Subgroups — C₅×#82580 #14330637

Order 1920.

1376 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×#120	C₂	1	1	yes	maximal, characteristic
2	C₅×#5247	C₂	1	2	yes	maximal, complemented
3	C₅×#5221	C₂	1	3	yes	maximal, complemented
4	C₅×#5220	C₂	1	3	yes	maximal, complemented
5	C₅×#4951	C₂	1	6	yes	maximal, complemented
6	C₅×#2375	—	3	1	no	maximal, Hall
7	C₂⁵×C₃×C₅	C₂²	1	1	yes	characteristic
8	C₂²×C₃×C₅×D₄	C₂²	1	3	yes	—
9	C₂³×C₅×Dic₃	C₂²	1	1	yes	characteristic, complemented
10	C₃×C₅×#120	C₄	1	8	yes	complemented
11	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	3	yes	—
12	C₂×C₅×#217	C₂²	1	6	yes	—
13	C₂×C₅×#217	C₂²	1	6	yes	—
14	C₂×C₅×#217	C₂²	1	3	yes	complemented
15	C₂×C₅×C₄⋊Dic₃	C₂²	1	3	yes	complemented
16	C₂×C₄×C₅×Dic₃	C₂²	1	3	yes	complemented
17	C₅×C₃⋊#95	C₂²	1	6	yes	complemented
18	#82580	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
19	C₂×C₅×#120	D₃	1	1	yes	characteristic
20	C₅×#390	—	3	3	no	—
21	C₅×#388	—	3	6	no	—
22	C₅×#386	—	3	3	no	—
23	C₅×#379	—	3	2	no	—
24	C₂⁴×C₃×C₅	—	4	3	no	—
25	C₂⁴×C₃×C₅	—	4	3	no	—
26	C₂⁴×C₃×C₅	C₂×C₄	1	4	yes	complemented
27	C₂⁴×C₃×C₅	C₂³	1	3	yes	—
28	C₂⁴×C₃×C₅	C₂³	1	1	yes	characteristic
29	C₂×C₃×C₅×D₄	—	4	3	no	—
30	C₂×C₃×C₅×D₄	—	4	3	no	—
31	C₂×C₃×C₅×D₄	C₂×C₄	1	12	yes	—
32	C₂²×C₃×C₄×C₅	C₂³	1	3	yes	—
33	C₂²×C₅×Dic₃	—	4	3	no	—
34	C₂²×C₅×Dic₃	C₂³	1	6	yes	—
35	C₂²×C₅×Dic₃	C₂³	1	2	yes	—
36	C₃×C₅×C₂²⋊C₄	C₂×C₄	1	12	yes	complemented
37	C₅×#217	—	4	6	no	—
38	C₅×#217	—	4	6	no	—
39	C₅×#217	—	4	3	no	—
40	C₅×C₄⋊Dic₃	—	4	3	no	—
41	C₄×C₅×Dic₃	—	4	3	no	—
42	C₂×C₃×#120	C₂×C₅	1	1	yes	characteristic
43	#5247	C₂×C₅	1	2	yes	complemented
44	#5221	C₂×C₅	1	3	yes	complemented
45	#5220	C₂×C₅	1	3	yes	complemented
46	#4951	C₂×C₅	1	6	yes	complemented
47	C₂⁵×C₅	C₂×D₃	1	1	yes	characteristic
48	C₂²×C₅×D₄	C₂×D₃	1	3	yes	—
49	C₂³×C₄×C₅	—	3	1	no	—
50	C₅×#120	Dic₃	1	8	yes	complemented
51	C₂×C₅×C₄⋊C₄	—	3	3	no	—
52	C₂×C₅×C₂²⋊C₄	—	3	6	no	—
53	C₂×C₅×C₂²⋊C₄	—	3	6	no	—
54	C₂×C₅×C₂²⋊C₄	—	3	3	no	—
55	C₂×C₅×C₂²⋊C₄	C₂×D₃	1	3	yes	—
56	C₂×C₄²×C₅	—	3	3	no	—
57	C₅×#95	—	3	6	no	—
58	#2375	—	3	1	no	Sylow
59	C₂³×C₃×C₅	—	2	3	no	—
60	C₂³×C₃×C₅	—	2	6	no	—
61	C₂³×C₃×C₅	—	2	6	no	—
62	C₂³×C₃×C₅	—	2	3	no	—
63	C₂³×C₃×C₅	—	2	3	no	—
64	C₂³×C₃×C₅	—	2	3	no	—
65	C₂³×C₃×C₅	—	4	12	no	—
66	C₂³×C₃×C₅	—	4	12	no	—
67	C₂³×C₃×C₅	—	4	4	no	—
68	C₂³×C₃×C₅	C₂²×C₄	1	6	yes	—
69	C₂³×C₃×C₅	C₂²×C₄	1	2	yes	—
70	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
71	C₃×C₅×D₄	—	4	12	no	—
72	C₂×C₃×C₄×C₅	—	2	3	no	—
73	C₂×C₃×C₄×C₅	—	2	3	no	—
74	C₂×C₃×C₄×C₅	C₂²×C₄	1	6	yes	—
75	C₂×C₅×Dic₃	—	2	6	no	—
76	C₂×C₅×Dic₃	—	2	6	no	—
77	C₂×C₅×Dic₃	—	2	6	no	—
78	C₂×C₅×Dic₃	—	2	6	no	—
79	C₂×C₅×Dic₃	—	4	4	no	—
80	C₂⁵×C₃	C₂²×C₅	1	1	yes	characteristic
81	C₂²×C₃×D₄	C₂²×C₅	1	3	yes	—
82	C₂³×Dic₃	C₂²×C₅	1	1	yes	characteristic, complemented
83	C₃×#120	C₄×C₅	1	8	yes	complemented
84	C₂×C₃×C₂²⋊C₄	C₂²×C₅	1	3	yes	—
85	C₂×#217	C₂²×C₅	1	6	yes	—
86	C₂×#217	C₂²×C₅	1	6	yes	—
87	C₂×#217	C₂²×C₅	1	3	yes	complemented
88	C₂×C₄⋊Dic₃	C₂²×C₅	1	3	yes	complemented
89	C₂×C₄×Dic₃	C₂²×C₅	1	3	yes	complemented
90	C₃⋊#95	C₂²×C₅	1	6	yes	complemented
91	C₂⁴×C₅	—	4	3	no	—
92	C₂⁴×C₅	—	4	3	no	—
93	C₂⁴×C₅	C₂×Dic₃	1	4	yes	complemented
94	C₂⁴×C₅	C₂²×D₃	1	3	yes	—
95	C₂⁴×C₅	C₂²×D₃	1	1	yes	characteristic
96	C₂×C₅×D₄	—	4	3	no	—
97	C₂×C₅×D₄	—	4	3	no	—
98	C₂×C₅×D₄	C₂×Dic₃	1	12	yes	—
99	C₂²×C₄×C₅	—	3	6	no	—
100	C₂²×C₄×C₅	—	3	2	no	—
101	C₂²×C₄×C₅	—	12	3	no	—
102	C₂²×C₄×C₅	C₂²×D₃	1	3	yes	—
103	C₅×C₄⋊C₄	—	12	3	no	—
104	C₅×C₂²⋊C₄	—	12	6	no	—
105	C₅×C₂²⋊C₄	—	12	3	no	—
106	C₅×C₂²⋊C₄	—	12	6	no	—
107	C₅×C₂²⋊C₄	C₂×Dic₃	1	12	yes	complemented
108	C₄²×C₅	—	12	3	no	—
109	C₂×#120	C₅×D₃	1	1	yes	characteristic
110	#390	—	3	3	no	—
111	#388	—	3	6	no	—
112	#386	—	3	3	no	—
113	#379	—	3	2	no	—
114	C₂²×C₃×C₅	—	2	6	no	—
115	C₂²×C₃×C₅	—	2	6	no	—
116	C₂²×C₃×C₅	—	2	12	no	—
117	C₂²×C₃×C₅	—	4	16	no	—
118	C₂²×C₃×C₅	—	4	3	no	—
119	C₂²×C₃×C₅	—	4	3	no	—
120	C₂²×C₃×C₅	—	4	6	no	—
121	C₂²×C₃×C₅	—	4	1	no	—
122	C₂²×C₃×C₅	—	4	3	no	—
123	C₂²×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
124	C₂²×C₃×C₅	H(5,𝔽₂)	1	3	yes	—
125	C₂²×C₃×C₅	H(5,𝔽₂)	1	3	yes	—
126	C₃×C₄×C₅	—	2	6	no	—
127	C₅×Dic₃	—	4	6	no	—
128	C₅×Dic₃	—	4	2	no	—
129	C₂⁴×C₃	—	4	3	no	—
130	C₂⁴×C₃	—	4	3	no	—
131	C₂⁴×C₃	C₂×C₄×C₅	1	4	yes	complemented
132	C₂⁴×C₃	C₂³×C₅	1	3	yes	—
133	C₂⁴×C₃	C₂³×C₅	1	1	yes	characteristic
134	C₂×C₃×D₄	—	4	3	no	—
135	C₂×C₃×D₄	—	4	3	no	—
136	C₂×C₃×D₄	C₂×C₄×C₅	1	12	yes	—
137	C₂²×C₃×C₄	C₂³×C₅	1	3	yes	—
138	C₂²×Dic₃	—	4	3	no	—
139	C₂²×Dic₃	C₂³×C₅	1	2	yes	—
140	C₂²×Dic₃	C₂³×C₅	1	6	yes	—
141	C₃×C₂²⋊C₄	C₂×C₄×C₅	1	12	yes	complemented
142	#217	—	4	6	no	—
143	#217	—	4	6	no	—
144	#217	—	4	3	no	—
145	C₄⋊Dic₃	—	4	3	no	—
146	C₄×Dic₃	—	4	3	no	—
147	C₂³×C₅	—	2	3	no	—
148	C₂³×C₅	—	2	6	no	—
149	C₂³×C₅	—	2	6	no	—
150	C₂³×C₅	—	2	3	no	—
151	C₂³×C₅	—	2	3	no	—
152	C₂³×C₅	—	2	3	no	—
153	C₂³×C₅	—	4	12	no	—
154	C₂³×C₅	—	4	4	no	—
155	C₂³×C₅	—	4	12	no	—
156	C₂³×C₅	C₂²×Dic₃	1	6	yes	—
157	C₂³×C₅	C₂²×Dic₃	1	2	yes	—
158	C₂³×C₅	C₂³×D₃	1	1	yes	characteristic
159	C₅×D₄	—	4	12	no	—
160	C₂×C₄×C₅	—	2	3	no	—
161	C₂×C₄×C₅	—	2	3	no	—
162	C₂×C₄×C₅	—	6	6	no	—
163	C₂×C₄×C₅	—	6	6	no	—
164	C₂×C₄×C₅	—	6	6	no	—
165	C₂×C₄×C₅	—	6	6	no	—
166	C₂×C₄×C₅	—	12	4	no	—
167	C₂×C₄×C₅	C₂²×Dic₃	1	6	yes	—
168	C₂⁵	C₂×C₅×D₃	1	1	yes	characteristic
169	C₂²×D₄	C₂×C₅×D₃	1	3	yes	—
170	C₂³×C₄	—	3	1	no	—
171	#120	C₅×Dic₃	1	8	yes	complemented
172	C₂×C₄⋊C₄	—	3	3	no	—
173	C₂×C₂²⋊C₄	—	3	6	no	—
174	C₂×C₂²⋊C₄	—	3	6	no	—
175	C₂×C₂²⋊C₄	—	3	3	no	—
176	C₂×C₂²⋊C₄	C₂×C₅×D₃	1	3	yes	—
177	C₂×C₄²	—	3	3	no	—
178	#95	—	3	6	no	—
179	C₂×C₃×C₅	—	4	2	no	—
180	C₂×C₃×C₅	—	4	6	no	—
181	C₂×C₃×C₅	#518	1	3	yes	—
182	C₂×C₃×C₅	#560	1	3	yes	—
183	C₂×C₃×C₅	H(3,𝔽₄)	1	1	yes	characteristic
184	C₂³×C₃	—	2	3	no	—
185	C₂³×C₃	—	2	6	no	—
186	C₂³×C₃	—	2	6	no	—
187	C₂³×C₃	—	2	3	no	—
188	C₂³×C₃	—	2	3	no	—
189	C₂³×C₃	—	2	3	no	—
190	C₂³×C₃	—	4	12	no	—
191	C₂³×C₃	—	4	12	no	—
192	C₂³×C₃	—	4	4	no	—
193	C₂³×C₃	C₂²×C₄×C₅	1	6	yes	—
194	C₂³×C₃	C₂²×C₄×C₅	1	2	yes	—
195	C₂³×C₃	C₂⁴×C₅	1	1	yes	characteristic
196	C₃×D₄	—	4	12	no	—
197	C₂×C₃×C₄	—	2	3	no	—
198	C₂×C₃×C₄	—	2	3	no	—
199	C₂×C₃×C₄	C₂²×C₄×C₅	1	6	yes	—
200	C₂×Dic₃	—	2	6	no	—
201	C₂×Dic₃	—	2	6	no	—
202	C₂×Dic₃	—	2	6	no	—
203	C₂×Dic₃	—	2	6	no	—
204	C₂×Dic₃	—	4	4	no	—
205	C₂²×C₅	—	2	6	no	—
206	C₂²×C₅	—	2	6	no	—
207	C₂²×C₅	—	2	12	no	—
208	C₂²×C₅	—	4	16	no	—
209	C₂²×C₅	—	4	1	no	—
210	C₂²×C₅	—	4	3	no	—
211	C₂²×C₅	—	4	3	no	—
212	C₂²×C₅	—	4	6	no	—
213	C₂²×C₅	—	4	3	no	—
214	C₂²×C₅	#1004	1	3	yes	—
215	C₂²×C₅	#1004	1	3	yes	—
216	C₂²×C₅	C₂³×Dic₃	1	1	yes	characteristic
217	C₄×C₅	—	2	6	no	—
218	C₄×C₅	—	12	2	no	—
219	C₄×C₅	—	12	6	no	—
220	C₂⁴	—	4	3	no	—
221	C₂⁴	—	4	3	no	—
222	C₂⁴	C₂×C₅×Dic₃	1	4	yes	complemented
223	C₂⁴	C₂²×C₅×D₃	1	3	yes	—
224	C₂⁴	C₂²×C₅×D₃	1	1	yes	characteristic
225	C₂×D₄	—	4	3	no	—
226	C₂×D₄	—	4	3	no	—
227	C₂×D₄	C₂×C₅×Dic₃	1	12	yes	—
228	C₂²×C₄	—	3	2	no	—
229	C₂²×C₄	—	3	6	no	—
230	C₂²×C₄	—	12	3	no	—
231	C₂²×C₄	C₂²×C₅×D₃	1	3	yes	—
232	C₄⋊C₄	—	12	3	no	—
233	C₂²⋊C₄	—	12	6	no	—
234	C₂²⋊C₄	—	12	3	no	—
235	C₂²⋊C₄	—	12	6	no	—
236	C₂²⋊C₄	C₂×C₅×Dic₃	1	12	yes	complemented
237	C₄²	—	12	3	no	—
238	C₃×C₅	#2375	1	1	yes	characteristic, complemented, Hall
239	C₂²×C₃	—	2	6	no	—
240	C₂²×C₃	—	2	6	no	—
241	C₂²×C₃	—	2	12	no	—
242	C₂²×C₃	—	4	16	no	—
243	C₂²×C₃	—	4	3	no	—
244	C₂²×C₃	—	4	3	no	—
245	C₂²×C₃	—	4	6	no	—
246	C₂²×C₃	—	4	1	no	—
247	C₂²×C₃	—	4	3	no	—
248	C₂²×C₃	C₂³×C₄×C₅	1	1	yes	characteristic
249	C₂²×C₃	C₅×H(5,𝔽₂)	1	3	yes	—
250	C₂²×C₃	C₅×H(5,𝔽₂)	1	3	yes	—
251	C₃×C₄	—	2	6	no	—
252	Dic₃	—	4	6	no	—
253	Dic₃	—	4	2	no	—
254	C₂×C₅	—	4	6	no	—
255	C₂×C₅	—	4	2	no	—
256	C₂×C₅	#5594	1	1	yes	characteristic
257	C₂×C₅	#5595	1	3	yes	—
258	C₂×C₅	#5798	1	3	yes	—
259	C₂³	—	2	3	no	—
260	C₂³	—	2	6	no	—
261	C₂³	—	2	6	no	—
262	C₂³	—	2	3	no	—
263	C₂³	—	2	3	no	—
264	C₂³	—	2	3	no	—
265	C₂³	—	4	12	no	—
266	C₂³	—	4	4	no	—
267	C₂³	—	4	12	no	—
268	C₂³	C₂²×C₅×Dic₃	1	6	yes	—
269	C₂³	C₂²×C₅×Dic₃	1	2	yes	—
270	C₂³	C₂³×C₅×D₃	1	1	yes	characteristic
271	D₄	—	4	12	no	—
272	C₂×C₄	—	2	3	no	—
273	C₂×C₄	—	2	3	no	—
274	C₂×C₄	—	6	6	no	—
275	C₂×C₄	—	6	6	no	—
276	C₂×C₄	—	6	6	no	—
277	C₂×C₄	—	6	6	no	—
278	C₂×C₄	—	12	4	no	—
279	C₂×C₄	C₂²×C₅×Dic₃	1	6	yes	—
280	C₂×C₃	—	4	6	no	—
281	C₂×C₃	—	4	2	no	—
282	C₂×C₃	C₅×#518	1	3	yes	—
283	C₂×C₃	C₅×#560	1	3	yes	—
284	C₂×C₃	C₅×H(3,𝔽₄)	1	1	yes	characteristic
285	C₅	#82580	1	1	yes	characteristic, direct product factor, Sylow
286	C₂²	—	2	6	no	—
287	C₂²	—	2	6	no	—
288	C₂²	—	2	12	no	—
289	C₂²	—	4	16	no	—
290	C₂²	—	4	1	no	—
291	C₂²	—	4	3	no	—
292	C₂²	—	4	6	no	—
293	C₂²	—	4	3	no	—
294	C₂²	—	4	3	no	—
295	C₂²	C₅×#1004	1	3	yes	—
296	C₂²	C₅×#1004	1	3	yes	—
297	C₂²	C₂³×C₅×Dic₃	1	1	yes	characteristic
298	C₄	—	2	6	no	—
299	C₄	—	12	2	no	—
300	C₄	—	12	6	no	—
301	C₃	C₅×#2375	1	1	yes	characteristic, complemented, Sylow
302	C₂	—	4	6	no	—
303	C₂	—	4	2	no	—
304	C₂	C₅×#5594	1	1	yes	characteristic
305	C₂	C₅×#5595	1	3	yes	—
306	C₂	C₅×#5798	1	3	yes	—

	subgroup	G/H
center	C₂³×C₅	C₂³×D₃
commutator	C₂²×C₃	C₂³×C₄×C₅
Frattini	C₂³	C₂³×C₅×D₃
Fitting, prime radical	C₂×C₃×C₅×#120	C₂
socle	C₂³×C₃×C₅	C₂⁴
cosocle	C₂³×C₃	C₂⁴×C₅
hypercenter	C₂×C₅×#120	D₃
hypocenter	C₃	C₅×#2375
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₅ ◁ C₂×C₅×#120

Complements

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

N	H ≅ G/N	type
2. C₅×#5247	C₂	semidirect
3. C₅×#5221	C₂	semidirect
4. C₅×#5220	C₂	semidirect
5. C₅×#4951	C₂	semidirect
9. C₂³×C₅×Dic₃	C₂²	semidirect
10. C₃×C₅×#120	C₄	semidirect
14. C₂×C₅×#217	C₂²	semidirect
15. C₂×C₅×C₄⋊Dic₃	C₂²	semidirect
16. C₂×C₄×C₅×Dic₃	C₂²	semidirect
17. C₅×C₃⋊#95	C₂²	semidirect
18. #82580	C₅	direct (N × H)
26. C₂⁴×C₃×C₅	C₂×C₄	semidirect
36. C₃×C₅×C₂²⋊C₄	C₂×C₄	semidirect
43. #5247	C₂×C₅	semidirect
44. #5221	C₂×C₅	semidirect
45. #5220	C₂×C₅	semidirect
46. #4951	C₂×C₅	semidirect
50. C₅×#120	Dic₃	semidirect
82. C₂³×Dic₃	C₂²×C₅	semidirect
83. C₃×#120	C₄×C₅	semidirect
87. C₂×#217	C₂²×C₅	semidirect
88. C₂×C₄⋊Dic₃	C₂²×C₅	semidirect
89. C₂×C₄×Dic₃	C₂²×C₅	semidirect
90. C₃⋊#95	C₂²×C₅	semidirect
93. C₂⁴×C₅	C₂×Dic₃	semidirect
107. C₅×C₂²⋊C₄	C₂×Dic₃	semidirect
131. C₂⁴×C₃	C₂×C₄×C₅	semidirect
141. C₃×C₂²⋊C₄	C₂×C₄×C₅	semidirect
171. #120	C₅×Dic₃	semidirect
222. C₂⁴	C₂×C₅×Dic₃	semidirect
236. C₂²⋊C₄	C₂×C₅×Dic₃	semidirect
238. C₃×C₅	#2375	semidirect
285. C₅	#82580	direct (N × H)
301. C₃	C₅×#2375	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×#120	2
C₅×#5247	2
C₅×#5221	2
C₅×#5220	2
C₅×#4951	2
C₅×#2375	3
#82580	5

Subgroups — C5×#82580 #14330637

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#82580 #14330637