Subgroups — C₅×#71370 #14290564

Order 1920.

400 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#5732	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×#500	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#5094	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#5083	C₂	1	1	yes	maximal, characteristic
5	C₅×#4547	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#4459	C₂	1	1	yes	maximal, characteristic
7	C₅×#4456	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#1650	—	3	1	no	maximal, Hall
9	C₃×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
10	C₃×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	C₅×#947	—	2	1	no	—
12	C₅×Q₈×Dic₃	—	2	1	no	—
13	C₂×C₅×#215	—	2	1	no	—
14	C₅×#898	—	2	2	no	—
15	C₅×#897	—	2	1	no	—
16	C₅×#892	—	2	1	no	—
17	C₄×C₅×D₁₂	C₂²	1	1	yes	characteristic
18	C₄×C₅×Dic₆	C₂²	1	1	yes	characteristic
19	C₃×C₅×#107	—	2	2	no	—
20	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
21	C₅×#835	—	2	1	no	—
22	C₅×#810	—	2	1	no	—
23	C₅×#808	—	2	1	no	—
24	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
25	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
26	#71370	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
27	C₅×#556	—	3	1	no	—
28	C₅×#500	D₃	1	1	yes	characteristic, complemented
29	C₅×#475	—	3	1	no	—
30	C₅×#461	—	3	1	no	—
31	C₅×#332	—	3	1	no	—
32	C₅×Q₈⋊C₈	—	3	1	no	—
33	C₅×D₄⋊C₈	—	3	1	no	—
34	C₂×C₃×C₅×Q₈	—	2	1	no	—
35	C₂×C₃×C₅×Q₈	—	2	1	no	—
36	C₂×C₅×D₁₂	D₄	1	1	yes	characteristic
37	C₂×C₄×C₅×D₃	—	2	1	no	—
38	C₂×C₅×Dic₆	D₄	1	1	yes	characteristic
39	C₂×C₃×C₅×C₈	—	2	1	no	—
40	C₃×C₅×C₄⋊C₄	—	2	1	no	—
41	C₃×C₅×C₄⋊C₄	—	2	1	no	—
42	C₃×C₅×C₄⋊C₄	—	2	1	no	—
43	C₃×C₅×C₄⋊C₄	D₄	1	2	yes	—
44	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
45	C₅×#215	—	4	2	no	—
46	C₅×(C₂×D₃)⋊C₄	—	2	2	no	—
47	C₅×(C₂×D₃)⋊C₄	—	2	2	no	—
48	C₅×(C₂×D₃)⋊C₄	—	2	1	no	—
49	C₅×C₄⋊Dic₃	—	2	2	no	—
50	C₅×C₄⋊Dic₃	D₄	1	1	yes	characteristic
51	C₅×C₄⋊Dic₃	D₄	1	1	yes	characteristic
52	C₅×Dic₃⋊C₄	—	2	1	no	—
53	C₄×C₅×Dic₃	—	2	2	no	—
54	C₄×C₅×Dic₃	—	2	1	no	—
55	C₂×C₅×C₃⋊C₈	—	2	1	no	—
56	C₂×C₅×C₃⋊C₈	—	2	1	no	—
57	#5732	C₂×C₅	1	1	yes	characteristic
58	C₃×#500	C₂×C₅	1	1	yes	characteristic, complemented
59	#5094	C₂×C₅	1	1	yes	characteristic, complemented
60	#5083	C₂×C₅	1	1	yes	characteristic
61	#4547	C₂×C₅	1	1	yes	characteristic, complemented
62	#4459	C₂×C₅	1	1	yes	characteristic
63	#4456	C₂×C₅	1	1	yes	characteristic, complemented
64	C₂×C₅×SD₁₆	—	6	1	no	—
65	C₅×C₄⋊Q₈	C₂×D₃	1	1	yes	characteristic
66	C₅×C₄⋊Q₈	C₂×D₃	1	1	yes	characteristic
67	C₅×#126	—	6	2	no	—
68	C₅×#124	—	6	1	no	—
69	C₅×C₂²⋊Q₈	—	6	1	no	—
70	C₄×C₅×Q₈	—	3	1	no	—
71	C₄×C₅×Q₈	—	6	1	no	—
72	C₄×C₅×D₄	—	3	1	no	—
73	C₅×#117	—	6	1	no	—
74	C₅×#107	—	2	2	no	—
75	C₅×#106	—	6	1	no	—
76	C₅×C₄⋊C₈	—	3	1	no	—
77	C₅×C₄⋊C₈	—	3	1	no	—
78	C₅×#103	—	6	1	no	—
79	C₅×#103	—	6	1	no	—
80	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
81	#1650	—	3	1	no	Sylow
82	C₂²×C₅×D₃	—	2	1	no	—
83	C₃×C₅×Q₈	—	4	1	no	—
84	C₃×C₅×Q₈	—	4	1	no	—
85	C₃×C₅×Q₈	—	4	1	no	—
86	C₃×C₅×Q₈	—	4	1	no	—
87	C₂×C₃×C₄×C₅	—	2	2	no	—
88	C₂×C₃×C₄×C₅	—	4	1	no	—
89	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
90	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
91	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
92	C₂×C₅×Dic₃	—	2	1	no	—
93	C₂×C₅×Dic₃	—	2	1	no	—
94	C₂×C₅×Dic₃	—	2	1	no	—
95	C₅×D₁₂	—	2	1	no	—
96	C₅×D₁₂	—	2	1	no	—
97	C₄×C₅×D₃	—	4	1	no	—
98	C₄×C₅×D₃	—	4	1	no	—
99	C₅×Dic₆	—	2	1	no	—
100	C₅×Dic₆	—	2	1	no	—
101	C₃×C₅×C₈	—	2	2	no	—
102	C₅×C₃⋊C₈	—	4	1	no	—
103	C₅×C₃⋊C₈	—	4	1	no	—
104	C₃×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
105	C₃×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
106	#947	—	2	1	no	—
107	Q₈×Dic₃	—	2	1	no	—
108	C₂×#215	—	2	1	no	—
109	#898	—	2	2	no	—
110	#897	—	2	1	no	—
111	#892	—	2	1	no	—
112	C₄×D₁₂	C₂²×C₅	1	1	yes	characteristic
113	C₄×Dic₆	C₂²×C₅	1	1	yes	characteristic
114	C₃×#107	—	2	2	no	—
115	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
116	#835	—	2	1	no	—
117	#810	—	2	1	no	—
118	#808	—	2	1	no	—
119	C₄⋊(C₃⋊C₈)	C₂²×C₅	1	1	yes	characteristic
120	C₄⋊(C₃⋊C₈)	C₂²×C₅	1	1	yes	characteristic
121	C₂×C₅×Q₈	—	2	1	no	—
122	C₂×C₅×Q₈	—	2	1	no	—
123	C₂×C₅×Q₈	—	3	1	no	—
124	C₂×C₅×D₄	—	3	1	no	—
125	C₂²×C₄×C₅	—	6	1	no	—
126	C₅×SD₁₆	—	12	2	no	—
127	C₂×C₅×C₈	—	2	1	no	—
128	C₂×C₅×C₈	—	6	1	no	—
129	C₂×C₅×C₈	—	6	1	no	—
130	C₅×C₄⋊C₄	—	2	1	no	—
131	C₅×C₄⋊C₄	—	2	1	no	—
132	C₅×C₄⋊C₄	—	2	1	no	—
133	C₅×C₄⋊C₄	—	3	1	no	—
134	C₅×C₄⋊C₄	—	3	1	no	—
135	C₅×C₄⋊C₄	—	6	2	no	—
136	C₅×C₄⋊C₄	—	6	1	no	—
137	C₅×C₄⋊C₄	#67	1	2	yes	—
138	C₅×C₂²⋊C₄	—	6	2	no	—
139	C₅×C₂²⋊C₄	—	6	2	no	—
140	C₅×C₂²⋊C₄	—	6	1	no	—
141	C₄²×C₅	—	6	2	no	—
142	C₄²×C₅	—	6	1	no	—
143	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
144	#556	—	3	1	no	—
145	#500	C₅×D₃	1	1	yes	characteristic, complemented
146	#475	—	3	1	no	—
147	#461	—	3	1	no	—
148	#332	—	3	1	no	—
149	Q₈⋊C₈	—	3	1	no	—
150	D₄⋊C₈	—	3	1	no	—
151	C₂²×C₃×C₅	#120	1	1	yes	characteristic
152	C₂×C₅×D₃	—	4	1	no	—
153	C₂×C₅×D₃	—	4	1	no	—
154	C₂×C₅×D₃	—	4	1	no	—
155	C₃×C₄×C₅	—	2	1	no	—
156	C₃×C₄×C₅	—	4	2	no	—
157	C₃×C₄×C₅	—	8	1	no	—
158	C₃×C₄×C₅	#135	1	1	yes	characteristic
159	C₃×C₄×C₅	#135	1	1	yes	characteristic
160	C₃×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
161	C₃×C₄×C₅	#137	1	1	yes	characteristic
162	C₅×Dic₃	—	4	1	no	—
163	C₅×Dic₃	—	4	1	no	—
164	C₅×Dic₃	—	4	1	no	—
165	C₂×C₃×Q₈	—	2	1	no	—
166	C₂×C₃×Q₈	—	2	1	no	—
167	C₂×D₁₂	C₅×D₄	1	1	yes	characteristic
168	C₂×C₄×D₃	—	2	1	no	—
169	C₂×Dic₆	C₅×D₄	1	1	yes	characteristic
170	C₂×C₃×C₈	—	2	1	no	—
171	C₃×C₄⋊C₄	—	2	1	no	—
172	C₃×C₄⋊C₄	—	2	1	no	—
173	C₃×C₄⋊C₄	—	2	1	no	—
174	C₃×C₄⋊C₄	C₅×D₄	1	2	yes	—
175	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
176	#215	—	4	2	no	—
177	(C₂×D₃)⋊C₄	—	2	2	no	—
178	(C₂×D₃)⋊C₄	—	2	2	no	—
179	(C₂×D₃)⋊C₄	—	2	1	no	—
180	C₄⋊Dic₃	—	2	2	no	—
181	C₄⋊Dic₃	C₅×D₄	1	1	yes	characteristic
182	C₄⋊Dic₃	C₅×D₄	1	1	yes	characteristic
183	Dic₃⋊C₄	—	2	1	no	—
184	C₄×Dic₃	—	2	2	no	—
185	C₄×Dic₃	—	2	1	no	—
186	C₂×C₃⋊C₈	—	2	1	no	—
187	C₂×C₃⋊C₈	—	2	1	no	—
188	C₂³×C₅	—	6	1	no	—
189	C₅×Q₈	—	4	1	no	—
190	C₅×Q₈	—	4	1	no	—
191	C₅×Q₈	—	4	1	no	—
192	C₅×Q₈	—	4	1	no	—
193	C₅×Q₈	—	6	1	no	—
194	C₅×Q₈	—	6	1	no	—
195	C₅×D₄	—	6	1	no	—
196	C₅×D₄	—	6	1	no	—
197	C₂×C₄×C₅	—	2	2	no	—
198	C₂×C₄×C₅	—	4	1	no	—
199	C₂×C₄×C₅	—	6	1	no	—
200	C₂×C₄×C₅	—	6	1	no	—
201	C₂×C₄×C₅	—	6	1	no	—
202	C₂×C₄×C₅	—	12	1	no	—
203	C₂×C₄×C₅	—	12	1	no	—
204	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
205	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
206	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
207	C₅×C₈	—	2	2	no	—
208	C₅×C₈	—	12	1	no	—
209	C₅×C₈	—	12	1	no	—
210	C₂×SD₁₆	—	6	1	no	—
211	C₄⋊Q₈	C₂×C₅×D₃	1	1	yes	characteristic
212	C₄⋊Q₈	C₂×C₅×D₃	1	1	yes	characteristic
213	#126	—	6	2	no	—
214	#124	—	6	1	no	—
215	C₂²⋊Q₈	—	6	1	no	—
216	C₄×Q₈	—	3	1	no	—
217	C₄×Q₈	—	6	1	no	—
218	C₄×D₄	—	3	1	no	—
219	#117	—	6	1	no	—
220	#107	—	2	2	no	—
221	#106	—	6	1	no	—
222	C₄⋊C₈	—	3	1	no	—
223	C₄⋊C₈	—	3	1	no	—
224	#103	—	6	1	no	—
225	#103	—	6	1	no	—
226	C₄×C₈	C₂×C₅×D₃	1	1	yes	characteristic
227	C₂×C₃×C₅	#449	1	1	yes	characteristic
228	C₂×C₃×C₅	#452	1	1	yes	characteristic
229	C₂×C₃×C₅	#456	1	1	yes	characteristic
230	C₅×D₃	—	8	1	no	—
231	C₂²×D₃	—	2	1	no	—
232	C₃×Q₈	—	4	1	no	—
233	C₃×Q₈	—	4	1	no	—
234	C₃×Q₈	—	4	1	no	—
235	C₃×Q₈	—	4	1	no	—
236	C₂×C₃×C₄	—	2	2	no	—
237	C₂×C₃×C₄	—	4	1	no	—
238	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
239	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
240	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
241	C₂×Dic₃	—	2	1	no	—
242	C₂×Dic₃	—	2	1	no	—
243	C₂×Dic₃	—	2	1	no	—
244	D₁₂	—	2	1	no	—
245	D₁₂	—	2	1	no	—
246	C₄×D₃	—	4	1	no	—
247	C₄×D₃	—	4	1	no	—
248	Dic₆	—	2	1	no	—
249	Dic₆	—	2	1	no	—
250	C₃×C₈	—	2	2	no	—
251	C₃⋊C₈	—	4	1	no	—
252	C₃⋊C₈	—	4	1	no	—
253	C₂²×C₅	—	12	1	no	—
254	C₂²×C₅	—	12	1	no	—
255	C₂²×C₅	—	12	1	no	—
256	C₂²×C₅	#937	1	1	yes	characteristic
257	C₄×C₅	—	2	1	no	—
258	C₄×C₅	—	4	2	no	—
259	C₄×C₅	—	8	1	no	—
260	C₄×C₅	—	12	1	no	—
261	C₄×C₅	—	12	1	no	—
262	C₄×C₅	—	12	1	no	—
263	C₄×C₅	#911	1	1	yes	characteristic
264	C₄×C₅	#912	1	1	yes	characteristic
265	C₄×C₅	#918	1	1	yes	characteristic
266	C₄×C₅	#919	1	1	yes	characteristic
267	C₂×Q₈	—	2	1	no	—
268	C₂×Q₈	—	2	1	no	—
269	C₂×Q₈	—	3	1	no	—
270	C₂×D₄	—	3	1	no	—
271	C₂²×C₄	—	6	1	no	—
272	SD₁₆	—	12	2	no	—
273	C₂×C₈	—	2	1	no	—
274	C₂×C₈	—	6	1	no	—
275	C₂×C₈	—	6	1	no	—
276	C₄⋊C₄	—	2	1	no	—
277	C₄⋊C₄	—	2	1	no	—
278	C₄⋊C₄	—	2	1	no	—
279	C₄⋊C₄	—	3	1	no	—
280	C₄⋊C₄	—	3	1	no	—
281	C₄⋊C₄	—	6	2	no	—
282	C₄⋊C₄	—	6	1	no	—
283	C₄⋊C₄	C₅×#67	1	2	yes	—
284	C₂²⋊C₄	—	6	2	no	—
285	C₂²⋊C₄	—	6	2	no	—
286	C₂²⋊C₄	—	6	1	no	—
287	C₄²	—	6	2	no	—
288	C₄²	—	6	1	no	—
289	C₄²	C₂²×C₅×D₃	1	1	yes	characteristic
290	C₃×C₅	#1650	1	1	yes	characteristic, complemented, Hall
291	C₂²×C₃	C₅×#120	1	1	yes	characteristic
292	C₂×D₃	—	4	1	no	—
293	C₂×D₃	—	4	1	no	—
294	C₂×D₃	—	4	1	no	—
295	C₃×C₄	—	2	1	no	—
296	C₃×C₄	—	4	2	no	—
297	C₃×C₄	—	8	1	no	—
298	C₃×C₄	C₅×#135	1	1	yes	characteristic
299	C₃×C₄	C₅×#135	1	1	yes	characteristic
300	C₃×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
301	C₃×C₄	C₅×#137	1	1	yes	characteristic
302	Dic₃	—	4	1	no	—
303	Dic₃	—	4	1	no	—
304	Dic₃	—	4	1	no	—
305	C₂×C₅	—	24	1	no	—
306	C₂×C₅	#5095	1	1	yes	characteristic
307	C₂×C₅	#5158	1	1	yes	characteristic
308	C₂×C₅	#5187	1	1	yes	characteristic
309	C₂³	—	6	1	no	—
310	Q₈	—	4	1	no	—
311	Q₈	—	4	1	no	—
312	Q₈	—	4	1	no	—
313	Q₈	—	4	1	no	—
314	Q₈	—	6	1	no	—
315	Q₈	—	6	1	no	—
316	D₄	—	6	1	no	—
317	D₄	—	6	1	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	4	1	no	—
320	C₂×C₄	—	6	1	no	—
321	C₂×C₄	—	6	1	no	—
322	C₂×C₄	—	6	1	no	—
323	C₂×C₄	—	12	1	no	—
324	C₂×C₄	—	12	1	no	—
325	C₂×C₄	C₅×D₃×D₄	1	1	yes	characteristic
326	C₂×C₄	C₅×D₃×D₄	1	1	yes	characteristic
327	C₂×C₄	C₂×C₅×#67	1	1	yes	characteristic
328	C₈	—	2	2	no	—
329	C₈	—	12	1	no	—
330	C₈	—	12	1	no	—
331	C₂×C₃	C₅×#449	1	1	yes	characteristic
332	C₂×C₃	C₅×#452	1	1	yes	characteristic
333	C₂×C₃	C₅×#456	1	1	yes	characteristic
334	D₃	—	8	1	no	—
335	C₅	#71370	1	1	yes	characteristic, direct product factor, Sylow
336	C₂²	—	12	1	no	—
337	C₂²	—	12	1	no	—
338	C₂²	—	12	1	no	—
339	C₂²	C₅×#937	1	1	yes	characteristic
340	C₄	—	2	1	no	—
341	C₄	—	4	2	no	—
342	C₄	—	8	1	no	—
343	C₄	—	12	1	no	—
344	C₄	—	12	1	no	—
345	C₄	—	12	1	no	—
346	C₄	C₅×#911	1	1	yes	characteristic
347	C₄	C₅×#912	1	1	yes	characteristic
348	C₄	C₅×#918	1	1	yes	characteristic
349	C₄	C₅×#919	1	1	yes	characteristic
350	C₃	C₅×#1650	1	1	yes	characteristic, complemented, Sylow
351	C₂	—	24	1	no	—
352	C₂	C₅×#5095	1	1	yes	characteristic
353	C₂	C₅×#5158	1	1	yes	characteristic
354	C₂	C₅×#5187	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅	#937
commutator, cosocle	C₃×C₄²	C₂³×C₅
Frattini	C₄²	C₂²×C₅×D₃
Fitting	C₃×C₅×#500	C₂
socle	C₂²×C₃×C₅	#120
prime radical	C₂×C₅×D₁₂	D₄
hypercenter	C₅×#500	D₃
hypocenter	C₃	C₅×#1650
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ C₅×#500

Complements

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

N	H ≅ G/N	type
2. C₃×C₅×#500	C₂	semidirect
3. C₅×#5094	C₂	semidirect
5. C₅×#4547	C₂	semidirect
7. C₅×#4456	C₂	semidirect
26. #71370	C₅	direct (N × H)
28. C₅×#500	D₃	semidirect
58. C₃×#500	C₂×C₅	semidirect
59. #5094	C₂×C₅	semidirect
61. #4547	C₂×C₅	semidirect
63. #4456	C₂×C₅	semidirect
145. #500	C₅×D₃	semidirect
290. C₃×C₅	#1650	semidirect
335. C₅	#71370	direct (N × H)
350. C₃	C₅×#1650	semidirect

Maximal subgroups

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

subgroup	index
C₅×#5732	2
C₃×C₅×#500	2
C₅×#5094	2
C₅×#5083	2
C₅×#4547	2
C₅×#4459	2
C₅×#4456	2
C₅×#1650	3
#71370	5

Subgroups — C5×#71370 #14290564

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#71370 #14290564