Subgroups — C₅×#65390 #13102362

Order 1600.

336 conjugacy classes of subgroups.

The subgroup lattice has 162 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅²×C₈	C₂	1	1	yes	maximal, characteristic
2	C₅×#4072	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#681	C₂	1	1	yes	maximal, characteristic
4	C₅×#3947	C₂	1	2	yes	maximal, complemented
5	C₅×C₈×Dic₅	C₂	1	2	yes	maximal, complemented
6	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
7	C₂×C₅²×C₈	C₄	1	4	yes	complemented
8	C₂×C₅²×C₈	C₂²	1	1	yes	characteristic
9	C₂×C₅²×C₈	C₂²	1	1	yes	characteristic
10	C₅×C₂²⋊Dic₅	C₄	1	2	yes	—
11	C₅×C₄⋊Dic₅	C₄	1	2	yes	—
12	C₄×C₅×Dic₅	C₂²	1	2	yes	—
13	C₅×#681	C₄	1	4	yes	complemented
14	C₂×C₅×#168	C₂²	1	2	yes	—
15	C₅×#405	—	5	1	no	maximal
16	#65390	C₅	1	1	yes	maximal, characteristic, direct product factor
17	C₂³×C₅²	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×C₅²	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×C₅²	C₂×C₄	1	2	yes	—
20	C₂×C₄×C₅²	C₂×C₄	1	2	yes	—
21	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
22	C₂×C₅×Dic₅	C₂×C₄	1	4	yes	—
23	C₅²×C₈	C₂×C₄	1	2	yes	—
24	C₅²×C₈	C₂×C₄	1	2	yes	—
25	C₅×#168	C₂×C₄	1	4	yes	—
26	C₂×C₅×M₁₆	—	5	1	no	—
27	C₂²×C₅×C₈	—	2	2	no	—
28	C₂²×C₅×C₈	D₅	1	1	yes	characteristic
29	C₂²×C₅×C₈	C₂×C₅	1	1	yes	characteristic
30	C₅×#117	—	5	1	no	—
31	#4072	C₂×C₅	1	1	yes	characteristic
32	C₂×#681	C₂×C₅	1	1	yes	characteristic
33	C₅×#97	—	5	2	no	—
34	C₄×C₅×C₈	—	5	2	no	—
35	#3947	C₂×C₅	1	2	yes	complemented
36	C₈×Dic₅	C₂×C₅	1	2	yes	complemented
37	C₂²×C₅²	—	2	2	no	—
38	C₂²×C₅²	C₄²	1	2	yes	—
39	C₂²×C₅²	C₂²×C₄	1	1	yes	characteristic
40	C₄×C₅²	C₄²	1	2	yes	—
41	C₄×C₅²	C₂²×C₄	1	1	yes	characteristic
42	C₄×C₅²	C₂²×C₄	1	1	yes	characteristic
43	C₅×Dic₅	—	2	4	no	—
44	C₂²×C₄×C₅	—	2	2	no	—
45	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
46	C₂²×C₄×C₅	C₂²×C₅	1	1	yes	characteristic
47	C₅×M₁₆	—	5	4	no	—
48	C₂×C₅×C₈	—	2	8	no	—
49	C₂×C₅×C₈	—	2	2	no	—
50	C₂×C₅×C₈	—	2	2	no	—
51	C₂×C₅×C₈	—	5	2	no	—
52	C₂×C₅×C₈	Dic₅	1	4	yes	complemented
53	C₂×C₅×C₈	C₄×C₅	1	4	yes	complemented
54	C₂×C₅×C₈	C₂×D₅	1	1	yes	characteristic
55	C₂×C₅×C₈	C₂×D₅	1	1	yes	characteristic
56	C₂×C₅×C₈	C₂²×C₅	1	1	yes	characteristic
57	C₂×C₅×C₈	C₂²×C₅	1	1	yes	characteristic
58	C₅×C₄⋊C₄	—	5	2	no	—
59	C₅×C₂²⋊C₄	—	5	2	no	—
60	C₄²×C₅	—	5	2	no	—
61	C₂²⋊Dic₅	C₄×C₅	1	2	yes	—
62	C₄⋊Dic₅	C₄×C₅	1	2	yes	—
63	C₄×Dic₅	C₂²×C₅	1	2	yes	—
64	#681	C₄×C₅	1	4	yes	complemented
65	C₂×#168	C₂²×C₅	1	2	yes	—
66	#405	—	5	1	no	Sylow
67	C₂×C₅²	—	2	2	no	—
68	C₂×C₅²	C₂×C₄²	1	1	yes	characteristic
69	C₂×C₅²	#131	1	2	yes	—
70	C₂³×C₅	—	2	2	no	—
71	C₂³×C₅	C₄×D₅	1	1	yes	characteristic
72	C₂³×C₅	C₂×C₄×C₅	1	1	yes	characteristic
73	C₂×C₄×C₅	—	2	4	no	—
74	C₂×C₄×C₅	—	2	2	no	—
75	C₂×C₄×C₅	—	2	4	no	—
76	C₂×C₄×C₅	—	2	2	no	—
77	C₂×C₄×C₅	—	5	4	no	—
78	C₂×C₄×C₅	C₄×D₅	1	1	yes	characteristic
79	C₂×C₄×C₅	C₄×D₅	1	2	yes	—
80	C₂×C₄×C₅	C₂×Dic₅	1	2	yes	—
81	C₂×C₄×C₅	C₂×C₄×C₅	1	1	yes	characteristic
82	C₂×C₄×C₅	C₂×C₄×C₅	1	2	yes	—
83	C₂×C₄×C₅	C₂×C₄×C₅	1	2	yes	—
84	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
85	C₂×C₄×C₅	C₂³×C₅	1	1	yes	characteristic
86	C₂×Dic₅	C₂×C₄×C₅	1	4	yes	—
87	C₅×C₈	—	2	4	no	—
88	C₅×C₈	—	2	4	no	—
89	C₅×C₈	—	5	4	no	—
90	C₅×C₈	C₂×Dic₅	1	2	yes	—
91	C₅×C₈	C₂×Dic₅	1	2	yes	—
92	C₅×C₈	C₂×C₄×C₅	1	2	yes	—
93	C₅×C₈	C₂×C₄×C₅	1	2	yes	—
94	#168	C₂×C₄×C₅	1	4	yes	—
95	C₂×M₁₆	—	5	1	no	—
96	C₂²×C₈	C₅×D₅	1	1	yes	characteristic
97	#117	—	5	1	no	—
98	#97	—	5	2	no	—
99	C₄×C₈	—	5	2	no	—
100	C₅²	#405	1	1	yes	characteristic, complemented, Sylow
101	C₂²×C₅	—	2	8	no	—
102	C₂²×C₅	—	2	2	no	—
103	C₂²×C₅	—	2	2	no	—
104	C₂²×C₅	—	2	2	no	—
105	C₂²×C₅	—	2	4	no	—
106	C₂²×C₅	C₄×Dic₅	1	2	yes	—
107	C₂²×C₅	C₄²×C₅	1	2	yes	—
108	C₂²×C₅	C₂×C₄×D₅	1	1	yes	characteristic
109	C₂²×C₅	C₂²×C₄×C₅	1	1	yes	characteristic
110	C₄×C₅	—	2	2	no	—
111	C₄×C₅	—	2	2	no	—
112	C₄×C₅	—	2	4	no	—
113	C₄×C₅	—	10	4	no	—
114	C₄×C₅	C₄×Dic₅	1	2	yes	—
115	C₄×C₅	C₄²×C₅	1	2	yes	—
116	C₄×C₅	C₂×C₄×D₅	1	1	yes	characteristic
117	C₄×C₅	C₂²×Dic₅	1	1	yes	characteristic
118	C₄×C₅	C₂²×C₄×C₅	1	1	yes	characteristic
119	C₄×C₅	C₂²×C₄×C₅	1	1	yes	characteristic
120	Dic₅	—	2	4	no	—
121	C₂²×C₄	C₂×C₅×D₅	1	1	yes	characteristic
122	M₁₆	—	5	4	no	—
123	C₂×C₈	—	5	2	no	—
124	C₂×C₈	C₅×Dic₅	1	4	yes	complemented
125	C₂×C₈	C₂×C₅×D₅	1	1	yes	characteristic
126	C₂×C₈	C₂×C₅×D₅	1	1	yes	characteristic
127	C₄⋊C₄	—	5	2	no	—
128	C₂²⋊C₄	—	5	2	no	—
129	C₄²	—	5	2	no	—
130	C₂×C₅	—	2	8	no	—
131	C₂×C₅	—	2	2	no	—
132	C₂×C₅	—	2	2	no	—
133	C₂×C₅	—	2	4	no	—
134	C₂×C₅	—	2	2	no	—
135	C₂×C₅	#4047	1	2	yes	—
136	C₂×C₅	C₂×C₄×Dic₅	1	1	yes	characteristic
137	C₂×C₅	C₂×C₄²×C₅	1	1	yes	characteristic
138	C₂×C₅	C₅×#131	1	2	yes	—
139	C₂³	C₄×C₅×D₅	1	1	yes	characteristic
140	C₂×C₄	—	5	4	no	—
141	C₂×C₄	C₄×C₅×D₅	1	2	yes	—
142	C₂×C₄	C₄×C₅×D₅	1	1	yes	characteristic
143	C₂×C₄	C₂×C₅×Dic₅	1	2	yes	—
144	C₂×C₄	C₂²×C₅×D₅	1	1	yes	characteristic
145	C₈	—	5	4	no	—
146	C₈	C₂×C₅×Dic₅	1	2	yes	—
147	C₈	C₂×C₅×Dic₅	1	2	yes	—
148	C₅	—	2	2	no	—
149	C₅	#65390	1	1	yes	characteristic, direct product factor
150	C₅	C₅×#405	1	1	yes	characteristic, complemented
151	C₂²	—	2	2	no	—
152	C₂²	C₄×C₅×Dic₅	1	2	yes	—
153	C₂²	C₂×C₄×C₅×D₅	1	1	yes	characteristic
154	C₄	—	10	4	no	—
155	C₄	C₄×C₅×Dic₅	1	2	yes	—
156	C₄	C₂×C₄×C₅×D₅	1	1	yes	characteristic
157	C₄	C₂²×C₅×Dic₅	1	1	yes	characteristic
158	C₂	—	2	2	no	—
159	C₂	C₅×#4047	1	2	yes	—
160	C₂	C₂×C₄×C₅×Dic₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₅×C₈	C₂×D₅
commutator	C₂×C₅	C₂×C₄²×C₅
Frattini	C₂×C₄	C₂²×C₅×D₅
Fitting	C₂²×C₅²×C₈	C₂
socle	C₂²×C₅²	C₂²×C₄
prime radical	C₂³×C₅²	C₂×C₄
cosocle	C₂×C₄×C₅	C₂³×C₅
hypercenter	C₂²×C₅×C₈	D₅
hypocenter	C₅	C₅×#405
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₅×C₈ ◁ C₂²×C₅×C₈

Complements

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

N	H ≅ G/N	type
4. C₅×#3947	C₂	semidirect
5. C₅×C₈×Dic₅	C₂	semidirect
7. C₂×C₅²×C₈	C₄	semidirect
13. C₅×#681	C₄	semidirect
16. #65390	C₅	direct (N × H)
35. #3947	C₂×C₅	semidirect
36. C₈×Dic₅	C₂×C₅	semidirect
52. C₂×C₅×C₈	Dic₅	semidirect
53. C₂×C₅×C₈	C₄×C₅	semidirect
64. #681	C₄×C₅	semidirect
100. C₅²	#405	semidirect
124. C₂×C₈	C₅×Dic₅	semidirect
149. C₅	#65390	direct (N × H)
150. C₅	C₅×#405	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₅²×C₈	2
C₅×#4072	2
C₂×C₅×#681	2
C₅×#3947	2
C₅×C₈×Dic₅	2
C₅×#405	5
#65390	5

Subgroups — C5×#65390 #13102362

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#65390 #13102362