Group #27271263 — subgroups

Subgroups — #27271263

Order 13568.

238 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 111 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×#16810444	C₂	1	2	yes	maximal
2	C₅₃×#1844	C₂	1	1	yes	maximal, characteristic
3	C₂×#13133157	—	2	2	no	—
4	#16810444	—	2	2	no	—
5	#16810444	C₄	1	4	yes	complemented
6	C₂×#106×C₅₃	—	2	2	no	—
7	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
8	C₅₃×#384	C₄	1	2	yes	—
9	#13133157	—	2	4	no	—
10	#13133157	—	4	2	no	—
11	C₂×#11719908	—	4	2	no	—
12	C₂²×C₈×C₅₃	D₄	1	2	yes	—
13	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
14	C₂×C₄⋊C₄×C₅₃	—	4	2	no	—
15	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
16	#106×C₅₃	—	4	2	no	—
17	#106×C₅₃	—	4	2	no	—
18	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
19	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
20	C₄×C₈×C₅₃	Q₈	1	1	yes	characteristic
21	C₂²×C₄×C₅₃	—	2	1	no	—
22	C₂²×C₄×C₅₃	—	4	2	no	—
23	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
24	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
25	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
26	C₂×C₈×C₅₃	D₈	1	2	yes	—
27	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
28	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
29	C₂×C₈×C₅₃	Q₁₆	1	2	yes	—
30	C₄⋊C₄×C₅₃	—	4	4	no	—
31	C₄⋊C₄×C₅₃	—	8	4	no	—
32	C₄²×C₅₃	—	2	1	no	—
33	C₄²×C₅₃	C₄²	1	1	yes	characteristic
34	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
35	#11719908	—	4	4	no	—
36	C₂³×C₅₃	#100	1	1	yes	characteristic
37	C₂×C₄×C₅₃	—	2	4	no	—
38	C₂×C₄×C₅₃	—	2	1	no	—
39	C₂×C₄×C₅₃	—	2	1	no	—
40	C₂×C₄×C₅₃	—	8	4	no	—
41	C₂×C₄×C₅₃	—	8	2	no	—
42	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
43	C₂×C₄×C₅₃	#101	1	1	yes	characteristic
44	C₂×C₄×C₅₃	#102	1	2	yes	—
45	C₂×C₄×C₅₃	#103	1	2	yes	—
46	C₈×C₅₃	#102	1	2	yes	—
47	C₈×C₅₃	#103	1	2	yes	—
48	C₈×C₅₃	#106	1	2	yes	—
49	C₈×C₅₃	#107	1	2	yes	—
50	#7268	—	53	1	no	maximal, Sylow
51	C₂²×C₅₃	—	2	2	no	—
52	C₂²×C₅₃	#331	1	1	yes	characteristic
53	C₂²×C₅₃	#333	1	1	yes	characteristic
54	C₂²×C₅₃	#343	1	1	yes	characteristic
55	C₄×C₅₃	—	2	2	no	—
56	C₄×C₅₃	—	2	2	no	—
57	C₄×C₅₃	—	16	2	no	—
58	C₄×C₅₃	#332	1	2	yes	—
59	C₄×C₅₃	#340	1	2	yes	—
60	C₂×#364	—	53	2	no	—
61	#1844	D₅₃	1	1	yes	characteristic
62	C₂×C₅₃	—	2	1	no	—
63	C₂×C₅₃	—	2	1	no	—
64	C₂×C₅₃	#1296	1	1	yes	characteristic
65	C₂×C₅₃	#1359	1	2	yes	—
66	C₂×M₃₂	—	106	2	no	—
67	C₂×#106	—	2	2	no	—
68	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
69	#384	Dic₅₃	1	2	yes	—
70	#364	—	53	4	no	—
71	#364	—	106	2	no	—
72	C₅₃	#7268	1	1	yes	characteristic, complemented, Sylow
73	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
74	C₂×C₄⋊C₄	—	2	2	no	—
75	C₂×C₄⋊C₄	—	4	2	no	—
76	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
77	M₃₂	—	106	4	no	—
78	M₃₂	—	212	2	no	—
79	C₂×C₁₆	—	212	2	no	—
80	#106	—	4	2	no	—
81	#106	—	4	2	no	—
82	C₄×C₈	Dic₁₀₆	1	1	yes	characteristic
83	C₄×C₈	C₄×D₅₃	1	2	yes	—
84	C₄×C₈	D₂₁₂	1	1	yes	characteristic
85	C₂²×C₄	—	2	1	no	—
86	C₂²×C₄	—	4	2	no	—
87	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
88	C₂×C₈	#11719919	1	2	yes	—
89	C₂×C₈	#11719921	1	2	yes	—
90	C₂×C₈	#11719922	1	2	yes	—
91	C₂×C₈	#11719923	1	2	yes	—
92	C₂×C₈	#11719924	1	2	yes	—
93	C₂×C₈	#11719925	1	2	yes	—
94	C₄⋊C₄	—	4	4	no	—
95	C₄⋊C₄	—	8	4	no	—
96	C₄²	—	2	1	no	—
97	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
98	C₄²	C₄⋊Dic₅₃	1	1	yes	characteristic
99	C₁₆	—	212	4	no	—
100	C₂³	#13133134	1	1	yes	characteristic
101	C₂×C₄	—	2	4	no	—
102	C₂×C₄	—	2	1	no	—
103	C₂×C₄	—	2	1	no	—
104	C₂×C₄	—	8	4	no	—
105	C₂×C₄	—	8	2	no	—
106	C₂×C₄	#13133124	1	1	yes	characteristic
107	C₂×C₄	#13133136	1	1	yes	characteristic
108	C₂×C₄	#13133138	1	2	yes	—
109	C₂×C₄	#13133141	1	2	yes	—
110	C₈	#13133137	1	2	yes	—
111	C₈	#13133140	1	2	yes	—
112	C₈	#13133148	1	2	yes	—
113	C₈	#13133150	1	2	yes	—
114	C₂²	—	2	2	no	—
115	C₂²	#16810367	1	1	yes	characteristic
116	C₂²	#16810372	1	1	yes	characteristic
117	C₂²	#16810392	1	1	yes	characteristic
118	C₄	—	2	2	no	—
119	C₄	—	2	2	no	—
120	C₄	—	16	2	no	—
121	C₄	#16810369	1	2	yes	—
122	C₄	#16810385	1	2	yes	—
123	C₂	—	2	1	no	—
124	C₂	—	2	1	no	—
125	C₂	#20272598	1	1	yes	characteristic
126	C₂	#20272756	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16810444

C₂

yes

maximal

C₅₃×#1844

C₂

yes

maximal, characteristic

C₂×#13133157

—

#16810444

—

#16810444

C₄

yes

complemented

C₂×#106×C₅₃

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×#384

C₄

yes

—

#13133157

—

#13133157

—

C₂×#11719908

—

C₂²×C₈×C₅₃

D₄

yes

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄²×C₅₃

C₂×C₄

yes

characteristic

#106×C₅₃

—

#106×C₅₃

—

C₄×C₈×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

D₄

yes

characteristic

C₄×C₈×C₅₃

Q₈

yes

characteristic

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₅₃

C₂²⋊C₄

yes

—

C₂×C₈×C₅₃

C₄⋊C₄

yes

—

C₂×C₈×C₅₃

D₈

yes

—

C₂×C₈×C₅₃

SD₁₆

yes

—

C₂×C₈×C₅₃

SD₁₆

yes

—

C₂×C₈×C₅₃

Q₁₆

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

C₄²

yes

characteristic

C₄²×C₅₃

C₄⋊C₄

yes

characteristic

#11719908

—

C₂³×C₅₃

#100

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

—

C₂×C₄×C₅₃

#103

yes

—

C₈×C₅₃

#102

yes

—

C₈×C₅₃

#103

yes

—

C₈×C₅₃

#106

yes

—

C₈×C₅₃

#107

yes

—

#7268

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#332

yes

—

C₄×C₅₃

#340

yes

—

C₂×#364

—

#1844

D₅₃

yes

characteristic

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#1296

yes

characteristic

C₂×C₅₃

#1359

yes

—

C₂×M₃₂

—

106

—

C₂×#106

—

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

#384

Dic₅₃

yes

—

#364

—

#364

—

106

—

C₅₃

#7268

yes

characteristic, complemented, Sylow

C₂²×C₈

C₂²⋊D₅₃

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×Dic₅₃

yes

characteristic

M₃₂

—

106

—

M₃₂

—

212

—

C₂×C₁₆

—

212

—

#106

—

#106

—

C₄×C₈

Dic₁₀₆

yes

characteristic

C₄×C₈

C₄×D₅₃

yes

—

C₄×C₈

D₂₁₂

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊Dic₅₃

yes

characteristic

C₂×C₈

#11719919

yes

—

C₂×C₈

#11719921

yes

—

C₂×C₈

#11719922

yes

—

C₂×C₈

#11719923

yes

—

C₂×C₈

#11719924

yes

—

C₂×C₈

#11719925

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

C₄×Dic₅₃

yes

characteristic

C₄²

C₄⋊Dic₅₃

yes

characteristic

C₁₆

—

212

—

100

C₂³

#13133134

yes

characteristic

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

#13133124

yes

characteristic

107

C₂×C₄

#13133136

yes

characteristic

108

C₂×C₄

#13133138

yes

—

109

C₂×C₄

#13133141

yes

—

110

C₈

#13133137

yes

—

111

C₈

#13133140

yes

—

112

C₈

#13133148

yes

—

113

C₈

#13133150

yes

—

114

C₂²

—

115

C₂²

#16810367

yes

characteristic

116

C₂²

#16810372

yes

characteristic

117

C₂²

#16810392

yes

characteristic

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

#16810369

yes

—

122

C₄

#16810385

yes

—

123

C₂

—

124

C₂

—

125

C₂

#20272598

yes

characteristic

126

C₂

#20272756

yes

—

	subgroup	G/H
center	C₂²	#16810367
commutator	C₄²×C₅₃	C₄²
Frattini	C₂×C₄×C₈	C₂×D₅₃
Fitting	C₅₃×#1844	C₂
socle	C₂²×C₅₃	#331
prime radical	C₂³×C₅₃	#100
cosocle	C₂×C₄×C₈×C₅₃	C₂²
hypercenter	#1844	D₅₃
hypocenter	C₅₃	#7268
LCS	G ▷ C₄²×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1844

subgroup

G/H

center

C₂²

#16810367

commutator

C₄²×C₅₃

C₄²

Frattini

C₂×C₄×C₈

C₂×D₅₃

Fitting

C₅₃×#1844

C₂

C₂²×C₅₃

C₂³×C₅₃

C₂×C₄×C₈×C₅₃

C₂²

D₅₃

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1844

N	H ≅ G/N	type
5. #16810444	C₄	semidirect
72. C₅₃	#7268	semidirect

H ≅ G/N

type

5. #16810444

C₄

semidirect

72. C₅₃

#7268

semidirect

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

subgroup	index
C₂×#16810444	2
C₅₃×#1844	2
#7268	53

subgroup

index

C₂×#16810444

C₅₃×#1844

#7268

Subgroups — #27271263

Conjugacy classes of subgroups

Complements

Maximal subgroups