Group #20288082 — subgroups

Subgroups — C₂×#16811234 #20288082

Order 6784.

1004 conjugacy classes of subgroups; only the 485 normal subgroups are shown.

normal subgroups only

This group has 1004 conjugacy classes of subgroups — too many to draw. Only its 485 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	#16811234	C₂	1	16	yes	maximal, direct product factor
2	C₂×#13133212	C₂	1	4	yes	maximal
3	C₂×#13133209	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#13133207	C₂	1	4	yes	maximal, complemented
5	C₂×#13133193	C₂	1	2	yes	maximal
6	C₂×Q₈×Dic₅₃	C₂	1	2	yes	maximal, complemented
7	C₂×#13133174	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₄×D₅₃	C₂²	1	1	yes	characteristic
10	#13133212	C₂²	1	32	yes	—
11	#13133209	C₂²	1	8	yes	complemented
12	#13133207	C₂²	1	32	yes	complemented
13	#13133193	C₂²	1	16	yes	—
14	Q₈×Dic₅₃	C₂²	1	16	yes	complemented
15	#13133174	C₂²	1	8	yes	—
16	C₂×C₄⋊Dic₅₃	C₂²	1	2	yes	—
17	C₂×C₄⋊Dic₅₃	C₂²	1	4	yes	—
18	C₂×#11719919	C₂²	1	2	yes	—
19	C₂×#11719919	C₂²	1	4	yes	—
20	C₂×#11719919	C₂²	1	4	yes	—
21	C₂×#11719921	C₂²	1	2	yes	—
22	C₂×#11719921	C₂²	1	4	yes	—
23	C₂×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
24	C₂×C₄×Dic₅₃	C₂²	1	4	yes	—
25	C₂²×Q₈×C₅₃	C₂²	1	2	yes	—
26	C₄⋊Q₈×C₅₃	C₂²	1	8	yes	complemented
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	C₂×Q₈×C₅₃	—	2	8	no	—
31	C₂×Q₈×C₅₃	C₂³	1	8	yes	—
32	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
33	C₂²×C₄×C₅₃	C₂³	1	4	yes	—
34	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
35	C₂²×Dic₅₃	C₂³	1	2	yes	—
36	C₂²×Dic₅₃	C₂³	1	4	yes	—
37	C₂²×Dic₅₃	C₂³	1	1	yes	characteristic
38	C₂×C₄×D₅₃	—	2	4	no	—
39	C₂×C₄×D₅₃	C₂³	1	4	yes	—
40	C₄⋊C₄×C₅₃	C₂³	1	16	yes	—
41	C₄²×C₅₃	C₂³	1	4	yes	—
42	#11719921	C₂³	1	8	yes	—
43	#11719921	C₂³	1	16	yes	—
44	C₄⋊Dic₅₃	C₂³	1	8	yes	—
45	C₄⋊Dic₅₃	C₂³	1	16	yes	—
46	#11719919	C₂³	1	8	yes	—
47	#11719919	C₂³	1	16	yes	—
48	#11719919	C₂³	1	16	yes	—
49	C₄×Dic₅₃	C₂³	1	4	yes	—
50	C₄×Dic₅₃	C₂³	1	16	yes	—
51	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
52	C₂²×D₅₃	—	2	2	no	—
53	C₂²×D₅₃	—	2	4	no	—
54	C₂²×D₅₃	C₂⁴	1	2	yes	—
55	Q₈×C₅₃	—	2	16	no	—
56	C₂×C₄×C₅₃	—	2	8	no	—
57	C₂×C₄×C₅₃	—	2	4	no	—
58	C₂×C₄×C₅₃	#41	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	2	no	—
63	C₂×Dic₅₃	—	2	4	no	—
64	C₂×Dic₅₃	—	2	8	no	—
65	C₂×Dic₅₃	C₂⁴	1	2	yes	—
66	C₂×Dic₅₃	C₂⁴	1	4	yes	—
67	C₂×Dic₅₃	C₂⁴	1	8	yes	—
68	C₄×D₅₃	—	2	8	no	—
69	C₂²×C₅₃	C₂×#41	1	2	yes	—
70	C₂²×C₅₃	2₋¹⁺⁴	1	4	yes	—
71	C₂²×C₅₃	C₂⁵	1	1	yes	characteristic
72	C₂×D₅₃	—	2	2	no	—
73	C₂×D₅₃	—	2	4	no	—
74	C₂×D₅₃	—	4	4	no	—
75	C₄×C₅₃	—	2	8	no	—
76	C₄×C₅₃	—	2	4	no	—
77	C₄×C₅₃	C₂×#41	1	4	yes	—
78	Dic₅₃	—	2	2	no	—
79	Dic₅₃	—	2	4	no	—
80	Dic₅₃	—	2	8	no	—
81	C₂×#541	—	53	1	no	maximal, Sylow
82	C₂×C₅₃	#541	1	4	yes	complemented
83	C₂×C₅₃	C₂²×#41	1	1	yes	characteristic
84	C₂×C₅₃	C₂×2₋¹⁺⁴	1	2	yes	—
85	D₅₃	—	4	2	no	—
86	#541	—	53	16	no	—
87	C₂×C₄⋊Q₈	D₅₃	1	1	yes	characteristic, complemented
88	C₂×#126	—	53	4	no	—
89	C₂×#125	—	53	4	no	—
90	C₂×#125	—	53	1	no	—
91	C₂×C₂²⋊Q₈	—	53	2	no	—
92	C₂×C₄×Q₈	—	53	2	no	—
93	C₂×#117	—	53	1	no	—
94	C₅₃	C₂×#541	1	1	yes	characteristic, complemented, Sylow
95	C₂²×Q₈	C₂×D₅₃	1	2	yes	—
96	C₂³×C₄	—	53	1	no	—
97	C₄⋊Q₈	C₂×D₅₃	1	8	yes	complemented
98	#126	—	53	32	no	—
99	#125	—	53	32	no	—
100	#125	—	53	8	no	—
101	C₂²⋊Q₈	—	53	16	no	—
102	C₄×Q₈	—	53	16	no	—
103	#117	—	53	8	no	—
104	C₂×C₄⋊C₄	—	53	2	no	—
105	C₂×C₄⋊C₄	—	53	2	no	—
106	C₂×C₄⋊C₄	—	53	4	no	—
107	C₂×C₄⋊C₄	—	53	4	no	—
108	C₂×C₄⋊C₄	—	53	4	no	—
109	C₂×C₄⋊C₄	C₂×D₅₃	1	4	yes	—
110	C₂×C₂²⋊C₄	—	53	2	no	—
111	C₂×C₂²⋊C₄	—	53	4	no	—
112	C₂×C₄²	—	53	1	no	—
113	C₂×C₄²	—	53	4	no	—
114	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
115	C₂⁴	—	53	1	no	—
116	C₂×Q₈	—	2	8	no	—
117	C₂×Q₈	C₂²×D₅₃	1	8	yes	—
118	C₂²×C₄	—	53	2	no	—
119	C₂²×C₄	—	53	4	no	—
120	C₂²×C₄	—	53	1	no	—
121	C₂²×C₄	—	53	4	no	—
122	C₂²×C₄	—	106	4	no	—
123	C₂²×C₄	C₂²×D₅₃	1	4	yes	—
124	C₂²×C₄	C₂²×D₅₃	1	2	yes	—
125	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
126	C₄⋊C₄	—	53	8	no	—
127	C₄⋊C₄	—	53	8	no	—
128	C₄⋊C₄	—	53	16	no	—
129	C₄⋊C₄	—	53	16	no	—
130	C₄⋊C₄	—	53	16	no	—
131	C₄⋊C₄	C₂²×D₅₃	1	16	yes	—
132	C₂²⋊C₄	—	53	8	no	—
133	C₂²⋊C₄	—	53	16	no	—
134	C₄²	—	53	4	no	—
135	C₄²	—	53	16	no	—
136	C₄²	C₂²×D₅₃	1	4	yes	—
137	C₂³	—	53	2	no	—
138	C₂³	—	106	2	no	—
139	C₂³	—	106	4	no	—
140	C₂³	C₂³×D₅₃	1	1	yes	characteristic
141	Q₈	—	2	16	no	—
142	C₂×C₄	—	2	8	no	—
143	C₂×C₄	—	2	4	no	—
144	C₂×C₄	—	53	4	no	—
145	C₂×C₄	—	53	8	no	—
146	C₂×C₄	—	53	2	no	—
147	C₂×C₄	—	106	2	no	—
148	C₂×C₄	—	106	4	no	—
149	C₂×C₄	—	106	8	no	—
150	C₂×C₄	—	106	8	no	—
151	C₂×C₄	#11719947	1	4	yes	—
152	C₂×C₄	C₂³×D₅₃	1	8	yes	—
153	C₂×C₄	C₂³×D₅₃	1	4	yes	—
154	C₂×C₄	C₂³×D₅₃	1	2	yes	—
155	C₂²	—	106	2	no	—
156	C₂²	—	106	4	no	—
157	C₂²	—	212	4	no	—
158	C₂²	C₂×#11719947	1	2	yes	—
159	C₂²	#13133265	1	4	yes	—
160	C₂²	C₂⁴×D₅₃	1	1	yes	characteristic
161	C₄	—	2	8	no	—
162	C₄	—	2	4	no	—
163	C₄	—	106	2	no	—
164	C₄	—	106	4	no	—
165	C₄	—	106	8	no	—
166	C₄	C₂×#11719947	1	4	yes	—
167	C₂	—	212	2	no	—
168	C₂	#16811234	1	4	yes	direct product factor
169	C₂	C₂²×#11719947	1	1	yes	characteristic
170	C₂	C₂×#13133265	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16811234

C₂

yes

maximal, direct product factor

C₂×#13133212

C₂

yes

maximal

C₂×#13133209

C₂

yes

maximal, characteristic, complemented

C₂×#13133207

C₂

yes

maximal, complemented

C₂×#13133193

C₂

yes

maximal

C₂×Q₈×Dic₅₃

C₂

yes

maximal, complemented

C₂×#13133174

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₅₃

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₅₃

C₂²

yes

characteristic

#13133212

C₂²

yes

—

#13133209

C₂²

yes

complemented

#13133207

C₂²

yes

complemented

#13133193

C₂²

yes

—

Q₈×Dic₅₃

C₂²

yes

complemented

#13133174

C₂²

yes

—

C₂×C₄⋊Dic₅₃

C₂²

yes

—

C₂×C₄⋊Dic₅₃

C₂²

yes

—

C₂×#11719919

C₂²

yes

—

C₂×#11719919

C₂²

yes

—

C₂×#11719919

C₂²

yes

—

C₂×#11719921

C₂²

yes

—

C₂×#11719921

C₂²

yes

—

C₂×C₄×Dic₅₃

C₂²

yes

characteristic

C₂×C₄×Dic₅₃

C₂²

yes

—

C₂²×Q₈×C₅₃

C₂²

yes

—

C₄⋊Q₈×C₅₃

C₂²

yes

complemented

C₂×C₄⋊C₄×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂³×D₅₃

C₂³

yes

characteristic

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

C₂³

yes

—

C₂²×C₄×C₅₃

C₂³

yes

—

C₂²×C₄×C₅₃

C₂³

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

C₂³

yes

—

C₂²×Dic₅₃

C₂³

yes

—

C₂²×Dic₅₃

C₂³

yes

characteristic

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

C₂³

yes

—

C₄⋊C₄×C₅₃

C₂³

yes

—

C₄²×C₅₃

C₂³

yes

—

#11719921

C₂³

yes

—

#11719921

C₂³

yes

—

C₄⋊Dic₅₃

C₂³

yes

—

C₄⋊Dic₅₃

C₂³

yes

—

#11719919

C₂³

yes

—

#11719919

C₂³

yes

—

#11719919

C₂³

yes

—

C₄×Dic₅₃

C₂³

yes

—

C₄×Dic₅₃

C₂³

yes

—

C₂³×C₅₃

C₂⁴

yes

characteristic

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

C₂⁴

yes

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#41

yes

—

C₂×C₄×C₅₃

C₂⁴

yes

—

C₂×C₄×C₅₃

C₂⁴

yes

—

C₂×C₄×C₅₃

C₂⁴

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂⁴

yes

—

C₂×Dic₅₃

C₂⁴

yes

—

C₂×Dic₅₃

C₂⁴

yes

—

C₄×D₅₃

—

C₂²×C₅₃

C₂×#41

yes

—

C₂²×C₅₃

2₋¹⁺⁴

yes

—

C₂²×C₅₃

C₂⁵

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×#41

yes

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

C₂²×#41

yes

characteristic

C₂×C₅₃

C₂×2₋¹⁺⁴

yes

—

D₅₃

—

#541

—

C₂×C₄⋊Q₈

D₅₃

yes

characteristic, complemented

C₂×#126

—

C₂×#125

—

C₂×#125

—

C₂×C₂²⋊Q₈

—

C₂×C₄×Q₈

—

C₂×#117

—

C₅₃

C₂×#541

yes

characteristic, complemented, Sylow

C₂²×Q₈

C₂×D₅₃

yes

—

C₂³×C₄

—

C₄⋊Q₈

C₂×D₅₃

yes

complemented

#126

—

#125

—

100

#125

—

101

C₂²⋊Q₈

—

102

C₄×Q₈

—

103

#117

—

104

C₂×C₄⋊C₄

—

105

C₂×C₄⋊C₄

—

106

C₂×C₄⋊C₄

—

107

C₂×C₄⋊C₄

—

108

C₂×C₄⋊C₄

—

109

C₂×C₄⋊C₄

C₂×D₅₃

yes

—

110

C₂×C₂²⋊C₄

—

111

C₂×C₂²⋊C₄

—

112

C₂×C₄²

—

113

C₂×C₄²

—

114

C₂×C₄²

C₂×D₅₃

yes

characteristic

115

C₂⁴

—

116

C₂×Q₈

—

117

C₂×Q₈

C₂²×D₅₃

yes

—

118

C₂²×C₄

—

119

C₂²×C₄

—

120

C₂²×C₄

—

121

C₂²×C₄

—

122

C₂²×C₄

—

106

—

123

C₂²×C₄

C₂²×D₅₃

yes

—

124

C₂²×C₄

C₂²×D₅₃

yes

—

125

C₂²×C₄

C₂²×D₅₃

yes

characteristic

126

C₄⋊C₄

—

127

C₄⋊C₄

—

128

C₄⋊C₄

—

129

C₄⋊C₄

—

130

C₄⋊C₄

—

131

C₄⋊C₄

C₂²×D₅₃

yes

—

132

C₂²⋊C₄

—

133

C₂²⋊C₄

—

134

C₄²

—

135

C₄²

—

136

C₄²

C₂²×D₅₃

yes

—

137

C₂³

—

138

C₂³

—

106

—

139

C₂³

—

106

—

140

C₂³

C₂³×D₅₃

yes

characteristic

141

Q₈

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

106

—

148

C₂×C₄

—

106

—

149

C₂×C₄

—

106

—

150

C₂×C₄

—

106

—

151

C₂×C₄

#11719947

yes

—

152

C₂×C₄

C₂³×D₅₃

yes

—

153

C₂×C₄

C₂³×D₅₃

yes

—

154

C₂×C₄

C₂³×D₅₃

yes

—

155

C₂²

—

106

—

156

C₂²

—

106

—

157

C₂²

—

212

—

158

C₂²

C₂×#11719947

yes

—

159

C₂²

#13133265

yes

—

160

C₂²

C₂⁴×D₅₃

yes

characteristic

161

C₄

—

162

C₄

—

163

C₄

—

106

—

164

C₄

—

106

—

165

C₄

—

106

—

166

C₄

C₂×#11719947

yes

—

167

C₂

—

212

—

168

C₂

#16811234

yes

direct product factor

169

C₂

C₂²×#11719947

yes

characteristic

170

C₂

C₂×#13133265

yes

—

	subgroup	G/H
center	C₂³	C₂³×D₅₃
commutator	C₂²×C₅₃	C₂⁵
Frattini	C₂²	C₂⁴×D₅₃
Fitting	C₂×C₄⋊Q₈×C₅₃	C₂
socle	C₂³×C₅₃	C₂⁴
hypercenter	C₂×C₄⋊Q₈	D₅₃
hypocenter	C₅₃	C₂×#541
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂³×D₅₃

commutator

C₂²×C₅₃

C₂⁵

Frattini

C₂²

C₂⁴×D₅₃

Fitting

C₂×C₄⋊Q₈×C₅₃

C₂

socle

C₂³×C₅₃

C₂⁴

hypercenter

C₂×C₄⋊Q₈

D₅₃

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
1. #16811234	C₂	direct (N × H)
3. C₂×#13133209	C₂	semidirect
4. C₂×#13133207	C₂	semidirect
6. C₂×Q₈×Dic₅₃	C₂	semidirect
8. C₂×C₄⋊Q₈×C₅₃	C₂	semidirect
11. #13133209	C₂²	semidirect
12. #13133207	C₂²	semidirect
14. Q₈×Dic₅₃	C₂²	semidirect
26. C₄⋊Q₈×C₅₃	C₂²	semidirect
82. C₂×C₅₃	#541	semidirect
87. C₂×C₄⋊Q₈	D₅₃	semidirect
94. C₅₃	C₂×#541	semidirect
97. C₄⋊Q₈	C₂×D₅₃	semidirect
168. C₂	#16811234	direct (N × H)

H ≅ G/N

type

1. #16811234

C₂

direct (N × H)

3. C₂×#13133209

C₂

semidirect

4. C₂×#13133207

C₂

semidirect

6. C₂×Q₈×Dic₅₃

C₂

semidirect

8. C₂×C₄⋊Q₈×C₅₃

C₂

semidirect

11. #13133209

C₂²

semidirect

12. #13133207

C₂²

semidirect

14. Q₈×Dic₅₃

C₂²

semidirect

26. C₄⋊Q₈×C₅₃

C₂²

semidirect

82. C₂×C₅₃

#541

semidirect

87. C₂×C₄⋊Q₈

D₅₃

semidirect

94. C₅₃

C₂×#541

semidirect

97. C₄⋊Q₈

C₂×D₅₃

semidirect

168. C₂

#16811234

direct (N × H)

Subgroups — C₂×#16811234 #20288082

Conjugacy classes of subgroups

Complements

Subgroups — C2×#16811234 #20288082

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16811234 #20288082