Group #20273016 — subgroups

Subgroups — #20273016

Order 6784.

178 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 178 conjugacy classes of subgroups — too many to draw. Only its 89 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	#16810674	C₂	1	1	yes	maximal, characteristic
2	#16810674	C₂	1	1	yes	maximal, characteristic
3	#16810674	C₂	1	1	yes	maximal, characteristic
4	#16810345	C₂	1	1	yes	maximal, characteristic, complemented
5	#16810344	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈×#88322	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#322	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄²×D₅₃	C₂²	1	1	yes	characteristic
9	#13133146	C₄	1	2	yes	—
10	#13133146	C₄	1	2	yes	—
11	#13133146	C₄	1	2	yes	—
12	#13133129	C₄	1	2	yes	—
13	#13133129	C₄	1	2	yes	—
14	#13133129	C₄	1	2	yes	—
15	C₄×#88322	C₂²	1	1	yes	characteristic
16	C₄×#88322	C₂²	1	1	yes	characteristic
17	C₄×#88322	C₂²	1	1	yes	characteristic
18	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
25	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
26	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
27	C₄²×C₅₃	C₂³	1	1	yes	characteristic
28	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
29	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
30	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
31	C₂×#88322	C₂×C₄	1	2	yes	—
32	C₂×#88322	C₂×C₄	1	2	yes	—
33	C₂×#88322	C₂×C₄	1	2	yes	—
34	C₂²×D₅₃	C₄²	1	1	yes	characteristic
35	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
38	C₂×Dic₅₃	C₄²	1	1	yes	characteristic
39	C₂×Dic₅₃	C₄²	1	1	yes	characteristic
40	C₂×Dic₅₃	C₄²	1	1	yes	characteristic
41	C₄×D₅₃	—	2	2	no	—
42	C₄×D₅₃	—	2	2	no	—
43	C₄×D₅₃	—	2	2	no	—
44	C₈×C₅₃	—	2	2	no	—
45	C₈×C₅₃	—	2	2	no	—
46	C₈×C₅₃	—	2	2	no	—
47	#88322	—	2	2	no	—
48	#88322	—	2	2	no	—
49	#88322	—	2	2	no	—
50	C₂²×C₅₃	C₂×C₄²	1	1	yes	characteristic
51	C₂×D₅₃	—	2	1	no	—
52	C₂×D₅₃	—	2	1	no	—
53	C₂×D₅₃	—	2	1	no	—
54	C₄×C₅₃	#131	1	2	yes	—
55	C₄×C₅₃	#131	1	2	yes	—
56	C₄×C₅₃	#131	1	2	yes	—
57	Dic₅₃	—	2	1	no	—
58	Dic₅₃	—	2	1	no	—
59	Dic₅₃	—	2	1	no	—
60	#1454	—	53	1	no	maximal, Sylow
61	C₂×C₅₃	#405	1	1	yes	characteristic
62	C₂×C₅₃	#405	1	1	yes	characteristic
63	C₂×C₅₃	#405	1	1	yes	characteristic
64	D₅₃	—	4	1	no	—
65	#431	—	53	1	no	—
66	#431	—	53	1	no	—
67	#431	—	53	1	no	—
68	#322	—	53	1	no	—
69	#322	—	53	1	no	—
70	#322	D₅₃	1	1	yes	characteristic, complemented
71	C₈²	—	53	1	no	—
72	C₅₃	#1454	1	1	yes	characteristic, complemented, Sylow
73	C₂×C₄²	—	53	1	no	—
74	C₄⋊C₈	—	53	2	no	—
75	C₄⋊C₈	—	53	2	no	—
76	C₄⋊C₈	—	53	2	no	—
77	C₂²⋊C₈	—	53	2	no	—
78	C₂²⋊C₈	—	53	2	no	—
79	C₂²⋊C₈	—	53	2	no	—
80	C₄×C₈	—	53	1	no	—
81	C₄×C₈	—	53	1	no	—
82	C₄×C₈	—	53	1	no	—
83	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
84	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
85	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
86	C₂²×C₄	—	53	1	no	—
87	C₂²×C₄	—	53	1	no	—
88	C₂²×C₄	—	53	1	no	—
89	C₂×C₈	—	53	2	no	—
90	C₂×C₈	—	53	2	no	—
91	C₂×C₈	—	53	2	no	—
92	C₂×C₈	C₄×D₅₃	1	2	yes	—
93	C₂×C₈	C₄×D₅₃	1	2	yes	—
94	C₂×C₈	C₄×D₅₃	1	2	yes	—
95	C₄²	—	53	1	no	—
96	C₄²	—	53	1	no	—
97	C₄²	—	53	1	no	—
98	C₄²	C₂²×D₅₃	1	1	yes	characteristic
99	C₂³	—	53	1	no	—
100	C₂×C₄	—	53	1	no	—
101	C₂×C₄	—	53	1	no	—
102	C₂×C₄	—	53	1	no	—
103	C₂×C₄	—	106	2	no	—
104	C₂×C₄	—	106	2	no	—
105	C₂×C₄	—	106	2	no	—
106	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
107	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
108	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
109	C₈	—	2	2	no	—
110	C₈	—	2	2	no	—
111	C₈	—	2	2	no	—
112	C₈	—	106	2	no	—
113	C₈	—	106	2	no	—
114	C₈	—	106	2	no	—
115	C₂²	—	106	1	no	—
116	C₂²	—	106	1	no	—
117	C₂²	—	106	1	no	—
118	C₂²	C₄²×D₅₃	1	1	yes	characteristic
119	C₄	—	106	1	no	—
120	C₄	—	106	1	no	—
121	C₄	—	106	1	no	—
122	C₄	#13133223	1	2	yes	—
123	C₄	#13133225	1	2	yes	—
124	C₄	#13133225	1	2	yes	—
125	C₂	—	212	1	no	—
126	C₂	#16810577	1	1	yes	characteristic
127	C₂	#16810580	1	1	yes	characteristic
128	C₂	#16810580	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810674

C₂

yes

maximal, characteristic

#16810674

C₂

yes

maximal, characteristic

#16810674

C₂

yes

maximal, characteristic

#16810345

C₂

yes

maximal, characteristic, complemented

#16810344

C₂

yes

maximal, characteristic, complemented

C₈×#88322

C₂

yes

maximal, characteristic, complemented

C₅₃×#322

C₂

yes

maximal, characteristic, complemented

C₄²×D₅₃

C₂²

yes

characteristic

#13133146

C₄

yes

—

#13133146

C₄

yes

—

#13133146

C₄

yes

—

#13133129

C₄

yes

—

#13133129

C₄

yes

—

#13133129

C₄

yes

—

C₄×#88322

C₂²

yes

characteristic

C₄×#88322

C₂²

yes

characteristic

C₄×#88322

C₂²

yes

characteristic

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂×C₄×D₅₃

C₂×C₄

yes

characteristic

C₂×C₄×D₅₃

C₂×C₄

yes

characteristic

C₂×C₄×D₅₃

C₂×C₄

yes

characteristic

C₂×C₈×C₅₃

C₂×C₄

yes

—

C₂×C₈×C₅₃

C₂×C₄

yes

—

C₂×C₈×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

C₂³

yes

characteristic

C₄×Dic₅₃

C₂×C₄

yes

characteristic

C₄×Dic₅₃

C₂×C₄

yes

characteristic

C₄×Dic₅₃

C₂×C₄

yes

characteristic

C₂×#88322

C₂×C₄

yes

—

C₂×#88322

C₂×C₄

yes

—

C₂×#88322

C₂×C₄

yes

—

C₂²×D₅₃

C₄²

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×Dic₅₃

C₄²

yes

characteristic

C₂×Dic₅₃

C₄²

yes

characteristic

C₂×Dic₅₃

C₄²

yes

characteristic

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

#88322

—

#88322

—

#88322

—

C₂²×C₅₃

C₂×C₄²

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

#131

yes

—

C₄×C₅₃

#131

yes

—

C₄×C₅₃

#131

yes

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

—

—

—

—

—

—

D₅₃

yes

characteristic, complemented

C₈²

—

C₅₃

#1454

yes

characteristic, complemented, Sylow

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₅₃

yes

characteristic

C₄×C₈

C₂×D₅₃

yes

characteristic

C₄×C₈

C₂×D₅₃

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×D₅₃

yes

—

C₂×C₈

C₄×D₅₃

yes

—

C₂×C₈

C₄×D₅₃

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂²×D₅₃

yes

characteristic

C₂³

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

106

—

104

C₂×C₄

—

106

—

105

C₂×C₄

—

106

—

106

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

107

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

108

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

109

C₈

—

110

C₈

—

111

C₈

—

112

C₈

—

106

—

113

C₈

—

106

—

114

C₈

—

106

—

115

C₂²

—

106

—

116

C₂²

—

106

—

117

C₂²

—

106

—

118

C₂²

C₄²×D₅₃

yes

characteristic

119

C₄

—

106

—

120

C₄

—

106

—

121

C₄

—

106

—

122

C₄

#13133223

yes

—

123

C₄

#13133225

yes

—

124

C₄

#13133225

yes

—

125

C₂

—

212

—

126

C₂

#16810577

yes

characteristic

127

C₂

#16810580

yes

characteristic

128

C₂

#16810580

yes

characteristic

	subgroup	G/H
center, Frattini	C₄²	C₂²×D₅₃
commutator, socle	C₂²×C₅₃	C₂×C₄²
Fitting	C₅₃×#322	C₂
hypercenter	#322	D₅₃
hypocenter	C₅₃	#1454
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄² ◁ #322

subgroup

G/H

center, Frattini

C₄²

C₂²×D₅₃

commutator, socle

C₂²×C₅₃

C₂×C₄²

Fitting

C₅₃×#322

C₂

D₅₃

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₄² ◁ #322

N	H ≅ G/N	type
4. #16810345	C₂	semidirect
5. #16810344	C₂	semidirect
6. C₈×#88322	C₂	semidirect
7. C₅₃×#322	C₂	semidirect
70. #322	D₅₃	semidirect
72. C₅₃	#1454	semidirect

H ≅ G/N

type

4. #16810345

C₂

semidirect

5. #16810344

C₂

semidirect

6. C₈×#88322

C₂

semidirect

7. C₅₃×#322

C₂

semidirect

70. #322

D₅₃

semidirect

72. C₅₃

#1454

semidirect

Subgroups — #20273016

Conjugacy classes of subgroups

Complements