Subgroups — C₅×#12590 #11935310

Order 1280.

414 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	C₂×C₄×C₅×C₁₆	C₂	1	2	yes	maximal
2	C₄²×C₅×C₈	C₂	1	1	yes	maximal, characteristic
3	C₅×#1374	C₂	1	4	yes	maximal
4	C₂²×C₅×C₁₆	C₂²	1	4	yes	—
5	C₂×C₄×C₅×C₈	C₄	1	2	yes	—
6	C₂×C₄×C₅×C₈	C₄	1	1	yes	characteristic
7	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
8	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
9	C₄³×C₅	C₄	1	1	yes	characteristic
10	C₄×C₅×C₁₆	C₄	1	8	yes	complemented
11	#12590	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
12	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
13	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
15	C₂×C₄²×C₅	C₈	1	2	yes	—
16	C₂×C₄²×C₅	C₈	1	2	yes	—
17	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
18	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
19	C₂×C₅×C₁₆	—	2	8	no	—
20	C₂×C₅×C₁₆	C₂×C₄	1	8	yes	—
21	C₄×C₅×C₈	—	2	4	no	—
22	C₄×C₅×C₈	—	2	4	no	—
23	C₄×C₅×C₈	C₈	1	4	yes	—
24	C₄×C₅×C₈	C₂×C₄	1	4	yes	—
25	C₂×C₄×C₁₆	C₂×C₅	1	2	yes	—
26	C₄²×C₈	C₂×C₅	1	1	yes	characteristic
27	#1374	C₂×C₅	1	4	yes	—
28	C₂²×C₄×C₅	C₂×C₈	1	4	yes	—
29	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
30	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
31	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
32	C₂×C₅×C₈	—	2	2	no	—
33	C₂×C₅×C₈	—	2	2	no	—
34	C₂×C₅×C₈	C₄²	1	2	yes	—
35	C₂×C₅×C₈	C₂×C₈	1	4	yes	—
36	C₂×C₅×C₈	M₁₆	1	4	yes	—
37	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
38	C₂×C₅×C₈	#41	1	4	yes	—
39	C₄²×C₅	—	2	4	no	—
40	C₄²×C₅	—	2	2	no	—
41	C₄²×C₅	—	2	2	no	—
42	C₄²×C₅	C₁₆	1	8	yes	complemented
43	C₄²×C₅	C₄²	1	2	yes	—
44	C₄²×C₅	C₂×C₈	1	2	yes	—
45	C₅×C₁₆	—	2	8	no	—
46	C₂²×C₁₆	C₂²×C₅	1	4	yes	—
47	C₂×C₄×C₈	C₄×C₅	1	2	yes	—
48	C₂×C₄×C₈	C₄×C₅	1	1	yes	characteristic
49	C₂×C₄×C₈	C₂²×C₅	1	2	yes	—
50	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
51	C₄³	C₄×C₅	1	1	yes	characteristic
52	C₄×C₁₆	C₄×C₅	1	8	yes	complemented
53	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
54	C₂×C₄×C₅	—	2	4	no	—
55	C₂×C₄×C₅	—	2	4	no	—
56	C₂×C₄×C₅	C₄×C₈	1	2	yes	—
57	C₂×C₄×C₅	C₄×C₈	1	2	yes	—
58	C₂×C₄×C₅	C₂×C₁₆	1	8	yes	—
59	C₂×C₄×C₅	C₂×C₁₆	1	4	yes	—
60	C₂×C₄×C₅	M₃₂	1	4	yes	—
61	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
62	C₂×C₄×C₅	#117	1	2	yes	—
63	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
64	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
65	C₅×C₈	—	2	4	no	—
66	C₅×C₈	#97	1	4	yes	—
67	C₅×C₈	#117	1	4	yes	—
68	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
69	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
70	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
71	C₂×C₄²	C₅×C₈	1	2	yes	—
72	C₂×C₄²	C₅×C₈	1	2	yes	—
73	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
74	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
75	C₂×C₁₆	—	2	8	no	—
76	C₂×C₁₆	C₂×C₄×C₅	1	8	yes	—
77	C₄×C₈	—	2	4	no	—
78	C₄×C₈	—	2	4	no	—
79	C₄×C₈	C₅×C₈	1	4	yes	—
80	C₄×C₈	C₂×C₄×C₅	1	4	yes	—
81	C₂²×C₅	C₂×C₄×C₈	1	1	yes	characteristic
82	C₂²×C₅	#431	1	2	yes	—
83	C₂²×C₅	C₂²×C₁₆	1	2	yes	—
84	C₂²×C₅	C₂×M₃₂	1	2	yes	—
85	C₄×C₅	—	2	8	no	—
86	C₄×C₅	C₄×C₁₆	1	4	yes	—
87	C₄×C₅	#346	1	4	yes	—
88	C₄×C₅	#376	1	1	yes	characteristic
89	C₄×C₅	C₂×#97	1	1	yes	characteristic
90	C₄×C₅	#431	1	2	yes	—
91	C₂²×C₄	C₂×C₅×C₈	1	4	yes	—
92	C₂²×C₄	C₂×C₅×C₈	1	1	yes	characteristic
93	C₂²×C₄	C₂×C₅×C₈	1	1	yes	characteristic
94	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
95	C₂×C₈	—	2	2	no	—
96	C₂×C₈	—	2	2	no	—
97	C₂×C₈	C₄²×C₅	1	2	yes	—
98	C₂×C₈	C₂×C₅×C₈	1	4	yes	—
99	C₂×C₈	C₅×M₁₆	1	4	yes	—
100	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
101	C₂×C₈	C₅×#41	1	4	yes	—
102	C₄²	—	2	2	no	—
103	C₄²	—	2	2	no	—
104	C₄²	—	2	4	no	—
105	C₄²	C₅×C₁₆	1	8	yes	complemented
106	C₄²	C₄²×C₅	1	2	yes	—
107	C₄²	C₂×C₅×C₈	1	2	yes	—
108	C₁₆	—	2	8	no	—
109	C₂×C₅	#1744	1	1	yes	characteristic
110	C₂×C₅	C₂×C₄×C₁₆	1	1	yes	characteristic
111	C₂×C₅	C₂×#346	1	1	yes	characteristic
112	C₂×C₅	#2162	1	4	yes	—
113	C₂³	C₂²×C₅×C₈	1	1	yes	characteristic
114	C₂×C₄	—	2	4	no	—
115	C₂×C₄	—	2	4	no	—
116	C₂×C₄	C₄×C₅×C₈	1	2	yes	—
117	C₂×C₄	C₄×C₅×C₈	1	2	yes	—
118	C₂×C₄	C₂×C₅×C₁₆	1	8	yes	—
119	C₂×C₄	C₂×C₅×C₁₆	1	4	yes	—
120	C₂×C₄	C₅×M₃₂	1	4	yes	—
121	C₂×C₄	C₂×C₄²×C₅	1	1	yes	characteristic
122	C₂×C₄	C₅×#117	1	2	yes	—
123	C₂×C₄	C₂²×C₅×C₈	1	1	yes	characteristic
124	C₂×C₄	C₂×C₅×M₁₆	1	2	yes	—
125	C₈	—	2	4	no	—
126	C₈	C₅×#97	1	4	yes	—
127	C₈	C₅×#117	1	4	yes	—
128	C₅	#12590	1	1	yes	characteristic, direct product factor, Sylow
129	C₂²	C₂×C₄×C₅×C₈	1	1	yes	characteristic
130	C₂²	C₅×#431	1	2	yes	—
131	C₂²	C₂²×C₅×C₁₆	1	2	yes	—
132	C₂²	C₂×C₅×M₃₂	1	2	yes	—
133	C₄	—	2	8	no	—
134	C₄	C₄×C₅×C₁₆	1	4	yes	—
135	C₄	C₅×#346	1	4	yes	—
136	C₄	C₅×#376	1	1	yes	characteristic
137	C₄	C₂×C₅×#97	1	1	yes	characteristic
138	C₄	C₅×#431	1	2	yes	—
139	C₂	C₅×#1744	1	1	yes	characteristic
140	C₂	C₂×C₄×C₅×C₁₆	1	1	yes	characteristic
141	C₂	C₂×C₅×#346	1	1	yes	characteristic
142	C₂	C₅×#2162	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
10. C₄×C₅×C₁₆	C₄	semidirect
11. #12590	C₅	direct (N × H)
42. C₄²×C₅	C₁₆	semidirect
52. C₄×C₁₆	C₄×C₅	semidirect
105. C₄²	C₅×C₁₆	semidirect
128. C₅	#12590	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×C₄×C₅×C₁₆	2
C₄²×C₅×C₈	2
C₅×#1374	2
#12590	5

Subgroups — C5×#12590 #11935310

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#12590 #11935310