Subgroups — C₂×C₇×C₉×D₅₅ #28357461

Order 13860.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×C₉×D₅₅	C₂	1	2	yes	maximal, direct product factor
2	C₂×C₅×C₇×C₉×C₁₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×C₇×D₅₅	C₃	1	1	yes	maximal, characteristic
4	C₅×C₇×C₉×C₁₁	C₂²	1	1	yes	characteristic, complemented, Hall
5	C₂×C₇×C₉×D₁₁	—	5	1	no	maximal, Hall
6	C₃×C₇×D₅₅	C₂×C₃	1	2	yes	—
7	C₂×C₃×C₅×C₇×C₁₁	C₂×C₃	1	1	yes	characteristic
8	C₂×C₉×D₅₅	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂×C₇×D₅₅	C₉	1	1	yes	characteristic, direct product factor, Hall
10	C₂×C₇×C₉×C₁₁	D₅	1	1	yes	characteristic, complemented
11	C₇×C₉×D₁₁	—	5	2	no	—
12	C₂×C₇×C₉×D₅	—	11	1	no	maximal, Hall
13	C₃×C₅×C₇×C₁₁	C₂²×C₃	1	1	yes	characteristic
14	C₂×C₅×C₉×C₁₁	C₂×C₇	1	1	yes	characteristic, complemented
15	C₉×D₅₅	C₂×C₇	1	2	yes	direct product factor
16	C₂×C₃×C₇×D₁₁	—	5	1	no	—
17	C₂×C₅×C₇×C₁₁	C₂×C₉	1	1	yes	characteristic, complemented
18	C₇×D₅₅	C₂×C₉	1	2	yes	direct product factor
19	C₇×C₉×C₁₁	C₂×D₅	1	1	yes	characteristic, complemented, Hall
20	C₂×C₃×D₅₅	C₃×C₇	1	1	yes	characteristic
21	C₂×C₅×C₇×C₉	D₁₁	1	1	yes	characteristic, complemented
22	C₇×C₉×D₅	—	11	2	no	—
23	C₅×C₉×C₁₁	C₂²×C₇	1	1	yes	characteristic, complemented, Hall
24	C₂×C₃×C₇×C₁₁	C₃×D₅	1	1	yes	characteristic
25	C₃×C₇×D₁₁	—	5	2	no	—
26	C₂×C₃×C₇×D₅	—	11	1	no	—
27	C₂×C₉×D₁₁	—	5	1	no	Hall
28	C₅×C₇×C₁₁	C₂²×C₉	1	1	yes	characteristic, complemented, Hall
29	C₂×C₃×C₅×C₁₁	C₂×C₃×C₇	1	1	yes	characteristic
30	C₃×D₅₅	C₂×C₃×C₇	1	2	yes	—
31	C₅×C₇×C₉	C₂×D₁₁	1	1	yes	characteristic, complemented, Hall
32	C₂×C₇×D₁₁	—	5	1	no	Hall
33	C₂²×C₇×C₉	—	55	1	no	Hall
34	C₃×C₇×C₁₁	C₂×C₃×D₅	1	1	yes	characteristic
35	C₂×D₅₅	C₇×C₉	1	1	yes	characteristic, direct product factor, Hall
36	C₂×C₃×C₅×C₇	C₃×D₁₁	1	1	yes	characteristic
37	C₃×C₇×D₅	—	11	2	no	—
38	C₂×C₉×C₁₁	C₇×D₅	1	1	yes	characteristic, complemented
39	C₉×D₁₁	—	5	2	no	—
40	C₂×C₉×D₅	—	11	1	no	Hall
41	C₃×C₅×C₁₁	C₂²×C₃×C₇	1	1	yes	characteristic
42	C₂×C₇×C₁₁	C₉×D₅	1	1	yes	characteristic, complemented
43	C₇×D₁₁	—	5	2	no	—
44	C₂×C₇×D₅	—	11	1	no	Hall
45	C₂×C₃×D₁₁	—	5	1	no	—
46	C₂×C₇×C₉	—	55	2	no	—
47	C₂×C₇×C₉	D₅₅	1	1	yes	characteristic, direct product factor
48	C₂×C₅×C₁₁	C₂×C₇×C₉	1	1	yes	characteristic, complemented
49	D₅₅	C₂×C₇×C₉	1	2	yes	direct product factor
50	C₃×C₅×C₇	C₂×C₃×D₁₁	1	1	yes	characteristic
51	C₉×C₁₁	C₂×C₇×D₅	1	1	yes	characteristic, complemented, Hall
52	C₂×C₅×C₉	C₇×D₁₁	1	1	yes	characteristic, complemented
53	C₉×D₅	—	11	2	no	—
54	C₂²×C₃×C₇	—	55	1	no	—
55	C₇×C₁₁	C₂×C₉×D₅	1	1	yes	characteristic, complemented, Hall
56	C₂×C₅×C₇	C₉×D₁₁	1	1	yes	characteristic, complemented
57	C₇×D₅	—	11	2	no	—
58	C₂×C₃×C₁₁	C₃×C₇×D₅	1	1	yes	characteristic
59	C₃×D₁₁	—	5	2	no	—
60	C₇×C₉	C₂×D₅₅	1	1	yes	characteristic, direct product factor, Hall
61	C₂×C₃×D₅	—	11	1	no	—
62	C₅×C₁₁	C₂²×C₇×C₉	1	1	yes	characteristic, complemented, Hall
63	C₅×C₉	C₂×C₇×D₁₁	1	1	yes	characteristic, complemented, Hall
64	C₂×D₁₁	—	5	1	no	Hall
65	C₂×C₃×C₇	—	55	2	no	—
66	C₂×C₃×C₇	C₃×D₅₅	1	1	yes	characteristic
67	C₂²×C₉	—	55	1	no	Hall
68	C₅×C₇	C₂×C₉×D₁₁	1	1	yes	characteristic, complemented, Hall
69	C₃×C₁₁	C₂×C₃×C₇×D₅	1	1	yes	characteristic
70	C₂×C₃×C₅	C₃×C₇×D₁₁	1	1	yes	characteristic
71	C₃×D₅	—	11	2	no	—
72	C₂²×C₇	—	55	1	no	Hall
73	C₂×C₁₁	C₇×C₉×D₅	1	1	yes	characteristic, complemented
74	D₁₁	—	5	2	no	—
75	C₃×C₇	C₂×C₃×D₅₅	1	1	yes	characteristic
76	C₂×D₅	—	11	1	no	Hall
77	C₂×C₉	—	55	2	no	—
78	C₂×C₉	C₇×D₅₅	1	1	yes	characteristic, direct product factor
79	C₃×C₅	C₂×C₃×C₇×D₁₁	1	1	yes	characteristic
80	C₂×C₇	—	55	2	no	—
81	C₂×C₇	C₉×D₅₅	1	1	yes	characteristic, direct product factor
82	C₂²×C₃	—	55	1	no	—
83	C₁₁	C₂×C₇×C₉×D₅	1	1	yes	characteristic, complemented, Sylow
84	C₂×C₅	C₇×C₉×D₁₁	1	1	yes	characteristic, complemented
85	D₅	—	11	2	no	—
86	C₉	C₂×C₇×D₅₅	1	1	yes	characteristic, direct product factor, Sylow
87	C₇	C₂×C₉×D₅₅	1	1	yes	characteristic, direct product factor, Sylow
88	C₂×C₃	—	55	2	no	—
89	C₂×C₃	C₃×C₇×D₅₅	1	1	yes	characteristic
90	C₅	C₂×C₇×C₉×D₁₁	1	1	yes	characteristic, complemented, Sylow
91	C₂²	—	55	1	no	Sylow
92	C₃	C₂×C₃×C₇×D₅₅	1	1	yes	characteristic
93	C₂	—	55	2	no	—
94	C₂	C₇×C₉×D₅₅	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₇×C₉	D₅₅
commutator, hypocenter	C₅×C₁₁	C₂²×C₇×C₉
Frattini	C₃	C₂×C₃×C₇×D₅₅
Fitting	C₂×C₅×C₇×C₉×C₁₁	C₂
socle	C₂×C₃×C₅×C₇×C₁₁	C₂×C₃
prime radical	C₂×C₃×C₇×D₅₅	C₃
cosocle	C₃×C₅×C₁₁	C₂²×C₃×C₇
LCS	G ▷ C₅×C₁₁
UCS	C₁ ◁ C₂×C₇×C₉

Complements

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

N	H ≅ G/N	type
1. C₇×C₉×D₅₅	C₂	direct (N × H)
2. C₂×C₅×C₇×C₉×C₁₁	C₂	semidirect
4. C₅×C₇×C₉×C₁₁	C₂²	semidirect
8. C₂×C₉×D₅₅	C₇	direct (N × H)
9. C₂×C₇×D₅₅	C₉	direct (N × H)
10. C₂×C₇×C₉×C₁₁	D₅	semidirect
14. C₂×C₅×C₉×C₁₁	C₂×C₇	semidirect
15. C₉×D₅₅	C₂×C₇	direct (N × H)
17. C₂×C₅×C₇×C₁₁	C₂×C₉	semidirect
18. C₇×D₅₅	C₂×C₉	direct (N × H)
19. C₇×C₉×C₁₁	C₂×D₅	semidirect
21. C₂×C₅×C₇×C₉	D₁₁	semidirect
23. C₅×C₉×C₁₁	C₂²×C₇	semidirect
28. C₅×C₇×C₁₁	C₂²×C₉	semidirect
31. C₅×C₇×C₉	C₂×D₁₁	semidirect
35. C₂×D₅₅	C₇×C₉	direct (N × H)
38. C₂×C₉×C₁₁	C₇×D₅	semidirect
42. C₂×C₇×C₁₁	C₉×D₅	semidirect
47. C₂×C₇×C₉	D₅₅	direct (N × H)
48. C₂×C₅×C₁₁	C₂×C₇×C₉	semidirect
49. D₅₅	C₂×C₇×C₉	direct (N × H)
51. C₉×C₁₁	C₂×C₇×D₅	semidirect
52. C₂×C₅×C₉	C₇×D₁₁	semidirect
55. C₇×C₁₁	C₂×C₉×D₅	semidirect
56. C₂×C₅×C₇	C₉×D₁₁	semidirect
60. C₇×C₉	C₂×D₅₅	direct (N × H)
62. C₅×C₁₁	C₂²×C₇×C₉	semidirect
63. C₅×C₉	C₂×C₇×D₁₁	semidirect
68. C₅×C₇	C₂×C₉×D₁₁	semidirect
73. C₂×C₁₁	C₇×C₉×D₅	semidirect
78. C₂×C₉	C₇×D₅₅	direct (N × H)
81. C₂×C₇	C₉×D₅₅	direct (N × H)
83. C₁₁	C₂×C₇×C₉×D₅	semidirect
84. C₂×C₅	C₇×C₉×D₁₁	semidirect
86. C₉	C₂×C₇×D₅₅	direct (N × H)
87. C₇	C₂×C₉×D₅₅	direct (N × H)
90. C₅	C₂×C₇×C₉×D₁₁	semidirect
94. C₂	C₇×C₉×D₅₅	direct (N × H)

Maximal subgroups

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

subgroup	index
C₇×C₉×D₅₅	2
C₂×C₅×C₇×C₉×C₁₁	2
C₂×C₃×C₇×D₅₅	3
C₂×C₇×C₉×D₁₁	5
C₂×C₉×D₅₅	7
C₂×C₇×C₉×D₅	11

Subgroups — C2×C7×C9×D55 #28357461

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×C₉×D₅₅ #28357461