Subgroups — C₂×C₃×(C₁₂₇⋊C₃)⋊D₇ #44544864

Order 32004.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₇×C₁₂₇⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×(C₁₂₇⋊C₃)⋊D₇	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₃×D₈₈₉	C₃	1	1	yes	maximal, characteristic, complemented
4	C₂×(C₁₂₇⋊C₃)⋊D₇	C₃	1	3	yes	maximal, direct product factor
5	C₃×C₇×C₁₂₇⋊C₃	C₂²	1	1	yes	characteristic, complemented, Hall
6	(C₁₂₇⋊C₃)⋊D₇	C₂×C₃	1	6	yes	direct product factor
7	C₃×D₈₈₉	C₂×C₃	1	2	yes	complemented
8	C₂×C₇×C₁₂₇⋊C₃	C₂×C₃	1	3	yes	complemented
9	C₂×C₃×C₇×C₁₂₇	C₂×C₃	1	1	yes	characteristic, complemented
10	C₂×C₃×D₁₂₇⋊C₃	—	7	1	no	maximal, Hall
11	C₂×D₈₈₉	C₃²	1	1	yes	characteristic, complemented, Hall
12	C₃×C₇×C₁₂₇	C₂²×C₃	1	1	yes	characteristic, complemented
13	C₇×C₁₂₇⋊C₃	C₂²×C₃	1	3	yes	complemented
14	C₂×C₃×C₁₂₇⋊C₃	D₇	1	1	yes	characteristic, complemented
15	C₃×D₁₂₇⋊C₃	—	7	2	no	—
16	C₂×C₇×C₁₂₇	C₂×C₃²	1	1	yes	characteristic, complemented
17	D₈₈₉	C₂×C₃²	1	2	yes	complemented
18	C₂×C₃×D₁₂₇	—	7	1	no	—
19	C₂×D₁₂₇⋊C₃	—	7	3	no	—
20	C₃×C₁₂₇⋊C₃	C₂×D₇	1	1	yes	characteristic, complemented, Hall
21	C₇×C₁₂₇	C₂²×C₃²	1	1	yes	characteristic, complemented, Hall
22	C₂×C₃×C₁₂₇	C₃×D₇	1	1	yes	characteristic, complemented
23	C₃×D₁₂₇	—	7	2	no	—
24	C₂×C₁₂₇⋊C₃	C₃×D₇	1	3	yes	complemented
25	D₁₂₇⋊C₃	—	7	6	no	—
26	C₂×D₁₂₇	—	7	1	no	Hall
27	C₃×C₁₂₇	C₂×C₃×D₇	1	1	yes	characteristic, complemented
28	C₁₂₇⋊C₃	C₂×C₃×D₇	1	3	yes	complemented
29	C₂×C₁₂₇	C₃²×D₇	1	1	yes	characteristic, complemented
30	D₁₂₇	—	7	2	no	—
31	C₂×C₃²×D₇	—	127	1	no	maximal, Hall
32	C₁₂₇	C₂×C₃²×D₇	1	1	yes	characteristic, complemented, Sylow
33	C₂×C₃²×C₇	—	127	1	no	—
34	C₃²×D₇	—	127	2	no	—
35	C₂×C₃×D₇	—	127	3	no	—
36	C₂×C₃×D₇	—	127	1	no	—
37	C₃²×C₇	—	127	1	no	Hall
38	C₂×C₃×C₇	—	127	3	no	—
39	C₂×C₃×C₇	D₁₂₇⋊C₃	1	1	yes	characteristic, complemented
40	C₃×D₇	—	127	6	no	—
41	C₃×D₇	—	127	2	no	—
42	C₂²×C₃²	—	889	1	no	Hall
43	C₂×D₇	—	127	1	no	Hall
44	C₃×C₇	—	127	3	no	—
45	C₃×C₇	C₂×D₁₂₇⋊C₃	1	1	yes	characteristic, complemented
46	C₂×C₃²	—	889	2	no	—
47	C₂×C₃²	—	127	1	no	—
48	C₂×C₇	C₃×D₁₂₇⋊C₃	1	1	yes	characteristic, complemented
49	D₇	—	127	2	no	—
50	C₂²×C₃	—	889	3	no	—
51	C₂²×C₃	—	889	1	no	—
52	C₃²	—	127	1	no	Sylow
53	C₇	C₂×C₃×D₁₂₇⋊C₃	1	1	yes	characteristic, complemented, Sylow
54	C₂×C₃	—	889	6	no	—
55	C₂×C₃	—	889	2	no	—
56	C₂×C₃	—	127	3	no	—
57	C₂×C₃	(C₁₂₇⋊C₃)⋊D₇	1	1	yes	characteristic, direct product factor
58	C₂²	—	889	1	no	Sylow
59	C₃	—	127	3	no	—
60	C₃	C₂×(C₁₂₇⋊C₃)⋊D₇	1	1	yes	characteristic, direct product factor
61	C₂	—	889	2	no	—
62	C₂	C₃×(C₁₂₇⋊C₃)⋊D₇	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₃	(C₁₂₇⋊C₃)⋊D₇
commutator, cosocle, hypocenter	C₇×C₁₂₇	C₂²×C₃²
Frattini	C₁	G
Fitting, socle	C₂×C₃×C₇×C₁₂₇	C₂×C₃
LCS	G ▷ C₇×C₁₂₇
UCS	C₁ ◁ C₂×C₃

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₇×C₁₂₇⋊C₃	C₂	semidirect
2. C₃×(C₁₂₇⋊C₃)⋊D₇	C₂	direct (N × H)
3. C₂×C₃×D₈₈₉	C₃	semidirect
4. C₂×(C₁₂₇⋊C₃)⋊D₇	C₃	direct (N × H)
5. C₃×C₇×C₁₂₇⋊C₃	C₂²	semidirect
6. (C₁₂₇⋊C₃)⋊D₇	C₂×C₃	direct (N × H)
7. C₃×D₈₈₉	C₂×C₃	semidirect
8. C₂×C₇×C₁₂₇⋊C₃	C₂×C₃	semidirect
9. C₂×C₃×C₇×C₁₂₇	C₂×C₃	semidirect
11. C₂×D₈₈₉	C₃²	semidirect
12. C₃×C₇×C₁₂₇	C₂²×C₃	semidirect
13. C₇×C₁₂₇⋊C₃	C₂²×C₃	semidirect
14. C₂×C₃×C₁₂₇⋊C₃	D₇	semidirect
16. C₂×C₇×C₁₂₇	C₂×C₃²	semidirect
17. D₈₈₉	C₂×C₃²	semidirect
20. C₃×C₁₂₇⋊C₃	C₂×D₇	semidirect
21. C₇×C₁₂₇	C₂²×C₃²	semidirect
22. C₂×C₃×C₁₂₇	C₃×D₇	semidirect
24. C₂×C₁₂₇⋊C₃	C₃×D₇	semidirect
27. C₃×C₁₂₇	C₂×C₃×D₇	semidirect
28. C₁₂₇⋊C₃	C₂×C₃×D₇	semidirect
29. C₂×C₁₂₇	C₃²×D₇	semidirect
32. C₁₂₇	C₂×C₃²×D₇	semidirect
39. C₂×C₃×C₇	D₁₂₇⋊C₃	semidirect
45. C₃×C₇	C₂×D₁₂₇⋊C₃	semidirect
48. C₂×C₇	C₃×D₁₂₇⋊C₃	semidirect
53. C₇	C₂×C₃×D₁₂₇⋊C₃	semidirect
57. C₂×C₃	(C₁₂₇⋊C₃)⋊D₇	direct (N × H)
60. C₃	C₂×(C₁₂₇⋊C₃)⋊D₇	direct (N × H)
62. C₂	C₃×(C₁₂₇⋊C₃)⋊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
C₂×C₃×C₇×C₁₂₇⋊C₃	2
C₃×(C₁₂₇⋊C₃)⋊D₇	2
C₂×C₃×D₈₈₉	3
C₂×(C₁₂₇⋊C₃)⋊D₇	3
C₂×C₃×D₁₂₇⋊C₃	7
C₂×C₃²×D₇	127

Subgroups — C2×C3×(C127⋊C3)⋊D7 #44544864

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×(C₁₂₇⋊C₃)⋊D₇ #44544864