Subgroups — C₃×C₅×#1554 #14274588

Order 1920.

396 conjugacy classes of subgroups.

The subgroup lattice has 292 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×#432	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#431	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×#1554	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₃×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
8	C₃×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	C₃×C₅×#125	—	2	1	no	—
10	C₃×C₅×C₂²⋊Q₈	—	2	1	no	—
11	C₃×C₅×C₂²⋊Q₈	C₄	1	2	yes	—
12	C₃×C₄×C₅×Q₈	—	2	1	no	—
13	C₃×C₄×C₅×Q₈	C₄	1	2	yes	—
14	C₃×C₅×#117	—	2	1	no	—
15	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
16	C₃×C₅×C₄⋊C₈	C₂²	1	2	yes	—
17	C₃×C₅×C₄⋊C₈	C₂²	1	2	yes	—
18	C₃×C₅×C₂²⋊C₈	—	2	1	no	—
19	C₃×C₅×C₂²⋊C₈	—	2	1	no	—
20	C₃×C₅×#97	—	2	1	no	—
21	C₃×C₄×C₅×C₈	—	2	1	no	—
22	C₃×#1554	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
23	C₅×#533	C₂×C₃	1	1	yes	characteristic
24	C₅×#432	C₂×C₃	1	1	yes	characteristic
25	C₅×#431	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	C₂×C₃×C₅×C₈	—	2	2	no	—
34	C₂×C₃×C₅×C₈	—	2	2	no	—
35	C₃×C₅×C₄⋊C₄	—	2	2	no	—
36	C₃×C₅×C₄⋊C₄	—	2	2	no	—
37	C₃×C₅×C₄⋊C₄	—	2	1	no	—
38	C₃×C₅×C₄⋊C₄	—	2	1	no	—
39	C₃×C₅×C₄⋊C₄	—	2	1	no	—
40	C₃×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
41	C₃×C₅×C₂²⋊C₄	—	2	2	no	—
42	C₃×C₄²×C₅	—	2	2	no	—
43	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
44	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
45	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
46	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
47	C₃×#533	C₂×C₅	1	1	yes	characteristic
48	C₃×#432	C₂×C₅	1	1	yes	characteristic
49	C₃×#431	C₂×C₅	1	1	yes	characteristic
50	C₃×#333	C₂×C₅	1	2	yes	complemented
51	C₃×#332	C₂×C₅	1	2	yes	complemented
52	C₅×C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
53	C₅×C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
54	C₅×#125	—	2	1	no	—
55	C₅×C₂²⋊Q₈	—	2	1	no	—
56	C₅×C₂²⋊Q₈	C₃×C₄	1	2	yes	—
57	C₄×C₅×Q₈	—	2	1	no	—
58	C₄×C₅×Q₈	C₃×C₄	1	2	yes	—
59	C₅×#117	—	2	1	no	—
60	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
61	C₅×C₄⋊C₈	C₂²×C₃	1	2	yes	—
62	C₅×C₄⋊C₈	C₂²×C₃	1	2	yes	—
63	C₅×C₂²⋊C₈	—	2	1	no	—
64	C₅×C₂²⋊C₈	—	2	1	no	—
65	C₅×#97	—	2	1	no	—
66	C₄×C₅×C₈	—	2	1	no	—
67	#1554	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
68	C₂³×C₃×C₅	C₂²⋊C₄	1	1	yes	characteristic
69	C₃×C₅×Q₈	—	4	2	no	—
70	C₃×C₅×Q₈	—	4	2	no	—
71	C₂×C₃×C₄×C₅	—	2	2	no	—
72	C₂×C₃×C₄×C₅	—	2	2	no	—
73	C₂×C₃×C₄×C₅	—	2	2	no	—
74	C₂×C₃×C₄×C₅	—	2	2	no	—
75	C₂×C₃×C₄×C₅	—	4	1	no	—
76	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
81	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
82	C₃×C₅×C₈	—	4	2	no	—
83	C₃×C₅×C₈	—	4	2	no	—
84	C₃×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
85	C₃×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
86	C₃×#125	—	2	1	no	—
87	C₃×C₂²⋊Q₈	—	2	1	no	—
88	C₃×C₂²⋊Q₈	C₄×C₅	1	2	yes	—
89	C₃×C₄×Q₈	—	2	1	no	—
90	C₃×C₄×Q₈	C₄×C₅	1	2	yes	—
91	C₃×#117	—	2	1	no	—
92	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
93	C₃×C₄⋊C₈	C₂²×C₅	1	2	yes	—
94	C₃×C₄⋊C₈	C₂²×C₅	1	2	yes	—
95	C₃×C₂²⋊C₈	—	2	1	no	—
96	C₃×C₂²⋊C₈	—	2	1	no	—
97	C₃×#97	—	2	1	no	—
98	C₃×C₄×C₈	—	2	1	no	—
99	C₂×C₅×Q₈	—	2	1	no	—
100	C₂×C₅×Q₈	C₂×C₃×C₄	1	2	yes	—
101	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
102	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
103	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
104	C₂×C₅×C₈	—	2	2	no	—
105	C₂×C₅×C₈	—	2	2	no	—
106	C₅×C₄⋊C₄	—	2	2	no	—
107	C₅×C₄⋊C₄	—	2	2	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	C₂×C₃×C₄	1	2	yes	—
112	C₅×C₂²⋊C₄	—	2	2	no	—
113	C₄²×C₅	—	2	2	no	—
114	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
115	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
116	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
117	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
118	#533	C₂×C₃×C₅	1	1	yes	characteristic
119	#432	C₂×C₃×C₅	1	1	yes	characteristic
120	#431	C₂×C₃×C₅	1	1	yes	characteristic
121	#333	C₂×C₃×C₅	1	2	yes	complemented
122	#332	C₂×C₃×C₅	1	2	yes	complemented
123	C₂²×C₃×C₅	—	2	1	no	—
124	C₂²×C₃×C₅	—	2	1	no	—
125	C₂²×C₃×C₅	—	2	1	no	—
126	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
127	C₃×C₄×C₅	—	2	1	no	—
128	C₃×C₄×C₅	—	2	1	no	—
129	C₃×C₄×C₅	—	2	1	no	—
130	C₃×C₄×C₅	—	2	1	no	—
131	C₃×C₄×C₅	—	4	2	no	—
132	C₃×C₄×C₅	—	4	2	no	—
133	C₃×C₄×C₅	#135	1	2	yes	—
134	C₃×C₄×C₅	#137	1	2	yes	—
135	C₂×C₃×Q₈	—	2	1	no	—
136	C₂×C₃×Q₈	C₂×C₄×C₅	1	2	yes	—
137	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
138	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
139	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
140	C₂×C₃×C₈	—	2	2	no	—
141	C₂×C₃×C₈	—	2	2	no	—
142	C₃×C₄⋊C₄	—	2	2	no	—
143	C₃×C₄⋊C₄	—	2	2	no	—
144	C₃×C₄⋊C₄	—	2	1	no	—
145	C₃×C₄⋊C₄	—	2	1	no	—
146	C₃×C₄⋊C₄	—	2	1	no	—
147	C₃×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
148	C₃×C₂²⋊C₄	—	2	2	no	—
149	C₃×C₄²	—	2	2	no	—
150	C₃×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
151	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
152	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
153	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
154	C₂³×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
155	C₅×Q₈	—	4	2	no	—
156	C₅×Q₈	—	4	2	no	—
157	C₂×C₄×C₅	—	2	2	no	—
158	C₂×C₄×C₅	—	2	2	no	—
159	C₂×C₄×C₅	—	2	2	no	—
160	C₂×C₄×C₅	—	2	2	no	—
161	C₂×C₄×C₅	—	4	1	no	—
162	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
163	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
164	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
165	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
166	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
167	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
168	C₅×C₈	—	4	2	no	—
169	C₅×C₈	—	4	2	no	—
170	C₄⋊Q₈	C₂²×C₃×C₅	1	1	yes	characteristic
171	C₄⋊Q₈	C₂²×C₃×C₅	1	1	yes	characteristic
172	#125	—	2	1	no	—
173	C₂²⋊Q₈	—	2	1	no	—
174	C₂²⋊Q₈	C₃×C₄×C₅	1	2	yes	—
175	C₄×Q₈	—	2	1	no	—
176	C₄×Q₈	C₃×C₄×C₅	1	2	yes	—
177	#117	—	2	1	no	—
178	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
179	C₄⋊C₈	C₂²×C₃×C₅	1	2	yes	—
180	C₄⋊C₈	C₂²×C₃×C₅	1	2	yes	—
181	C₂²⋊C₈	—	2	1	no	—
182	C₂²⋊C₈	—	2	1	no	—
183	#97	—	2	1	no	—
184	C₄×C₈	—	2	1	no	—
185	C₂×C₃×C₅	—	4	1	no	—
186	C₂×C₃×C₅	#413	1	1	yes	characteristic
187	C₂×C₃×C₅	#416	1	1	yes	characteristic
188	C₂×C₃×C₅	#419	1	1	yes	characteristic
189	C₂³×C₃	C₅×C₂²⋊C₄	1	1	yes	characteristic
190	C₃×Q₈	—	4	2	no	—
191	C₃×Q₈	—	4	2	no	—
192	C₂×C₃×C₄	—	2	2	no	—
193	C₂×C₃×C₄	—	2	2	no	—
194	C₂×C₃×C₄	—	2	2	no	—
195	C₂×C₃×C₄	—	2	2	no	—
196	C₂×C₃×C₄	—	4	1	no	—
197	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
198	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
199	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
200	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
201	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
202	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
203	C₃×C₈	—	4	2	no	—
204	C₃×C₈	—	4	2	no	—
205	C₂²×C₅	—	2	1	no	—
206	C₂²×C₅	—	2	1	no	—
207	C₂²×C₅	—	2	1	no	—
208	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
209	C₄×C₅	—	2	1	no	—
210	C₄×C₅	—	2	1	no	—
211	C₄×C₅	—	2	1	no	—
212	C₄×C₅	—	2	1	no	—
213	C₄×C₅	—	4	2	no	—
214	C₄×C₅	—	4	2	no	—
215	C₄×C₅	C₃×#135	1	2	yes	—
216	C₄×C₅	C₃×#137	1	2	yes	—
217	C₂×Q₈	—	2	1	no	—
218	C₂×Q₈	C₂×C₃×C₄×C₅	1	2	yes	—
219	C₂²×C₄	C₂×C₃×C₄×C₅	1	1	yes	characteristic
220	C₂²×C₄	C₃×C₅×D₄	1	1	yes	characteristic
221	C₂²×C₄	C₃×C₅×D₄	1	1	yes	characteristic
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	2	2	no	—
224	C₄⋊C₄	—	2	2	no	—
225	C₄⋊C₄	—	2	2	no	—
226	C₄⋊C₄	—	2	1	no	—
227	C₄⋊C₄	—	2	1	no	—
228	C₄⋊C₄	—	2	1	no	—
229	C₄⋊C₄	C₂×C₃×C₄×C₅	1	2	yes	—
230	C₂²⋊C₄	—	2	2	no	—
231	C₄²	—	2	2	no	—
232	C₄²	C₂×C₃×C₄×C₅	1	1	yes	characteristic
233	C₄²	C₃×C₅×D₄	1	1	yes	characteristic
234	C₄²	C₃×C₅×D₄	1	1	yes	characteristic
235	C₄²	C₂³×C₃×C₅	1	1	yes	characteristic
236	C₃×C₅	#1554	1	1	yes	characteristic, direct product factor, Hall
237	C₂²×C₃	—	2	1	no	—
238	C₂²×C₃	—	2	1	no	—
239	C₂²×C₃	—	2	1	no	—
240	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
241	C₃×C₄	—	2	1	no	—
242	C₃×C₄	—	2	1	no	—
243	C₃×C₄	—	2	1	no	—
244	C₃×C₄	—	2	1	no	—
245	C₃×C₄	—	4	2	no	—
246	C₃×C₄	—	4	2	no	—
247	C₃×C₄	C₅×#135	1	2	yes	—
248	C₃×C₄	C₅×#137	1	2	yes	—
249	C₂×C₅	—	4	1	no	—
250	C₂×C₅	C₃×#413	1	1	yes	characteristic
251	C₂×C₅	C₃×#416	1	1	yes	characteristic
252	C₂×C₅	C₃×#419	1	1	yes	characteristic
253	C₂³	C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
254	Q₈	—	4	2	no	—
255	Q₈	—	4	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	4	1	no	—
261	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
262	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
263	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
264	C₂×C₄	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
265	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
266	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
267	C₈	—	4	2	no	—
268	C₈	—	4	2	no	—
269	C₂×C₃	—	4	1	no	—
270	C₂×C₃	C₅×#413	1	1	yes	characteristic
271	C₂×C₃	C₅×#416	1	1	yes	characteristic
272	C₂×C₃	C₅×#419	1	1	yes	characteristic
273	C₅	C₃×#1554	1	1	yes	characteristic, direct product factor, Sylow
274	C₂²	—	2	1	no	—
275	C₂²	—	2	1	no	—
276	C₂²	—	2	1	no	—
277	C₂²	C₂×C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
278	C₄	—	2	1	no	—
279	C₄	—	2	1	no	—
280	C₄	—	2	1	no	—
281	C₄	—	2	1	no	—
282	C₄	—	4	2	no	—
283	C₄	—	4	2	no	—
284	C₄	C₃×C₅×#135	1	2	yes	—
285	C₄	C₃×C₅×#137	1	2	yes	—
286	C₃	C₅×#1554	1	1	yes	characteristic, direct product factor, Sylow
287	C₂	—	4	1	no	—
288	C₂	C₃×C₅×#413	1	1	yes	characteristic
289	C₂	C₃×C₅×#416	1	1	yes	characteristic
290	C₂	C₃×C₅×#419	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×#1554	C₃	direct (N × H)
22. C₃×#1554	C₅	direct (N × H)
26. C₅×#333	C₂×C₃	semidirect
27. C₅×#332	C₂×C₃	semidirect
50. C₃×#333	C₂×C₅	semidirect
51. C₃×#332	C₂×C₅	semidirect
67. #1554	C₃×C₅	direct (N × H)
121. #333	C₂×C₃×C₅	semidirect
122. #332	C₂×C₃×C₅	semidirect
236. C₃×C₅	#1554	direct (N × H)
273. C₅	C₃×#1554	direct (N × H)
286. C₃	C₅×#1554	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×#432	2
C₃×C₅×#431	2
C₃×C₅×#333	2
C₃×C₅×#332	2
C₅×#1554	3
C₃×#1554	5

Subgroups — C3×C5×#1554 #14274588

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#1554 #14274588