Subgroups — C₅×#71855 #14303040

Order 1920.

680 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×#5237	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅×#833	C₂	1	2	yes	maximal, complemented
5	C₅×#4533	C₂	1	2	yes	maximal, complemented
6	C₅×#1914	—	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	—	2	1	no	—
10	C₃×C₅×#124	—	2	1	no	—
11	C₃×C₅×#121	—	2	2	no	—
12	C₃×C₅×#120	C₄	1	4	yes	—
13	C₃×C₅×#117	C₂²	1	1	yes	characteristic
14	C₅×#948	—	2	4	no	—
15	C₂×C₅×#208	—	2	2	no	—
16	C₂×C₅×#208	C₂²	1	2	yes	—
17	C₂×C₅×#208	C₂²	1	2	yes	—
18	C₅×#836	—	2	1	no	—
19	C₅×#833	—	2	1	no	—
20	C₅×#833	—	2	4	no	—
21	#71855	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
22	C₅×#568	—	3	1	no	—
23	C₅×#535	D₃	1	1	yes	characteristic
24	C₅×#413	—	3	1	no	—
25	C₂×C₅×#100	—	3	2	no	—
26	C₅×#344	—	3	2	no	—
27	C₂⁴×C₃×C₅	C₂×C₄	1	2	yes	—
28	C₃×C₅×#41	—	2	2	no	—
29	C₃×C₅×#41	—	2	2	no	—
30	C₂×C₃×C₅×Q₈	D₄	1	1	yes	characteristic
31	C₂×C₃×C₅×D₄	—	2	4	no	—
32	C₂×C₃×C₅×D₄	—	2	2	no	—
33	C₂×C₃×C₅×D₄	—	2	1	no	—
34	C₂×C₃×C₅×D₄	—	2	1	no	—
35	C₂×C₃×C₅×D₄	—	2	2	no	—
36	C₂×C₃×C₅×D₄	C₂×C₄	1	2	yes	—
37	C₂×C₃×C₅×D₄	D₄	1	1	yes	characteristic
38	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
39	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
40	C₃×C₅×C₄⋊C₄	—	2	1	no	—
41	C₃×C₅×C₂²⋊C₄	—	2	4	no	—
42	C₃×C₅×C₂²⋊C₄	C₂×C₄	1	2	yes	—
43	C₃×C₄²×C₅	—	2	1	no	—
44	C₅×#208	—	2	4	no	—
45	C₅×#208	—	2	2	no	—
46	C₅×#208	—	2	2	no	—
47	C₅×#208	—	2	2	no	—
48	C₅×#208	D₄	1	4	yes	complemented
49	C₂×C₅×C₃⋊C₈	—	2	4	no	—
50	C₂×C₅×C₃⋊C₈	—	2	2	no	—
51	C₂×C₅×C₃⋊C₈	—	2	2	no	—
52	C₃×#535	C₂×C₅	1	1	yes	characteristic
53	#5819	C₂×C₅	1	1	yes	characteristic, complemented
54	#5237	C₂×C₅	1	1	yes	characteristic, complemented
55	C₂×#833	C₂×C₅	1	2	yes	complemented
56	#4533	C₂×C₅	1	2	yes	complemented
57	C₂×C₅×#41	C₂×D₃	1	1	yes	characteristic
58	C₂²×C₅×D₄	C₂×D₃	1	1	yes	characteristic
59	C₅×#131	—	6	4	no	—
60	C₂×C₅×M₁₆	—	3	2	no	—
61	C₂×C₅×M₁₆	—	3	2	no	—
62	C₂×C₅×M₁₆	—	6	2	no	—
63	C₅×#127	—	2	1	no	—
64	C₅×#124	—	2	1	no	—
65	C₅×#121	—	2	2	no	—
66	C₅×#120	Dic₃	1	4	yes	—
67	C₅×#117	C₂×D₃	1	1	yes	characteristic
68	C₅×#101	—	6	1	no	—
69	C₅×#100	—	6	4	no	—
70	C₅×#100	—	6	1	no	—
71	#1914	—	3	1	no	Sylow
72	C₂³×C₃×C₅	—	2	2	no	—
73	C₂³×C₃×C₅	—	2	4	no	—
74	C₂³×C₃×C₅	—	8	2	no	—
75	C₂³×C₃×C₅	C₂²⋊C₄	1	2	yes	—
76	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
77	C₃×C₅×Q₈	—	2	2	no	—
78	C₃×C₅×D₄	—	2	4	no	—
79	C₃×C₅×D₄	—	2	2	no	—
80	C₃×C₅×D₄	—	4	1	no	—
81	C₃×C₅×D₄	—	4	2	no	—
82	C₃×C₅×D₄	—	4	1	no	—
83	C₃×C₅×D₄	—	8	4	no	—
84	C₂×C₃×C₄×C₅	—	2	2	no	—
85	C₂×C₃×C₄×C₅	—	2	2	no	—
86	C₂×C₃×C₄×C₅	—	2	2	no	—
87	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
88	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
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	#41	1	2	yes	—
92	C₅×C₃⋊C₈	—	2	4	no	—
93	C₅×C₃⋊C₈	—	4	2	no	—
94	C₂×C₃×#41	C₂²×C₅	1	1	yes	characteristic
95	C₂²×C₃×D₄	C₂²×C₅	1	1	yes	characteristic
96	C₃×#127	—	2	1	no	—
97	C₃×#124	—	2	1	no	—
98	C₃×#121	—	2	2	no	—
99	C₃×#120	C₄×C₅	1	4	yes	—
100	C₃×#117	C₂²×C₅	1	1	yes	characteristic
101	#948	—	2	4	no	—
102	C₂×#208	—	2	2	no	—
103	C₂×#208	C₂²×C₅	1	2	yes	—
104	C₂×#208	C₂²×C₅	1	2	yes	—
105	#836	—	2	1	no	—
106	#833	—	2	1	no	—
107	#833	—	2	4	no	—
108	C₂⁴×C₅	C₂×Dic₃	1	2	yes	—
109	C₅×#41	—	2	2	no	—
110	C₅×#41	—	2	2	no	—
111	C₂×C₅×Q₈	#67	1	1	yes	characteristic
112	C₂×C₅×D₄	—	2	4	no	—
113	C₂×C₅×D₄	—	2	2	no	—
114	C₂×C₅×D₄	—	2	2	no	—
115	C₂×C₅×D₄	—	2	1	no	—
116	C₂×C₅×D₄	—	2	1	no	—
117	C₂×C₅×D₄	C₂×Dic₃	1	2	yes	—
118	C₂×C₅×D₄	#67	1	1	yes	characteristic
119	C₂²×C₄×C₅	#67	1	2	yes	—
120	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
121	C₅×M₁₆	—	3	4	no	—
122	C₅×M₁₆	—	6	4	no	—
123	C₅×M₁₆	—	6	2	no	—
124	C₅×M₁₆	—	6	2	no	—
125	C₅×M₁₆	—	6	2	no	—
126	C₂×C₅×C₈	—	6	4	no	—
127	C₂×C₅×C₈	—	6	2	no	—
128	C₂×C₅×C₈	—	6	2	no	—
129	C₅×C₄⋊C₄	—	2	1	no	—
130	C₅×C₂²⋊C₄	—	2	4	no	—
131	C₅×C₂²⋊C₄	C₂×Dic₃	1	2	yes	—
132	C₄²×C₅	—	2	1	no	—
133	#568	—	3	1	no	—
134	#535	C₅×D₃	1	1	yes	characteristic
135	#413	—	3	1	no	—
136	C₂×#100	—	3	2	no	—
137	#344	—	3	2	no	—
138	C₂²×C₃×C₅	—	2	2	no	—
139	C₂²×C₃×C₅	—	4	1	no	—
140	C₂²×C₃×C₅	—	4	2	no	—
141	C₂²×C₃×C₅	—	4	4	no	—
142	C₂²×C₃×C₅	—	4	2	no	—
143	C₂²×C₃×C₅	—	8	4	no	—
144	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
145	C₂²×C₃×C₅	C₄×D₄	1	2	yes	—
146	C₃×C₄×C₅	—	2	2	no	—
147	C₃×C₄×C₅	—	4	2	no	—
148	C₃×C₄×C₅	#120	1	1	yes	characteristic
149	C₃×C₄×C₅	#121	1	2	yes	—
150	C₃×C₄×C₅	#123	1	1	yes	characteristic
151	C₂⁴×C₃	C₂×C₄×C₅	1	2	yes	—
152	C₃×#41	—	2	2	no	—
153	C₃×#41	—	2	2	no	—
154	C₂×C₃×Q₈	C₅×D₄	1	1	yes	characteristic
155	C₂×C₃×D₄	—	2	4	no	—
156	C₂×C₃×D₄	—	2	2	no	—
157	C₂×C₃×D₄	—	2	1	no	—
158	C₂×C₃×D₄	—	2	1	no	—
159	C₂×C₃×D₄	—	2	2	no	—
160	C₂×C₃×D₄	C₂×C₄×C₅	1	2	yes	—
161	C₂×C₃×D₄	C₅×D₄	1	1	yes	characteristic
162	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
163	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
164	C₃×C₄⋊C₄	—	2	1	no	—
165	C₃×C₂²⋊C₄	—	2	4	no	—
166	C₃×C₂²⋊C₄	C₂×C₄×C₅	1	2	yes	—
167	C₃×C₄²	—	2	1	no	—
168	#208	—	2	4	no	—
169	#208	—	2	2	no	—
170	#208	—	2	2	no	—
171	#208	—	2	2	no	—
172	#208	C₅×D₄	1	4	yes	complemented
173	C₂×C₃⋊C₈	—	2	4	no	—
174	C₂×C₃⋊C₈	—	2	2	no	—
175	C₂×C₃⋊C₈	—	2	2	no	—
176	C₂³×C₅	—	2	2	no	—
177	C₂³×C₅	—	2	4	no	—
178	C₂³×C₅	—	8	2	no	—
179	C₂³×C₅	#217	1	2	yes	—
180	C₂³×C₅	C₂²×Dic₃	1	1	yes	characteristic
181	C₅×Q₈	—	2	2	no	—
182	C₅×D₄	—	2	4	no	—
183	C₅×D₄	—	2	2	no	—
184	C₅×D₄	—	4	1	no	—
185	C₅×D₄	—	4	1	no	—
186	C₅×D₄	—	4	2	no	—
187	C₅×D₄	—	8	4	no	—
188	C₂×C₄×C₅	—	2	2	no	—
189	C₂×C₄×C₅	—	2	2	no	—
190	C₂×C₄×C₅	—	2	2	no	—
191	C₂×C₄×C₅	#217	1	2	yes	—
192	C₂×C₄×C₅	D₃×D₄	1	2	yes	—
193	C₂×C₄×C₅	#237	1	2	yes	—
194	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
195	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
196	C₅×C₈	—	6	4	no	—
197	C₅×C₈	—	12	2	no	—
198	C₂×#41	C₂×C₅×D₃	1	1	yes	characteristic
199	C₂²×D₄	C₂×C₅×D₃	1	1	yes	characteristic
200	#131	—	6	4	no	—
201	C₂×M₁₆	—	3	2	no	—
202	C₂×M₁₆	—	3	2	no	—
203	C₂×M₁₆	—	6	2	no	—
204	#127	—	2	1	no	—
205	#124	—	2	1	no	—
206	#121	—	2	2	no	—
207	#120	C₅×Dic₃	1	4	yes	—
208	#117	C₂×C₅×D₃	1	1	yes	characteristic
209	#101	—	6	1	no	—
210	#100	—	6	4	no	—
211	#100	—	6	1	no	—
212	C₂×C₃×C₅	—	2	1	no	—
213	C₂×C₃×C₅	—	2	2	no	—
214	C₂×C₃×C₅	—	4	2	no	—
215	C₂×C₃×C₅	—	8	2	no	—
216	C₂×C₃×C₅	#386	1	1	yes	characteristic
217	C₂³×C₃	—	2	2	no	—
218	C₂³×C₃	—	2	4	no	—
219	C₂³×C₃	—	8	2	no	—
220	C₂³×C₃	C₅×C₂²⋊C₄	1	2	yes	—
221	C₂³×C₃	C₂²×C₄×C₅	1	1	yes	characteristic
222	C₃×Q₈	—	2	2	no	—
223	C₃×D₄	—	2	4	no	—
224	C₃×D₄	—	2	2	no	—
225	C₃×D₄	—	4	1	no	—
226	C₃×D₄	—	4	2	no	—
227	C₃×D₄	—	4	1	no	—
228	C₃×D₄	—	8	4	no	—
229	C₂×C₃×C₄	—	2	2	no	—
230	C₂×C₃×C₄	—	2	2	no	—
231	C₂×C₃×C₄	—	2	2	no	—
232	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	2	yes	—
233	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
234	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
235	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
236	C₂×C₃×C₄	C₅×#41	1	2	yes	—
237	C₃⋊C₈	—	2	4	no	—
238	C₃⋊C₈	—	4	2	no	—
239	C₂²×C₅	—	2	2	no	—
240	C₂²×C₅	—	4	1	no	—
241	C₂²×C₅	—	4	2	no	—
242	C₂²×C₅	—	4	4	no	—
243	C₂²×C₅	—	4	2	no	—
244	C₂²×C₅	—	8	4	no	—
245	C₂²×C₅	D₄×Dic₃	1	2	yes	—
246	C₂²×C₅	C₂×#217	1	1	yes	characteristic
247	C₄×C₅	—	2	2	no	—
248	C₄×C₅	—	4	2	no	—
249	C₄×C₅	#935	1	1	yes	characteristic
250	C₄×C₅	#937	1	1	yes	characteristic
251	C₄×C₅	#939	1	2	yes	—
252	C₂⁴	C₂×C₅×Dic₃	1	2	yes	—
253	#41	—	2	2	no	—
254	#41	—	2	2	no	—
255	C₂×Q₈	C₅×#67	1	1	yes	characteristic
256	C₂×D₄	—	2	4	no	—
257	C₂×D₄	—	2	2	no	—
258	C₂×D₄	—	2	2	no	—
259	C₂×D₄	—	2	1	no	—
260	C₂×D₄	—	2	1	no	—
261	C₂×D₄	C₂×C₅×Dic₃	1	2	yes	—
262	C₂×D₄	C₅×#67	1	1	yes	characteristic
263	C₂²×C₄	C₅×#67	1	2	yes	—
264	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
265	M₁₆	—	3	4	no	—
266	M₁₆	—	6	4	no	—
267	M₁₆	—	6	2	no	—
268	M₁₆	—	6	2	no	—
269	M₁₆	—	6	2	no	—
270	C₂×C₈	—	6	4	no	—
271	C₂×C₈	—	6	2	no	—
272	C₂×C₈	—	6	2	no	—
273	C₄⋊C₄	—	2	1	no	—
274	C₂²⋊C₄	—	2	4	no	—
275	C₂²⋊C₄	C₂×C₅×Dic₃	1	2	yes	—
276	C₄²	—	2	1	no	—
277	C₃×C₅	#1914	1	1	yes	characteristic, complemented, Hall
278	C₂²×C₃	—	2	2	no	—
279	C₂²×C₃	—	4	1	no	—
280	C₂²×C₃	—	4	2	no	—
281	C₂²×C₃	—	4	4	no	—
282	C₂²×C₃	—	4	2	no	—
283	C₂²×C₃	—	8	4	no	—
284	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
285	C₂²×C₃	C₄×C₅×D₄	1	2	yes	—
286	C₃×C₄	—	2	2	no	—
287	C₃×C₄	—	4	2	no	—
288	C₃×C₄	C₅×#120	1	1	yes	characteristic
289	C₃×C₄	C₅×#121	1	2	yes	—
290	C₃×C₄	C₅×#123	1	1	yes	characteristic
291	C₂×C₅	—	2	1	no	—
292	C₂×C₅	—	2	2	no	—
293	C₂×C₅	—	4	2	no	—
294	C₂×C₅	—	8	2	no	—
295	C₂×C₅	#5220	1	1	yes	characteristic
296	C₂³	—	2	2	no	—
297	C₂³	—	2	4	no	—
298	C₂³	—	8	2	no	—
299	C₂³	C₅×#217	1	2	yes	—
300	C₂³	C₂²×C₅×Dic₃	1	1	yes	characteristic
301	Q₈	—	2	2	no	—
302	D₄	—	2	4	no	—
303	D₄	—	2	2	no	—
304	D₄	—	4	1	no	—
305	D₄	—	4	1	no	—
306	D₄	—	4	2	no	—
307	D₄	—	8	4	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	2	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	C₂×C₅×#67	1	1	yes	characteristic
315	C₂×C₄	C₂×C₅×#67	1	1	yes	characteristic
316	C₈	—	6	4	no	—
317	C₈	—	12	2	no	—
318	C₂×C₃	—	2	1	no	—
319	C₂×C₃	—	2	2	no	—
320	C₂×C₃	—	4	2	no	—
321	C₂×C₃	—	8	2	no	—
322	C₂×C₃	C₅×#386	1	1	yes	characteristic
323	C₅	#71855	1	1	yes	characteristic, direct product factor, Sylow
324	C₂²	—	2	2	no	—
325	C₂²	—	4	1	no	—
326	C₂²	—	4	2	no	—
327	C₂²	—	4	4	no	—
328	C₂²	—	4	2	no	—
329	C₂²	—	8	4	no	—
330	C₂²	C₅×D₄×Dic₃	1	2	yes	—
331	C₂²	C₂×C₅×#217	1	1	yes	characteristic
332	C₄	—	2	2	no	—
333	C₄	—	4	2	no	—
334	C₄	C₅×#935	1	1	yes	characteristic
335	C₄	C₅×#937	1	1	yes	characteristic
336	C₄	C₅×#939	1	2	yes	—
337	C₃	C₅×#1914	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	C₂³×C₃	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×#1914
LCS	G ▷ C₂³×C₃ ▷ 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×#5237	C₂	semidirect
4. C₂×C₅×#833	C₂	semidirect
5. C₅×#4533	C₂	semidirect
21. #71855	C₅	direct (N × H)
48. C₅×#208	D₄	semidirect
53. #5819	C₂×C₅	semidirect
54. #5237	C₂×C₅	semidirect
55. C₂×#833	C₂×C₅	semidirect
56. #4533	C₂×C₅	semidirect
172. #208	C₅×D₄	semidirect
277. C₃×C₅	#1914	semidirect
323. C₅	#71855	direct (N × H)
337. C₃	C₅×#1914	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×#5237	2
C₂×C₅×#833	2
C₅×#4533	2
C₅×#1914	3
#71855	5

Subgroups — C5×#71855 #14303040

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#71855 #14303040