Subgroups — C₅×#64978 #13101950

Order 1600.

304 conjugacy classes of subgroups.

The subgroup lattice has 183 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#676	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×C₈×D₅	C₂	1	1	yes	maximal, characteristic
3	C₅×#4018	C₂	1	1	yes	maximal, characteristic
4	C₄×C₅²×C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#3947	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×C₈×Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#3935	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×C₅×D₅	C₂²	1	1	yes	characteristic
9	C₂×C₅²×C₈	C₂²	1	1	yes	characteristic
10	C₂×C₅²×C₈	C₂²	1	1	yes	characteristic
11	C₄²×C₅²	C₂²	1	1	yes	characteristic
12	C₅×#685	C₄	1	2	yes	—
13	C₅×#683	C₄	1	2	yes	—
14	C₄×C₅×Dic₅	C₂²	1	1	yes	characteristic
15	C₂×C₅×#168	C₂²	1	1	yes	characteristic
16	C₂×C₅×#168	C₂²	1	1	yes	characteristic
17	C₅×#676	C₄	1	4	yes	complemented
18	C₅×C₈×D₅	C₄	1	4	yes	complemented
19	C₅×#405	—	5	1	no	maximal
20	#64978	C₅	1	1	yes	maximal, characteristic, direct product factor
21	C₂²×C₅×D₅	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×C₅²	C₂×C₄	1	2	yes	—
23	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
24	C₂×C₅×Dic₅	C₂×C₄	1	2	yes	—
25	C₂×C₅×Dic₅	C₂×C₄	1	1	yes	characteristic
26	C₄×C₅×D₅	C₂×C₄	1	2	yes	—
27	C₄×C₅×D₅	C₂×C₄	1	2	yes	—
28	C₅²×C₈	C₂×C₄	1	2	yes	—
29	C₅²×C₈	C₂×C₄	1	2	yes	—
30	C₅×#168	C₂×C₄	1	2	yes	—
31	C₅×#168	C₂×C₄	1	2	yes	—
32	C₂×C₅×M₁₆	—	5	1	no	—
33	C₂²×C₅×C₈	—	5	1	no	—
34	C₅×#117	—	5	1	no	—
35	C₂×#676	C₂×C₅	1	1	yes	characteristic
36	C₂×C₈×D₅	C₂×C₅	1	1	yes	characteristic
37	#4018	C₂×C₅	1	1	yes	characteristic
38	C₅×#97	—	5	1	no	—
39	C₅×#97	—	5	1	no	—
40	C₄×C₅×C₈	—	2	2	no	—
41	C₄×C₅×C₈	—	5	1	no	—
42	C₄×C₅×C₈	D₅	1	1	yes	characteristic, complemented
43	C₄×C₅×C₈	C₂×C₅	1	1	yes	characteristic, complemented
44	#3947	C₂×C₅	1	1	yes	characteristic, complemented
45	C₈×Dic₅	C₂×C₅	1	1	yes	characteristic, complemented
46	#3935	C₂×C₅	1	1	yes	characteristic, complemented
47	C₂²×C₅²	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₅×D₅	—	2	2	no	—
49	C₂×C₅×D₅	C₄²	1	2	yes	—
50	C₄×C₅²	—	2	2	no	—
51	C₄×C₅²	C₂²×C₄	1	1	yes	characteristic
52	C₄×C₅²	C₂²×C₄	1	1	yes	characteristic
53	C₅×Dic₅	—	2	2	no	—
54	C₅×Dic₅	C₄²	1	2	yes	—
55	C₂²×C₄×C₅	—	5	1	no	—
56	C₂×C₄×D₅	C₂²×C₅	1	1	yes	characteristic
57	C₅×M₁₆	—	5	4	no	—
58	C₂×C₅×C₈	—	2	2	no	—
59	C₂×C₅×C₈	—	2	2	no	—
60	C₂×C₅×C₈	—	5	1	no	—
61	C₂×C₅×C₈	—	5	1	no	—
62	C₂×C₅×C₈	—	5	4	no	—
63	C₂×C₅×C₈	C₂×D₅	1	1	yes	characteristic
64	C₂×C₅×C₈	C₂×D₅	1	1	yes	characteristic
65	C₂×C₅×C₈	C₂²×C₅	1	1	yes	characteristic
66	C₂×C₅×C₈	C₂²×C₅	1	1	yes	characteristic
67	C₅×C₄⋊C₄	—	5	2	no	—
68	C₅×C₂²⋊C₄	—	5	2	no	—
69	C₄²×C₅	—	2	2	no	—
70	C₄²×C₅	—	5	1	no	—
71	C₄²×C₅	C₂×D₅	1	1	yes	characteristic
72	C₄²×C₅	C₂²×C₅	1	1	yes	characteristic
73	#685	C₄×C₅	1	2	yes	—
74	#683	C₄×C₅	1	2	yes	—
75	C₄×Dic₅	C₂²×C₅	1	1	yes	characteristic
76	C₂×#168	C₂²×C₅	1	1	yes	characteristic
77	C₂×#168	C₂²×C₅	1	1	yes	characteristic
78	#676	C₄×C₅	1	4	yes	complemented
79	C₈×D₅	C₄×C₅	1	4	yes	complemented
80	#405	—	5	1	no	Sylow
81	C₂×C₅²	C₂×C₄²	1	1	yes	characteristic
82	C₂×C₅²	#131	1	2	yes	—
83	C₅×D₅	—	2	2	no	—
84	C₂³×C₅	—	5	1	no	—
85	C₂²×D₅	C₂×C₄×C₅	1	1	yes	characteristic
86	C₂×C₄×C₅	—	2	4	no	—
87	C₂×C₄×C₅	—	2	2	no	—
88	C₂×C₄×C₅	—	5	1	no	—
89	C₂×C₄×C₅	—	5	2	no	—
90	C₂×C₄×C₅	—	5	2	no	—
91	C₂×C₄×C₅	—	5	2	no	—
92	C₂×C₄×C₅	C₄×D₅	1	2	yes	—
93	C₂×C₄×C₅	C₂×C₄×C₅	1	2	yes	—
94	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
95	C₂×C₄×C₅	C₂³×C₅	1	1	yes	characteristic
96	C₂×Dic₅	C₂×C₄×C₅	1	2	yes	—
97	C₂×Dic₅	C₂×C₄×C₅	1	1	yes	characteristic
98	C₄×D₅	C₂×C₄×C₅	1	2	yes	—
99	C₄×D₅	C₂×C₄×C₅	1	2	yes	—
100	C₅×C₈	—	2	4	no	—
101	C₅×C₈	—	2	4	no	—
102	C₅×C₈	—	5	2	no	—
103	C₅×C₈	—	5	2	no	—
104	C₅×C₈	C₄×D₅	1	2	yes	—
105	C₅×C₈	C₄×D₅	1	2	yes	—
106	C₅×C₈	C₂×C₄×C₅	1	2	yes	—
107	C₅×C₈	C₂×C₄×C₅	1	2	yes	—
108	#168	C₂×C₄×C₅	1	2	yes	—
109	#168	C₂×C₄×C₅	1	2	yes	—
110	C₂×M₁₆	—	5	1	no	—
111	C₂²×C₈	—	5	1	no	—
112	#117	—	5	1	no	—
113	#97	—	5	1	no	—
114	#97	—	5	1	no	—
115	C₄×C₈	—	5	1	no	—
116	C₄×C₈	C₅×D₅	1	1	yes	characteristic, complemented
117	C₅²	#405	1	1	yes	characteristic, complemented, Sylow
118	C₂²×C₅	—	2	2	no	—
119	C₂²×C₅	—	5	2	no	—
120	C₂²×C₅	—	10	2	no	—
121	C₂²×C₅	C₂×C₄×D₅	1	1	yes	characteristic
122	C₂²×C₅	C₂²×C₄×C₅	1	1	yes	characteristic
123	C₂×D₅	—	2	2	no	—
124	C₂×D₅	C₄²×C₅	1	2	yes	—
125	C₄×C₅	—	2	8	no	—
126	C₄×C₅	—	2	2	no	—
127	C₄×C₅	—	2	2	no	—
128	C₄×C₅	—	2	2	no	—
129	C₄×C₅	—	2	2	no	—
130	C₄×C₅	—	5	2	no	—
131	C₄×C₅	—	10	2	no	—
132	C₄×C₅	C₂×C₄×D₅	1	1	yes	characteristic
133	C₄×C₅	C₂×C₄×D₅	1	1	yes	characteristic
134	C₄×C₅	C₂²×C₄×C₅	1	1	yes	characteristic
135	C₄×C₅	C₂²×C₄×C₅	1	1	yes	characteristic
136	Dic₅	—	2	2	no	—
137	Dic₅	C₄²×C₅	1	2	yes	—
138	C₂²×C₄	—	5	1	no	—
139	M₁₆	—	5	4	no	—
140	C₂×C₈	—	5	1	no	—
141	C₂×C₈	—	5	1	no	—
142	C₂×C₈	—	5	4	no	—
143	C₂×C₈	C₂×C₅×D₅	1	1	yes	characteristic
144	C₂×C₈	C₂×C₅×D₅	1	1	yes	characteristic
145	C₄⋊C₄	—	5	2	no	—
146	C₂²⋊C₄	—	5	2	no	—
147	C₄²	—	5	1	no	—
148	C₄²	C₂×C₅×D₅	1	1	yes	characteristic
149	C₂×C₅	—	2	4	no	—
150	C₂×C₅	—	2	2	no	—
151	C₂×C₅	—	10	2	no	—
152	C₂×C₅	C₄²×D₅	1	1	yes	characteristic
153	C₂×C₅	#4047	1	2	yes	—
154	C₂×C₅	C₂×C₄²×C₅	1	1	yes	characteristic
155	C₂×C₅	C₅×#131	1	2	yes	—
156	D₅	—	2	2	no	—
157	C₂³	—	5	1	no	—
158	C₂×C₄	—	5	1	no	—
159	C₂×C₄	—	5	2	no	—
160	C₂×C₄	—	5	2	no	—
161	C₂×C₄	—	5	2	no	—
162	C₂×C₄	C₄×C₅×D₅	1	2	yes	—
163	C₂×C₄	C₂²×C₅×D₅	1	1	yes	characteristic
164	C₈	—	5	2	no	—
165	C₈	—	5	2	no	—
166	C₈	C₄×C₅×D₅	1	2	yes	—
167	C₈	C₄×C₅×D₅	1	2	yes	—
168	C₅	—	2	2	no	—
169	C₅	#64978	1	1	yes	characteristic, direct product factor
170	C₅	C₅×#405	1	1	yes	characteristic, complemented
171	C₂²	—	5	2	no	—
172	C₂²	—	10	2	no	—
173	C₂²	C₂×C₄×C₅×D₅	1	1	yes	characteristic
174	C₄	—	2	2	no	—
175	C₄	—	5	2	no	—
176	C₄	—	10	2	no	—
177	C₄	C₂×C₄×C₅×D₅	1	1	yes	characteristic
178	C₄	C₂×C₄×C₅×D₅	1	1	yes	characteristic
179	C₂	—	10	2	no	—
180	C₂	C₄²×C₅×D₅	1	1	yes	characteristic
181	C₂	C₅×#4047	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
4. C₄×C₅²×C₈	C₂	semidirect
5. C₅×#3947	C₂	semidirect
6. C₅×C₈×Dic₅	C₂	semidirect
7. C₅×#3935	C₂	semidirect
17. C₅×#676	C₄	semidirect
18. C₅×C₈×D₅	C₄	semidirect
20. #64978	C₅	direct (N × H)
42. C₄×C₅×C₈	D₅	semidirect
43. C₄×C₅×C₈	C₂×C₅	semidirect
44. #3947	C₂×C₅	semidirect
45. C₈×Dic₅	C₂×C₅	semidirect
46. #3935	C₂×C₅	semidirect
78. #676	C₄×C₅	semidirect
79. C₈×D₅	C₄×C₅	semidirect
116. C₄×C₈	C₅×D₅	semidirect
117. C₅²	#405	semidirect
169. C₅	#64978	direct (N × H)
170. 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
C₂×C₅×#676	2
C₂×C₅×C₈×D₅	2
C₅×#4018	2
C₄×C₅²×C₈	2
C₅×#3947	2
C₅×C₈×Dic₅	2
C₅×#3935	2
C₅×#405	5
#64978	5

Subgroups — C5×#64978 #13101950

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#64978 #13101950