Subgroups — C₂²×C₄×C₅×D₇ #11765422

Order 1120.

472 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₅×D₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×C₅×C₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₅×Dic₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₄×C₅×D₇	C₂	1	12	yes	maximal, direct product factor
5	C₂³×C₅×C₇	C₂²	1	1	yes	characteristic
6	C₂²×C₅×D₇	C₄	1	8	yes	direct product factor
7	C₂²×C₅×D₇	C₂²	1	6	yes	—
8	C₂×C₄×C₅×C₇	C₂²	1	6	yes	complemented
9	C₂×C₅×Dic₇	C₂²	1	6	yes	complemented
10	C₄×C₅×D₇	C₂²	1	16	yes	direct product factor
11	C₂²×C₄×D₇	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
12	C₂³×C₄×C₅	—	7	1	no	maximal, Hall
13	C₂²×C₅×C₇	C₂×C₄	1	4	yes	complemented
14	C₂²×C₅×C₇	C₂³	1	3	yes	—
15	C₂×C₅×D₇	C₂×C₄	1	24	yes	direct product factor
16	C₂×C₅×D₇	C₂³	1	4	yes	—
17	C₄×C₅×C₇	C₂³	1	4	yes	complemented
18	C₅×Dic₇	C₂³	1	4	yes	complemented
19	C₂³×D₇	C₂×C₅	1	1	yes	characteristic
20	C₂²×C₄×C₇	C₂×C₅	1	1	yes	characteristic, complemented
21	C₂²×Dic₇	C₂×C₅	1	1	yes	characteristic, complemented
22	C₂×C₄×D₇	C₂×C₅	1	12	yes	direct product factor
23	C₂⁴×C₅	—	7	1	no	—
24	C₂²×C₄×C₅	—	7	1	no	—
25	C₂²×C₄×C₅	—	7	12	no	—
26	C₂²×C₄×C₅	D₇	1	1	yes	characteristic, direct product factor
27	C₂×C₅×C₇	C₂²×C₄	1	6	yes	complemented
28	C₂×C₅×C₇	C₂⁴	1	1	yes	characteristic
29	C₅×D₇	C₂²×C₄	1	8	yes	direct product factor
30	C₂³×C₇	C₂²×C₅	1	1	yes	characteristic
31	C₂²×D₇	C₄×C₅	1	8	yes	direct product factor
32	C₂²×D₇	C₂²×C₅	1	6	yes	—
33	C₂×C₄×C₇	C₂²×C₅	1	6	yes	complemented
34	C₂×Dic₇	C₂²×C₅	1	6	yes	complemented
35	C₄×D₇	C₂²×C₅	1	16	yes	direct product factor
36	C₂³×C₅	—	7	8	no	—
37	C₂³×C₅	—	7	6	no	—
38	C₂³×C₅	C₂×D₇	1	1	yes	characteristic
39	C₂×C₄×C₅	—	7	6	no	—
40	C₂×C₄×C₅	—	7	16	no	—
41	C₂×C₄×C₅	C₂×D₇	1	6	yes	direct product factor
42	C₅×C₇	C₂³×C₄	1	1	yes	characteristic, complemented, Hall
43	C₂³×C₄	—	7	1	no	Sylow
44	C₂²×C₇	C₂×C₄×C₅	1	4	yes	complemented
45	C₂²×C₇	C₂³×C₅	1	3	yes	—
46	C₂×D₇	C₂×C₄×C₅	1	24	yes	direct product factor
47	C₂×D₇	C₂³×C₅	1	4	yes	—
48	C₄×C₇	C₂³×C₅	1	4	yes	complemented
49	Dic₇	C₂³×C₅	1	4	yes	complemented
50	C₂²×C₅	—	7	24	no	—
51	C₂²×C₅	—	7	4	no	—
52	C₂²×C₅	C₄×D₇	1	4	yes	direct product factor
53	C₂²×C₅	C₂²×D₇	1	3	yes	—
54	C₄×C₅	—	7	4	no	—
55	C₄×C₅	C₂²×D₇	1	4	yes	direct product factor
56	C₂⁴	—	7	1	no	—
57	C₂²×C₄	—	7	1	no	—
58	C₂²×C₄	—	7	12	no	—
59	C₂²×C₄	C₅×D₇	1	1	yes	characteristic, direct product factor
60	C₂×C₇	C₂²×C₄×C₅	1	6	yes	complemented
61	C₂×C₇	C₂⁴×C₅	1	1	yes	characteristic
62	D₇	C₂²×C₄×C₅	1	8	yes	direct product factor
63	C₂×C₅	—	7	8	no	—
64	C₂×C₅	C₂×C₄×D₇	1	6	yes	direct product factor
65	C₂×C₅	C₂³×D₇	1	1	yes	characteristic
66	C₂³	—	7	8	no	—
67	C₂³	—	7	6	no	—
68	C₂³	C₂×C₅×D₇	1	1	yes	characteristic
69	C₂×C₄	—	7	6	no	—
70	C₂×C₄	—	7	16	no	—
71	C₂×C₄	C₂×C₅×D₇	1	6	yes	direct product factor
72	C₇	C₂³×C₄×C₅	1	1	yes	characteristic, complemented, Sylow
73	C₅	C₂²×C₄×D₇	1	1	yes	characteristic, direct product factor, Sylow
74	C₂²	—	7	24	no	—
75	C₂²	—	7	4	no	—
76	C₂²	C₄×C₅×D₇	1	4	yes	direct product factor
77	C₂²	C₂²×C₅×D₇	1	3	yes	—
78	C₄	—	7	4	no	—
79	C₄	C₂²×C₅×D₇	1	4	yes	direct product factor
80	C₂	—	7	8	no	—
81	C₂	C₂×C₄×C₅×D₇	1	6	yes	direct product factor
82	C₂	C₂³×C₅×D₇	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂²×C₄×C₅	D₇
commutator, hypocenter	C₇	C₂³×C₄×C₅
Frattini	C₂	C₂³×C₅×D₇
Fitting	C₂²×C₄×C₅×C₇	C₂
socle	C₂³×C₅×C₇	C₂²
prime radical	C₂³×C₅×D₇	C₂
cosocle	C₂×C₇	C₂⁴×C₅
LCS	G ▷ C₇
UCS	C₁ ◁ C₂²×C₄×C₅

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×C₅×Dic₇	C₂	semidirect
4. C₂×C₄×C₅×D₇	C₂	direct (N × H)
6. C₂²×C₅×D₇	C₄	direct (N × H)
8. C₂×C₄×C₅×C₇	C₂²	semidirect
9. C₂×C₅×Dic₇	C₂²	semidirect
10. C₄×C₅×D₇	C₂²	direct (N × H)
11. C₂²×C₄×D₇	C₅	direct (N × H)
13. C₂²×C₅×C₇	C₂×C₄	semidirect
15. C₂×C₅×D₇	C₂×C₄	direct (N × H)
17. C₄×C₅×C₇	C₂³	semidirect
18. C₅×Dic₇	C₂³	semidirect
20. C₂²×C₄×C₇	C₂×C₅	semidirect
21. C₂²×Dic₇	C₂×C₅	semidirect
22. C₂×C₄×D₇	C₂×C₅	direct (N × H)
26. C₂²×C₄×C₅	D₇	direct (N × H)
27. C₂×C₅×C₇	C₂²×C₄	semidirect
29. C₅×D₇	C₂²×C₄	direct (N × H)
31. C₂²×D₇	C₄×C₅	direct (N × H)
33. C₂×C₄×C₇	C₂²×C₅	semidirect
34. C₂×Dic₇	C₂²×C₅	semidirect
35. C₄×D₇	C₂²×C₅	direct (N × H)
41. C₂×C₄×C₅	C₂×D₇	direct (N × H)
42. C₅×C₇	C₂³×C₄	semidirect
44. C₂²×C₇	C₂×C₄×C₅	semidirect
46. C₂×D₇	C₂×C₄×C₅	direct (N × H)
48. C₄×C₇	C₂³×C₅	semidirect
49. Dic₇	C₂³×C₅	semidirect
52. C₂²×C₅	C₄×D₇	direct (N × H)
55. C₄×C₅	C₂²×D₇	direct (N × H)
59. C₂²×C₄	C₅×D₇	direct (N × H)
60. C₂×C₇	C₂²×C₄×C₅	semidirect
62. D₇	C₂²×C₄×C₅	direct (N × H)
64. C₂×C₅	C₂×C₄×D₇	direct (N × H)
71. C₂×C₄	C₂×C₅×D₇	direct (N × H)
72. C₇	C₂³×C₄×C₅	semidirect
73. C₅	C₂²×C₄×D₇	direct (N × H)
76. C₂²	C₄×C₅×D₇	direct (N × H)
79. C₄	C₂²×C₅×D₇	direct (N × H)
81. C₂	C₂×C₄×C₅×D₇	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×C₅×D₇	2
C₂²×C₄×C₅×C₇	2
C₂²×C₅×Dic₇	2
C₂×C₄×C₅×D₇	2
C₂²×C₄×D₇	5
C₂³×C₄×C₅	7

Subgroups — C22×C4×C5×D7 #11765422

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₄×C₅×D₇ #11765422