Subgroups — #30660694

Order 16758.

52 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22552721	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₇×(C₁₉⋊C₃)⋊D₇	C₃	1	1	yes	maximal, characteristic
3	C₇²×C₁₉⋊C₃	C₂×C₃	1	1	yes	characteristic
4	#14558807	—	7	1	no	maximal
5	#14558802	—	7	1	no	maximal
6	C₇×D₁₃₃	C₉	1	1	yes	characteristic, complemented, Hall
7	#11924781	—	7	1	no	—
8	#11924780	—	7	1	no	—
9	C₇²×C₁₉	C₂×C₉	1	1	yes	characteristic, complemented, Hall
10	#11725075	—	19	1	no	maximal, Hall
11	(C₁₉⋊C₃)⋊D₇	ΓL(1,𝔽₈)	1	1	yes	characteristic
12	C₇×D₁₉⋊C₃	—	7	1	no	—
13	#89205	—	19	1	no	Hall
14	C₇×C₁₉⋊C₃	—	6	1	no	—
15	C₇×C₁₉⋊C₃	AGL(1,𝔽₇)	1	1	yes	characteristic
16	C₇×C₁₉⋊C₃	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic
17	AGL(1,𝔽₁₉)	—	49	1	no	Hall
18	C₃×C₇×D₇	—	19	1	no	—
19	D₁₃₃	C₇⋊C₉	1	1	yes	characteristic, complemented
20	C₇×D₁₉	—	7	1	no	—
21	#4296	—	49	1	no	Hall
22	C₃×C₇²	—	19	1	no	—
23	C₇×C₁₉	—	6	1	no	—
24	C₇×C₁₉	D₇⋊C₉	1	1	yes	characteristic, complemented
25	C₇×C₁₉	C₂×C₇⋊C₉	1	1	yes	characteristic, complemented
26	C₂×C₇⋊C₉	—	133	1	no	—
27	D₇⋊C₉	—	133	1	no	—
28	D₁₉⋊C₃	—	21	1	no	—
29	C₇×D₇	—	171	1	no	Hall
30	C₇⋊C₉	—	133	1	no	—
31	C₇⋊C₉	—	133	1	no	—
32	C₁₉⋊C₃	#64422	1	1	yes	characteristic
33	C₇²	AGL(1,𝔽₁₉)	1	1	yes	characteristic, complemented, Sylow
34	C₂×C₃×C₇	—	399	1	no	—
35	C₃×D₇	—	19	1	no	—
36	D₁₉	—	21	1	no	Hall
37	C₃×C₇	—	19	1	no	—
38	C₃×C₇	—	19	1	no	—
39	C₃×C₇	—	114	1	no	—
40	C₁₉	#11725075	1	1	yes	characteristic, complemented, Sylow
41	C₂×C₉	—	931	1	no	Hall
42	C₂×C₇	—	1197	1	no	—
43	D₇	—	171	1	no	—
44	C₉	—	931	1	no	Sylow
45	C₇	—	6	1	no	—
46	C₇	#14558802	1	1	yes	characteristic, complemented
47	C₇	#14558807	1	1	yes	characteristic, complemented
48	C₂×C₃	—	399	1	no	—
49	C₃	—	19	1	no	—
50	C₂	—	1197	1	no	Sylow

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₇²×C₁₉	C₂×C₉
prime radical	C₇×(C₁₉⋊C₃)⋊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. #22552721	C₂	semidirect
6. C₇×D₁₃₃	C₉	semidirect
9. C₇²×C₁₉	C₂×C₉	semidirect
19. D₁₃₃	C₇⋊C₉	semidirect
24. C₇×C₁₉	D₇⋊C₉	semidirect
25. C₇×C₁₉	C₂×C₇⋊C₉	semidirect
33. C₇²	AGL(1,𝔽₁₉)	semidirect
40. C₁₉	#11725075	semidirect
46. C₇	#14558802	semidirect
47. C₇	#14558807	semidirect

Maximal subgroups

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

subgroup	index
#22552721	2
C₇×(C₁₉⋊C₃)⋊D₇	3
#14558807	7
#14558802	7
#11725075	19