Subgroups — C₇×#6180 #13094179

Order 1512.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₇×H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic
2	C₃×C₇×C₃⋊C₈	—	3	4	no	maximal
3	C₂×C₇×H(3,𝔽₃)	C₄	1	1	yes	characteristic
4	C₃²×C₄×C₇	D₃	1	4	yes	—
5	#6180	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₇×H(3,𝔽₃)	C₈	1	1	yes	characteristic, complemented, Hall
7	C₃×C₇×C₈	—	9	1	no	—
8	C₇×C₃⋊C₈	—	3	4	no	—
9	C₂×C₃²×C₇	Dic₃	1	4	yes	—
10	C₄×H(3,𝔽₃)	C₂×C₇	1	1	yes	characteristic
11	C₃×C₄×C₇	—	3	4	no	—
12	C₃×C₄×C₇	C₃⋊D₃	1	1	yes	characteristic
13	C₃×C₃⋊C₈	—	3	4	no	—
14	C₃²×C₇	C₃⋊C₈	1	4	yes	complemented
15	C₇×C₈	—	9	1	no	Hall
16	C₂×H(3,𝔽₃)	C₄×C₇	1	1	yes	characteristic
17	C₂×C₃×C₇	—	3	4	no	—
18	C₂×C₃×C₇	C₃⋊Dic₃	1	1	yes	characteristic
19	C₃²×C₄	D₃×C₇	1	4	yes	—
20	C₄×C₇	#272	1	1	yes	characteristic
21	H(3,𝔽₃)	C₇×C₈	1	1	yes	characteristic, complemented, Sylow
22	C₃×C₈	—	9	1	no	—
23	C₃⋊C₈	—	3	4	no	—
24	C₃×C₇	—	3	4	no	—
25	C₃×C₇	C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
26	C₂×C₃²	C₇×Dic₃	1	4	yes	—
27	C₂×C₇	#1084	1	1	yes	characteristic
28	C₃×C₄	—	3	4	no	—
29	C₃×C₄	C₇×C₃⋊D₃	1	1	yes	characteristic
30	C₃²	C₇×C₃⋊C₈	1	4	yes	complemented
31	C₈	—	9	1	no	Sylow
32	C₇	#6180	1	1	yes	characteristic, direct product factor, Sylow
33	C₂×C₃	—	3	4	no	—
34	C₂×C₃	C₇×C₃⋊Dic₃	1	1	yes	characteristic
35	C₄	C₇×#272	1	1	yes	characteristic
36	C₃	—	3	4	no	—
37	C₃	C₇×C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
38	C₂	C₇×#1084	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₃×C₄×C₇	C₃⋊D₃
commutator, hypocenter	H(3,𝔽₃)	C₇×C₈
Frattini	C₃×C₄	C₇×C₃⋊D₃
Fitting	C₄×C₇×H(3,𝔽₃)	C₂
socle	C₂×C₃×C₇	C₃⋊Dic₃
prime radical	C₂×C₇×H(3,𝔽₃)	C₄
cosocle	C₄×H(3,𝔽₃)	C₂×C₇
LCS	G ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₃×C₄×C₇

Complements

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

N	H ≅ G/N	type
5. #6180	C₇	direct (N × H)
6. C₇×H(3,𝔽₃)	C₈	semidirect
14. C₃²×C₇	C₃⋊C₈	semidirect
21. H(3,𝔽₃)	C₇×C₈	semidirect
30. C₃²	C₇×C₃⋊C₈	semidirect
32. C₇	#6180	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₇×H(3,𝔽₃)	2
C₃×C₇×C₃⋊C₈	3
#6180	7

Subgroups — C7×#6180 #13094179

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#6180 #13094179