD₁₆

yes

characteristic

C₂×C₈×C₅₃

SD₁₆

yes

characteristic

C₄²×C₅₃

C₂²⋊C₄

yes

characteristic

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₁₆×C₅₃

—

#11719908

—

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

—

C₂²×C₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

—

#1710

—

#1367

—

#1367

D₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

—

—

106

—

#487

—

106

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

106

—

C₈²

C₂×D₅₃

yes

characteristic

C₅₃

#7093

yes

characteristic, complemented, Sylow

C₄⋊Q₈

—

106

—

C₄⋊Q₈

—

106

—

C₂×C₁₆

—

C₂×C₁₆

—

212

—

#107

—

212

—

#107

—

212

—

#103

—

212

—

#103

—

212

—

#103

—

212

—

#103

—

212

—

C₄×C₈

C₄×D₅₃

yes

characteristic

C₄×C₈

D₂₁₂

yes

characteristic

C₄×C₈

C₂²⋊D₅₃

yes

characteristic

C₂×Q₈

—

212

—

C₂×Q₈

—

212

—

C₂×C₈

—

C₂×C₈

#11719913

yes

characteristic

C₂×C₈

D₄₂₄

yes

C₂×C₈

yes

C₂×C₈

yes

C₄⋊C₄

—

212

—

C₄⋊C₄

—

212

—

C₄⋊C₄

—

212

—

C₄⋊C₄

—

212

—

C₄²

#11719921

yes

characteristic

100

C₁₆

—

101

C₁₆

—

424

—

102

Q₈

—

424

—

103

Q₈

—

424

—

104

Q₈

—

424

—

105

Q₈

—

424

—

106

C₂×C₄

—

424

—

107

C₂×C₄

—

424

—

108

C₂×C₄

#13133133

yes

characteristic

109

C₂×C₄

#13133137

yes

characteristic

110

C₂×C₄

#13133139

yes

characteristic

111

C₈

—

112

C₈

—

113

C₈

—

114

C₈

—

115

C₈

—

116

C₂²

#16810366

yes

characteristic

117

C₄

—

118

C₄

—

848

—

119

C₄

—

848

—

120

C₄

#16810429

yes

characteristic

121

C₄

#16810431

yes

characteristic

122

C₄

#16810432

yes

characteristic

123

C₄

#16810434

yes

characteristic

124

C₂

#20272753

yes

characteristic

125

C₂

#20272757

yes

characteristic

126

C₂

#20272759

yes

characteristic

	subgroup	G/H
center	C₂²	#16810366
commutator	C₄×C₈×C₅₃	C₂×C₄
Frattini	C₈²	C₂×D₅₃
Fitting	C₅₃×#1367	C₂
socle, prime radical	C₂²×C₅₃	#331
cosocle	C₈²×C₅₃	C₂²
hypercenter	#1367	D₅₃
hypocenter	C₅₃	#7093
LCS	G ▷ C₄×C₈×C₅₃ ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1367

subgroup

G/H

center

C₂²

#16810366

commutator

C₄×C₈×C₅₃

C₂×C₄

Frattini

C₈²

C₂×D₅₃

Fitting

C₅₃×#1367

C₂

socle, prime radical

C₂²×C₅₃

#331

cosocle

C₈²×C₅₃

C₂²

D₅₃

C₅₃

G ▷ C₄×C₈×C₅₃ ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1367

N	H ≅ G/N	type
74. C₅₃	#7093	semidirect

H ≅ G/N

type

74. C₅₃

#7093

semidirect

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

subgroup	index
#20274325	2
#20272779	2
C₅₃×#1367	2
#7093	53

subgroup

index

#20274325

#20272779

C₅₃×#1367

#7093

Subgroups — #27270814

Conjugacy classes of subgroups

Complements

Maximal subgroups