Group #20273091 — subgroups

Subgroups — C₂×#16810353 #20273091

Order 6784.

280 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 125 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	C₂×C₄×#88322	C₂	1	1	yes	maximal, characteristic
3	#16810353	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₄⋊C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₁₀₆	C₄	1	1	yes	characteristic
6	C₂²×#88322	—	2	1	no	—
7	C₄×Dic₁₀₆	—	2	1	no	—
8	C₄×Dic₁₀₆	—	2	1	no	—
9	C₄×Dic₁₀₆	C₄	1	2	yes	—
10	C₄×Dic₁₀₆	C₂²	1	2	yes	—
11	C₂×C₄⋊Dic₅₃	C₄	1	1	yes	characteristic
12	C₂×#11719919	—	2	1	no	—
13	C₂×C₄×Dic₅₃	—	2	1	no	—
14	C₄×#88322	—	2	1	no	—
15	C₄×#88322	C₂²	1	2	yes	—
16	C₂²×C₈×C₅₃	—	2	1	no	—
17	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
18	C₄⋊C₈×C₅₃	—	2	1	no	—
19	C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
20	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
22	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
23	C₂²×Dic₅₃	—	2	1	no	—
24	C₂²×Dic₅₃	—	2	1	no	—
25	C₂×Dic₁₀₆	—	2	1	no	—
26	C₂×Dic₁₀₆	—	2	2	no	—
27	C₂×Dic₁₀₆	C₈	1	4	yes	complemented
28	C₂×Dic₁₀₆	C₂×C₄	1	2	yes	—
29	C₂×C₈×C₅₃	—	2	2	no	—
30	C₂×C₈×C₅₃	—	4	2	no	—
31	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₄²×C₅₃	D₄	1	1	yes	characteristic
33	C₄²×C₅₃	D₄	1	1	yes	characteristic
34	C₄²×C₅₃	C₂³	1	1	yes	characteristic
35	C₄⋊Dic₅₃	—	2	1	no	—
36	C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
37	#11719919	—	2	2	no	—
38	#11719919	—	2	2	no	—
39	C₄×Dic₅₃	—	2	2	no	—
40	C₄×Dic₅₃	—	2	2	no	—
41	C₂×#88322	—	2	4	no	—
42	C₂×#88322	—	2	2	no	—
43	C₂³×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₅₃	—	2	2	no	—
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	SD₁₆	1	2	yes	—
51	C₂×C₄×C₅₃	Q₁₆	1	2	yes	—
52	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
55	C₂×Dic₅₃	—	2	4	no	—
56	C₂×Dic₅₃	—	2	2	no	—
57	C₂×Dic₅₃	—	2	2	no	—
58	C₂×Dic₅₃	—	4	2	no	—
59	Dic₁₀₆	—	2	4	no	—
60	Dic₁₀₆	—	2	2	no	—
61	Dic₁₀₆	C₂×C₈	1	4	yes	complemented
62	C₈×C₅₃	—	4	2	no	—
63	#88322	—	2	4	no	—
64	C₂²×C₅₃	C₂²⋊C₈	1	2	yes	—
65	C₂²×C₅₃	#103	1	2	yes	—
66	C₂²×C₅₃	#104	1	2	yes	—
67	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	C₄×C₅₃	—	2	1	no	—
69	C₄×C₅₃	—	2	1	no	—
70	C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
71	C₄×C₅₃	#103	1	2	yes	—
72	C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
73	C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
74	C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
75	C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
76	Dic₅₃	—	2	4	no	—
77	Dic₅₃	—	4	2	no	—
78	C₂×Q₈⋊C₈	—	53	1	no	maximal, Sylow
79	C₂×C₅₃	Q₈⋊C₈	1	4	yes	complemented
80	C₂×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
81	C₂×C₅₃	C₂×#103	1	1	yes	characteristic
82	C₂×C₅₃	C₂×#104	1	1	yes	characteristic
83	C₂×C₄×Q₈	—	53	1	no	—
84	C₂×C₄⋊C₈	D₅₃	1	1	yes	characteristic
85	C₂×C₄×C₈	—	53	1	no	—
86	Q₈⋊C₈	—	53	4	no	—
87	C₅₃	C₂×Q₈⋊C₈	1	1	yes	characteristic, complemented, Sylow
88	C₂²×Q₈	—	53	1	no	—
89	C₂²×C₈	—	2	1	no	—
90	C₂²×C₈	—	106	1	no	—
91	C₄×Q₈	—	53	2	no	—
92	C₄×Q₈	—	53	2	no	—
93	C₄×Q₈	—	106	1	no	—
94	C₄×Q₈	—	106	1	no	—
95	C₂×C₄⋊C₄	—	53	1	no	—
96	C₂×C₄⋊C₄	—	106	1	no	—
97	C₂×C₄²	—	106	1	no	—
98	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
99	C₄⋊C₈	—	2	1	no	—
100	C₄⋊C₈	C₂×D₅₃	1	2	yes	—
101	C₄×C₈	—	53	2	no	—
102	C₄×C₈	—	106	1	no	—
103	C₂×Q₈	—	53	2	no	—
104	C₂×Q₈	—	53	4	no	—
105	C₂×Q₈	—	106	2	no	—
106	C₂×Q₈	—	106	1	no	—
107	C₂²×C₄	—	106	1	no	—
108	C₂²×C₄	—	106	1	no	—
109	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
110	C₂²×C₄	D₂₁₂	1	1	yes	characteristic
111	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
112	C₂×C₈	—	2	2	no	—
113	C₂×C₈	—	4	2	no	—
114	C₂×C₈	—	106	4	no	—
115	C₂×C₈	—	106	2	no	—
116	C₄⋊C₄	—	53	2	no	—
117	C₄⋊C₄	—	106	2	no	—
118	C₄⋊C₄	—	106	1	no	—
119	C₄⋊C₄	—	106	2	no	—
120	C₄²	—	106	2	no	—
121	C₄²	—	106	2	no	—
122	C₄²	C₄×D₅₃	1	1	yes	characteristic
123	C₄²	D₂₁₂	1	1	yes	characteristic
124	C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
125	C₄²	C₂²×D₅₃	1	1	yes	characteristic
126	C₂³	#11719921	1	1	yes	characteristic
127	Q₈	—	53	4	no	—
128	Q₈	—	106	4	no	—
129	Q₈	—	106	2	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	106	4	no	—
132	C₂×C₄	—	106	2	no	—
133	C₂×C₄	—	106	2	no	—
134	C₂×C₄	—	212	2	no	—
135	C₂×C₄	C₈×D₅₃	1	2	yes	—
136	C₂×C₄	#11719912	1	2	yes	—
137	C₂×C₄	#11719921	1	1	yes	characteristic
138	C₂×C₄	#11719921	1	1	yes	characteristic
139	C₂×C₄	#11719921	1	1	yes	characteristic
140	C₂×C₄	#11719923	1	2	yes	—
141	C₂×C₄	#11719925	1	2	yes	—
142	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
143	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
144	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
145	C₈	—	4	2	no	—
146	C₈	—	106	4	no	—
147	C₂²	#13133129	1	2	yes	—
148	C₂²	#13133140	1	2	yes	—
149	C₂²	#13133143	1	2	yes	—
150	C₂²	C₂×#11719921	1	1	yes	characteristic
151	C₄	—	2	1	no	—
152	C₄	—	2	1	no	—
153	C₄	—	106	4	no	—
154	C₄	—	212	2	no	—
155	C₄	#13133129	1	2	yes	—
156	C₄	#13133140	1	2	yes	—
157	C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
158	C₄	C₂×#11719912	1	1	yes	characteristic
159	C₄	C₂×#11719923	1	1	yes	characteristic
160	C₄	C₂×#11719925	1	1	yes	characteristic
161	C₂	#16810353	1	4	yes	direct product factor
162	C₂	C₂×#13133129	1	1	yes	characteristic
163	C₂	C₂×#13133140	1	1	yes	characteristic
164	C₂	C₂×#13133143	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Dic₁₀₆

C₂

yes

maximal, characteristic

C₂×C₄×#88322

C₂

yes

maximal, characteristic

#16810353

C₂

yes

maximal, direct product factor

C₂×C₄⋊C₈×C₅₃

C₂

yes

maximal, characteristic

C₂²×Dic₁₀₆

C₄

yes

characteristic

C₂²×#88322

—

C₄×Dic₁₀₆

—

C₄×Dic₁₀₆

—

C₄×Dic₁₀₆

C₄

yes

—

C₄×Dic₁₀₆

C₂²

yes

—

C₂×C₄⋊Dic₅₃

C₄

yes

characteristic

C₂×#11719919

—

C₂×C₄×Dic₅₃

—

C₄×#88322

—

C₄×#88322

C₂²

yes

—

C₂²×C₈×C₅₃

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

—

C₄⋊C₈×C₅₃

C₂²

yes

—

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×Dic₅₃

—

C₂²×Dic₅₃

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

C₈

yes

complemented

C₂×Dic₁₀₆

C₂×C₄

yes

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

C₂×C₄

yes

—

#11719919

—

#11719919

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₂×#88322

—

C₂×#88322

—

C₂³×C₅₃

C₂²⋊C₄

yes

characteristic

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

C₂×C₈

yes

—

C₂×C₄×C₅₃

M₁₆

yes

—

C₂×C₄×C₅₃

SD₁₆

yes

—

C₂×C₄×C₅₃

Q₁₆

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

C₂×C₈

yes

complemented

C₈×C₅₃

—

#88322

—

C₂²×C₅₃

C₂²⋊C₈

yes

—

C₂²×C₅₃

#103

yes

—

C₂²×C₅₃

#104

yes

—

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂²⋊C₈

yes

—

C₄×C₅₃

#103

yes

—

C₄×C₅₃

C₂²×C₈

yes

characteristic

C₄×C₅₃

C₂×M₁₆

yes

characteristic

C₄×C₅₃

C₂×SD₁₆

yes

characteristic

C₄×C₅₃

C₂×Q₁₆

yes

characteristic

Dic₅₃

—

Dic₅₃

—

C₂×Q₈⋊C₈

—

maximal, Sylow

C₂×C₅₃

Q₈⋊C₈

yes

complemented

C₂×C₅₃

C₂×C₂²⋊C₈

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₄×Q₈

—

C₂×C₄⋊C₈

D₅₃

yes

characteristic

C₂×C₄×C₈

—

Q₈⋊C₈

—

C₅₃

C₂×Q₈⋊C₈

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×C₈

—

C₂²×C₈

—

106

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

106

—

C₄×Q₈

—

106

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

106

—

C₂×C₄²

—

106

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₄⋊C₈

—

100

C₄⋊C₈

C₂×D₅₃

yes

—

101

C₄×C₈

—

102

C₄×C₈

—

106

—

103

C₂×Q₈

—

104

C₂×Q₈

—

105

C₂×Q₈

—

106

—

106

C₂×Q₈

—

106

—

107

C₂²×C₄

—

106

—

108

C₂²×C₄

—

106

—

109

C₂²×C₄

C₄×D₅₃

yes

characteristic

110

C₂²×C₄

D₂₁₂

yes

characteristic

111

C₂²×C₄

C₂²⋊D₅₃

yes

characteristic

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

106

—

115

C₂×C₈

—

106

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

106

—

118

C₄⋊C₄

—

106

—

119

C₄⋊C₄

—

106

—

120

C₄²

—

106

—

121

C₄²

—

106

—

122

C₄²

C₄×D₅₃

yes

characteristic

123

C₄²

D₂₁₂

yes

characteristic

124

C₄²

C₂²⋊D₅₃

yes

characteristic

125

C₄²

C₂²×D₅₃

yes

characteristic

126

C₂³

#11719921

yes

characteristic

127

Q₈

—

128

Q₈

—

106

—

129

Q₈

—

106

—

130

C₂×C₄

—

131

C₂×C₄

—

106

—

132

C₂×C₄

—

106

—

133

C₂×C₄

—

106

—

134

C₂×C₄

—

212

—

135

C₂×C₄

C₈×D₅₃

yes

—

136

C₂×C₄

#11719912

yes

—

137

C₂×C₄

#11719921

yes

characteristic

138

C₂×C₄

#11719921

yes

characteristic

139

C₂×C₄

#11719921

yes

characteristic

140

C₂×C₄

#11719923

yes

—

141

C₂×C₄

#11719925

yes

—

142

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

143

C₂×C₄

C₂×D₂₁₂

yes

characteristic

144

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

145

C₈

—

146

C₈

—

106

—

147

C₂²

#13133129

yes

—

148

C₂²

#13133140

yes

—

149

C₂²

#13133143

yes

—

150

C₂²

C₂×#11719921

yes

characteristic

151

C₄

—

152

C₄

—

153

C₄

—

106

—

154

C₄

—

212

—

155

C₄

#13133129

yes

—

156

C₄

#13133140

yes

—

157

C₄

C₂×C₈×D₅₃

yes

characteristic

158

C₄

C₂×#11719912

yes

characteristic

159

C₄

C₂×#11719923

yes

characteristic

160

C₄

C₂×#11719925

yes

characteristic

161

C₂

#16810353

yes

direct product factor

162

C₂

C₂×#13133129

yes

characteristic

163

C₂

C₂×#13133140

yes

characteristic

164

C₂

C₂×#13133143

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²⋊D₅₃
commutator	C₄×C₅₃	C₂²×C₈
Frattini	C₄²	C₂²×D₅₃
Fitting	C₂×C₄⋊C₈×C₅₃	C₂
socle	C₂³×C₅₃	C₂²⋊C₄
hypercenter	C₂×C₄⋊C₈	D₅₃
hypocenter	C₅₃	C₂×Q₈⋊C₈
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂²×C₄

C₂²⋊D₅₃

commutator

C₄×C₅₃

C₂²×C₈

Frattini

C₄²

C₂²×D₅₃

Fitting

C₂×C₄⋊C₈×C₅₃

C₂

socle

C₂³×C₅₃

C₂²⋊C₄

hypercenter

C₂×C₄⋊C₈

D₅₃

hypocenter

C₅₃

C₂×Q₈⋊C₈

LCS

G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
3. #16810353	C₂	direct (N × H)
27. C₂×Dic₁₀₆	C₈	semidirect
61. Dic₁₀₆	C₂×C₈	semidirect
79. C₂×C₅₃	Q₈⋊C₈	semidirect
87. C₅₃	C₂×Q₈⋊C₈	semidirect
161. C₂	#16810353	direct (N × H)

H ≅ G/N

type

3. #16810353

C₂

direct (N × H)

27. C₂×Dic₁₀₆

C₈

semidirect

61. Dic₁₀₆

C₂×C₈

semidirect

79. C₂×C₅₃

Q₈⋊C₈

semidirect

87. C₅₃

C₂×Q₈⋊C₈

semidirect

161. C₂

#16810353

direct (N × H)

Subgroups — C₂×#16810353 #20273091

Conjugacy classes of subgroups

Complements

Subgroups — C2×#16810353 #20273091

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16810353 #20273091