Subgroups — C₇²×Dic₃×C₁₃⋊C₃ #37524751

Order 22932.

280 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₇²×C₁₃⋊C₃	C₂	1	1	yes	maximal, characteristic
2	C₇²×Dic₃×C₁₃	C₃	1	1	yes	maximal, characteristic, complemented
3	C₄×C₇²×C₁₃⋊C₃	—	3	1	no	maximal
4	C₃×C₇²×C₁₃⋊C₃	C₄	1	1	yes	characteristic, complemented, Hall
5	C₂×C₇²×C₁₃⋊C₃	—	2	1	no	—
6	C₂×C₇²×C₁₃⋊C₃	D₃	1	1	yes	characteristic
7	C₂×C₃×C₇²×C₁₃	C₂×C₃	1	1	yes	characteristic
8	C₇×Dic₃×C₁₃⋊C₃	C₇	1	8	yes	maximal, direct product factor
9	C₄×C₇²×C₁₃	—	3	1	no	Hall
10	C₃×C₇²×C₁₃	C₃×C₄	1	1	yes	characteristic, complemented
11	C₇²×C₁₃⋊C₃	—	2	1	no	—
12	C₇²×C₁₃⋊C₃	Dic₃	1	1	yes	characteristic, direct product factor
13	C₃×C₇²×Dic₃	—	13	1	no	maximal, Hall
14	C₂×C₃×C₇×C₁₃⋊C₃	C₂×C₇	1	8	yes	—
15	C₂×C₇²×C₁₃	C₃×D₃	1	1	yes	characteristic
16	C₇×Dic₃×C₁₃	C₃×C₇	1	8	yes	complemented
17	C₄×C₇×C₁₃⋊C₃	—	3	8	no	—
18	C₂×C₃²×C₇²	—	13	1	no	—
19	C₃×C₇×C₁₃⋊C₃	C₄×C₇	1	8	yes	complemented
20	C₇²×C₁₃	C₃×Dic₃	1	1	yes	characteristic, complemented, Hall
21	C₃×C₄×C₇²	—	39	1	no	—
22	C₇²×Dic₃	C₁₃⋊C₃	1	1	yes	characteristic, direct product factor
23	C₂×C₃×C₇×C₁₃	C₂×C₃×C₇	1	8	yes	—
24	C₂×C₇×C₁₃⋊C₃	—	2	8	no	—
25	C₂×C₇×C₁₃⋊C₃	D₃×C₇	1	8	yes	—
26	Dic₃×C₁₃⋊C₃	C₇²	1	1	yes	characteristic, direct product factor, Hall
27	C₃²×C₇²	—	13	1	no	Hall
28	C₄×C₇×C₁₃	—	3	8	no	—
29	C₂×C₃×C₇²	—	13	1	no	—
30	C₂×C₃×C₇²	—	26	1	no	—
31	C₂×C₃×C₇²	C₂×C₁₃⋊C₃	1	1	yes	characteristic
32	C₃×C₇×C₁₃	C₃×C₄×C₇	1	8	yes	complemented
33	C₇×C₁₃⋊C₃	—	2	8	no	—
34	C₇×C₁₃⋊C₃	C₇×Dic₃	1	8	yes	direct product factor
35	C₃×C₇×Dic₃	—	13	8	no	—
36	C₂×C₃×C₁₃⋊C₃	C₂×C₇²	1	1	yes	characteristic
37	C₄×C₇²	—	3	1	no	Hall
38	C₂×C₇×C₁₃	C₃×D₃×C₇	1	8	yes	—
39	Dic₃×C₁₃	C₃×C₇²	1	1	yes	characteristic, complemented
40	C₄×C₁₃⋊C₃	—	3	1	no	—
41	C₃×C₇²	—	13	1	no	—
42	C₃×C₇²	—	26	1	no	—
43	C₃×C₇²	C₄×C₁₃⋊C₃	1	1	yes	characteristic, complemented
44	C₂×C₃²×C₇	—	13	8	no	—
45	C₃×C₁₃⋊C₃	C₄×C₇²	1	1	yes	characteristic, complemented, Hall
46	C₂×C₇²	D₃×C₁₃⋊C₃	1	1	yes	characteristic
47	C₇×C₁₃	C₃×C₇×Dic₃	1	8	yes	complemented
48	C₃×C₄×C₇	—	39	8	no	—
49	C₇×Dic₃	C₇×C₁₃⋊C₃	1	8	yes	direct product factor
50	C₂×C₃×C₁₃	C₂×C₃×C₇²	1	1	yes	characteristic
51	C₂×C₁₃⋊C₃	—	2	1	no	—
52	C₂×C₁₃⋊C₃	D₃×C₇²	1	1	yes	characteristic
53	C₃²×C₇	—	13	8	no	—
54	C₄×C₁₃	—	3	1	no	Hall
55	C₇²	Dic₃×C₁₃⋊C₃	1	1	yes	characteristic, direct product factor, Sylow
56	C₂×C₃×C₇	—	13	8	no	—
57	C₂×C₃×C₇	—	26	8	no	—
58	C₂×C₃×C₇	C₂×C₇×C₁₃⋊C₃	1	8	yes	—
59	C₃×C₁₃	C₃×C₄×C₇²	1	1	yes	characteristic, complemented
60	C₁₃⋊C₃	—	2	1	no	—
61	C₁₃⋊C₃	C₇²×Dic₃	1	1	yes	characteristic, direct product factor
62	C₃×Dic₃	—	13	1	no	Hall
63	C₄×C₇	—	3	8	no	—
64	C₂×C₁₃	C₃×D₃×C₇²	1	1	yes	characteristic
65	C₃×C₇	—	13	8	no	—
66	C₃×C₇	—	26	8	no	—
67	C₃×C₇	C₄×C₇×C₁₃⋊C₃	1	8	yes	complemented
68	C₂×C₃²	—	13	1	no	—
69	C₂×C₇	D₃×C₇×C₁₃⋊C₃	1	8	yes	—
70	C₁₃	C₃×C₇²×Dic₃	1	1	yes	characteristic, complemented, Sylow
71	C₃×C₄	—	39	1	no	—
72	Dic₃	C₇²×C₁₃⋊C₃	1	1	yes	characteristic, direct product factor
73	C₃²	—	13	1	no	Sylow
74	C₇	C₇×Dic₃×C₁₃⋊C₃	1	8	yes	direct product factor
75	C₂×C₃	—	13	1	no	—
76	C₂×C₃	—	26	1	no	—
77	C₂×C₃	C₂×C₇²×C₁₃⋊C₃	1	1	yes	characteristic
78	C₄	—	3	1	no	Sylow
79	C₃	—	13	1	no	—
80	C₃	—	26	1	no	—
81	C₃	C₄×C₇²×C₁₃⋊C₃	1	1	yes	characteristic, complemented
82	C₂	D₃×C₇²×C₁₃⋊C₃	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₇²×Dic₃×C₁₃	C₃	semidirect
4. C₃×C₇²×C₁₃⋊C₃	C₄	semidirect
8. C₇×Dic₃×C₁₃⋊C₃	C₇	direct (N × H)
10. C₃×C₇²×C₁₃	C₃×C₄	semidirect
12. C₇²×C₁₃⋊C₃	Dic₃	direct (N × H)
16. C₇×Dic₃×C₁₃	C₃×C₇	semidirect
19. C₃×C₇×C₁₃⋊C₃	C₄×C₇	semidirect
20. C₇²×C₁₃	C₃×Dic₃	semidirect
22. C₇²×Dic₃	C₁₃⋊C₃	direct (N × H)
26. Dic₃×C₁₃⋊C₃	C₇²	direct (N × H)
32. C₃×C₇×C₁₃	C₃×C₄×C₇	semidirect
34. C₇×C₁₃⋊C₃	C₇×Dic₃	direct (N × H)
39. Dic₃×C₁₃	C₃×C₇²	semidirect
43. C₃×C₇²	C₄×C₁₃⋊C₃	semidirect
45. C₃×C₁₃⋊C₃	C₄×C₇²	semidirect
47. C₇×C₁₃	C₃×C₇×Dic₃	semidirect
49. C₇×Dic₃	C₇×C₁₃⋊C₃	direct (N × H)
55. C₇²	Dic₃×C₁₃⋊C₃	direct (N × H)
59. C₃×C₁₃	C₃×C₄×C₇²	semidirect
61. C₁₃⋊C₃	C₇²×Dic₃	direct (N × H)
67. C₃×C₇	C₄×C₇×C₁₃⋊C₃	semidirect
70. C₁₃	C₃×C₇²×Dic₃	semidirect
72. Dic₃	C₇²×C₁₃⋊C₃	direct (N × H)
74. C₇	C₇×Dic₃×C₁₃⋊C₃	direct (N × H)
81. C₃	C₄×C₇²×C₁₃⋊C₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₇²×C₁₃⋊C₃	2
C₇²×Dic₃×C₁₃	3
C₄×C₇²×C₁₃⋊C₃	3
C₇×Dic₃×C₁₃⋊C₃	7
C₃×C₇²×Dic₃	13

Subgroups — C72×Dic3×C13⋊C3 #37524751

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇²×Dic₃×C₁₃⋊C₃ #37524751