Subgroups — C₂×C₄×Dic₅₅ #13183221

Order 1760.

216 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×Dic₅₅	C₂	1	2	yes	maximal
3	C₄×Dic₅₅	C₂	1	4	yes	maximal, direct product factor
4	C₂³×C₅×C₁₁	C₂²	1	1	yes	characteristic
5	C₂×C₄×C₅×C₁₁	C₄	1	4	yes	complemented
6	C₂×C₄×C₅×C₁₁	C₂²	1	2	yes	—
7	C₂×Dic₅₅	C₄	1	8	yes	direct product factor
8	C₂×Dic₅₅	C₂²	1	4	yes	—
9	C₂×C₄×Dic₁₁	—	5	1	no	maximal, Hall
10	C₂²×C₅×C₁₁	C₂×C₄	1	4	yes	—
11	C₂²×C₅×C₁₁	C₂×C₄	1	2	yes	—
12	C₂²×C₅×C₁₁	C₂³	1	1	yes	characteristic
13	C₄×C₅×C₁₁	C₂×C₄	1	4	yes	complemented
14	Dic₅₅	C₂×C₄	1	8	yes	direct product factor
15	C₂²×C₄×C₁₁	D₅	1	1	yes	characteristic
16	C₂²×Dic₁₁	—	5	2	no	—
17	C₄×Dic₁₁	—	5	4	no	—
18	C₂×C₄×Dic₅	—	11	1	no	maximal, Hall
19	C₂×C₅×C₁₁	C₄²	1	4	yes	complemented
20	C₂×C₅×C₁₁	C₂²×C₄	1	2	yes	—
21	C₂×C₅×C₁₁	C₂²×C₄	1	1	yes	characteristic
22	C₂³×C₁₁	C₂×D₅	1	1	yes	characteristic
23	C₂×C₄×C₁₁	Dic₅	1	4	yes	complemented
24	C₂×C₄×C₁₁	C₂×D₅	1	2	yes	—
25	C₂×Dic₁₁	—	5	8	no	—
26	C₂×Dic₁₁	—	5	4	no	—
27	C₂²×C₄×C₅	D₁₁	1	1	yes	characteristic
28	C₂²×Dic₅	—	11	2	no	—
29	C₄×Dic₅	—	11	4	no	—
30	C₅×C₁₁	C₂×C₄²	1	1	yes	characteristic, complemented, Hall
31	C₂²×C₁₁	C₄×D₅	1	4	yes	—
32	C₂²×C₁₁	C₂×Dic₅	1	2	yes	—
33	C₂²×C₁₁	C₂²×D₅	1	1	yes	characteristic
34	C₄×C₁₁	C₂×Dic₅	1	4	yes	complemented
35	Dic₁₁	—	5	8	no	—
36	C₂³×C₅	C₂×D₁₁	1	1	yes	characteristic
37	C₂×C₄×C₅	Dic₁₁	1	4	yes	complemented
38	C₂×C₄×C₅	C₂×D₁₁	1	2	yes	—
39	C₂×Dic₅	—	11	4	no	—
40	C₂×Dic₅	—	11	8	no	—
41	C₂×C₄²	—	55	1	no	Sylow
42	C₂×C₁₁	C₄×Dic₅	1	4	yes	complemented
43	C₂×C₁₁	C₂×C₄×D₅	1	2	yes	—
44	C₂×C₁₁	C₂²×Dic₅	1	1	yes	characteristic
45	C₂²×C₅	C₄×D₁₁	1	4	yes	—
46	C₂²×C₅	C₂×Dic₁₁	1	2	yes	—
47	C₂²×C₅	C₂²×D₁₁	1	1	yes	characteristic
48	C₄×C₅	C₂×Dic₁₁	1	4	yes	complemented
49	Dic₅	—	11	8	no	—
50	C₂²×C₄	—	55	2	no	—
51	C₂²×C₄	D₅₅	1	1	yes	characteristic
52	C₄²	—	55	4	no	—
53	C₁₁	C₂×C₄×Dic₅	1	1	yes	characteristic, complemented, Sylow
54	C₂×C₅	C₄×Dic₁₁	1	4	yes	complemented
55	C₂×C₅	C₂×C₄×D₁₁	1	2	yes	—
56	C₂×C₅	C₂²×Dic₁₁	1	1	yes	characteristic
57	C₂³	C₂×D₅₅	1	1	yes	characteristic
58	C₂×C₄	—	55	8	no	—
59	C₂×C₄	—	55	4	no	—
60	C₂×C₄	Dic₅₅	1	4	yes	direct product factor
61	C₂×C₄	C₂×D₅₅	1	2	yes	—
62	C₅	C₂×C₄×Dic₁₁	1	1	yes	characteristic, complemented, Sylow
63	C₂²	C₄×D₅₅	1	4	yes	—
64	C₂²	C₂×Dic₅₅	1	2	yes	—
65	C₂²	C₂²×D₅₅	1	1	yes	characteristic
66	C₄	—	55	8	no	—
67	C₄	C₂×Dic₅₅	1	4	yes	direct product factor
68	C₂	C₄×Dic₅₅	1	4	yes	direct product factor
69	C₂	C₂×C₄×D₅₅	1	2	yes	—
70	C₂	C₂²×Dic₅₅	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂²×C₄	D₅₅
commutator, hypocenter	C₅×C₁₁	C₂×C₄²
Frattini	C₂²	C₂²×D₅₅
Fitting	C₂²×C₄×C₅×C₁₁	C₂
socle, prime radical	C₂³×C₅×C₁₁	C₂²
cosocle	C₂²×C₅×C₁₁	C₂³
LCS	G ▷ C₅×C₁₁
UCS	C₁ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
3. C₄×Dic₅₅	C₂	direct (N × H)
5. C₂×C₄×C₅×C₁₁	C₄	semidirect
7. C₂×Dic₅₅	C₄	direct (N × H)
13. C₄×C₅×C₁₁	C₂×C₄	semidirect
14. Dic₅₅	C₂×C₄	direct (N × H)
19. C₂×C₅×C₁₁	C₄²	semidirect
23. C₂×C₄×C₁₁	Dic₅	semidirect
30. C₅×C₁₁	C₂×C₄²	semidirect
34. C₄×C₁₁	C₂×Dic₅	semidirect
37. C₂×C₄×C₅	Dic₁₁	semidirect
42. C₂×C₁₁	C₄×Dic₅	semidirect
48. C₄×C₅	C₂×Dic₁₁	semidirect
53. C₁₁	C₂×C₄×Dic₅	semidirect
54. C₂×C₅	C₄×Dic₁₁	semidirect
60. C₂×C₄	Dic₅₅	direct (N × H)
62. C₅	C₂×C₄×Dic₁₁	semidirect
67. C₄	C₂×Dic₅₅	direct (N × H)
68. C₂	C₄×Dic₅₅	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×Dic₅₅	2
C₄×Dic₅₅	2
C₂×C₄×Dic₁₁	5
C₂×C₄×Dic₅	11

Subgroups — C2×C4×Dic55 #13183221

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×Dic₅₅ #13183221