Group #20289758 — subgroups

Subgroups — #20289758

Order 6784.

148 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 148 conjugacy classes of subgroups — too many to draw. Only its 54 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	1	yes	maximal, characteristic
2	C₂×C₈×#6157	C₂	1	1	yes	maximal, characteristic
3	M₃₂×D₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×D₅₃⋊C₈	C₄	1	1	yes	characteristic
5	C₂×C₄×#6157	C₄	1	1	yes	characteristic
6	#13133292	—	2	1	no	—
7	#13133291	C₄	1	2	yes	complemented
8	D₅₃⋊C₁₆	—	2	1	no	—
9	C₈×#6157	—	2	1	no	—
10	C₈×#6157	—	2	1	no	—
11	C₂×C₈×D₅₃	C₂²	1	1	yes	characteristic
12	#13133157	C₄	1	1	yes	characteristic, complemented
13	#13133156	—	2	1	no	—
14	C₁₆×D₅₃	—	2	1	no	—
15	M₃₂×C₅₃	C₄	1	1	yes	characteristic, complemented
16	C₂²×#6157	—	2	1	no	—
17	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
18	C₂×#88324	—	2	1	no	—
19	C₄×#6157	—	2	1	no	—
20	C₄×#6157	C₈	1	2	yes	—
21	D₅₃⋊C₈	—	2	1	no	—
22	D₅₃⋊C₈	C₈	1	2	yes	—
23	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂×#88322	C₂×C₄	1	1	yes	characteristic
25	C₈×D₅₃	D₄	1	1	yes	characteristic
26	C₈×D₅₃	D₄	1	1	yes	characteristic
27	C₈×D₅₃	D₄	1	1	yes	characteristic
28	C₈×D₅₃	Q₈	1	1	yes	characteristic
29	#11719910	—	2	2	no	—
30	C₁₆×C₅₃	—	2	1	no	—
31	#11719908	—	2	1	no	—
32	C₂²×D₅₃	M₁₆	1	1	yes	characteristic
33	C₂×#6157	—	2	4	no	—
34	C₂×#6157	—	2	2	no	—
35	C₂×C₄×C₅₃	C₄²	1	1	yes	characteristic
36	C₂×Dic₅₃	M₁₆	1	1	yes	characteristic
37	C₄×D₅₃	C₂²⋊C₄	1	1	yes	characteristic
38	C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
39	C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
40	C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
41	#88324	—	2	2	no	—
42	C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
43	C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
44	#88322	C₂²⋊C₄	1	1	yes	characteristic
45	#88322	C₄⋊C₄	1	1	yes	characteristic
46	C₂²×C₅₃	#97	1	1	yes	characteristic
47	C₂×D₅₃	—	2	2	no	—
48	C₂×D₅₃	C₂²⋊C₈	1	1	yes	characteristic
49	C₂×D₅₃	C₄⋊C₈	1	1	yes	characteristic
50	#6157	—	2	4	no	—
51	C₄×C₅₃	#95	1	1	yes	characteristic
52	C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
53	Dic₅₃	C₂²⋊C₈	1	1	yes	characteristic
54	Dic₅₃	C₄⋊C₈	1	1	yes	characteristic
55	#1376	—	53	1	no	maximal, Sylow
56	C₂×C₅₃	—	2	1	no	—
57	C₂×C₅₃	#336	1	1	yes	characteristic
58	D₅₃	—	2	1	no	—
59	D₅₃	#364	1	2	yes	—
60	C₂×M₃₂	—	53	1	no	—
61	C₂×M₃₂	—	53	1	no	—
62	C₂×C₄×C₈	—	53	1	no	—
63	C₅₃	#1376	1	1	yes	characteristic, complemented, Sylow
64	C₂²×C₈	—	53	1	no	—
65	C₂²×C₈	—	53	1	no	—
66	C₂×C₄²	—	53	1	no	—
67	M₃₂	—	53	2	no	—
68	M₃₂	—	53	1	no	—
69	M₃₂	—	106	1	no	—
70	M₃₂	—	106	1	no	—
71	M₃₂	#6157	1	1	yes	characteristic, complemented
72	C₂×C₁₆	—	106	1	no	—
73	C₂×C₁₆	—	106	1	no	—
74	C₄×C₈	—	106	1	no	—
75	C₄×C₈	—	106	1	no	—
76	C₂²×C₄	—	53	1	no	—
77	C₂²×C₄	—	106	1	no	—
78	C₂×C₈	—	53	1	no	—
79	C₂×C₈	—	53	1	no	—
80	C₂×C₈	—	53	1	no	—
81	C₂×C₈	—	53	1	no	—
82	C₂×C₈	—	53	1	no	—
83	C₂×C₈	—	53	2	no	—
84	C₂×C₈	—	106	1	no	—
85	C₂×C₈	—	106	1	no	—
86	C₂×C₈	C₂×#6157	1	1	yes	characteristic
87	C₄²	—	53	2	no	—
88	C₄²	—	106	1	no	—
89	C₁₆	—	2	1	no	—
90	C₁₆	—	106	2	no	—
91	C₁₆	—	106	1	no	—
92	C₂³	—	53	1	no	—
93	C₂×C₄	—	53	1	no	—
94	C₂×C₄	—	53	1	no	—
95	C₂×C₄	—	53	1	no	—
96	C₂×C₄	—	53	1	no	—
97	C₂×C₄	—	53	1	no	—
98	C₂×C₄	—	106	2	no	—
99	C₂×C₄	—	106	4	no	—
100	C₂×C₄	C₄×#6157	1	1	yes	characteristic
101	C₈	—	53	1	no	—
102	C₈	—	53	1	no	—
103	C₈	—	106	2	no	—
104	C₈	C₄⋊#6157	1	1	yes	characteristic
105	C₈	C₂²⋊#6157	1	1	yes	characteristic
106	C₂²	—	53	1	no	—
107	C₂²	—	53	1	no	—
108	C₂²	—	106	2	no	—
109	C₂²	#13133271	1	1	yes	characteristic
110	C₄	—	53	1	no	—
111	C₄	—	53	1	no	—
112	C₄	—	106	4	no	—
113	C₄	(C₂×#6157)⋊C₄	1	1	yes	characteristic
114	C₄	C₈×#6157	1	1	yes	characteristic
115	C₂	—	2	1	no	—
116	C₂	—	53	2	no	—
117	C₂	—	106	1	no	—
118	C₂	(C₂×#6157)⋊C₈	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

D₅₃⋊M₃₂

C₂

yes

maximal, characteristic

C₂×C₈×#6157

C₂

yes

maximal, characteristic

M₃₂×D₅₃

C₂

yes

maximal, characteristic

C₂×D₅₃⋊C₈

C₄

yes

characteristic

C₂×C₄×#6157

C₄

yes

characteristic

#13133292

—

#13133291

C₄

yes

complemented

D₅₃⋊C₁₆

—

C₈×#6157

—

C₈×#6157

—

C₂×C₈×D₅₃

C₂²

yes

characteristic

#13133157

C₄

yes

characteristic, complemented

#13133156

—

C₁₆×D₅₃

—

M₃₂×C₅₃

C₄

yes

characteristic, complemented

C₂²×#6157

—

C₂×C₄×D₅₃

C₂×C₄

yes

—

—

C₈

yes

—

D₅₃⋊C₈

—

D₅₃⋊C₈

C₈

yes

—

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×#88322

C₂×C₄

yes

characteristic

C₈×D₅₃

D₄

yes

characteristic

C₈×D₅₃

D₄

yes

characteristic

C₈×D₅₃

D₄

yes

characteristic

C₈×D₅₃

Q₈

yes

characteristic

#11719910

—

C₁₆×C₅₃

—

#11719908

—

C₂²×D₅₃

M₁₆

yes

characteristic

C₂×#6157

—

C₂×#6157

—

C₂×C₄×C₅₃

C₄²

yes

characteristic

C₂×Dic₅₃

M₁₆

yes

characteristic

C₄×D₅₃

C₂²⋊C₄

yes

characteristic

C₄×D₅₃

C₄⋊C₄

yes

characteristic

C₄×D₅₃

C₂×C₈

yes

characteristic

C₄×D₅₃

C₂×C₈

yes

characteristic

#88324

—

C₈×C₅₃

C₂²⋊C₄

yes

characteristic

C₈×C₅₃

C₄⋊C₄

yes

characteristic

#88322

C₂²⋊C₄

yes

characteristic

#88322

C₄⋊C₄

yes

characteristic

C₂²×C₅₃

#97

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

C₂²⋊C₈

yes

characteristic

C₂×D₅₃

C₄⋊C₈

yes

characteristic

#6157

—

C₄×C₅₃

#95

yes

characteristic

C₄×C₅₃

C₄×C₈

yes

characteristic

Dic₅₃

C₂²⋊C₈

yes

characteristic

Dic₅₃

C₄⋊C₈

yes

characteristic

#1376

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

#336

yes

characteristic

D₅₃

—

D₅₃

#364

yes

—

C₂×M₃₂

—

C₂×M₃₂

—

C₂×C₄×C₈

—

C₅₃

#1376

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

M₃₂

—

M₃₂

—

M₃₂

—

106

—

M₃₂

—

106

—

M₃₂

#6157

yes

characteristic, complemented

C₂×C₁₆

—

106

—

C₂×C₁₆

—

106

—

C₄×C₈

—

106

—

C₄×C₈

—

106

—

C₂²×C₄

—

C₂²×C₄

—

106

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₂×C₈

C₂×#6157

yes

characteristic

C₄²

—

C₄²

—

106

—

C₁₆

—

C₁₆

—

106

—

C₁₆

—

106

—

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

106

—

C₂×C₄

—

106

—

100

C₂×C₄

C₄×#6157

yes

characteristic

101

C₈

—

102

C₈

—

103

C₈

—

106

—

104

C₈

C₄⋊#6157

yes

characteristic

105

C₈

C₂²⋊#6157

yes

characteristic

106

C₂²

—

107

C₂²

—

108

C₂²

—

106

—

109

C₂²

#13133271

yes

characteristic

110

C₄

—

111

C₄

—

112

C₄

—

106

—

113

C₄

(C₂×#6157)⋊C₄

yes

characteristic

114

C₄

C₈×#6157

yes

characteristic

115

C₂

—

116

C₂

—

117

C₂

—

106

—

118

C₂

(C₂×#6157)⋊C₈

yes

characteristic

	subgroup	G/H
center	C₈	C₂²⋊#6157
commutator	C₄×C₅₃	C₄×C₈
Frattini	C₂×C₈	C₂×#6157
Fitting	M₃₂×C₅₃	C₄
socle	C₂×C₅₃	#336
hypercenter	M₃₂	#6157
hypocenter	C₅₃	#1376
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ M₃₂

subgroup

G/H

center

C₈

C₂²⋊#6157

commutator

C₄×C₅₃

C₄×C₈

Frattini

C₂×C₈

C₂×#6157

Fitting

M₃₂×C₅₃

C₄

C₂×C₅₃

M₃₂

C₅₃

G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₈ ◁ C₂×C₈ ◁ M₃₂

N	H ≅ G/N	type
7. #13133291	C₄	semidirect
12. #13133157	C₄	semidirect
15. M₃₂×C₅₃	C₄	semidirect
63. C₅₃	#1376	semidirect
71. M₃₂	#6157	semidirect

H ≅ G/N

type

7. #13133291

C₄

semidirect

12. #13133157

C₄

semidirect

15. M₃₂×C₅₃

C₄

semidirect

63. C₅₃

#1376

semidirect

71. M₃₂

#6157

semidirect

Subgroups — #20289758

Conjugacy classes of subgroups

Complements