Subgroups — C₃×C₅×ΓL(1,𝔽₈)⋊#261 #30655315

Order 16380.

144 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×ΓL(1,𝔽₈)×D₁₃	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×C₇⋊#261	C₃	1	1	yes	maximal, characteristic, complemented
3	C₅×ΓL(1,𝔽₈)⋊#261	C₃	1	3	yes	maximal, direct product factor
4	C₃×C₅×C₁₃×ΓL(1,𝔽₈)	C₄	1	1	yes	characteristic, complemented, Hall
5	C₃×ΓL(1,𝔽₈)⋊#261	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₅×ΓL(1,𝔽₈)×D₁₃	C₂×C₃	1	3	yes	—
7	C₃×C₅×C₇×D₁₃	C₂×C₃	1	1	yes	characteristic
8	C₃²×C₅×#261	—	7	1	no	maximal, Hall
9	C₅×C₇⋊#261	C₃²	1	1	yes	characteristic, complemented, Hall
10	C₃×ΓL(1,𝔽₈)×D₁₃	C₂×C₅	1	1	yes	characteristic
11	C₃×C₅×C₇×C₁₃	C₃×C₄	1	1	yes	characteristic, complemented
12	C₅×C₁₃×ΓL(1,𝔽₈)	C₃×C₄	1	3	yes	complemented
13	C₃×C₅×ΓL(1,𝔽₈)⋊C₄	—	13	1	no	maximal, Hall
14	C₃²×C₅×D₁₃	—	7	1	no	—
15	C₃×C₇⋊#261	C₃×C₅	1	1	yes	characteristic, complemented
16	ΓL(1,𝔽₈)⋊#261	C₃×C₅	1	3	yes	direct product factor
17	C₅×C₇×D₁₃	C₂×C₃²	1	1	yes	characteristic
18	C₃×C₁₃×ΓL(1,𝔽₈)	C₄×C₅	1	1	yes	characteristic, complemented, Hall
19	C₃×C₅×#261	—	7	3	no	—
20	C₃×C₅×#261	—	21	1	no	—
21	C₂×C₃×C₅×ΓL(1,𝔽₈)	—	13	1	no	—
22	C₃²×C₅×C₁₃	—	7	1	no	Hall
23	C₃×C₇×D₁₃	C₂×C₃×C₅	1	1	yes	characteristic
24	ΓL(1,𝔽₈)×D₁₃	C₂×C₃×C₅	1	3	yes	—
25	C₃²×#261	—	7	1	no	Hall
26	C₅×C₇×C₁₃	C₃²×C₄	1	1	yes	characteristic, complemented, Hall
27	C₃×C₅×Dic₇	—	13	1	no	—
28	C₅×ΓL(1,𝔽₈)⋊C₄	—	13	3	no	—
29	C₃×C₅×D₁₃	—	7	3	no	—
30	C₃×C₅×D₁₃	AGL(1,𝔽₇)	1	1	yes	characteristic
31	C₇⋊#261	C₃²×C₅	1	1	yes	characteristic, complemented, Hall
32	C₃×C₅×ΓL(1,𝔽₈)	#261	1	1	yes	characteristic, complemented, Hall
33	C₃×C₇×C₁₃	C₃×C₄×C₅	1	1	yes	characteristic, complemented
34	C₁₃×ΓL(1,𝔽₈)	C₃×C₄×C₅	1	3	yes	complemented
35	C₅×#261	—	21	1	no	Hall
36	C₃×ΓL(1,𝔽₈)⋊C₄	—	13	1	no	Hall
37	C₃²×D₁₃	—	7	1	no	—
38	C₂×C₃×C₅×C₇	—	13	1	no	—
39	C₂×C₅×ΓL(1,𝔽₈)	—	13	3	no	—
40	C₃×C₅×C₁₃	—	7	3	no	—
41	C₃×C₅×C₁₃	ΓL(1,𝔽₈)⋊C₄	1	1	yes	characteristic, complemented
42	C₇×D₁₃	C₂×C₃²×C₅	1	1	yes	characteristic
43	C₃²×C₄×C₅	—	91	1	no	Hall
44	C₃×#261	—	7	3	no	—
45	C₃×#261	—	21	1	no	—
46	C₅×Dic₇	—	13	1	no	Hall
47	C₅×D₁₃	C₃×AGL(1,𝔽₇)	1	1	yes	characteristic
48	C₂×C₃×ΓL(1,𝔽₈)	—	13	1	no	—
49	C₃²×C₁₃	—	7	1	no	Hall
50	C₃×C₅×C₇	C₃×#261	1	1	yes	characteristic, complemented
51	C₅×ΓL(1,𝔽₈)	C₃×#261	1	3	yes	complemented
52	C₇×C₁₃	C₃²×C₄×C₅	1	1	yes	characteristic, complemented, Hall
53	C₂×C₃²×C₅	—	91	1	no	—
54	C₃×Dic₇	—	13	1	no	—
55	ΓL(1,𝔽₈)⋊C₄	—	13	3	no	—
56	C₃×D₁₃	—	7	3	no	—
57	C₃×D₁₃	C₅×AGL(1,𝔽₇)	1	1	yes	characteristic
58	C₂×C₅×C₇	—	13	1	no	—
59	C₅×C₁₃	C₃×ΓL(1,𝔽₈)⋊C₄	1	1	yes	characteristic, complemented, Hall
60	C₃×ΓL(1,𝔽₈)	C₅×#261	1	1	yes	characteristic, complemented, Hall
61	C₃×C₄×C₅	—	273	1	no	—
62	C₃×C₄×C₅	—	91	3	no	—
63	#261	—	21	1	no	Hall
64	C₃²×C₅	—	7	1	no	Hall
65	C₂×C₃×C₇	—	13	1	no	—
66	C₂×ΓL(1,𝔽₈)	—	13	3	no	—
67	C₃×C₁₃	—	7	3	no	—
68	C₃×C₁₃	C₅×ΓL(1,𝔽₈)⋊C₄	1	1	yes	characteristic, complemented
69	C₃²×C₄	—	91	1	no	Hall
70	C₅×C₇	C₃²×#261	1	1	yes	characteristic, complemented, Hall
71	C₂×C₃×C₅	—	13	1	no	—
72	C₂×C₃×C₅	—	91	3	no	—
73	Dic₇	—	13	1	no	Hall
74	D₁₃	C₃×C₅×AGL(1,𝔽₇)	1	1	yes	characteristic
75	C₃×C₇	C₃×C₅×#261	1	1	yes	characteristic, complemented
76	ΓL(1,𝔽₈)	C₃×C₅×#261	1	3	yes	complemented
77	C₄×C₅	—	273	1	no	Hall
78	C₂×C₃²	—	91	1	no	—
79	C₃×C₅	—	7	3	no	—
80	C₃×C₅	ΓL(1,𝔽₈)⋊#261	1	1	yes	characteristic, direct product factor
81	C₂×C₇	—	13	1	no	—
82	C₁₃	C₃×C₅×ΓL(1,𝔽₈)⋊C₄	1	1	yes	characteristic, complemented, Sylow
83	C₃×C₄	—	91	1	no	—
84	C₃×C₄	—	91	3	no	—
85	C₂×C₅	—	13	1	no	—
86	C₃²	—	7	1	no	Sylow
87	C₇	C₃²×C₅×#261	1	1	yes	characteristic, complemented, Sylow
88	C₂×C₃	—	13	1	no	—
89	C₂×C₃	—	91	3	no	—
90	C₅	C₃×ΓL(1,𝔽₈)⋊#261	1	1	yes	characteristic, direct product factor, Sylow
91	C₄	—	91	1	no	Sylow
92	C₃	—	7	3	no	—
93	C₃	C₅×ΓL(1,𝔽₈)⋊#261	1	1	yes	characteristic, direct product factor
94	C₂	—	13	1	no	—

	subgroup	G/H
center, hypercenter	C₃×C₅	ΓL(1,𝔽₈)⋊#261
commutator, hypocenter	C₇×C₁₃	C₃²×C₄×C₅
Frattini	C₁	G
Fitting, socle	C₃×C₅×C₇×C₁₃	C₃×C₄
prime radical	C₃×C₅×ΓL(1,𝔽₈)×D₁₃	C₂
cosocle	C₇×D₁₃	C₂×C₃²×C₅
LCS	G ▷ C₇×C₁₃
UCS	C₁ ◁ C₃×C₅

Complements

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

N	H ≅ G/N	type
2. C₃×C₅×C₇⋊#261	C₃	semidirect
3. C₅×ΓL(1,𝔽₈)⋊#261	C₃	direct (N × H)
4. C₃×C₅×C₁₃×ΓL(1,𝔽₈)	C₄	semidirect
5. C₃×ΓL(1,𝔽₈)⋊#261	C₅	direct (N × H)
9. C₅×C₇⋊#261	C₃²	semidirect
11. C₃×C₅×C₇×C₁₃	C₃×C₄	semidirect
12. C₅×C₁₃×ΓL(1,𝔽₈)	C₃×C₄	semidirect
15. C₃×C₇⋊#261	C₃×C₅	semidirect
16. ΓL(1,𝔽₈)⋊#261	C₃×C₅	direct (N × H)
18. C₃×C₁₃×ΓL(1,𝔽₈)	C₄×C₅	semidirect
26. C₅×C₇×C₁₃	C₃²×C₄	semidirect
31. C₇⋊#261	C₃²×C₅	semidirect
32. C₃×C₅×ΓL(1,𝔽₈)	#261	semidirect
33. C₃×C₇×C₁₃	C₃×C₄×C₅	semidirect
34. C₁₃×ΓL(1,𝔽₈)	C₃×C₄×C₅	semidirect
41. C₃×C₅×C₁₃	ΓL(1,𝔽₈)⋊C₄	semidirect
50. C₃×C₅×C₇	C₃×#261	semidirect
51. C₅×ΓL(1,𝔽₈)	C₃×#261	semidirect
52. C₇×C₁₃	C₃²×C₄×C₅	semidirect
59. C₅×C₁₃	C₃×ΓL(1,𝔽₈)⋊C₄	semidirect
60. C₃×ΓL(1,𝔽₈)	C₅×#261	semidirect
68. C₃×C₁₃	C₅×ΓL(1,𝔽₈)⋊C₄	semidirect
70. C₅×C₇	C₃²×#261	semidirect
75. C₃×C₇	C₃×C₅×#261	semidirect
76. ΓL(1,𝔽₈)	C₃×C₅×#261	semidirect
80. C₃×C₅	ΓL(1,𝔽₈)⋊#261	direct (N × H)
82. C₁₃	C₃×C₅×ΓL(1,𝔽₈)⋊C₄	semidirect
87. C₇	C₃²×C₅×#261	semidirect
90. C₅	C₃×ΓL(1,𝔽₈)⋊#261	direct (N × H)
93. C₃	C₅×ΓL(1,𝔽₈)⋊#261	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×C₅×ΓL(1,𝔽₈)×D₁₃	2
C₃×C₅×C₇⋊#261	3
C₅×ΓL(1,𝔽₈)⋊#261	3
C₃×ΓL(1,𝔽₈)⋊#261	5
C₃²×C₅×#261	7
C₃×C₅×ΓL(1,𝔽₈)⋊C₄	13

Subgroups — C3×C5×ΓL(1,𝔽8)⋊#261 #30655315

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×ΓL(1,𝔽₈)⋊#261 #30655315