Group #20288005 — subgroups

Subgroups — C₂×#16811179 #20288005

Order 6784.

1168 conjugacy classes of subgroups; only the 541 normal subgroups are shown.

normal subgroups only

This group has 1168 conjugacy classes of subgroups — too many to draw. Only its 541 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	#16811179	C₂	1	16	yes	maximal, direct product factor
2	C₂×#13133216	C₂	1	2	yes	maximal, complemented
3	C₂×#13133209	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#13133195	C₂	1	4	yes	maximal
5	C₂×#13133182	C₂	1	4	yes	maximal, complemented
6	C₂×#13133175	C₂	1	2	yes	maximal
7	C₂×C₄²×D₅₃	C₂	1	1	yes	maximal, characteristic
8	C₂×#125×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×Dic₁₀₆	C₂²	1	4	yes	—
10	C₂²×C₄×D₅₃	C₂²	1	2	yes	—
11	C₂²×C₄×D₅₃	C₂²	1	1	yes	characteristic
12	#13133216	C₂²	1	16	yes	complemented
13	#13133209	C₂²	1	8	yes	complemented
14	#13133195	C₂²	1	32	yes	—
15	#13133182	C₂²	1	32	yes	complemented
16	#13133175	C₂²	1	16	yes	—
17	C₂×C₄⋊Dic₅₃	C₂²	1	2	yes	—
18	C₂×#11719919	C₂²	1	4	yes	—
19	C₂×#11719919	C₂²	1	4	yes	—
20	C₂×#11719921	C₂²	1	4	yes	—
21	C₂×C₄×Dic₅₃	C₂²	1	2	yes	—
22	C₂×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
23	C₂×C₄×Dic₅₃	C₂²	1	4	yes	—
24	C₄²×D₅₃	C₂²	1	8	yes	—
25	#125×C₅₃	C₂²	1	8	yes	complemented
26	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
27	C₂×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
28	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
29	C₂³×D₅₃	C₂³	1	1	yes	characteristic
30	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
31	C₂²×C₄×C₅₃	C₂³	1	4	yes	—
32	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
33	C₂²×Dic₅₃	C₂³	1	4	yes	—
34	C₂²×Dic₅₃	C₂³	1	1	yes	characteristic
35	C₂²×Dic₅₃	C₂³	1	2	yes	—
36	C₂×C₄×D₅₃	—	2	8	no	—
37	C₂×C₄×D₅₃	Q₈	1	8	yes	complemented
38	C₂×C₄×D₅₃	C₂³	1	8	yes	—
39	C₂×C₄×D₅₃	C₂³	1	4	yes	—
40	C₂×Dic₁₀₆	—	2	16	no	—
41	C₂×Dic₁₀₆	C₂³	1	16	yes	—
42	C₄⋊C₄×C₅₃	C₂³	1	8	yes	—
43	C₄⋊C₄×C₅₃	C₂³	1	16	yes	—
44	C₄²×C₅₃	C₂³	1	4	yes	—
45	#11719921	C₂³	1	16	yes	—
46	C₄⋊Dic₅₃	C₂³	1	8	yes	—
47	#11719919	C₂³	1	16	yes	—
48	#11719919	C₂³	1	16	yes	—
49	C₄×Dic₅₃	C₂³	1	8	yes	—
50	C₄×Dic₅₃	C₂³	1	4	yes	—
51	C₄×Dic₅₃	C₂³	1	16	yes	—
52	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
53	C₂²×D₅₃	—	2	4	no	—
54	C₂²×D₅₃	C₂×Q₈	1	4	yes	—
55	C₂²×D₅₃	C₂⁴	1	2	yes	—
56	C₂×C₄×C₅₃	—	2	8	no	—
57	C₂×C₄×C₅₃	—	2	4	no	—
58	C₂×C₄×C₅₃	C₂×Q₈	1	4	yes	—
59	C₂×C₄×C₅₃	C₂⁴	1	8	yes	—
60	C₂×C₄×C₅₃	C₂⁴	1	4	yes	—
61	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
62	C₂×Dic₅₃	—	2	8	no	—
63	C₂×Dic₅₃	C₂×Q₈	1	4	yes	—
64	C₂×Dic₅₃	#41	1	8	yes	—
65	C₂×Dic₅₃	C₂⁴	1	8	yes	—
66	C₂×Dic₅₃	C₂⁴	1	2	yes	—
67	C₂×Dic₅₃	C₂⁴	1	4	yes	—
68	C₄×D₅₃	—	2	16	no	—
69	C₄×D₅₃	C₂×Q₈	1	16	yes	complemented
70	Dic₁₀₆	—	2	32	no	—
71	C₂²×C₅₃	C₂²×Q₈	1	2	yes	—
72	C₂²×C₅₃	C₂×#41	1	4	yes	—
73	C₂²×C₅₃	C₂⁵	1	1	yes	characteristic
74	C₂×D₅₃	—	2	8	no	—
75	C₂×D₅₃	—	2	4	no	—
76	C₂×D₅₃	C₂²×Q₈	1	4	yes	—
77	C₄×C₅₃	—	2	8	no	—
78	C₄×C₅₃	—	2	4	no	—
79	C₄×C₅₃	C₂²×Q₈	1	4	yes	—
80	Dic₅₃	—	2	8	no	—
81	Dic₅₃	C₂²×Q₈	1	4	yes	—
82	Dic₅₃	C₂×#41	1	8	yes	—
83	C₂×#533	—	53	1	no	maximal, Sylow
84	C₂×C₅₃	#533	1	4	yes	complemented
85	C₂×C₅₃	C₂³×Q₈	1	1	yes	characteristic
86	C₂×C₅₃	C₂²×#41	1	2	yes	—
87	D₅₃	—	2	4	no	—
88	#533	—	53	16	no	—
89	C₂×C₄⋊Q₈	—	53	2	no	—
90	C₂×#125	—	53	1	no	—
91	C₂×#125	D₅₃	1	1	yes	characteristic, complemented
92	C₂×C₂²⋊Q₈	—	53	4	no	—
93	C₂×C₄×Q₈	—	53	4	no	—
94	C₂×#117	—	53	2	no	—
95	C₂²×C₄²	—	53	1	no	—
96	C₅₃	C₂×#533	1	1	yes	characteristic, complemented, Sylow
97	C₂²×Q₈	—	53	4	no	—
98	C₂³×C₄	—	53	1	no	—
99	C₂³×C₄	—	53	2	no	—
100	C₄⋊Q₈	—	53	16	no	—
101	#125	—	53	8	no	—
102	#125	C₂×D₅₃	1	8	yes	complemented
103	C₂²⋊Q₈	—	53	32	no	—
104	C₄×Q₈	—	53	32	no	—
105	#117	—	53	16	no	—
106	C₂×C₄⋊C₄	—	53	2	no	—
107	C₂×C₄⋊C₄	—	53	4	no	—
108	C₂×C₄⋊C₄	—	53	4	no	—
109	C₂×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
110	C₂×C₄⋊C₄	C₂×D₅₃	1	4	yes	—
111	C₂×C₂²⋊C₄	—	53	4	no	—
112	C₂×C₄²	—	53	4	no	—
113	C₂×C₄²	—	53	2	no	—
114	C₂×C₄²	—	53	1	no	—
115	C₂×C₄²	—	53	8	no	—
116	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
117	C₂⁴	—	53	1	no	—
118	C₂×Q₈	—	53	16	no	—
119	C₂×Q₈	—	106	16	no	—
120	C₂²×C₄	—	53	1	no	—
121	C₂²×C₄	—	53	2	no	—
122	C₂²×C₄	—	53	8	no	—
123	C₂²×C₄	—	53	4	no	—
124	C₂²×C₄	—	53	4	no	—
125	C₂²×C₄	—	53	8	no	—
126	C₂²×C₄	—	106	8	no	—
127	C₂²×C₄	C₂²×D₅₃	1	2	yes	—
128	C₂²×C₄	C₂²×D₅₃	1	4	yes	—
129	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
130	C₄⋊C₄	—	53	8	no	—
131	C₄⋊C₄	—	53	16	no	—
132	C₄⋊C₄	—	53	16	no	—
133	C₄⋊C₄	C₂²×D₅₃	1	8	yes	—
134	C₄⋊C₄	C₂²×D₅₃	1	16	yes	—
135	C₂²⋊C₄	—	53	16	no	—
136	C₄²	—	53	8	no	—
137	C₄²	—	53	4	no	—
138	C₄²	—	53	16	no	—
139	C₄²	C₂²×D₅₃	1	4	yes	—
140	C₂³	—	53	4	no	—
141	C₂³	—	53	2	no	—
142	C₂³	—	106	4	no	—
143	C₂³	C₂³×D₅₃	1	1	yes	characteristic
144	Q₈	—	106	32	no	—
145	C₂×C₄	—	2	4	no	—
146	C₂×C₄	—	2	8	no	—
147	C₂×C₄	—	53	4	no	—
148	C₂×C₄	—	53	8	no	—
149	C₂×C₄	—	53	2	no	—
150	C₂×C₄	—	53	4	no	—
151	C₂×C₄	—	53	16	no	—
152	C₂×C₄	—	53	8	no	—
153	C₂×C₄	—	106	16	no	—
154	C₂×C₄	—	106	8	no	—
155	C₂×C₄	Q₈×D₅₃	1	4	yes	—
156	C₂×C₄	C₂³×D₅₃	1	4	yes	—
157	C₂×C₄	C₂³×D₅₃	1	8	yes	—
158	C₂×C₄	C₂³×D₅₃	1	2	yes	—
159	C₂²	—	53	4	no	—
160	C₂²	—	106	8	no	—
161	C₂²	—	106	4	no	—
162	C₂²	C₂×Q₈×D₅₃	1	2	yes	—
163	C₂²	#41×D₅₃	1	4	yes	—
164	C₂²	C₂⁴×D₅₃	1	1	yes	characteristic
165	C₄	—	2	4	no	—
166	C₄	—	2	8	no	—
167	C₄	—	53	4	no	—
168	C₄	—	53	8	no	—
169	C₄	—	106	8	no	—
170	C₄	C₂×Q₈×D₅₃	1	4	yes	—
171	C₂	—	106	4	no	—
172	C₂	#16811179	1	4	yes	direct product factor
173	C₂	C₂²×Q₈×D₅₃	1	1	yes	characteristic
174	C₂	C₂×#41×D₅₃	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16811179

C₂

yes

maximal, direct product factor

C₂×#13133216

C₂

yes

maximal, complemented

C₂×#13133209

C₂

yes

maximal, characteristic, complemented

C₂×#13133195

C₂

yes

maximal

C₂×#13133182

C₂

yes

maximal, complemented

C₂×#13133175

C₂

yes

maximal

C₂×C₄²×D₅₃

C₂

yes

maximal, characteristic

C₂×#125×C₅₃

C₂

yes

maximal, characteristic, complemented

C₂²×Dic₁₀₆

C₂²

yes

—

C₂²×C₄×D₅₃

C₂²

yes

—

C₂²×C₄×D₅₃

C₂²

yes

characteristic

#13133216

C₂²

yes

complemented

#13133209

C₂²

yes

complemented

#13133195

C₂²

yes

—

#13133182

C₂²

yes

complemented

#13133175

C₂²

yes

—

C₂×C₄⋊Dic₅₃

C₂²

yes

—

C₂×#11719919

C₂²

yes

—

C₂×#11719919

C₂²

yes

—

C₂×#11719921

C₂²

yes

—

C₂×C₄×Dic₅₃

C₂²

yes

—

C₂×C₄×Dic₅₃

C₂²

yes

characteristic

C₂×C₄×Dic₅₃

C₂²

yes

—

C₄²×D₅₃

C₂²

yes

—

#125×C₅₃

C₂²

yes

complemented

C₂×C₄⋊C₄×C₅₃

C₂²

yes

—

C₂×C₄⋊C₄×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂³×D₅₃

C₂³

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

—

C₂²×C₄×C₅₃

C₂³

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

C₂³

yes

—

C₂²×Dic₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

C₂³

yes

—

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

Q₈

yes

complemented

C₂×C₄×D₅₃

C₂³

yes

—

C₂×C₄×D₅₃

C₂³

yes

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

C₂³

yes

—

C₄⋊C₄×C₅₃

C₂³

yes

—

C₄⋊C₄×C₅₃

C₂³

yes

—

C₄²×C₅₃

C₂³

yes

—

#11719921

C₂³

yes

—

C₄⋊Dic₅₃

C₂³

yes

—

#11719919

C₂³

yes

—

#11719919

C₂³

yes

—

C₄×Dic₅₃

C₂³

yes

—

C₄×Dic₅₃

C₂³

yes

—

C₄×Dic₅₃

C₂³

yes

—

C₂³×C₅₃

C₂⁴

yes

characteristic

C₂²×D₅₃

—

C₂²×D₅₃

C₂×Q₈

yes

—

C₂²×D₅₃

C₂⁴

yes

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×Q₈

yes

—

C₂×C₄×C₅₃

C₂⁴

yes

—

C₂×C₄×C₅₃

C₂⁴

yes

—

C₂×C₄×C₅₃

C₂⁴

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂×Q₈

yes

—

C₂×Dic₅₃

#41

yes

—

C₂×Dic₅₃

C₂⁴

yes

—

C₂×Dic₅₃

C₂⁴

yes

—

C₂×Dic₅₃

C₂⁴

yes

—

C₄×D₅₃

—

C₄×D₅₃

C₂×Q₈

yes

complemented

Dic₁₀₆

—

C₂²×C₅₃

C₂²×Q₈

yes

—

C₂²×C₅₃

C₂×#41

yes

—

C₂²×C₅₃

C₂⁵

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

C₂²×Q₈

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂²×Q₈

yes

—

Dic₅₃

—

Dic₅₃

C₂²×Q₈

yes

—

Dic₅₃

C₂×#41

yes

—

—

C₂×C₅₃

yes

C₂×C₅₃

C₂³×Q₈

yes

characteristic

C₂×C₅₃

C₂²×#41

yes

—

D₅₃

—

#533

—

C₂×C₄⋊Q₈

—

C₂×#125

—

C₂×#125

D₅₃

yes

characteristic, complemented

C₂×C₂²⋊Q₈

—

C₂×C₄×Q₈

—

C₂×#117

—

C₂²×C₄²

—

C₅₃

C₂×#533

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂³×C₄

—

C₂³×C₄

—

100

C₄⋊Q₈

—

101

#125

—

102

#125

C₂×D₅₃

yes

complemented

103

C₂²⋊Q₈

—

104

C₄×Q₈

—

105

#117

—

106

C₂×C₄⋊C₄

—

107

C₂×C₄⋊C₄

—

108

C₂×C₄⋊C₄

—

109

C₂×C₄⋊C₄

C₂×D₅₃

yes

—

110

C₂×C₄⋊C₄

C₂×D₅₃

yes

—

111

C₂×C₂²⋊C₄

—

112

C₂×C₄²

—

113

C₂×C₄²

—

114

C₂×C₄²

—

115

C₂×C₄²

—

116

C₂×C₄²

C₂×D₅₃

yes

characteristic

117

C₂⁴

—

118

C₂×Q₈

—

119

C₂×Q₈

—

106

—

120

C₂²×C₄

—

121

C₂²×C₄

—

122

C₂²×C₄

—

123

C₂²×C₄

—

124

C₂²×C₄

—

125

C₂²×C₄

—

126

C₂²×C₄

—

106

—

127

C₂²×C₄

C₂²×D₅₃

yes

—

128

C₂²×C₄

C₂²×D₅₃

yes

—

129

C₂²×C₄

C₂²×D₅₃

yes

characteristic

130

C₄⋊C₄

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₄⋊C₄

C₂²×D₅₃

yes

—

134

C₄⋊C₄

C₂²×D₅₃

yes

—

135

C₂²⋊C₄

—

136

C₄²

—

137

C₄²

—

138

C₄²

—

139

C₄²

C₂²×D₅₃

yes

—

140

C₂³

—

141

C₂³

—

142

C₂³

—

106

—

143

C₂³

C₂³×D₅₃

yes

characteristic

144

Q₈

—

106

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

106

—

154

C₂×C₄

—

106

—

155

C₂×C₄

Q₈×D₅₃

yes

—

156

C₂×C₄

C₂³×D₅₃

yes

—

157

C₂×C₄

C₂³×D₅₃

yes

—

158

C₂×C₄

C₂³×D₅₃

yes

—

159

C₂²

—

160

C₂²

—

106

—

161

C₂²

—

106

—

162

C₂²

C₂×Q₈×D₅₃

yes

—

163

C₂²

#41×D₅₃

yes

—

164

C₂²

C₂⁴×D₅₃

yes

characteristic

165

C₄

—

166

C₄

—

167

C₄

—

168

C₄

—

169

C₄

—

106

—

170

C₄

C₂×Q₈×D₅₃

yes

—

171

C₂

—

106

—

172

C₂

#16811179

yes

direct product factor

173

C₂

C₂²×Q₈×D₅₃

yes

characteristic

174

C₂

C₂×#41×D₅₃

yes

—

	subgroup	G/H
center	C₂³	C₂³×D₅₃
commutator	C₂²×C₅₃	C₂⁵
Frattini	C₂²	C₂⁴×D₅₃
Fitting	C₂×#125×C₅₃	C₂
socle	C₂³×C₅₃	C₂⁴
hypercenter	C₂×#125	D₅₃
hypocenter	C₅₃	C₂×#533
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×#125

subgroup

G/H

center

C₂³

C₂³×D₅₃

commutator

C₂²×C₅₃

C₂⁵

Frattini

C₂²

C₂⁴×D₅₃

Fitting

C₂×#125×C₅₃

C₂

socle

C₂³×C₅₃

C₂⁴

D₅₃

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₂³ ◁ C₂×#125

N	H ≅ G/N	type
1. #16811179	C₂	direct (N × H)
2. C₂×#13133216	C₂	semidirect
3. C₂×#13133209	C₂	semidirect
5. C₂×#13133182	C₂	semidirect
8. C₂×#125×C₅₃	C₂	semidirect
12. #13133216	C₂²	semidirect
13. #13133209	C₂²	semidirect
15. #13133182	C₂²	semidirect
25. #125×C₅₃	C₂²	semidirect
37. C₂×C₄×D₅₃	Q₈	semidirect
69. C₄×D₅₃	C₂×Q₈	semidirect
84. C₂×C₅₃	#533	semidirect
91. C₂×#125	D₅₃	semidirect
96. C₅₃	C₂×#533	semidirect
102. #125	C₂×D₅₃	semidirect
172. C₂	#16811179	direct (N × H)

H ≅ G/N

type

1. #16811179

C₂

direct (N × H)

2. C₂×#13133216

C₂

semidirect

3. C₂×#13133209

C₂

semidirect

5. C₂×#13133182

C₂

semidirect

8. C₂×#125×C₅₃

C₂

semidirect

12. #13133216

C₂²

semidirect

13. #13133209

C₂²

semidirect

15. #13133182

C₂²

semidirect

25. #125×C₅₃

C₂²

semidirect

37. C₂×C₄×D₅₃

Q₈

semidirect

69. C₄×D₅₃

C₂×Q₈

semidirect

84. C₂×C₅₃

#533

semidirect

91. C₂×#125

D₅₃

semidirect

96. C₅₃

C₂×#533

semidirect

102. #125

C₂×D₅₃

semidirect

172. C₂

#16811179

direct (N × H)

Subgroups — C₂×#16811179 #20288005

Conjugacy classes of subgroups

Complements

Subgroups — C2×#16811179 #20288005

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16811179 #20288005