Subgroups — C₅×#81916 #14439843

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×#5688	C₂	1	2	yes	maximal
2	C₃×C₅×#499	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#5065	C₂	1	2	yes	maximal, complemented
4	C₅×#4880	C₂	1	2	yes	maximal
5	C₅×#4874	C₂	1	2	yes	maximal, complemented
6	C₅×#4860	C₂	1	2	yes	maximal
7	C₅×#4855	C₂	1	2	yes	maximal, complemented
8	C₄×C₅×D₃×C₈	C₂	1	1	yes	maximal, characteristic
9	C₅×#4683	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅×#3160	—	3	1	no	maximal, Hall
11	C₃×C₅×#125	C₂²	1	2	yes	—
12	C₂×C₅×D₃×C₈	C₂²	1	1	yes	characteristic
13	C₂×C₅×D₃×C₈	C₂²	1	1	yes	characteristic
14	C₅×#896	—	2	4	no	—
15	C₅×#892	C₂²	1	4	yes	—
16	C₅×#888	C₂²	1	4	yes	—
17	C₅×#887	—	2	4	no	—
18	C₄²×C₅×D₃	C₂²	1	1	yes	characteristic
19	C₅×#870	C₂²	1	2	yes	—
20	C₃×C₅×#107	C₂²	1	2	yes	—
21	C₃×C₅×#106	C₂²	1	2	yes	—
22	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
23	C₅×#818	C₂²	1	2	yes	—
24	C₅×#817	C₂²	1	2	yes	—
25	C₅×C₈×Dic₃	C₂²	1	1	yes	characteristic
26	C₅×C₈×Dic₃	C₂²	1	1	yes	characteristic
27	C₅×#808	C₂²	1	4	yes	—
28	C₅×#807	C₂²	1	4	yes	—
29	C₄×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
30	#81916	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
31	C₅×#533	—	3	2	no	—
32	C₅×#500	—	3	2	no	—
33	C₅×#499	—	3	1	no	—
34	C₅×#499	—	3	2	no	—
35	C₅×#499	D₃	1	1	yes	characteristic, complemented
36	C₅×#498	—	3	2	no	—
37	C₅×#427	—	3	2	no	—
38	C₅×#427	—	3	2	no	—
39	C₂×C₄×C₅×C₈	—	3	1	no	—
40	C₂×C₄×C₅×D₃	D₄	1	2	yes	—
41	C₂×C₄×C₅×D₃	C₂³	1	1	yes	characteristic
42	C₂×C₅×Dic₆	—	2	4	no	—
43	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
44	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
45	C₃×C₅×C₄⋊C₄	—	2	4	no	—
46	C₃×C₅×C₄⋊C₄	C₂³	1	4	yes	—
47	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
48	C₅×(C₂×D₃)⋊C₄	—	2	4	no	—
49	C₅×C₄⋊Dic₃	C₂³	1	4	yes	—
50	C₅×Dic₃⋊C₄	—	2	4	no	—
51	C₅×Dic₃⋊C₄	—	2	4	no	—
52	C₄×C₅×Dic₃	—	2	4	no	—
53	C₄×C₅×Dic₃	D₄	1	2	yes	—
54	C₄×C₅×Dic₃	C₂³	1	1	yes	characteristic
55	C₂×C₅×C₃⋊C₈	C₂³	1	1	yes	characteristic
56	C₂×C₅×C₃⋊C₈	C₂³	1	1	yes	characteristic
57	C₅×D₃×C₈	Q₈	1	4	yes	complemented
58	C₅×D₃×C₈	Q₈	1	4	yes	complemented
59	#5688	C₂×C₅	1	2	yes	—
60	C₃×#499	C₂×C₅	1	1	yes	characteristic, complemented
61	#5065	C₂×C₅	1	2	yes	complemented
62	#4880	C₂×C₅	1	2	yes	—
63	#4874	C₂×C₅	1	2	yes	complemented
64	#4860	C₂×C₅	1	2	yes	—
65	#4855	C₂×C₅	1	2	yes	complemented
66	C₄×D₃×C₈	C₂×C₅	1	1	yes	characteristic
67	#4683	C₂×C₅	1	1	yes	characteristic, complemented
68	C₂²×C₅×C₈	—	3	1	no	—
69	C₂²×C₅×C₈	—	3	1	no	—
70	C₅×C₄⋊Q₈	—	3	4	no	—
71	C₅×#125	—	3	2	no	—
72	C₅×#125	C₂×D₃	1	2	yes	—
73	C₅×C₂²⋊Q₈	—	6	4	no	—
74	C₄×C₅×Q₈	—	6	4	no	—
75	C₅×#117	—	3	4	no	—
76	C₂×C₄²×C₅	—	3	1	no	—
77	C₅×#107	—	3	2	no	—
78	C₅×#107	—	3	4	no	—
79	C₅×#107	C₂×D₃	1	2	yes	—
80	C₅×#106	—	3	2	no	—
81	C₅×#106	—	3	4	no	—
82	C₅×#106	C₂×D₃	1	2	yes	—
83	C₄×C₅×C₈	—	3	1	no	—
84	C₄×C₅×C₈	—	3	1	no	—
85	C₄×C₅×C₈	—	3	1	no	—
86	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
87	#3160	—	3	1	no	Sylow
88	C₂²×C₅×D₃	C₂×D₄	1	1	yes	characteristic
89	C₂×C₃×C₄×C₅	—	2	4	no	—
90	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
91	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
92	C₂×C₅×Dic₃	—	2	4	no	—
93	C₂×C₅×Dic₃	C₂×D₄	1	1	yes	characteristic
94	C₂×C₅×Dic₃	C₂×D₄	1	2	yes	—
95	C₄×C₅×D₃	—	2	4	no	—
96	C₄×C₅×D₃	C₂×Q₈	1	2	yes	—
97	C₄×C₅×D₃	C₂×Q₈	1	2	yes	—
98	C₅×Dic₆	—	4	8	no	—
99	C₃×C₅×C₈	C₂×Q₈	1	2	yes	—
100	C₃×C₅×C₈	C₂×Q₈	1	2	yes	—
101	C₅×C₃⋊C₈	C₂×Q₈	1	2	yes	—
102	C₅×C₃⋊C₈	C₂×Q₈	1	2	yes	—
103	C₃×#125	C₂²×C₅	1	2	yes	—
104	C₂×D₃×C₈	C₂²×C₅	1	1	yes	characteristic
105	C₂×D₃×C₈	C₂²×C₅	1	1	yes	characteristic
106	#896	—	2	4	no	—
107	#892	C₂²×C₅	1	4	yes	—
108	#888	C₂²×C₅	1	4	yes	—
109	#887	—	2	4	no	—
110	C₄²×D₃	C₂²×C₅	1	1	yes	characteristic
111	#870	C₂²×C₅	1	2	yes	—
112	C₃×#107	C₂²×C₅	1	2	yes	—
113	C₃×#106	C₂²×C₅	1	2	yes	—
114	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
115	#818	C₂²×C₅	1	2	yes	—
116	#817	C₂²×C₅	1	2	yes	—
117	C₈×Dic₃	C₂²×C₅	1	1	yes	characteristic
118	C₈×Dic₃	C₂²×C₅	1	1	yes	characteristic
119	#808	C₂²×C₅	1	4	yes	—
120	#807	C₂²×C₅	1	4	yes	—
121	C₄×C₃⋊C₈	C₂²×C₅	1	1	yes	characteristic
122	C₂×C₅×Q₈	—	6	4	no	—
123	C₂²×C₄×C₅	—	3	1	no	—
124	C₂²×C₄×C₅	—	3	2	no	—
125	C₂×C₅×C₈	—	3	1	no	—
126	C₂×C₅×C₈	—	3	1	no	—
127	C₂×C₅×C₈	—	3	4	no	—
128	C₂×C₅×C₈	—	3	4	no	—
129	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
130	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
131	C₅×C₄⋊C₄	—	2	4	no	—
132	C₅×C₄⋊C₄	—	3	4	no	—
133	C₅×C₄⋊C₄	—	6	4	no	—
134	C₅×C₄⋊C₄	—	6	4	no	—
135	C₅×C₄⋊C₄	C₂²×D₃	1	4	yes	—
136	C₅×C₂²⋊C₄	—	6	4	no	—
137	C₄²×C₅	—	3	2	no	—
138	C₄²×C₅	—	3	1	no	—
139	C₄²×C₅	—	6	4	no	—
140	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
141	#533	—	3	2	no	—
142	#500	—	3	2	no	—
143	#499	—	3	2	no	—
144	#499	—	3	1	no	—
145	#499	C₅×D₃	1	1	yes	characteristic, complemented
146	#498	—	3	2	no	—
147	#427	—	3	2	no	—
148	#427	—	3	2	no	—
149	C₂×C₄×C₈	—	3	1	no	—
150	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
151	C₂×C₅×D₃	—	2	2	no	—
152	C₂×C₅×D₃	C₄⋊Q₈	1	2	yes	—
153	C₃×C₄×C₅	—	2	2	no	—
154	C₃×C₄×C₅	—	4	4	no	—
155	C₃×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
156	C₃×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
157	C₅×Dic₃	—	4	4	no	—
158	C₅×Dic₃	C₄⋊Q₈	1	2	yes	—
159	C₅×Dic₃	#135	1	4	yes	—
160	C₂×C₄×D₃	C₅×D₄	1	2	yes	—
161	C₂×C₄×D₃	C₂³×C₅	1	1	yes	characteristic
162	C₂×Dic₆	—	2	4	no	—
163	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
164	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
165	C₃×C₄⋊C₄	—	2	4	no	—
166	C₃×C₄⋊C₄	C₂³×C₅	1	4	yes	—
167	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
168	(C₂×D₃)⋊C₄	—	2	4	no	—
169	C₄⋊Dic₃	C₂³×C₅	1	4	yes	—
170	Dic₃⋊C₄	—	2	4	no	—
171	Dic₃⋊C₄	—	2	4	no	—
172	C₄×Dic₃	—	2	4	no	—
173	C₄×Dic₃	C₅×D₄	1	2	yes	—
174	C₄×Dic₃	C₂³×C₅	1	1	yes	characteristic
175	C₂×C₃⋊C₈	C₂³×C₅	1	1	yes	characteristic
176	C₂×C₃⋊C₈	C₂³×C₅	1	1	yes	characteristic
177	D₃×C₈	C₅×Q₈	1	4	yes	complemented
178	D₃×C₈	C₅×Q₈	1	4	yes	complemented
179	C₂³×C₅	—	3	1	no	—
180	C₅×Q₈	—	12	8	no	—
181	C₂×C₄×C₅	—	2	4	no	—
182	C₂×C₄×C₅	—	3	1	no	—
183	C₂×C₄×C₅	—	3	2	no	—
184	C₂×C₄×C₅	—	3	2	no	—
185	C₂×C₄×C₅	—	3	2	no	—
186	C₂×C₄×C₅	—	6	4	no	—
187	C₂×C₄×C₅	—	6	4	no	—
188	C₂×C₄×C₅	D₃×D₄	1	2	yes	—
189	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
190	C₅×C₈	—	3	2	no	—
191	C₅×C₈	—	3	2	no	—
192	C₅×C₈	D₃×Q₈	1	2	yes	—
193	C₅×C₈	D₃×Q₈	1	2	yes	—
194	C₂²×C₈	—	3	1	no	—
195	C₂²×C₈	—	3	1	no	—
196	C₄⋊Q₈	—	3	4	no	—
197	#125	—	3	2	no	—
198	#125	C₂×C₅×D₃	1	2	yes	—
199	C₂²⋊Q₈	—	6	4	no	—
200	C₄×Q₈	—	6	4	no	—
201	#117	—	3	4	no	—
202	C₂×C₄²	—	3	1	no	—
203	#107	—	3	2	no	—
204	#107	—	3	4	no	—
205	#107	C₂×C₅×D₃	1	2	yes	—
206	#106	—	3	2	no	—
207	#106	—	3	4	no	—
208	#106	C₂×C₅×D₃	1	2	yes	—
209	C₄×C₈	—	3	1	no	—
210	C₄×C₈	—	3	1	no	—
211	C₄×C₈	—	3	1	no	—
212	C₄×C₈	C₂×C₅×D₃	1	1	yes	characteristic
213	C₂×C₃×C₅	C₂×C₄⋊Q₈	1	1	yes	characteristic
214	C₂×C₃×C₅	C₂×#135	1	2	yes	—
215	C₅×D₃	—	2	2	no	—
216	C₂²×D₃	C₂×C₅×D₄	1	1	yes	characteristic
217	C₂×C₃×C₄	—	2	4	no	—
218	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
219	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
220	C₂×Dic₃	—	2	4	no	—
221	C₂×Dic₃	C₂×C₅×D₄	1	1	yes	characteristic
222	C₂×Dic₃	C₂×C₅×D₄	1	2	yes	—
223	C₄×D₃	—	2	4	no	—
224	C₄×D₃	C₂×C₅×Q₈	1	2	yes	—
225	C₄×D₃	C₂×C₅×Q₈	1	2	yes	—
226	Dic₆	—	4	8	no	—
227	C₃×C₈	C₂×C₅×Q₈	1	2	yes	—
228	C₃×C₈	C₂×C₅×Q₈	1	2	yes	—
229	C₃⋊C₈	C₂×C₅×Q₈	1	2	yes	—
230	C₃⋊C₈	C₂×C₅×Q₈	1	2	yes	—
231	C₂²×C₅	—	3	2	no	—
232	C₂²×C₅	—	6	2	no	—
233	C₂²×C₅	C₂×D₃×D₄	1	1	yes	characteristic
234	C₄×C₅	—	2	2	no	—
235	C₄×C₅	—	3	2	no	—
236	C₄×C₅	—	3	4	no	—
237	C₄×C₅	—	4	4	no	—
238	C₄×C₅	—	12	4	no	—
239	C₄×C₅	C₂×D₃×Q₈	1	1	yes	characteristic
240	C₄×C₅	C₂×D₃×Q₈	1	1	yes	characteristic
241	C₂×Q₈	—	6	4	no	—
242	C₂²×C₄	—	3	1	no	—
243	C₂²×C₄	—	3	2	no	—
244	C₂×C₈	—	3	1	no	—
245	C₂×C₈	—	3	1	no	—
246	C₂×C₈	—	3	4	no	—
247	C₂×C₈	—	3	4	no	—
248	C₂×C₈	C₂²×C₅×D₃	1	1	yes	characteristic
249	C₂×C₈	C₂²×C₅×D₃	1	1	yes	characteristic
250	C₄⋊C₄	—	2	4	no	—
251	C₄⋊C₄	—	3	4	no	—
252	C₄⋊C₄	—	6	4	no	—
253	C₄⋊C₄	—	6	4	no	—
254	C₄⋊C₄	C₂²×C₅×D₃	1	4	yes	—
255	C₂²⋊C₄	—	6	4	no	—
256	C₄²	—	3	2	no	—
257	C₄²	—	3	1	no	—
258	C₄²	—	6	4	no	—
259	C₄²	C₂²×C₅×D₃	1	1	yes	characteristic
260	C₃×C₅	#3160	1	1	yes	characteristic, complemented, Hall
261	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
262	C₂×D₃	—	2	2	no	—
263	C₂×D₃	C₅×C₄⋊Q₈	1	2	yes	—
264	C₃×C₄	—	2	2	no	—
265	C₃×C₄	—	4	4	no	—
266	C₃×C₄	C₂²×C₅×Q₈	1	1	yes	characteristic
267	C₃×C₄	C₂²×C₅×Q₈	1	1	yes	characteristic
268	Dic₃	—	4	4	no	—
269	Dic₃	C₅×C₄⋊Q₈	1	2	yes	—
270	Dic₃	C₅×#135	1	4	yes	—
271	C₂×C₅	—	6	2	no	—
272	C₂×C₅	D₃×C₄⋊Q₈	1	1	yes	characteristic
273	C₂×C₅	D₃×#135	1	2	yes	—
274	C₂³	—	3	1	no	—
275	Q₈	—	12	8	no	—
276	C₂×C₄	—	2	4	no	—
277	C₂×C₄	—	3	1	no	—
278	C₂×C₄	—	3	2	no	—
279	C₂×C₄	—	3	2	no	—
280	C₂×C₄	—	3	2	no	—
281	C₂×C₄	—	6	4	no	—
282	C₂×C₄	—	6	4	no	—
283	C₂×C₄	C₅×D₃×D₄	1	2	yes	—
284	C₂×C₄	C₂³×C₅×D₃	1	1	yes	characteristic
285	C₈	—	3	2	no	—
286	C₈	—	3	2	no	—
287	C₈	C₅×D₃×Q₈	1	2	yes	—
288	C₈	C₅×D₃×Q₈	1	2	yes	—
289	C₂×C₃	C₂×C₅×C₄⋊Q₈	1	1	yes	characteristic
290	C₂×C₃	C₂×C₅×#135	1	2	yes	—
291	D₃	—	2	2	no	—
292	C₅	#81916	1	1	yes	characteristic, direct product factor, Sylow
293	C₂²	—	3	2	no	—
294	C₂²	—	6	2	no	—
295	C₂²	C₂×C₅×D₃×D₄	1	1	yes	characteristic
296	C₄	—	2	2	no	—
297	C₄	—	3	2	no	—
298	C₄	—	3	4	no	—
299	C₄	—	4	4	no	—
300	C₄	—	12	4	no	—
301	C₄	C₂×C₅×D₃×Q₈	1	1	yes	characteristic
302	C₄	C₂×C₅×D₃×Q₈	1	1	yes	characteristic
303	C₃	C₅×#3160	1	1	yes	characteristic, complemented, Sylow
304	C₂	—	6	2	no	—
305	C₂	C₅×D₃×C₄⋊Q₈	1	1	yes	characteristic
306	C₂	C₅×D₃×#135	1	2	yes	—

	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×#499	C₂
socle	C₂²×C₃×C₅	C₂²×D₄
prime radical	C₂²×C₅×D₃	C₂×D₄
hypercenter	C₅×#499	D₃
hypocenter	C₃	C₅×#3160
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ C₅×#499

Complements

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

N	H ≅ G/N	type
2. C₃×C₅×#499	C₂	semidirect
3. C₅×#5065	C₂	semidirect
5. C₅×#4874	C₂	semidirect
7. C₅×#4855	C₂	semidirect
9. C₅×#4683	C₂	semidirect
30. #81916	C₅	direct (N × H)
35. C₅×#499	D₃	semidirect
57. C₅×D₃×C₈	Q₈	semidirect
58. C₅×D₃×C₈	Q₈	semidirect
60. C₃×#499	C₂×C₅	semidirect
61. #5065	C₂×C₅	semidirect
63. #4874	C₂×C₅	semidirect
65. #4855	C₂×C₅	semidirect
67. #4683	C₂×C₅	semidirect
145. #499	C₅×D₃	semidirect
177. D₃×C₈	C₅×Q₈	semidirect
178. D₃×C₈	C₅×Q₈	semidirect
260. C₃×C₅	#3160	semidirect
292. C₅	#81916	direct (N × H)
303. 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×#5688	2
C₃×C₅×#499	2
C₅×#5065	2
C₅×#4880	2
C₅×#4874	2
C₅×#4860	2
C₅×#4855	2
C₄×C₅×D₃×C₈	2
C₅×#4683	2
C₅×#3160	3
#81916	5

Subgroups — C5×#81916 #14439843

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#81916 #14439843