Subgroups — C₅×A₄×#63271 #30655552

Order 16380.

112 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×C₂²⋊#63271	C₃	1	2	yes	maximal, complemented
2	C₅×C₇×A₄×C₁₃	C₃	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₅×#63271	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃×C₅×#63271	—	4	1	no	maximal, Hall
5	A₄×#63271	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂×C₅×#63271	—	3	1	no	—
7	C₅×A₄×C₁₃⋊C₃	—	7	1	no	maximal, Hall
8	C₂²×C₅×C₇×C₁₃	C₃²	1	1	yes	characteristic, complemented, Hall
9	C₃×C₅×C₇×C₁₃	—	4	1	no	—
10	C₅×#63271	—	4	2	no	—
11	C₅×#63271	A₄	1	1	yes	characteristic, direct product factor
12	C₅×A₄×ΓL(1,𝔽₈)	—	13	1	no	maximal, Hall
13	C₂²⋊#63271	C₃×C₅	1	2	yes	complemented
14	C₇×A₄×C₁₃	C₃×C₅	1	1	yes	characteristic, complemented
15	C₂²×#63271	C₃×C₅	1	1	yes	characteristic, complemented
16	C₂×C₅×C₇×C₁₃	—	3	1	no	—
17	C₃×#63271	—	4	1	no	Hall
18	C₅×C₂²⋊(C₁₃⋊C₃)	—	7	2	no	—
19	C₅×A₄×C₁₃	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
20	C₂²×C₅×C₁₃⋊C₃	—	7	1	no	—
21	C₃×C₅×C₁₃⋊C₃	—	28	1	no	Hall
22	C₂×#63271	—	3	1	no	—
23	A₄×C₁₃⋊C₃	—	7	1	no	Hall
24	C₅×C₇×C₁₃	C₃×A₄	1	1	yes	characteristic, complemented, Hall
25	C₅×C₂²⋊ΓL(1,𝔽₈)	—	13	2	no	—
26	C₅×C₇×A₄	C₁₃⋊C₃	1	1	yes	characteristic, complemented
27	C₂²×C₅×ΓL(1,𝔽₈)	—	13	1	no	—
28	C₂×C₅×C₁₃⋊C₃	—	21	1	no	—
29	C₂²×C₇×C₁₃	C₃²×C₅	1	1	yes	characteristic, complemented, Hall
30	C₃×C₅×ΓL(1,𝔽₈)	—	52	1	no	Hall
31	C₃×C₇×C₁₃	—	4	1	no	—
32	#63271	—	4	2	no	—
33	#63271	C₅×A₄	1	1	yes	characteristic, direct product factor
34	C₂²×C₅×C₁₃	C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
35	A₄×ΓL(1,𝔽₈)	—	13	1	no	Hall
36	C₂×C₅×ΓL(1,𝔽₈)	—	39	1	no	—
37	C₃×C₅×C₁₃	—	4	1	no	—
38	C₅×C₁₃⋊C₃	—	7	1	no	—
39	C₅×C₁₃⋊C₃	—	28	2	no	—
40	C₂×C₇×C₁₃	—	3	1	no	—
41	C₃×C₅×A₄	—	91	1	no	Hall
42	C₂²⋊(C₁₃⋊C₃)	—	7	2	no	—
43	A₄×C₁₃	C₅×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
44	C₂²×C₁₃⋊C₃	—	7	1	no	—
45	C₂²×C₅×C₇	C₃×C₁₃⋊C₃	1	1	yes	characteristic, complemented, Hall
46	C₂×C₅×C₁₃	—	3	1	no	—
47	C₃×C₁₃⋊C₃	—	28	1	no	Hall
48	C₃×C₅×C₇	—	4	1	no	—
49	C₅×ΓL(1,𝔽₈)	—	13	1	no	—
50	C₅×ΓL(1,𝔽₈)	—	52	2	no	—
51	C₇×C₁₃	C₃×C₅×A₄	1	1	yes	characteristic, complemented, Hall
52	C₂²⋊ΓL(1,𝔽₈)	—	13	2	no	—
53	C₇×A₄	C₅×C₁₃⋊C₃	1	1	yes	characteristic, complemented
54	C₂²×ΓL(1,𝔽₈)	—	13	1	no	—
55	C₂×C₁₃⋊C₃	—	21	1	no	—
56	C₂×C₅×C₇	—	3	1	no	—
57	C₅×C₁₃	A₄×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
58	C₃×ΓL(1,𝔽₈)	—	52	1	no	Hall
59	C₂²×C₃×C₅	—	91	1	no	—
60	C₅×A₄	—	91	2	no	—
61	C₅×A₄	#63271	1	1	yes	characteristic, direct product factor
62	C₂²×C₁₃	C₃×C₅×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
63	C₃²×C₅	—	364	1	no	Hall
64	C₂×ΓL(1,𝔽₈)	—	39	1	no	—
65	C₃×C₁₃	—	4	1	no	—
66	C₁₃⋊C₃	—	7	1	no	—
67	C₁₃⋊C₃	—	28	2	no	—
68	C₃×A₄	—	91	1	no	Hall
69	C₅×C₇	A₄×C₁₃⋊C₃	1	1	yes	characteristic, complemented, Hall
70	C₂×C₃×C₅	—	273	1	no	—
71	C₂²×C₇	C₃×C₅×C₁₃⋊C₃	1	1	yes	characteristic, complemented, Hall
72	C₂×C₁₃	—	3	1	no	—
73	C₃×C₇	—	4	1	no	—
74	ΓL(1,𝔽₈)	—	13	1	no	—
75	ΓL(1,𝔽₈)	—	52	2	no	—
76	C₂²×C₅	C₃×#63271	1	1	yes	characteristic, complemented, Hall
77	C₃×C₅	—	4	1	no	—
78	C₃×C₅	—	91	1	no	—
79	C₃×C₅	—	364	2	no	—
80	C₂×C₇	—	3	1	no	—
81	C₁₃	C₅×A₄×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
82	C₂²×C₃	—	91	1	no	—
83	A₄	—	91	2	no	—
84	A₄	C₅×#63271	1	1	yes	characteristic, direct product factor
85	C₂×C₅	—	3	1	no	—
86	C₃²	—	364	1	no	Sylow
87	C₇	C₅×A₄×C₁₃⋊C₃	1	1	yes	characteristic, complemented, Sylow
88	C₂×C₃	—	273	1	no	—
89	C₅	A₄×#63271	1	1	yes	characteristic, direct product factor, Sylow
90	C₂²	C₃×C₅×#63271	1	1	yes	characteristic, complemented, Sylow
91	C₃	—	4	1	no	—
92	C₃	—	91	1	no	—
93	C₃	—	364	2	no	—
94	C₂	—	3	1	no	—

	subgroup	G/H
center, hypercenter	C₅	A₄×#63271
commutator, cosocle, hypocenter	C₂²×C₇×C₁₃	C₃²×C₅
Frattini	C₁	G
Fitting, socle	C₂²×C₅×C₇×C₁₃	C₃²
LCS	G ▷ C₂²×C₇×C₁₃
UCS	C₁ ◁ C₅

Complements

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

N	H ≅ G/N	type
1. C₅×C₂²⋊#63271	C₃	semidirect
2. C₅×C₇×A₄×C₁₃	C₃	semidirect
3. C₂²×C₅×#63271	C₃	semidirect
5. A₄×#63271	C₅	direct (N × H)
8. C₂²×C₅×C₇×C₁₃	C₃²	semidirect
11. C₅×#63271	A₄	direct (N × H)
13. C₂²⋊#63271	C₃×C₅	semidirect
14. C₇×A₄×C₁₃	C₃×C₅	semidirect
15. C₂²×#63271	C₃×C₅	semidirect
19. C₅×A₄×C₁₃	ΓL(1,𝔽₈)	semidirect
24. C₅×C₇×C₁₃	C₃×A₄	semidirect
26. C₅×C₇×A₄	C₁₃⋊C₃	semidirect
29. C₂²×C₇×C₁₃	C₃²×C₅	semidirect
33. #63271	C₅×A₄	direct (N × H)
34. C₂²×C₅×C₁₃	C₃×ΓL(1,𝔽₈)	semidirect
43. A₄×C₁₃	C₅×ΓL(1,𝔽₈)	semidirect
45. C₂²×C₅×C₇	C₃×C₁₃⋊C₃	semidirect
51. C₇×C₁₃	C₃×C₅×A₄	semidirect
53. C₇×A₄	C₅×C₁₃⋊C₃	semidirect
57. C₅×C₁₃	A₄×ΓL(1,𝔽₈)	semidirect
61. C₅×A₄	#63271	direct (N × H)
62. C₂²×C₁₃	C₃×C₅×ΓL(1,𝔽₈)	semidirect
69. C₅×C₇	A₄×C₁₃⋊C₃	semidirect
71. C₂²×C₇	C₃×C₅×C₁₃⋊C₃	semidirect
76. C₂²×C₅	C₃×#63271	semidirect
81. C₁₃	C₅×A₄×ΓL(1,𝔽₈)	semidirect
84. A₄	C₅×#63271	direct (N × H)
87. C₇	C₅×A₄×C₁₃⋊C₃	semidirect
89. C₅	A₄×#63271	direct (N × H)
90. C₂²	C₃×C₅×#63271	semidirect

Maximal subgroups

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

subgroup	index
C₅×C₂²⋊#63271	3
C₅×C₇×A₄×C₁₃	3
C₂²×C₅×#63271	3
C₃×C₅×#63271	4
A₄×#63271	5
C₅×A₄×C₁₃⋊C₃	7
C₅×A₄×ΓL(1,𝔽₈)	13

Subgroups — C5×A4×#63271 #30655552

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×A₄×#63271 #30655552