Subgroups — C₅×#71650 #14301528

Order 1920.

888 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#518	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅×#940	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#833	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#5001	C₂	1	2	yes	maximal, complemented
5	C₅×#4532	C₂	1	2	yes	maximal, complemented
6	C₅×#1887	—	3	1	no	maximal, Hall
7	C₂²×C₃×C₅×D₄	C₂²	1	1	yes	characteristic
8	C₂²×C₅×#67	—	2	2	no	—
9	C₂²×C₅×D₁₂	C₂²	1	1	yes	characteristic
10	C₃×C₄×C₅×D₄	—	2	4	no	—
11	C₃×C₅×#117	C₂²	1	2	yes	—
12	C₂×C₃×C₅×C₂²⋊C₄	—	2	2	no	—
13	C₅×#940	—	2	2	no	—
14	C₅×#940	C₄	1	4	yes	complemented
15	C₂×C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
16	C₂×C₅×#208	C₂²	1	2	yes	—
17	C₅×#833	—	2	2	no	—
18	C₅×#809	—	2	4	no	—
19	#71650	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
20	C₂×C₅×#127	—	3	1	no	—
21	C₅×#518	D₃	1	1	yes	characteristic, complemented
22	C₅×#418	—	3	2	no	—
23	C₂×C₅×#100	—	3	1	no	—
24	C₅×#339	—	3	2	no	—
25	C₂⁴×C₃×C₅	D₄	1	2	yes	—
26	C₂³×C₅×D₃	—	2	1	no	—
27	C₂×C₃×C₅×D₄	—	2	2	no	—
28	C₂×C₃×C₅×D₄	—	2	2	no	—
29	C₂×C₃×C₅×D₄	C₂×C₄	1	2	yes	—
30	C₂×C₃×C₅×D₄	D₄	1	2	yes	—
31	C₂²×C₃×C₄×C₅	—	2	4	no	—
32	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
33	C₂×C₅×#67	—	2	8	no	—
34	C₂×C₅×#67	—	4	8	no	—
35	C₂²×C₅×Dic₃	—	2	1	no	—
36	C₂×C₅×D₁₂	—	2	2	no	—
37	C₂×C₅×D₁₂	—	2	2	no	—
38	C₂×C₅×D₁₂	—	2	1	no	—
39	C₂×C₅×D₁₂	C₂×C₄	1	2	yes	complemented
40	C₃×C₅×C₄⋊C₄	D₄	1	4	yes	complemented
41	C₃×C₅×C₂²⋊C₄	—	2	4	no	—
42	C₃×C₅×C₂²⋊C₄	—	2	2	no	—
43	C₃×C₄²×C₅	—	2	2	no	—
44	C₄×C₅×Dic₃	—	2	1	no	—
45	C₄×C₅×Dic₃	C₂×C₄	1	2	yes	complemented
46	C₅×#208	—	2	2	no	—
47	C₅×#208	—	2	2	no	—
48	C₂×C₅×C₃⋊C₈	—	2	2	no	—
49	C₃×#518	C₂×C₅	1	1	yes	characteristic, complemented
50	C₂×#940	C₂×C₅	1	1	yes	characteristic
51	C₂×#833	C₂×C₅	1	1	yes	characteristic, complemented
52	#5001	C₂×C₅	1	2	yes	complemented
53	#4532	C₂×C₅	1	2	yes	complemented
54	C₂²×C₅×D₄	—	3	1	no	—
55	C₂²×C₅×D₄	—	6	2	no	—
56	C₂²×C₅×D₄	C₂×D₃	1	1	yes	characteristic
57	C₂×C₅×M₁₆	—	3	2	no	—
58	C₅×#127	—	3	4	no	—
59	C₅×#127	—	6	2	no	—
60	C₄×C₅×D₄	—	2	4	no	—
61	C₅×#117	C₂×D₃	1	2	yes	—
62	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
63	C₂×C₄²×C₅	—	3	1	no	—
64	C₅×#102	—	6	4	no	—
65	C₅×#100	—	6	2	no	—
66	#1887	—	3	1	no	Sylow
67	C₂³×C₃×C₅	—	2	2	no	—
68	C₂³×C₃×C₅	—	2	2	no	—
69	C₂³×C₃×C₅	—	4	4	no	—
70	C₂³×C₃×C₅	C₂²⋊C₄	1	4	yes	—
71	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₅×D₃	—	2	2	no	—
73	C₂²×C₅×D₃	—	4	4	no	—
74	C₂²×C₅×D₃	—	4	1	no	—
75	C₂²×C₅×D₃	—	4	1	no	—
76	C₃×C₅×D₄	—	2	4	no	—
77	C₃×C₅×D₄	—	2	4	no	—
78	C₂×C₃×C₄×C₅	—	2	4	no	—
79	C₂×C₃×C₄×C₅	—	8	4	no	—
80	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
82	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
83	C₂×C₃×C₄×C₅	#41	1	2	yes	—
84	C₅×#67	—	4	16	no	—
85	C₂×C₅×Dic₃	—	2	4	no	—
86	C₂×C₅×Dic₃	—	2	2	no	—
87	C₅×D₁₂	—	2	4	no	—
88	C₅×D₁₂	—	4	2	no	—
89	C₅×C₃⋊C₈	—	4	2	no	—
90	C₂²×C₃×D₄	C₂²×C₅	1	1	yes	characteristic
91	C₂²×#67	—	2	2	no	—
92	C₂²×D₁₂	C₂²×C₅	1	1	yes	characteristic
93	C₃×C₄×D₄	—	2	4	no	—
94	C₃×#117	C₂²×C₅	1	2	yes	—
95	C₂×C₃×C₂²⋊C₄	—	2	2	no	—
96	#940	—	2	2	no	—
97	#940	C₄×C₅	1	4	yes	complemented
98	C₂×C₄×Dic₃	C₂²×C₅	1	1	yes	characteristic
99	C₂×#208	C₂²×C₅	1	2	yes	—
100	#833	—	2	2	no	—
101	#809	—	2	4	no	—
102	C₂⁴×C₅	—	6	1	no	—
103	C₂⁴×C₅	#67	1	2	yes	—
104	C₂×C₅×D₄	—	2	2	no	—
105	C₂×C₅×D₄	—	2	2	no	—
106	C₂×C₅×D₄	—	3	2	no	—
107	C₂×C₅×D₄	—	6	8	no	—
108	C₂×C₅×D₄	—	6	1	no	—
109	C₂×C₅×D₄	—	6	2	no	—
110	C₂×C₅×D₄	—	6	2	no	—
111	C₂×C₅×D₄	—	12	8	no	—
112	C₂×C₅×D₄	C₄×D₃	1	2	yes	—
113	C₂×C₅×D₄	D₁₂	1	2	yes	—
114	C₂²×C₄×C₅	—	2	4	no	—
115	C₂²×C₄×C₅	—	6	1	no	—
116	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
117	C₅×M₁₆	—	6	2	no	—
118	C₅×M₁₆	—	6	2	no	—
119	C₂×C₅×C₈	—	6	2	no	—
120	C₅×C₄⋊C₄	#67	1	4	yes	complemented
121	C₅×C₂²⋊C₄	—	2	4	no	—
122	C₅×C₂²⋊C₄	—	2	2	no	—
123	C₄²×C₅	—	2	2	no	—
124	C₄²×C₅	—	3	2	no	—
125	C₄²×C₅	—	6	1	no	—
126	C₂×#127	—	3	1	no	—
127	#518	C₅×D₃	1	1	yes	characteristic, complemented
128	#418	—	3	2	no	—
129	C₂×#100	—	3	1	no	—
130	#339	—	3	2	no	—
131	C₂²×C₃×C₅	—	2	2	no	—
132	C₂²×C₃×C₅	—	2	4	no	—
133	C₂²×C₃×C₅	—	4	4	no	—
134	C₂²×C₃×C₅	—	4	4	no	—
135	C₂²×C₃×C₅	—	8	2	no	—
136	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
137	C₂²×C₃×C₅	#120	1	1	yes	characteristic
138	C₂²×C₃×C₅	#123	1	1	yes	characteristic
139	C₂×C₅×D₃	—	4	4	no	—
140	C₂×C₅×D₃	—	4	2	no	—
141	C₂×C₅×D₃	—	8	2	no	—
142	C₂×C₅×D₃	—	8	2	no	—
143	C₃×C₄×C₅	—	4	4	no	—
144	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
145	C₃×C₄×C₅	#121	1	2	yes	—
146	C₅×Dic₃	—	2	4	no	—
147	C₂⁴×C₃	C₅×D₄	1	2	yes	—
148	C₂³×D₃	—	2	1	no	—
149	C₂×C₃×D₄	—	2	2	no	—
150	C₂×C₃×D₄	—	2	2	no	—
151	C₂×C₃×D₄	C₂×C₄×C₅	1	2	yes	—
152	C₂×C₃×D₄	C₅×D₄	1	2	yes	—
153	C₂²×C₃×C₄	—	2	4	no	—
154	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
155	C₂×#67	—	2	8	no	—
156	C₂×#67	—	4	8	no	—
157	C₂²×Dic₃	—	2	1	no	—
158	C₂×D₁₂	—	2	2	no	—
159	C₂×D₁₂	—	2	2	no	—
160	C₂×D₁₂	—	2	1	no	—
161	C₂×D₁₂	C₂×C₄×C₅	1	2	yes	complemented
162	C₃×C₄⋊C₄	C₅×D₄	1	4	yes	complemented
163	C₃×C₂²⋊C₄	—	2	4	no	—
164	C₃×C₂²⋊C₄	—	2	2	no	—
165	C₃×C₄²	—	2	2	no	—
166	C₄×Dic₃	—	2	1	no	—
167	C₄×Dic₃	C₂×C₄×C₅	1	2	yes	complemented
168	#208	—	2	2	no	—
169	#208	—	2	2	no	—
170	C₂×C₃⋊C₈	—	2	2	no	—
171	C₂³×C₅	—	2	2	no	—
172	C₂³×C₅	—	2	2	no	—
173	C₂³×C₅	—	4	4	no	—
174	C₂³×C₅	—	6	2	no	—
175	C₂³×C₅	—	12	4	no	—
176	C₂³×C₅	—	12	1	no	—
177	C₂³×C₅	—	12	1	no	—
178	C₂³×C₅	(C₂×D₃)⋊C₄	1	4	yes	—
179	C₂³×C₅	C₂×#67	1	1	yes	characteristic
180	C₅×D₄	—	2	4	no	—
181	C₅×D₄	—	2	4	no	—
182	C₅×D₄	—	6	4	no	—
183	C₅×D₄	—	12	16	no	—
184	C₅×D₄	—	12	2	no	—
185	C₂×C₄×C₅	—	2	4	no	—
186	C₂×C₄×C₅	—	6	4	no	—
187	C₂×C₄×C₅	—	6	2	no	—
188	C₂×C₄×C₅	—	8	4	no	—
189	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
190	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
191	C₂×C₄×C₅	#235	1	2	yes	—
192	C₂×C₄×C₅	C₂×#67	1	2	yes	—
193	C₅×C₈	—	12	2	no	—
194	C₂²×D₄	—	3	1	no	—
195	C₂²×D₄	—	6	2	no	—
196	C₂²×D₄	C₂×C₅×D₃	1	1	yes	characteristic
197	C₂×M₁₆	—	3	2	no	—
198	#127	—	3	4	no	—
199	#127	—	6	2	no	—
200	C₄×D₄	—	2	4	no	—
201	#117	C₂×C₅×D₃	1	2	yes	—
202	C₂×C₂²⋊C₄	—	2	2	no	—
203	C₂×C₄²	—	3	1	no	—
204	#102	—	6	4	no	—
205	#100	—	6	2	no	—
206	C₂×C₃×C₅	—	2	1	no	—
207	C₂×C₃×C₅	—	2	1	no	—
208	C₂×C₃×C₅	—	4	4	no	—
209	C₂×C₃×C₅	#386	1	1	yes	characteristic
210	C₂×C₃×C₅	#453	1	2	yes	—
211	C₅×D₃	—	8	2	no	—
212	C₂³×C₃	—	2	2	no	—
213	C₂³×C₃	—	2	2	no	—
214	C₂³×C₃	—	4	4	no	—
215	C₂³×C₃	C₅×C₂²⋊C₄	1	4	yes	—
216	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
217	C₂²×D₃	—	2	2	no	—
218	C₂²×D₃	—	4	4	no	—
219	C₂²×D₃	—	4	1	no	—
220	C₂²×D₃	—	4	1	no	—
221	C₃×D₄	—	2	4	no	—
222	C₃×D₄	—	2	4	no	—
223	C₂×C₃×C₄	—	2	4	no	—
224	C₂×C₃×C₄	—	8	4	no	—
225	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
226	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
227	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
228	C₂×C₃×C₄	C₅×#41	1	2	yes	—
229	#67	—	4	16	no	—
230	C₂×Dic₃	—	2	4	no	—
231	C₂×Dic₃	—	2	2	no	—
232	D₁₂	—	2	4	no	—
233	D₁₂	—	4	2	no	—
234	C₃⋊C₈	—	4	2	no	—
235	C₂²×C₅	—	2	2	no	—
236	C₂²×C₅	—	2	4	no	—
237	C₂²×C₅	—	4	4	no	—
238	C₂²×C₅	—	4	4	no	—
239	C₂²×C₅	—	8	2	no	—
240	C₂²×C₅	—	12	4	no	—
241	C₂²×C₅	—	12	2	no	—
242	C₂²×C₅	—	24	2	no	—
243	C₂²×C₅	—	24	2	no	—
244	C₂²×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
245	C₂²×C₅	#929	1	1	yes	characteristic
246	C₂²×C₅	#953	1	1	yes	characteristic
247	C₄×C₅	—	4	4	no	—
248	C₄×C₅	—	6	4	no	—
249	C₄×C₅	C₄×#67	1	2	yes	—
250	C₄×C₅	#930	1	2	yes	—
251	C₂⁴	—	6	1	no	—
252	C₂⁴	C₅×#67	1	2	yes	—
253	C₂×D₄	—	2	2	no	—
254	C₂×D₄	—	2	2	no	—
255	C₂×D₄	—	3	2	no	—
256	C₂×D₄	—	6	8	no	—
257	C₂×D₄	—	6	1	no	—
258	C₂×D₄	—	6	2	no	—
259	C₂×D₄	—	6	2	no	—
260	C₂×D₄	—	12	8	no	—
261	C₂×D₄	C₄×C₅×D₃	1	2	yes	—
262	C₂×D₄	C₅×D₁₂	1	2	yes	—
263	C₂²×C₄	—	2	4	no	—
264	C₂²×C₄	—	6	1	no	—
265	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
266	M₁₆	—	6	2	no	—
267	M₁₆	—	6	2	no	—
268	C₂×C₈	—	6	2	no	—
269	C₄⋊C₄	C₅×#67	1	4	yes	complemented
270	C₂²⋊C₄	—	2	4	no	—
271	C₂²⋊C₄	—	2	2	no	—
272	C₄²	—	2	2	no	—
273	C₄²	—	3	2	no	—
274	C₄²	—	6	1	no	—
275	C₃×C₅	#1887	1	1	yes	characteristic, complemented, Hall
276	C₂²×C₃	—	2	2	no	—
277	C₂²×C₃	—	2	4	no	—
278	C₂²×C₃	—	4	4	no	—
279	C₂²×C₃	—	4	4	no	—
280	C₂²×C₃	—	8	2	no	—
281	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
282	C₂²×C₃	C₅×#120	1	1	yes	characteristic
283	C₂²×C₃	C₅×#123	1	1	yes	characteristic
284	C₂×D₃	—	4	4	no	—
285	C₂×D₃	—	4	2	no	—
286	C₂×D₃	—	8	2	no	—
287	C₂×D₃	—	8	2	no	—
288	C₃×C₄	—	4	4	no	—
289	C₃×C₄	C₄×C₅×D₄	1	2	yes	—
290	C₃×C₄	C₅×#121	1	2	yes	—
291	Dic₃	—	2	4	no	—
292	C₂×C₅	—	2	1	no	—
293	C₂×C₅	—	2	1	no	—
294	C₂×C₅	—	4	4	no	—
295	C₂×C₅	—	24	2	no	—
296	C₂×C₅	#5213	1	1	yes	characteristic
297	C₂×C₅	#5241	1	2	yes	—
298	C₂³	—	2	2	no	—
299	C₂³	—	2	2	no	—
300	C₂³	—	4	4	no	—
301	C₂³	—	6	2	no	—
302	C₂³	—	12	4	no	—
303	C₂³	—	12	1	no	—
304	C₂³	—	12	1	no	—
305	C₂³	C₅×(C₂×D₃)⋊C₄	1	4	yes	—
306	C₂³	C₂×C₅×#67	1	1	yes	characteristic
307	D₄	—	2	4	no	—
308	D₄	—	2	4	no	—
309	D₄	—	6	4	no	—
310	D₄	—	12	16	no	—
311	D₄	—	12	2	no	—
312	C₂×C₄	—	2	4	no	—
313	C₂×C₄	—	6	4	no	—
314	C₂×C₄	—	6	2	no	—
315	C₂×C₄	—	8	4	no	—
316	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
317	C₂×C₄	C₂×C₅×D₁₂	1	1	yes	characteristic
318	C₂×C₄	C₅×#235	1	2	yes	—
319	C₂×C₄	C₂×C₅×#67	1	2	yes	—
320	C₈	—	12	2	no	—
321	C₂×C₃	—	2	1	no	—
322	C₂×C₃	—	2	1	no	—
323	C₂×C₃	—	4	4	no	—
324	C₂×C₃	C₅×#386	1	1	yes	characteristic
325	C₂×C₃	C₅×#453	1	2	yes	—
326	D₃	—	8	2	no	—
327	C₅	#71650	1	1	yes	characteristic, direct product factor, Sylow
328	C₂²	—	2	2	no	—
329	C₂²	—	2	4	no	—
330	C₂²	—	4	4	no	—
331	C₂²	—	4	4	no	—
332	C₂²	—	8	2	no	—
333	C₂²	—	12	4	no	—
334	C₂²	—	12	2	no	—
335	C₂²	—	24	2	no	—
336	C₂²	—	24	2	no	—
337	C₂²	C₂×C₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic
338	C₂²	C₅×#929	1	1	yes	characteristic
339	C₂²	C₅×#953	1	1	yes	characteristic
340	C₄	—	4	4	no	—
341	C₄	—	6	4	no	—
342	C₄	C₄×C₅×#67	1	2	yes	—
343	C₄	C₅×#930	1	2	yes	—
344	C₃	C₅×#1887	1	1	yes	characteristic, complemented, Sylow
345	C₂	—	2	1	no	—
346	C₂	—	2	1	no	—
347	C₂	—	4	4	no	—
348	C₂	—	24	2	no	—
349	C₂	C₅×#5213	1	1	yes	characteristic
350	C₂	C₅×#5241	1	2	yes	—

	subgroup	G/H
center	C₂²×C₅	#953
commutator	C₂×C₃×C₄	C₂²×C₄×C₅
Frattini	C₂²×C₄	C₂²×C₅×D₃
Fitting	C₃×C₅×#518	C₂
socle	C₂²×C₃×C₅	#120
prime radical	C₂×C₅×#940	C₂
cosocle	C₂²×C₃×C₄	C₂³×C₅
hypercenter	C₅×#518	D₃
hypocenter	C₃	C₅×#1887
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ C₅×#518

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×#518	C₂	semidirect
3. C₂×C₅×#833	C₂	semidirect
4. C₅×#5001	C₂	semidirect
5. C₅×#4532	C₂	semidirect
14. C₅×#940	C₄	semidirect
19. #71650	C₅	direct (N × H)
21. C₅×#518	D₃	semidirect
39. C₂×C₅×D₁₂	C₂×C₄	semidirect
40. C₃×C₅×C₄⋊C₄	D₄	semidirect
45. C₄×C₅×Dic₃	C₂×C₄	semidirect
49. C₃×#518	C₂×C₅	semidirect
51. C₂×#833	C₂×C₅	semidirect
52. #5001	C₂×C₅	semidirect
53. #4532	C₂×C₅	semidirect
97. #940	C₄×C₅	semidirect
120. C₅×C₄⋊C₄	#67	semidirect
127. #518	C₅×D₃	semidirect
161. C₂×D₁₂	C₂×C₄×C₅	semidirect
162. C₃×C₄⋊C₄	C₅×D₄	semidirect
167. C₄×Dic₃	C₂×C₄×C₅	semidirect
269. C₄⋊C₄	C₅×#67	semidirect
275. C₃×C₅	#1887	semidirect
327. C₅	#71650	direct (N × H)
344. C₃	C₅×#1887	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₅×#518	2
C₂×C₅×#940	2
C₂×C₅×#833	2
C₅×#5001	2
C₅×#4532	2
C₅×#1887	3
#71650	5

Subgroups — C5×#71650 #14301528

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#71650 #14301528