Group #20276548 — subgroups

Subgroups — #20276548

Order 6784.

162 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 162 conjugacy classes of subgroups — too many to draw. Only its 73 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	C₂×#13133156	C₂	1	2	yes	maximal, complemented
2	C₈×D₂₁₂	C₂	1	1	yes	maximal, characteristic
3	C₄⋊#11719908	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810402	C₂	1	2	yes	maximal, complemented
5	C₄×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₈×D₅₃	C₂²	1	2	yes	—
7	C₄×D₂₁₂	C₄	1	1	yes	characteristic
8	#13133156	—	2	4	no	—
9	C₂×#11719908	C₂²	1	2	yes	—
10	C₄⋊#88322	C₄	1	1	yes	characteristic
11	#13133129	C₄	1	2	yes	—
12	C₂×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×D₂₁₂	C₈	1	1	yes	characteristic
16	C₂×C₄×D₅₃	C₂×C₄	1	2	yes	—
17	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
18	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
19	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
20	#11719921	C₈	1	2	yes	—
21	C₄⋊Dic₅₃	C₈	1	1	yes	characteristic
22	C₂×#88322	C₂×C₄	1	2	yes	—
23	C₈×D₅₃	—	2	2	no	—
24	C₈×D₅₃	—	2	2	no	—
25	C₁₆×C₅₃	—	2	1	no	—
26	C₁₆×C₅₃	D₄	1	2	yes	complemented
27	#11719908	—	2	2	no	—
28	C₂²×D₅₃	C₂×C₈	1	2	yes	—
29	C₂×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
30	C₂×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
31	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
32	C₂×Dic₅₃	C₂×C₈	1	2	yes	—
33	D₂₁₂	—	2	2	no	—
34	C₄×D₅₃	—	2	2	no	—
35	C₄×D₅₃	—	2	2	no	—
36	C₈×C₅₃	—	2	1	no	—
37	C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
38	C₈×C₅₃	#41	1	1	yes	characteristic
39	#88322	—	2	2	no	—
40	C₂²×C₅₃	C₂²×C₈	1	1	yes	characteristic
41	C₂×D₅₃	—	2	2	no	—
42	C₂×D₅₃	—	2	2	no	—
43	C₂×D₅₃	—	2	2	no	—
44	C₄×C₅₃	—	2	1	no	—
45	C₄×C₅₃	M₃₂	1	2	yes	complemented
46	C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
47	C₄×C₅₃	#131	1	1	yes	characteristic
48	Dic₅₃	—	2	2	no	—
49	#2169	—	53	1	no	maximal, Sylow
50	C₂×C₅₃	C₈×D₄	1	1	yes	characteristic
51	C₂×C₅₃	C₂×M₃₂	1	1	yes	characteristic
52	C₂×C₅₃	#504	1	1	yes	characteristic
53	D₅₃	—	4	2	no	—
54	C₂×M₃₂	—	53	2	no	—
55	C₈×D₄	—	53	1	no	—
56	C₄⋊C₁₆	—	53	1	no	—
57	C₂²⋊C₁₆	—	53	2	no	—
58	C₄×C₁₆	D₅₃	1	1	yes	characteristic, complemented
59	C₅₃	#2169	1	1	yes	characteristic, complemented, Sylow
60	C₂²×C₈	—	53	2	no	—
61	C₄×D₄	—	53	1	no	—
62	M₃₂	—	106	4	no	—
63	C₂×C₁₆	—	53	2	no	—
64	C₂×C₁₆	C₂×D₅₃	1	1	yes	characteristic
65	C₂×C₁₆	C₂×D₅₃	1	1	yes	characteristic
66	C₄⋊C₈	—	53	1	no	—
67	C₂²⋊C₈	—	53	2	no	—
68	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
69	C₂×D₄	—	53	1	no	—
70	C₂²×C₄	—	53	2	no	—
71	C₂×C₈	—	53	2	no	—
72	C₂×C₈	—	106	2	no	—
73	C₂×C₈	—	106	2	no	—
74	C₂×C₈	C₄×D₅₃	1	1	yes	characteristic
75	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
76	C₄⋊C₄	—	53	1	no	—
77	C₂²⋊C₄	—	53	2	no	—
78	C₄²	C₄×D₅₃	1	1	yes	characteristic
79	C₁₆	—	2	1	no	—
80	C₁₆	—	106	2	no	—
81	C₁₆	D₂₁₂	1	2	yes	complemented
82	C₂³	—	53	2	no	—
83	D₄	—	106	2	no	—
84	C₂×C₄	—	53	2	no	—
85	C₂×C₄	—	106	2	no	—
86	C₂×C₄	—	106	2	no	—
87	C₂×C₄	C₈×D₅₃	1	1	yes	characteristic
88	C₂×C₄	C₈×D₅₃	1	1	yes	characteristic
89	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
90	C₈	—	2	1	no	—
91	C₈	—	106	2	no	—
92	C₈	C₂×D₂₁₂	1	1	yes	characteristic
93	C₈	#11719945	1	1	yes	characteristic
94	C₂²	—	106	2	no	—
95	C₂²	—	106	2	no	—
96	C₂²	—	106	2	no	—
97	C₂²	C₂×C₈×D₅₃	1	1	yes	characteristic
98	C₄	—	2	1	no	—
99	C₄	—	106	2	no	—
100	C₄	#13133156	1	2	yes	complemented
101	C₄	C₄×D₂₁₂	1	1	yes	characteristic
102	C₄	#13133223	1	1	yes	characteristic
103	C₂	—	212	2	no	—
104	C₂	C₈×D₂₁₂	1	1	yes	characteristic
105	C₂	C₂×#13133156	1	1	yes	characteristic
106	C₂	#16811039	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133156

C₂

yes

maximal, complemented

C₈×D₂₁₂

C₂

yes

maximal, characteristic

C₄⋊#11719908

C₂

yes

maximal, characteristic, complemented

#16810402

C₂

yes

maximal, complemented

C₄×C₁₆×C₅₃

C₂

yes

maximal, characteristic, complemented

C₂×C₈×D₅₃

C₂²

yes

—

C₄×D₂₁₂

C₄

yes

characteristic

#13133156

—

C₂×#11719908

C₂²

yes

—

C₄⋊#88322

C₄

yes

characteristic

#13133129

C₄

yes

—

C₂×C₁₆×C₅₃

C₂²

yes

characteristic

C₂×C₁₆×C₅₃

C₂²

yes

characteristic

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂×D₂₁₂

C₈

yes

characteristic

C₂×C₄×D₅₃

C₂×C₄

yes

—

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₄²×C₅₃

C₂×C₄

yes

characteristic

#11719921

C₈

yes

—

C₄⋊Dic₅₃

C₈

yes

characteristic

C₂×#88322

C₂×C₄

yes

—

C₈×D₅₃

—

C₈×D₅₃

—

C₁₆×C₅₃

—

C₁₆×C₅₃

D₄

yes

complemented

#11719908

—

C₂²×D₅₃

C₂×C₈

yes

—

C₂×C₄×C₅₃

C₂×C₈

yes

characteristic

C₂×C₄×C₅₃

C₂×C₈

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×Dic₅₃

C₂×C₈

yes

—

D₂₁₂

—

C₄×D₅₃

—

C₄×D₅₃

—

C₈×C₅₃

—

C₈×C₅₃

C₂×D₄

yes

C₈×C₅₃

yes

—

C₂²×C₅₃

C₂²×C₈

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

M₃₂

yes

complemented

C₄×C₅₃

C₄×D₄

yes

characteristic

C₄×C₅₃

#131

yes

characteristic

Dic₅₃

—

#2169

—

maximal, Sylow

C₂×C₅₃

C₈×D₄

yes

characteristic

C₂×C₅₃

C₂×M₃₂

yes

characteristic

C₂×C₅₃

#504

yes

characteristic

D₅₃

—

C₂×M₃₂

—

C₈×D₄

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

C₄×C₁₆

D₅₃

yes

characteristic, complemented

C₅₃

#2169

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₄×D₄

—

M₃₂

—

106

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₅₃

yes

characteristic

C₂×C₁₆

C₂×D₅₃

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

C₂×D₅₃

yes

characteristic

C₂×D₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₂×C₈

C₄×D₅₃

yes

characteristic

C₂×C₈

C₂²×D₅₃

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

C₄×D₅₃

yes

characteristic

C₁₆

—

C₁₆

—

106

—

C₁₆

D₂₁₂

yes

complemented

C₂³

—

D₄

—

106

—

C₂×C₄

—

C₂×C₄

—

106

—

C₂×C₄

—

106

—

C₂×C₄

C₈×D₅₃

yes

characteristic

C₂×C₄

C₈×D₅₃

yes

characteristic

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

C₈

—

C₈

—

106

—

C₈

C₂×D₂₁₂

yes

characteristic

C₈

#11719945

yes

characteristic

C₂²

—

106

—

C₂²

—

106

—

C₂²

—

106

—

C₂²

C₂×C₈×D₅₃

yes

characteristic

C₄

—

C₄

—

106

—

100

C₄

#13133156

yes

complemented

101

C₄

C₄×D₂₁₂

yes

characteristic

102

C₄

#13133223

yes

characteristic

103

C₂

—

212

—

104

C₂

C₈×D₂₁₂

yes

characteristic

105

C₂

C₂×#13133156

yes

characteristic

106

C₂

#16811039

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₅₃
commutator, socle	C₂²×C₅₃	C₂²×C₈
Fitting	C₄×C₁₆×C₅₃	C₂
hypercenter	C₄×C₁₆	D₅₃
hypocenter	C₅₃	#2169
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₁₆

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×D₅₃

commutator, socle

C₂²×C₅₃

C₂²×C₈

Fitting

C₄×C₁₆×C₅₃

C₂

hypercenter

C₄×C₁₆

D₅₃

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₂×C₈ ◁ C₄×C₁₆

N	H ≅ G/N	type
1. C₂×#13133156	C₂	semidirect
3. C₄⋊#11719908	C₂	semidirect
4. #16810402	C₂	semidirect
5. C₄×C₁₆×C₅₃	C₂	semidirect
26. C₁₆×C₅₃	D₄	semidirect
45. C₄×C₅₃	M₃₂	semidirect
58. C₄×C₁₆	D₅₃	semidirect
59. C₅₃	#2169	semidirect
81. C₁₆	D₂₁₂	semidirect
100. C₄	#13133156	semidirect

H ≅ G/N

type

1. C₂×#13133156

C₂

semidirect

3. C₄⋊#11719908

C₂

semidirect

4. #16810402

C₂

semidirect

5. C₄×C₁₆×C₅₃

C₂

semidirect

26. C₁₆×C₅₃

D₄

semidirect

45. C₄×C₅₃

M₃₂

semidirect

58. C₄×C₁₆

D₅₃

semidirect

59. C₅₃

#2169

semidirect

81. C₁₆

D₂₁₂

semidirect

100. C₄

#13133156

semidirect

Subgroups — #20276548

Conjugacy classes of subgroups

Complements