Subgroups — C₇×C₃⋊((C₁₂₇⋊C₃)⋊C₄) #44544739

Order 32004.

56 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
2	C₇×(C₁₂₇⋊C₃)⋊C₄	—	3	1	no	maximal
3	C₇×Dic₃₈₁	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃×C₇×C₁₂₇⋊C₃	C₄	1	1	yes	characteristic, complemented, Hall
5	C₂×C₇×C₁₂₇⋊C₃	—	2	1	no	—
6	C₂×C₇×C₁₂₇⋊C₃	D₃	1	1	yes	characteristic
7	C₂×C₃×C₇×C₁₂₇	C₂×C₃	1	1	yes	characteristic
8	C₃⋊((C₁₂₇⋊C₃)⋊C₄)	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₇×Dic₁₂₇	—	3	1	no	Hall
10	C₃×C₇×C₁₂₇	C₃×C₄	1	1	yes	characteristic, complemented
11	C₇×C₁₂₇⋊C₃	—	2	1	no	—
12	C₇×C₁₂₇⋊C₃	Dic₃	1	1	yes	characteristic, complemented
13	C₂×C₃×C₁₂₇⋊C₃	C₂×C₇	1	1	yes	characteristic
14	C₂×C₇×C₁₂₇	C₃×D₃	1	1	yes	characteristic
15	Dic₃₈₁	C₃×C₇	1	1	yes	characteristic, complemented
16	(C₁₂₇⋊C₃)⋊C₄	—	3	1	no	—
17	C₃×C₁₂₇⋊C₃	C₄×C₇	1	1	yes	characteristic, complemented, Hall
18	C₇×C₁₂₇	C₃×Dic₃	1	1	yes	characteristic, complemented, Hall
19	C₂×C₃×C₁₂₇	C₂×C₃×C₇	1	1	yes	characteristic
20	C₂×C₁₂₇⋊C₃	—	2	1	no	—
21	C₂×C₁₂₇⋊C₃	D₃×C₇	1	1	yes	characteristic
22	Dic₁₂₇	—	3	1	no	Hall
23	C₃×C₁₂₇	C₃×C₄×C₇	1	1	yes	characteristic, complemented
24	C₁₂₇⋊C₃	—	2	1	no	—
25	C₁₂₇⋊C₃	C₇×Dic₃	1	1	yes	characteristic, complemented
26	C₂×C₁₂₇	C₃×D₃×C₇	1	1	yes	characteristic
27	C₃×C₇×Dic₃	—	127	1	no	maximal, Hall
28	C₁₂₇	C₃×C₇×Dic₃	1	1	yes	characteristic, complemented, Sylow
29	C₂×C₃²×C₇	—	127	1	no	—
30	C₃×C₄×C₇	—	381	1	no	—
31	C₇×Dic₃	—	127	1	no	—
32	C₃²×C₇	—	127	1	no	Hall
33	C₂×C₃×C₇	—	127	1	no	—
34	C₂×C₃×C₇	—	254	1	no	—
35	C₂×C₃×C₇	D₁₂₇⋊C₃	1	1	yes	characteristic
36	C₃×Dic₃	—	127	1	no	Hall
37	C₄×C₇	—	381	1	no	Hall
38	C₃×C₇	—	127	1	no	—
39	C₃×C₇	—	254	1	no	—
40	C₃×C₇	(C₁₂₇⋊C₃)⋊C₄	1	1	yes	characteristic, complemented
41	C₂×C₃²	—	127	1	no	—
42	C₂×C₇	C₃⋊(D₁₂₇⋊C₃)	1	1	yes	characteristic
43	C₃×C₄	—	381	1	no	—
44	Dic₃	—	127	1	no	—
45	C₃²	—	127	1	no	Sylow
46	C₇	C₃⋊((C₁₂₇⋊C₃)⋊C₄)	1	1	yes	characteristic, direct product factor, Sylow
47	C₂×C₃	—	127	1	no	—
48	C₂×C₃	—	254	1	no	—
49	C₂×C₃	C₇×D₁₂₇⋊C₃	1	1	yes	characteristic
50	C₄	—	381	1	no	Sylow
51	C₃	—	127	1	no	—
52	C₃	—	254	1	no	—
53	C₃	C₇×(C₁₂₇⋊C₃)⋊C₄	1	1	yes	characteristic, complemented
54	C₂	C₇×C₃⋊(D₁₂₇⋊C₃)	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₇	C₃⋊(D₁₂₇⋊C₃)
commutator, hypocenter	C₃×C₁₂₇	C₃×C₄×C₇
Frattini	C₂	C₇×C₃⋊(D₁₂₇⋊C₃)
Fitting, socle	C₂×C₃×C₇×C₁₂₇	C₂×C₃
prime radical	C₂×C₃×C₇×C₁₂₇⋊C₃	C₂
cosocle	C₂×C₃×C₁₂₇	C₂×C₃×C₇
LCS	G ▷ C₃×C₁₂₇
UCS	C₁ ◁ C₂×C₇

Complements

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

N	H ≅ G/N	type
3. C₇×Dic₃₈₁	C₃	semidirect
4. C₃×C₇×C₁₂₇⋊C₃	C₄	semidirect
8. C₃⋊((C₁₂₇⋊C₃)⋊C₄)	C₇	direct (N × H)
10. C₃×C₇×C₁₂₇	C₃×C₄	semidirect
12. C₇×C₁₂₇⋊C₃	Dic₃	semidirect
15. Dic₃₈₁	C₃×C₇	semidirect
17. C₃×C₁₂₇⋊C₃	C₄×C₇	semidirect
18. C₇×C₁₂₇	C₃×Dic₃	semidirect
23. C₃×C₁₂₇	C₃×C₄×C₇	semidirect
25. C₁₂₇⋊C₃	C₇×Dic₃	semidirect
28. C₁₂₇	C₃×C₇×Dic₃	semidirect
40. C₃×C₇	(C₁₂₇⋊C₃)⋊C₄	semidirect
46. C₇	C₃⋊((C₁₂₇⋊C₃)⋊C₄)	direct (N × H)
53. C₃	C₇×(C₁₂₇⋊C₃)⋊C₄	semidirect

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
C₇×(C₁₂₇⋊C₃)⋊C₄	3
C₇×Dic₃₈₁	3
C₃⋊((C₁₂₇⋊C₃)⋊C₄)	7
C₃×C₇×Dic₃	127

Subgroups — C7×C3⋊((C127⋊C3)⋊C4) #44544739

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×C₃⋊((C₁₂₇⋊C₃)⋊C₄) #44544739