Group #20273038 — subgroups

Subgroups — #20273038

Order 6784.

248 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 97 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×C₄×Dic₁₀₆	C₂	1	1	yes	maximal, characteristic
2	#16810676	C₂	1	1	yes	maximal, characteristic
3	#16810348	C₂	1	4	yes	maximal
4	C₅₃×#431	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₁₀₆	C₄	1	1	yes	characteristic
6	C₄×Dic₁₀₆	—	2	1	no	—
7	C₄×Dic₁₀₆	—	2	1	no	—
8	C₄×Dic₁₀₆	—	2	1	no	—
9	C₄×Dic₁₀₆	—	2	1	no	—
10	C₂×C₄⋊Dic₅₃	C₄	1	1	yes	characteristic
11	C₂×#11719919	C₂²	1	2	yes	—
12	C₂×C₄×Dic₅₃	C₄	1	2	yes	—
13	C₄⋊#88322	—	2	1	no	—
14	C₂²⋊#88322	C₂²	1	2	yes	—
15	C₄×#88322	—	2	1	no	—
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	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
23	C₂²×Dic₅₃	C₂×C₄	1	2	yes	—
24	C₂²×Dic₅₃	C₂×C₄	1	2	yes	—
25	C₂×Dic₁₀₆	—	2	1	no	—
26	C₂×Dic₁₀₆	—	2	1	no	—
27	C₂×Dic₁₀₆	—	2	2	no	—
28	C₂×Dic₁₀₆	C₈	1	4	yes	complemented
29	C₂×C₈×C₅₃	—	2	2	no	—
30	C₄²×C₅₃	D₄	1	1	yes	characteristic
31	C₄²×C₅₃	D₄	1	1	yes	characteristic
32	C₄²×C₅₃	D₄	1	1	yes	characteristic
33	C₄²×C₅₃	D₄	1	1	yes	characteristic
34	C₄⋊Dic₅₃	—	2	1	no	—
35	C₄⋊Dic₅₃	—	2	1	no	—
36	#11719919	—	2	2	no	—
37	#11719919	—	2	2	no	—
38	C₄×Dic₅₃	—	2	2	no	—
39	C₄×Dic₅₃	—	2	2	no	—
40	C₂×#88322	—	2	2	no	—
41	C₂³×C₅₃	C₂²×C₄	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	C₂×C₈	1	2	yes	—
49	C₂×C₄×C₅₃	M₁₆	1	2	yes	—
50	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
52	C₂×Dic₅₃	—	2	2	no	—
53	C₂×Dic₅₃	—	2	2	no	—
54	C₂×Dic₅₃	—	2	2	no	—
55	C₂×Dic₅₃	—	4	2	no	—
56	C₂×Dic₅₃	C₂×C₈	1	4	yes	—
57	Dic₁₀₆	—	2	4	no	—
58	Dic₁₀₆	—	2	4	no	—
59	C₈×C₅₃	—	4	2	no	—
60	#88322	—	4	2	no	—
61	C₂²×C₅₃	—	2	1	no	—
62	C₂²×C₅₃	—	2	1	no	—
63	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₅₃	C₂²×C₈	1	1	yes	characteristic
65	C₂²×C₅₃	C₂×M₁₆	1	1	yes	characteristic
66	C₄×C₅₃	—	2	1	no	—
67	C₄×C₅₃	—	2	1	no	—
68	C₄×C₅₃	—	2	1	no	—
69	C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
70	C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
71	C₄×C₅₃	#101	1	2	yes	—
72	Dic₅₃	—	2	4	no	—
73	Dic₅₃	—	4	2	no	—
74	#1461	—	53	1	no	maximal, Sylow
75	C₂×C₅₃	—	2	1	no	—
76	C₂×C₅₃	—	2	1	no	—
77	C₂×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
78	C₂×C₅₃	#410	1	1	yes	characteristic
79	C₂×C₅₃	C₂×#101	1	1	yes	characteristic
80	C₂×C₄×Q₈	—	53	1	no	—
81	#431	—	53	1	no	—
82	#431	D₅₃	1	1	yes	characteristic
83	#324	—	53	4	no	—
84	C₅₃	#1461	1	1	yes	characteristic, complemented, Sylow
85	C₂²×Q₈	—	53	1	no	—
86	C₄×Q₈	—	106	1	no	—
87	C₄×Q₈	—	106	1	no	—
88	C₄×Q₈	—	106	1	no	—
89	C₄×Q₈	—	106	1	no	—
90	C₂×C₄⋊C₄	—	53	1	no	—
91	C₂×C₄⋊C₄	—	53	2	no	—
92	C₂×C₄²	—	53	2	no	—
93	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
94	C₄⋊C₈	—	2	1	no	—
95	C₄⋊C₈	—	106	1	no	—
96	C₂²⋊C₈	—	53	2	no	—
97	C₂²⋊C₈	C₂×D₅₃	1	2	yes	—
98	C₄×C₈	—	2	1	no	—
99	C₄×C₈	—	106	1	no	—
100	C₂×Q₈	—	53	4	no	—
101	C₂×Q₈	—	106	2	no	—
102	C₂×Q₈	—	106	1	no	—
103	C₂×Q₈	—	106	1	no	—
104	C₂²×C₄	—	53	2	no	—
105	C₂²×C₄	—	53	2	no	—
106	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
107	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
108	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
109	C₂×C₈	—	2	2	no	—
110	C₂×C₈	—	106	2	no	—
111	C₄⋊C₄	—	106	2	no	—
112	C₄⋊C₄	—	106	2	no	—
113	C₄⋊C₄	—	106	1	no	—
114	C₄⋊C₄	—	106	1	no	—
115	C₄²	—	106	2	no	—
116	C₄²	—	106	2	no	—
117	C₄²	D₂₁₂	1	1	yes	characteristic
118	C₄²	D₂₁₂	1	1	yes	characteristic
119	C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
120	C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
121	C₂³	C₂×C₄×D₅₃	1	1	yes	characteristic
122	Q₈	—	106	4	no	—
123	Q₈	—	106	4	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	4	1	no	—
126	C₂×C₄	—	53	4	no	—
127	C₂×C₄	—	106	2	no	—
128	C₂×C₄	—	106	2	no	—
129	C₂×C₄	—	106	2	no	—
130	C₂×C₄	—	212	2	no	—
131	C₂×C₄	C₈×D₅₃	1	2	yes	—
132	C₂×C₄	#11719912	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×D₂₁₂	1	1	yes	characteristic
138	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
139	C₈	—	4	2	no	—
140	C₈	—	212	2	no	—
141	C₂²	—	2	1	no	—
142	C₂²	—	2	1	no	—
143	C₂²	C₂×#11719921	1	1	yes	characteristic
144	C₂²	C₂×C₈×D₅₃	1	1	yes	characteristic
145	C₂²	C₂×#11719912	1	1	yes	characteristic
146	C₄	—	2	1	no	—
147	C₄	—	2	1	no	—
148	C₄	—	2	1	no	—
149	C₄	—	106	4	no	—
150	C₄	—	212	2	no	—
151	C₄	#13133129	1	2	yes	—
152	C₄	#13133129	1	2	yes	—
153	C₄	#13133135	1	2	yes	—
154	C₂	—	2	1	no	—
155	C₂	—	2	1	no	—
156	C₂	C₂×#13133129	1	1	yes	characteristic
157	C₂	#16810596	1	1	yes	characteristic
158	C₂	C₂×#13133135	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Dic₁₀₆

C₂

yes

maximal, characteristic

#16810676

C₂

yes

maximal, characteristic

#16810348

C₂

yes

maximal

C₅₃×#431

C₂

yes

maximal, characteristic

C₂²×Dic₁₀₆

C₄

yes

characteristic

C₄×Dic₁₀₆

—

C₄×Dic₁₀₆

—

C₄×Dic₁₀₆

—

C₄×Dic₁₀₆

—

C₂×C₄⋊Dic₅₃

C₄

yes

characteristic

C₂×#11719919

C₂²

yes

—

C₂×C₄×Dic₅₃

C₄

yes

—

C₄⋊#88322

—

C₂²⋊#88322

C₂²

yes

—

C₄×#88322

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

—

C₂²⋊C₈×C₅₃

C₂²

yes

—

C₄×C₈×C₅₃

—

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

C₂×C₄

yes

—

C₂²×Dic₅₃

C₂×C₄

yes

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

C₈

yes

complemented

C₂×C₈×C₅₃

—

C₄²×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

D₄

yes

characteristic

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

#11719919

—

#11719919

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₂×#88322

—

C₂³×C₅₃

C₂²×C₄

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

C₂×C₈

yes

—

C₂×C₄×C₅₃

M₁₆

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂×C₈

yes

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₅₃

C₂²×C₈

yes

characteristic

C₂²×C₅₃

C₂×M₁₆

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂²⋊C₈

yes

—

C₄×C₅₃

C₂²⋊C₈

yes

—

C₄×C₅₃

#101

yes

—

Dic₅₃

—

Dic₅₃

—

#1461

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

C₂×C₂²⋊C₈

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₄×Q₈

—

#431

—

#431

D₅₃

yes

characteristic

#324

—

C₅₃

#1461

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₄×Q₈

—

106

—

C₄×Q₈

—

106

—

C₄×Q₈

—

106

—

C₄×Q₈

—

106

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

106

—

C₂²⋊C₈

—

C₂²⋊C₈

C₂×D₅₃

yes

—

C₄×C₈

—

C₄×C₈

—

106

—

100

C₂×Q₈

—

101

C₂×Q₈

—

106

—

102

C₂×Q₈

—

106

—

103

C₂×Q₈

—

106

—

104

C₂²×C₄

—

105

C₂²×C₄

—

106

C₂²×C₄

C₄×D₅₃

yes

characteristic

107

C₂²×C₄

C₄×D₅₃

yes

characteristic

108

C₂²×C₄

C₂²×D₅₃

yes

characteristic

109

C₂×C₈

—

110

C₂×C₈

—

106

—

111

C₄⋊C₄

—

106

—

112

C₄⋊C₄

—

106

—

113

C₄⋊C₄

—

106

—

114

C₄⋊C₄

—

106

—

115

C₄²

—

106

—

116

C₄²

—

106

—

117

C₄²

D₂₁₂

yes

characteristic

118

C₄²

D₂₁₂

yes

characteristic

119

C₄²

C₂²⋊D₅₃

yes

characteristic

120

C₄²

C₂²⋊D₅₃

yes

characteristic

121

C₂³

C₂×C₄×D₅₃

yes

characteristic

122

Q₈

—

106

—

123

Q₈

—

106

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

106

—

128

C₂×C₄

—

106

—

129

C₂×C₄

—

106

—

130

C₂×C₄

—

212

—

131

C₂×C₄

C₈×D₅₃

yes

—

132

C₂×C₄

#11719912

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×D₂₁₂

yes

characteristic

138

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

139

C₈

—

140

C₈

—

212

—

141

C₂²

—

142

C₂²

—

143

C₂²

C₂×#11719921

yes

characteristic

144

C₂²

C₂×C₈×D₅₃

yes

characteristic

145

C₂²

C₂×#11719912

yes

characteristic

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

—

106

—

150

C₄

—

212

—

151

C₄

#13133129

yes

—

152

C₄

#13133129

yes

—

153

C₄

#13133135

yes

—

154

C₂

—

155

C₂

—

156

C₂

C₂×#13133129

yes

characteristic

157

C₂

#16810596

yes

characteristic

158

C₂

C₂×#13133135

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#11719921
commutator	C₂²×C₅₃	C₂²×C₈
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₅₃×#431	C₂
socle	C₂²×C₅₃	C₂×C₂²⋊C₄
hypercenter	#431	D₅₃
hypocenter	C₅₃	#1461
LCS	G ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

subgroup

G/H

center

C₂×C₄

#11719921

commutator

C₂²×C₅₃

C₂²×C₈

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₅₃×#431

C₂

socle

C₂²×C₅₃

C₂×C₂²⋊C₄

D₅₃

C₅₃

G ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

N	H ≅ G/N	type
28. C₂×Dic₁₀₆	C₈	semidirect
84. C₅₃	#1461	semidirect

H ≅ G/N

type

28. C₂×Dic₁₀₆

C₈

semidirect

84. C₅₃

#1461

semidirect

Subgroups — #20273038

Conjugacy classes of subgroups

Complements