Subgroups — C₅×#81713 #14437995

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×#5716	C₂	1	1	yes	maximal, characteristic
2	C₅×#5688	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₅×#490	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#5071	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#5069	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#4772	C₂	1	4	yes	maximal, complemented
7	C₅×#4765	C₂	1	4	yes	maximal, complemented
8	C₅×D₃×#97	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅×#4703	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅×#3135	—	3	1	no	maximal, Hall
11	C₂×C₅×#237	C₂²	1	2	yes	—
12	C₃×C₅×#127	C₂²	1	1	yes	characteristic
13	C₃×C₅×#125	C₂²	1	1	yes	characteristic
14	C₅×#938	—	2	2	no	—
15	C₅×#936	C₂²	1	2	yes	—
16	C₅×D₄×Dic₃	—	2	2	no	—
17	C₂×C₅×D₃×C₈	C₂²	1	2	yes	—
18	C₅×#896	—	2	2	no	—
19	C₅×#892	C₂²	1	2	yes	—
20	C₅×#888	C₂²	1	2	yes	—
21	C₅×#887	—	2	2	no	—
22	C₄²×C₅×D₃	C₂²	1	1	yes	characteristic
23	C₅×#870	C₂²	1	1	yes	characteristic
24	C₅×#869	C₂²	1	1	yes	characteristic
25	C₃×C₅×#102	C₂²	1	4	yes	—
26	C₃×C₅×#97	C₂²	1	1	yes	characteristic
27	C₅×#832	C₂²	1	4	yes	—
28	C₅×#816	C₂²	1	4	yes	—
29	C₅×#815	C₂²	1	2	yes	—
30	C₅×#810	C₂²	1	4	yes	—
31	C₅×#803	C₂²	1	1	yes	characteristic
32	#81713	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	—	3	1	no	—
37	C₅×#490	—	3	1	no	—
38	C₅×#490	D₃	1	1	yes	characteristic, complemented
39	C₅×#489	—	3	4	no	—
40	C₅×#416	—	3	4	no	—
41	C₂×C₅×#97	—	3	1	no	—
42	C₂×C₃×C₅×D₄	—	2	2	no	—
43	C₂×C₃×C₅×D₄	C₂³	1	2	yes	—
44	C₂×C₅×#67	—	2	2	no	—
45	C₂²×C₅×Dic₃	—	2	2	no	—
46	C₅×#237	—	2	8	no	—
47	C₂×C₄×C₅×D₃	D₄	1	2	yes	—
48	C₂×C₄×C₅×D₃	C₂³	1	1	yes	characteristic
49	C₂×C₅×Dic₆	—	2	2	no	—
50	C₂×C₅×Dic₆	C₂³	1	2	yes	—
51	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
52	C₃×C₅×C₄⋊C₄	—	2	2	no	—
53	C₃×C₅×C₄⋊C₄	C₂³	1	2	yes	—
54	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
55	C₅×#217	—	2	4	no	—
56	C₅×(C₂×D₃)⋊C₄	—	2	2	no	—
57	C₅×C₄⋊Dic₃	—	2	2	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	#5716	C₂×C₅	1	1	yes	characteristic
67	#5688	C₂×C₅	1	1	yes	characteristic, complemented
68	C₃×#490	C₂×C₅	1	1	yes	characteristic, complemented
69	#5071	C₂×C₅	1	1	yes	characteristic, complemented
70	#5069	C₂×C₅	1	1	yes	characteristic, complemented
71	#4772	C₂×C₅	1	4	yes	complemented
72	#4765	C₂×C₅	1	4	yes	complemented
73	D₃×#97	C₂×C₅	1	1	yes	characteristic, complemented
74	#4703	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	—	3	1	no	—
79	C₅×#127	C₂×D₃	1	1	yes	characteristic
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	—	3	4	no	—
91	C₅×#102	—	3	4	no	—
92	C₅×#102	C₂×D₃	1	4	yes	—
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	C₂³×C₃×C₅	—	2	2	no	—
98	C₂²×C₅×D₃	C₂×D₄	1	1	yes	characteristic
99	C₃×C₅×D₄	—	2	4	no	—
100	C₃×C₅×D₄	—	4	4	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	C₅×#67	—	4	4	no	—
105	C₂×C₅×Dic₃	—	2	2	no	—
106	C₂×C₅×Dic₃	—	2	2	no	—
107	C₂×C₅×Dic₃	—	4	4	no	—
108	C₂×C₅×Dic₃	C₂×D₄	1	2	yes	—
109	C₂×C₅×Dic₃	C₂×D₄	1	1	yes	characteristic
110	C₄×C₅×D₃	—	4	2	no	—
111	C₄×C₅×D₃	#41	1	4	yes	—
112	C₅×Dic₆	—	2	4	no	—
113	C₅×Dic₆	—	4	4	no	—
114	C₃×C₅×C₈	—	2	2	no	—
115	C₅×C₃⋊C₈	—	2	2	no	—
116	C₂×#237	C₂²×C₅	1	2	yes	—
117	C₃×#127	C₂²×C₅	1	1	yes	characteristic
118	C₃×#125	C₂²×C₅	1	1	yes	characteristic
119	#938	—	2	2	no	—
120	#936	C₂²×C₅	1	2	yes	—
121	D₄×Dic₃	—	2	2	no	—
122	C₂×D₃×C₈	C₂²×C₅	1	2	yes	—
123	#896	—	2	2	no	—
124	#892	C₂²×C₅	1	2	yes	—
125	#888	C₂²×C₅	1	2	yes	—
126	#887	—	2	2	no	—
127	C₄²×D₃	C₂²×C₅	1	1	yes	characteristic
128	#870	C₂²×C₅	1	1	yes	characteristic
129	#869	C₂²×C₅	1	1	yes	characteristic
130	C₃×#102	C₂²×C₅	1	4	yes	—
131	C₃×#97	C₂²×C₅	1	1	yes	characteristic
132	#832	C₂²×C₅	1	4	yes	—
133	#816	C₂²×C₅	1	4	yes	—
134	#815	C₂²×C₅	1	2	yes	—
135	#810	C₂²×C₅	1	4	yes	—
136	#803	C₂²×C₅	1	1	yes	characteristic
137	C₅×#41	—	6	8	no	—
138	C₂×C₅×Q₈	—	3	2	no	—
139	C₂×C₅×Q₈	—	6	2	no	—
140	C₂×C₅×D₄	—	2	2	no	—
141	C₂×C₅×D₄	—	6	2	no	—
142	C₂×C₅×D₄	C₂²×D₃	1	2	yes	—
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	—	3	2	no	—
151	C₅×C₄⋊C₄	—	6	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	2	no	—
156	C₅×C₂²⋊C₄	—	6	4	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	—	3	1	no	—
165	#490	—	3	1	no	—
166	#490	C₅×D₃	1	1	yes	characteristic, complemented
167	#489	—	3	4	no	—
168	#416	—	3	4	no	—
169	C₂×#97	—	3	1	no	—
170	C₂²×C₃×C₅	—	4	2	no	—
171	C₂²×C₃×C₅	—	4	4	no	—
172	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
173	C₂×C₅×D₃	—	2	2	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	C₂×#41	1	2	yes	—
178	C₅×Dic₃	—	2	2	no	—
179	C₅×Dic₃	—	4	2	no	—
180	C₅×Dic₃	—	4	2	no	—
181	C₅×Dic₃	#124	1	2	yes	—
182	C₂×C₃×D₄	—	2	2	no	—
183	C₂×C₃×D₄	C₂³×C₅	1	2	yes	—
184	C₂×#67	—	2	2	no	—
185	C₂²×Dic₃	—	2	2	no	—
186	#237	—	2	8	no	—
187	C₂×C₄×D₃	C₅×D₄	1	2	yes	—
188	C₂×C₄×D₃	C₂³×C₅	1	1	yes	characteristic
189	C₂×Dic₆	—	2	2	no	—
190	C₂×Dic₆	C₂³×C₅	1	2	yes	—
191	C₂×C₃×C₈	C₂³×C₅	1	2	yes	—
192	C₃×C₄⋊C₄	—	2	2	no	—
193	C₃×C₄⋊C₄	C₂³×C₅	1	2	yes	—
194	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
195	#217	—	2	4	no	—
196	(C₂×D₃)⋊C₄	—	2	2	no	—
197	C₄⋊Dic₃	—	2	2	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	—	2	2	no	—
207	C₂³×C₅	—	3	1	no	—
208	C₅×Q₈	—	6	4	no	—
209	C₅×Q₈	—	12	4	no	—
210	C₅×D₄	—	2	4	no	—
211	C₅×D₄	—	4	4	no	—
212	C₅×D₄	—	12	4	no	—
213	C₂×C₄×C₅	—	2	2	no	—
214	C₂×C₄×C₅	—	3	1	no	—
215	C₂×C₄×C₅	—	3	4	no	—
216	C₂×C₄×C₅	—	3	2	no	—
217	C₂×C₄×C₅	—	6	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	—	3	1	no	—
229	#127	C₂×C₅×D₃	1	1	yes	characteristic
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	—	3	4	no	—
241	#102	—	3	4	no	—
242	#102	C₂×C₅×D₃	1	4	yes	—
243	#97	—	3	2	no	—
244	#97	—	3	1	no	—
245	#97	C₂×C₅×D₃	1	1	yes	characteristic
246	C₂×C₃×C₅	—	8	2	no	—
247	C₂×C₃×C₅	C₂×#124	1	1	yes	characteristic
248	C₂×C₃×C₅	#575	1	2	yes	—
249	C₅×D₃	—	2	2	no	—
250	C₂³×C₃	—	2	2	no	—
251	C₂²×D₃	C₂×C₅×D₄	1	1	yes	characteristic
252	C₃×D₄	—	2	4	no	—
253	C₃×D₄	—	4	4	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	#67	—	4	4	no	—
258	C₂×Dic₃	—	2	2	no	—
259	C₂×Dic₃	—	2	2	no	—
260	C₂×Dic₃	—	4	4	no	—
261	C₂×Dic₃	C₂×C₅×D₄	1	2	yes	—
262	C₂×Dic₃	C₂×C₅×D₄	1	1	yes	characteristic
263	C₄×D₃	—	4	2	no	—
264	C₄×D₃	C₅×#41	1	4	yes	—
265	Dic₆	—	2	4	no	—
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	—	4	2	no	—
271	C₂²×C₅	—	4	4	no	—
272	C₂²×C₅	—	6	2	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	—	6	2	no	—
278	C₄×C₅	—	12	2	no	—
279	C₄×C₅	—	12	2	no	—
280	C₄×C₅	D₃×#41	1	2	yes	—
281	#41	—	6	8	no	—
282	C₂×Q₈	—	3	2	no	—
283	C₂×Q₈	—	6	2	no	—
284	C₂×D₄	—	2	2	no	—
285	C₂×D₄	—	6	2	no	—
286	C₂×D₄	C₂²×C₅×D₃	1	2	yes	—
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	—	3	2	no	—
295	C₄⋊C₄	—	6	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	2	no	—
300	C₂²⋊C₄	—	6	4	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	—	4	2	no	—
307	C₂²×C₃	—	4	4	no	—
308	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
309	C₂×D₃	—	2	2	no	—
310	C₂×D₃	C₅×#124	1	2	yes	—
311	C₃×C₄	—	2	2	no	—
312	C₃×C₄	—	4	2	no	—
313	C₃×C₄	C₂×C₅×#41	1	2	yes	—
314	Dic₃	—	2	2	no	—
315	Dic₃	—	4	2	no	—
316	Dic₃	—	4	2	no	—
317	Dic₃	C₅×#124	1	2	yes	—
318	C₂×C₅	—	6	2	no	—
319	C₂×C₅	—	8	2	no	—
320	C₂×C₅	D₃×#124	1	1	yes	characteristic
321	C₂×C₅	#5773	1	2	yes	—
322	C₂³	—	2	2	no	—
323	C₂³	—	3	1	no	—
324	Q₈	—	6	4	no	—
325	Q₈	—	12	4	no	—
326	D₄	—	2	4	no	—
327	D₄	—	4	4	no	—
328	D₄	—	12	4	no	—
329	C₂×C₄	—	2	2	no	—
330	C₂×C₄	—	3	1	no	—
331	C₂×C₄	—	3	4	no	—
332	C₂×C₄	—	3	2	no	—
333	C₂×C₄	—	6	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	—	8	2	no	—
342	C₂×C₃	C₂×C₅×#124	1	1	yes	characteristic
343	C₂×C₃	C₅×#575	1	2	yes	—
344	D₃	—	2	2	no	—
345	C₅	#81713	1	1	yes	characteristic, direct product factor, Sylow
346	C₂²	—	3	2	no	—
347	C₂²	—	4	2	no	—
348	C₂²	—	4	4	no	—
349	C₂²	—	6	2	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	—	6	2	no	—
355	C₄	—	12	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	—	8	2	no	—
361	C₂	C₅×D₃×#124	1	1	yes	characteristic
362	C₂	C₅×#5773	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×#490	C₂
socle	C₂²×C₃×C₅	C₂²×D₄
prime radical	C₅×#5716	C₂
hypercenter	C₅×#490	D₃
hypocenter	C₃	C₅×#3135
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ C₅×#490

Complements

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

N	H ≅ G/N	type
2. C₅×#5688	C₂	semidirect
3. C₃×C₅×#490	C₂	semidirect
4. C₅×#5071	C₂	semidirect
5. C₅×#5069	C₂	semidirect
6. C₅×#4772	C₂	semidirect
7. C₅×#4765	C₂	semidirect
8. C₅×D₃×#97	C₂	semidirect
9. C₅×#4703	C₂	semidirect
32. #81713	C₅	direct (N × H)
38. C₅×#490	D₃	semidirect
67. #5688	C₂×C₅	semidirect
68. C₃×#490	C₂×C₅	semidirect
69. #5071	C₂×C₅	semidirect
70. #5069	C₂×C₅	semidirect
71. #4772	C₂×C₅	semidirect
72. #4765	C₂×C₅	semidirect
73. D₃×#97	C₂×C₅	semidirect
74. #4703	C₂×C₅	semidirect
166. #490	C₅×D₃	semidirect
305. C₃×C₅	#3135	semidirect
345. C₅	#81713	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×#5716	2
C₅×#5688	2
C₃×C₅×#490	2
C₅×#5071	2
C₅×#5069	2
C₅×#4772	2
C₅×#4765	2
C₅×D₃×#97	2
C₅×#4703	2
C₅×#3135	3
#81713	5

Subgroups — C5×#81713 #14437995

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#81713 #14437995