Group #20273450 — subgroups

Subgroups — #20273450

Order 6784.

254 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 85 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	#16810647	C₂	1	2	yes	maximal, complemented
2	#16810372	C₂	1	2	yes	maximal, complemented
3	#16810367	C₂	1	2	yes	maximal, complemented
4	C₅₃×#532	C₂	1	1	yes	maximal, characteristic
5	C₂×#11719917	—	2	2	no	—
6	C₄⋊#88322	—	2	2	no	—
7	C₄⋊#88322	C₂²	1	4	yes	—
8	C₂×#41×C₅₃	C₄	1	2	yes	—
9	C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
10	#127×C₅₃	C₂²	1	1	yes	characteristic
11	#124×C₅₃	C₄	1	2	yes	—
12	#121×C₅₃	—	2	2	no	—
13	C₄×D₄×C₅₃	—	2	2	no	—
14	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
15	#41×C₅₃	—	4	4	no	—
16	C₂×Q₈×C₅₃	C₂×C₄	1	2	yes	—
17	C₂×D₄×C₅₃	—	2	2	no	—
18	C₂×D₄×C₅₃	—	2	2	no	—
19	C₂×D₄×C₅₃	C₂×C₄	1	2	yes	—
20	C₂²×C₄×C₅₃	—	2	2	no	—
21	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₅₃	D₄	1	2	yes	—
23	C₄⋊C₄×C₅₃	—	2	2	no	—
24	C₂²⋊C₄×C₅₃	—	2	4	no	—
25	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₄²×C₅₃	D₄	1	2	yes	—
27	C₄²×C₅₃	C₂³	1	1	yes	characteristic
28	#11719917	—	4	4	no	—
29	C₂×#88322	—	2	4	no	—
30	C₂³×C₅₃	—	2	2	no	—
31	C₂³×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
32	Q₈×C₅₃	—	4	2	no	—
33	D₄×C₅₃	—	4	2	no	—
34	D₄×C₅₃	—	4	4	no	—
35	D₄×C₅₃	—	4	2	no	—
36	D₄×C₅₃	—	4	2	no	—
37	C₂×C₄×C₅₃	—	2	2	no	—
38	C₂×C₄×C₅₃	—	2	2	no	—
39	C₂×C₄×C₅₃	—	2	2	no	—
40	C₂×C₄×C₅₃	—	2	2	no	—
41	C₂×C₄×C₅₃	—	4	4	no	—
42	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
43	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
46	#88322	—	4	4	no	—
47	C₂²×C₅₃	—	2	1	no	—
48	C₂²×C₅₃	—	2	2	no	—
49	C₂²×C₅₃	—	4	2	no	—
50	C₂²×C₅₃	—	4	4	no	—
51	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
52	C₄×C₅₃	—	2	1	no	—
53	C₄×C₅₃	—	2	2	no	—
54	C₄×C₅₃	—	4	2	no	—
55	C₄×C₅₃	#100	1	2	yes	—
56	C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
57	C₄×C₅₃	#137	1	2	yes	—
58	#1560	—	53	1	no	maximal, Sylow
59	C₂×C₅₃	—	4	1	no	—
60	C₂×C₅₃	—	8	2	no	—
61	C₂×C₅₃	C₂×#100	1	1	yes	characteristic
62	C₂×C₅₃	#417	1	2	yes	—
63	#532	D₅₃	1	1	yes	characteristic
64	#423	—	53	2	no	—
65	#333	—	53	2	no	—
66	#331	—	53	2	no	—
67	C₅₃	#1560	1	1	yes	characteristic, complemented, Sylow
68	C₂×#41	Dic₅₃	1	2	yes	—
69	C₂×M₁₆	—	106	2	no	—
70	C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic
71	#127	C₂×D₅₃	1	1	yes	characteristic
72	#124	Dic₅₃	1	2	yes	—
73	#121	—	2	2	no	—
74	C₄×D₄	—	2	2	no	—
75	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
76	C₄⋊C₈	—	53	4	no	—
77	C₄⋊C₈	—	106	2	no	—
78	#41	—	4	4	no	—
79	C₂×Q₈	C₂×Dic₅₃	1	2	yes	—
80	C₂×D₄	—	2	2	no	—
81	C₂×D₄	—	2	2	no	—
82	C₂×D₄	C₂×Dic₅₃	1	2	yes	—
83	C₂²×C₄	—	2	2	no	—
84	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
85	C₂²×C₄	C₂²⋊D₅₃	1	2	yes	—
86	M₁₆	—	212	4	no	—
87	C₂×C₈	—	106	4	no	—
88	C₄⋊C₄	—	2	2	no	—
89	C₂²⋊C₄	—	2	4	no	—
90	C₄²	C₂×Dic₅₃	1	1	yes	characteristic
91	C₄²	C₂²⋊D₅₃	1	2	yes	—
92	C₄²	C₂²×D₅₃	1	1	yes	characteristic
93	C₂³	—	2	2	no	—
94	C₂³	C₂²⋊Dic₅₃	1	1	yes	characteristic
95	Q₈	—	4	2	no	—
96	D₄	—	4	2	no	—
97	D₄	—	4	4	no	—
98	D₄	—	4	2	no	—
99	D₄	—	4	2	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	2	2	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	4	4	no	—
105	C₂×C₄	C₂²⋊Dic₅₃	1	2	yes	—
106	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
107	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
108	C₂×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
109	C₈	—	212	4	no	—
110	C₂²	—	2	1	no	—
111	C₂²	—	2	2	no	—
112	C₂²	—	4	2	no	—
113	C₂²	—	4	4	no	—
114	C₂²	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
115	C₄	—	2	1	no	—
116	C₄	—	2	2	no	—
117	C₄	—	4	2	no	—
118	C₄	#13133134	1	2	yes	—
119	C₄	#13133242	1	2	yes	—
120	C₄	#13133247	1	2	yes	—
121	C₂	—	4	1	no	—
122	C₂	—	8	2	no	—
123	C₂	C₂×#13133134	1	1	yes	characteristic
124	C₂	#16810624	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16810647

C₂

yes

maximal, complemented

#16810372

C₂

yes

maximal, complemented

#16810367

C₂

yes

maximal, complemented

C₅₃×#532

C₂

yes

maximal, characteristic

C₂×#11719917

—

C₄⋊#88322

—

C₄⋊#88322

C₂²

yes

—

C₂×#41×C₅₃

C₄

yes

—

C₄⋊Q₈×C₅₃

C₂²

yes

characteristic

#127×C₅₃

C₂²

yes

characteristic

#124×C₅₃

C₄

yes

—

#121×C₅₃

—

C₄×D₄×C₅₃

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

#41×C₅₃

—

C₂×Q₈×C₅₃

C₂×C₄

yes

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

—

C₄⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

D₄

yes

—

C₄²×C₅₃

C₂³

yes

characteristic

#11719917

—

C₂×#88322

—

C₂³×C₅₃

—

C₂³×C₅₃

C₂²⋊C₄

yes

characteristic

Q₈×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×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×C₄×C₅₃

C₂×D₄

yes

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#100

yes

—

C₄×C₅₃

AGL(1,ℤ/8)

yes

—

C₄×C₅₃

#137

yes

—

#1560

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

C₂×#100

yes

characteristic

C₂×C₅₃

#417

yes

—

#532

D₅₃

yes

—

—

—

C₅₃

#1560

yes

characteristic, complemented, Sylow

C₂×#41

Dic₅₃

yes

—

C₂×M₁₆

—

106

—

C₄⋊Q₈

C₂×D₅₃

yes

characteristic

#127

C₂×D₅₃

yes

characteristic

#124

Dic₅₃

yes

—

#121

—

C₄×D₄

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

106

—

#41

—

C₂×Q₈

C₂×Dic₅₃

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×Dic₅₃

yes

—

C₂²×C₄

—

C₂²×C₄

C₂×Dic₅₃

yes

characteristic

C₂²×C₄

C₂²⋊D₅₃

yes

—

M₁₆

—

212

—

C₂×C₈

—

106

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

C₂×Dic₅₃

yes

characteristic

C₄²

C₂²⋊D₅₃

yes

—

C₄²

C₂²×D₅₃

yes

characteristic

C₂³

—

C₂³

C₂²⋊Dic₅₃

yes

characteristic

Q₈

—

D₄

—

D₄

—

D₄

—

D₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

C₂²⋊Dic₅₃

yes

—

106

C₂×C₄

C₂²⋊Dic₅₃

yes

characteristic

107

C₂×C₄

C₂²×Dic₅₃

yes

characteristic

108

C₂×C₄

C₂×C₂²⋊D₅₃

yes

—

109

C₈

—

212

—

110

C₂²

—

111

C₂²

—

112

C₂²

—

113

C₂²

—

114

C₂²

C₂×C₂²⋊Dic₅₃

yes

characteristic

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

#13133134

yes

—

119

C₄

#13133242

yes

—

120

C₄

#13133247

yes

—

121

C₂

—

122

C₂

—

123

C₂

C₂×#13133134

yes

characteristic

124

C₂

#16810624

yes

—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₄²	C₂²×D₅₃
Fitting	C₅₃×#532	C₂
socle	C₂²×C₅₃	C₂×C₂²⋊C₄
hypercenter	#532	D₅₃
hypocenter	C₅₃	#1560
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #532

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₅₃

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₄²

C₂²×D₅₃

Fitting

C₅₃×#532

C₂

socle

C₂²×C₅₃

C₂×C₂²⋊C₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #532

N	H ≅ G/N	type
1. #16810647	C₂	semidirect
2. #16810372	C₂	semidirect
3. #16810367	C₂	semidirect
67. C₅₃	#1560	semidirect

H ≅ G/N

type

1. #16810647

C₂

semidirect

2. #16810372

C₂

semidirect

3. #16810367

C₂

semidirect

67. C₅₃

#1560

semidirect

Subgroups — #20273450

Conjugacy classes of subgroups

Complements