Subgroups — #30660650

Order 16758.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22552719	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₁₉⋊C₃×C₇⋊D₇	C₃	1	1	yes	maximal, characteristic
3	C₇²×C₁₉⋊C₃	C₂×C₃	1	1	yes	characteristic
4	#14558804	—	7	1	no	maximal
5	#14558804	—	7	7	no	maximal
6	C₁₉×C₇⋊D₇	C₉	1	1	yes	characteristic, complemented, Hall
7	#11924781	—	7	1	no	—
8	#11924781	—	7	7	no	—
9	C₇²×C₁₉	C₂×C₉	1	1	yes	characteristic, complemented, Hall
10	#11725077	—	19	1	no	maximal, Hall
11	D₇×C₁₉⋊C₃	—	7	1	no	—
12	D₇×C₁₉⋊C₃	—	21	7	no	—
13	#89204	—	19	1	no	Hall
14	C₇×C₁₉⋊C₃	AGL(1,𝔽₇)	1	1	yes	characteristic
15	C₇×C₁₉⋊C₃	AGL(1,𝔽₇)	1	7	yes	—
16	C₂×#4296	—	49	1	no	Hall
17	C₃×C₇⋊D₇	—	19	1	no	—
18	D₇×C₁₉	—	7	1	no	—
19	D₇×C₁₉	—	21	7	no	—
20	#4296	—	49	1	no	Hall
21	C₃×C₇²	—	19	1	no	—
22	C₇×C₁₉	D₇⋊C₉	1	1	yes	characteristic, complemented
23	C₇×C₁₉	D₇⋊C₉	1	7	yes	complemented
24	D₇⋊C₉	—	133	1	no	—
25	D₇⋊C₉	—	133	7	no	—
26	C₂×C₁₉⋊C₃	—	147	1	no	—
27	C₇⋊D₇	#4296	1	1	yes	characteristic, complemented, Hall
28	C₇⋊C₉	—	133	1	no	—
29	C₇⋊C₉	—	133	7	no	—
30	C₁₉⋊C₃	#64424	1	1	yes	characteristic
31	C₇²	C₂×#4296	1	1	yes	characteristic, complemented, Sylow
32	C₃×D₇	—	399	1	no	—
33	C₃×D₇	—	399	7	no	—
34	C₂×C₁₉	—	147	1	no	Hall
35	C₃×C₇	—	19	1	no	—
36	C₃×C₇	—	19	7	no	—
37	C₁₉	#11725077	1	1	yes	characteristic, complemented, Sylow
38	C₂×C₉	—	931	1	no	Hall
39	D₇	—	21	1	no	—
40	D₇	—	21	7	no	—
41	C₉	—	931	1	no	Sylow
42	C₇	#14558804	1	1	yes	characteristic, complemented
43	C₇	#14558804	1	7	yes	complemented
44	C₂×C₃	—	2793	1	no	—
45	C₃	—	19	1	no	—
46	C₂	—	147	1	no	Sylow

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₇²×C₁₉	C₂×C₉
prime radical	C₁₉⋊C₃×C₇⋊D₇	C₃
cosocle	C₇²×C₁₉⋊C₃	C₂×C₃
LCS	G ▷ C₇²×C₁₉
UCS	C₁

Complements

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

N	H ≅ G/N	type
1. #22552719	C₂	semidirect
6. C₁₉×C₇⋊D₇	C₉	semidirect
9. C₇²×C₁₉	C₂×C₉	semidirect
22. C₇×C₁₉	D₇⋊C₉	semidirect
23. C₇×C₁₉	D₇⋊C₉	semidirect
27. C₇⋊D₇	#4296	semidirect
31. C₇²	C₂×#4296	semidirect
37. C₁₉	#11725077	semidirect
42. C₇	#14558804	semidirect
43. C₇	#14558804	semidirect

Maximal subgroups

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

subgroup	index
#22552719	2
C₁₉⋊C₃×C₇⋊D₇	3
#14558804	7
#11725077	19