Group #20272837 — subgroups

Subgroups — #20272837

Order 6784.

158 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 158 conjugacy classes of subgroups — too many to draw. Only its 53 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	#16810607	C₂	1	1	yes	maximal, characteristic
2	#16810590	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#413	C₂	1	1	yes	maximal, characteristic
4	C₂×#11719917	—	2	1	no	—
5	C₂×#11719917	C₄	1	1	yes	characteristic
6	C₂²×#88322	C₄	1	1	yes	characteristic
7	#13133136	—	2	1	no	—
8	#13133134	C₄	1	2	yes	—
9	C₂²⋊#88322	—	2	1	no	—
10	C₂²⋊#88322	—	2	1	no	—
11	C₂×#41×C₅₃	C₂²	1	1	yes	characteristic
12	C₂×M₁₆×C₅₃	—	2	1	no	—
13	#101×C₅₃	—	2	1	no	—
14	#100×C₅₃	C₄	1	2	yes	—
15	#41×C₅₃	—	4	1	no	—
16	#41×C₅₃	—	4	1	no	—
17	C₂×Q₈×C₅₃	D₄	1	1	yes	characteristic
18	C₂×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
19	C₂×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
20	C₂×D₄×C₅₃	Q₈	1	1	yes	characteristic
21	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
23	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
24	M₁₆×C₅₃	—	2	2	no	—
25	M₁₆×C₅₃	—	4	1	no	—
26	C₂×C₈×C₅₃	—	4	1	no	—
27	#11719917	—	2	2	no	—
28	#11719917	—	4	1	no	—
29	#11719917	—	4	1	no	—
30	C₂×#88322	—	2	1	no	—
31	C₂×#88322	—	2	1	no	—
32	C₂×#88322	—	4	1	no	—
33	C₂×#88322	—	4	1	no	—
34	C₂³×C₅₃	C₄²	1	1	yes	characteristic
35	C₂³×C₅₃	C₄⋊C₄	1	1	yes	characteristic
36	C₂³×C₅₃	C₄⋊C₄	1	1	yes	characteristic
37	Q₈×C₅₃	—	4	1	no	—
38	D₄×C₅₃	—	4	1	no	—
39	D₄×C₅₃	—	4	1	no	—
40	D₄×C₅₃	—	4	1	no	—
41	C₂×C₄×C₅₃	—	2	1	no	—
42	C₂×C₄×C₅₃	—	2	1	no	—
43	C₂×C₄×C₅₃	—	2	1	no	—
44	C₂×C₄×C₅₃	—	2	1	no	—
45	C₂×C₄×C₅₃	—	2	1	no	—
46	C₂×C₄×C₅₃	—	2	1	no	—
47	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
51	C₈×C₅₃	—	4	2	no	—
52	#88322	—	4	2	no	—
53	#88322	—	4	1	no	—
54	#88322	—	4	1	no	—
55	C₂²×C₅₃	—	2	1	no	—
56	C₂²×C₅₃	—	2	1	no	—
57	C₂²×C₅₃	—	2	1	no	—
58	C₂²×C₅₃	—	4	1	no	—
59	C₂²×C₅₃	—	4	1	no	—
60	C₂²×C₅₃	—	4	1	no	—
61	C₂²×C₅₃	#95	1	1	yes	characteristic
62	C₄×C₅₃	—	2	1	no	—
63	C₄×C₅₃	—	2	1	no	—
64	C₄×C₅₃	—	2	1	no	—
65	C₄×C₅₃	#99	1	1	yes	characteristic
66	C₄×C₅₃	#99	1	1	yes	characteristic
67	#1392	—	53	1	no	maximal, Sylow
68	C₂×C₅₃	—	2	1	no	—
69	C₂×C₅₃	—	4	1	no	—
70	C₂×C₅₃	—	4	1	no	—
71	C₂×C₅₃	—	4	1	no	—
72	C₂×C₅₃	#342	1	1	yes	characteristic
73	#413	—	53	1	no	—
74	#413	D₅₃	1	1	yes	characteristic
75	#408	—	53	1	no	—
76	C₅₃	#1392	1	1	yes	characteristic, complemented, Sylow
77	C₂×#41	C₂×D₅₃	1	1	yes	characteristic
78	C₂×M₁₆	—	2	1	no	—
79	C₂×M₁₆	—	53	1	no	—
80	C₂×M₁₆	—	106	1	no	—
81	C₂²×C₈	—	53	1	no	—
82	#101	—	2	1	no	—
83	#101	—	106	1	no	—
84	#100	—	53	2	no	—
85	#100	Dic₅₃	1	2	yes	—
86	C₂²⋊C₈	—	106	1	no	—
87	C₂²⋊C₈	—	106	1	no	—
88	#41	—	4	1	no	—
89	#41	—	4	1	no	—
90	C₂×Q₈	C₂²⋊D₅₃	1	1	yes	characteristic
91	C₂×D₄	Dic₁₀₆	1	1	yes	characteristic
92	C₂×D₄	C₄×D₅₃	1	1	yes	characteristic
93	C₂×D₄	C₂×Dic₅₃	1	1	yes	characteristic
94	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
95	C₂²×C₄	D₂₁₂	1	1	yes	characteristic
96	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
97	M₁₆	—	2	2	no	—
98	M₁₆	—	4	1	no	—
99	M₁₆	—	106	2	no	—
100	M₁₆	—	212	1	no	—
101	M₁₆	—	212	1	no	—
102	C₂×C₈	—	4	1	no	—
103	C₂×C₈	—	106	1	no	—
104	C₂×C₈	—	106	1	no	—
105	C₂×C₈	—	212	1	no	—
106	C₂×C₈	—	212	1	no	—
107	C₂³	C₄×Dic₅₃	1	1	yes	characteristic
108	C₂³	#11719919	1	1	yes	characteristic
109	C₂³	C₄⋊Dic₅₃	1	1	yes	characteristic
110	Q₈	—	4	1	no	—
111	D₄	—	4	1	no	—
112	D₄	—	4	1	no	—
113	D₄	—	4	1	no	—
114	C₂×C₄	—	2	1	no	—
115	C₂×C₄	—	2	1	no	—
116	C₂×C₄	—	2	1	no	—
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	—	2	1	no	—
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	#11719919	1	1	yes	characteristic
121	C₂×C₄	#11719921	1	1	yes	characteristic
122	C₂×C₄	#11719921	1	1	yes	characteristic
123	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
124	C₈	—	4	2	no	—
125	C₈	—	212	2	no	—
126	C₈	—	212	1	no	—
127	C₈	—	212	1	no	—
128	C₂²	—	2	1	no	—
129	C₂²	—	2	1	no	—
130	C₂²	—	2	1	no	—
131	C₂²	—	4	1	no	—
132	C₂²	—	4	1	no	—
133	C₂²	—	4	1	no	—
134	C₂²	#13133124	1	1	yes	characteristic
135	C₄	—	2	1	no	—
136	C₄	—	2	1	no	—
137	C₄	—	2	1	no	—
138	C₄	#13133131	1	1	yes	characteristic
139	C₄	#13133131	1	1	yes	characteristic
140	C₂	—	2	1	no	—
141	C₂	—	4	1	no	—
142	C₂	—	4	1	no	—
143	C₂	—	4	1	no	—
144	C₂	#16810389	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810607

C₂

yes

maximal, characteristic

#16810590

C₂

yes

maximal, characteristic

C₅₃×#413

C₂

yes

maximal, characteristic

C₂×#11719917

—

C₂×#11719917

C₄

yes

characteristic

C₂²×#88322

C₄

yes

characteristic

#13133136

—

#13133134

C₄

yes

—

C₂²⋊#88322

—

C₂²⋊#88322

—

C₂×#41×C₅₃

C₂²

yes

characteristic

C₂×M₁₆×C₅₃

—

#101×C₅₃

—

#100×C₅₃

C₄

yes

—

#41×C₅₃

—

#41×C₅₃

—

C₂×Q₈×C₅₃

D₄

yes

characteristic

C₂×D₄×C₅₃

C₂×C₄

yes

characteristic

C₂×D₄×C₅₃

C₂×C₄

yes

characteristic

C₂×D₄×C₅₃

Q₈

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

M₁₆×C₅₃

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

—

—

—

—

—

—

—

C₂³×C₅₃

C₄²

yes

characteristic

C₂³×C₅₃

C₄⋊C₄

yes

characteristic

C₂³×C₅₃

C₄⋊C₄

yes

characteristic

Q₈×C₅₃

—

D₄×C₅₃

—

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

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄⋊C₄

yes

characteristic

C₈×C₅₃

—

#88322

—

#88322

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#95

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

yes

—

D₅₃

yes

characteristic

#408

—

C₅₃

#1392

yes

characteristic, complemented, Sylow

C₂×#41

C₂×D₅₃

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

106

—

C₂²×C₈

—

#101

—

#101

—

106

—

#100

—

#100

Dic₅₃

yes

—

C₂²⋊C₈

—

106

—

C₂²⋊C₈

—

106

—

#41

—

#41

—

C₂×Q₈

C₂²⋊D₅₃

yes

characteristic

C₂×D₄

Dic₁₀₆

yes

characteristic

C₂×D₄

C₄×D₅₃

yes

characteristic

C₂×D₄

C₂×Dic₅₃

yes

characteristic

C₂²×C₄

C₄×D₅₃

yes

characteristic

C₂²×C₄

D₂₁₂

yes

characteristic

C₂²×C₄

C₂²⋊D₅₃

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

106

—

100

M₁₆

—

212

—

101

M₁₆

—

212

—

102

C₂×C₈

—

103

C₂×C₈

—

106

—

104

C₂×C₈

—

106

—

105

C₂×C₈

—

212

—

106

C₂×C₈

—

212

—

107

C₂³

C₄×Dic₅₃

yes

characteristic

108

C₂³

#11719919

yes

characteristic

109

C₂³

C₄⋊Dic₅₃

yes

characteristic

110

Q₈

—

111

D₄

—

112

D₄

—

113

D₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

#11719919

yes

characteristic

121

C₂×C₄

#11719921

yes

characteristic

122

C₂×C₄

#11719921

yes

characteristic

123

C₂×C₄

C₂²⋊Dic₅₃

yes

characteristic

124

C₈

—

125

C₈

—

212

—

126

C₈

—

212

—

127

C₈

—

212

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

—

133

C₂²

—

134

C₂²

#13133124

yes

characteristic

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

#13133131

yes

characteristic

139

C₄

#13133131

yes

characteristic

140

C₂

—

141

C₂

—

142

C₂

—

143

C₂

—

144

C₂

#16810389

yes

characteristic

	subgroup	G/H
center	C₄	#13133131
commutator	C₂³×C₅₃	C₄²
Frattini	C₂×#41	C₂×D₅₃
Fitting	C₅₃×#413	C₂
socle	C₂×C₅₃	#342
hypercenter	#413	D₅₃
hypocenter	C₅₃	#1392
LCS	G ▷ C₂³×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×#41 ◁ #413

subgroup

G/H

center

C₄

#13133131

commutator

C₂³×C₅₃

C₄²

Frattini

C₂×#41

C₂×D₅₃

Fitting

C₅₃×#413

C₂

C₂×C₅₃

D₅₃

C₅₃

G ▷ C₂³×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×#41 ◁ #413

N	H ≅ G/N	type
76. C₅₃	#1392	semidirect

H ≅ G/N

type

76. C₅₃

#1392

semidirect

Subgroups — #20272837

Conjugacy classes of subgroups

Complements