Group #20285129 — subgroups

Subgroups — C₂×#16811001 #20285129

Order 6784.

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

normal subgroups only

This group has 552 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	C₂²×Dic₂₁₂	C₂	1	1	yes	maximal, characteristic
2	C₂²×#11719925	C₂	1	2	yes	maximal
3	C₂×#13133215	C₂	1	2	yes	maximal
4	#16811001	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₁₀₆	C₂²	1	2	yes	—
8	C₂×Dic₂₁₂	—	2	4	no	—
9	C₂×Dic₂₁₂	C₂²	1	4	yes	—
10	C₂×#11719925	—	2	8	no	—
11	C₂×#11719925	C₂²	1	8	yes	—
12	C₂²×#88322	C₂²	1	1	yes	characteristic
13	#13133215	—	2	4	no	—
14	#13133215	C₂²	1	8	yes	—
15	C₂×#11719919	—	2	4	no	—
16	C₂×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
17	C₈×Dic₅₃	C₂²	1	4	yes	—
18	C₂²×Q₈×C₅₃	C₂²	1	2	yes	—
19	C₂×Q₁₆×C₅₃	—	2	4	no	—
20	C₂×Q₁₆×C₅₃	C₂²	1	4	yes	—
21	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
22	C₂×Q₈×C₅₃	—	2	2	no	—
23	C₂×Q₈×C₅₃	—	2	4	no	—
24	C₂×Q₈×C₅₃	—	2	4	no	—
25	C₂×Q₈×C₅₃	C₂³	1	4	yes	—
26	C₂²×C₄×C₅₃	—	2	2	no	—
27	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
28	C₂²×Dic₅₃	—	2	2	no	—
29	C₂²×Dic₅₃	D₄	1	2	yes	—
30	C₂×Dic₁₀₆	—	2	4	no	—
31	C₂×Dic₁₀₆	—	2	4	no	—
32	C₂×Dic₁₀₆	—	2	2	no	—
33	C₂×Dic₁₀₆	C₂³	1	4	yes	—
34	Q₁₆×C₅₃	—	2	8	no	—
35	C₂×C₈×C₅₃	D₄	1	4	yes	—
36	C₂×C₈×C₅₃	C₂³	1	2	yes	—
37	#11719925	—	2	16	no	—
38	#11719919	—	2	16	no	—
39	C₄×Dic₅₃	D₄	1	2	yes	—
40	C₄×Dic₅₃	C₂³	1	2	yes	—
41	C₂×#88322	D₄	1	4	yes	—
42	C₂×#88322	C₂³	1	2	yes	—
43	Dic₂₁₂	—	2	8	no	—
44	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
45	Q₈×C₅₃	—	2	4	no	—
46	Q₈×C₅₃	—	2	4	no	—
47	Q₈×C₅₃	—	4	4	no	—
48	C₂×C₄×C₅₃	—	2	4	no	—
49	C₂×C₄×C₅₃	—	4	4	no	—
50	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
53	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
54	C₂×Dic₅₃	—	2	4	no	—
55	C₂×Dic₅₃	—	4	4	no	—
56	C₂×Dic₅₃	Q₁₆	1	8	yes	complemented
57	C₂×Dic₅₃	C₂×D₄	1	4	yes	—
58	Dic₁₀₆	—	2	4	no	—
59	Dic₁₀₆	—	2	4	no	—
60	Dic₁₀₆	—	4	4	no	—
61	C₈×C₅₃	C₂×D₄	1	4	yes	—
62	#88322	C₂×D₄	1	4	yes	—
63	C₂²×C₅₃	#127	1	2	yes	—
64	C₂²×C₅₃	C₂×Q₁₆	1	4	yes	—
65	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
66	C₄×C₅₃	—	4	4	no	—
67	C₄×C₅₃	#127	1	2	yes	—
68	C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
69	C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
70	Dic₅₃	—	4	4	no	—
71	Dic₅₃	C₂×Q₁₆	1	8	yes	complemented
72	C₂×#494	—	53	1	no	maximal, Sylow
73	C₂×C₅₃	#494	1	4	yes	complemented
74	C₂×C₅₃	C₂×#127	1	1	yes	characteristic
75	C₂×C₅₃	C₂²×Q₁₆	1	2	yes	—
76	C₂²×Q₁₆	—	53	2	no	—
77	C₂²×Q₁₆	—	53	1	no	—
78	C₂²×Q₁₆	D₅₃	1	1	yes	characteristic
79	C₂×C₄⋊Q₈	—	53	2	no	—
80	#494	—	53	8	no	—
81	C₂×C₄×C₈	—	53	1	no	—
82	C₅₃	C₂×#494	1	1	yes	characteristic, complemented, Sylow
83	C₂²×Q₈	—	53	2	no	—
84	C₂²×Q₈	C₂×D₅₃	1	2	yes	—
85	C₂×Q₁₆	—	2	4	no	—
86	C₂×Q₁₆	—	53	8	no	—
87	C₂×Q₁₆	—	53	4	no	—
88	C₂×Q₁₆	—	106	8	no	—
89	C₂×Q₁₆	—	106	4	no	—
90	C₂×Q₁₆	C₂×D₅₃	1	4	yes	—
91	C₂²×C₈	—	53	1	no	—
92	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
93	C₄⋊Q₈	—	53	8	no	—
94	C₄⋊Q₈	—	106	4	no	—
95	C₂×C₄⋊C₄	—	106	4	no	—
96	C₂×C₄²	—	53	1	no	—
97	C₄×C₈	—	53	4	no	—
98	C₂×Q₈	—	2	2	no	—
99	C₂×Q₈	—	2	4	no	—
100	C₂×Q₈	—	2	4	no	—
101	C₂×Q₈	—	53	4	no	—
102	C₂×Q₈	—	106	2	no	—
103	C₂×Q₈	—	106	4	no	—
104	C₂×Q₈	—	106	4	no	—
105	C₂×Q₈	C₂²×D₅₃	1	4	yes	—
106	C₂²×C₄	—	2	2	no	—
107	C₂²×C₄	—	53	2	no	—
108	C₂²×C₄	—	106	2	no	—
109	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
110	Q₁₆	—	2	8	no	—
111	Q₁₆	—	106	16	no	—
112	Q₁₆	—	106	8	no	—
113	C₂×C₈	—	53	4	no	—
114	C₂×C₈	—	53	2	no	—
115	C₂×C₈	C₂²⋊D₅₃	1	4	yes	—
116	C₂×C₈	C₂²×D₅₃	1	2	yes	—
117	C₄⋊C₄	—	106	16	no	—
118	C₄²	—	53	2	no	—
119	C₄²	—	53	2	no	—
120	C₂³	D₄×D₅₃	1	1	yes	characteristic
121	Q₈	—	2	4	no	—
122	Q₈	—	2	4	no	—
123	Q₈	—	4	4	no	—
124	Q₈	—	106	4	no	—
125	Q₈	—	106	4	no	—
126	Q₈	—	212	4	no	—
127	C₂×C₄	—	2	4	no	—
128	C₂×C₄	—	4	4	no	—
129	C₂×C₄	—	53	8	no	—
130	C₂×C₄	—	53	4	no	—
131	C₂×C₄	—	106	4	no	—
132	C₂×C₄	—	212	4	no	—
133	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
134	C₂×C₄	D₄×D₅₃	1	2	yes	—
135	C₂×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
136	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
137	C₈	—	53	4	no	—
138	C₈	C₂×C₂²⋊D₅₃	1	4	yes	—
139	C₂²	#13133213	1	2	yes	—
140	C₂²	Q₁₆×D₅₃	1	4	yes	—
141	C₂²	C₂×D₄×D₅₃	1	1	yes	characteristic
142	C₄	—	4	4	no	—
143	C₄	—	53	8	no	—
144	C₄	—	212	4	no	—
145	C₄	#13133213	1	2	yes	—
146	C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
147	C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
148	C₂	#16811001	1	4	yes	direct product factor
149	C₂	C₂×#13133213	1	1	yes	characteristic
150	C₂	C₂×Q₁₆×D₅₃	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₂₁₂

C₂

yes

maximal, characteristic

C₂²×#11719925

C₂

yes

maximal

C₂×#13133215

C₂

yes

maximal

#16811001

C₂

yes

maximal, direct product factor

C₂×C₈×Dic₅₃

C₂

yes

maximal, characteristic

C₂²×Q₁₆×C₅₃

C₂

yes

maximal, characteristic

C₂²×Dic₁₀₆

C₂²

yes

—

C₂×Dic₂₁₂

—

C₂×Dic₂₁₂

C₂²

yes

—

C₂×#11719925

—

C₂×#11719925

C₂²

yes

—

C₂²×#88322

C₂²

yes

characteristic

#13133215

—

#13133215

C₂²

yes

—

C₂×#11719919

—

C₂×C₄×Dic₅₃

C₂²

yes

characteristic

C₈×Dic₅₃

C₂²

yes

—

C₂²×Q₈×C₅₃

C₂²

yes

—

C₂×Q₁₆×C₅₃

—

C₂×Q₁₆×C₅₃

C₂²

yes

—

C₂²×C₈×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

C₂³

yes

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

—

C₂²×Dic₅₃

D₄

yes

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

C₂³

yes

—

Q₁₆×C₅₃

—

C₂×C₈×C₅₃

D₄

yes

—

C₂×C₈×C₅₃

C₂³

yes

—

#11719925

—

#11719919

—

C₄×Dic₅₃

D₄

yes

—

C₄×Dic₅₃

C₂³

yes

—

C₂×#88322

D₄

yes

—

C₂×#88322

C₂³

yes

—

Dic₂₁₂

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

Q₈×C₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂⁴

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

Q₁₆

yes

complemented

C₂×Dic₅₃

C₂×D₄

yes

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

C₂×D₄

yes

—

#88322

C₂×D₄

yes

—

C₂²×C₅₃

#127

yes

—

C₂²×C₅₃

C₂×Q₁₆

yes

—

C₂²×C₅₃

C₂²×D₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

#127

yes

—

C₄×C₅₃

C₂²×D₄

yes

characteristic

C₄×C₅₃

C₂²×D₄

yes

characteristic

Dic₅₃

—

Dic₅₃

C₂×Q₁₆

yes

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

C₂²×Q₁₆

yes

—

C₂²×Q₁₆

—

C₂²×Q₁₆

—

C₂²×Q₁₆

D₅₃

yes

characteristic

C₂×C₄⋊Q₈

—

#494

—

C₂×C₄×C₈

—

C₅₃

C₂×#494

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×Q₈

C₂×D₅₃

yes

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

106

—

C₂×Q₁₆

—

106

—

C₂×Q₁₆

C₂×D₅₃

yes

—

C₂²×C₈

—

C₂²×C₈

C₂×D₅₃

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

106

—

C₂×C₄⋊C₄

—

106

—

C₂×C₄²

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂×Q₈

—

102

C₂×Q₈

—

106

—

103

C₂×Q₈

—

106

—

104

C₂×Q₈

—

106

—

105

C₂×Q₈

C₂²×D₅₃

yes

—

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₂²×C₄

—

106

—

109

C₂²×C₄

C₂²×D₅₃

yes

characteristic

110

Q₁₆

—

111

Q₁₆

—

106

—

112

Q₁₆

—

106

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

C₂²⋊D₅₃

yes

—

116

C₂×C₈

C₂²×D₅₃

yes

—

117

C₄⋊C₄

—

106

—

118

C₄²

—

119

C₄²

—

120

C₂³

D₄×D₅₃

yes

characteristic

121

Q₈

—

122

Q₈

—

123

Q₈

—

124

Q₈

—

106

—

125

Q₈

—

106

—

126

Q₈

—

212

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

106

—

132

C₂×C₄

—

212

—

133

C₂×C₄

D₄×D₅₃

yes

characteristic

134

C₂×C₄

D₄×D₅₃

yes

—

135

C₂×C₄

C₂×C₂²⋊D₅₃

yes

—

136

C₂×C₄

C₂³×D₅₃

yes

characteristic

137

C₈

—

138

C₈

C₂×C₂²⋊D₅₃

yes

—

139

C₂²

#13133213

yes

—

140

C₂²

Q₁₆×D₅₃

yes

—

141

C₂²

C₂×D₄×D₅₃

yes

characteristic

142

C₄

—

143

C₄

—

144

C₄

—

212

—

145

C₄

#13133213

yes

—

146

C₄

C₂×D₄×D₅₃

yes

characteristic

147

C₄

C₂²×C₂²⋊D₅₃

yes

characteristic

148

C₂

#16811001

yes

direct product factor

149

C₂

C₂×#13133213

yes

characteristic

150

C₂

C₂×Q₁₆×D₅₃

yes

—

	subgroup	G/H
center	C₂³	D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₂²×Q₁₆×C₅₃	C₂
socle	C₂³×C₅₃	C₂×D₄
hypercenter	C₂²×Q₁₆	D₅₃
hypocenter	C₅₃	C₂×#494
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×Q₁₆

subgroup

G/H

center

C₂³

D₄×D₅₃

commutator

C₂×C₄×C₅₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₃

Fitting

C₂²×Q₁₆×C₅₃

C₂

socle

C₂³×C₅₃

C₂×D₄

hypercenter

C₂²×Q₁₆

D₅₃

hypocenter

C₅₃

C₂×#494

LCS

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×Q₁₆

N	H ≅ G/N	type
4. #16811001	C₂	direct (N × H)
56. C₂×Dic₅₃	Q₁₆	semidirect
71. Dic₅₃	C₂×Q₁₆	semidirect
73. C₂×C₅₃	#494	semidirect
82. C₅₃	C₂×#494	semidirect
148. C₂	#16811001	direct (N × H)

H ≅ G/N

type

4. #16811001

C₂

direct (N × H)

56. C₂×Dic₅₃

Q₁₆

semidirect

71. Dic₅₃

C₂×Q₁₆

semidirect

73. C₂×C₅₃

#494

semidirect

82. C₅₃

C₂×#494

semidirect

148. C₂

#16811001

direct (N × H)

Subgroups — C₂×#16811001 #20285129

Conjugacy classes of subgroups

Complements

Subgroups — C2×#16811001 #20285129

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16811001 #20285129