Group #20291113 — subgroups

Subgroups — #20291113

Order 6784.

172 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 48 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	D₈×#6157	C₂	1	2	yes	maximal, complemented
2	#16811550	C₂	1	1	yes	maximal, characteristic, complemented
3	#16811535	C₂	1	2	yes	maximal, complemented
4	#16811510	C₂	1	1	yes	maximal, characteristic, complemented
5	D₁₆×D₅₃	C₂	1	1	yes	maximal, characteristic
6	D₄×#6157	—	2	2	no	—
7	D₅₃⋊C₁₆	C₂²	1	1	yes	characteristic
8	#13133286	C₂²	1	2	yes	—
9	#13133279	—	2	2	no	—
10	C₈×#6157	C₂²	1	1	yes	characteristic
11	D₈×D₅₃	C₂²	1	2	yes	—
12	D₈₄₈	C₄	1	1	yes	characteristic, complemented
13	#13133158	C₄	1	2	yes	complemented
14	C₁₆×D₅₃	C₂²	1	1	yes	characteristic
15	D₁₆×C₅₃	C₄	1	1	yes	characteristic, complemented
16	C₂²×#6157	—	4	2	no	—
17	D₄×D₅₃	—	2	2	no	—
18	C₂²⋊#6157	—	4	2	no	—
19	C₄⋊#6157	—	2	2	no	—
20	C₄×#6157	D₄	1	1	yes	characteristic
21	D₅₃⋊C₈	D₄	1	1	yes	characteristic
22	D₈×C₅₃	C₂×C₄	1	2	yes	complemented
23	#11719922	—	2	2	no	—
24	D₄₂₄	C₂×C₄	1	2	yes	complemented
25	C₈×D₅₃	C₂³	1	1	yes	characteristic
26	#11719910	—	2	1	no	—
27	C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic, complemented
28	#11719908	C₂×C₄	1	1	yes	characteristic, complemented
29	C₂²×D₅₃	—	4	2	no	—
30	C₂×#6157	—	4	2	no	—
31	C₂×#6157	—	8	4	no	—
32	C₂×#6157	D₈	1	2	yes	—
33	D₄×C₅₃	—	2	2	no	—
34	C₂²⋊D₅₃	—	4	2	no	—
35	D₂₁₂	—	2	2	no	—
36	C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
37	#88324	—	2	1	no	—
38	C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
39	#88322	#41	1	1	yes	characteristic
40	C₂²×C₅₃	—	4	2	no	—
41	C₂×D₅₃	—	4	2	no	—
42	C₂×D₅₃	—	8	2	no	—
43	C₂×D₅₃	—	8	2	no	—
44	C₂×D₅₃	C₂×D₈	1	1	yes	characteristic
45	#6157	—	2	2	no	—
46	#6157	—	8	2	no	—
47	C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
48	Dic₅₃	#135	1	1	yes	characteristic
49	#2177	—	53	1	no	maximal, Sylow
50	C₂×C₅₃	—	8	2	no	—
51	C₂×C₅₃	C₄×D₈	1	1	yes	characteristic
52	D₅₃	—	8	2	no	—
53	D₅₃	#509	1	2	yes	—
54	C₂×D₁₆	—	53	1	no	—
55	C₄×D₈	—	53	2	no	—
56	#367	—	53	1	no	—
57	#357	—	53	2	no	—
58	#346	—	53	1	no	—
59	C₅₃	#2177	1	1	yes	characteristic, complemented, Sylow
60	C₂×D₈	—	53	2	no	—
61	C₄×D₄	—	106	2	no	—
62	D₁₆	—	53	2	no	—
63	D₁₆	—	53	1	no	—
64	D₁₆	#6157	1	1	yes	characteristic, complemented
65	C₂×C₁₆	—	53	1	no	—
66	C₂×C₁₆	—	53	1	no	—
67	#107	—	53	2	no	—
68	#102	—	106	2	no	—
69	C₄×C₈	—	53	1	no	—
70	C₂×D₄	—	106	2	no	—
71	C₂²×C₄	—	212	2	no	—
72	D₈	—	53	2	no	—
73	D₈	—	106	2	no	—
74	D₈	C₂×#6157	1	2	yes	complemented
75	C₂×C₈	—	53	1	no	—
76	C₂×C₈	—	53	1	no	—
77	C₄⋊C₄	—	106	2	no	—
78	C₂²⋊C₄	—	212	2	no	—
79	C₄²	—	53	1	no	—
80	C₁₆	—	53	1	no	—
81	C₁₆	—	106	1	no	—
82	C₁₆	C₂×#6157	1	1	yes	characteristic, complemented
83	C₂³	—	212	2	no	—
84	D₄	—	2	2	no	—
85	D₄	—	106	2	no	—
86	D₄	—	212	2	no	—
87	C₂×C₄	—	53	1	no	—
88	C₂×C₄	—	53	2	no	—
89	C₂×C₄	—	424	4	no	—
90	C₂×C₄	—	212	2	no	—
91	C₈	—	53	1	no	—
92	C₈	—	106	1	no	—
93	C₈	C₂²×#6157	1	1	yes	characteristic
94	C₂²	—	4	2	no	—
95	C₂²	—	53	1	no	—
96	C₂²	—	424	2	no	—
97	C₂²	—	424	2	no	—
98	C₂²	—	212	2	no	—
99	C₄	—	53	1	no	—
100	C₄	—	106	2	no	—
101	C₄	—	424	2	no	—
102	C₄	D₄×#6157	1	1	yes	characteristic
103	C₂	—	8	2	no	—
104	C₂	—	53	2	no	—
105	C₂	—	424	2	no	—
106	C₂	D₈×#6157	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

D₈×#6157

C₂

yes

maximal, complemented

#16811550

C₂

yes

maximal, characteristic, complemented

#16811535

C₂

yes

maximal, complemented

#16811510

C₂

yes

maximal, characteristic, complemented

D₁₆×D₅₃

C₂

yes

maximal, characteristic

D₄×#6157

—

D₅₃⋊C₁₆

C₂²

yes

characteristic

#13133286

C₂²

yes

—

#13133279

—

C₈×#6157

C₂²

yes

characteristic

D₈×D₅₃

C₂²

yes

—

D₈₄₈

C₄

yes

characteristic, complemented

#13133158

C₄

yes

complemented

C₁₆×D₅₃

C₂²

yes

characteristic

D₁₆×C₅₃

C₄

yes

characteristic, complemented

C₂²×#6157

—

D₄×D₅₃

—

C₂²⋊#6157

—

C₄⋊#6157

—

C₄×#6157

D₄

yes

characteristic

D₅₃⋊C₈

D₄

yes

characteristic

D₈×C₅₃

C₂×C₄

yes

complemented

#11719922

—

D₄₂₄

C₂×C₄

yes

complemented

C₈×D₅₃

C₂³

yes

characteristic

#11719910

—

C₁₆×C₅₃

C₂×C₄

yes

characteristic, complemented

#11719908

C₂×C₄

yes

characteristic, complemented

C₂²×D₅₃

—

C₂×#6157

—

C₂×#6157

—

C₂×#6157

D₈

yes

—

D₄×C₅₃

—

C₂²⋊D₅₃

—

D₂₁₂

—

C₄×D₅₃

C₂×D₄

yes

characteristic

#88324

—

C₈×C₅₃

C₂²×C₄

yes

yes

C₂²×C₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

C₂×D₈

yes

characteristic

#6157

—

#6157

—

C₄×C₅₃

C₄×D₄

yes

Dic₅₃

yes

—

C₂×C₅₃

—

C₂×C₅₃

C₄×D₈

yes

characteristic

D₅₃

—

D₅₃

#509

yes

—

C₂×D₁₆

—

C₄×D₈

—

#367

—

#357

—

#346

—

C₅₃

#2177

yes

characteristic, complemented, Sylow

C₂×D₈

—

C₄×D₄

—

106

—

D₁₆

—

D₁₆

—

D₁₆

#6157

yes

characteristic, complemented

C₂×C₁₆

—

C₂×C₁₆

—

#107

—

#102

—

106

—

C₄×C₈

—

C₂×D₄

—

106

—

C₂²×C₄

—

212

—

D₈

—

D₈

—

106

—

D₈

C₂×#6157

yes

complemented

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

106

—

C₂²⋊C₄

—

212

—

C₄²

—

C₁₆

—

C₁₆

—

106

—

C₁₆

C₂×#6157

yes

characteristic, complemented

C₂³

—

212

—

D₄

—

D₄

—

106

—

D₄

—

212

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

424

—

C₂×C₄

—

212

—

C₈

—

C₈

—

106

—

C₈

C₂²×#6157

yes

characteristic

C₂²

—

C₂²

—

C₂²

—

424

—

C₂²

—

424

—

C₂²

—

212

—

C₄

—

100

C₄

—

106

—

101

C₄

—

424

—

102

C₄

D₄×#6157

yes

characteristic

103

C₂

—

104

C₂

—

105

C₂

—

424

—

106

C₂

D₈×#6157

yes

characteristic

	subgroup	G/H
center	C₂	D₈×#6157
commutator	C₈×C₅₃	C₂²×C₄
Frattini	C₈	C₂²×#6157
Fitting	D₁₆×C₅₃	C₄
socle	C₂×C₅₃	C₄×D₈
hypercenter	D₁₆	#6157
hypocenter	C₅₃	#2177
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ D₁₆

subgroup

G/H

center

C₂

D₈×#6157

commutator

C₈×C₅₃

C₂²×C₄

Frattini

C₈

C₂²×#6157

Fitting

D₁₆×C₅₃

C₄

socle

C₂×C₅₃

C₄×D₈

D₁₆

C₅₃

G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ D₁₆

N	H ≅ G/N	type
1. D₈×#6157	C₂	semidirect
2. #16811550	C₂	semidirect
3. #16811535	C₂	semidirect
4. #16811510	C₂	semidirect
12. D₈₄₈	C₄	semidirect
13. #13133158	C₄	semidirect
15. D₁₆×C₅₃	C₄	semidirect
22. D₈×C₅₃	C₂×C₄	semidirect
24. D₄₂₄	C₂×C₄	semidirect
27. C₁₆×C₅₃	C₂×C₄	semidirect
28. #11719908	C₂×C₄	semidirect
59. C₅₃	#2177	semidirect
64. D₁₆	#6157	semidirect
74. D₈	C₂×#6157	semidirect
82. C₁₆	C₂×#6157	semidirect

H ≅ G/N

type

1. D₈×#6157

C₂

semidirect

2. #16811550

C₂

semidirect

3. #16811535

C₂

semidirect

4. #16811510

C₂

semidirect

12. D₈₄₈

C₄

semidirect

13. #13133158

C₄

semidirect

15. D₁₆×C₅₃

C₄

semidirect

22. D₈×C₅₃

C₂×C₄

semidirect

24. D₄₂₄

C₂×C₄

semidirect

27. C₁₆×C₅₃

C₂×C₄

semidirect

28. #11719908

C₂×C₄

semidirect

59. C₅₃

#2177

semidirect

64. D₁₆

#6157

semidirect

74. D₈

C₂×#6157

semidirect

82. C₁₆

C₂×#6157

semidirect

Subgroups — #20291113

Conjugacy classes of subgroups

Complements