Group #20274459 — subgroups

Subgroups — #20274459

Order 6784.

296 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 149 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×C₈×Dic₅₃	C₂	1	2	yes	maximal
2	C₄²×Dic₅₃	C₂	1	1	yes	maximal, characteristic
3	#16810378	C₂	1	1	yes	maximal, characteristic
4	#16810377	C₂	1	2	yes	maximal
5	C₅₃×#336	C₂	1	1	yes	maximal, characteristic
6	C₂²×#88322	C₂²	1	2	yes	—
7	C₂×C₄×Dic₅₃	C₄	1	2	yes	—
8	C₂×C₄×Dic₅₃	C₄	1	2	yes	—
9	C₂×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
10	C₂×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
11	C₈×Dic₅₃	C₄	1	8	yes	complemented
12	C₂²×C₈×C₅₃	C₂²	1	2	yes	—
13	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
14	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
15	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
16	C₂²×Dic₅₃	C₂×C₄	1	2	yes	—
17	C₂²×Dic₅₃	C₂×C₄	1	1	yes	characteristic
18	C₂²×Dic₅₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₈×C₅₃	—	2	2	no	—
20	C₂×C₈×C₅₃	—	2	2	no	—
21	C₂×C₈×C₅₃	C₂×C₄	1	4	yes	—
22	C₄²×C₅₃	—	2	2	no	—
23	C₄×Dic₅₃	—	2	1	no	—
24	C₄×Dic₅₃	—	2	1	no	—
25	C₄×Dic₅₃	—	2	4	no	—
26	C₄×Dic₅₃	C₈	1	8	yes	complemented
27	C₄×Dic₅₃	C₂×C₄	1	4	yes	—
28	C₂×#88322	—	2	2	no	—
29	C₂×#88322	—	2	2	no	—
30	C₂×#88322	C₂×C₄	1	4	yes	—
31	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
32	C₂×C₄×C₅₃	—	2	2	no	—
33	C₂×C₄×C₅₃	—	2	2	no	—
34	C₂×C₄×C₅₃	C₂×C₈	1	4	yes	—
35	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
36	C₂×C₄×C₅₃	#41	1	2	yes	—
37	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
38	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
39	C₂×Dic₅₃	—	2	2	no	—
40	C₂×Dic₅₃	—	2	2	no	—
41	C₂×Dic₅₃	C₄²	1	2	yes	—
42	C₂×Dic₅₃	C₄²	1	2	yes	—
43	C₂×Dic₅₃	C₂×C₈	1	4	yes	—
44	C₂×Dic₅₃	C₂×C₈	1	4	yes	—
45	C₂×Dic₅₃	M₁₆	1	4	yes	—
46	C₈×C₅₃	—	2	4	no	—
47	#88322	—	2	4	no	—
48	C₂²×C₅₃	C₂×C₄²	1	1	yes	characteristic
49	C₂²×C₅₃	#117	1	1	yes	characteristic
50	C₂²×C₅₃	#117	1	1	yes	characteristic
51	C₂²×C₅₃	C₂²×C₈	1	2	yes	—
52	C₂²×C₅₃	C₂×M₁₆	1	2	yes	—
53	C₄×C₅₃	—	2	4	no	—
54	C₄×C₅₃	#117	1	2	yes	—
55	C₄×C₅₃	#117	1	2	yes	—
56	Dic₅₃	—	2	4	no	—
57	Dic₅₃	C₄×C₈	1	4	yes	—
58	Dic₅₃	#97	1	4	yes	—
59	#1744	—	53	1	no	maximal, Sylow
60	C₂×C₅₃	#376	1	1	yes	characteristic
61	C₂×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
62	C₂×C₅₃	C₂×#97	1	1	yes	characteristic
63	C₂×C₅₃	#431	1	2	yes	—
64	C₂×C₅₃	#431	1	2	yes	—
65	C₂×C₄×C₈	—	53	2	no	—
66	C₄³	—	53	1	no	—
67	#336	—	53	2	no	—
68	#336	—	53	1	no	—
69	#336	D₅₃	1	1	yes	characteristic
70	C₅₃	#1744	1	1	yes	characteristic, complemented, Sylow
71	C₂²×C₈	—	53	2	no	—
72	C₂²×C₈	C₂×D₅₃	1	2	yes	—
73	C₂×C₄²	—	53	2	no	—
74	C₂×C₄²	—	53	1	no	—
75	C₂×C₄²	—	53	2	no	—
76	C₂×C₄²	—	53	1	no	—
77	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
78	C₄×C₈	—	53	8	no	—
79	C₂²×C₄	—	53	1	no	—
80	C₂²×C₄	—	53	1	no	—
81	C₂²×C₄	—	53	2	no	—
82	C₂²×C₄	C₄×D₅₃	1	2	yes	—
83	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
84	C₂×C₈	—	2	2	no	—
85	C₂×C₈	—	2	2	no	—
86	C₂×C₈	—	53	4	no	—
87	C₂×C₈	—	106	2	no	—
88	C₂×C₈	—	106	2	no	—
89	C₂×C₈	C₄×D₅₃	1	4	yes	—
90	C₄²	—	2	2	no	—
91	C₄²	—	53	4	no	—
92	C₄²	—	53	8	no	—
93	C₄²	—	106	4	no	—
94	C₄²	—	106	1	no	—
95	C₄²	—	106	1	no	—
96	C₂³	C₂×C₄×D₅₃	1	1	yes	characteristic
97	C₂×C₄	—	2	2	no	—
98	C₂×C₄	—	2	2	no	—
99	C₂×C₄	—	53	2	no	—
100	C₂×C₄	—	53	2	no	—
101	C₂×C₄	—	53	4	no	—
102	C₂×C₄	—	53	4	no	—
103	C₂×C₄	—	53	4	no	—
104	C₂×C₄	—	106	2	no	—
105	C₂×C₄	—	106	2	no	—
106	C₂×C₄	C₈×D₅₃	1	4	yes	—
107	C₂×C₄	C₂×C₄×D₅₃	1	2	yes	—
108	C₂×C₄	#11719947	1	1	yes	characteristic
109	C₂×C₄	#11719947	1	2	yes	—
110	C₂×C₄	#11719949	1	1	yes	characteristic
111	C₈	—	2	4	no	—
112	C₈	—	106	4	no	—
113	C₂²	C₄²×D₅₃	1	1	yes	characteristic
114	C₂²	#13133173	1	1	yes	characteristic
115	C₂²	#13133175	1	1	yes	characteristic
116	C₂²	C₂×C₈×D₅₃	1	2	yes	—
117	C₂²	M₁₆×D₅₃	1	2	yes	—
118	C₄	—	2	4	no	—
119	C₄	—	53	4	no	—
120	C₄	—	53	4	no	—
121	C₄	—	106	4	no	—
122	C₄	#13133173	1	2	yes	—
123	C₄	#13133175	1	2	yes	—
124	C₂	#16810468	1	1	yes	characteristic
125	C₂	C₄×C₈×D₅₃	1	1	yes	characteristic
126	C₂	#97×D₅₃	1	1	yes	characteristic
127	C₂	#16810673	1	2	yes	—
128	C₂	#16810675	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₈×Dic₅₃

C₂

yes

maximal

C₄²×Dic₅₃

C₂

yes

maximal, characteristic

#16810378

C₂

yes

maximal, characteristic

#16810377

C₂

yes

maximal

C₅₃×#336

C₂

yes

maximal, characteristic

C₂²×#88322

C₂²

yes

—

C₂×C₄×Dic₅₃

C₄

yes

—

C₂×C₄×Dic₅₃

C₄

yes

—

C₂×C₄×Dic₅₃

C₂²

yes

characteristic

C₂×C₄×Dic₅₃

C₂²

yes

characteristic

C₈×Dic₅₃

C₄

yes

complemented

C₂²×C₈×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

C₂×C₄

yes

—

C₂²×Dic₅₃

C₂×C₄

yes

characteristic

C₂²×Dic₅₃

C₂×C₄

yes

characteristic

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₄×Dic₅₃

C₈

yes

complemented

C₄×Dic₅₃

C₂×C₄

yes

—

C₂×#88322

—

C₂×#88322

—

C₂×#88322

C₂×C₄

yes

—

C₂³×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×C₈

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

#41

yes

—

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

C₄²

yes

—

C₂×Dic₅₃

C₄²

yes

—

C₂×Dic₅₃

C₂×C₈

yes

—

C₂×Dic₅₃

C₂×C₈

yes

—

C₂×Dic₅₃

M₁₆

yes

—

C₈×C₅₃

—

#88322

—

C₂²×C₅₃

C₂×C₄²

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

C₂²×C₈

yes

—

C₂²×C₅₃

C₂×M₁₆

yes

—

C₄×C₅₃

—

C₄×C₅₃

#117

yes

—

C₄×C₅₃

#117

yes

—

Dic₅₃

—

Dic₅₃

C₄×C₈

yes

—

Dic₅₃

#97

yes

—

—

C₂×C₅₃

yes

C₂×C₅₃

C₂×C₄×C₈

yes

characteristic

C₂×C₅₃

C₂×#97

yes

characteristic

C₂×C₅₃

#431

yes

—

C₂×C₅₃

#431

yes

—

C₂×C₄×C₈

—

C₄³

—

#336

—

#336

—

#336

D₅₃

yes

characteristic

C₅₃

#1744

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

C₂×D₅₃

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₄×C₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₄×D₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₂×C₈

C₄×D₅₃

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

106

—

C₄²

—

106

—

C₄²

—

106

—

C₂³

C₂×C₄×D₅₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

106

—

105

C₂×C₄

—

106

—

106

C₂×C₄

C₈×D₅₃

yes

—

107

C₂×C₄

C₂×C₄×D₅₃

yes

—

108

C₂×C₄

#11719947

yes

characteristic

109

C₂×C₄

#11719947

yes

—

110

C₂×C₄

#11719949

yes

characteristic

111

C₈

—

112

C₈

—

106

—

113

C₂²

C₄²×D₅₃

yes

characteristic

114

C₂²

#13133173

yes

characteristic

115

C₂²

#13133175

yes

characteristic

116

C₂²

C₂×C₈×D₅₃

yes

—

117

C₂²

M₁₆×D₅₃

yes

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

—

106

—

122

C₄

#13133173

yes

—

123

C₄

#13133175

yes

—

124

C₂

#16810468

yes

characteristic

125

C₂

C₄×C₈×D₅₃

yes

characteristic

126

C₂

#97×D₅₃

yes

characteristic

127

C₂

#16810673

yes

—

128

C₂

#16810675

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₅₃
commutator	C₂×C₅₃	C₂×C₄×C₈
Fitting	C₅₃×#336	C₂
socle	C₂³×C₅₃	C₂²×C₄
hypercenter	#336	D₅₃
hypocenter	C₅₃	#1744
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ #336

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₅₃

commutator

C₂×C₅₃

C₂×C₄×C₈

Fitting

C₅₃×#336

C₂

socle

C₂³×C₅₃

C₂²×C₄

D₅₃

C₅₃

G ▷ C₂×C₅₃ ▷ C₅₃

C₁ ◁ C₂²×C₄ ◁ #336

N	H ≅ G/N	type
11. C₈×Dic₅₃	C₄	semidirect
26. C₄×Dic₅₃	C₈	semidirect
70. C₅₃	#1744	semidirect

H ≅ G/N

type

11. C₈×Dic₅₃

C₄

semidirect

26. C₄×Dic₅₃

C₈

semidirect

70. C₅₃

#1744

semidirect

Subgroups — #20274459

Conjugacy classes of subgroups

Complements