Group #20273215 — subgroups

Subgroups — #20273215

Order 6784.

240 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 89 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	C₄×C₄⋊Dic₅₃	C₂	1	1	yes	maximal, characteristic
3	#16810361	C₂	1	4	yes	maximal
4	C₅₃×#431	C₂	1	1	yes	maximal, characteristic
5	#13133217	C₄	1	1	yes	characteristic, complemented
6	#13133217	C₄	1	1	yes	characteristic, complemented
7	#13133209	—	2	1	no	—
8	#13133196	—	2	1	no	—
9	#13133196	C₂²	1	2	yes	—
10	C₄×Dic₁₀₆	—	2	1	no	—
11	C₄×Dic₁₀₆	C₄	1	2	yes	complemented
12	#13133176	—	2	1	no	—
13	C₂×C₄⋊Dic₅₃	—	2	1	no	—
14	C₂×C₄×Dic₅₃	—	2	1	no	—
15	#13133124	C₂²	1	2	yes	—
16	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
17	C₄⋊C₈×C₅₃	—	2	1	no	—
18	C₂²⋊C₈×C₅₃	C₂²	1	2	yes	—
19	C₄×C₈×C₅₃	—	2	1	no	—
20	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
21	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
22	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
23	C₂²×Dic₅₃	—	2	2	no	—
24	C₂×Dic₁₀₆	—	2	1	no	—
25	C₂×Dic₁₀₆	C₂×C₄	1	2	yes	—
26	C₂×C₈×C₅₃	—	2	2	no	—
27	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₄²×C₅₃	D₄	1	1	yes	characteristic
30	C₄²×C₅₃	D₄	1	1	yes	characteristic
31	C₂²⋊Dic₅₃	—	2	2	no	—
32	C₄⋊Dic₅₃	—	2	4	no	—
33	C₄⋊Dic₅₃	—	2	1	no	—
34	C₄⋊Dic₅₃	—	2	1	no	—
35	C₄⋊Dic₅₃	—	2	1	no	—
36	C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
37	#11719919	—	2	2	no	—
38	#11719919	—	2	2	no	—
39	C₄×Dic₅₃	—	2	2	no	—
40	C₄×Dic₅₃	—	4	2	no	—
41	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₅₃	—	2	2	no	—
43	C₂×C₄×C₅₃	—	4	1	no	—
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	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₅₃	SD₁₆	1	2	yes	—
49	C₂×C₄×C₅₃	Q₁₆	1	2	yes	—
50	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
52	C₂×Dic₅₃	—	2	4	no	—
53	C₂×Dic₅₃	—	2	2	no	—
54	C₂×Dic₅₃	—	2	2	no	—
55	C₂×Dic₅₃	—	4	2	no	—
56	C₂×Dic₅₃	—	4	2	no	—
57	Dic₁₀₆	—	4	2	no	—
58	Dic₁₀₆	—	4	2	no	—
59	C₈×C₅₃	—	4	2	no	—
60	C₂²×C₅₃	—	2	1	no	—
61	C₂²×C₅₃	—	2	1	no	—
62	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
64	C₂²×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
65	C₄×C₅₃	—	2	1	no	—
66	C₄×C₅₃	—	2	1	no	—
67	C₄×C₅₃	—	2	1	no	—
68	C₄×C₅₃	#103	1	2	yes	—
69	C₄×C₅₃	#103	1	2	yes	—
70	C₄×C₅₃	#104	1	2	yes	—
71	Dic₅₃	—	4	4	no	—
72	Dic₅₃	—	4	2	no	—
73	Dic₅₃	—	4	2	no	—
74	#1503	—	53	1	no	maximal, Sylow
75	C₂×C₅₃	—	2	1	no	—
76	C₂×C₅₃	—	2	1	no	—
77	C₂×C₅₃	#410	1	1	yes	characteristic
78	C₂×C₅₃	C₂×#103	1	1	yes	characteristic
79	C₂×C₅₃	C₂×#104	1	1	yes	characteristic
80	#533	—	53	1	no	—
81	#431	D₅₃	1	1	yes	characteristic
82	C₄×C₄⋊C₄	—	53	1	no	—
83	#328	—	53	4	no	—
84	C₅₃	#1503	1	1	yes	characteristic, complemented, Sylow
85	C₄⋊Q₈	—	53	1	no	—
86	C₄⋊Q₈	—	53	1	no	—
87	#125	—	106	1	no	—
88	C₂²⋊Q₈	—	53	2	no	—
89	C₂²⋊Q₈	—	106	1	no	—
90	C₄×Q₈	—	53	2	no	—
91	C₄×Q₈	—	106	1	no	—
92	#117	—	106	1	no	—
93	C₂×C₄⋊C₄	—	106	1	no	—
94	C₂×C₄²	—	106	1	no	—
95	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
96	C₄⋊C₈	—	2	1	no	—
97	C₂²⋊C₈	C₂×D₅₃	1	2	yes	—
98	C₄×C₈	—	2	1	no	—
99	#95	—	53	2	no	—
100	C₂×Q₈	—	53	2	no	—
101	C₂×Q₈	—	106	1	no	—
102	C₂²×C₄	—	106	2	no	—
103	C₂²×C₄	D₂₁₂	1	1	yes	characteristic
104	C₂²×C₄	D₂₁₂	1	1	yes	characteristic
105	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
106	C₂×C₈	—	2	2	no	—
107	C₄⋊C₄	—	53	2	no	—
108	C₄⋊C₄	—	106	4	no	—
109	C₄⋊C₄	—	106	2	no	—
110	C₄⋊C₄	—	106	1	no	—
111	C₄⋊C₄	—	106	1	no	—
112	C₄⋊C₄	—	106	2	no	—
113	C₄⋊C₄	—	106	1	no	—
114	C₂²⋊C₄	—	106	2	no	—
115	C₄²	—	106	2	no	—
116	C₄²	—	212	2	no	—
117	C₄²	C₄×D₅₃	1	1	yes	characteristic
118	C₄²	C₄×D₅₃	1	1	yes	characteristic
119	C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
120	C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
121	C₂³	C₂×D₂₁₂	1	1	yes	characteristic
122	Q₈	—	212	2	no	—
123	Q₈	—	212	2	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	4	1	no	—
126	C₂×C₄	—	106	4	no	—
127	C₂×C₄	—	106	2	no	—
128	C₂×C₄	—	106	2	no	—
129	C₂×C₄	—	212	2	no	—
130	C₂×C₄	—	212	2	no	—
131	C₂×C₄	#11719913	1	2	yes	—
132	C₂×C₄	Dic₂₁₂	1	2	yes	—
133	C₂×C₄	#11719921	1	1	yes	characteristic
134	C₂×C₄	#11719921	1	1	yes	characteristic
135	C₂×C₄	#11719921	1	1	yes	characteristic
136	C₂×C₄	#11719921	1	1	yes	characteristic
137	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
138	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
139	C₈	—	4	2	no	—
140	C₂²	—	2	1	no	—
141	C₂²	—	2	1	no	—
142	C₂²	C₂×#11719921	1	1	yes	characteristic
143	C₂²	C₂×#11719913	1	1	yes	characteristic
144	C₂²	C₂×Dic₂₁₂	1	1	yes	characteristic
145	C₄	—	2	1	no	—
146	C₄	—	2	1	no	—
147	C₄	—	2	1	no	—
148	C₄	—	212	4	no	—
149	C₄	—	212	2	no	—
150	C₄	—	212	2	no	—
151	C₄	#13133142	1	2	yes	—
152	C₄	#13133142	1	2	yes	—
153	C₄	#13133143	1	2	yes	—
154	C₂	—	2	1	no	—
155	C₂	—	2	1	no	—
156	C₂	#16810596	1	1	yes	characteristic
157	C₂	C₂×#13133142	1	1	yes	characteristic
158	C₂	C₂×#13133143	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811180

C₂

yes

maximal, characteristic

C₄×C₄⋊Dic₅₃

C₂

yes

maximal, characteristic

#16810361

C₂

yes

maximal

C₅₃×#431

C₂

yes

maximal, characteristic

#13133217

C₄

yes

characteristic, complemented

#13133217

C₄

yes

characteristic, complemented

#13133209

—

#13133196

—

#13133196

C₂²

yes

—

C₄×Dic₁₀₆

—

C₄×Dic₁₀₆

C₄

yes

complemented

#13133176

—

C₂×C₄⋊Dic₅₃

—

C₂×C₄×Dic₅₃

—

#13133124

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

—

C₂²⋊C₈×C₅₃

C₂²

yes

—

C₄×C₈×C₅₃

—

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

C₂×C₄

yes

—

C₂×C₈×C₅₃

—

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

D₄

yes

characteristic

C₂²⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

C₂×C₄

yes

—

#11719919

—

#11719919

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

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

SD₁₆

yes

—

C₂×C₄×C₅₃

Q₁₆

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₅₃

C₂×SD₁₆

yes

characteristic

C₂²×C₅₃

C₂×Q₁₆

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#103

yes

—

C₄×C₅₃

#103

yes

—

C₄×C₅₃

#104

yes

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

#1503

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

D₅₃

yes

characteristic

C₄×C₄⋊C₄

—

#328

—

C₅₃

#1503

yes

characteristic, complemented, Sylow

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

106

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

106

—

C₄×Q₈

—

C₄×Q₈

—

106

—

#117

—

106

—

C₂×C₄⋊C₄

—

106

—

C₂×C₄²

—

106

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

C₂×D₅₃

yes

—

C₄×C₈

—

#95

—

100

C₂×Q₈

—

101

C₂×Q₈

—

106

—

102

C₂²×C₄

—

106

—

103

C₂²×C₄

D₂₁₂

yes

characteristic

104

C₂²×C₄

D₂₁₂

yes

characteristic

105

C₂²×C₄

C₂²×D₅₃

yes

characteristic

106

C₂×C₈

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

106

—

109

C₄⋊C₄

—

106

—

110

C₄⋊C₄

—

106

—

111

C₄⋊C₄

—

106

—

112

C₄⋊C₄

—

106

—

113

C₄⋊C₄

—

106

—

114

C₂²⋊C₄

—

106

—

115

C₄²

—

106

—

116

C₄²

—

212

—

117

C₄²

C₄×D₅₃

yes

characteristic

118

C₄²

C₄×D₅₃

yes

characteristic

119

C₄²

C₂²⋊D₅₃

yes

characteristic

120

C₄²

C₂²⋊D₅₃

yes

characteristic

121

C₂³

C₂×D₂₁₂

yes

characteristic

122

Q₈

—

212

—

123

Q₈

—

212

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

106

—

127

C₂×C₄

—

106

—

128

C₂×C₄

—

106

—

129

C₂×C₄

—

212

—

130

C₂×C₄

—

212

—

131

C₂×C₄

#11719913

yes

—

132

C₂×C₄

Dic₂₁₂

yes

—

133

C₂×C₄

#11719921

yes

characteristic

134

C₂×C₄

#11719921

yes

characteristic

135

C₂×C₄

#11719921

yes

characteristic

136

C₂×C₄

#11719921

yes

characteristic

137

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

138

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

139

C₈

—

140

C₂²

—

141

C₂²

—

142

C₂²

C₂×#11719921

yes

characteristic

143

C₂²

C₂×#11719913

yes

characteristic

144

C₂²

C₂×Dic₂₁₂

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

212

—

149

C₄

—

212

—

150

C₄

—

212

—

151

C₄

#13133142

yes

—

152

C₄

#13133142

yes

—

153

C₄

#13133143

yes

—

154

C₂

—

155

C₂

—

156

C₂

#16810596

yes

characteristic

157

C₂

C₂×#13133142

yes

characteristic

158

C₂

C₂×#13133143

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#11719921
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₅₃×#431	C₂
socle	C₂²×C₅₃	C₂×C₂²⋊C₄
hypercenter	#431	D₅₃
hypocenter	C₅₃	#1503
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

subgroup

G/H

center

C₂×C₄

#11719921

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₅₃×#431

C₂

socle

C₂²×C₅₃

C₂×C₂²⋊C₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

N	H ≅ G/N	type
5. #13133217	C₄	semidirect
6. #13133217	C₄	semidirect
11. C₄×Dic₁₀₆	C₄	semidirect
84. C₅₃	#1503	semidirect

H ≅ G/N

type

5. #13133217

C₄

semidirect

6. #13133217

C₄

semidirect

11. C₄×Dic₁₀₆

C₄

semidirect

84. C₅₃

#1503

semidirect

Subgroups — #20273215

Conjugacy classes of subgroups

Complements