Group #20273297 — subgroups

Subgroups — #330×D₅₃ #20273297

Order 6784.

232 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 67 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	#125×D₅₃	C₂	1	1	yes	maximal, characteristic
2	#97×D₅₃	C₂	1	2	yes	maximal
3	#16810365	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810364	C₂	1	2	yes	maximal, complemented
5	C₅₃×#330	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₈×D₅₃	—	2	2	no	—
7	#13133209	C₂²	1	1	yes	characteristic
8	#13133208	C₄	1	2	yes	—
9	#13133207	—	2	2	no	—
10	C₄⋊C₄×D₅₃	—	2	2	no	—
11	C₄⋊C₄×D₅₃	C₄	1	2	yes	—
12	C₄²×D₅₃	C₂²	1	1	yes	characteristic
13	#13133128	C₂²	1	2	yes	—
14	#13133127	—	2	2	no	—
15	#125×C₅₃	C₂²	1	1	yes	characteristic
16	#97×C₅₃	C₂²	1	2	yes	—
17	C₂×C₄×D₅₃	—	2	2	no	—
18	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×D₅₃	D₄	1	2	yes	—
20	C₂×C₈×C₅₃	—	2	2	no	—
21	C₄⋊C₄×C₅₃	—	2	2	no	—
22	C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
23	C₄²×C₅₃	C₂³	1	1	yes	characteristic
24	C₄⋊Dic₅₃	—	2	2	no	—
25	C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
26	#11719919	—	2	2	no	—
27	#11719919	—	2	2	no	—
28	#11719919	—	2	2	no	—
29	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
30	C₄×Dic₅₃	D₄	1	2	yes	—
31	C₂×#88322	—	2	2	no	—
32	C₈×D₅₃	—	2	8	no	—
33	C₂²×D₅₃	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₅₃	—	2	2	no	—
35	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
37	C₂×Dic₅₃	—	2	2	no	—
38	C₂×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
39	C₂×Dic₅₃	C₂²⋊C₄	1	2	yes	—
40	C₄×D₅₃	—	2	2	no	—
41	C₄×D₅₃	—	2	4	no	—
42	C₄×D₅₃	—	4	4	no	—
43	C₈×C₅₃	—	2	4	no	—
44	#88322	—	2	4	no	—
45	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
46	C₂×D₅₃	#101	1	2	yes	—
47	C₂×D₅₃	#104	1	4	yes	—
48	C₄×C₅₃	—	2	1	no	—
49	C₄×C₅₃	—	2	2	no	—
50	C₄×C₅₃	—	4	2	no	—
51	Dic₅₃	—	2	1	no	—
52	Dic₅₃	—	2	2	no	—
53	Dic₅₃	—	4	2	no	—
54	C₂×#330	—	53	1	no	maximal, Sylow
55	C₂×C₅₃	C₂×#101	1	1	yes	characteristic
56	C₂×C₅₃	C₂×#104	1	2	yes	—
57	D₅₃	#330	1	4	yes	direct product factor
58	C₂×#125	—	53	1	no	—
59	C₂×#97	—	53	2	no	—
60	#330	—	53	1	no	—
61	#330	—	53	2	no	—
62	#330	D₅₃	1	1	yes	characteristic, direct product factor
63	C₅₃	C₂×#330	1	1	yes	characteristic, complemented, Sylow
64	C₂²×C₈	—	106	2	no	—
65	#125	—	53	2	no	—
66	#125	—	53	1	no	—
67	#125	—	106	2	no	—
68	#125	C₂×D₅₃	1	1	yes	characteristic
69	C₂×C₄⋊C₄	—	53	2	no	—
70	C₂×C₄⋊C₄	—	106	2	no	—
71	C₂×C₄²	—	53	1	no	—
72	#97	—	53	2	no	—
73	#97	—	106	2	no	—
74	#97	C₂×D₅₃	1	2	yes	—
75	C₂²×C₄	—	53	1	no	—
76	C₂²×C₄	—	53	2	no	—
77	C₂²×C₄	—	106	2	no	—
78	C₂×C₈	—	2	2	no	—
79	C₂×C₈	—	106	8	no	—
80	C₂×C₈	—	106	2	no	—
81	C₄⋊C₄	—	2	2	no	—
82	C₄⋊C₄	—	53	2	no	—
83	C₄⋊C₄	—	106	2	no	—
84	C₄⋊C₄	—	106	2	no	—
85	C₄⋊C₄	—	106	2	no	—
86	C₄⋊C₄	—	106	2	no	—
87	C₄⋊C₄	C₄×D₅₃	1	2	yes	—
88	C₄²	—	53	1	no	—
89	C₄²	—	53	2	no	—
90	C₄²	C₂²×D₅₃	1	1	yes	characteristic
91	C₂³	—	53	1	no	—
92	C₂×C₄	—	2	2	no	—
93	C₂×C₄	—	53	1	no	—
94	C₂×C₄	—	53	2	no	—
95	C₂×C₄	—	106	4	no	—
96	C₂×C₄	—	106	2	no	—
97	C₂×C₄	—	106	2	no	—
98	C₂×C₄	—	212	4	no	—
99	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
100	C₂×C₄	D₄×D₅₃	1	2	yes	—
101	C₈	—	2	4	no	—
102	C₈	—	106	4	no	—
103	C₂²	—	53	2	no	—
104	C₂²	—	53	4	no	—
105	C₂²	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
106	C₄	—	2	1	no	—
107	C₄	—	2	2	no	—
108	C₄	—	4	2	no	—
109	C₄	—	106	1	no	—
110	C₄	—	106	2	no	—
111	C₄	—	212	2	no	—
112	C₂	—	53	4	no	—
113	C₂	#101×D₅₃	1	1	yes	characteristic
114	C₂	#104×D₅₃	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#125×D₅₃

C₂

yes

maximal, characteristic

#97×D₅₃

C₂

yes

maximal

#16810365

C₂

yes

maximal, characteristic, complemented

#16810364

C₂

yes

maximal, complemented

C₅₃×#330

C₂

yes

maximal, characteristic, complemented

C₂×C₈×D₅₃

—

#13133209

C₂²

yes

characteristic

#13133208

C₄

yes

—

#13133207

—

C₄⋊C₄×D₅₃

—

C₄⋊C₄×D₅₃

C₄

yes

—

C₄²×D₅₃

C₂²

yes

characteristic

#13133128

C₂²

yes

—

#13133127

—

#125×C₅₃

C₂²

yes

characteristic

#97×C₅₃

C₂²

yes

—

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

C₂×C₄

yes

characteristic

C₂×C₄×D₅₃

D₄

yes

—

C₂×C₈×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

C₂×C₄

yes

—

#11719919

—

#11719919

—

#11719919

—

C₄×Dic₅₃

C₂×C₄

yes

characteristic

C₄×Dic₅₃

D₄

yes

—

C₂×#88322

—

C₈×D₅₃

—

C₂²×D₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂²⋊C₄

yes

characteristic

C₂×Dic₅₃

C₂²⋊C₄

yes

—

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

—

C₈×C₅₃

—

#88322

—

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₅₃

#101

yes

—

C₂×D₅₃

#104

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

—

D₅₃

#330

yes

direct product factor

C₂×#125

—

C₂×#97

—

#330

—

#330

—

#330

D₅₃

yes

characteristic, direct product factor

C₅₃

C₂×#330

yes

characteristic, complemented, Sylow

C₂²×C₈

—

106

—

#125

—

#125

—

#125

—

106

—

#125

C₂×D₅₃

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

106

—

C₂×C₄²

—

#97

—

#97

—

106

—

#97

C₂×D₅₃

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

106

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

106

—

C₄⋊C₄

—

106

—

C₄⋊C₄

—

106

—

C₄⋊C₄

—

106

—

C₄⋊C₄

C₄×D₅₃

yes

—

C₄²

—

C₄²

—

C₄²

C₂²×D₅₃

yes

characteristic

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

106

—

C₂×C₄

—

106

—

C₂×C₄

—

106

—

C₂×C₄

—

212

—

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

100

C₂×C₄

D₄×D₅₃

yes

—

101

C₈

—

102

C₈

—

106

—

103

C₂²

—

104

C₂²

—

105

C₂²

C₂²⋊C₄×D₅₃

yes

characteristic

106

C₄

—

107

C₄

—

108

C₄

—

109

C₄

—

106

—

110

C₄

—

106

—

111

C₄

—

212

—

112

C₂

—

113

C₂

#101×D₅₃

yes

characteristic

114

C₂

#104×D₅₃

yes

—

	subgroup	G/H
center	C₂²	C₂²⋊C₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₄²	C₂²×D₅₃
Fitting	C₅₃×#330	C₂
socle	C₂²×C₅₃	C₂×C₂²⋊C₄
hypercenter	#330	D₅₃
hypocenter	C₅₃	C₂×#330
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #330

subgroup

G/H

center

C₂²

C₂²⋊C₄×D₅₃

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₄²

C₂²×D₅₃

Fitting

C₅₃×#330

C₂

socle

C₂²×C₅₃

C₂×C₂²⋊C₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #330

N	H ≅ G/N	type
3. #16810365	C₂	semidirect
4. #16810364	C₂	semidirect
5. C₅₃×#330	C₂	semidirect
57. D₅₃	#330	direct (N × H)
62. #330	D₅₃	direct (N × H)
63. C₅₃	C₂×#330	semidirect

H ≅ G/N

type

3. #16810365

C₂

semidirect

4. #16810364

C₂

semidirect

5. C₅₃×#330

C₂

semidirect

57. D₅₃

#330

direct (N × H)

62. #330

D₅₃

direct (N × H)

63. C₅₃

C₂×#330

semidirect

Subgroups — #330×D₅₃ #20273297

Conjugacy classes of subgroups

Complements

Subgroups — #330×D53 #20273297

Conjugacy classes of subgroups

Complements

Subgroups — #330×D₅₃ #20273297