Subgroups — C₂²×C₄×C₇×D₅ #11765437

Order 1120.

472 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₇×D₅	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×C₅×C₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₇×Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₄×C₇×D₅	C₂	1	12	yes	maximal, direct product factor
5	C₂³×C₅×C₇	C₂²	1	1	yes	characteristic
6	C₂²×C₇×D₅	C₄	1	8	yes	direct product factor
7	C₂²×C₇×D₅	C₂²	1	6	yes	—
8	C₂×C₄×C₅×C₇	C₂²	1	6	yes	complemented
9	C₂×C₇×Dic₅	C₂²	1	6	yes	complemented
10	C₄×C₇×D₅	C₂²	1	16	yes	direct product factor
11	C₂³×C₄×C₇	—	5	1	no	maximal, Hall
12	C₂²×C₄×D₅	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
13	C₂²×C₅×C₇	C₂×C₄	1	4	yes	complemented
14	C₂²×C₅×C₇	C₂³	1	3	yes	—
15	C₂×C₇×D₅	C₂×C₄	1	24	yes	direct product factor
16	C₂×C₇×D₅	C₂³	1	4	yes	—
17	C₄×C₅×C₇	C₂³	1	4	yes	complemented
18	C₇×Dic₅	C₂³	1	4	yes	complemented
19	C₂⁴×C₇	—	5	1	no	—
20	C₂²×C₄×C₇	—	5	1	no	—
21	C₂²×C₄×C₇	—	5	12	no	—
22	C₂²×C₄×C₇	D₅	1	1	yes	characteristic, direct product factor
23	C₂³×D₅	C₂×C₇	1	1	yes	characteristic
24	C₂²×C₄×C₅	C₂×C₇	1	1	yes	characteristic, complemented
25	C₂²×Dic₅	C₂×C₇	1	1	yes	characteristic, complemented
26	C₂×C₄×D₅	C₂×C₇	1	12	yes	direct product factor
27	C₂×C₅×C₇	C₂²×C₄	1	6	yes	complemented
28	C₂×C₅×C₇	C₂⁴	1	1	yes	characteristic
29	C₇×D₅	C₂²×C₄	1	8	yes	direct product factor
30	C₂³×C₇	—	5	8	no	—
31	C₂³×C₇	—	5	6	no	—
32	C₂³×C₇	C₂×D₅	1	1	yes	characteristic
33	C₂×C₄×C₇	—	5	6	no	—
34	C₂×C₄×C₇	—	5	16	no	—
35	C₂×C₄×C₇	C₂×D₅	1	6	yes	direct product factor
36	C₂³×C₅	C₂²×C₇	1	1	yes	characteristic
37	C₂²×D₅	C₄×C₇	1	8	yes	direct product factor
38	C₂²×D₅	C₂²×C₇	1	6	yes	—
39	C₂×C₄×C₅	C₂²×C₇	1	6	yes	complemented
40	C₂×Dic₅	C₂²×C₇	1	6	yes	complemented
41	C₄×D₅	C₂²×C₇	1	16	yes	direct product factor
42	C₅×C₇	C₂³×C₄	1	1	yes	characteristic, complemented, Hall
43	C₂³×C₄	—	5	1	no	Sylow
44	C₂²×C₇	—	5	24	no	—
45	C₂²×C₇	—	5	4	no	—
46	C₂²×C₇	C₄×D₅	1	4	yes	direct product factor
47	C₂²×C₇	C₂²×D₅	1	3	yes	—
48	C₄×C₇	—	5	4	no	—
49	C₄×C₇	C₂²×D₅	1	4	yes	direct product factor
50	C₂²×C₅	C₂×C₄×C₇	1	4	yes	complemented
51	C₂²×C₅	C₂³×C₇	1	3	yes	—
52	C₂×D₅	C₂×C₄×C₇	1	24	yes	direct product factor
53	C₂×D₅	C₂³×C₇	1	4	yes	—
54	C₄×C₅	C₂³×C₇	1	4	yes	complemented
55	Dic₅	C₂³×C₇	1	4	yes	complemented
56	C₂⁴	—	5	1	no	—
57	C₂²×C₄	—	5	1	no	—
58	C₂²×C₄	—	5	12	no	—
59	C₂²×C₄	C₇×D₅	1	1	yes	characteristic, direct product factor
60	C₂×C₇	—	5	8	no	—
61	C₂×C₇	C₂×C₄×D₅	1	6	yes	direct product factor
62	C₂×C₇	C₂³×D₅	1	1	yes	characteristic
63	C₂×C₅	C₂²×C₄×C₇	1	6	yes	complemented
64	C₂×C₅	C₂⁴×C₇	1	1	yes	characteristic
65	D₅	C₂²×C₄×C₇	1	8	yes	direct product factor
66	C₂³	—	5	8	no	—
67	C₂³	—	5	6	no	—
68	C₂³	C₂×C₇×D₅	1	1	yes	characteristic
69	C₂×C₄	—	5	6	no	—
70	C₂×C₄	—	5	16	no	—
71	C₂×C₄	C₂×C₇×D₅	1	6	yes	direct product factor
72	C₇	C₂²×C₄×D₅	1	1	yes	characteristic, direct product factor, Sylow
73	C₅	C₂³×C₄×C₇	1	1	yes	characteristic, complemented, Sylow
74	C₂²	—	5	24	no	—
75	C₂²	—	5	4	no	—
76	C₂²	C₄×C₇×D₅	1	4	yes	direct product factor
77	C₂²	C₂²×C₇×D₅	1	3	yes	—
78	C₄	—	5	4	no	—
79	C₄	C₂²×C₇×D₅	1	4	yes	direct product factor
80	C₂	—	5	8	no	—
81	C₂	C₂×C₄×C₇×D₅	1	6	yes	direct product factor
82	C₂	C₂³×C₇×D₅	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂²×C₄×C₇	D₅
commutator, hypocenter	C₅	C₂³×C₄×C₇
Frattini	C₂	C₂³×C₇×D₅
Fitting	C₂²×C₄×C₅×C₇	C₂
socle	C₂³×C₅×C₇	C₂²
prime radical	C₂³×C₇×D₅	C₂
cosocle	C₂×C₅	C₂⁴×C₇
LCS	G ▷ C₅
UCS	C₁ ◁ C₂²×C₄×C₇

Complements

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

N	H ≅ G/N	type
2. C₂²×C₄×C₅×C₇	C₂	semidirect
3. C₂²×C₇×Dic₅	C₂	semidirect
4. C₂×C₄×C₇×D₅	C₂	direct (N × H)
6. C₂²×C₇×D₅	C₄	direct (N × H)
8. C₂×C₄×C₅×C₇	C₂²	semidirect
9. C₂×C₇×Dic₅	C₂²	semidirect
10. C₄×C₇×D₅	C₂²	direct (N × H)
12. C₂²×C₄×D₅	C₇	direct (N × H)
13. C₂²×C₅×C₇	C₂×C₄	semidirect
15. C₂×C₇×D₅	C₂×C₄	direct (N × H)
17. C₄×C₅×C₇	C₂³	semidirect
18. C₇×Dic₅	C₂³	semidirect
22. C₂²×C₄×C₇	D₅	direct (N × H)
24. C₂²×C₄×C₅	C₂×C₇	semidirect
25. C₂²×Dic₅	C₂×C₇	semidirect
26. C₂×C₄×D₅	C₂×C₇	direct (N × H)
27. C₂×C₅×C₇	C₂²×C₄	semidirect
29. C₇×D₅	C₂²×C₄	direct (N × H)
35. C₂×C₄×C₇	C₂×D₅	direct (N × H)
37. C₂²×D₅	C₄×C₇	direct (N × H)
39. C₂×C₄×C₅	C₂²×C₇	semidirect
40. C₂×Dic₅	C₂²×C₇	semidirect
41. C₄×D₅	C₂²×C₇	direct (N × H)
42. C₅×C₇	C₂³×C₄	semidirect
46. C₂²×C₇	C₄×D₅	direct (N × H)
49. C₄×C₇	C₂²×D₅	direct (N × H)
50. C₂²×C₅	C₂×C₄×C₇	semidirect
52. C₂×D₅	C₂×C₄×C₇	direct (N × H)
54. C₄×C₅	C₂³×C₇	semidirect
55. Dic₅	C₂³×C₇	semidirect
59. C₂²×C₄	C₇×D₅	direct (N × H)
61. C₂×C₇	C₂×C₄×D₅	direct (N × H)
63. C₂×C₅	C₂²×C₄×C₇	semidirect
65. D₅	C₂²×C₄×C₇	direct (N × H)
71. C₂×C₄	C₂×C₇×D₅	direct (N × H)
72. C₇	C₂²×C₄×D₅	direct (N × H)
73. C₅	C₂³×C₄×C₇	semidirect
76. C₂²	C₄×C₇×D₅	direct (N × H)
79. C₄	C₂²×C₇×D₅	direct (N × H)
81. C₂	C₂×C₄×C₇×D₅	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₇×D₅	2
C₂²×C₄×C₅×C₇	2
C₂²×C₇×Dic₅	2
C₂×C₄×C₇×D₅	2
C₂³×C₄×C₇	5
C₂²×C₄×D₅	7

Subgroups — C22×C4×C7×D5 #11765437

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₄×C₇×D₅ #11765437