Subgroups — C₇×A₄×C₁₂₇⋊C₃ #44544839

Order 32004.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₇×C₁₂₇⋊C₃	C₃	1	1	yes	maximal, characteristic, complemented
2	C₇×A₄×C₁₂₇	C₃	1	1	yes	maximal, characteristic, complemented
3	C₇×C₂²⋊(C₁₂₇⋊C₃)	C₃	1	2	yes	maximal, complemented
4	C₃×C₇×C₁₂₇⋊C₃	—	4	1	no	maximal, Hall
5	C₂×C₇×C₁₂₇⋊C₃	—	3	1	no	—
6	A₄×C₁₂₇⋊C₃	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂²×C₇×C₁₂₇	C₃²	1	1	yes	characteristic, complemented, Hall
8	C₃×C₇×C₁₂₇	—	4	1	no	—
9	C₇×C₁₂₇⋊C₃	—	4	2	no	—
10	C₇×C₁₂₇⋊C₃	A₄	1	1	yes	characteristic, direct product factor
11	C₂×C₇×C₁₂₇	—	3	1	no	—
12	C₂²⋊(C₁₂₇⋊C₃)	C₃×C₇	1	2	yes	complemented
13	A₄×C₁₂₇	C₃×C₇	1	1	yes	characteristic, complemented
14	C₂²×C₁₂₇⋊C₃	C₃×C₇	1	1	yes	characteristic, complemented
15	C₃×C₁₂₇⋊C₃	—	4	1	no	Hall
16	C₇×C₁₂₇	C₃×A₄	1	1	yes	characteristic, complemented, Hall
17	C₂×C₁₂₇⋊C₃	—	3	1	no	—
18	C₂²×C₁₂₇	C₃²×C₇	1	1	yes	characteristic, complemented, Hall
19	C₃×C₁₂₇	—	4	1	no	—
20	C₁₂₇⋊C₃	—	4	2	no	—
21	C₁₂₇⋊C₃	C₇×A₄	1	1	yes	characteristic, direct product factor
22	C₂×C₁₂₇	—	3	1	no	—
23	C₃×C₇×A₄	—	127	1	no	maximal, Hall
24	C₁₂₇	C₃×C₇×A₄	1	1	yes	characteristic, complemented, Sylow
25	C₂²×C₃×C₇	—	127	1	no	—
26	C₇×A₄	—	127	2	no	—
27	C₇×A₄	C₁₂₇⋊C₃	1	1	yes	characteristic, direct product factor
28	C₃²×C₇	—	508	1	no	Hall
29	C₂×C₃×C₇	—	381	1	no	—
30	C₃×A₄	—	127	1	no	Hall
31	C₂²×C₇	C₃×C₁₂₇⋊C₃	1	1	yes	characteristic, complemented, Hall
32	C₃×C₇	—	4	1	no	—
33	C₃×C₇	—	127	1	no	—
34	C₃×C₇	—	508	2	no	—
35	C₂×C₇	—	3	1	no	—
36	C₂²×C₃	—	127	1	no	—
37	A₄	—	127	2	no	—
38	A₄	C₇×C₁₂₇⋊C₃	1	1	yes	characteristic, direct product factor
39	C₃²	—	508	1	no	Sylow
40	C₇	A₄×C₁₂₇⋊C₃	1	1	yes	characteristic, direct product factor, Sylow
41	C₂×C₃	—	381	1	no	—
42	C₂²	C₃×C₇×C₁₂₇⋊C₃	1	1	yes	characteristic, complemented, Sylow
43	C₃	—	4	1	no	—
44	C₃	—	127	1	no	—
45	C₃	—	508	2	no	—
46	C₂	—	3	1	no	—

	subgroup	G/H
center, hypercenter	C₇	A₄×C₁₂₇⋊C₃
commutator, cosocle, hypocenter	C₂²×C₁₂₇	C₃²×C₇
Frattini	C₁	G
Fitting, socle	C₂²×C₇×C₁₂₇	C₃²
LCS	G ▷ C₂²×C₁₂₇
UCS	C₁ ◁ C₇

Complements

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

N	H ≅ G/N	type
1. C₂²×C₇×C₁₂₇⋊C₃	C₃	semidirect
2. C₇×A₄×C₁₂₇	C₃	semidirect
3. C₇×C₂²⋊(C₁₂₇⋊C₃)	C₃	semidirect
6. A₄×C₁₂₇⋊C₃	C₇	direct (N × H)
7. C₂²×C₇×C₁₂₇	C₃²	semidirect
10. C₇×C₁₂₇⋊C₃	A₄	direct (N × H)
12. C₂²⋊(C₁₂₇⋊C₃)	C₃×C₇	semidirect
13. A₄×C₁₂₇	C₃×C₇	semidirect
14. C₂²×C₁₂₇⋊C₃	C₃×C₇	semidirect
16. C₇×C₁₂₇	C₃×A₄	semidirect
18. C₂²×C₁₂₇	C₃²×C₇	semidirect
21. C₁₂₇⋊C₃	C₇×A₄	direct (N × H)
24. C₁₂₇	C₃×C₇×A₄	semidirect
27. C₇×A₄	C₁₂₇⋊C₃	direct (N × H)
31. C₂²×C₇	C₃×C₁₂₇⋊C₃	semidirect
38. A₄	C₇×C₁₂₇⋊C₃	direct (N × H)
40. C₇	A₄×C₁₂₇⋊C₃	direct (N × H)
42. C₂²	C₃×C₇×C₁₂₇⋊C₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₇×C₁₂₇⋊C₃	3
C₇×A₄×C₁₂₇	3
C₇×C₂²⋊(C₁₂₇⋊C₃)	3
C₃×C₇×C₁₂₇⋊C₃	4
A₄×C₁₂₇⋊C₃	7
C₃×C₇×A₄	127

Subgroups — C7×A4×C127⋊C3 #44544839

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×A₄×C₁₂₇⋊C₃ #44544839