Group #27274521 — subgroups

Subgroups — #27274521

Order 13568.

446 conjugacy classes of subgroups; only the 243 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 243 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	2	yes	maximal
2	#20272542	C₂	1	4	yes	maximal
3	C₄²×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×#88322	C₂²	1	4	yes	—
5	#16810378	C₄	1	8	yes	—
6	C₂×C₄×C₈×C₅₃	C₄	1	2	yes	—
7	C₂×C₄×C₈×C₅₃	C₄	1	2	yes	—
8	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
9	C₄³×C₅₃	C₂²	1	1	yes	characteristic
10	C₂²×#88322	C₂×C₄	1	8	yes	—
11	C₄×#88322	—	2	8	no	—
12	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
13	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
14	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
15	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
17	C₄×C₈×C₅₃	—	2	4	no	—
18	C₄×C₈×C₅₃	—	2	4	no	—
19	C₄×C₈×C₅₃	C₈	1	8	yes	complemented
20	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
21	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
22	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
23	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
24	C₂×C₈×C₅₃	—	2	4	no	—
25	C₂×C₈×C₅₃	—	2	4	no	—
26	C₂×C₈×C₅₃	C₂×C₈	1	8	yes	—
27	C₄²×C₅₃	—	2	8	no	—
28	C₄²×C₅₃	C₂×C₈	1	4	yes	—
29	C₄²×C₅₃	M₁₆	1	4	yes	—
30	C₄²×C₅₃	#41	1	4	yes	—
31	C₂×#88322	—	2	8	no	—
32	C₂×#88322	—	2	16	no	—
33	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
34	C₂×C₄×C₅₃	—	2	4	no	—
35	C₂×C₄×C₅₃	—	2	4	no	—
36	C₂×C₄×C₅₃	C₄×C₈	1	4	yes	—
37	C₂×C₄×C₅₃	#97	1	4	yes	—
38	C₂×C₄×C₅₃	#117	1	4	yes	—
39	C₂×C₄×C₅₃	#117	1	2	yes	—
40	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
41	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
42	C₂×C₄×C₅₃	#131	1	8	yes	—
43	C₈×C₅₃	—	2	8	no	—
44	#88322	—	4	8	no	—
45	#8133	—	53	1	no	maximal, Sylow
46	C₂²×C₅₃	#376	1	1	yes	characteristic
47	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
48	C₂²×C₅₃	C₂×#97	1	1	yes	characteristic
49	C₂²×C₅₃	#405	1	4	yes	—
50	C₄×C₅₃	—	2	4	no	—
51	C₄×C₅₃	—	2	4	no	—
52	C₄×C₅₃	#431	1	4	yes	—
53	C₄×C₅₃	#433	1	8	yes	—
54	#1744	—	53	2	no	—
55	C₄²×C₈	D₅₃	1	1	yes	characteristic
56	#1273	—	53	4	no	—
57	C₂×C₅₃	#1450	1	2	yes	—
58	C₂×C₅₃	#1452	1	2	yes	—
59	C₂×C₅₃	#1744	1	1	yes	characteristic
60	C₂×C₅₃	#1746	1	2	yes	—
61	C₂×C₄×C₈	—	53	4	no	—
62	C₂×C₄×C₈	Dic₅₃	1	2	yes	—
63	C₂×C₄×C₈	Dic₅₃	1	2	yes	—
64	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
65	C₄³	C₂×D₅₃	1	1	yes	characteristic
66	#336	—	53	8	no	—
67	C₅₃	#8133	1	1	yes	characteristic, complemented, Sylow
68	C₂²×C₈	—	53	8	no	—
69	C₂²×C₈	C₂×Dic₅₃	1	4	yes	—
70	C₂×C₄²	C₄×D₅₃	1	4	yes	—
71	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
72	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
73	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
74	C₄×C₈	—	2	4	no	—
75	C₄×C₈	—	2	4	no	—
76	C₄×C₈	—	106	8	no	—
77	C₄×C₈	#88322	1	8	yes	complemented
78	C₂²×C₄	C₄×Dic₅₃	1	2	yes	—
79	C₂²×C₄	C₄×Dic₅₃	1	2	yes	—
80	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
81	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
82	C₂×C₈	—	2	4	no	—
83	C₂×C₈	—	2	4	no	—
84	C₂×C₈	—	106	8	no	—
85	C₂×C₈	—	106	16	no	—
86	C₂×C₈	C₂×#88322	1	8	yes	—
87	C₄²	—	2	8	no	—
88	C₄²	C₂×#88322	1	4	yes	—
89	C₄²	#11719917	1	4	yes	—
90	C₄²	#11719945	1	4	yes	—
91	C₂³	C₂×C₄×Dic₅₃	1	1	yes	characteristic
92	C₂×C₄	—	2	4	no	—
93	C₂×C₄	—	2	4	no	—
94	C₂×C₄	C₄×#88322	1	4	yes	—
95	C₂×C₄	#13133128	1	4	yes	—
96	C₂×C₄	#13133174	1	4	yes	—
97	C₂×C₄	#13133176	1	2	yes	—
98	C₂×C₄	C₂²×#88322	1	2	yes	—
99	C₂×C₄	C₂×#11719917	1	2	yes	—
100	C₂×C₄	#13133223	1	8	yes	—
101	C₈	—	2	8	no	—
102	C₈	—	212	8	no	—
103	C₂²	#16810469	1	1	yes	characteristic
104	C₂²	C₂×C₄×#88322	1	1	yes	characteristic
105	C₂²	C₂×#13133128	1	1	yes	characteristic
106	C₂²	#16810578	1	4	yes	—
107	C₄	—	2	4	no	—
108	C₄	—	2	4	no	—
109	C₄	#16810676	1	4	yes	—
110	C₄	#16810682	1	8	yes	—
111	C₂	#20272998	1	2	yes	—
112	C₂	#20273007	1	2	yes	—
113	C₂	#20274461	1	1	yes	characteristic
114	C₂	#20274466	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20274461

C₂

yes

maximal

#20272542

C₂

yes

maximal

C₄²×C₈×C₅₃

C₂

yes

maximal, characteristic

C₂×C₄×#88322

C₂²

yes

—

#16810378

C₄

yes

—

C₂×C₄×C₈×C₅₃

C₄

yes

—

C₂×C₄×C₈×C₅₃

C₄

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

—

C₄³×C₅₃

C₂²

yes

characteristic

C₂²×#88322

C₂×C₄

yes

—

C₄×#88322

—

C₂²×C₈×C₅₃

C₂×C₄

yes

—

C₂×C₄²×C₅₃

C₂×C₄

yes

—

C₂×C₄²×C₅₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₅₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄×C₈×C₅₃

—

C₄×C₈×C₅₃

—

C₄×C₈×C₅₃

C₈

yes

complemented

C₂²×C₄×C₅₃

C₄²

yes

—

C₂²×C₄×C₅₃

C₄²

yes

—

C₂²×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂×C₈

yes

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂×C₈

yes

—

C₄²×C₅₃

M₁₆

yes

—

C₄²×C₅₃

#41

yes

—

C₂×#88322

—

C₂×#88322

—

C₂³×C₅₃

C₂×C₄²

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₄×C₈

yes

—

C₂×C₄×C₅₃

#97

yes

—

C₂×C₄×C₅₃

#117

yes

—

C₂×C₄×C₅₃

#117

yes

—

C₂×C₄×C₅₃

C₂²×C₈

yes

—

C₂×C₄×C₅₃

C₂×M₁₆

yes

—

C₂×C₄×C₅₃

#131

yes

—

C₈×C₅₃

—

—

—

C₂²×C₅₃

yes

C₂²×C₅₃

C₂×C₄×C₈

yes

characteristic

C₂²×C₅₃

C₂×#97

yes

characteristic

C₂²×C₅₃

#405

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#431

yes

—

C₄×C₅₃

#433

yes

—

#1744

—

C₄²×C₈

D₅₃

yes

characteristic

#1273

—

C₂×C₅₃

#1450

yes

—

C₂×C₅₃

#1452

yes

—

C₂×C₅₃

#1744

yes

characteristic

C₂×C₅₃

#1746

yes

—

C₂×C₄×C₈

—

C₂×C₄×C₈

Dic₅₃

yes

—

C₂×C₄×C₈

Dic₅₃

yes

—

C₂×C₄×C₈

C₂×D₅₃

yes

—

C₄³

C₂×D₅₃

yes

characteristic

#336

—

C₅₃

#8133

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

C₂×Dic₅₃

yes

—

C₂×C₄²

C₄×D₅₃

yes

—

C₂×C₄²

C₂×Dic₅₃

yes

characteristic

C₂×C₄²

C₂×Dic₅₃

yes

characteristic

C₂×C₄²

C₂²×D₅₃

yes

characteristic

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

106

—

C₄×C₈

#88322

yes

complemented

C₂²×C₄

C₄×Dic₅₃

yes

—

C₂²×C₄

C₄×Dic₅₃

yes

—

C₂²×C₄

C₂×C₄×D₅₃

yes

—

C₂²×C₄

C₂²×Dic₅₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₂×C₈

C₂×#88322

yes

—

C₄²

—

C₄²

C₂×#88322

yes

—

C₄²

#11719917

yes

—

C₄²

#11719945

yes

—

C₂³

C₂×C₄×Dic₅₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×#88322

yes

—

C₂×C₄

#13133128

yes

—

C₂×C₄

#13133174

yes

—

C₂×C₄

#13133176

yes

—

C₂×C₄

C₂²×#88322

yes

—

C₂×C₄

C₂×#11719917

yes

—

100

C₂×C₄

#13133223

yes

—

101

C₈

—

102

C₈

—

212

—

103

C₂²

#16810469

yes

characteristic

104

C₂²

C₂×C₄×#88322

yes

characteristic

105

C₂²

C₂×#13133128

yes

characteristic

106

C₂²

#16810578

yes

—

107

C₄

—

108

C₄

—

109

C₄

#16810676

yes

—

110

C₄

#16810682

yes

—

111

C₂

#20272998

yes

—

112

C₂

#20273007

yes

—

113

C₂

#20274461

yes

characteristic

114

C₂

#20274466

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂×C₄×C₈
Fitting	C₄²×C₈×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₄²
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₄²×C₈	D₅₃
hypocenter	C₅₃	#8133
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄² ◁ C₄²×C₈

subgroup

G/H

center, Frattini

C₂×C₄²

C₂²×D₅₃

commutator

C₂²×C₅₃

C₂×C₄×C₈

Fitting

C₄²×C₈×C₅₃

C₂

socle, prime radical

C₂³×C₅₃

C₂×C₄²

cosocle

C₂×C₄²×C₅₃

C₂³

hypercenter

C₄²×C₈

D₅₃

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₂×C₄² ◁ C₄²×C₈

N	H ≅ G/N	type
19. C₄×C₈×C₅₃	C₈	semidirect
67. C₅₃	#8133	semidirect
77. C₄×C₈	#88322	semidirect

H ≅ G/N

type

19. C₄×C₈×C₅₃

C₈

semidirect

67. C₅₃

#8133

semidirect

77. C₄×C₈

#88322

semidirect

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

subgroup	index
#20274461	2
#20272542	2
C₄²×C₈×C₅₃	2
#8133	53

subgroup

index

#20274461

#20272542

C₄²×C₈×C₅₃

#8133

Subgroups — #27274521

Conjugacy classes of subgroups

Complements

Maximal subgroups