Group #20290364 — subgroups

Subgroups — #20290364

Order 6784.

176 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 56 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	#16811657	C₂	1	2	yes	maximal, complemented
2	#16811484	C₂	1	1	yes	maximal, characteristic, complemented
3	#16811471	C₂	1	2	yes	maximal, complemented
4	#16811464	C₂	1	1	yes	maximal, characteristic, complemented
5	#16810705	C₂	1	1	yes	maximal, characteristic
6	C₂×#11719936	—	2	2	no	—
7	C₄⋊#88324	C₂²	1	2	yes	—
8	#13133274	—	2	2	no	—
9	C₄×#88324	C₂²	1	1	yes	characteristic
10	C₄×#88324	C₂²	1	1	yes	characteristic
11	C₂×D₄₂₄	C₄	1	1	yes	characteristic
12	#13133179	C₂²	1	2	yes	—
13	#13133137	C₄	1	2	yes	—
14	C₈×Dic₅₃	C₂²	1	1	yes	characteristic
15	#107×C₅₃	C₄	1	1	yes	characteristic
16	C₂×D₂₁₂	C₂×C₄	1	2	yes	—
17	C₂×C₄×D₅₃	—	2	2	no	—
18	C₂×#88324	—	2	2	no	—
19	C₂×#88324	—	2	1	no	—
20	C₂×#88324	D₄	1	2	yes	—
21	#11719936	—	4	4	no	—
22	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
23	C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
24	#11719921	—	2	2	no	—
25	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
26	C₂×#88322	C₂×C₄	1	1	yes	characteristic
27	D₄₂₄	—	2	2	no	—
28	C₂²×D₅₃	—	2	2	no	—
29	C₂×C₄×C₅₃	—	2	2	no	—
30	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
31	C₂×Dic₅₃	C₂×D₄	1	1	yes	characteristic
32	C₂×Dic₅₃	#41	1	1	yes	characteristic
33	D₂₁₂	—	2	2	no	—
34	D₂₁₂	—	2	2	no	—
35	C₄×D₅₃	—	4	2	no	—
36	C₄×D₅₃	—	4	2	no	—
37	#88324	—	2	2	no	—
38	#88324	—	4	1	no	—
39	#88324	—	4	2	no	—
40	C₈×C₅₃	M₁₆	1	2	yes	complemented
41	#88322	—	2	1	no	—
42	C₂²×C₅₃	C₄×D₄	1	1	yes	characteristic
43	C₂×D₅₃	—	4	2	no	—
44	C₂×D₅₃	—	4	2	no	—
45	C₂×D₅₃	—	4	2	no	—
46	C₄×C₅₃	—	4	2	no	—
47	C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
48	C₄×C₅₃	#131	1	1	yes	characteristic
49	Dic₅₃	—	2	1	no	—
50	Dic₅₃	AGL(1,ℤ/8)	1	2	yes	—
51	#1594	—	53	1	no	maximal, Sylow
52	C₂×C₅₃	#436	1	1	yes	characteristic
53	C₂×C₅₃	#442	1	1	yes	characteristic
54	C₂×C₅₃	#444	1	1	yes	characteristic
55	D₅₃	—	8	2	no	—
56	C₄×D₈	—	53	1	no	—
57	#436	—	53	2	no	—
58	#334	—	53	1	no	—
59	D₄⋊C₈	—	53	2	no	—
60	#322	—	53	1	no	—
61	C₅₃	#1594	1	1	yes	characteristic, complemented, Sylow
62	C₂×D₈	—	53	1	no	—
63	C₂×M₁₆	—	106	2	no	—
64	C₄×D₄	—	53	2	no	—
65	#107	#6157	1	1	yes	characteristic
66	C₄⋊C₈	—	53	2	no	—
67	#102	—	53	2	no	—
68	C₂²⋊C₈	—	106	2	no	—
69	C₄×C₈	—	53	1	no	—
70	C₄×C₈	—	53	1	no	—
71	C₄×C₈	—	53	1	no	—
72	C₂×D₄	—	53	2	no	—
73	C₂²×C₄	—	106	2	no	—
74	D₈	—	106	2	no	—
75	M₁₆	—	212	4	no	—
76	C₂×C₈	—	53	2	no	—
77	C₂×C₈	—	53	1	no	—
78	C₂×C₈	—	106	1	no	—
79	C₂×C₈	—	106	2	no	—
80	C₂×C₈	C₂×#6157	1	1	yes	characteristic
81	C₄⋊C₄	C₂×#6157	1	2	yes	—
82	C₂²⋊C₄	—	106	2	no	—
83	C₄²	—	53	1	no	—
84	C₂³	—	106	2	no	—
85	D₄	—	106	2	no	—
86	D₄	—	106	2	no	—
87	C₂×C₄	—	2	2	no	—
88	C₂×C₄	—	53	1	no	—
89	C₂×C₄	—	53	1	no	—
90	C₂×C₄	—	212	2	no	—
91	C₂×C₄	—	212	2	no	—
92	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
93	C₈	—	106	2	no	—
94	C₈	—	106	1	no	—
95	C₈	—	212	2	no	—
96	C₈	—	212	1	no	—
97	C₈	#11719936	1	2	yes	complemented
98	C₂²	—	212	2	no	—
99	C₂²	—	212	2	no	—
100	C₂²	—	212	2	no	—
101	C₂²	D₄×#6157	1	1	yes	characteristic
102	C₄	—	4	2	no	—
103	C₄	—	53	2	no	—
104	C₄	—	106	1	no	—
105	C₄	C₂×#11719936	1	1	yes	characteristic
106	C₄	#13133305	1	1	yes	characteristic
107	C₂	—	424	2	no	—
108	C₂	#16811657	1	1	yes	characteristic
109	C₂	#16811664	1	1	yes	characteristic
110	C₂	#16811669	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811657

C₂

yes

maximal, complemented

#16811484

C₂

yes

maximal, characteristic, complemented

#16811471

C₂

yes

maximal, complemented

#16811464

C₂

yes

maximal, characteristic, complemented

#16810705

C₂

yes

maximal, characteristic

C₂×#11719936

—

C₄⋊#88324

C₂²

yes

—

#13133274

—

C₄×#88324

C₂²

yes

characteristic

C₄×#88324

C₂²

yes

characteristic

C₂×D₄₂₄

C₄

yes

characteristic

#13133179

C₂²

yes

—

#13133137

C₄

yes

—

C₈×Dic₅₃

C₂²

yes

characteristic

#107×C₅₃

C₄

yes

characteristic

C₂×D₂₁₂

C₂×C₄

yes

—

C₂×C₄×D₅₃

—

C₂×#88324

—

C₂×#88324

—

C₂×#88324

D₄

yes

—

#11719936

—

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄⋊C₄×C₅₃

C₂×C₄

yes

—

#11719921

—

C₄×Dic₅₃

C₂³

yes

characteristic

C₂×#88322

C₂×C₄

yes

characteristic

D₄₂₄

—

C₂²×D₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×Dic₅₃

C₂×D₄

yes

characteristic

C₂×Dic₅₃

#41

yes

characteristic

D₂₁₂

—

D₂₁₂

—

C₄×D₅₃

—

C₄×D₅₃

—

#88324

—

#88324

—

#88324

—

C₈×C₅₃

M₁₆

yes

complemented

#88322

—

C₂²×C₅₃

C₄×D₄

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×M₁₆

yes

characteristic

C₄×C₅₃

#131

yes

characteristic

Dic₅₃

—

Dic₅₃

AGL(1,ℤ/8)

yes

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

—

C₄×D₈

—

#436

—

#334

—

D₄⋊C₈

—

#322

—

C₅₃

#1594

yes

characteristic, complemented, Sylow

C₂×D₈

—

C₂×M₁₆

—

106

—

C₄×D₄

—

#107

#6157

yes

characteristic

C₄⋊C₈

—

#102

—

C₂²⋊C₈

—

106

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₂×D₄

—

C₂²×C₄

—

106

—

D₈

—

106

—

M₁₆

—

212

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₂×C₈

C₂×#6157

yes

characteristic

C₄⋊C₄

C₂×#6157

yes

—

C₂²⋊C₄

—

106

—

C₄²

—

C₂³

—

106

—

D₄

—

106

—

D₄

—

106

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

212

—

C₂×C₄

—

212

—

C₂×C₄

C₂²×#6157

yes

characteristic

C₈

—

106

—

C₈

—

106

—

C₈

—

212

—

C₈

—

212

—

C₈

#11719936

yes

complemented

C₂²

—

212

—

C₂²

—

212

—

100

C₂²

—

212

—

101

C₂²

D₄×#6157

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

—

106

—

105

C₄

C₂×#11719936

yes

characteristic

106

C₄

#13133305

yes

characteristic

107

C₂

—

424

—

108

C₂

#16811657

yes

characteristic

109

C₂

#16811664

yes

characteristic

110

C₂

#16811669

yes

characteristic

	subgroup	G/H
center	C₂²	D₄×#6157
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×#6157
Fitting	#107×C₅₃	C₄
socle	C₂²×C₅₃	C₄×D₄
hypercenter	#107	#6157
hypocenter	C₅₃	#1594
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #107

subgroup

G/H

center

C₂²

D₄×#6157

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₂×C₄

C₂²×#6157

Fitting

#107×C₅₃

C₄

socle

C₂²×C₅₃

C₄×D₄

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ #107

N	H ≅ G/N	type
1. #16811657	C₂	semidirect
2. #16811484	C₂	semidirect
3. #16811471	C₂	semidirect
4. #16811464	C₂	semidirect
40. C₈×C₅₃	M₁₆	semidirect
61. C₅₃	#1594	semidirect
97. C₈	#11719936	semidirect

H ≅ G/N

type

1. #16811657

C₂

semidirect

2. #16811484

C₂

semidirect

3. #16811471

C₂

semidirect

4. #16811464

C₂

semidirect

40. C₈×C₅₃

M₁₆

semidirect

61. C₅₃

#1594

semidirect

97. C₈

#11719936

semidirect

Subgroups — #20290364

Conjugacy classes of subgroups

Complements