Subgroups — C₃×C₅×#1550 #14274584

Order 1920.

432 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×#533	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×#423	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₅×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₃×C₅×#333	C₂	1	2	yes	maximal, complemented
5	C₃×C₅×#332	C₂	1	2	yes	maximal, complemented
6	C₅×#1550	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×C₅×M₁₆	—	2	1	no	—
8	C₂²×C₃×C₅×C₈	—	2	1	no	—
9	C₃×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
10	C₃×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	C₃×C₅×#125	—	2	1	no	—
12	C₃×C₅×C₂²⋊Q₈	—	2	1	no	—
13	C₃×C₅×C₂²⋊Q₈	C₄	1	2	yes	—
14	C₃×C₄×C₅×Q₈	—	2	1	no	—
15	C₃×C₄×C₅×Q₈	C₄	1	2	yes	—
16	C₃×C₅×#117	—	2	1	no	—
17	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
18	C₃×C₅×C₄⋊C₈	—	2	1	no	—
19	C₃×C₅×C₄⋊C₈	—	2	1	no	—
20	C₃×C₅×C₄⋊C₈	C₂²	1	2	yes	—
21	C₃×C₅×C₄⋊C₈	C₂²	1	2	yes	—
22	C₃×#1550	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
23	C₅×#533	C₂×C₃	1	1	yes	characteristic
24	C₅×#423	C₂×C₃	1	1	yes	characteristic
25	C₂×C₅×C₄⋊C₈	C₂×C₃	1	1	yes	characteristic
26	C₅×#333	C₂×C₃	1	2	yes	complemented
27	C₅×#332	C₂×C₃	1	2	yes	complemented
28	C₂×C₃×C₅×Q₈	—	2	1	no	—
29	C₂×C₃×C₅×Q₈	C₂×C₄	1	2	yes	—
30	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
32	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
33	C₃×C₅×M₁₆	—	4	2	no	—
34	C₂×C₃×C₅×C₈	—	2	2	no	—
35	C₂×C₃×C₅×C₈	—	2	2	no	—
36	C₂×C₃×C₅×C₈	—	4	2	no	—
37	C₃×C₅×C₄⋊C₄	—	2	2	no	—
38	C₃×C₅×C₄⋊C₄	—	2	2	no	—
39	C₃×C₅×C₄⋊C₄	—	2	1	no	—
40	C₃×C₅×C₄⋊C₄	—	2	1	no	—
41	C₃×C₅×C₄⋊C₄	—	2	1	no	—
42	C₃×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
43	C₃×C₅×C₂²⋊C₄	—	2	2	no	—
44	C₃×C₄²×C₅	—	2	2	no	—
45	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
46	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
47	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
48	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
49	C₃×#533	C₂×C₅	1	1	yes	characteristic
50	C₃×#423	C₂×C₅	1	1	yes	characteristic
51	C₂×C₃×C₄⋊C₈	C₂×C₅	1	1	yes	characteristic
52	C₃×#333	C₂×C₅	1	2	yes	complemented
53	C₃×#332	C₂×C₅	1	2	yes	complemented
54	C₂×C₅×M₁₆	—	2	1	no	—
55	C₂²×C₅×C₈	—	2	1	no	—
56	C₅×C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
57	C₅×C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
58	C₅×#125	—	2	1	no	—
59	C₅×C₂²⋊Q₈	—	2	1	no	—
60	C₅×C₂²⋊Q₈	C₃×C₄	1	2	yes	—
61	C₄×C₅×Q₈	—	2	1	no	—
62	C₄×C₅×Q₈	C₃×C₄	1	2	yes	—
63	C₅×#117	—	2	1	no	—
64	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
65	C₅×C₄⋊C₈	—	2	1	no	—
66	C₅×C₄⋊C₈	—	2	1	no	—
67	C₅×C₄⋊C₈	C₂²×C₃	1	2	yes	—
68	C₅×C₄⋊C₈	C₂²×C₃	1	2	yes	—
69	#1550	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
70	C₂³×C₃×C₅	C₂²⋊C₄	1	1	yes	characteristic
71	C₃×C₅×Q₈	—	4	2	no	—
72	C₃×C₅×Q₈	—	4	1	no	—
73	C₃×C₅×Q₈	—	4	1	no	—
74	C₂×C₃×C₄×C₅	—	2	1	no	—
75	C₂×C₃×C₄×C₅	—	2	1	no	—
76	C₂×C₃×C₄×C₅	—	2	2	no	—
77	C₂×C₃×C₄×C₅	—	2	2	no	—
78	C₂×C₃×C₄×C₅	—	4	1	no	—
79	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₃×C₄×C₅	SD₁₆	1	2	yes	—
83	C₂×C₃×C₄×C₅	Q₁₆	1	2	yes	—
84	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
86	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
87	C₃×C₅×C₈	—	4	2	no	—
88	C₃×C₅×C₈	—	4	2	no	—
89	C₂×C₃×M₁₆	—	2	1	no	—
90	C₂²×C₃×C₈	—	2	1	no	—
91	C₃×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
92	C₃×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
93	C₃×#125	—	2	1	no	—
94	C₃×C₂²⋊Q₈	—	2	1	no	—
95	C₃×C₂²⋊Q₈	C₄×C₅	1	2	yes	—
96	C₃×C₄×Q₈	—	2	1	no	—
97	C₃×C₄×Q₈	C₄×C₅	1	2	yes	—
98	C₃×#117	—	2	1	no	—
99	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
100	C₃×C₄⋊C₈	—	2	1	no	—
101	C₃×C₄⋊C₈	—	2	1	no	—
102	C₃×C₄⋊C₈	C₂²×C₅	1	2	yes	—
103	C₃×C₄⋊C₈	C₂²×C₅	1	2	yes	—
104	C₂×C₅×Q₈	—	2	1	no	—
105	C₂×C₅×Q₈	C₂×C₃×C₄	1	2	yes	—
106	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
107	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
108	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
109	C₅×M₁₆	—	4	2	no	—
110	C₂×C₅×C₈	—	2	2	no	—
111	C₂×C₅×C₈	—	2	2	no	—
112	C₂×C₅×C₈	—	4	2	no	—
113	C₅×C₄⋊C₄	—	2	2	no	—
114	C₅×C₄⋊C₄	—	2	2	no	—
115	C₅×C₄⋊C₄	—	2	1	no	—
116	C₅×C₄⋊C₄	—	2	1	no	—
117	C₅×C₄⋊C₄	—	2	1	no	—
118	C₅×C₄⋊C₄	C₂×C₃×C₄	1	2	yes	—
119	C₅×C₂²⋊C₄	—	2	2	no	—
120	C₄²×C₅	—	2	2	no	—
121	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
122	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
123	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
124	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
125	#533	C₂×C₃×C₅	1	1	yes	characteristic
126	#423	C₂×C₃×C₅	1	1	yes	characteristic
127	C₂×C₄⋊C₈	C₂×C₃×C₅	1	1	yes	characteristic
128	#333	C₂×C₃×C₅	1	2	yes	complemented
129	#332	C₂×C₃×C₅	1	2	yes	complemented
130	C₂²×C₃×C₅	—	2	1	no	—
131	C₂²×C₃×C₅	—	2	1	no	—
132	C₂²×C₃×C₅	#103	1	2	yes	—
133	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
134	C₃×C₄×C₅	—	2	1	no	—
135	C₃×C₄×C₅	—	2	1	no	—
136	C₃×C₄×C₅	—	2	1	no	—
137	C₃×C₄×C₅	—	4	2	no	—
138	C₃×C₄×C₅	—	4	2	no	—
139	C₃×C₄×C₅	#103	1	2	yes	—
140	C₃×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
141	C₃×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
142	C₃×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
143	C₃×C₄×C₅	#137	1	1	yes	characteristic
144	C₂×C₃×Q₈	—	2	1	no	—
145	C₂×C₃×Q₈	C₂×C₄×C₅	1	2	yes	—
146	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
147	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
148	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
149	C₃×M₁₆	—	4	2	no	—
150	C₂×C₃×C₈	—	2	2	no	—
151	C₂×C₃×C₈	—	2	2	no	—
152	C₂×C₃×C₈	—	4	2	no	—
153	C₃×C₄⋊C₄	—	2	2	no	—
154	C₃×C₄⋊C₄	—	2	2	no	—
155	C₃×C₄⋊C₄	—	2	1	no	—
156	C₃×C₄⋊C₄	—	2	1	no	—
157	C₃×C₄⋊C₄	—	2	1	no	—
158	C₃×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
159	C₃×C₂²⋊C₄	—	2	2	no	—
160	C₃×C₄²	—	2	2	no	—
161	C₃×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
162	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
163	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
164	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
165	C₂³×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
166	C₅×Q₈	—	4	2	no	—
167	C₅×Q₈	—	4	1	no	—
168	C₅×Q₈	—	4	1	no	—
169	C₂×C₄×C₅	—	2	1	no	—
170	C₂×C₄×C₅	—	2	1	no	—
171	C₂×C₄×C₅	—	2	2	no	—
172	C₂×C₄×C₅	—	2	2	no	—
173	C₂×C₄×C₅	—	4	1	no	—
174	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
175	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
176	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
177	C₂×C₄×C₅	C₃×SD₁₆	1	2	yes	—
178	C₂×C₄×C₅	C₃×Q₁₆	1	2	yes	—
179	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
180	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
181	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
182	C₅×C₈	—	4	2	no	—
183	C₅×C₈	—	4	2	no	—
184	C₂×M₁₆	—	2	1	no	—
185	C₂²×C₈	—	2	1	no	—
186	C₄⋊Q₈	C₂²×C₃×C₅	1	1	yes	characteristic
187	C₄⋊Q₈	C₂²×C₃×C₅	1	1	yes	characteristic
188	#125	—	2	1	no	—
189	C₂²⋊Q₈	—	2	1	no	—
190	C₂²⋊Q₈	C₃×C₄×C₅	1	2	yes	—
191	C₄×Q₈	—	2	1	no	—
192	C₄×Q₈	C₃×C₄×C₅	1	2	yes	—
193	#117	—	2	1	no	—
194	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
195	C₄⋊C₈	—	2	1	no	—
196	C₄⋊C₈	—	2	1	no	—
197	C₄⋊C₈	C₂²×C₃×C₅	1	2	yes	—
198	C₄⋊C₈	C₂²×C₃×C₅	1	2	yes	—
199	C₂×C₃×C₅	—	2	2	no	—
200	C₂×C₃×C₅	#413	1	1	yes	characteristic
201	C₂×C₃×C₅	C₂×#103	1	1	yes	characteristic
202	C₂×C₃×C₅	#417	1	1	yes	characteristic
203	C₂³×C₃	C₅×C₂²⋊C₄	1	1	yes	characteristic
204	C₃×Q₈	—	4	2	no	—
205	C₃×Q₈	—	4	1	no	—
206	C₃×Q₈	—	4	1	no	—
207	C₂×C₃×C₄	—	2	1	no	—
208	C₂×C₃×C₄	—	2	1	no	—
209	C₂×C₃×C₄	—	2	2	no	—
210	C₂×C₃×C₄	—	2	2	no	—
211	C₂×C₃×C₄	—	4	1	no	—
212	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
213	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
214	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
215	C₂×C₃×C₄	C₅×SD₁₆	1	2	yes	—
216	C₂×C₃×C₄	C₅×Q₁₆	1	2	yes	—
217	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
218	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
219	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
220	C₃×C₈	—	4	2	no	—
221	C₃×C₈	—	4	2	no	—
222	C₂²×C₅	—	2	1	no	—
223	C₂²×C₅	—	2	1	no	—
224	C₂²×C₅	C₃×#103	1	2	yes	—
225	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
226	C₄×C₅	—	2	1	no	—
227	C₄×C₅	—	2	1	no	—
228	C₄×C₅	—	2	1	no	—
229	C₄×C₅	—	4	2	no	—
230	C₄×C₅	—	4	2	no	—
231	C₄×C₅	C₃×#103	1	2	yes	—
232	C₄×C₅	C₂×C₃×SD₁₆	1	1	yes	characteristic
233	C₄×C₅	C₂×C₃×Q₁₆	1	1	yes	characteristic
234	C₄×C₅	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
235	C₄×C₅	C₃×#137	1	1	yes	characteristic
236	C₂×Q₈	—	2	1	no	—
237	C₂×Q₈	C₂×C₃×C₄×C₅	1	2	yes	—
238	C₂²×C₄	C₂×C₃×C₄×C₅	1	1	yes	characteristic
239	C₂²×C₄	C₃×C₅×D₄	1	1	yes	characteristic
240	C₂²×C₄	C₃×C₅×D₄	1	1	yes	characteristic
241	M₁₆	—	4	2	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	4	2	no	—
245	C₄⋊C₄	—	2	2	no	—
246	C₄⋊C₄	—	2	2	no	—
247	C₄⋊C₄	—	2	1	no	—
248	C₄⋊C₄	—	2	1	no	—
249	C₄⋊C₄	—	2	1	no	—
250	C₄⋊C₄	C₂×C₃×C₄×C₅	1	2	yes	—
251	C₂²⋊C₄	—	2	2	no	—
252	C₄²	—	2	2	no	—
253	C₄²	C₂×C₃×C₄×C₅	1	1	yes	characteristic
254	C₄²	C₃×C₅×D₄	1	1	yes	characteristic
255	C₄²	C₃×C₅×D₄	1	1	yes	characteristic
256	C₄²	C₂³×C₃×C₅	1	1	yes	characteristic
257	C₃×C₅	#1550	1	1	yes	characteristic, direct product factor, Hall
258	C₂²×C₃	—	2	1	no	—
259	C₂²×C₃	—	2	1	no	—
260	C₂²×C₃	C₅×#103	1	2	yes	—
261	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
262	C₃×C₄	—	2	1	no	—
263	C₃×C₄	—	2	1	no	—
264	C₃×C₄	—	2	1	no	—
265	C₃×C₄	—	4	2	no	—
266	C₃×C₄	—	4	2	no	—
267	C₃×C₄	C₅×#103	1	2	yes	—
268	C₃×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
269	C₃×C₄	C₂×C₅×Q₁₆	1	1	yes	characteristic
270	C₃×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
271	C₃×C₄	C₅×#137	1	1	yes	characteristic
272	C₂×C₅	—	2	2	no	—
273	C₂×C₅	C₃×#413	1	1	yes	characteristic
274	C₂×C₅	C₂×C₃×#103	1	1	yes	characteristic
275	C₂×C₅	C₃×#417	1	1	yes	characteristic
276	C₂³	C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
277	Q₈	—	4	2	no	—
278	Q₈	—	4	1	no	—
279	Q₈	—	4	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	4	1	no	—
285	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
286	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
287	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
288	C₂×C₄	C₃×C₅×SD₁₆	1	2	yes	—
289	C₂×C₄	C₃×C₅×Q₁₆	1	2	yes	—
290	C₂×C₄	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
291	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
292	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
293	C₈	—	4	2	no	—
294	C₈	—	4	2	no	—
295	C₂×C₃	—	2	2	no	—
296	C₂×C₃	C₅×#413	1	1	yes	characteristic
297	C₂×C₃	C₂×C₅×#103	1	1	yes	characteristic
298	C₂×C₃	C₅×#417	1	1	yes	characteristic
299	C₅	C₃×#1550	1	1	yes	characteristic, direct product factor, Sylow
300	C₂²	—	2	1	no	—
301	C₂²	—	2	1	no	—
302	C₂²	C₃×C₅×#103	1	2	yes	—
303	C₂²	C₂×C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
304	C₄	—	2	1	no	—
305	C₄	—	2	1	no	—
306	C₄	—	2	1	no	—
307	C₄	—	4	2	no	—
308	C₄	—	4	2	no	—
309	C₄	C₃×C₅×#103	1	2	yes	—
310	C₄	C₂×C₃×C₅×SD₁₆	1	1	yes	characteristic
311	C₄	C₂×C₃×C₅×Q₁₆	1	1	yes	characteristic
312	C₄	C₃×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
313	C₄	C₃×C₅×#137	1	1	yes	characteristic
314	C₃	C₅×#1550	1	1	yes	characteristic, direct product factor, Sylow
315	C₂	—	2	2	no	—
316	C₂	C₃×C₅×#413	1	1	yes	characteristic
317	C₂	C₂×C₃×C₅×#103	1	1	yes	characteristic
318	C₂	C₃×C₅×#417	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₃×C₄×C₅
Frattini, cosocle	C₄²	C₂³×C₃×C₅
prime radical	C₂³×C₃×C₅	C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₅ ◁ C₂×C₃×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₃×C₅×#333	C₂	semidirect
5. C₃×C₅×#332	C₂	semidirect
6. C₅×#1550	C₃	direct (N × H)
22. C₃×#1550	C₅	direct (N × H)
26. C₅×#333	C₂×C₃	semidirect
27. C₅×#332	C₂×C₃	semidirect
52. C₃×#333	C₂×C₅	semidirect
53. C₃×#332	C₂×C₅	semidirect
69. #1550	C₃×C₅	direct (N × H)
128. #333	C₂×C₃×C₅	semidirect
129. #332	C₂×C₃×C₅	semidirect
257. C₃×C₅	#1550	direct (N × H)
299. C₅	C₃×#1550	direct (N × H)
314. C₃	C₅×#1550	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×#533	2
C₃×C₅×#423	2
C₂×C₃×C₅×C₄⋊C₈	2
C₃×C₅×#333	2
C₃×C₅×#332	2
C₅×#1550	3
C₃×#1550	5

Subgroups — C3×C5×#1550 #14274584

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#1550 #14274584