Subgroups — C₃×C₅×#1666 #14274700

Order 1920.

344 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	C₃×C₅×#492	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×#475	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₅×#460	C₂	1	1	yes	maximal, characteristic
4	C₃×C₅×C₈×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₅×#334	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×C₅×#332	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×C₅×#331	C₂	1	1	yes	maximal, characteristic
8	C₅×#1666	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂×C₃×C₅×SD₁₆	—	2	1	no	—
10	C₂×C₃×C₅×SD₁₆	—	2	1	no	—
11	C₂×C₃×C₅×SD₁₆	—	2	1	no	—
12	C₃×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
13	C₃×C₅×#127	C₂²	1	1	yes	characteristic
14	C₃×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
15	C₃×C₅×#106	—	2	1	no	—
16	C₃×C₅×C₄⋊C₈	—	2	1	no	—
17	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
18	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
19	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
20	C₃×C₅×#103	—	2	1	no	—
21	C₃×C₅×#102	—	2	1	no	—
22	C₃×C₄×C₅×C₈	—	2	1	no	—
23	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
24	C₃×#1666	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
25	C₅×#492	C₂×C₃	1	1	yes	characteristic
26	C₅×#475	C₂×C₃	1	1	yes	characteristic, complemented
27	C₅×#460	C₂×C₃	1	1	yes	characteristic
28	C₅×C₈×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
29	C₅×#334	C₂×C₃	1	1	yes	characteristic, complemented
30	C₅×#332	C₂×C₃	1	1	yes	characteristic, complemented
31	C₅×#331	C₂×C₃	1	1	yes	characteristic
32	C₂×C₃×C₅×Q₈	—	2	1	no	—
33	C₂×C₃×C₅×Q₈	D₄	1	1	yes	characteristic
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	C₃×C₅×SD₁₆	—	4	2	no	—
38	C₃×C₅×SD₁₆	—	4	2	no	—
39	C₃×C₅×SD₁₆	—	4	2	no	—
40	C₂×C₃×C₅×C₈	—	2	1	no	—
41	C₂×C₃×C₅×C₈	—	2	1	no	—
42	C₂×C₃×C₅×C₈	—	2	1	no	—
43	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
44	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
45	C₃×C₅×C₄⋊C₄	—	2	1	no	—
46	C₃×C₅×C₄⋊C₄	—	2	1	no	—
47	C₃×C₅×C₄⋊C₄	—	2	1	no	—
48	C₃×C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
49	C₃×C₄²×C₅	—	2	1	no	—
50	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
51	C₃×#492	C₂×C₅	1	1	yes	characteristic
52	C₃×#475	C₂×C₅	1	1	yes	characteristic, complemented
53	C₃×#460	C₂×C₅	1	1	yes	characteristic
54	C₃×C₈×Q₈	C₂×C₅	1	1	yes	characteristic, complemented
55	C₃×#334	C₂×C₅	1	1	yes	characteristic, complemented
56	C₃×#332	C₂×C₅	1	1	yes	characteristic, complemented
57	C₃×#331	C₂×C₅	1	1	yes	characteristic
58	C₂×C₅×SD₁₆	—	2	1	no	—
59	C₂×C₅×SD₁₆	—	2	1	no	—
60	C₂×C₅×SD₁₆	—	2	1	no	—
61	C₅×C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
62	C₅×#127	C₂²×C₃	1	1	yes	characteristic
63	C₄×C₅×Q₈	C₂²×C₃	1	1	yes	characteristic
64	C₅×#106	—	2	1	no	—
65	C₅×C₄⋊C₈	—	2	1	no	—
66	C₅×C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
67	C₅×C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
68	C₅×C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
69	C₅×#103	—	2	1	no	—
70	C₅×#102	—	2	1	no	—
71	C₄×C₅×C₈	—	2	1	no	—
72	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
73	#1666	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
74	C₂³×C₃×C₅	—	4	1	no	—
75	C₃×C₅×Q₈	—	2	1	no	—
76	C₃×C₅×Q₈	—	4	1	no	—
77	C₃×C₅×Q₈	—	4	1	no	—
78	C₃×C₅×Q₈	SD₁₆	1	2	yes	complemented
79	C₃×C₅×D₄	—	4	1	no	—
80	C₃×C₅×D₄	—	4	1	no	—
81	C₃×C₅×D₄	—	4	1	no	—
82	C₃×C₅×D₄	—	4	1	no	—
83	C₃×C₅×D₄	—	8	1	no	—
84	C₂×C₃×C₄×C₅	—	2	1	no	—
85	C₂×C₃×C₄×C₅	—	2	1	no	—
86	C₂×C₃×C₄×C₅	—	4	1	no	—
87	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
88	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
89	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
90	C₃×C₅×C₈	—	2	1	no	—
91	C₃×C₅×C₈	—	4	1	no	—
92	C₃×C₅×C₈	—	4	1	no	—
93	C₃×C₅×C₈	—	4	1	no	—
94	C₃×C₅×C₈	SD₁₆	1	2	yes	complemented
95	C₂×C₃×SD₁₆	—	2	1	no	—
96	C₂×C₃×SD₁₆	—	2	1	no	—
97	C₂×C₃×SD₁₆	—	2	1	no	—
98	C₃×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
99	C₃×#127	C₂²×C₅	1	1	yes	characteristic
100	C₃×C₄×Q₈	C₂²×C₅	1	1	yes	characteristic
101	C₃×#106	—	2	1	no	—
102	C₃×C₄⋊C₈	—	2	1	no	—
103	C₃×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
104	C₃×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
105	C₃×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
106	C₃×#103	—	2	1	no	—
107	C₃×#102	—	2	1	no	—
108	C₃×C₄×C₈	—	2	1	no	—
109	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
110	C₂×C₅×Q₈	—	2	1	no	—
111	C₂×C₅×Q₈	C₃×D₄	1	1	yes	characteristic
112	C₂×C₅×D₄	—	2	1	no	—
113	C₂×C₅×D₄	—	2	1	no	—
114	C₂×C₅×D₄	—	2	1	no	—
115	C₅×SD₁₆	—	4	2	no	—
116	C₅×SD₁₆	—	4	2	no	—
117	C₅×SD₁₆	—	4	2	no	—
118	C₂×C₅×C₈	—	2	1	no	—
119	C₂×C₅×C₈	—	2	1	no	—
120	C₂×C₅×C₈	—	2	1	no	—
121	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
122	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
123	C₅×C₄⋊C₄	—	2	1	no	—
124	C₅×C₄⋊C₄	—	2	1	no	—
125	C₅×C₄⋊C₄	—	2	1	no	—
126	C₅×C₄⋊C₄	C₃×D₄	1	1	yes	characteristic
127	C₄²×C₅	—	2	1	no	—
128	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
129	#492	C₂×C₃×C₅	1	1	yes	characteristic
130	#475	C₂×C₃×C₅	1	1	yes	characteristic, complemented
131	#460	C₂×C₃×C₅	1	1	yes	characteristic
132	C₈×Q₈	C₂×C₃×C₅	1	1	yes	characteristic, complemented
133	#334	C₂×C₃×C₅	1	1	yes	characteristic, complemented
134	#332	C₂×C₃×C₅	1	1	yes	characteristic, complemented
135	#331	C₂×C₃×C₅	1	1	yes	characteristic
136	C₂²×C₃×C₅	—	8	1	no	—
137	C₂²×C₃×C₅	—	8	1	no	—
138	C₂²×C₃×C₅	—	8	1	no	—
139	C₂²×C₃×C₅	#121	1	1	yes	characteristic
140	C₃×C₄×C₅	—	2	2	no	—
141	C₃×C₄×C₅	—	2	1	no	—
142	C₃×C₄×C₅	—	4	1	no	—
143	C₃×C₄×C₅	—	8	1	no	—
144	C₃×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
145	C₃×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
146	C₃×C₄×C₅	#135	1	1	yes	characteristic
147	C₃×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
148	C₂×C₃×Q₈	—	2	1	no	—
149	C₂×C₃×Q₈	C₅×D₄	1	1	yes	characteristic
150	C₂×C₃×D₄	—	2	1	no	—
151	C₂×C₃×D₄	—	2	1	no	—
152	C₂×C₃×D₄	—	2	1	no	—
153	C₃×SD₁₆	—	4	2	no	—
154	C₃×SD₁₆	—	4	2	no	—
155	C₃×SD₁₆	—	4	2	no	—
156	C₂×C₃×C₈	—	2	1	no	—
157	C₂×C₃×C₈	—	2	1	no	—
158	C₂×C₃×C₈	—	2	1	no	—
159	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
160	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
161	C₃×C₄⋊C₄	—	2	1	no	—
162	C₃×C₄⋊C₄	—	2	1	no	—
163	C₃×C₄⋊C₄	—	2	1	no	—
164	C₃×C₄⋊C₄	C₅×D₄	1	1	yes	characteristic
165	C₃×C₄²	—	2	1	no	—
166	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
167	C₂³×C₅	—	4	1	no	—
168	C₅×Q₈	—	2	1	no	—
169	C₅×Q₈	—	4	1	no	—
170	C₅×Q₈	—	4	1	no	—
171	C₅×Q₈	C₃×SD₁₆	1	2	yes	complemented
172	C₅×D₄	—	4	1	no	—
173	C₅×D₄	—	4	1	no	—
174	C₅×D₄	—	4	1	no	—
175	C₅×D₄	—	4	1	no	—
176	C₅×D₄	—	8	1	no	—
177	C₂×C₄×C₅	—	2	1	no	—
178	C₂×C₄×C₅	—	2	1	no	—
179	C₂×C₄×C₅	—	4	1	no	—
180	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
181	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
182	C₂×C₄×C₅	C₃×#41	1	1	yes	characteristic
183	C₅×C₈	—	2	1	no	—
184	C₅×C₈	—	4	1	no	—
185	C₅×C₈	—	4	1	no	—
186	C₅×C₈	—	4	1	no	—
187	C₅×C₈	C₃×SD₁₆	1	2	yes	complemented
188	C₂×SD₁₆	—	2	1	no	—
189	C₂×SD₁₆	—	2	1	no	—
190	C₂×SD₁₆	—	2	1	no	—
191	C₄⋊Q₈	C₂²×C₃×C₅	1	1	yes	characteristic
192	#127	C₂²×C₃×C₅	1	1	yes	characteristic
193	C₄×Q₈	C₂²×C₃×C₅	1	1	yes	characteristic
194	#106	—	2	1	no	—
195	C₄⋊C₈	—	2	1	no	—
196	C₄⋊C₈	C₂²×C₃×C₅	1	1	yes	characteristic
197	C₄⋊C₈	C₂²×C₃×C₅	1	1	yes	characteristic
198	C₄⋊C₈	C₂²×C₃×C₅	1	1	yes	characteristic
199	#103	—	2	1	no	—
200	#102	—	2	1	no	—
201	C₄×C₈	—	2	1	no	—
202	C₄×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
203	C₂×C₃×C₅	—	16	1	no	—
204	C₂×C₃×C₅	#460	1	1	yes	characteristic
205	C₂×C₃×C₅	#465	1	1	yes	characteristic
206	C₂×C₃×C₅	#471	1	1	yes	characteristic
207	C₂³×C₃	—	4	1	no	—
208	C₃×Q₈	—	2	1	no	—
209	C₃×Q₈	—	4	1	no	—
210	C₃×Q₈	—	4	1	no	—
211	C₃×Q₈	C₅×SD₁₆	1	2	yes	complemented
212	C₃×D₄	—	4	1	no	—
213	C₃×D₄	—	4	1	no	—
214	C₃×D₄	—	4	1	no	—
215	C₃×D₄	—	4	1	no	—
216	C₃×D₄	—	8	1	no	—
217	C₂×C₃×C₄	—	2	1	no	—
218	C₂×C₃×C₄	—	2	1	no	—
219	C₂×C₃×C₄	—	4	1	no	—
220	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
221	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
222	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
223	C₃×C₈	—	2	1	no	—
224	C₃×C₈	—	4	1	no	—
225	C₃×C₈	—	4	1	no	—
226	C₃×C₈	—	4	1	no	—
227	C₃×C₈	C₅×SD₁₆	1	2	yes	complemented
228	C₂²×C₅	—	8	1	no	—
229	C₂²×C₅	—	8	1	no	—
230	C₂²×C₅	—	8	1	no	—
231	C₂²×C₅	C₃×#121	1	1	yes	characteristic
232	C₄×C₅	—	2	2	no	—
233	C₄×C₅	—	2	1	no	—
234	C₄×C₅	—	4	1	no	—
235	C₄×C₅	—	8	1	no	—
236	C₄×C₅	C₂×C₃×SD₁₆	1	1	yes	characteristic
237	C₄×C₅	C₂×C₃×SD₁₆	1	1	yes	characteristic
238	C₄×C₅	C₃×#135	1	1	yes	characteristic
239	C₄×C₅	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
240	C₂×Q₈	—	2	1	no	—
241	C₂×Q₈	C₃×C₅×D₄	1	1	yes	characteristic
242	C₂×D₄	—	2	1	no	—
243	C₂×D₄	—	2	1	no	—
244	C₂×D₄	—	2	1	no	—
245	SD₁₆	—	4	2	no	—
246	SD₁₆	—	4	2	no	—
247	SD₁₆	—	4	2	no	—
248	C₂×C₈	—	2	1	no	—
249	C₂×C₈	—	2	1	no	—
250	C₂×C₈	—	2	1	no	—
251	C₂×C₈	C₃×C₅×D₄	1	1	yes	characteristic
252	C₂×C₈	C₃×C₅×D₄	1	1	yes	characteristic
253	C₄⋊C₄	—	2	1	no	—
254	C₄⋊C₄	—	2	1	no	—
255	C₄⋊C₄	—	2	1	no	—
256	C₄⋊C₄	C₃×C₅×D₄	1	1	yes	characteristic
257	C₄²	—	2	1	no	—
258	C₄²	C₂³×C₃×C₅	1	1	yes	characteristic
259	C₃×C₅	#1666	1	1	yes	characteristic, direct product factor, Hall
260	C₂²×C₃	—	8	1	no	—
261	C₂²×C₃	—	8	1	no	—
262	C₂²×C₃	—	8	1	no	—
263	C₂²×C₃	C₅×#121	1	1	yes	characteristic
264	C₃×C₄	—	2	2	no	—
265	C₃×C₄	—	2	1	no	—
266	C₃×C₄	—	4	1	no	—
267	C₃×C₄	—	8	1	no	—
268	C₃×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
269	C₃×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
270	C₃×C₄	C₅×#135	1	1	yes	characteristic
271	C₃×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
272	C₂×C₅	—	16	1	no	—
273	C₂×C₅	C₃×#460	1	1	yes	characteristic
274	C₂×C₅	C₃×#465	1	1	yes	characteristic
275	C₂×C₅	C₃×#471	1	1	yes	characteristic
276	C₂³	—	4	1	no	—
277	Q₈	—	2	1	no	—
278	Q₈	—	4	1	no	—
279	Q₈	—	4	1	no	—
280	Q₈	C₃×C₅×SD₁₆	1	2	yes	complemented
281	D₄	—	4	1	no	—
282	D₄	—	4	1	no	—
283	D₄	—	4	1	no	—
284	D₄	—	4	1	no	—
285	D₄	—	8	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	4	1	no	—
289	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
290	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
291	C₂×C₄	C₃×C₅×#41	1	1	yes	characteristic
292	C₈	—	2	1	no	—
293	C₈	—	4	1	no	—
294	C₈	—	4	1	no	—
295	C₈	—	4	1	no	—
296	C₈	C₃×C₅×SD₁₆	1	2	yes	complemented
297	C₂×C₃	—	16	1	no	—
298	C₂×C₃	C₅×#460	1	1	yes	characteristic
299	C₂×C₃	C₅×#465	1	1	yes	characteristic
300	C₂×C₃	C₅×#471	1	1	yes	characteristic
301	C₅	C₃×#1666	1	1	yes	characteristic, direct product factor, Sylow
302	C₂²	—	8	1	no	—
303	C₂²	—	8	1	no	—
304	C₂²	—	8	1	no	—
305	C₂²	C₃×C₅×#121	1	1	yes	characteristic
306	C₄	—	2	2	no	—
307	C₄	—	2	1	no	—
308	C₄	—	4	1	no	—
309	C₄	—	8	1	no	—
310	C₄	C₂×C₃×C₅×SD₁₆	1	1	yes	characteristic
311	C₄	C₂×C₃×C₅×SD₁₆	1	1	yes	characteristic
312	C₄	C₃×C₅×#135	1	1	yes	characteristic
313	C₄	C₃×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
314	C₃	C₅×#1666	1	1	yes	characteristic, direct product factor, Sylow
315	C₂	—	16	1	no	—
316	C₂	C₃×C₅×#460	1	1	yes	characteristic
317	C₂	C₃×C₅×#465	1	1	yes	characteristic
318	C₂	C₃×C₅×#471	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃×C₅	#121
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
2. C₃×C₅×#475	C₂	semidirect
4. C₃×C₅×C₈×Q₈	C₂	semidirect
5. C₃×C₅×#334	C₂	semidirect
6. C₃×C₅×#332	C₂	semidirect
8. C₅×#1666	C₃	direct (N × H)
24. C₃×#1666	C₅	direct (N × H)
26. C₅×#475	C₂×C₃	semidirect
28. C₅×C₈×Q₈	C₂×C₃	semidirect
29. C₅×#334	C₂×C₃	semidirect
30. C₅×#332	C₂×C₃	semidirect
52. C₃×#475	C₂×C₅	semidirect
54. C₃×C₈×Q₈	C₂×C₅	semidirect
55. C₃×#334	C₂×C₅	semidirect
56. C₃×#332	C₂×C₅	semidirect
73. #1666	C₃×C₅	direct (N × H)
78. C₃×C₅×Q₈	SD₁₆	semidirect
94. C₃×C₅×C₈	SD₁₆	semidirect
130. #475	C₂×C₃×C₅	semidirect
132. C₈×Q₈	C₂×C₃×C₅	semidirect
133. #334	C₂×C₃×C₅	semidirect
134. #332	C₂×C₃×C₅	semidirect
171. C₅×Q₈	C₃×SD₁₆	semidirect
187. C₅×C₈	C₃×SD₁₆	semidirect
211. C₃×Q₈	C₅×SD₁₆	semidirect
227. C₃×C₈	C₅×SD₁₆	semidirect
259. C₃×C₅	#1666	direct (N × H)
280. Q₈	C₃×C₅×SD₁₆	semidirect
296. C₈	C₃×C₅×SD₁₆	semidirect
301. C₅	C₃×#1666	direct (N × H)
314. C₃	C₅×#1666	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×#492	2
C₃×C₅×#475	2
C₃×C₅×#460	2
C₃×C₅×C₈×Q₈	2
C₃×C₅×#334	2
C₃×C₅×#332	2
C₃×C₅×#331	2
C₅×#1666	3
C₃×#1666	5

Subgroups — C3×C5×#1666 #14274700

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#1666 #14274700