Subgroups — C₅×#65014 #13101987

Order 1600.

318 conjugacy classes of subgroups.

The subgroup lattice has 247 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₅×C₂²⋊D₅	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₅×#168	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×C₈×D₅	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅²×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#3952	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#3946	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×C₈×Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
9	C₂×C₅×C₂²⋊D₅	C₄	1	1	yes	characteristic
10	C₂×C₄×C₅×D₅	C₂²	1	1	yes	characteristic
11	C₂×C₅²×C₈	C₂²	1	1	yes	characteristic
12	C₂×C₅²×C₈	C₂²	1	1	yes	characteristic
13	C₅×C₂²⋊Dic₅	C₄	1	1	yes	characteristic
14	C₅×#685	C₄	1	1	yes	characteristic
15	C₅×#683	C₄	1	1	yes	characteristic
16	C₄×C₅×Dic₅	C₂²	1	1	yes	characteristic
17	C₂×C₅×#168	—	2	2	no	—
18	C₂×C₅×#168	C₂²	1	1	yes	characteristic
19	C₂×C₅×#168	C₂²	1	1	yes	characteristic
20	C₅×C₈×D₅	—	2	2	no	—
21	C₅×C₈×D₄	—	5	1	no	maximal
22	#65014	C₅	1	1	yes	maximal, characteristic, direct product factor
23	C₂³×C₅²	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₅×D₅	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×C₅²	—	2	1	no	—
26	C₂×C₄×C₅²	—	2	1	no	—
27	C₂×C₄×C₅²	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
29	C₅×C₂²⋊D₅	C₈	1	4	yes	complemented
30	C₂×C₅×Dic₅	C₂×C₄	1	1	yes	characteristic
31	C₂×C₅×Dic₅	C₂×C₄	1	1	yes	characteristic
32	C₂×C₅×Dic₅	C₂×C₄	1	1	yes	characteristic
33	C₄×C₅×D₅	—	2	1	no	—
34	C₄×C₅×D₅	—	2	1	no	—
35	C₅²×C₈	—	2	1	no	—
36	C₅²×C₈	—	2	1	no	—
37	C₅×#168	—	2	1	no	—
38	C₅×#168	D₄	1	2	yes	complemented
39	C₂²×C₅×C₈	—	5	1	no	—
40	C₂²×C₅×C₈	—	5	1	no	—
41	C₄×C₅×D₄	—	5	1	no	—
42	C₄×C₂²⋊D₅	C₂×C₅	1	1	yes	characteristic
43	C₂²×#168	C₂×C₅	1	1	yes	characteristic, complemented
44	C₂×C₈×D₅	C₂×C₅	1	1	yes	characteristic, complemented
45	C₅×C₄⋊C₈	—	5	1	no	—
46	C₅×C₂²⋊C₈	—	2	2	no	—
47	C₅×C₂²⋊C₈	—	5	1	no	—
48	C₅×C₂²⋊C₈	D₅	1	1	yes	characteristic, complemented
49	C₅×C₂²⋊C₈	C₂×C₅	1	1	yes	characteristic, complemented
50	C₄×C₅×C₈	—	5	1	no	—
51	#3952	C₂×C₅	1	1	yes	characteristic, complemented
52	#3946	C₂×C₅	1	1	yes	characteristic, complemented
53	C₈×Dic₅	C₂×C₅	1	1	yes	characteristic, complemented
54	C₂²×C₅²	—	2	1	no	—
55	C₂²×C₅²	—	2	1	no	—
56	C₂²×C₅²	C₂×C₈	1	2	yes	complemented
57	C₂²×C₅²	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₅×D₅	—	2	1	no	—
59	C₂×C₅×D₅	—	2	1	no	—
60	C₂×C₅×D₅	C₂×C₈	1	2	yes	complemented
61	C₄×C₅²	—	2	1	no	—
62	C₄×C₅²	C₂×D₄	1	1	yes	characteristic
63	C₄×C₅²	#41	1	1	yes	characteristic
64	C₅×Dic₅	—	2	1	no	—
65	C₅×Dic₅	—	2	1	no	—
66	C₅×Dic₅	C₂×C₈	1	2	yes	complemented
67	C₂×C₅×D₄	—	5	1	no	—
68	C₂²×C₄×C₅	—	2	2	no	—
69	C₂²×C₄×C₅	—	5	1	no	—
70	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
71	C₂²×C₄×C₅	C₂²×C₅	1	1	yes	characteristic
72	C₂×C₂²⋊D₅	C₄×C₅	1	1	yes	characteristic
73	C₂×C₄×D₅	C₂²×C₅	1	1	yes	characteristic
74	C₂×C₅×C₈	—	2	2	no	—
75	C₂×C₅×C₈	—	2	2	no	—
76	C₂×C₅×C₈	—	5	1	no	—
77	C₂×C₅×C₈	—	5	1	no	—
78	C₂×C₅×C₈	—	10	2	no	—
79	C₂×C₅×C₈	—	10	2	no	—
80	C₂×C₅×C₈	C₂×D₅	1	1	yes	characteristic
81	C₂×C₅×C₈	C₂×D₅	1	1	yes	characteristic
82	C₂×C₅×C₈	C₂²×C₅	1	1	yes	characteristic
83	C₂×C₅×C₈	C₂²×C₅	1	1	yes	characteristic
84	C₅×C₄⋊C₄	—	5	1	no	—
85	C₅×C₂²⋊C₄	—	5	1	no	—
86	C₅×C₂²⋊C₄	—	5	1	no	—
87	C₄²×C₅	—	5	1	no	—
88	C₂²⋊Dic₅	C₄×C₅	1	1	yes	characteristic
89	#685	C₄×C₅	1	1	yes	characteristic
90	#683	C₄×C₅	1	1	yes	characteristic
91	C₄×Dic₅	C₂²×C₅	1	1	yes	characteristic
92	C₂×#168	—	2	2	no	—
93	C₂×#168	C₂²×C₅	1	1	yes	characteristic
94	C₂×#168	C₂²×C₅	1	1	yes	characteristic
95	C₈×D₅	—	2	2	no	—
96	C₈×D₄	—	5	1	no	Sylow
97	C₂×C₅²	—	2	2	no	—
98	C₂×C₅²	C₄×D₄	1	1	yes	characteristic
99	C₂×C₅²	C₂²×C₈	1	1	yes	characteristic
100	C₂×C₅²	#131	1	1	yes	characteristic
101	C₅×D₅	—	2	2	no	—
102	C₂³×C₅	—	2	2	no	—
103	C₂³×C₅	—	5	1	no	—
104	C₂³×C₅	C₄×D₅	1	1	yes	characteristic
105	C₂³×C₅	C₂×C₄×C₅	1	1	yes	characteristic
106	C₂²×D₅	C₂×C₄×C₅	1	1	yes	characteristic
107	C₅×D₄	—	5	4	no	—
108	C₂×C₄×C₅	—	2	1	no	—
109	C₂×C₄×C₅	—	2	1	no	—
110	C₂×C₄×C₅	—	2	1	no	—
111	C₂×C₄×C₅	—	2	1	no	—
112	C₂×C₄×C₅	—	2	2	no	—
113	C₂×C₄×C₅	—	2	2	no	—
114	C₂×C₄×C₅	—	4	2	no	—
115	C₂×C₄×C₅	—	4	2	no	—
116	C₂×C₄×C₅	—	5	1	no	—
117	C₂×C₄×C₅	—	5	1	no	—
118	C₂×C₄×C₅	—	5	1	no	—
119	C₂×C₄×C₅	—	10	1	no	—
120	C₂×C₄×C₅	—	10	1	no	—
121	C₂×C₄×C₅	C₄×D₅	1	1	yes	characteristic
122	C₂×C₄×C₅	C₂×C₄×C₅	1	1	yes	characteristic
123	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
124	C₂×C₄×C₅	C₂³×C₅	1	1	yes	characteristic
125	C₂²⋊D₅	C₅×C₈	1	4	yes	complemented
126	C₂×Dic₅	C₂×C₄×C₅	1	1	yes	characteristic
127	C₂×Dic₅	C₂×C₄×C₅	1	1	yes	characteristic
128	C₂×Dic₅	C₂×C₄×C₅	1	1	yes	characteristic
129	C₄×D₅	—	2	1	no	—
130	C₄×D₅	—	2	1	no	—
131	C₅×C₈	—	2	1	no	—
132	C₅×C₈	—	2	1	no	—
133	C₅×C₈	—	2	1	no	—
134	C₅×C₈	—	2	1	no	—
135	C₅×C₈	—	4	2	no	—
136	C₅×C₈	—	4	2	no	—
137	C₅×C₈	—	5	2	no	—
138	C₅×C₈	—	10	1	no	—
139	#168	—	2	1	no	—
140	#168	C₅×D₄	1	2	yes	complemented
141	C₂²×C₈	—	5	1	no	—
142	C₂²×C₈	—	5	1	no	—
143	C₄×D₄	—	5	1	no	—
144	C₄⋊C₈	—	5	1	no	—
145	C₂²⋊C₈	—	5	1	no	—
146	C₂²⋊C₈	C₅×D₅	1	1	yes	characteristic, complemented
147	C₄×C₈	—	5	1	no	—
148	C₅²	C₈×D₄	1	1	yes	characteristic, complemented, Sylow
149	C₂²×C₅	—	2	1	no	—
150	C₂²×C₅	—	2	1	no	—
151	C₂²×C₅	—	2	1	no	—
152	C₂²×C₅	—	2	1	no	—
153	C₂²×C₅	—	2	2	no	—
154	C₂²×C₅	—	2	4	no	—
155	C₂²×C₅	—	4	2	no	—
156	C₂²×C₅	—	4	2	no	—
157	C₂²×C₅	—	5	2	no	—
158	C₂²×C₅	—	10	1	no	—
159	C₂²×C₅	—	10	1	no	—
160	C₂²×C₅	C₈×D₅	1	2	yes	complemented
161	C₂²×C₅	C₂×C₅×C₈	1	2	yes	complemented
162	C₂²×C₅	C₂×C₄×D₅	1	1	yes	characteristic
163	C₂²×C₅	C₂²×C₄×C₅	1	1	yes	characteristic
164	C₂×D₅	—	2	1	no	—
165	C₂×D₅	—	2	1	no	—
166	C₂×D₅	C₂×C₅×C₈	1	2	yes	complemented
167	C₄×C₅	—	2	1	no	—
168	C₄×C₅	—	2	1	no	—
169	C₄×C₅	—	2	2	no	—
170	C₄×C₅	—	2	2	no	—
171	C₄×C₅	—	4	2	no	—
172	C₄×C₅	—	5	2	no	—
173	C₄×C₅	—	10	1	no	—
174	C₄×C₅	—	10	1	no	—
175	C₄×C₅	D₄×D₅	1	1	yes	characteristic
176	C₄×C₅	#711	1	1	yes	characteristic
177	C₄×C₅	C₂×C₅×D₄	1	1	yes	characteristic
178	C₄×C₅	C₅×#41	1	1	yes	characteristic
179	Dic₅	—	2	1	no	—
180	Dic₅	—	2	1	no	—
181	Dic₅	C₂×C₅×C₈	1	2	yes	complemented
182	C₂×D₄	—	5	1	no	—
183	C₂²×C₄	—	5	1	no	—
184	C₂²×C₄	C₂×C₅×D₅	1	1	yes	characteristic
185	C₂×C₈	—	5	1	no	—
186	C₂×C₈	—	5	1	no	—
187	C₂×C₈	—	10	2	no	—
188	C₂×C₈	—	10	2	no	—
189	C₂×C₈	C₂×C₅×D₅	1	1	yes	characteristic
190	C₂×C₈	C₂×C₅×D₅	1	1	yes	characteristic
191	C₄⋊C₄	—	5	1	no	—
192	C₂²⋊C₄	—	5	1	no	—
193	C₂²⋊C₄	—	5	1	no	—
194	C₄²	—	5	1	no	—
195	C₂×C₅	—	2	2	no	—
196	C₂×C₅	—	2	2	no	—
197	C₂×C₅	—	2	2	no	—
198	C₂×C₅	—	2	2	no	—
199	C₂×C₅	—	2	2	no	—
200	C₂×C₅	—	4	4	no	—
201	C₂×C₅	—	10	2	no	—
202	C₂×C₅	#4027	1	1	yes	characteristic
203	C₂×C₅	C₂×C₈×D₅	1	1	yes	characteristic
204	C₂×C₅	#4053	1	1	yes	characteristic
205	C₂×C₅	C₄×C₅×D₄	1	1	yes	characteristic
206	C₂×C₅	C₂²×C₅×C₈	1	1	yes	characteristic
207	C₂×C₅	C₅×#131	1	1	yes	characteristic
208	D₅	—	2	2	no	—
209	C₂³	—	5	1	no	—
210	C₂³	C₄×C₅×D₅	1	1	yes	characteristic
211	D₄	—	5	4	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	5	1	no	—
215	C₂×C₄	—	5	1	no	—
216	C₂×C₄	—	5	1	no	—
217	C₂×C₄	—	10	1	no	—
218	C₂×C₄	—	10	1	no	—
219	C₂×C₄	C₄×C₅×D₅	1	1	yes	characteristic
220	C₂×C₄	C₂²×C₅×D₅	1	1	yes	characteristic
221	C₈	—	2	1	no	—
222	C₈	—	2	1	no	—
223	C₈	—	5	2	no	—
224	C₈	—	10	1	no	—
225	C₅	—	2	2	no	—
226	C₅	#65014	1	1	yes	characteristic, direct product factor
227	C₅	C₅×C₈×D₄	1	1	yes	characteristic, complemented
228	C₂²	—	2	1	no	—
229	C₂²	—	2	1	no	—
230	C₂²	—	5	2	no	—
231	C₂²	—	10	1	no	—
232	C₂²	—	10	1	no	—
233	C₂²	C₅×C₈×D₅	1	2	yes	complemented
234	C₂²	C₂×C₄×C₅×D₅	1	1	yes	characteristic
235	C₄	—	2	1	no	—
236	C₄	—	5	2	no	—
237	C₄	—	10	1	no	—
238	C₄	—	10	1	no	—
239	C₄	C₅×D₄×D₅	1	1	yes	characteristic
240	C₄	C₅×#711	1	1	yes	characteristic
241	C₂	—	2	2	no	—
242	C₂	—	10	2	no	—
243	C₂	C₅×#4027	1	1	yes	characteristic
244	C₂	C₂×C₅×C₈×D₅	1	1	yes	characteristic
245	C₂	C₅×#4053	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	C₂²×D₅
commutator	C₂×C₅	C₂²×C₅×C₈
Frattini	C₂×C₄	C₂²×C₅×D₅
Fitting	C₅²×C₂²⋊C₈	C₂
socle	C₂²×C₅²	C₂²×C₄
prime radical	C₂×C₅×C₂²⋊D₅	C₄
cosocle	C₂×C₄×C₅	C₂³×C₅
hypercenter	C₅×C₂²⋊C₈	D₅
hypocenter	C₅	C₅×C₈×D₄
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₅×C₂²⋊C₈

Complements

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

N	H ≅ G/N	type
2. C₂²×C₅×#168	C₂	semidirect
3. C₂×C₅×C₈×D₅	C₂	semidirect
4. C₅²×C₂²⋊C₈	C₂	semidirect
5. C₅×#3952	C₂	semidirect
6. C₅×#3946	C₂	semidirect
7. C₅×C₈×Dic₅	C₂	semidirect
22. #65014	C₅	direct (N × H)
29. C₅×C₂²⋊D₅	C₈	semidirect
38. C₅×#168	D₄	semidirect
43. C₂²×#168	C₂×C₅	semidirect
44. C₂×C₈×D₅	C₂×C₅	semidirect
48. C₅×C₂²⋊C₈	D₅	semidirect
49. C₅×C₂²⋊C₈	C₂×C₅	semidirect
51. #3952	C₂×C₅	semidirect
52. #3946	C₂×C₅	semidirect
53. C₈×Dic₅	C₂×C₅	semidirect
56. C₂²×C₅²	C₂×C₈	semidirect
60. C₂×C₅×D₅	C₂×C₈	semidirect
66. C₅×Dic₅	C₂×C₈	semidirect
125. C₂²⋊D₅	C₅×C₈	semidirect
140. #168	C₅×D₄	semidirect
146. C₂²⋊C₈	C₅×D₅	semidirect
148. C₅²	C₈×D₄	semidirect
160. C₂²×C₅	C₈×D₅	semidirect
161. C₂²×C₅	C₂×C₅×C₈	semidirect
166. C₂×D₅	C₂×C₅×C₈	semidirect
181. Dic₅	C₂×C₅×C₈	semidirect
226. C₅	#65014	direct (N × H)
227. C₅	C₅×C₈×D₄	semidirect
233. C₂²	C₅×C₈×D₅	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₅×C₂²⋊D₅	2
C₂²×C₅×#168	2
C₂×C₅×C₈×D₅	2
C₅²×C₂²⋊C₈	2
C₅×#3952	2
C₅×#3946	2
C₅×C₈×Dic₅	2
C₅×C₈×D₄	5
#65014	5

Subgroups — C5×#65014 #13101987

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#65014 #13101987