Subgroups — C₇²×#90865 #37525560

Order 22932.

200 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇²×C₁₃×C₃⋊D₃	C₂	1	1	yes	maximal, characteristic
2	C₃²×C₇²×C₁₃	C₄	1	1	yes	characteristic, complemented, Hall
3	D₃×C₇²×C₁₃	—	6	2	no	—
4	C₇×#90865	C₇	1	8	yes	maximal, direct product factor
5	C₇²×Dic₁₃	—	9	1	no	maximal, Hall
6	C₃×C₇²×C₁₃	—	2	2	no	—
7	C₇²×#157	—	13	1	no	maximal, Hall
8	C₇×C₁₃×C₃⋊D₃	C₂×C₇	1	8	yes	—
9	C₂×C₇²×C₁₃	—	9	1	no	—
10	C₇²×C₃⋊D₃	D₁₃	1	1	yes	characteristic
11	C₃²×C₇×C₁₃	C₄×C₇	1	8	yes	complemented
12	C₇²×C₁₃	#157	1	1	yes	characteristic, complemented, Hall
13	D₃×C₇×C₁₃	—	6	16	no	—
14	#90865	C₇²	1	1	yes	characteristic, direct product factor, Hall
15	C₃²×C₇²	Dic₁₃	1	1	yes	characteristic, complemented, Hall
16	C₇×Dic₁₃	—	9	8	no	—
17	D₃×C₇²	—	6	2	no	—
18	C₃×C₇×C₁₃	—	2	16	no	—
19	C₇×#157	—	13	8	no	—
20	C₁₃×C₃⋊D₃	C₂×C₇²	1	1	yes	characteristic
21	C₄×C₇²	—	117	1	no	Hall
22	C₂×C₇×C₁₃	—	9	8	no	—
23	C₃×C₇²	—	2	2	no	—
24	C₇×C₃⋊D₃	C₇×D₁₃	1	8	yes	—
25	C₃²×C₁₃	C₄×C₇²	1	1	yes	characteristic, complemented, Hall
26	C₂×C₇²	—	9	1	no	—
27	C₇×C₁₃	C₇×#157	1	8	yes	complemented
28	D₃×C₁₃	—	6	2	no	—
29	C₃²×C₇	C₇×Dic₁₃	1	8	yes	complemented
30	Dic₁₃	—	9	1	no	Hall
31	C₇²	#90865	1	1	yes	characteristic, direct product factor, Sylow
32	D₃×C₇	—	6	16	no	—
33	C₃×C₁₃	—	2	2	no	—
34	#157	—	13	1	no	Hall
35	C₄×C₇	—	117	8	no	—
36	C₂×C₁₃	—	9	1	no	—
37	C₃×C₇	—	2	16	no	—
38	C₃⋊D₃	C₇²×D₁₃	1	1	yes	characteristic
39	C₂×C₇	—	9	8	no	—
40	C₁₃	C₇²×#157	1	1	yes	characteristic, complemented, Sylow
41	C₃²	C₇²×Dic₁₃	1	1	yes	characteristic, complemented, Sylow
42	C₇	C₇×#90865	1	8	yes	direct product factor
43	D₃	—	6	2	no	—
44	C₄	—	117	1	no	Sylow
45	C₃	—	2	2	no	—
46	C₂	—	9	1	no	—

	subgroup	G/H
center, hypercenter	C₇²	#90865
commutator, hypocenter	C₃²×C₁₃	C₄×C₇²
Frattini	C₁	G
Fitting, socle	C₃²×C₇²×C₁₃	C₄
prime radical	C₇²×C₁₃×C₃⋊D₃	C₂
cosocle	C₁₃×C₃⋊D₃	C₂×C₇²
LCS	G ▷ C₃²×C₁₃
UCS	C₁ ◁ C₇²

Complements

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

N	H ≅ G/N	type
2. C₃²×C₇²×C₁₃	C₄	semidirect
4. C₇×#90865	C₇	direct (N × H)
11. C₃²×C₇×C₁₃	C₄×C₇	semidirect
12. C₇²×C₁₃	#157	semidirect
14. #90865	C₇²	direct (N × H)
15. C₃²×C₇²	Dic₁₃	semidirect
25. C₃²×C₁₃	C₄×C₇²	semidirect
27. C₇×C₁₃	C₇×#157	semidirect
29. C₃²×C₇	C₇×Dic₁₃	semidirect
31. C₇²	#90865	direct (N × H)
40. C₁₃	C₇²×#157	semidirect
41. C₃²	C₇²×Dic₁₃	semidirect
42. C₇	C₇×#90865	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×C₁₃×C₃⋊D₃	2
C₇×#90865	7
C₇²×Dic₁₃	9
C₇²×#157	13

Subgroups — C72×#90865 #37525560

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇²×#90865 #37525560