Subgroups — C₅×#71854 #14302836

Order 1920.

688 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#535	C₂	1	1	yes	maximal, characteristic
2	C₅×#5819	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#5240	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#5215	C₂	1	2	yes	maximal, complemented
5	C₅×#4531	C₂	1	2	yes	maximal, complemented
6	C₅×#1910	—	3	1	no	maximal, Hall
7	C₂×C₃×C₅×#41	C₂²	1	1	yes	characteristic
8	C₂²×C₃×C₅×D₄	C₂²	1	1	yes	characteristic
9	C₃×C₅×#127	C₄	1	2	yes	complemented
10	C₃×C₅×#124	C₄	1	2	yes	complemented
11	C₃×C₅×#121	—	2	2	no	—
12	C₃×C₅×#120	—	2	2	no	—
13	C₃×C₅×#117	C₂²	1	1	yes	characteristic
14	C₅×#948	—	2	4	no	—
15	C₅×#926	C₂²	1	2	yes	—
16	C₂×C₅×#208	—	2	2	no	—
17	C₂×C₅×#208	C₂²	1	2	yes	—
18	C₅×#834	—	2	2	no	—
19	C₅×#832	—	2	4	no	—
20	#71854	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
21	C₅×#568	—	3	1	no	—
22	C₅×#535	D₃	1	1	yes	characteristic
23	C₅×#418	—	3	2	no	—
24	C₅×#410	—	3	1	no	—
25	C₅×H(3,ℤ/4)	—	3	2	no	—
26	C₂⁴×C₃×C₅	—	2	1	no	—
27	C₃×C₅×#41	—	2	2	no	—
28	C₃×C₅×#41	—	2	2	no	—
29	C₂×C₃×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
30	C₂×C₃×C₅×D₄	—	2	4	no	—
31	C₂×C₃×C₅×D₄	—	2	2	no	—
32	C₂×C₃×C₅×D₄	—	2	1	no	—
33	C₂×C₃×C₅×D₄	—	2	2	no	—
34	C₂×C₃×C₅×D₄	C₂×C₄	1	2	yes	—
35	C₂×C₃×C₅×D₄	C₂×C₄	1	1	yes	characteristic
36	C₂×C₃×C₅×D₄	D₄	1	2	yes	—
37	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
38	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
39	C₃×C₅×C₄⋊C₄	—	2	1	no	—
40	C₃×C₅×C₂²⋊C₄	—	2	1	no	—
41	C₃×C₅×C₂²⋊C₄	—	2	4	no	—
42	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
43	C₅×#217	—	2	2	no	—
44	C₅×C₄⋊Dic₃	—	2	2	no	—
45	C₄×C₅×Dic₃	—	2	2	no	—
46	C₅×#208	—	2	4	no	—
47	C₅×#208	—	2	2	no	—
48	C₅×#208	—	2	2	no	—
49	C₂×C₅×C₃⋊C₈	—	2	4	no	—
50	C₂×C₅×C₃⋊C₈	D₄	1	4	yes	complemented
51	C₃×#535	C₂×C₅	1	1	yes	characteristic
52	#5819	C₂×C₅	1	1	yes	characteristic, complemented
53	#5240	C₂×C₅	1	1	yes	characteristic, complemented
54	#5215	C₂×C₅	1	2	yes	complemented
55	#4531	C₂×C₅	1	2	yes	complemented
56	C₂×C₅×#41	C₂×D₃	1	1	yes	characteristic
57	C₂²×C₅×D₄	C₂×D₃	1	1	yes	characteristic
58	C₅×#131	—	6	4	no	—
59	C₂×C₅×M₁₆	—	3	2	no	—
60	C₂×C₅×M₁₆	—	6	2	no	—
61	C₅×#127	Dic₃	1	2	yes	complemented
62	C₅×#124	Dic₃	1	2	yes	complemented
63	C₅×#121	—	2	2	no	—
64	C₅×#120	—	2	2	no	—
65	C₅×#117	—	3	2	no	—
66	C₅×#117	C₂×D₃	1	1	yes	characteristic
67	C₅×#102	—	6	4	no	—
68	C₅×#99	—	6	2	no	—
69	#1910	—	3	1	no	Sylow
70	C₂³×C₃×C₅	—	2	2	no	—
71	C₂³×C₃×C₅	—	4	2	no	—
72	C₂³×C₃×C₅	—	8	2	no	—
73	C₂³×C₃×C₅	C₂²⋊C₄	1	2	yes	—
74	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
75	C₃×C₅×Q₈	—	2	2	no	—
76	C₃×C₅×D₄	—	2	4	no	—
77	C₃×C₅×D₄	—	2	2	no	—
78	C₃×C₅×D₄	—	2	4	no	—
79	C₃×C₅×D₄	—	4	1	no	—
80	C₃×C₅×D₄	—	4	1	no	—
81	C₃×C₅×D₄	—	8	4	no	—
82	C₂×C₃×C₄×C₅	—	2	2	no	—
83	C₂×C₃×C₄×C₅	—	2	2	no	—
84	C₂×C₃×C₄×C₅	—	2	2	no	—
85	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
86	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
88	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
89	C₂×C₃×C₄×C₅	#41	1	2	yes	—
90	C₂×C₅×Dic₃	—	4	2	no	—
91	C₅×C₃⋊C₈	—	2	4	no	—
92	C₂×C₃×#41	C₂²×C₅	1	1	yes	characteristic
93	C₂²×C₃×D₄	C₂²×C₅	1	1	yes	characteristic
94	C₃×#127	C₄×C₅	1	2	yes	complemented
95	C₃×#124	C₄×C₅	1	2	yes	complemented
96	C₃×#121	—	2	2	no	—
97	C₃×#120	—	2	2	no	—
98	C₃×#117	C₂²×C₅	1	1	yes	characteristic
99	#948	—	2	4	no	—
100	#926	C₂²×C₅	1	2	yes	—
101	C₂×#208	—	2	2	no	—
102	C₂×#208	C₂²×C₅	1	2	yes	—
103	#834	—	2	2	no	—
104	#832	—	2	4	no	—
105	C₂⁴×C₅	—	2	1	no	—
106	C₅×#41	—	2	2	no	—
107	C₅×#41	—	2	2	no	—
108	C₂×C₅×Q₈	C₂×Dic₃	1	1	yes	characteristic
109	C₂×C₅×D₄	—	2	4	no	—
110	C₂×C₅×D₄	—	2	2	no	—
111	C₂×C₅×D₄	—	2	2	no	—
112	C₂×C₅×D₄	—	2	1	no	—
113	C₂×C₅×D₄	C₂×Dic₃	1	2	yes	—
114	C₂×C₅×D₄	C₂×Dic₃	1	1	yes	characteristic
115	C₂×C₅×D₄	#67	1	2	yes	—
116	C₂²×C₄×C₅	#67	1	2	yes	—
117	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
118	C₅×M₁₆	—	6	4	no	—
119	C₅×M₁₆	—	6	2	no	—
120	C₅×M₁₆	—	6	2	no	—
121	C₂×C₅×C₈	—	3	4	no	—
122	C₂×C₅×C₈	—	6	4	no	—
123	C₅×C₄⋊C₄	—	2	1	no	—
124	C₅×C₄⋊C₄	—	6	2	no	—
125	C₅×C₂²⋊C₄	—	2	1	no	—
126	C₅×C₂²⋊C₄	—	2	4	no	—
127	C₅×C₂²⋊C₄	—	6	2	no	—
128	C₄²×C₅	—	6	2	no	—
129	C₄²×C₅	C₂×Dic₃	1	2	yes	—
130	#568	—	3	1	no	—
131	#535	C₅×D₃	1	1	yes	characteristic
132	#418	—	3	2	no	—
133	#410	—	3	1	no	—
134	H(3,ℤ/4)	—	3	2	no	—
135	C₂²×C₃×C₅	—	2	2	no	—
136	C₂²×C₃×C₅	—	4	1	no	—
137	C₂²×C₃×C₅	—	4	2	no	—
138	C₂²×C₃×C₅	—	4	4	no	—
139	C₂²×C₃×C₅	—	4	2	no	—
140	C₂²×C₃×C₅	—	8	4	no	—
141	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
142	C₂²×C₃×C₅	#121	1	2	yes	—
143	C₃×C₄×C₅	—	2	2	no	—
144	C₃×C₄×C₅	—	4	2	no	—
145	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
146	C₃×C₄×C₅	#120	1	1	yes	characteristic
147	C₃×C₄×C₅	#123	1	1	yes	characteristic
148	C₅×Dic₃	—	8	2	no	—
149	C₂⁴×C₃	—	2	1	no	—
150	C₃×#41	—	2	2	no	—
151	C₃×#41	—	2	2	no	—
152	C₂×C₃×Q₈	C₂×C₄×C₅	1	1	yes	characteristic
153	C₂×C₃×D₄	—	2	4	no	—
154	C₂×C₃×D₄	—	2	2	no	—
155	C₂×C₃×D₄	—	2	1	no	—
156	C₂×C₃×D₄	—	2	2	no	—
157	C₂×C₃×D₄	C₂×C₄×C₅	1	2	yes	—
158	C₂×C₃×D₄	C₂×C₄×C₅	1	1	yes	characteristic
159	C₂×C₃×D₄	C₅×D₄	1	2	yes	—
160	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
161	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
162	C₃×C₄⋊C₄	—	2	1	no	—
163	C₃×C₂²⋊C₄	—	2	1	no	—
164	C₃×C₂²⋊C₄	—	2	4	no	—
165	C₃×C₄²	C₂×C₄×C₅	1	2	yes	—
166	#217	—	2	2	no	—
167	C₄⋊Dic₃	—	2	2	no	—
168	C₄×Dic₃	—	2	2	no	—
169	#208	—	2	4	no	—
170	#208	—	2	2	no	—
171	#208	—	2	2	no	—
172	C₂×C₃⋊C₈	—	2	4	no	—
173	C₂×C₃⋊C₈	C₅×D₄	1	4	yes	complemented
174	C₂³×C₅	—	2	2	no	—
175	C₂³×C₅	—	4	2	no	—
176	C₂³×C₅	—	8	2	no	—
177	C₂³×C₅	#217	1	2	yes	—
178	C₂³×C₅	C₂×#67	1	1	yes	characteristic
179	C₅×Q₈	—	2	2	no	—
180	C₅×D₄	—	2	4	no	—
181	C₅×D₄	—	2	2	no	—
182	C₅×D₄	—	2	4	no	—
183	C₅×D₄	—	4	1	no	—
184	C₅×D₄	—	4	1	no	—
185	C₅×D₄	—	8	4	no	—
186	C₂×C₄×C₅	—	2	2	no	—
187	C₂×C₄×C₅	—	2	2	no	—
188	C₂×C₄×C₅	—	2	2	no	—
189	C₂×C₄×C₅	—	12	2	no	—
190	C₂×C₄×C₅	#217	1	2	yes	—
191	C₂×C₄×C₅	D₃×D₄	1	2	yes	—
192	C₂×C₄×C₅	#237	1	2	yes	—
193	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
194	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
195	C₅×C₈	—	6	4	no	—
196	C₂×#41	C₂×C₅×D₃	1	1	yes	characteristic
197	C₂²×D₄	C₂×C₅×D₃	1	1	yes	characteristic
198	#131	—	6	4	no	—
199	C₂×M₁₆	—	3	2	no	—
200	C₂×M₁₆	—	6	2	no	—
201	#127	C₅×Dic₃	1	2	yes	complemented
202	#124	C₅×Dic₃	1	2	yes	complemented
203	#121	—	2	2	no	—
204	#120	—	2	2	no	—
205	#117	—	3	2	no	—
206	#117	C₂×C₅×D₃	1	1	yes	characteristic
207	#102	—	6	4	no	—
208	#99	—	6	2	no	—
209	C₂×C₃×C₅	—	2	2	no	—
210	C₂×C₃×C₅	—	2	1	no	—
211	C₂×C₃×C₅	—	4	2	no	—
212	C₂×C₃×C₅	—	8	2	no	—
213	C₂×C₃×C₅	#386	1	1	yes	characteristic
214	C₂³×C₃	—	2	2	no	—
215	C₂³×C₃	—	4	2	no	—
216	C₂³×C₃	—	8	2	no	—
217	C₂³×C₃	C₅×C₂²⋊C₄	1	2	yes	—
218	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
219	C₃×Q₈	—	2	2	no	—
220	C₃×D₄	—	2	4	no	—
221	C₃×D₄	—	2	2	no	—
222	C₃×D₄	—	2	4	no	—
223	C₃×D₄	—	4	1	no	—
224	C₃×D₄	—	4	1	no	—
225	C₃×D₄	—	8	4	no	—
226	C₂×C₃×C₄	—	2	2	no	—
227	C₂×C₃×C₄	—	2	2	no	—
228	C₂×C₃×C₄	—	2	2	no	—
229	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	2	yes	—
230	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
231	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
232	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
233	C₂×C₃×C₄	C₅×#41	1	2	yes	—
234	C₂×Dic₃	—	4	2	no	—
235	C₃⋊C₈	—	2	4	no	—
236	C₂²×C₅	—	2	2	no	—
237	C₂²×C₅	—	4	1	no	—
238	C₂²×C₅	—	4	2	no	—
239	C₂²×C₅	—	4	4	no	—
240	C₂²×C₅	—	4	2	no	—
241	C₂²×C₅	—	8	4	no	—
242	C₂²×C₅	#939	1	2	yes	—
243	C₂²×C₅	C₂×#217	1	1	yes	characteristic
244	C₄×C₅	—	2	2	no	—
245	C₄×C₅	—	4	2	no	—
246	C₄×C₅	—	24	2	no	—
247	C₄×C₅	D₄×Dic₃	1	2	yes	—
248	C₄×C₅	#935	1	1	yes	characteristic
249	C₄×C₅	#937	1	1	yes	characteristic
250	C₂⁴	—	2	1	no	—
251	#41	—	2	2	no	—
252	#41	—	2	2	no	—
253	C₂×Q₈	C₂×C₅×Dic₃	1	1	yes	characteristic
254	C₂×D₄	—	2	4	no	—
255	C₂×D₄	—	2	2	no	—
256	C₂×D₄	—	2	2	no	—
257	C₂×D₄	—	2	1	no	—
258	C₂×D₄	C₂×C₅×Dic₃	1	2	yes	—
259	C₂×D₄	C₂×C₅×Dic₃	1	1	yes	characteristic
260	C₂×D₄	C₅×#67	1	2	yes	—
261	C₂²×C₄	C₅×#67	1	2	yes	—
262	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
263	M₁₆	—	6	4	no	—
264	M₁₆	—	6	2	no	—
265	M₁₆	—	6	2	no	—
266	C₂×C₈	—	3	4	no	—
267	C₂×C₈	—	6	4	no	—
268	C₄⋊C₄	—	2	1	no	—
269	C₄⋊C₄	—	6	2	no	—
270	C₂²⋊C₄	—	2	1	no	—
271	C₂²⋊C₄	—	2	4	no	—
272	C₂²⋊C₄	—	6	2	no	—
273	C₄²	—	6	2	no	—
274	C₄²	C₂×C₅×Dic₃	1	2	yes	—
275	C₃×C₅	#1910	1	1	yes	characteristic, complemented, Hall
276	C₂²×C₃	—	2	2	no	—
277	C₂²×C₃	—	4	1	no	—
278	C₂²×C₃	—	4	2	no	—
279	C₂²×C₃	—	4	4	no	—
280	C₂²×C₃	—	4	2	no	—
281	C₂²×C₃	—	8	4	no	—
282	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
283	C₂²×C₃	C₅×#121	1	2	yes	—
284	C₃×C₄	—	2	2	no	—
285	C₃×C₄	—	4	2	no	—
286	C₃×C₄	C₄×C₅×D₄	1	2	yes	—
287	C₃×C₄	C₅×#120	1	1	yes	characteristic
288	C₃×C₄	C₅×#123	1	1	yes	characteristic
289	Dic₃	—	8	2	no	—
290	C₂×C₅	—	2	1	no	—
291	C₂×C₅	—	2	2	no	—
292	C₂×C₅	—	4	2	no	—
293	C₂×C₅	—	8	2	no	—
294	C₂×C₅	#5220	1	1	yes	characteristic
295	C₂³	—	2	2	no	—
296	C₂³	—	4	2	no	—
297	C₂³	—	8	2	no	—
298	C₂³	C₅×#217	1	2	yes	—
299	C₂³	C₂×C₅×#67	1	1	yes	characteristic
300	Q₈	—	2	2	no	—
301	D₄	—	2	4	no	—
302	D₄	—	2	2	no	—
303	D₄	—	2	4	no	—
304	D₄	—	4	1	no	—
305	D₄	—	4	1	no	—
306	D₄	—	8	4	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	12	2	no	—
311	C₂×C₄	C₅×#217	1	2	yes	—
312	C₂×C₄	C₅×D₃×D₄	1	2	yes	—
313	C₂×C₄	C₅×#237	1	2	yes	—
314	C₂×C₄	C₂²×C₅×Dic₃	1	1	yes	characteristic
315	C₂×C₄	C₂×C₅×#67	1	1	yes	characteristic
316	C₈	—	6	4	no	—
317	C₂×C₃	—	2	2	no	—
318	C₂×C₃	—	2	1	no	—
319	C₂×C₃	—	4	2	no	—
320	C₂×C₃	—	8	2	no	—
321	C₂×C₃	C₅×#386	1	1	yes	characteristic
322	C₅	#71854	1	1	yes	characteristic, direct product factor, Sylow
323	C₂²	—	2	2	no	—
324	C₂²	—	4	1	no	—
325	C₂²	—	4	2	no	—
326	C₂²	—	4	4	no	—
327	C₂²	—	4	2	no	—
328	C₂²	—	8	4	no	—
329	C₂²	C₅×#939	1	2	yes	—
330	C₂²	C₂×C₅×#217	1	1	yes	characteristic
331	C₄	—	2	2	no	—
332	C₄	—	4	2	no	—
333	C₄	—	24	2	no	—
334	C₄	C₅×D₄×Dic₃	1	2	yes	—
335	C₄	C₅×#935	1	1	yes	characteristic
336	C₄	C₅×#937	1	1	yes	characteristic
337	C₃	C₅×#1910	1	1	yes	characteristic, complemented, Sylow
338	C₂	—	2	1	no	—
339	C₂	—	2	2	no	—
340	C₂	—	4	2	no	—
341	C₂	—	8	2	no	—
342	C₂	C₅×#5220	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₅	#5220
commutator	C₂×C₃×C₄	C₂²×C₄×C₅
Frattini	C₂²×C₄	C₂²×C₅×D₃
Fitting, prime radical	C₃×C₅×#535	C₂
socle	C₂×C₃×C₅	#386
cosocle	C₂²×C₃×C₄	C₂³×C₅
hypercenter	C₅×#535	D₃
hypocenter	C₃	C₅×#1910
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₅ ◁ C₂²×C₄×C₅ ◁ C₅×#535

Complements

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

N	H ≅ G/N	type
2. C₅×#5819	C₂	semidirect
3. C₅×#5240	C₂	semidirect
4. C₅×#5215	C₂	semidirect
5. C₅×#4531	C₂	semidirect
9. C₃×C₅×#127	C₄	semidirect
10. C₃×C₅×#124	C₄	semidirect
20. #71854	C₅	direct (N × H)
50. C₂×C₅×C₃⋊C₈	D₄	semidirect
52. #5819	C₂×C₅	semidirect
53. #5240	C₂×C₅	semidirect
54. #5215	C₂×C₅	semidirect
55. #4531	C₂×C₅	semidirect
61. C₅×#127	Dic₃	semidirect
62. C₅×#124	Dic₃	semidirect
94. C₃×#127	C₄×C₅	semidirect
95. C₃×#124	C₄×C₅	semidirect
173. C₂×C₃⋊C₈	C₅×D₄	semidirect
201. #127	C₅×Dic₃	semidirect
202. #124	C₅×Dic₃	semidirect
275. C₃×C₅	#1910	semidirect
322. C₅	#71854	direct (N × H)
337. C₃	C₅×#1910	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₅×#535	2
C₅×#5819	2
C₅×#5240	2
C₅×#5215	2
C₅×#4531	2
C₅×#1910	3
#71854	5

Subgroups — C5×#71854 #14302836

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#71854 #14302836