Subgroups — C₃×#261×#3862 #37525118

Order 22932.

168 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×D₁₃×#3862	C₂	1	1	yes	maximal, characteristic
2	C₃×C₇²×#261	C₃	1	1	yes	maximal, characteristic, complemented
3	#261×#3862	C₃	1	3	yes	maximal, direct product factor
4	C₃×C₁₃×#3862	C₄	1	1	yes	characteristic, complemented, Hall
5	C₃×C₇²×D₁₃	C₂×C₃	1	1	yes	characteristic
6	D₁₃×#3862	C₂×C₃	1	3	yes	—
7	C₃×ΓL(1,𝔽₈)×#261	—	7	2	no	maximal
8	C₇²×#261	C₃²	1	1	yes	characteristic, complemented, Hall
9	C₃×C₇²×C₁₃	C₃×C₄	1	1	yes	characteristic, complemented
10	C₁₃×#3862	C₃×C₄	1	3	yes	complemented
11	C₃×C₄×#3862	—	13	1	no	maximal, Hall
12	C₃×ΓL(1,𝔽₈)×D₁₃	—	7	2	no	—
13	C₇²×D₁₃	C₂×C₃²	1	1	yes	characteristic
14	C₃×C₇×#261	—	3	2	no	—
15	C₃×C₇×#261	ΓL(1,𝔽₈)	1	2	yes	complemented
16	ΓL(1,𝔽₈)×#261	—	7	6	no	—
17	C₂×C₃×#3862	—	13	1	no	—
18	C₃×C₁₃×ΓL(1,𝔽₈)	—	7	2	no	—
19	C₇²×C₁₃	C₃²×C₄	1	1	yes	characteristic, complemented, Hall
20	C₃×C₄×C₇²	—	13	1	no	—
21	C₄×#3862	—	13	3	no	—
22	C₃×C₇×D₁₃	—	3	2	no	—
23	C₃×C₇×D₁₃	C₂×ΓL(1,𝔽₈)	1	2	yes	—
24	ΓL(1,𝔽₈)×D₁₃	—	7	6	no	—
25	C₃²×#261	—	49	1	no	Hall
26	C₃×#3862	#261	1	1	yes	characteristic, direct product factor, Hall
27	C₇×#261	—	3	2	no	—
28	C₇×#261	C₃×ΓL(1,𝔽₈)	1	2	yes	complemented
29	C₂×C₃×C₇²	—	13	1	no	—
30	C₂×#3862	—	13	3	no	—
31	C₃×C₇×C₁₃	—	3	2	no	—
32	C₃×C₇×C₁₃	C₄×ΓL(1,𝔽₈)	1	2	yes	complemented
33	C₁₃×ΓL(1,𝔽₈)	—	7	6	no	—
34	C₃×C₄×ΓL(1,𝔽₈)	—	91	2	no	—
35	C₃²×D₁₃	—	49	1	no	—
36	C₄×C₇²	—	13	1	no	Hall
37	C₇×D₁₃	—	3	2	no	—
38	C₇×D₁₃	C₂×C₃×ΓL(1,𝔽₈)	1	2	yes	—
39	C₃×#261	—	49	3	no	—
40	C₃×#261	#3862	1	1	yes	characteristic, direct product factor
41	C₃×C₇²	C₃×#261	1	1	yes	characteristic, complemented
42	#3862	C₃×#261	1	3	yes	direct product factor
43	C₂×C₃×ΓL(1,𝔽₈)	—	91	2	no	—
44	C₃²×C₁₃	—	49	1	no	Hall
45	C₂×C₇²	—	13	1	no	—
46	C₇×C₁₃	—	3	2	no	—
47	C₇×C₁₃	C₃×C₄×ΓL(1,𝔽₈)	1	2	yes	complemented
48	C₃×C₄×C₇	—	13	2	no	—
49	C₃×C₄×C₇	—	39	2	no	—
50	C₄×ΓL(1,𝔽₈)	—	91	6	no	—
51	C₃×D₁₃	—	49	3	no	—
52	C₃×D₁₃	C₂×#3862	1	1	yes	characteristic
53	C₃×ΓL(1,𝔽₈)	—	7	2	no	—
54	#261	C₃×#3862	1	1	yes	characteristic, direct product factor, Hall
55	C₇²	C₃²×#261	1	1	yes	characteristic, complemented, Sylow
56	C₂×C₃×C₇	—	13	2	no	—
57	C₂×C₃×C₇	—	39	2	no	—
58	C₂×ΓL(1,𝔽₈)	—	91	6	no	—
59	C₃×C₁₃	—	49	3	no	—
60	C₃×C₁₃	C₄×#3862	1	1	yes	characteristic, complemented
61	C₃²×C₄	—	637	1	no	Hall
62	C₄×C₇	—	13	2	no	—
63	C₄×C₇	—	39	2	no	—
64	D₁₃	C₂×C₃×#3862	1	1	yes	characteristic
65	C₃×C₇	—	3	2	no	—
66	C₃×C₇	ΓL(1,𝔽₈)×#261	1	2	yes	complemented
67	ΓL(1,𝔽₈)	—	7	6	no	—
68	C₂×C₃²	—	637	1	no	—
69	C₂×C₇	—	13	2	no	—
70	C₂×C₇	—	39	2	no	—
71	C₁₃	C₃×C₄×#3862	1	1	yes	characteristic, complemented, Sylow
72	C₃×C₄	—	13	1	no	—
73	C₃×C₄	—	637	3	no	—
74	C₃²	—	49	1	no	Sylow
75	C₇	—	3	2	no	—
76	C₇	C₃×ΓL(1,𝔽₈)×#261	1	2	yes	complemented
77	C₂×C₃	—	13	1	no	—
78	C₂×C₃	—	637	3	no	—
79	C₄	—	13	1	no	Sylow
80	C₃	—	49	3	no	—
81	C₃	#261×#3862	1	1	yes	characteristic, direct product factor
82	C₂	—	13	1	no	—

	subgroup	G/H
center, hypercenter	C₃	#261×#3862
commutator, hypocenter	C₇²×C₁₃	C₃²×C₄
Frattini	C₁	G
Fitting, socle	C₃×C₇²×C₁₃	C₃×C₄
prime radical	C₃×D₁₃×#3862	C₂
cosocle	C₇²×D₁₃	C₂×C₃²
LCS	G ▷ C₇²×C₁₃
UCS	C₁ ◁ C₃

Complements

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

N	H ≅ G/N	type
2. C₃×C₇²×#261	C₃	semidirect
3. #261×#3862	C₃	direct (N × H)
4. C₃×C₁₃×#3862	C₄	semidirect
8. C₇²×#261	C₃²	semidirect
9. C₃×C₇²×C₁₃	C₃×C₄	semidirect
10. C₁₃×#3862	C₃×C₄	semidirect
15. C₃×C₇×#261	ΓL(1,𝔽₈)	semidirect
19. C₇²×C₁₃	C₃²×C₄	semidirect
26. C₃×#3862	#261	direct (N × H)
28. C₇×#261	C₃×ΓL(1,𝔽₈)	semidirect
32. C₃×C₇×C₁₃	C₄×ΓL(1,𝔽₈)	semidirect
40. C₃×#261	#3862	direct (N × H)
41. C₃×C₇²	C₃×#261	semidirect
42. #3862	C₃×#261	direct (N × H)
47. C₇×C₁₃	C₃×C₄×ΓL(1,𝔽₈)	semidirect
54. #261	C₃×#3862	direct (N × H)
55. C₇²	C₃²×#261	semidirect
60. C₃×C₁₃	C₄×#3862	semidirect
66. C₃×C₇	ΓL(1,𝔽₈)×#261	semidirect
71. C₁₃	C₃×C₄×#3862	semidirect
76. C₇	C₃×ΓL(1,𝔽₈)×#261	semidirect
81. C₃	#261×#3862	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×D₁₃×#3862	2
C₃×C₇²×#261	3
#261×#3862	3
C₃×ΓL(1,𝔽₈)×#261	7
C₃×C₄×#3862	13

Subgroups — C3×#261×#3862 #37525118

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#261×#3862 #37525118