Subgroups — D₅×#4481 #14284635

Order 1920.

512 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₅×#869	C₂	1	1	yes	maximal, characteristic
2	C₃×D₅×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
3	D₅×C₄⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
4	#11746302	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#4481	C₂	1	1	yes	maximal, characteristic, complemented
6	#11745863	C₂	1	1	yes	maximal, characteristic, complemented
7	#11745862	C₂	1	1	yes	maximal, characteristic, complemented
8	D₅×#333	—	3	1	no	maximal, Hall
9	C₂×D₅×Dic₆	C₄	1	2	yes	—
10	#91758	C₂²	1	1	yes	characteristic
11	C₅×#869	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₈×D₅	—	2	1	no	—
13	C₃×C₄²×D₅	C₂²	1	1	yes	characteristic
14	#91468	—	2	1	no	—
15	#91468	—	2	1	no	—
16	D₅×C₄⋊Dic₃	—	2	1	no	—
17	D₅×C₄⋊Dic₃	—	2	1	no	—
18	#91376	C₄	1	2	yes	—
19	C₂×D₅×C₃⋊C₈	—	2	1	no	—
20	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
21	C₄⋊(C₃⋊#168)	C₂²	1	1	yes	characteristic
22	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
23	C₃×#3946	—	2	1	no	—
24	C₃×C₄⋊#168	C₂²	1	1	yes	characteristic
25	#90982	—	2	1	no	—
26	C₂×#4481	—	5	1	no	maximal, Hall
27	D₅×C₄⋊Q₈	—	3	1	no	—
28	D₅×C₄⋊C₈	—	3	1	no	—
29	D₅×C₄⋊C₈	D₃	1	1	yes	characteristic
30	C₅×#333	—	3	1	no	—
31	#64766	—	3	1	no	—
32	#64701	—	3	1	no	—
33	#64701	—	3	1	no	—
34	C₂×Dic₃₀	C₂×C₄	1	2	yes	—
35	C₂×C₅×Dic₆	C₂×C₄	1	2	yes	—
36	C₂×C₃×C₄×D₅	C₂×C₄	1	1	yes	characteristic
37	C₂×C₃×C₄×D₅	D₄	1	1	yes	characteristic
38	C₂×C₃×C₄×D₅	D₄	1	1	yes	characteristic
39	C₂×#1204	—	2	2	no	—
40	C₂×D₅×Dic₃	—	2	2	no	—
41	D₅×Dic₆	—	4	4	no	—
42	C₂×C₃×C₅×C₈	—	2	1	no	—
43	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
44	C₄⋊Dic₁₅	—	2	1	no	—
45	C₄⋊Dic₁₅	—	2	1	no	—
46	C₂×C₃⋊#168	—	2	1	no	—
47	C₅×C₄⋊Dic₃	—	2	1	no	—
48	C₅×C₄⋊Dic₃	—	2	1	no	—
49	C₂×C₅×C₃⋊C₈	—	2	1	no	—
50	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
51	C₃×C₄×Dic₅	D₄	1	1	yes	characteristic
52	C₃×C₄×Dic₅	D₄	1	1	yes	characteristic
53	C₂×C₃×#168	—	2	1	no	—
54	C₃×C₈×D₅	—	4	2	no	—
55	#6668	—	2	2	no	—
56	#6668	—	2	2	no	—
57	D₅×C₃⋊C₈	—	4	2	no	—
58	C₂×#869	—	5	1	no	—
59	C₂×C₃×C₄⋊C₈	—	5	1	no	—
60	C₂×C₄⋊(C₃⋊C₈)	—	5	1	no	—
61	#4481	—	5	1	no	—
62	#4481	—	5	1	no	—
63	#4481	—	5	1	no	—
64	#4481	D₅	1	1	yes	characteristic, direct product factor
65	C₂×Q₈×D₅	—	3	2	no	—
66	C₅×C₄⋊Q₈	—	3	1	no	—
67	#4090	—	3	2	no	—
68	C₂×C₈×D₅	—	2	1	no	—
69	C₂×C₈×D₅	—	6	1	no	—
70	D₅×C₄⋊C₄	—	6	1	no	—
71	D₅×C₄⋊C₄	—	6	1	no	—
72	#4034	—	6	1	no	—
73	#4034	—	6	1	no	—
74	C₄²×D₅	C₂×D₃	1	1	yes	characteristic
75	#4015	—	3	1	no	—
76	C₅×C₄⋊C₈	—	3	1	no	—
77	C₅×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
78	#3946	—	2	1	no	—
79	#3946	—	6	1	no	—
80	C₄⋊#168	—	3	1	no	—
81	C₄⋊#168	C₂×D₃	1	1	yes	characteristic
82	C₂×#333	—	15	1	no	Sylow
83	C₂²×C₃×D₅	C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
86	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
87	C₂×Dic₁₅	—	2	2	no	—
88	Dic₃₀	—	4	2	no	—
89	C₂×C₅×Dic₃	—	2	2	no	—
90	C₅×Dic₆	—	4	2	no	—
91	C₂×C₃×Dic₅	C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₃×Dic₅	C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₃×Dic₅	C₂²⋊C₄	1	1	yes	characteristic
94	C₃×C₄×D₅	—	2	2	no	—
95	C₃×C₄×D₅	SD₁₆	1	2	yes	—
96	C₃×C₄×D₅	SD₁₆	1	2	yes	—
97	C₃×C₄×D₅	Q₁₆	1	2	yes	—
98	C₃×C₄×D₅	Q₁₆	1	2	yes	—
99	#1204	—	4	4	no	—
100	D₅×Dic₃	—	4	4	no	—
101	C₃×C₅×C₈	—	4	1	no	—
102	C₃⋊#168	—	4	1	no	—
103	C₃×#168	—	4	1	no	—
104	C₅×C₃⋊C₈	—	4	1	no	—
105	C₂²×Dic₆	—	5	2	no	—
106	C₂²×C₃×C₈	—	10	1	no	—
107	C₂×C₃×C₄²	—	5	1	no	—
108	C₂×C₄⋊Dic₃	—	10	1	no	—
109	C₂×C₄⋊Dic₃	—	10	1	no	—
110	C₂²×C₃⋊C₈	—	10	1	no	—
111	#869	—	5	2	no	—
112	#869	—	5	1	no	—
113	#869	—	10	1	no	—
114	#869	—	10	1	no	—
115	#869	C₂×D₅	1	1	yes	characteristic
116	C₃×C₄⋊C₈	—	5	1	no	—
117	C₃×C₄⋊C₈	—	10	1	no	—
118	C₃×C₄⋊C₈	C₂×D₅	1	1	yes	characteristic
119	C₄⋊(C₃⋊C₈)	—	5	1	no	—
120	C₄⋊(C₃⋊C₈)	—	10	1	no	—
121	C₄⋊(C₃⋊C₈)	C₂×D₅	1	1	yes	characteristic
122	C₂×C₅×Q₈	—	3	2	no	—
123	Q₈×D₅	—	12	4	no	—
124	C₂×C₄×D₅	—	6	2	no	—
125	C₂×C₄×D₅	C₄×D₃	1	1	yes	characteristic
126	C₂×C₄×D₅	D₁₂	1	1	yes	characteristic
127	C₂×C₄×D₅	#67	1	1	yes	characteristic
128	C₂×Dic₁₀	—	3	2	no	—
129	C₂×Dic₁₀	—	6	2	no	—
130	C₂×C₅×C₈	—	2	1	no	—
131	C₂×C₅×C₈	—	6	1	no	—
132	C₅×C₄⋊C₄	—	6	1	no	—
133	C₅×C₄⋊C₄	—	6	1	no	—
134	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
135	C₄⋊Dic₅	—	6	1	no	—
136	C₄⋊Dic₅	—	6	1	no	—
137	#683	—	6	2	no	—
138	#683	—	6	2	no	—
139	C₄×Dic₅	C₄×D₃	1	1	yes	characteristic
140	C₄×Dic₅	D₁₂	1	1	yes	characteristic
141	C₄×Dic₅	#67	1	1	yes	characteristic
142	C₂×#168	—	2	1	no	—
143	C₂×#168	—	6	1	no	—
144	C₈×D₅	—	4	2	no	—
145	C₈×D₅	—	12	2	no	—
146	C₂×C₄⋊Q₈	—	15	1	no	—
147	C₂×C₄⋊C₈	—	5	1	no	—
148	C₂×C₄⋊C₈	—	15	1	no	—
149	#333	—	3	1	no	—
150	#333	—	15	1	no	—
151	#333	—	15	1	no	—
152	#333	—	15	1	no	—
153	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
154	C₂×C₃×D₅	#100	1	2	yes	—
155	C₂×C₃×D₅	#103	1	2	yes	—
156	C₂×C₃×D₅	#103	1	2	yes	—
157	C₃×C₄×C₅	—	2	1	no	—
158	C₃×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
159	C₃×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
160	C₃×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
161	C₃×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
162	Dic₁₅	—	4	2	no	—
163	C₃×Dic₅	—	2	1	no	—
164	C₃×Dic₅	#103	1	2	yes	—
165	C₃×Dic₅	#103	1	2	yes	—
166	C₅×Dic₃	—	4	2	no	—
167	C₂²×C₃×C₄	—	5	1	no	—
168	C₂²×C₃×C₄	—	5	1	no	—
169	C₂²×C₃×C₄	—	5	1	no	—
170	C₂²×Dic₃	—	10	2	no	—
171	C₂×Dic₆	—	5	2	no	—
172	C₂×Dic₆	—	10	2	no	—
173	C₂×Dic₆	—	20	4	no	—
174	C₂×Dic₆	C₄×D₅	1	2	yes	—
175	C₂×C₃×C₈	—	2	1	no	—
176	C₂×C₃×C₈	—	10	1	no	—
177	C₂×C₃×C₈	—	20	2	no	—
178	C₃×C₄²	—	5	1	no	—
179	C₃×C₄²	—	5	1	no	—
180	C₃×C₄²	—	5	1	no	—
181	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
182	C₄⋊Dic₃	—	2	1	no	—
183	C₄⋊Dic₃	—	2	1	no	—
184	C₄⋊Dic₃	—	10	2	no	—
185	C₄⋊Dic₃	—	10	2	no	—
186	C₄⋊Dic₃	—	10	1	no	—
187	C₄⋊Dic₃	—	10	1	no	—
188	C₂×C₃⋊C₈	—	2	1	no	—
189	C₂×C₃⋊C₈	—	10	1	no	—
190	C₂×C₃⋊C₈	—	20	2	no	—
191	C₂²×D₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
192	C₅×Q₈	—	12	2	no	—
193	C₂×C₄×C₅	—	6	2	no	—
194	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
195	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
196	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
197	C₂×Dic₅	—	6	2	no	—
198	C₂×Dic₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
199	C₂×Dic₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
200	C₂×Dic₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
201	C₄×D₅	—	2	2	no	—
202	C₄×D₅	—	12	4	no	—
203	C₄×D₅	#204	1	2	yes	—
204	C₄×D₅	Dic₁₂	1	2	yes	—
205	C₄×D₅	#214	1	2	yes	—
206	C₄×D₅	#216	1	2	yes	—
207	Dic₁₀	—	12	4	no	—
208	Dic₁₀	—	12	2	no	—
209	C₅×C₈	—	4	1	no	—
210	C₅×C₈	—	12	1	no	—
211	#168	—	4	1	no	—
212	#168	—	12	1	no	—
213	C₂²×Q₈	—	15	2	no	—
214	C₂²×C₈	—	10	1	no	—
215	C₂²×C₈	—	30	1	no	—
216	C₄⋊Q₈	—	3	1	no	—
217	C₄⋊Q₈	—	15	2	no	—
218	C₄⋊Q₈	—	15	1	no	—
219	C₄⋊Q₈	—	30	1	no	—
220	C₄⋊Q₈	—	30	1	no	—
221	C₂×C₄⋊C₄	—	30	1	no	—
222	C₂×C₄⋊C₄	—	30	1	no	—
223	C₂×C₄²	—	5	1	no	—
224	C₄⋊C₈	—	3	1	no	—
225	C₄⋊C₈	—	5	1	no	—
226	C₄⋊C₈	—	10	1	no	—
227	C₄⋊C₈	—	15	1	no	—
228	C₄⋊C₈	—	30	1	no	—
229	C₄⋊C₈	D₃×D₅	1	1	yes	characteristic
230	C₂×C₃×C₅	C₂×#100	1	1	yes	characteristic
231	C₂×C₃×C₅	C₂×#103	1	1	yes	characteristic
232	C₂×C₃×C₅	C₂×#103	1	1	yes	characteristic
233	C₃×D₅	#333	1	4	yes	complemented
234	C₂³×C₃	—	5	1	no	—
235	C₂×C₃×C₄	—	5	1	no	—
236	C₂×C₃×C₄	—	5	1	no	—
237	C₂×C₃×C₄	—	5	2	no	—
238	C₂×C₃×C₄	—	5	2	no	—
239	C₂×C₃×C₄	—	5	2	no	—
240	C₂×C₃×C₄	—	5	2	no	—
241	C₂×C₃×C₄	—	5	1	no	—
242	C₂×C₃×C₄	—	10	2	no	—
243	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
244	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
245	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
246	C₂×Dic₃	—	2	2	no	—
247	C₂×Dic₃	—	10	2	no	—
248	C₂×Dic₃	—	20	4	no	—
249	Dic₆	—	4	2	no	—
250	Dic₆	—	20	4	no	—
251	Dic₆	—	20	2	no	—
252	C₃×C₈	—	4	1	no	—
253	C₃×C₈	—	20	1	no	—
254	C₃⋊C₈	—	4	1	no	—
255	C₃⋊C₈	—	20	1	no	—
256	C₂²×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
257	C₂×D₅	#810	1	2	yes	—
258	C₂×D₅	#816	1	2	yes	—
259	C₂×D₅	#823	1	2	yes	—
260	C₄×C₅	—	2	1	no	—
261	C₄×C₅	—	12	2	no	—
262	C₄×C₅	C₂×#204	1	1	yes	characteristic
263	C₄×C₅	C₂×Dic₁₂	1	1	yes	characteristic
264	C₄×C₅	C₂×#214	1	1	yes	characteristic
265	C₄×C₅	C₂×#216	1	1	yes	characteristic
266	Dic₅	—	2	1	no	—
267	Dic₅	—	12	2	no	—
268	Dic₅	#810	1	2	yes	—
269	Dic₅	#816	1	2	yes	—
270	C₂×Q₈	—	3	2	no	—
271	C₂×Q₈	—	15	2	no	—
272	C₂×Q₈	—	30	2	no	—
273	C₂×Q₈	—	60	4	no	—
274	C₂²×C₄	—	5	1	no	—
275	C₂²×C₄	—	5	1	no	—
276	C₂²×C₄	—	5	1	no	—
277	C₂²×C₄	—	30	2	no	—
278	C₂×C₈	—	2	1	no	—
279	C₂×C₈	—	6	1	no	—
280	C₂×C₈	—	10	1	no	—
281	C₂×C₈	—	20	2	no	—
282	C₂×C₈	—	30	1	no	—
283	C₂×C₈	—	60	2	no	—
284	C₄⋊C₄	—	6	1	no	—
285	C₄⋊C₄	—	6	1	no	—
286	C₄⋊C₄	—	30	2	no	—
287	C₄⋊C₄	—	30	2	no	—
288	C₄⋊C₄	—	30	1	no	—
289	C₄⋊C₄	—	30	1	no	—
290	C₄²	—	5	1	no	—
291	C₄²	—	5	1	no	—
292	C₄²	—	5	1	no	—
293	C₄²	C₂×D₃×D₅	1	1	yes	characteristic
294	C₃×C₅	C₂×#333	1	1	yes	characteristic, complemented, Hall
295	C₂²×C₃	—	5	2	no	—
296	C₂²×C₃	—	5	2	no	—
297	C₂²×C₃	—	5	2	no	—
298	C₂²×C₃	D₅×C₂²⋊C₄	1	1	yes	characteristic
299	C₃×C₄	—	2	1	no	—
300	C₃×C₄	—	5	2	no	—
301	C₃×C₄	—	5	2	no	—
302	C₃×C₄	—	10	1	no	—
303	C₃×C₄	D₅×SD₁₆	1	1	yes	characteristic
304	C₃×C₄	D₅×SD₁₆	1	1	yes	characteristic
305	C₃×C₄	D₅×Q₁₆	1	1	yes	characteristic
306	C₃×C₄	D₅×Q₁₆	1	1	yes	characteristic
307	Dic₃	—	4	2	no	—
308	Dic₃	—	20	2	no	—
309	C₂×C₅	C₂×#810	1	1	yes	characteristic
310	C₂×C₅	C₂×#816	1	1	yes	characteristic
311	C₂×C₅	C₂×#823	1	1	yes	characteristic
312	D₅	#4481	1	4	yes	direct product factor
313	C₂³	—	5	1	no	—
314	Q₈	—	12	2	no	—
315	Q₈	—	60	2	no	—
316	Q₈	—	60	4	no	—
317	C₂×C₄	—	5	1	no	—
318	C₂×C₄	—	5	1	no	—
319	C₂×C₄	—	5	2	no	—
320	C₂×C₄	—	5	2	no	—
321	C₂×C₄	—	5	2	no	—
322	C₂×C₄	—	5	2	no	—
323	C₂×C₄	—	5	1	no	—
324	C₂×C₄	—	6	2	no	—
325	C₂×C₄	—	10	2	no	—
326	C₂×C₄	—	30	2	no	—
327	C₂×C₄	—	60	4	no	—
328	C₂×C₄	C₄×D₃×D₅	1	1	yes	characteristic
329	C₂×C₄	D₅×D₁₂	1	1	yes	characteristic
330	C₂×C₄	D₅×#67	1	1	yes	characteristic
331	C₈	—	4	1	no	—
332	C₈	—	12	1	no	—
333	C₈	—	20	1	no	—
334	C₈	—	60	1	no	—
335	C₂×C₃	—	5	4	no	—
336	C₂×C₃	D₅×#100	1	1	yes	characteristic
337	C₂×C₃	D₅×#103	1	1	yes	characteristic
338	C₂×C₃	D₅×#103	1	1	yes	characteristic
339	C₅	C₂×#4481	1	1	yes	characteristic, complemented, Sylow
340	C₂²	—	5	2	no	—
341	C₂²	—	5	2	no	—
342	C₂²	—	5	2	no	—
343	C₂²	D₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic
344	C₄	—	2	1	no	—
345	C₄	—	5	2	no	—
346	C₄	—	5	2	no	—
347	C₄	—	10	1	no	—
348	C₄	—	12	2	no	—
349	C₄	—	60	2	no	—
350	C₄	D₅×#204	1	1	yes	characteristic
351	C₄	D₅×Dic₁₂	1	1	yes	characteristic
352	C₄	D₅×#214	1	1	yes	characteristic
353	C₄	D₅×#216	1	1	yes	characteristic
354	C₃	D₅×#333	1	1	yes	characteristic, complemented, Sylow
355	C₂	—	5	4	no	—
356	C₂	D₅×#810	1	1	yes	characteristic
357	C₂	D₅×#816	1	1	yes	characteristic
358	C₂	D₅×#823	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₅×(C₂×D₃)⋊C₄
commutator	C₂×C₃×C₄×C₅	C₂²×C₄
Frattini	C₄²	C₂×D₃×D₅
Fitting	C₃×C₅×C₄⋊C₈	C₂²
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₂²×C₃×D₅	C₂²⋊C₄
cosocle	C₃×C₄²×C₅	C₂³
hypercenter	C₄⋊C₈	D₃×D₅
hypocenter	C₃×C₅	C₂×#333
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
4. #11746302	C₂	semidirect
5. C₅×#4481	C₂	semidirect
6. #11745863	C₂	semidirect
7. #11745862	C₂	semidirect
64. #4481	D₅	direct (N × H)
233. C₃×D₅	#333	semidirect
294. C₃×C₅	C₂×#333	semidirect
312. D₅	#4481	direct (N × H)
339. C₅	C₂×#4481	semidirect
354. C₃	D₅×#333	semidirect

Maximal subgroups

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

subgroup	index
D₅×#869	2
C₃×D₅×C₄⋊C₈	2
D₅×C₄⋊(C₃⋊C₈)	2
#11746302	2
C₅×#4481	2
#11745863	2
#11745862	2
D₅×#333	3
C₂×#4481	5

Subgroups — D5×#4481 #14284635

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₅×#4481 #14284635