Group #27274527 — subgroups

Subgroups — #27274527

Order 13568.

382 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 179 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	#20274461	C₂	1	1	yes	maximal, characteristic
2	#20274461	C₂	1	1	yes	maximal, characteristic
3	#20272542	C₂	1	4	yes	maximal
4	C₅₃×#1744	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×#88322	C₂²	1	2	yes	—
6	C₂×C₄×#88322	C₂²	1	2	yes	—
7	#16810378	C₄	1	4	yes	—
8	#16810378	C₄	1	4	yes	—
9	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
10	C₄³×C₅₃	C₂²	1	1	yes	characteristic
11	C₅₃×#336	C₄	1	4	yes	—
12	C₂²×#88322	C₂×C₄	1	4	yes	—
13	C₂²×#88322	C₂×C₄	1	4	yes	—
14	C₄×#88322	—	2	4	no	—
15	C₄×#88322	—	2	4	no	—
16	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
17	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
18	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
19	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
20	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
21	C₄×C₈×C₅₃	—	2	4	no	—
22	C₂²×C₄×C₅₃	C₄²	1	4	yes	—
23	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
25	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
26	C₂×C₈×C₅₃	—	2	4	no	—
27	C₂×C₈×C₅₃	—	2	8	no	—
28	C₄²×C₅₃	—	2	4	no	—
29	C₄²×C₅₃	—	2	4	no	—
30	C₄²×C₅₃	—	2	4	no	—
31	C₄²×C₅₃	#41	1	4	yes	—
32	C₂×#88322	—	2	4	no	—
33	C₂×#88322	—	2	4	no	—
34	C₂×#88322	—	2	8	no	—
35	C₂×#88322	—	2	8	no	—
36	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
37	C₂×C₄×C₅₃	—	2	4	no	—
38	C₂×C₄×C₅₃	—	2	4	no	—
39	C₂×C₄×C₅₃	—	2	4	no	—
40	C₂×C₄×C₅₃	#117	1	2	yes	—
41	C₂×C₄×C₅₃	#117	1	2	yes	—
42	C₂×C₄×C₅₃	#117	1	2	yes	—
43	C₂×C₄×C₅₃	#131	1	4	yes	—
44	C₂×C₄×C₅₃	#131	1	4	yes	—
45	C₂×C₄×C₅₃	#131	1	4	yes	—
46	C₈×C₅₃	—	4	4	no	—
47	#88322	—	4	4	no	—
48	#88322	—	4	4	no	—
49	#8135	—	53	1	no	maximal, Sylow
50	C₂²×C₅₃	#376	1	1	yes	characteristic
51	C₂²×C₅₃	#405	1	2	yes	—
52	C₂²×C₅₃	#405	1	2	yes	—
53	C₂²×C₅₃	#405	1	2	yes	—
54	C₄×C₅₃	—	4	4	no	—
55	C₄×C₅₃	#433	1	4	yes	—
56	C₄×C₅₃	#433	1	4	yes	—
57	C₄×C₅₃	#433	1	4	yes	—
58	#1744	—	53	1	no	—
59	#1744	—	53	1	no	—
60	#1744	D₅₃	1	1	yes	characteristic
61	#1273	—	53	4	no	—
62	C₂×C₅₃	#1454	1	4	yes	—
63	C₂×C₅₃	#1746	1	1	yes	characteristic
64	C₂×C₅₃	#1746	1	1	yes	characteristic
65	C₂×C₅₃	#1746	1	1	yes	characteristic
66	C₂×C₄×C₈	—	53	2	no	—
67	C₂×C₄×C₈	—	53	2	no	—
68	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
69	C₄³	C₂×D₅₃	1	1	yes	characteristic
70	#336	—	53	4	no	—
71	#336	—	53	4	no	—
72	#336	Dic₅₃	1	4	yes	—
73	C₅₃	#8135	1	1	yes	characteristic, complemented, Sylow
74	C₂²×C₈	—	53	4	no	—
75	C₂²×C₈	—	53	4	no	—
76	C₂²×C₈	C₂×Dic₅₃	1	4	yes	—
77	C₂×C₄²	C₄×D₅₃	1	2	yes	—
78	C₂×C₄²	C₄×D₅₃	1	2	yes	—
79	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
80	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
81	C₄×C₈	—	2	4	no	—
82	C₄×C₈	—	106	4	no	—
83	C₄×C₈	—	106	4	no	—
84	C₂²×C₄	C₄×Dic₅₃	1	4	yes	—
85	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
86	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
87	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
88	C₂×C₈	—	2	4	no	—
89	C₂×C₈	—	2	8	no	—
90	C₂×C₈	—	106	4	no	—
91	C₂×C₈	—	106	4	no	—
92	C₂×C₈	—	106	8	no	—
93	C₂×C₈	—	106	8	no	—
94	C₄²	—	2	4	no	—
95	C₄²	—	2	4	no	—
96	C₄²	—	2	4	no	—
97	C₄²	#11719945	1	4	yes	—
98	C₂³	C₂×C₄×Dic₅₃	1	1	yes	characteristic
99	C₂×C₄	—	2	4	no	—
100	C₂×C₄	—	2	4	no	—
101	C₂×C₄	—	2	4	no	—
102	C₂×C₄	#13133174	1	2	yes	—
103	C₂×C₄	#13133174	1	2	yes	—
104	C₂×C₄	#13133176	1	2	yes	—
105	C₂×C₄	#13133223	1	4	yes	—
106	C₂×C₄	#13133224	1	4	yes	—
107	C₂×C₄	#13133225	1	4	yes	—
108	C₈	—	4	4	no	—
109	C₈	—	212	4	no	—
110	C₈	—	212	4	no	—
111	C₂²	#16810469	1	1	yes	characteristic
112	C₂²	#16810578	1	2	yes	—
113	C₂²	#16810579	1	2	yes	—
114	C₂²	#16810581	1	2	yes	—
115	C₄	—	4	4	no	—
116	C₄	#16810682	1	4	yes	—
117	C₄	#16810684	1	4	yes	—
118	C₄	#16810685	1	4	yes	—
119	C₂	#20273015	1	4	yes	—
120	C₂	#20274466	1	1	yes	characteristic
121	C₂	#20274467	1	1	yes	characteristic
122	C₂	#20274468	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20274461

C₂

yes

maximal, characteristic

#20274461

C₂

yes

maximal, characteristic

#20272542

C₂

yes

maximal

C₅₃×#1744

C₂

yes

maximal, characteristic

C₂×C₄×#88322

C₂²

yes

—

C₂×C₄×#88322

C₂²

yes

—

#16810378

C₄

yes

—

#16810378

C₄

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

—

C₄³×C₅₃

C₂²

yes

characteristic

C₅₃×#336

C₄

yes

—

C₂²×#88322

C₂×C₄

yes

—

C₂²×#88322

C₂×C₄

yes

—

C₄×#88322

—

C₄×#88322

—

C₂²×C₈×C₅₃

C₂×C₄

yes

—

C₂×C₄²×C₅₃

C₂×C₄

yes

—

C₂×C₄²×C₅₃

C₂×C₄

yes

—

C₂×C₄²×C₅₃

C₂×C₄

yes

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄×C₈×C₅₃

—

C₂²×C₄×C₅₃

C₄²

yes

—

C₂²×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

#41

yes

—

—

—

—

—

C₂³×C₅₃

C₂×C₄²

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#117

yes

—

C₂×C₄×C₅₃

#117

yes

—

C₂×C₄×C₅₃

#117

yes

—

C₂×C₄×C₅₃

#131

yes

—

C₂×C₄×C₅₃

#131

yes

—

C₂×C₄×C₅₃

#131

yes

—

C₈×C₅₃

—

—

—

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

—

C₂²×C₅₃

#405

yes

—

C₂²×C₅₃

#405

yes

—

C₄×C₅₃

—

C₄×C₅₃

#433

yes

—

C₄×C₅₃

#433

yes

—

C₄×C₅₃

#433

yes

—

#1744

—

#1744

—

#1744

D₅₃

yes

characteristic

#1273

—

C₂×C₅₃

#1454

yes

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₅₃

yes

—

C₄³

C₂×D₅₃

yes

—

—

Dic₅₃

yes

—

C₅₃

#8135

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×Dic₅₃

yes

—

C₂×C₄²

C₄×D₅₃

yes

—

C₂×C₄²

C₄×D₅₃

yes

—

C₂×C₄²

C₂×Dic₅₃

yes

—

C₂×C₄²

C₂²×D₅₃

yes

characteristic

C₄×C₈

—

C₄×C₈

—

106

—

C₄×C₈

—

106

—

C₂²×C₄

C₄×Dic₅₃

yes

—

C₂²×C₄

C₂×C₄×D₅₃

yes

characteristic

C₂²×C₄

C₂×C₄×D₅₃

yes

characteristic

C₂²×C₄

C₂²×Dic₅₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₄²

—

C₄²

—

C₄²

—

C₄²

#11719945

yes

—

C₂³

C₂×C₄×Dic₅₃

yes

characteristic

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

#13133174

yes

—

103

C₂×C₄

#13133174

yes

—

104

C₂×C₄

#13133176

yes

—

105

C₂×C₄

#13133223

yes

—

106

C₂×C₄

#13133224

yes

—

107

C₂×C₄

#13133225

yes

—

108

C₈

—

109

C₈

—

212

—

110

C₈

—

212

—

111

C₂²

#16810469

yes

characteristic

112

C₂²

#16810578

yes

—

113

C₂²

#16810579

yes

—

114

C₂²

#16810581

yes

—

115

C₄

—

116

C₄

#16810682

yes

—

117

C₄

#16810684

yes

—

118

C₄

#16810685

yes

—

119

C₂

#20273015

yes

—

120

C₂

#20274466

yes

characteristic

121

C₂

#20274467

yes

characteristic

122

C₂

#20274468

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₅₃
commutator, socle, prime radical	C₂³×C₅₃	C₂×C₄²
Fitting	C₅₃×#1744	C₂
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1744	D₅₃
hypocenter	C₅₃	#8135
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄² ◁ #1744

subgroup

G/H

center, Frattini

C₂×C₄²

C₂²×D₅₃

commutator, socle, prime radical

C₂³×C₅₃

C₂×C₄²

Fitting

C₅₃×#1744

C₂

cosocle

C₂×C₄²×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₂³×C₅₃ ▷ C₅₃

C₁ ◁ C₂×C₄² ◁ #1744

N	H ≅ G/N	type
73. C₅₃	#8135	semidirect

H ≅ G/N

type

73. C₅₃

#8135

semidirect

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

subgroup	index
#20274461	2
#20272542	2
C₅₃×#1744	2
#8135	53

subgroup

index

#20274461

#20272542

C₅₃×#1744

#8135

Subgroups — #27274527

Conjugacy classes of subgroups

Complements

Maximal subgroups