Subgroups — D₅×#4050 #13105766

Order 1600.

594 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₅×#709	C₂	1	2	yes	maximal, complemented
2	#11717224	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#4050	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅×C₈×D₅	C₂	1	1	yes	maximal, characteristic, complemented
5	#11716947	C₂	1	2	yes	maximal, complemented
6	#11716926	C₂	1	1	yes	maximal, characteristic, complemented
7	#11716925	C₂	1	1	yes	maximal, characteristic, complemented
8	#11716924	C₂	1	2	yes	maximal, complemented
9	D₅×Dic₂₀	C₂	1	1	yes	maximal, characteristic, complemented
10	D₅×D₄₀	C₂	1	1	yes	maximal, characteristic, complemented
11	D₅×#677	C₂	1	2	yes	maximal, complemented
12	#87900	C₂²	1	2	yes	complemented
13	C₅×#709	C₂²	1	2	yes	complemented
14	C₂×C₄×C₅×D₅	C₂²	1	1	yes	characteristic
15	D₅×C₂²⋊D₅	—	2	2	no	—
16	#87879	—	2	2	no	—
17	D₅×D₂₀	C₂²	1	2	yes	—
18	C₄×D₅²	—	2	2	no	—
19	#87874	C₂²	1	2	yes	complemented
20	#87873	—	2	2	no	—
21	#87870	C₂²	1	2	yes	complemented
22	D₅×Dic₁₀	C₂²	1	2	yes	—
23	C₂×C₅²×C₈	C₂²	1	1	yes	characteristic, complemented
24	#87802	C₂²	1	1	yes	characteristic, complemented
25	#87801	C₂²	1	1	yes	characteristic, complemented
26	#87800	C₂²	1	2	yes	complemented
27	C₂×C₅×#168	C₂²	1	1	yes	characteristic, complemented
28	C₅×Dic₂₀	C₂²	1	1	yes	characteristic, complemented
29	C₅×D₄₀	C₂²	1	1	yes	characteristic, complemented
30	C₅×#677	C₂²	1	2	yes	complemented
31	C₅×C₈×D₅	C₂²	1	1	yes	characteristic
32	C₅×C₈×D₅	C₂²	1	1	yes	characteristic
33	C₅×C₈×D₅	C₂²	1	1	yes	characteristic, complemented
34	C₅×C₈×D₅	C₂²	1	1	yes	characteristic, complemented
35	#87776	C₂²	1	2	yes	complemented
36	#87774	C₂²	1	2	yes	complemented
37	#87773	C₂²	1	2	yes	complemented
38	#87771	C₂²	1	2	yes	complemented
39	D₅×#135	—	5	1	no	maximal
40	C₂×#4050	—	5	1	no	maximal
41	C₂²×C₅×D₅	D₄	1	1	yes	characteristic
42	C₂×D₅²	—	2	2	no	—
43	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
44	C₂²⋊(C₅⋊D₅)	—	2	2	no	—
45	C₄⋊(C₅⋊D₅)	C₂³	1	2	yes	—
46	C₄×C₅⋊D₅	—	2	2	no	—
47	#6045	C₂³	1	2	yes	—
48	C₅×C₂²⋊D₅	—	2	2	no	—
49	C₂×C₅×Dic₅	D₄	1	1	yes	characteristic
50	C₅×D₂₀	C₂³	1	2	yes	—
51	C₄×C₅×D₅	—	2	2	no	—
52	C₄×C₅×D₅	D₄	1	1	yes	characteristic
53	C₄×C₅×D₅	D₄	1	1	yes	characteristic
54	C₄×C₅×D₅	C₂³	1	1	yes	characteristic
55	C₄×C₅×D₅	C₂³	1	1	yes	characteristic
56	C₅×Dic₁₀	C₂³	1	2	yes	—
57	#6039	—	2	2	no	—
58	#6038	—	2	2	no	—
59	#6038	—	2	2	no	—
60	#6037	—	2	2	no	—
61	#6036	—	2	2	no	—
62	D₅×Dic₅	—	2	2	no	—
63	D₅×Dic₅	—	2	2	no	—
64	C₅²×C₈	C₂³	1	1	yes	characteristic
65	C₅²×C₈	C₂³	1	1	yes	characteristic
66	C₅×#168	C₂³	1	1	yes	characteristic
67	C₅×#168	C₂³	1	1	yes	characteristic
68	D₅×#41	—	5	2	no	—
69	C₂×#709	—	5	2	no	—
70	C₅×#135	—	5	1	no	—
71	C₂²×C₅×C₈	—	5	1	no	—
72	#4096	—	5	2	no	—
73	#4065	—	5	1	no	—
74	D₅×Q₁₆	—	5	1	no	—
75	#4062	—	5	2	no	—
76	D₅×SD₁₆	—	5	2	no	—
77	#4058	—	5	1	no	—
78	D₅×D₈	—	5	1	no	—
79	C₂×Dic₂₀	—	5	1	no	—
80	#4050	—	5	2	no	—
81	#4050	—	5	1	no	—
82	#4050	—	5	1	no	—
83	#4050	—	5	1	no	—
84	#4050	—	5	1	no	—
85	#4050	—	5	2	no	—
86	#4050	—	10	2	no	—
87	#4050	D₅	1	1	yes	characteristic, direct product factor
88	C₂×D₄₀	—	5	1	no	—
89	C₂×#677	—	5	2	no	—
90	C₂×C₈×D₅	D₅	1	1	yes	characteristic, complemented
91	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
92	C₂×C₅⋊D₅	—	2	2	no	—
93	C₂×C₅×D₅	—	2	2	no	—
94	C₂×C₅×D₅	—	2	2	no	—
95	C₂×C₅×D₅	C₂×D₄	1	1	yes	characteristic
96	C₂×C₅×D₅	C₂×D₄	1	1	yes	characteristic
97	D₅²	—	4	4	no	—
98	C₄×C₅²	C₂×D₄	1	1	yes	characteristic
99	C₄×C₅²	C₂⁴	1	1	yes	characteristic
100	#1039	—	2	2	no	—
101	C₅×Dic₅	—	2	2	no	—
102	C₅×Dic₅	C₂×D₄	1	1	yes	characteristic
103	C₅×Dic₅	C₂×D₄	1	1	yes	characteristic
104	C₅×#41	—	5	2	no	—
105	C₂²×C₄×C₅	—	5	1	no	—
106	C₂×C₂²⋊D₅	—	10	2	no	—
107	#713	—	5	2	no	—
108	Q₈×D₅	—	5	2	no	—
109	#711	—	5	2	no	—
110	#711	—	10	2	no	—
111	D₄×D₅	—	5	2	no	—
112	D₄×D₅	—	10	2	no	—
113	#709	—	5	2	no	—
114	#709	—	5	2	no	—
115	#709	—	5	2	no	—
116	#709	—	5	2	no	—
117	#709	—	10	4	no	—
118	#709	—	10	2	no	—
119	#709	—	10	2	no	—
120	#709	—	10	2	no	—
121	#709	C₂×D₅	1	2	yes	complemented
122	C₂×D₂₀	—	5	2	no	—
123	C₂×C₄×D₅	—	10	2	no	—
124	C₂×C₄×D₅	—	10	2	no	—
125	C₂×C₄×D₅	C₂×D₅	1	1	yes	characteristic
126	C₂×Dic₁₀	—	5	2	no	—
127	C₅×Q₁₆	—	5	1	no	—
128	C₅×SD₁₆	—	5	2	no	—
129	C₅×D₈	—	5	1	no	—
130	C₂×C₅×C₈	—	2	2	no	—
131	C₂×C₅×C₈	—	5	1	no	—
132	C₂×C₅×C₈	—	5	1	no	—
133	C₂×C₅×C₈	—	5	1	no	—
134	C₂×C₅×C₈	—	5	1	no	—
135	C₂×C₅×C₈	—	5	1	no	—
136	C₂×C₅×C₈	C₂×D₅	1	1	yes	characteristic, complemented
137	C₂×C₅×C₈	C₂×D₅	1	1	yes	characteristic, complemented
138	#689	—	5	2	no	—
139	#688	—	5	2	no	—
140	#687	—	5	2	no	—
141	#686	—	5	2	no	—
142	C₂×#168	C₂×D₅	1	1	yes	characteristic, complemented
143	Dic₂₀	—	5	2	no	—
144	Dic₂₀	—	5	1	no	—
145	Dic₂₀	—	5	1	no	—
146	Dic₂₀	—	10	2	no	—
147	Dic₂₀	C₂×D₅	1	1	yes	characteristic, complemented
148	D₄₀	—	5	2	no	—
149	D₄₀	—	5	1	no	—
150	D₄₀	—	5	1	no	—
151	D₄₀	—	10	2	no	—
152	D₄₀	C₂×D₅	1	1	yes	characteristic, complemented
153	#677	—	5	2	no	—
154	#677	—	5	2	no	—
155	#677	—	5	2	no	—
156	#677	—	5	2	no	—
157	#677	—	10	4	no	—
158	#677	C₂×D₅	1	2	yes	complemented
159	C₈×D₅	C₂×D₅	1	1	yes	characteristic
160	C₈×D₅	C₂×D₅	1	1	yes	characteristic
161	C₈×D₅	C₂×D₅	1	1	yes	characteristic, complemented
162	C₈×D₅	C₂×D₅	1	1	yes	characteristic, complemented
163	C₂×#135	—	25	1	no	Sylow
164	C₂×C₅²	—	2	1	no	—
165	C₂×C₅²	C₂²×D₄	1	1	yes	characteristic
166	C₅⋊D₅	—	4	2	no	—
167	C₅×D₅	—	2	1	no	—
168	C₅×D₅	—	4	2	no	—
169	C₅×D₅	#135	1	2	yes	complemented
170	C₂³×C₅	—	5	1	no	—
171	C₂²×D₅	—	10	2	no	—
172	C₂²×D₅	—	10	2	no	—
173	C₂²×D₅	D₂₀	1	1	yes	characteristic
174	C₅×Q₈	—	5	2	no	—
175	C₅×D₄	—	5	2	no	—
176	C₅×D₄	—	10	2	no	—
177	C₂×C₄×C₅	—	2	2	no	—
178	C₂×C₄×C₅	—	5	1	no	—
179	C₂×C₄×C₅	—	5	1	no	—
180	C₂×C₄×C₅	—	5	1	no	—
181	C₂×C₄×C₅	—	5	1	no	—
182	C₂×C₄×C₅	—	5	1	no	—
183	C₂×C₄×C₅	—	10	2	no	—
184	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
185	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
186	C₂²⋊D₅	—	2	2	no	—
187	C₂²⋊D₅	—	10	2	no	—
188	C₂²⋊D₅	—	10	2	no	—
189	C₂²⋊D₅	—	10	2	no	—
190	C₂²⋊D₅	—	10	2	no	—
191	C₂²⋊D₅	—	10	2	no	—
192	C₂²⋊D₅	—	10	2	no	—
193	C₂²⋊D₅	—	20	4	no	—
194	C₂×Dic₅	—	10	2	no	—
195	C₂×Dic₅	—	10	2	no	—
196	C₂×Dic₅	D₂₀	1	1	yes	characteristic
197	D₂₀	—	5	2	no	—
198	D₂₀	—	5	2	no	—
199	D₂₀	—	10	4	no	—
200	D₂₀	—	10	2	no	—
201	D₂₀	—	10	2	no	—
202	D₂₀	C₂²×D₅	1	2	yes	—
203	C₄×D₅	—	2	2	no	—
204	C₄×D₅	—	10	2	no	—
205	C₄×D₅	—	10	2	no	—
206	C₄×D₅	—	10	2	no	—
207	C₄×D₅	—	10	2	no	—
208	C₄×D₅	—	10	2	no	—
209	C₄×D₅	—	10	2	no	—
210	C₄×D₅	—	20	4	no	—
211	C₄×D₅	D₂₀	1	1	yes	characteristic
212	C₄×D₅	D₂₀	1	1	yes	characteristic
213	C₄×D₅	C₂²×D₅	1	1	yes	characteristic
214	C₄×D₅	C₂²×D₅	1	1	yes	characteristic
215	Dic₁₀	—	5	2	no	—
216	Dic₁₀	—	5	2	no	—
217	Dic₁₀	—	10	4	no	—
218	Dic₁₀	—	10	2	no	—
219	Dic₁₀	—	10	2	no	—
220	Dic₁₀	C₂²×D₅	1	2	yes	—
221	C₅×C₈	—	2	2	no	—
222	C₅×C₈	—	2	2	no	—
223	C₅×C₈	—	5	1	no	—
224	C₅×C₈	—	5	1	no	—
225	C₅×C₈	C₂²×D₅	1	1	yes	characteristic
226	C₅×C₈	C₂²×D₅	1	1	yes	characteristic
227	C₅×C₈	C₂²×D₅	1	1	yes	characteristic
228	C₅×C₈	C₂²×D₅	1	1	yes	characteristic
229	#168	C₂²×D₅	1	1	yes	characteristic
230	#168	C₂²×D₅	1	1	yes	characteristic
231	C₂×#41	—	25	2	no	—
232	#135	—	5	1	no	—
233	#135	—	25	1	no	—
234	#135	—	25	2	no	—
235	#135	—	25	1	no	—
236	#135	—	25	2	no	—
237	#135	—	25	1	no	—
238	C₂×Q₁₆	—	25	1	no	—
239	C₂×SD₁₆	—	25	2	no	—
240	C₂×D₈	—	25	1	no	—
241	C₂²×C₈	—	5	1	no	—
242	C₅²	C₂×#135	1	1	yes	characteristic, complemented, Sylow
243	C₂²×C₅	—	2	2	no	—
244	C₂²×C₅	—	5	1	no	—
245	C₂²×C₅	—	5	1	no	—
246	C₂²×C₅	—	10	2	no	—
247	C₂²×C₅	—	10	2	no	—
248	C₂²×C₅	C₂×D₂₀	1	1	yes	characteristic
249	C₂²×C₅	D₄×D₅	1	1	yes	characteristic
250	C₂×D₅	—	2	2	no	—
251	C₂×D₅	—	2	2	no	—
252	C₂×D₅	—	10	2	no	—
253	C₂×D₅	—	10	2	no	—
254	C₂×D₅	—	20	4	no	—
255	C₂×D₅	—	20	4	no	—
256	C₂×D₅	—	20	4	no	—
257	C₂×D₅	C₂×D₂₀	1	1	yes	characteristic
258	C₂×D₅	C₂×D₂₀	1	1	yes	characteristic
259	C₄×C₅	—	2	2	no	—
260	C₄×C₅	—	2	2	no	—
261	C₄×C₅	—	5	1	no	—
262	C₄×C₅	—	5	1	no	—
263	C₄×C₅	—	10	2	no	—
264	C₄×C₅	C₂×D₂₀	1	1	yes	characteristic
265	C₄×C₅	D₄×D₅	1	1	yes	characteristic
266	C₄×C₅	C₂³×D₅	1	1	yes	characteristic
267	C₄×C₅	C₂³×D₅	1	1	yes	characteristic
268	Dic₅	—	2	2	no	—
269	Dic₅	—	10	2	no	—
270	Dic₅	—	10	2	no	—
271	Dic₅	—	20	4	no	—
272	Dic₅	C₂×D₂₀	1	1	yes	characteristic
273	Dic₅	C₂×D₂₀	1	1	yes	characteristic
274	#41	—	5	2	no	—
275	#41	—	25	2	no	—
276	#41	—	25	2	no	—
277	#41	—	25	2	no	—
278	#41	—	50	2	no	—
279	#41	—	50	2	no	—
280	C₂×Q₈	—	25	2	no	—
281	C₂×D₄	—	25	2	no	—
282	C₂×D₄	—	50	2	no	—
283	C₂²×C₄	—	5	1	no	—
284	C₂²×C₄	—	50	2	no	—
285	Q₁₆	—	5	1	no	—
286	Q₁₆	—	25	2	no	—
287	Q₁₆	—	25	1	no	—
288	SD₁₆	—	5	2	no	—
289	SD₁₆	—	25	2	no	—
290	SD₁₆	—	25	2	no	—
291	SD₁₆	—	25	2	no	—
292	D₈	—	5	1	no	—
293	D₈	—	25	2	no	—
294	D₈	—	25	1	no	—
295	C₂×C₈	—	5	1	no	—
296	C₂×C₈	—	5	1	no	—
297	C₂×C₈	—	5	1	no	—
298	C₂×C₈	—	5	1	no	—
299	C₂×C₈	—	5	1	no	—
300	C₂×C₈	D₅²	1	1	yes	characteristic, complemented
301	C₂×C₅	—	2	1	no	—
302	C₂×C₅	—	2	1	no	—
303	C₂×C₅	—	2	2	no	—
304	C₂×C₅	—	4	2	no	—
305	C₂×C₅	—	5	2	no	—
306	C₂×C₅	—	10	1	no	—
307	C₂×C₅	—	20	2	no	—
308	C₂×C₅	C₂²×D₂₀	1	1	yes	characteristic
309	C₂×C₅	C₂×D₄×D₅	1	1	yes	characteristic
310	D₅	—	2	1	no	—
311	D₅	—	4	2	no	—
312	D₅	—	20	2	no	—
313	D₅	—	20	2	no	—
314	D₅	—	40	4	no	—
315	D₅	#4050	1	2	yes	direct product factor
316	C₂³	—	5	1	no	—
317	C₂³	—	50	2	no	—
318	Q₈	—	5	2	no	—
319	Q₈	—	25	2	no	—
320	Q₈	—	50	2	no	—
321	D₄	—	5	2	no	—
322	D₄	—	10	2	no	—
323	D₄	—	25	2	no	—
324	D₄	—	50	2	no	—
325	D₄	—	50	2	no	—
326	D₄	—	50	2	no	—
327	D₄	—	50	2	no	—
328	C₂×C₄	—	5	1	no	—
329	C₂×C₄	—	5	1	no	—
330	C₂×C₄	—	5	1	no	—
331	C₂×C₄	—	5	1	no	—
332	C₂×C₄	—	5	1	no	—
333	C₂×C₄	—	10	2	no	—
334	C₂×C₄	—	50	2	no	—
335	C₂×C₄	—	50	2	no	—
336	C₂×C₄	—	50	2	no	—
337	C₂×C₄	—	50	2	no	—
338	C₂×C₄	C₂×D₅²	1	1	yes	characteristic
339	C₈	—	5	1	no	—
340	C₈	—	5	1	no	—
341	C₈	C₂×D₅²	1	1	yes	characteristic
342	C₈	C₂×D₅²	1	1	yes	characteristic
343	C₅	—	2	2	no	—
344	C₅	C₂×#4050	1	1	yes	characteristic, complemented
345	C₅	D₅×#135	1	1	yes	characteristic, complemented
346	C₂²	—	5	1	no	—
347	C₂²	—	5	1	no	—
348	C₂²	—	10	2	no	—
349	C₂²	—	10	2	no	—
350	C₂²	—	50	2	no	—
351	C₂²	—	100	4	no	—
352	C₂²	D₅×D₂₀	1	1	yes	characteristic
353	C₄	—	5	1	no	—
354	C₄	—	5	1	no	—
355	C₄	—	10	2	no	—
356	C₄	—	50	2	no	—
357	C₄	D₅×D₂₀	1	1	yes	characteristic
358	C₄	C₂²×D₅²	1	1	yes	characteristic
359	C₂	—	2	1	no	—
360	C₂	—	5	2	no	—
361	C₂	—	10	1	no	—
362	C₂	—	20	2	no	—
363	C₂	—	100	2	no	—
364	C₂	C₂×D₅×D₂₀	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₅×D₂₀
commutator, cosocle	C₄×C₅²	C₂⁴
Frattini	C₄	C₂²×D₅²
Fitting	C₂×C₅²×C₈	C₂²
socle	C₂×C₅²	C₂²×D₄
hypercenter	C₂×C₈	D₅²
hypocenter	C₅²	C₂×#135
LCS	G ▷ C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
1. D₅×#709	C₂	semidirect
2. #11717224	C₂	semidirect
3. C₅×#4050	C₂	semidirect
4. C₂×C₅×C₈×D₅	C₂	semidirect
5. #11716947	C₂	semidirect
6. #11716926	C₂	semidirect
7. #11716925	C₂	semidirect
8. #11716924	C₂	semidirect
9. D₅×Dic₂₀	C₂	semidirect
10. D₅×D₄₀	C₂	semidirect
11. D₅×#677	C₂	semidirect
12. #87900	C₂²	semidirect
13. C₅×#709	C₂²	semidirect
19. #87874	C₂²	semidirect
21. #87870	C₂²	semidirect
23. C₂×C₅²×C₈	C₂²	semidirect
24. #87802	C₂²	semidirect
25. #87801	C₂²	semidirect
26. #87800	C₂²	semidirect
27. C₂×C₅×#168	C₂²	semidirect
28. C₅×Dic₂₀	C₂²	semidirect
29. C₅×D₄₀	C₂²	semidirect
30. C₅×#677	C₂²	semidirect
33. C₅×C₈×D₅	C₂²	semidirect
34. C₅×C₈×D₅	C₂²	semidirect
35. #87776	C₂²	semidirect
36. #87774	C₂²	semidirect
37. #87773	C₂²	semidirect
38. #87771	C₂²	semidirect
87. #4050	D₅	direct (N × H)
90. C₂×C₈×D₅	D₅	semidirect
121. #709	C₂×D₅	semidirect
136. C₂×C₅×C₈	C₂×D₅	semidirect
137. C₂×C₅×C₈	C₂×D₅	semidirect
142. C₂×#168	C₂×D₅	semidirect
147. Dic₂₀	C₂×D₅	semidirect
152. D₄₀	C₂×D₅	semidirect
158. #677	C₂×D₅	semidirect
161. C₈×D₅	C₂×D₅	semidirect
162. C₈×D₅	C₂×D₅	semidirect
169. C₅×D₅	#135	semidirect
242. C₅²	C₂×#135	semidirect
300. C₂×C₈	D₅²	semidirect
315. D₅	#4050	direct (N × H)
344. C₅	C₂×#4050	semidirect
345. C₅	D₅×#135	semidirect

Maximal subgroups

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

subgroup	index
D₅×#709	2
#11717224	2
C₅×#4050	2
C₂×C₅×C₈×D₅	2
#11716947	2
#11716926	2
#11716925	2
#11716924	2
D₅×Dic₂₀	2
D₅×D₄₀	2
D₅×#677	2
D₅×#135	5
C₂×#4050	5

Subgroups — D5×#4050 #13105766

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₅×#4050 #13105766