Subgroups — C₃×C₅×#1717 #14274751

Order 1920.

416 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#493	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×#492	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₅×#487	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₅×#486	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₅×#486	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×C₅×#486	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×C₅×#322	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#1717	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂×C₃×C₅×SD₁₆	—	2	2	no	—
10	C₂×C₃×C₅×D₈	—	2	2	no	—
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×#127	C₂²	1	1	yes	characteristic
15	C₃×C₅×#103	—	2	2	no	—
16	C₃×C₅×#102	—	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×#1717	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
23	C₅×#493	C₂×C₃	1	1	yes	characteristic
24	C₅×#492	C₂×C₃	1	1	yes	characteristic, complemented
25	C₅×#487	C₂×C₃	1	1	yes	characteristic, complemented
26	C₅×#486	C₂×C₃	1	1	yes	characteristic, complemented
27	C₅×#486	C₂×C₃	1	1	yes	characteristic, complemented
28	C₅×#486	C₂×C₃	1	1	yes	characteristic, complemented
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×D₄	—	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	C₂×C₃×C₅×D₄	—	2	1	no	—
37	C₂×C₃×C₅×D₄	—	2	1	no	—
38	C₃×C₅×SD₁₆	—	4	4	no	—
39	C₃×C₅×D₈	—	4	4	no	—
40	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
41	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
42	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
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×C₅×C₄⋊C₄	—	2	1	no	—
47	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
48	C₃×#493	C₂×C₅	1	1	yes	characteristic
49	C₃×#492	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, complemented
52	C₃×#486	C₂×C₅	1	1	yes	characteristic, complemented
53	C₃×#486	C₂×C₅	1	1	yes	characteristic, complemented
54	C₃×#322	C₂×C₅	1	1	yes	characteristic, complemented
55	C₂×C₅×SD₁₆	—	2	2	no	—
56	C₂×C₅×D₈	—	2	2	no	—
57	C₅×C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
58	C₅×C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
59	C₅×#127	C₂²×C₃	1	1	yes	characteristic
60	C₅×#127	C₂²×C₃	1	1	yes	characteristic
61	C₅×#103	—	2	2	no	—
62	C₅×#102	—	2	2	no	—
63	C₅×#102	—	2	2	no	—
64	C₅×#102	—	2	2	no	—
65	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
66	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
67	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
68	#1717	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
69	C₂³×C₃×C₅	—	4	1	no	—
70	C₂³×C₃×C₅	—	4	1	no	—
71	C₃×C₅×Q₈	—	4	2	no	—
72	C₃×C₅×Q₈	—	4	2	no	—
73	C₃×C₅×D₄	—	4	2	no	—
74	C₃×C₅×D₄	—	4	2	no	—
75	C₃×C₅×D₄	—	4	2	no	—
76	C₃×C₅×D₄	—	4	2	no	—
77	C₃×C₅×D₄	—	4	2	no	—
78	C₃×C₅×D₄	—	4	2	no	—
79	C₂×C₃×C₄×C₅	—	4	1	no	—
80	C₂×C₃×C₄×C₅	—	4	1	no	—
81	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
82	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
83	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
84	C₃×C₅×C₈	—	2	2	no	—
85	C₃×C₅×C₈	—	2	2	no	—
86	C₃×C₅×C₈	—	2	2	no	—
87	C₂×C₃×SD₁₆	—	2	2	no	—
88	C₂×C₃×D₈	—	2	2	no	—
89	C₃×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
90	C₃×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
91	C₃×#127	C₂²×C₅	1	1	yes	characteristic
92	C₃×#127	C₂²×C₅	1	1	yes	characteristic
93	C₃×#103	—	2	2	no	—
94	C₃×#102	—	2	2	no	—
95	C₃×#102	—	2	2	no	—
96	C₃×#102	—	2	2	no	—
97	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
98	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
99	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
100	C₂×C₅×Q₈	—	2	1	no	—
101	C₂×C₅×Q₈	—	2	1	no	—
102	C₂×C₅×D₄	—	2	1	no	—
103	C₂×C₅×D₄	—	2	1	no	—
104	C₂×C₅×D₄	—	2	1	no	—
105	C₂×C₅×D₄	—	2	1	no	—
106	C₂×C₅×D₄	—	2	1	no	—
107	C₂×C₅×D₄	—	2	1	no	—
108	C₅×SD₁₆	—	4	4	no	—
109	C₅×D₈	—	4	4	no	—
110	C₂×C₅×C₈	C₃×D₄	1	2	yes	—
111	C₂×C₅×C₈	C₃×D₄	1	2	yes	—
112	C₂×C₅×C₈	C₃×D₄	1	2	yes	—
113	C₅×C₄⋊C₄	—	2	1	no	—
114	C₅×C₄⋊C₄	—	2	1	no	—
115	C₅×C₄⋊C₄	—	2	1	no	—
116	C₅×C₄⋊C₄	—	2	1	no	—
117	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
118	#493	C₂×C₃×C₅	1	1	yes	characteristic
119	#492	C₂×C₃×C₅	1	1	yes	characteristic, complemented
120	#487	C₂×C₃×C₅	1	1	yes	characteristic, complemented
121	#486	C₂×C₃×C₅	1	1	yes	characteristic, complemented
122	#486	C₂×C₃×C₅	1	1	yes	characteristic, complemented
123	#486	C₂×C₃×C₅	1	1	yes	characteristic, complemented
124	#322	C₂×C₃×C₅	1	1	yes	characteristic, complemented
125	C₂²×C₃×C₅	—	8	1	no	—
126	C₂²×C₃×C₅	—	8	1	no	—
127	C₂²×C₃×C₅	—	8	1	no	—
128	C₂²×C₃×C₅	—	8	1	no	—
129	C₂²×C₃×C₅	—	8	1	no	—
130	C₂²×C₃×C₅	—	8	1	no	—
131	C₂²×C₃×C₅	#127	1	1	yes	characteristic
132	C₃×C₄×C₅	—	8	1	no	—
133	C₃×C₄×C₅	—	8	1	no	—
134	C₃×C₄×C₅	#135	1	2	yes	—
135	C₃×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
136	C₃×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
137	C₂×C₃×Q₈	—	2	1	no	—
138	C₂×C₃×Q₈	—	2	1	no	—
139	C₂×C₃×D₄	—	2	1	no	—
140	C₂×C₃×D₄	—	2	1	no	—
141	C₂×C₃×D₄	—	2	1	no	—
142	C₂×C₃×D₄	—	2	1	no	—
143	C₂×C₃×D₄	—	2	1	no	—
144	C₂×C₃×D₄	—	2	1	no	—
145	C₃×SD₁₆	—	4	4	no	—
146	C₃×D₈	—	4	4	no	—
147	C₂×C₃×C₈	C₅×D₄	1	2	yes	—
148	C₂×C₃×C₈	C₅×D₄	1	2	yes	—
149	C₂×C₃×C₈	C₅×D₄	1	2	yes	—
150	C₃×C₄⋊C₄	—	2	1	no	—
151	C₃×C₄⋊C₄	—	2	1	no	—
152	C₃×C₄⋊C₄	—	2	1	no	—
153	C₃×C₄⋊C₄	—	2	1	no	—
154	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
155	C₂³×C₅	—	4	1	no	—
156	C₂³×C₅	—	4	1	no	—
157	C₅×Q₈	—	4	2	no	—
158	C₅×Q₈	—	4	2	no	—
159	C₅×D₄	—	4	2	no	—
160	C₅×D₄	—	4	2	no	—
161	C₅×D₄	—	4	2	no	—
162	C₅×D₄	—	4	2	no	—
163	C₅×D₄	—	4	2	no	—
164	C₅×D₄	—	4	2	no	—
165	C₂×C₄×C₅	—	4	1	no	—
166	C₂×C₄×C₅	—	4	1	no	—
167	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
168	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
169	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
170	C₅×C₈	—	2	2	no	—
171	C₅×C₈	—	2	2	no	—
172	C₅×C₈	—	2	2	no	—
173	C₂×SD₁₆	—	2	2	no	—
174	C₂×D₈	—	2	2	no	—
175	C₄⋊Q₈	C₂²×C₃×C₅	1	1	yes	characteristic
176	C₄⋊Q₈	C₂²×C₃×C₅	1	1	yes	characteristic
177	#127	C₂²×C₃×C₅	1	1	yes	characteristic
178	#127	C₂²×C₃×C₅	1	1	yes	characteristic
179	#103	—	2	2	no	—
180	#102	—	2	2	no	—
181	#102	—	2	2	no	—
182	#102	—	2	2	no	—
183	C₄×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
184	C₄×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
185	C₄×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
186	C₂×C₃×C₅	—	16	1	no	—
187	C₂×C₃×C₅	—	16	1	no	—
188	C₂×C₃×C₅	#495	1	1	yes	characteristic
189	C₂×C₃×C₅	#496	1	1	yes	characteristic
190	C₂×C₃×C₅	#496	1	1	yes	characteristic
191	C₂³×C₃	—	4	1	no	—
192	C₂³×C₃	—	4	1	no	—
193	C₃×Q₈	—	4	2	no	—
194	C₃×Q₈	—	4	2	no	—
195	C₃×D₄	—	4	2	no	—
196	C₃×D₄	—	4	2	no	—
197	C₃×D₄	—	4	2	no	—
198	C₃×D₄	—	4	2	no	—
199	C₃×D₄	—	4	2	no	—
200	C₃×D₄	—	4	2	no	—
201	C₂×C₃×C₄	—	4	1	no	—
202	C₂×C₃×C₄	—	4	1	no	—
203	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
204	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
205	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
206	C₃×C₈	—	2	2	no	—
207	C₃×C₈	—	2	2	no	—
208	C₃×C₈	—	2	2	no	—
209	C₂²×C₅	—	8	1	no	—
210	C₂²×C₅	—	8	1	no	—
211	C₂²×C₅	—	8	1	no	—
212	C₂²×C₅	—	8	1	no	—
213	C₂²×C₅	—	8	1	no	—
214	C₂²×C₅	—	8	1	no	—
215	C₂²×C₅	C₃×#127	1	1	yes	characteristic
216	C₄×C₅	—	8	1	no	—
217	C₄×C₅	—	8	1	no	—
218	C₄×C₅	C₃×#135	1	2	yes	—
219	C₄×C₅	C₃×AGL(1,ℤ/8)	1	2	yes	—
220	C₄×C₅	C₃×AGL(1,ℤ/8)	1	2	yes	—
221	C₂×Q₈	—	2	1	no	—
222	C₂×Q₈	—	2	1	no	—
223	C₂×D₄	—	2	1	no	—
224	C₂×D₄	—	2	1	no	—
225	C₂×D₄	—	2	1	no	—
226	C₂×D₄	—	2	1	no	—
227	C₂×D₄	—	2	1	no	—
228	C₂×D₄	—	2	1	no	—
229	SD₁₆	—	4	4	no	—
230	D₈	—	4	4	no	—
231	C₂×C₈	C₃×C₅×D₄	1	2	yes	—
232	C₂×C₈	C₃×C₅×D₄	1	2	yes	—
233	C₂×C₈	C₃×C₅×D₄	1	2	yes	—
234	C₄⋊C₄	—	2	1	no	—
235	C₄⋊C₄	—	2	1	no	—
236	C₄⋊C₄	—	2	1	no	—
237	C₄⋊C₄	—	2	1	no	—
238	C₄²	C₂³×C₃×C₅	1	1	yes	characteristic
239	C₃×C₅	#1717	1	1	yes	characteristic, direct product factor, Hall
240	C₂²×C₃	—	8	1	no	—
241	C₂²×C₃	—	8	1	no	—
242	C₂²×C₃	—	8	1	no	—
243	C₂²×C₃	—	8	1	no	—
244	C₂²×C₃	—	8	1	no	—
245	C₂²×C₃	—	8	1	no	—
246	C₂²×C₃	C₅×#127	1	1	yes	characteristic
247	C₃×C₄	—	8	1	no	—
248	C₃×C₄	—	8	1	no	—
249	C₃×C₄	C₅×#135	1	2	yes	—
250	C₃×C₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
251	C₃×C₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
252	C₂×C₅	—	16	1	no	—
253	C₂×C₅	—	16	1	no	—
254	C₂×C₅	C₃×#495	1	1	yes	characteristic
255	C₂×C₅	C₃×#496	1	1	yes	characteristic
256	C₂×C₅	C₃×#496	1	1	yes	characteristic
257	C₂³	—	4	1	no	—
258	C₂³	—	4	1	no	—
259	Q₈	—	4	2	no	—
260	Q₈	—	4	2	no	—
261	D₄	—	4	2	no	—
262	D₄	—	4	2	no	—
263	D₄	—	4	2	no	—
264	D₄	—	4	2	no	—
265	D₄	—	4	2	no	—
266	D₄	—	4	2	no	—
267	C₂×C₄	—	4	1	no	—
268	C₂×C₄	—	4	1	no	—
269	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
270	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
271	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
272	C₈	—	2	2	no	—
273	C₈	—	2	2	no	—
274	C₈	—	2	2	no	—
275	C₂×C₃	—	16	1	no	—
276	C₂×C₃	—	16	1	no	—
277	C₂×C₃	C₅×#495	1	1	yes	characteristic
278	C₂×C₃	C₅×#496	1	1	yes	characteristic
279	C₂×C₃	C₅×#496	1	1	yes	characteristic
280	C₅	C₃×#1717	1	1	yes	characteristic, direct product factor, Sylow
281	C₂²	—	8	1	no	—
282	C₂²	—	8	1	no	—
283	C₂²	—	8	1	no	—
284	C₂²	—	8	1	no	—
285	C₂²	—	8	1	no	—
286	C₂²	—	8	1	no	—
287	C₂²	C₃×C₅×#127	1	1	yes	characteristic
288	C₄	—	8	1	no	—
289	C₄	—	8	1	no	—
290	C₄	C₃×C₅×#135	1	2	yes	—
291	C₄	C₃×C₅×AGL(1,ℤ/8)	1	2	yes	—
292	C₄	C₃×C₅×AGL(1,ℤ/8)	1	2	yes	—
293	C₃	C₅×#1717	1	1	yes	characteristic, direct product factor, Sylow
294	C₂	—	16	1	no	—
295	C₂	—	16	1	no	—
296	C₂	C₃×C₅×#495	1	1	yes	characteristic
297	C₂	C₃×C₅×#496	1	1	yes	characteristic
298	C₂	C₃×C₅×#496	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃×C₅	#127
commutator, Frattini, cosocle	C₄²	C₂³×C₃×C₅
prime radical	C₃×C₅×#493	C₂
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
2. C₃×C₅×#492	C₂	semidirect
3. C₃×C₅×#487	C₂	semidirect
4. C₃×C₅×#486	C₂	semidirect
5. C₃×C₅×#486	C₂	semidirect
6. C₃×C₅×#486	C₂	semidirect
7. C₃×C₅×#322	C₂	semidirect
8. C₅×#1717	C₃	direct (N × H)
22. C₃×#1717	C₅	direct (N × H)
24. C₅×#492	C₂×C₃	semidirect
25. C₅×#487	C₂×C₃	semidirect
26. C₅×#486	C₂×C₃	semidirect
27. C₅×#486	C₂×C₃	semidirect
28. C₅×#486	C₂×C₃	semidirect
29. C₅×#322	C₂×C₃	semidirect
49. C₃×#492	C₂×C₅	semidirect
50. C₃×#487	C₂×C₅	semidirect
51. C₃×#486	C₂×C₅	semidirect
52. C₃×#486	C₂×C₅	semidirect
53. C₃×#486	C₂×C₅	semidirect
54. C₃×#322	C₂×C₅	semidirect
68. #1717	C₃×C₅	direct (N × H)
119. #492	C₂×C₃×C₅	semidirect
120. #487	C₂×C₃×C₅	semidirect
121. #486	C₂×C₃×C₅	semidirect
122. #486	C₂×C₃×C₅	semidirect
123. #486	C₂×C₃×C₅	semidirect
124. #322	C₂×C₃×C₅	semidirect
239. C₃×C₅	#1717	direct (N × H)
280. C₅	C₃×#1717	direct (N × H)
293. C₃	C₅×#1717	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×#493	2
C₃×C₅×#492	2
C₃×C₅×#487	2
C₃×C₅×#486	2
C₃×C₅×#322	2
C₅×#1717	3
C₃×#1717	5

Subgroups — C3×C5×#1717 #14274751

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#1717 #14274751