Group #20276390 — subgroups

Subgroups — #20276390

Order 6784.

238 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 73 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	#16810597	C₂	1	2	yes	maximal
2	#16810418	C₂	1	4	yes	maximal, complemented
3	C₅₃×#536	C₂	1	1	yes	maximal, characteristic
4	#13133176	—	2	2	no	—
5	#13133132	—	2	2	no	—
6	#13133132	C₂²	1	4	yes	—
7	C₂×#41×C₅₃	C₂²	1	1	yes	characteristic
8	C₂²×Q₈×C₅₃	C₄	1	1	yes	characteristic, complemented
9	C₄⋊Q₈×C₅₃	C₄	1	2	yes	complemented
10	#124×C₅₃	C₂²	1	2	yes	—
11	#123×C₅₃	—	2	2	no	—
12	C₂²⋊Q₈×C₅₃	—	2	2	no	—
13	#117×C₅₃	C₄	1	1	yes	characteristic, complemented
14	#41×C₅₃	—	4	2	no	—
15	C₂×Q₈×C₅₃	—	2	1	no	—
16	C₂×Q₈×C₅₃	—	4	2	no	—
17	C₂×Q₈×C₅₃	C₂×C₄	1	2	yes	—
18	C₂×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
19	C₂×D₄×C₅₃	D₄	1	2	yes	—
20	C₂×D₄×C₅₃	C₂³	1	1	yes	characteristic
21	C₂²×C₄×C₅₃	—	2	1	no	—
22	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₄×C₅₃	D₄	1	2	yes	—
24	C₄⋊C₄×C₅₃	—	2	1	no	—
25	C₄⋊C₄×C₅₃	—	2	4	no	—
26	C₂²⋊C₄×C₅₃	—	2	1	no	—
27	C₂²⋊C₄×C₅₃	—	2	4	no	—
28	C₄²×C₅₃	C₂×C₄	1	2	yes	—
29	C₂²⋊Dic₅₃	—	2	4	no	—
30	C₄⋊Dic₅₃	—	4	2	no	—
31	C₄×Dic₅₃	—	4	2	no	—
32	C₂³×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
33	C₂³×C₅₃	C₂×D₄	1	2	yes	—
34	Q₈×C₅₃	—	4	1	no	—
35	Q₈×C₅₃	—	4	2	no	—
36	Q₈×C₅₃	—	4	2	no	—
37	D₄×C₅₃	—	4	2	no	—
38	D₄×C₅₃	—	4	1	no	—
39	C₂×C₄×C₅₃	—	2	2	no	—
40	C₂×C₄×C₅₃	—	2	2	no	—
41	C₂×C₄×C₅₃	—	2	2	no	—
42	C₂×C₄×C₅₃	—	2	2	no	—
43	C₂×C₄×C₅₃	—	2	2	no	—
44	C₂×C₄×C₅₃	—	4	2	no	—
45	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
46	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
48	C₂×Dic₅₃	—	4	4	no	—
49	C₂²×C₅₃	—	2	2	no	—
50	C₂²×C₅₃	—	2	1	no	—
51	C₂²×C₅₃	—	4	1	no	—
52	C₂²×C₅₃	—	4	2	no	—
53	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
54	C₄×C₅₃	—	2	2	no	—
55	C₄×C₅₃	—	2	1	no	—
56	C₄×C₅₃	—	4	2	no	—
57	C₄×C₅₃	—	4	2	no	—
58	C₄×C₅₃	#99	1	2	yes	—
59	Dic₅₃	—	8	4	no	—
60	#2129	—	53	1	no	maximal, Sylow
61	C₂×C₅₃	—	2	1	no	—
62	C₂×C₅₃	—	4	2	no	—
63	C₂×C₅₃	—	4	1	no	—
64	C₂×C₅₃	C₂×#99	1	1	yes	characteristic
65	#536	D₅₃	1	1	yes	characteristic
66	#410	—	53	2	no	—
67	#354	—	53	4	no	—
68	C₅₃	#2129	1	1	yes	characteristic, complemented, Sylow
69	C₂×#41	C₂×D₅₃	1	1	yes	characteristic
70	C₂²×Q₈	Dic₅₃	1	1	yes	characteristic, complemented
71	C₄⋊Q₈	Dic₅₃	1	2	yes	complemented
72	#124	C₂×D₅₃	1	2	yes	—
73	#123	—	2	2	no	—
74	C₂²⋊Q₈	—	2	2	no	—
75	#117	—	106	2	no	—
76	#117	Dic₅₃	1	1	yes	characteristic, complemented
77	#99	—	53	4	no	—
78	#99	—	106	2	no	—
79	#41	—	4	2	no	—
80	C₂×Q₈	—	2	1	no	—
81	C₂×Q₈	—	4	2	no	—
82	C₂×Q₈	C₂×Dic₅₃	1	2	yes	—
83	C₂×Q₈	C₂×Dic₅₃	1	1	yes	characteristic
84	C₂×D₄	C₂²⋊D₅₃	1	2	yes	—
85	C₂×D₄	C₂²×D₅₃	1	1	yes	characteristic
86	C₂²×C₄	—	2	1	no	—
87	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
88	C₂²×C₄	C₂²⋊D₅₃	1	2	yes	—
89	C₄⋊C₄	—	2	4	no	—
90	C₄⋊C₄	—	2	1	no	—
91	C₄⋊C₄	—	212	2	no	—
92	C₂²⋊C₄	—	2	4	no	—
93	C₂²⋊C₄	—	2	1	no	—
94	C₂²⋊C₄	—	106	4	no	—
95	C₄²	—	212	2	no	—
96	C₄²	C₂×Dic₅₃	1	2	yes	—
97	C₂³	C₂²⋊Dic₅₃	1	1	yes	characteristic
98	C₂³	C₂×C₂²⋊D₅₃	1	2	yes	—
99	Q₈	—	4	1	no	—
100	Q₈	—	4	2	no	—
101	Q₈	—	4	2	no	—
102	D₄	—	4	2	no	—
103	D₄	—	4	1	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	4	2	no	—
110	C₂×C₄	—	212	4	no	—
111	C₂×C₄	C₂²⋊Dic₅₃	1	2	yes	—
112	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
113	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
114	C₂²	—	2	1	no	—
115	C₂²	—	2	2	no	—
116	C₂²	—	4	2	no	—
117	C₂²	—	4	1	no	—
118	C₂²	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
119	C₄	—	2	1	no	—
120	C₄	—	2	2	no	—
121	C₄	—	4	2	no	—
122	C₄	—	4	2	no	—
123	C₄	—	424	4	no	—
124	C₄	#13133132	1	2	yes	—
125	C₂	—	2	1	no	—
126	C₂	—	4	2	no	—
127	C₂	—	4	1	no	—
128	C₂	C₂×#13133132	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810597

C₂

yes

maximal

#16810418

C₂

yes

maximal, complemented

C₅₃×#536

C₂

yes

maximal, characteristic

#13133176

—

#13133132

—

#13133132

C₂²

yes

—

C₂×#41×C₅₃

C₂²

yes

characteristic

C₂²×Q₈×C₅₃

C₄

yes

characteristic, complemented

C₄⋊Q₈×C₅₃

C₄

yes

complemented

#124×C₅₃

C₂²

yes

—

#123×C₅₃

—

C₂²⋊Q₈×C₅₃

—

#117×C₅₃

C₄

yes

characteristic, complemented

#41×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

C₂×C₄

yes

—

C₂×Q₈×C₅₃

C₂×C₄

yes

characteristic

C₂×D₄×C₅₃

D₄

yes

—

C₂×D₄×C₅₃

C₂³

yes

characteristic

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

C₂×C₄

yes

—

C₂²⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄×Dic₅₃

—

C₂³×C₅₃

C₂²⋊C₄

yes

characteristic

C₂³×C₅₃

C₂×D₄

yes

—

Q₈×C₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×Dic₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#99

yes

—

Dic₅₃

—

#2129

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

C₂×#99

yes

characteristic

#536

D₅₃

yes

characteristic

#410

—

#354

—

C₅₃

#2129

yes

characteristic, complemented, Sylow

C₂×#41

C₂×D₅₃

yes

characteristic

C₂²×Q₈

Dic₅₃

yes

characteristic, complemented

C₄⋊Q₈

Dic₅₃

yes

complemented

#124

C₂×D₅₃

yes

—

#123

—

C₂²⋊Q₈

—

#117

—

106

—

#117

Dic₅₃

yes

characteristic, complemented

#99

—

#99

—

106

—

#41

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×Dic₅₃

yes

—

C₂×Q₈

C₂×Dic₅₃

yes

characteristic

C₂×D₄

C₂²⋊D₅₃

yes

—

C₂×D₄

C₂²×D₅₃

yes

characteristic

C₂²×C₄

—

C₂²×C₄

C₂×Dic₅₃

yes

characteristic

C₂²×C₄

C₂²⋊D₅₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

212

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

106

—

C₄²

—

212

—

C₄²

C₂×Dic₅₃

yes

—

C₂³

C₂²⋊Dic₅₃

yes

characteristic

C₂³

C₂×C₂²⋊D₅₃

yes

—

Q₈

—

100

Q₈

—

101

Q₈

—

102

D₄

—

103

D₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

212

—

111

C₂×C₄

C₂²⋊Dic₅₃

yes

—

112

C₂×C₄

C₂²⋊Dic₅₃

yes

characteristic

113

C₂×C₄

C₂²×Dic₅₃

yes

characteristic

114

C₂²

—

115

C₂²

—

116

C₂²

—

117

C₂²

—

118

C₂²

C₂×C₂²⋊Dic₅₃

yes

characteristic

119

C₄

—

120

C₄

—

121

C₄

—

122

C₄

—

123

C₄

—

424

—

124

C₄

#13133132

yes

—

125

C₂

—

126

C₂

—

127

C₂

—

128

C₂

C₂×#13133132

yes

characteristic

	subgroup	G/H
center	C₂	C₂×#13133132
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂×D₄	C₂²×D₅₃
Fitting	C₅₃×#536	C₂
socle	C₂×C₅₃	C₂×#99
hypercenter	#536	D₅₃
hypocenter	C₅₃	#2129
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#41 ◁ #536

subgroup

G/H

center

C₂

C₂×#13133132

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₂×D₄

C₂²×D₅₃

Fitting

C₅₃×#536

C₂

socle

C₂×C₅₃

C₂×#99

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#41 ◁ #536

N	H ≅ G/N	type
2. #16810418	C₂	semidirect
8. C₂²×Q₈×C₅₃	C₄	semidirect
9. C₄⋊Q₈×C₅₃	C₄	semidirect
13. #117×C₅₃	C₄	semidirect
68. C₅₃	#2129	semidirect
70. C₂²×Q₈	Dic₅₃	semidirect
71. C₄⋊Q₈	Dic₅₃	semidirect
76. #117	Dic₅₃	semidirect

H ≅ G/N

type

2. #16810418

C₂

semidirect

8. C₂²×Q₈×C₅₃

C₄

semidirect

9. C₄⋊Q₈×C₅₃

C₄

semidirect

13. #117×C₅₃

C₄

semidirect

68. C₅₃

#2129

semidirect

70. C₂²×Q₈

Dic₅₃

semidirect

71. C₄⋊Q₈

Dic₅₃

semidirect

76. #117

Dic₅₃

semidirect

Subgroups — #20276390

Conjugacy classes of subgroups

Complements