Subgroups — C₇×C₂²⋊#87705 #26014032

Order 11172.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₇²×C₁₉	C₃	1	1	yes	maximal, characteristic, complemented, Hall
2	C₇×#87705	—	4	1	no	maximal, Hall
3	C₂×C₇²×C₁₉	—	3	1	no	—
4	C₂²⋊#87705	C₇	1	1	yes	maximal, characteristic, direct product factor
5	C₇×C₂²⋊(C₁₉⋊C₃)	—	7	1	no	maximal
6	C₇²×C₁₉	A₄	1	1	yes	characteristic, complemented, Hall
7	C₇×C₂²⋊ΓL(1,𝔽₈)	—	19	1	no	maximal, Hall
8	C₂²×C₇×C₁₉	—	3	2	no	—
9	C₂²×C₇×C₁₉	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
10	C₂²×C₇×C₁₉	C₃×C₇	1	1	yes	characteristic, complemented
11	#87705	—	4	1	no	—
12	C₇×C₁₉⋊C₃	—	28	1	no	—
13	C₂×C₇×C₁₉	—	3	6	no	—
14	C₂×C₇×C₁₉	—	3	1	no	—
15	C₂×C₇×C₁₉	—	3	1	no	—
16	C₂²⋊(C₁₉⋊C₃)	—	7	1	no	Hall
17	C₂²×C₇²	C₁₉⋊C₃	1	1	yes	characteristic, complemented, Hall
18	C₇×ΓL(1,𝔽₈)	—	76	1	no	Hall
19	C₇×C₁₉	—	3	2	no	—
20	C₇×C₁₉	C₇×A₄	1	1	yes	characteristic, complemented
21	C₇×C₁₉	C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
22	C₂×C₇²	—	3	1	no	—
23	C₂²⋊ΓL(1,𝔽₈)	—	19	1	no	—
24	C₇×A₄	—	133	1	no	—
25	C₂²×C₁₉	C₇×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
26	C₁₉⋊C₃	—	28	1	no	Hall
27	C₇²	C₂²⋊(C₁₉⋊C₃)	1	1	yes	characteristic, complemented, Sylow
28	C₂×C₁₉	—	3	1	no	—
29	C₂²×C₇	—	3	2	no	—
30	C₂²×C₇	C₇×C₁₉⋊C₃	1	1	yes	characteristic, complemented
31	C₂²×C₇	#87705	1	1	yes	characteristic, complemented
32	C₃×C₇	—	532	1	no	—
33	ΓL(1,𝔽₈)	—	76	1	no	—
34	C₁₉	C₇×C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
35	C₂×C₇	—	3	6	no	—
36	C₂×C₇	—	3	1	no	—
37	C₂×C₇	—	3	1	no	—
38	A₄	—	133	1	no	Hall
39	C₇	—	3	2	no	—
40	C₇	C₇×C₂²⋊(C₁₉⋊C₃)	1	1	yes	characteristic, complemented
41	C₇	C₂²⋊#87705	1	1	yes	characteristic, direct product factor
42	C₂²	C₇×#87705	1	1	yes	characteristic, complemented, Sylow
43	C₃	—	532	1	no	Sylow
44	C₂	—	3	1	no	—

	subgroup	G/H
center, hypercenter	C₇	C₂²⋊#87705
commutator, cosocle, hypocenter	C₂²×C₇×C₁₉	C₃×C₇
Frattini	C₁	G
Fitting, socle	C₂²×C₇²×C₁₉	C₃
LCS	G ▷ C₂²×C₇×C₁₉
UCS	C₁ ◁ C₇

Complements

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

N	H ≅ G/N	type
1. C₂²×C₇²×C₁₉	C₃	semidirect
4. C₂²⋊#87705	C₇	direct (N × H)
6. C₇²×C₁₉	A₄	semidirect
9. C₂²×C₇×C₁₉	ΓL(1,𝔽₈)	semidirect
10. C₂²×C₇×C₁₉	C₃×C₇	semidirect
17. C₂²×C₇²	C₁₉⋊C₃	semidirect
20. C₇×C₁₉	C₇×A₄	semidirect
21. C₇×C₁₉	C₂²⋊ΓL(1,𝔽₈)	semidirect
25. C₂²×C₁₉	C₇×ΓL(1,𝔽₈)	semidirect
27. C₇²	C₂²⋊(C₁₉⋊C₃)	semidirect
30. C₂²×C₇	C₇×C₁₉⋊C₃	semidirect
31. C₂²×C₇	#87705	semidirect
34. C₁₉	C₇×C₂²⋊ΓL(1,𝔽₈)	semidirect
40. C₇	C₇×C₂²⋊(C₁₉⋊C₃)	semidirect
41. C₇	C₂²⋊#87705	direct (N × H)
42. C₂²	C₇×#87705	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₇²×C₁₉	3
C₇×#87705	4
C₂²⋊#87705	7
C₇×C₂²⋊(C₁₉⋊C₃)	7
C₇×C₂²⋊ΓL(1,𝔽₈)	19

Subgroups — C7×C22⋊#87705 #26014032

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×C₂²⋊#87705 #26014032