Group #20285021 — subgroups

Subgroups — C₂×#16810970 #20285021

Order 6784.

528 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 181 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×#13133217	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#13133214	C₂	1	1	yes	maximal, characteristic
3	#16810970	C₂	1	8	yes	maximal, direct product factor
4	C₂×#13133139	C₂	1	4	yes	maximal, complemented
5	C₂×C₄×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×Dic₁₀₆	—	2	1	no	—
7	C₂²×D₂₁₂	—	2	1	no	—
8	C₂²×D₂₁₂	—	2	1	no	—
9	C₂²×D₂₁₂	C₂²	1	2	yes	—
10	#13133217	—	2	2	no	—
11	#13133217	C₂²	1	4	yes	complemented
12	#13133214	—	2	2	no	—
13	#13133214	C₂²	1	4	yes	—
14	C₂×C₄⋊Dic₅₃	—	2	1	no	—
15	C₂×C₄⋊Dic₅₃	C₂²	1	2	yes	—
16	#13133139	C₂²	1	16	yes	complemented
17	C₂²×C₈×C₅₃	C₂²	1	2	yes	—
18	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
19	C₄×C₈×C₅₃	C₂²	1	4	yes	complemented
20	C₂³×D₅₃	—	2	2	no	—
21	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
22	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
23	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
24	C₂²×Dic₅₃	—	2	2	no	—
25	C₂×D₂₁₂	—	2	8	no	—
26	C₂×D₂₁₂	—	2	2	no	—
27	C₂×D₂₁₂	—	2	4	no	—
28	C₂×D₂₁₂	—	2	4	no	—
29	C₂×D₂₁₂	—	4	4	no	—
30	C₂×D₂₁₂	—	4	4	no	—
31	C₂×D₂₁₂	C₂³	1	4	yes	—
32	C₂×Dic₁₀₆	—	2	4	no	—
33	C₂×Dic₁₀₆	—	4	4	no	—
34	C₂×C₈×C₅₃	—	2	4	no	—
35	C₂×C₈×C₅₃	C₂³	1	4	yes	—
36	C₄²×C₅₃	D₄	1	2	yes	—
37	C₄²×C₅₃	C₂³	1	2	yes	—
38	C₄⋊Dic₅₃	—	2	4	no	—
39	C₄⋊Dic₅₃	—	2	2	no	—
40	C₄⋊Dic₅₃	C₂³	1	4	yes	—
41	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
42	C₂²×D₅₃	—	2	4	no	—
43	C₂²×D₅₃	—	4	4	no	—
44	C₂²×D₅₃	—	4	4	no	—
45	C₂²×D₅₃	—	4	4	no	—
46	C₂×C₄×C₅₃	D₈	1	4	yes	—
47	C₂×C₄×C₅₃	SD₁₆	1	4	yes	—
48	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₅₃	#41	1	4	yes	—
52	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
53	C₂×Dic₅₃	—	2	4	no	—
54	C₂×Dic₅₃	—	4	4	no	—
55	D₂₁₂	—	2	8	no	—
56	D₂₁₂	—	4	4	no	—
57	D₂₁₂	—	4	8	no	—
58	D₂₁₂	—	4	8	no	—
59	Dic₁₀₆	—	4	8	no	—
60	C₈×C₅₃	—	2	4	no	—
61	C₂²×C₅₃	#124	1	2	yes	—
62	C₂²×C₅₃	C₂×D₈	1	2	yes	—
63	C₂²×C₅₃	C₂×SD₁₆	1	2	yes	—
64	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
65	C₂×D₅₃	—	4	4	no	—
66	C₂×D₅₃	—	4	4	no	—
67	C₂×D₅₃	—	4	4	no	—
68	C₂×D₅₃	—	8	8	no	—
69	C₄×C₅₃	#124	1	2	yes	—
70	C₄×C₅₃	C₂×D₈	1	4	yes	—
71	C₄×C₅₃	C₂×SD₁₆	1	4	yes	—
72	C₄×C₅₃	C₂×#41	1	2	yes	—
73	Dic₅₃	—	4	4	no	—
74	C₂×#486	—	53	1	no	maximal, Sylow
75	C₂×C₅₃	#486	1	4	yes	complemented
76	C₂×C₅₃	C₂×#124	1	1	yes	characteristic
77	C₂×C₅₃	C₂²×D₈	1	1	yes	characteristic
78	C₂×C₅₃	C₂²×SD₁₆	1	1	yes	characteristic
79	D₅₃	—	8	4	no	—
80	C₂×C₄⋊Q₈	—	53	1	no	—
81	C₂×#127	—	53	1	no	—
82	#486	—	53	8	no	—
83	C₂×#102	—	53	4	no	—
84	C₂×C₄×C₈	D₅₃	1	1	yes	characteristic, complemented
85	C₅₃	C₂×#486	1	1	yes	characteristic, complemented, Sylow
86	C₂²×Q₈	—	106	1	no	—
87	C₂²×D₄	—	53	2	no	—
88	C₂²×D₄	—	106	1	no	—
89	C₂²×D₄	—	106	1	no	—
90	C₂²×C₈	C₂×D₅₃	1	2	yes	—
91	C₄⋊Q₈	—	53	4	no	—
92	C₄⋊Q₈	—	106	2	no	—
93	#127	—	53	4	no	—
94	#127	—	106	2	no	—
95	C₂×C₄⋊C₄	—	53	2	no	—
96	C₂×C₄⋊C₄	—	106	1	no	—
97	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
98	#102	—	53	16	no	—
99	C₄×C₈	C₂×D₅₃	1	4	yes	complemented
100	C₂⁴	—	106	2	no	—
101	C₂×Q₈	—	106	4	no	—
102	C₂×Q₈	—	212	4	no	—
103	C₂×D₄	—	53	4	no	—
104	C₂×D₄	—	106	4	no	—
105	C₂×D₄	—	106	4	no	—
106	C₂×D₄	—	106	2	no	—
107	C₂×D₄	—	106	8	no	—
108	C₂×D₄	—	212	4	no	—
109	C₂×D₄	—	212	4	no	—
110	C₂²×C₄	—	106	2	no	—
111	C₂²×C₄	D₂₁₂	1	1	yes	characteristic
112	C₂²×C₄	D₂₁₂	1	1	yes	characteristic
113	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
114	C₂×C₈	—	2	4	no	—
115	C₂×C₈	C₂²×D₅₃	1	4	yes	—
116	C₄⋊C₄	—	53	4	no	—
117	C₄⋊C₄	—	106	2	no	—
118	C₄⋊C₄	—	106	4	no	—
119	C₄²	D₂₁₂	1	2	yes	—
120	C₄²	C₂²×D₅₃	1	2	yes	—
121	C₂³	—	106	4	no	—
122	C₂³	—	212	4	no	—
123	C₂³	—	212	4	no	—
124	C₂³	—	212	4	no	—
125	C₂³	C₂×D₂₁₂	1	1	yes	characteristic
126	Q₈	—	212	8	no	—
127	D₄	—	106	8	no	—
128	D₄	—	212	4	no	—
129	D₄	—	212	8	no	—
130	D₄	—	212	8	no	—
131	C₂×C₄	—	106	4	no	—
132	C₂×C₄	—	212	4	no	—
133	C₂×C₄	#11719913	1	4	yes	—
134	C₂×C₄	D₄₂₄	1	4	yes	—
135	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
136	C₂×C₄	C₂×D₂₁₂	1	2	yes	—
137	C₂×C₄	C₂×D₂₁₂	1	2	yes	—
138	C₂×C₄	#11719945	1	4	yes	—
139	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
140	C₈	—	2	4	no	—
141	C₂²	—	424	8	no	—
142	C₂²	—	212	4	no	—
143	C₂²	—	212	4	no	—
144	C₂²	—	212	4	no	—
145	C₂²	#13133205	1	2	yes	—
146	C₂²	C₂×D₄₂₄	1	2	yes	—
147	C₂²	C₂×#11719913	1	2	yes	—
148	C₂²	C₂²×D₂₁₂	1	1	yes	characteristic
149	C₄	—	212	4	no	—
150	C₄	#13133205	1	2	yes	—
151	C₄	C₂×D₄₂₄	1	4	yes	—
152	C₄	C₂×#11719913	1	4	yes	—
153	C₄	C₂×#11719945	1	2	yes	—
154	C₂	—	424	4	no	—
155	C₂	#16810970	1	4	yes	direct product factor
156	C₂	C₂×#13133205	1	1	yes	characteristic
157	C₂	C₂²×D₄₂₄	1	1	yes	characteristic
158	C₂	C₂²×#11719913	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133217

C₂

yes

maximal, characteristic, complemented

C₂×#13133214

C₂

yes

maximal, characteristic

#16810970

C₂

yes

maximal, direct product factor

C₂×#13133139

C₂

yes

maximal, complemented

C₂×C₄×C₈×C₅₃

C₂

yes

maximal, characteristic, complemented

C₂²×Dic₁₀₆

—

C₂²×D₂₁₂

—

C₂²×D₂₁₂

—

C₂²×D₂₁₂

C₂²

yes

—

#13133217

—

#13133217

C₂²

yes

complemented

#13133214

—

#13133214

C₂²

yes

—

C₂×C₄⋊Dic₅₃

—

C₂×C₄⋊Dic₅₃

C₂²

yes

—

#13133139

C₂²

yes

complemented

C₂²×C₈×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₄×C₈×C₅₃

C₂²

yes

complemented

C₂³×D₅₃

—

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

C₂³

yes

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂³

yes

—

C₄²×C₅₃

D₄

yes

—

C₄²×C₅₃

C₂³

yes

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

C₂³

yes

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂×C₄×C₅₃

D₈

yes

—

C₂×C₄×C₅₃

SD₁₆

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

#41

yes

—

C₂×C₄×C₅₃

C₂⁴

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

Dic₁₀₆

—

C₈×C₅₃

—

C₂²×C₅₃

#124

yes

—

C₂²×C₅₃

C₂×D₈

yes

—

C₂²×C₅₃

C₂×SD₁₆

yes

—

C₂²×C₅₃

C₂²×D₄

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

#124

yes

—

C₄×C₅₃

C₂×D₈

yes

—

C₄×C₅₃

C₂×SD₁₆

yes

—

C₄×C₅₃

C₂×#41

yes

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

C₂²×D₈

yes

characteristic

C₂×C₅₃

C₂²×SD₁₆

yes

characteristic

D₅₃

—

C₂×C₄⋊Q₈

—

C₂×#127

—

#486

—

C₂×#102

—

C₂×C₄×C₈

D₅₃

yes

characteristic, complemented

C₅₃

C₂×#486

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

106

—

C₂²×D₄

—

C₂²×D₄

—

106

—

C₂²×D₄

—

106

—

C₂²×C₈

C₂×D₅₃

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

106

—

#127

—

#127

—

106

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

106

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

#102

—

C₄×C₈

C₂×D₅₃

yes

complemented

100

C₂⁴

—

106

—

101

C₂×Q₈

—

106

—

102

C₂×Q₈

—

212

—

103

C₂×D₄

—

104

C₂×D₄

—

106

—

105

C₂×D₄

—

106

—

106

C₂×D₄

—

106

—

107

C₂×D₄

—

106

—

108

C₂×D₄

—

212

—

109

C₂×D₄

—

212

—

110

C₂²×C₄

—

106

—

111

C₂²×C₄

D₂₁₂

yes

characteristic

112

C₂²×C₄

D₂₁₂

yes

characteristic

113

C₂²×C₄

C₂²×D₅₃

yes

characteristic

114

C₂×C₈

—

115

C₂×C₈

C₂²×D₅₃

yes

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

106

—

118

C₄⋊C₄

—

106

—

119

C₄²

D₂₁₂

yes

—

120

C₄²

C₂²×D₅₃

yes

—

121

C₂³

—

106

—

122

C₂³

—

212

—

123

C₂³

—

212

—

124

C₂³

—

212

—

125

C₂³

C₂×D₂₁₂

yes

characteristic

126

Q₈

—

212

—

127

D₄

—

106

—

128

D₄

—

212

—

129

D₄

—

212

—

130

D₄

—

212

—

131

C₂×C₄

—

106

—

132

C₂×C₄

—

212

—

133

C₂×C₄

#11719913

yes

—

134

C₂×C₄

D₄₂₄

yes

—

135

C₂×C₄

C₂×D₂₁₂

yes

characteristic

136

C₂×C₄

C₂×D₂₁₂

yes

—

137

C₂×C₄

C₂×D₂₁₂

yes

—

138

C₂×C₄

#11719945

yes

—

139

C₂×C₄

C₂³×D₅₃

yes

characteristic

140

C₈

—

141

C₂²

—

424

—

142

C₂²

—

212

—

143

C₂²

—

212

—

144

C₂²

—

212

—

145

C₂²

#13133205

yes

—

146

C₂²

C₂×D₄₂₄

yes

—

147

C₂²

C₂×#11719913

yes

—

148

C₂²

C₂²×D₂₁₂

yes

characteristic

149

C₄

—

212

—

150

C₄

#13133205

yes

—

151

C₄

C₂×D₄₂₄

yes

—

152

C₄

C₂×#11719913

yes

—

153

C₄

C₂×#11719945

yes

—

154

C₂

—

424

—

155

C₂

#16810970

yes

direct product factor

156

C₂

C₂×#13133205

yes

characteristic

157

C₂

C₂²×D₄₂₄

yes

characteristic

158

C₂

C₂²×#11719913

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₂×C₄×C₈×C₅₃	C₂
socle	C₂³×C₅₃	C₂×D₄
hypercenter	C₂×C₄×C₈	D₅₃
hypocenter	C₅₃	C₂×#486
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂³

C₂×D₂₁₂

commutator

C₂×C₄×C₅₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₃

Fitting

C₂×C₄×C₈×C₅₃

C₂

socle

C₂³×C₅₃

C₂×D₄

hypercenter

C₂×C₄×C₈

D₅₃

hypocenter

C₅₃

C₂×#486

LCS

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
1. C₂×#13133217	C₂	semidirect
3. #16810970	C₂	direct (N × H)
4. C₂×#13133139	C₂	semidirect
5. C₂×C₄×C₈×C₅₃	C₂	semidirect
11. #13133217	C₂²	semidirect
16. #13133139	C₂²	semidirect
19. C₄×C₈×C₅₃	C₂²	semidirect
75. C₂×C₅₃	#486	semidirect
84. C₂×C₄×C₈	D₅₃	semidirect
85. C₅₃	C₂×#486	semidirect
99. C₄×C₈	C₂×D₅₃	semidirect
155. C₂	#16810970	direct (N × H)

H ≅ G/N

type

1. C₂×#13133217

C₂

semidirect

3. #16810970

C₂

direct (N × H)

4. C₂×#13133139

C₂

semidirect

5. C₂×C₄×C₈×C₅₃

C₂

semidirect

11. #13133217

C₂²

semidirect

16. #13133139

C₂²

semidirect

19. C₄×C₈×C₅₃

C₂²

semidirect

75. C₂×C₅₃

#486

semidirect

84. C₂×C₄×C₈

D₅₃

semidirect

85. C₅₃

C₂×#486

semidirect

99. C₄×C₈

C₂×D₅₃

semidirect

155. C₂

#16810970

direct (N × H)

Subgroups — C₂×#16810970 #20285021

Conjugacy classes of subgroups

Complements

Subgroups — C2×#16810970 #20285021

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16810970 #20285021