Group #20287584 — subgroups

Subgroups — #20287584

Order 6784.

352 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 352 conjugacy classes of subgroups — too many to draw. Only its 125 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	#16811334	C₂	1	2	yes	maximal, complemented
2	#16811024	C₂	1	2	yes	maximal, complemented
3	#16810926	C₂	1	4	yes	maximal, complemented
4	#16810910	C₂	1	2	yes	maximal, complemented
5	C₈×Dic₁₀₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×D₄₂₄	C₂	1	1	yes	maximal, characteristic
7	#16810705	C₂	1	2	yes	maximal
8	C₅₃×#500	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×D₄₂₄	C₂²	1	1	yes	characteristic
10	#13133208	C₂²	1	4	yes	—
11	#13133197	—	2	2	no	—
12	#13133193	—	2	4	no	—
13	C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic
14	C₄×D₂₁₂	C₂²	1	2	yes	—
15	#13133179	C₂²	1	4	yes	—
16	C₄⋊C₄×D₅₃	—	2	2	no	—
17	#13133151	C₂²	1	1	yes	characteristic
18	#13133148	C₂²	1	4	yes	—
19	C₄⋊#88322	C₂²	1	1	yes	characteristic
20	#13133146	C₂²	1	2	yes	—
21	#13133139	C₂²	1	2	yes	—
22	#13133137	C₂²	1	4	yes	—
23	C₈×Dic₅₃	C₂²	1	2	yes	—
24	C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
25	#107×C₅₃	C₂²	1	2	yes	—
26	#107×C₅₃	C₂²	1	2	yes	—
27	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
28	C₂×Q₈×C₅₃	—	2	2	no	—
29	C₂×D₂₁₂	C₂³	1	2	yes	—
30	C₂×C₄×D₅₃	—	2	2	no	—
31	C₂×C₄×D₅₃	—	2	4	no	—
32	C₂×Dic₁₀₆	D₄	1	1	yes	characteristic
33	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
34	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
35	C₄⋊C₄×C₅₃	—	2	2	no	—
36	C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
37	C₄²×C₅₃	C₂³	1	1	yes	characteristic
38	#11719921	—	2	2	no	—
39	#11719921	—	2	4	no	—
40	C₄⋊Dic₅₃	—	2	4	no	—
41	C₄⋊Dic₅₃	D₄	1	1	yes	characteristic
42	C₄⋊Dic₅₃	C₂³	1	2	yes	—
43	#11719919	—	2	4	no	—
44	#11719919	D₄	1	2	yes	—
45	C₄×Dic₅₃	C₂³	1	2	yes	—
46	C₂×#88322	C₂³	1	2	yes	—
47	D₄₂₄	Q₈	1	4	yes	complemented
48	C₂²×D₅₃	—	2	2	no	—
49	Q₈×C₅₃	—	8	2	no	—
50	C₂×C₄×C₅₃	—	2	4	no	—
51	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
54	C₂×Dic₅₃	—	2	2	no	—
55	C₂×Dic₅₃	C₂×D₄	1	2	yes	—
56	C₂×Dic₅₃	C₂×D₄	1	2	yes	—
57	D₂₁₂	—	2	2	no	—
58	D₂₁₂	C₂×Q₈	1	4	yes	—
59	C₄×D₅₃	—	4	4	no	—
60	C₄×D₅₃	—	8	4	no	—
61	Dic₁₀₆	—	2	2	no	—
62	C₈×C₅₃	—	2	1	no	—
63	C₈×C₅₃	C₂×Q₈	1	2	yes	—
64	#88322	—	2	2	no	—
65	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
66	C₂×D₅₃	—	2	4	no	—
67	C₂×D₅₃	—	4	2	no	—
68	C₂×D₅₃	—	4	2	no	—
69	C₄×C₅₃	—	2	1	no	—
70	C₄×C₅₃	—	4	4	no	—
71	C₄×C₅₃	#135	1	2	yes	—
72	C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
73	C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
74	Dic₅₃	—	2	2	no	—
75	Dic₅₃	—	2	2	no	—
76	Dic₅₃	—	4	2	no	—
77	#3380	—	53	1	no	maximal, Sylow
78	C₂×C₅₃	D₄×Q₈	1	1	yes	characteristic
79	C₂×C₅₃	C₂×#135	1	1	yes	characteristic
80	C₂×C₅₃	#576	1	1	yes	characteristic
81	D₅₃	—	4	4	no	—
82	#554	—	53	2	no	—
83	#500	D₅₃	1	1	yes	characteristic, complemented
84	#499	—	53	2	no	—
85	#478	—	53	4	no	—
86	#476	—	53	2	no	—
87	C₈×Q₈	—	53	1	no	—
88	C₄×D₈	—	53	2	no	—
89	C₄×D₈	—	53	1	no	—
90	C₅₃	#3380	1	1	yes	characteristic, complemented, Sylow
91	C₂×D₈	—	53	1	no	—
92	C₄⋊Q₈	C₂×D₅₃	1	2	yes	—
93	#125	—	53	4	no	—
94	C₂²⋊Q₈	—	106	2	no	—
95	C₂²⋊Q₈	—	106	4	no	—
96	C₄×Q₈	—	53	1	no	—
97	C₄×D₄	—	53	4	no	—
98	C₄×D₄	—	53	2	no	—
99	C₂×C₄⋊C₄	—	106	2	no	—
100	#107	—	53	1	no	—
101	#107	C₂×D₅₃	1	2	yes	—
102	#107	C₂×D₅₃	1	2	yes	—
103	#106	—	53	4	no	—
104	C₄⋊C₈	—	53	2	no	—
105	C₄⋊C₈	—	53	1	no	—
106	#102	—	53	2	no	—
107	#102	—	53	4	no	—
108	C₄×C₈	—	53	2	no	—
109	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
110	C₂×Q₈	—	2	2	no	—
111	C₂×Q₈	—	53	1	no	—
112	C₂×D₄	—	53	2	no	—
113	C₂²×C₄	—	106	4	no	—
114	C₂²×C₄	—	106	2	no	—
115	D₈	—	53	4	no	—
116	C₂×C₈	—	53	2	no	—
117	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
118	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
119	C₄⋊C₄	—	2	2	no	—
120	C₄⋊C₄	—	53	1	no	—
121	C₄⋊C₄	—	53	2	no	—
122	C₄⋊C₄	—	53	2	no	—
123	C₄⋊C₄	—	106	4	no	—
124	C₄⋊C₄	—	106	4	no	—
125	C₄⋊C₄	C₂²×D₅₃	1	4	yes	—
126	C₂²⋊C₄	—	106	4	no	—
127	C₂²⋊C₄	—	106	2	no	—
128	C₄²	—	53	2	no	—
129	C₄²	C₂²×D₅₃	1	1	yes	characteristic
130	C₂³	—	106	2	no	—
131	Q₈	—	8	2	no	—
132	Q₈	—	106	2	no	—
133	D₄	—	53	4	no	—
134	D₄	—	106	2	no	—
135	C₂×C₄	—	2	4	no	—
136	C₂×C₄	—	53	2	no	—
137	C₂×C₄	—	53	2	no	—
138	C₂×C₄	—	106	2	no	—
139	C₂×C₄	—	424	4	no	—
140	C₂×C₄	—	212	4	no	—
141	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
142	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
143	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
144	C₈	—	2	1	no	—
145	C₈	—	106	2	no	—
146	C₈	Q₈×D₅₃	1	2	yes	—
147	C₂²	—	106	4	no	—
148	C₂²	—	212	2	no	—
149	C₂²	—	212	2	no	—
150	C₂²	C₂×D₄×D₅₃	1	1	yes	characteristic
151	C₄	—	2	1	no	—
152	C₄	—	4	4	no	—
153	C₄	—	106	2	no	—
154	C₄	—	106	2	no	—
155	C₄	—	212	2	no	—
156	C₄	#13133240	1	2	yes	—
157	C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
158	C₄	#13133265	1	1	yes	characteristic
159	C₂	—	212	4	no	—
160	C₂	#16811288	1	1	yes	characteristic
161	C₂	C₂×#13133240	1	1	yes	characteristic
162	C₂	#16811424	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811334

C₂

yes

maximal, complemented

#16811024

C₂

yes

maximal, complemented

#16810926

C₂

yes

maximal, complemented

#16810910

C₂

yes

maximal, complemented

C₈×Dic₁₀₆

C₂

yes

maximal, characteristic, complemented

C₄×D₄₂₄

C₂

yes

maximal, characteristic

#16810705

C₂

yes

maximal

C₅₃×#500

C₂

yes

maximal, characteristic, complemented

C₂×D₄₂₄

C₂²

yes

characteristic

#13133208

C₂²

yes

—

#13133197

—

#13133193

—

C₄×Dic₁₀₆

C₂²

yes

characteristic

C₄×D₂₁₂

C₂²

yes

—

#13133179

C₂²

yes

—

C₄⋊C₄×D₅₃

—

#13133151

C₂²

yes

characteristic

#13133148

C₂²

yes

—

C₄⋊#88322

C₂²

yes

characteristic

#13133146

C₂²

yes

—

#13133139

C₂²

yes

—

#13133137

C₂²

yes

—

C₈×Dic₅₃

C₂²

yes

—

C₄⋊Q₈×C₅₃

C₂²

yes

—

#107×C₅₃

C₂²

yes

—

#107×C₅₃

C₂²

yes

—

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

—

C₂×D₂₁₂

C₂³

yes

—

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

—

C₂×Dic₁₀₆

D₄

yes

characteristic

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

C₂³

yes

—

C₄²×C₅₃

C₂³

yes

characteristic

#11719921

—

#11719921

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

D₄

yes

characteristic

C₄⋊Dic₅₃

C₂³

yes

—

#11719919

—

#11719919

D₄

yes

—

C₄×Dic₅₃

C₂³

yes

—

C₂×#88322

C₂³

yes

—

D₄₂₄

Q₈

yes

complemented

C₂²×D₅₃

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂⁴

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂×D₄

yes

—

C₂×Dic₅₃

C₂×D₄

yes

—

D₂₁₂

—

D₂₁₂

C₂×Q₈

yes

—

C₄×D₅₃

—

C₄×D₅₃

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

C₂×Q₈

yes

—

#88322

—

C₂²×C₅₃

C₂²×D₄

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#135

yes

—

C₄×C₅₃

C₂²×Q₈

yes

characteristic

C₄×C₅₃

2₋¹⁺⁴

yes

characteristic

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

#3380

—

maximal, Sylow

C₂×C₅₃

D₄×Q₈

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

—

#554

—

#500

D₅₃

yes

characteristic, complemented

#499

—

#478

—

#476

—

C₈×Q₈

—

C₄×D₈

—

C₄×D₈

—

C₅₃

#3380

yes

characteristic, complemented, Sylow

C₂×D₈

—

C₄⋊Q₈

C₂×D₅₃

yes

—

#125

—

C₂²⋊Q₈

—

106

—

C₂²⋊Q₈

—

106

—

C₄×Q₈

—

C₄×D₄

—

C₄×D₄

—

C₂×C₄⋊C₄

—

106

—

100

#107

—

101

#107

C₂×D₅₃

yes

—

102

#107

C₂×D₅₃

yes

—

103

#106

—

104

C₄⋊C₈

—

105

C₄⋊C₈

—

106

#102

—

107

#102

—

108

C₄×C₈

—

109

C₄×C₈

C₂×D₅₃

yes

characteristic

110

C₂×Q₈

—

111

C₂×Q₈

—

112

C₂×D₄

—

113

C₂²×C₄

—

106

—

114

C₂²×C₄

—

106

—

115

D₈

—

116

C₂×C₈

—

117

C₂×C₈

C₂²×D₅₃

yes

characteristic

118

C₂×C₈

C₂²×D₅₃

yes

characteristic

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

106

—

124

C₄⋊C₄

—

106

—

125

C₄⋊C₄

C₂²×D₅₃

yes

—

126

C₂²⋊C₄

—

106

—

127

C₂²⋊C₄

—

106

—

128

C₄²

—

129

C₄²

C₂²×D₅₃

yes

characteristic

130

C₂³

—

106

—

131

Q₈

—

132

Q₈

—

106

—

133

D₄

—

134

D₄

—

106

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

106

—

139

C₂×C₄

—

424

—

140

C₂×C₄

—

212

—

141

C₂×C₄

D₄×D₅₃

yes

characteristic

142

C₂×C₄

D₄×D₅₃

yes

characteristic

143

C₂×C₄

C₂³×D₅₃

yes

characteristic

144

C₈

—

145

C₈

—

106

—

146

C₈

Q₈×D₅₃

yes

—

147

C₂²

—

106

—

148

C₂²

—

212

—

149

C₂²

—

212

—

150

C₂²

C₂×D₄×D₅₃

yes

characteristic

151

C₄

—

152

C₄

—

153

C₄

—

106

—

154

C₄

—

106

—

155

C₄

—

212

—

156

C₄

#13133240

yes

—

157

C₄

C₂×Q₈×D₅₃

yes

characteristic

158

C₄

#13133265

yes

characteristic

159

C₂

—

212

—

160

C₂

#16811288

yes

characteristic

161

C₂

C₂×#13133240

yes

characteristic

162

C₂

#16811424

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₅₃×#500	C₂
socle	C₂²×C₅₃	C₂²×D₄
hypercenter	#500	D₅₃
hypocenter	C₅₃	#3380
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #500

subgroup

G/H

center

C₂²

C₂×D₄×D₅₃

commutator

C₂×C₄×C₅₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₃

Fitting

C₅₃×#500

C₂

socle

C₂²×C₅₃

C₂²×D₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #500

N	H ≅ G/N	type
1. #16811334	C₂	semidirect
2. #16811024	C₂	semidirect
3. #16810926	C₂	semidirect
4. #16810910	C₂	semidirect
5. C₈×Dic₁₀₆	C₂	semidirect
8. C₅₃×#500	C₂	semidirect
47. D₄₂₄	Q₈	semidirect
83. #500	D₅₃	semidirect
90. C₅₃	#3380	semidirect

H ≅ G/N

type

1. #16811334

C₂

semidirect

2. #16811024

C₂

semidirect

3. #16810926

C₂

semidirect

4. #16810910

C₂

semidirect

5. C₈×Dic₁₀₆

C₂

semidirect

8. C₅₃×#500

C₂

semidirect

47. D₄₂₄

Q₈

semidirect

83. #500

D₅₃

semidirect

90. C₅₃

#3380

semidirect

Subgroups — #20287584

Conjugacy classes of subgroups

Complements