Subgroups — C₂²×C₄⁵×C₅ #72198101

Order 20480.

13111542 conjugacy classes of subgroups.

The subgroup lattice has 126 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄⁴×C₅	C₂	1	31	yes	maximal
2	C₂×C₄⁵×C₅	C₂	1	96	yes	maximal, direct product factor
3	C₂⁴×C₄³×C₅	C₂²	1	155	yes	—
4	C₂²×C₄⁴×C₅	C₄	1	1984	yes	direct product factor
5	C₂²×C₄⁴×C₅	C₂²	1	1488	yes	—
6	C₄⁵×C₅	C₂²	1	1024	yes	direct product factor
7	C₂²×C₄⁵	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
8	C₂⁵×C₄²×C₅	C₂³	1	155	yes	—
9	C₂³×C₄³×C₅	C₂×C₄	1	14880	yes	—
10	C₂³×C₄³×C₅	C₂³	1	3720	yes	—
11	C₂×C₄⁴×C₅	C₂×C₄	1	47616	yes	direct product factor
12	C₂×C₄⁴×C₅	C₂³	1	7936	yes	—
13	C₂³×C₄⁴	C₂×C₅	1	31	yes	—
14	C₂×C₄⁵	C₂×C₅	1	96	yes	direct product factor
15	C₂⁴×C₄³	C₂²×C₅	1	155	yes	—
16	C₂²×C₄⁴	C₄×C₅	1	1984	yes	direct product factor
17	C₂²×C₄⁴	C₂²×C₅	1	1488	yes	—
18	C₄⁵	C₂²×C₅	1	1024	yes	direct product factor
19	C₂⁶×C₄×C₅	C₂⁴	1	31	yes	—
20	C₂⁴×C₄²×C₅	C₂²×C₄	1	17360	yes	—
21	C₂⁴×C₄²×C₅	C₂⁴	1	1860	yes	—
22	C₂²×C₄³×C₅	C₄²	1	158720	yes	direct product factor
23	C₂²×C₄³×C₅	C₂²×C₄	1	178560	yes	—
24	C₂²×C₄³×C₅	C₂⁴	1	9920	yes	—
25	C₄⁴×C₅	C₂²×C₄	1	126976	yes	direct product factor
26	C₂⁷×C₅	C₂⁵	1	1	yes	characteristic
27	C₂⁵×C₄×C₅	C₂³×C₄	1	3720	yes	—
28	C₂⁵×C₄×C₅	C₂⁵	1	186	yes	—
29	C₂³×C₄²×C₅	C₂×C₄²	1	277760	yes	—
30	C₂³×C₄²×C₅	C₂³×C₄	1	104160	yes	—
31	C₂³×C₄²×C₅	C₂⁵	1	2480	yes	—
32	C₂×C₄³×C₅	C₂×C₄²	1	952320	yes	direct product factor
33	C₂×C₄³×C₅	C₂³×C₄	1	238080	yes	—
34	C₂⁵×C₄²	C₂³×C₅	1	155	yes	—
35	C₂³×C₄³	C₂×C₄×C₅	1	14880	yes	—
36	C₂³×C₄³	C₂³×C₅	1	3720	yes	—
37	C₂×C₄⁴	C₂×C₄×C₅	1	47616	yes	direct product factor
38	C₂×C₄⁴	C₂³×C₅	1	7936	yes	—
39	C₂⁶×C₅	C₂⁴×C₄	1	124	yes	—
40	C₂⁶×C₅	C₂⁶	1	3	yes	—
41	C₂⁴×C₄×C₅	C₂²×C₄²	1	69440	yes	—
42	C₂⁴×C₄×C₅	C₂⁴×C₄	1	11160	yes	—
43	C₂⁴×C₄×C₅	C₂⁶	1	124	yes	—
44	C₂²×C₄²×C₅	C₄³	1	634880	yes	direct product factor
45	C₂²×C₄²×C₅	C₂²×C₄²	1	833280	yes	—
46	C₂²×C₄²×C₅	C₂⁴×C₄	1	69440	yes	—
47	C₄³×C₅	C₂²×C₄²	1	634880	yes	direct product factor
48	C₂⁶×C₄	C₂⁴×C₅	1	31	yes	—
49	C₂⁴×C₄²	C₂²×C₄×C₅	1	17360	yes	—
50	C₂⁴×C₄²	C₂⁴×C₅	1	1860	yes	—
51	C₂²×C₄³	C₄²×C₅	1	158720	yes	direct product factor
52	C₂²×C₄³	C₂²×C₄×C₅	1	178560	yes	—
53	C₂²×C₄³	C₂⁴×C₅	1	9920	yes	—
54	C₄⁴	C₂²×C₄×C₅	1	126976	yes	direct product factor
55	C₂⁵×C₅	C₂³×C₄²	1	2480	yes	—
56	C₂⁵×C₅	C₂⁵×C₄	1	186	yes	—
57	C₂⁵×C₅	C₂⁷	1	1	yes	characteristic
58	C₂³×C₄×C₅	C₂×C₄³	1	238080	yes	—
59	C₂³×C₄×C₅	C₂³×C₄²	1	104160	yes	—
60	C₂³×C₄×C₅	C₂⁵×C₄	1	3720	yes	—
61	C₂×C₄²×C₅	C₂×C₄³	1	952320	yes	direct product factor
62	C₂×C₄²×C₅	C₂³×C₄²	1	277760	yes	—
63	C₂⁷	C₂⁵×C₅	1	1	yes	characteristic
64	C₂⁵×C₄	C₂³×C₄×C₅	1	3720	yes	—
65	C₂⁵×C₄	C₂⁵×C₅	1	186	yes	—
66	C₂³×C₄²	C₂×C₄²×C₅	1	277760	yes	—
67	C₂³×C₄²	C₂³×C₄×C₅	1	104160	yes	—
68	C₂³×C₄²	C₂⁵×C₅	1	2480	yes	—
69	C₂×C₄³	C₂×C₄²×C₅	1	952320	yes	direct product factor
70	C₂×C₄³	C₂³×C₄×C₅	1	238080	yes	—
71	C₂⁴×C₅	C₂²×C₄³	1	9920	yes	—
72	C₂⁴×C₅	C₂⁴×C₄²	1	1860	yes	—
73	C₂⁴×C₅	C₂⁶×C₄	1	31	yes	—
74	C₂²×C₄×C₅	C₄⁴	1	126976	yes	direct product factor
75	C₂²×C₄×C₅	C₂²×C₄³	1	178560	yes	—
76	C₂²×C₄×C₅	C₂⁴×C₄²	1	17360	yes	—
77	C₄²×C₅	C₂²×C₄³	1	158720	yes	direct product factor
78	C₂⁶	C₂⁴×C₄×C₅	1	124	yes	—
79	C₂⁶	C₂⁶×C₅	1	3	yes	—
80	C₂⁴×C₄	C₂²×C₄²×C₅	1	69440	yes	—
81	C₂⁴×C₄	C₂⁴×C₄×C₅	1	11160	yes	—
82	C₂⁴×C₄	C₂⁶×C₅	1	124	yes	—
83	C₂²×C₄²	C₄³×C₅	1	634880	yes	direct product factor
84	C₂²×C₄²	C₂²×C₄²×C₅	1	833280	yes	—
85	C₂²×C₄²	C₂⁴×C₄×C₅	1	69440	yes	—
86	C₄³	C₂²×C₄²×C₅	1	634880	yes	direct product factor
87	C₂³×C₅	C₂×C₄⁴	1	7936	yes	—
88	C₂³×C₅	C₂³×C₄³	1	3720	yes	—
89	C₂³×C₅	C₂⁵×C₄²	1	155	yes	—
90	C₂×C₄×C₅	C₂×C₄⁴	1	47616	yes	direct product factor
91	C₂×C₄×C₅	C₂³×C₄³	1	14880	yes	—
92	C₂⁵	C₂³×C₄²×C₅	1	2480	yes	—
93	C₂⁵	C₂⁵×C₄×C₅	1	186	yes	—
94	C₂⁵	C₂⁷×C₅	1	1	yes	characteristic
95	C₂³×C₄	C₂×C₄³×C₅	1	238080	yes	—
96	C₂³×C₄	C₂³×C₄²×C₅	1	104160	yes	—
97	C₂³×C₄	C₂⁵×C₄×C₅	1	3720	yes	—
98	C₂×C₄²	C₂×C₄³×C₅	1	952320	yes	direct product factor
99	C₂×C₄²	C₂³×C₄²×C₅	1	277760	yes	—
100	C₂²×C₅	C₄⁵	1	1024	yes	direct product factor
101	C₂²×C₅	C₂²×C₄⁴	1	1488	yes	—
102	C₂²×C₅	C₂⁴×C₄³	1	155	yes	—
103	C₄×C₅	C₂²×C₄⁴	1	1984	yes	direct product factor
104	C₂⁴	C₂²×C₄³×C₅	1	9920	yes	—
105	C₂⁴	C₂⁴×C₄²×C₅	1	1860	yes	—
106	C₂⁴	C₂⁶×C₄×C₅	1	31	yes	—
107	C₂²×C₄	C₄⁴×C₅	1	126976	yes	direct product factor
108	C₂²×C₄	C₂²×C₄³×C₅	1	178560	yes	—
109	C₂²×C₄	C₂⁴×C₄²×C₅	1	17360	yes	—
110	C₄²	C₂²×C₄³×C₅	1	158720	yes	direct product factor
111	C₂×C₅	C₂×C₄⁵	1	96	yes	direct product factor
112	C₂×C₅	C₂³×C₄⁴	1	31	yes	—
113	C₂³	C₂×C₄⁴×C₅	1	7936	yes	—
114	C₂³	C₂³×C₄³×C₅	1	3720	yes	—
115	C₂³	C₂⁵×C₄²×C₅	1	155	yes	—
116	C₂×C₄	C₂×C₄⁴×C₅	1	47616	yes	direct product factor
117	C₂×C₄	C₂³×C₄³×C₅	1	14880	yes	—
118	C₅	C₂²×C₄⁵	1	1	yes	characteristic, direct product factor, Sylow
119	C₂²	C₄⁵×C₅	1	1024	yes	direct product factor
120	C₂²	C₂²×C₄⁴×C₅	1	1488	yes	—
121	C₂²	C₂⁴×C₄³×C₅	1	155	yes	—
122	C₄	C₂²×C₄⁴×C₅	1	1984	yes	direct product factor
123	C₂	C₂×C₄⁵×C₅	1	96	yes	direct product factor
124	C₂	C₂³×C₄⁴×C₅	1	31	yes	—

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂⁵	C₂⁷×C₅
socle, prime radical	C₂⁷×C₅	C₂⁵
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₄⁵×C₅	C₂	direct (N × H)
4. C₂²×C₄⁴×C₅	C₄	direct (N × H)
6. C₄⁵×C₅	C₂²	direct (N × H)
7. C₂²×C₄⁵	C₅	direct (N × H)
11. C₂×C₄⁴×C₅	C₂×C₄	direct (N × H)
14. C₂×C₄⁵	C₂×C₅	direct (N × H)
16. C₂²×C₄⁴	C₄×C₅	direct (N × H)
18. C₄⁵	C₂²×C₅	direct (N × H)
22. C₂²×C₄³×C₅	C₄²	direct (N × H)
25. C₄⁴×C₅	C₂²×C₄	direct (N × H)
32. C₂×C₄³×C₅	C₂×C₄²	direct (N × H)
37. C₂×C₄⁴	C₂×C₄×C₅	direct (N × H)
44. C₂²×C₄²×C₅	C₄³	direct (N × H)
47. C₄³×C₅	C₂²×C₄²	direct (N × H)
51. C₂²×C₄³	C₄²×C₅	direct (N × H)
54. C₄⁴	C₂²×C₄×C₅	direct (N × H)
61. C₂×C₄²×C₅	C₂×C₄³	direct (N × H)
69. C₂×C₄³	C₂×C₄²×C₅	direct (N × H)
74. C₂²×C₄×C₅	C₄⁴	direct (N × H)
77. C₄²×C₅	C₂²×C₄³	direct (N × H)
83. C₂²×C₄²	C₄³×C₅	direct (N × H)
86. C₄³	C₂²×C₄²×C₅	direct (N × H)
90. C₂×C₄×C₅	C₂×C₄⁴	direct (N × H)
98. C₂×C₄²	C₂×C₄³×C₅	direct (N × H)
100. C₂²×C₅	C₄⁵	direct (N × H)
103. C₄×C₅	C₂²×C₄⁴	direct (N × H)
107. C₂²×C₄	C₄⁴×C₅	direct (N × H)
110. C₄²	C₂²×C₄³×C₅	direct (N × H)
111. C₂×C₅	C₂×C₄⁵	direct (N × H)
116. C₂×C₄	C₂×C₄⁴×C₅	direct (N × H)
118. C₅	C₂²×C₄⁵	direct (N × H)
119. C₂²	C₄⁵×C₅	direct (N × H)
122. C₄	C₂²×C₄⁴×C₅	direct (N × H)
123. C₂	C₂×C₄⁵×C₅	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₄⁴×C₅	2
C₂×C₄⁵×C₅	2
C₂²×C₄⁵	5

Subgroups — C22×C45×C5 #72198101

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₄⁵×C₅ #72198101