Group #20272571 — subgroups

Subgroups — #20272571

Order 6784.

164 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 164 conjugacy classes of subgroups — too many to draw. Only its 61 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	#16810680	C₂	1	1	yes	maximal, characteristic
2	#16810680	C₂	1	1	yes	maximal, characteristic
3	C₄×C₄⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₄⋊#88322	—	2	1	no	—
5	C₄⋊#88322	C₄	1	2	yes	complemented
6	C₂²⋊#88322	—	2	1	no	—
7	C₂²⋊#88322	—	2	1	no	—
8	#13133128	—	2	1	no	—
9	#13133128	C₄	1	2	yes	complemented
10	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
11	C₂×C₄²×C₅₃	C₄	1	2	yes	—
12	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
13	#95×C₅₃	—	2	1	no	—
14	C₂²×C₄×C₅₃	—	2	2	no	—
15	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
17	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
18	C₄⋊C₄×C₅₃	—	2	4	no	—
19	C₄²×C₅₃	—	4	2	no	—
20	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
21	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₄²×C₅₃	D₄	1	1	yes	characteristic
23	C₄²×C₅₃	Q₈	1	1	yes	characteristic
24	C₂×#88322	—	2	2	no	—
25	C₂×#88322	—	2	2	no	—
26	C₂³×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
27	C₂×C₄×C₅₃	—	2	1	no	—
28	C₂×C₄×C₅₃	—	2	1	no	—
29	C₂×C₄×C₅₃	—	2	2	no	—
30	C₂×C₄×C₅₃	—	2	4	no	—
31	C₂×C₄×C₅₃	—	4	1	no	—
32	C₂×C₄×C₅₃	—	4	2	no	—
33	C₂×C₄×C₅₃	—	4	2	no	—
34	C₂×C₄×C₅₃	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	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
40	#88322	—	4	2	no	—
41	#88322	—	4	2	no	—
42	C₂²×C₅₃	—	2	1	no	—
43	C₂²×C₅₃	—	2	1	no	—
44	C₂²×C₅₃	#95	1	1	yes	characteristic
45	C₂²×C₅₃	#100	1	1	yes	characteristic
46	C₂²×C₅₃	#101	1	1	yes	characteristic
47	C₄×C₅₃	—	2	1	no	—
48	C₄×C₅₃	—	2	1	no	—
49	C₄×C₅₃	—	2	1	no	—
50	C₄×C₅₃	—	2	1	no	—
51	C₄×C₅₃	—	2	1	no	—
52	C₄×C₅₃	—	4	4	no	—
53	C₄×C₅₃	#104	1	2	yes	—
54	#1285	—	53	1	no	maximal, Sylow
55	C₂×C₅₃	—	2	1	no	—
56	C₂×C₅₃	—	2	1	no	—
57	C₂×C₅₃	H(3,ℤ/4)	1	1	yes	characteristic
58	C₂×C₅₃	#339	1	1	yes	characteristic
59	C₂×C₅₃	#343	1	1	yes	characteristic
60	#432	—	53	1	no	—
61	#432	—	53	1	no	—
62	C₄×C₄⋊C₄	D₅₃	1	1	yes	characteristic
63	C₅₃	#1285	1	1	yes	characteristic, complemented, Sylow
64	C₂×C₄⋊C₄	—	2	1	no	—
65	C₂×C₄²	Dic₅₃	1	2	yes	—
66	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
67	C₄⋊C₈	—	53	2	no	—
68	C₄⋊C₈	—	106	1	no	—
69	C₂²⋊C₈	—	106	1	no	—
70	C₂²⋊C₈	—	106	1	no	—
71	#97	—	53	2	no	—
72	#97	—	106	1	no	—
73	#95	—	2	1	no	—
74	C₂²×C₄	—	2	2	no	—
75	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
76	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
77	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
78	C₂×C₈	—	106	2	no	—
79	C₂×C₈	—	106	2	no	—
80	C₄⋊C₄	—	2	4	no	—
81	C₄²	—	4	2	no	—
82	C₄²	Dic₁₀₆	1	1	yes	characteristic
83	C₄²	C₄×D₅₃	1	1	yes	characteristic
84	C₄²	C₄×D₅₃	1	1	yes	characteristic
85	C₄²	D₂₁₂	1	1	yes	characteristic
86	C₂³	C₂²⋊Dic₅₃	1	1	yes	characteristic
87	C₂×C₄	—	2	1	no	—
88	C₂×C₄	—	2	1	no	—
89	C₂×C₄	—	2	2	no	—
90	C₂×C₄	—	2	4	no	—
91	C₂×C₄	—	4	1	no	—
92	C₂×C₄	—	4	2	no	—
93	C₂×C₄	—	4	2	no	—
94	C₂×C₄	C₄×Dic₅₃	1	1	yes	characteristic
95	C₂×C₄	#11719919	1	1	yes	characteristic
96	C₂×C₄	#11719919	1	1	yes	characteristic
97	C₂×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
98	C₂×C₄	#11719921	1	1	yes	characteristic
99	C₂×C₄	#11719921	1	1	yes	characteristic
100	C₈	—	212	2	no	—
101	C₈	—	212	2	no	—
102	C₂²	—	2	1	no	—
103	C₂²	—	2	1	no	—
104	C₂²	#13133124	1	1	yes	characteristic
105	C₂²	#13133134	1	1	yes	characteristic
106	C₂²	#13133136	1	1	yes	characteristic
107	C₄	—	2	1	no	—
108	C₄	—	2	1	no	—
109	C₄	—	2	1	no	—
110	C₄	—	2	1	no	—
111	C₄	—	2	1	no	—
112	C₄	—	4	4	no	—
113	C₄	#13133145	1	2	yes	—
114	C₂	—	2	1	no	—
115	C₂	—	2	1	no	—
116	C₂	#16810379	1	1	yes	characteristic
117	C₂	#16810383	1	1	yes	characteristic
118	C₂	#16810392	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810680

C₂

yes

maximal, characteristic

#16810680

C₂

yes

maximal, characteristic

C₄×C₄⋊C₄×C₅₃

C₂

yes

maximal, characteristic

C₄⋊#88322

—

C₄⋊#88322

C₄

yes

—

—

—

C₄

yes

complemented

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄²×C₅₃

C₄

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

#95×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

Q₈

yes

characteristic

C₂×#88322

—

C₂×#88322

—

C₂³×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₄²

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄⋊C₄

yes

characteristic

#88322

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#104

yes

—

#1285

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

—

C₄×C₄⋊C₄

D₅₃

yes

characteristic

C₅₃

#1285

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₄

—

C₂×C₄²

Dic₅₃

yes

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

106

—

C₂²⋊C₈

—

106

—

C₂²⋊C₈

—

106

—

#97

—

#97

—

106

—

#95

—

C₂²×C₄

—

C₂²×C₄

C₂×Dic₅₃

yes

characteristic

C₂²×C₄

C₂²⋊D₅₃

yes

characteristic

C₂²×C₄

C₂²⋊D₅₃

yes

characteristic

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₄⋊C₄

—

C₄²

—

C₄²

Dic₁₀₆

yes

characteristic

C₄²

C₄×D₅₃

yes

characteristic

C₄²

C₄×D₅₃

yes

characteristic

C₄²

D₂₁₂

yes

characteristic

C₂³

C₂²⋊Dic₅₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×Dic₅₃

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₄⋊Dic₅₃

yes

C₂×C₄

yes

C₂×C₄

yes

100

C₈

—

212

—

101

C₈

—

212

—

102

C₂²

—

103

C₂²

—

104

C₂²

#13133124

yes

characteristic

105

C₂²

#13133134

yes

characteristic

106

C₂²

#13133136

yes

characteristic

107

C₄

—

108

C₄

—

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

#13133145

yes

—

114

C₂

—

115

C₂

—

116

C₂

#16810379

yes

characteristic

117

C₂

#16810383

yes

characteristic

118

C₂

#16810392

yes

characteristic

	subgroup	G/H
center	C₂²	#13133124
commutator	C₂×C₄×C₅₃	C₄²
Frattini	C₂×C₄²	C₂×D₅₃
Fitting	C₄×C₄⋊C₄×C₅₃	C₂
socle	C₂²×C₅₃	#95
hypercenter	C₄×C₄⋊C₄	D₅₃
hypocenter	C₅₃	#1285
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×C₄⋊C₄

subgroup

G/H

center

C₂²

#13133124

commutator

C₂×C₄×C₅₃

C₄²

Frattini

C₂×C₄²

C₂×D₅₃

Fitting

C₄×C₄⋊C₄×C₅₃

C₂

socle

C₂²×C₅₃

#95

hypercenter

C₄×C₄⋊C₄

D₅₃

hypocenter

C₅₃

#1285

LCS

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×C₄⋊C₄

N	H ≅ G/N	type
5. C₄⋊#88322	C₄	semidirect
9. #13133128	C₄	semidirect
63. C₅₃	#1285	semidirect

H ≅ G/N

type

5. C₄⋊#88322

C₄

semidirect

9. #13133128

C₄

semidirect

63. C₅₃

#1285

semidirect

Subgroups — #20272571

Conjugacy classes of subgroups

Complements