Subgroups — C₃×D₅×#493 #14438764

Order 1920.

1520 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×D₅×D₈	C₂	1	4	yes	maximal, complemented
2	C₃×D₅×#127	C₂	1	2	yes	maximal, complemented
3	C₃×C₅×#493	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#65440	C₂	1	4	yes	maximal, complemented
5	C₃×#65362	C₂	1	2	yes	maximal, complemented
6	C₃×#64984	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×C₄×C₈×D₅	C₂	1	1	yes	maximal, characteristic, complemented
8	D₅×#493	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂×C₃×D₄×D₅	—	2	4	no	—
10	C₂×C₃×D₄×D₅	C₂²	1	4	yes	—
11	C₂×C₃×C₅×D₈	C₂²	1	4	yes	complemented
12	C₃×C₅×#127	C₂²	1	2	yes	complemented
13	C₃×#4086	—	2	4	no	—
14	C₃×#4086	C₂²	1	4	yes	complemented
15	C₂×C₃×#686	C₂²	1	8	yes	complemented
16	C₃×D₅×D₈	—	2	16	no	—
17	C₂×C₃×D₄₀	C₂²	1	4	yes	complemented
18	C₂×C₃×C₈×D₅	C₂²	1	2	yes	—
19	C₃×#4020	C₂²	1	2	yes	complemented
20	C₃×C₄²×D₅	C₂²	1	1	yes	characteristic
21	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic, complemented
22	C₃×C₈×Dic₅	C₂²	1	2	yes	complemented
23	C₃×C₄×#168	C₂²	1	1	yes	characteristic, complemented
24	C₂×C₃×#493	—	5	1	no	maximal, Hall
25	C₂×D₅×D₈	C₂×C₃	1	4	yes	complemented
26	D₅×#127	C₂×C₃	1	2	yes	complemented
27	C₅×#493	C₂×C₃	1	1	yes	characteristic, complemented
28	#65440	C₂×C₃	1	4	yes	complemented
29	#65362	C₂×C₃	1	2	yes	complemented
30	#64984	C₂×C₃	1	1	yes	characteristic, complemented
31	C₄×C₈×D₅	C₂×C₃	1	1	yes	characteristic, complemented
32	C₂³×C₃×D₅	—	2	4	no	—
33	C₂×C₃×C₅×D₄	—	2	4	no	—
34	C₂×C₃×C₅×D₄	C₂³	1	4	yes	—
35	C₂×C₃×C₂²⋊D₅	—	2	4	no	—
36	C₂×C₃×C₂²⋊D₅	—	2	8	no	—
37	C₃×D₄×D₅	—	2	16	no	—
38	C₃×D₄×D₅	—	4	16	no	—
39	C₂×C₃×D₂₀	—	2	4	no	—
40	C₂×C₃×D₂₀	C₂³	1	4	yes	—
41	C₂×C₃×C₄×D₅	D₄	1	2	yes	—
42	C₂×C₃×C₄×D₅	C₂³	1	1	yes	characteristic
43	C₃×C₅×D₈	—	2	8	no	—
44	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
45	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
46	C₃×#686	—	2	16	no	—
47	C₃×C₄×Dic₅	D₄	1	2	yes	—
48	C₃×C₄×Dic₅	C₂³	1	1	yes	characteristic
49	C₂×C₃×#168	C₂³	1	2	yes	—
50	C₃×D₄₀	—	2	8	no	—
51	C₃×C₈×D₅	D₄	1	8	yes	complemented
52	C₂²×C₃×D₈	—	5	4	no	—
53	C₂×C₃×#127	—	5	2	no	—
54	C₃×#493	—	5	4	no	—
55	C₃×#493	—	5	2	no	—
56	C₃×#493	—	5	1	no	—
57	C₃×#493	D₅	1	1	yes	characteristic, direct product factor
58	C₂×C₃×C₄×C₈	—	5	1	no	—
59	C₂×D₄×D₅	—	2	4	no	—
60	C₂×D₄×D₅	C₂²×C₃	1	4	yes	—
61	C₂×C₅×D₈	C₂²×C₃	1	4	yes	complemented
62	C₅×#127	C₂²×C₃	1	2	yes	complemented
63	#4086	—	2	4	no	—
64	#4086	C₂²×C₃	1	4	yes	complemented
65	C₂×#686	C₂²×C₃	1	8	yes	complemented
66	D₅×D₈	—	2	16	no	—
67	C₂×D₄₀	C₂²×C₃	1	4	yes	complemented
68	C₂×C₈×D₅	C₂²×C₃	1	2	yes	—
69	#4020	C₂²×C₃	1	2	yes	complemented
70	C₄²×D₅	C₂²×C₃	1	1	yes	characteristic
71	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic, complemented
72	C₈×Dic₅	C₂²×C₃	1	2	yes	complemented
73	C₄×#168	C₂²×C₃	1	1	yes	characteristic, complemented
74	C₂×#493	—	5	1	no	Sylow
75	C₂³×C₃×C₅	—	2	4	no	—
76	C₂²×C₃×D₅	—	2	4	no	—
77	C₂²×C₃×D₅	—	4	8	no	—
78	C₂²×C₃×D₅	—	4	16	no	—
79	C₂²×C₃×D₅	C₂×D₄	1	1	yes	characteristic
80	C₃×C₅×D₄	—	2	8	no	—
81	C₃×C₅×D₄	—	4	8	no	—
82	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
83	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
84	C₃×C₂²⋊D₅	—	4	8	no	—
85	C₃×C₂²⋊D₅	—	4	16	no	—
86	C₂×C₃×Dic₅	C₂×D₄	1	1	yes	characteristic
87	C₂×C₃×Dic₅	C₂×D₄	1	2	yes	—
88	C₃×D₂₀	—	2	8	no	—
89	C₃×D₂₀	—	4	8	no	—
90	C₃×C₄×D₅	D₈	1	8	yes	complemented
91	C₃×C₄×D₅	C₂×D₄	1	4	yes	—
92	C₃×C₅×C₈	C₂×D₄	1	4	yes	complemented
93	C₃×#168	C₂×D₄	1	4	yes	complemented
94	C₂²×C₃×D₄	—	5	4	no	—
95	C₂²×C₃×D₄	—	10	4	no	—
96	C₂×C₃×D₈	—	5	8	no	—
97	C₂×C₃×D₈	—	5	4	no	—
98	C₂×C₃×D₈	—	10	16	no	—
99	C₂×C₃×D₈	C₂×D₅	1	4	yes	complemented
100	C₂²×C₃×C₈	—	5	2	no	—
101	C₃×#127	—	5	4	no	—
102	C₃×#127	—	5	2	no	—
103	C₃×#127	—	10	4	no	—
104	C₃×#127	C₂×D₅	1	2	yes	complemented
105	C₂×C₃×C₄²	—	5	1	no	—
106	C₃×C₄×C₈	—	5	2	no	—
107	C₃×C₄×C₈	—	5	1	no	—
108	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic, complemented
109	C₂³×D₅	—	2	4	no	—
110	C₂×C₅×D₄	—	2	4	no	—
111	C₂×C₅×D₄	C₂³×C₃	1	4	yes	—
112	C₂×C₂²⋊D₅	—	2	8	no	—
113	C₂×C₂²⋊D₅	—	2	4	no	—
114	D₄×D₅	—	2	16	no	—
115	D₄×D₅	—	4	16	no	—
116	C₂×D₂₀	—	2	4	no	—
117	C₂×D₂₀	C₂³×C₃	1	4	yes	—
118	C₂×C₄×D₅	C₃×D₄	1	2	yes	—
119	C₂×C₄×D₅	C₂³×C₃	1	1	yes	characteristic
120	C₅×D₈	—	2	8	no	—
121	C₂×C₅×C₈	C₂³×C₃	1	2	yes	—
122	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
123	#686	—	2	16	no	—
124	C₄×Dic₅	C₃×D₄	1	2	yes	—
125	C₄×Dic₅	C₂³×C₃	1	1	yes	characteristic
126	C₂×#168	C₂³×C₃	1	2	yes	—
127	D₄₀	—	2	8	no	—
128	C₈×D₅	C₃×D₄	1	8	yes	complemented
129	C₂²×D₈	—	5	4	no	—
130	C₂×#127	—	5	2	no	—
131	#493	—	5	4	no	—
132	#493	—	5	2	no	—
133	#493	—	5	1	no	—
134	#493	C₃×D₅	1	1	yes	characteristic, direct product factor
135	C₂×C₄×C₈	—	5	1	no	—
136	C₂²×C₃×C₅	—	4	4	no	—
137	C₂²×C₃×C₅	—	4	8	no	—
138	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
139	C₂×C₃×D₅	—	4	4	no	—
140	C₂×C₃×D₅	—	4	8	no	—
141	C₂×C₃×D₅	—	8	16	no	—
142	C₂×C₃×D₅	#127	1	2	yes	—
143	C₂×C₃×D₅	C₂×D₈	1	4	yes	—
144	C₃×C₄×C₅	C₂×D₈	1	4	yes	complemented
145	C₃×C₄×C₅	C₂²×D₄	1	2	yes	—
146	C₃×Dic₅	#127	1	2	yes	—
147	C₃×Dic₅	C₂×D₈	1	4	yes	complemented
148	C₂⁴×C₃	—	10	4	no	—
149	C₂×C₃×D₄	—	2	4	no	—
150	C₂×C₃×D₄	—	5	4	no	—
151	C₂×C₃×D₄	—	10	16	no	—
152	C₂×C₃×D₄	—	10	8	no	—
153	C₂×C₃×D₄	—	10	4	no	—
154	C₂×C₃×D₄	—	10	4	no	—
155	C₂×C₃×D₄	—	20	16	no	—
156	C₂×C₃×D₄	C₂²×D₅	1	4	yes	—
157	C₂²×C₃×C₄	—	5	2	no	—
158	C₂²×C₃×C₄	—	5	1	no	—
159	C₃×D₈	—	2	8	no	—
160	C₃×D₈	—	10	16	no	—
161	C₃×D₈	—	10	8	no	—
162	C₂×C₃×C₈	—	5	2	no	—
163	C₂×C₃×C₈	—	5	8	no	—
164	C₂×C₃×C₈	C₂²×D₅	1	2	yes	—
165	C₃×C₄²	—	5	2	no	—
166	C₃×C₄²	—	5	1	no	—
167	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
168	C₂³×C₅	—	2	4	no	—
169	C₂²×D₅	—	2	4	no	—
170	C₂²×D₅	—	4	8	no	—
171	C₂²×D₅	—	4	16	no	—
172	C₂²×D₅	C₂×C₃×D₄	1	1	yes	characteristic
173	C₅×D₄	—	2	8	no	—
174	C₅×D₄	—	4	8	no	—
175	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
176	C₂×C₄×C₅	C₂⁴×C₃	1	1	yes	characteristic
177	C₂²⋊D₅	—	4	8	no	—
178	C₂²⋊D₅	—	4	16	no	—
179	C₂×Dic₅	C₂×C₃×D₄	1	1	yes	characteristic
180	C₂×Dic₅	C₂×C₃×D₄	1	2	yes	—
181	D₂₀	—	2	8	no	—
182	D₂₀	—	4	8	no	—
183	C₄×D₅	C₃×D₈	1	8	yes	complemented
184	C₄×D₅	C₂×C₃×D₄	1	4	yes	—
185	C₅×C₈	C₂×C₃×D₄	1	4	yes	complemented
186	#168	C₂×C₃×D₄	1	4	yes	complemented
187	C₂²×D₄	—	5	4	no	—
188	C₂²×D₄	—	10	4	no	—
189	C₂×D₈	—	5	8	no	—
190	C₂×D₈	—	5	4	no	—
191	C₂×D₈	—	10	16	no	—
192	C₂×D₈	C₂×C₃×D₅	1	4	yes	complemented
193	C₂²×C₈	—	5	2	no	—
194	#127	—	5	4	no	—
195	#127	—	5	2	no	—
196	#127	—	10	4	no	—
197	#127	C₂×C₃×D₅	1	2	yes	complemented
198	C₂×C₄²	—	5	1	no	—
199	C₄×C₈	—	5	2	no	—
200	C₄×C₈	—	5	1	no	—
201	C₄×C₈	C₂×C₃×D₅	1	1	yes	characteristic, complemented
202	C₂×C₃×C₅	—	8	4	no	—
203	C₂×C₃×C₅	C₂×#127	1	1	yes	characteristic
204	C₂×C₃×C₅	C₂²×D₈	1	2	yes	—
205	C₃×D₅	—	8	4	no	—
206	C₃×D₅	#493	1	4	yes	direct product factor
207	C₂³×C₃	—	2	4	no	—
208	C₂³×C₃	—	5	1	no	—
209	C₂³×C₃	—	10	4	no	—
210	C₂³×C₃	—	20	8	no	—
211	C₂³×C₃	—	20	16	no	—
212	C₃×D₄	—	2	8	no	—
213	C₃×D₄	—	4	8	no	—
214	C₃×D₄	—	10	8	no	—
215	C₃×D₄	—	20	8	no	—
216	C₃×D₄	—	20	16	no	—
217	C₃×D₄	—	20	8	no	—
218	C₂×C₃×C₄	—	5	1	no	—
219	C₂×C₃×C₄	—	5	2	no	—
220	C₂×C₃×C₄	—	5	4	no	—
221	C₂×C₃×C₄	—	5	8	no	—
222	C₂×C₃×C₄	D₄×D₅	1	2	yes	—
223	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
224	C₃×C₈	—	5	4	no	—
225	C₃×C₈	D₄×D₅	1	4	yes	complemented
226	C₂²×C₅	—	4	8	no	—
227	C₂²×C₅	—	4	4	no	—
228	C₂²×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
229	C₂×D₅	—	4	8	no	—
230	C₂×D₅	—	4	4	no	—
231	C₂×D₅	—	8	16	no	—
232	C₂×D₅	C₃×#127	1	2	yes	—
233	C₂×D₅	C₂×C₃×D₈	1	4	yes	—
234	C₄×C₅	C₂×C₃×D₈	1	4	yes	complemented
235	C₄×C₅	C₂²×C₃×D₄	1	2	yes	—
236	Dic₅	C₃×#127	1	2	yes	—
237	Dic₅	C₂×C₃×D₈	1	4	yes	complemented
238	C₂⁴	—	10	4	no	—
239	C₂×D₄	—	2	4	no	—
240	C₂×D₄	—	5	4	no	—
241	C₂×D₄	—	10	16	no	—
242	C₂×D₄	—	10	8	no	—
243	C₂×D₄	—	10	4	no	—
244	C₂×D₄	—	10	4	no	—
245	C₂×D₄	—	20	16	no	—
246	C₂×D₄	C₂²×C₃×D₅	1	4	yes	—
247	C₂²×C₄	—	5	2	no	—
248	C₂²×C₄	—	5	1	no	—
249	D₈	—	2	8	no	—
250	D₈	—	10	16	no	—
251	D₈	—	10	8	no	—
252	C₂×C₈	—	5	2	no	—
253	C₂×C₈	—	5	8	no	—
254	C₂×C₈	C₂²×C₃×D₅	1	2	yes	—
255	C₄²	—	5	2	no	—
256	C₄²	—	5	1	no	—
257	C₄²	C₂²×C₃×D₅	1	1	yes	characteristic
258	C₃×C₅	C₂×#493	1	1	yes	characteristic, complemented, Hall
259	C₂²×C₃	—	4	4	no	—
260	C₂²×C₃	—	4	8	no	—
261	C₂²×C₃	—	5	2	no	—
262	C₂²×C₃	—	5	4	no	—
263	C₂²×C₃	—	20	4	no	—
264	C₂²×C₃	—	20	8	no	—
265	C₂²×C₃	—	40	16	no	—
266	C₂²×C₃	C₂×D₄×D₅	1	1	yes	characteristic
267	C₃×C₄	—	5	2	no	—
268	C₃×C₄	—	5	4	no	—
269	C₃×C₄	D₅×D₈	1	4	yes	complemented
270	C₃×C₄	C₂×D₄×D₅	1	2	yes	—
271	C₂×C₅	—	8	4	no	—
272	C₂×C₅	C₂×C₃×#127	1	1	yes	characteristic
273	C₂×C₅	C₂²×C₃×D₈	1	2	yes	—
274	D₅	—	8	4	no	—
275	D₅	C₃×#493	1	4	yes	direct product factor
276	C₂³	—	2	4	no	—
277	C₂³	—	5	1	no	—
278	C₂³	—	10	4	no	—
279	C₂³	—	20	8	no	—
280	C₂³	—	20	16	no	—
281	D₄	—	2	8	no	—
282	D₄	—	4	8	no	—
283	D₄	—	10	8	no	—
284	D₄	—	20	8	no	—
285	D₄	—	20	16	no	—
286	D₄	—	20	8	no	—
287	C₂×C₄	—	5	1	no	—
288	C₂×C₄	—	5	2	no	—
289	C₂×C₄	—	5	4	no	—
290	C₂×C₄	—	5	8	no	—
291	C₂×C₄	C₃×D₄×D₅	1	2	yes	—
292	C₂×C₄	C₂³×C₃×D₅	1	1	yes	characteristic
293	C₈	—	5	4	no	—
294	C₈	C₃×D₄×D₅	1	4	yes	complemented
295	C₂×C₃	—	5	4	no	—
296	C₂×C₃	—	8	4	no	—
297	C₂×C₃	—	40	4	no	—
298	C₂×C₃	D₅×#127	1	1	yes	characteristic
299	C₂×C₃	C₂×D₅×D₈	1	2	yes	—
300	C₅	C₂×C₃×#493	1	1	yes	characteristic, complemented, Sylow
301	C₂²	—	4	4	no	—
302	C₂²	—	4	8	no	—
303	C₂²	—	5	2	no	—
304	C₂²	—	5	4	no	—
305	C₂²	—	20	4	no	—
306	C₂²	—	20	8	no	—
307	C₂²	—	40	16	no	—
308	C₂²	C₂×C₃×D₄×D₅	1	1	yes	characteristic
309	C₄	—	5	2	no	—
310	C₄	—	5	4	no	—
311	C₄	C₃×D₅×D₈	1	4	yes	complemented
312	C₄	C₂×C₃×D₄×D₅	1	2	yes	—
313	C₃	D₅×#493	1	1	yes	characteristic, direct product factor, Sylow
314	C₂	—	5	4	no	—
315	C₂	—	8	4	no	—
316	C₂	—	40	4	no	—
317	C₂	C₃×D₅×#127	1	1	yes	characteristic
318	C₂	C₂×C₃×D₅×D₈	1	2	yes	—

	subgroup	G/H
center	C₂²×C₃	C₂×D₄×D₅
commutator, cosocle	C₂×C₄×C₅	C₂⁴×C₃
Frattini	C₂×C₄	C₂³×C₃×D₅
Fitting	C₃×C₅×#493	C₂
socle	C₂²×C₃×C₅	C₂²×D₄
hypercenter	C₃×#493	D₅
hypocenter	C₅	C₂×C₃×#493
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ C₃×#493

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×D₅×D₈	C₂	semidirect
2. C₃×D₅×#127	C₂	semidirect
3. C₃×C₅×#493	C₂	semidirect
4. C₃×#65440	C₂	semidirect
5. C₃×#65362	C₂	semidirect
6. C₃×#64984	C₂	semidirect
7. C₃×C₄×C₈×D₅	C₂	semidirect
8. D₅×#493	C₃	direct (N × H)
11. C₂×C₃×C₅×D₈	C₂²	semidirect
12. C₃×C₅×#127	C₂²	semidirect
14. C₃×#4086	C₂²	semidirect
15. C₂×C₃×#686	C₂²	semidirect
17. C₂×C₃×D₄₀	C₂²	semidirect
19. C₃×#4020	C₂²	semidirect
21. C₃×C₄×C₅×C₈	C₂²	semidirect
22. C₃×C₈×Dic₅	C₂²	semidirect
23. C₃×C₄×#168	C₂²	semidirect
25. C₂×D₅×D₈	C₂×C₃	semidirect
26. D₅×#127	C₂×C₃	semidirect
27. C₅×#493	C₂×C₃	semidirect
28. #65440	C₂×C₃	semidirect
29. #65362	C₂×C₃	semidirect
30. #64984	C₂×C₃	semidirect
31. C₄×C₈×D₅	C₂×C₃	semidirect
51. C₃×C₈×D₅	D₄	semidirect
57. C₃×#493	D₅	direct (N × H)
61. C₂×C₅×D₈	C₂²×C₃	semidirect
62. C₅×#127	C₂²×C₃	semidirect
64. #4086	C₂²×C₃	semidirect
65. C₂×#686	C₂²×C₃	semidirect
67. C₂×D₄₀	C₂²×C₃	semidirect
69. #4020	C₂²×C₃	semidirect
71. C₄×C₅×C₈	C₂²×C₃	semidirect
72. C₈×Dic₅	C₂²×C₃	semidirect
73. C₄×#168	C₂²×C₃	semidirect
90. C₃×C₄×D₅	D₈	semidirect
92. C₃×C₅×C₈	C₂×D₄	semidirect
93. C₃×#168	C₂×D₄	semidirect
99. C₂×C₃×D₈	C₂×D₅	semidirect
104. C₃×#127	C₂×D₅	semidirect
108. C₃×C₄×C₈	C₂×D₅	semidirect
128. C₈×D₅	C₃×D₄	semidirect
134. #493	C₃×D₅	direct (N × H)
144. C₃×C₄×C₅	C₂×D₈	semidirect
147. C₃×Dic₅	C₂×D₈	semidirect
183. C₄×D₅	C₃×D₈	semidirect
185. C₅×C₈	C₂×C₃×D₄	semidirect
186. #168	C₂×C₃×D₄	semidirect
192. C₂×D₈	C₂×C₃×D₅	semidirect
197. #127	C₂×C₃×D₅	semidirect
201. C₄×C₈	C₂×C₃×D₅	semidirect
206. C₃×D₅	#493	direct (N × H)
225. C₃×C₈	D₄×D₅	semidirect
234. C₄×C₅	C₂×C₃×D₈	semidirect
237. Dic₅	C₂×C₃×D₈	semidirect
258. C₃×C₅	C₂×#493	semidirect
269. C₃×C₄	D₅×D₈	semidirect
275. D₅	C₃×#493	direct (N × H)
294. C₈	C₃×D₄×D₅	semidirect
300. C₅	C₂×C₃×#493	semidirect
311. C₄	C₃×D₅×D₈	semidirect
313. C₃	D₅×#493	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×D₅×D₈	2
C₃×D₅×#127	2
C₃×C₅×#493	2
C₃×#65440	2
C₃×#65362	2
C₃×#64984	2
C₃×C₄×C₈×D₅	2
D₅×#493	3
C₂×C₃×#493	5

Subgroups — C3×D5×#493 #14438764

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×D₅×#493 #14438764