Group #20290932 — subgroups

Subgroups — #20290932

Order 6784.

168 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 80 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	D₅₃⋊M₃₂	C₂	1	2	yes	maximal
2	#16811516	C₂	1	2	yes	maximal, complemented
3	#16811515	C₂	1	2	yes	maximal, complemented
4	#97×D₅₃	C₂	1	1	yes	maximal, characteristic
5	#13133292	C₄	1	4	yes	complemented
6	#13133291	C₂²	1	4	yes	—
7	D₅₃⋊C₁₆	C₄	1	4	yes	complemented
8	C₂×C₈×D₅₃	C₄	1	1	yes	characteristic
9	C₂×C₈×D₅₃	C₂²	1	1	yes	characteristic
10	C₄²×D₅₃	C₄	1	1	yes	characteristic
11	#13133128	C₂²	1	1	yes	characteristic
12	#13133127	C₄	1	1	yes	characteristic
13	#13133127	C₂²	1	1	yes	characteristic
14	#97×C₅₃	C₄	1	1	yes	characteristic
15	C₂×C₄×D₅₃	—	2	1	no	—
16	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
17	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
18	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
19	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
20	C₄×Dic₅₃	—	2	1	no	—
21	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
22	C₂×#88322	C₂×C₄	1	1	yes	characteristic
23	C₂×#88322	C₂³	1	1	yes	characteristic
24	C₈×D₅₃	—	2	1	no	—
25	C₈×D₅₃	—	2	1	no	—
26	C₈×D₅₃	C₂×C₄	1	2	yes	—
27	C₈×D₅₃	C₂×C₄	1	2	yes	—
28	#11719910	C₂×C₄	1	8	yes	complemented
29	C₂²×D₅₃	M₁₆	1	1	yes	characteristic
30	C₂×C₄×C₅₃	—	2	1	no	—
31	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
32	C₂×Dic₅₃	—	2	1	no	—
33	C₂×Dic₅₃	M₁₆	1	1	yes	characteristic
34	C₄×D₅₃	—	4	2	no	—
35	C₄×D₅₃	C₄²	1	1	yes	characteristic
36	C₄×D₅₃	C₄²	1	1	yes	characteristic
37	C₄×D₅₃	M₁₆	1	1	yes	characteristic
38	C₄×D₅₃	M₁₆	1	1	yes	characteristic
39	C₈×C₅₃	—	2	1	no	—
40	C₈×C₅₃	C₄²	1	2	yes	—
41	#88322	—	2	1	no	—
42	#88322	C₂²×C₄	1	2	yes	—
43	C₂²×C₅₃	C₂×M₁₆	1	1	yes	characteristic
44	C₂×D₅₃	—	2	2	no	—
45	C₂×D₅₃	#97	1	1	yes	characteristic
46	C₂×D₅₃	#97	1	1	yes	characteristic
47	C₄×C₅₃	—	4	1	no	—
48	C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
49	C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
50	Dic₅₃	—	4	1	no	—
51	Dic₅₃	#97	1	1	yes	characteristic
52	Dic₅₃	#97	1	1	yes	characteristic
53	C₂×#347	—	53	1	no	maximal, Sylow
54	C₂×C₅₃	—	2	1	no	—
55	C₂×C₅₃	C₂×#97	1	1	yes	characteristic
56	D₅₃	—	2	1	no	—
57	D₅₃	#347	1	2	yes	complemented
58	C₂×M₃₂	—	53	2	no	—
59	C₂×#97	—	53	1	no	—
60	#347	—	53	2	no	—
61	#347	—	53	2	no	—
62	C₅₃	C₂×#347	1	1	yes	characteristic, complemented, Sylow
63	C₂²×C₈	—	53	1	no	—
64	C₂²×C₈	—	53	1	no	—
65	C₂×C₄²	—	53	1	no	—
66	M₃₂	—	53	4	no	—
67	M₃₂	—	53	4	no	—
68	C₂×C₁₆	—	53	4	no	—
69	#97	—	53	1	no	—
70	#97	—	53	1	no	—
71	#97	—	53	1	no	—
72	#97	#6157	1	1	yes	characteristic
73	C₂²×C₄	—	53	1	no	—
74	C₂²×C₄	—	106	1	no	—
75	C₂×C₈	—	53	1	no	—
76	C₂×C₈	—	53	2	no	—
77	C₂×C₈	—	53	2	no	—
78	C₂×C₈	—	53	1	no	—
79	C₂×C₈	—	106	1	no	—
80	C₂×C₈	—	106	1	no	—
81	C₂×C₈	C₂×#6157	1	1	yes	characteristic
82	C₂×C₈	C₂×#6157	1	1	yes	characteristic
83	C₄²	—	53	1	no	—
84	C₄²	—	106	1	no	—
85	C₄²	C₂×#6157	1	1	yes	characteristic
86	C₁₆	—	53	8	no	—
87	C₂³	—	53	1	no	—
88	C₂×C₄	—	2	1	no	—
89	C₂×C₄	—	53	1	no	—
90	C₂×C₄	—	53	1	no	—
91	C₂×C₄	—	53	1	no	—
92	C₂×C₄	—	53	1	no	—
93	C₂×C₄	—	53	1	no	—
94	C₂×C₄	—	106	1	no	—
95	C₂×C₄	—	212	2	no	—
96	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
97	C₈	—	2	1	no	—
98	C₈	—	53	2	no	—
99	C₈	—	106	1	no	—
100	C₈	C₄×#6157	1	2	yes	—
101	C₂²	—	53	1	no	—
102	C₂²	—	53	1	no	—
103	C₂²	—	106	2	no	—
104	C₂²	D₅₃⋊M₁₆	1	1	yes	characteristic
105	C₄	—	4	1	no	—
106	C₄	—	53	1	no	—
107	C₄	—	53	1	no	—
108	C₄	—	212	1	no	—
109	C₄	C₂×C₄×#6157	1	1	yes	characteristic
110	C₄	D₅₃⋊M₁₆	1	1	yes	characteristic
111	C₂	—	2	1	no	—
112	C₂	—	53	2	no	—
113	C₂	—	106	1	no	—
114	C₂	#16811582	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

D₅₃⋊M₃₂

C₂

yes

maximal

#16811516

C₂

yes

maximal, complemented

#16811515

C₂

yes

maximal, complemented

#97×D₅₃

C₂

yes

maximal, characteristic

#13133292

C₄

yes

complemented

#13133291

C₂²

yes

—

D₅₃⋊C₁₆

C₄

yes

complemented

C₂×C₈×D₅₃

C₄

yes

characteristic

C₂×C₈×D₅₃

C₂²

yes

characteristic

C₄²×D₅₃

C₄

yes

characteristic

#13133128

C₂²

yes

characteristic

#13133127

C₄

yes

characteristic

#13133127

C₂²

yes

characteristic

#97×C₅₃

C₄

yes

characteristic

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

C₂×C₄

yes

characteristic

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄×Dic₅₃

—

C₄×Dic₅₃

C₂×C₄

yes

characteristic

C₂×#88322

C₂×C₄

yes

characteristic

C₂×#88322

C₂³

yes

characteristic

C₈×D₅₃

—

C₈×D₅₃

—

C₈×D₅₃

C₂×C₄

yes

—

C₈×D₅₃

C₂×C₄

yes

—

#11719910

C₂×C₄

yes

complemented

C₂²×D₅₃

M₁₆

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

M₁₆

yes

characteristic

C₄×D₅₃

—

C₄×D₅₃

C₄²

yes

characteristic

C₄×D₅₃

C₄²

yes

characteristic

C₄×D₅₃

M₁₆

yes

characteristic

C₄×D₅₃

M₁₆

yes

characteristic

C₈×C₅₃

—

C₈×C₅₃

C₄²

yes

—

#88322

—

#88322

C₂²×C₄

yes

—

C₂²×C₅₃

C₂×M₁₆

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

yes

C₂×D₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

C₂×C₄²

yes

characteristic

C₄×C₅₃

C₂×M₁₆

yes

characteristic

Dic₅₃

—

Dic₅₃

yes

Dic₅₃

yes

—

C₂×C₅₃

—

C₂×C₅₃

C₂×#97

yes

characteristic

D₅₃

—

D₅₃

#347

yes

complemented

C₂×M₃₂

—

C₂×#97

—

#347

—

#347

—

C₅₃

C₂×#347

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

—

—

—

yes

C₂²×C₄

—

C₂²×C₄

—

106

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₂×C₈

yes

C₂×C₈

yes

C₄²

—

C₄²

—

106

—

C₄²

C₂×#6157

yes

characteristic

C₁₆

—

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

106

—

C₂×C₄

—

212

—

C₂×C₄

C₂²×#6157

yes

characteristic

C₈

—

C₈

—

C₈

—

106

—

100

C₈

C₄×#6157

yes

—

101

C₂²

—

102

C₂²

—

103

C₂²

—

106

—

104

C₂²

D₅₃⋊M₁₆

yes

characteristic

105

C₄

—

106

C₄

—

107

C₄

—

108

C₄

—

212

—

109

C₄

C₂×C₄×#6157

yes

characteristic

110

C₄

D₅₃⋊M₁₆

yes

characteristic

111

C₂

—

112

C₂

—

113

C₂

—

106

—

114

C₂

#16811582

yes

characteristic

	subgroup	G/H
center	C₄	C₂×C₄×#6157
commutator	C₄×C₅₃	C₂×C₄²
Frattini	C₂×C₄	C₂²×#6157
Fitting	#97×C₅₃	C₄
socle	C₂×C₅₃	C₂×#97
hypercenter	#97	#6157
hypocenter	C₅₃	C₂×#347
LCS	G ▷ C₄×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ #97

subgroup

G/H

center

C₄

C₂×C₄×#6157

commutator

C₄×C₅₃

C₂×C₄²

Frattini

C₂×C₄

C₂²×#6157

Fitting

#97×C₅₃

C₄

socle

C₂×C₅₃

C₂×#97

C₅₃

G ▷ C₄×C₅₃ ▷ C₅₃

C₁ ◁ C₄ ◁ #97

N	H ≅ G/N	type
2. #16811516	C₂	semidirect
3. #16811515	C₂	semidirect
5. #13133292	C₄	semidirect
7. D₅₃⋊C₁₆	C₄	semidirect
28. #11719910	C₂×C₄	semidirect
57. D₅₃	#347	semidirect
62. C₅₃	C₂×#347	semidirect

H ≅ G/N

type

2. #16811516

C₂

semidirect

3. #16811515

C₂

semidirect

5. #13133292

C₄

semidirect

7. D₅₃⋊C₁₆

C₄

semidirect

28. #11719910

C₂×C₄

semidirect

57. D₅₃

#347

semidirect

62. C₅₃

C₂×#347

semidirect

Subgroups — #20290932

Conjugacy classes of subgroups

Complements