Group #20288110 — subgroups

Subgroups — C₂×#16811249 #20288110

Order 6784.

1296 conjugacy classes of subgroups; only the 565 normal subgroups are shown.

normal subgroups only

This group has 1296 conjugacy classes of subgroups — too many to draw. Only its 565 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	#16811249	C₂	1	16	yes	maximal, direct product factor
2	C₂×#13133196	C₂	1	4	yes	maximal, complemented
3	C₂×#13133194	C₂	1	8	yes	maximal, complemented
4	C₂²×C₂²⋊Dic₅₃	C₂	1	2	yes	maximal
5	C₂²×C₂²⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₀₆	C₂²	1	4	yes	complemented
7	C₂³×Dic₅₃	C₂²	1	4	yes	—
8	#13133196	C₂²	1	32	yes	complemented
9	#13133194	C₂²	1	64	yes	complemented
10	C₂×C₄⋊Dic₅₃	C₂²	1	4	yes	complemented
11	C₂×#11719919	C₂²	1	8	yes	complemented
12	C₂×C₂²⋊Dic₅₃	C₂²	1	4	yes	—
13	C₂×C₂²⋊Dic₅₃	C₂²	1	4	yes	—
14	C₂×C₂²⋊Dic₅₃	C₂²	1	16	yes	—
15	C₂⁵×C₅₃	C₂²	1	1	yes	characteristic
16	C₂³×C₄×C₅₃	C₂²	1	2	yes	—
17	C₂×C₂²⋊C₄×C₅₃	C₂²	1	4	yes	—
18	C₂×C₂²⋊C₄×C₅₃	C₂²	1	8	yes	—
19	C₂⁴×C₅₃	—	4	4	no	—
20	C₂⁴×C₅₃	Q₈	1	8	yes	complemented
21	C₂⁴×C₅₃	C₂³	1	2	yes	—
22	C₂⁴×C₅₃	C₂³	1	1	yes	characteristic
23	C₂⁴×C₅₃	C₂³	1	4	yes	—
24	C₂²×C₄×C₅₃	—	4	4	no	—
25	C₂²×C₄×C₅₃	C₂³	1	4	yes	—
26	C₂²×C₄×C₅₃	C₂³	1	8	yes	—
27	C₂²×Dic₅₃	—	4	8	no	—
28	C₂²×Dic₅₃	C₂³	1	8	yes	—
29	C₂²×Dic₅₃	C₂³	1	16	yes	—
30	C₂×Dic₁₀₆	—	4	8	no	—
31	C₂×Dic₁₀₆	C₂³	1	16	yes	complemented
32	C₂²⋊C₄×C₅₃	C₂³	1	16	yes	—
33	C₂²⋊Dic₅₃	C₂³	1	16	yes	—
34	C₂²⋊Dic₅₃	C₂³	1	16	yes	—
35	C₄⋊Dic₅₃	C₂³	1	16	yes	complemented
36	#11719919	C₂³	1	32	yes	complemented
37	C₂³×C₅₃	—	2	8	no	—
38	C₂³×C₅₃	—	2	4	no	—
39	C₂³×C₅₃	—	4	16	no	—
40	C₂³×C₅₃	—	4	8	no	—
41	C₂³×C₅₃	C₂×Q₈	1	8	yes	—
42	C₂³×C₅₃	C₂×Q₈	1	4	yes	—
43	C₂³×C₅₃	C₂×Q₈	1	16	yes	complemented
44	C₂³×C₅₃	C₂⁴	1	4	yes	—
45	C₂³×C₅₃	C₂⁴	1	2	yes	—
46	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
47	C₂×C₄×C₅₃	—	2	8	no	—
48	C₂×C₄×C₅₃	—	4	8	no	—
49	C₂×C₄×C₅₃	C₂⁴	1	8	yes	—
50	C₂×Dic₅₃	—	2	16	no	—
51	C₂×Dic₅₃	—	4	16	no	—
52	C₂×Dic₅₃	C₂⁴	1	16	yes	—
53	Dic₁₀₆	—	4	16	no	—
54	C₂²×C₅₃	—	2	16	no	—
55	C₂²×C₅₃	—	2	8	no	—
56	C₂²×C₅₃	—	2	8	no	—
57	C₂²×C₅₃	—	2	4	no	—
58	C₂²×C₅₃	—	4	16	no	—
59	C₂²×C₅₃	C₂²×Q₈	1	8	yes	—
60	C₂²×C₅₃	C₂²×Q₈	1	4	yes	—
61	C₂²×C₅₃	C₂²×Q₈	1	2	yes	—
62	C₂²×C₅₃	H(5,𝔽₂)	1	4	yes	—
63	C₂²×C₅₃	C₂⁵	1	1	yes	characteristic
64	C₄×C₅₃	—	2	8	no	—
65	Dic₅₃	—	2	16	no	—
66	C₂×#543	—	53	1	no	maximal, Sylow
67	C₂×C₅₃	—	2	8	no	—
68	C₂×C₅₃	—	2	4	no	—
69	C₂×C₅₃	#543	1	4	yes	complemented
70	C₂×C₅₃	C₂³×Q₈	1	1	yes	characteristic
71	C₂×C₅₃	C₂×H(5,𝔽₂)	1	2	yes	—
72	#543	—	53	16	no	—
73	C₂×C₂²⋊Q₈	—	53	4	no	—
74	C₂×C₂²⋊Q₈	—	53	8	no	—
75	C₂²×C₂²⋊C₄	—	53	2	no	—
76	C₂²×C₂²⋊C₄	D₅₃	1	1	yes	characteristic
77	C₅₃	C₂×#543	1	1	yes	characteristic, complemented, Sylow
78	C₂⁵	C₂×D₅₃	1	1	yes	characteristic
79	C₂²×Q₈	—	53	4	no	—
80	C₂³×C₄	—	53	4	no	—
81	C₂³×C₄	C₂×D₅₃	1	2	yes	—
82	C₂²⋊Q₈	—	53	32	no	—
83	C₂²⋊Q₈	—	53	64	no	—
84	C₂×C₄⋊C₄	—	53	8	no	—
85	C₂×C₄⋊C₄	—	53	4	no	—
86	C₂×C₂²⋊C₄	—	53	4	no	—
87	C₂×C₂²⋊C₄	—	53	4	no	—
88	C₂×C₂²⋊C₄	—	53	16	no	—
89	C₂×C₂²⋊C₄	C₂×D₅₃	1	4	yes	—
90	C₂×C₂²⋊C₄	C₂×D₅₃	1	8	yes	—
91	C₂⁴	—	4	4	no	—
92	C₂⁴	Dic₁₀₆	1	8	yes	complemented
93	C₂⁴	C₂²×D₅₃	1	1	yes	characteristic
94	C₂⁴	C₂²×D₅₃	1	2	yes	—
95	C₂⁴	C₂²×D₅₃	1	4	yes	—
96	C₂×Q₈	—	53	16	no	—
97	C₂×Q₈	—	212	8	no	—
98	C₂²×C₄	—	4	4	no	—
99	C₂²×C₄	—	53	8	no	—
100	C₂²×C₄	—	53	16	no	—
101	C₂²×C₄	—	212	8	no	—
102	C₂²×C₄	C₂²×D₅₃	1	4	yes	—
103	C₂²×C₄	C₂²×D₅₃	1	8	yes	—
104	C₄⋊C₄	—	53	32	no	—
105	C₄⋊C₄	—	53	16	no	—
106	C₂²⋊C₄	—	53	16	no	—
107	C₂²⋊C₄	—	53	16	no	—
108	C₂²⋊C₄	C₂²×D₅₃	1	16	yes	—
109	C₂³	—	2	4	no	—
110	C₂³	—	2	8	no	—
111	C₂³	—	4	16	no	—
112	C₂³	—	4	8	no	—
113	C₂³	C₂×Dic₁₀₆	1	4	yes	—
114	C₂³	C₂×Dic₁₀₆	1	8	yes	—
115	C₂³	C₂×Dic₁₀₆	1	16	yes	complemented
116	C₂³	C₂³×D₅₃	1	2	yes	—
117	C₂³	C₂³×D₅₃	1	4	yes	—
118	C₂³	C₂³×D₅₃	1	1	yes	characteristic
119	Q₈	—	212	16	no	—
120	C₂×C₄	—	2	8	no	—
121	C₂×C₄	—	4	8	no	—
122	C₂×C₄	—	53	16	no	—
123	C₂×C₄	—	106	16	no	—
124	C₂×C₄	—	212	16	no	—
125	C₂×C₄	C₂³×D₅₃	1	8	yes	—
126	C₂²	—	2	4	no	—
127	C₂²	—	2	8	no	—
128	C₂²	—	2	8	no	—
129	C₂²	—	2	16	no	—
130	C₂²	—	4	16	no	—
131	C₂²	C₂²×Dic₁₀₆	1	4	yes	—
132	C₂²	C₂²×Dic₁₀₆	1	8	yes	—
133	C₂²	C₂²×Dic₁₀₆	1	2	yes	—
134	C₂²	#13133262	1	4	yes	—
135	C₂²	C₂⁴×D₅₃	1	1	yes	characteristic
136	C₄	—	2	8	no	—
137	C₄	—	106	16	no	—
138	C₂	—	2	4	no	—
139	C₂	—	2	8	no	—
140	C₂	#16811249	1	4	yes	direct product factor
141	C₂	C₂³×Dic₁₀₆	1	1	yes	characteristic
142	C₂	C₂×#13133262	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16811249

C₂

yes

maximal, direct product factor

C₂×#13133196

C₂

yes

maximal, complemented

C₂×#13133194

C₂

yes

maximal, complemented

C₂²×C₂²⋊Dic₅₃

C₂

yes

maximal

C₂²×C₂²⋊C₄×C₅₃

C₂

yes

maximal, characteristic

C₂²×Dic₁₀₆

C₂²

yes

complemented

C₂³×Dic₅₃

C₂²

yes

—

#13133196

C₂²

yes

complemented

#13133194

C₂²

yes

complemented

C₂×C₄⋊Dic₅₃

C₂²

yes

complemented

C₂×#11719919

C₂²

yes

complemented

C₂×C₂²⋊Dic₅₃

C₂²

yes

—

C₂×C₂²⋊Dic₅₃

C₂²

yes

—

C₂×C₂²⋊Dic₅₃

C₂²

yes

—

C₂⁵×C₅₃

C₂²

yes

characteristic

C₂³×C₄×C₅₃

C₂²

yes

—

C₂×C₂²⋊C₄×C₅₃

C₂²

yes

—

C₂×C₂²⋊C₄×C₅₃

C₂²

yes

—

C₂⁴×C₅₃

—

C₂⁴×C₅₃

Q₈

yes

complemented

C₂⁴×C₅₃

C₂³

yes

—

C₂⁴×C₅₃

C₂³

yes

characteristic

C₂⁴×C₅₃

C₂³

yes

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂³

yes

—

C₂²×C₄×C₅₃

C₂³

yes

—

C₂²×Dic₅₃

—

C₂²×Dic₅₃

C₂³

yes

—

C₂²×Dic₅₃

C₂³

yes

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

C₂³

yes

complemented

C₂²⋊C₄×C₅₃

C₂³

yes

—

C₂²⋊Dic₅₃

C₂³

yes

—

C₂²⋊Dic₅₃

C₂³

yes

—

C₄⋊Dic₅₃

C₂³

yes

complemented

#11719919

C₂³

yes

complemented

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

C₂×Q₈

yes

—

C₂³×C₅₃

C₂×Q₈

yes

—

C₂³×C₅₃

C₂×Q₈

yes

complemented

C₂³×C₅₃

C₂⁴

yes

—

C₂³×C₅₃

C₂⁴

yes

—

C₂³×C₅₃

C₂⁴

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂⁴

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂⁴

yes

—

Dic₁₀₆

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂²×Q₈

yes

—

C₂²×C₅₃

C₂²×Q₈

yes

—

C₂²×C₅₃

C₂²×Q₈

yes

—

C₂²×C₅₃

H(5,𝔽₂)

yes

—

C₂²×C₅₃

C₂⁵

yes

characteristic

C₄×C₅₃

—

Dic₅₃

—

C₂×#543

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#543

yes

complemented

C₂×C₅₃

C₂³×Q₈

yes

characteristic

C₂×C₅₃

C₂×H(5,𝔽₂)

yes

—

#543

—

C₂×C₂²⋊Q₈

—

C₂×C₂²⋊Q₈

—

C₂²×C₂²⋊C₄

—

C₂²×C₂²⋊C₄

D₅₃

yes

characteristic

C₅₃

C₂×#543

yes

characteristic, complemented, Sylow

C₂⁵

C₂×D₅₃

yes

characteristic

C₂²×Q₈

—

C₂³×C₄

—

C₂³×C₄

C₂×D₅₃

yes

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

C₂×D₅₃

yes

—

C₂×C₂²⋊C₄

C₂×D₅₃

yes

—

C₂⁴

—

C₂⁴

Dic₁₀₆

yes

complemented

C₂⁴

C₂²×D₅₃

yes

characteristic

C₂⁴

C₂²×D₅₃

yes

—

C₂⁴

C₂²×D₅₃

yes

—

C₂×Q₈

—

C₂×Q₈

—

212

—

C₂²×C₄

—

C₂²×C₄

—

100

C₂²×C₄

—

101

C₂²×C₄

—

212

—

102

C₂²×C₄

C₂²×D₅₃

yes

—

103

C₂²×C₄

C₂²×D₅₃

yes

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₂²⋊C₄

—

107

C₂²⋊C₄

—

108

C₂²⋊C₄

C₂²×D₅₃

yes

—

109

C₂³

—

110

C₂³

—

111

C₂³

—

112

C₂³

—

113

C₂³

C₂×Dic₁₀₆

yes

—

114

C₂³

C₂×Dic₁₀₆

yes

—

115

C₂³

C₂×Dic₁₀₆

yes

complemented

116

C₂³

C₂³×D₅₃

yes

—

117

C₂³

C₂³×D₅₃

yes

—

118

C₂³

C₂³×D₅₃

yes

characteristic

119

Q₈

—

212

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

106

—

124

C₂×C₄

—

212

—

125

C₂×C₄

C₂³×D₅₃

yes

—

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

C₂²×Dic₁₀₆

yes

—

132

C₂²

C₂²×Dic₁₀₆

yes

—

133

C₂²

C₂²×Dic₁₀₆

yes

—

134

C₂²

#13133262

yes

—

135

C₂²

C₂⁴×D₅₃

yes

characteristic

136

C₄

—

137

C₄

—

106

—

138

C₂

—

139

C₂

—

140

C₂

#16811249

yes

direct product factor

141

C₂

C₂³×Dic₁₀₆

yes

characteristic

142

C₂

C₂×#13133262

yes

—

	subgroup	G/H
center	C₂³	C₂³×D₅₃
commutator	C₂²×C₅₃	C₂⁵
Frattini	C₂²	C₂⁴×D₅₃
Fitting	C₂²×C₂²⋊C₄×C₅₃	C₂
socle	C₂³×C₅₃	C₂⁴
hypercenter	C₂²×C₂²⋊C₄	D₅₃
hypocenter	C₅₃	C₂×#543
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₂²⋊C₄

subgroup

G/H

center

C₂³

C₂³×D₅₃

commutator

C₂²×C₅₃

C₂⁵

Frattini

C₂²

C₂⁴×D₅₃

Fitting

C₂²×C₂²⋊C₄×C₅₃

C₂

socle

C₂³×C₅₃

C₂⁴

hypercenter

C₂²×C₂²⋊C₄

D₅₃

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₂³ ◁ C₂²×C₂²⋊C₄

N	H ≅ G/N	type
1. #16811249	C₂	direct (N × H)
2. C₂×#13133196	C₂	semidirect
3. C₂×#13133194	C₂	semidirect
6. C₂²×Dic₁₀₆	C₂²	semidirect
8. #13133196	C₂²	semidirect
9. #13133194	C₂²	semidirect
10. C₂×C₄⋊Dic₅₃	C₂²	semidirect
11. C₂×#11719919	C₂²	semidirect
20. C₂⁴×C₅₃	Q₈	semidirect
31. C₂×Dic₁₀₆	C₂³	semidirect
35. C₄⋊Dic₅₃	C₂³	semidirect
36. #11719919	C₂³	semidirect
43. C₂³×C₅₃	C₂×Q₈	semidirect
69. C₂×C₅₃	#543	semidirect
77. C₅₃	C₂×#543	semidirect
92. C₂⁴	Dic₁₀₆	semidirect
115. C₂³	C₂×Dic₁₀₆	semidirect
140. C₂	#16811249	direct (N × H)

H ≅ G/N

type

1. #16811249

C₂

direct (N × H)

2. C₂×#13133196

C₂

semidirect

3. C₂×#13133194

C₂

semidirect

6. C₂²×Dic₁₀₆

C₂²

semidirect

8. #13133196

C₂²

semidirect

9. #13133194

C₂²

semidirect

10. C₂×C₄⋊Dic₅₃

C₂²

semidirect

11. C₂×#11719919

C₂²

semidirect

20. C₂⁴×C₅₃

Q₈

semidirect

31. C₂×Dic₁₀₆

C₂³

semidirect

35. C₄⋊Dic₅₃

C₂³

semidirect

36. #11719919

C₂³

semidirect

43. C₂³×C₅₃

C₂×Q₈

semidirect

69. C₂×C₅₃

#543

semidirect

77. C₅₃

C₂×#543

semidirect

92. C₂⁴

Dic₁₀₆

semidirect

115. C₂³

C₂×Dic₁₀₆

semidirect

140. C₂

#16811249

direct (N × H)

Subgroups — C₂×#16811249 #20288110

Conjugacy classes of subgroups

Complements

Subgroups — C2×#16811249 #20288110

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16811249 #20288110