Subgroups — C₇²×#10597480 #44472912

Order 29988.

200 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇²×C₁₇×C₃⋊D₃	C₂	1	1	yes	maximal, characteristic
2	C₃²×C₇²×C₁₇	C₄	1	1	yes	characteristic, complemented, Hall
3	D₃×C₇²×C₁₇	—	6	2	no	—
4	C₇×#10597480	C₇	1	8	yes	maximal, direct product factor
5	C₇²×Dic₁₇	—	9	1	no	maximal, Hall
6	C₃×C₇²×C₁₇	—	2	2	no	—
7	C₇×C₁₇×C₃⋊D₃	C₂×C₇	1	8	yes	—
8	C₇²×#157	—	17	1	no	maximal, Hall
9	C₂×C₇²×C₁₇	—	9	1	no	—
10	C₃²×C₇×C₁₇	C₄×C₇	1	8	yes	complemented
11	C₇²×C₃⋊D₃	D₁₇	1	1	yes	characteristic
12	C₇²×C₁₇	#157	1	1	yes	characteristic, complemented, Hall
13	D₃×C₇×C₁₇	—	6	16	no	—
14	#10597480	C₇²	1	1	yes	characteristic, direct product factor, Hall
15	C₇×Dic₁₇	—	9	8	no	—
16	C₃²×C₇²	Dic₁₇	1	1	yes	characteristic, complemented, Hall
17	C₃×C₇×C₁₇	—	2	16	no	—
18	C₁₇×C₃⋊D₃	C₂×C₇²	1	1	yes	characteristic
19	D₃×C₇²	—	6	2	no	—
20	C₇×#157	—	17	8	no	—
21	C₂×C₇×C₁₇	—	9	8	no	—
22	C₄×C₇²	—	153	1	no	Hall
23	C₃²×C₁₇	C₄×C₇²	1	1	yes	characteristic, complemented, Hall
24	C₃×C₇²	—	2	2	no	—
25	C₇×C₃⋊D₃	C₇×D₁₇	1	8	yes	—
26	C₇×C₁₇	C₇×#157	1	8	yes	complemented
27	D₃×C₁₇	—	6	2	no	—
28	C₂×C₇²	—	9	1	no	—
29	Dic₁₇	—	9	1	no	Hall
30	C₃²×C₇	C₇×Dic₁₇	1	8	yes	complemented
31	C₃×C₁₇	—	2	2	no	—
32	C₇²	#10597480	1	1	yes	characteristic, direct product factor, Sylow
33	D₃×C₇	—	6	16	no	—
34	#157	—	17	1	no	Hall
35	C₂×C₁₇	—	9	1	no	—
36	C₄×C₇	—	153	8	no	—
37	C₃×C₇	—	2	16	no	—
38	C₃⋊D₃	C₇²×D₁₇	1	1	yes	characteristic
39	C₁₇	C₇²×#157	1	1	yes	characteristic, complemented, Sylow
40	C₂×C₇	—	9	8	no	—
41	C₃²	C₇²×Dic₁₇	1	1	yes	characteristic, complemented, Sylow
42	C₇	C₇×#10597480	1	8	yes	direct product factor
43	D₃	—	6	2	no	—
44	C₄	—	153	1	no	Sylow
45	C₃	—	2	2	no	—
46	C₂	—	9	1	no	—

	subgroup	G/H
center, hypercenter	C₇²	#10597480
commutator, hypocenter	C₃²×C₁₇	C₄×C₇²
Frattini	C₁	G
Fitting, socle	C₃²×C₇²×C₁₇	C₄
prime radical	C₇²×C₁₇×C₃⋊D₃	C₂
cosocle	C₁₇×C₃⋊D₃	C₂×C₇²
LCS	G ▷ C₃²×C₁₇
UCS	C₁ ◁ C₇²

Complements

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

N	H ≅ G/N	type
2. C₃²×C₇²×C₁₇	C₄	semidirect
4. C₇×#10597480	C₇	direct (N × H)
10. C₃²×C₇×C₁₇	C₄×C₇	semidirect
12. C₇²×C₁₇	#157	semidirect
14. #10597480	C₇²	direct (N × H)
16. C₃²×C₇²	Dic₁₇	semidirect
23. C₃²×C₁₇	C₄×C₇²	semidirect
26. C₇×C₁₇	C₇×#157	semidirect
30. C₃²×C₇	C₇×Dic₁₇	semidirect
32. C₇²	#10597480	direct (N × H)
39. C₁₇	C₇²×#157	semidirect
41. C₃²	C₇²×Dic₁₇	semidirect
42. C₇	C₇×#10597480	direct (N × H)

Maximal subgroups

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

subgroup	index
C₇²×C₁₇×C₃⋊D₃	2
C₇×#10597480	7
C₇²×Dic₁₇	9
C₇²×#157	17

Subgroups — C72×#10597480 #44472912

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇²×#10597480 #44472912