Group #20272536 — subgroups

Subgroups — #20272536

Order 6784.

202 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 202 conjugacy classes of subgroups — too many to draw. Only its 65 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	#16810378	C₂	1	2	yes	maximal, complemented
2	C₅₃×#390	C₂	1	1	yes	maximal, characteristic
3	C₂²×#88322	—	2	2	no	—
4	C₂²×D₄×C₅₃	C₄	1	1	yes	characteristic
5	C₂×C₄⋊C₄×C₅₃	C₄	1	1	yes	characteristic
6	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
7	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
8	C₂⁴×C₅₃	—	2	1	no	—
9	C₂×D₄×C₅₃	—	2	1	no	—
10	C₂×D₄×C₅₃	—	4	1	no	—
11	C₂×D₄×C₅₃	—	4	1	no	—
12	C₂×D₄×C₅₃	C₈	1	2	yes	complemented
13	C₂²×C₄×C₅₃	—	2	1	no	—
14	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₄×C₅₃	D₄	1	2	yes	—
16	C₄⋊C₄×C₅₃	—	4	1	no	—
17	C₂²⋊C₄×C₅₃	—	4	2	no	—
18	C₂²⋊C₄×C₅₃	—	4	2	no	—
19	C₄²×C₅₃	—	4	1	no	—
20	C₂×#88322	—	2	4	no	—
21	C₂×#88322	—	4	2	no	—
22	C₂×#88322	—	4	2	no	—
23	C₂³×C₅₃	—	2	2	no	—
24	C₂³×C₅₃	—	4	1	no	—
25	C₂³×C₅₃	—	4	1	no	—
26	C₂³×C₅₃	—	4	2	no	—
27	C₂³×C₅₃	—	4	2	no	—
28	C₂³×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
29	D₄×C₅₃	—	4	2	no	—
30	D₄×C₅₃	—	4	2	no	—
31	C₂×C₄×C₅₃	—	2	1	no	—
32	C₂×C₄×C₅₃	—	2	2	no	—
33	C₂×C₄×C₅₃	—	2	1	no	—
34	C₂×C₄×C₅₃	—	2	2	no	—
35	C₂×C₄×C₅₃	—	4	1	no	—
36	C₂×C₄×C₅₃	—	4	2	no	—
37	C₂×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
38	C₂×C₄×C₅₃	M₁₆	1	1	yes	characteristic
39	C₂×C₄×C₅₃	D₈	1	2	yes	—
40	C₂×C₄×C₅₃	SD₁₆	1	2	yes	—
41	#88322	—	4	4	no	—
42	C₂²×C₅₃	—	4	2	no	—
43	C₂²×C₅₃	—	4	4	no	—
44	C₂²×C₅₃	—	8	2	no	—
45	C₂²×C₅₃	—	8	1	no	—
46	C₂²×C₅₃	—	8	1	no	—
47	C₂²×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
48	C₂²×C₅₃	#99	1	1	yes	characteristic
49	C₂²×C₅₃	#100	1	1	yes	characteristic
50	C₂²×C₅₃	#102	1	2	yes	—
51	C₂²×C₅₃	#104	1	2	yes	—
52	C₄×C₅₃	—	2	1	no	—
53	C₄×C₅₃	—	2	1	no	—
54	C₄×C₅₃	—	2	2	no	—
55	C₄×C₅₃	—	2	2	no	—
56	C₄×C₅₃	—	8	1	no	—
57	#1270	—	53	1	no	maximal, Sylow
58	C₂×C₅₃	—	8	2	no	—
59	C₂×C₅₃	#323	1	1	yes	characteristic
60	C₂×C₅₃	D₄⋊C₈	1	2	yes	—
61	C₂×C₅₃	#327	1	2	yes	—
62	C₂×C₅₃	#329	1	1	yes	characteristic
63	C₂×C₅₃	#331	1	1	yes	characteristic
64	#390	D₅₃	1	1	yes	characteristic
65	#336	—	53	2	no	—
66	C₅₃	#1270	1	1	yes	characteristic, complemented, Sylow
67	C₂²×D₄	Dic₅₃	1	1	yes	characteristic
68	C₂²×C₈	—	106	2	no	—
69	C₂×C₄⋊C₄	Dic₅₃	1	1	yes	characteristic
70	C₂×C₂²⋊C₄	—	2	2	no	—
71	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
72	C₂⁴	—	2	1	no	—
73	C₂×D₄	—	2	1	no	—
74	C₂×D₄	—	4	1	no	—
75	C₂×D₄	—	4	1	no	—
76	C₂×D₄	#88322	1	2	yes	complemented
77	C₂²×C₄	—	2	1	no	—
78	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
79	C₂²×C₄	C₂²⋊D₅₃	1	2	yes	—
80	C₂×C₈	—	106	4	no	—
81	C₂×C₈	—	212	2	no	—
82	C₂×C₈	—	212	2	no	—
83	C₄⋊C₄	—	4	1	no	—
84	C₂²⋊C₄	—	4	2	no	—
85	C₂²⋊C₄	—	4	2	no	—
86	C₄²	—	4	1	no	—
87	C₂³	—	2	2	no	—
88	C₂³	—	4	1	no	—
89	C₂³	—	4	1	no	—
90	C₂³	—	4	2	no	—
91	C₂³	—	4	2	no	—
92	C₂³	C₂²⋊Dic₅₃	1	1	yes	characteristic
93	D₄	—	4	2	no	—
94	D₄	—	4	2	no	—
95	C₂×C₄	—	2	2	no	—
96	C₂×C₄	—	2	1	no	—
97	C₂×C₄	—	2	1	no	—
98	C₂×C₄	—	2	2	no	—
99	C₂×C₄	—	4	1	no	—
100	C₂×C₄	—	4	2	no	—
101	C₂×C₄	C₂×#88322	1	1	yes	characteristic
102	C₂×C₄	#11719917	1	1	yes	characteristic
103	C₂×C₄	#11719922	1	2	yes	—
104	C₂×C₄	#11719923	1	2	yes	—
105	C₈	—	212	4	no	—
106	C₂²	—	4	2	no	—
107	C₂²	—	4	4	no	—
108	C₂²	—	8	2	no	—
109	C₂²	—	8	1	no	—
110	C₂²	—	8	1	no	—
111	C₂²	C₂²⋊#88322	1	1	yes	characteristic
112	C₂²	#13133132	1	1	yes	characteristic
113	C₂²	#13133134	1	1	yes	characteristic
114	C₂²	#13133138	1	2	yes	—
115	C₂²	#13133144	1	2	yes	—
116	C₄	—	2	1	no	—
117	C₄	—	2	1	no	—
118	C₄	—	2	2	no	—
119	C₄	—	2	2	no	—
120	C₄	—	8	1	no	—
121	C₂	—	8	2	no	—
122	C₂	#16810347	1	1	yes	characteristic
123	C₂	D₄⋊#88322	1	2	yes	—
124	C₂	#16810357	1	2	yes	—
125	C₂	#16810363	1	1	yes	characteristic
126	C₂	#16810367	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810378

C₂

yes

maximal, complemented

C₅₃×#390

C₂

yes

maximal, characteristic

C₂²×#88322

—

C₂²×D₄×C₅₃

C₄

yes

characteristic

C₂×C₄⋊C₄×C₅₃

C₄

yes

characteristic

C₂×C₂²⋊C₄×C₅₃

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂⁴×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

C₈

yes

complemented

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

—

C₄⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₂×#88322

—

C₂×#88322

—

C₂×#88322

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

C₂²⋊C₄

yes

characteristic

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×C₈

yes

characteristic

C₂×C₄×C₅₃

M₁₆

yes

characteristic

C₂×C₄×C₅₃

D₈

yes

—

C₂×C₄×C₅₃

SD₁₆

yes

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂²⋊C₈

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

—

C₂²×C₅₃

#104

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

#1270

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

#323

yes

characteristic

C₂×C₅₃

D₄⋊C₈

yes

—

C₂×C₅₃

#327

yes

—

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

yes

characteristic

#336

—

C₅₃

#1270

yes

characteristic, complemented, Sylow

C₂²×D₄

Dic₅₃

yes

characteristic

C₂²×C₈

—

106

—

C₂×C₄⋊C₄

Dic₅₃

yes

characteristic

C₂×C₂²⋊C₄

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

#88322

yes

complemented

C₂²×C₄

—

C₂²×C₄

C₂×Dic₅₃

yes

characteristic

C₂²×C₄

C₂²⋊D₅₃

yes

—

C₂×C₈

—

106

—

C₂×C₈

—

212

—

C₂×C₈

—

212

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂²⋊Dic₅₃

yes

characteristic

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

C₂×#88322

yes

characteristic

102

C₂×C₄

#11719917

yes

characteristic

103

C₂×C₄

#11719922

yes

—

104

C₂×C₄

#11719923

yes

—

105

C₈

—

212

—

106

C₂²

—

107

C₂²

—

108

C₂²

—

109

C₂²

—

110

C₂²

—

111

C₂²

C₂²⋊#88322

yes

characteristic

112

C₂²

#13133132

yes

characteristic

113

C₂²

#13133134

yes

characteristic

114

C₂²

#13133138

yes

—

115

C₂²

#13133144

yes

—

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₂

—

122

C₂

#16810347

yes

characteristic

123

C₂

D₄⋊#88322

yes

—

124

C₂

#16810357

yes

—

125

C₂

#16810363

yes

characteristic

126

C₂

#16810367

yes

characteristic

	subgroup	G/H
center	C₂³	C₂²⋊Dic₅₃
commutator	C₂×C₄×C₅₃	C₂×C₈
Frattini	C₂×C₄²	C₂×D₅₃
Fitting	C₅₃×#390	C₂
socle	C₂³×C₅₃	C₂²⋊C₄
hypercenter	#390	D₅₃
hypocenter	C₅₃	#1270
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #390

subgroup

G/H

center

C₂³

C₂²⋊Dic₅₃

commutator

C₂×C₄×C₅₃

C₂×C₈

Frattini

C₂×C₄²

C₂×D₅₃

Fitting

C₅₃×#390

C₂

socle

C₂³×C₅₃

C₂²⋊C₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #390

N	H ≅ G/N	type
1. #16810378	C₂	semidirect
12. C₂×D₄×C₅₃	C₈	semidirect
66. C₅₃	#1270	semidirect
76. C₂×D₄	#88322	semidirect

H ≅ G/N

type

1. #16810378

C₂

semidirect

12. C₂×D₄×C₅₃

C₈

semidirect

66. C₅₃

#1270

semidirect

76. C₂×D₄

#88322

semidirect

Subgroups — #20272536

Conjugacy classes of subgroups

Complements