Subgroups — C₉²×C₁₆×C₂₅ #72209111

Order 32400.

345 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×C₉²×C₂₅	C₂	1	1	yes	maximal, characteristic
2	C₃×C₉×C₁₆×C₂₅	C₃	1	4	yes	maximal
3	C₄×C₉²×C₂₅	C₄	1	1	yes	characteristic
4	C₅×C₉²×C₁₆	C₅	1	1	yes	maximal, characteristic
5	C₃×C₈×C₉×C₂₅	C₂×C₃	1	4	yes	—
6	C₂×C₉²×C₂₅	C₈	1	1	yes	characteristic
7	C₃²×C₁₆×C₂₅	C₃²	1	1	yes	characteristic
8	C₉×C₁₆×C₂₅	C₉	1	12	yes	direct product factor
9	C₅×C₈×C₉²	C₂×C₅	1	1	yes	characteristic
10	C₃×C₄×C₉×C₂₅	C₃×C₄	1	4	yes	—
11	C₃×C₅×C₉×C₁₆	C₃×C₅	1	4	yes	—
12	C₉²×C₂₅	C₁₆	1	1	yes	characteristic, direct product factor, Hall
13	C₃²×C₈×C₂₅	C₂×C₃²	1	1	yes	characteristic
14	C₈×C₉×C₂₅	C₂×C₉	1	12	yes	—
15	C₄×C₅×C₉²	C₄×C₅	1	1	yes	characteristic
16	C₂×C₃×C₉×C₂₅	C₃×C₈	1	4	yes	—
17	C₉²×C₁₆	C₂₅	1	1	yes	characteristic, direct product factor, Hall
18	C₃×C₁₆×C₂₅	C₃×C₉	1	4	yes	—
19	C₃×C₅×C₈×C₉	C₂×C₃×C₅	1	4	yes	—
20	C₃²×C₄×C₂₅	C₃²×C₄	1	1	yes	characteristic
21	C₄×C₉×C₂₅	C₄×C₉	1	12	yes	—
22	C₂×C₅×C₉²	C₅×C₈	1	1	yes	characteristic
23	C₃²×C₅×C₁₆	C₃²×C₅	1	1	yes	characteristic
24	C₅×C₉×C₁₆	C₅×C₉	1	12	yes	—
25	C₃×C₉×C₂₅	C₃×C₁₆	1	4	yes	—
26	C₈×C₉²	C₂×C₂₅	1	1	yes	characteristic
27	C₃×C₈×C₂₅	C₂×C₃×C₉	1	4	yes	—
28	C₃×C₄×C₅×C₉	C₃×C₄×C₅	1	4	yes	—
29	C₂×C₃²×C₂₅	C₃²×C₈	1	1	yes	characteristic
30	C₂×C₉×C₂₅	C₈×C₉	1	12	yes	—
31	C₃×C₉×C₁₆	C₃×C₂₅	1	4	yes	—
32	C₅×C₉²	C₅×C₁₆	1	1	yes	characteristic
33	C₁₆×C₂₅	C₉²	1	1	yes	characteristic, direct product factor, Hall
34	C₃²×C₅×C₈	C₂×C₃²×C₅	1	1	yes	characteristic
35	C₅×C₈×C₉	C₂×C₅×C₉	1	12	yes	—
36	C₄×C₉²	C₄×C₂₅	1	1	yes	characteristic
37	C₃×C₄×C₂₅	C₃×C₄×C₉	1	4	yes	—
38	C₂×C₃×C₅×C₉	C₃×C₅×C₈	1	4	yes	—
39	C₃×C₅×C₁₆	C₃×C₅×C₉	1	4	yes	—
40	C₃²×C₂₅	C₃²×C₁₆	1	1	yes	characteristic
41	C₉×C₂₅	C₉×C₁₆	1	12	yes	direct product factor
42	C₃×C₈×C₉	C₂×C₃×C₂₅	1	4	yes	—
43	C₈×C₂₅	C₂×C₉²	1	1	yes	characteristic
44	C₃²×C₄×C₅	C₃²×C₄×C₅	1	1	yes	characteristic
45	C₄×C₅×C₉	C₄×C₅×C₉	1	12	yes	—
46	C₂×C₉²	C₈×C₂₅	1	1	yes	characteristic
47	C₂×C₃×C₂₅	C₃×C₈×C₉	1	4	yes	—
48	C₃²×C₁₆	C₃²×C₂₅	1	1	yes	characteristic
49	C₉×C₁₆	C₉×C₂₅	1	12	yes	direct product factor
50	C₃×C₅×C₉	C₃×C₅×C₁₆	1	4	yes	—
51	C₃×C₅×C₈	C₂×C₃×C₅×C₉	1	4	yes	—
52	C₃×C₄×C₉	C₃×C₄×C₂₅	1	4	yes	—
53	C₄×C₂₅	C₄×C₉²	1	1	yes	characteristic
54	C₂×C₃²×C₅	C₃²×C₅×C₈	1	1	yes	characteristic
55	C₂×C₅×C₉	C₅×C₈×C₉	1	12	yes	—
56	C₉²	C₁₆×C₂₅	1	1	yes	characteristic, direct product factor, Sylow
57	C₅×C₁₆	C₅×C₉²	1	1	yes	characteristic
58	C₃×C₂₅	C₃×C₉×C₁₆	1	4	yes	—
59	C₃²×C₈	C₂×C₃²×C₂₅	1	1	yes	characteristic
60	C₈×C₉	C₂×C₉×C₂₅	1	12	yes	—
61	C₃×C₄×C₅	C₃×C₄×C₅×C₉	1	4	yes	—
62	C₂×C₃×C₉	C₃×C₈×C₂₅	1	4	yes	—
63	C₂×C₂₅	C₈×C₉²	1	1	yes	characteristic
64	C₃×C₁₆	C₃×C₉×C₂₅	1	4	yes	—
65	C₃²×C₅	C₃²×C₅×C₁₆	1	1	yes	characteristic
66	C₅×C₉	C₅×C₉×C₁₆	1	12	yes	—
67	C₅×C₈	C₂×C₅×C₉²	1	1	yes	characteristic
68	C₃²×C₄	C₃²×C₄×C₂₅	1	1	yes	characteristic
69	C₄×C₉	C₄×C₉×C₂₅	1	12	yes	—
70	C₂×C₃×C₅	C₃×C₅×C₈×C₉	1	4	yes	—
71	C₃×C₉	C₃×C₁₆×C₂₅	1	4	yes	—
72	C₂₅	C₉²×C₁₆	1	1	yes	characteristic, direct product factor, Sylow
73	C₃×C₈	C₂×C₃×C₉×C₂₅	1	4	yes	—
74	C₄×C₅	C₄×C₅×C₉²	1	1	yes	characteristic
75	C₂×C₃²	C₃²×C₈×C₂₅	1	1	yes	characteristic
76	C₂×C₉	C₈×C₉×C₂₅	1	12	yes	—
77	C₁₆	C₉²×C₂₅	1	1	yes	characteristic, direct product factor, Sylow
78	C₃×C₅	C₃×C₅×C₉×C₁₆	1	4	yes	—
79	C₃×C₄	C₃×C₄×C₉×C₂₅	1	4	yes	—
80	C₂×C₅	C₅×C₈×C₉²	1	1	yes	characteristic
81	C₃²	C₃²×C₁₆×C₂₅	1	1	yes	characteristic
82	C₉	C₉×C₁₆×C₂₅	1	12	yes	direct product factor
83	C₈	C₂×C₉²×C₂₅	1	1	yes	characteristic
84	C₂×C₃	C₃×C₈×C₉×C₂₅	1	4	yes	—
85	C₅	C₅×C₉²×C₁₆	1	1	yes	characteristic
86	C₄	C₄×C₉²×C₂₅	1	1	yes	characteristic
87	C₃	C₃×C₉×C₁₆×C₂₅	1	4	yes	—
88	C₂	C₈×C₉²×C₂₅	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₃²×C₅×C₈	C₂×C₃²×C₅
socle, prime radical	C₂×C₃²×C₅	C₃²×C₅×C₈
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
8. C₉×C₁₆×C₂₅	C₉	direct (N × H)
12. C₉²×C₂₅	C₁₆	direct (N × H)
17. C₉²×C₁₆	C₂₅	direct (N × H)
33. C₁₆×C₂₅	C₉²	direct (N × H)
41. C₉×C₂₅	C₉×C₁₆	direct (N × H)
49. C₉×C₁₆	C₉×C₂₅	direct (N × H)
56. C₉²	C₁₆×C₂₅	direct (N × H)
72. C₂₅	C₉²×C₁₆	direct (N × H)
77. C₁₆	C₉²×C₂₅	direct (N × H)
82. C₉	C₉×C₁₆×C₂₅	direct (N × H)

Maximal subgroups

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

subgroup	index
C₈×C₉²×C₂₅	2
C₃×C₉×C₁₆×C₂₅	3
C₅×C₉²×C₁₆	5

Subgroups — C92×C16×C25 #72209111

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉²×C₁₆×C₂₅ #72209111