Subgroups — C₅×#65509 #13102480

Order 1600.

808 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅²×D₄	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₅×Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×C₂²⋊Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅×C₂²⋊Dic₅	C₂	1	2	yes	maximal, complemented
5	C₅×#3963	C₂	1	2	yes	maximal, complemented
6	C₂⁴×C₅²	C₂²	1	1	yes	characteristic
7	C₂⁴×C₅²	C₂²	1	1	yes	characteristic
8	C₂×C₅²×D₄	—	2	2	no	—
9	C₂×C₅²×D₄	—	2	2	no	—
10	C₂×C₅²×D₄	C₄	1	4	yes	complemented
11	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
12	C₂²×C₅×Dic₅	—	2	2	no	—
13	C₂²×C₅×Dic₅	—	2	4	no	—
14	C₂²×C₅×Dic₅	C₂²	1	2	yes	—
15	C₂²×C₅×Dic₅	C₂²	1	2	yes	—
16	C₅×C₂²⋊Dic₅	—	2	2	no	—
17	C₅×C₂²⋊Dic₅	—	2	4	no	—
18	C₅×#386	—	5	1	no	maximal
19	#65509	C₅	1	1	yes	maximal, characteristic, direct product factor
20	C₂³×C₅²	—	2	2	no	—
21	C₂³×C₅²	—	2	2	no	—
22	C₂³×C₅²	—	2	1	no	—
23	C₂³×C₅²	—	2	1	no	—
24	C₂³×C₅²	—	2	2	no	—
25	C₂³×C₅²	—	2	2	no	—
26	C₂³×C₅²	C₂×C₄	1	2	yes	—
27	C₂³×C₅²	C₂×C₄	1	2	yes	complemented
28	C₂³×C₅²	D₄	1	2	yes	—
29	C₂³×C₅²	D₄	1	2	yes	—
30	C₂³×C₅²	C₂³	1	1	yes	characteristic
31	C₅²×D₄	—	2	8	no	—
32	C₂×C₄×C₅²	—	2	1	no	—
33	C₂×C₄×C₅²	—	2	1	no	—
34	C₂×C₄×C₅²	C₂×C₄	1	2	yes	complemented
35	C₂×C₅×Dic₅	—	2	8	no	—
36	C₂×C₅×Dic₅	—	2	2	no	—
37	C₂×C₅×Dic₅	—	2	2	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	D₄	1	4	yes	complemented
42	C₂²×C₅×D₄	—	2	2	no	—
43	C₂²×C₅×D₄	D₅	1	1	yes	characteristic
44	C₂²×C₅×D₄	C₂×C₅	1	1	yes	characteristic
45	C₂³×C₄×C₅	—	5	1	no	—
46	C₂³×Dic₅	C₂×C₅	1	1	yes	characteristic, complemented
47	C₂×C₅×C₂²⋊C₄	—	5	2	no	—
48	C₂×C₅×C₂²⋊C₄	—	5	1	no	—
49	C₂×C₂²⋊Dic₅	C₂×C₅	1	1	yes	characteristic, complemented
50	C₂×C₂²⋊Dic₅	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	—	2	2	no	—
58	C₂²×C₅²	—	2	4	no	—
59	C₂²×C₅²	—	4	1	no	—
60	C₂²×C₅²	—	4	2	no	—
61	C₂²×C₅²	—	4	1	no	—
62	C₂²×C₅²	C₂²⋊C₄	1	4	yes	complemented
63	C₂²×C₅²	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₅²	C₂×D₄	1	2	yes	—
67	C₂²×C₅²	#41	1	2	yes	—
68	C₄×C₅²	—	2	2	no	—
69	C₅×Dic₅	—	2	4	no	—
70	C₅×Dic₅	—	4	2	no	—
71	C₂⁴×C₅	—	2	2	no	—
72	C₂⁴×C₅	—	2	2	no	—
73	C₂⁴×C₅	C₂×D₅	1	1	yes	characteristic
74	C₂⁴×C₅	C₂×D₅	1	1	yes	characteristic
75	C₂⁴×C₅	C₂²×C₅	1	1	yes	characteristic
76	C₂⁴×C₅	C₂²×C₅	1	1	yes	characteristic
77	C₂×C₅×D₄	—	2	8	no	—
78	C₂×C₅×D₄	—	2	8	no	—
79	C₂×C₅×D₄	—	2	2	no	—
80	C₂×C₅×D₄	—	2	2	no	—
81	C₂×C₅×D₄	—	2	2	no	—
82	C₂×C₅×D₄	—	2	2	no	—
83	C₂×C₅×D₄	—	2	8	no	—
84	C₂×C₅×D₄	Dic₅	1	4	yes	complemented
85	C₂×C₅×D₄	C₄×C₅	1	4	yes	complemented
86	C₂²×C₄×C₅	—	2	2	no	—
87	C₂²×C₄×C₅	—	5	2	no	—
88	C₂²×C₄×C₅	—	5	2	no	—
89	C₂²×C₄×C₅	—	10	4	no	—
90	C₂²×C₄×C₅	—	10	2	no	—
91	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
92	C₂²×C₄×C₅	C₂²×C₅	1	1	yes	characteristic
93	C₂²×Dic₅	—	2	2	no	—
94	C₂²×Dic₅	—	2	4	no	—
95	C₂²×Dic₅	C₂²×C₅	1	2	yes	—
96	C₂²×Dic₅	C₂²×C₅	1	2	yes	—
97	C₅×C₂²⋊C₄	—	10	4	no	—
98	C₅×C₂²⋊C₄	—	10	2	no	—
99	C₂²⋊Dic₅	—	2	2	no	—
100	C₂²⋊Dic₅	—	2	4	no	—
101	#386	—	5	1	no	Sylow
102	C₂×C₅²	—	2	4	no	—
103	C₂×C₅²	—	4	2	no	—
104	C₂×C₅²	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₅²	C₄×D₄	1	2	yes	—
106	C₂×C₅²	#120	1	1	yes	characteristic
107	C₂×C₅²	#121	1	2	yes	—
108	C₂×C₅²	#123	1	1	yes	characteristic
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	2	no	—
116	C₂³×C₅	—	2	2	no	—
117	C₂³×C₅	—	2	1	no	—
118	C₂³×C₅	—	2	1	no	—
119	C₂³×C₅	—	2	2	no	—
120	C₂³×C₅	—	2	1	no	—
121	C₂³×C₅	—	2	1	no	—
122	C₂³×C₅	—	2	2	no	—
123	C₂³×C₅	—	2	4	no	—
124	C₂³×C₅	—	2	4	no	—
125	C₂³×C₅	—	2	4	no	—
126	C₂³×C₅	—	2	4	no	—
127	C₂³×C₅	—	2	2	no	—
128	C₂³×C₅	—	4	4	no	—
129	C₂³×C₅	—	4	4	no	—
130	C₂³×C₅	—	4	4	no	—
131	C₂³×C₅	—	4	4	no	—
132	C₂³×C₅	C₂×Dic₅	1	2	yes	—
133	C₂³×C₅	C₂×Dic₅	1	2	yes	complemented
134	C₂³×C₅	C₂²⋊D₅	1	2	yes	—
135	C₂³×C₅	C₂²⋊D₅	1	2	yes	—
136	C₂³×C₅	C₂×C₄×C₅	1	2	yes	—
137	C₂³×C₅	C₂×C₄×C₅	1	2	yes	complemented
138	C₂³×C₅	C₅×D₄	1	2	yes	—
139	C₂³×C₅	C₅×D₄	1	2	yes	—
140	C₂³×C₅	C₂²×D₅	1	1	yes	characteristic
141	C₂³×C₅	C₂³×C₅	1	1	yes	characteristic
142	C₅×D₄	—	2	32	no	—
143	C₅×D₄	—	2	8	no	—
144	C₅×D₄	—	2	8	no	—
145	C₂×C₄×C₅	—	2	4	no	—
146	C₂×C₄×C₅	—	2	4	no	—
147	C₂×C₄×C₅	—	2	1	no	—
148	C₂×C₄×C₅	—	2	1	no	—
149	C₂×C₄×C₅	—	2	1	no	—
150	C₂×C₄×C₅	—	2	1	no	—
151	C₂×C₄×C₅	—	2	4	no	—
152	C₂×C₄×C₅	—	5	4	no	—
153	C₂×C₄×C₅	—	10	8	no	—
154	C₂×C₄×C₅	—	10	2	no	—
155	C₂×C₄×C₅	—	10	2	no	—
156	C₂×C₄×C₅	—	10	2	no	—
157	C₂×C₄×C₅	—	10	2	no	—
158	C₂×C₄×C₅	—	10	2	no	—
159	C₂×C₄×C₅	C₂×Dic₅	1	2	yes	complemented
160	C₂×C₄×C₅	C₂×C₄×C₅	1	2	yes	complemented
161	C₂×Dic₅	—	2	8	no	—
162	C₂×Dic₅	—	2	2	no	—
163	C₂×Dic₅	—	2	2	no	—
164	C₂×Dic₅	—	2	2	no	—
165	C₂×Dic₅	—	2	2	no	—
166	C₂×Dic₅	—	2	2	no	—
167	C₂×Dic₅	C₅×D₄	1	4	yes	complemented
168	C₂²×D₄	C₅×D₅	1	1	yes	characteristic
169	C₂³×C₄	—	5	1	no	—
170	C₂×C₂²⋊C₄	—	5	2	no	—
171	C₂×C₂²⋊C₄	—	5	1	no	—
172	#95	—	5	2	no	—
173	C₅²	#386	1	1	yes	characteristic, complemented, Sylow
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	8	no	—
183	C₂²×C₅	—	2	2	no	—
184	C₂²×C₅	—	2	2	no	—
185	C₂²×C₅	—	2	4	no	—
186	C₂²×C₅	—	2	4	no	—
187	C₂²×C₅	—	2	4	no	—
188	C₂²×C₅	—	2	4	no	—
189	C₂²×C₅	—	2	2	no	—
190	C₂²×C₅	—	2	2	no	—
191	C₂²×C₅	—	2	2	no	—
192	C₂²×C₅	—	2	8	no	—
193	C₂²×C₅	—	4	8	no	—
194	C₂²×C₅	—	4	4	no	—
195	C₂²×C₅	—	4	4	no	—
196	C₂²×C₅	—	4	8	no	—
197	C₂²×C₅	—	4	8	no	—
198	C₂²×C₅	—	4	4	no	—
199	C₂²×C₅	—	4	4	no	—
200	C₂²×C₅	—	4	8	no	—
201	C₂²×C₅	—	4	1	no	—
202	C₂²×C₅	—	4	2	no	—
203	C₂²×C₅	—	4	1	no	—
204	C₂²×C₅	—	4	1	no	—
205	C₂²×C₅	—	4	1	no	—
206	C₂²×C₅	—	4	2	no	—
207	C₂²×C₅	C₂²⋊Dic₅	1	4	yes	complemented
208	C₂²×C₅	C₅×C₂²⋊C₄	1	4	yes	complemented
209	C₂²×C₅	D₄×D₅	1	2	yes	—
210	C₂²×C₅	#711	1	2	yes	—
211	C₂²×C₅	C₂²×Dic₅	1	1	yes	characteristic
212	C₂²×C₅	C₂×C₂²⋊D₅	1	1	yes	characteristic
213	C₂²×C₅	C₂×C₂²⋊D₅	1	1	yes	characteristic
214	C₂²×C₅	C₂²×C₄×C₅	1	1	yes	characteristic
215	C₂²×C₅	C₂×C₅×D₄	1	2	yes	—
216	C₂²×C₅	C₂×C₅×D₄	1	1	yes	characteristic
217	C₂²×C₅	C₂×C₅×D₄	1	1	yes	characteristic
218	C₂²×C₅	C₅×#41	1	2	yes	—
219	C₄×C₅	—	2	8	no	—
220	C₄×C₅	—	2	2	no	—
221	C₄×C₅	—	2	2	no	—
222	C₄×C₅	—	10	4	no	—
223	C₄×C₅	—	20	2	no	—
224	Dic₅	—	2	4	no	—
225	Dic₅	—	4	2	no	—
226	C₂⁴	C₂×C₅×D₅	1	1	yes	characteristic
227	C₂⁴	C₂×C₅×D₅	1	1	yes	characteristic
228	C₂×D₄	—	2	2	no	—
229	C₂×D₄	—	2	2	no	—
230	C₂×D₄	C₅×Dic₅	1	4	yes	complemented
231	C₂²×C₄	—	5	2	no	—
232	C₂²×C₄	—	5	2	no	—
233	C₂²×C₄	—	10	4	no	—
234	C₂²×C₄	—	10	2	no	—
235	C₂²×C₄	C₂×C₅×D₅	1	1	yes	characteristic
236	C₂²⋊C₄	—	10	4	no	—
237	C₂²⋊C₄	—	10	2	no	—
238	C₂×C₅	—	2	4	no	—
239	C₂×C₅	—	2	4	no	—
240	C₂×C₅	—	2	4	no	—
241	C₂×C₅	—	2	2	no	—
242	C₂×C₅	—	2	2	no	—
243	C₂×C₅	—	2	4	no	—
244	C₂×C₅	—	2	2	no	—
245	C₂×C₅	—	4	8	no	—
246	C₂×C₅	—	4	8	no	—
247	C₂×C₅	—	4	2	no	—
248	C₂×C₅	—	4	2	no	—
249	C₂×C₅	D₄×Dic₅	1	2	yes	—
250	C₂×C₅	#4081	1	1	yes	characteristic
251	C₂×C₅	#4083	1	1	yes	characteristic
252	C₂×C₅	#4085	1	2	yes	—
253	C₂×C₅	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
254	C₂×C₅	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
255	C₂×C₅	C₄×C₅×D₄	1	2	yes	—
256	C₂×C₅	C₅×#120	1	1	yes	characteristic
257	C₂×C₅	C₅×#121	1	2	yes	—
258	C₂×C₅	C₅×#123	1	1	yes	characteristic
259	C₂³	—	2	2	no	—
260	C₂³	—	2	2	no	—
261	C₂³	—	2	2	no	—
262	C₂³	—	2	2	no	—
263	C₂³	—	2	1	no	—
264	C₂³	—	2	1	no	—
265	C₂³	C₂×C₅×Dic₅	1	2	yes	—
266	C₂³	C₂×C₅×Dic₅	1	2	yes	complemented
267	C₂³	C₅×C₂²⋊D₅	1	2	yes	—
268	C₂³	C₅×C₂²⋊D₅	1	2	yes	—
269	C₂³	C₂²×C₅×D₅	1	1	yes	characteristic
270	D₄	—	2	8	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	5	4	no	—
274	C₂×C₄	—	10	8	no	—
275	C₂×C₄	—	10	2	no	—
276	C₂×C₄	—	10	2	no	—
277	C₂×C₄	—	10	2	no	—
278	C₂×C₄	—	10	2	no	—
279	C₂×C₄	—	10	2	no	—
280	C₂×C₄	C₂×C₅×Dic₅	1	2	yes	complemented
281	C₅	—	2	2	no	—
282	C₅	#65509	1	1	yes	characteristic, direct product factor
283	C₅	C₅×#386	1	1	yes	characteristic, complemented
284	C₂²	—	2	4	no	—
285	C₂²	—	2	2	no	—
286	C₂²	—	2	2	no	—
287	C₂²	—	2	4	no	—
288	C₂²	—	2	4	no	—
289	C₂²	—	2	2	no	—
290	C₂²	—	4	1	no	—
291	C₂²	—	4	1	no	—
292	C₂²	—	4	2	no	—
293	C₂²	C₅×C₂²⋊Dic₅	1	4	yes	complemented
294	C₂²	C₅×D₄×D₅	1	2	yes	—
295	C₂²	C₅×#711	1	2	yes	—
296	C₂²	C₂²×C₅×Dic₅	1	1	yes	characteristic
297	C₂²	C₂×C₅×C₂²⋊D₅	1	1	yes	characteristic
298	C₂²	C₂×C₅×C₂²⋊D₅	1	1	yes	characteristic
299	C₄	—	2	2	no	—
300	C₄	—	10	4	no	—
301	C₄	—	20	2	no	—
302	C₂	—	2	4	no	—
303	C₂	—	4	2	no	—
304	C₂	C₅×D₄×Dic₅	1	2	yes	—
305	C₂	C₅×#4081	1	1	yes	characteristic
306	C₂	C₅×#4083	1	1	yes	characteristic
307	C₂	C₅×#4085	1	2	yes	—
308	C₂	C₂×C₅×C₂²⋊Dic₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₅	C₂²×D₅
commutator	C₂²×C₅	C₂²×C₄×C₅
Frattini	C₂³	C₂²×C₅×D₅
Fitting, prime radical	C₂²×C₅²×D₄	C₂
socle	C₂³×C₅²	C₂³
cosocle	C₂³×C₅	C₂³×C₅
hypercenter	C₂²×C₅×D₄	D₅
hypocenter	C₅	C₅×#386
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³×C₅ ◁ C₂²×C₅×D₄

Complements

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

N	H ≅ G/N	type
2. C₂³×C₅×Dic₅	C₂	semidirect
3. C₂×C₅×C₂²⋊Dic₅	C₂	semidirect
4. C₂×C₅×C₂²⋊Dic₅	C₂	semidirect
5. C₅×#3963	C₂	semidirect
10. C₂×C₅²×D₄	C₄	semidirect
19. #65509	C₅	direct (N × H)
27. C₂³×C₅²	C₂×C₄	semidirect
34. C₂×C₄×C₅²	C₂×C₄	semidirect
41. C₂×C₅×Dic₅	D₄	semidirect
46. C₂³×Dic₅	C₂×C₅	semidirect
49. C₂×C₂²⋊Dic₅	C₂×C₅	semidirect
50. C₂×C₂²⋊Dic₅	C₂×C₅	semidirect
52. #3963	C₂×C₅	semidirect
62. C₂²×C₅²	C₂²⋊C₄	semidirect
84. C₂×C₅×D₄	Dic₅	semidirect
85. C₂×C₅×D₄	C₄×C₅	semidirect
133. C₂³×C₅	C₂×Dic₅	semidirect
137. C₂³×C₅	C₂×C₄×C₅	semidirect
159. C₂×C₄×C₅	C₂×Dic₅	semidirect
160. C₂×C₄×C₅	C₂×C₄×C₅	semidirect
167. C₂×Dic₅	C₅×D₄	semidirect
173. C₅²	#386	semidirect
207. C₂²×C₅	C₂²⋊Dic₅	semidirect
208. C₂²×C₅	C₅×C₂²⋊C₄	semidirect
230. C₂×D₄	C₅×Dic₅	semidirect
266. C₂³	C₂×C₅×Dic₅	semidirect
280. C₂×C₄	C₂×C₅×Dic₅	semidirect
282. C₅	#65509	direct (N × H)
283. C₅	C₅×#386	semidirect
293. C₂²	C₅×C₂²⋊Dic₅	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₅²×D₄	2
C₂³×C₅×Dic₅	2
C₂×C₅×C₂²⋊Dic₅	2
C₅×#3963	2
C₅×#386	5
#65509	5

Subgroups — C5×#65509 #13102480

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#65509 #13102480