Subgroups — C₂⁴×C₄×C₉×C₂₅ #72192574

Order 14400.

6129 conjugacy classes of subgroups.

The subgroup lattice has 135 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁵×C₉×C₂₅	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₄×C₉×C₂₅	C₂	1	30	yes	maximal, direct product factor
3	C₂⁴×C₃×C₄×C₂₅	C₃	1	1	yes	maximal, characteristic
4	C₂⁴×C₉×C₂₅	C₄	1	16	yes	direct product factor
5	C₂⁴×C₉×C₂₅	C₂²	1	15	yes	—
6	C₂²×C₄×C₉×C₂₅	C₂²	1	140	yes	direct product factor
7	C₂⁴×C₄×C₅×C₉	C₅	1	1	yes	maximal, characteristic
8	C₂⁵×C₃×C₂₅	C₂×C₃	1	1	yes	characteristic
9	C₂³×C₃×C₄×C₂₅	C₂×C₃	1	30	yes	—
10	C₂³×C₉×C₂₅	C₂×C₄	1	120	yes	direct product factor
11	C₂³×C₉×C₂₅	C₂³	1	35	yes	—
12	C₂×C₄×C₉×C₂₅	C₂³	1	120	yes	direct product factor
13	C₂⁴×C₄×C₂₅	C₉	1	1	yes	characteristic, direct product factor, Hall
14	C₂⁵×C₅×C₉	C₂×C₅	1	1	yes	characteristic
15	C₂³×C₄×C₅×C₉	C₂×C₅	1	30	yes	—
16	C₂⁴×C₃×C₂₅	C₃×C₄	1	16	yes	—
17	C₂⁴×C₃×C₂₅	C₂²×C₃	1	15	yes	—
18	C₂²×C₃×C₄×C₂₅	C₂²×C₃	1	140	yes	—
19	C₂⁴×C₃×C₄×C₅	C₃×C₅	1	1	yes	characteristic
20	C₂²×C₉×C₂₅	C₂²×C₄	1	140	yes	direct product factor
21	C₂²×C₉×C₂₅	C₂⁴	1	15	yes	—
22	C₄×C₉×C₂₅	C₂⁴	1	16	yes	direct product factor
23	C₂⁵×C₂₅	C₂×C₉	1	1	yes	characteristic
24	C₂³×C₄×C₂₅	C₂×C₉	1	30	yes	direct product factor
25	C₂⁴×C₅×C₉	C₄×C₅	1	16	yes	—
26	C₂⁴×C₅×C₉	C₂²×C₅	1	15	yes	—
27	C₂²×C₄×C₅×C₉	C₂²×C₅	1	140	yes	—
28	C₂³×C₃×C₂₅	C₂×C₃×C₄	1	120	yes	—
29	C₂³×C₃×C₂₅	C₂³×C₃	1	35	yes	—
30	C₂×C₃×C₄×C₂₅	C₂³×C₃	1	120	yes	—
31	C₂⁴×C₄×C₉	C₂₅	1	1	yes	characteristic, direct product factor, Hall
32	C₂⁵×C₃×C₅	C₂×C₃×C₅	1	1	yes	characteristic
33	C₂³×C₃×C₄×C₅	C₂×C₃×C₅	1	30	yes	—
34	C₂×C₉×C₂₅	C₂³×C₄	1	30	yes	direct product factor
35	C₂×C₉×C₂₅	C₂⁵	1	1	yes	characteristic
36	C₂⁴×C₂₅	C₄×C₉	1	16	yes	direct product factor
37	C₂⁴×C₂₅	C₂²×C₉	1	15	yes	—
38	C₂²×C₄×C₂₅	C₂²×C₉	1	140	yes	direct product factor
39	C₂³×C₅×C₉	C₂×C₄×C₅	1	120	yes	—
40	C₂³×C₅×C₉	C₂³×C₅	1	35	yes	—
41	C₂×C₄×C₅×C₉	C₂³×C₅	1	120	yes	—
42	C₂⁴×C₄×C₅	C₅×C₉	1	1	yes	characteristic
43	C₂²×C₃×C₂₅	C₂²×C₃×C₄	1	140	yes	—
44	C₂²×C₃×C₂₅	C₂⁴×C₃	1	15	yes	—
45	C₃×C₄×C₂₅	C₂⁴×C₃	1	16	yes	—
46	C₂⁵×C₉	C₂×C₂₅	1	1	yes	characteristic
47	C₂³×C₄×C₉	C₂×C₂₅	1	30	yes	direct product factor
48	C₂⁴×C₃×C₅	C₃×C₄×C₅	1	16	yes	—
49	C₂⁴×C₃×C₅	C₂²×C₃×C₅	1	15	yes	—
50	C₂²×C₃×C₄×C₅	C₂²×C₃×C₅	1	140	yes	—
51	C₉×C₂₅	C₂⁴×C₄	1	1	yes	characteristic, direct product factor, Hall
52	C₂³×C₂₅	C₂×C₄×C₉	1	120	yes	direct product factor
53	C₂³×C₂₅	C₂³×C₉	1	35	yes	—
54	C₂×C₄×C₂₅	C₂³×C₉	1	120	yes	direct product factor
55	C₂⁴×C₃×C₄	C₃×C₂₅	1	1	yes	characteristic
56	C₂²×C₅×C₉	C₂²×C₄×C₅	1	140	yes	—
57	C₂²×C₅×C₉	C₂⁴×C₅	1	15	yes	—
58	C₄×C₅×C₉	C₂⁴×C₅	1	16	yes	—
59	C₂⁵×C₅	C₂×C₅×C₉	1	1	yes	characteristic
60	C₂³×C₄×C₅	C₂×C₅×C₉	1	30	yes	—
61	C₂×C₃×C₂₅	C₂³×C₃×C₄	1	30	yes	—
62	C₂×C₃×C₂₅	C₂⁵×C₃	1	1	yes	characteristic
63	C₂⁴×C₉	C₄×C₂₅	1	16	yes	direct product factor
64	C₂⁴×C₉	C₂²×C₂₅	1	15	yes	—
65	C₂²×C₄×C₉	C₂²×C₂₅	1	140	yes	direct product factor
66	C₂³×C₃×C₅	C₂×C₃×C₄×C₅	1	120	yes	—
67	C₂³×C₃×C₅	C₂³×C₃×C₅	1	35	yes	—
68	C₂×C₃×C₄×C₅	C₂³×C₃×C₅	1	120	yes	—
69	C₂²×C₂₅	C₂²×C₄×C₉	1	140	yes	direct product factor
70	C₂²×C₂₅	C₂⁴×C₉	1	15	yes	—
71	C₄×C₂₅	C₂⁴×C₉	1	16	yes	direct product factor
72	C₂⁵×C₃	C₂×C₃×C₂₅	1	1	yes	characteristic
73	C₂³×C₃×C₄	C₂×C₃×C₂₅	1	30	yes	—
74	C₂×C₅×C₉	C₂³×C₄×C₅	1	30	yes	—
75	C₂×C₅×C₉	C₂⁵×C₅	1	1	yes	characteristic
76	C₂⁴×C₅	C₄×C₅×C₉	1	16	yes	—
77	C₂⁴×C₅	C₂²×C₅×C₉	1	15	yes	—
78	C₂²×C₄×C₅	C₂²×C₅×C₉	1	140	yes	—
79	C₃×C₂₅	C₂⁴×C₃×C₄	1	1	yes	characteristic
80	C₂³×C₉	C₂×C₄×C₂₅	1	120	yes	direct product factor
81	C₂³×C₉	C₂³×C₂₅	1	35	yes	—
82	C₂×C₄×C₉	C₂³×C₂₅	1	120	yes	direct product factor
83	C₂⁴×C₄	C₉×C₂₅	1	1	yes	characteristic, direct product factor, Sylow
84	C₂²×C₃×C₅	C₂²×C₃×C₄×C₅	1	140	yes	—
85	C₂²×C₃×C₅	C₂⁴×C₃×C₅	1	15	yes	—
86	C₃×C₄×C₅	C₂⁴×C₃×C₅	1	16	yes	—
87	C₂×C₂₅	C₂³×C₄×C₉	1	30	yes	direct product factor
88	C₂×C₂₅	C₂⁵×C₉	1	1	yes	characteristic
89	C₂⁴×C₃	C₃×C₄×C₂₅	1	16	yes	—
90	C₂⁴×C₃	C₂²×C₃×C₂₅	1	15	yes	—
91	C₂²×C₃×C₄	C₂²×C₃×C₂₅	1	140	yes	—
92	C₅×C₉	C₂⁴×C₄×C₅	1	1	yes	characteristic
93	C₂³×C₅	C₂×C₄×C₅×C₉	1	120	yes	—
94	C₂³×C₅	C₂³×C₅×C₉	1	35	yes	—
95	C₂×C₄×C₅	C₂³×C₅×C₉	1	120	yes	—
96	C₂²×C₉	C₂²×C₄×C₂₅	1	140	yes	direct product factor
97	C₂²×C₉	C₂⁴×C₂₅	1	15	yes	—
98	C₄×C₉	C₂⁴×C₂₅	1	16	yes	direct product factor
99	C₂⁵	C₂×C₉×C₂₅	1	1	yes	characteristic
100	C₂³×C₄	C₂×C₉×C₂₅	1	30	yes	direct product factor
101	C₂×C₃×C₅	C₂³×C₃×C₄×C₅	1	30	yes	—
102	C₂×C₃×C₅	C₂⁵×C₃×C₅	1	1	yes	characteristic
103	C₂₅	C₂⁴×C₄×C₉	1	1	yes	characteristic, direct product factor, Sylow
104	C₂³×C₃	C₂×C₃×C₄×C₂₅	1	120	yes	—
105	C₂³×C₃	C₂³×C₃×C₂₅	1	35	yes	—
106	C₂×C₃×C₄	C₂³×C₃×C₂₅	1	120	yes	—
107	C₂²×C₅	C₂²×C₄×C₅×C₉	1	140	yes	—
108	C₂²×C₅	C₂⁴×C₅×C₉	1	15	yes	—
109	C₄×C₅	C₂⁴×C₅×C₉	1	16	yes	—
110	C₂×C₉	C₂³×C₄×C₂₅	1	30	yes	direct product factor
111	C₂×C₉	C₂⁵×C₂₅	1	1	yes	characteristic
112	C₂⁴	C₄×C₉×C₂₅	1	16	yes	direct product factor
113	C₂⁴	C₂²×C₉×C₂₅	1	15	yes	—
114	C₂²×C₄	C₂²×C₉×C₂₅	1	140	yes	direct product factor
115	C₃×C₅	C₂⁴×C₃×C₄×C₅	1	1	yes	characteristic
116	C₂²×C₃	C₂²×C₃×C₄×C₂₅	1	140	yes	—
117	C₂²×C₃	C₂⁴×C₃×C₂₅	1	15	yes	—
118	C₃×C₄	C₂⁴×C₃×C₂₅	1	16	yes	—
119	C₂×C₅	C₂³×C₄×C₅×C₉	1	30	yes	—
120	C₂×C₅	C₂⁵×C₅×C₉	1	1	yes	characteristic
121	C₉	C₂⁴×C₄×C₂₅	1	1	yes	characteristic, direct product factor, Sylow
122	C₂³	C₂×C₄×C₉×C₂₅	1	120	yes	direct product factor
123	C₂³	C₂³×C₉×C₂₅	1	35	yes	—
124	C₂×C₄	C₂³×C₉×C₂₅	1	120	yes	direct product factor
125	C₂×C₃	C₂³×C₃×C₄×C₂₅	1	30	yes	—
126	C₂×C₃	C₂⁵×C₃×C₂₅	1	1	yes	characteristic
127	C₅	C₂⁴×C₄×C₅×C₉	1	1	yes	characteristic
128	C₂²	C₂²×C₄×C₉×C₂₅	1	140	yes	direct product factor
129	C₂²	C₂⁴×C₉×C₂₅	1	15	yes	—
130	C₄	C₂⁴×C₉×C₂₅	1	16	yes	direct product factor
131	C₃	C₂⁴×C₃×C₄×C₂₅	1	1	yes	characteristic
132	C₂	C₂³×C₄×C₉×C₂₅	1	30	yes	direct product factor
133	C₂	C₂⁵×C₉×C₂₅	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂×C₃×C₅	C₂⁵×C₃×C₅
socle, prime radical	C₂⁵×C₃×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
2. C₂³×C₄×C₉×C₂₅	C₂	direct (N × H)
4. C₂⁴×C₉×C₂₅	C₄	direct (N × H)
6. C₂²×C₄×C₉×C₂₅	C₂²	direct (N × H)
10. C₂³×C₉×C₂₅	C₂×C₄	direct (N × H)
12. C₂×C₄×C₉×C₂₅	C₂³	direct (N × H)
13. C₂⁴×C₄×C₂₅	C₉	direct (N × H)
20. C₂²×C₉×C₂₅	C₂²×C₄	direct (N × H)
22. C₄×C₉×C₂₅	C₂⁴	direct (N × H)
24. C₂³×C₄×C₂₅	C₂×C₉	direct (N × H)
31. C₂⁴×C₄×C₉	C₂₅	direct (N × H)
34. C₂×C₉×C₂₅	C₂³×C₄	direct (N × H)
36. C₂⁴×C₂₅	C₄×C₉	direct (N × H)
38. C₂²×C₄×C₂₅	C₂²×C₉	direct (N × H)
47. C₂³×C₄×C₉	C₂×C₂₅	direct (N × H)
51. C₉×C₂₅	C₂⁴×C₄	direct (N × H)
52. C₂³×C₂₅	C₂×C₄×C₉	direct (N × H)
54. C₂×C₄×C₂₅	C₂³×C₉	direct (N × H)
63. C₂⁴×C₉	C₄×C₂₅	direct (N × H)
65. C₂²×C₄×C₉	C₂²×C₂₅	direct (N × H)
69. C₂²×C₂₅	C₂²×C₄×C₉	direct (N × H)
71. C₄×C₂₅	C₂⁴×C₉	direct (N × H)
80. C₂³×C₉	C₂×C₄×C₂₅	direct (N × H)
82. C₂×C₄×C₉	C₂³×C₂₅	direct (N × H)
83. C₂⁴×C₄	C₉×C₂₅	direct (N × H)
87. C₂×C₂₅	C₂³×C₄×C₉	direct (N × H)
96. C₂²×C₉	C₂²×C₄×C₂₅	direct (N × H)
98. C₄×C₉	C₂⁴×C₂₅	direct (N × H)
100. C₂³×C₄	C₂×C₉×C₂₅	direct (N × H)
103. C₂₅	C₂⁴×C₄×C₉	direct (N × H)
110. C₂×C₉	C₂³×C₄×C₂₅	direct (N × H)
112. C₂⁴	C₄×C₉×C₂₅	direct (N × H)
114. C₂²×C₄	C₂²×C₉×C₂₅	direct (N × H)
121. C₉	C₂⁴×C₄×C₂₅	direct (N × H)
122. C₂³	C₂×C₄×C₉×C₂₅	direct (N × H)
124. C₂×C₄	C₂³×C₉×C₂₅	direct (N × H)
128. C₂²	C₂²×C₄×C₉×C₂₅	direct (N × H)
130. C₄	C₂⁴×C₉×C₂₅	direct (N × H)
132. C₂	C₂³×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
C₂⁵×C₉×C₂₅	2
C₂³×C₄×C₉×C₂₅	2
C₂⁴×C₃×C₄×C₂₅	3
C₂⁴×C₄×C₅×C₉	5

Subgroups — C24×C4×C9×C25 #72192574

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×C₄×C₉×C₂₅ #72192574