Subgroups — #30660729

Order 16758.

64 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×ΓL(1,𝔽₈)²	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₁₉⋊#64418	C₃	1	1	yes	maximal, characteristic, complemented
3	#19150298	—	3	1	no	maximal
4	C₁₉×#3862	C₂×C₃	1	1	yes	characteristic, complemented
5	C₁₉×#3861	D₃	1	1	yes	characteristic, complemented
6	C₇×C₁₉×ΓL(1,𝔽₈)	—	2	1	no	—
7	C₇×D₁₃₃	—	3	1	no	Hall
8	C₃×C₁₉×ΓL(1,𝔽₈)	—	14	1	no	—
9	C₇²×C₁₉	C₃×D₃	1	1	yes	characteristic, complemented, Hall
10	#64418⋊C₃	—	19	1	no	maximal, Hall
11	ΓL(1,𝔽₈)⋊D₁₉	—	21	1	no	—
12	ΓL(1,𝔽₈)×D₁₉	—	21	1	no	—
13	ΓL(1,𝔽₈)²	D₁₉	1	1	yes	characteristic, complemented, Hall
14	C₃×C₇×C₁₉	—	14	1	no	—
15	C₁₉×ΓL(1,𝔽₈)	—	2	1	no	—
16	C₁₉×ΓL(1,𝔽₈)	—	14	1	no	—
17	C₁₉×ΓL(1,𝔽₈)	—	14	1	no	—
18	C₁₉×ΓL(1,𝔽₈)	—	21	1	no	—
19	C₁₉×ΓL(1,𝔽₈)	—	42	1	no	—
20	C₃×D₅₇	—	49	1	no	maximal, Hall
21	#64423	—	57	1	no	—
22	#64418	—	19	1	no	—
23	D₁₃₃	—	3	1	no	—
24	C₇×D₁₉	—	21	1	no	—
25	C₃²×C₁₉	—	98	1	no	Hall
26	#3862	C₃×D₁₉	1	1	yes	characteristic, complemented
27	#3861	D₅₇	1	1	yes	characteristic, complemented
28	C₇×ΓL(1,𝔽₈)	—	2	1	no	—
29	C₇×C₁₉	—	2	1	no	—
30	C₇×C₁₉	—	6	1	no	—
31	C₇×C₁₉	—	6	1	no	—
32	D₅₇	—	49	1	no	—
33	C₃×D₁₉	—	147	1	no	—
34	C₇×D₇	—	57	1	no	Hall
35	C₃×ΓL(1,𝔽₈)	—	14	1	no	—
36	C₃×C₁₉	—	14	1	no	—
37	C₃×C₁₉	—	98	1	no	—
38	C₃×C₁₉	—	98	1	no	—
39	C₇²	C₃×D₅₇	1	1	yes	characteristic, complemented, Sylow
40	C₂×ΓL(1,𝔽₈)	—	399	1	no	—
41	AGL(1,𝔽₇)	—	399	1	no	—
42	D₁₉	—	21	1	no	Hall
43	C₃×C₇	—	14	1	no	—
44	ΓL(1,𝔽₈)	—	2	1	no	—
45	ΓL(1,𝔽₈)	—	14	1	no	—
46	ΓL(1,𝔽₈)	—	14	1	no	—
47	ΓL(1,𝔽₈)	—	21	1	no	—
48	ΓL(1,𝔽₈)	—	42	1	no	—
49	C₁₉	#64418⋊C₃	1	1	yes	characteristic, complemented, Sylow
50	C₃×D₃	—	931	1	no	Hall
51	C₂×C₇	—	399	1	no	—
52	D₇	—	57	1	no	—
53	C₃²	—	98	1	no	Sylow
54	C₇	—	2	1	no	—
55	C₇	—	3	1	no	—
56	C₇	—	6	1	no	—
57	C₂×C₃	—	2793	1	no	—
58	D₃	—	931	1	no	—
59	C₃	—	14	1	no	—
60	C₃	—	98	1	no	—
61	C₃	—	98	1	no	—
62	C₂	—	399	1	no	Sylow

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₁₉×#3862	C₂×C₃
Fitting, socle	C₇²×C₁₉	C₃×D₃
LCS	G ▷ C₁₉×#3862
UCS	C₁

Complements

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

N	H ≅ G/N	type
1. C₁₉×ΓL(1,𝔽₈)²	C₂	semidirect
2. C₁₉⋊#64418	C₃	semidirect
4. C₁₉×#3862	C₂×C₃	semidirect
5. C₁₉×#3861	D₃	semidirect
9. C₇²×C₁₉	C₃×D₃	semidirect
13. ΓL(1,𝔽₈)²	D₁₉	semidirect
26. #3862	C₃×D₁₉	semidirect
27. #3861	D₅₇	semidirect
39. C₇²	C₃×D₅₇	semidirect
49. C₁₉	#64418⋊C₃	semidirect

Maximal subgroups

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

subgroup	index
C₁₉×ΓL(1,𝔽₈)²	2
C₁₉⋊#64418	3
#19150298	3
#64418⋊C₃	19
C₃×D₅₇	49