Subgroups — C₂×C₃×C₇×D₃₂₁ #41004156

Order 26964.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₇×D₃₂₁	C₂	1	2	yes	maximal, direct product factor
2	C₂×C₃²×C₇×C₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×C₇×D₁₀₇	—	3	1	no	maximal
4	C₂×C₇×D₃₂₁	C₃	1	1	yes	maximal, characteristic, direct product factor
5	C₃²×C₇×C₁₀₇	C₂²	1	1	yes	characteristic, complemented, Hall
6	C₇×D₃₂₁	C₂×C₃	1	2	yes	direct product factor
7	C₃×C₇×D₁₀₇	—	3	2	no	—
8	C₂×C₃×C₇×C₁₀₇	—	2	1	no	—
9	C₂×C₃×C₇×C₁₀₇	D₃	1	1	yes	characteristic, complemented
10	C₂×C₃×C₇×C₁₀₇	C₂×C₃	1	1	yes	characteristic, complemented
11	C₂×C₃×D₃₂₁	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
12	C₂×C₇×D₁₀₇	—	3	1	no	Hall
13	C₃×C₇×C₁₀₇	—	2	1	no	—
14	C₃×C₇×C₁₀₇	C₂×D₃	1	1	yes	characteristic, complemented
15	C₃×C₇×C₁₀₇	C₂²×C₃	1	1	yes	characteristic, complemented
16	C₂×C₃²×C₁₀₇	C₂×C₇	1	1	yes	characteristic, complemented
17	C₃×D₃₂₁	C₂×C₇	1	2	yes	direct product factor
18	C₂×C₇×C₁₀₇	C₃×D₃	1	1	yes	characteristic, complemented
19	C₇×D₁₀₇	—	3	2	no	—
20	C₂×D₃₂₁	C₃×C₇	1	1	yes	characteristic, direct product factor
21	C₂×C₃×D₁₀₇	—	3	1	no	—
22	C₃²×C₁₀₇	C₂²×C₇	1	1	yes	characteristic, complemented, Hall
23	C₇×C₁₀₇	C₂×C₃×D₃	1	1	yes	characteristic, complemented, Hall
24	C₂×C₃×C₁₀₇	—	2	1	no	—
25	C₂×C₃×C₁₀₇	D₃×C₇	1	1	yes	characteristic, complemented
26	C₂×C₃×C₁₀₇	C₂×C₃×C₇	1	1	yes	characteristic, complemented
27	D₃₂₁	C₂×C₃×C₇	1	2	yes	direct product factor
28	C₃×D₁₀₇	—	3	2	no	—
29	C₂×D₁₀₇	—	3	1	no	Hall
30	C₃×C₁₀₇	—	2	1	no	—
31	C₃×C₁₀₇	C₂×D₃×C₇	1	1	yes	characteristic, complemented
32	C₃×C₁₀₇	C₂²×C₃×C₇	1	1	yes	characteristic, complemented
33	C₂×C₃×D₃×C₇	—	107	1	no	maximal, Hall
34	C₂×C₁₀₇	C₃×D₃×C₇	1	1	yes	characteristic, complemented
35	D₁₀₇	—	3	2	no	—
36	C₂×C₃²×C₇	D₁₀₇	1	1	yes	characteristic, complemented
37	C₃×D₃×C₇	—	107	2	no	—
38	C₁₀₇	C₂×C₃×D₃×C₇	1	1	yes	characteristic, complemented, Sylow
39	C₂²×C₃×C₇	—	321	1	no	—
40	C₂×D₃×C₇	—	107	1	no	—
41	C₃²×C₇	C₂×D₁₀₇	1	1	yes	characteristic, complemented, Hall
42	C₂×C₃×C₇	—	2	1	no	—
43	C₂×C₃×C₇	—	321	2	no	—
44	C₂×C₃×C₇	C₃×D₁₀₇	1	1	yes	characteristic, complemented
45	C₂×C₃×C₇	D₃₂₁	1	1	yes	characteristic, direct product factor
46	D₃×C₇	—	107	2	no	—
47	C₂×C₃×D₃	—	107	1	no	Hall
48	C₂²×C₇	—	321	1	no	Hall
49	C₃×C₇	—	2	1	no	—
50	C₃×C₇	C₂×C₃×D₁₀₇	1	1	yes	characteristic, complemented
51	C₃×C₇	C₂×D₃₂₁	1	1	yes	characteristic, direct product factor
52	C₂×C₃²	C₇×D₁₀₇	1	1	yes	characteristic, complemented
53	C₃×D₃	—	107	2	no	—
54	C₂×C₇	—	321	2	no	—
55	C₂×C₇	C₃×D₃₂₁	1	1	yes	characteristic, direct product factor
56	C₂²×C₃	—	321	1	no	—
57	C₂×D₃	—	107	1	no	—
58	C₃²	C₂×C₇×D₁₀₇	1	1	yes	characteristic, complemented, Sylow
59	C₇	C₂×C₃×D₃₂₁	1	1	yes	characteristic, direct product factor, Sylow
60	C₂×C₃	—	2	1	no	—
61	C₂×C₃	—	321	2	no	—
62	C₂×C₃	C₃×C₇×D₁₀₇	1	1	yes	characteristic, complemented
63	C₂×C₃	C₇×D₃₂₁	1	1	yes	characteristic, direct product factor
64	D₃	—	107	2	no	—
65	C₂²	—	321	1	no	Sylow
66	C₃	—	2	1	no	—
67	C₃	C₂×C₇×D₃₂₁	1	1	yes	characteristic, direct product factor
68	C₃	C₂×C₃×C₇×D₁₀₇	1	1	yes	characteristic, complemented
69	C₂	—	321	2	no	—
70	C₂	C₃×C₇×D₃₂₁	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₃×C₇	D₃₂₁
commutator, cosocle, hypocenter	C₃×C₁₀₇	C₂²×C₃×C₇
Frattini	C₁	G
Fitting, socle	C₂×C₃²×C₇×C₁₀₇	C₂
LCS	G ▷ C₃×C₁₀₇
UCS	C₁ ◁ C₂×C₃×C₇

Complements

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

N	H ≅ G/N	type
1. C₃×C₇×D₃₂₁	C₂	direct (N × H)
2. C₂×C₃²×C₇×C₁₀₇	C₂	semidirect
4. C₂×C₇×D₃₂₁	C₃	direct (N × H)
5. C₃²×C₇×C₁₀₇	C₂²	semidirect
6. C₇×D₃₂₁	C₂×C₃	direct (N × H)
9. C₂×C₃×C₇×C₁₀₇	D₃	semidirect
10. C₂×C₃×C₇×C₁₀₇	C₂×C₃	semidirect
11. C₂×C₃×D₃₂₁	C₇	direct (N × H)
14. C₃×C₇×C₁₀₇	C₂×D₃	semidirect
15. C₃×C₇×C₁₀₇	C₂²×C₃	semidirect
16. C₂×C₃²×C₁₀₇	C₂×C₇	semidirect
17. C₃×D₃₂₁	C₂×C₇	direct (N × H)
18. C₂×C₇×C₁₀₇	C₃×D₃	semidirect
20. C₂×D₃₂₁	C₃×C₇	direct (N × H)
22. C₃²×C₁₀₇	C₂²×C₇	semidirect
23. C₇×C₁₀₇	C₂×C₃×D₃	semidirect
25. C₂×C₃×C₁₀₇	D₃×C₇	semidirect
26. C₂×C₃×C₁₀₇	C₂×C₃×C₇	semidirect
27. D₃₂₁	C₂×C₃×C₇	direct (N × H)
31. C₃×C₁₀₇	C₂×D₃×C₇	semidirect
32. C₃×C₁₀₇	C₂²×C₃×C₇	semidirect
34. C₂×C₁₀₇	C₃×D₃×C₇	semidirect
36. C₂×C₃²×C₇	D₁₀₇	semidirect
38. C₁₀₇	C₂×C₃×D₃×C₇	semidirect
41. C₃²×C₇	C₂×D₁₀₇	semidirect
44. C₂×C₃×C₇	C₃×D₁₀₇	semidirect
45. C₂×C₃×C₇	D₃₂₁	direct (N × H)
50. C₃×C₇	C₂×C₃×D₁₀₇	semidirect
51. C₃×C₇	C₂×D₃₂₁	direct (N × H)
52. C₂×C₃²	C₇×D₁₀₇	semidirect
55. C₂×C₇	C₃×D₃₂₁	direct (N × H)
58. C₃²	C₂×C₇×D₁₀₇	semidirect
59. C₇	C₂×C₃×D₃₂₁	direct (N × H)
62. C₂×C₃	C₃×C₇×D₁₀₇	semidirect
63. C₂×C₃	C₇×D₃₂₁	direct (N × H)
67. C₃	C₂×C₇×D₃₂₁	direct (N × H)
68. C₃	C₂×C₃×C₇×D₁₀₇	semidirect
70. C₂	C₃×C₇×D₃₂₁	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×C₇×D₃₂₁	2
C₂×C₃²×C₇×C₁₀₇	2
C₂×C₃×C₇×D₁₀₇	3
C₂×C₇×D₃₂₁	3
C₂×C₃×D₃₂₁	7
C₂×C₃×D₃×C₇	107

Subgroups — C2×C3×C7×D321 #41004156

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₇×D₃₂₁ #41004156