Subgroups — C₅×#68534 #14285945

Order 1920.

320 conjugacy classes of subgroups.

The subgroup lattice has 304 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×D₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₅×#4685	C₂	1	1	yes	maximal, characteristic
4	C₃×C₅×#334	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#4480	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#4479	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#4458	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#1573	—	3	1	no	maximal, Hall
9	C₂×C₅×D₃×C₈	—	2	1	no	—
10	C₄²×C₅×D₃	C₂²	1	1	yes	characteristic
11	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
12	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
13	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
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	—	2	1	no	—
19	C₅×Dic₃⋊C₈	C₄	1	1	yes	characteristic
20	C₅×Dic₃⋊C₈	C₄	1	1	yes	characteristic
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	#68534	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
25	C₅×#432	—	3	1	no	—
26	C₅×#431	—	3	1	no	—
27	C₂×C₅×C₄⋊C₈	—	3	1	no	—
28	C₅×#335	—	3	1	no	—
29	C₅×#335	—	3	1	no	—
30	C₅×#334	—	3	1	no	—
31	C₅×#334	D₃	1	1	yes	characteristic, complemented
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	C₅×D₃×C₈	—	4	2	no	—
48	#4835	C₂×C₅	1	1	yes	characteristic
49	D₃×C₄⋊C₈	C₂×C₅	1	1	yes	characteristic
50	#4685	C₂×C₅	1	1	yes	characteristic
51	C₃×#334	C₂×C₅	1	1	yes	characteristic, complemented
52	#4480	C₂×C₅	1	1	yes	characteristic, complemented
53	#4479	C₂×C₅	1	1	yes	characteristic, complemented
54	#4458	C₂×C₅	1	1	yes	characteristic, complemented
55	C₂²×C₅×C₈	—	6	1	no	—
56	C₂×C₄²×C₅	—	3	1	no	—
57	C₅×C₄⋊C₈	—	3	1	no	—
58	C₅×C₄⋊C₈	—	3	1	no	—
59	C₅×C₄⋊C₈	—	3	1	no	—
60	C₅×C₄⋊C₈	—	3	1	no	—
61	C₅×C₄⋊C₈	—	6	1	no	—
62	C₅×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
63	C₅×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
64	C₅×C₂²⋊C₈	—	3	1	no	—
65	C₅×C₂²⋊C₈	—	3	1	no	—
66	C₅×C₂²⋊C₈	—	6	1	no	—
67	C₅×#97	—	6	1	no	—
68	C₄×C₅×C₈	—	3	1	no	—
69	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
70	#1573	—	3	1	no	Sylow
71	C₂²×C₅×D₃	C₄⋊C₄	1	1	yes	characteristic
72	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
74	C₂×C₃×C₄×C₅	C₂×Q₈	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	C₂×C₅×Dic₃	C₄⋊C₄	1	1	yes	characteristic
78	C₄×C₅×D₃	—	2	2	no	—
79	C₄×C₅×D₃	—	2	1	no	—
80	C₄×C₅×D₃	—	2	1	no	—
81	C₄×C₅×D₃	—	2	1	no	—
82	C₄×C₅×D₃	—	2	1	no	—
83	C₃×C₅×C₈	—	2	1	no	—
84	C₃×C₅×C₈	—	2	1	no	—
85	C₃×C₅×C₈	—	4	1	no	—
86	C₃×C₅×C₈	—	4	1	no	—
87	C₅×C₃⋊C₈	—	2	1	no	—
88	C₅×C₃⋊C₈	—	2	1	no	—
89	C₅×C₃⋊C₈	—	4	1	no	—
90	C₅×C₃⋊C₈	—	4	1	no	—
91	C₂×D₃×C₈	—	2	1	no	—
92	C₄²×D₃	C₂²×C₅	1	1	yes	characteristic
93	C₃×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
94	C₃×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
95	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
96	(C₂×D₃)⋊C₈	—	2	1	no	—
97	(C₂×D₃)⋊C₈	C₄×C₅	1	1	yes	characteristic
98	(C₂×D₃)⋊C₈	C₄×C₅	1	1	yes	characteristic
99	#815	—	2	1	no	—
100	Dic₃⋊C₈	—	2	1	no	—
101	Dic₃⋊C₈	C₄×C₅	1	1	yes	characteristic
102	Dic₃⋊C₈	C₄×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	C₄×C₃⋊C₈	C₂²×C₅	1	1	yes	characteristic
106	C₂²×C₄×C₅	—	3	1	no	—
107	C₂²×C₄×C₅	—	3	1	no	—
108	C₂²×C₄×C₅	—	3	1	no	—
109	C₂×C₅×C₈	—	2	1	no	—
110	C₂×C₅×C₈	—	2	1	no	—
111	C₂×C₅×C₈	—	3	1	no	—
112	C₂×C₅×C₈	—	3	1	no	—
113	C₂×C₅×C₈	—	6	1	no	—
114	C₂×C₅×C₈	—	6	1	no	—
115	C₂×C₅×C₈	—	12	2	no	—
116	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
117	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
118	C₄²×C₅	—	3	1	no	—
119	C₄²×C₅	—	3	1	no	—
120	C₄²×C₅	—	3	1	no	—
121	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
122	#432	—	3	1	no	—
123	#431	—	3	1	no	—
124	C₂×C₄⋊C₈	—	3	1	no	—
125	#335	—	3	1	no	—
126	#335	—	3	1	no	—
127	#334	—	3	1	no	—
128	#334	C₅×D₃	1	1	yes	characteristic, complemented
129	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
130	C₂×C₅×D₃	—	2	1	no	—
131	C₂×C₅×D₃	—	2	1	no	—
132	C₂×C₅×D₃	#108	1	2	yes	—
133	C₃×C₄×C₅	—	2	1	no	—
134	C₃×C₄×C₅	#131	1	1	yes	characteristic
135	C₃×C₄×C₅	#131	1	1	yes	characteristic
136	C₃×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
137	C₃×C₄×C₅	#137	1	1	yes	characteristic
138	C₅×Dic₃	—	2	1	no	—
139	C₅×Dic₃	—	2	1	no	—
140	C₅×Dic₃	—	2	1	no	—
141	C₂×C₄×D₃	C₂×C₄×C₅	1	1	yes	characteristic
142	C₂×C₄×D₃	C₅×D₄	1	1	yes	characteristic
143	C₂×C₄×D₃	C₅×Q₈	1	1	yes	characteristic
144	C₂×C₃×C₈	—	2	1	no	—
145	C₂×C₃×C₈	—	2	1	no	—
146	C₂×C₃×C₈	C₂×C₄×C₅	1	1	yes	characteristic
147	C₂×C₃×C₈	C₂×C₄×C₅	1	1	yes	characteristic
148	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
149	C₄×Dic₃	C₂×C₄×C₅	1	1	yes	characteristic
150	C₄×Dic₃	C₅×D₄	1	1	yes	characteristic
151	C₄×Dic₃	C₅×Q₈	1	1	yes	characteristic
152	C₂×C₃⋊C₈	—	2	1	no	—
153	C₂×C₃⋊C₈	—	2	1	no	—
154	C₂×C₃⋊C₈	C₂×C₄×C₅	1	1	yes	characteristic
155	C₂×C₃⋊C₈	C₂×C₄×C₅	1	1	yes	characteristic
156	D₃×C₈	—	4	2	no	—
157	C₂³×C₅	—	3	1	no	—
158	C₂×C₄×C₅	—	3	1	no	—
159	C₂×C₄×C₅	—	3	1	no	—
160	C₂×C₄×C₅	—	3	1	no	—
161	C₂×C₄×C₅	—	6	2	no	—
162	C₂×C₄×C₅	—	6	1	no	—
163	C₂×C₄×C₅	—	6	1	no	—
164	C₂×C₄×C₅	—	6	1	no	—
165	C₂×C₄×C₅	—	6	1	no	—
166	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
167	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
168	C₂×C₄×C₅	D₃×Q₈	1	1	yes	characteristic
169	C₅×C₈	—	2	1	no	—
170	C₅×C₈	—	2	1	no	—
171	C₅×C₈	—	4	1	no	—
172	C₅×C₈	—	4	1	no	—
173	C₅×C₈	—	6	1	no	—
174	C₅×C₈	—	6	1	no	—
175	C₅×C₈	—	12	1	no	—
176	C₅×C₈	—	12	1	no	—
177	C₂²×C₈	—	6	1	no	—
178	C₂×C₄²	—	3	1	no	—
179	C₄⋊C₈	—	3	1	no	—
180	C₄⋊C₈	—	3	1	no	—
181	C₄⋊C₈	—	3	1	no	—
182	C₄⋊C₈	—	3	1	no	—
183	C₄⋊C₈	—	6	1	no	—
184	C₄⋊C₈	C₂×C₅×D₃	1	1	yes	characteristic
185	C₄⋊C₈	C₂×C₅×D₃	1	1	yes	characteristic
186	C₂²⋊C₈	—	3	1	no	—
187	C₂²⋊C₈	—	3	1	no	—
188	C₂²⋊C₈	—	6	1	no	—
189	#97	—	6	1	no	—
190	C₄×C₈	—	3	1	no	—
191	C₄×C₈	C₂×C₅×D₃	1	1	yes	characteristic
192	C₂×C₃×C₅	#424	1	1	yes	characteristic
193	C₂×C₃×C₅	#428	1	1	yes	characteristic
194	C₂×C₃×C₅	C₂×#108	1	1	yes	characteristic
195	C₅×D₃	—	2	2	no	—
196	C₂²×D₃	C₅×C₄⋊C₄	1	1	yes	characteristic
197	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
198	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
199	C₂×C₃×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
200	C₂×Dic₃	C₅×C₄⋊C₄	1	1	yes	characteristic
201	C₂×Dic₃	C₅×C₄⋊C₄	1	1	yes	characteristic
202	C₂×Dic₃	C₅×C₄⋊C₄	1	1	yes	characteristic
203	C₄×D₃	—	2	2	no	—
204	C₄×D₃	—	2	1	no	—
205	C₄×D₃	—	2	1	no	—
206	C₄×D₃	—	2	1	no	—
207	C₄×D₃	—	2	1	no	—
208	C₃×C₈	—	2	1	no	—
209	C₃×C₈	—	2	1	no	—
210	C₃×C₈	—	4	1	no	—
211	C₃×C₈	—	4	1	no	—
212	C₃⋊C₈	—	2	1	no	—
213	C₃⋊C₈	—	2	1	no	—
214	C₃⋊C₈	—	4	1	no	—
215	C₃⋊C₈	—	4	1	no	—
216	C₂²×C₅	—	3	2	no	—
217	C₂²×C₅	—	6	1	no	—
218	C₂²×C₅	—	6	1	no	—
219	C₂²×C₅	D₃×C₄⋊C₄	1	1	yes	characteristic
220	C₄×C₅	—	2	1	no	—
221	C₄×C₅	—	6	1	no	—
222	C₄×C₅	—	6	1	no	—
223	C₄×C₅	—	6	1	no	—
224	C₄×C₅	#901	1	1	yes	characteristic
225	C₄×C₅	#907	1	1	yes	characteristic
226	C₄×C₅	#914	1	1	yes	characteristic
227	C₄×C₅	#915	1	1	yes	characteristic
228	C₂²×C₄	—	3	1	no	—
229	C₂²×C₄	—	3	1	no	—
230	C₂²×C₄	—	3	1	no	—
231	C₂×C₈	—	2	1	no	—
232	C₂×C₈	—	2	1	no	—
233	C₂×C₈	—	3	1	no	—
234	C₂×C₈	—	3	1	no	—
235	C₂×C₈	—	6	1	no	—
236	C₂×C₈	—	6	1	no	—
237	C₂×C₈	—	12	2	no	—
238	C₂×C₈	C₄×C₅×D₃	1	1	yes	characteristic
239	C₂×C₈	C₄×C₅×D₃	1	1	yes	characteristic
240	C₄²	—	3	1	no	—
241	C₄²	—	3	1	no	—
242	C₄²	—	3	1	no	—
243	C₄²	C₂²×C₅×D₃	1	1	yes	characteristic
244	C₃×C₅	#1573	1	1	yes	characteristic, complemented, Hall
245	C₂²×C₃	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
246	C₂×D₃	—	2	1	no	—
247	C₂×D₃	—	2	1	no	—
248	C₂×D₃	C₅×#108	1	2	yes	—
249	C₃×C₄	—	2	1	no	—
250	C₃×C₄	C₅×#131	1	1	yes	characteristic
251	C₃×C₄	C₅×#131	1	1	yes	characteristic
252	C₃×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
253	C₃×C₄	C₅×#137	1	1	yes	characteristic
254	Dic₃	—	2	1	no	—
255	Dic₃	—	2	1	no	—
256	Dic₃	—	2	1	no	—
257	C₂×C₅	—	6	2	no	—
258	C₂×C₅	#4839	1	1	yes	characteristic
259	C₂×C₅	#4861	1	1	yes	characteristic
260	C₂×C₅	D₃×#108	1	1	yes	characteristic
261	C₂³	—	3	1	no	—
262	C₂×C₄	—	3	1	no	—
263	C₂×C₄	—	3	1	no	—
264	C₂×C₄	—	3	1	no	—
265	C₂×C₄	—	6	2	no	—
266	C₂×C₄	—	6	1	no	—
267	C₂×C₄	—	6	1	no	—
268	C₂×C₄	—	6	1	no	—
269	C₂×C₄	—	6	1	no	—
270	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
271	C₂×C₄	C₅×D₃×D₄	1	1	yes	characteristic
272	C₂×C₄	C₅×D₃×Q₈	1	1	yes	characteristic
273	C₈	—	2	1	no	—
274	C₈	—	2	1	no	—
275	C₈	—	4	1	no	—
276	C₈	—	4	1	no	—
277	C₈	—	6	1	no	—
278	C₈	—	6	1	no	—
279	C₈	—	12	1	no	—
280	C₈	—	12	1	no	—
281	C₂×C₃	C₅×#424	1	1	yes	characteristic
282	C₂×C₃	C₅×#428	1	1	yes	characteristic
283	C₂×C₃	C₂×C₅×#108	1	1	yes	characteristic
284	D₃	—	2	2	no	—
285	C₅	#68534	1	1	yes	characteristic, direct product factor, Sylow
286	C₂²	—	3	2	no	—
287	C₂²	—	6	1	no	—
288	C₂²	—	6	1	no	—
289	C₂²	C₅×D₃×C₄⋊C₄	1	1	yes	characteristic
290	C₄	—	2	1	no	—
291	C₄	—	6	1	no	—
292	C₄	—	6	1	no	—
293	C₄	—	6	1	no	—
294	C₄	C₅×#901	1	1	yes	characteristic
295	C₄	C₅×#907	1	1	yes	characteristic
296	C₄	C₅×#914	1	1	yes	characteristic
297	C₄	C₅×#915	1	1	yes	characteristic
298	C₃	C₅×#1573	1	1	yes	characteristic, complemented, Sylow
299	C₂	—	6	2	no	—
300	C₂	C₅×#4839	1	1	yes	characteristic
301	C₂	C₅×#4861	1	1	yes	characteristic
302	C₂	C₅×D₃×#108	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×#334	C₂
socle	C₂²×C₃×C₅	C₂×C₄⋊C₄
prime radical	C₂²×C₅×D₃	C₄⋊C₄
cosocle	C₃×C₄²	C₂³×C₅
hypercenter	C₅×#334	D₃
hypocenter	C₃	C₅×#1573
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₄²×C₅ ◁ C₅×#334

Complements

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

N	H ≅ G/N	type
4. C₃×C₅×#334	C₂	semidirect
5. C₅×#4480	C₂	semidirect
6. C₅×#4479	C₂	semidirect
7. C₅×#4458	C₂	semidirect
24. #68534	C₅	direct (N × H)
31. C₅×#334	D₃	semidirect
51. C₃×#334	C₂×C₅	semidirect
52. #4480	C₂×C₅	semidirect
53. #4479	C₂×C₅	semidirect
54. #4458	C₂×C₅	semidirect
128. #334	C₅×D₃	semidirect
244. C₃×C₅	#1573	semidirect
285. C₅	#68534	direct (N × H)
298. C₃	C₅×#1573	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×D₃×C₄⋊C₈	2
C₅×#4685	2
C₃×C₅×#334	2
C₅×#4480	2
C₅×#4479	2
C₅×#4458	2
C₅×#1573	3
#68534	5

Subgroups — C5×#68534 #14285945

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#68534 #14285945