Group #27277746 — subgroups

Subgroups — #27277746

Order 13568.

262 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 115 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	#20273463	C₂	1	2	yes	maximal
2	#20272560	C₂	1	2	yes	maximal
3	#20272544	C₂	1	2	yes	maximal
4	C₅₃×#1746	C₂	1	1	yes	maximal, characteristic
5	#16810676	C₂²	1	4	yes	—
6	#16810376	—	2	2	no	—
7	#16810374	—	2	2	no	—
8	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
10	C₅₃×#376	C₂²	1	1	yes	characteristic
11	C₅₃×#336	C₄	1	4	yes	—
12	C₄⋊#88322	—	2	4	no	—
13	C₂²⋊#88322	—	2	4	no	—
14	C₄×#88322	—	2	4	no	—
15	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
16	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
17	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
18	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
19	#97×C₅₃	—	2	2	no	—
20	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
21	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
22	C₄×C₈×C₅₃	Q₈	1	2	yes	—
23	#95×C₅₃	—	2	2	no	—
24	C₂²×C₄×C₅₃	—	2	2	no	—
25	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₅₃	#41	1	2	yes	—
27	C₂×C₈×C₅₃	—	2	4	no	—
28	C₂×C₈×C₅₃	—	2	2	no	—
29	C₂×C₈×C₅₃	—	2	4	no	—
30	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
31	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
32	C₄²×C₅₃	—	2	4	no	—
33	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
34	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
35	C₄²×C₅₃	#41	1	2	yes	—
36	C₂×#88322	—	4	4	no	—
37	C₂³×C₅₃	#117	1	1	yes	characteristic
38	C₂×C₄×C₅₃	—	4	2	no	—
39	C₂×C₄×C₅₃	—	4	4	no	—
40	C₂×C₄×C₅₃	—	4	2	no	—
41	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
43	C₂×C₄×C₅₃	#124	1	2	yes	—
44	C₂×C₄×C₅₃	#125	1	2	yes	—
45	C₂×C₄×C₅₃	#131	1	4	yes	—
46	C₈×C₅₃	—	2	2	no	—
47	C₈×C₅₃	—	2	2	no	—
48	C₈×C₅₃	—	4	2	no	—
49	#88322	—	8	4	no	—
50	#8782	—	53	1	no	maximal, Sylow
51	C₂²×C₅₃	—	2	2	no	—
52	C₂²×C₅₃	#382	1	1	yes	characteristic
53	C₂²×C₅₃	#433	1	2	yes	—
54	C₄×C₅₃	—	2	2	no	—
55	C₄×C₅₃	—	2	2	no	—
56	C₄×C₅₃	—	8	2	no	—
57	C₄×C₅₃	#424	1	2	yes	—
58	C₄×C₅₃	#433	1	2	yes	—
59	#1746	D₅₃	1	1	yes	characteristic
60	#1564	—	53	2	no	—
61	#1281	—	53	2	no	—
62	#1274	—	53	2	no	—
63	C₂×C₅₃	—	2	2	no	—
64	C₂×C₅₃	#1832	1	1	yes	characteristic
65	C₂×C₅₃	#1833	1	2	yes	—
66	#431	—	53	4	no	—
67	C₂×#97	C₂×D₅₃	1	1	yes	characteristic
68	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
69	#376	C₂×D₅₃	1	1	yes	characteristic
70	#336	Dic₅₃	1	4	yes	—
71	#335	—	106	2	no	—
72	#334	—	106	2	no	—
73	C₅₃	#8782	1	1	yes	characteristic, complemented, Sylow
74	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
75	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
76	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
77	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
78	C₄⋊C₈	—	106	4	no	—
79	C₂²⋊C₈	—	106	4	no	—
80	#97	—	2	2	no	—
81	C₄×C₈	—	106	4	no	—
82	C₄×C₈	Dic₁₀₆	1	2	yes	—
83	C₄×C₈	D₂₁₂	1	1	yes	characteristic
84	C₄×C₈	D₂₁₂	1	1	yes	characteristic
85	#95	—	2	2	no	—
86	C₂²×C₄	—	2	2	no	—
87	C₂²×C₄	#11719945	1	2	yes	—
88	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
89	C₂×C₈	—	2	4	no	—
90	C₂×C₈	—	2	2	no	—
91	C₂×C₈	—	2	4	no	—
92	C₂×C₈	—	212	4	no	—
93	C₂×C₈	C₄⋊Dic₅₃	1	2	yes	—
94	C₂×C₈	C₄⋊Dic₅₃	1	2	yes	—
95	C₄²	—	2	4	no	—
96	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
97	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
98	C₄²	#11719945	1	2	yes	—
99	C₂³	#13133176	1	1	yes	characteristic
100	C₂×C₄	—	4	2	no	—
101	C₂×C₄	—	4	4	no	—
102	C₂×C₄	—	4	2	no	—
103	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
104	C₂×C₄	#13133176	1	1	yes	characteristic
105	C₂×C₄	#13133205	1	2	yes	—
106	C₂×C₄	#13133209	1	2	yes	—
107	C₂×C₄	#13133224	1	4	yes	—
108	C₈	—	2	2	no	—
109	C₈	—	2	2	no	—
110	C₈	—	4	2	no	—
111	C₈	—	424	4	no	—
112	C₂²	—	2	2	no	—
113	C₂²	#16810489	1	1	yes	characteristic
114	C₂²	#16810684	1	2	yes	—
115	C₄	—	2	2	no	—
116	C₄	—	2	2	no	—
117	C₄	—	8	2	no	—
118	C₄	#16810650	1	2	yes	—
119	C₄	#16810684	1	2	yes	—
120	C₂	—	2	2	no	—
121	C₂	#20274835	1	1	yes	characteristic
122	C₂	#20274839	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20273463

C₂

yes

maximal

#20272560

C₂

yes

maximal

#20272544

C₂

yes

maximal

C₅₃×#1746

C₂

yes

maximal, characteristic

#16810676

C₂²

yes

—

#16810376

—

#16810374

—

C₂×#97×C₅₃

C₂²

yes

characteristic

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×#376

C₂²

yes

characteristic

C₅₃×#336

C₄

yes

—

C₄⋊#88322

—

C₂²⋊#88322

—

C₄×#88322

—

C₂²×C₈×C₅₃

C₂×C₄

yes

—

C₂²×C₈×C₅₃

C₂×C₄

yes

—

C₂×C₄²×C₅₃

C₂×C₄

yes

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

#97×C₅₃

—

C₄×C₈×C₅₃

D₄

yes

characteristic

C₄×C₈×C₅₃

D₄

yes

characteristic

C₄×C₈×C₅₃

Q₈

yes

—

#95×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₅₃

#41

yes

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₄⋊C₄

yes

—

C₂×C₈×C₅₃

C₄⋊C₄

yes

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂×D₄

yes

characteristic

C₄²×C₅₃

C₂×Q₈

yes

characteristic

C₄²×C₅₃

#41

yes

—

C₂×#88322

—

C₂³×C₅₃

#117

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

—

C₂×C₄×C₅₃

#125

yes

—

C₂×C₄×C₅₃

#131

yes

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

#88322

—

#8782

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

#382

yes

characteristic

C₂²×C₅₃

#433

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#424

yes

—

C₄×C₅₃

#433

yes

—

#1746

D₅₃

yes

—

—

—

C₂×C₅₃

—

C₂×C₅₃

#1832

yes

characteristic

C₂×C₅₃

#1833

yes

—

#431

—

C₂×#97

C₂×D₅₃

yes

characteristic

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

#376

C₂×D₅₃

yes

characteristic

#336

Dic₅₃

yes

—

#335

—

106

—

#334

—

106

—

C₅₃

#8782

yes

characteristic, complemented, Sylow

C₂²×C₈

C₂×Dic₅₃

yes

—

C₂²×C₈

C₂×Dic₅₃

yes

—

C₂×C₄²

C₂×Dic₅₃

yes

—

C₂×C₄²

C₂²×D₅₃

yes

characteristic

C₄⋊C₈

—

106

—

C₂²⋊C₈

—

106

—

#97

—

C₄×C₈

—

106

—

C₄×C₈

Dic₁₀₆

yes

—

C₄×C₈

D₂₁₂

yes

characteristic

C₄×C₈

D₂₁₂

yes

characteristic

#95

—

C₂²×C₄

—

C₂²×C₄

#11719945

yes

—

C₂²×C₄

C₂²×Dic₅₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

212

—

C₂×C₈

C₄⋊Dic₅₃

yes

—

C₂×C₈

C₄⋊Dic₅₃

yes

—

C₄²

—

C₄²

C₂×Dic₁₀₆

yes

characteristic

C₄²

C₂×D₂₁₂

yes

characteristic

C₄²

#11719945

yes

—

C₂³

#13133176

yes

characteristic

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

C₂×C₄⋊Dic₅₃

yes

characteristic

104

C₂×C₄

#13133176

yes

characteristic

105

C₂×C₄

#13133205

yes

—

106

C₂×C₄

#13133209

yes

—

107

C₂×C₄

#13133224

yes

—

108

C₈

—

109

C₈

—

110

C₈

—

111

C₈

—

424

—

112

C₂²

—

113

C₂²

#16810489

yes

characteristic

114

C₂²

#16810684

yes

—

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

#16810650

yes

—

119

C₄

#16810684

yes

—

120

C₂

—

121

C₂

#20274835

yes

characteristic

122

C₂

#20274839

yes

—

	subgroup	G/H
center	C₂×C₄	#13133176
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1746	C₂
socle	C₂²×C₅₃	#382
prime radical	C₂³×C₅₃	#117
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1746	D₅₃
hypocenter	C₅₃	#8782
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #376 ◁ #1746

subgroup

G/H

center

C₂×C₄

#13133176

commutator

C₂²×C₄×C₅₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₅₃×#1746

C₂

C₂²×C₅₃

C₂³×C₅₃

C₂×C₄²×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ #376 ◁ #1746

N	H ≅ G/N	type
73. C₅₃	#8782	semidirect

H ≅ G/N

type

73. C₅₃

#8782

semidirect

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

subgroup	index
#20273463	2
#20272560	2
#20272544	2
C₅₃×#1746	2
#8782	53

subgroup

index

C₅₃×#1746

Subgroups — #27277746

Conjugacy classes of subgroups

Complements

Maximal subgroups