Group #20273854 — subgroups

Subgroups — #20273854

Order 6784.

206 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 59 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	Q₈×Dic₁₀₆	C₂	1	1	yes	maximal, characteristic
2	#16811002	C₂	1	1	yes	maximal, characteristic
3	#16810915	C₂	1	1	yes	maximal, characteristic
4	#16810371	C₂	1	1	yes	maximal, characteristic
5	Q₈⋊#88322	C₂	1	1	yes	maximal, characteristic
6	#16810353	C₂	1	1	yes	maximal, characteristic
7	C₅₃×#477	C₂	1	1	yes	maximal, characteristic
8	C₂×#11719925	—	2	2	no	—
9	#13133217	—	2	1	no	—
10	#13133217	C₂²	1	1	yes	characteristic
11	#13133216	—	2	1	no	—
12	#13133216	—	2	2	no	—
13	Q₈×Dic₅₃	—	2	1	no	—
14	C₄×Dic₁₀₆	—	2	1	no	—
15	C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic
16	#13133150	—	2	1	no	—
17	C₄⋊#88322	C₂²	1	1	yes	characteristic
18	#13133140	—	2	1	no	—
19	C₄×#88322	C₂²	1	1	yes	characteristic
20	C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
22	#107×C₅₃	—	2	1	no	—
23	C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
24	#103×C₅₃	—	2	1	no	—
25	C₂×Q₈×C₅₃	—	2	1	no	—
26	C₂×Q₈×C₅₃	D₄	1	1	yes	characteristic
27	C₂×Dic₁₀₆	—	2	2	no	—
28	C₂×Dic₁₀₆	D₄	1	1	yes	characteristic
29	C₂×Dic₁₀₆	D₄	1	2	yes	—
30	C₂×C₈×C₅₃	—	2	1	no	—
31	C₄⋊C₄×C₅₃	—	2	1	no	—
32	C₄⋊C₄×C₅₃	—	2	1	no	—
33	C₄⋊C₄×C₅₃	—	2	1	no	—
34	C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
35	C₄²×C₅₃	—	2	1	no	—
36	C₄²×C₅₃	C₂³	1	1	yes	characteristic
37	#11719925	—	4	4	no	—
38	C₄⋊Dic₅₃	—	2	2	no	—
39	C₄⋊Dic₅₃	—	2	1	no	—
40	C₄⋊Dic₅₃	—	2	1	no	—
41	C₄⋊Dic₅₃	D₄	1	1	yes	characteristic
42	#11719919	—	2	2	no	—
43	#11719919	—	2	1	no	—
44	C₄×Dic₅₃	—	2	2	no	—
45	C₄×Dic₅₃	—	2	1	no	—
46	C₂×#88322	—	2	1	no	—
47	C₂×#88322	—	2	1	no	—
48	Q₈×C₅₃	—	2	1	no	—
49	Q₈×C₅₃	—	4	2	no	—
50	Q₈×C₅₃	Q₁₆	1	2	yes	complemented
51	C₂×C₄×C₅₃	—	2	1	no	—
52	C₂×C₄×C₅₃	—	2	1	no	—
53	C₂×C₄×C₅₃	—	4	1	no	—
54	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
57	C₂×Dic₅₃	—	2	1	no	—
58	C₂×Dic₅₃	—	2	1	no	—
59	C₂×Dic₅₃	—	2	2	no	—
60	Dic₁₀₆	—	2	1	no	—
61	Dic₁₀₆	—	4	2	no	—
62	Dic₁₀₆	—	8	2	no	—
63	Dic₁₀₆	Q₁₆	1	2	yes	complemented
64	C₈×C₅₃	—	4	1	no	—
65	#88322	—	2	2	no	—
66	#88322	—	4	1	no	—
67	C₂²×C₅₃	#120	1	1	yes	characteristic
68	C₄×C₅₃	—	2	2	no	—
69	C₄×C₅₃	—	2	1	no	—
70	C₄×C₅₃	—	4	1	no	—
71	C₄×C₅₃	—	8	1	no	—
72	C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
73	C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
74	C₄×C₅₃	#137	1	1	yes	characteristic
75	C₄×C₅₃	#137	1	1	yes	characteristic
76	Dic₅₃	—	2	2	no	—
77	Dic₅₃	—	4	2	no	—
78	Dic₅₃	—	4	1	no	—
79	#1633	—	53	1	no	maximal, Sylow
80	C₂×C₅₃	#451	1	1	yes	characteristic
81	C₂×C₅₃	#451	1	1	yes	characteristic
82	C₂×C₅₃	#453	1	1	yes	characteristic
83	Q₈²	—	53	1	no	—
84	#494	—	53	1	no	—
85	#477	—	53	1	no	—
86	#477	D₅₃	1	1	yes	characteristic
87	#333	—	53	1	no	—
88	Q₈⋊C₈	—	53	1	no	—
89	Q₈⋊C₈	—	53	1	no	—
90	C₅₃	#1633	1	1	yes	characteristic, complemented, Sylow
91	C₂×Q₁₆	—	106	2	no	—
92	C₄⋊Q₈	—	53	1	no	—
93	C₄⋊Q₈	—	106	1	no	—
94	C₄⋊Q₈	—	106	2	no	—
95	C₄⋊Q₈	—	106	1	no	—
96	C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic
97	C₄×Q₈	—	53	1	no	—
98	C₄×Q₈	—	106	1	no	—
99	C₄×Q₈	—	106	1	no	—
100	C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
101	#107	—	2	1	no	—
102	#107	—	106	1	no	—
103	C₄⋊C₈	—	53	1	no	—
104	C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
105	#103	—	2	1	no	—
106	#103	—	106	1	no	—
107	C₄×C₈	—	53	1	no	—
108	C₂×Q₈	—	2	1	no	—
109	C₂×Q₈	—	53	1	no	—
110	C₂×Q₈	—	53	2	no	—
111	C₂×Q₈	—	106	2	no	—
112	C₂×Q₈	C₂²⋊D₅₃	1	1	yes	characteristic
113	Q₁₆	—	212	4	no	—
114	C₂×C₈	—	2	1	no	—
115	C₂×C₈	—	106	1	no	—
116	C₂×C₈	—	106	1	no	—
117	C₄⋊C₄	—	2	1	no	—
118	C₄⋊C₄	—	2	1	no	—
119	C₄⋊C₄	—	2	1	no	—
120	C₄⋊C₄	—	53	1	no	—
121	C₄⋊C₄	—	106	2	no	—
122	C₄⋊C₄	—	106	2	no	—
123	C₄⋊C₄	—	106	1	no	—
124	C₄⋊C₄	—	106	1	no	—
125	C₄⋊C₄	—	106	1	no	—
126	C₄⋊C₄	C₂²⋊D₅₃	1	1	yes	characteristic
127	C₄²	—	2	1	no	—
128	C₄²	—	106	2	no	—
129	C₄²	—	106	1	no	—
130	C₄²	C₂²×D₅₃	1	1	yes	characteristic
131	Q₈	—	2	1	no	—
132	Q₈	—	4	2	no	—
133	Q₈	—	53	2	no	—
134	Q₈	—	106	1	no	—
135	Q₈	—	424	2	no	—
136	Q₈	—	212	2	no	—
137	Q₈	#11719925	1	2	yes	complemented
138	C₂×C₄	—	2	1	no	—
139	C₂×C₄	—	2	1	no	—
140	C₂×C₄	—	4	1	no	—
141	C₂×C₄	—	106	1	no	—
142	C₂×C₄	—	106	1	no	—
143	C₂×C₄	—	106	2	no	—
144	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
145	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
146	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
147	C₈	—	4	1	no	—
148	C₈	—	106	2	no	—
149	C₈	—	212	1	no	—
150	C₂²	#13133185	1	1	yes	characteristic
151	C₄	—	2	2	no	—
152	C₄	—	2	1	no	—
153	C₄	—	4	1	no	—
154	C₄	—	8	1	no	—
155	C₄	—	106	2	no	—
156	C₄	—	212	1	no	—
157	C₄	—	212	2	no	—
158	C₄	Q₁₆×D₅₃	1	1	yes	characteristic
159	C₄	C₂×#11719925	1	1	yes	characteristic
160	C₄	#13133247	1	1	yes	characteristic
161	C₄	#13133249	1	1	yes	characteristic
162	C₂	#16810777	1	1	yes	characteristic
163	C₂	#16810778	1	1	yes	characteristic
164	C₂	#16810789	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×Dic₁₀₆

C₂

yes

maximal, characteristic

#16811002

C₂

yes

maximal, characteristic

#16810915

C₂

yes

maximal, characteristic

#16810371

C₂

yes

maximal, characteristic

Q₈⋊#88322

C₂

yes

maximal, characteristic

#16810353

C₂

yes

maximal, characteristic

C₅₃×#477

C₂

yes

maximal, characteristic

C₂×#11719925

—

#13133217

—

#13133217

C₂²

yes

characteristic

#13133216

—

#13133216

—

Q₈×Dic₅₃

—

C₄×Dic₁₀₆

—

C₄×Dic₁₀₆

C₂²

yes

characteristic

#13133150

—

C₄⋊#88322

C₂²

yes

characteristic

#13133140

—

C₄×#88322

C₂²

yes

characteristic

C₄⋊Q₈×C₅₃

C₂²

yes

characteristic

C₄×Q₈×C₅₃

C₂²

yes

characteristic

#107×C₅₃

—

C₄⋊C₈×C₅₃

C₂²

yes

characteristic

#103×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

D₄

yes

characteristic

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

D₄

yes

characteristic

C₂×Dic₁₀₆

D₄

yes

—

C₂×C₈×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

—

C₄²×C₅₃

C₂³

yes

characteristic

#11719925

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

D₄

yes

characteristic

#11719919

—

#11719919

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₂×#88322

—

C₂×#88322

—

Q₈×C₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

Q₁₆

yes

complemented

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

Q₁₆

yes

complemented

C₈×C₅₃

—

#88322

—

#88322

—

C₂²×C₅₃

#120

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×Q₁₆

yes

characteristic

C₄×C₅₃

C₂×Q₁₆

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

Q₈²

—

#494

—

#477

—

#477

D₅₃

yes

characteristic

#333

—

Q₈⋊C₈

—

Q₈⋊C₈

—

C₅₃

#1633

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

106

—

C₄⋊Q₈

—

C₄⋊Q₈

—

106

—

C₄⋊Q₈

—

106

—

C₄⋊Q₈

—

106

—

C₄⋊Q₈

C₂×D₅₃

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

106

—

C₄×Q₈

—

106

—

100

C₄×Q₈

C₂×D₅₃

yes

characteristic

101

#107

—

102

#107

—

106

—

103

C₄⋊C₈

—

104

C₄⋊C₈

C₂×D₅₃

yes

characteristic

105

#103

—

106

#103

—

106

—

107

C₄×C₈

—

108

C₂×Q₈

—

109

C₂×Q₈

—

110

C₂×Q₈

—

111

C₂×Q₈

—

106

—

112

C₂×Q₈

C₂²⋊D₅₃

yes

characteristic

113

Q₁₆

—

212

—

114

C₂×C₈

—

115

C₂×C₈

—

106

—

116

C₂×C₈

—

106

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

106

—

122

C₄⋊C₄

—

106

—

123

C₄⋊C₄

—

106

—

124

C₄⋊C₄

—

106

—

125

C₄⋊C₄

—

106

—

126

C₄⋊C₄

C₂²⋊D₅₃

yes

characteristic

127

C₄²

—

128

C₄²

—

106

—

129

C₄²

—

106

—

130

C₄²

C₂²×D₅₃

yes

characteristic

131

Q₈

—

132

Q₈

—

133

Q₈

—

134

Q₈

—

106

—

135

Q₈

—

424

—

136

Q₈

—

212

—

137

Q₈

#11719925

yes

complemented

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

106

—

142

C₂×C₄

—

106

—

143

C₂×C₄

—

106

—

144

C₂×C₄

D₄×D₅₃

yes

characteristic

145

C₂×C₄

D₄×D₅₃

yes

characteristic

146

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

147

C₈

—

148

C₈

—

106

—

149

C₈

—

212

—

150

C₂²

#13133185

yes

characteristic

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

—

106

—

156

C₄

—

212

—

157

C₄

—

212

—

158

C₄

Q₁₆×D₅₃

yes

characteristic

159

C₄

C₂×#11719925

yes

characteristic

160

C₄

#13133247

yes

characteristic

161

C₄

#13133249

yes

characteristic

162

C₂

#16810777

yes

characteristic

163

C₂

#16810778

yes

characteristic

164

C₂

#16810789

yes

characteristic

	subgroup	G/H
center	C₂²	#13133185
commutator	C₄²×C₅₃	C₂³
Frattini	C₄²	C₂²×D₅₃
Fitting	C₅₃×#477	C₂
socle	C₂²×C₅₃	#120
hypercenter	#477	D₅₃
hypocenter	C₅₃	#1633
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #477

subgroup

G/H

center

C₂²

#13133185

commutator

C₄²×C₅₃

C₂³

Frattini

C₄²

C₂²×D₅₃

Fitting

C₅₃×#477

C₂

C₂²×C₅₃

D₅₃

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #477

N	H ≅ G/N	type
50. Q₈×C₅₃	Q₁₆	semidirect
63. Dic₁₀₆	Q₁₆	semidirect
90. C₅₃	#1633	semidirect
137. Q₈	#11719925	semidirect

H ≅ G/N

type

50. Q₈×C₅₃

Q₁₆

semidirect

63. Dic₁₀₆

Q₁₆

semidirect

90. C₅₃

#1633

semidirect

137. Q₈

#11719925

semidirect

Subgroups — #20273854

Conjugacy classes of subgroups

Complements