Subgroups — C₂²×#14584116 #26014044

Order 11172.

200 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14584116	C₂	1	3	yes	maximal, direct product factor
2	C₂²×C₇²×C₁₉	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	#14584116	C₂²	1	1	yes	characteristic, direct product factor, Hall
4	C₂×C₇²×C₁₉	C₂×C₃	1	3	yes	complemented
5	C₂²×#87705	—	7	8	no	maximal
6	C₇²×C₁₉	C₂²×C₃	1	1	yes	characteristic, complemented, Hall
7	C₂×#87705	—	7	24	no	—
8	C₂²×#3861	—	19	1	no	maximal, Hall
9	C₂²×C₇×C₁₉	ΓL(1,𝔽₈)	1	8	yes	complemented
10	#87705	—	7	8	no	—
11	C₂×#3861	—	19	3	no	—
12	C₂×C₇×C₁₉	C₂×ΓL(1,𝔽₈)	1	24	yes	complemented
13	C₂²×C₁₉⋊C₃	—	49	1	no	Hall
14	C₂²×C₇²	C₁₉⋊C₃	1	1	yes	characteristic, complemented, Hall
15	#3861	—	19	1	no	Hall
16	C₇×C₁₉	C₂²×ΓL(1,𝔽₈)	1	8	yes	complemented
17	C₂×C₁₉⋊C₃	—	49	3	no	—
18	C₂×C₇²	C₂×C₁₉⋊C₃	1	3	yes	complemented
19	C₂²×ΓL(1,𝔽₈)	—	133	8	no	—
20	C₂²×C₁₉	#3861	1	1	yes	characteristic, complemented, Hall
21	C₁₉⋊C₃	—	49	1	no	Hall
22	C₇²	C₂²×C₁₉⋊C₃	1	1	yes	characteristic, complemented, Sylow
23	C₂×ΓL(1,𝔽₈)	—	133	24	no	—
24	C₂×C₁₉	C₂×#3861	1	3	yes	complemented
25	C₂²×C₇	#87705	1	8	yes	complemented
26	ΓL(1,𝔽₈)	—	133	8	no	—
27	C₁₉	C₂²×#3861	1	1	yes	characteristic, complemented, Sylow
28	C₂×C₇	C₂×#87705	1	24	yes	complemented
29	C₂²×C₃	—	931	1	no	Hall
30	C₇	C₂²×#87705	1	8	yes	complemented
31	C₂×C₃	—	931	3	no	—
32	C₂²	#14584116	1	1	yes	characteristic, direct product factor, Sylow
33	C₃	—	931	1	no	Sylow
34	C₂	C₂×#14584116	1	3	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂²	#14584116
commutator, cosocle, hypocenter	C₇²×C₁₉	C₂²×C₃
Frattini	C₁	G
Fitting, socle	C₂²×C₇²×C₁₉	C₃
LCS	G ▷ C₇²×C₁₉
UCS	C₁ ◁ C₂²

Complements

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

N	H ≅ G/N	type
1. C₂×#14584116	C₂	direct (N × H)
2. C₂²×C₇²×C₁₉	C₃	semidirect
3. #14584116	C₂²	direct (N × H)
4. C₂×C₇²×C₁₉	C₂×C₃	semidirect
6. C₇²×C₁₉	C₂²×C₃	semidirect
9. C₂²×C₇×C₁₉	ΓL(1,𝔽₈)	semidirect
12. C₂×C₇×C₁₉	C₂×ΓL(1,𝔽₈)	semidirect
14. C₂²×C₇²	C₁₉⋊C₃	semidirect
16. C₇×C₁₉	C₂²×ΓL(1,𝔽₈)	semidirect
18. C₂×C₇²	C₂×C₁₉⋊C₃	semidirect
20. C₂²×C₁₉	#3861	semidirect
22. C₇²	C₂²×C₁₉⋊C₃	semidirect
24. C₂×C₁₉	C₂×#3861	semidirect
25. C₂²×C₇	#87705	semidirect
27. C₁₉	C₂²×#3861	semidirect
28. C₂×C₇	C₂×#87705	semidirect
30. C₇	C₂²×#87705	semidirect
32. C₂²	#14584116	direct (N × H)
34. C₂	C₂×#14584116	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#14584116	2
C₂²×C₇²×C₁₉	3
C₂²×#87705	7
C₂²×#3861	19

Subgroups — C22×#14584116 #26014044

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#14584116 #26014044