Subgroups — C₅×#83017 #14430675

Order 1920.

768 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	C₂×C₃×C₅×#125	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×C₅×#235	C₂	1	1	yes	maximal, characteristic
3	C₅×#5066	C₂	1	4	yes	maximal, complemented
4	C₅×#5063	C₂	1	4	yes	maximal, complemented
5	C₅×#4966	C₂	1	2	yes	maximal
6	C₅×#4964	C₂	1	2	yes	maximal, complemented
7	C₅×#4924	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#3080	—	3	1	no	maximal, Hall
9	C₂×C₅×#235	C₂²	1	1	yes	characteristic
10	C₃×C₅×#125	—	2	1	no	—
11	C₃×C₅×#125	—	2	1	no	—
12	C₃×C₅×#125	C₂²	1	4	yes	—
13	C₂×C₃×C₅×C₄⋊C₄	—	2	2	no	—
14	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
15	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
16	C₄×C₅×#67	—	2	2	no	—
17	C₅×#926	C₂²	1	1	yes	characteristic
18	C₂×C₅×#208	C₂²	1	2	yes	—
19	C₄×C₅×D₁₂	C₂²	1	2	yes	—
20	C₅×#872	—	2	1	no	—
21	C₄²×C₅×D₃	—	2	1	no	—
22	C₄×C₅×Dic₆	C₂²	1	2	yes	—
23	C₅×#810	C₂²	1	4	yes	—
24	C₅×#809	C₂²	1	4	yes	—
25	C₅×#808	C₂²	1	4	yes	—
26	C₅×#807	C₂²	1	4	yes	—
27	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	4	yes	—
28	#83017	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
29	C₂×C₅×#125	D₃	1	1	yes	characteristic, complemented
30	C₄×C₅×#41	—	3	1	no	—
31	C₅×#479	—	3	4	no	—
32	C₅×#478	—	3	4	no	—
33	C₅×#428	—	3	2	no	—
34	C₅×#423	—	3	1	no	—
35	C₅×#417	—	3	2	no	—
36	C₂²×C₃×C₄×C₅	—	2	2	no	—
37	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
38	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
39	C₂×C₅×#67	—	2	1	no	—
40	C₅×#235	—	2	2	no	—
41	C₅×#235	Q₈	1	4	yes	—
42	C₂×C₅×D₁₂	C₂³	1	1	yes	characteristic
43	C₂×C₄×C₅×D₃	—	2	1	no	—
44	C₂×C₄×C₅×D₃	—	2	2	no	—
45	C₂×C₅×Dic₆	C₂³	1	1	yes	characteristic
46	C₃×C₅×C₄⋊C₄	—	2	4	no	—
47	C₃×C₅×C₄⋊C₄	—	2	4	no	—
48	C₃×C₅×C₄⋊C₄	—	2	2	no	—
49	C₃×C₅×C₄⋊C₄	C₂³	1	4	yes	—
50	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
51	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
52	C₃×C₄²×C₅	C₂³	1	2	yes	—
53	C₅×#217	—	2	1	no	—
54	C₅×(C₂×D₃)⋊C₄	—	2	2	no	—
55	C₅×C₄⋊Dic₃	C₂³	1	2	yes	—
56	C₅×Dic₃⋊C₄	—	2	2	no	—
57	C₄×C₅×Dic₃	—	2	2	no	—
58	C₄×C₅×Dic₃	—	2	1	no	—
59	C₅×#208	—	2	4	no	—
60	C₂×C₅×C₃⋊C₈	C₂³	1	4	yes	—
61	C₂×C₃×#125	C₂×C₅	1	1	yes	characteristic, complemented
62	C₄×#235	C₂×C₅	1	1	yes	characteristic
63	#5066	C₂×C₅	1	4	yes	complemented
64	#5063	C₂×C₅	1	4	yes	complemented
65	#4966	C₂×C₅	1	2	yes	—
66	#4964	C₂×C₅	1	2	yes	complemented
67	#4924	C₂×C₅	1	1	yes	characteristic, complemented
68	C₂×C₅×#41	—	3	1	no	—
69	C₂×C₅×M₁₆	—	3	2	no	—
70	C₅×#125	—	2	1	no	—
71	C₅×#125	—	2	1	no	—
72	C₅×#125	C₂×D₃	1	4	yes	—
73	C₄×C₅×Q₈	—	3	2	no	—
74	C₄×C₅×D₄	—	3	2	no	—
75	C₄×C₅×D₄	—	6	2	no	—
76	C₅×#117	—	3	1	no	—
77	C₅×#117	—	6	1	no	—
78	C₂×C₅×C₄⋊C₄	—	2	2	no	—
79	C₂×C₅×C₄⋊C₄	C₂×D₃	1	2	yes	—
80	C₂×C₄²×C₅	—	6	1	no	—
81	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
82	C₅×#107	—	3	4	no	—
83	C₅×#106	—	3	4	no	—
84	C₅×C₄⋊C₈	—	3	4	no	—
85	C₅×#103	—	3	4	no	—
86	C₅×#102	—	3	4	no	—
87	#3080	—	3	1	no	Sylow
88	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₅×D₃	—	2	1	no	—
90	C₂×C₃×C₄×C₅	—	2	4	no	—
91	C₂×C₃×C₄×C₅	—	4	2	no	—
92	C₂×C₃×C₄×C₅	—	4	4	no	—
93	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
94	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
95	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
96	C₂×C₃×C₄×C₅	C₂×Q₈	1	2	yes	—
97	C₂×C₃×C₄×C₅	#41	1	2	yes	—
98	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
99	C₅×#67	—	2	4	no	—
100	C₂×C₅×Dic₃	—	2	1	no	—
101	C₂×C₅×Dic₃	—	2	2	no	—
102	C₅×D₁₂	—	2	1	no	—
103	C₅×D₁₂	C₂×Q₈	1	2	yes	—
104	C₄×C₅×D₃	—	2	4	no	—
105	C₄×C₅×D₃	—	4	4	no	—
106	C₅×Dic₆	—	2	1	no	—
107	C₅×Dic₆	C₂×Q₈	1	2	yes	—
108	C₅×C₃⋊C₈	—	2	4	no	—
109	C₂×#235	C₂²×C₅	1	1	yes	characteristic
110	C₃×#125	—	2	1	no	—
111	C₃×#125	—	2	1	no	—
112	C₃×#125	C₂²×C₅	1	4	yes	—
113	C₂×C₃×C₄⋊C₄	—	2	2	no	—
114	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
115	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
116	C₄×#67	—	2	2	no	—
117	#926	C₂²×C₅	1	1	yes	characteristic
118	C₂×#208	C₂²×C₅	1	2	yes	—
119	C₄×D₁₂	C₂²×C₅	1	2	yes	—
120	#872	—	2	1	no	—
121	C₄²×D₃	—	2	1	no	—
122	C₄×Dic₆	C₂²×C₅	1	2	yes	—
123	#810	C₂²×C₅	1	4	yes	—
124	#809	C₂²×C₅	1	4	yes	—
125	#808	C₂²×C₅	1	4	yes	—
126	#807	C₂²×C₅	1	4	yes	—
127	C₄⋊(C₃⋊C₈)	C₂²×C₅	1	4	yes	—
128	C₅×#41	—	3	4	no	—
129	C₅×#41	—	6	2	no	—
130	C₂×C₅×Q₈	—	3	1	no	—
131	C₂×C₅×D₄	—	3	1	no	—
132	C₂×C₅×D₄	—	6	1	no	—
133	C₂²×C₄×C₅	—	2	2	no	—
134	C₂²×C₄×C₅	—	6	1	no	—
135	C₂²×C₄×C₅	—	6	2	no	—
136	C₂²×C₄×C₅	#67	1	2	yes	—
137	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
138	C₅×M₁₆	—	6	4	no	—
139	C₂×C₅×C₈	—	3	4	no	—
140	C₅×C₄⋊C₄	—	2	4	no	—
141	C₅×C₄⋊C₄	—	2	4	no	—
142	C₅×C₄⋊C₄	—	2	2	no	—
143	C₅×C₄⋊C₄	—	3	2	no	—
144	C₅×C₄⋊C₄	—	6	2	no	—
145	C₅×C₄⋊C₄	C₂²×D₃	1	4	yes	—
146	C₅×C₂²⋊C₄	—	6	1	no	—
147	C₅×C₂²⋊C₄	—	6	2	no	—
148	C₄²×C₅	—	6	1	no	—
149	C₄²×C₅	—	6	2	no	—
150	C₄²×C₅	#67	1	1	yes	characteristic
151	C₄²×C₅	#67	1	1	yes	characteristic
152	C₄²×C₅	C₂²×D₃	1	2	yes	—
153	C₂×#125	C₅×D₃	1	1	yes	characteristic, complemented
154	C₄×#41	—	3	1	no	—
155	#479	—	3	4	no	—
156	#478	—	3	4	no	—
157	#428	—	3	2	no	—
158	#423	—	3	1	no	—
159	#417	—	3	2	no	—
160	C₂²×C₃×C₅	—	2	2	no	—
161	C₂²×C₃×C₅	C₂²⋊Q₈	1	2	yes	—
162	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
163	C₂×C₅×D₃	—	2	2	no	—
164	C₂×C₅×D₃	—	4	2	no	—
165	C₃×C₄×C₅	—	2	2	no	—
166	C₃×C₄×C₅	—	2	2	no	—
167	C₃×C₄×C₅	—	4	4	no	—
168	C₃×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
169	C₃×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
170	C₃×C₄×C₅	C₂×#41	1	1	yes	characteristic
171	C₅×Dic₃	—	2	2	no	—
172	C₅×Dic₃	—	4	2	no	—
173	C₂²×C₃×C₄	—	2	2	no	—
174	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
175	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
176	C₂×#67	—	2	1	no	—
177	#235	—	2	2	no	—
178	#235	C₅×Q₈	1	4	yes	—
179	C₂×D₁₂	C₂³×C₅	1	1	yes	characteristic
180	C₂×C₄×D₃	—	2	1	no	—
181	C₂×C₄×D₃	—	2	2	no	—
182	C₂×Dic₆	C₂³×C₅	1	1	yes	characteristic
183	C₃×C₄⋊C₄	—	2	4	no	—
184	C₃×C₄⋊C₄	—	2	4	no	—
185	C₃×C₄⋊C₄	—	2	2	no	—
186	C₃×C₄⋊C₄	C₂³×C₅	1	4	yes	—
187	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
188	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
189	C₃×C₄²	C₂³×C₅	1	2	yes	—
190	#217	—	2	1	no	—
191	(C₂×D₃)⋊C₄	—	2	2	no	—
192	C₄⋊Dic₃	C₂³×C₅	1	2	yes	—
193	Dic₃⋊C₄	—	2	2	no	—
194	C₄×Dic₃	—	2	2	no	—
195	C₄×Dic₃	—	2	1	no	—
196	#208	—	2	4	no	—
197	C₂×C₃⋊C₈	C₂³×C₅	1	4	yes	—
198	C₂³×C₅	—	6	1	no	—
199	C₂³×C₅	C₂×#67	1	1	yes	characteristic
200	C₅×Q₈	—	3	2	no	—
201	C₅×Q₈	—	6	1	no	—
202	C₅×D₄	—	3	2	no	—
203	C₅×D₄	—	6	4	no	—
204	C₅×D₄	—	6	1	no	—
205	C₂×C₄×C₅	—	2	4	no	—
206	C₂×C₄×C₅	—	4	2	no	—
207	C₂×C₄×C₅	—	4	4	no	—
208	C₂×C₄×C₅	—	6	1	no	—
209	C₂×C₄×C₅	—	6	4	no	—
210	C₂×C₄×C₅	—	6	2	no	—
211	C₂×C₄×C₅	—	12	4	no	—
212	C₂×C₄×C₅	D₃×Q₈	1	2	yes	—
213	C₂×C₄×C₅	#239	1	2	yes	—
214	C₂×C₄×C₅	C₂×#67	1	2	yes	—
215	C₂×C₄×C₅	C₂×#67	1	2	yes	—
216	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
217	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
218	C₅×C₈	—	6	4	no	—
219	C₂×#41	—	3	1	no	—
220	C₂×M₁₆	—	3	2	no	—
221	#125	—	2	1	no	—
222	#125	—	2	1	no	—
223	#125	C₂×C₅×D₃	1	4	yes	—
224	C₄×Q₈	—	3	2	no	—
225	C₄×D₄	—	3	2	no	—
226	C₄×D₄	—	6	2	no	—
227	#117	—	3	1	no	—
228	#117	—	6	1	no	—
229	C₂×C₄⋊C₄	—	2	2	no	—
230	C₂×C₄⋊C₄	C₂×C₅×D₃	1	2	yes	—
231	C₂×C₄²	—	6	1	no	—
232	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
233	#107	—	3	4	no	—
234	#106	—	3	4	no	—
235	C₄⋊C₈	—	3	4	no	—
236	#103	—	3	4	no	—
237	#102	—	3	4	no	—
238	C₂×C₃×C₅	—	2	2	no	—
239	C₂×C₃×C₅	C₂×C₂²⋊Q₈	1	1	yes	characteristic
240	C₂×C₃×C₅	#575	1	2	yes	—
241	C₅×D₃	—	4	2	no	—
242	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
243	C₂²×D₃	—	2	1	no	—
244	C₂×C₃×C₄	—	2	4	no	—
245	C₂×C₃×C₄	—	4	2	no	—
246	C₂×C₃×C₄	—	4	4	no	—
247	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
248	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
249	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
250	C₂×C₃×C₄	C₂×C₅×Q₈	1	2	yes	—
251	C₂×C₃×C₄	C₅×#41	1	2	yes	—
252	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
253	#67	—	2	4	no	—
254	C₂×Dic₃	—	2	1	no	—
255	C₂×Dic₃	—	2	2	no	—
256	D₁₂	—	2	1	no	—
257	D₁₂	C₂×C₅×Q₈	1	2	yes	—
258	C₄×D₃	—	2	4	no	—
259	C₄×D₃	—	4	4	no	—
260	Dic₆	—	2	1	no	—
261	Dic₆	C₂×C₅×Q₈	1	2	yes	—
262	C₃⋊C₈	—	2	4	no	—
263	C₂²×C₅	—	2	2	no	—
264	C₂²×C₅	—	6	2	no	—
265	C₂²×C₅	—	12	2	no	—
266	C₂²×C₅	#946	1	2	yes	—
267	C₂²×C₅	C₂²×#67	1	1	yes	characteristic
268	C₄×C₅	—	2	2	no	—
269	C₄×C₅	—	2	2	no	—
270	C₄×C₅	—	4	4	no	—
271	C₄×C₅	—	6	2	no	—
272	C₄×C₅	—	12	2	no	—
273	C₄×C₅	#946	1	2	yes	—
274	C₄×C₅	C₂×D₃×Q₈	1	1	yes	characteristic
275	C₄×C₅	C₂×#239	1	1	yes	characteristic
276	#41	—	3	4	no	—
277	#41	—	6	2	no	—
278	C₂×Q₈	—	3	1	no	—
279	C₂×D₄	—	3	1	no	—
280	C₂×D₄	—	6	1	no	—
281	C₂²×C₄	—	2	2	no	—
282	C₂²×C₄	—	6	1	no	—
283	C₂²×C₄	—	6	2	no	—
284	C₂²×C₄	C₅×#67	1	2	yes	—
285	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
286	M₁₆	—	6	4	no	—
287	C₂×C₈	—	3	4	no	—
288	C₄⋊C₄	—	2	4	no	—
289	C₄⋊C₄	—	2	4	no	—
290	C₄⋊C₄	—	2	2	no	—
291	C₄⋊C₄	—	3	2	no	—
292	C₄⋊C₄	—	6	2	no	—
293	C₄⋊C₄	C₂²×C₅×D₃	1	4	yes	—
294	C₂²⋊C₄	—	6	1	no	—
295	C₂²⋊C₄	—	6	2	no	—
296	C₄²	—	6	1	no	—
297	C₄²	—	6	2	no	—
298	C₄²	C₅×#67	1	1	yes	characteristic
299	C₄²	C₅×#67	1	1	yes	characteristic
300	C₄²	C₂²×C₅×D₃	1	2	yes	—
301	C₃×C₅	#3080	1	1	yes	characteristic, complemented, Hall
302	C₂²×C₃	—	2	2	no	—
303	C₂²×C₃	C₅×C₂²⋊Q₈	1	2	yes	—
304	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
305	C₂×D₃	—	2	2	no	—
306	C₂×D₃	—	4	2	no	—
307	C₃×C₄	—	2	2	no	—
308	C₃×C₄	—	2	2	no	—
309	C₃×C₄	—	4	4	no	—
310	C₃×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
311	C₃×C₄	C₂²×C₅×Q₈	1	1	yes	characteristic
312	C₃×C₄	C₂×C₅×#41	1	1	yes	characteristic
313	Dic₃	—	2	2	no	—
314	Dic₃	—	4	2	no	—
315	C₂×C₅	—	2	2	no	—
316	C₂×C₅	—	12	2	no	—
317	C₂×C₅	C₂×#946	1	1	yes	characteristic
318	C₂×C₅	#5822	1	2	yes	—
319	C₂³	—	6	1	no	—
320	C₂³	C₂×C₅×#67	1	1	yes	characteristic
321	Q₈	—	3	2	no	—
322	Q₈	—	6	1	no	—
323	D₄	—	3	2	no	—
324	D₄	—	6	4	no	—
325	D₄	—	6	1	no	—
326	C₂×C₄	—	2	4	no	—
327	C₂×C₄	—	4	2	no	—
328	C₂×C₄	—	4	4	no	—
329	C₂×C₄	—	6	1	no	—
330	C₂×C₄	—	6	4	no	—
331	C₂×C₄	—	6	2	no	—
332	C₂×C₄	—	12	4	no	—
333	C₂×C₄	C₅×D₃×Q₈	1	2	yes	—
334	C₂×C₄	C₅×#239	1	2	yes	—
335	C₂×C₄	C₂×C₅×#67	1	2	yes	—
336	C₂×C₄	C₂×C₅×#67	1	2	yes	—
337	C₂×C₄	C₂×C₅×#67	1	1	yes	characteristic
338	C₂×C₄	C₂³×C₅×D₃	1	1	yes	characteristic
339	C₈	—	6	4	no	—
340	C₂×C₃	—	2	2	no	—
341	C₂×C₃	C₂×C₅×C₂²⋊Q₈	1	1	yes	characteristic
342	C₂×C₃	C₅×#575	1	2	yes	—
343	D₃	—	4	2	no	—
344	C₅	#83017	1	1	yes	characteristic, direct product factor, Sylow
345	C₂²	—	2	2	no	—
346	C₂²	—	6	2	no	—
347	C₂²	—	12	2	no	—
348	C₂²	C₅×#946	1	2	yes	—
349	C₂²	C₂²×C₅×#67	1	1	yes	characteristic
350	C₄	—	2	2	no	—
351	C₄	—	2	2	no	—
352	C₄	—	4	4	no	—
353	C₄	—	6	2	no	—
354	C₄	—	12	2	no	—
355	C₄	C₅×#946	1	2	yes	—
356	C₄	C₂×C₅×D₃×Q₈	1	1	yes	characteristic
357	C₄	C₂×C₅×#239	1	1	yes	characteristic
358	C₃	C₅×#3080	1	1	yes	characteristic, complemented, Sylow
359	C₂	—	2	2	no	—
360	C₂	—	12	2	no	—
361	C₂	C₂×C₅×#946	1	1	yes	characteristic
362	C₂	C₅×#5822	1	2	yes	—

	subgroup	G/H
center	C₂²×C₅	C₂²×#67
commutator, cosocle	C₂×C₃×C₄	C₂⁴×C₅
Frattini	C₂×C₄	C₂³×C₅×D₃
Fitting	C₂×C₃×C₅×#125	C₂
socle	C₂²×C₃×C₅	C₂²×D₄
prime radical	C₂×C₅×#235	C₂²
hypercenter	C₂×C₅×#125	D₃
hypocenter	C₃	C₅×#3080
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ C₂×C₅×#125

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₅×#125	C₂	semidirect
3. C₅×#5066	C₂	semidirect
4. C₅×#5063	C₂	semidirect
6. C₅×#4964	C₂	semidirect
7. C₅×#4924	C₂	semidirect
28. #83017	C₅	direct (N × H)
29. C₂×C₅×#125	D₃	semidirect
61. C₂×C₃×#125	C₂×C₅	semidirect
63. #5066	C₂×C₅	semidirect
64. #5063	C₂×C₅	semidirect
66. #4964	C₂×C₅	semidirect
67. #4924	C₂×C₅	semidirect
153. C₂×#125	C₅×D₃	semidirect
301. C₃×C₅	#3080	semidirect
344. C₅	#83017	direct (N × H)
358. C₃	C₅×#3080	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₅×#125	2
C₄×C₅×#235	2
C₅×#5066	2
C₅×#5063	2
C₅×#4966	2
C₅×#4964	2
C₅×#4924	2
C₅×#3080	3
#83017	5

Subgroups — C5×#83017 #14430675

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#83017 #14430675