Group #20290635 — subgroups

Subgroups — #20290635

Order 6784.

172 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 62 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	D₅₃⋊#108	C₂	1	1	yes	maximal, characteristic
2	#16811633	C₂	1	1	yes	maximal, characteristic
3	#16811505	C₂	1	2	yes	maximal
4	#16811490	C₂	1	2	yes	maximal
5	#16810580	C₂	1	1	yes	maximal, characteristic
6	D₅₃⋊M₁₆	C₂²	1	2	yes	—
7	#13133296	C₂²	1	2	yes	—
8	#13133288	—	2	1	no	—
9	#13133287	—	2	1	no	—
10	#13133286	—	2	1	no	—
11	#13133285	—	2	1	no	—
12	C₂×#11719912	C₂²	1	1	yes	characteristic
13	C₂×C₈×D₅₃	C₂²	1	1	yes	characteristic
14	#13133174	C₂²	1	1	yes	characteristic
15	#13133127	C₄	1	1	yes	characteristic, complemented
16	C₄×#88322	C₄	1	1	yes	characteristic, complemented
17	C₈×Dic₅₃	C₄	1	1	yes	characteristic, complemented
18	#97×C₅₃	C₄	1	1	yes	characteristic, complemented
19	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
20	C₂²⋊#6157	—	2	2	no	—
21	#11719940	—	2	2	no	—
22	C₄⋊#6157	—	2	2	no	—
23	C₄×#6157	—	2	2	no	—
24	#11719936	—	2	2	no	—
25	D₅₃⋊C₈	—	2	2	no	—
26	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
27	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
29	#11719921	—	2	1	no	—
30	#11719919	—	2	1	no	—
31	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂×#88322	C₂×C₄	1	1	yes	characteristic
33	C₂×#88322	C₂×C₄	1	1	yes	characteristic
34	#11719912	—	2	1	no	—
35	#11719912	—	2	1	no	—
36	C₈×D₅₃	D₄	1	1	yes	characteristic
37	C₈×D₅₃	D₄	1	1	yes	characteristic
38	C₈×D₅₃	Q₈	1	1	yes	characteristic
39	C₈×D₅₃	Q₈	1	1	yes	characteristic
40	C₂²×D₅₃	#41	1	1	yes	characteristic
41	C₂×#6157	—	4	2	no	—
42	C₂×C₄×C₅₃	—	2	1	no	—
43	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
44	C₂×Dic₅₃	—	2	1	no	—
45	C₂×Dic₅₃	#41	1	1	yes	characteristic
46	C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
47	C₄×D₅₃	C₂×Q₈	1	1	yes	characteristic
48	C₄×D₅₃	#41	1	1	yes	characteristic
49	C₄×D₅₃	#41	1	1	yes	characteristic
50	#88324	—	4	2	no	—
51	C₈×C₅₃	—	2	1	no	—
52	C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
53	C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
54	#88322	—	2	1	no	—
55	#88322	C₄⋊C₄	1	1	yes	characteristic
56	#88322	C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₅₃	#117	1	1	yes	characteristic
58	C₂×D₅₃	—	4	1	no	—
59	C₂×D₅₃	#124	1	1	yes	characteristic
60	C₂×D₅₃	#125	1	1	yes	characteristic
61	#6157	—	8	2	no	—
62	C₄×C₅₃	—	4	1	no	—
63	C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₄×C₅₃	#117	1	1	yes	characteristic
65	Dic₅₃	—	4	1	no	—
66	Dic₅₃	#124	1	1	yes	characteristic
67	Dic₅₃	#125	1	1	yes	characteristic
68	#1833	—	53	1	no	maximal, Sylow
69	C₂×C₅₃	—	2	1	no	—
70	C₂×C₅₃	#382	1	1	yes	characteristic
71	D₅₃	—	2	1	no	—
72	D₅₃	—	2	1	no	—
73	C₂×#108	—	53	1	no	—
74	#427	—	53	1	no	—
75	#405	—	53	1	no	—
76	#344	—	53	2	no	—
77	H(3,ℤ/4)	—	53	2	no	—
78	C₅₃	#1833	1	1	yes	characteristic, complemented, Sylow
79	C₂×M₁₆	—	53	1	no	—
80	C₂×M₁₆	—	53	2	no	—
81	C₂²×C₈	—	53	1	no	—
82	#117	—	53	1	no	—
83	#117	—	53	2	no	—
84	#108	—	106	1	no	—
85	#108	—	106	1	no	—
86	#107	—	106	1	no	—
87	#106	—	106	1	no	—
88	#97	—	53	1	no	—
89	#97	#6157	1	1	yes	characteristic, complemented
90	C₄×C₈	—	53	1	no	—
91	C₄×C₈	—	53	1	no	—
92	C₂²×C₄	—	53	1	no	—
93	M₁₆	—	106	1	no	—
94	M₁₆	—	106	1	no	—
95	M₁₆	—	106	2	no	—
96	M₁₆	—	106	2	no	—
97	C₂×C₈	—	53	1	no	—
98	C₂×C₈	—	53	1	no	—
99	C₂×C₈	—	53	1	no	—
100	C₂×C₈	—	53	1	no	—
101	C₂×C₈	—	53	1	no	—
102	C₂×C₈	—	53	1	no	—
103	C₂×C₈	—	106	2	no	—
104	C₂×C₈	C₂×#6157	1	1	yes	characteristic
105	C₂×C₈	C₂×#6157	1	1	yes	characteristic
106	C₄⋊C₄	—	106	1	no	—
107	C₄⋊C₄	—	106	2	no	—
108	C₂²⋊C₄	—	106	1	no	—
109	C₂²⋊C₄	—	106	2	no	—
110	C₄²	—	53	1	no	—
111	C₄²	—	106	2	no	—
112	C₄²	C₂×#6157	1	1	yes	characteristic
113	C₂³	—	53	1	no	—
114	C₂×C₄	—	2	1	no	—
115	C₂×C₄	—	53	1	no	—
116	C₂×C₄	—	53	1	no	—
117	C₂×C₄	—	53	1	no	—
118	C₂×C₄	—	53	1	no	—
119	C₂×C₄	—	53	1	no	—
120	C₂×C₄	—	106	1	no	—
121	C₂×C₄	—	212	2	no	—
122	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
123	C₈	—	2	1	no	—
124	C₈	—	53	1	no	—
125	C₈	—	53	1	no	—
126	C₈	—	106	1	no	—
127	C₈	—	212	2	no	—
128	C₈	C₄⋊#6157	1	1	yes	characteristic
129	C₈	C₄⋊#6157	1	1	yes	characteristic
130	C₂²	—	53	1	no	—
131	C₂²	—	53	1	no	—
132	C₂²	—	212	1	no	—
133	C₂²	#13133296	1	1	yes	characteristic
134	C₄	—	4	1	no	—
135	C₄	—	53	1	no	—
136	C₄	—	53	1	no	—
137	C₄	—	424	2	no	—
138	C₄	—	212	1	no	—
139	C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
140	C₄	#13133296	1	1	yes	characteristic
141	C₂	—	2	1	no	—
142	C₂	—	106	1	no	—
143	C₂	—	106	1	no	—
144	C₂	#16811567	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

D₅₃⋊#108

C₂

yes

maximal, characteristic

#16811633

C₂

yes

maximal, characteristic

#16811505

C₂

yes

maximal

#16811490

C₂

yes

maximal

#16810580

C₂

yes

maximal, characteristic

D₅₃⋊M₁₆

C₂²

yes

—

#13133296

C₂²

yes

—

—

—

—

—

C₂²

yes

characteristic

C₂×C₈×D₅₃

C₂²

yes

characteristic

#13133174

C₂²

yes

characteristic

#13133127

C₄

yes

characteristic, complemented

C₄×#88322

C₄

yes

characteristic, complemented

C₈×Dic₅₃

C₄

yes

characteristic, complemented

#97×C₅₃

C₄

yes

characteristic, complemented

C₂×C₄×D₅₃

C₂³

yes

characteristic

C₂²⋊#6157

—

—

—

—

—

D₅₃⋊C₈

—

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

C₂×C₄

yes

characteristic

#11719921

—

#11719919

—

C₄×Dic₅₃

C₂×C₄

yes

characteristic

C₂×#88322

C₂×C₄

yes

characteristic

C₂×#88322

C₂×C₄

yes

characteristic

#11719912

—

#11719912

—

C₈×D₅₃

D₄

yes

characteristic

C₈×D₅₃

D₄

yes

characteristic

C₈×D₅₃

Q₈

yes

characteristic

C₈×D₅₃

Q₈

yes

C₂²×D₅₃

yes

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

#41

yes

characteristic

C₄×D₅₃

C₂×D₄

yes

characteristic

C₄×D₅₃

C₂×Q₈

yes

C₄×D₅₃

yes

C₄×D₅₃

yes

—

C₈×C₅₃

—

C₈×C₅₃

C₄⋊C₄

yes

characteristic

C₈×C₅₃

C₄⋊C₄

yes

characteristic

#88322

—

#88322

C₄⋊C₄

yes

characteristic

#88322

C₄⋊C₄

yes

characteristic

C₂²×C₅₃

#117

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

yes

C₂×D₅₃

yes

—

C₄×C₅₃

—

C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₅₃

#117

yes

characteristic

Dic₅₃

—

Dic₅₃

yes

Dic₅₃

yes

—

C₂×C₅₃

—

C₂×C₅₃

#382

yes

characteristic

D₅₃

—

D₅₃

—

—

—

—

—

—

C₅₃

#1833

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

#117

—

#117

—

#108

—

106

—

#108

—

106

—

#107

—

106

—

#106

—

106

—

#97

—

#97

#6157

yes

characteristic, complemented

C₄×C₈

—

C₄×C₈

—

C₂²×C₄

—

M₁₆

—

106

—

M₁₆

—

106

—

M₁₆

—

106

—

M₁₆

—

106

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

106

—

104

C₂×C₈

C₂×#6157

yes

characteristic

105

C₂×C₈

C₂×#6157

yes

characteristic

106

C₄⋊C₄

—

106

—

107

C₄⋊C₄

—

106

—

108

C₂²⋊C₄

—

106

—

109

C₂²⋊C₄

—

106

—

110

C₄²

—

111

C₄²

—

106

—

112

C₄²

C₂×#6157

yes

characteristic

113

C₂³

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

106

—

121

C₂×C₄

—

212

—

122

C₂×C₄

C₂²×#6157

yes

characteristic

123

C₈

—

124

C₈

—

125

C₈

—

126

C₈

—

106

—

127

C₈

—

212

—

128

C₈

C₄⋊#6157

yes

characteristic

129

C₈

C₄⋊#6157

yes

characteristic

130

C₂²

—

131

C₂²

—

132

C₂²

—

212

—

133

C₂²

#13133296

yes

characteristic

134

C₄

—

135

C₄

—

136

C₄

—

137

C₄

—

424

—

138

C₄

—

212

—

139

C₄

C₂×C₄⋊#6157

yes

characteristic

140

C₄

#13133296

yes

characteristic

141

C₂

—

142

C₂

—

106

—

143

C₂

—

106

—

144

C₂

#16811567

yes

characteristic

	subgroup	G/H
center	C₄	#13133296
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×#6157
Fitting	#97×C₅₃	C₄
socle	C₂×C₅₃	#382
hypercenter	#97	#6157
hypocenter	C₅₃	#1833
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₄² ◁ #97

subgroup

G/H

center

C₄

#13133296

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₂×C₄

C₂²×#6157

Fitting

#97×C₅₃

C₄

C₂×C₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₄ ◁ C₄² ◁ #97

N	H ≅ G/N	type
15. #13133127	C₄	semidirect
16. C₄×#88322	C₄	semidirect
17. C₈×Dic₅₃	C₄	semidirect
18. #97×C₅₃	C₄	semidirect
78. C₅₃	#1833	semidirect
89. #97	#6157	semidirect

H ≅ G/N

type

15. #13133127

C₄

semidirect

16. C₄×#88322

C₄

semidirect

17. C₈×Dic₅₃

C₄

semidirect

18. #97×C₅₃

C₄

semidirect

78. C₅₃

#1833

semidirect

89. #97

#6157

semidirect

Subgroups — #20290635

Conjugacy classes of subgroups

Complements