Subgroups — C₅×#68590 #14292168

Order 1920.

332 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#4849	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#4848	C₂	1	1	yes	maximal, characteristic
3	C₅×#4694	C₂	1	1	yes	maximal, characteristic
4	C₅×C₈×Dic₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₅×#335	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#4485	C₂	1	1	yes	maximal, characteristic
7	C₅×#4482	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#1682	—	3	1	no	maximal, Hall
9	C₂×C₅×Dic₁₂	—	2	1	no	—
10	C₂×C₅×#204	—	2	1	no	—
11	C₅×#874	C₂²	1	1	yes	characteristic
12	C₅×#869	C₂²	1	1	yes	characteristic
13	C₄×C₅×Dic₆	C₂²	1	1	yes	characteristic
14	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
15	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
16	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
17	C₅×#821	—	2	1	no	—
18	C₅×#821	—	2	1	no	—
19	C₅×#821	—	2	1	no	—
20	C₅×#816	—	2	1	no	—
21	C₅×Dic₃⋊C₈	—	2	1	no	—
22	C₅×C₈×Dic₃	—	2	1	no	—
23	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
24	#68590	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
25	C₅×#486	—	3	1	no	—
26	C₅×#462	—	3	1	no	—
27	C₅×#460	—	3	1	no	—
28	C₅×C₈×Q₈	—	3	1	no	—
29	C₅×#335	D₃	1	1	yes	characteristic, complemented
30	C₅×#332	—	3	1	no	—
31	C₅×#331	—	3	1	no	—
32	C₂×C₅×D₁₂	—	2	1	no	—
33	C₂×C₅×D₁₂	—	2	1	no	—
34	C₂×C₅×D₁₂	—	2	1	no	—
35	C₂×C₅×Dic₆	—	2	1	no	—
36	C₂×C₅×Dic₆	D₄	1	1	yes	characteristic
37	C₂×C₃×C₅×C₈	—	2	1	no	—
38	C₂×C₃×C₅×C₈	—	2	1	no	—
39	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
40	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
41	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
42	C₅×C₄⋊Dic₃	—	2	1	no	—
43	C₅×C₄⋊Dic₃	—	2	1	no	—
44	C₅×C₄⋊Dic₃	D₄	1	1	yes	characteristic
45	C₅×Dic₃⋊C₄	—	2	1	no	—
46	C₄×C₅×Dic₃	—	2	1	no	—
47	C₂×C₅×C₃⋊C₈	—	2	1	no	—
48	C₅×Dic₁₂	—	4	2	no	—
49	C₅×#204	—	4	2	no	—
50	#4849	C₂×C₅	1	1	yes	characteristic, complemented
51	#4848	C₂×C₅	1	1	yes	characteristic
52	#4694	C₂×C₅	1	1	yes	characteristic
53	C₈×Dic₆	C₂×C₅	1	1	yes	characteristic, complemented
54	C₃×#335	C₂×C₅	1	1	yes	characteristic, complemented
55	#4485	C₂×C₅	1	1	yes	characteristic
56	#4482	C₂×C₅	1	1	yes	characteristic, complemented
57	C₂×C₅×Q₁₆	—	6	1	no	—
58	C₂×C₅×SD₁₆	—	6	1	no	—
59	C₅×C₄⋊Q₈	—	3	1	no	—
60	C₅×#127	—	3	1	no	—
61	C₄×C₅×Q₈	—	3	1	no	—
62	C₅×C₄⋊C₈	—	3	1	no	—
63	C₅×C₄⋊C₈	—	6	1	no	—
64	C₅×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
65	C₅×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
66	C₅×#103	—	6	1	no	—
67	C₅×#102	—	6	1	no	—
68	C₅×#102	—	6	1	no	—
69	C₅×#102	—	6	1	no	—
70	C₄×C₅×C₈	—	6	1	no	—
71	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
72	#1682	—	3	1	no	Sylow
73	C₂²×C₅×D₃	—	4	1	no	—
74	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
75	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
76	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
77	C₂×C₅×Dic₃	—	2	1	no	—
78	C₂×C₅×Dic₃	—	2	1	no	—
79	C₂×C₅×Dic₃	—	4	1	no	—
80	C₅×D₁₂	—	4	1	no	—
81	C₅×D₁₂	—	4	1	no	—
82	C₅×D₁₂	—	4	1	no	—
83	C₅×D₁₂	—	4	1	no	—
84	C₅×D₁₂	—	8	1	no	—
85	C₅×Dic₆	—	2	1	no	—
86	C₅×Dic₆	—	4	1	no	—
87	C₅×Dic₆	—	4	1	no	—
88	C₅×Dic₆	D₈	1	2	yes	—
89	C₃×C₅×C₈	—	2	1	no	—
90	C₃×C₅×C₈	—	2	1	no	—
91	C₃×C₅×C₈	—	4	1	no	—
92	C₃×C₅×C₈	—	4	1	no	—
93	C₅×C₃⋊C₈	—	4	1	no	—
94	C₂×Dic₁₂	—	2	1	no	—
95	C₂×#204	—	2	1	no	—
96	#874	C₂²×C₅	1	1	yes	characteristic
97	#869	C₂²×C₅	1	1	yes	characteristic
98	C₄×Dic₆	C₂²×C₅	1	1	yes	characteristic
99	C₃×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
100	C₃×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
101	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
102	#821	—	2	1	no	—
103	#821	—	2	1	no	—
104	#821	—	2	1	no	—
105	#816	—	2	1	no	—
106	Dic₃⋊C₈	—	2	1	no	—
107	C₈×Dic₃	—	2	1	no	—
108	C₄⋊(C₃⋊C₈)	C₂²×C₅	1	1	yes	characteristic
109	C₂×C₅×Q₈	—	3	1	no	—
110	C₂×C₅×Q₈	—	6	1	no	—
111	C₂×C₅×D₄	—	6	1	no	—
112	C₂×C₅×D₄	—	6	1	no	—
113	C₂×C₅×D₄	—	6	1	no	—
114	C₅×Q₁₆	—	12	2	no	—
115	C₅×SD₁₆	—	12	2	no	—
116	C₂×C₅×C₈	—	2	1	no	—
117	C₂×C₅×C₈	—	2	1	no	—
118	C₂×C₅×C₈	—	6	1	no	—
119	C₂×C₅×C₈	D₁₂	1	1	yes	characteristic
120	C₂×C₅×C₈	D₁₂	1	1	yes	characteristic
121	C₅×C₄⋊C₄	—	3	1	no	—
122	C₅×C₄⋊C₄	—	6	1	no	—
123	C₅×C₄⋊C₄	—	6	1	no	—
124	C₅×C₄⋊C₄	—	6	1	no	—
125	C₄²×C₅	—	6	1	no	—
126	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
127	#486	—	3	1	no	—
128	#462	—	3	1	no	—
129	#460	—	3	1	no	—
130	C₈×Q₈	—	3	1	no	—
131	#335	C₅×D₃	1	1	yes	characteristic, complemented
132	#332	—	3	1	no	—
133	#331	—	3	1	no	—
134	C₂²×C₃×C₅	#121	1	1	yes	characteristic
135	C₂×C₅×D₃	—	8	1	no	—
136	C₂×C₅×D₃	—	8	1	no	—
137	C₂×C₅×D₃	—	8	1	no	—
138	C₃×C₄×C₅	—	2	1	no	—
139	C₃×C₄×C₅	C₂×D₈	1	1	yes	characteristic
140	C₃×C₄×C₅	#135	1	1	yes	characteristic
141	C₃×C₄×C₅	#135	1	1	yes	characteristic
142	C₃×C₄×C₅	#137	1	1	yes	characteristic
143	C₅×Dic₃	—	2	2	no	—
144	C₅×Dic₃	—	4	1	no	—
145	C₅×Dic₃	—	8	1	no	—
146	C₂×D₁₂	—	2	1	no	—
147	C₂×D₁₂	—	2	1	no	—
148	C₂×D₁₂	—	2	1	no	—
149	C₂×Dic₆	—	2	1	no	—
150	C₂×Dic₆	C₅×D₄	1	1	yes	characteristic
151	C₂×C₃×C₈	—	2	1	no	—
152	C₂×C₃×C₈	—	2	1	no	—
153	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
154	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
155	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
156	C₄⋊Dic₃	—	2	1	no	—
157	C₄⋊Dic₃	—	2	1	no	—
158	C₄⋊Dic₃	C₅×D₄	1	1	yes	characteristic
159	Dic₃⋊C₄	—	2	1	no	—
160	C₄×Dic₃	—	2	1	no	—
161	C₂×C₃⋊C₈	—	2	1	no	—
162	Dic₁₂	—	4	2	no	—
163	#204	—	4	2	no	—
164	C₂³×C₅	—	12	1	no	—
165	C₅×Q₈	—	3	2	no	—
166	C₅×Q₈	—	6	1	no	—
167	C₅×Q₈	—	12	1	no	—
168	C₅×Q₈	—	12	1	no	—
169	C₅×D₄	—	12	1	no	—
170	C₅×D₄	—	12	1	no	—
171	C₅×D₄	—	12	1	no	—
172	C₅×D₄	—	12	1	no	—
173	C₅×D₄	—	24	1	no	—
174	C₂×C₄×C₅	—	6	1	no	—
175	C₂×C₄×C₅	—	6	1	no	—
176	C₂×C₄×C₅	—	12	1	no	—
177	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
178	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
179	C₂×C₄×C₅	#239	1	1	yes	characteristic
180	C₅×C₈	—	2	1	no	—
181	C₅×C₈	—	2	1	no	—
182	C₅×C₈	—	4	1	no	—
183	C₅×C₈	—	4	1	no	—
184	C₅×C₈	—	12	1	no	—
185	C₂×Q₁₆	—	6	1	no	—
186	C₂×SD₁₆	—	6	1	no	—
187	C₄⋊Q₈	—	3	1	no	—
188	#127	—	3	1	no	—
189	C₄×Q₈	—	3	1	no	—
190	C₄⋊C₈	—	3	1	no	—
191	C₄⋊C₈	—	6	1	no	—
192	C₄⋊C₈	C₂×C₅×D₃	1	1	yes	characteristic
193	C₄⋊C₈	C₂×C₅×D₃	1	1	yes	characteristic
194	#103	—	6	1	no	—
195	#102	—	6	1	no	—
196	#102	—	6	1	no	—
197	#102	—	6	1	no	—
198	C₄×C₈	—	6	1	no	—
199	C₄×C₈	C₂×C₅×D₃	1	1	yes	characteristic
200	C₂×C₃×C₅	#464	1	1	yes	characteristic
201	C₂×C₃×C₅	#466	1	1	yes	characteristic
202	C₂×C₃×C₅	#471	1	1	yes	characteristic
203	C₅×D₃	—	16	1	no	—
204	C₂²×D₃	—	4	1	no	—
205	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
206	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
207	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
208	C₂×Dic₃	—	2	1	no	—
209	C₂×Dic₃	—	2	1	no	—
210	C₂×Dic₃	—	4	1	no	—
211	D₁₂	—	4	1	no	—
212	D₁₂	—	4	1	no	—
213	D₁₂	—	4	1	no	—
214	D₁₂	—	4	1	no	—
215	D₁₂	—	8	1	no	—
216	Dic₆	—	2	1	no	—
217	Dic₆	—	4	1	no	—
218	Dic₆	—	4	1	no	—
219	Dic₆	C₅×D₈	1	2	yes	—
220	C₃×C₈	—	2	1	no	—
221	C₃×C₈	—	2	1	no	—
222	C₃×C₈	—	4	1	no	—
223	C₃×C₈	—	4	1	no	—
224	C₃⋊C₈	—	4	1	no	—
225	C₂²×C₅	—	24	1	no	—
226	C₂²×C₅	—	24	1	no	—
227	C₂²×C₅	—	24	1	no	—
228	C₂²×C₅	#895	1	1	yes	characteristic
229	C₄×C₅	—	2	1	no	—
230	C₄×C₅	—	6	2	no	—
231	C₄×C₅	—	12	1	no	—
232	C₄×C₅	—	24	1	no	—
233	C₄×C₅	#904	1	1	yes	characteristic
234	C₄×C₅	#909	1	1	yes	characteristic
235	C₄×C₅	D₃×D₈	1	1	yes	characteristic
236	C₄×C₅	#919	1	1	yes	characteristic
237	C₂×Q₈	—	3	1	no	—
238	C₂×Q₈	—	6	1	no	—
239	C₂×D₄	—	6	1	no	—
240	C₂×D₄	—	6	1	no	—
241	C₂×D₄	—	6	1	no	—
242	Q₁₆	—	12	2	no	—
243	SD₁₆	—	12	2	no	—
244	C₂×C₈	—	2	1	no	—
245	C₂×C₈	—	2	1	no	—
246	C₂×C₈	—	6	1	no	—
247	C₂×C₈	C₅×D₁₂	1	1	yes	characteristic
248	C₂×C₈	C₅×D₁₂	1	1	yes	characteristic
249	C₄⋊C₄	—	3	1	no	—
250	C₄⋊C₄	—	6	1	no	—
251	C₄⋊C₄	—	6	1	no	—
252	C₄⋊C₄	—	6	1	no	—
253	C₄²	—	6	1	no	—
254	C₄²	C₂²×C₅×D₃	1	1	yes	characteristic
255	C₃×C₅	#1682	1	1	yes	characteristic, complemented, Hall
256	C₂²×C₃	C₅×#121	1	1	yes	characteristic
257	C₂×D₃	—	8	1	no	—
258	C₂×D₃	—	8	1	no	—
259	C₂×D₃	—	8	1	no	—
260	C₃×C₄	—	2	1	no	—
261	C₃×C₄	C₂×C₅×D₈	1	1	yes	characteristic
262	C₃×C₄	C₅×#135	1	1	yes	characteristic
263	C₃×C₄	C₅×#135	1	1	yes	characteristic
264	C₃×C₄	C₅×#137	1	1	yes	characteristic
265	Dic₃	—	2	2	no	—
266	Dic₃	—	4	1	no	—
267	Dic₃	—	8	1	no	—
268	C₂×C₅	—	48	1	no	—
269	C₂×C₅	#4845	1	1	yes	characteristic
270	C₂×C₅	#4883	1	1	yes	characteristic
271	C₂×C₅	#4898	1	1	yes	characteristic
272	C₂³	—	12	1	no	—
273	Q₈	—	3	2	no	—
274	Q₈	—	6	1	no	—
275	Q₈	—	12	1	no	—
276	Q₈	—	12	1	no	—
277	D₄	—	12	1	no	—
278	D₄	—	12	1	no	—
279	D₄	—	12	1	no	—
280	D₄	—	12	1	no	—
281	D₄	—	24	1	no	—
282	C₂×C₄	—	6	1	no	—
283	C₂×C₄	—	6	1	no	—
284	C₂×C₄	—	12	1	no	—
285	C₂×C₄	C₂×C₅×D₁₂	1	1	yes	characteristic
286	C₂×C₄	C₅×D₃×D₄	1	1	yes	characteristic
287	C₂×C₄	C₅×#239	1	1	yes	characteristic
288	C₈	—	2	1	no	—
289	C₈	—	2	1	no	—
290	C₈	—	4	1	no	—
291	C₈	—	4	1	no	—
292	C₈	—	12	1	no	—
293	C₂×C₃	C₅×#464	1	1	yes	characteristic
294	C₂×C₃	C₅×#466	1	1	yes	characteristic
295	C₂×C₃	C₅×#471	1	1	yes	characteristic
296	D₃	—	16	1	no	—
297	C₅	#68590	1	1	yes	characteristic, direct product factor, Sylow
298	C₂²	—	24	1	no	—
299	C₂²	—	24	1	no	—
300	C₂²	—	24	1	no	—
301	C₂²	C₅×#895	1	1	yes	characteristic
302	C₄	—	2	1	no	—
303	C₄	—	6	2	no	—
304	C₄	—	12	1	no	—
305	C₄	—	24	1	no	—
306	C₄	C₅×#904	1	1	yes	characteristic
307	C₄	C₅×#909	1	1	yes	characteristic
308	C₄	C₅×D₃×D₈	1	1	yes	characteristic
309	C₄	C₅×#919	1	1	yes	characteristic
310	C₃	C₅×#1682	1	1	yes	characteristic, complemented, Sylow
311	C₂	—	48	1	no	—
312	C₂	C₅×#4845	1	1	yes	characteristic
313	C₂	C₅×#4883	1	1	yes	characteristic
314	C₂	C₅×#4898	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅	#895
commutator, cosocle	C₃×C₄²	C₂³×C₅
Frattini	C₄²	C₂²×C₅×D₃
Fitting	C₃×C₅×#335	C₂
socle	C₂²×C₃×C₅	#121
prime radical	C₅×#874	C₂²
hypercenter	C₅×#335	D₃
hypocenter	C₃	C₅×#1682
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ C₅×#335

Complements

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

N	H ≅ G/N	type
1. C₅×#4849	C₂	semidirect
4. C₅×C₈×Dic₆	C₂	semidirect
5. C₃×C₅×#335	C₂	semidirect
7. C₅×#4482	C₂	semidirect
24. #68590	C₅	direct (N × H)
29. C₅×#335	D₃	semidirect
50. #4849	C₂×C₅	semidirect
53. C₈×Dic₆	C₂×C₅	semidirect
54. C₃×#335	C₂×C₅	semidirect
56. #4482	C₂×C₅	semidirect
131. #335	C₅×D₃	semidirect
255. C₃×C₅	#1682	semidirect
297. C₅	#68590	direct (N × H)
310. C₃	C₅×#1682	semidirect

Maximal subgroups

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

subgroup	index
C₅×#4849	2
C₅×#4848	2
C₅×#4694	2
C₅×C₈×Dic₆	2
C₃×C₅×#335	2
C₅×#4485	2
C₅×#4482	2
C₅×#1682	3
#68590	5

Subgroups — C5×#68590 #14292168

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#68590 #14292168