Group #20291065 — subgroups

Subgroups — C₂×#16811550 #20291065

Order 6784.

168 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 74 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×#13133286	C₂	1	2	yes	maximal
2	#16811550	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₁₆×D₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊#6157	—	2	2	no	—
5	#13133286	—	2	2	no	—
6	#13133286	C₂²	1	4	yes	—
7	C₂×C₈×D₅₃	C₂²	1	1	yes	characteristic
8	C₂×#11719908	C₄	1	1	yes	characteristic, complemented
9	C₁₆×D₅₃	C₄	1	2	yes	complemented
10	C₁₆×D₅₃	C₂²	1	2	yes	—
11	C₂×C₁₆×C₅₃	C₄	1	1	yes	characteristic, complemented
12	C₂²×#6157	—	4	2	no	—
13	C₂×C₄×D₅₃	D₄	1	1	yes	characteristic
14	C₄⋊#6157	—	2	4	no	—
15	C₄⋊#6157	—	4	2	no	—
16	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
17	C₂×#88322	Q₈	1	1	yes	characteristic
18	C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
19	C₈×D₅₃	D₄	1	1	yes	characteristic
20	C₈×D₅₃	Q₈	1	1	yes	characteristic
21	C₈×D₅₃	C₂³	1	1	yes	characteristic
22	C₁₆×C₅₃	C₂×C₄	1	2	yes	complemented
23	#11719908	C₂×C₄	1	2	yes	complemented
24	C₂²×D₅₃	D₈	1	1	yes	characteristic
25	C₂×#6157	—	4	4	no	—
26	C₂×#6157	—	8	4	no	—
27	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
28	C₂×Dic₅₃	Q₁₆	1	1	yes	characteristic
29	C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
30	C₄×D₅₃	D₈	1	1	yes	characteristic
31	C₄×D₅₃	Q₁₆	1	1	yes	characteristic
32	C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
33	C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
34	C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
35	#88322	C₄⋊C₄	1	1	yes	characteristic
36	#88322	C₂×Q₈	1	1	yes	characteristic
37	C₂²×C₅₃	#107	1	1	yes	characteristic
38	C₂×D₅₃	#107	1	1	yes	characteristic
39	C₂×D₅₃	SD₃₂	1	4	yes	—
40	C₂×D₅₃	C₂×D₈	1	1	yes	characteristic
41	#6157	—	8	4	no	—
42	C₄×C₅₃	#107	1	1	yes	characteristic
43	C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
44	Dic₅₃	#107	1	1	yes	characteristic
45	Dic₅₃	C₂×Q₁₆	1	1	yes	characteristic
46	C₂×#367	—	53	1	no	maximal, Sylow
47	C₂×C₅₃	#367	1	2	yes	complemented
48	C₂×C₅₃	C₂×#107	1	1	yes	characteristic
49	D₅₃	#367	1	2	yes	complemented
50	D₅₃	C₂×SD₃₂	1	2	yes	—
51	C₂²×C₁₆	—	53	1	no	—
52	C₂×#107	—	53	2	no	—
53	#367	—	53	4	no	—
54	C₅₃	C₂×#367	1	1	yes	characteristic, complemented, Sylow
55	C₂²×C₈	—	53	1	no	—
56	C₂×C₄⋊C₄	—	106	2	no	—
57	C₂×C₁₆	—	53	1	no	—
58	C₂×C₁₆	—	53	2	no	—
59	C₂×C₁₆	—	53	2	no	—
60	C₂×C₁₆	#6157	1	1	yes	characteristic, complemented
61	#107	—	53	4	no	—
62	#107	—	106	2	no	—
63	C₂²×C₄	—	53	1	no	—
64	C₂²×C₄	—	212	2	no	—
65	C₂×C₈	—	53	1	no	—
66	C₂×C₈	—	53	1	no	—
67	C₂×C₈	—	53	1	no	—
68	C₂×C₈	—	53	1	no	—
69	C₂×C₈	—	53	1	no	—
70	C₂×C₈	C₂×#6157	1	1	yes	characteristic
71	C₄⋊C₄	—	106	4	no	—
72	C₄⋊C₄	—	212	2	no	—
73	C₁₆	—	53	2	no	—
74	C₁₆	C₂×#6157	1	2	yes	complemented
75	C₂³	—	53	1	no	—
76	C₂×C₄	—	53	1	no	—
77	C₂×C₄	—	53	1	no	—
78	C₂×C₄	—	53	1	no	—
79	C₂×C₄	—	53	1	no	—
80	C₂×C₄	—	53	1	no	—
81	C₂×C₄	—	424	4	no	—
82	C₂×C₄	—	212	4	no	—
83	C₂×C₄	C₄⋊#6157	1	1	yes	characteristic
84	C₈	—	53	1	no	—
85	C₈	—	53	1	no	—
86	C₈	C₄⋊#6157	1	1	yes	characteristic
87	C₈	C₂²×#6157	1	1	yes	characteristic
88	C₂²	—	53	4	no	—
89	C₂²	—	53	1	no	—
90	C₂²	—	53	1	no	—
91	C₂²	#13133286	1	1	yes	characteristic
92	C₄	—	53	1	no	—
93	C₄	—	53	1	no	—
94	C₄	—	424	4	no	—
95	C₄	#13133286	1	1	yes	characteristic
96	C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
97	C₂	—	53	2	no	—
98	C₂	—	53	2	no	—
99	C₂	#16811550	1	2	yes	direct product factor
100	C₂	C₂×#13133286	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133286

C₂

yes

maximal

#16811550

C₂

yes

maximal, direct product factor

C₂×C₁₆×D₅₃

C₂

yes

maximal, characteristic

C₂×C₄⋊#6157

—

#13133286

—

#13133286

C₂²

yes

—

C₂×C₈×D₅₃

C₂²

yes

characteristic

C₂×#11719908

C₄

yes

characteristic, complemented

C₁₆×D₅₃

C₄

yes

complemented

C₁₆×D₅₃

C₂²

yes

—

C₂×C₁₆×C₅₃

C₄

yes

characteristic, complemented

C₂²×#6157

—

C₂×C₄×D₅₃

D₄

yes

characteristic

C₄⋊#6157

—

C₄⋊#6157

—

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×#88322

Q₈

yes

characteristic

C₈×D₅₃

C₂×C₄

yes

characteristic

C₈×D₅₃

D₄

yes

characteristic

C₈×D₅₃

Q₈

yes

characteristic

C₈×D₅₃

C₂³

yes

characteristic

C₁₆×C₅₃

C₂×C₄

yes

complemented

#11719908

C₂×C₄

yes

complemented

C₂²×D₅₃

D₈

yes

characteristic

C₂×#6157

—

C₂×#6157

—

C₂×C₄×C₅₃

C₄⋊C₄

yes

characteristic

C₂×Dic₅₃

Q₁₆

yes

characteristic

C₄×D₅₃

C₄⋊C₄

yes

characteristic

C₄×D₅₃

D₈

yes

characteristic

C₄×D₅₃

Q₁₆

yes

characteristic

C₄×D₅₃

C₂×D₄

yes

characteristic

C₈×C₅₃

C₄⋊C₄

yes

characteristic

C₈×C₅₃

C₂²×C₄

yes

characteristic

#88322

C₄⋊C₄

yes

characteristic

#88322

C₂×Q₈

yes

C₂²×C₅₃

yes

C₂×D₅₃

yes

C₂×D₅₃

SD₃₂

yes

—

C₂×D₅₃

C₂×D₈

yes

characteristic

#6157

—

C₄×C₅₃

#107

yes

characteristic

C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

Dic₅₃

#107

yes

characteristic

Dic₅₃

C₂×Q₁₆

yes

—

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

yes

D₅₃

C₂×SD₃₂

yes

—

C₂²×C₁₆

—

C₂×#107

—

#367

—

C₅₃

C₂×#367

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂×C₄⋊C₄

—

106

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

#6157

yes

characteristic, complemented

#107

—

#107

—

106

—

C₂²×C₄

—

C₂²×C₄

—

212

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×#6157

yes

characteristic

C₄⋊C₄

—

106

—

C₄⋊C₄

—

212

—

C₁₆

—

C₁₆

C₂×#6157

yes

complemented

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

424

—

C₂×C₄

—

212

—

C₂×C₄

C₄⋊#6157

yes

characteristic

C₈

—

C₈

—

C₈

C₄⋊#6157

yes

characteristic

C₈

C₂²×#6157

yes

characteristic

C₂²

—

C₂²

—

C₂²

—

C₂²

#13133286

yes

characteristic

C₄

—

C₄

—

C₄

—

424

—

C₄

#13133286

yes

characteristic

C₄

C₂×C₄⋊#6157

yes

characteristic

C₂

—

C₂

—

C₂

#16811550

yes

direct product factor

100

C₂

C₂×#13133286

yes

characteristic

	subgroup	G/H
center	C₂²	#13133286
commutator	C₈×C₅₃	C₂²×C₄
Frattini	C₈	C₂²×#6157
Fitting	C₂×C₁₆×C₅₃	C₄
socle	C₂²×C₅₃	#107
hypercenter	C₂×C₁₆	#6157
hypocenter	C₅₃	C₂×#367
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆

subgroup

G/H

center

C₂²

#13133286

commutator

C₈×C₅₃

C₂²×C₄

Frattini

C₈

C₂²×#6157

Fitting

C₂×C₁₆×C₅₃

C₄

C₂²×C₅₃

C₂×C₁₆

C₅₃

G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆

N	H ≅ G/N	type
2. #16811550	C₂	direct (N × H)
8. C₂×#11719908	C₄	semidirect
9. C₁₆×D₅₃	C₄	semidirect
11. C₂×C₁₆×C₅₃	C₄	semidirect
22. C₁₆×C₅₃	C₂×C₄	semidirect
23. #11719908	C₂×C₄	semidirect
47. C₂×C₅₃	#367	semidirect
49. D₅₃	#367	semidirect
54. C₅₃	C₂×#367	semidirect
60. C₂×C₁₆	#6157	semidirect
74. C₁₆	C₂×#6157	semidirect
99. C₂	#16811550	direct (N × H)

H ≅ G/N

type

2. #16811550

C₂

direct (N × H)

8. C₂×#11719908

C₄

semidirect

9. C₁₆×D₅₃

C₄

semidirect

11. C₂×C₁₆×C₅₃

C₄

semidirect

22. C₁₆×C₅₃

C₂×C₄

semidirect

23. #11719908

C₂×C₄

semidirect

47. C₂×C₅₃

#367

semidirect

49. D₅₃

#367

semidirect

54. C₅₃

C₂×#367

semidirect

60. C₂×C₁₆

#6157

semidirect

74. C₁₆

C₂×#6157

semidirect

99. C₂

#16811550

direct (N × H)

Subgroups — C₂×#16811550 #20291065

Conjugacy classes of subgroups

Complements

Subgroups — C2×#16811550 #20291065

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16811550 #20291065