Group #20276040 — subgroups

Subgroups — #20276040

Order 6784.

282 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 99 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×#13133150	C₂	1	2	yes	maximal
2	#16810521	C₂	1	1	yes	maximal, characteristic
3	#16810385	C₂	1	2	yes	maximal
4	#16810378	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×#88322	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
9	C₂×#11719919	—	2	1	no	—
10	#13133150	—	2	4	no	—
11	#13133124	—	2	1	no	—
12	C₂²×Q₈×C₅₃	—	2	1	no	—
13	C₄⋊Q₈×C₅₃	—	2	4	no	—
14	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
15	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
16	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×Q₈×C₅₃	—	2	4	no	—
18	C₂×Q₈×C₅₃	—	4	2	no	—
19	C₂×Q₈×C₅₃	—	4	2	no	—
20	C₂²×C₄×C₅₃	—	2	2	no	—
21	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
22	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
23	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
24	C₂²×Dic₅₃	—	2	1	no	—
25	C₂²×Dic₅₃	—	2	1	no	—
26	C₄⋊C₄×C₅₃	—	2	4	no	—
27	C₄⋊C₄×C₅₃	—	2	2	no	—
28	C₄⋊C₄×C₅₃	D₄	1	4	yes	—
29	C₄²×C₅₃	—	2	2	no	—
30	C₄⋊Dic₅₃	—	2	1	no	—
31	C₄⋊Dic₅₃	—	2	2	no	—
32	C₄⋊Dic₅₃	Q₈	1	2	yes	complemented
33	#11719919	—	4	2	no	—
34	C₂×#88322	—	2	2	no	—
35	C₂×#88322	—	2	2	no	—
36	C₂×#88322	Q₈	1	4	yes	complemented
37	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
38	Q₈×C₅₃	—	4	8	no	—
39	C₂×C₄×C₅₃	—	2	1	no	—
40	C₂×C₄×C₅₃	—	2	1	no	—
41	C₂×C₄×C₅₃	—	2	1	no	—
42	C₂×C₄×C₅₃	—	2	1	no	—
43	C₂×C₄×C₅₃	—	2	4	no	—
44	C₂×C₄×C₅₃	—	4	2	no	—
45	C₂×C₄×C₅₃	—	4	2	no	—
46	C₂×C₄×C₅₃	D₈	1	2	yes	—
47	C₂×C₄×C₅₃	Q₁₆	1	2	yes	—
48	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
49	C₂×C₄×C₅₃	C₂×Q₈	1	2	yes	—
50	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
52	C₂×Dic₅₃	—	2	2	no	—
53	C₂×Dic₅₃	—	4	1	no	—
54	C₂×Dic₅₃	—	4	1	no	—
55	C₂×Dic₅₃	—	4	1	no	—
56	C₂×Dic₅₃	—	4	1	no	—
57	C₂×Dic₅₃	—	4	1	no	—
58	#88322	—	2	4	no	—
59	C₂²×C₅₃	#120	1	1	yes	characteristic
60	C₂²×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
61	C₂²×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
62	C₂²×C₅₃	C₂×D₈	1	1	yes	characteristic
63	C₂²×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
64	C₂²×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
65	C₂²×C₅₃	#137	1	1	yes	characteristic
66	C₄×C₅₃	—	2	2	no	—
67	C₄×C₅₃	—	2	2	no	—
68	C₄×C₅₃	—	4	4	no	—
69	C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
70	C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
71	Dic₅₃	—	4	2	no	—
72	Dic₅₃	—	8	1	no	—
73	#2057	—	53	1	no	maximal, Sylow
74	C₂×C₅₃	#395	1	1	yes	characteristic
75	C₂×C₅₃	#447	1	1	yes	characteristic
76	C₂×C₅₃	#451	1	1	yes	characteristic
77	C₂×C₅₃	#474	1	1	yes	characteristic
78	C₂×C₅₃	#477	1	1	yes	characteristic
79	C₂×C₅₃	#500	1	1	yes	characteristic
80	C₂×C₅₃	#501	1	1	yes	characteristic
81	C₂×C₄⋊Q₈	D₅₃	1	1	yes	characteristic
82	C₂×#107	—	53	2	no	—
83	#389	—	53	1	no	—
84	#340	—	53	2	no	—
85	#336	—	53	1	no	—
86	C₅₃	#2057	1	1	yes	characteristic, complemented, Sylow
87	C₂²×Q₈	—	2	1	no	—
88	C₂²×C₈	—	53	2	no	—
89	C₄⋊Q₈	—	2	4	no	—
90	C₂×C₄⋊C₄	—	2	1	no	—
91	C₂×C₄⋊C₄	—	53	1	no	—
92	C₂×C₄⋊C₄	—	53	1	no	—
93	C₂×C₄⋊C₄	—	106	1	no	—
94	C₂×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
95	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
96	#107	—	106	4	no	—
97	#95	—	106	1	no	—
98	C₂×Q₈	—	2	4	no	—
99	C₂×Q₈	—	4	2	no	—
100	C₂×Q₈	—	4	2	no	—
101	C₂²×C₄	—	2	2	no	—
102	C₂²×C₄	—	106	1	no	—
103	C₂²×C₄	—	106	1	no	—
104	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
105	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
106	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
107	C₂×C₈	—	53	4	no	—
108	C₂×C₈	—	106	2	no	—
109	C₂×C₈	—	106	2	no	—
110	C₄⋊C₄	—	2	4	no	—
111	C₄⋊C₄	—	2	2	no	—
112	C₄⋊C₄	—	53	2	no	—
113	C₄⋊C₄	—	106	1	no	—
114	C₄⋊C₄	—	106	2	no	—
115	C₄⋊C₄	—	212	2	no	—
116	C₄⋊C₄	C₂²⋊D₅₃	1	4	yes	—
117	C₄²	—	2	2	no	—
118	C₂³	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
119	Q₈	—	4	8	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	—	2	1	no	—
124	C₂×C₄	—	2	4	no	—
125	C₂×C₄	—	4	2	no	—
126	C₂×C₄	—	4	2	no	—
127	C₂×C₄	—	106	2	no	—
128	C₂×C₄	—	212	1	no	—
129	C₂×C₄	—	212	1	no	—
130	C₂×C₄	—	212	1	no	—
131	C₂×C₄	—	212	1	no	—
132	C₂×C₄	—	212	1	no	—
133	C₂×C₄	#11719922	1	2	yes	—
134	C₂×C₄	#11719925	1	2	yes	—
135	C₂×C₄	Q₈×D₅₃	1	2	yes	—
136	C₂×C₄	Q₈×D₅₃	1	1	yes	characteristic
137	C₂×C₄	#11719949	1	1	yes	characteristic
138	C₂×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
139	C₈	—	106	4	no	—
140	C₂²	#13133186	1	1	yes	characteristic
141	C₂²	#13133193	1	1	yes	characteristic
142	C₂²	#13133215	1	1	yes	characteristic
143	C₂²	C₂×#11719922	1	1	yes	characteristic
144	C₂²	C₂×#11719925	1	1	yes	characteristic
145	C₂²	#13133243	1	1	yes	characteristic
146	C₂²	#13133248	1	1	yes	characteristic
147	C₄	—	2	2	no	—
148	C₄	—	2	2	no	—
149	C₄	—	4	4	no	—
150	C₄	—	424	1	no	—
151	C₄	—	212	2	no	—
152	C₄	#13133193	1	2	yes	—
153	C₄	#13133215	1	2	yes	—
154	C₂	#16810542	1	1	yes	characteristic
155	C₂	#16810757	1	1	yes	characteristic
156	C₂	#16810779	1	1	yes	characteristic
157	C₂	#16810900	1	1	yes	characteristic
158	C₂	#16810915	1	1	yes	characteristic
159	C₂	#16811026	1	1	yes	characteristic
160	C₂	#16811029	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133150

C₂

yes

maximal

#16810521

C₂

yes

maximal, characteristic

#16810385

C₂

yes

maximal

#16810378

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₅₃

C₂

yes

maximal, characteristic

C₂²×#88322

C₂²

yes

—

C₂×C₄⋊Dic₅₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₅₃

C₂²

yes

—

—

—

C₂²×Q₈×C₅₃

—

C₄⋊Q₈×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

—

C₂²×Dic₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

D₄

yes

—

C₄²×C₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

Q₈

yes

—

—

—

Q₈

yes

complemented

C₂³×C₅₃

C₂×D₄

yes

characteristic

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

D₈

yes

—

C₂×C₄×C₅₃

Q₁₆

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×Q₈

yes

—

C₂×C₄×C₅₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₅₃

#41

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

#88322

—

C₂²×C₅₃

#120

yes

characteristic

C₂²×C₅₃

C₂²⋊Q₈

yes

characteristic

C₂²×C₅₃

C₄⋊Q₈

yes

characteristic

C₂²×C₅₃

C₂×D₈

yes

characteristic

C₂²×C₅₃

C₂×Q₁₆

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂²⋊Q₈

yes

—

C₄×C₅₃

C₄⋊Q₈

yes

—

Dic₅₃

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₄⋊Q₈

D₅₃

yes

—

—

—

—

C₅₃

#2057

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×C₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

106

—

C₂×C₄⋊C₄

C₂×D₅₃

yes

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

#107

—

106

—

#95

—

106

—

C₂×Q₈

—

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂²×C₄

—

102

C₂²×C₄

—

106

—

103

C₂²×C₄

—

106

—

104

C₂²×C₄

C₂²⋊D₅₃

yes

characteristic

105

C₂²×C₄

C₂²⋊D₅₃

yes

characteristic

106

C₂²×C₄

C₂²×D₅₃

yes

characteristic

107

C₂×C₈

—

108

C₂×C₈

—

106

—

109

C₂×C₈

—

106

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

106

—

114

C₄⋊C₄

—

106

—

115

C₄⋊C₄

—

212

—

116

C₄⋊C₄

C₂²⋊D₅₃

yes

—

117

C₄²

—

118

C₂³

C₂×C₂²⋊D₅₃

yes

characteristic

119

Q₈

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

106

—

128

C₂×C₄

—

212

—

129

C₂×C₄

—

212

—

130

C₂×C₄

—

212

—

131

C₂×C₄

—

212

—

132

C₂×C₄

—

212

—

133

C₂×C₄

#11719922

yes

—

134

C₂×C₄

#11719925

yes

—

135

C₂×C₄

Q₈×D₅₃

yes

—

136

C₂×C₄

Q₈×D₅₃

yes

characteristic

137

C₂×C₄

#11719949

yes

characteristic

138

C₂×C₄

C₂×C₂²⋊D₅₃

yes

—

139

C₈

—

106

—

140

C₂²

#13133186

yes

characteristic

141

C₂²

#13133193

yes

characteristic

142

C₂²

#13133215

yes

characteristic

143

C₂²

C₂×#11719922

yes

characteristic

144

C₂²

C₂×#11719925

yes

characteristic

145

C₂²

#13133243

yes

characteristic

146

C₂²

#13133248

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

—

424

—

151

C₄

—

212

—

152

C₄

#13133193

yes

—

153

C₄

#13133215

yes

—

154

C₂

#16810542

yes

characteristic

155

C₂

#16810757

yes

characteristic

156

C₂

#16810779

yes

characteristic

157

C₂

#16810900

yes

characteristic

158

C₂

#16810915

yes

characteristic

159

C₂

#16811026

yes

characteristic

160

C₂

#16811029

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₅₃
commutator	C₂²×C₄×C₅₃	C₂³
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₂×C₄⋊Q₈×C₅₃	C₂
socle	C₂³×C₅₃	C₂×D₄
hypercenter	C₂×C₄⋊Q₈	D₅₃
hypocenter	C₅₃	#2057
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₅₃

commutator

C₂²×C₄×C₅₃

C₂³

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₂×C₄⋊Q₈×C₅₃

C₂

socle

C₂³×C₅₃

C₂×D₄

hypercenter

C₂×C₄⋊Q₈

D₅₃

hypocenter

C₅₃

#2057

LCS

G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
32. C₄⋊Dic₅₃	Q₈	semidirect
36. C₂×#88322	Q₈	semidirect
86. C₅₃	#2057	semidirect

H ≅ G/N

type

32. C₄⋊Dic₅₃

Q₈

semidirect

36. C₂×#88322

Q₈

semidirect

86. C₅₃

#2057

semidirect

Subgroups — #20276040

Conjugacy classes of subgroups

Complements