Group #20285137 — subgroups

Subgroups — C₂×#16811008 #20285137

Order 6784.

592 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 592 conjugacy classes of subgroups — too many to draw. Only its 165 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×#11719924	C₂	1	2	yes	maximal, complemented
2	C₂²×D₄₂₄	C₂	1	1	yes	maximal, characteristic
3	C₂×#13133206	C₂	1	2	yes	maximal, complemented
4	#16811008	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×Q₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₂₁₂	C₂²	1	2	yes	—
8	C₂×#11719924	—	2	8	no	—
9	C₂×#11719924	C₂²	1	8	yes	complemented
10	C₂×D₄₂₄	—	2	4	no	—
11	C₂×D₄₂₄	C₂²	1	4	yes	—
12	C₂²×#88322	C₂²	1	1	yes	characteristic
13	#13133206	—	2	4	no	—
14	#13133206	C₂²	1	8	yes	complemented
15	C₂×#11719921	—	2	4	no	—
16	C₂×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
17	C₈×Dic₅₃	C₂²	1	4	yes	complemented
18	C₂²×Q₈×C₅₃	C₂²	1	2	yes	—
19	C₂×Q₁₆×C₅₃	—	2	4	no	—
20	C₂×Q₁₆×C₅₃	C₂²	1	4	yes	complemented
21	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
22	C₂³×D₅₃	—	2	2	no	—
23	C₂×Q₈×C₅₃	—	2	2	no	—
24	C₂×Q₈×C₅₃	—	2	4	no	—
25	C₂×Q₈×C₅₃	—	2	4	no	—
26	C₂×Q₈×C₅₃	C₂³	1	4	yes	—
27	C₂²×C₄×C₅₃	—	2	2	no	—
28	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
29	C₂²×Dic₅₃	D₄	1	2	yes	—
30	C₂×D₂₁₂	—	2	4	no	—
31	C₂×D₂₁₂	—	2	4	no	—
32	C₂×D₂₁₂	—	2	2	no	—
33	C₂×D₂₁₂	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	#11719924	—	2	16	no	—
38	#11719921	—	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	D₄₂₄	—	2	8	no	—
44	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
45	C₂²×D₅₃	—	2	4	no	—
46	C₂²×D₅₃	—	4	4	no	—
47	C₂²×D₅₃	—	4	8	no	—
48	Q₈×C₅₃	—	2	4	no	—
49	Q₈×C₅₃	—	2	4	no	—
50	Q₈×C₅₃	—	4	4	no	—
51	C₂×C₄×C₅₃	—	2	4	no	—
52	C₂×C₄×C₅₃	—	4	4	no	—
53	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
55	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
56	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
57	C₂×Dic₅₃	—	2	4	no	—
58	C₂×Dic₅₃	C₂×D₄	1	4	yes	—
59	D₂₁₂	—	2	4	no	—
60	D₂₁₂	—	2	4	no	—
61	D₂₁₂	—	4	4	no	—
62	C₈×C₅₃	C₂×D₄	1	4	yes	—
63	#88322	C₂×D₄	1	4	yes	—
64	C₂²×C₅₃	#127	1	2	yes	—
65	C₂²×C₅₃	#135	1	4	yes	—
66	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
67	C₂×D₅₃	—	4	4	no	—
68	C₂×D₅₃	—	4	8	no	—
69	C₂×D₅₃	—	8	8	no	—
70	C₄×C₅₃	—	4	4	no	—
71	C₄×C₅₃	#127	1	2	yes	—
72	C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
73	C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
74	Dic₅₃	—	2	4	no	—
75	C₂×#495	—	53	1	no	maximal, Sylow
76	C₂×C₅₃	#495	1	4	yes	complemented
77	C₂×C₅₃	C₂×#127	1	1	yes	characteristic
78	C₂×C₅₃	C₂×#135	1	2	yes	—
79	D₅₃	—	8	4	no	—
80	C₂²×Q₁₆	D₅₃	1	1	yes	characteristic, complemented
81	C₂²×SD₁₆	—	53	2	no	—
82	C₂²×D₈	—	53	1	no	—
83	C₂×#124	—	53	2	no	—
84	#495	—	53	8	no	—
85	C₂×C₄×C₈	—	53	1	no	—
86	C₅₃	C₂×#495	1	1	yes	characteristic, complemented, Sylow
87	C₂²×Q₈	C₂×D₅₃	1	2	yes	—
88	C₂²×D₄	—	53	2	no	—
89	C₂×Q₁₆	—	2	4	no	—
90	C₂×Q₁₆	C₂×D₅₃	1	4	yes	complemented
91	C₂×SD₁₆	—	53	8	no	—
92	C₂×SD₁₆	—	106	8	no	—
93	C₂×D₈	—	53	4	no	—
94	C₂×D₈	—	106	4	no	—
95	C₂²×C₈	—	53	1	no	—
96	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
97	#124	—	53	8	no	—
98	#124	—	106	4	no	—
99	C₂×C₂²⋊C₄	—	106	4	no	—
100	C₂×C₄²	—	53	1	no	—
101	C₄×C₈	—	53	4	no	—
102	C₂⁴	—	106	2	no	—
103	C₂×Q₈	—	2	2	no	—
104	C₂×Q₈	—	2	4	no	—
105	C₂×Q₈	—	2	4	no	—
106	C₂×Q₈	C₂²×D₅₃	1	4	yes	—
107	C₂×D₄	—	53	4	no	—
108	C₂×D₄	—	106	4	no	—
109	C₂×D₄	—	106	2	no	—
110	C₂×D₄	—	106	4	no	—
111	C₂²×C₄	—	2	2	no	—
112	C₂²×C₄	—	53	2	no	—
113	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
114	Q₁₆	—	2	8	no	—
115	SD₁₆	—	106	16	no	—
116	D₈	—	106	8	no	—
117	C₂×C₈	—	53	4	no	—
118	C₂×C₈	—	53	2	no	—
119	C₂×C₈	C₂²⋊D₅₃	1	4	yes	—
120	C₂×C₈	C₂²×D₅₃	1	2	yes	—
121	C₂²⋊C₄	—	106	16	no	—
122	C₄²	—	53	2	no	—
123	C₄²	—	53	2	no	—
124	C₂³	—	106	4	no	—
125	C₂³	—	212	4	no	—
126	C₂³	—	212	8	no	—
127	C₂³	D₄×D₅₃	1	1	yes	characteristic
128	Q₈	—	2	4	no	—
129	Q₈	—	2	4	no	—
130	Q₈	—	4	4	no	—
131	D₄	—	106	4	no	—
132	D₄	—	106	4	no	—
133	D₄	—	212	4	no	—
134	C₂×C₄	—	2	4	no	—
135	C₂×C₄	—	4	4	no	—
136	C₂×C₄	—	53	4	no	—
137	C₂×C₄	—	106	4	no	—
138	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
139	C₂×C₄	D₄×D₅₃	1	2	yes	—
140	C₂×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
141	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
142	C₈	—	53	4	no	—
143	C₈	C₂×C₂²⋊D₅₃	1	4	yes	—
144	C₂²	—	424	8	no	—
145	C₂²	—	212	8	no	—
146	C₂²	—	212	4	no	—
147	C₂²	#13133213	1	2	yes	—
148	C₂²	#13133240	1	4	yes	—
149	C₂²	C₂×D₄×D₅₃	1	1	yes	characteristic
150	C₄	—	4	4	no	—
151	C₄	—	106	4	no	—
152	C₄	#13133213	1	2	yes	—
153	C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
154	C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
155	C₂	—	424	4	no	—
156	C₂	#16811008	1	4	yes	direct product factor
157	C₂	C₂×#13133213	1	1	yes	characteristic
158	C₂	C₂×#13133240	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#11719924

C₂

yes

maximal, complemented

C₂²×D₄₂₄

C₂

yes

maximal, characteristic

C₂×#13133206

C₂

yes

maximal, complemented

#16811008

C₂

yes

maximal, direct product factor

C₂×C₈×Dic₅₃

C₂

yes

maximal, characteristic, complemented

C₂²×Q₁₆×C₅₃

C₂

yes

maximal, characteristic, complemented

C₂²×D₂₁₂

C₂²

yes

—

C₂×#11719924

—

C₂×#11719924

C₂²

yes

complemented

C₂×D₄₂₄

—

C₂×D₄₂₄

C₂²

yes

—

C₂²×#88322

C₂²

yes

characteristic

#13133206

—

#13133206

C₂²

yes

complemented

C₂×#11719921

—

C₂×C₄×Dic₅₃

C₂²

yes

characteristic

C₈×Dic₅₃

C₂²

yes

complemented

C₂²×Q₈×C₅₃

C₂²

yes

—

C₂×Q₁₆×C₅₃

—

C₂×Q₁₆×C₅₃

C₂²

yes

complemented

C₂²×C₈×C₅₃

C₂²

yes

characteristic

C₂³×D₅₃

—

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

D₄

yes

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

C₂³

yes

—

Q₁₆×C₅₃

—

C₂×C₈×C₅₃

D₄

yes

—

C₂×C₈×C₅₃

C₂³

yes

—

#11719924

—

#11719921

—

C₄×Dic₅₃

D₄

yes

—

C₄×Dic₅₃

C₂³

yes

—

C₂×#88322

D₄

yes

—

C₂×#88322

C₂³

yes

—

D₄₂₄

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

—

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×D₄

yes

—

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

C₈×C₅₃

C₂×D₄

yes

—

#88322

C₂×D₄

yes

—

C₂²×C₅₃

#127

yes

—

C₂²×C₅₃

#135

yes

—

C₂²×C₅₃

C₂²×D₄

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#127

yes

—

C₄×C₅₃

C₂²×D₄

yes

characteristic

C₄×C₅₃

C₂²×D₄

yes

characteristic

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

D₅₃

—

C₂²×Q₁₆

D₅₃

yes

characteristic, complemented

C₂²×SD₁₆

—

C₂²×D₈

—

C₂×#124

—

#495

—

C₂×C₄×C₈

—

C₅₃

C₂×#495

yes

characteristic, complemented, Sylow

C₂²×Q₈

C₂×D₅₃

yes

—

C₂²×D₄

—

C₂×Q₁₆

—

C₂×Q₁₆

C₂×D₅₃

yes

complemented

C₂×SD₁₆

—

C₂×SD₁₆

—

106

—

C₂×D₈

—

C₂×D₈

—

106

—

C₂²×C₈

—

C₂²×C₈

C₂×D₅₃

yes

characteristic

#124

—

#124

—

106

—

C₂×C₂²⋊C₄

—

106

—

100

C₂×C₄²

—

101

C₄×C₈

—

102

C₂⁴

—

106

—

103

C₂×Q₈

—

104

C₂×Q₈

—

105

C₂×Q₈

—

106

C₂×Q₈

C₂²×D₅₃

yes

—

107

C₂×D₄

—

108

C₂×D₄

—

106

—

109

C₂×D₄

—

106

—

110

C₂×D₄

—

106

—

111

C₂²×C₄

—

112

C₂²×C₄

—

113

C₂²×C₄

C₂²×D₅₃

yes

characteristic

114

Q₁₆

—

115

SD₁₆

—

106

—

116

D₈

—

106

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

C₂²⋊D₅₃

yes

—

120

C₂×C₈

C₂²×D₅₃

yes

—

121

C₂²⋊C₄

—

106

—

122

C₄²

—

123

C₄²

—

124

C₂³

—

106

—

125

C₂³

—

212

—

126

C₂³

—

212

—

127

C₂³

D₄×D₅₃

yes

characteristic

128

Q₈

—

129

Q₈

—

130

Q₈

—

131

D₄

—

106

—

132

D₄

—

106

—

133

D₄

—

212

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

106

—

138

C₂×C₄

D₄×D₅₃

yes

characteristic

139

C₂×C₄

D₄×D₅₃

yes

—

140

C₂×C₄

C₂×C₂²⋊D₅₃

yes

—

141

C₂×C₄

C₂³×D₅₃

yes

characteristic

142

C₈

—

143

C₈

C₂×C₂²⋊D₅₃

yes

—

144

C₂²

—

424

—

145

C₂²

—

212

—

146

C₂²

—

212

—

147

C₂²

#13133213

yes

—

148

C₂²

#13133240

yes

—

149

C₂²

C₂×D₄×D₅₃

yes

characteristic

150

C₄

—

151

C₄

—

106

—

152

C₄

#13133213

yes

—

153

C₄

C₂×D₄×D₅₃

yes

characteristic

154

C₄

C₂²×C₂²⋊D₅₃

yes

characteristic

155

C₂

—

424

—

156

C₂

#16811008

yes

direct product factor

157

C₂

C₂×#13133213

yes

characteristic

158

C₂

C₂×#13133240

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×#495
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×#495

LCS

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×Q₁₆

N	H ≅ G/N	type
1. C₂²×#11719924	C₂	semidirect
3. C₂×#13133206	C₂	semidirect
4. #16811008	C₂	direct (N × H)
5. C₂×C₈×Dic₅₃	C₂	semidirect
6. C₂²×Q₁₆×C₅₃	C₂	semidirect
9. C₂×#11719924	C₂²	semidirect
14. #13133206	C₂²	semidirect
17. C₈×Dic₅₃	C₂²	semidirect
20. C₂×Q₁₆×C₅₃	C₂²	semidirect
76. C₂×C₅₃	#495	semidirect
80. C₂²×Q₁₆	D₅₃	semidirect
86. C₅₃	C₂×#495	semidirect
90. C₂×Q₁₆	C₂×D₅₃	semidirect
156. C₂	#16811008	direct (N × H)

H ≅ G/N

type

1. C₂²×#11719924

C₂

semidirect

3. C₂×#13133206

C₂

semidirect

4. #16811008

C₂

direct (N × H)

5. C₂×C₈×Dic₅₃

C₂

semidirect

6. C₂²×Q₁₆×C₅₃

C₂

semidirect

9. C₂×#11719924

C₂²

semidirect

14. #13133206

C₂²

semidirect

17. C₈×Dic₅₃

C₂²

semidirect

20. C₂×Q₁₆×C₅₃

C₂²

semidirect

76. C₂×C₅₃

#495

semidirect

80. C₂²×Q₁₆

D₅₃

semidirect

86. C₅₃

C₂×#495

semidirect

90. C₂×Q₁₆

C₂×D₅₃

semidirect

156. C₂

#16811008

direct (N × H)

Subgroups — C₂×#16811008 #20285137

Conjugacy classes of subgroups

Complements

Subgroups — C2×#16811008 #20285137

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16811008 #20285137