Group #20289685 — subgroups

Subgroups — #20289685

Order 6784.

114 conjugacy classes of subgroups; only the 26 normal subgroups are shown.

normal subgroups only

This group has 114 conjugacy classes of subgroups — too many to draw. Only its 26 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	#16811520	C₂	1	1	yes	maximal, characteristic, complemented
2	#16811491	C₂	1	1	yes	maximal, characteristic, complemented
3	#16810873	C₂	1	1	yes	maximal, characteristic
4	#13133296	—	2	1	no	—
5	#13133291	—	2	1	no	—
6	C₂×#11719912	C₂²	1	1	yes	characteristic
7	#13133188	—	2	1	no	—
8	#13133149	C₄	1	1	yes	characteristic, complemented
9	#13133138	—	2	1	no	—
10	C₂×D₈×C₅₃	C₄	1	1	yes	characteristic, complemented
11	C₂×D₄×C₅₃	—	2	1	no	—
12	C₂×C₂²⋊D₅₃	—	4	1	no	—
13	C₂×C₄×D₅₃	D₄	1	1	yes	characteristic
14	C₂²⋊#6157	—	4	1	no	—
15	C₄⋊#6157	—	4	1	no	—
16	C₄×#6157	—	2	2	no	—
17	D₈×C₅₃	—	4	1	no	—
18	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
19	C₂²⋊Dic₅₃	—	4	1	no	—
20	C₄⋊Dic₅₃	—	2	1	no	—
21	C₂×#88322	D₄	1	1	yes	characteristic
22	#11719912	—	4	1	no	—
23	#11719910	—	4	1	no	—
24	C₂³×C₅₃	—	4	1	no	—
25	C₂²×D₅₃	D₈	1	1	yes	characteristic
26	C₂×#6157	—	4	2	no	—
27	D₄×C₅₃	—	4	1	no	—
28	D₄×C₅₃	—	4	1	no	—
29	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
30	C₂²⋊D₅₃	—	8	2	no	—
31	C₂×Dic₅₃	—	4	1	no	—
32	C₂×Dic₅₃	SD₁₆	1	1	yes	characteristic
33	C₄×D₅₃	—	2	1	no	—
34	C₄×D₅₃	—	2	1	no	—
35	C₈×C₅₃	—	2	1	no	—
36	#88322	—	2	1	no	—
37	C₂²×C₅₃	—	8	1	no	—
38	C₂²×C₅₃	—	8	1	no	—
39	C₂²×C₅₃	—	8	1	no	—
40	C₂²×C₅₃	#102	1	1	yes	characteristic
41	C₂×D₅₃	—	2	1	no	—
42	C₂×D₅₃	—	4	1	no	—
43	#6157	—	8	2	no	—
44	C₄×C₅₃	#99	1	1	yes	characteristic
45	C₄×C₅₃	#104	1	1	yes	characteristic
46	Dic₅₃	—	2	1	no	—
47	Dic₅₃	—	8	1	no	—
48	#1339	—	53	1	no	maximal, Sylow
49	C₂×C₅₃	—	2	1	no	—
50	C₂×C₅₃	—	16	1	no	—
51	C₂×C₅₃	#327	1	1	yes	characteristic
52	D₅₃	—	4	1	no	—
53	#469	—	53	1	no	—
54	#349	—	53	1	no	—
55	H(3,ℤ/4)	—	53	1	no	—
56	C₅₃	#1339	1	1	yes	characteristic, complemented, Sylow
57	C₂×D₈	#6157	1	1	yes	characteristic, complemented
58	C₂×M₁₆	—	53	1	no	—
59	#121	—	106	1	no	—
60	#117	—	106	1	no	—
61	M₃₂	—	106	1	no	—
62	#106	—	53	1	no	—
63	#102	—	106	1	no	—
64	C₂×D₄	—	2	1	no	—
65	C₂×D₄	—	212	1	no	—
66	C₂²×C₄	—	53	1	no	—
67	D₈	—	4	1	no	—
68	M₁₆	—	212	1	no	—
69	C₂×C₈	—	53	1	no	—
70	C₂×C₈	C₂×#6157	1	1	yes	characteristic
71	C₄⋊C₄	—	106	1	no	—
72	C₄⋊C₄	—	212	1	no	—
73	C₂²⋊C₄	—	212	1	no	—
74	C₂²⋊C₄	—	212	1	no	—
75	C₄²	—	106	2	no	—
76	C₁₆	—	212	1	no	—
77	C₂³	—	4	1	no	—
78	C₂³	—	53	1	no	—
79	D₄	—	4	1	no	—
80	D₄	—	4	1	no	—
81	D₄	—	424	2	no	—
82	C₂×C₄	—	53	1	no	—
83	C₂×C₄	—	106	1	no	—
84	C₂×C₄	—	106	1	no	—
85	C₂×C₄	—	212	2	no	—
86	C₂×C₄	—	212	1	no	—
87	C₂×C₄	C₂²⋊#6157	1	1	yes	characteristic
88	C₈	—	2	1	no	—
89	C₈	—	106	1	no	—
90	C₂²	—	8	1	no	—
91	C₂²	—	8	1	no	—
92	C₂²	—	8	1	no	—
93	C₂²	—	106	1	no	—
94	C₂²	—	212	1	no	—
95	C₂²	#13133279	1	1	yes	characteristic
96	C₄	—	106	1	no	—
97	C₄	—	424	1	no	—
98	C₄	—	424	2	no	—
99	C₄	#13133275	1	1	yes	characteristic
100	C₄	#13133281	1	1	yes	characteristic
101	C₂	—	2	1	no	—
102	C₂	—	16	1	no	—
103	C₂	—	212	1	no	—
104	C₂	#16811474	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811520

C₂

yes

maximal, characteristic, complemented

#16811491

C₂

yes

maximal, characteristic, complemented

#16810873

C₂

yes

maximal, characteristic

#13133296

—

#13133291

—

C₂×#11719912

C₂²

yes

characteristic

#13133188

—

#13133149

C₄

yes

characteristic, complemented

#13133138

—

C₂×D₈×C₅₃

C₄

yes

characteristic, complemented

C₂×D₄×C₅₃

—

C₂×C₂²⋊D₅₃

—

C₂×C₄×D₅₃

D₄

yes

characteristic

C₂²⋊#6157

—

C₄⋊#6157

—

C₄×#6157

—

D₈×C₅₃

—

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂²⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₂×#88322

D₄

yes

characteristic

#11719912

—

#11719910

—

C₂³×C₅₃

—

C₂²×D₅₃

D₈

yes

characteristic

C₂×#6157

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂²⋊D₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

SD₁₆

yes

characteristic

C₄×D₅₃

—

C₄×D₅₃

—

C₈×C₅₃

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#102

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

#6157

—

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

—

#1339

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#327

yes

characteristic

D₅₃

—

#469

—

#349

—

H(3,ℤ/4)

—

C₅₃

#1339

yes

characteristic, complemented, Sylow

C₂×D₈

#6157

yes

characteristic, complemented

C₂×M₁₆

—

#121

—

106

—

#117

—

106

—

M₃₂

—

106

—

#106

—

#102

—

106

—

C₂×D₄

—

C₂×D₄

—

212

—

C₂²×C₄

—

D₈

—

M₁₆

—

212

—

C₂×C₈

—

C₂×C₈

C₂×#6157

yes

characteristic

C₄⋊C₄

—

106

—

C₄⋊C₄

—

212

—

C₂²⋊C₄

—

212

—

C₂²⋊C₄

—

212

—

C₄²

—

106

—

C₁₆

—

212

—

C₂³

—

C₂³

—

D₄

—

D₄

—

D₄

—

424

—

C₂×C₄

—

C₂×C₄

—

106

—

C₂×C₄

—

106

—

C₂×C₄

—

212

—

C₂×C₄

—

212

—

C₂×C₄

C₂²⋊#6157

yes

characteristic

C₈

—

C₈

—

106

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

106

—

C₂²

—

212

—

C₂²

#13133279

yes

characteristic

C₄

—

106

—

C₄

—

424

—

C₄

—

424

—

C₄

#13133275

yes

characteristic

100

C₄

#13133281

yes

characteristic

101

C₂

—

102

C₂

—

103

C₂

—

212

—

104

C₂

#16811474

yes

characteristic

	subgroup	G/H
center	C₂	#16811474
commutator	C₂×C₈×C₅₃	C₂×C₄
Frattini	C₂×C₈	C₂×#6157
Fitting	C₂×D₈×C₅₃	C₄
socle	C₂×C₅₃	#327
hypercenter	C₂×D₈	#6157
hypocenter	C₅₃	#1339
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×D₈

subgroup

G/H

center

C₂

#16811474

commutator

C₂×C₈×C₅₃

C₂×C₄

Frattini

C₂×C₈

C₂×#6157

Fitting

C₂×D₈×C₅₃

C₄

C₂×C₅₃

C₂×D₈

C₅₃

G ▷ C₂×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×D₈

N	H ≅ G/N	type
1. #16811520	C₂	semidirect
2. #16811491	C₂	semidirect
8. #13133149	C₄	semidirect
10. C₂×D₈×C₅₃	C₄	semidirect
56. C₅₃	#1339	semidirect
57. C₂×D₈	#6157	semidirect

H ≅ G/N

type

1. #16811520

C₂

semidirect

2. #16811491

C₂

semidirect

8. #13133149

C₄

semidirect

10. C₂×D₈×C₅₃

C₄

semidirect

56. C₅₃

#1339

semidirect

57. C₂×D₈

#6157

semidirect

Subgroups — #20289685

Conjugacy classes of subgroups

Complements