Group #20290813 — subgroups

Subgroups — #20290813

Order 6784.

260 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 110 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×#13133284	C₂	1	2	yes	maximal
2	C₂×C₄⋊#88324	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×#88324	C₂	1	1	yes	maximal, characteristic
4	#16811489	C₂	1	2	yes	maximal
5	C₄⋊C₄×Dic₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×#88324	C₂²	1	2	yes	—
7	C₂²×#88324	C₂²	1	1	yes	characteristic
8	C₂²×#88324	C₂²	1	1	yes	characteristic
9	#13133284	—	2	4	no	—
10	C₄⋊#88324	—	2	2	no	—
11	C₄×#88324	—	2	2	no	—
12	C₂×C₄⋊Dic₅₃	C₄	1	1	yes	characteristic
13	C₂×C₄×Dic₅₃	C₂²	1	2	yes	—
14	C₂×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
15	#13133124	C₄	1	2	yes	—
16	C₂×C₄⋊C₄×C₅₃	C₄	1	1	yes	characteristic
17	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
18	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×Dic₅₃	C₂×C₄	1	1	yes	characteristic
20	C₂²×Dic₅₃	C₂×C₄	1	2	yes	—
21	C₂²×Dic₅₃	C₂³	1	1	yes	characteristic
22	C₂×#88324	—	2	1	no	—
23	C₂×#88324	—	2	1	no	—
24	C₂×#88324	—	2	1	no	—
25	C₂×#88324	—	2	1	no	—
26	C₂×#88324	—	2	2	no	—
27	C₂×#88324	—	2	2	no	—
28	C₂×#88324	—	2	2	no	—
29	C₂×#88324	D₄	1	2	yes	—
30	C₂×#88324	Q₈	1	2	yes	—
31	C₄⋊C₄×C₅₃	—	2	2	no	—
32	C₄⋊Dic₅₃	C₈	1	4	yes	complemented
33	C₄×Dic₅₃	—	2	2	no	—
34	C₄×Dic₅₃	—	2	2	no	—
35	C₄×Dic₅₃	D₄	1	2	yes	—
36	C₄×Dic₅₃	Q₈	1	2	yes	—
37	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₅₃	—	2	2	no	—
39	C₂×C₄×C₅₃	—	2	2	no	—
40	C₂×C₄×C₅₃	—	2	2	no	—
41	C₂×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
42	C₂×C₄×C₅₃	C₂×C₈	1	2	yes	—
43	C₂×C₄×C₅₃	M₁₆	1	2	yes	—
44	C₂×Dic₅₃	—	2	2	no	—
45	C₂×Dic₅₃	—	2	1	no	—
46	C₂×Dic₅₃	—	2	1	no	—
47	C₂×Dic₅₃	—	2	2	no	—
48	C₂×Dic₅₃	C₄⋊C₄	1	2	yes	—
49	C₂×Dic₅₃	C₂×C₈	1	4	yes	—
50	C₂×Dic₅₃	C₂×D₄	1	1	yes	characteristic
51	C₂×Dic₅₃	C₂×D₄	1	1	yes	characteristic
52	C₂×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
53	C₂×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
54	C₂×Dic₅₃	#41	1	1	yes	characteristic
55	C₂×Dic₅₃	#41	1	1	yes	characteristic
56	#88324	—	2	2	no	—
57	#88324	—	2	2	no	—
58	#88324	—	4	2	no	—
59	C₂²×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₅₃	C₄×D₄	1	1	yes	characteristic
61	C₂²×C₅₃	C₄×Q₈	1	1	yes	characteristic
62	C₂²×C₅₃	C₂²×C₈	1	1	yes	characteristic
63	C₂²×C₅₃	C₂×M₁₆	1	1	yes	characteristic
64	C₂²×C₅₃	#131	1	1	yes	characteristic
65	C₂²×C₅₃	#131	1	1	yes	characteristic
66	C₄×C₅₃	—	4	2	no	—
67	C₄×C₅₃	C₄⋊C₈	1	4	yes	complemented
68	Dic₅₃	—	2	2	no	—
69	Dic₅₃	—	2	4	no	—
70	Dic₅₃	#121	1	1	yes	characteristic
71	Dic₅₃	C₂²⋊Q₈	1	1	yes	characteristic
72	Dic₅₃	#125	1	1	yes	characteristic
73	Dic₅₃	C₄⋊Q₈	1	1	yes	characteristic
74	#1939	—	53	1	no	maximal, Sylow
75	C₂×C₅₃	#389	1	1	yes	characteristic
76	C₂×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
77	C₂×C₅₃	#424	1	1	yes	characteristic
78	C₂×C₅₃	C₈×D₄	1	1	yes	characteristic
79	C₂×C₅₃	#436	1	1	yes	characteristic
80	C₂×C₅₃	C₈×Q₈	1	1	yes	characteristic
81	C₂×C₅₃	#446	1	1	yes	characteristic
82	C₂×C₄⋊C₈	—	53	1	no	—
83	C₂×C₄⋊C₈	—	53	2	no	—
84	C₂×C₄×C₈	—	53	1	no	—
85	C₄×C₄⋊C₄	—	53	1	no	—
86	#336	—	53	2	no	—
87	C₅₃	#1939	1	1	yes	characteristic, complemented, Sylow
88	C₂²×C₈	—	53	1	no	—
89	C₂²×C₈	—	53	1	no	—
90	C₂²×C₈	—	53	2	no	—
91	C₂×C₄⋊C₄	—	53	1	no	—
92	C₂×C₄⋊C₄	#6157	1	1	yes	characteristic
93	C₂×C₄²	—	53	1	no	—
94	C₂×C₄²	—	53	2	no	—
95	C₄⋊C₈	—	106	2	no	—
96	C₄⋊C₈	—	106	4	no	—
97	C₄×C₈	—	106	2	no	—
98	#95	—	53	2	no	—
99	C₂²×C₄	—	53	1	no	—
100	C₂²×C₄	—	53	1	no	—
101	C₂²×C₄	—	53	2	no	—
102	C₂²×C₄	C₂×#6157	1	2	yes	—
103	C₂²×C₄	C₂×#6157	1	1	yes	characteristic
104	C₂×C₈	—	53	2	no	—
105	C₂×C₈	—	53	2	no	—
106	C₂×C₈	—	106	1	no	—
107	C₂×C₈	—	106	1	no	—
108	C₂×C₈	—	106	1	no	—
109	C₂×C₈	—	106	1	no	—
110	C₂×C₈	—	106	2	no	—
111	C₂×C₈	—	106	2	no	—
112	C₂×C₈	—	106	2	no	—
113	C₄⋊C₄	—	2	2	no	—
114	C₄⋊C₄	—	53	4	no	—
115	C₄²	—	53	2	no	—
116	C₄²	—	53	2	no	—
117	C₄²	—	106	2	no	—
118	C₄²	—	106	2	no	—
119	C₂³	C₂²×#6157	1	1	yes	characteristic
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	53	1	no	—
124	C₂×C₄	—	53	1	no	—
125	C₂×C₄	—	53	1	no	—
126	C₂×C₄	—	53	1	no	—
127	C₂×C₄	—	53	1	no	—
128	C₂×C₄	—	53	1	no	—
129	C₂×C₄	—	53	2	no	—
130	C₂×C₄	—	53	4	no	—
131	C₂×C₄	—	106	2	no	—
132	C₂×C₄	—	106	1	no	—
133	C₂×C₄	—	106	1	no	—
134	C₂×C₄	—	106	2	no	—
135	C₂×C₄	D₅₃⋊C₈	1	2	yes	—
136	C₂×C₄	C₄⋊#6157	1	2	yes	—
137	C₂×C₄	#11719940	1	2	yes	—
138	C₈	—	106	2	no	—
139	C₈	—	106	2	no	—
140	C₈	—	212	2	no	—
141	C₂²	C₂×C₄⋊#6157	1	1	yes	characteristic
142	C₂²	D₄×#6157	1	1	yes	characteristic
143	C₂²	Q₈×#6157	1	1	yes	characteristic
144	C₂²	C₂×D₅₃⋊C₈	1	1	yes	characteristic
145	C₂²	C₂×#11719940	1	1	yes	characteristic
146	C₂²	#13133304	1	1	yes	characteristic
147	C₂²	#13133305	1	1	yes	characteristic
148	C₄	—	4	2	no	—
149	C₄	—	53	1	no	—
150	C₄	—	53	1	no	—
151	C₄	—	53	1	no	—
152	C₄	—	53	1	no	—
153	C₄	—	106	2	no	—
154	C₄	—	106	4	no	—
155	C₄	#13133284	1	4	yes	complemented
156	C₂	#16811574	1	1	yes	characteristic
157	C₂	C₂×#13133284	1	1	yes	characteristic
158	C₂	#16811630	1	1	yes	characteristic
159	C₂	#16811651	1	1	yes	characteristic
160	C₂	#16811658	1	1	yes	characteristic
161	C₂	#16811672	1	1	yes	characteristic
162	C₂	#16811674	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133284

C₂

yes

maximal

C₂×C₄⋊#88324

C₂

yes

maximal, characteristic

C₂×C₄×#88324

C₂

yes

maximal, characteristic

#16811489

C₂

yes

maximal

C₄⋊C₄×Dic₅₃

C₂

yes

maximal, characteristic

C₂²×#88324

C₂²

yes

—

C₂²×#88324

C₂²

yes

characteristic

C₂²×#88324

C₂²

yes

—

—

—

C₂×C₄⋊Dic₅₃

C₄

yes

characteristic

C₂×C₄×Dic₅₃

C₂²

yes

—

C₂×C₄×Dic₅₃

C₂²

yes

characteristic

#13133124

C₄

yes

—

C₂×C₄⋊C₄×C₅₃

C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×Dic₅₃

C₂×C₄

yes

characteristic

C₂²×Dic₅₃

C₂×C₄

yes

—

C₂²×Dic₅₃

C₂³

yes

—

—

—

—

—

—

—

D₄

yes

—

C₂×#88324

Q₈

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊Dic₅₃

C₈

yes

complemented

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₄×Dic₅₃

D₄

yes

—

C₄×Dic₅₃

Q₈

yes

—

C₂³×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₄⋊C₄

yes

—

C₂×C₄×C₅₃

C₂×C₈

yes

—

C₂×C₄×C₅₃

M₁₆

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

C₄⋊C₄

yes

—

C₂×Dic₅₃

C₂×C₈

yes

—

C₂×Dic₅₃

C₂×D₄

yes

characteristic

C₂×Dic₅₃

C₂×D₄

yes

characteristic

C₂×Dic₅₃

C₂×Q₈

yes

characteristic

C₂×Dic₅₃

C₂×Q₈

yes

C₂×Dic₅₃

yes

C₂×Dic₅₃

yes

—

—

—

C₂²×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₅₃

C₄×D₄

yes

characteristic

C₂²×C₅₃

C₄×Q₈

yes

characteristic

C₂²×C₅₃

C₂²×C₈

yes

characteristic

C₂²×C₅₃

C₂×M₁₆

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

C₄⋊C₈

yes

complemented

Dic₅₃

—

Dic₅₃

—

Dic₅₃

#121

yes

characteristic

Dic₅₃

C₂²⋊Q₈

yes

characteristic

Dic₅₃

#125

yes

characteristic

Dic₅₃

C₄⋊Q₈

yes

—

C₂×C₅₃

yes

C₂×C₅₃

C₂×C₄⋊C₈

yes

characteristic

C₂×C₅₃

#424

yes

characteristic

C₂×C₅₃

C₈×D₄

yes

characteristic

C₂×C₅₃

#436

yes

characteristic

C₂×C₅₃

C₈×Q₈

yes

characteristic

C₂×C₅₃

#446

yes

characteristic

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄×C₈

—

C₄×C₄⋊C₄

—

#336

—

C₅₃

#1939

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

#6157

yes

characteristic

C₂×C₄²

—

C₂×C₄²

—

C₄⋊C₈

—

106

—

C₄⋊C₈

—

106

—

C₄×C₈

—

106

—

#95

—

C₂²×C₄

—

100

C₂²×C₄

—

101

C₂²×C₄

—

102

C₂²×C₄

C₂×#6157

yes

—

103

C₂²×C₄

C₂×#6157

yes

characteristic

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

106

—

107

C₂×C₈

—

106

—

108

C₂×C₈

—

106

—

109

C₂×C₈

—

106

—

110

C₂×C₈

—

106

—

111

C₂×C₈

—

106

—

112

C₂×C₈

—

106

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄²

—

116

C₄²

—

117

C₄²

—

106

—

118

C₄²

—

106

—

119

C₂³

C₂²×#6157

yes

characteristic

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

106

—

132

C₂×C₄

—

106

—

133

C₂×C₄

—

106

—

134

C₂×C₄

—

106

—

135

C₂×C₄

D₅₃⋊C₈

yes

—

136

C₂×C₄

C₄⋊#6157

yes

—

137

C₂×C₄

#11719940

yes

—

138

C₈

—

106

—

139

C₈

—

106

—

140

C₈

—

212

—

141

C₂²

C₂×C₄⋊#6157

yes

characteristic

142

C₂²

D₄×#6157

yes

characteristic

143

C₂²

Q₈×#6157

yes

characteristic

144

C₂²

C₂×D₅₃⋊C₈

yes

characteristic

145

C₂²

C₂×#11719940

yes

characteristic

146

C₂²

#13133304

yes

characteristic

147

C₂²

#13133305

yes

characteristic

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

—

106

—

154

C₄

—

106

—

155

C₄

#13133284

yes

complemented

156

C₂

#16811574

yes

characteristic

157

C₂

C₂×#13133284

yes

characteristic

158

C₂

#16811630

yes

characteristic

159

C₂

#16811651

yes

characteristic

160

C₂

#16811658

yes

characteristic

161

C₂

#16811672

yes

characteristic

162

C₂

#16811674

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×#6157
commutator	C₂²×C₅₃	C₂²×C₈
Fitting	C₂×C₄⋊C₄×C₅₃	C₄
socle	C₂³×C₅₃	C₂²×C₄
hypercenter	C₂×C₄⋊C₄	#6157
hypocenter	C₅₃	#1939
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

subgroup

G/H

center, Frattini

C₂³

C₂²×#6157

commutator

C₂²×C₅₃

C₂²×C₈

Fitting

C₂×C₄⋊C₄×C₅₃

C₄

socle

C₂³×C₅₃

C₂²×C₄

C₂×C₄⋊C₄

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

N	H ≅ G/N	type
32. C₄⋊Dic₅₃	C₈	semidirect
67. C₄×C₅₃	C₄⋊C₈	semidirect
87. C₅₃	#1939	semidirect
155. C₄	#13133284	semidirect

H ≅ G/N

type

32. C₄⋊Dic₅₃

C₈

semidirect

67. C₄×C₅₃

C₄⋊C₈

semidirect

87. C₅₃

#1939

semidirect

155. C₄

#13133284

semidirect

Subgroups — #20290813

Conjugacy classes of subgroups

Complements