Subgroups — #30660578

Order 16758.

64 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22552712	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₃×C₇⋊D₁₃₃	C₃	1	1	yes	maximal, characteristic
3	C₃×C₇²×C₁₉	C₂×C₃	1	1	yes	characteristic
4	#14558787	—	7	1	no	maximal
5	#14558786	—	7	1	no	maximal
6	C₇⋊D₁₃₃	C₉	1	1	yes	characteristic, complemented, Hall
7	#11924777	—	7	1	no	—
8	#11924776	—	7	1	no	—
9	C₇²×C₁₉	C₂×C₉	1	1	yes	characteristic, complemented, Hall
10	#11725078	—	19	1	no	maximal, Hall
11	C₃×D₁₃₃	—	21	2	no	—
12	C₃×D₁₃₃	—	21	1	no	—
13	C₃×D₁₃₃	—	21	1	no	—
14	#89205	—	19	1	no	Hall
15	C₃×C₇×C₁₉	—	3	2	no	—
16	C₃×C₇×C₁₉	AGL(1,𝔽₇)	1	1	yes	characteristic
17	C₃×C₇×C₁₉	AGL(1,𝔽₇)	1	1	yes	characteristic
18	#4294⋊C₂	—	49	1	no	Hall
19	C₃×C₇⋊D₇	—	57	1	no	—
20	D₁₃₃	—	21	2	no	—
21	D₁₃₃	—	21	1	no	—
22	D₁₃₃	—	21	1	no	—
23	#4294	—	49	1	no	Hall
24	C₃×C₇²	D₁₉⋊C₃	1	1	yes	characteristic
25	C₇×C₁₉	—	3	2	no	—
26	C₇×C₁₉	D₇⋊C₉	1	1	yes	characteristic, complemented
27	C₇×C₁₉	D₇⋊C₉	1	1	yes	characteristic, complemented
28	D₇⋊C₉	—	133	1	no	—
29	D₇⋊C₉	—	133	1	no	—
30	C₃×D₁₉	—	147	1	no	—
31	C₇⋊D₇	—	57	1	no	Hall
32	C₇⋊C₉	—	133	1	no	—
33	C₇⋊C₉	—	133	1	no	—
34	C₃×C₁₉	#64425	1	1	yes	characteristic
35	C₇²	#4294⋊C₂	1	1	yes	characteristic, complemented, Sylow
36	C₃×D₇	—	399	2	no	—
37	C₃×D₇	—	399	1	no	—
38	C₃×D₇	—	399	1	no	—
39	D₁₉	—	147	1	no	Hall
40	C₃×C₇	—	3	2	no	—
41	C₃×C₇	#11716327	1	1	yes	characteristic
42	C₃×C₇	#11716328	1	1	yes	characteristic
43	C₁₉	#11725078	1	1	yes	characteristic, complemented, Sylow
44	C₂×C₉	—	931	1	no	Hall
45	D₇	—	399	2	no	—
46	D₇	—	399	1	no	—
47	D₇	—	399	1	no	—
48	C₉	—	931	1	no	Sylow
49	C₇	—	3	2	no	—
50	C₇	#14558786	1	1	yes	characteristic, complemented
51	C₇	#14558787	1	1	yes	characteristic, complemented
52	C₂×C₃	—	2793	1	no	—
53	C₃	#19150290	1	1	yes	characteristic
54	C₂	—	2793	1	no	Sylow

	subgroup	G/H
center, Frattini, hypercenter	C₃	#19150290
commutator, hypocenter	C₇²×C₁₉	C₂×C₉
Fitting, socle, cosocle	C₃×C₇²×C₁₉	C₂×C₃
prime radical	C₃×C₇⋊D₁₃₃	C₃
LCS	G ▷ C₇²×C₁₉
UCS	C₁ ◁ C₃

Complements

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

N	H ≅ G/N	type
1. #22552712	C₂	semidirect
6. C₇⋊D₁₃₃	C₉	semidirect
9. C₇²×C₁₉	C₂×C₉	semidirect
26. C₇×C₁₉	D₇⋊C₉	semidirect
27. C₇×C₁₉	D₇⋊C₉	semidirect
35. C₇²	#4294⋊C₂	semidirect
43. C₁₉	#11725078	semidirect
50. C₇	#14558786	semidirect
51. C₇	#14558787	semidirect

Maximal subgroups

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

subgroup	index
#22552712	2
C₃×C₇⋊D₁₃₃	3
#14558787	7
#14558786	7
#11725078	19