Subgroups — C₃³×C₄×C₉×C₂₅ #72201574

Order 24300.

3564 conjugacy classes of subgroups.

The subgroup lattice has 108 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃³×C₉×C₂₅	C₂	1	1	yes	maximal, characteristic
2	C₃⁴×C₄×C₂₅	C₃	1	1	yes	maximal, characteristic
3	C₃²×C₄×C₉×C₂₅	C₃	1	39	yes	maximal, direct product factor
4	C₃³×C₉×C₂₅	C₄	1	1	yes	characteristic, direct product factor, Hall
5	C₃³×C₄×C₅×C₉	C₅	1	1	yes	maximal, characteristic
6	C₂×C₃⁴×C₂₅	C₂×C₃	1	1	yes	characteristic
7	C₂×C₃²×C₉×C₂₅	C₂×C₃	1	39	yes	—
8	C₃³×C₄×C₂₅	C₉	1	27	yes	direct product factor
9	C₃³×C₄×C₂₅	C₃²	1	13	yes	—
10	C₃×C₄×C₉×C₂₅	C₃²	1	117	yes	direct product factor
11	C₂×C₃³×C₅×C₉	C₂×C₅	1	1	yes	characteristic
12	C₃⁴×C₂₅	C₃×C₄	1	1	yes	characteristic
13	C₃²×C₉×C₂₅	C₃×C₄	1	39	yes	direct product factor
14	C₃⁴×C₄×C₅	C₃×C₅	1	1	yes	characteristic
15	C₃²×C₄×C₅×C₉	C₃×C₅	1	39	yes	—
16	C₂×C₃³×C₂₅	C₂×C₉	1	27	yes	—
17	C₂×C₃³×C₂₅	C₂×C₃²	1	13	yes	—
18	C₂×C₃×C₉×C₂₅	C₂×C₃²	1	117	yes	—
19	C₃³×C₅×C₉	C₄×C₅	1	1	yes	characteristic
20	C₃³×C₄×C₉	C₂₅	1	1	yes	characteristic, direct product factor, Hall
21	C₃²×C₄×C₂₅	C₃×C₉	1	117	yes	direct product factor
22	C₃²×C₄×C₂₅	C₃³	1	13	yes	—
23	C₄×C₉×C₂₅	C₃³	1	27	yes	direct product factor
24	C₂×C₃⁴×C₅	C₂×C₃×C₅	1	1	yes	characteristic
25	C₂×C₃²×C₅×C₉	C₂×C₃×C₅	1	39	yes	—
26	C₃³×C₂₅	C₄×C₉	1	27	yes	direct product factor
27	C₃³×C₂₅	C₃²×C₄	1	13	yes	—
28	C₃×C₉×C₂₅	C₃²×C₄	1	117	yes	direct product factor
29	C₃³×C₄×C₅	C₅×C₉	1	27	yes	—
30	C₃³×C₄×C₅	C₃²×C₅	1	13	yes	—
31	C₃×C₄×C₅×C₉	C₃²×C₅	1	117	yes	—
32	C₂×C₃³×C₉	C₂×C₂₅	1	1	yes	characteristic
33	C₂×C₃²×C₂₅	C₂×C₃×C₉	1	117	yes	—
34	C₂×C₃²×C₂₅	C₂×C₃³	1	13	yes	—
35	C₂×C₉×C₂₅	C₂×C₃³	1	27	yes	—
36	C₃⁴×C₅	C₃×C₄×C₅	1	1	yes	characteristic
37	C₃²×C₅×C₉	C₃×C₄×C₅	1	39	yes	—
38	C₃⁴×C₄	C₃×C₂₅	1	1	yes	characteristic
39	C₃²×C₄×C₉	C₃×C₂₅	1	39	yes	direct product factor
40	C₃×C₄×C₂₅	C₃²×C₉	1	39	yes	direct product factor
41	C₃×C₄×C₂₅	C₃⁴	1	1	yes	characteristic
42	C₂×C₃³×C₅	C₂×C₅×C₉	1	27	yes	—
43	C₂×C₃³×C₅	C₂×C₃²×C₅	1	13	yes	—
44	C₂×C₃×C₅×C₉	C₂×C₃²×C₅	1	117	yes	—
45	C₃³×C₉	C₄×C₂₅	1	1	yes	characteristic, direct product factor, Sylow
46	C₃²×C₂₅	C₃×C₄×C₉	1	117	yes	direct product factor
47	C₃²×C₂₅	C₃³×C₄	1	13	yes	—
48	C₉×C₂₅	C₃³×C₄	1	27	yes	direct product factor
49	C₃²×C₄×C₅	C₃×C₅×C₉	1	117	yes	—
50	C₃²×C₄×C₅	C₃³×C₅	1	13	yes	—
51	C₄×C₅×C₉	C₃³×C₅	1	27	yes	—
52	C₂×C₃⁴	C₂×C₃×C₂₅	1	1	yes	characteristic
53	C₂×C₃²×C₉	C₂×C₃×C₂₅	1	39	yes	—
54	C₂×C₃×C₂₅	C₂×C₃²×C₉	1	39	yes	—
55	C₂×C₃×C₂₅	C₂×C₃⁴	1	1	yes	characteristic
56	C₃³×C₅	C₄×C₅×C₉	1	27	yes	—
57	C₃³×C₅	C₃²×C₄×C₅	1	13	yes	—
58	C₃×C₅×C₉	C₃²×C₄×C₅	1	117	yes	—
59	C₃³×C₄	C₉×C₂₅	1	27	yes	direct product factor
60	C₃³×C₄	C₃²×C₂₅	1	13	yes	—
61	C₃×C₄×C₉	C₃²×C₂₅	1	117	yes	direct product factor
62	C₄×C₂₅	C₃³×C₉	1	1	yes	characteristic, direct product factor, Hall
63	C₂×C₃²×C₅	C₂×C₃×C₅×C₉	1	117	yes	—
64	C₂×C₃²×C₅	C₂×C₃³×C₅	1	13	yes	—
65	C₂×C₅×C₉	C₂×C₃³×C₅	1	27	yes	—
66	C₃⁴	C₃×C₄×C₂₅	1	1	yes	characteristic
67	C₃²×C₉	C₃×C₄×C₂₅	1	39	yes	direct product factor
68	C₃×C₂₅	C₃²×C₄×C₉	1	39	yes	direct product factor
69	C₃×C₂₅	C₃⁴×C₄	1	1	yes	characteristic
70	C₃×C₄×C₅	C₃²×C₅×C₉	1	39	yes	—
71	C₃×C₄×C₅	C₃⁴×C₅	1	1	yes	characteristic
72	C₂×C₃³	C₂×C₉×C₂₅	1	27	yes	—
73	C₂×C₃³	C₂×C₃²×C₂₅	1	13	yes	—
74	C₂×C₃×C₉	C₂×C₃²×C₂₅	1	117	yes	—
75	C₂×C₂₅	C₂×C₃³×C₉	1	1	yes	characteristic
76	C₃²×C₅	C₃×C₄×C₅×C₉	1	117	yes	—
77	C₃²×C₅	C₃³×C₄×C₅	1	13	yes	—
78	C₅×C₉	C₃³×C₄×C₅	1	27	yes	—
79	C₃²×C₄	C₃×C₉×C₂₅	1	117	yes	direct product factor
80	C₃²×C₄	C₃³×C₂₅	1	13	yes	—
81	C₄×C₉	C₃³×C₂₅	1	27	yes	direct product factor
82	C₂×C₃×C₅	C₂×C₃²×C₅×C₉	1	39	yes	—
83	C₂×C₃×C₅	C₂×C₃⁴×C₅	1	1	yes	characteristic
84	C₃³	C₄×C₉×C₂₅	1	27	yes	direct product factor
85	C₃³	C₃²×C₄×C₂₅	1	13	yes	—
86	C₃×C₉	C₃²×C₄×C₂₅	1	117	yes	direct product factor
87	C₂₅	C₃³×C₄×C₉	1	1	yes	characteristic, direct product factor, Sylow
88	C₄×C₅	C₃³×C₅×C₉	1	1	yes	characteristic
89	C₂×C₃²	C₂×C₃×C₉×C₂₅	1	117	yes	—
90	C₂×C₃²	C₂×C₃³×C₂₅	1	13	yes	—
91	C₂×C₉	C₂×C₃³×C₂₅	1	27	yes	—
92	C₃×C₅	C₃²×C₄×C₅×C₉	1	39	yes	—
93	C₃×C₅	C₃⁴×C₄×C₅	1	1	yes	characteristic
94	C₃×C₄	C₃²×C₉×C₂₅	1	39	yes	direct product factor
95	C₃×C₄	C₃⁴×C₂₅	1	1	yes	characteristic
96	C₂×C₅	C₂×C₃³×C₅×C₉	1	1	yes	characteristic
97	C₃²	C₃×C₄×C₉×C₂₅	1	117	yes	direct product factor
98	C₃²	C₃³×C₄×C₂₅	1	13	yes	—
99	C₉	C₃³×C₄×C₂₅	1	27	yes	direct product factor
100	C₂×C₃	C₂×C₃²×C₉×C₂₅	1	39	yes	—
101	C₂×C₃	C₂×C₃⁴×C₂₅	1	1	yes	characteristic
102	C₅	C₃³×C₄×C₅×C₉	1	1	yes	characteristic
103	C₄	C₃³×C₉×C₂₅	1	1	yes	characteristic, direct product factor, Sylow
104	C₃	C₃²×C₄×C₉×C₂₅	1	39	yes	direct product factor
105	C₃	C₃⁴×C₄×C₂₅	1	1	yes	characteristic
106	C₂	C₂×C₃³×C₉×C₂₅	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂×C₃×C₅	C₂×C₃⁴×C₅
socle, prime radical	C₂×C₃⁴×C₅	C₂×C₃×C₅
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃²×C₄×C₉×C₂₅	C₃	direct (N × H)
4. C₃³×C₉×C₂₅	C₄	direct (N × H)
8. C₃³×C₄×C₂₅	C₉	direct (N × H)
10. C₃×C₄×C₉×C₂₅	C₃²	direct (N × H)
13. C₃²×C₉×C₂₅	C₃×C₄	direct (N × H)
20. C₃³×C₄×C₉	C₂₅	direct (N × H)
21. C₃²×C₄×C₂₅	C₃×C₉	direct (N × H)
23. C₄×C₉×C₂₅	C₃³	direct (N × H)
26. C₃³×C₂₅	C₄×C₉	direct (N × H)
28. C₃×C₉×C₂₅	C₃²×C₄	direct (N × H)
39. C₃²×C₄×C₉	C₃×C₂₅	direct (N × H)
40. C₃×C₄×C₂₅	C₃²×C₉	direct (N × H)
45. C₃³×C₉	C₄×C₂₅	direct (N × H)
46. C₃²×C₂₅	C₃×C₄×C₉	direct (N × H)
48. C₉×C₂₅	C₃³×C₄	direct (N × H)
59. C₃³×C₄	C₉×C₂₅	direct (N × H)
61. C₃×C₄×C₉	C₃²×C₂₅	direct (N × H)
62. C₄×C₂₅	C₃³×C₉	direct (N × H)
67. C₃²×C₉	C₃×C₄×C₂₅	direct (N × H)
68. C₃×C₂₅	C₃²×C₄×C₉	direct (N × H)
79. C₃²×C₄	C₃×C₉×C₂₅	direct (N × H)
81. C₄×C₉	C₃³×C₂₅	direct (N × H)
84. C₃³	C₄×C₉×C₂₅	direct (N × H)
86. C₃×C₉	C₃²×C₄×C₂₅	direct (N × H)
87. C₂₅	C₃³×C₄×C₉	direct (N × H)
94. C₃×C₄	C₃²×C₉×C₂₅	direct (N × H)
97. C₃²	C₃×C₄×C₉×C₂₅	direct (N × H)
99. C₉	C₃³×C₄×C₂₅	direct (N × H)
103. C₄	C₃³×C₉×C₂₅	direct (N × H)
104. C₃	C₃²×C₄×C₉×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×C₃³×C₉×C₂₅	2
C₃⁴×C₄×C₂₅	3
C₃²×C₄×C₉×C₂₅	3
C₃³×C₄×C₅×C₉	5

Subgroups — C33×C4×C9×C25 #72201574

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃³×C₄×C₉×C₂₅ #72201574