Subgroups — C₅×#65813 #13106633

Order 1600.

576 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	C₅×#4076	C₂	1	1	yes	maximal, characteristic
3	C₅×#4075	C₂	1	2	yes	maximal, complemented
4	C₄×C₅×C₂²⋊D₅	C₂	1	2	yes	maximal, complemented
5	C₅×#4070	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#4022	C₂	1	2	yes	maximal, complemented
7	C₅×#4021	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×C₅×D₂₀	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅×#4018	C₂	1	2	yes	maximal, complemented
10	C₅×#4016	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₅×Dic₁₀	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
13	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
14	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
15	C₂×C₅×C₂²⋊D₅	C₂²	1	2	yes	—
16	C₂×C₅×D₂₀	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₅×D₅	C₂²	1	2	yes	—
18	C₂×C₅×Dic₁₀	C₂²	1	1	yes	characteristic
19	C₄²×C₅²	C₂²	1	1	yes	characteristic
20	C₄²×C₅²	C₂²	1	1	yes	characteristic
21	C₄²×C₅²	C₂²	1	1	yes	characteristic
22	C₄²×C₅²	C₂²	1	1	yes	characteristic
23	C₅×C₂²⋊Dic₅	C₂²	1	2	yes	—
24	C₅×#685	C₂²	1	2	yes	—
25	C₅×#685	C₂²	1	2	yes	—
26	C₅×#685	C₂²	1	2	yes	—
27	C₅×#685	C₂²	1	2	yes	—
28	C₅×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
29	C₅×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
30	C₅×#683	C₂²	1	2	yes	—
31	C₅×#683	C₂²	1	2	yes	—
32	C₅×#683	C₂²	1	2	yes	—
33	C₅×#683	C₂²	1	2	yes	—
34	C₄×C₅×Dic₅	C₂²	1	2	yes	—
35	C₅×#529	—	5	1	no	maximal
36	#65813	C₅	1	1	yes	maximal, characteristic, direct product factor
37	C₂³×C₅²	C₂³	1	1	yes	characteristic
38	C₂²×C₅×D₅	C₂³	1	2	yes	—
39	C₂×C₄×C₅²	—	2	2	no	—
40	C₂×C₄×C₅²	—	2	2	no	—
41	C₂×C₄×C₅²	—	2	2	no	—
42	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
43	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
44	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
45	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
46	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
47	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
48	C₅×C₂²⋊D₅	—	2	4	no	—
49	C₂×C₅×Dic₅	C₂³	1	2	yes	—
50	C₂×C₅×Dic₅	C₂³	1	2	yes	—
51	C₂×C₅×Dic₅	C₂³	1	2	yes	—
52	C₅×D₂₀	—	2	2	no	—
53	C₄×C₅×D₅	—	2	4	no	—
54	C₅×Dic₁₀	—	2	2	no	—
55	C₅×#126	—	5	2	no	—
56	C₅×#125	—	5	1	no	—
57	C₅×#124	—	5	1	no	—
58	C₅×#123	—	5	2	no	—
59	C₅×C₂²⋊Q₈	—	5	1	no	—
60	C₅×#121	—	5	1	no	—
61	C₄×C₅×Q₈	—	5	1	no	—
62	C₄×C₅×D₄	—	5	2	no	—
63	C₄×C₅×D₄	—	5	1	no	—
64	C₅×#117	—	5	2	no	—
65	C₂×C₄²×C₅	—	2	2	no	—
66	C₂×C₄²×C₅	D₅	1	1	yes	characteristic, complemented
67	C₂×C₄²×C₅	C₂×C₅	1	1	yes	characteristic, complemented
68	#4076	C₂×C₅	1	1	yes	characteristic
69	#4075	C₂×C₅	1	2	yes	complemented
70	C₄×C₂²⋊D₅	C₂×C₅	1	2	yes	complemented
71	#4070	C₂×C₅	1	1	yes	characteristic, complemented
72	#4022	C₂×C₅	1	2	yes	complemented
73	#4021	C₂×C₅	1	1	yes	characteristic, complemented
74	C₄×D₂₀	C₂×C₅	1	1	yes	characteristic, complemented
75	#4018	C₂×C₅	1	2	yes	complemented
76	#4016	C₂×C₅	1	1	yes	characteristic, complemented
77	C₄×Dic₁₀	C₂×C₅	1	1	yes	characteristic, complemented
78	C₂²×C₅²	—	2	1	no	—
79	C₂²×C₅²	—	2	1	no	—
80	C₂²×C₅²	#41	1	2	yes	—
81	C₂²×C₅²	C₂⁴	1	1	yes	characteristic
82	C₂×C₅×D₅	—	2	2	no	—
83	C₂×C₅×D₅	—	2	2	no	—
84	C₂×C₅×D₅	—	2	2	no	—
85	C₄×C₅²	—	2	1	no	—
86	C₄×C₅²	—	2	1	no	—
87	C₄×C₅²	—	2	1	no	—
88	C₄×C₅²	—	2	1	no	—
89	C₄×C₅²	#41	1	2	yes	—
90	C₄×C₅²	#41	1	2	yes	—
91	C₅×Dic₅	—	2	2	no	—
92	C₅×Dic₅	—	2	2	no	—
93	C₅×Dic₅	—	2	2	no	—
94	C₂×C₅×Q₈	—	5	1	no	—
95	C₂×C₅×D₄	—	5	2	no	—
96	C₂×C₅×D₄	—	5	1	no	—
97	C₂²×C₄×C₅	—	2	2	no	—
98	C₂²×C₄×C₅	—	2	2	no	—
99	C₂²×C₄×C₅	—	2	2	no	—
100	C₂²×C₄×C₅	—	5	2	no	—
101	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
102	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
103	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
104	C₂²×C₄×C₅	C₂²×C₅	1	1	yes	characteristic
105	C₂²×C₄×C₅	C₂²×C₅	1	1	yes	characteristic
106	C₂²×C₄×C₅	C₂²×C₅	1	1	yes	characteristic
107	C₂×C₂²⋊D₅	C₂²×C₅	1	2	yes	—
108	C₂×D₂₀	C₂²×C₅	1	1	yes	characteristic
109	C₂×C₄×D₅	C₂²×C₅	1	2	yes	—
110	C₂×Dic₁₀	C₂²×C₅	1	1	yes	characteristic
111	C₅×C₄⋊C₄	—	5	2	no	—
112	C₅×C₄⋊C₄	—	5	2	no	—
113	C₅×C₄⋊C₄	—	5	1	no	—
114	C₅×C₄⋊C₄	—	5	2	no	—
115	C₅×C₄⋊C₄	—	5	1	no	—
116	C₅×C₄⋊C₄	—	5	2	no	—
117	C₅×C₂²⋊C₄	—	5	2	no	—
118	C₅×C₂²⋊C₄	—	5	2	no	—
119	C₅×C₂²⋊C₄	—	5	2	no	—
120	C₅×C₂²⋊C₄	—	5	2	no	—
121	C₅×C₂²⋊C₄	—	5	2	no	—
122	C₄²×C₅	—	2	2	no	—
123	C₄²×C₅	—	2	2	no	—
124	C₄²×C₅	—	2	2	no	—
125	C₄²×C₅	—	2	2	no	—
126	C₄²×C₅	—	5	2	no	—
127	C₄²×C₅	C₂×D₅	1	1	yes	characteristic
128	C₄²×C₅	C₂×D₅	1	1	yes	characteristic
129	C₄²×C₅	C₂×D₅	1	1	yes	characteristic
130	C₄²×C₅	C₂×D₅	1	1	yes	characteristic
131	C₄²×C₅	C₂²×C₅	1	1	yes	characteristic
132	C₄²×C₅	C₂²×C₅	1	1	yes	characteristic
133	C₄²×C₅	C₂²×C₅	1	1	yes	characteristic
134	C₄²×C₅	C₂²×C₅	1	1	yes	characteristic
135	C₂²⋊Dic₅	C₂²×C₅	1	2	yes	—
136	#685	C₂²×C₅	1	2	yes	—
137	#685	C₂²×C₅	1	2	yes	—
138	#685	C₂²×C₅	1	2	yes	—
139	#685	C₂²×C₅	1	2	yes	—
140	C₄⋊Dic₅	C₂²×C₅	1	1	yes	characteristic
141	C₄⋊Dic₅	C₂²×C₅	1	1	yes	characteristic
142	#683	C₂²×C₅	1	2	yes	—
143	#683	C₂²×C₅	1	2	yes	—
144	#683	C₂²×C₅	1	2	yes	—
145	#683	C₂²×C₅	1	2	yes	—
146	C₄×Dic₅	C₂²×C₅	1	2	yes	—
147	#529	—	5	1	no	Sylow
148	C₂×C₅²	—	2	2	no	—
149	C₂×C₅²	C₂×#41	1	1	yes	characteristic
150	C₂×C₅²	C₂×#41	1	1	yes	characteristic
151	C₂×C₅²	C₂×#41	1	1	yes	characteristic
152	C₅×D₅	—	4	2	no	—
153	C₂³×C₅	—	2	2	no	—
154	C₂³×C₅	—	5	2	no	—
155	C₂³×C₅	C₂²×D₅	1	1	yes	characteristic
156	C₂³×C₅	C₂³×C₅	1	1	yes	characteristic
157	C₂²×D₅	C₂³×C₅	1	2	yes	—
158	C₅×Q₈	—	10	2	no	—
159	C₅×D₄	—	10	4	no	—
160	C₅×D₄	—	10	2	no	—
161	C₂×C₄×C₅	—	2	8	no	—
162	C₂×C₄×C₅	—	2	8	no	—
163	C₂×C₄×C₅	—	2	8	no	—
164	C₂×C₄×C₅	—	2	2	no	—
165	C₂×C₄×C₅	—	2	2	no	—
166	C₂×C₄×C₅	—	2	2	no	—
167	C₂×C₄×C₅	—	2	2	no	—
168	C₂×C₄×C₅	—	2	2	no	—
169	C₂×C₄×C₅	—	2	2	no	—
170	C₂×C₄×C₅	—	2	2	no	—
171	C₂×C₄×C₅	—	2	2	no	—
172	C₂×C₄×C₅	—	2	2	no	—
173	C₂×C₄×C₅	—	2	2	no	—
174	C₂×C₄×C₅	—	2	2	no	—
175	C₂×C₄×C₅	—	2	2	no	—
176	C₂×C₄×C₅	—	5	2	no	—
177	C₂×C₄×C₅	—	5	2	no	—
178	C₂×C₄×C₅	—	5	2	no	—
179	C₂×C₄×C₅	—	10	4	no	—
180	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
181	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
182	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
183	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
184	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
185	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
186	C₂×C₄×C₅	C₂³×C₅	1	1	yes	characteristic
187	C₂×C₄×C₅	C₂³×C₅	1	1	yes	characteristic
188	C₂×C₄×C₅	C₂³×C₅	1	1	yes	characteristic
189	C₂×C₄×C₅	C₂³×C₅	1	1	yes	characteristic
190	C₂×C₄×C₅	C₂³×C₅	1	1	yes	characteristic
191	C₂×C₄×C₅	C₂³×C₅	1	1	yes	characteristic
192	C₂²⋊D₅	—	2	4	no	—
193	C₂×Dic₅	C₂³×C₅	1	2	yes	—
194	C₂×Dic₅	C₂³×C₅	1	2	yes	—
195	C₂×Dic₅	C₂³×C₅	1	2	yes	—
196	D₂₀	—	2	2	no	—
197	C₄×D₅	—	2	4	no	—
198	Dic₁₀	—	2	2	no	—
199	#126	—	5	2	no	—
200	#125	—	5	1	no	—
201	#124	—	5	1	no	—
202	#123	—	5	2	no	—
203	C₂²⋊Q₈	—	5	1	no	—
204	#121	—	5	1	no	—
205	C₄×Q₈	—	5	1	no	—
206	C₄×D₄	—	5	2	no	—
207	C₄×D₄	—	5	1	no	—
208	#117	—	5	2	no	—
209	C₂×C₄²	C₅×D₅	1	1	yes	characteristic, complemented
210	C₅²	#529	1	1	yes	characteristic, complemented, Sylow
211	C₂²×C₅	—	2	4	no	—
212	C₂²×C₅	—	2	4	no	—
213	C₂²×C₅	—	2	1	no	—
214	C₂²×C₅	—	2	1	no	—
215	C₂²×C₅	—	2	1	no	—
216	C₂²×C₅	—	2	1	no	—
217	C₂²×C₅	—	2	4	no	—
218	C₂²×C₅	—	2	2	no	—
219	C₂²×C₅	—	10	2	no	—
220	C₂²×C₅	—	10	2	no	—
221	C₂²×C₅	—	10	2	no	—
222	C₂²×C₅	#709	1	2	yes	—
223	C₂²×C₅	C₅×#41	1	2	yes	—
224	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
225	C₂²×C₅	C₂⁴×C₅	1	1	yes	characteristic
226	C₂×D₅	—	2	2	no	—
227	C₂×D₅	—	2	2	no	—
228	C₂×D₅	—	2	2	no	—
229	C₄×C₅	—	2	4	no	—
230	C₄×C₅	—	2	4	no	—
231	C₄×C₅	—	2	4	no	—
232	C₄×C₅	—	2	4	no	—
233	C₄×C₅	—	2	1	no	—
234	C₄×C₅	—	2	1	no	—
235	C₄×C₅	—	2	1	no	—
236	C₄×C₅	—	2	1	no	—
237	C₄×C₅	—	2	1	no	—
238	C₄×C₅	—	2	1	no	—
239	C₄×C₅	—	2	1	no	—
240	C₄×C₅	—	2	1	no	—
241	C₄×C₅	—	2	4	no	—
242	C₄×C₅	—	2	4	no	—
243	C₄×C₅	—	10	2	no	—
244	C₄×C₅	—	10	2	no	—
245	C₄×C₅	—	10	2	no	—
246	C₄×C₅	#709	1	2	yes	—
247	C₄×C₅	#709	1	2	yes	—
248	C₄×C₅	C₅×#41	1	2	yes	—
249	C₄×C₅	C₅×#41	1	2	yes	—
250	Dic₅	—	2	2	no	—
251	Dic₅	—	2	2	no	—
252	Dic₅	—	2	2	no	—
253	C₂×Q₈	—	5	1	no	—
254	C₂×D₄	—	5	2	no	—
255	C₂×D₄	—	5	1	no	—
256	C₂²×C₄	—	5	2	no	—
257	C₂²×C₄	C₂×C₅×D₅	1	1	yes	characteristic
258	C₂²×C₄	C₂×C₅×D₅	1	1	yes	characteristic
259	C₂²×C₄	C₂×C₅×D₅	1	1	yes	characteristic
260	C₄⋊C₄	—	5	2	no	—
261	C₄⋊C₄	—	5	2	no	—
262	C₄⋊C₄	—	5	1	no	—
263	C₄⋊C₄	—	5	2	no	—
264	C₄⋊C₄	—	5	1	no	—
265	C₄⋊C₄	—	5	2	no	—
266	C₂²⋊C₄	—	5	2	no	—
267	C₂²⋊C₄	—	5	2	no	—
268	C₂²⋊C₄	—	5	2	no	—
269	C₂²⋊C₄	—	5	2	no	—
270	C₂²⋊C₄	—	5	2	no	—
271	C₄²	—	5	2	no	—
272	C₄²	C₂×C₅×D₅	1	1	yes	characteristic
273	C₄²	C₂×C₅×D₅	1	1	yes	characteristic
274	C₄²	C₂×C₅×D₅	1	1	yes	characteristic
275	C₄²	C₂×C₅×D₅	1	1	yes	characteristic
276	C₂×C₅	—	2	8	no	—
277	C₂×C₅	—	2	2	no	—
278	C₂×C₅	—	2	2	no	—
279	C₂×C₅	—	2	2	no	—
280	C₂×C₅	—	2	2	no	—
281	C₂×C₅	—	2	2	no	—
282	C₂×C₅	—	20	2	no	—
283	C₂×C₅	C₂×#709	1	1	yes	characteristic
284	C₂×C₅	C₂×#709	1	1	yes	characteristic
285	C₂×C₅	C₂×#709	1	1	yes	characteristic
286	C₂×C₅	C₂×C₅×#41	1	1	yes	characteristic
287	C₂×C₅	C₂×C₅×#41	1	1	yes	characteristic
288	C₂×C₅	C₂×C₅×#41	1	1	yes	characteristic
289	D₅	—	4	2	no	—
290	C₂³	—	5	2	no	—
291	C₂³	C₂²×C₅×D₅	1	1	yes	characteristic
292	Q₈	—	10	2	no	—
293	D₄	—	10	4	no	—
294	D₄	—	10	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	5	2	no	—
299	C₂×C₄	—	5	2	no	—
300	C₂×C₄	—	5	2	no	—
301	C₂×C₄	—	10	4	no	—
302	C₂×C₄	C₂²×C₅×D₅	1	1	yes	characteristic
303	C₂×C₄	C₂²×C₅×D₅	1	1	yes	characteristic
304	C₂×C₄	C₂²×C₅×D₅	1	1	yes	characteristic
305	C₂×C₄	C₂²×C₅×D₅	1	1	yes	characteristic
306	C₂×C₄	C₂²×C₅×D₅	1	1	yes	characteristic
307	C₂×C₄	C₂²×C₅×D₅	1	1	yes	characteristic
308	C₅	—	2	2	no	—
309	C₅	#65813	1	1	yes	characteristic, direct product factor
310	C₅	C₅×#529	1	1	yes	characteristic, complemented
311	C₂²	—	2	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	10	2	no	—
314	C₂²	—	10	2	no	—
315	C₂²	—	10	2	no	—
316	C₂²	C₅×#709	1	2	yes	—
317	C₂²	C₂³×C₅×D₅	1	1	yes	characteristic
318	C₄	—	2	1	no	—
319	C₄	—	2	1	no	—
320	C₄	—	2	1	no	—
321	C₄	—	2	1	no	—
322	C₄	—	10	2	no	—
323	C₄	—	10	2	no	—
324	C₄	—	10	2	no	—
325	C₄	C₅×#709	1	2	yes	—
326	C₄	C₅×#709	1	2	yes	—
327	C₂	—	2	2	no	—
328	C₂	—	20	2	no	—
329	C₂	C₂×C₅×#709	1	1	yes	characteristic
330	C₂	C₂×C₅×#709	1	1	yes	characteristic
331	C₂	C₂×C₅×#709	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	C₂²×D₅
commutator, cosocle	C₂²×C₅	C₂⁴×C₅
Frattini	C₂²	C₂³×C₅×D₅
Fitting	C₂×C₄²×C₅²	C₂
socle	C₂²×C₅²	C₂⁴
prime radical	C₅×#4076	C₂
hypercenter	C₂×C₄²×C₅	D₅
hypocenter	C₅	C₅×#529
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄×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
3. C₅×#4075	C₂	semidirect
4. C₄×C₅×C₂²⋊D₅	C₂	semidirect
5. C₅×#4070	C₂	semidirect
6. C₅×#4022	C₂	semidirect
7. C₅×#4021	C₂	semidirect
8. C₄×C₅×D₂₀	C₂	semidirect
9. C₅×#4018	C₂	semidirect
10. C₅×#4016	C₂	semidirect
11. C₄×C₅×Dic₁₀	C₂	semidirect
36. #65813	C₅	direct (N × H)
66. C₂×C₄²×C₅	D₅	semidirect
67. C₂×C₄²×C₅	C₂×C₅	semidirect
69. #4075	C₂×C₅	semidirect
70. C₄×C₂²⋊D₅	C₂×C₅	semidirect
71. #4070	C₂×C₅	semidirect
72. #4022	C₂×C₅	semidirect
73. #4021	C₂×C₅	semidirect
74. C₄×D₂₀	C₂×C₅	semidirect
75. #4018	C₂×C₅	semidirect
76. #4016	C₂×C₅	semidirect
77. C₄×Dic₁₀	C₂×C₅	semidirect
209. C₂×C₄²	C₅×D₅	semidirect
210. C₅²	#529	semidirect
309. C₅	#65813	direct (N × H)
310. C₅	C₅×#529	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
C₅×#4076	2
C₅×#4075	2
C₄×C₅×C₂²⋊D₅	2
C₅×#4070	2
C₅×#4022	2
C₅×#4021	2
C₄×C₅×D₂₀	2
C₅×#4018	2
C₅×#4016	2
C₄×C₅×Dic₁₀	2
C₅×#529	5
#65813	5

Subgroups — C5×#65813 #13106633

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#65813 #13106633