Subgroups — C₅×#68573 #14285896

Order 1920.

300 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#4835	C₂	1	1	yes	maximal, characteristic
2	C₅×#4833	C₂	1	1	yes	maximal, characteristic
3	C₅×#4685	C₂	1	1	yes	maximal, characteristic
4	C₃×C₅×#335	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#4480	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#4480	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#4458	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#1572	—	3	1	no	maximal, Hall
9	C₄²×C₅×D₃	C₂²	1	1	yes	characteristic
10	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
11	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
12	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
13	C₅×(C₂×D₃)⋊C₈	—	2	1	no	—
14	C₅×(C₂×D₃)⋊C₈	—	2	1	no	—
15	C₅×(C₂×D₃)⋊C₈	C₄	1	1	yes	characteristic
16	C₅×(C₂×D₃)⋊C₈	C₄	1	1	yes	characteristic
17	C₅×#815	—	2	1	no	—
18	C₅×Dic₃⋊C₈	C₄	1	1	yes	characteristic
19	C₅×Dic₃⋊C₈	C₄	1	1	yes	characteristic
20	C₅×C₈×Dic₃	—	2	1	no	—
21	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
22	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
23	C₄×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
24	#68573	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
25	C₅×#432	—	3	1	no	—
26	C₅×#431	—	3	1	no	—
27	C₅×#431	—	3	1	no	—
28	C₅×#335	—	3	1	no	—
29	C₅×#335	D₃	1	1	yes	characteristic, complemented
30	C₅×#334	—	3	1	no	—
31	C₅×#334	—	3	1	no	—
32	C₂×C₄×C₅×D₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄×C₅×D₃	D₄	1	1	yes	characteristic
34	C₂×C₄×C₅×D₃	Q₈	1	1	yes	characteristic
35	C₂×C₃×C₅×C₈	—	2	1	no	—
36	C₂×C₃×C₅×C₈	—	2	1	no	—
37	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
38	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
39	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
40	C₄×C₅×Dic₃	C₂×C₄	1	1	yes	characteristic
41	C₄×C₅×Dic₃	D₄	1	1	yes	characteristic
42	C₄×C₅×Dic₃	Q₈	1	1	yes	characteristic
43	C₂×C₅×C₃⋊C₈	—	2	1	no	—
44	C₂×C₅×C₃⋊C₈	—	2	1	no	—
45	C₂×C₅×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
46	C₂×C₅×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
47	#4835	C₂×C₅	1	1	yes	characteristic
48	#4833	C₂×C₅	1	1	yes	characteristic
49	#4685	C₂×C₅	1	1	yes	characteristic
50	C₃×#335	C₂×C₅	1	1	yes	characteristic, complemented
51	#4480	C₂×C₅	1	1	yes	characteristic, complemented
52	#4480	C₂×C₅	1	1	yes	characteristic, complemented
53	#4458	C₂×C₅	1	1	yes	characteristic, complemented
54	C₂×C₄²×C₅	—	3	1	no	—
55	C₅×C₄⋊C₈	—	3	1	no	—
56	C₅×C₄⋊C₈	—	3	1	no	—
57	C₅×C₄⋊C₈	—	3	1	no	—
58	C₅×C₄⋊C₈	—	3	1	no	—
59	C₅×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
60	C₅×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
61	C₅×C₂²⋊C₈	—	3	1	no	—
62	C₅×C₂²⋊C₈	—	3	1	no	—
63	C₅×C₂²⋊C₈	—	6	1	no	—
64	C₅×C₂²⋊C₈	—	6	1	no	—
65	C₅×#97	—	6	1	no	—
66	C₄×C₅×C₈	—	3	1	no	—
67	C₄×C₅×C₈	—	6	1	no	—
68	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
69	#1572	—	3	1	no	Sylow
70	C₂²×C₅×D₃	C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
73	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
74	C₂×C₅×Dic₃	C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₅×Dic₃	C₄⋊C₄	1	1	yes	characteristic
76	C₂×C₅×Dic₃	C₄⋊C₄	1	1	yes	characteristic
77	C₄×C₅×D₃	—	2	1	no	—
78	C₄×C₅×D₃	—	2	1	no	—
79	C₄×C₅×D₃	—	2	1	no	—
80	C₄×C₅×D₃	—	2	1	no	—
81	C₄×C₅×D₃	—	4	1	no	—
82	C₃×C₅×C₈	—	2	1	no	—
83	C₃×C₅×C₈	—	2	1	no	—
84	C₃×C₅×C₈	—	4	1	no	—
85	C₃×C₅×C₈	—	4	1	no	—
86	C₅×C₃⋊C₈	—	2	1	no	—
87	C₅×C₃⋊C₈	—	2	1	no	—
88	C₅×C₃⋊C₈	—	4	1	no	—
89	C₅×C₃⋊C₈	—	4	1	no	—
90	C₄²×D₃	C₂²×C₅	1	1	yes	characteristic
91	C₃×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
92	C₃×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
93	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
94	(C₂×D₃)⋊C₈	—	2	1	no	—
95	(C₂×D₃)⋊C₈	—	2	1	no	—
96	(C₂×D₃)⋊C₈	C₄×C₅	1	1	yes	characteristic
97	(C₂×D₃)⋊C₈	C₄×C₅	1	1	yes	characteristic
98	#815	—	2	1	no	—
99	Dic₃⋊C₈	C₄×C₅	1	1	yes	characteristic
100	Dic₃⋊C₈	C₄×C₅	1	1	yes	characteristic
101	C₈×Dic₃	—	2	1	no	—
102	C₄⋊(C₃⋊C₈)	C₂²×C₅	1	1	yes	characteristic
103	C₄⋊(C₃⋊C₈)	C₂²×C₅	1	1	yes	characteristic
104	C₄×C₃⋊C₈	C₂²×C₅	1	1	yes	characteristic
105	C₂²×C₄×C₅	—	3	1	no	—
106	C₂²×C₄×C₅	—	3	1	no	—
107	C₂²×C₄×C₅	—	3	1	no	—
108	C₂×C₅×C₈	—	2	1	no	—
109	C₂×C₅×C₈	—	2	1	no	—
110	C₂×C₅×C₈	—	3	1	no	—
111	C₂×C₅×C₈	—	3	1	no	—
112	C₂×C₅×C₈	—	6	1	no	—
113	C₂×C₅×C₈	—	6	1	no	—
114	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
115	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
116	C₄²×C₅	—	3	1	no	—
117	C₄²×C₅	—	3	1	no	—
118	C₄²×C₅	—	3	1	no	—
119	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
120	#432	—	3	1	no	—
121	#431	—	3	1	no	—
122	#431	—	3	1	no	—
123	#335	—	3	1	no	—
124	#335	C₅×D₃	1	1	yes	characteristic, complemented
125	#334	—	3	1	no	—
126	#334	—	3	1	no	—
127	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
128	C₂×C₅×D₃	—	2	1	no	—
129	C₂×C₅×D₃	—	2	1	no	—
130	C₂×C₅×D₃	—	2	1	no	—
131	C₃×C₄×C₅	—	2	1	no	—
132	C₃×C₄×C₅	#131	1	1	yes	characteristic
133	C₃×C₄×C₅	#131	1	1	yes	characteristic
134	C₃×C₄×C₅	#135	1	1	yes	characteristic
135	C₃×C₄×C₅	#135	1	1	yes	characteristic
136	C₅×Dic₃	—	2	1	no	—
137	C₅×Dic₃	—	2	1	no	—
138	C₅×Dic₃	—	2	1	no	—
139	C₂×C₄×D₃	C₂×C₄×C₅	1	1	yes	characteristic
140	C₂×C₄×D₃	C₅×D₄	1	1	yes	characteristic
141	C₂×C₄×D₃	C₅×Q₈	1	1	yes	characteristic
142	C₂×C₃×C₈	—	2	1	no	—
143	C₂×C₃×C₈	—	2	1	no	—
144	C₂×C₃×C₈	C₂×C₄×C₅	1	1	yes	characteristic
145	C₂×C₃×C₈	C₂×C₄×C₅	1	1	yes	characteristic
146	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
147	C₄×Dic₃	C₂×C₄×C₅	1	1	yes	characteristic
148	C₄×Dic₃	C₅×D₄	1	1	yes	characteristic
149	C₄×Dic₃	C₅×Q₈	1	1	yes	characteristic
150	C₂×C₃⋊C₈	—	2	1	no	—
151	C₂×C₃⋊C₈	—	2	1	no	—
152	C₂×C₃⋊C₈	C₂×C₄×C₅	1	1	yes	characteristic
153	C₂×C₃⋊C₈	C₂×C₄×C₅	1	1	yes	characteristic
154	C₂³×C₅	—	3	1	no	—
155	C₂×C₄×C₅	—	3	1	no	—
156	C₂×C₄×C₅	—	3	1	no	—
157	C₂×C₄×C₅	—	3	1	no	—
158	C₂×C₄×C₅	—	6	1	no	—
159	C₂×C₄×C₅	—	6	1	no	—
160	C₂×C₄×C₅	—	6	1	no	—
161	C₂×C₄×C₅	—	6	1	no	—
162	C₂×C₄×C₅	—	12	1	no	—
163	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
164	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
165	C₂×C₄×C₅	D₃×Q₈	1	1	yes	characteristic
166	C₅×C₈	—	2	1	no	—
167	C₅×C₈	—	2	1	no	—
168	C₅×C₈	—	4	1	no	—
169	C₅×C₈	—	4	1	no	—
170	C₅×C₈	—	6	1	no	—
171	C₅×C₈	—	6	1	no	—
172	C₅×C₈	—	12	1	no	—
173	C₅×C₈	—	12	1	no	—
174	C₂×C₄²	—	3	1	no	—
175	C₄⋊C₈	—	3	1	no	—
176	C₄⋊C₈	—	3	1	no	—
177	C₄⋊C₈	—	3	1	no	—
178	C₄⋊C₈	—	3	1	no	—
179	C₄⋊C₈	C₂×C₅×D₃	1	1	yes	characteristic
180	C₄⋊C₈	C₂×C₅×D₃	1	1	yes	characteristic
181	C₂²⋊C₈	—	3	1	no	—
182	C₂²⋊C₈	—	3	1	no	—
183	C₂²⋊C₈	—	6	1	no	—
184	C₂²⋊C₈	—	6	1	no	—
185	#97	—	6	1	no	—
186	C₄×C₈	—	3	1	no	—
187	C₄×C₈	—	6	1	no	—
188	C₄×C₈	C₂×C₅×D₃	1	1	yes	characteristic
189	C₂×C₃×C₅	#424	1	1	yes	characteristic
190	C₂×C₃×C₅	#427	1	1	yes	characteristic
191	C₂×C₃×C₅	#430	1	1	yes	characteristic
192	C₅×D₃	—	4	1	no	—
193	C₂²×D₃	C₅×C₄⋊C₄	1	1	yes	characteristic
194	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
195	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
196	C₂×C₃×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
197	C₂×Dic₃	C₅×C₄⋊C₄	1	1	yes	characteristic
198	C₂×Dic₃	C₅×C₄⋊C₄	1	1	yes	characteristic
199	C₂×Dic₃	C₅×C₄⋊C₄	1	1	yes	characteristic
200	C₄×D₃	—	2	1	no	—
201	C₄×D₃	—	2	1	no	—
202	C₄×D₃	—	2	1	no	—
203	C₄×D₃	—	2	1	no	—
204	C₄×D₃	—	4	1	no	—
205	C₃×C₈	—	2	1	no	—
206	C₃×C₈	—	2	1	no	—
207	C₃×C₈	—	4	1	no	—
208	C₃×C₈	—	4	1	no	—
209	C₃⋊C₈	—	2	1	no	—
210	C₃⋊C₈	—	2	1	no	—
211	C₃⋊C₈	—	4	1	no	—
212	C₃⋊C₈	—	4	1	no	—
213	C₂²×C₅	—	6	1	no	—
214	C₂²×C₅	—	6	1	no	—
215	C₂²×C₅	—	6	1	no	—
216	C₂²×C₅	D₃×C₄⋊C₄	1	1	yes	characteristic
217	C₄×C₅	—	2	1	no	—
218	C₄×C₅	—	6	1	no	—
219	C₄×C₅	—	6	1	no	—
220	C₄×C₅	—	6	1	no	—
221	C₄×C₅	#901	1	1	yes	characteristic
222	C₄×C₅	#907	1	1	yes	characteristic
223	C₄×C₅	#912	1	1	yes	characteristic
224	C₄×C₅	#919	1	1	yes	characteristic
225	C₂²×C₄	—	3	1	no	—
226	C₂²×C₄	—	3	1	no	—
227	C₂²×C₄	—	3	1	no	—
228	C₂×C₈	—	2	1	no	—
229	C₂×C₈	—	2	1	no	—
230	C₂×C₈	—	3	1	no	—
231	C₂×C₈	—	3	1	no	—
232	C₂×C₈	—	6	1	no	—
233	C₂×C₈	—	6	1	no	—
234	C₂×C₈	C₄×C₅×D₃	1	1	yes	characteristic
235	C₂×C₈	C₄×C₅×D₃	1	1	yes	characteristic
236	C₄²	—	3	1	no	—
237	C₄²	—	3	1	no	—
238	C₄²	—	3	1	no	—
239	C₄²	C₂²×C₅×D₃	1	1	yes	characteristic
240	C₃×C₅	#1572	1	1	yes	characteristic, complemented, Hall
241	C₂²×C₃	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
242	C₂×D₃	—	2	1	no	—
243	C₂×D₃	—	2	1	no	—
244	C₂×D₃	—	2	1	no	—
245	C₃×C₄	—	2	1	no	—
246	C₃×C₄	C₅×#131	1	1	yes	characteristic
247	C₃×C₄	C₅×#131	1	1	yes	characteristic
248	C₃×C₄	C₅×#135	1	1	yes	characteristic
249	C₃×C₄	C₅×#135	1	1	yes	characteristic
250	Dic₃	—	2	1	no	—
251	Dic₃	—	2	1	no	—
252	Dic₃	—	2	1	no	—
253	C₂×C₅	—	12	1	no	—
254	C₂×C₅	#4839	1	1	yes	characteristic
255	C₂×C₅	#4880	1	1	yes	characteristic
256	C₂×C₅	#4893	1	1	yes	characteristic
257	C₂³	—	3	1	no	—
258	C₂×C₄	—	3	1	no	—
259	C₂×C₄	—	3	1	no	—
260	C₂×C₄	—	3	1	no	—
261	C₂×C₄	—	6	1	no	—
262	C₂×C₄	—	6	1	no	—
263	C₂×C₄	—	6	1	no	—
264	C₂×C₄	—	6	1	no	—
265	C₂×C₄	—	12	1	no	—
266	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
267	C₂×C₄	C₅×D₃×D₄	1	1	yes	characteristic
268	C₂×C₄	C₅×D₃×Q₈	1	1	yes	characteristic
269	C₈	—	2	1	no	—
270	C₈	—	2	1	no	—
271	C₈	—	4	1	no	—
272	C₈	—	4	1	no	—
273	C₈	—	6	1	no	—
274	C₈	—	6	1	no	—
275	C₈	—	12	1	no	—
276	C₈	—	12	1	no	—
277	C₂×C₃	C₅×#424	1	1	yes	characteristic
278	C₂×C₃	C₅×#427	1	1	yes	characteristic
279	C₂×C₃	C₅×#430	1	1	yes	characteristic
280	D₃	—	4	1	no	—
281	C₅	#68573	1	1	yes	characteristic, direct product factor, Sylow
282	C₂²	—	6	1	no	—
283	C₂²	—	6	1	no	—
284	C₂²	—	6	1	no	—
285	C₂²	C₅×D₃×C₄⋊C₄	1	1	yes	characteristic
286	C₄	—	2	1	no	—
287	C₄	—	6	1	no	—
288	C₄	—	6	1	no	—
289	C₄	—	6	1	no	—
290	C₄	C₅×#901	1	1	yes	characteristic
291	C₄	C₅×#907	1	1	yes	characteristic
292	C₄	C₅×#912	1	1	yes	characteristic
293	C₄	C₅×#919	1	1	yes	characteristic
294	C₃	C₅×#1572	1	1	yes	characteristic, complemented, Sylow
295	C₂	—	12	1	no	—
296	C₂	C₅×#4839	1	1	yes	characteristic
297	C₂	C₅×#4880	1	1	yes	characteristic
298	C₂	C₅×#4893	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	D₃×D₄
commutator	C₂×C₃×C₄	C₂²×C₄×C₅
Frattini	C₄²	C₂²×C₅×D₃
Fitting	C₃×C₅×#335	C₂
socle	C₂²×C₃×C₅	C₂×C₄⋊C₄
prime radical	C₂²×C₅×D₃	C₄⋊C₄
cosocle	C₃×C₄²	C₂³×C₅
hypercenter	C₅×#335	D₃
hypocenter	C₃	C₅×#1572
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₄²×C₅ ◁ C₅×#335

Complements

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

N	H ≅ G/N	type
4. C₃×C₅×#335	C₂	semidirect
5. C₅×#4480	C₂	semidirect
6. C₅×#4480	C₂	semidirect
7. C₅×#4458	C₂	semidirect
24. #68573	C₅	direct (N × H)
29. C₅×#335	D₃	semidirect
50. C₃×#335	C₂×C₅	semidirect
51. #4480	C₂×C₅	semidirect
52. #4480	C₂×C₅	semidirect
53. #4458	C₂×C₅	semidirect
124. #335	C₅×D₃	semidirect
240. C₃×C₅	#1572	semidirect
281. C₅	#68573	direct (N × H)
294. C₃	C₅×#1572	semidirect

Maximal subgroups

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

subgroup	index
C₅×#4835	2
C₅×#4833	2
C₅×#4685	2
C₃×C₅×#335	2
C₅×#4480	2
C₅×#4458	2
C₅×#1572	3
#68573	5

Subgroups — C5×#68573 #14285896

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#68573 #14285896