Subgroups — D₅×#4018 #13105082

Order 1600.

736 conjugacy classes of subgroups.

The subgroup lattice has 339 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₅²	C₂	1	1	yes	maximal, characteristic
2	#11717192	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#4018	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄²×C₅×D₅	C₂	1	1	yes	maximal, characteristic, complemented
5	D₅×#685	C₂	1	2	yes	maximal
6	#11716995	C₂	1	2	yes	maximal, complemented
7	D₅×#683	C₂	1	2	yes	maximal, complemented
8	C₄×D₅×Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
9	#11716992	C₂	1	1	yes	maximal, characteristic, complemented
10	#11716973	C₂	1	1	yes	maximal, characteristic, complemented
11	#11716954	C₂	1	2	yes	maximal, complemented
12	C₂²×D₅²	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₅⋊D₅	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₅×D₅	C₂²	1	2	yes	—
15	C₂×C₄×C₅×D₅	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₅×D₅	C₂²	1	1	yes	characteristic
17	C₂×#6036	C₂²	1	1	yes	characteristic
18	C₂×D₅×Dic₅	C₂²	1	2	yes	—
19	C₂×D₅×Dic₅	C₂²	1	1	yes	characteristic
20	C₂×D₅×Dic₅	C₂²	1	1	yes	characteristic
21	C₄×D₅²	C₄	1	8	yes	complemented
22	C₄²×C₅²	C₂²	1	1	yes	characteristic, complemented
23	#87808	C₂²	1	2	yes	—
24	#87806	C₂²	1	2	yes	complemented
25	C₄×#1039	C₂²	1	1	yes	characteristic, complemented
26	C₅×#685	C₂²	1	2	yes	—
27	C₅×#683	C₂²	1	2	yes	complemented
28	C₄×C₅×Dic₅	C₂²	1	2	yes	complemented
29	C₄×C₅×Dic₅	C₂²	1	1	yes	characteristic, complemented
30	C₄×C₅×Dic₅	C₂²	1	1	yes	characteristic, complemented
31	#87781	C₂²	1	2	yes	complemented
32	#87780	C₂²	1	2	yes	complemented
33	#87779	C₂²	1	2	yes	—
34	#87778	C₂²	1	2	yes	—
35	Dic₅²	C₂²	1	2	yes	complemented
36	D₅×#117	—	5	1	no	maximal
37	C₂×#4018	—	5	1	no	maximal
38	C₂²×C₅⋊D₅	C₂³	1	1	yes	characteristic
39	C₂²×C₅×D₅	C₂³	1	1	yes	characteristic
40	C₂²×C₅×D₅	C₂³	1	1	yes	characteristic
41	C₂×D₅²	—	2	4	no	—
42	C₂×D₅²	C₂×C₄	1	4	yes	—
43	C₂×C₄×C₅²	C₂³	1	2	yes	—
44	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
45	C₂×#1039	C₂³	1	2	yes	—
46	C₂×#1039	C₂³	1	1	yes	characteristic
47	C₄×C₅⋊D₅	C₂×C₄	1	4	yes	complemented
48	C₂×C₅×Dic₅	C₂³	1	2	yes	—
49	C₂×C₅×Dic₅	C₂³	1	2	yes	—
50	C₂×C₅×Dic₅	C₂³	1	1	yes	characteristic
51	C₂×C₅×Dic₅	C₂³	1	1	yes	characteristic
52	C₄×C₅×D₅	—	2	4	no	—
53	C₄×C₅×D₅	C₂×C₄	1	4	yes	—
54	C₄×C₅×D₅	C₂×C₄	1	4	yes	complemented
55	#6036	C₂×C₄	1	4	yes	complemented
56	D₅×Dic₅	—	2	4	no	—
57	D₅×Dic₅	C₂×C₄	1	4	yes	—
58	D₅×Dic₅	C₂×C₄	1	4	yes	complemented
59	C₂²×C₄×D₅	—	5	1	no	—
60	C₂²×C₄×D₅	—	5	1	no	—
61	C₅×#117	—	5	1	no	—
62	C₂×C₄²×C₅	—	5	1	no	—
63	C₂×#685	—	5	2	no	—
64	#4072	—	5	1	no	—
65	C₂×#683	—	5	2	no	—
66	C₂×C₄×Dic₅	—	5	1	no	—
67	#4038	—	5	2	no	—
68	D₅×C₄⋊C₄	—	5	2	no	—
69	D₅×C₂²⋊C₄	—	5	2	no	—
70	#4023	—	5	2	no	—
71	#4018	—	5	1	no	—
72	#4018	—	5	1	no	—
73	#4018	—	5	1	no	—
74	#4018	—	5	2	no	—
75	#4018	—	5	2	no	—
76	#4018	—	5	1	no	—
77	#4018	—	5	1	no	—
78	#4018	—	10	2	no	—
79	#4018	D₅	1	1	yes	characteristic, direct product factor
80	C₄²×D₅	—	5	1	no	—
81	C₄²×D₅	D₅	1	1	yes	characteristic, complemented
82	C₂²×C₅²	C₂⁴	1	1	yes	characteristic
83	C₂×C₅⋊D₅	—	2	2	no	—
84	C₂×C₅⋊D₅	C₂²×C₄	1	2	yes	—
85	C₂×C₅×D₅	—	2	2	no	—
86	C₂×C₅×D₅	C₂²×C₄	1	2	yes	—
87	C₂×C₅×D₅	C₂²×C₄	1	2	yes	—
88	C₂×C₅×D₅	#41	1	4	yes	—
89	D₅²	—	2	8	no	—
90	C₄×C₅²	—	2	2	no	—
91	C₄×C₅²	C₂²×C₄	1	2	yes	—
92	#1039	—	2	2	no	—
93	#1039	C₂²×C₄	1	2	yes	—
94	C₅×Dic₅	—	2	2	no	—
95	C₅×Dic₅	—	2	2	no	—
96	C₅×Dic₅	C₂²×C₄	1	2	yes	—
97	C₅×Dic₅	C₂²×C₄	1	2	yes	—
98	C₂³×D₅	—	5	1	no	—
99	C₂³×D₅	—	5	1	no	—
100	C₂²×C₄×C₅	—	5	1	no	—
101	C₂²×C₄×C₅	—	5	1	no	—
102	C₂²×C₄×C₅	—	5	2	no	—
103	C₂²×Dic₅	—	5	1	no	—
104	C₂²×Dic₅	—	5	1	no	—
105	C₂²×Dic₅	—	5	2	no	—
106	C₂×C₄×D₅	—	5	1	no	—
107	C₂×C₄×D₅	—	5	1	no	—
108	C₂×C₄×D₅	—	5	1	no	—
109	C₂×C₄×D₅	—	5	1	no	—
110	C₂×C₄×D₅	—	5	1	no	—
111	C₂×C₄×D₅	—	5	1	no	—
112	C₂×C₄×D₅	—	5	8	no	—
113	C₂×C₄×D₅	—	5	8	no	—
114	C₂×C₄×D₅	—	5	2	no	—
115	C₂×C₄×D₅	—	10	2	no	—
116	C₂×C₄×D₅	C₂×D₅	1	2	yes	—
117	C₂×C₄×D₅	C₂×D₅	1	1	yes	characteristic
118	C₂×C₄×D₅	C₂×D₅	1	1	yes	characteristic
119	C₅×C₄⋊C₄	—	5	2	no	—
120	C₅×C₂²⋊C₄	—	5	2	no	—
121	C₄²×C₅	—	2	2	no	—
122	C₄²×C₅	—	5	2	no	—
123	C₄²×C₅	—	5	1	no	—
124	C₄²×C₅	—	5	1	no	—
125	C₄²×C₅	C₂×D₅	1	1	yes	characteristic, complemented
126	C₄²×C₅	C₂×D₅	1	1	yes	characteristic, complemented
127	C₂²⋊Dic₅	—	5	2	no	—
128	#685	—	5	2	no	—
129	#685	—	5	2	no	—
130	#685	—	5	2	no	—
131	#685	—	5	2	no	—
132	#685	—	5	2	no	—
133	#685	—	10	4	no	—
134	#685	C₂×D₅	1	2	yes	—
135	C₄⋊Dic₅	—	5	2	no	—
136	#683	—	5	2	no	—
137	#683	—	5	2	no	—
138	#683	—	5	2	no	—
139	#683	—	5	2	no	—
140	#683	—	5	2	no	—
141	#683	—	10	4	no	—
142	#683	C₂×D₅	1	2	yes	complemented
143	C₄×Dic₅	—	5	1	no	—
144	C₄×Dic₅	—	5	1	no	—
145	C₄×Dic₅	—	5	2	no	—
146	C₄×Dic₅	—	5	2	no	—
147	C₄×Dic₅	—	10	2	no	—
148	C₄×Dic₅	C₂×D₅	1	1	yes	characteristic, complemented
149	C₄×Dic₅	C₂×D₅	1	2	yes	complemented
150	C₄×Dic₅	C₂×D₅	1	1	yes	characteristic, complemented
151	C₂×#117	—	25	1	no	Sylow
152	C₂×C₅²	C₂³×C₄	1	1	yes	characteristic
153	C₂×C₅²	C₂×#41	1	2	yes	—
154	C₅⋊D₅	—	2	2	no	—
155	C₅×D₅	—	2	2	no	—
156	C₅×D₅	#117	1	4	yes	complemented
157	C₂³×C₅	—	5	1	no	—
158	C₂³×C₅	—	5	1	no	—
159	C₂²×D₅	—	5	1	no	—
160	C₂²×D₅	—	5	1	no	—
161	C₂²×D₅	—	5	4	no	—
162	C₂²×D₅	—	5	4	no	—
163	C₂²×D₅	—	10	2	no	—
164	C₂²×D₅	—	10	4	no	—
165	C₂²×D₅	—	10	4	no	—
166	C₂²×D₅	C₂²×D₅	1	1	yes	characteristic
167	C₂²×D₅	C₂²×D₅	1	1	yes	characteristic
168	C₂×C₄×C₅	—	2	4	no	—
169	C₂×C₄×C₅	—	2	2	no	—
170	C₂×C₄×C₅	—	5	1	no	—
171	C₂×C₄×C₅	—	5	1	no	—
172	C₂×C₄×C₅	—	5	4	no	—
173	C₂×C₄×C₅	—	5	4	no	—
174	C₂×C₄×C₅	—	5	2	no	—
175	C₂×C₄×C₅	—	5	2	no	—
176	C₂×C₄×C₅	—	10	4	no	—
177	C₂×C₄×C₅	C₂²×D₅	1	2	yes	—
178	C₂×C₄×C₅	C₂²×D₅	1	2	yes	—
179	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
180	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
181	C₂×Dic₅	—	5	1	no	—
182	C₂×Dic₅	—	5	1	no	—
183	C₂×Dic₅	—	5	4	no	—
184	C₂×Dic₅	—	5	4	no	—
185	C₂×Dic₅	—	5	2	no	—
186	C₂×Dic₅	—	5	2	no	—
187	C₂×Dic₅	—	10	2	no	—
188	C₂×Dic₅	—	10	4	no	—
189	C₂×Dic₅	—	10	4	no	—
190	C₂×Dic₅	C₂²×D₅	1	2	yes	—
191	C₂×Dic₅	C₂²×D₅	1	2	yes	—
192	C₂×Dic₅	C₂²×D₅	1	1	yes	characteristic
193	C₂×Dic₅	C₂²×D₅	1	1	yes	characteristic
194	C₄×D₅	—	2	4	no	—
195	C₄×D₅	—	5	4	no	—
196	C₄×D₅	—	5	4	no	—
197	C₄×D₅	—	5	4	no	—
198	C₄×D₅	—	5	4	no	—
199	C₄×D₅	—	5	4	no	—
200	C₄×D₅	—	5	4	no	—
201	C₄×D₅	—	10	8	no	—
202	C₄×D₅	—	10	4	no	—
203	C₄×D₅	C₄×D₅	1	4	yes	—
204	C₄×D₅	C₄×D₅	1	4	yes	complemented
205	C₂³×C₄	—	25	1	no	—
206	#117	—	5	1	no	—
207	#117	—	25	1	no	—
208	#117	—	25	2	no	—
209	#117	—	25	2	no	—
210	#117	—	25	1	no	—
211	#117	—	25	1	no	—
212	C₂×C₄⋊C₄	—	25	2	no	—
213	C₂×C₂²⋊C₄	—	25	2	no	—
214	C₂×C₄²	—	5	1	no	—
215	C₂×C₄²	—	25	1	no	—
216	C₅²	C₂×#117	1	1	yes	characteristic, complemented, Sylow
217	C₂²×C₅	—	2	2	no	—
218	C₂²×C₅	—	5	2	no	—
219	C₂²×C₅	—	5	4	no	—
220	C₂²×C₅	—	5	2	no	—
221	C₂²×C₅	—	10	2	no	—
222	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
223	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
224	C₂×D₅	—	2	2	no	—
225	C₂×D₅	—	5	2	no	—
226	C₂×D₅	—	5	2	no	—
227	C₂×D₅	—	10	4	no	—
228	C₂×D₅	—	10	8	no	—
229	C₂×D₅	—	10	2	no	—
230	C₂×D₅	—	10	2	no	—
231	C₂×D₅	—	10	8	no	—
232	C₂×D₅	—	20	4	no	—
233	C₂×D₅	C₂×C₄×D₅	1	2	yes	—
234	C₂×D₅	C₂×C₄×D₅	1	2	yes	—
235	C₂×D₅	#709	1	4	yes	—
236	C₄×C₅	—	2	2	no	—
237	C₄×C₅	—	2	2	no	—
238	C₄×C₅	—	2	4	no	—
239	C₄×C₅	—	4	4	no	—
240	C₄×C₅	—	5	2	no	—
241	C₄×C₅	—	5	2	no	—
242	C₄×C₅	—	10	2	no	—
243	C₄×C₅	—	10	2	no	—
244	C₄×C₅	C₂×C₄×D₅	1	2	yes	—
245	C₄×C₅	C₂×C₄×D₅	1	2	yes	—
246	Dic₅	—	2	2	no	—
247	Dic₅	—	2	2	no	—
248	Dic₅	—	5	2	no	—
249	Dic₅	—	5	2	no	—
250	Dic₅	—	10	4	no	—
251	Dic₅	—	10	2	no	—
252	Dic₅	—	10	2	no	—
253	Dic₅	—	20	4	no	—
254	Dic₅	C₂×C₄×D₅	1	2	yes	—
255	Dic₅	C₂×C₄×D₅	1	2	yes	—
256	C₂⁴	—	25	1	no	—
257	C₂²×C₄	—	5	1	no	—
258	C₂²×C₄	—	5	1	no	—
259	C₂²×C₄	—	5	2	no	—
260	C₂²×C₄	—	25	1	no	—
261	C₂²×C₄	—	25	8	no	—
262	C₂²×C₄	—	25	1	no	—
263	C₂²×C₄	—	25	1	no	—
264	C₂²×C₄	—	25	2	no	—
265	C₂²×C₄	—	25	1	no	—
266	C₄⋊C₄	—	5	2	no	—
267	C₄⋊C₄	—	25	2	no	—
268	C₄⋊C₄	—	25	2	no	—
269	C₄⋊C₄	—	25	2	no	—
270	C₂²⋊C₄	—	5	2	no	—
271	C₂²⋊C₄	—	25	2	no	—
272	C₂²⋊C₄	—	25	2	no	—
273	C₂²⋊C₄	—	25	2	no	—
274	C₄²	—	5	2	no	—
275	C₄²	—	5	1	no	—
276	C₄²	—	5	1	no	—
277	C₄²	—	25	2	no	—
278	C₄²	—	25	1	no	—
279	C₄²	D₅²	1	1	yes	characteristic, complemented
280	C₂×C₅	—	2	2	no	—
281	C₂×C₅	—	2	4	no	—
282	C₂×C₅	—	5	4	no	—
283	C₂×C₅	—	10	2	no	—
284	C₂×C₅	C₂²×C₄×D₅	1	1	yes	characteristic
285	C₂×C₅	C₂²×C₄×D₅	1	1	yes	characteristic
286	C₂×C₅	C₂×#709	1	2	yes	—
287	C₂×C₅	D₅×#41	1	2	yes	—
288	D₅	—	2	2	no	—
289	D₅	—	10	2	no	—
290	D₅	—	10	2	no	—
291	D₅	—	20	4	no	—
292	D₅	#4018	1	4	yes	direct product factor
293	C₂³	—	5	1	no	—
294	C₂³	—	5	1	no	—
295	C₂³	—	25	4	no	—
296	C₂³	—	25	1	no	—
297	C₂³	—	50	4	no	—
298	C₂×C₄	—	5	1	no	—
299	C₂×C₄	—	5	1	no	—
300	C₂×C₄	—	5	4	no	—
301	C₂×C₄	—	5	4	no	—
302	C₂×C₄	—	5	2	no	—
303	C₂×C₄	—	5	2	no	—
304	C₂×C₄	—	10	4	no	—
305	C₂×C₄	—	25	4	no	—
306	C₂×C₄	—	25	4	no	—
307	C₂×C₄	—	25	4	no	—
308	C₂×C₄	—	25	1	no	—
309	C₂×C₄	—	25	4	no	—
310	C₂×C₄	—	25	2	no	—
311	C₂×C₄	—	50	4	no	—
312	C₂×C₄	C₂×D₅²	1	2	yes	—
313	C₂×C₄	C₂×D₅²	1	1	yes	characteristic
314	C₅	—	2	2	no	—
315	C₅	C₂×#4018	1	1	yes	characteristic, complemented
316	C₅	D₅×#117	1	1	yes	characteristic, complemented
317	C₂²	—	5	2	no	—
318	C₂²	—	5	4	no	—
319	C₂²	—	5	2	no	—
320	C₂²	—	10	2	no	—
321	C₂²	—	25	2	no	—
322	C₂²	—	50	8	no	—
323	C₂²	—	50	2	no	—
324	C₂²	C₂²×D₅²	1	1	yes	characteristic
325	C₄	—	2	2	no	—
326	C₄	—	5	2	no	—
327	C₄	—	5	2	no	—
328	C₄	—	10	2	no	—
329	C₄	—	10	2	no	—
330	C₄	—	25	2	no	—
331	C₄	—	50	2	no	—
332	C₄	C₄×D₅²	1	2	yes	—
333	C₂	—	5	4	no	—
334	C₂	—	10	2	no	—
335	C₂	—	50	2	no	—
336	C₂	C₂×C₄×D₅²	1	1	yes	characteristic
337	C₂	D₅×#709	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₅²
commutator	C₂×C₅²	C₂³×C₄
Frattini	C₂²	C₂²×D₅²
Fitting	C₄²×C₅²	C₂²
socle, cosocle	C₂²×C₅²	C₂⁴
prime radical	C₂²×D₅²	C₂²
hypercenter	C₄²	D₅²
hypocenter	C₅²	C₂×#117
LCS	G ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂×C₄ ◁ C₄²

Complements

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

N	H ≅ G/N	type
2. #11717192	C₂	semidirect
3. C₅×#4018	C₂	semidirect
4. C₄²×C₅×D₅	C₂	semidirect
6. #11716995	C₂	semidirect
7. D₅×#683	C₂	semidirect
8. C₄×D₅×Dic₅	C₂	semidirect
9. #11716992	C₂	semidirect
10. #11716973	C₂	semidirect
11. #11716954	C₂	semidirect
21. C₄×D₅²	C₄	semidirect
22. C₄²×C₅²	C₂²	semidirect
24. #87806	C₂²	semidirect
25. C₄×#1039	C₂²	semidirect
27. C₅×#683	C₂²	semidirect
28. C₄×C₅×Dic₅	C₂²	semidirect
29. C₄×C₅×Dic₅	C₂²	semidirect
30. C₄×C₅×Dic₅	C₂²	semidirect
31. #87781	C₂²	semidirect
32. #87780	C₂²	semidirect
35. Dic₅²	C₂²	semidirect
47. C₄×C₅⋊D₅	C₂×C₄	semidirect
54. C₄×C₅×D₅	C₂×C₄	semidirect
55. #6036	C₂×C₄	semidirect
58. D₅×Dic₅	C₂×C₄	semidirect
79. #4018	D₅	direct (N × H)
81. C₄²×D₅	D₅	semidirect
125. C₄²×C₅	C₂×D₅	semidirect
126. C₄²×C₅	C₂×D₅	semidirect
142. #683	C₂×D₅	semidirect
148. C₄×Dic₅	C₂×D₅	semidirect
149. C₄×Dic₅	C₂×D₅	semidirect
150. C₄×Dic₅	C₂×D₅	semidirect
156. C₅×D₅	#117	semidirect
204. C₄×D₅	C₄×D₅	semidirect
216. C₅²	C₂×#117	semidirect
279. C₄²	D₅²	semidirect
292. D₅	#4018	direct (N × H)
315. C₅	C₂×#4018	semidirect
316. C₅	D₅×#117	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₅²	2
#11717192	2
C₅×#4018	2
C₄²×C₅×D₅	2
D₅×#685	2
#11716995	2
D₅×#683	2
C₄×D₅×Dic₅	2
#11716992	2
#11716973	2
#11716954	2
D₅×#117	5
C₂×#4018	5

Subgroups — D5×#4018 #13105082

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₅×#4018 #13105082