Group #20284117 — subgroups

Subgroups — #20284117

Order 6784.

772 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 772 conjugacy classes of subgroups — too many to draw. Only its 161 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	#16811187	C₂	1	3	yes	maximal, complemented
2	#16810810	C₂	1	8	yes	maximal, complemented
3	#16810597	C₂	1	3	yes	maximal, complemented
4	C₅₃×#585	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₂₁₂	C₂²	1	1	yes	characteristic, complemented
6	#13133213	—	2	3	no	—
7	#13133206	—	2	3	no	—
8	#13133191	—	2	6	no	—
9	#13133185	C₂²	1	12	yes	complemented
10	#13133176	C₂²	1	3	yes	complemented
11	#13133132	C₂²	1	12	yes	complemented
12	2₋¹⁺⁴×C₅₃	—	2	3	no	—
13	H(5,𝔽₂)×C₅₃	—	2	3	no	—
14	H(5,𝔽₂)×C₅₃	C₂²	1	4	yes	—
15	C₂×#41×C₅₃	—	2	6	no	—
16	C₂×#41×C₅₃	C₂²	1	3	yes	—
17	C₂³×D₅₃	C₂³	1	2	yes	—
18	#41×C₅₃	—	2	4	no	—
19	#41×C₅₃	—	2	12	no	—
20	#41×C₅₃	—	2	6	no	—
21	#41×C₅₃	—	2	12	no	—
22	#41×C₅₃	—	2	6	no	—
23	C₂×Q₈×C₅₃	—	2	6	no	—
24	C₂×Q₈×C₅₃	D₄	1	3	yes	complemented
25	C₂×D₄×C₅₃	—	2	6	no	—
26	C₂×D₄×C₅₃	—	2	12	no	—
27	C₂×D₄×C₅₃	D₄	1	3	yes	complemented
28	C₂×D₄×C₅₃	C₂³	1	6	yes	—
29	C₂²×C₄×C₅₃	—	2	4	no	—
30	C₂²×C₄×C₅₃	D₄	1	6	yes	complemented
31	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
32	C₂×C₂²⋊D₅₃	—	2	12	no	—
33	C₂×D₂₁₂	—	2	3	no	—
34	C₂×D₂₁₂	—	2	3	no	—
35	C₂²⋊Dic₅₃	C₂³	1	6	yes	—
36	#11719921	—	2	12	no	—
37	C₄⋊Dic₅₃	—	2	3	no	—
38	C₄×Dic₅₃	—	2	3	no	—
39	C₂³×C₅₃	—	2	4	no	—
40	C₂³×C₅₃	C₂×D₄	1	6	yes	—
41	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
42	C₂²×D₅₃	—	2	6	no	—
43	C₂²×D₅₃	—	8	2	no	—
44	Q₈×C₅₃	—	2	4	no	—
45	Q₈×C₅₃	—	2	6	no	—
46	D₄×C₅₃	—	2	12	no	—
47	D₄×C₅₃	—	2	12	no	—
48	D₄×C₅₃	—	2	6	no	—
49	C₂×C₄×C₅₃	—	2	12	no	—
50	C₂×C₄×C₅₃	—	2	6	no	—
51	C₂×C₄×C₅₃	—	2	6	no	—
52	C₂×C₄×C₅₃	C₂×D₄	1	6	yes	—
53	C₂×C₄×C₅₃	C₂×D₄	1	3	yes	—
54	C₂×C₄×C₅₃	C₂×D₄	1	3	yes	—
55	C₂²⋊D₅₃	—	8	12	no	—
56	C₂×Dic₅₃	—	2	6	no	—
57	D₂₁₂	—	4	1	no	—
58	D₂₁₂	—	4	3	no	—
59	C₂²×C₅₃	—	2	6	no	—
60	C₂²×C₅₃	—	4	1	no	—
61	C₂²×C₅₃	—	4	6	no	—
62	C₂²×C₅₃	—	8	4	no	—
63	C₂²×C₅₃	C₂²×D₄	1	3	yes	—
64	C₂×D₅₃	—	4	12	no	—
65	C₂×D₅₃	—	4	2	no	—
66	C₄×C₅₃	—	2	6	no	—
67	C₄×C₅₃	#120	1	1	yes	characteristic
68	C₄×C₅₃	#120	1	3	yes	—
69	Dic₅₃	—	4	6	no	—
70	#3027	—	53	1	no	maximal, Sylow
71	C₂×C₅₃	—	2	3	no	—
72	C₂×C₅₃	—	4	6	no	—
73	C₂×C₅₃	C₂×#120	1	1	yes	characteristic
74	D₅₃	—	8	2	no	—
75	#585	D₅₃	1	1	yes	characteristic, complemented
76	#535	—	53	3	no	—
77	B(4,𝔽₂)	—	53	8	no	—
78	#410	—	53	3	no	—
79	C₅₃	#3027	1	1	yes	characteristic, complemented, Sylow
80	2₋¹⁺⁴	—	2	3	no	—
81	H(5,𝔽₂)	—	2	3	no	—
82	H(5,𝔽₂)	C₂×D₅₃	1	4	yes	—
83	C₂×#41	—	2	6	no	—
84	C₂×#41	C₂×D₅₃	1	3	yes	—
85	C₂²×D₄	—	53	1	no	—
86	#127	—	106	3	no	—
87	#124	—	106	3	no	—
88	#121	—	106	6	no	—
89	#120	—	53	12	no	—
90	#117	—	53	3	no	—
91	#99	—	53	12	no	—
92	C₂⁴	—	53	2	no	—
93	#41	—	2	4	no	—
94	#41	—	2	12	no	—
95	#41	—	2	6	no	—
96	#41	—	2	12	no	—
97	#41	—	2	6	no	—
98	C₂×Q₈	—	2	6	no	—
99	C₂×Q₈	C₂²⋊D₅₃	1	3	yes	complemented
100	C₂×D₄	—	2	6	no	—
101	C₂×D₄	—	2	12	no	—
102	C₂×D₄	—	106	12	no	—
103	C₂×D₄	—	106	3	no	—
104	C₂×D₄	—	106	3	no	—
105	C₂×D₄	C₂²⋊D₅₃	1	3	yes	complemented
106	C₂×D₄	C₂²×D₅₃	1	6	yes	—
107	C₂²×C₄	—	2	4	no	—
108	C₂²×C₄	C₂²⋊D₅₃	1	6	yes	complemented
109	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
110	C₄⋊C₄	—	106	3	no	—
111	C₂²⋊C₄	—	53	6	no	—
112	C₂²⋊C₄	—	106	12	no	—
113	C₄²	—	106	3	no	—
114	C₂³	—	2	4	no	—
115	C₂³	—	106	6	no	—
116	C₂³	—	424	2	no	—
117	C₂³	C₂×C₂²⋊D₅₃	1	6	yes	—
118	C₂³	C₂³×D₅₃	1	1	yes	characteristic
119	Q₈	—	2	4	no	—
120	Q₈	—	2	6	no	—
121	D₄	—	2	12	no	—
122	D₄	—	2	12	no	—
123	D₄	—	2	6	no	—
124	D₄	—	424	12	no	—
125	D₄	—	212	1	no	—
126	D₄	—	212	3	no	—
127	C₂×C₄	—	2	12	no	—
128	C₂×C₄	—	2	6	no	—
129	C₂×C₄	—	2	6	no	—
130	C₂×C₄	—	106	6	no	—
131	C₂×C₄	C₂×C₂²⋊D₅₃	1	6	yes	—
132	C₂×C₄	C₂×C₂²⋊D₅₃	1	3	yes	—
133	C₂×C₄	C₂×C₂²⋊D₅₃	1	3	yes	—
134	C₂²	—	2	6	no	—
135	C₂²	—	4	1	no	—
136	C₂²	—	4	6	no	—
137	C₂²	—	8	4	no	—
138	C₂²	—	212	12	no	—
139	C₂²	—	212	2	no	—
140	C₂²	C₂²×C₂²⋊D₅₃	1	3	yes	—
141	C₄	—	2	6	no	—
142	C₄	—	212	6	no	—
143	C₄	#13133186	1	1	yes	characteristic
144	C₄	#13133186	1	3	yes	—
145	C₂	—	2	3	no	—
146	C₂	—	4	6	no	—
147	C₂	—	424	2	no	—
148	C₂	C₂×#13133186	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811187

C₂

yes

maximal, complemented

#16810810

C₂

yes

maximal, complemented

#16810597

C₂

yes

maximal, complemented

C₅₃×#585

C₂

yes

maximal, characteristic, complemented

C₂²×D₂₁₂

C₂²

yes

characteristic, complemented

—

—

—

C₂²

yes

complemented

#13133176

C₂²

yes

complemented

#13133132

C₂²

yes

complemented

2₋¹⁺⁴×C₅₃

—

H(5,𝔽₂)×C₅₃

—

H(5,𝔽₂)×C₅₃

C₂²

yes

—

C₂×#41×C₅₃

—

C₂×#41×C₅₃

C₂²

yes

—

C₂³×D₅₃

C₂³

yes

—

#41×C₅₃

—

#41×C₅₃

—

#41×C₅₃

—

#41×C₅₃

—

#41×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

D₄

yes

complemented

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

D₄

yes

complemented

C₂×D₄×C₅₃

C₂³

yes

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

D₄

yes

complemented

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂×C₂²⋊D₅₃

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂²⋊Dic₅₃

C₂³

yes

—

#11719921

—

C₄⋊Dic₅₃

—

C₄×Dic₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

C₂×D₄

yes

—

C₂³×C₅₃

C₂⁴

yes

characteristic

C₂²×D₅₃

—

C₂²×D₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂²⋊D₅₃

—

C₂×Dic₅₃

—

D₂₁₂

—

D₂₁₂

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂²×D₄

yes

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#120

yes

characteristic

C₄×C₅₃

#120

yes

—

Dic₅₃

—

#3027

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

C₂×#120

yes

characteristic

D₅₃

—

#585

D₅₃

yes

characteristic, complemented

#535

—

B(4,𝔽₂)

—

#410

—

C₅₃

#3027

yes

characteristic, complemented, Sylow

2₋¹⁺⁴

—

H(5,𝔽₂)

—

H(5,𝔽₂)

C₂×D₅₃

yes

—

C₂×#41

—

C₂×#41

C₂×D₅₃

yes

—

C₂²×D₄

—

#127

—

106

—

#124

—

106

—

#121

—

106

—

#120

—

#117

—

#99

—

C₂⁴

—

—

—

—

—

—

C₂×Q₈

—

C₂×Q₈

C₂²⋊D₅₃

yes

complemented

100

C₂×D₄

—

101

C₂×D₄

—

102

C₂×D₄

—

106

—

103

C₂×D₄

—

106

—

104

C₂×D₄

—

106

—

105

C₂×D₄

C₂²⋊D₅₃

yes

complemented

106

C₂×D₄

C₂²×D₅₃

yes

—

107

C₂²×C₄

—

108

C₂²×C₄

C₂²⋊D₅₃

yes

complemented

109

C₂²×C₄

C₂²×D₅₃

yes

characteristic

110

C₄⋊C₄

—

106

—

111

C₂²⋊C₄

—

112

C₂²⋊C₄

—

106

—

113

C₄²

—

106

—

114

C₂³

—

115

C₂³

—

106

—

116

C₂³

—

424

—

117

C₂³

C₂×C₂²⋊D₅₃

yes

—

118

C₂³

C₂³×D₅₃

yes

characteristic

119

Q₈

—

120

Q₈

—

121

D₄

—

122

D₄

—

123

D₄

—

124

D₄

—

424

—

125

D₄

—

212

—

126

D₄

—

212

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

106

—

131

C₂×C₄

C₂×C₂²⋊D₅₃

yes

—

132

C₂×C₄

C₂×C₂²⋊D₅₃

yes

—

133

C₂×C₄

C₂×C₂²⋊D₅₃

yes

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

—

138

C₂²

—

212

—

139

C₂²

—

212

—

140

C₂²

C₂²×C₂²⋊D₅₃

yes

—

141

C₄

—

142

C₄

—

212

—

143

C₄

#13133186

yes

characteristic

144

C₄

#13133186

yes

—

145

C₂

—

146

C₂

—

147

C₂

—

424

—

148

C₂

C₂×#13133186

yes

characteristic

	subgroup	G/H
center	C₂	C₂×#13133186
commutator	C₂³×C₅₃	C₂⁴
Frattini	C₂³	C₂³×D₅₃
Fitting	C₅₃×#585	C₂
socle	C₂×C₅₃	C₂×#120
hypercenter	#585	D₅₃
hypocenter	C₅₃	#3027
LCS	G ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #585

subgroup

G/H

center

C₂

C₂×#13133186

commutator

C₂³×C₅₃

C₂⁴

Frattini

C₂³

C₂³×D₅₃

Fitting

C₅₃×#585

C₂

C₂×C₅₃

D₅₃

C₅₃

G ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #585

N	H ≅ G/N	type
1. #16811187	C₂	semidirect
2. #16810810	C₂	semidirect
3. #16810597	C₂	semidirect
4. C₅₃×#585	C₂	semidirect
5. C₂²×D₂₁₂	C₂²	semidirect
9. #13133185	C₂²	semidirect
10. #13133176	C₂²	semidirect
11. #13133132	C₂²	semidirect
24. C₂×Q₈×C₅₃	D₄	semidirect
27. C₂×D₄×C₅₃	D₄	semidirect
30. C₂²×C₄×C₅₃	D₄	semidirect
75. #585	D₅₃	semidirect
79. C₅₃	#3027	semidirect
99. C₂×Q₈	C₂²⋊D₅₃	semidirect
105. C₂×D₄	C₂²⋊D₅₃	semidirect
108. C₂²×C₄	C₂²⋊D₅₃	semidirect

H ≅ G/N

type

1. #16811187

C₂

semidirect

2. #16810810

C₂

semidirect

3. #16810597

C₂

semidirect

4. C₅₃×#585

C₂

semidirect

5. C₂²×D₂₁₂

C₂²

semidirect

9. #13133185

C₂²

semidirect

10. #13133176

C₂²

semidirect

11. #13133132

C₂²

semidirect

24. C₂×Q₈×C₅₃

D₄

semidirect

27. C₂×D₄×C₅₃

D₄

semidirect

30. C₂²×C₄×C₅₃

D₄

semidirect

75. #585

D₅₃

semidirect

79. C₅₃

#3027

semidirect

99. C₂×Q₈

C₂²⋊D₅₃

semidirect

105. C₂×D₄

C₂²⋊D₅₃

semidirect

108. C₂²×C₄

C₂²⋊D₅₃

semidirect

Subgroups — #20284117

Conjugacy classes of subgroups

Complements