Subgroups — C₅×#65229 #13102201

Order 1600.

488 conjugacy classes of subgroups.

The subgroup lattice has 333 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄²×C₅²	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅×#685	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#685	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅×#685	C₂	1	1	yes	maximal, characteristic
5	C₂×C₅×#683	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#3963	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#3963	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×C₅×D₅	C₂²	1	1	yes	characteristic
9	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
10	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
11	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
12	C₂²×C₅×Dic₅	C₂²	1	1	yes	characteristic
13	C₂²×C₅×Dic₅	C₂²	1	1	yes	characteristic
14	C₂²×C₅×Dic₅	C₂²	1	1	yes	characteristic
15	C₄²×C₅²	—	2	2	no	—
16	C₅×#685	—	2	2	no	—
17	C₅×#685	—	2	2	no	—
18	C₅×#685	C₄	1	4	yes	complemented
19	C₅×#683	—	2	2	no	—
20	C₅×#388	—	5	1	no	maximal
21	#65229	C₅	1	1	yes	maximal, characteristic, direct product factor
22	C₂³×C₅²	C₂³	1	1	yes	characteristic
23	C₂²×C₅×D₅	—	2	1	no	—
24	C₂²×C₅×D₅	—	2	1	no	—
25	C₂²×C₅×D₅	—	2	1	no	—
26	C₂²×C₅×D₅	—	2	1	no	—
27	C₂²×C₅×D₅	—	2	1	no	—
28	C₂²×C₅×D₅	—	2	1	no	—
29	C₂²×C₅×D₅	C₂×C₄	1	2	yes	—
30	C₂×C₄×C₅²	—	2	1	no	—
31	C₂×C₄×C₅²	—	2	1	no	—
32	C₂×C₄×C₅²	—	2	1	no	—
33	C₂×C₄×C₅²	—	2	1	no	—
34	C₂×C₄×C₅²	—	2	1	no	—
35	C₂×C₄×C₅²	—	2	1	no	—
36	C₂×C₄×C₅²	C₂×C₄	1	2	yes	—
37	C₂×C₄×C₅²	D₄	1	2	yes	—
38	C₂×C₄×C₅²	D₄	1	2	yes	—
39	C₂×C₅×Dic₅	—	2	1	no	—
40	C₂×C₅×Dic₅	—	2	1	no	—
41	C₂×C₅×Dic₅	—	2	1	no	—
42	C₂×C₅×Dic₅	—	2	1	no	—
43	C₂×C₅×Dic₅	—	2	1	no	—
44	C₂×C₅×Dic₅	—	2	1	no	—
45	C₂×C₅×Dic₅	—	2	1	no	—
46	C₂×C₅×Dic₅	—	2	1	no	—
47	C₂×C₅×Dic₅	C₂×C₄	1	2	yes	—
48	C₂×C₅×C₄⋊C₄	—	5	1	no	—
49	C₂×C₅×C₂²⋊C₄	—	5	1	no	—
50	C₂×C₅×C₂²⋊C₄	—	5	1	no	—
51	C₂×C₅×C₂²⋊C₄	—	5	1	no	—
52	C₂×C₄²×C₅	—	2	2	no	—
53	C₂×C₄²×C₅	D₅	1	1	yes	characteristic, complemented
54	C₂×C₄²×C₅	C₂×C₅	1	1	yes	characteristic, complemented
55	C₂×#685	C₂×C₅	1	1	yes	characteristic
56	C₂×#685	C₂×C₅	1	1	yes	characteristic
57	C₂×#685	C₂×C₅	1	1	yes	characteristic
58	C₂×#683	C₂×C₅	1	1	yes	characteristic, complemented
59	C₅×#95	—	5	1	no	—
60	C₅×#95	—	5	1	no	—
61	#3963	C₂×C₅	1	1	yes	characteristic, complemented
62	#3963	C₂×C₅	1	1	yes	characteristic, 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	#41	1	1	yes	characteristic
67	C₂²×C₅²	#41	1	1	yes	characteristic
68	C₂²×C₅²	#41	1	1	yes	characteristic
69	C₂²×C₅²	#41	1	1	yes	characteristic
70	C₂×C₅×D₅	—	2	2	no	—
71	C₂×C₅×D₅	—	2	2	no	—
72	C₂×C₅×D₅	—	2	2	no	—
73	C₂×C₅×D₅	—	4	1	no	—
74	C₂×C₅×D₅	—	4	1	no	—
75	C₂×C₅×D₅	—	4	1	no	—
76	C₂×C₅×D₅	—	4	1	no	—
77	C₄×C₅²	—	2	2	no	—
78	C₄×C₅²	—	2	2	no	—
79	C₄×C₅²	—	2	2	no	—
80	C₅×Dic₅	—	2	2	no	—
81	C₅×Dic₅	—	4	1	no	—
82	C₅×Dic₅	—	4	1	no	—
83	C₂⁴×C₅	—	5	1	no	—
84	C₂³×D₅	C₂²×C₅	1	1	yes	characteristic
85	C₂²×C₄×C₅	—	2	2	no	—
86	C₂²×C₄×C₅	—	2	2	no	—
87	C₂²×C₄×C₅	—	2	2	no	—
88	C₂²×C₄×C₅	—	5	1	no	—
89	C₂²×C₄×C₅	—	5	1	no	—
90	C₂²×C₄×C₅	—	5	1	no	—
91	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
92	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
93	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
94	C₂²×C₄×C₅	C₂²×C₅	1	1	yes	characteristic
95	C₂²×C₄×C₅	C₂²×C₅	1	1	yes	characteristic
96	C₂²×C₄×C₅	C₂²×C₅	1	1	yes	characteristic
97	C₂²×Dic₅	C₂²×C₅	1	1	yes	characteristic
98	C₂²×Dic₅	C₂²×C₅	1	1	yes	characteristic
99	C₂²×Dic₅	C₂²×C₅	1	1	yes	characteristic
100	C₅×C₄⋊C₄	—	10	2	no	—
101	C₅×C₂²⋊C₄	—	5	4	no	—
102	C₅×C₂²⋊C₄	—	10	2	no	—
103	C₅×C₂²⋊C₄	—	10	2	no	—
104	C₄²×C₅	—	2	8	no	—
105	C₄²×C₅	—	2	2	no	—
106	C₄²×C₅	—	2	2	no	—
107	#685	—	2	2	no	—
108	#685	—	2	2	no	—
109	#685	C₄×C₅	1	4	yes	complemented
110	#683	—	2	2	no	—
111	#388	—	5	1	no	Sylow
112	C₂×C₅²	#117	1	1	yes	characteristic
113	C₂×C₅²	C₄×D₄	1	1	yes	characteristic
114	C₂×C₅²	C₄×D₄	1	1	yes	characteristic
115	C₂×C₅²	#121	1	1	yes	characteristic
116	C₂×C₅²	#123	1	1	yes	characteristic
117	C₂×C₅²	#124	1	1	yes	characteristic
118	C₂×C₅²	#126	1	1	yes	characteristic
119	C₅×D₅	—	4	2	no	—
120	C₂³×C₅	—	2	2	no	—
121	C₂³×C₅	—	5	2	no	—
122	C₂³×C₅	—	10	1	no	—
123	C₂³×C₅	—	10	1	no	—
124	C₂³×C₅	—	10	1	no	—
125	C₂³×C₅	—	10	1	no	—
126	C₂³×C₅	—	10	1	no	—
127	C₂³×C₅	—	10	1	no	—
128	C₂³×C₅	C₂²×D₅	1	1	yes	characteristic
129	C₂³×C₅	C₂³×C₅	1	1	yes	characteristic
130	C₂²×D₅	—	2	1	no	—
131	C₂²×D₅	—	2	1	no	—
132	C₂²×D₅	—	2	1	no	—
133	C₂²×D₅	—	2	1	no	—
134	C₂²×D₅	—	2	1	no	—
135	C₂²×D₅	—	2	1	no	—
136	C₂²×D₅	C₂×C₄×C₅	1	2	yes	—
137	C₂×C₄×C₅	—	2	4	no	—
138	C₂×C₄×C₅	—	2	4	no	—
139	C₂×C₄×C₅	—	2	4	no	—
140	C₂×C₄×C₅	—	2	4	no	—
141	C₂×C₄×C₅	—	2	4	no	—
142	C₂×C₄×C₅	—	2	4	no	—
143	C₂×C₄×C₅	—	2	1	no	—
144	C₂×C₄×C₅	—	2	1	no	—
145	C₂×C₄×C₅	—	2	1	no	—
146	C₂×C₄×C₅	—	2	1	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	1	no	—
152	C₂×C₄×C₅	—	2	1	no	—
153	C₂×C₄×C₅	—	2	1	no	—
154	C₂×C₄×C₅	—	2	1	no	—
155	C₂×C₄×C₅	—	2	4	no	—
156	C₂×C₄×C₅	—	2	4	no	—
157	C₂×C₄×C₅	—	2	4	no	—
158	C₂×C₄×C₅	—	5	2	no	—
159	C₂×C₄×C₅	—	10	1	no	—
160	C₂×C₄×C₅	—	10	1	no	—
161	C₂×C₄×C₅	—	10	1	no	—
162	C₂×C₄×C₅	—	10	1	no	—
163	C₂×C₄×C₅	—	10	1	no	—
164	C₂×C₄×C₅	—	10	1	no	—
165	C₂×C₄×C₅	—	10	1	no	—
166	C₂×C₄×C₅	—	10	1	no	—
167	C₂×C₄×C₅	C₄×D₅	1	2	yes	—
168	C₂×C₄×C₅	D₂₀	1	2	yes	—
169	C₂×C₄×C₅	C₂²⋊D₅	1	2	yes	—
170	C₂×C₄×C₅	C₂×C₄×C₅	1	2	yes	—
171	C₂×C₄×C₅	C₅×D₄	1	2	yes	—
172	C₂×C₄×C₅	C₅×D₄	1	2	yes	—
173	C₂×Dic₅	—	2	1	no	—
174	C₂×Dic₅	—	2	1	no	—
175	C₂×Dic₅	—	2	1	no	—
176	C₂×Dic₅	—	2	1	no	—
177	C₂×Dic₅	—	2	1	no	—
178	C₂×Dic₅	—	2	1	no	—
179	C₂×Dic₅	—	2	1	no	—
180	C₂×Dic₅	—	2	1	no	—
181	C₂×Dic₅	C₂×C₄×C₅	1	2	yes	—
182	C₂×C₄⋊C₄	—	5	1	no	—
183	C₂×C₂²⋊C₄	—	5	1	no	—
184	C₂×C₂²⋊C₄	—	5	1	no	—
185	C₂×C₂²⋊C₄	—	5	1	no	—
186	C₂×C₄²	C₅×D₅	1	1	yes	characteristic, complemented
187	#95	—	5	1	no	—
188	#95	—	5	1	no	—
189	C₅²	#388	1	1	yes	characteristic, complemented, Sylow
190	C₂²×C₅	—	2	2	no	—
191	C₂²×C₅	—	2	2	no	—
192	C₂²×C₅	—	2	2	no	—
193	C₂²×C₅	—	2	2	no	—
194	C₂²×C₅	—	2	2	no	—
195	C₂²×C₅	—	2	2	no	—
196	C₂²×C₅	—	2	2	no	—
197	C₂²×C₅	—	10	2	no	—
198	C₂²×C₅	—	10	2	no	—
199	C₂²×C₅	—	10	2	no	—
200	C₂²×C₅	—	20	1	no	—
201	C₂²×C₅	—	20	1	no	—
202	C₂²×C₅	—	20	1	no	—
203	C₂²×C₅	—	20	1	no	—
204	C₂²×C₅	C₂×C₄×D₅	1	1	yes	characteristic
205	C₂²×C₅	C₂×D₂₀	1	1	yes	characteristic
206	C₂²×C₅	#709	1	1	yes	characteristic
207	C₂²×C₅	#709	1	1	yes	characteristic
208	C₂²×C₅	#709	1	1	yes	characteristic
209	C₂²×C₅	#709	1	1	yes	characteristic
210	C₂²×C₅	C₂×C₂²⋊D₅	1	1	yes	characteristic
211	C₂²×C₅	C₂²×C₄×C₅	1	1	yes	characteristic
212	C₂²×C₅	C₂×C₅×D₄	1	1	yes	characteristic
213	C₂²×C₅	C₂×C₅×D₄	1	1	yes	characteristic
214	C₂²×C₅	C₅×#41	1	1	yes	characteristic
215	C₂²×C₅	C₅×#41	1	1	yes	characteristic
216	C₂²×C₅	C₅×#41	1	1	yes	characteristic
217	C₂²×C₅	C₅×#41	1	1	yes	characteristic
218	C₂×D₅	—	2	2	no	—
219	C₂×D₅	—	2	2	no	—
220	C₂×D₅	—	2	2	no	—
221	C₂×D₅	—	4	1	no	—
222	C₂×D₅	—	4	1	no	—
223	C₂×D₅	—	4	1	no	—
224	C₂×D₅	—	4	1	no	—
225	C₄×C₅	—	2	8	no	—
226	C₄×C₅	—	2	8	no	—
227	C₄×C₅	—	2	8	no	—
228	C₄×C₅	—	2	2	no	—
229	C₄×C₅	—	2	2	no	—
230	C₄×C₅	—	2	2	no	—
231	C₄×C₅	—	2	2	no	—
232	C₄×C₅	—	2	2	no	—
233	C₄×C₅	—	2	2	no	—
234	C₄×C₅	—	10	2	no	—
235	C₄×C₅	—	20	1	no	—
236	C₄×C₅	—	20	1	no	—
237	Dic₅	—	2	2	no	—
238	Dic₅	—	4	1	no	—
239	Dic₅	—	4	1	no	—
240	C₂⁴	—	5	1	no	—
241	C₂²×C₄	—	5	1	no	—
242	C₂²×C₄	—	5	1	no	—
243	C₂²×C₄	—	5	1	no	—
244	C₂²×C₄	C₂×C₅×D₅	1	1	yes	characteristic
245	C₂²×C₄	C₂×C₅×D₅	1	1	yes	characteristic
246	C₂²×C₄	C₂×C₅×D₅	1	1	yes	characteristic
247	C₄⋊C₄	—	10	2	no	—
248	C₂²⋊C₄	—	5	4	no	—
249	C₂²⋊C₄	—	10	2	no	—
250	C₂²⋊C₄	—	10	2	no	—
251	C₄²	—	2	2	no	—
252	C₂×C₅	—	2	2	no	—
253	C₂×C₅	—	2	2	no	—
254	C₂×C₅	—	2	2	no	—
255	C₂×C₅	—	2	2	no	—
256	C₂×C₅	—	2	2	no	—
257	C₂×C₅	—	2	2	no	—
258	C₂×C₅	—	2	2	no	—
259	C₂×C₅	—	20	2	no	—
260	C₂×C₅	#4018	1	1	yes	characteristic
261	C₂×C₅	C₄×D₂₀	1	1	yes	characteristic
262	C₂×C₅	#4021	1	1	yes	characteristic
263	C₂×C₅	#4022	1	1	yes	characteristic
264	C₂×C₅	C₄×C₂²⋊D₅	1	1	yes	characteristic
265	C₂×C₅	#4075	1	1	yes	characteristic
266	C₂×C₅	#4076	1	1	yes	characteristic
267	C₂×C₅	C₅×#117	1	1	yes	characteristic
268	C₂×C₅	C₄×C₅×D₄	1	1	yes	characteristic
269	C₂×C₅	C₄×C₅×D₄	1	1	yes	characteristic
270	C₂×C₅	C₅×#121	1	1	yes	characteristic
271	C₂×C₅	C₅×#123	1	1	yes	characteristic
272	C₂×C₅	C₅×#124	1	1	yes	characteristic
273	C₂×C₅	C₅×#126	1	1	yes	characteristic
274	D₅	—	4	2	no	—
275	C₂³	—	5	2	no	—
276	C₂³	—	10	1	no	—
277	C₂³	—	10	1	no	—
278	C₂³	—	10	1	no	—
279	C₂³	—	10	1	no	—
280	C₂³	—	10	1	no	—
281	C₂³	—	10	1	no	—
282	C₂³	C₂²×C₅×D₅	1	1	yes	characteristic
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	5	2	no	—
290	C₂×C₄	—	10	1	no	—
291	C₂×C₄	—	10	1	no	—
292	C₂×C₄	—	10	1	no	—
293	C₂×C₄	—	10	1	no	—
294	C₂×C₄	—	10	1	no	—
295	C₂×C₄	—	10	1	no	—
296	C₂×C₄	—	10	1	no	—
297	C₂×C₄	—	10	1	no	—
298	C₂×C₄	C₄×C₅×D₅	1	2	yes	—
299	C₂×C₄	C₅×D₂₀	1	2	yes	—
300	C₂×C₄	C₅×C₂²⋊D₅	1	2	yes	—
301	C₅	—	2	2	no	—
302	C₅	#65229	1	1	yes	characteristic, direct product factor
303	C₅	C₅×#388	1	1	yes	characteristic, complemented
304	C₂²	—	10	2	no	—
305	C₂²	—	10	2	no	—
306	C₂²	—	10	2	no	—
307	C₂²	—	20	1	no	—
308	C₂²	—	20	1	no	—
309	C₂²	—	20	1	no	—
310	C₂²	—	20	1	no	—
311	C₂²	C₂×C₄×C₅×D₅	1	1	yes	characteristic
312	C₂²	C₂×C₅×D₂₀	1	1	yes	characteristic
313	C₂²	C₅×#709	1	1	yes	characteristic
314	C₂²	C₅×#709	1	1	yes	characteristic
315	C₂²	C₅×#709	1	1	yes	characteristic
316	C₂²	C₅×#709	1	1	yes	characteristic
317	C₂²	C₂×C₅×C₂²⋊D₅	1	1	yes	characteristic
318	C₄	—	2	2	no	—
319	C₄	—	2	2	no	—
320	C₄	—	2	2	no	—
321	C₄	—	10	2	no	—
322	C₄	—	20	1	no	—
323	C₄	—	20	1	no	—
324	C₂	—	20	2	no	—
325	C₂	C₅×#4018	1	1	yes	characteristic
326	C₂	C₄×C₅×D₂₀	1	1	yes	characteristic
327	C₂	C₅×#4021	1	1	yes	characteristic
328	C₂	C₅×#4022	1	1	yes	characteristic
329	C₂	C₄×C₅×C₂²⋊D₅	1	1	yes	characteristic
330	C₂	C₅×#4075	1	1	yes	characteristic
331	C₂	C₅×#4076	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₅	C₂²×D₅
commutator	C₂²×C₅	C₂²×C₄×C₅
Frattini	C₂³	C₂²×C₅×D₅
Fitting	C₂×C₄²×C₅²	C₂
socle	C₂³×C₅²	C₂³
prime radical	C₂³×C₅×D₅	C₂²
cosocle	C₂³×C₅	C₂³×C₅
hypercenter	C₂×C₄²×C₅	D₅
hypocenter	C₅	C₅×#388
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³×C₅ ◁ C₂×C₄²×C₅

Complements

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

N	H ≅ G/N	type
1. C₂×C₄²×C₅²	C₂	semidirect
5. C₂×C₅×#683	C₂	semidirect
6. C₅×#3963	C₂	semidirect
7. C₅×#3963	C₂	semidirect
18. C₅×#685	C₄	semidirect
21. #65229	C₅	direct (N × H)
53. C₂×C₄²×C₅	D₅	semidirect
54. C₂×C₄²×C₅	C₂×C₅	semidirect
58. C₂×#683	C₂×C₅	semidirect
61. #3963	C₂×C₅	semidirect
62. #3963	C₂×C₅	semidirect
109. #685	C₄×C₅	semidirect
186. C₂×C₄²	C₅×D₅	semidirect
189. C₅²	#388	semidirect
302. C₅	#65229	direct (N × H)
303. C₅	C₅×#388	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄²×C₅²	2
C₂×C₅×#685	2
C₂×C₅×#683	2
C₅×#3963	2
C₅×#388	5
#65229	5

Subgroups — C5×#65229 #13102201

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#65229 #13102201