Subgroups — C₂²×C₅×#504 #11985342

Order 1280.

2118 conjugacy classes of subgroups.

The subgroup lattice has 144 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×#131	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#504	C₂	1	24	yes	maximal, direct product factor
3	C₂²×C₅×M₃₂	C₂	1	3	yes	maximal, complemented
4	C₂³×C₅×C₁₆	C₂	1	3	yes	maximal, complemented
5	C₂²×C₅×#41	C₄	1	1	yes	characteristic
6	C₂×C₅×#131	C₄	1	12	yes	—
7	C₂×C₅×#131	C₂²	1	12	yes	—
8	C₂²×C₅×M₁₆	C₄	1	3	yes	—
9	C₂³×C₅×C₈	C₂²	1	3	yes	—
10	C₅×#504	C₂²	1	64	yes	direct product factor
11	C₂×C₅×M₃₂	C₂²	1	36	yes	complemented
12	C₂²×C₅×C₁₆	C₂²	1	3	yes	—
13	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
14	C₂²×C₅×C₁₆	C₂²	1	36	yes	complemented
15	C₂²×#504	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
16	C₂×C₅×#41	C₈	1	12	yes	—
17	C₂×C₅×#41	C₂×C₄	1	12	yes	—
18	C₂²×C₅×Q₈	C₈	1	1	yes	characteristic
19	C₂²×C₅×D₄	C₈	1	3	yes	—
20	C₂³×C₄×C₅	C₂×C₄	1	3	yes	—
21	C₅×#131	C₂×C₄	1	48	yes	—
22	C₅×#131	C₂³	1	16	yes	—
23	C₂×C₅×M₁₆	C₂×C₄	1	36	yes	—
24	C₂²×C₅×C₈	C₂×C₄	1	3	yes	—
25	C₂²×C₅×C₈	C₂×C₄	1	18	yes	—
26	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
27	C₂²×C₅×C₈	C₂³	1	18	yes	—
28	C₅×M₃₂	C₂³	1	48	yes	complemented
29	C₂×C₅×C₁₆	C₂³	1	18	yes	—
30	C₂×C₅×C₁₆	C₂³	1	6	yes	—
31	C₂×C₅×C₁₆	C₂³	1	48	yes	complemented
32	C₂²×#131	C₂×C₅	1	1	yes	characteristic
33	C₂×#504	C₂×C₅	1	24	yes	direct product factor
34	C₂²×M₃₂	C₂×C₅	1	3	yes	complemented
35	C₂³×C₁₆	C₂×C₅	1	3	yes	complemented
36	C₂⁴×C₅	C₂×C₈	1	3	yes	—
37	C₅×#41	C₂×C₈	1	48	yes	—
38	C₅×#41	C₂²×C₄	1	16	yes	—
39	C₂×C₅×Q₈	C₂×C₈	1	12	yes	—
40	C₂×C₅×D₄	C₂×C₈	1	36	yes	—
41	C₂²×C₄×C₅	C₂×C₈	1	3	yes	—
42	C₂²×C₄×C₅	C₂×C₈	1	18	yes	—
43	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₅	C₂²×C₄	1	18	yes	—
45	C₅×M₁₆	C₂²×C₄	1	48	yes	—
46	C₂×C₅×C₈	C₂²×C₄	1	3	yes	—
47	C₂×C₅×C₈	C₂²×C₄	1	18	yes	—
48	C₂×C₅×C₈	C₂²×C₄	1	36	yes	—
49	C₂×C₅×C₈	C₂⁴	1	3	yes	—
50	C₂×C₅×C₈	C₂⁴	1	12	yes	—
51	C₅×C₁₆	C₂⁴	1	4	yes	—
52	C₅×C₁₆	C₂⁴	1	12	yes	—
53	C₂²×#41	C₄×C₅	1	1	yes	characteristic
54	C₂×#131	C₄×C₅	1	12	yes	—
55	C₂×#131	C₂²×C₅	1	12	yes	—
56	C₂²×M₁₆	C₄×C₅	1	3	yes	—
57	C₂³×C₈	C₂²×C₅	1	3	yes	—
58	#504	C₂²×C₅	1	64	yes	direct product factor
59	C₂×M₃₂	C₂²×C₅	1	36	yes	complemented
60	C₂²×C₁₆	C₂²×C₅	1	1	yes	characteristic
61	C₂²×C₁₆	C₂²×C₅	1	3	yes	—
62	C₂²×C₁₆	C₂²×C₅	1	36	yes	complemented
63	C₂³×C₅	—	2	12	no	—
64	C₂³×C₅	C₂²×C₈	1	18	yes	—
65	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
66	C₅×Q₈	C₂²×C₈	1	16	yes	—
67	C₅×D₄	C₂²×C₈	1	48	yes	—
68	C₂×C₄×C₅	C₂²×C₈	1	3	yes	—
69	C₂×C₄×C₅	C₂²×C₈	1	18	yes	—
70	C₂×C₄×C₅	C₂²×C₈	1	36	yes	—
71	C₂×C₄×C₅	C₂³×C₄	1	3	yes	—
72	C₂×C₄×C₅	C₂³×C₄	1	12	yes	—
73	C₅×C₈	C₂³×C₄	1	3	yes	—
74	C₅×C₈	C₂³×C₄	1	12	yes	—
75	C₅×C₈	C₂⁵	1	1	yes	characteristic
76	C₂×#41	C₅×C₈	1	12	yes	—
77	C₂×#41	C₂×C₄×C₅	1	12	yes	—
78	C₂²×Q₈	C₅×C₈	1	1	yes	characteristic
79	C₂²×D₄	C₅×C₈	1	3	yes	—
80	C₂³×C₄	C₂×C₄×C₅	1	3	yes	—
81	#131	C₂×C₄×C₅	1	48	yes	—
82	#131	C₂³×C₅	1	16	yes	—
83	C₂×M₁₆	C₂×C₄×C₅	1	36	yes	—
84	C₂²×C₈	C₂×C₄×C₅	1	3	yes	—
85	C₂²×C₈	C₂×C₄×C₅	1	18	yes	—
86	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
87	C₂²×C₈	C₂³×C₅	1	18	yes	—
88	M₃₂	C₂³×C₅	1	48	yes	complemented
89	C₂×C₁₆	C₂³×C₅	1	6	yes	—
90	C₂×C₁₆	C₂³×C₅	1	18	yes	—
91	C₂×C₁₆	C₂³×C₅	1	48	yes	complemented
92	C₂²×C₅	—	2	36	no	—
93	C₂²×C₅	#504	1	4	yes	direct product factor
94	C₂²×C₅	C₂³×C₈	1	12	yes	—
95	C₂²×C₅	C₂³×C₈	1	3	yes	—
96	C₄×C₅	C₂³×C₈	1	3	yes	—
97	C₄×C₅	C₂³×C₈	1	12	yes	—
98	C₄×C₅	C₂⁴×C₄	1	1	yes	characteristic
99	C₂⁴	C₂×C₅×C₈	1	3	yes	—
100	#41	C₂×C₅×C₈	1	48	yes	—
101	#41	C₂²×C₄×C₅	1	16	yes	—
102	C₂×Q₈	C₂×C₅×C₈	1	12	yes	—
103	C₂×D₄	C₂×C₅×C₈	1	36	yes	—
104	C₂²×C₄	C₂×C₅×C₈	1	3	yes	—
105	C₂²×C₄	C₂×C₅×C₈	1	18	yes	—
106	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
107	C₂²×C₄	C₂²×C₄×C₅	1	18	yes	—
108	M₁₆	C₂²×C₄×C₅	1	48	yes	—
109	C₂×C₈	C₂²×C₄×C₅	1	3	yes	—
110	C₂×C₈	C₂²×C₄×C₅	1	18	yes	—
111	C₂×C₈	C₂²×C₄×C₅	1	36	yes	—
112	C₂×C₈	C₂⁴×C₅	1	3	yes	—
113	C₂×C₈	C₂⁴×C₅	1	12	yes	—
114	C₁₆	C₂⁴×C₅	1	4	yes	—
115	C₁₆	C₂⁴×C₅	1	12	yes	—
116	C₂×C₅	—	2	12	no	—
117	C₂×C₅	C₂×#504	1	6	yes	direct product factor
118	C₂×C₅	C₂⁴×C₈	1	1	yes	characteristic
119	C₂³	—	2	12	no	—
120	C₂³	C₂²×C₅×C₈	1	18	yes	—
121	C₂³	C₂²×C₅×C₈	1	1	yes	characteristic
122	Q₈	C₂²×C₅×C₈	1	16	yes	—
123	D₄	C₂²×C₅×C₈	1	48	yes	—
124	C₂×C₄	C₂²×C₅×C₈	1	3	yes	—
125	C₂×C₄	C₂²×C₅×C₈	1	18	yes	—
126	C₂×C₄	C₂²×C₅×C₈	1	36	yes	—
127	C₂×C₄	C₂³×C₄×C₅	1	3	yes	—
128	C₂×C₄	C₂³×C₄×C₅	1	12	yes	—
129	C₈	C₂³×C₄×C₅	1	3	yes	—
130	C₈	C₂³×C₄×C₅	1	12	yes	—
131	C₈	C₂⁵×C₅	1	1	yes	characteristic
132	C₅	C₂²×#504	1	1	yes	characteristic, direct product factor, Sylow
133	C₂²	—	2	36	no	—
134	C₂²	C₅×#504	1	4	yes	direct product factor
135	C₂²	C₂³×C₅×C₈	1	12	yes	—
136	C₂²	C₂³×C₅×C₈	1	3	yes	—
137	C₄	C₂³×C₅×C₈	1	3	yes	—
138	C₄	C₂³×C₅×C₈	1	12	yes	—
139	C₄	C₂⁴×C₄×C₅	1	1	yes	characteristic
140	C₂	—	2	12	no	—
141	C₂	C₂×C₅×#504	1	6	yes	direct product factor
142	C₂	C₂⁴×C₅×C₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅×C₁₆	C₂²
commutator	C₂	C₂⁴×C₅×C₈
Frattini, cosocle	C₈	C₂⁵×C₅
socle	C₂³×C₅	C₂²×C₈
prime radical	C₂²×C₅×#41	C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅×C₁₆ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×#504	C₂	direct (N × H)
3. C₂²×C₅×M₃₂	C₂	semidirect
4. C₂³×C₅×C₁₆	C₂	semidirect
10. C₅×#504	C₂²	direct (N × H)
11. C₂×C₅×M₃₂	C₂²	semidirect
14. C₂²×C₅×C₁₆	C₂²	semidirect
15. C₂²×#504	C₅	direct (N × H)
28. C₅×M₃₂	C₂³	semidirect
31. C₂×C₅×C₁₆	C₂³	semidirect
33. C₂×#504	C₂×C₅	direct (N × H)
34. C₂²×M₃₂	C₂×C₅	semidirect
35. C₂³×C₁₆	C₂×C₅	semidirect
58. #504	C₂²×C₅	direct (N × H)
59. C₂×M₃₂	C₂²×C₅	semidirect
62. C₂²×C₁₆	C₂²×C₅	semidirect
88. M₃₂	C₂³×C₅	semidirect
91. C₂×C₁₆	C₂³×C₅	semidirect
93. C₂²×C₅	#504	direct (N × H)
117. C₂×C₅	C₂×#504	direct (N × H)
132. C₅	C₂²×#504	direct (N × H)
134. C₂²	C₅×#504	direct (N × H)
141. C₂	C₂×C₅×#504	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₅×#131	2
C₂×C₅×#504	2
C₂²×C₅×M₃₂	2
C₂³×C₅×C₁₆	2
C₂²×#504	5

Subgroups — C22×C5×#504 #11985342

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₅×#504 #11985342