Subgroups — C₅×#81737 #14438004

Order 1920.

784 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#5725	C₂	1	1	yes	maximal, characteristic
2	C₅×#5688	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₅×#491	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#5082	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#5067	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#4804	C₂	1	4	yes	maximal, complemented
7	C₅×#4794	C₂	1	4	yes	maximal, complemented
8	C₅×#4713	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅×D₃×#97	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅×#3135	—	3	1	no	maximal, Hall
11	C₂×C₅×#239	C₂²	1	2	yes	—
12	C₃×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
13	C₃×C₅×#125	C₂²	1	1	yes	characteristic
14	C₅×#947	C₂²	1	2	yes	—
15	C₂×C₅×D₃×C₈	C₂²	1	2	yes	—
16	C₅×#896	—	2	2	no	—
17	C₅×#895	—	2	2	no	—
18	C₅×#893	—	2	2	no	—
19	C₅×#892	C₂²	1	2	yes	—
20	C₅×#888	C₂²	1	2	yes	—
21	C₅×#887	—	2	2	no	—
22	C₅×#874	C₂²	1	1	yes	characteristic
23	C₄²×C₅×D₃	C₂²	1	1	yes	characteristic
24	C₅×#870	C₂²	1	1	yes	characteristic
25	C₃×C₅×#103	C₂²	1	4	yes	—
26	C₃×C₅×#97	C₂²	1	1	yes	characteristic
27	C₅×#835	C₂²	1	4	yes	—
28	C₅×#821	C₂²	1	4	yes	—
29	C₅×#815	C₂²	1	2	yes	—
30	C₅×#809	C₂²	1	4	yes	—
31	C₅×#803	C₂²	1	1	yes	characteristic
32	#81737	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
33	C₅×#533	—	3	1	no	—
34	C₅×#532	—	3	1	no	—
35	C₅×#491	—	3	1	no	—
36	C₅×#491	D₃	1	1	yes	characteristic, complemented
37	C₅×#490	—	3	1	no	—
38	C₅×#490	—	3	1	no	—
39	C₅×#489	—	3	4	no	—
40	C₅×#416	—	3	4	no	—
41	C₂×C₅×#97	—	3	1	no	—
42	C₂×C₃×C₅×Q₈	C₂³	1	2	yes	—
43	C₅×#239	—	2	8	no	—
44	C₂×C₅×D₁₂	—	2	2	no	—
45	C₂×C₅×D₁₂	—	2	2	no	—
46	C₂×C₅×D₁₂	C₂³	1	2	yes	—
47	C₂×C₄×C₅×D₃	—	2	2	no	—
48	C₂×C₄×C₅×D₃	D₄	1	2	yes	—
49	C₂×C₄×C₅×D₃	C₂³	1	1	yes	characteristic
50	C₂×C₅×Dic₆	—	2	2	no	—
51	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
52	C₃×C₅×C₄⋊C₄	—	2	2	no	—
53	C₃×C₅×C₄⋊C₄	—	2	2	no	—
54	C₃×C₅×C₄⋊C₄	C₂³	1	2	yes	—
55	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
56	C₅×(C₂×D₃)⋊C₄	—	2	2	no	—
57	C₅×(C₂×D₃)⋊C₄	—	2	4	no	—
58	C₅×C₄⋊Dic₃	C₂³	1	2	yes	—
59	C₅×Dic₃⋊C₄	—	2	2	no	—
60	C₅×Dic₃⋊C₄	—	2	2	no	—
61	C₄×C₅×Dic₃	—	2	2	no	—
62	C₄×C₅×Dic₃	D₄	1	2	yes	—
63	C₄×C₅×Dic₃	C₂³	1	1	yes	characteristic
64	C₂×C₅×C₃⋊C₈	C₂³	1	2	yes	—
65	C₅×D₃×C₈	—	2	4	no	—
66	#5725	C₂×C₅	1	1	yes	characteristic
67	#5688	C₂×C₅	1	1	yes	characteristic, complemented
68	C₃×#491	C₂×C₅	1	1	yes	characteristic, complemented
69	#5082	C₂×C₅	1	1	yes	characteristic, complemented
70	#5067	C₂×C₅	1	1	yes	characteristic, complemented
71	#4804	C₂×C₅	1	4	yes	complemented
72	#4794	C₂×C₅	1	4	yes	complemented
73	#4713	C₂×C₅	1	1	yes	characteristic, complemented
74	D₃×#97	C₂×C₅	1	1	yes	characteristic, complemented
75	C₂×C₅×#41	—	3	2	no	—
76	C₂²×C₅×C₈	—	3	2	no	—
77	C₅×C₄⋊Q₈	—	3	2	no	—
78	C₅×C₄⋊Q₈	C₂×D₃	1	1	yes	characteristic
79	C₅×#127	—	3	1	no	—
80	C₅×#125	—	3	1	no	—
81	C₅×#125	C₂×D₃	1	1	yes	characteristic
82	C₅×#124	—	3	2	no	—
83	C₅×C₂²⋊Q₈	—	6	2	no	—
84	C₅×#121	—	6	2	no	—
85	C₄×C₅×Q₈	—	6	2	no	—
86	C₄×C₅×D₄	—	6	2	no	—
87	C₅×#117	—	3	2	no	—
88	C₂×C₄²×C₅	—	3	1	no	—
89	C₅×#103	—	3	4	no	—
90	C₅×#103	C₂×D₃	1	4	yes	—
91	C₅×#102	—	3	4	no	—
92	C₅×#102	—	3	4	no	—
93	C₅×#97	—	3	2	no	—
94	C₅×#97	—	3	1	no	—
95	C₅×#97	C₂×D₃	1	1	yes	characteristic
96	#3135	—	3	1	no	Sylow
97	C₂²×C₅×D₃	—	2	2	no	—
98	C₂²×C₅×D₃	C₂×D₄	1	1	yes	characteristic
99	C₃×C₅×Q₈	—	2	4	no	—
100	C₂×C₃×C₄×C₅	—	2	2	no	—
101	C₂×C₃×C₄×C₅	—	2	2	no	—
102	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
103	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
104	C₂×C₅×Dic₃	—	2	2	no	—
105	C₂×C₅×Dic₃	C₂×D₄	1	2	yes	—
106	C₂×C₅×Dic₃	C₂×D₄	1	1	yes	characteristic
107	C₅×D₁₂	—	2	4	no	—
108	C₅×D₁₂	—	4	4	no	—
109	C₅×D₁₂	—	4	4	no	—
110	C₄×C₅×D₃	—	4	2	no	—
111	C₄×C₅×D₃	—	4	4	no	—
112	C₄×C₅×D₃	#41	1	4	yes	—
113	C₅×Dic₆	—	4	4	no	—
114	C₃×C₅×C₈	—	2	2	no	—
115	C₅×C₃⋊C₈	—	2	2	no	—
116	C₂×#239	C₂²×C₅	1	2	yes	—
117	C₃×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
118	C₃×#125	C₂²×C₅	1	1	yes	characteristic
119	#947	C₂²×C₅	1	2	yes	—
120	C₂×D₃×C₈	C₂²×C₅	1	2	yes	—
121	#896	—	2	2	no	—
122	#895	—	2	2	no	—
123	#893	—	2	2	no	—
124	#892	C₂²×C₅	1	2	yes	—
125	#888	C₂²×C₅	1	2	yes	—
126	#887	—	2	2	no	—
127	#874	C₂²×C₅	1	1	yes	characteristic
128	C₄²×D₃	C₂²×C₅	1	1	yes	characteristic
129	#870	C₂²×C₅	1	1	yes	characteristic
130	C₃×#103	C₂²×C₅	1	4	yes	—
131	C₃×#97	C₂²×C₅	1	1	yes	characteristic
132	#835	C₂²×C₅	1	4	yes	—
133	#821	C₂²×C₅	1	4	yes	—
134	#815	C₂²×C₅	1	2	yes	—
135	#809	C₂²×C₅	1	4	yes	—
136	#803	C₂²×C₅	1	1	yes	characteristic
137	C₅×#41	—	6	8	no	—
138	C₂×C₅×Q₈	—	6	2	no	—
139	C₂×C₅×Q₈	C₂²×D₃	1	2	yes	—
140	C₂×C₅×D₄	—	3	2	no	—
141	C₂×C₅×D₄	—	6	2	no	—
142	C₂×C₅×D₄	—	6	2	no	—
143	C₂²×C₄×C₅	—	3	1	no	—
144	C₂²×C₄×C₅	—	3	2	no	—
145	C₂²×C₄×C₅	—	6	2	no	—
146	C₂×C₅×C₈	—	3	2	no	—
147	C₂×C₅×C₈	—	6	4	no	—
148	C₂×C₅×C₈	C₂²×D₃	1	2	yes	—
149	C₅×C₄⋊C₄	—	2	2	no	—
150	C₅×C₄⋊C₄	—	2	2	no	—
151	C₅×C₄⋊C₄	—	3	2	no	—
152	C₅×C₄⋊C₄	—	6	2	no	—
153	C₅×C₄⋊C₄	—	6	2	no	—
154	C₅×C₄⋊C₄	C₂²×D₃	1	2	yes	—
155	C₅×C₂²⋊C₄	—	6	4	no	—
156	C₅×C₂²⋊C₄	—	6	2	no	—
157	C₄²×C₅	—	3	2	no	—
158	C₄²×C₅	—	3	1	no	—
159	C₄²×C₅	—	6	2	no	—
160	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
161	#533	—	3	1	no	—
162	#532	—	3	1	no	—
163	#491	—	3	1	no	—
164	#491	C₅×D₃	1	1	yes	characteristic, complemented
165	#490	—	3	1	no	—
166	#490	—	3	1	no	—
167	#489	—	3	4	no	—
168	#416	—	3	4	no	—
169	C₂×#97	—	3	1	no	—
170	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
171	C₂×C₅×D₃	—	2	2	no	—
172	C₂×C₅×D₃	—	4	2	no	—
173	C₂×C₅×D₃	—	4	4	no	—
174	C₂×C₅×D₃	#124	1	2	yes	—
175	C₃×C₄×C₅	—	2	2	no	—
176	C₃×C₄×C₅	—	4	2	no	—
177	C₃×C₄×C₅	—	4	2	no	—
178	C₃×C₄×C₅	C₂×#41	1	2	yes	—
179	C₅×Dic₃	—	2	2	no	—
180	C₅×Dic₃	—	4	2	no	—
181	C₅×Dic₃	#124	1	2	yes	—
182	C₂×C₃×Q₈	C₂³×C₅	1	2	yes	—
183	#239	—	2	8	no	—
184	C₂×D₁₂	—	2	2	no	—
185	C₂×D₁₂	—	2	2	no	—
186	C₂×D₁₂	C₂³×C₅	1	2	yes	—
187	C₂×C₄×D₃	—	2	2	no	—
188	C₂×C₄×D₃	C₅×D₄	1	2	yes	—
189	C₂×C₄×D₃	C₂³×C₅	1	1	yes	characteristic
190	C₂×Dic₆	—	2	2	no	—
191	C₂×C₃×C₈	C₂³×C₅	1	2	yes	—
192	C₃×C₄⋊C₄	—	2	2	no	—
193	C₃×C₄⋊C₄	—	2	2	no	—
194	C₃×C₄⋊C₄	C₂³×C₅	1	2	yes	—
195	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
196	(C₂×D₃)⋊C₄	—	2	2	no	—
197	(C₂×D₃)⋊C₄	—	2	4	no	—
198	C₄⋊Dic₃	C₂³×C₅	1	2	yes	—
199	Dic₃⋊C₄	—	2	2	no	—
200	Dic₃⋊C₄	—	2	2	no	—
201	C₄×Dic₃	—	2	2	no	—
202	C₄×Dic₃	C₅×D₄	1	2	yes	—
203	C₄×Dic₃	C₂³×C₅	1	1	yes	characteristic
204	C₂×C₃⋊C₈	C₂³×C₅	1	2	yes	—
205	D₃×C₈	—	2	4	no	—
206	C₂³×C₅	—	3	1	no	—
207	C₂³×C₅	—	6	2	no	—
208	C₅×Q₈	—	2	4	no	—
209	C₅×Q₈	—	12	4	no	—
210	C₅×D₄	—	6	4	no	—
211	C₅×D₄	—	12	4	no	—
212	C₅×D₄	—	12	4	no	—
213	C₂×C₄×C₅	—	2	2	no	—
214	C₂×C₄×C₅	—	2	2	no	—
215	C₂×C₄×C₅	—	3	1	no	—
216	C₂×C₄×C₅	—	3	4	no	—
217	C₂×C₄×C₅	—	3	2	no	—
218	C₂×C₄×C₅	—	6	2	no	—
219	C₂×C₄×C₅	—	12	2	no	—
220	C₂×C₄×C₅	—	12	4	no	—
221	C₂×C₄×C₅	D₃×D₄	1	2	yes	—
222	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
223	C₅×C₈	—	2	2	no	—
224	C₅×C₈	—	6	2	no	—
225	C₂×#41	—	3	2	no	—
226	C₂²×C₈	—	3	2	no	—
227	C₄⋊Q₈	—	3	2	no	—
228	C₄⋊Q₈	C₂×C₅×D₃	1	1	yes	characteristic
229	#127	—	3	1	no	—
230	#125	—	3	1	no	—
231	#125	C₂×C₅×D₃	1	1	yes	characteristic
232	#124	—	3	2	no	—
233	C₂²⋊Q₈	—	6	2	no	—
234	#121	—	6	2	no	—
235	C₄×Q₈	—	6	2	no	—
236	C₄×D₄	—	6	2	no	—
237	#117	—	3	2	no	—
238	C₂×C₄²	—	3	1	no	—
239	#103	—	3	4	no	—
240	#103	C₂×C₅×D₃	1	4	yes	—
241	#102	—	3	4	no	—
242	#102	—	3	4	no	—
243	#97	—	3	2	no	—
244	#97	—	3	1	no	—
245	#97	C₂×C₅×D₃	1	1	yes	characteristic
246	C₂×C₃×C₅	C₂×#124	1	1	yes	characteristic
247	C₂×C₃×C₅	#575	1	2	yes	—
248	C₅×D₃	—	2	2	no	—
249	C₅×D₃	—	8	2	no	—
250	C₂²×D₃	—	2	2	no	—
251	C₂²×D₃	C₂×C₅×D₄	1	1	yes	characteristic
252	C₃×Q₈	—	2	4	no	—
253	C₂×C₃×C₄	—	2	2	no	—
254	C₂×C₃×C₄	—	2	2	no	—
255	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
256	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
257	C₂×Dic₃	—	2	2	no	—
258	C₂×Dic₃	C₂×C₅×D₄	1	2	yes	—
259	C₂×Dic₃	C₂×C₅×D₄	1	1	yes	characteristic
260	D₁₂	—	2	4	no	—
261	D₁₂	—	4	4	no	—
262	D₁₂	—	4	4	no	—
263	C₄×D₃	—	4	2	no	—
264	C₄×D₃	—	4	4	no	—
265	C₄×D₃	C₅×#41	1	4	yes	—
266	Dic₆	—	4	4	no	—
267	C₃×C₈	—	2	2	no	—
268	C₃⋊C₈	—	2	2	no	—
269	C₂²×C₅	—	3	2	no	—
270	C₂²×C₅	—	6	2	no	—
271	C₂²×C₅	—	12	2	no	—
272	C₂²×C₅	—	12	4	no	—
273	C₂²×C₅	C₂×D₃×D₄	1	1	yes	characteristic
274	C₄×C₅	—	2	2	no	—
275	C₄×C₅	—	3	2	no	—
276	C₄×C₅	—	4	2	no	—
277	C₄×C₅	—	4	2	no	—
278	C₄×C₅	—	6	2	no	—
279	C₄×C₅	—	12	2	no	—
280	C₄×C₅	D₃×#41	1	2	yes	—
281	#41	—	6	8	no	—
282	C₂×Q₈	—	6	2	no	—
283	C₂×Q₈	C₂²×C₅×D₃	1	2	yes	—
284	C₂×D₄	—	3	2	no	—
285	C₂×D₄	—	6	2	no	—
286	C₂×D₄	—	6	2	no	—
287	C₂²×C₄	—	3	1	no	—
288	C₂²×C₄	—	3	2	no	—
289	C₂²×C₄	—	6	2	no	—
290	C₂×C₈	—	3	2	no	—
291	C₂×C₈	—	6	4	no	—
292	C₂×C₈	C₂²×C₅×D₃	1	2	yes	—
293	C₄⋊C₄	—	2	2	no	—
294	C₄⋊C₄	—	2	2	no	—
295	C₄⋊C₄	—	3	2	no	—
296	C₄⋊C₄	—	6	2	no	—
297	C₄⋊C₄	—	6	2	no	—
298	C₄⋊C₄	C₂²×C₅×D₃	1	2	yes	—
299	C₂²⋊C₄	—	6	4	no	—
300	C₂²⋊C₄	—	6	2	no	—
301	C₄²	—	3	2	no	—
302	C₄²	—	3	1	no	—
303	C₄²	—	6	2	no	—
304	C₄²	C₂²×C₅×D₃	1	1	yes	characteristic
305	C₃×C₅	#3135	1	1	yes	characteristic, complemented, Hall
306	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
307	C₂×D₃	—	2	2	no	—
308	C₂×D₃	—	4	2	no	—
309	C₂×D₃	—	4	4	no	—
310	C₂×D₃	C₅×#124	1	2	yes	—
311	C₃×C₄	—	2	2	no	—
312	C₃×C₄	—	4	2	no	—
313	C₃×C₄	—	4	2	no	—
314	C₃×C₄	C₂×C₅×#41	1	2	yes	—
315	Dic₃	—	2	2	no	—
316	Dic₃	—	4	2	no	—
317	Dic₃	C₅×#124	1	2	yes	—
318	C₂×C₅	—	6	2	no	—
319	C₂×C₅	—	24	2	no	—
320	C₂×C₅	D₃×#124	1	1	yes	characteristic
321	C₂×C₅	#5777	1	2	yes	—
322	C₂³	—	3	1	no	—
323	C₂³	—	6	2	no	—
324	Q₈	—	2	4	no	—
325	Q₈	—	12	4	no	—
326	D₄	—	6	4	no	—
327	D₄	—	12	4	no	—
328	D₄	—	12	4	no	—
329	C₂×C₄	—	2	2	no	—
330	C₂×C₄	—	2	2	no	—
331	C₂×C₄	—	3	1	no	—
332	C₂×C₄	—	3	4	no	—
333	C₂×C₄	—	3	2	no	—
334	C₂×C₄	—	6	2	no	—
335	C₂×C₄	—	12	2	no	—
336	C₂×C₄	—	12	4	no	—
337	C₂×C₄	C₅×D₃×D₄	1	2	yes	—
338	C₂×C₄	C₂³×C₅×D₃	1	1	yes	characteristic
339	C₈	—	2	2	no	—
340	C₈	—	6	2	no	—
341	C₂×C₃	C₂×C₅×#124	1	1	yes	characteristic
342	C₂×C₃	C₅×#575	1	2	yes	—
343	D₃	—	2	2	no	—
344	D₃	—	8	2	no	—
345	C₅	#81737	1	1	yes	characteristic, direct product factor, Sylow
346	C₂²	—	3	2	no	—
347	C₂²	—	6	2	no	—
348	C₂²	—	12	2	no	—
349	C₂²	—	12	4	no	—
350	C₂²	C₂×C₅×D₃×D₄	1	1	yes	characteristic
351	C₄	—	2	2	no	—
352	C₄	—	3	2	no	—
353	C₄	—	4	2	no	—
354	C₄	—	4	2	no	—
355	C₄	—	6	2	no	—
356	C₄	—	12	2	no	—
357	C₄	C₅×D₃×#41	1	2	yes	—
358	C₃	C₅×#3135	1	1	yes	characteristic, complemented, Sylow
359	C₂	—	6	2	no	—
360	C₂	—	24	2	no	—
361	C₂	C₅×D₃×#124	1	1	yes	characteristic
362	C₂	C₅×#5777	1	2	yes	—

	subgroup	G/H
center	C₂²×C₅	C₂×D₃×D₄
commutator, cosocle	C₂×C₃×C₄	C₂⁴×C₅
Frattini	C₂×C₄	C₂³×C₅×D₃
Fitting	C₃×C₅×#491	C₂
socle	C₂²×C₃×C₅	C₂²×D₄
prime radical	C₅×#5725	C₂
hypercenter	C₅×#491	D₃
hypocenter	C₃	C₅×#3135
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ C₅×#491

Complements

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

N	H ≅ G/N	type
2. C₅×#5688	C₂	semidirect
3. C₃×C₅×#491	C₂	semidirect
4. C₅×#5082	C₂	semidirect
5. C₅×#5067	C₂	semidirect
6. C₅×#4804	C₂	semidirect
7. C₅×#4794	C₂	semidirect
8. C₅×#4713	C₂	semidirect
9. C₅×D₃×#97	C₂	semidirect
32. #81737	C₅	direct (N × H)
36. C₅×#491	D₃	semidirect
67. #5688	C₂×C₅	semidirect
68. C₃×#491	C₂×C₅	semidirect
69. #5082	C₂×C₅	semidirect
70. #5067	C₂×C₅	semidirect
71. #4804	C₂×C₅	semidirect
72. #4794	C₂×C₅	semidirect
73. #4713	C₂×C₅	semidirect
74. D₃×#97	C₂×C₅	semidirect
164. #491	C₅×D₃	semidirect
305. C₃×C₅	#3135	semidirect
345. C₅	#81737	direct (N × H)
358. C₃	C₅×#3135	semidirect

Maximal subgroups

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

subgroup	index
C₅×#5725	2
C₅×#5688	2
C₃×C₅×#491	2
C₅×#5082	2
C₅×#5067	2
C₅×#4804	2
C₅×#4794	2
C₅×#4713	2
C₅×D₃×#97	2
C₅×#3135	3
#81737	5

Subgroups — C5×#81737 #14438004

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#81737 #14438004