Group #20273187 — subgroups

Subgroups — #20273187

Order 6784.

202 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 202 conjugacy classes of subgroups — too many to draw. Only its 73 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	#16811180	C₂	1	1	yes	maximal, characteristic
2	#16810676	C₂	1	1	yes	maximal, characteristic
3	#16810353	C₂	1	4	yes	maximal, complemented
4	C₅₃×#432	C₂	1	1	yes	maximal, characteristic
5	#13133217	C₄	1	1	yes	characteristic
6	#13133217	C₄	1	1	yes	characteristic
7	#13133209	—	2	1	no	—
8	#13133196	—	2	1	no	—
9	#13133196	C₄	1	2	yes	—
10	C₄×Dic₁₀₆	—	2	1	no	—
11	C₄×Dic₁₀₆	C₂²	1	2	yes	—
12	#13133176	—	2	1	no	—
13	C₄⋊#88322	—	2	1	no	—
14	C₂²⋊#88322	—	2	1	no	—
15	C₄×#88322	C₂²	1	2	yes	—
16	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
17	C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
18	C₂²⋊C₈×C₅₃	—	2	1	no	—
19	#97×C₅₃	—	2	1	no	—
20	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
22	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
23	C₂×Dic₁₀₆	—	2	1	no	—
24	C₂×Dic₁₀₆	C₂×C₄	1	2	yes	—
25	C₂×C₈×C₅₃	—	2	2	no	—
26	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
27	C₄²×C₅₃	D₄	1	1	yes	characteristic
28	C₄²×C₅₃	D₄	1	1	yes	characteristic
29	C₄²×C₅₃	C₂³	1	1	yes	characteristic
30	C₂²⋊Dic₅₃	—	2	2	no	—
31	C₄⋊Dic₅₃	—	2	1	no	—
32	C₄⋊Dic₅₃	—	2	1	no	—
33	C₄⋊Dic₅₃	—	2	1	no	—
34	C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
35	#11719919	—	2	2	no	—
36	#11719919	—	2	2	no	—
37	C₄×Dic₅₃	—	2	2	no	—
38	C₂×#88322	—	2	2	no	—
39	C₂³×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
40	C₂×C₄×C₅₃	—	2	2	no	—
41	C₂×C₄×C₅₃	—	2	2	no	—
42	C₂×C₄×C₅₃	—	4	1	no	—
43	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
49	C₂×Dic₅₃	—	2	2	no	—
50	C₂×Dic₅₃	—	2	2	no	—
51	Dic₁₀₆	—	2	4	no	—
52	Dic₁₀₆	—	4	2	no	—
53	C₈×C₅₃	—	4	2	no	—
54	#88322	—	4	2	no	—
55	C₂²×C₅₃	—	2	1	no	—
56	C₂²×C₅₃	—	2	1	no	—
57	C₂²×C₅₃	—	2	1	no	—
58	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
59	C₄×C₅₃	—	2	1	no	—
60	C₄×C₅₃	—	2	1	no	—
61	C₄×C₅₃	—	2	1	no	—
62	C₄×C₅₃	—	2	1	no	—
63	C₄×C₅₃	#131	1	2	yes	—
64	C₄×C₅₃	#137	1	2	yes	—
65	Dic₅₃	—	4	2	no	—
66	Dic₅₃	—	4	2	no	—
67	#1497	—	53	1	no	maximal, Sylow
68	C₂×C₅₃	—	4	1	no	—
69	C₂×C₅₃	#408	1	1	yes	characteristic
70	C₂×C₅₃	#419	1	1	yes	characteristic
71	C₂×C₅₃	#421	1	1	yes	characteristic
72	#533	—	53	1	no	—
73	#432	D₅₃	1	1	yes	characteristic
74	#431	—	53	1	no	—
75	Q₈⋊C₈	—	53	4	no	—
76	C₅₃	#1497	1	1	yes	characteristic, complemented, Sylow
77	C₄⋊Q₈	—	53	1	no	—
78	C₄⋊Q₈	—	53	1	no	—
79	#125	—	106	1	no	—
80	C₂²⋊Q₈	—	53	2	no	—
81	C₂²⋊Q₈	—	106	1	no	—
82	C₄×Q₈	—	53	2	no	—
83	C₄×Q₈	—	106	1	no	—
84	#117	—	106	1	no	—
85	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
86	C₄⋊C₈	—	106	1	no	—
87	C₄⋊C₈	C₂×D₅₃	1	2	yes	—
88	C₂²⋊C₈	—	2	1	no	—
89	C₂²⋊C₈	—	106	1	no	—
90	#97	—	2	1	no	—
91	C₄×C₈	—	53	2	no	—
92	C₂×Q₈	—	53	2	no	—
93	C₂×Q₈	—	106	1	no	—
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	C₂×C₈	—	2	2	no	—
98	C₂×C₈	—	106	2	no	—
99	C₄⋊C₄	—	53	2	no	—
100	C₄⋊C₄	—	106	2	no	—
101	C₄⋊C₄	—	106	1	no	—
102	C₄⋊C₄	—	106	1	no	—
103	C₄⋊C₄	—	106	2	no	—
104	C₄⋊C₄	—	106	1	no	—
105	C₂²⋊C₄	—	106	2	no	—
106	C₄²	—	106	2	no	—
107	C₄²	C₄×D₅₃	1	1	yes	characteristic
108	C₄²	D₂₁₂	1	1	yes	characteristic
109	C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
110	C₄²	C₂²×D₅₃	1	1	yes	characteristic
111	C₂³	#11719921	1	1	yes	characteristic
112	Q₈	—	106	4	no	—
113	Q₈	—	212	2	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	4	1	no	—
117	C₂×C₄	—	106	2	no	—
118	C₂×C₄	—	106	2	no	—
119	C₂×C₄	#11719921	1	1	yes	characteristic
120	C₂×C₄	#11719921	1	1	yes	characteristic
121	C₂×C₄	#11719921	1	1	yes	characteristic
122	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
123	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
124	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
125	C₈	—	4	2	no	—
126	C₈	—	212	2	no	—
127	C₂²	—	2	1	no	—
128	C₂²	—	2	1	no	—
129	C₂²	—	2	1	no	—
130	C₂²	C₂×#11719921	1	1	yes	characteristic
131	C₄	—	2	1	no	—
132	C₄	—	2	1	no	—
133	C₄	—	2	1	no	—
134	C₄	—	2	1	no	—
135	C₄	—	212	2	no	—
136	C₄	—	212	2	no	—
137	C₄	#13133225	1	2	yes	—
138	C₄	#13133247	1	2	yes	—
139	C₂	—	4	1	no	—
140	C₂	#16810591	1	1	yes	characteristic
141	C₂	#16810631	1	1	yes	characteristic
142	C₂	#16810639	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811180

C₂

yes

maximal, characteristic

#16810676

C₂

yes

maximal, characteristic

#16810353

C₂

yes

maximal, complemented

C₅₃×#432

C₂

yes

maximal, characteristic

#13133217

C₄

yes

characteristic

#13133217

C₄

yes

—

—

C₄

yes

—

C₄×Dic₁₀₆

—

C₄×Dic₁₀₆

C₂²

yes

—

—

—

—

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

C₂²

yes

—

C₂²⋊C₈×C₅₃

—

#97×C₅₃

—

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

C₂×C₄

yes

—

C₂×C₈×C₅₃

—

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

C₂³

yes

characteristic

C₂²⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

C₂×C₄

yes

—

#11719919

—

#11719919

—

C₄×Dic₅₃

—

C₂×#88322

—

C₂³×C₅₃

C₂²⋊C₄

yes

characteristic

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

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#131

yes

—

C₄×C₅₃

#137

yes

—

Dic₅₃

—

Dic₅₃

—

#1497

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

D₅₃

yes

characteristic

#431

—

Q₈⋊C₈

—

C₅₃

#1497

yes

characteristic, complemented, Sylow

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

106

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

106

—

C₄×Q₈

—

C₄×Q₈

—

106

—

#117

—

106

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₄⋊C₈

—

106

—

C₄⋊C₈

C₂×D₅₃

yes

—

C₂²⋊C₈

—

C₂²⋊C₈

—

106

—

#97

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

106

—

C₂²×C₄

C₄×D₅₃

yes

characteristic

C₂²×C₄

D₂₁₂

yes

characteristic

C₂²×C₄

C₂²⋊D₅₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

106

—

C₄⋊C₄

—

100

C₄⋊C₄

—

106

—

101

C₄⋊C₄

—

106

—

102

C₄⋊C₄

—

106

—

103

C₄⋊C₄

—

106

—

104

C₄⋊C₄

—

106

—

105

C₂²⋊C₄

—

106

—

106

C₄²

—

106

—

107

C₄²

C₄×D₅₃

yes

characteristic

108

C₄²

D₂₁₂

yes

characteristic

109

C₄²

C₂²⋊D₅₃

yes

characteristic

110

C₄²

C₂²×D₅₃

yes

characteristic

111

C₂³

#11719921

yes

characteristic

112

Q₈

—

106

—

113

Q₈

—

212

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

106

—

118

C₂×C₄

—

106

—

119

C₂×C₄

#11719921

yes

characteristic

120

C₂×C₄

#11719921

yes

characteristic

121

C₂×C₄

#11719921

yes

characteristic

122

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

123

C₂×C₄

C₂×D₂₁₂

yes

characteristic

124

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

125

C₈

—

126

C₈

—

212

—

127

C₂²

—

128

C₂²

—

129

C₂²

—

130

C₂²

C₂×#11719921

yes

characteristic

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

—

212

—

136

C₄

—

212

—

137

C₄

#13133225

yes

—

138

C₄

#13133247

yes

—

139

C₂

—

140

C₂

#16810591

yes

characteristic

141

C₂

#16810631

yes

characteristic

142

C₂

#16810639

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₄²	C₂²×D₅₃
Fitting	C₅₃×#432	C₂
socle	C₂²×C₅₃	C₂×C₂²⋊C₄
hypercenter	#432	D₅₃
hypocenter	C₅₃	#1497
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #432

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₅₃

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₄²

C₂²×D₅₃

Fitting

C₅₃×#432

C₂

socle

C₂²×C₅₃

C₂×C₂²⋊C₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #432

N	H ≅ G/N	type
3. #16810353	C₂	semidirect
76. C₅₃	#1497	semidirect

H ≅ G/N

type

3. #16810353

C₂

semidirect

76. C₅₃

#1497

semidirect

Subgroups — #20273187

Conjugacy classes of subgroups

Complements