Subgroups — C₂²×C₃×C₉²×C₂₅ #72201585

Order 24300.

1890 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×C₃×C₉²×C₂₅	C₂	1	3	yes	maximal, direct product factor
2	C₂²×C₃²×C₉×C₂₅	C₃	1	4	yes	maximal
3	C₂²×C₉²×C₂₅	C₃	1	9	yes	maximal, direct product factor
4	C₃×C₉²×C₂₅	C₂²	1	1	yes	characteristic, direct product factor, Hall
5	C₂²×C₃×C₅×C₉²	C₅	1	1	yes	maximal, characteristic
6	C₂×C₃²×C₉×C₂₅	C₂×C₃	1	12	yes	—
7	C₂×C₉²×C₂₅	C₂×C₃	1	27	yes	direct product factor
8	C₂²×C₃³×C₂₅	C₃²	1	1	yes	characteristic
9	C₂²×C₃×C₉×C₂₅	C₉	1	36	yes	direct product factor
10	C₂²×C₃×C₉×C₂₅	C₃²	1	12	yes	—
11	C₂×C₃×C₅×C₉²	C₂×C₅	1	3	yes	—
12	C₃²×C₉×C₂₅	C₂²×C₃	1	4	yes	—
13	C₉²×C₂₅	C₂²×C₃	1	9	yes	direct product factor
14	C₂²×C₃²×C₅×C₉	C₃×C₅	1	4	yes	—
15	C₂²×C₅×C₉²	C₃×C₅	1	9	yes	—
16	C₂×C₃³×C₂₅	C₂×C₃²	1	3	yes	—
17	C₂×C₃×C₉×C₂₅	C₂×C₉	1	108	yes	direct product factor
18	C₂×C₃×C₉×C₂₅	C₂×C₃²	1	36	yes	—
19	C₃×C₅×C₉²	C₂²×C₅	1	1	yes	characteristic
20	C₂²×C₃×C₉²	C₂₅	1	1	yes	characteristic, direct product factor, Hall
21	C₂²×C₃²×C₂₅	C₃×C₉	1	12	yes	—
22	C₂²×C₃²×C₂₅	C₃³	1	1	yes	characteristic
23	C₂²×C₉×C₂₅	C₃×C₉	1	36	yes	direct product factor
24	C₂×C₃²×C₅×C₉	C₂×C₃×C₅	1	12	yes	—
25	C₂×C₅×C₉²	C₂×C₃×C₅	1	27	yes	—
26	C₃³×C₂₅	C₂²×C₃²	1	1	yes	characteristic
27	C₃×C₉×C₂₅	C₂²×C₉	1	36	yes	direct product factor
28	C₃×C₉×C₂₅	C₂²×C₃²	1	12	yes	—
29	C₂²×C₃³×C₅	C₃²×C₅	1	1	yes	characteristic
30	C₂²×C₃×C₅×C₉	C₅×C₉	1	36	yes	—
31	C₂²×C₃×C₅×C₉	C₃²×C₅	1	12	yes	—
32	C₂×C₃×C₉²	C₂×C₂₅	1	3	yes	direct product factor
33	C₂×C₃²×C₂₅	C₂×C₃×C₉	1	36	yes	—
34	C₂×C₃²×C₂₅	C₂×C₃³	1	3	yes	—
35	C₂×C₉×C₂₅	C₂×C₃×C₉	1	108	yes	direct product factor
36	C₃²×C₅×C₉	C₂²×C₃×C₅	1	4	yes	—
37	C₅×C₉²	C₂²×C₃×C₅	1	9	yes	—
38	C₂²×C₃²×C₉	C₃×C₂₅	1	4	yes	—
39	C₂²×C₉²	C₃×C₂₅	1	9	yes	direct product factor
40	C₂²×C₃×C₂₅	C₉²	1	9	yes	direct product factor
41	C₂²×C₃×C₂₅	C₃²×C₉	1	4	yes	—
42	C₂×C₃³×C₅	C₂×C₃²×C₅	1	3	yes	—
43	C₂×C₃×C₅×C₉	C₂×C₅×C₉	1	108	yes	—
44	C₂×C₃×C₅×C₉	C₂×C₃²×C₅	1	36	yes	—
45	C₃×C₉²	C₂²×C₂₅	1	1	yes	characteristic, direct product factor, Sylow
46	C₃²×C₂₅	C₂²×C₃×C₉	1	12	yes	—
47	C₃²×C₂₅	C₂²×C₃³	1	1	yes	characteristic
48	C₉×C₂₅	C₂²×C₃×C₉	1	36	yes	direct product factor
49	C₂²×C₃²×C₅	C₃×C₅×C₉	1	12	yes	—
50	C₂²×C₃²×C₅	C₃³×C₅	1	1	yes	characteristic
51	C₂²×C₅×C₉	C₃×C₅×C₉	1	36	yes	—
52	C₂×C₃²×C₉	C₂×C₃×C₂₅	1	12	yes	—
53	C₂×C₉²	C₂×C₃×C₂₅	1	27	yes	direct product factor
54	C₂×C₃×C₂₅	C₂×C₉²	1	27	yes	direct product factor
55	C₂×C₃×C₂₅	C₂×C₃²×C₉	1	12	yes	—
56	C₃³×C₅	C₂²×C₃²×C₅	1	1	yes	characteristic
57	C₃×C₅×C₉	C₂²×C₅×C₉	1	36	yes	—
58	C₃×C₅×C₉	C₂²×C₃²×C₅	1	12	yes	—
59	C₂²×C₃³	C₃²×C₂₅	1	1	yes	characteristic
60	C₂²×C₃×C₉	C₉×C₂₅	1	36	yes	direct product factor
61	C₂²×C₃×C₉	C₃²×C₂₅	1	12	yes	—
62	C₂²×C₂₅	C₃×C₉²	1	1	yes	characteristic, direct product factor, Hall
63	C₂×C₃²×C₅	C₂×C₃×C₅×C₉	1	36	yes	—
64	C₂×C₃²×C₅	C₂×C₃³×C₅	1	3	yes	—
65	C₂×C₅×C₉	C₂×C₃×C₅×C₉	1	108	yes	—
66	C₃²×C₉	C₂²×C₃×C₂₅	1	4	yes	—
67	C₉²	C₂²×C₃×C₂₅	1	9	yes	direct product factor
68	C₃×C₂₅	C₂²×C₉²	1	9	yes	direct product factor
69	C₃×C₂₅	C₂²×C₃²×C₉	1	4	yes	—
70	C₂²×C₃×C₅	C₅×C₉²	1	9	yes	—
71	C₂²×C₃×C₅	C₃²×C₅×C₉	1	4	yes	—
72	C₂×C₃³	C₂×C₃²×C₂₅	1	3	yes	—
73	C₂×C₃×C₉	C₂×C₉×C₂₅	1	108	yes	direct product factor
74	C₂×C₃×C₉	C₂×C₃²×C₂₅	1	36	yes	—
75	C₂×C₂₅	C₂×C₃×C₉²	1	3	yes	direct product factor
76	C₃²×C₅	C₂²×C₃×C₅×C₉	1	12	yes	—
77	C₃²×C₅	C₂²×C₃³×C₅	1	1	yes	characteristic
78	C₅×C₉	C₂²×C₃×C₅×C₉	1	36	yes	—
79	C₂²×C₃²	C₃×C₉×C₂₅	1	12	yes	—
80	C₂²×C₃²	C₃³×C₂₅	1	1	yes	characteristic
81	C₂²×C₉	C₃×C₉×C₂₅	1	36	yes	direct product factor
82	C₂×C₃×C₅	C₂×C₅×C₉²	1	27	yes	—
83	C₂×C₃×C₅	C₂×C₃²×C₅×C₉	1	12	yes	—
84	C₃³	C₂²×C₃²×C₂₅	1	1	yes	characteristic
85	C₃×C₉	C₂²×C₉×C₂₅	1	36	yes	direct product factor
86	C₃×C₉	C₂²×C₃²×C₂₅	1	12	yes	—
87	C₂₅	C₂²×C₃×C₉²	1	1	yes	characteristic, direct product factor, Sylow
88	C₂²×C₅	C₃×C₅×C₉²	1	1	yes	characteristic
89	C₂×C₃²	C₂×C₃×C₉×C₂₅	1	36	yes	—
90	C₂×C₃²	C₂×C₃³×C₂₅	1	3	yes	—
91	C₂×C₉	C₂×C₃×C₉×C₂₅	1	108	yes	direct product factor
92	C₃×C₅	C₂²×C₅×C₉²	1	9	yes	—
93	C₃×C₅	C₂²×C₃²×C₅×C₉	1	4	yes	—
94	C₂²×C₃	C₉²×C₂₅	1	9	yes	direct product factor
95	C₂²×C₃	C₃²×C₉×C₂₅	1	4	yes	—
96	C₂×C₅	C₂×C₃×C₅×C₉²	1	3	yes	—
97	C₃²	C₂²×C₃×C₉×C₂₅	1	12	yes	—
98	C₃²	C₂²×C₃³×C₂₅	1	1	yes	characteristic
99	C₉	C₂²×C₃×C₉×C₂₅	1	36	yes	direct product factor
100	C₂×C₃	C₂×C₉²×C₂₅	1	27	yes	direct product factor
101	C₂×C₃	C₂×C₃²×C₉×C₂₅	1	12	yes	—
102	C₅	C₂²×C₃×C₅×C₉²	1	1	yes	characteristic
103	C₂²	C₃×C₉²×C₂₅	1	1	yes	characteristic, direct product factor, Sylow
104	C₃	C₂²×C₉²×C₂₅	1	9	yes	direct product factor
105	C₃	C₂²×C₃²×C₉×C₂₅	1	4	yes	—
106	C₂	C₂×C₃×C₉²×C₂₅	1	3	yes	direct product factor

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₃²×C₅	C₂²×C₃³×C₅
socle, prime radical	C₂²×C₃³×C₅	C₃²×C₅
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₉²×C₂₅	C₂	direct (N × H)
3. C₂²×C₉²×C₂₅	C₃	direct (N × H)
4. C₃×C₉²×C₂₅	C₂²	direct (N × H)
7. C₂×C₉²×C₂₅	C₂×C₃	direct (N × H)
9. C₂²×C₃×C₉×C₂₅	C₉	direct (N × H)
13. C₉²×C₂₅	C₂²×C₃	direct (N × H)
17. C₂×C₃×C₉×C₂₅	C₂×C₉	direct (N × H)
20. C₂²×C₃×C₉²	C₂₅	direct (N × H)
23. C₂²×C₉×C₂₅	C₃×C₉	direct (N × H)
27. C₃×C₉×C₂₅	C₂²×C₉	direct (N × H)
32. C₂×C₃×C₉²	C₂×C₂₅	direct (N × H)
35. C₂×C₉×C₂₅	C₂×C₃×C₉	direct (N × H)
39. C₂²×C₉²	C₃×C₂₅	direct (N × H)
40. C₂²×C₃×C₂₅	C₉²	direct (N × H)
45. C₃×C₉²	C₂²×C₂₅	direct (N × H)
48. C₉×C₂₅	C₂²×C₃×C₉	direct (N × H)
53. C₂×C₉²	C₂×C₃×C₂₅	direct (N × H)
54. C₂×C₃×C₂₅	C₂×C₉²	direct (N × H)
60. C₂²×C₃×C₉	C₉×C₂₅	direct (N × H)
62. C₂²×C₂₅	C₃×C₉²	direct (N × H)
67. C₉²	C₂²×C₃×C₂₅	direct (N × H)
68. C₃×C₂₅	C₂²×C₉²	direct (N × H)
73. C₂×C₃×C₉	C₂×C₉×C₂₅	direct (N × H)
75. C₂×C₂₅	C₂×C₃×C₉²	direct (N × H)
81. C₂²×C₉	C₃×C₉×C₂₅	direct (N × H)
85. C₃×C₉	C₂²×C₉×C₂₅	direct (N × H)
87. C₂₅	C₂²×C₃×C₉²	direct (N × H)
91. C₂×C₉	C₂×C₃×C₉×C₂₅	direct (N × H)
94. C₂²×C₃	C₉²×C₂₅	direct (N × H)
99. C₉	C₂²×C₃×C₉×C₂₅	direct (N × H)
100. C₂×C₃	C₂×C₉²×C₂₅	direct (N × H)
103. C₂²	C₃×C₉²×C₂₅	direct (N × H)
104. C₃	C₂²×C₉²×C₂₅	direct (N × H)
106. C₂	C₂×C₃×C₉²×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×C₉²×C₂₅	2
C₂²×C₃²×C₉×C₂₅	3
C₂²×C₉²×C₂₅	3
C₂²×C₃×C₅×C₉²	5

Subgroups — C22×C3×C92×C25 #72201585

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×C₉²×C₂₅ #72201585