Group #20283123 — subgroups

Subgroups — C₂×#16810575 #20283123

Order 6784.

656 conjugacy classes of subgroups; only the 341 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 341 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×#11719912	C₂	1	2	yes	maximal
2	C₂×C₄²×D₅₃	C₂	1	1	yes	maximal, characteristic
3	#16810575	C₂	1	8	yes	maximal, direct product factor
4	C₂×#13133128	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₈×Dic₅₃	C₂	1	2	yes	maximal, complemented
6	C₂×#97×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₄×D₅₃	C₄	1	2	yes	—
8	C₂²×C₄×D₅₃	C₂²	1	1	yes	characteristic
9	C₂×#11719912	C₄	1	16	yes	complemented
10	C₂×#11719912	C₂²	1	8	yes	—
11	C₂²×#88322	C₂²	1	2	yes	—
12	C₂×C₄×Dic₅₃	C₄	1	2	yes	—
13	C₂×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
14	C₄²×D₅₃	C₄	1	4	yes	—
15	C₄²×D₅₃	C₂²	1	4	yes	—
16	#13133128	C₂²	1	4	yes	complemented
17	C₈×Dic₅₃	C₂²	1	8	yes	complemented
18	C₂²×C₈×C₅₃	C₂²	1	2	yes	—
19	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
20	#97×C₅₃	C₂²	1	4	yes	complemented
21	C₂³×D₅₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
24	C₂²×Dic₅₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×Dic₅₃	C₂×C₄	1	2	yes	—
26	C₂×C₄×D₅₃	—	2	8	no	—
27	C₂×C₄×D₅₃	C₂×C₄	1	8	yes	—
28	C₂×C₄×D₅₃	C₂×C₄	1	2	yes	—
29	C₂×C₄×D₅₃	C₂×C₄	1	8	yes	—
30	C₂×C₄×D₅₃	C₂³	1	2	yes	—
31	C₂×C₈×C₅₃	C₂×C₄	1	8	yes	complemented
32	C₂×C₈×C₅₃	C₂³	1	4	yes	—
33	C₄²×C₅₃	C₂×C₄	1	2	yes	—
34	C₄²×C₅₃	C₂³	1	2	yes	—
35	C₄×Dic₅₃	C₂×C₄	1	2	yes	—
36	C₄×Dic₅₃	C₂×C₄	1	8	yes	—
37	C₄×Dic₅₃	C₂³	1	2	yes	—
38	C₂×#88322	C₂×C₄	1	8	yes	complemented
39	C₂×#88322	C₂³	1	4	yes	—
40	#11719912	C₂×C₄	1	32	yes	complemented
41	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
42	C₂²×D₅₃	—	2	4	no	—
43	C₂²×D₅₃	C₄²	1	4	yes	—
44	C₂²×D₅₃	C₂²×C₄	1	2	yes	—
45	C₂×C₄×C₅₃	—	2	4	no	—
46	C₂×C₄×C₅₃	C₂²×C₄	1	4	yes	—
47	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₄×C₅₃	C₂²×C₄	1	4	yes	—
49	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
50	C₂×Dic₅₃	C₄²	1	4	yes	—
51	C₂×Dic₅₃	M₁₆	1	8	yes	complemented
52	C₂×Dic₅₃	C₂²×C₄	1	2	yes	—
53	C₂×Dic₅₃	C₂²×C₄	1	4	yes	—
54	C₄×D₅₃	—	2	16	no	—
55	C₄×D₅₃	C₄²	1	8	yes	—
56	C₄×D₅₃	C₂²×C₄	1	8	yes	—
57	C₈×C₅₃	C₂²×C₄	1	8	yes	complemented
58	#88322	C₂²×C₄	1	8	yes	complemented
59	C₂²×C₅₃	C₂×C₄²	1	2	yes	—
60	C₂²×C₅₃	C₂×M₁₆	1	4	yes	—
61	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
62	C₂×D₅₃	—	2	4	no	—
63	C₂×D₅₃	—	2	8	no	—
64	C₂×D₅₃	C₂×C₄²	1	4	yes	—
65	C₄×C₅₃	—	2	4	no	—
66	C₄×C₅₃	C₂×C₄²	1	2	yes	—
67	C₄×C₅₃	C₂³×C₄	1	2	yes	—
68	Dic₅₃	C₂×C₄²	1	4	yes	—
69	Dic₅₃	C₂×M₁₆	1	8	yes	complemented
70	C₂×C₄×M₁₆	—	53	1	no	maximal, Sylow
71	C₂×C₅₃	C₄×M₁₆	1	4	yes	complemented
72	C₂×C₅₃	C₂²×C₄²	1	1	yes	characteristic
73	C₂×C₅₃	C₂²×M₁₆	1	2	yes	—
74	D₅₃	—	2	4	no	—
75	C₂²×M₁₆	—	53	2	no	—
76	C₂²×C₄²	—	53	1	no	—
77	C₄×M₁₆	—	53	8	no	—
78	C₂×#97	—	53	1	no	—
79	C₂×#97	D₅₃	1	1	yes	characteristic, complemented
80	C₂×C₄×C₈	—	53	2	no	—
81	C₅₃	C₂×C₄×M₁₆	1	1	yes	characteristic, complemented, Sylow
82	C₂³×C₄	—	53	1	no	—
83	C₂³×C₄	—	53	2	no	—
84	C₂×M₁₆	—	53	16	no	—
85	C₂×M₁₆	—	53	8	no	—
86	C₂²×C₈	—	53	2	no	—
87	C₂²×C₈	C₂×D₅₃	1	2	yes	—
88	C₂×C₄²	—	53	2	no	—
89	C₂×C₄²	—	53	1	no	—
90	C₂×C₄²	—	53	4	no	—
91	C₂×C₄²	—	53	4	no	—
92	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
93	#97	—	53	4	no	—
94	#97	C₂×D₅₃	1	4	yes	complemented
95	C₄×C₈	—	53	8	no	—
96	C₂⁴	—	53	1	no	—
97	C₂²×C₄	—	53	1	no	—
98	C₂²×C₄	—	53	2	no	—
99	C₂²×C₄	—	53	2	no	—
100	C₂²×C₄	—	53	8	no	—
101	C₂²×C₄	—	53	2	no	—
102	C₂²×C₄	—	53	8	no	—
103	C₂²×C₄	—	106	8	no	—
104	C₂²×C₄	C₄×D₅₃	1	2	yes	—
105	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
106	M₁₆	—	53	32	no	—
107	C₂×C₈	—	53	8	no	—
108	C₂×C₈	—	53	4	no	—
109	C₂×C₈	C₄×D₅₃	1	8	yes	complemented
110	C₂×C₈	C₂²×D₅₃	1	4	yes	—
111	C₄²	—	53	8	no	—
112	C₄²	—	53	2	no	—
113	C₄²	—	53	2	no	—
114	C₄²	C₄×D₅₃	1	2	yes	—
115	C₄²	C₂²×D₅₃	1	2	yes	—
116	C₂³	—	53	4	no	—
117	C₂³	—	53	2	no	—
118	C₂³	—	106	4	no	—
119	C₂³	C₂×C₄×D₅₃	1	1	yes	characteristic
120	C₂×C₄	—	2	4	no	—
121	C₂×C₄	—	53	4	no	—
122	C₂×C₄	—	53	8	no	—
123	C₂×C₄	—	53	2	no	—
124	C₂×C₄	—	53	4	no	—
125	C₂×C₄	—	53	8	no	—
126	C₂×C₄	—	53	8	no	—
127	C₂×C₄	—	106	16	no	—
128	C₂×C₄	C₂×C₄×D₅₃	1	4	yes	—
129	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
130	C₂×C₄	C₂×C₄×D₅₃	1	4	yes	—
131	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
132	C₈	—	53	8	no	—
133	C₈	C₂×C₄×D₅₃	1	8	yes	complemented
134	C₂²	—	53	4	no	—
135	C₂²	—	106	8	no	—
136	C₂²	—	106	4	no	—
137	C₂²	C₄²×D₅₃	1	2	yes	—
138	C₂²	M₁₆×D₅₃	1	4	yes	—
139	C₂²	C₂²×C₄×D₅₃	1	1	yes	characteristic
140	C₄	—	2	4	no	—
141	C₄	—	53	4	no	—
142	C₄	—	53	8	no	—
143	C₄	C₄²×D₅₃	1	2	yes	—
144	C₄	C₂²×C₄×D₅₃	1	2	yes	—
145	C₂	—	106	4	no	—
146	C₂	#16810575	1	4	yes	direct product factor
147	C₂	C₂×C₄²×D₅₃	1	1	yes	characteristic
148	C₂	C₂×M₁₆×D₅₃	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#11719912

C₂

yes

maximal

C₂×C₄²×D₅₃

C₂

yes

maximal, characteristic

#16810575

C₂

yes

maximal, direct product factor

C₂×#13133128

C₂

yes

maximal, characteristic, complemented

C₂×C₈×Dic₅₃

C₂

yes

maximal, complemented

C₂×#97×C₅₃

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₅₃

C₄

yes

—

C₂²×C₄×D₅₃

C₂²

yes

characteristic

C₂×#11719912

C₄

yes

complemented

C₂×#11719912

C₂²

yes

—

C₂²×#88322

C₂²

yes

—

C₂×C₄×Dic₅₃

C₄

yes

—

C₂×C₄×Dic₅₃

C₂²

yes

characteristic

C₄²×D₅₃

C₄

yes

—

C₄²×D₅₃

C₂²

yes

—

#13133128

C₂²

yes

complemented

C₈×Dic₅₃

C₂²

yes

complemented

C₂²×C₈×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

#97×C₅₃

C₂²

yes

complemented

C₂³×D₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

C₂×C₄

yes

characteristic

C₂²×Dic₅₃

C₂×C₄

yes

—

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

C₂×C₄

yes

—

C₂×C₄×D₅₃

C₂×C₄

yes

—

C₂×C₄×D₅₃

C₂×C₄

yes

—

C₂×C₄×D₅₃

C₂³

yes

—

C₂×C₈×C₅₃

C₂×C₄

yes

complemented

C₂×C₈×C₅₃

C₂³

yes

—

C₄²×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

C₂³

yes

—

C₄×Dic₅₃

C₂×C₄

yes

—

C₄×Dic₅₃

C₂×C₄

yes

—

C₄×Dic₅₃

C₂³

yes

—

C₂×#88322

C₂×C₄

yes

complemented

C₂×#88322

C₂³

yes

—

#11719912

C₂×C₄

yes

complemented

C₂³×C₅₃

C₂²×C₄

yes

characteristic

C₂²×D₅₃

—

C₂²×D₅₃

C₄²

yes

—

C₂²×D₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂⁴

yes

characteristic

C₂×Dic₅₃

C₄²

yes

—

C₂×Dic₅₃

M₁₆

yes

complemented

C₂×Dic₅₃

C₂²×C₄

yes

—

C₂×Dic₅₃

C₂²×C₄

yes

—

C₄×D₅₃

—

C₄×D₅₃

C₄²

yes

—

C₄×D₅₃

C₂²×C₄

yes

—

C₈×C₅₃

C₂²×C₄

yes

complemented

#88322

C₂²×C₄

yes

complemented

C₂²×C₅₃

C₂×C₄²

yes

—

C₂²×C₅₃

C₂×M₁₆

yes

—

C₂²×C₅₃

C₂³×C₄

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

C₂×C₄²

yes

—

C₄×C₅₃

—

C₄×C₅₃

C₂×C₄²

yes

—

C₄×C₅₃

C₂³×C₄

yes

—

Dic₅₃

C₂×C₄²

yes

—

Dic₅₃

C₂×M₁₆

yes

complemented

C₂×C₄×M₁₆

—

maximal, Sylow

C₂×C₅₃

C₄×M₁₆

yes

complemented

C₂×C₅₃

C₂²×C₄²

yes

characteristic

C₂×C₅₃

C₂²×M₁₆

yes

—

D₅₃

—

C₂²×M₁₆

—

C₂²×C₄²

—

C₄×M₁₆

—

C₂×#97

—

C₂×#97

D₅₃

yes

characteristic, complemented

C₂×C₄×C₈

—

C₅₃

C₂×C₄×M₁₆

yes

characteristic, complemented, Sylow

C₂³×C₄

—

C₂³×C₄

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂×D₅₃

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

#97

—

#97

C₂×D₅₃

yes

complemented

C₄×C₈

—

C₂⁴

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

100

C₂²×C₄

—

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

—

106

—

104

C₂²×C₄

C₄×D₅₃

yes

—

105

C₂²×C₄

C₂²×D₅₃

yes

characteristic

106

M₁₆

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

C₄×D₅₃

yes

complemented

110

C₂×C₈

C₂²×D₅₃

yes

—

111

C₄²

—

112

C₄²

—

113

C₄²

—

114

C₄²

C₄×D₅₃

yes

—

115

C₄²

C₂²×D₅₃

yes

—

116

C₂³

—

117

C₂³

—

118

C₂³

—

106

—

119

C₂³

C₂×C₄×D₅₃

yes

characteristic

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

106

—

128

C₂×C₄

C₂×C₄×D₅₃

yes

—

129

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

130

C₂×C₄

C₂×C₄×D₅₃

yes

—

131

C₂×C₄

C₂³×D₅₃

yes

characteristic

132

C₈

—

133

C₈

C₂×C₄×D₅₃

yes

complemented

134

C₂²

—

135

C₂²

—

106

—

136

C₂²

—

106

—

137

C₂²

C₄²×D₅₃

yes

—

138

C₂²

M₁₆×D₅₃

yes

—

139

C₂²

C₂²×C₄×D₅₃

yes

characteristic

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

C₄²×D₅₃

yes

—

144

C₄

C₂²×C₄×D₅₃

yes

—

145

C₂

—

106

—

146

C₂

#16810575

yes

direct product factor

147

C₂

C₂×C₄²×D₅₃

yes

characteristic

148

C₂

C₂×M₁₆×D₅₃

yes

—

	subgroup	G/H
center	C₂²×C₄	C₂²×D₅₃
commutator	C₂×C₅₃	C₂²×C₄²
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₂×#97×C₅₃	C₂
socle	C₂³×C₅₃	C₂²×C₄
hypercenter	C₂×#97	D₅₃
hypocenter	C₅₃	C₂×C₄×M₁₆
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#97

subgroup

G/H

center

C₂²×C₄

C₂²×D₅₃

commutator

C₂×C₅₃

C₂²×C₄²

Frattini

C₂×C₄

C₂³×D₅₃

Fitting

C₂×#97×C₅₃

C₂

socle

C₂³×C₅₃

C₂²×C₄

hypercenter

C₂×#97

D₅₃

hypocenter

C₅₃

C₂×C₄×M₁₆

LCS

G ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂²×C₄ ◁ C₂×#97

N	H ≅ G/N	type
3. #16810575	C₂	direct (N × H)
4. C₂×#13133128	C₂	semidirect
5. C₂×C₈×Dic₅₃	C₂	semidirect
6. C₂×#97×C₅₃	C₂	semidirect
9. C₂×#11719912	C₄	semidirect
16. #13133128	C₂²	semidirect
17. C₈×Dic₅₃	C₂²	semidirect
20. #97×C₅₃	C₂²	semidirect
31. C₂×C₈×C₅₃	C₂×C₄	semidirect
38. C₂×#88322	C₂×C₄	semidirect
40. #11719912	C₂×C₄	semidirect
51. C₂×Dic₅₃	M₁₆	semidirect
57. C₈×C₅₃	C₂²×C₄	semidirect
58. #88322	C₂²×C₄	semidirect
69. Dic₅₃	C₂×M₁₆	semidirect
71. C₂×C₅₃	C₄×M₁₆	semidirect
79. C₂×#97	D₅₃	semidirect
81. C₅₃	C₂×C₄×M₁₆	semidirect
94. #97	C₂×D₅₃	semidirect
109. C₂×C₈	C₄×D₅₃	semidirect
133. C₈	C₂×C₄×D₅₃	semidirect
146. C₂	#16810575	direct (N × H)

H ≅ G/N

type

3. #16810575

C₂

direct (N × H)

4. C₂×#13133128

C₂

semidirect

5. C₂×C₈×Dic₅₃

C₂

semidirect

6. C₂×#97×C₅₃

C₂

semidirect

9. C₂×#11719912

C₄

semidirect

16. #13133128

C₂²

semidirect

17. C₈×Dic₅₃

C₂²

semidirect

20. #97×C₅₃

C₂²

semidirect

31. C₂×C₈×C₅₃

C₂×C₄

semidirect

38. C₂×#88322

C₂×C₄

semidirect

40. #11719912

C₂×C₄

semidirect

51. C₂×Dic₅₃

M₁₆

semidirect

57. C₈×C₅₃

C₂²×C₄

semidirect

58. #88322

C₂²×C₄

semidirect

69. Dic₅₃

C₂×M₁₆

semidirect

71. C₂×C₅₃

C₄×M₁₆

semidirect

79. C₂×#97

D₅₃

semidirect

81. C₅₃

C₂×C₄×M₁₆

semidirect

94. #97

C₂×D₅₃

semidirect

109. C₂×C₈

C₄×D₅₃

semidirect

133. C₈

C₂×C₄×D₅₃

semidirect

146. C₂

#16810575

direct (N × H)

Subgroups — C₂×#16810575 #20283123

Conjugacy classes of subgroups

Complements

Subgroups — C2×#16810575 #20283123

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16810575 #20283123