Subgroups — C₅×#81938 #14439837

Order 1920.

712 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	C₅×#5728	C₂	1	2	yes	maximal
2	C₃×C₅×#500	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#5080	C₂	1	2	yes	maximal, complemented
4	C₅×#4880	C₂	1	4	yes	maximal
5	C₅×#4878	C₂	1	4	yes	maximal, complemented
6	C₄×C₅×D₃×C₈	C₂	1	1	yes	maximal, characteristic
7	C₅×#4682	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#3160	—	3	1	no	maximal, Hall
9	C₃×C₅×C₄⋊Q₈	C₂²	1	2	yes	—
10	C₅×#946	—	2	2	no	—
11	C₅×Q₈×Dic₃	—	2	2	no	—
12	C₂×C₅×D₃×C₈	C₂²	1	2	yes	—
13	C₅×#897	—	2	2	no	—
14	C₅×#892	C₂²	1	4	yes	—
15	C₅×#890	C₂²	1	4	yes	—
16	C₅×#887	—	2	2	no	—
17	C₄²×C₅×D₃	C₂²	1	1	yes	characteristic
18	C₅×#869	C₂²	1	2	yes	—
19	C₃×C₅×#107	C₂²	1	4	yes	—
20	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
21	C₅×#818	C₂²	1	4	yes	—
22	C₅×C₈×Dic₃	C₂²	1	2	yes	—
23	C₅×#808	C₂²	1	8	yes	—
24	C₄×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
25	#81938	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
26	C₅×#533	—	3	2	no	—
27	C₅×#500	—	3	1	no	—
28	C₅×#500	D₃	1	1	yes	characteristic, complemented
29	C₅×#499	—	3	4	no	—
30	C₅×#498	—	3	2	no	—
31	C₅×#427	—	3	4	no	—
32	C₂×C₄×C₅×C₈	—	3	1	no	—
33	C₂×C₃×C₅×Q₈	—	2	2	no	—
34	C₂×C₄×C₅×D₃	D₄	1	1	yes	characteristic
35	C₂×C₄×C₅×D₃	D₄	1	1	yes	characteristic
36	C₂×C₄×C₅×D₃	C₂³	1	1	yes	characteristic
37	C₂×C₅×Dic₆	—	2	2	no	—
38	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
39	C₃×C₅×C₄⋊C₄	—	2	2	no	—
40	C₃×C₅×C₄⋊C₄	C₂³	1	4	yes	—
41	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
42	C₅×(C₂×D₃)⋊C₄	—	2	4	no	—
43	C₅×C₄⋊Dic₃	—	2	2	no	—
44	C₅×C₄⋊Dic₃	—	2	4	no	—
45	C₅×C₄⋊Dic₃	C₂³	1	4	yes	—
46	C₅×Dic₃⋊C₄	—	2	4	no	—
47	C₄×C₅×Dic₃	—	2	4	no	—
48	C₄×C₅×Dic₃	D₄	1	1	yes	characteristic
49	C₄×C₅×Dic₃	D₄	1	1	yes	characteristic
50	C₄×C₅×Dic₃	C₂³	1	1	yes	characteristic
51	C₂×C₅×C₃⋊C₈	C₂³	1	2	yes	—
52	C₅×D₃×C₈	Q₈	1	8	yes	complemented
53	#5728	C₂×C₅	1	2	yes	—
54	C₃×#500	C₂×C₅	1	1	yes	characteristic, complemented
55	#5080	C₂×C₅	1	2	yes	complemented
56	#4880	C₂×C₅	1	4	yes	—
57	#4878	C₂×C₅	1	4	yes	complemented
58	C₄×D₃×C₈	C₂×C₅	1	1	yes	characteristic
59	#4682	C₂×C₅	1	1	yes	characteristic, complemented
60	C₂²×C₅×C₈	—	3	2	no	—
61	C₅×C₄⋊Q₈	—	3	2	no	—
62	C₅×C₄⋊Q₈	C₂×D₃	1	2	yes	—
63	C₅×#125	—	3	4	no	—
64	C₅×C₂²⋊Q₈	—	6	2	no	—
65	C₅×C₂²⋊Q₈	—	6	2	no	—
66	C₄×C₅×Q₈	—	6	2	no	—
67	C₄×C₅×Q₈	—	6	2	no	—
68	C₅×#117	—	3	4	no	—
69	C₂×C₄²×C₅	—	3	1	no	—
70	C₅×#107	—	3	4	no	—
71	C₅×#107	C₂×D₃	1	4	yes	—
72	C₅×#106	—	3	8	no	—
73	C₄×C₅×C₈	—	3	2	no	—
74	C₄×C₅×C₈	—	3	1	no	—
75	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
76	#3160	—	3	1	no	Sylow
77	C₂²×C₅×D₃	C₂×D₄	1	1	yes	characteristic
78	C₃×C₅×Q₈	—	4	4	no	—
79	C₂×C₃×C₄×C₅	—	2	4	no	—
80	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
81	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
82	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
83	C₂×C₅×Dic₃	—	2	4	no	—
84	C₂×C₅×Dic₃	C₂×D₄	1	1	yes	characteristic
85	C₂×C₅×Dic₃	C₂×D₄	1	1	yes	characteristic
86	C₂×C₅×Dic₃	C₂×D₄	1	1	yes	characteristic
87	C₄×C₅×D₃	—	2	2	no	—
88	C₄×C₅×D₃	—	2	2	no	—
89	C₄×C₅×D₃	C₂×Q₈	1	4	yes	—
90	C₅×Dic₆	—	4	4	no	—
91	C₃×C₅×C₈	C₂×Q₈	1	4	yes	—
92	C₅×C₃⋊C₈	C₂×Q₈	1	4	yes	—
93	C₃×C₄⋊Q₈	C₂²×C₅	1	2	yes	—
94	#946	—	2	2	no	—
95	Q₈×Dic₃	—	2	2	no	—
96	C₂×D₃×C₈	C₂²×C₅	1	2	yes	—
97	#897	—	2	2	no	—
98	#892	C₂²×C₅	1	4	yes	—
99	#890	C₂²×C₅	1	4	yes	—
100	#887	—	2	2	no	—
101	C₄²×D₃	C₂²×C₅	1	1	yes	characteristic
102	#869	C₂²×C₅	1	2	yes	—
103	C₃×#107	C₂²×C₅	1	4	yes	—
104	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
105	#818	C₂²×C₅	1	4	yes	—
106	C₈×Dic₃	C₂²×C₅	1	2	yes	—
107	#808	C₂²×C₅	1	8	yes	—
108	C₄×C₃⋊C₈	C₂²×C₅	1	1	yes	characteristic
109	C₂×C₅×Q₈	—	2	2	no	—
110	C₂×C₅×Q₈	—	6	2	no	—
111	C₂²×C₄×C₅	—	3	1	no	—
112	C₂²×C₄×C₅	—	3	1	no	—
113	C₂²×C₄×C₅	—	3	1	no	—
114	C₂×C₅×C₈	—	3	2	no	—
115	C₂×C₅×C₈	—	3	8	no	—
116	C₂×C₅×C₈	C₂²×D₃	1	2	yes	—
117	C₅×C₄⋊C₄	—	2	2	no	—
118	C₅×C₄⋊C₄	—	3	4	no	—
119	C₅×C₄⋊C₄	—	6	4	no	—
120	C₅×C₄⋊C₄	—	6	4	no	—
121	C₅×C₄⋊C₄	—	6	2	no	—
122	C₅×C₄⋊C₄	C₂²×D₃	1	4	yes	—
123	C₅×C₂²⋊C₄	—	6	4	no	—
124	C₄²×C₅	—	3	1	no	—
125	C₄²×C₅	—	3	1	no	—
126	C₄²×C₅	—	3	1	no	—
127	C₄²×C₅	—	6	4	no	—
128	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
129	#533	—	3	2	no	—
130	#500	—	3	1	no	—
131	#500	C₅×D₃	1	1	yes	characteristic, complemented
132	#499	—	3	4	no	—
133	#498	—	3	2	no	—
134	#427	—	3	4	no	—
135	C₂×C₄×C₈	—	3	1	no	—
136	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
137	C₂×C₅×D₃	—	2	1	no	—
138	C₂×C₅×D₃	—	2	1	no	—
139	C₂×C₅×D₃	C₄⋊Q₈	1	2	yes	—
140	C₃×C₄×C₅	—	4	4	no	—
141	C₃×C₄×C₅	#135	1	2	yes	—
142	C₃×C₄×C₅	#135	1	2	yes	—
143	C₃×C₄×C₅	C₂²×Q₈	1	2	yes	—
144	C₅×Dic₃	—	2	1	no	—
145	C₅×Dic₃	—	2	1	no	—
146	C₅×Dic₃	—	4	4	no	—
147	C₅×Dic₃	C₄⋊Q₈	1	2	yes	—
148	C₂×C₃×Q₈	—	2	2	no	—
149	C₂×C₄×D₃	C₅×D₄	1	1	yes	characteristic
150	C₂×C₄×D₃	C₅×D₄	1	1	yes	characteristic
151	C₂×C₄×D₃	C₂³×C₅	1	1	yes	characteristic
152	C₂×Dic₆	—	2	2	no	—
153	C₂×C₃×C₈	C₂³×C₅	1	2	yes	—
154	C₃×C₄⋊C₄	—	2	2	no	—
155	C₃×C₄⋊C₄	C₂³×C₅	1	4	yes	—
156	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
157	(C₂×D₃)⋊C₄	—	2	4	no	—
158	C₄⋊Dic₃	—	2	2	no	—
159	C₄⋊Dic₃	—	2	4	no	—
160	C₄⋊Dic₃	C₂³×C₅	1	4	yes	—
161	Dic₃⋊C₄	—	2	4	no	—
162	C₄×Dic₃	—	2	4	no	—
163	C₄×Dic₃	C₅×D₄	1	1	yes	characteristic
164	C₄×Dic₃	C₅×D₄	1	1	yes	characteristic
165	C₄×Dic₃	C₂³×C₅	1	1	yes	characteristic
166	C₂×C₃⋊C₈	C₂³×C₅	1	2	yes	—
167	D₃×C₈	C₅×Q₈	1	8	yes	complemented
168	C₂³×C₅	—	3	1	no	—
169	C₅×Q₈	—	4	4	no	—
170	C₅×Q₈	—	12	4	no	—
171	C₂×C₄×C₅	—	2	4	no	—
172	C₂×C₄×C₅	—	3	1	no	—
173	C₂×C₄×C₅	—	3	4	no	—
174	C₂×C₄×C₅	—	3	1	no	—
175	C₂×C₄×C₅	—	3	1	no	—
176	C₂×C₄×C₅	—	6	2	no	—
177	C₂×C₄×C₅	—	6	2	no	—
178	C₂×C₄×C₅	—	6	4	no	—
179	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
180	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
181	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
182	C₅×C₈	—	3	4	no	—
183	C₅×C₈	D₃×Q₈	1	4	yes	—
184	C₂²×C₈	—	3	2	no	—
185	C₄⋊Q₈	—	3	2	no	—
186	C₄⋊Q₈	C₂×C₅×D₃	1	2	yes	—
187	#125	—	3	4	no	—
188	C₂²⋊Q₈	—	6	2	no	—
189	C₂²⋊Q₈	—	6	2	no	—
190	C₄×Q₈	—	6	2	no	—
191	C₄×Q₈	—	6	2	no	—
192	#117	—	3	4	no	—
193	C₂×C₄²	—	3	1	no	—
194	#107	—	3	4	no	—
195	#107	C₂×C₅×D₃	1	4	yes	—
196	#106	—	3	8	no	—
197	C₄×C₈	—	3	2	no	—
198	C₄×C₈	—	3	1	no	—
199	C₄×C₈	C₂×C₅×D₃	1	1	yes	characteristic
200	C₂×C₃×C₅	C₂×C₄⋊Q₈	1	1	yes	characteristic
201	C₂×C₃×C₅	C₂×#135	1	1	yes	characteristic
202	C₂×C₃×C₅	C₂×#135	1	1	yes	characteristic
203	C₅×D₃	—	2	2	no	—
204	C₂²×D₃	C₂×C₅×D₄	1	1	yes	characteristic
205	C₃×Q₈	—	4	4	no	—
206	C₂×C₃×C₄	—	2	4	no	—
207	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
208	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
209	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
210	C₂×Dic₃	—	2	4	no	—
211	C₂×Dic₃	C₂×C₅×D₄	1	1	yes	characteristic
212	C₂×Dic₃	C₂×C₅×D₄	1	1	yes	characteristic
213	C₂×Dic₃	C₂×C₅×D₄	1	1	yes	characteristic
214	C₄×D₃	—	2	2	no	—
215	C₄×D₃	—	2	2	no	—
216	C₄×D₃	C₂×C₅×Q₈	1	4	yes	—
217	Dic₆	—	4	4	no	—
218	C₃×C₈	C₂×C₅×Q₈	1	4	yes	—
219	C₃⋊C₈	C₂×C₅×Q₈	1	4	yes	—
220	C₂²×C₅	—	3	2	no	—
221	C₂²×C₅	—	6	1	no	—
222	C₂²×C₅	—	6	1	no	—
223	C₂²×C₅	C₂×D₃×D₄	1	1	yes	characteristic
224	C₄×C₅	—	3	2	no	—
225	C₄×C₅	—	4	4	no	—
226	C₄×C₅	—	6	1	no	—
227	C₄×C₅	—	6	1	no	—
228	C₄×C₅	—	12	4	no	—
229	C₄×C₅	#912	1	2	yes	—
230	C₄×C₅	#919	1	2	yes	—
231	C₄×C₅	C₂×D₃×Q₈	1	2	yes	—
232	C₂×Q₈	—	2	2	no	—
233	C₂×Q₈	—	6	2	no	—
234	C₂²×C₄	—	3	1	no	—
235	C₂²×C₄	—	3	1	no	—
236	C₂²×C₄	—	3	1	no	—
237	C₂×C₈	—	3	2	no	—
238	C₂×C₈	—	3	8	no	—
239	C₂×C₈	C₂²×C₅×D₃	1	2	yes	—
240	C₄⋊C₄	—	2	2	no	—
241	C₄⋊C₄	—	3	4	no	—
242	C₄⋊C₄	—	6	4	no	—
243	C₄⋊C₄	—	6	4	no	—
244	C₄⋊C₄	—	6	2	no	—
245	C₄⋊C₄	C₂²×C₅×D₃	1	4	yes	—
246	C₂²⋊C₄	—	6	4	no	—
247	C₄²	—	3	1	no	—
248	C₄²	—	3	1	no	—
249	C₄²	—	3	1	no	—
250	C₄²	—	6	4	no	—
251	C₄²	C₂²×C₅×D₃	1	1	yes	characteristic
252	C₃×C₅	#3160	1	1	yes	characteristic, complemented, Hall
253	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
254	C₂×D₃	—	2	1	no	—
255	C₂×D₃	—	2	1	no	—
256	C₂×D₃	C₅×C₄⋊Q₈	1	2	yes	—
257	C₃×C₄	—	4	4	no	—
258	C₃×C₄	C₅×#135	1	2	yes	—
259	C₃×C₄	C₅×#135	1	2	yes	—
260	C₃×C₄	C₂²×C₅×Q₈	1	2	yes	—
261	Dic₃	—	2	1	no	—
262	Dic₃	—	2	1	no	—
263	Dic₃	—	4	4	no	—
264	Dic₃	C₅×C₄⋊Q₈	1	2	yes	—
265	C₂×C₅	—	6	2	no	—
266	C₂×C₅	D₃×C₄⋊Q₈	1	1	yes	characteristic
267	C₂×C₅	C₂×#912	1	1	yes	characteristic
268	C₂×C₅	C₂×#919	1	1	yes	characteristic
269	C₂³	—	3	1	no	—
270	Q₈	—	4	4	no	—
271	Q₈	—	12	4	no	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	3	1	no	—
274	C₂×C₄	—	3	4	no	—
275	C₂×C₄	—	3	1	no	—
276	C₂×C₄	—	3	1	no	—
277	C₂×C₄	—	6	2	no	—
278	C₂×C₄	—	6	2	no	—
279	C₂×C₄	—	6	4	no	—
280	C₂×C₄	C₅×D₃×D₄	1	1	yes	characteristic
281	C₂×C₄	C₅×D₃×D₄	1	1	yes	characteristic
282	C₂×C₄	C₂³×C₅×D₃	1	1	yes	characteristic
283	C₈	—	3	4	no	—
284	C₈	C₅×D₃×Q₈	1	4	yes	—
285	C₂×C₃	C₂×C₅×C₄⋊Q₈	1	1	yes	characteristic
286	C₂×C₃	C₂×C₅×#135	1	1	yes	characteristic
287	C₂×C₃	C₂×C₅×#135	1	1	yes	characteristic
288	D₃	—	2	2	no	—
289	C₅	#81938	1	1	yes	characteristic, direct product factor, Sylow
290	C₂²	—	3	2	no	—
291	C₂²	—	6	1	no	—
292	C₂²	—	6	1	no	—
293	C₂²	C₂×C₅×D₃×D₄	1	1	yes	characteristic
294	C₄	—	3	2	no	—
295	C₄	—	4	4	no	—
296	C₄	—	6	1	no	—
297	C₄	—	6	1	no	—
298	C₄	—	12	4	no	—
299	C₄	C₅×#912	1	2	yes	—
300	C₄	C₅×#919	1	2	yes	—
301	C₄	C₂×C₅×D₃×Q₈	1	2	yes	—
302	C₃	C₅×#3160	1	1	yes	characteristic, complemented, Sylow
303	C₂	—	6	2	no	—
304	C₂	C₅×D₃×C₄⋊Q₈	1	1	yes	characteristic
305	C₂	C₂×C₅×#912	1	1	yes	characteristic
306	C₂	C₂×C₅×#919	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅	C₂×D₃×D₄
commutator, cosocle	C₂×C₃×C₄	C₂⁴×C₅
Frattini	C₂×C₄	C₂³×C₅×D₃
Fitting	C₃×C₅×#500	C₂
socle	C₂²×C₃×C₅	C₂²×D₄
prime radical	C₂²×C₅×D₃	C₂×D₄
hypercenter	C₅×#500	D₃
hypocenter	C₃	C₅×#3160
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ C₅×#500

Complements

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

N	H ≅ G/N	type
2. C₃×C₅×#500	C₂	semidirect
3. C₅×#5080	C₂	semidirect
5. C₅×#4878	C₂	semidirect
7. C₅×#4682	C₂	semidirect
25. #81938	C₅	direct (N × H)
28. C₅×#500	D₃	semidirect
52. C₅×D₃×C₈	Q₈	semidirect
54. C₃×#500	C₂×C₅	semidirect
55. #5080	C₂×C₅	semidirect
57. #4878	C₂×C₅	semidirect
59. #4682	C₂×C₅	semidirect
131. #500	C₅×D₃	semidirect
167. D₃×C₈	C₅×Q₈	semidirect
252. C₃×C₅	#3160	semidirect
289. C₅	#81938	direct (N × H)
302. C₃	C₅×#3160	semidirect

Maximal subgroups

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

subgroup	index
C₅×#5728	2
C₃×C₅×#500	2
C₅×#5080	2
C₅×#4880	2
C₅×#4878	2
C₄×C₅×D₃×C₈	2
C₅×#4682	2
C₅×#3160	3
#81938	5

Subgroups — C5×#81938 #14439837

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#81938 #14439837