Group #20285089 — subgroups

Subgroups — #20285089

Order 6784.

484 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 141 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	#16811180	C₂	1	1	yes	maximal, characteristic, complemented
2	#16810987	C₂	1	4	yes	maximal, complemented
3	#16810969	C₂	1	4	yes	maximal, complemented
4	#16810628	C₂	1	4	yes	maximal, complemented
5	C₄×#11719917	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#127×C₅₃	C₂	1	1	yes	maximal, characteristic
7	C₂×#11719917	C₂²	1	2	yes	—
8	#13133217	C₂²	1	2	yes	complemented
9	#13133209	C₂²	1	2	yes	complemented
10	#13133196	—	2	2	no	—
11	C₄×Dic₁₀₆	—	2	2	no	—
12	#13133176	C₂²	1	2	yes	—
13	#13133138	C₂²	1	16	yes	complemented
14	#13133128	C₂²	1	2	yes	complemented
15	C₄×#88322	C₂²	1	2	yes	complemented
16	C₂²×D₄×C₅₃	—	2	2	no	—
17	C₂²×D₄×C₅₃	C₂²	1	2	yes	—
18	#127×C₅₃	—	2	2	no	—
19	#127×C₅₃	C₂²	1	2	yes	—
20	#127×C₅₃	C₂²	1	2	yes	—
21	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
22	C₂⁴×C₅₃	—	2	2	no	—
23	C₂×D₄×C₅₃	—	2	2	no	—
24	C₂×D₄×C₅₃	—	2	4	no	—
25	C₂×D₄×C₅₃	—	2	4	no	—
26	C₂×D₄×C₅₃	—	2	8	no	—
27	C₂×D₄×C₅₃	—	4	8	no	—
28	C₂×D₄×C₅₃	C₂³	1	4	yes	—
29	C₂²×C₄×C₅₃	D₄	1	2	yes	—
30	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
31	C₂×Dic₁₀₆	—	2	2	no	—
32	C₄²×C₅₃	D₄	1	2	yes	—
33	C₄²×C₅₃	C₂³	1	1	yes	characteristic
34	C₄²×C₅₃	C₂³	1	1	yes	characteristic
35	C₂²⋊Dic₅₃	—	2	2	no	—
36	C₄⋊Dic₅₃	—	2	2	no	—
37	C₄⋊Dic₅₃	C₂³	1	4	yes	—
38	#11719919	—	2	4	no	—
39	C₄×Dic₅₃	—	2	2	no	—
40	#11719917	—	2	4	no	—
41	C₂×#88322	C₂³	1	4	yes	—
42	C₂³×C₅₃	—	2	4	no	—
43	C₂³×C₅₃	—	4	8	no	—
44	C₂³×C₅₃	—	4	4	no	—
45	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
46	D₄×C₅₃	—	2	8	no	—
47	D₄×C₅₃	—	4	8	no	—
48	D₄×C₅₃	—	4	4	no	—
49	D₄×C₅₃	—	4	8	no	—
50	C₂×C₄×C₅₃	—	2	4	no	—
51	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
54	C₂×C₄×C₅₃	#41	1	4	yes	—
55	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
56	C₂×Dic₅₃	—	2	4	no	—
57	Dic₁₀₆	—	4	4	no	—
58	#88322	—	2	4	no	—
59	C₂²×C₅₃	—	2	2	no	—
60	C₂²×C₅₃	—	4	4	no	—
61	C₂²×C₅₃	—	4	8	no	—
62	C₂²×C₅₃	—	8	8	no	—
63	C₂²×C₅₃	#124	1	2	yes	—
64	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
65	C₄×C₅₃	—	2	2	no	—
66	C₄×C₅₃	#124	1	2	yes	—
67	C₄×C₅₃	AGL(1,ℤ/8)	1	4	yes	—
68	C₄×C₅₃	C₂×#41	1	2	yes	—
69	Dic₅₃	—	4	4	no	—
70	#3138	—	53	1	no	maximal, Sylow
71	C₂×C₅₃	—	2	2	no	—
72	C₂×C₅₃	—	8	4	no	—
73	C₂×C₅₃	C₂×#124	1	1	yes	characteristic
74	C₂×C₅₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
75	#533	—	53	1	no	—
76	C₂×#127	D₅₃	1	1	yes	characteristic
77	#490	—	53	4	no	—
78	#486	—	53	4	no	—
79	#418	—	53	4	no	—
80	C₄×M₁₆	—	53	1	no	—
81	C₅₃	#3138	1	1	yes	characteristic, complemented, Sylow
82	C₂²×D₄	—	2	2	no	—
83	C₂²×D₄	C₂×D₅₃	1	2	yes	—
84	C₂×M₁₆	—	53	2	no	—
85	C₄⋊Q₈	—	53	2	no	—
86	#127	—	2	2	no	—
87	#127	C₂×D₅₃	1	2	yes	—
88	#127	C₂×D₅₃	1	2	yes	—
89	#125	—	53	2	no	—
90	C₂²⋊Q₈	—	106	2	no	—
91	C₄×Q₈	—	106	2	no	—
92	#117	—	53	2	no	—
93	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
94	#102	—	53	16	no	—
95	#97	—	53	2	no	—
96	C₄×C₈	—	53	2	no	—
97	C₂⁴	—	2	2	no	—
98	C₂×Q₈	—	106	2	no	—
99	C₂×D₄	—	2	8	no	—
100	C₂×D₄	—	2	2	no	—
101	C₂×D₄	—	2	4	no	—
102	C₂×D₄	—	2	4	no	—
103	C₂×D₄	—	4	8	no	—
104	C₂×D₄	C₂²×D₅₃	1	4	yes	—
105	C₂²×C₄	C₂²⋊D₅₃	1	2	yes	—
106	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
107	M₁₆	—	106	4	no	—
108	C₂×C₈	—	53	4	no	—
109	C₄⋊C₄	—	53	4	no	—
110	C₄⋊C₄	—	106	4	no	—
111	C₄⋊C₄	—	106	2	no	—
112	C₂²⋊C₄	—	106	2	no	—
113	C₄²	—	106	2	no	—
114	C₄²	C₂²⋊D₅₃	1	2	yes	—
115	C₄²	C₂²×D₅₃	1	1	yes	characteristic
116	C₄²	C₂²×D₅₃	1	1	yes	characteristic
117	C₂³	—	2	4	no	—
118	C₂³	—	4	8	no	—
119	C₂³	—	4	4	no	—
120	C₂³	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
121	Q₈	—	212	4	no	—
122	D₄	—	2	8	no	—
123	D₄	—	4	8	no	—
124	D₄	—	4	4	no	—
125	D₄	—	4	8	no	—
126	C₂×C₄	—	2	4	no	—
127	C₂×C₄	—	106	4	no	—
128	C₂×C₄	#11719947	1	4	yes	—
129	C₂×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
130	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
131	C₂×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
132	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
133	C₈	—	106	4	no	—
134	C₂²	—	2	2	no	—
135	C₂²	—	4	4	no	—
136	C₂²	—	4	8	no	—
137	C₂²	—	8	8	no	—
138	C₂²	#13133204	1	2	yes	—
139	C₂²	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
140	C₄	—	2	2	no	—
141	C₄	—	212	4	no	—
142	C₄	#13133204	1	2	yes	—
143	C₄	#13133243	1	4	yes	—
144	C₄	C₂×#11719947	1	2	yes	—
145	C₂	—	2	2	no	—
146	C₂	—	8	4	no	—
147	C₂	C₂×#13133204	1	1	yes	characteristic
148	C₂	C₂×#13133243	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16811180

C₂

yes

maximal, characteristic, complemented

#16810987

C₂

yes

maximal, complemented

#16810969

C₂

yes

maximal, complemented

#16810628

C₂

yes

maximal, complemented

C₄×#11719917

C₂

yes

maximal, characteristic, complemented

C₂×#127×C₅₃

C₂

yes

maximal, characteristic

C₂×#11719917

C₂²

yes

—

#13133217

C₂²

yes

complemented

#13133209

C₂²

yes

complemented

#13133196

—

C₄×Dic₁₀₆

—

#13133176

C₂²

yes

—

#13133138

C₂²

yes

complemented

#13133128

C₂²

yes

complemented

C₄×#88322

C₂²

yes

complemented

C₂²×D₄×C₅₃

—

C₂²×D₄×C₅₃

C₂²

yes

—

#127×C₅₃

—

#127×C₅₃

C₂²

yes

—

#127×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂⁴×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

C₂³

yes

—

C₂²×C₄×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂×Dic₁₀₆

—

C₄²×C₅₃

D₄

yes

—

C₄²×C₅₃

C₂³

yes

characteristic

C₄²×C₅₃

C₂³

yes

characteristic

C₂²⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

C₂³

yes

—

#11719919

—

C₄×Dic₅₃

—

#11719917

—

C₂×#88322

C₂³

yes

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

#41

yes

—

C₂×C₄×C₅₃

C₂⁴

yes

characteristic

C₂×Dic₅₃

—

Dic₁₀₆

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#124

yes

—

C₂²×C₅₃

C₂²×D₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

#124

yes

—

C₄×C₅₃

AGL(1,ℤ/8)

yes

—

C₄×C₅₃

C₂×#41

yes

—

Dic₅₃

—

#3138

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

C₂×#124

yes

characteristic

C₂×C₅₃

C₂×AGL(1,ℤ/8)

yes

—

#533

—

C₂×#127

D₅₃

yes

—

—

—

C₄×M₁₆

—

C₅₃

#3138

yes

characteristic, complemented, Sylow

C₂²×D₄

—

C₂²×D₄

C₂×D₅₃

yes

—

C₂×M₁₆

—

C₄⋊Q₈

—

#127

—

#127

C₂×D₅₃

yes

—

#127

C₂×D₅₃

yes

—

#125

—

C₂²⋊Q₈

—

106

—

C₄×Q₈

—

106

—

#117

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

#102

—

#97

—

C₄×C₈

—

C₂⁴

—

C₂×Q₈

—

106

—

C₂×D₄

—

100

C₂×D₄

—

101

C₂×D₄

—

102

C₂×D₄

—

103

C₂×D₄

—

104

C₂×D₄

C₂²×D₅₃

yes

—

105

C₂²×C₄

C₂²⋊D₅₃

yes

—

106

C₂²×C₄

C₂²×D₅₃

yes

characteristic

107

M₁₆

—

106

—

108

C₂×C₈

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

106

—

111

C₄⋊C₄

—

106

—

112

C₂²⋊C₄

—

106

—

113

C₄²

—

106

—

114

C₄²

C₂²⋊D₅₃

yes

—

115

C₄²

C₂²×D₅₃

yes

characteristic

116

C₄²

C₂²×D₅₃

yes

characteristic

117

C₂³

—

118

C₂³

—

119

C₂³

—

120

C₂³

C₂×C₂²⋊D₅₃

yes

characteristic

121

Q₈

—

212

—

122

D₄

—

123

D₄

—

124

D₄

—

125

D₄

—

126

C₂×C₄

—

127

C₂×C₄

—

106

—

128

C₂×C₄

#11719947

yes

—

129

C₂×C₄

C₂×C₂²⋊D₅₃

yes

—

130

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

131

C₂×C₄

C₂×C₂²⋊D₅₃

yes

—

132

C₂×C₄

C₂³×D₅₃

yes

characteristic

133

C₈

—

106

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

—

138

C₂²

#13133204

yes

—

139

C₂²

C₂²×C₂²⋊D₅₃

yes

characteristic

140

C₄

—

141

C₄

—

212

—

142

C₄

#13133204

yes

—

143

C₄

#13133243

yes

—

144

C₄

C₂×#11719947

yes

—

145

C₂

—

146

C₂

—

147

C₂

C₂×#13133204

yes

characteristic

148

C₂

C₂×#13133243

yes

—

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₂×#127×C₅₃	C₂
socle	C₂²×C₅₃	C₂²×D₄
hypercenter	C₂×#127	D₅₃
hypocenter	C₅₃	#3138
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#127

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₅₃

commutator

C₂×C₄×C₅₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₃

Fitting

C₂×#127×C₅₃

C₂

socle

C₂²×C₅₃

C₂²×D₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#127

N	H ≅ G/N	type
1. #16811180	C₂	semidirect
2. #16810987	C₂	semidirect
3. #16810969	C₂	semidirect
4. #16810628	C₂	semidirect
5. C₄×#11719917	C₂	semidirect
8. #13133217	C₂²	semidirect
9. #13133209	C₂²	semidirect
13. #13133138	C₂²	semidirect
14. #13133128	C₂²	semidirect
15. C₄×#88322	C₂²	semidirect
81. C₅₃	#3138	semidirect

H ≅ G/N

type

1. #16811180

C₂

semidirect

2. #16810987

C₂

semidirect

3. #16810969

C₂

semidirect

4. #16810628

C₂

semidirect

5. C₄×#11719917

C₂

semidirect

8. #13133217

C₂²

semidirect

9. #13133209

C₂²

semidirect

13. #13133138

C₂²

semidirect

14. #13133128

C₂²

semidirect

15. C₄×#88322

C₂²

semidirect

81. C₅₃

#3138

semidirect

Subgroups — #20285089

Conjugacy classes of subgroups

Complements