Subgroups — 3₋¹⁺²⋊(C₇⋊C₈) #13094177

Order 1512.

54 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₇×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic
2	C₃×C₃⋊(C₇⋊C₈)	—	3	1	no	maximal
3	C₇⋊(C₉⋊C₈)	C₃	1	1	yes	maximal, characteristic, complemented
4	C₂×C₇×3₋¹⁺²	C₄	1	1	yes	characteristic
5	C₃²×C₄×C₇	D₃	1	1	yes	characteristic
6	C₄×C₇×C₉	—	2	1	no	—
7	C₄×C₇×C₉	C₂×C₃	1	1	yes	characteristic
8	3₋¹⁺²⋊C₈	—	7	1	no	maximal, Hall
9	C₇×3₋¹⁺²	C₈	1	1	yes	characteristic, complemented, Hall
10	C₃⋊(C₇⋊C₈)	—	3	1	no	—
11	C₃×C₇⋊C₈	—	9	1	no	—
12	C₂×C₃²×C₇	Dic₃	1	1	yes	characteristic
13	C₂×C₇×C₉	—	2	1	no	—
14	C₂×C₇×C₉	C₃×C₄	1	1	yes	characteristic
15	C₄×3₋¹⁺²	D₇	1	1	yes	characteristic
16	C₃×C₄×C₇	—	3	1	no	—
17	C₃×C₄×C₇	C₃×D₃	1	1	yes	characteristic
18	C₃×C₃⋊C₈	—	21	1	no	—
19	C₉⋊C₈	—	7	1	no	—
20	C₃²×C₇	C₃⋊C₈	1	1	yes	characteristic
21	C₇×C₉	—	2	1	no	—
22	C₇×C₉	C₃×C₈	1	1	yes	characteristic, complemented
23	C₇⋊C₈	—	9	1	no	Hall
24	C₂×3₋¹⁺²	Dic₇	1	1	yes	characteristic
25	C₂×C₃×C₇	—	3	1	no	—
26	C₂×C₃×C₇	C₃×Dic₃	1	1	yes	characteristic
27	C₃²×C₄	D₂₁	1	1	yes	characteristic
28	C₄×C₉	—	2	1	no	—
29	C₄×C₉	C₃×D₇	1	1	yes	characteristic
30	C₄×C₇	AGL(1,ℤ/9)	1	1	yes	characteristic
31	3₋¹⁺²	C₇⋊C₈	1	1	yes	characteristic, complemented, Sylow
32	C₃×C₈	—	63	1	no	—
33	C₃⋊C₈	—	21	1	no	—
34	C₃×C₇	—	3	1	no	—
35	C₃×C₇	C₃×C₃⋊C₈	1	1	yes	characteristic
36	C₂×C₃²	Dic₂₁	1	1	yes	characteristic
37	C₂×C₉	—	2	1	no	—
38	C₂×C₉	C₃×Dic₇	1	1	yes	characteristic
39	C₂×C₇	3₋¹⁺²⋊C₄	1	1	yes	characteristic
40	C₃×C₄	—	3	1	no	—
41	C₃×C₄	C₃×D₂₁	1	1	yes	characteristic
42	C₃²	C₃⋊(C₇⋊C₈)	1	1	yes	characteristic
43	C₉	—	2	1	no	—
44	C₉	C₃×C₇⋊C₈	1	1	yes	characteristic, complemented
45	C₈	—	63	1	no	Sylow
46	C₇	3₋¹⁺²⋊C₈	1	1	yes	characteristic, complemented, Sylow
47	C₂×C₃	—	3	1	no	—
48	C₂×C₃	C₃×Dic₂₁	1	1	yes	characteristic
49	C₄	3₋¹⁺²⋊D₇	1	1	yes	characteristic
50	C₃	—	3	1	no	—
51	C₃	C₃×C₃⋊(C₇⋊C₈)	1	1	yes	characteristic
52	C₂	3₋¹⁺²⋊Dic₇	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₄	3₋¹⁺²⋊D₇
commutator, hypocenter	C₇×C₉	C₃×C₈
Frattini	C₃×C₄	C₃×D₂₁
Fitting	C₄×C₇×3₋¹⁺²	C₂
socle	C₂×C₃×C₇	C₃×Dic₃
prime radical	C₂×C₃²×C₇	Dic₃
cosocle	C₄×C₇×C₉	C₂×C₃
LCS	G ▷ C₇×C₉
UCS	C₁ ◁ C₄

Complements

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

N	H ≅ G/N	type
3. C₇⋊(C₉⋊C₈)	C₃	semidirect
9. C₇×3₋¹⁺²	C₈	semidirect
22. C₇×C₉	C₃×C₈	semidirect
31. 3₋¹⁺²	C₇⋊C₈	semidirect
44. C₉	C₃×C₇⋊C₈	semidirect
46. C₇	3₋¹⁺²⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₇×3₋¹⁺²	2
C₃×C₃⋊(C₇⋊C₈)	3
C₇⋊(C₉⋊C₈)	3
3₋¹⁺²⋊C₈	7

Subgroups — 3−1+2⋊(C7⋊C8) #13094177

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — 3₋¹⁺²⋊(C₇⋊C₈) #13094177