Subgroups — C₃×C₄³×C₅×C₂₅ #72201284

Order 24000.

3612 conjugacy classes of subgroups.

The subgroup lattice has 120 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄²×C₅×C₂₅	C₂	1	7	yes	maximal
2	C₄³×C₅×C₂₅	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
3	C₂²×C₃×C₄×C₅×C₂₅	C₂²	1	7	yes	—
4	C₃×C₄²×C₅×C₂₅	C₄	1	28	yes	direct product factor
5	C₃×C₄³×C₅²	C₅	1	1	yes	maximal, characteristic
6	C₃×C₄³×C₂₅	C₅	1	5	yes	maximal, direct product factor
7	C₂×C₄²×C₅×C₂₅	C₂×C₃	1	7	yes	—
8	C₂³×C₃×C₅×C₂₅	C₂³	1	1	yes	characteristic
9	C₂×C₃×C₄×C₅×C₂₅	C₂×C₄	1	42	yes	—
10	C₂×C₃×C₄²×C₅²	C₂×C₅	1	7	yes	—
11	C₂×C₃×C₄²×C₂₅	C₂×C₅	1	35	yes	—
12	C₂²×C₄×C₅×C₂₅	C₂²×C₃	1	7	yes	—
13	C₄²×C₅×C₂₅	C₃×C₄	1	28	yes	direct product factor
14	C₄³×C₅²	C₃×C₅	1	1	yes	characteristic
15	C₄³×C₂₅	C₃×C₅	1	5	yes	direct product factor
16	C₂²×C₃×C₅×C₂₅	C₂²×C₄	1	7	yes	—
17	C₃×C₄×C₅×C₂₅	C₄²	1	28	yes	direct product factor
18	C₂²×C₃×C₄×C₅²	C₂²×C₅	1	7	yes	—
19	C₃×C₄²×C₅²	C₄×C₅	1	28	yes	—
20	C₂²×C₃×C₄×C₂₅	C₂²×C₅	1	35	yes	—
21	C₃×C₄²×C₂₅	C₄×C₅	1	140	yes	direct product factor
22	C₂³×C₅×C₂₅	C₂³×C₃	1	1	yes	characteristic
23	C₂×C₄×C₅×C₂₅	C₂×C₃×C₄	1	42	yes	—
24	C₃×C₄³×C₅	C₂₅	1	5	yes	direct product factor
25	C₃×C₄³×C₅	C₅²	1	1	yes	characteristic
26	C₂×C₄²×C₅²	C₂×C₃×C₅	1	7	yes	—
27	C₂×C₄²×C₂₅	C₂×C₃×C₅	1	35	yes	—
28	C₂×C₃×C₅×C₂₅	C₂×C₄²	1	7	yes	—
29	C₂³×C₃×C₅²	C₂³×C₅	1	1	yes	characteristic
30	C₂×C₃×C₄×C₅²	C₂×C₄×C₅	1	42	yes	—
31	C₂³×C₃×C₂₅	C₂³×C₅	1	5	yes	—
32	C₂×C₃×C₄×C₂₅	C₂×C₄×C₅	1	210	yes	—
33	C₂²×C₅×C₂₅	C₂²×C₃×C₄	1	7	yes	—
34	C₄×C₅×C₂₅	C₃×C₄²	1	28	yes	direct product factor
35	C₂×C₃×C₄²×C₅	C₂×C₂₅	1	35	yes	—
36	C₂×C₃×C₄²×C₅	C₂×C₅²	1	7	yes	—
37	C₂²×C₄×C₅²	C₂²×C₃×C₅	1	7	yes	—
38	C₄²×C₅²	C₃×C₄×C₅	1	28	yes	—
39	C₂²×C₄×C₂₅	C₂²×C₃×C₅	1	35	yes	—
40	C₄²×C₂₅	C₃×C₄×C₅	1	140	yes	direct product factor
41	C₃×C₅×C₂₅	C₄³	1	1	yes	characteristic, direct product factor, Hall
42	C₄³×C₅	C₃×C₂₅	1	5	yes	direct product factor
43	C₄³×C₅	C₃×C₅²	1	1	yes	characteristic
44	C₂²×C₃×C₅²	C₂²×C₄×C₅	1	7	yes	—
45	C₃×C₄×C₅²	C₄²×C₅	1	28	yes	—
46	C₂²×C₃×C₂₅	C₂²×C₄×C₅	1	35	yes	—
47	C₃×C₄×C₂₅	C₄²×C₅	1	140	yes	direct product factor
48	C₂×C₅×C₂₅	C₂×C₃×C₄²	1	7	yes	—
49	C₂²×C₃×C₄×C₅	C₂²×C₂₅	1	35	yes	—
50	C₂²×C₃×C₄×C₅	C₂²×C₅²	1	7	yes	—
51	C₃×C₄²×C₅	C₄×C₂₅	1	140	yes	direct product factor
52	C₃×C₄²×C₅	C₄×C₅²	1	28	yes	—
53	C₂³×C₅²	C₂³×C₃×C₅	1	1	yes	characteristic
54	C₂×C₄×C₅²	C₂×C₃×C₄×C₅	1	42	yes	—
55	C₂³×C₂₅	C₂³×C₃×C₅	1	5	yes	—
56	C₂×C₄×C₂₅	C₂×C₃×C₄×C₅	1	210	yes	—
57	C₃×C₄³	C₅×C₂₅	1	1	yes	characteristic, direct product factor, Hall
58	C₂×C₄²×C₅	C₂×C₃×C₂₅	1	35	yes	—
59	C₂×C₄²×C₅	C₂×C₃×C₅²	1	7	yes	—
60	C₂×C₃×C₅²	C₂×C₄²×C₅	1	7	yes	—
61	C₂×C₃×C₂₅	C₂×C₄²×C₅	1	35	yes	—
62	C₅×C₂₅	C₃×C₄³	1	1	yes	characteristic, direct product factor, Sylow
63	C₂³×C₃×C₅	C₂³×C₂₅	1	5	yes	—
64	C₂³×C₃×C₅	C₂³×C₅²	1	1	yes	characteristic
65	C₂×C₃×C₄×C₅	C₂×C₄×C₂₅	1	210	yes	—
66	C₂×C₃×C₄×C₅	C₂×C₄×C₅²	1	42	yes	—
67	C₂²×C₅²	C₂²×C₃×C₄×C₅	1	7	yes	—
68	C₄×C₅²	C₃×C₄²×C₅	1	28	yes	—
69	C₂²×C₂₅	C₂²×C₃×C₄×C₅	1	35	yes	—
70	C₄×C₂₅	C₃×C₄²×C₅	1	140	yes	direct product factor
71	C₂×C₃×C₄²	C₂×C₅×C₂₅	1	7	yes	—
72	C₂²×C₄×C₅	C₂²×C₃×C₂₅	1	35	yes	—
73	C₂²×C₄×C₅	C₂²×C₃×C₅²	1	7	yes	—
74	C₄²×C₅	C₃×C₄×C₂₅	1	140	yes	direct product factor
75	C₄²×C₅	C₃×C₄×C₅²	1	28	yes	—
76	C₃×C₅²	C₄³×C₅	1	1	yes	characteristic
77	C₃×C₂₅	C₄³×C₅	1	5	yes	direct product factor
78	C₄³	C₃×C₅×C₂₅	1	1	yes	characteristic, direct product factor, Sylow
79	C₂²×C₃×C₅	C₂²×C₄×C₂₅	1	35	yes	—
80	C₂²×C₃×C₅	C₂²×C₄×C₅²	1	7	yes	—
81	C₃×C₄×C₅	C₄²×C₂₅	1	140	yes	direct product factor
82	C₃×C₄×C₅	C₄²×C₅²	1	28	yes	—
83	C₂×C₅²	C₂×C₃×C₄²×C₅	1	7	yes	—
84	C₂×C₂₅	C₂×C₃×C₄²×C₅	1	35	yes	—
85	C₂²×C₃×C₄	C₂²×C₅×C₂₅	1	7	yes	—
86	C₃×C₄²	C₄×C₅×C₂₅	1	28	yes	direct product factor
87	C₂³×C₅	C₂³×C₃×C₂₅	1	5	yes	—
88	C₂³×C₅	C₂³×C₃×C₅²	1	1	yes	characteristic
89	C₂×C₄×C₅	C₂×C₃×C₄×C₂₅	1	210	yes	—
90	C₂×C₄×C₅	C₂×C₃×C₄×C₅²	1	42	yes	—
91	C₂×C₄²	C₂×C₃×C₅×C₂₅	1	7	yes	—
92	C₂×C₃×C₅	C₂×C₄²×C₂₅	1	35	yes	—
93	C₂×C₃×C₅	C₂×C₄²×C₅²	1	7	yes	—
94	C₅²	C₃×C₄³×C₅	1	1	yes	characteristic
95	C₂₅	C₃×C₄³×C₅	1	5	yes	direct product factor
96	C₂³×C₃	C₂³×C₅×C₂₅	1	1	yes	characteristic
97	C₂×C₃×C₄	C₂×C₄×C₅×C₂₅	1	42	yes	—
98	C₂²×C₅	C₂²×C₃×C₄×C₂₅	1	35	yes	—
99	C₂²×C₅	C₂²×C₃×C₄×C₅²	1	7	yes	—
100	C₄×C₅	C₃×C₄²×C₂₅	1	140	yes	direct product factor
101	C₄×C₅	C₃×C₄²×C₅²	1	28	yes	—
102	C₂²×C₄	C₂²×C₃×C₅×C₂₅	1	7	yes	—
103	C₄²	C₃×C₄×C₅×C₂₅	1	28	yes	direct product factor
104	C₃×C₅	C₄³×C₂₅	1	5	yes	direct product factor
105	C₃×C₅	C₄³×C₅²	1	1	yes	characteristic
106	C₂²×C₃	C₂²×C₄×C₅×C₂₅	1	7	yes	—
107	C₃×C₄	C₄²×C₅×C₂₅	1	28	yes	direct product factor
108	C₂×C₅	C₂×C₃×C₄²×C₂₅	1	35	yes	—
109	C₂×C₅	C₂×C₃×C₄²×C₅²	1	7	yes	—
110	C₂³	C₂³×C₃×C₅×C₂₅	1	1	yes	characteristic
111	C₂×C₄	C₂×C₃×C₄×C₅×C₂₅	1	42	yes	—
112	C₂×C₃	C₂×C₄²×C₅×C₂₅	1	7	yes	—
113	C₅	C₃×C₄³×C₂₅	1	5	yes	direct product factor
114	C₅	C₃×C₄³×C₅²	1	1	yes	characteristic
115	C₂²	C₂²×C₃×C₄×C₅×C₂₅	1	7	yes	—
116	C₄	C₃×C₄²×C₅×C₂₅	1	28	yes	direct product factor
117	C₃	C₄³×C₅×C₂₅	1	1	yes	characteristic, direct product factor, Sylow
118	C₂	C₂×C₃×C₄²×C₅×C₂₅	1	7	yes	—

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂³×C₅	C₂³×C₃×C₅²
socle, prime radical	C₂³×C₃×C₅²	C₂³×C₅
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄³×C₅×C₂₅	C₃	direct (N × H)
4. C₃×C₄²×C₅×C₂₅	C₄	direct (N × H)
6. C₃×C₄³×C₂₅	C₅	direct (N × H)
13. C₄²×C₅×C₂₅	C₃×C₄	direct (N × H)
15. C₄³×C₂₅	C₃×C₅	direct (N × H)
17. C₃×C₄×C₅×C₂₅	C₄²	direct (N × H)
21. C₃×C₄²×C₂₅	C₄×C₅	direct (N × H)
24. C₃×C₄³×C₅	C₂₅	direct (N × H)
34. C₄×C₅×C₂₅	C₃×C₄²	direct (N × H)
40. C₄²×C₂₅	C₃×C₄×C₅	direct (N × H)
41. C₃×C₅×C₂₅	C₄³	direct (N × H)
42. C₄³×C₅	C₃×C₂₅	direct (N × H)
47. C₃×C₄×C₂₅	C₄²×C₅	direct (N × H)
51. C₃×C₄²×C₅	C₄×C₂₅	direct (N × H)
57. C₃×C₄³	C₅×C₂₅	direct (N × H)
62. C₅×C₂₅	C₃×C₄³	direct (N × H)
70. C₄×C₂₅	C₃×C₄²×C₅	direct (N × H)
74. C₄²×C₅	C₃×C₄×C₂₅	direct (N × H)
77. C₃×C₂₅	C₄³×C₅	direct (N × H)
78. C₄³	C₃×C₅×C₂₅	direct (N × H)
81. C₃×C₄×C₅	C₄²×C₂₅	direct (N × H)
86. C₃×C₄²	C₄×C₅×C₂₅	direct (N × H)
95. C₂₅	C₃×C₄³×C₅	direct (N × H)
100. C₄×C₅	C₃×C₄²×C₂₅	direct (N × H)
103. C₄²	C₃×C₄×C₅×C₂₅	direct (N × H)
104. C₃×C₅	C₄³×C₂₅	direct (N × H)
107. C₃×C₄	C₄²×C₅×C₂₅	direct (N × H)
113. C₅	C₃×C₄³×C₂₅	direct (N × H)
116. C₄	C₃×C₄²×C₅×C₂₅	direct (N × H)
117. C₃	C₄³×C₅×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
C₂×C₃×C₄²×C₅×C₂₅	2
C₄³×C₅×C₂₅	3
C₃×C₄³×C₅²	5
C₃×C₄³×C₂₅	5

Subgroups — C3×C43×C5×C25 #72201284

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₄³×C₅×C₂₅ #72201284