Group #20274615 — subgroups

Subgroups — #20274615

Order 6784.

272 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 133 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×#13133151	C₂	1	1	yes	maximal, characteristic
2	C₂×#13133127	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₄×Dic₅₃	C₂	1	2	yes	maximal
4	#16810385	C₂	1	2	yes	maximal
5	C₂×#106×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×#88322	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₅₃	C₂²	1	2	yes	—
8	C₂×C₄×Dic₅₃	—	2	2	no	—
9	C₂×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
10	#13133151	C₄	1	4	yes	complemented
11	#13133127	C₄	1	4	yes	complemented
12	#13133124	—	2	2	no	—
13	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
15	#106×C₅₃	C₄	1	4	yes	complemented
16	C₂²×C₄×C₅₃	—	2	2	no	—
17	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
18	C₂²×Dic₅₃	—	2	2	no	—
19	C₂²×Dic₅₃	D₄	1	2	yes	—
20	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
21	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
22	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
23	C₄⋊C₄×C₅₃	—	2	2	no	—
24	C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
25	C₄⋊Dic₅₃	—	2	2	no	—
26	C₄⋊Dic₅₃	C₂×C₄	1	4	yes	—
27	C₄×Dic₅₃	—	4	4	no	—
28	C₄×Dic₅₃	C₂×C₄	1	2	yes	—
29	C₄×Dic₅₃	Q₈	1	2	yes	—
30	C₂×#88322	—	2	1	no	—
31	C₂×#88322	—	2	1	no	—
32	C₂×#88322	C₂×C₄	1	2	yes	—
33	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₅₃	—	2	4	no	—
35	C₂×C₄×C₅₃	—	4	2	no	—
36	C₂×C₄×C₅₃	—	4	2	no	—
37	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
42	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
43	C₂×Dic₅₃	—	2	2	no	—
44	C₂×Dic₅₃	—	2	2	no	—
45	C₂×Dic₅₃	—	2	4	no	—
46	C₂×Dic₅₃	—	4	2	no	—
47	C₂×Dic₅₃	—	4	2	no	—
48	C₂×Dic₅₃	C₄⋊C₄	1	4	yes	—
49	C₈×C₅₃	C₄²	1	4	yes	complemented
50	#88322	—	2	2	no	—
51	C₂²×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₂²×C₅₃	C₄×D₄	1	1	yes	characteristic
53	C₂²×C₅₃	C₄×D₄	1	1	yes	characteristic
54	C₂²×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
55	C₂²×C₅₃	#137	1	2	yes	—
56	C₄×C₅₃	—	4	4	no	—
57	C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
58	C₄×C₅₃	#117	1	1	yes	characteristic
59	C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
60	C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
61	Dic₅₃	—	2	4	no	—
62	Dic₅₃	—	4	4	no	—
63	#1776	—	53	1	no	maximal, Sylow
64	C₂×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₅₃	#428	1	2	yes	—
66	C₂×C₅₃	#440	1	2	yes	—
67	C₂×C₅₃	#441	1	1	yes	characteristic
68	C₂×C₅₃	#442	1	1	yes	characteristic
69	C₂×#107	—	53	1	no	—
70	C₂×#106	D₅₃	1	1	yes	characteristic
71	C₂×#97	—	53	1	no	—
72	C₄×C₄⋊C₄	—	53	2	no	—
73	#340	—	53	2	no	—
74	C₅₃	#1776	1	1	yes	characteristic, complemented, Sylow
75	C₂²×C₈	—	53	1	no	—
76	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
77	C₂×C₄⋊C₄	—	53	2	no	—
78	C₂×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
79	C₂×C₄²	—	53	1	no	—
80	C₂×C₄²	—	106	2	no	—
81	#107	—	53	4	no	—
82	#106	Dic₅₃	1	4	yes	complemented
83	#97	—	53	4	no	—
84	#95	—	106	2	no	—
85	C₂²×C₄	—	2	2	no	—
86	C₂²×C₄	—	53	2	no	—
87	C₂²×C₄	—	106	2	no	—
88	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
89	C₂×C₈	—	53	2	no	—
90	C₂×C₈	—	106	1	no	—
91	C₂×C₈	—	106	1	no	—
92	C₂×C₈	C₄×D₅₃	1	2	yes	—
93	C₂×C₈	C₄×D₅₃	1	2	yes	—
94	C₂×C₈	C₂×Dic₅₃	1	2	yes	—
95	C₄⋊C₄	—	2	2	no	—
96	C₄⋊C₄	—	53	4	no	—
97	C₄⋊C₄	—	106	2	no	—
98	C₄⋊C₄	C₂×Dic₅₃	1	4	yes	—
99	C₄²	—	53	2	no	—
100	C₄²	—	53	2	no	—
101	C₄²	—	212	4	no	—
102	C₂³	D₄×D₅₃	1	1	yes	characteristic
103	C₂×C₄	—	2	4	no	—
104	C₂×C₄	—	4	2	no	—
105	C₂×C₄	—	4	2	no	—
106	C₂×C₄	—	53	4	no	—
107	C₂×C₄	—	106	2	no	—
108	C₂×C₄	—	106	2	no	—
109	C₂×C₄	—	106	4	no	—
110	C₂×C₄	—	212	2	no	—
111	C₂×C₄	—	212	2	no	—
112	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
113	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
114	C₂×C₄	#11719947	1	1	yes	characteristic
115	C₂×C₄	Q₈×D₅₃	1	1	yes	characteristic
116	C₂×C₄	#11719949	1	1	yes	characteristic
117	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
118	C₈	—	106	2	no	—
119	C₈	C₄×Dic₅₃	1	4	yes	complemented
120	C₂²	C₄⋊C₄×D₅₃	1	1	yes	characteristic
121	C₂²	#13133179	1	1	yes	characteristic
122	C₂²	D₄×Dic₅₃	1	1	yes	characteristic
123	C₂²	#13133244	1	2	yes	—
124	C₂²	#13133246	1	2	yes	—
125	C₄	—	4	4	no	—
126	C₄	—	106	4	no	—
127	C₄	—	212	4	no	—
128	C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
129	C₄	#13133175	1	1	yes	characteristic
130	C₄	#13133182	1	1	yes	characteristic
131	C₄	Q₈×Dic₅₃	1	1	yes	characteristic
132	C₂	C₄⋊C₄×Dic₅₃	1	1	yes	characteristic
133	C₂	#16810662	1	2	yes	—
134	C₂	#16810724	1	2	yes	—
135	C₂	#16810729	1	1	yes	characteristic
136	C₂	#16810734	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133151

C₂

yes

maximal, characteristic

C₂×#13133127

C₂

yes

maximal, characteristic

C₄⋊C₄×Dic₅₃

C₂

yes

maximal

#16810385

C₂

yes

maximal

C₂×#106×C₅₃

C₂

yes

maximal, characteristic

C₂²×#88322

C₂²

yes

characteristic

C₂×C₄⋊Dic₅₃

C₂²

yes

—

C₂×C₄×Dic₅₃

—

C₂×C₄×Dic₅₃

C₂²

yes

characteristic

#13133151

C₄

yes

complemented

#13133127

C₄

yes

complemented

#13133124

—

C₂²×C₈×C₅₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₅₃

C₂²

yes

—

#106×C₅₃

C₄

yes

complemented

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

—

C₂²×Dic₅₃

D₄

yes

—

C₂×C₈×C₅₃

C₂×C₄

yes

—

C₂×C₈×C₅₃

C₂×C₄

yes

—

C₂×C₈×C₅₃

C₂×C₄

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

C₂×C₄

yes

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

C₂×C₄

yes

—

C₄×Dic₅₃

—

C₄×Dic₅₃

C₂×C₄

yes

—

C₄×Dic₅₃

Q₈

yes

—

C₂×#88322

—

C₂×#88322

—

C₂×#88322

C₂×C₄

yes

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×Q₈

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

C₄⋊C₄

yes

—

C₈×C₅₃

C₄²

yes

complemented

#88322

—

C₂²×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₅₃

C₄×D₄

yes

characteristic

C₂²×C₅₃

C₄×D₄

yes

characteristic

C₂²×C₅₃

AGL(1,ℤ/8)

yes

—

C₂²×C₅₃

#137

yes

—

C₄×C₅₃

—

C₄×C₅₃

C₂×C₄²

yes

characteristic

C₄×C₅₃

#117

yes

characteristic

C₄×C₅₃

C₄×Q₈

yes

characteristic

C₄×C₅₃

C₄×Q₈

yes

characteristic

Dic₅₃

—

Dic₅₃

—

#1776

—

maximal, Sylow

C₂×C₅₃

C₄×C₄⋊C₄

yes

characteristic

C₂×C₅₃

#428

yes

—

C₂×C₅₃

#440

yes

—

C₂×C₅₃

yes

C₂×C₅₃

yes

—

D₅₃

yes

characteristic

C₂×#97

—

C₄×C₄⋊C₄

—

#340

—

C₅₃

#1776

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

C₂×D₅₃

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₅₃

yes

—

C₂×C₄²

—

C₂×C₄²

—

106

—

#107

—

#106

Dic₅₃

yes

complemented

#97

—

#95

—

106

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

106

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₂×C₈

C₄×D₅₃

yes

—

C₂×C₈

C₄×D₅₃

yes

—

C₂×C₈

C₂×Dic₅₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

106

—

C₄⋊C₄

C₂×Dic₅₃

yes

—

C₄²

—

100

C₄²

—

101

C₄²

—

212

—

102

C₂³

D₄×D₅₃

yes

characteristic

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

106

—

108

C₂×C₄

—

106

—

109

C₂×C₄

—

106

—

110

C₂×C₄

—

212

—

111

C₂×C₄

—

212

—

112

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

113

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

114

C₂×C₄

#11719947

yes

characteristic

115

C₂×C₄

Q₈×D₅₃

yes

characteristic

116

C₂×C₄

#11719949

yes

characteristic

117

C₂×C₄

C₂²×Dic₅₃

yes

characteristic

118

C₈

—

106

—

119

C₈

C₄×Dic₅₃

yes

complemented

120

C₂²

C₄⋊C₄×D₅₃

yes

characteristic

121

C₂²

#13133179

yes

characteristic

122

C₂²

D₄×Dic₅₃

yes

characteristic

123

C₂²

#13133244

yes

—

124

C₂²

#13133246

yes

—

125

C₄

—

126

C₄

—

106

—

127

C₄

—

212

—

128

C₄

C₂×C₄×Dic₅₃

yes

characteristic

129

C₄

#13133175

yes

characteristic

130

C₄

#13133182

yes

characteristic

131

C₄

Q₈×Dic₅₃

yes

characteristic

132

C₂

C₄⋊C₄×Dic₅₃

yes

characteristic

133

C₂

#16810662

yes

—

134

C₂

#16810724

yes

—

135

C₂

#16810729

yes

characteristic

136

C₂

#16810734

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₅₃
commutator	C₄×C₅₃	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₂×#106×C₅₃	C₂
socle	C₂³×C₅₃	C₂×D₄
hypercenter	C₂×#106	D₅₃
hypocenter	C₅₃	#1776
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

subgroup

G/H

center

C₂³

D₄×D₅₃

commutator

C₄×C₅₃

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₂×#106×C₅₃

C₂

socle

C₂³×C₅₃

C₂×D₄

D₅₃

C₅₃

G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

N	H ≅ G/N	type
10. #13133151	C₄	semidirect
11. #13133127	C₄	semidirect
15. #106×C₅₃	C₄	semidirect
49. C₈×C₅₃	C₄²	semidirect
74. C₅₃	#1776	semidirect
82. #106	Dic₅₃	semidirect
119. C₈	C₄×Dic₅₃	semidirect

H ≅ G/N

type

10. #13133151

C₄

semidirect

11. #13133127

C₄

semidirect

15. #106×C₅₃

C₄

semidirect

49. C₈×C₅₃

C₄²

semidirect

74. C₅₃

#1776

semidirect

82. #106

Dic₅₃

semidirect

119. C₈

C₄×Dic₅₃

semidirect

Subgroups — #20274615

Conjugacy classes of subgroups

Complements