Subgroups — C₅×#65502 #13102474

Order 1600.

808 conjugacy classes of subgroups.

The subgroup lattice has 297 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄×C₅²	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₅×C₂²⋊D₅	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×C₂²⋊Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅×#685	C₂	1	2	yes	maximal, complemented
5	C₅×#3963	C₂	1	2	yes	maximal, complemented
6	C₂⁴×C₅²	C₂²	1	1	yes	characteristic
7	C₂³×C₅×D₅	C₂²	1	1	yes	characteristic
8	C₂²×C₄×C₅²	—	2	2	no	—
9	C₂²×C₄×C₅²	—	2	4	no	—
10	C₂²×C₄×C₅²	C₂²	1	2	yes	—
11	C₂×C₅×C₂²⋊D₅	—	2	2	no	—
12	C₂×C₅×C₂²⋊D₅	—	2	2	no	—
13	C₂×C₅×C₂²⋊D₅	C₄	1	4	yes	complemented
14	C₂²×C₅×Dic₅	C₂²	1	2	yes	—
15	C₂²×C₅×Dic₅	C₂²	1	1	yes	characteristic
16	C₅×C₂²⋊Dic₅	—	2	2	no	—
17	C₅×#685	—	2	4	no	—
18	C₅×#386	—	5	1	no	maximal
19	#65502	C₅	1	1	yes	maximal, characteristic, direct product factor
20	C₂³×C₅²	—	2	2	no	—
21	C₂³×C₅²	—	2	2	no	—
22	C₂³×C₅²	C₂×C₄	1	2	yes	—
23	C₂³×C₅²	D₄	1	2	yes	—
24	C₂³×C₅²	D₄	1	2	yes	—
25	C₂³×C₅²	C₂³	1	1	yes	characteristic
26	C₂²×C₅×D₅	—	2	1	no	—
27	C₂²×C₅×D₅	—	2	1	no	—
28	C₂²×C₅×D₅	—	2	2	no	—
29	C₂²×C₅×D₅	—	2	2	no	—
30	C₂²×C₅×D₅	C₂×C₄	1	2	yes	complemented
31	C₂×C₄×C₅²	—	2	8	no	—
32	C₂×C₄×C₅²	—	2	2	no	—
33	C₂×C₄×C₅²	—	2	2	no	—
34	C₂×C₄×C₅²	D₄	1	4	yes	complemented
35	C₅×C₂²⋊D₅	—	2	8	no	—
36	C₂×C₅×Dic₅	—	2	1	no	—
37	C₂×C₅×Dic₅	—	2	1	no	—
38	C₂×C₅×Dic₅	—	2	2	no	—
39	C₂×C₅×Dic₅	—	2	2	no	—
40	C₂×C₅×Dic₅	—	2	2	no	—
41	C₂×C₅×Dic₅	C₂×C₄	1	2	yes	complemented
42	C₂²×C₅×D₄	—	5	1	no	—
43	C₂³×C₄×C₅	—	2	2	no	—
44	C₂³×C₄×C₅	D₅	1	1	yes	characteristic, complemented
45	C₂³×C₄×C₅	C₂×C₅	1	1	yes	characteristic, complemented
46	C₂²×C₂²⋊D₅	C₂×C₅	1	1	yes	characteristic
47	C₂×C₅×C₂²⋊C₄	—	5	1	no	—
48	C₂×C₅×C₂²⋊C₄	—	5	2	no	—
49	C₂×C₂²⋊Dic₅	C₂×C₅	1	1	yes	characteristic, complemented
50	C₂×#685	C₂×C₅	1	2	yes	complemented
51	C₅×#95	—	5	2	no	—
52	#3963	C₂×C₅	1	2	yes	complemented
53	C₂²×C₅²	—	2	4	no	—
54	C₂²×C₅²	—	2	2	no	—
55	C₂²×C₅²	—	2	4	no	—
56	C₂²×C₅²	—	2	2	no	—
57	C₂²×C₅²	C₂²⋊C₄	1	4	yes	complemented
58	C₂²×C₅²	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₅²	C₂×D₄	1	2	yes	—
62	C₂²×C₅²	#41	1	2	yes	—
63	C₂×C₅×D₅	—	2	2	no	—
64	C₂×C₅×D₅	—	2	4	no	—
65	C₂×C₅×D₅	—	4	1	no	—
66	C₂×C₅×D₅	—	4	2	no	—
67	C₂×C₅×D₅	—	4	1	no	—
68	C₄×C₅²	—	2	4	no	—
69	C₅×Dic₅	—	2	2	no	—
70	C₅×Dic₅	—	4	2	no	—
71	C₂⁴×C₅	—	2	2	no	—
72	C₂⁴×C₅	—	5	1	no	—
73	C₂⁴×C₅	C₂×D₅	1	1	yes	characteristic
74	C₂⁴×C₅	C₂²×C₅	1	1	yes	characteristic
75	C₂³×D₅	C₂²×C₅	1	1	yes	characteristic
76	C₂×C₅×D₄	—	5	4	no	—
77	C₂×C₅×D₄	—	10	2	no	—
78	C₂×C₅×D₄	—	10	2	no	—
79	C₂²×C₄×C₅	—	2	8	no	—
80	C₂²×C₄×C₅	—	2	16	no	—
81	C₂²×C₄×C₅	—	2	2	no	—
82	C₂²×C₄×C₅	—	2	2	no	—
83	C₂²×C₄×C₅	—	2	4	no	—
84	C₂²×C₄×C₅	—	2	4	no	—
85	C₂²×C₄×C₅	—	2	4	no	—
86	C₂²×C₄×C₅	—	5	1	no	—
87	C₂²×C₄×C₅	—	5	2	no	—
88	C₂²×C₄×C₅	C₂×D₅	1	2	yes	—
89	C₂²×C₄×C₅	C₂²×C₅	1	2	yes	—
90	C₂×C₂²⋊D₅	—	2	2	no	—
91	C₂×C₂²⋊D₅	—	2	2	no	—
92	C₂×C₂²⋊D₅	C₄×C₅	1	4	yes	complemented
93	C₂²×Dic₅	C₂²×C₅	1	2	yes	—
94	C₂²×Dic₅	C₂²×C₅	1	1	yes	characteristic
95	C₅×C₂²⋊C₄	—	10	2	no	—
96	C₅×C₂²⋊C₄	—	10	4	no	—
97	C₂²⋊Dic₅	—	2	2	no	—
98	#685	—	2	4	no	—
99	#386	—	5	1	no	Sylow
100	C₂×C₅²	—	2	4	no	—
101	C₂×C₅²	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	C₂×C₅²	C₄×D₄	1	2	yes	—
103	C₂×C₅²	#120	1	1	yes	characteristic
104	C₂×C₅²	#121	1	2	yes	—
105	C₂×C₅²	#123	1	1	yes	characteristic
106	C₅×D₅	—	4	2	no	—
107	C₂³×C₅	—	2	8	no	—
108	C₂³×C₅	—	2	8	no	—
109	C₂³×C₅	—	2	2	no	—
110	C₂³×C₅	—	2	2	no	—
111	C₂³×C₅	—	2	2	no	—
112	C₂³×C₅	—	2	2	no	—
113	C₂³×C₅	—	2	4	no	—
114	C₂³×C₅	—	2	4	no	—
115	C₂³×C₅	—	2	4	no	—
116	C₂³×C₅	—	2	2	no	—
117	C₂³×C₅	—	5	2	no	—
118	C₂³×C₅	—	10	1	no	—
119	C₂³×C₅	—	10	2	no	—
120	C₂³×C₅	—	10	2	no	—
121	C₂³×C₅	—	10	1	no	—
122	C₂³×C₅	C₄×D₅	1	2	yes	—
123	C₂³×C₅	D₂₀	1	2	yes	—
124	C₂³×C₅	C₂²⋊D₅	1	2	yes	—
125	C₂³×C₅	C₂×C₄×C₅	1	2	yes	—
126	C₂³×C₅	C₅×D₄	1	2	yes	—
127	C₂³×C₅	C₅×D₄	1	2	yes	—
128	C₂³×C₅	C₂²×D₅	1	1	yes	characteristic
129	C₂³×C₅	C₂³×C₅	1	1	yes	characteristic
130	C₂²×D₅	—	2	2	no	—
131	C₂²×D₅	—	2	1	no	—
132	C₂²×D₅	—	2	1	no	—
133	C₂²×D₅	—	2	2	no	—
134	C₂²×D₅	C₂×C₄×C₅	1	2	yes	complemented
135	C₅×D₄	—	10	8	no	—
136	C₂×C₄×C₅	—	2	32	no	—
137	C₂×C₄×C₅	—	2	8	no	—
138	C₂×C₄×C₅	—	2	8	no	—
139	C₂×C₄×C₅	—	2	8	no	—
140	C₂×C₄×C₅	—	2	8	no	—
141	C₂×C₄×C₅	—	2	2	no	—
142	C₂×C₄×C₅	—	2	2	no	—
143	C₂×C₄×C₅	—	2	2	no	—
144	C₂×C₄×C₅	—	2	2	no	—
145	C₂×C₄×C₅	—	2	8	no	—
146	C₂×C₄×C₅	—	5	2	no	—
147	C₂×C₄×C₅	—	10	1	no	—
148	C₂×C₄×C₅	—	10	1	no	—
149	C₂×C₄×C₅	—	10	2	no	—
150	C₂×C₄×C₅	—	10	2	no	—
151	C₂×C₄×C₅	—	10	2	no	—
152	C₂×C₄×C₅	C₂²⋊D₅	1	4	yes	complemented
153	C₂×C₄×C₅	C₅×D₄	1	4	yes	complemented
154	C₂²⋊D₅	—	2	8	no	—
155	C₂×Dic₅	—	2	1	no	—
156	C₂×Dic₅	—	2	1	no	—
157	C₂×Dic₅	—	2	2	no	—
158	C₂×Dic₅	—	2	2	no	—
159	C₂×Dic₅	—	2	2	no	—
160	C₂×Dic₅	C₂×C₄×C₅	1	2	yes	complemented
161	C₂²×D₄	—	5	1	no	—
162	C₂³×C₄	C₅×D₅	1	1	yes	characteristic, complemented
163	C₂×C₂²⋊C₄	—	5	1	no	—
164	C₂×C₂²⋊C₄	—	5	2	no	—
165	#95	—	5	2	no	—
166	C₅²	#386	1	1	yes	characteristic, complemented, Sylow
167	C₂²×C₅	—	2	8	no	—
168	C₂²×C₅	—	2	16	no	—
169	C₂²×C₅	—	2	8	no	—
170	C₂²×C₅	—	2	16	no	—
171	C₂²×C₅	—	2	2	no	—
172	C₂²×C₅	—	2	4	no	—
173	C₂²×C₅	—	2	2	no	—
174	C₂²×C₅	—	2	4	no	—
175	C₂²×C₅	—	2	2	no	—
176	C₂²×C₅	—	2	4	no	—
177	C₂²×C₅	—	2	2	no	—
178	C₂²×C₅	—	2	4	no	—
179	C₂²×C₅	—	2	2	no	—
180	C₂²×C₅	—	2	2	no	—
181	C₂²×C₅	—	2	4	no	—
182	C₂²×C₅	—	2	2	no	—
183	C₂²×C₅	—	2	4	no	—
184	C₂²×C₅	—	2	8	no	—
185	C₂²×C₅	—	10	2	no	—
186	C₂²×C₅	—	10	4	no	—
187	C₂²×C₅	—	20	1	no	—
188	C₂²×C₅	—	20	1	no	—
189	C₂²×C₅	—	20	2	no	—
190	C₂²×C₅	#685	1	4	yes	complemented
191	C₂²×C₅	C₅×C₂²⋊C₄	1	4	yes	complemented
192	C₂²×C₅	C₂×C₄×D₅	1	1	yes	characteristic
193	C₂²×C₅	C₂×D₂₀	1	1	yes	characteristic
194	C₂²×C₅	#709	1	2	yes	—
195	C₂²×C₅	C₂×C₂²⋊D₅	1	1	yes	characteristic
196	C₂²×C₅	C₂×C₂²⋊D₅	1	2	yes	—
197	C₂²×C₅	C₂²×C₄×C₅	1	1	yes	characteristic
198	C₂²×C₅	C₂×C₅×D₄	1	1	yes	characteristic
199	C₂²×C₅	C₂×C₅×D₄	1	1	yes	characteristic
200	C₂²×C₅	C₂×C₅×D₄	1	2	yes	—
201	C₂²×C₅	C₅×#41	1	2	yes	—
202	C₂×D₅	—	2	2	no	—
203	C₂×D₅	—	2	4	no	—
204	C₂×D₅	—	4	1	no	—
205	C₂×D₅	—	4	2	no	—
206	C₂×D₅	—	4	1	no	—
207	C₄×C₅	—	2	16	no	—
208	C₄×C₅	—	2	4	no	—
209	C₄×C₅	—	2	4	no	—
210	C₄×C₅	—	10	2	no	—
211	C₄×C₅	—	20	2	no	—
212	Dic₅	—	2	2	no	—
213	Dic₅	—	4	2	no	—
214	C₂⁴	—	5	1	no	—
215	C₂⁴	C₂×C₅×D₅	1	1	yes	characteristic
216	C₂×D₄	—	5	4	no	—
217	C₂×D₄	—	10	2	no	—
218	C₂×D₄	—	10	2	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	2	4	no	—
221	C₂²×C₄	—	5	1	no	—
222	C₂²×C₄	—	5	2	no	—
223	C₂²×C₄	C₂×C₅×D₅	1	2	yes	—
224	C₂²⋊C₄	—	10	2	no	—
225	C₂²⋊C₄	—	10	4	no	—
226	C₂×C₅	—	2	16	no	—
227	C₂×C₅	—	2	4	no	—
228	C₂×C₅	—	2	4	no	—
229	C₂×C₅	—	2	2	no	—
230	C₂×C₅	—	2	4	no	—
231	C₂×C₅	—	2	2	no	—
232	C₂×C₅	—	2	4	no	—
233	C₂×C₅	—	2	2	no	—
234	C₂×C₅	—	20	2	no	—
235	C₂×C₅	C₂×#685	1	1	yes	characteristic
236	C₂×C₅	C₄×C₂²⋊D₅	1	2	yes	—
237	C₂×C₅	#4075	1	1	yes	characteristic
238	C₂×C₅	#4076	1	2	yes	—
239	C₂×C₅	#4099	1	1	yes	characteristic
240	C₂×C₅	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
241	C₂×C₅	C₄×C₅×D₄	1	2	yes	—
242	C₂×C₅	C₅×#120	1	1	yes	characteristic
243	C₂×C₅	C₅×#121	1	2	yes	—
244	C₂×C₅	C₅×#123	1	1	yes	characteristic
245	D₅	—	4	2	no	—
246	C₂³	—	2	2	no	—
247	C₂³	—	2	2	no	—
248	C₂³	—	5	2	no	—
249	C₂³	—	10	1	no	—
250	C₂³	—	10	2	no	—
251	C₂³	—	10	2	no	—
252	C₂³	—	10	1	no	—
253	C₂³	C₄×C₅×D₅	1	2	yes	—
254	C₂³	C₅×D₂₀	1	2	yes	—
255	C₂³	C₅×C₂²⋊D₅	1	2	yes	—
256	C₂³	C₂²×C₅×D₅	1	1	yes	characteristic
257	D₄	—	10	8	no	—
258	C₂×C₄	—	2	8	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	5	2	no	—
262	C₂×C₄	—	10	1	no	—
263	C₂×C₄	—	10	1	no	—
264	C₂×C₄	—	10	2	no	—
265	C₂×C₄	—	10	2	no	—
266	C₂×C₄	—	10	2	no	—
267	C₂×C₄	C₅×C₂²⋊D₅	1	4	yes	complemented
268	C₅	—	2	2	no	—
269	C₅	#65502	1	1	yes	characteristic, direct product factor
270	C₅	C₅×#386	1	1	yes	characteristic, complemented
271	C₂²	—	2	4	no	—
272	C₂²	—	2	2	no	—
273	C₂²	—	2	4	no	—
274	C₂²	—	2	2	no	—
275	C₂²	—	10	2	no	—
276	C₂²	—	10	4	no	—
277	C₂²	—	20	1	no	—
278	C₂²	—	20	2	no	—
279	C₂²	—	20	1	no	—
280	C₂²	C₅×#685	1	4	yes	complemented
281	C₂²	C₂×C₄×C₅×D₅	1	1	yes	characteristic
282	C₂²	C₂×C₅×D₂₀	1	1	yes	characteristic
283	C₂²	C₅×#709	1	2	yes	—
284	C₂²	C₂×C₅×C₂²⋊D₅	1	1	yes	characteristic
285	C₂²	C₂×C₅×C₂²⋊D₅	1	2	yes	—
286	C₄	—	2	4	no	—
287	C₄	—	10	2	no	—
288	C₄	—	20	2	no	—
289	C₂	—	2	4	no	—
290	C₂	—	20	2	no	—
291	C₂	C₂×C₅×#685	1	1	yes	characteristic
292	C₂	C₄×C₅×C₂²⋊D₅	1	2	yes	—
293	C₂	C₅×#4075	1	1	yes	characteristic
294	C₂	C₅×#4076	1	2	yes	—
295	C₂	C₅×#4099	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₅	C₂²×D₅
commutator	C₂²×C₅	C₂²×C₄×C₅
Frattini	C₂³	C₂²×C₅×D₅
Fitting	C₂³×C₄×C₅²	C₂
socle	C₂³×C₅²	C₂³
prime radical	C₂²×C₅×C₂²⋊D₅	C₂
cosocle	C₂³×C₅	C₂³×C₅
hypercenter	C₂³×C₄×C₅	D₅
hypocenter	C₅	C₅×#386
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³×C₅ ◁ C₂³×C₄×C₅

Complements

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

N	H ≅ G/N	type
1. C₂³×C₄×C₅²	C₂	semidirect
3. C₂×C₅×C₂²⋊Dic₅	C₂	semidirect
4. C₂×C₅×#685	C₂	semidirect
5. C₅×#3963	C₂	semidirect
13. C₂×C₅×C₂²⋊D₅	C₄	semidirect
19. #65502	C₅	direct (N × H)
30. C₂²×C₅×D₅	C₂×C₄	semidirect
34. C₂×C₄×C₅²	D₄	semidirect
41. C₂×C₅×Dic₅	C₂×C₄	semidirect
44. C₂³×C₄×C₅	D₅	semidirect
45. C₂³×C₄×C₅	C₂×C₅	semidirect
49. C₂×C₂²⋊Dic₅	C₂×C₅	semidirect
50. C₂×#685	C₂×C₅	semidirect
52. #3963	C₂×C₅	semidirect
57. C₂²×C₅²	C₂²⋊C₄	semidirect
92. C₂×C₂²⋊D₅	C₄×C₅	semidirect
134. C₂²×D₅	C₂×C₄×C₅	semidirect
152. C₂×C₄×C₅	C₂²⋊D₅	semidirect
153. C₂×C₄×C₅	C₅×D₄	semidirect
160. C₂×Dic₅	C₂×C₄×C₅	semidirect
162. C₂³×C₄	C₅×D₅	semidirect
166. C₅²	#386	semidirect
190. C₂²×C₅	#685	semidirect
191. C₂²×C₅	C₅×C₂²⋊C₄	semidirect
267. C₂×C₄	C₅×C₂²⋊D₅	semidirect
269. C₅	#65502	direct (N × H)
270. C₅	C₅×#386	semidirect
280. C₂²	C₅×#685	semidirect

Maximal subgroups

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

subgroup	index
C₂³×C₄×C₅²	2
C₂²×C₅×C₂²⋊D₅	2
C₂×C₅×C₂²⋊Dic₅	2
C₂×C₅×#685	2
C₅×#3963	2
C₅×#386	5
#65502	5

Subgroups — C5×#65502 #13102474

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#65502 #13102474