Subgroups — C₃×C₅×#1719 #14274753

Order 1920.

372 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#500	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×C₅×#492	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#487	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₅×#487	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₅×#486	C₂	1	1	yes	maximal, characteristic
6	C₃×C₅×#486	C₂	1	1	yes	maximal, characteristic
7	C₃×C₅×#322	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#1719	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂×C₃×C₅×SD₁₆	—	2	2	no	—
10	C₃×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	C₃×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
12	C₃×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
13	C₃×C₅×#127	C₂²	1	1	yes	characteristic
14	C₃×C₅×#107	—	2	2	no	—
15	C₃×C₅×#103	—	2	2	no	—
16	C₃×C₅×#103	—	2	2	no	—
17	C₃×C₅×#102	—	2	2	no	—
18	C₃×C₅×#102	—	2	2	no	—
19	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
20	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
21	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
22	C₃×#1719	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
23	C₅×#500	C₂×C₃	1	1	yes	characteristic, complemented
24	C₅×#492	C₂×C₃	1	1	yes	characteristic
25	C₅×#487	C₂×C₃	1	1	yes	characteristic, complemented
26	C₅×#487	C₂×C₃	1	1	yes	characteristic, complemented
27	C₅×#486	C₂×C₃	1	1	yes	characteristic
28	C₅×#486	C₂×C₃	1	1	yes	characteristic
29	C₅×#322	C₂×C₃	1	1	yes	characteristic, complemented
30	C₂×C₃×C₅×Q₈	—	2	1	no	—
31	C₂×C₃×C₅×Q₈	—	2	1	no	—
32	C₂×C₃×C₅×Q₈	—	2	1	no	—
33	C₂×C₃×C₅×D₄	—	2	1	no	—
34	C₂×C₃×C₅×D₄	—	2	1	no	—
35	C₂×C₃×C₅×D₄	—	2	1	no	—
36	C₃×C₅×SD₁₆	—	4	4	no	—
37	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
38	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
39	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
40	C₃×C₅×C₄⋊C₄	—	2	1	no	—
41	C₃×C₅×C₄⋊C₄	—	2	1	no	—
42	C₃×C₅×C₄⋊C₄	—	2	1	no	—
43	C₃×C₅×C₄⋊C₄	—	2	1	no	—
44	C₃×C₅×C₄⋊C₄	—	2	1	no	—
45	C₃×C₅×C₄⋊C₄	—	2	1	no	—
46	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
47	C₃×#500	C₂×C₅	1	1	yes	characteristic, complemented
48	C₃×#492	C₂×C₅	1	1	yes	characteristic
49	C₃×#487	C₂×C₅	1	1	yes	characteristic, complemented
50	C₃×#487	C₂×C₅	1	1	yes	characteristic, complemented
51	C₃×#486	C₂×C₅	1	1	yes	characteristic
52	C₃×#486	C₂×C₅	1	1	yes	characteristic
53	C₃×#322	C₂×C₅	1	1	yes	characteristic, complemented
54	C₂×C₅×SD₁₆	—	2	2	no	—
55	C₅×C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
56	C₅×C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
57	C₅×C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
58	C₅×#127	C₂²×C₃	1	1	yes	characteristic
59	C₅×#107	—	2	2	no	—
60	C₅×#103	—	2	2	no	—
61	C₅×#103	—	2	2	no	—
62	C₅×#102	—	2	2	no	—
63	C₅×#102	—	2	2	no	—
64	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
65	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
66	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
67	#1719	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
68	C₂³×C₃×C₅	—	4	1	no	—
69	C₃×C₅×Q₈	—	4	2	no	—
70	C₃×C₅×Q₈	—	4	2	no	—
71	C₃×C₅×Q₈	—	4	2	no	—
72	C₃×C₅×D₄	—	4	2	no	—
73	C₃×C₅×D₄	—	4	2	no	—
74	C₃×C₅×D₄	—	4	2	no	—
75	C₂×C₃×C₄×C₅	—	4	1	no	—
76	C₂×C₃×C₄×C₅	—	4	1	no	—
77	C₂×C₃×C₄×C₅	—	4	1	no	—
78	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
79	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
80	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
81	C₃×C₅×C₈	—	2	2	no	—
82	C₃×C₅×C₈	—	2	2	no	—
83	C₃×C₅×C₈	—	2	2	no	—
84	C₂×C₃×SD₁₆	—	2	2	no	—
85	C₃×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
86	C₃×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
87	C₃×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
88	C₃×#127	C₂²×C₅	1	1	yes	characteristic
89	C₃×#107	—	2	2	no	—
90	C₃×#103	—	2	2	no	—
91	C₃×#103	—	2	2	no	—
92	C₃×#102	—	2	2	no	—
93	C₃×#102	—	2	2	no	—
94	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
95	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
96	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
97	C₂×C₅×Q₈	—	2	1	no	—
98	C₂×C₅×Q₈	—	2	1	no	—
99	C₂×C₅×Q₈	—	2	1	no	—
100	C₂×C₅×D₄	—	2	1	no	—
101	C₂×C₅×D₄	—	2	1	no	—
102	C₂×C₅×D₄	—	2	1	no	—
103	C₅×SD₁₆	—	4	4	no	—
104	C₂×C₅×C₈	C₃×D₄	1	2	yes	—
105	C₂×C₅×C₈	C₃×D₄	1	2	yes	—
106	C₂×C₅×C₈	C₃×D₄	1	2	yes	—
107	C₅×C₄⋊C₄	—	2	1	no	—
108	C₅×C₄⋊C₄	—	2	1	no	—
109	C₅×C₄⋊C₄	—	2	1	no	—
110	C₅×C₄⋊C₄	—	2	1	no	—
111	C₅×C₄⋊C₄	—	2	1	no	—
112	C₅×C₄⋊C₄	—	2	1	no	—
113	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
114	#500	C₂×C₃×C₅	1	1	yes	characteristic, complemented
115	#492	C₂×C₃×C₅	1	1	yes	characteristic
116	#487	C₂×C₃×C₅	1	1	yes	characteristic, complemented
117	#487	C₂×C₃×C₅	1	1	yes	characteristic, complemented
118	#486	C₂×C₃×C₅	1	1	yes	characteristic
119	#486	C₂×C₃×C₅	1	1	yes	characteristic
120	#322	C₂×C₃×C₅	1	1	yes	characteristic, complemented
121	C₂²×C₃×C₅	—	8	1	no	—
122	C₂²×C₃×C₅	—	8	1	no	—
123	C₂²×C₃×C₅	—	8	1	no	—
124	C₂²×C₃×C₅	#127	1	1	yes	characteristic
125	C₃×C₄×C₅	—	8	1	no	—
126	C₃×C₄×C₅	—	8	1	no	—
127	C₃×C₄×C₅	—	8	1	no	—
128	C₃×C₄×C₅	#135	1	2	yes	—
129	C₃×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
130	C₃×C₄×C₅	#137	1	2	yes	—
131	C₂×C₃×Q₈	—	2	1	no	—
132	C₂×C₃×Q₈	—	2	1	no	—
133	C₂×C₃×Q₈	—	2	1	no	—
134	C₂×C₃×D₄	—	2	1	no	—
135	C₂×C₃×D₄	—	2	1	no	—
136	C₂×C₃×D₄	—	2	1	no	—
137	C₃×SD₁₆	—	4	4	no	—
138	C₂×C₃×C₈	C₅×D₄	1	2	yes	—
139	C₂×C₃×C₈	C₅×D₄	1	2	yes	—
140	C₂×C₃×C₈	C₅×D₄	1	2	yes	—
141	C₃×C₄⋊C₄	—	2	1	no	—
142	C₃×C₄⋊C₄	—	2	1	no	—
143	C₃×C₄⋊C₄	—	2	1	no	—
144	C₃×C₄⋊C₄	—	2	1	no	—
145	C₃×C₄⋊C₄	—	2	1	no	—
146	C₃×C₄⋊C₄	—	2	1	no	—
147	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
148	C₂³×C₅	—	4	1	no	—
149	C₅×Q₈	—	4	2	no	—
150	C₅×Q₈	—	4	2	no	—
151	C₅×Q₈	—	4	2	no	—
152	C₅×D₄	—	4	2	no	—
153	C₅×D₄	—	4	2	no	—
154	C₅×D₄	—	4	2	no	—
155	C₂×C₄×C₅	—	4	1	no	—
156	C₂×C₄×C₅	—	4	1	no	—
157	C₂×C₄×C₅	—	4	1	no	—
158	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
159	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
160	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
161	C₅×C₈	—	2	2	no	—
162	C₅×C₈	—	2	2	no	—
163	C₅×C₈	—	2	2	no	—
164	C₂×SD₁₆	—	2	2	no	—
165	C₄⋊Q₈	C₂²×C₃×C₅	1	1	yes	characteristic
166	C₄⋊Q₈	C₂²×C₃×C₅	1	1	yes	characteristic
167	C₄⋊Q₈	C₂²×C₃×C₅	1	1	yes	characteristic
168	#127	C₂²×C₃×C₅	1	1	yes	characteristic
169	#107	—	2	2	no	—
170	#103	—	2	2	no	—
171	#103	—	2	2	no	—
172	#102	—	2	2	no	—
173	#102	—	2	2	no	—
174	C₄×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
175	C₄×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
176	C₄×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
177	C₂×C₃×C₅	—	16	1	no	—
178	C₂×C₃×C₅	#495	1	1	yes	characteristic
179	C₂×C₃×C₅	#496	1	1	yes	characteristic
180	C₂×C₃×C₅	#497	1	1	yes	characteristic
181	C₂³×C₃	—	4	1	no	—
182	C₃×Q₈	—	4	2	no	—
183	C₃×Q₈	—	4	2	no	—
184	C₃×Q₈	—	4	2	no	—
185	C₃×D₄	—	4	2	no	—
186	C₃×D₄	—	4	2	no	—
187	C₃×D₄	—	4	2	no	—
188	C₂×C₃×C₄	—	4	1	no	—
189	C₂×C₃×C₄	—	4	1	no	—
190	C₂×C₃×C₄	—	4	1	no	—
191	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
192	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
193	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
194	C₃×C₈	—	2	2	no	—
195	C₃×C₈	—	2	2	no	—
196	C₃×C₈	—	2	2	no	—
197	C₂²×C₅	—	8	1	no	—
198	C₂²×C₅	—	8	1	no	—
199	C₂²×C₅	—	8	1	no	—
200	C₂²×C₅	C₃×#127	1	1	yes	characteristic
201	C₄×C₅	—	8	1	no	—
202	C₄×C₅	—	8	1	no	—
203	C₄×C₅	—	8	1	no	—
204	C₄×C₅	C₃×#135	1	2	yes	—
205	C₄×C₅	C₃×AGL(1,ℤ/8)	1	2	yes	—
206	C₄×C₅	C₃×#137	1	2	yes	—
207	C₂×Q₈	—	2	1	no	—
208	C₂×Q₈	—	2	1	no	—
209	C₂×Q₈	—	2	1	no	—
210	C₂×D₄	—	2	1	no	—
211	C₂×D₄	—	2	1	no	—
212	C₂×D₄	—	2	1	no	—
213	SD₁₆	—	4	4	no	—
214	C₂×C₈	C₃×C₅×D₄	1	2	yes	—
215	C₂×C₈	C₃×C₅×D₄	1	2	yes	—
216	C₂×C₈	C₃×C₅×D₄	1	2	yes	—
217	C₄⋊C₄	—	2	1	no	—
218	C₄⋊C₄	—	2	1	no	—
219	C₄⋊C₄	—	2	1	no	—
220	C₄⋊C₄	—	2	1	no	—
221	C₄⋊C₄	—	2	1	no	—
222	C₄⋊C₄	—	2	1	no	—
223	C₄²	C₂³×C₃×C₅	1	1	yes	characteristic
224	C₃×C₅	#1719	1	1	yes	characteristic, direct product factor, Hall
225	C₂²×C₃	—	8	1	no	—
226	C₂²×C₃	—	8	1	no	—
227	C₂²×C₃	—	8	1	no	—
228	C₂²×C₃	C₅×#127	1	1	yes	characteristic
229	C₃×C₄	—	8	1	no	—
230	C₃×C₄	—	8	1	no	—
231	C₃×C₄	—	8	1	no	—
232	C₃×C₄	C₅×#135	1	2	yes	—
233	C₃×C₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
234	C₃×C₄	C₅×#137	1	2	yes	—
235	C₂×C₅	—	16	1	no	—
236	C₂×C₅	C₃×#495	1	1	yes	characteristic
237	C₂×C₅	C₃×#496	1	1	yes	characteristic
238	C₂×C₅	C₃×#497	1	1	yes	characteristic
239	C₂³	—	4	1	no	—
240	Q₈	—	4	2	no	—
241	Q₈	—	4	2	no	—
242	Q₈	—	4	2	no	—
243	D₄	—	4	2	no	—
244	D₄	—	4	2	no	—
245	D₄	—	4	2	no	—
246	C₂×C₄	—	4	1	no	—
247	C₂×C₄	—	4	1	no	—
248	C₂×C₄	—	4	1	no	—
249	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
250	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
251	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
252	C₈	—	2	2	no	—
253	C₈	—	2	2	no	—
254	C₈	—	2	2	no	—
255	C₂×C₃	—	16	1	no	—
256	C₂×C₃	C₅×#495	1	1	yes	characteristic
257	C₂×C₃	C₅×#496	1	1	yes	characteristic
258	C₂×C₃	C₅×#497	1	1	yes	characteristic
259	C₅	C₃×#1719	1	1	yes	characteristic, direct product factor, Sylow
260	C₂²	—	8	1	no	—
261	C₂²	—	8	1	no	—
262	C₂²	—	8	1	no	—
263	C₂²	C₃×C₅×#127	1	1	yes	characteristic
264	C₄	—	8	1	no	—
265	C₄	—	8	1	no	—
266	C₄	—	8	1	no	—
267	C₄	C₃×C₅×#135	1	2	yes	—
268	C₄	C₃×C₅×AGL(1,ℤ/8)	1	2	yes	—
269	C₄	C₃×C₅×#137	1	2	yes	—
270	C₃	C₅×#1719	1	1	yes	characteristic, direct product factor, Sylow
271	C₂	—	16	1	no	—
272	C₂	C₃×C₅×#495	1	1	yes	characteristic
273	C₂	C₃×C₅×#496	1	1	yes	characteristic
274	C₂	C₃×C₅×#497	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃×C₅	#127
commutator, Frattini, cosocle	C₄²	C₂³×C₃×C₅
prime radical	C₃×C₅×#127	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₅ ◁ C₃×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×#500	C₂	semidirect
3. C₃×C₅×#487	C₂	semidirect
4. C₃×C₅×#487	C₂	semidirect
7. C₃×C₅×#322	C₂	semidirect
8. C₅×#1719	C₃	direct (N × H)
22. C₃×#1719	C₅	direct (N × H)
23. C₅×#500	C₂×C₃	semidirect
25. C₅×#487	C₂×C₃	semidirect
26. C₅×#487	C₂×C₃	semidirect
29. C₅×#322	C₂×C₃	semidirect
47. C₃×#500	C₂×C₅	semidirect
49. C₃×#487	C₂×C₅	semidirect
50. C₃×#487	C₂×C₅	semidirect
53. C₃×#322	C₂×C₅	semidirect
67. #1719	C₃×C₅	direct (N × H)
114. #500	C₂×C₃×C₅	semidirect
116. #487	C₂×C₃×C₅	semidirect
117. #487	C₂×C₃×C₅	semidirect
120. #322	C₂×C₃×C₅	semidirect
224. C₃×C₅	#1719	direct (N × H)
259. C₅	C₃×#1719	direct (N × H)
270. C₃	C₅×#1719	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×C₅×#500	2
C₃×C₅×#492	2
C₃×C₅×#487	2
C₃×C₅×#486	2
C₃×C₅×#322	2
C₅×#1719	3
C₃×#1719	5

Subgroups — C3×C5×#1719 #14274753

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#1719 #14274753