Subgroups — C₅²×C₃⋊#64425 #36348030

Order 22050.

352 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅²×#3862	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₅²×C₃⋊(C₇⋊D₇)	C₃	1	1	yes	maximal, characteristic, complemented
3	C₅²×#64425	—	3	1	no	maximal
4	C₅×C₃⋊#64425	C₅	1	6	yes	maximal, direct product factor
5	C₅²×#3862	—	2	1	no	—
6	C₅²×#3862	D₃	1	1	yes	characteristic, complemented
7	C₃×C₅²×C₇²	C₂×C₃	1	1	yes	characteristic, complemented
8	C₅²×C₃⋊AGL(1,𝔽₇)	—	7	2	no	maximal
9	C₅²×C₇⋊D₇	—	3	1	no	Hall
10	C₃×C₅×#3862	C₂×C₅	1	6	yes	complemented
11	C₃×C₅²×ΓL(1,𝔽₈)	—	7	2	no	—
12	C₅×C₃⋊(C₇⋊D₇)	C₃×C₅	1	6	yes	complemented
13	C₅×#64425	—	3	6	no	—
14	C₅²×C₇²	C₃×D₃	1	1	yes	characteristic, complemented, Hall
15	C₅²×D₂₁	—	7	2	no	—
16	C₅²×D₂₁	—	21	2	no	—
17	C₅²×AGL(1,𝔽₇)	—	21	2	no	—
18	C₃⋊#64425	C₅²	1	1	yes	characteristic, direct product factor, Hall
19	C₃×C₅×C₇²	C₂×C₃×C₅	1	6	yes	complemented
20	C₅×#3862	—	2	6	no	—
21	C₅×#3862	C₅×D₃	1	6	yes	complemented
22	C₅×C₃⋊AGL(1,𝔽₇)	—	7	12	no	—
23	C₃×C₅²×C₇	—	3	2	no	—
24	C₃×C₅²×C₇	AGL(1,𝔽₇)	1	2	yes	complemented
25	C₅²×ΓL(1,𝔽₈)	—	7	2	no	—
26	C₅²×ΓL(1,𝔽₈)	—	14	2	no	—
27	C₅×C₇⋊D₇	—	3	6	no	—
28	C₃×C₅²×D₃	—	49	1	no	Hall
29	C₃×#3862	C₂×C₅²	1	1	yes	characteristic, complemented, Hall
30	C₅²×D₇	—	21	2	no	—
31	C₅²×D₇	—	63	2	no	—
32	C₃×C₅×ΓL(1,𝔽₈)	—	7	12	no	—
33	C₃⋊(C₇⋊D₇)	C₃×C₅²	1	1	yes	characteristic, complemented
34	#64425	—	3	1	no	—
35	C₅×C₇²	C₃×C₅×D₃	1	6	yes	complemented
36	C₃²×C₅²	—	49	1	no	Hall
37	C₅×D₂₁	—	7	12	no	—
38	C₅×D₂₁	—	21	12	no	—
39	C₅×AGL(1,𝔽₇)	—	21	12	no	—
40	C₅²×C₇	—	3	2	no	—
41	C₅²×C₇	C₃⋊AGL(1,𝔽₇)	1	2	yes	complemented
42	C₂×C₃×C₅²	—	147	1	no	—
43	C₅²×D₃	—	49	1	no	—
44	C₃×C₇²	C₂×C₃×C₅²	1	1	yes	characteristic, complemented
45	#3862	—	2	1	no	—
46	#3862	C₅²×D₃	1	1	yes	characteristic, complemented
47	C₃⋊AGL(1,𝔽₇)	—	7	2	no	—
48	C₃×C₅×C₇	—	3	12	no	—
49	C₃×C₅×C₇	C₅×AGL(1,𝔽₇)	1	12	yes	complemented
50	C₅×ΓL(1,𝔽₈)	—	7	12	no	—
51	C₅×ΓL(1,𝔽₈)	—	14	12	no	—
52	C₇⋊D₇	—	3	1	no	Hall
53	C₃×C₅×D₃	—	49	6	no	—
54	C₃×C₅²	—	49	1	no	—
55	C₃×C₅²	—	98	1	no	—
56	C₃×C₅²	#64425	1	1	yes	characteristic, complemented
57	C₅×D₇	—	21	12	no	—
58	C₅×D₇	—	63	12	no	—
59	C₃×ΓL(1,𝔽₈)	—	7	2	no	—
60	C₂×C₅²	—	147	1	no	Hall
61	C₇²	C₃×C₅²×D₃	1	1	yes	characteristic, complemented, Sylow
62	C₃²×C₅	—	49	6	no	—
63	D₂₁	—	7	2	no	—
64	D₂₁	—	21	2	no	—
65	AGL(1,𝔽₇)	—	21	2	no	—
66	C₅×C₇	—	3	12	no	—
67	C₅×C₇	C₅×C₃⋊AGL(1,𝔽₇)	1	12	yes	complemented
68	C₂×C₃×C₅	—	147	6	no	—
69	C₅×D₃	—	49	6	no	—
70	C₅²	C₃⋊#64425	1	1	yes	characteristic, direct product factor, Sylow
71	C₃×C₇	—	3	2	no	—
72	C₃×C₇	C₅²×AGL(1,𝔽₇)	1	2	yes	complemented
73	ΓL(1,𝔽₈)	—	7	2	no	—
74	ΓL(1,𝔽₈)	—	14	2	no	—
75	C₃×D₃	—	49	1	no	Hall
76	C₃×C₅	—	49	6	no	—
77	C₃×C₅	—	98	6	no	—
78	C₃×C₅	C₅×#64425	1	6	yes	complemented
79	D₇	—	21	2	no	—
80	D₇	—	63	2	no	—
81	C₂×C₅	—	147	6	no	—
82	C₃²	—	49	1	no	Sylow
83	C₇	—	3	2	no	—
84	C₇	C₅²×C₃⋊AGL(1,𝔽₇)	1	2	yes	complemented
85	C₂×C₃	—	147	1	no	—
86	D₃	—	49	1	no	—
87	C₅	C₅×C₃⋊#64425	1	6	yes	direct product factor
88	C₃	—	49	1	no	—
89	C₃	—	98	1	no	—
90	C₃	C₅²×#64425	1	1	yes	characteristic, complemented
91	C₂	—	147	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₅²	C₃⋊#64425
commutator, cosocle, hypocenter	C₃×C₇²	C₂×C₃×C₅²
Frattini	C₁	G
Fitting, socle	C₃×C₅²×C₇²	C₂×C₃
LCS	G ▷ C₃×C₇²
UCS	C₁ ◁ C₅²

Complements

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

N	H ≅ G/N	type
1. C₃×C₅²×#3862	C₂	semidirect
2. C₅²×C₃⋊(C₇⋊D₇)	C₃	semidirect
4. C₅×C₃⋊#64425	C₅	direct (N × H)
6. C₅²×#3862	D₃	semidirect
7. C₃×C₅²×C₇²	C₂×C₃	semidirect
10. C₃×C₅×#3862	C₂×C₅	semidirect
12. C₅×C₃⋊(C₇⋊D₇)	C₃×C₅	semidirect
14. C₅²×C₇²	C₃×D₃	semidirect
18. C₃⋊#64425	C₅²	direct (N × H)
19. C₃×C₅×C₇²	C₂×C₃×C₅	semidirect
21. C₅×#3862	C₅×D₃	semidirect
24. C₃×C₅²×C₇	AGL(1,𝔽₇)	semidirect
29. C₃×#3862	C₂×C₅²	semidirect
33. C₃⋊(C₇⋊D₇)	C₃×C₅²	semidirect
35. C₅×C₇²	C₃×C₅×D₃	semidirect
41. C₅²×C₇	C₃⋊AGL(1,𝔽₇)	semidirect
44. C₃×C₇²	C₂×C₃×C₅²	semidirect
46. #3862	C₅²×D₃	semidirect
49. C₃×C₅×C₇	C₅×AGL(1,𝔽₇)	semidirect
56. C₃×C₅²	#64425	semidirect
61. C₇²	C₃×C₅²×D₃	semidirect
67. C₅×C₇	C₅×C₃⋊AGL(1,𝔽₇)	semidirect
70. C₅²	C₃⋊#64425	direct (N × H)
72. C₃×C₇	C₅²×AGL(1,𝔽₇)	semidirect
78. C₃×C₅	C₅×#64425	semidirect
84. C₇	C₅²×C₃⋊AGL(1,𝔽₇)	semidirect
87. C₅	C₅×C₃⋊#64425	direct (N × H)
90. C₃	C₅²×#64425	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₅²×#3862	2
C₅²×C₃⋊(C₇⋊D₇)	3
C₅²×#64425	3
C₅×C₃⋊#64425	5
C₅²×C₃⋊AGL(1,𝔽₇)	7

Subgroups — C52×C3⋊#64425 #36348030

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅²×C₃⋊#64425 #36348030