Subgroups — C₅×#5648 #11756033

Order 960.

540 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#237	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₅×Dic₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₅×#123	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#935	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#929	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#896	C₂	1	2	yes	maximal, complemented
7	C₅×#888	C₂	1	2	yes	maximal, complemented
8	C₅×#886	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅×#885	C₂	1	2	yes	maximal, complemented
10	C₅×#878	C₂	1	2	yes	maximal, complemented
11	C₅×#877	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅×#536	—	3	1	no	maximal, Hall
13	C₂×C₃×C₅×D₄	C₂²	1	1	yes	characteristic
14	C₂²×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
15	C₂×C₅×#67	C₂²	1	1	yes	characteristic
16	C₂×C₅×#67	C₂²	1	1	yes	characteristic
17	C₂²×C₅×Dic₃	C₂²	1	1	yes	characteristic
18	C₂²×C₅×Dic₃	C₂²	1	1	yes	characteristic
19	C₂²×C₅×Dic₃	C₂²	1	1	yes	characteristic
20	C₅×#237	—	2	4	no	—
21	C₂×C₄×C₅×D₃	C₂²	1	1	yes	characteristic
22	C₂×C₅×Dic₆	—	2	4	no	—
23	C₂×C₅×Dic₆	C₂²	1	2	yes	—
24	C₂×C₅×Dic₆	C₂²	1	2	yes	—
25	C₂×C₅×Dic₆	C₂²	1	1	yes	characteristic
26	C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
27	C₃×C₅×C₂²⋊C₄	C₂²	1	1	yes	characteristic
28	C₃×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
29	C₅×#217	C₂²	1	1	yes	characteristic
30	C₅×#217	C₂²	1	2	yes	—
31	C₅×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
32	C₅×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
33	C₅×C₄⋊Dic₃	C₂²	1	2	yes	—
34	C₅×Dic₃⋊C₄	C₂²	1	2	yes	—
35	C₅×Dic₃⋊C₄	C₂²	1	2	yes	—
36	C₅×Dic₃⋊C₄	C₂²	1	2	yes	—
37	C₄×C₅×Dic₃	C₂²	1	2	yes	—
38	#5648	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
39	C₂×C₅×#41	—	3	1	no	—
40	C₂²×C₅×Q₈	—	3	1	no	—
41	C₅×C₄⋊Q₈	—	3	2	no	—
42	C₅×#124	—	3	2	no	—
43	C₅×#123	—	3	1	no	—
44	C₅×#123	—	3	1	no	—
45	C₅×#123	—	3	1	no	—
46	C₅×#123	D₃	1	1	yes	characteristic, complemented
47	C₅×C₂²⋊Q₈	—	3	2	no	—
48	C₅×C₂²⋊Q₈	—	3	2	no	—
49	C₅×#117	—	3	1	no	—
50	C₂³×C₃×C₅	C₂³	1	1	yes	characteristic
51	C₂³×C₃×C₅	C₂³	1	1	yes	characteristic
52	C₂²×C₅×D₃	C₂³	1	1	yes	characteristic
53	C₃×C₅×D₄	—	2	2	no	—
54	C₂×C₃×C₄×C₅	—	2	2	no	—
55	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
56	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
57	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
58	C₅×#67	—	2	2	no	—
59	C₅×#67	—	2	2	no	—
60	C₂×C₅×Dic₃	—	2	2	no	—
61	C₂×C₅×Dic₃	—	2	2	no	—
62	C₂×C₅×Dic₃	D₄	1	4	yes	—
63	C₂×C₅×Dic₃	C₂³	1	2	yes	—
64	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
65	C₂×C₅×Dic₃	C₂³	1	2	yes	—
66	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
67	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
68	C₄×C₅×D₃	—	2	2	no	—
69	C₅×Dic₆	—	2	4	no	—
70	C₅×Dic₆	—	2	4	no	—
71	C₅×Dic₆	—	2	2	no	—
72	C₂×#237	C₂×C₅	1	1	yes	characteristic
73	C₂²×Dic₆	C₂×C₅	1	1	yes	characteristic, complemented
74	C₃×#123	C₂×C₅	1	1	yes	characteristic, complemented
75	#935	C₂×C₅	1	1	yes	characteristic, complemented
76	#929	C₂×C₅	1	1	yes	characteristic, complemented
77	#896	C₂×C₅	1	2	yes	complemented
78	#888	C₂×C₅	1	2	yes	complemented
79	#886	C₂×C₅	1	1	yes	characteristic, complemented
80	#885	C₂×C₅	1	2	yes	complemented
81	#878	C₂×C₅	1	2	yes	complemented
82	#877	C₂×C₅	1	1	yes	characteristic, complemented
83	C₅×#41	—	6	4	no	—
84	C₂×C₅×Q₈	—	3	2	no	—
85	C₂×C₅×Q₈	—	3	2	no	—
86	C₂×C₅×Q₈	—	3	1	no	—
87	C₂×C₅×Q₈	—	6	4	no	—
88	C₂×C₅×D₄	—	3	1	no	—
89	C₂×C₅×D₄	—	3	1	no	—
90	C₂×C₅×D₄	C₂×D₃	1	1	yes	characteristic
91	C₂²×C₄×C₅	—	3	1	no	—
92	C₂²×C₄×C₅	—	3	1	no	—
93	C₂²×C₄×C₅	—	3	1	no	—
94	C₂²×C₄×C₅	—	3	1	no	—
95	C₂²×C₄×C₅	C₂×D₃	1	1	yes	characteristic
96	C₅×C₄⋊C₄	—	3	2	no	—
97	C₅×C₄⋊C₄	—	3	2	no	—
98	C₅×C₄⋊C₄	—	3	2	no	—
99	C₅×C₄⋊C₄	—	3	2	no	—
100	C₅×C₄⋊C₄	C₂×D₃	1	2	yes	—
101	C₅×C₂²⋊C₄	—	3	1	no	—
102	C₅×C₂²⋊C₄	—	3	2	no	—
103	C₅×C₂²⋊C₄	—	3	2	no	—
104	C₅×C₂²⋊C₄	—	3	2	no	—
105	C₅×C₂²⋊C₄	C₂×D₃	1	1	yes	characteristic
106	C₅×C₂²⋊C₄	C₂×D₃	1	2	yes	—
107	C₄²×C₅	—	3	2	no	—
108	#536	—	3	1	no	Sylow
109	C₂²×C₃×C₅	—	2	2	no	—
110	C₂²×C₃×C₅	—	2	1	no	—
111	C₂²×C₃×C₅	—	2	2	no	—
112	C₂²×C₃×C₅	C₂×D₄	1	2	yes	—
113	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
114	C₂×C₅×D₃	—	2	1	no	—
115	C₂×C₅×D₃	—	2	2	no	—
116	C₃×C₄×C₅	—	2	2	no	—
117	C₃×C₄×C₅	—	2	1	no	—
118	C₃×C₄×C₅	—	2	2	no	—
119	C₅×Dic₃	—	2	2	no	—
120	C₅×Dic₃	—	2	1	no	—
121	C₅×Dic₃	—	2	2	no	—
122	C₅×Dic₃	C₂×D₄	1	2	yes	—
123	C₅×Dic₃	C₂×D₄	1	2	yes	—
124	C₂×C₃×D₄	C₂²×C₅	1	1	yes	characteristic
125	C₂²×C₃×C₄	C₂²×C₅	1	1	yes	characteristic
126	C₂×#67	C₂²×C₅	1	1	yes	characteristic
127	C₂×#67	C₂²×C₅	1	1	yes	characteristic
128	C₂²×Dic₃	C₂²×C₅	1	1	yes	characteristic
129	C₂²×Dic₃	C₂²×C₅	1	1	yes	characteristic
130	C₂²×Dic₃	C₂²×C₅	1	1	yes	characteristic
131	#237	—	2	4	no	—
132	C₂×C₄×D₃	C₂²×C₅	1	1	yes	characteristic
133	C₂×Dic₆	—	2	4	no	—
134	C₂×Dic₆	C₂²×C₅	1	2	yes	—
135	C₂×Dic₆	C₂²×C₅	1	2	yes	—
136	C₂×Dic₆	C₂²×C₅	1	1	yes	characteristic
137	C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
138	C₃×C₂²⋊C₄	C₂²×C₅	1	1	yes	characteristic
139	C₃×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
140	#217	C₂²×C₅	1	1	yes	characteristic
141	#217	C₂²×C₅	1	2	yes	—
142	(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
143	(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
144	C₄⋊Dic₃	C₂²×C₅	1	2	yes	—
145	Dic₃⋊C₄	C₂²×C₅	1	2	yes	—
146	Dic₃⋊C₄	C₂²×C₅	1	2	yes	—
147	Dic₃⋊C₄	C₂²×C₅	1	2	yes	—
148	C₄×Dic₃	C₂²×C₅	1	2	yes	—
149	C₂³×C₅	—	3	1	no	—
150	C₂³×C₅	C₂²×D₃	1	1	yes	characteristic
151	C₂³×C₅	C₂²×D₃	1	1	yes	characteristic
152	C₅×Q₈	—	6	4	no	—
153	C₅×Q₈	—	6	4	no	—
154	C₅×Q₈	—	6	2	no	—
155	C₅×D₄	—	2	2	no	—
156	C₅×D₄	—	6	2	no	—
157	C₅×D₄	—	6	2	no	—
158	C₂×C₄×C₅	—	2	2	no	—
159	C₂×C₄×C₅	—	3	4	no	—
160	C₂×C₄×C₅	—	3	1	no	—
161	C₂×C₄×C₅	—	3	1	no	—
162	C₂×C₄×C₅	—	3	2	no	—
163	C₂×C₄×C₅	—	3	2	no	—
164	C₂×C₄×C₅	—	3	1	no	—
165	C₂×C₄×C₅	—	6	2	no	—
166	C₂×C₄×C₅	—	6	2	no	—
167	C₂×C₄×C₅	—	6	2	no	—
168	C₂×C₄×C₅	C₂²×D₃	1	2	yes	—
169	C₂×C₄×C₅	C₂²×D₃	1	2	yes	—
170	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
171	C₂×#41	—	3	1	no	—
172	C₂²×Q₈	—	3	1	no	—
173	C₄⋊Q₈	—	3	2	no	—
174	#124	—	3	2	no	—
175	#123	—	3	1	no	—
176	#123	—	3	1	no	—
177	#123	—	3	1	no	—
178	#123	C₅×D₃	1	1	yes	characteristic, complemented
179	C₂²⋊Q₈	—	3	2	no	—
180	C₂²⋊Q₈	—	3	2	no	—
181	#117	—	3	1	no	—
182	C₂×C₃×C₅	—	2	2	no	—
183	C₂×C₃×C₅	—	4	1	no	—
184	C₂×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
185	C₂×C₃×C₅	2₋¹⁺⁴	1	2	yes	—
186	C₅×D₃	—	4	1	no	—
187	C₂³×C₃	C₂³×C₅	1	1	yes	characteristic
188	C₂³×C₃	C₂³×C₅	1	1	yes	characteristic
189	C₂²×D₃	C₂³×C₅	1	1	yes	characteristic
190	C₃×D₄	—	2	2	no	—
191	C₂×C₃×C₄	—	2	2	no	—
192	C₂×C₃×C₄	C₂³×C₅	1	2	yes	—
193	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
194	C₂×C₃×C₄	C₂³×C₅	1	2	yes	—
195	#67	—	2	2	no	—
196	#67	—	2	2	no	—
197	C₂×Dic₃	—	2	2	no	—
198	C₂×Dic₃	—	2	2	no	—
199	C₂×Dic₃	C₅×D₄	1	4	yes	—
200	C₂×Dic₃	C₂³×C₅	1	2	yes	—
201	C₂×Dic₃	C₂³×C₅	1	1	yes	characteristic
202	C₂×Dic₃	C₂³×C₅	1	2	yes	—
203	C₂×Dic₃	C₂³×C₅	1	1	yes	characteristic
204	C₂×Dic₃	C₂³×C₅	1	1	yes	characteristic
205	C₄×D₃	—	2	2	no	—
206	Dic₆	—	2	4	no	—
207	Dic₆	—	2	4	no	—
208	Dic₆	—	2	2	no	—
209	C₂²×C₅	—	2	2	no	—
210	C₂²×C₅	—	2	1	no	—
211	C₂²×C₅	—	2	2	no	—
212	C₂²×C₅	—	6	1	no	—
213	C₂²×C₅	—	6	2	no	—
214	C₂²×C₅	D₃×D₄	1	2	yes	—
215	C₂²×C₅	C₂³×D₃	1	1	yes	characteristic
216	C₄×C₅	—	2	2	no	—
217	C₄×C₅	—	2	1	no	—
218	C₄×C₅	—	2	2	no	—
219	C₄×C₅	—	3	2	no	—
220	C₄×C₅	—	3	2	no	—
221	C₄×C₅	—	6	1	no	—
222	C₄×C₅	—	6	2	no	—
223	C₄×C₅	—	6	2	no	—
224	#41	—	6	4	no	—
225	C₂×Q₈	—	3	2	no	—
226	C₂×Q₈	—	3	2	no	—
227	C₂×Q₈	—	3	1	no	—
228	C₂×Q₈	—	6	4	no	—
229	C₂×D₄	—	3	1	no	—
230	C₂×D₄	—	3	1	no	—
231	C₂×D₄	C₂×C₅×D₃	1	1	yes	characteristic
232	C₂²×C₄	—	3	1	no	—
233	C₂²×C₄	—	3	1	no	—
234	C₂²×C₄	—	3	1	no	—
235	C₂²×C₄	—	3	1	no	—
236	C₂²×C₄	C₂×C₅×D₃	1	1	yes	characteristic
237	C₄⋊C₄	—	3	2	no	—
238	C₄⋊C₄	—	3	2	no	—
239	C₄⋊C₄	—	3	2	no	—
240	C₄⋊C₄	—	3	2	no	—
241	C₄⋊C₄	C₂×C₅×D₃	1	2	yes	—
242	C₂²⋊C₄	—	3	1	no	—
243	C₂²⋊C₄	—	3	2	no	—
244	C₂²⋊C₄	—	3	2	no	—
245	C₂²⋊C₄	—	3	2	no	—
246	C₂²⋊C₄	C₂×C₅×D₃	1	1	yes	characteristic
247	C₂²⋊C₄	C₂×C₅×D₃	1	2	yes	—
248	C₄²	—	3	2	no	—
249	C₃×C₅	#536	1	1	yes	characteristic, complemented, Hall
250	C₂²×C₃	—	2	2	no	—
251	C₂²×C₃	—	2	1	no	—
252	C₂²×C₃	—	2	2	no	—
253	C₂²×C₃	C₂×C₅×D₄	1	2	yes	—
254	C₂²×C₃	C₂⁴×C₅	1	1	yes	characteristic
255	C₂×D₃	—	2	1	no	—
256	C₂×D₃	—	2	2	no	—
257	C₃×C₄	—	2	2	no	—
258	C₃×C₄	—	2	1	no	—
259	C₃×C₄	—	2	2	no	—
260	Dic₃	—	2	2	no	—
261	Dic₃	—	2	1	no	—
262	Dic₃	—	2	2	no	—
263	Dic₃	C₂×C₅×D₄	1	2	yes	—
264	Dic₃	C₂×C₅×D₄	1	2	yes	—
265	C₂×C₅	—	2	2	no	—
266	C₂×C₅	—	4	1	no	—
267	C₂×C₅	—	12	1	no	—
268	C₂×C₅	C₂×D₃×D₄	1	1	yes	characteristic
269	C₂×C₅	#1010	1	2	yes	—
270	C₂³	—	3	1	no	—
271	C₂³	C₂²×C₅×D₃	1	1	yes	characteristic
272	C₂³	C₂²×C₅×D₃	1	1	yes	characteristic
273	Q₈	—	6	4	no	—
274	Q₈	—	6	4	no	—
275	Q₈	—	6	2	no	—
276	D₄	—	2	2	no	—
277	D₄	—	6	2	no	—
278	D₄	—	6	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	3	4	no	—
281	C₂×C₄	—	3	1	no	—
282	C₂×C₄	—	3	1	no	—
283	C₂×C₄	—	3	2	no	—
284	C₂×C₄	—	3	2	no	—
285	C₂×C₄	—	3	1	no	—
286	C₂×C₄	—	6	2	no	—
287	C₂×C₄	—	6	2	no	—
288	C₂×C₄	—	6	2	no	—
289	C₂×C₄	C₂²×C₅×D₃	1	2	yes	—
290	C₂×C₄	C₂²×C₅×D₃	1	2	yes	—
291	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
292	C₂×C₃	—	2	2	no	—
293	C₂×C₃	—	4	1	no	—
294	C₂×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
295	C₂×C₃	C₅×2₋¹⁺⁴	1	2	yes	—
296	D₃	—	4	1	no	—
297	C₅	#5648	1	1	yes	characteristic, direct product factor, Sylow
298	C₂²	—	2	2	no	—
299	C₂²	—	2	1	no	—
300	C₂²	—	2	2	no	—
301	C₂²	—	6	1	no	—
302	C₂²	—	6	2	no	—
303	C₂²	C₅×D₃×D₄	1	2	yes	—
304	C₂²	C₂³×C₅×D₃	1	1	yes	characteristic
305	C₄	—	2	2	no	—
306	C₄	—	2	1	no	—
307	C₄	—	2	2	no	—
308	C₄	—	3	2	no	—
309	C₄	—	3	2	no	—
310	C₄	—	6	1	no	—
311	C₄	—	6	2	no	—
312	C₄	—	6	2	no	—
313	C₃	C₅×#536	1	1	yes	characteristic, complemented, Sylow
314	C₂	—	2	2	no	—
315	C₂	—	4	1	no	—
316	C₂	—	12	1	no	—
317	C₂	C₂×C₅×D₃×D₄	1	1	yes	characteristic
318	C₂	C₅×#1010	1	2	yes	—

	subgroup	G/H
center	C₂²×C₅	C₂³×D₃
commutator, cosocle	C₂²×C₃	C₂⁴×C₅
Frattini	C₂²	C₂³×C₅×D₃
Fitting	C₃×C₅×#123	C₂
socle	C₂²×C₃×C₅	C₂⁴
prime radical	C₂×C₅×#237	C₂
hypercenter	C₅×#123	D₃
hypocenter	C₃	C₅×#536
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₅×#123

Complements

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

N	H ≅ G/N	type
2. C₂²×C₅×Dic₆	C₂	semidirect
3. C₃×C₅×#123	C₂	semidirect
4. C₅×#935	C₂	semidirect
5. C₅×#929	C₂	semidirect
6. C₅×#896	C₂	semidirect
7. C₅×#888	C₂	semidirect
8. C₅×#886	C₂	semidirect
9. C₅×#885	C₂	semidirect
10. C₅×#878	C₂	semidirect
11. C₅×#877	C₂	semidirect
38. #5648	C₅	direct (N × H)
46. C₅×#123	D₃	semidirect
73. C₂²×Dic₆	C₂×C₅	semidirect
74. C₃×#123	C₂×C₅	semidirect
75. #935	C₂×C₅	semidirect
76. #929	C₂×C₅	semidirect
77. #896	C₂×C₅	semidirect
78. #888	C₂×C₅	semidirect
79. #886	C₂×C₅	semidirect
80. #885	C₂×C₅	semidirect
81. #878	C₂×C₅	semidirect
82. #877	C₂×C₅	semidirect
178. #123	C₅×D₃	semidirect
249. C₃×C₅	#536	semidirect
297. C₅	#5648	direct (N × H)
313. C₃	C₅×#536	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#237	2
C₂²×C₅×Dic₆	2
C₃×C₅×#123	2
C₅×#935	2
C₅×#929	2
C₅×#896	2
C₅×#888	2
C₅×#886	2
C₅×#885	2
C₅×#878	2
C₅×#877	2
C₅×#536	3
#5648	5

Subgroups — C5×#5648 #11756033

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#5648 #11756033