Subgroups — C₂²×C₅×Dic₂₂ #13183410

Order 1760.

312 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₅×C₁₁	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₅×Dic₁₁	C₂	1	2	yes	maximal
3	C₂×C₅×Dic₂₂	C₂	1	12	yes	maximal, direct product factor
4	C₂³×C₅×C₁₁	C₂²	1	1	yes	characteristic
5	C₂×C₄×C₅×C₁₁	C₂²	1	6	yes	—
6	C₂×C₅×Dic₁₁	C₂²	1	12	yes	—
7	C₅×Dic₂₂	C₂²	1	16	yes	direct product factor
8	C₂²×Dic₂₂	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂²×C₅×C₁₁	Q₈	1	4	yes	complemented
10	C₂²×C₅×C₁₁	C₂³	1	3	yes	—
11	C₄×C₅×C₁₁	C₂³	1	4	yes	—
12	C₅×Dic₁₁	C₂³	1	8	yes	—
13	C₂²×C₄×C₁₁	C₂×C₅	1	1	yes	characteristic
14	C₂²×Dic₁₁	C₂×C₅	1	2	yes	—
15	C₂×Dic₂₂	C₂×C₅	1	12	yes	direct product factor
16	C₂²×C₅×Q₈	—	11	1	no	maximal, Hall
17	C₂×C₅×C₁₁	C₂×Q₈	1	6	yes	complemented
18	C₂×C₅×C₁₁	C₂⁴	1	1	yes	characteristic
19	C₂³×C₁₁	C₂²×C₅	1	1	yes	characteristic
20	C₂×C₄×C₁₁	C₂²×C₅	1	6	yes	—
21	C₂×Dic₁₁	C₂²×C₅	1	12	yes	—
22	Dic₂₂	C₂²×C₅	1	16	yes	direct product factor
23	C₂×C₅×Q₈	—	11	12	no	—
24	C₂²×C₄×C₅	—	11	2	no	—
25	C₂²×C₄×C₅	D₁₁	1	1	yes	characteristic
26	C₅×C₁₁	C₂²×Q₈	1	1	yes	characteristic, complemented, Hall
27	C₂²×C₁₁	C₅×Q₈	1	4	yes	complemented
28	C₂²×C₁₁	C₂³×C₅	1	3	yes	—
29	C₄×C₁₁	C₂³×C₅	1	4	yes	—
30	Dic₁₁	C₂³×C₅	1	8	yes	—
31	C₂³×C₅	C₂×D₁₁	1	1	yes	characteristic
32	C₅×Q₈	—	11	16	no	—
33	C₂×C₄×C₅	—	11	12	no	—
34	C₂×C₄×C₅	C₂×D₁₁	1	6	yes	—
35	C₂²×Q₈	—	11	1	no	Sylow
36	C₂×C₁₁	C₂×C₅×Q₈	1	6	yes	complemented
37	C₂×C₁₁	C₂⁴×C₅	1	1	yes	characteristic
38	C₂²×C₅	Dic₂₂	1	4	yes	direct product factor
39	C₂²×C₅	C₂²×D₁₁	1	3	yes	—
40	C₄×C₅	—	11	8	no	—
41	C₄×C₅	C₂²×D₁₁	1	4	yes	—
42	C₂×Q₈	—	11	12	no	—
43	C₂²×C₄	—	11	2	no	—
44	C₂²×C₄	C₅×D₁₁	1	1	yes	characteristic
45	C₁₁	C₂²×C₅×Q₈	1	1	yes	characteristic, complemented, Sylow
46	C₂×C₅	C₂×Dic₂₂	1	6	yes	direct product factor
47	C₂×C₅	C₂³×D₁₁	1	1	yes	characteristic
48	C₂³	C₂×C₅×D₁₁	1	1	yes	characteristic
49	Q₈	—	11	16	no	—
50	C₂×C₄	—	11	12	no	—
51	C₂×C₄	C₂×C₅×D₁₁	1	6	yes	—
52	C₅	C₂²×Dic₂₂	1	1	yes	characteristic, direct product factor, Sylow
53	C₂²	C₅×Dic₂₂	1	4	yes	direct product factor
54	C₂²	C₂²×C₅×D₁₁	1	3	yes	—
55	C₄	—	11	8	no	—
56	C₄	C₂²×C₅×D₁₁	1	4	yes	—
57	C₂	C₂×C₅×Dic₂₂	1	6	yes	direct product factor
58	C₂	C₂³×C₅×D₁₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₅	C₂×D₁₁
commutator, cosocle	C₂×C₁₁	C₂⁴×C₅
Frattini	C₂	C₂³×C₅×D₁₁
Fitting	C₂²×C₄×C₅×C₁₁	C₂
socle, prime radical	C₂³×C₅×C₁₁	C₂²
hypercenter	C₂²×C₄×C₅	D₁₁
hypocenter	C₁₁	C₂²×C₅×Q₈
LCS	G ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₂²×C₄×C₅

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
3. C₂×C₅×Dic₂₂	C₂	direct (N × H)
7. C₅×Dic₂₂	C₂²	direct (N × H)
8. C₂²×Dic₂₂	C₅	direct (N × H)
9. C₂²×C₅×C₁₁	Q₈	semidirect
15. C₂×Dic₂₂	C₂×C₅	direct (N × H)
17. C₂×C₅×C₁₁	C₂×Q₈	semidirect
22. Dic₂₂	C₂²×C₅	direct (N × H)
26. C₅×C₁₁	C₂²×Q₈	semidirect
27. C₂²×C₁₁	C₅×Q₈	semidirect
36. C₂×C₁₁	C₂×C₅×Q₈	semidirect
38. C₂²×C₅	Dic₂₂	direct (N × H)
45. C₁₁	C₂²×C₅×Q₈	semidirect
46. C₂×C₅	C₂×Dic₂₂	direct (N × H)
52. C₅	C₂²×Dic₂₂	direct (N × H)
53. C₂²	C₅×Dic₂₂	direct (N × H)
57. C₂	C₂×C₅×Dic₂₂	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₄×C₅×C₁₁	2
C₂²×C₅×Dic₁₁	2
C₂×C₅×Dic₂₂	2
C₂²×Dic₂₂	5
C₂²×C₅×Q₈	11

Subgroups — C22×C5×Dic22 #13183410

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₅×Dic₂₂ #13183410