Subgroups — C₃×C₇²×Dic₅₅ #44568532

Order 32340.

240 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×C₇²×C₁₁	C₂	1	1	yes	maximal, characteristic
2	C₇²×Dic₅₅	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
3	C₃×C₅×C₇²×C₁₁	C₄	1	1	yes	characteristic, complemented, Hall
4	C₃×C₇²×Dic₁₁	—	5	1	no	maximal, Hall
5	C₂×C₅×C₇²×C₁₁	C₂×C₃	1	1	yes	characteristic
6	C₃×C₇×Dic₅₅	C₇	1	8	yes	maximal, direct product factor
7	C₂×C₃×C₇²×C₁₁	D₅	1	1	yes	characteristic
8	C₃×C₇²×Dic₅	—	11	1	no	maximal, Hall
9	C₅×C₇²×C₁₁	C₃×C₄	1	1	yes	characteristic, complemented, Hall
10	C₂×C₃×C₅×C₇×C₁₁	C₂×C₇	1	8	yes	—
11	C₇²×Dic₁₁	—	5	1	no	Hall
12	C₃×C₇²×C₁₁	Dic₅	1	1	yes	characteristic, complemented, Hall
13	C₇×Dic₅₅	C₃×C₇	1	8	yes	direct product factor
14	C₂×C₃×C₅×C₇²	D₁₁	1	1	yes	characteristic
15	C₃×C₅×C₇×C₁₁	C₄×C₇	1	8	yes	complemented
16	C₂×C₇²×C₁₁	C₃×D₅	1	1	yes	characteristic
17	C₇²×Dic₅	—	11	1	no	Hall
18	C₃×C₇×Dic₁₁	—	5	8	no	—
19	C₂×C₅×C₇×C₁₁	C₂×C₃×C₇	1	8	yes	—
20	C₃×C₅×C₇²	Dic₁₁	1	1	yes	characteristic, complemented, Hall
21	C₃×Dic₅₅	C₇²	1	1	yes	characteristic, direct product factor, Hall
22	C₃×C₄×C₇²	—	55	1	no	Hall
23	C₇²×C₁₁	C₃×Dic₅	1	1	yes	characteristic, complemented, Hall
24	C₂×C₅×C₇²	C₃×D₁₁	1	1	yes	characteristic
25	C₂×C₃×C₇×C₁₁	C₇×D₅	1	8	yes	—
26	C₃×C₇×Dic₅	—	11	8	no	—
27	C₅×C₇×C₁₁	C₃×C₄×C₇	1	8	yes	complemented
28	C₂×C₃×C₅×C₁₁	C₂×C₇²	1	1	yes	characteristic
29	C₇×Dic₁₁	—	5	8	no	—
30	C₂×C₃×C₇²	D₅₅	1	1	yes	characteristic
31	C₅×C₇²	C₃×Dic₁₁	1	1	yes	characteristic, complemented, Hall
32	C₃×C₇×C₁₁	C₇×Dic₅	1	8	yes	complemented
33	Dic₅₅	C₃×C₇²	1	1	yes	characteristic, direct product factor, Hall
34	C₂×C₃×C₅×C₇	C₇×D₁₁	1	8	yes	—
35	C₄×C₇²	—	55	1	no	Hall
36	C₃×C₅×C₁₁	C₄×C₇²	1	1	yes	characteristic, complemented, Hall
37	C₂×C₇×C₁₁	C₃×C₇×D₅	1	8	yes	—
38	C₃×C₇²	Dic₅₅	1	1	yes	characteristic, direct product factor, Hall
39	C₇×Dic₅	—	11	8	no	—
40	C₃×Dic₁₁	—	5	1	no	Hall
41	C₂×C₅×C₁₁	C₂×C₃×C₇²	1	1	yes	characteristic
42	C₃×C₅×C₇	C₇×Dic₁₁	1	8	yes	complemented
43	C₂×C₇²	C₃×D₅₅	1	1	yes	characteristic
44	C₃×C₄×C₇	—	55	8	no	—
45	C₇×C₁₁	C₃×C₇×Dic₅	1	8	yes	complemented
46	C₂×C₅×C₇	C₃×C₇×D₁₁	1	8	yes	—
47	C₂×C₃×C₁₁	C₇²×D₅	1	1	yes	characteristic
48	C₃×Dic₅	—	11	1	no	Hall
49	C₅×C₁₁	C₃×C₄×C₇²	1	1	yes	characteristic, complemented, Hall
50	C₇²	C₃×Dic₅₅	1	1	yes	characteristic, direct product factor, Sylow
51	Dic₁₁	—	5	1	no	Hall
52	C₂×C₃×C₇	C₇×D₅₅	1	8	yes	—
53	C₅×C₇	C₃×C₇×Dic₁₁	1	8	yes	complemented
54	C₃×C₁₁	C₇²×Dic₅	1	1	yes	characteristic, complemented, Hall
55	C₂×C₃×C₅	C₇²×D₁₁	1	1	yes	characteristic
56	C₄×C₇	—	55	8	no	—
57	C₂×C₁₁	C₃×C₇²×D₅	1	1	yes	characteristic
58	C₃×C₇	C₇×Dic₅₅	1	8	yes	direct product factor
59	Dic₅	—	11	1	no	Hall
60	C₃×C₅	C₇²×Dic₁₁	1	1	yes	characteristic, complemented, Hall
61	C₂×C₇	C₃×C₇×D₅₅	1	8	yes	—
62	C₃×C₄	—	55	1	no	Hall
63	C₁₁	C₃×C₇²×Dic₅	1	1	yes	characteristic, complemented, Sylow
64	C₂×C₅	C₃×C₇²×D₁₁	1	1	yes	characteristic
65	C₇	C₃×C₇×Dic₅₅	1	8	yes	direct product factor
66	C₂×C₃	C₇²×D₅₅	1	1	yes	characteristic
67	C₅	C₃×C₇²×Dic₁₁	1	1	yes	characteristic, complemented, Sylow
68	C₄	—	55	1	no	Sylow
69	C₃	C₇²×Dic₅₅	1	1	yes	characteristic, direct product factor, Sylow
70	C₂	C₃×C₇²×D₅₅	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₃×C₇²	D₅₅
commutator, hypocenter	C₅×C₁₁	C₃×C₄×C₇²
Frattini	C₂	C₃×C₇²×D₅₅
Fitting, socle, prime radical	C₂×C₃×C₅×C₇²×C₁₁	C₂
cosocle	C₂×C₅×C₁₁	C₂×C₃×C₇²
LCS	G ▷ C₅×C₁₁
UCS	C₁ ◁ C₂×C₃×C₇²

Complements

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

N	H ≅ G/N	type
2. C₇²×Dic₅₅	C₃	direct (N × H)
3. C₃×C₅×C₇²×C₁₁	C₄	semidirect
6. C₃×C₇×Dic₅₅	C₇	direct (N × H)
9. C₅×C₇²×C₁₁	C₃×C₄	semidirect
12. C₃×C₇²×C₁₁	Dic₅	semidirect
13. C₇×Dic₅₅	C₃×C₇	direct (N × H)
15. C₃×C₅×C₇×C₁₁	C₄×C₇	semidirect
20. C₃×C₅×C₇²	Dic₁₁	semidirect
21. C₃×Dic₅₅	C₇²	direct (N × H)
23. C₇²×C₁₁	C₃×Dic₅	semidirect
27. C₅×C₇×C₁₁	C₃×C₄×C₇	semidirect
31. C₅×C₇²	C₃×Dic₁₁	semidirect
32. C₃×C₇×C₁₁	C₇×Dic₅	semidirect
33. Dic₅₅	C₃×C₇²	direct (N × H)
36. C₃×C₅×C₁₁	C₄×C₇²	semidirect
38. C₃×C₇²	Dic₅₅	direct (N × H)
42. C₃×C₅×C₇	C₇×Dic₁₁	semidirect
45. C₇×C₁₁	C₃×C₇×Dic₅	semidirect
49. C₅×C₁₁	C₃×C₄×C₇²	semidirect
50. C₇²	C₃×Dic₅₅	direct (N × H)
53. C₅×C₇	C₃×C₇×Dic₁₁	semidirect
54. C₃×C₁₁	C₇²×Dic₅	semidirect
58. C₃×C₇	C₇×Dic₅₅	direct (N × H)
60. C₃×C₅	C₇²×Dic₁₁	semidirect
63. C₁₁	C₃×C₇²×Dic₅	semidirect
65. C₇	C₃×C₇×Dic₅₅	direct (N × H)
67. C₅	C₃×C₇²×Dic₁₁	semidirect
69. C₃	C₇²×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
C₂×C₃×C₅×C₇²×C₁₁	2
C₇²×Dic₅₅	3
C₃×C₇²×Dic₁₁	5
C₃×C₇×Dic₅₅	7
C₃×C₇²×Dic₅	11

Subgroups — C3×C72×Dic55 #44568532

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₇²×Dic₅₅ #44568532