Group #20285091 — subgroups

Subgroups — #20285091

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	C₂×#13133214	C₂	1	1	yes	maximal, characteristic
2	#16810989	C₂	1	4	yes	maximal, complemented
3	#16810971	C₂	1	4	yes	maximal, complemented
4	#16810627	C₂	1	4	yes	maximal, complemented
5	C₄×#11719917	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#533	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₂₁₂	—	2	2	no	—
8	C₂²×D₂₁₂	C₂²	1	2	yes	—
9	C₂×#11719917	C₂²	1	2	yes	—
10	#13133214	—	2	2	no	—
11	#13133214	C₂²	1	2	yes	—
12	#13133214	C₂²	1	2	yes	—
13	#13133137	C₂²	1	16	yes	complemented
14	#13133128	C₂²	1	2	yes	complemented
15	C₄×#88322	C₂²	1	2	yes	complemented
16	C₄⋊Q₈×C₅₃	C₂²	1	2	yes	complemented
17	#125×C₅₃	C₂²	1	2	yes	complemented
18	C₂²⋊Q₈×C₅₃	—	2	2	no	—
19	C₄×Q₈×C₅₃	—	2	2	no	—
20	#117×C₅₃	C₂²	1	2	yes	—
21	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
22	C₂³×D₅₃	—	2	2	no	—
23	C₂×Q₈×C₅₃	—	2	2	no	—
24	C₂²×C₄×C₅₃	D₄	1	2	yes	—
25	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
26	C₂×D₂₁₂	—	2	8	no	—
27	C₂×D₂₁₂	—	2	2	no	—
28	C₂×D₂₁₂	—	2	4	no	—
29	C₂×D₂₁₂	—	2	4	no	—
30	C₂×D₂₁₂	—	4	8	no	—
31	C₂×D₂₁₂	C₂³	1	4	yes	—
32	C₄⋊C₄×C₅₃	—	2	4	no	—
33	C₄⋊C₄×C₅₃	—	2	2	no	—
34	C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
35	C₂²⋊C₄×C₅₃	—	2	2	no	—
36	C₄²×C₅₃	—	2	2	no	—
37	C₄²×C₅₃	D₄	1	2	yes	—
38	C₄²×C₅₃	C₂³	1	1	yes	characteristic
39	C₄²×C₅₃	C₂³	1	1	yes	characteristic
40	#11719917	—	2	4	no	—
41	C₂×#88322	C₂³	1	4	yes	—
42	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
43	C₂²×D₅₃	—	2	4	no	—
44	C₂²×D₅₃	—	4	8	no	—
45	C₂²×D₅₃	—	4	4	no	—
46	Q₈×C₅₃	—	4	4	no	—
47	C₂×C₄×C₅₃	—	2	4	no	—
48	C₂×C₄×C₅₃	—	2	4	no	—
49	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₅₃	#41	1	4	yes	—
53	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
54	D₂₁₂	—	2	8	no	—
55	D₂₁₂	—	4	8	no	—
56	D₂₁₂	—	4	4	no	—
57	D₂₁₂	—	4	8	no	—
58	#88322	—	2	4	no	—
59	C₂²×C₅₃	—	2	2	no	—
60	C₂²×C₅₃	#124	1	2	yes	—
61	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
62	C₂×D₅₃	—	4	4	no	—
63	C₂×D₅₃	—	4	8	no	—
64	C₂×D₅₃	—	8	8	no	—
65	C₄×C₅₃	—	2	2	no	—
66	C₄×C₅₃	—	4	4	no	—
67	C₄×C₅₃	#124	1	2	yes	—
68	C₄×C₅₃	AGL(1,ℤ/8)	1	4	yes	—
69	C₄×C₅₃	C₂×#41	1	2	yes	—
70	#3138	—	53	1	no	maximal, Sylow
71	C₂×C₅₃	—	2	2	no	—
72	C₂×C₅₃	C₂×#124	1	1	yes	characteristic
73	C₂×C₅₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
74	D₅₃	—	8	4	no	—
75	#533	D₅₃	1	1	yes	characteristic, complemented
76	C₂×#127	—	53	1	no	—
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	—	53	2	no	—
83	C₂²×D₄	—	106	2	no	—
84	C₂×M₁₆	—	53	2	no	—
85	C₄⋊Q₈	C₂×D₅₃	1	2	yes	complemented
86	#127	—	53	2	no	—
87	#127	—	53	2	no	—
88	#127	—	106	2	no	—
89	#125	C₂×D₅₃	1	2	yes	complemented
90	C₂²⋊Q₈	—	2	2	no	—
91	C₄×Q₈	—	2	2	no	—
92	#117	C₂×D₅₃	1	2	yes	—
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	—	106	2	no	—
98	C₂×Q₈	—	2	2	no	—
99	C₂×D₄	—	53	4	no	—
100	C₂×D₄	—	106	8	no	—
101	C₂×D₄	—	106	4	no	—
102	C₂×D₄	—	106	4	no	—
103	C₂×D₄	—	106	2	no	—
104	C₂×D₄	—	212	8	no	—
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	—	2	4	no	—
110	C₄⋊C₄	—	2	2	no	—
111	C₄⋊C₄	C₂²×D₅₃	1	4	yes	—
112	C₂²⋊C₄	—	2	2	no	—
113	C₄²	—	2	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	—	106	4	no	—
118	C₂³	—	212	8	no	—
119	C₂³	—	212	4	no	—
120	C₂³	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
121	Q₈	—	4	4	no	—
122	D₄	—	106	8	no	—
123	D₄	—	212	8	no	—
124	D₄	—	212	4	no	—
125	D₄	—	212	8	no	—
126	C₂×C₄	—	2	4	no	—
127	C₂×C₄	—	2	4	no	—
128	C₂×C₄	#11719949	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	—	424	8	no	—
136	C₂²	—	212	4	no	—
137	C₂²	—	212	8	no	—
138	C₂²	#13133206	1	2	yes	—
139	C₂²	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
140	C₄	—	2	2	no	—
141	C₄	—	4	4	no	—
142	C₄	#13133206	1	2	yes	—
143	C₄	#13133242	1	4	yes	—
144	C₄	C₂×#11719949	1	2	yes	—
145	C₂	—	2	2	no	—
146	C₂	—	424	4	no	—
147	C₂	C₂×#13133206	1	1	yes	characteristic
148	C₂	C₂×#13133242	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133214

C₂

yes

maximal, characteristic

#16810989

C₂

yes

maximal, complemented

#16810971

C₂

yes

maximal, complemented

#16810627

C₂

yes

maximal, complemented

C₄×#11719917

C₂

yes

maximal, characteristic, complemented

C₅₃×#533

C₂

yes

maximal, characteristic, complemented

C₂²×D₂₁₂

—

C₂²×D₂₁₂

C₂²

yes

—

C₂×#11719917

C₂²

yes

—

#13133214

—

#13133214

C₂²

yes

—

#13133214

C₂²

yes

—

#13133137

C₂²

yes

complemented

#13133128

C₂²

yes

complemented

C₄×#88322

C₂²

yes

complemented

C₄⋊Q₈×C₅₃

C₂²

yes

complemented

#125×C₅₃

C₂²

yes

complemented

C₂²⋊Q₈×C₅₃

—

C₄×Q₈×C₅₃

—

#117×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂³×D₅₃

—

C₂×Q₈×C₅₃

—

C₂²×C₄×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

C₂³

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

C₂³

yes

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

D₄

yes

—

C₄²×C₅₃

C₂³

yes

characteristic

C₄²×C₅₃

C₂³

yes

characteristic

#11719917

—

C₂×#88322

C₂³

yes

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

—

Q₈×C₅₃

—

C₂×C₄×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

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

#124

yes

—

C₂²×C₅₃

C₂²×D₄

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#124

yes

—

C₄×C₅₃

AGL(1,ℤ/8)

yes

—

C₄×C₅₃

C₂×#41

yes

—

#3138

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

C₂×#124

yes

characteristic

C₂×C₅₃

C₂×AGL(1,ℤ/8)

yes

—

D₅₃

—

#533

D₅₃

yes

characteristic, complemented

—

—

—

—

C₄×M₁₆

—

C₅₃

#3138

yes

characteristic, complemented, Sylow

C₂²×D₄

—

C₂²×D₄

—

106

—

C₂×M₁₆

—

C₄⋊Q₈

C₂×D₅₃

yes

—

—

—

106

—

#125

C₂×D₅₃

yes

complemented

C₂²⋊Q₈

—

C₄×Q₈

—

#117

C₂×D₅₃

yes

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

#102

—

#97

—

C₄×C₈

—

C₂⁴

—

106

—

C₂×Q₈

—

C₂×D₄

—

100

C₂×D₄

—

106

—

101

C₂×D₄

—

106

—

102

C₂×D₄

—

106

—

103

C₂×D₄

—

106

—

104

C₂×D₄

—

212

—

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

—

111

C₄⋊C₄

C₂²×D₅₃

yes

—

112

C₂²⋊C₄

—

113

C₄²

—

114

C₄²

C₂²⋊D₅₃

yes

—

115

C₄²

C₂²×D₅₃

yes

characteristic

116

C₄²

C₂²×D₅₃

yes

characteristic

117

C₂³

—

106

—

118

C₂³

—

212

—

119

C₂³

—

212

—

120

C₂³

C₂×C₂²⋊D₅₃

yes

characteristic

121

Q₈

—

122

D₄

—

106

—

123

D₄

—

212

—

124

D₄

—

212

—

125

D₄

—

212

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

#11719949

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

—

424

—

136

C₂²

—

212

—

137

C₂²

—

212

—

138

C₂²

#13133206

yes

—

139

C₂²

C₂²×C₂²⋊D₅₃

yes

characteristic

140

C₄

—

141

C₄

—

142

C₄

#13133206

yes

—

143

C₄

#13133242

yes

—

144

C₄

C₂×#11719949

yes

—

145

C₂

—

146

C₂

—

424

—

147

C₂

C₂×#13133206

yes

characteristic

148

C₂

C₂×#13133242

yes

—

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₅₃×#533	C₂
socle	C₂²×C₅₃	C₂²×D₄
hypercenter	#533	D₅₃
hypocenter	C₅₃	#3138
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₅₃

commutator

C₂×C₄×C₅₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₃

Fitting

C₅₃×#533

C₂

socle

C₂²×C₅₃

C₂²×D₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
2. #16810989	C₂	semidirect
3. #16810971	C₂	semidirect
4. #16810627	C₂	semidirect
5. C₄×#11719917	C₂	semidirect
6. C₅₃×#533	C₂	semidirect
13. #13133137	C₂²	semidirect
14. #13133128	C₂²	semidirect
15. C₄×#88322	C₂²	semidirect
16. C₄⋊Q₈×C₅₃	C₂²	semidirect
17. #125×C₅₃	C₂²	semidirect
75. #533	D₅₃	semidirect
81. C₅₃	#3138	semidirect
85. C₄⋊Q₈	C₂×D₅₃	semidirect
89. #125	C₂×D₅₃	semidirect

H ≅ G/N

type

2. #16810989

C₂

semidirect

3. #16810971

C₂

semidirect

4. #16810627

C₂

semidirect

5. C₄×#11719917

C₂

semidirect

6. C₅₃×#533

C₂

semidirect

13. #13133137

C₂²

semidirect

14. #13133128

C₂²

semidirect

15. C₄×#88322

C₂²

semidirect

16. C₄⋊Q₈×C₅₃

C₂²

semidirect

17. #125×C₅₃

C₂²

semidirect

75. #533

D₅₃

semidirect

81. C₅₃

#3138

semidirect

85. C₄⋊Q₈

C₂×D₅₃

semidirect

89. #125

C₂×D₅₃

semidirect

Subgroups — #20285091

Conjugacy classes of subgroups

Complements