Group #20275866 — subgroups

Subgroups — #20275866

Order 6784.

258 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 85 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×#13133209	C₂	1	1	yes	maximal, characteristic
2	C₂×#13133141	C₂	1	2	yes	maximal
3	#16810385	C₂	1	2	yes	maximal
4	#16810378	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#391	C₂	1	1	yes	maximal, characteristic
6	C₂²×#88322	C₂²	1	2	yes	—
7	#13133209	—	2	4	no	—
8	C₂×C₄⋊Dic₅₃	C₂²	1	2	yes	—
9	C₂×#11719919	—	2	2	no	—
10	#13133141	—	2	4	no	—
11	C₂²×Q₈×C₅₃	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
14	#95×C₅₃	—	2	2	no	—
15	C₂×Q₈×C₅₃	—	2	1	no	—
16	C₂×Q₈×C₅₃	—	2	2	no	—
17	C₂×Q₈×C₅₃	—	2	2	no	—
18	C₂×Q₈×C₅₃	Q₈	1	2	yes	—
19	C₂²×C₄×C₅₃	—	2	1	no	—
20	C₂²×C₄×C₅₃	—	2	1	no	—
21	C₂²×C₄×C₅₃	D₄	1	2	yes	—
22	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
23	C₂²×Dic₅₃	—	2	2	no	—
24	C₄⋊C₄×C₅₃	—	2	2	no	—
25	C₄²×C₅₃	—	2	2	no	—
26	C₄⋊Dic₅₃	—	2	2	no	—
27	C₄⋊Dic₅₃	D₄	1	4	yes	—
28	#11719919	—	2	8	no	—
29	C₂×#88322	—	2	2	no	—
30	C₂×#88322	—	2	2	no	—
31	C₂×#88322	—	2	2	no	—
32	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
33	Q₈×C₅₃	—	2	4	no	—
34	Q₈×C₅₃	—	4	2	no	—
35	C₂×C₄×C₅₃	—	2	2	no	—
36	C₂×C₄×C₅₃	—	2	2	no	—
37	C₂×C₄×C₅₃	—	2	2	no	—
38	C₂×C₄×C₅₃	—	2	2	no	—
39	C₂×C₄×C₅₃	—	4	1	no	—
40	C₂×C₄×C₅₃	—	4	1	no	—
41	C₂×C₄×C₅₃	—	4	1	no	—
42	C₂×C₄×C₅₃	—	4	2	no	—
43	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
46	C₂×C₄×C₅₃	#41	1	2	yes	—
47	C₂×Dic₅₃	—	2	4	no	—
48	C₂×Dic₅₃	—	4	2	no	—
49	C₂×Dic₅₃	—	4	2	no	—
50	#88322	—	4	2	no	—
51	C₂²×C₅₃	#120	1	1	yes	characteristic
52	C₂²×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
53	C₂²×C₅₃	#124	1	1	yes	characteristic
54	C₂²×C₅₃	#135	1	2	yes	—
55	C₂²×C₅₃	#137	1	2	yes	—
56	C₄×C₅₃	—	4	2	no	—
57	C₄×C₅₃	—	4	2	no	—
58	C₄×C₅₃	—	8	1	no	—
59	C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
60	C₄×C₅₃	#124	1	2	yes	—
61	Dic₅₃	—	4	4	no	—
62	#2026	—	53	1	no	maximal, Sylow
63	C₂×C₅₃	#393	1	1	yes	characteristic
64	C₂×C₅₃	#452	1	2	yes	—
65	C₂×C₅₃	#479	1	2	yes	—
66	C₂×C₅₃	#488	1	1	yes	characteristic
67	C₂×C₅₃	#491	1	1	yes	characteristic
68	C₂×#125	—	53	1	no	—
69	C₂×#103	—	53	2	no	—
70	#391	D₅₃	1	1	yes	characteristic
71	#340	—	53	2	no	—
72	#336	—	53	1	no	—
73	C₅₃	#2026	1	1	yes	characteristic, complemented, Sylow
74	C₂²×Q₈	C₂×D₅₃	1	1	yes	characteristic
75	C₂²×C₈	—	53	2	no	—
76	#125	—	106	4	no	—
77	C₂×C₄⋊C₄	—	53	2	no	—
78	C₂×C₄⋊C₄	—	106	2	no	—
79	C₂×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
80	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
81	#103	—	106	4	no	—
82	#95	—	2	2	no	—
83	C₂×Q₈	—	2	1	no	—
84	C₂×Q₈	—	2	2	no	—
85	C₂×Q₈	—	2	2	no	—
86	C₂×Q₈	Dic₁₀₆	1	2	yes	—
87	C₂²×C₄	—	2	1	no	—
88	C₂²×C₄	—	2	1	no	—
89	C₂²×C₄	—	106	2	no	—
90	C₂²×C₄	C₂²⋊D₅₃	1	2	yes	—
91	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
92	C₂×C₈	—	106	2	no	—
93	C₂×C₈	—	106	2	no	—
94	C₂×C₈	—	106	2	no	—
95	C₄⋊C₄	—	2	2	no	—
96	C₄⋊C₄	—	53	4	no	—
97	C₄⋊C₄	—	106	8	no	—
98	C₄⋊C₄	—	106	2	no	—
99	C₄²	—	2	2	no	—
100	C₂³	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
101	Q₈	—	2	4	no	—
102	Q₈	—	4	2	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	4	1	no	—
108	C₂×C₄	—	4	1	no	—
109	C₂×C₄	—	4	1	no	—
110	C₂×C₄	—	4	2	no	—
111	C₂×C₄	—	106	4	no	—
112	C₂×C₄	—	212	2	no	—
113	C₂×C₄	—	212	2	no	—
114	C₂×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
115	C₂×C₄	#11719945	1	1	yes	characteristic
116	C₂×C₄	D₄×D₅₃	1	2	yes	—
117	C₂×C₄	#11719947	1	2	yes	—
118	C₈	—	212	2	no	—
119	C₂²	#13133185	1	1	yes	characteristic
120	C₂²	#13133196	1	1	yes	characteristic
121	C₂²	#13133204	1	1	yes	characteristic
122	C₂²	#13133237	1	2	yes	—
123	C₂²	#13133248	1	2	yes	—
124	C₄	—	4	2	no	—
125	C₄	—	4	2	no	—
126	C₄	—	8	1	no	—
127	C₄	—	212	4	no	—
128	C₄	#13133194	1	2	yes	—
129	C₄	#13133203	1	2	yes	—
130	C₂	#16810535	1	1	yes	characteristic
131	C₂	#16810783	1	2	yes	—
132	C₂	#16810930	1	2	yes	—
133	C₂	#16810977	1	1	yes	characteristic
134	C₂	#16810991	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133209

C₂

yes

maximal, characteristic

C₂×#13133141

C₂

yes

maximal

#16810385

C₂

yes

maximal

#16810378

C₂

yes

maximal, characteristic

C₅₃×#391

C₂

yes

maximal, characteristic

C₂²×#88322

C₂²

yes

—

#13133209

—

C₂×C₄⋊Dic₅₃

C₂²

yes

—

C₂×#11719919

—

#13133141

—

C₂²×Q₈×C₅₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₅₃

C₂²

yes

characteristic

C₂×C₄²×C₅₃

C₂²

yes

characteristic

#95×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

Q₈

yes

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

D₄

yes

—

—

—

—

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

Q₈×C₅₃

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×Q₈

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

#88322

—

C₂²×C₅₃

#120

yes

characteristic

C₂²×C₅₃

C₂²⋊Q₈

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

—

C₂²×C₅₃

#137

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂²⋊Q₈

yes

—

C₄×C₅₃

#124

yes

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

—

C₂×C₅₃

#479

yes

—

C₂×C₅₃

yes

C₂×C₅₃

yes

—

—

D₅₃

yes

characteristic

#340

—

#336

—

C₅₃

#2026

yes

characteristic, complemented, Sylow

C₂²×Q₈

C₂×D₅₃

yes

characteristic

C₂²×C₈

—

#125

—

106

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

106

—

C₂×C₄⋊C₄

C₂×D₅₃

yes

characteristic

C₂×C₄²

C₂×D₅₃

yes

characteristic

#103

—

106

—

#95

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

Dic₁₀₆

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

106

—

C₂²×C₄

C₂²⋊D₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

106

—

C₄⋊C₄

—

106

—

C₄²

—

100

C₂³

C₂×C₂²⋊D₅₃

yes

characteristic

101

Q₈

—

102

Q₈

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

106

—

112

C₂×C₄

—

212

—

113

C₂×C₄

—

212

—

114

C₂×C₄

C₂×Dic₁₀₆

yes

characteristic

115

C₂×C₄

#11719945

yes

characteristic

116

C₂×C₄

D₄×D₅₃

yes

—

117

C₂×C₄

#11719947

yes

—

118

C₈

—

212

—

119

C₂²

#13133185

yes

characteristic

120

C₂²

#13133196

yes

characteristic

121

C₂²

#13133204

yes

characteristic

122

C₂²

#13133237

yes

—

123

C₂²

#13133248

yes

—

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

—

212

—

128

C₄

#13133194

yes

—

129

C₄

#13133203

yes

—

130

C₂

#16810535

yes

characteristic

131

C₂

#16810783

yes

—

132

C₂

#16810930

yes

—

133

C₂

#16810977

yes

characteristic

134

C₂

#16810991

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₅₃
commutator	C₂²×C₄×C₅₃	C₂³
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₅₃×#391	C₂
socle	C₂³×C₅₃	C₂×D₄
hypercenter	#391	D₅₃
hypocenter	C₅₃	#2026
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #391

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₅₃

commutator

C₂²×C₄×C₅₃

C₂³

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₅₃×#391

C₂

socle

C₂³×C₅₃

C₂×D₄

D₅₃

C₅₃

G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #391

N	H ≅ G/N	type
73. C₅₃	#2026	semidirect

H ≅ G/N

type

73. C₅₃

#2026

semidirect

Subgroups — #20275866

Conjugacy classes of subgroups

Complements