Subgroups — C₃×C₅×#2646 #14275680

Order 1920.

872 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×#400	C₂	1	4	yes	maximal, complemented
2	C₃×C₅×#398	C₂	1	4	yes	maximal, complemented
3	C₃×C₅×#385	C₂	1	4	yes	maximal
4	C₃×C₅×#380	C₂	1	2	yes	maximal
5	C₂×C₃×C₅×#95	C₂	1	1	yes	maximal, characteristic
6	C₅×#2646	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂³×C₃×C₄×C₅	C₂²	1	2	yes	—
8	C₂³×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
9	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
10	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
11	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	4	yes	—
12	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	4	yes	—
13	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	4	yes	—
14	C₃×C₅×#95	C₂²	1	4	yes	—
15	C₃×C₅×#95	C₂²	1	4	yes	—
16	C₃×C₅×#95	C₂²	1	4	yes	—
17	C₃×C₅×#95	C₂²	1	4	yes	—
18	C₃×#2646	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
19	C₅×#400	C₂×C₃	1	4	yes	complemented
20	C₅×#398	C₂×C₃	1	4	yes	complemented
21	C₅×#385	C₂×C₃	1	4	yes	—
22	C₅×#380	C₂×C₃	1	2	yes	—
23	C₂×C₅×#95	C₂×C₃	1	1	yes	characteristic
24	C₂⁴×C₃×C₅	C₂³	1	1	yes	characteristic
25	C₂²×C₃×C₄×C₅	—	2	4	no	—
26	C₂²×C₃×C₄×C₅	—	4	2	no	—
27	C₂²×C₃×C₄×C₅	—	4	2	no	—
28	C₂²×C₃×C₄×C₅	—	4	2	no	—
29	C₂²×C₃×C₄×C₅	Q₈	1	4	yes	—
30	C₂²×C₃×C₄×C₅	C₂³	1	4	yes	—
31	C₂²×C₃×C₄×C₅	C₂³	1	4	yes	—
32	C₂²×C₃×C₄×C₅	C₂³	1	4	yes	—
33	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
34	C₃×C₅×C₄⋊C₄	—	4	2	no	—
35	C₃×C₅×C₄⋊C₄	—	4	2	no	—
36	C₃×C₅×C₄⋊C₄	—	4	4	no	—
37	C₃×C₅×C₄⋊C₄	—	4	4	no	—
38	C₃×C₅×C₂²⋊C₄	—	2	8	no	—
39	C₃×#400	C₂×C₅	1	4	yes	complemented
40	C₃×#398	C₂×C₅	1	4	yes	complemented
41	C₃×#385	C₂×C₅	1	4	yes	—
42	C₃×#380	C₂×C₅	1	2	yes	—
43	C₂×C₃×#95	C₂×C₅	1	1	yes	characteristic
44	C₂³×C₄×C₅	C₂²×C₃	1	2	yes	—
45	C₂³×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
46	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
47	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
48	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	4	yes	—
49	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	4	yes	—
50	C₂×C₅×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
51	C₅×#95	C₂²×C₃	1	4	yes	—
52	C₅×#95	C₂²×C₃	1	4	yes	—
53	C₅×#95	C₂²×C₃	1	4	yes	—
54	C₅×#95	C₂²×C₃	1	4	yes	—
55	#2646	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
56	C₂³×C₃×C₅	—	2	1	no	—
57	C₂³×C₃×C₅	—	2	2	no	—
58	C₂³×C₃×C₅	—	2	1	no	—
59	C₂³×C₃×C₅	C₂×Q₈	1	2	yes	—
60	C₂³×C₃×C₅	#41	1	4	yes	—
61	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
62	C₂×C₃×C₄×C₅	—	2	2	no	—
63	C₂×C₃×C₄×C₅	—	2	2	no	—
64	C₂×C₃×C₄×C₅	—	2	4	no	—
65	C₂×C₃×C₄×C₅	—	2	4	no	—
66	C₂×C₃×C₄×C₅	—	2	4	no	—
67	C₂×C₃×C₄×C₅	—	2	4	no	—
68	C₂×C₃×C₄×C₅	—	2	4	no	—
69	C₂×C₃×C₄×C₅	—	2	4	no	—
70	C₂×C₃×C₄×C₅	—	2	4	no	—
71	C₂×C₃×C₄×C₅	—	2	4	no	—
72	C₂×C₃×C₄×C₅	—	2	4	no	—
73	C₂×C₃×C₄×C₅	—	4	4	no	—
74	C₂×C₃×C₄×C₅	—	4	8	no	—
75	C₂×C₃×C₄×C₅	C₂×Q₈	1	4	yes	—
76	C₂³×C₃×C₄	C₂²×C₅	1	2	yes	—
77	C₂³×C₃×C₄	C₂²×C₅	1	1	yes	characteristic
78	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
79	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
80	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	4	yes	—
81	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	4	yes	—
82	C₂×C₃×C₂²⋊C₄	C₂²×C₅	1	4	yes	—
83	C₃×#95	C₂²×C₅	1	4	yes	—
84	C₃×#95	C₂²×C₅	1	4	yes	—
85	C₃×#95	C₂²×C₅	1	4	yes	—
86	C₃×#95	C₂²×C₅	1	4	yes	—
87	C₂⁴×C₅	C₂³×C₃	1	1	yes	characteristic
88	C₂²×C₄×C₅	—	2	4	no	—
89	C₂²×C₄×C₅	—	4	2	no	—
90	C₂²×C₄×C₅	—	4	2	no	—
91	C₂²×C₄×C₅	—	4	2	no	—
92	C₂²×C₄×C₅	C₃×Q₈	1	4	yes	—
93	C₂²×C₄×C₅	C₂³×C₃	1	4	yes	—
94	C₂²×C₄×C₅	C₂³×C₃	1	4	yes	—
95	C₂²×C₄×C₅	C₂³×C₃	1	4	yes	—
96	C₂²×C₄×C₅	C₂³×C₃	1	2	yes	—
97	C₅×C₄⋊C₄	—	4	2	no	—
98	C₅×C₄⋊C₄	—	4	2	no	—
99	C₅×C₄⋊C₄	—	4	4	no	—
100	C₅×C₄⋊C₄	—	4	4	no	—
101	C₅×C₂²⋊C₄	—	2	8	no	—
102	#400	C₂×C₃×C₅	1	4	yes	complemented
103	#398	C₂×C₃×C₅	1	4	yes	complemented
104	#385	C₂×C₃×C₅	1	4	yes	—
105	#380	C₂×C₃×C₅	1	2	yes	—
106	C₂×#95	C₂×C₃×C₅	1	1	yes	characteristic
107	C₂²×C₃×C₅	—	2	4	no	—
108	C₂²×C₃×C₅	—	2	4	no	—
109	C₂²×C₃×C₅	—	2	2	no	—
110	C₂²×C₃×C₅	—	2	2	no	—
111	C₂²×C₃×C₅	#125	1	4	yes	—
112	C₂²×C₃×C₅	C₂²×Q₈	1	1	yes	characteristic
113	C₂²×C₃×C₅	C₂×#41	1	2	yes	—
114	C₂²×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
115	C₂²×C₃×C₅	H(5,𝔽₂)	1	2	yes	—
116	C₂²×C₃×C₅	2₋¹⁺⁴	1	1	yes	characteristic
117	C₃×C₄×C₅	—	2	4	no	—
118	C₃×C₄×C₅	—	4	4	no	—
119	C₃×C₄×C₅	—	4	4	no	—
120	C₃×C₄×C₅	—	4	4	no	—
121	C₂⁴×C₃	C₂³×C₅	1	1	yes	characteristic
122	C₂²×C₃×C₄	—	2	4	no	—
123	C₂²×C₃×C₄	—	4	2	no	—
124	C₂²×C₃×C₄	—	4	2	no	—
125	C₂²×C₃×C₄	—	4	2	no	—
126	C₂²×C₃×C₄	C₅×Q₈	1	4	yes	—
127	C₂²×C₃×C₄	C₂³×C₅	1	4	yes	—
128	C₂²×C₃×C₄	C₂³×C₅	1	4	yes	—
129	C₂²×C₃×C₄	C₂³×C₅	1	4	yes	—
130	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
131	C₃×C₄⋊C₄	—	4	2	no	—
132	C₃×C₄⋊C₄	—	4	2	no	—
133	C₃×C₄⋊C₄	—	4	4	no	—
134	C₃×C₄⋊C₄	—	4	4	no	—
135	C₃×C₂²⋊C₄	—	2	8	no	—
136	C₂³×C₅	—	2	1	no	—
137	C₂³×C₅	—	2	2	no	—
138	C₂³×C₅	—	2	1	no	—
139	C₂³×C₅	C₂×C₃×Q₈	1	2	yes	—
140	C₂³×C₅	C₃×#41	1	4	yes	—
141	C₂³×C₅	C₂⁴×C₃	1	1	yes	characteristic
142	C₂×C₄×C₅	—	2	2	no	—
143	C₂×C₄×C₅	—	2	2	no	—
144	C₂×C₄×C₅	—	2	4	no	—
145	C₂×C₄×C₅	—	2	4	no	—
146	C₂×C₄×C₅	—	2	4	no	—
147	C₂×C₄×C₅	—	2	4	no	—
148	C₂×C₄×C₅	—	2	4	no	—
149	C₂×C₄×C₅	—	2	4	no	—
150	C₂×C₄×C₅	—	2	4	no	—
151	C₂×C₄×C₅	—	2	4	no	—
152	C₂×C₄×C₅	—	2	4	no	—
153	C₂×C₄×C₅	—	4	4	no	—
154	C₂×C₄×C₅	—	4	8	no	—
155	C₂×C₄×C₅	C₂×C₃×Q₈	1	4	yes	—
156	C₂³×C₄	C₂²×C₃×C₅	1	2	yes	—
157	C₂³×C₄	C₂²×C₃×C₅	1	1	yes	characteristic
158	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	2	yes	—
159	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	2	yes	—
160	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	4	yes	—
161	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	4	yes	—
162	C₂×C₂²⋊C₄	C₂²×C₃×C₅	1	4	yes	—
163	#95	C₂²×C₃×C₅	1	4	yes	—
164	#95	C₂²×C₃×C₅	1	4	yes	—
165	#95	C₂²×C₃×C₅	1	4	yes	—
166	#95	C₂²×C₃×C₅	1	4	yes	—
167	C₂×C₃×C₅	—	2	4	no	—
168	C₂×C₃×C₅	C₂×#125	1	1	yes	characteristic
169	C₂×C₃×C₅	#538	1	2	yes	—
170	C₂×C₃×C₅	#539	1	2	yes	—
171	C₂×C₃×C₅	#543	1	1	yes	characteristic
172	C₂×C₃×C₅	#544	1	1	yes	characteristic
173	C₂³×C₃	—	2	1	no	—
174	C₂³×C₃	—	2	2	no	—
175	C₂³×C₃	—	2	1	no	—
176	C₂³×C₃	C₂×C₅×Q₈	1	2	yes	—
177	C₂³×C₃	C₅×#41	1	4	yes	—
178	C₂³×C₃	C₂⁴×C₅	1	1	yes	characteristic
179	C₂×C₃×C₄	—	2	2	no	—
180	C₂×C₃×C₄	—	2	2	no	—
181	C₂×C₃×C₄	—	2	4	no	—
182	C₂×C₃×C₄	—	2	4	no	—
183	C₂×C₃×C₄	—	2	4	no	—
184	C₂×C₃×C₄	—	2	4	no	—
185	C₂×C₃×C₄	—	2	4	no	—
186	C₂×C₃×C₄	—	2	4	no	—
187	C₂×C₃×C₄	—	2	4	no	—
188	C₂×C₃×C₄	—	2	4	no	—
189	C₂×C₃×C₄	—	2	4	no	—
190	C₂×C₃×C₄	—	4	4	no	—
191	C₂×C₃×C₄	—	4	8	no	—
192	C₂×C₃×C₄	C₂×C₅×Q₈	1	4	yes	—
193	C₂²×C₅	—	2	4	no	—
194	C₂²×C₅	—	2	4	no	—
195	C₂²×C₅	—	2	2	no	—
196	C₂²×C₅	—	2	2	no	—
197	C₂²×C₅	C₃×#125	1	4	yes	—
198	C₂²×C₅	C₂²×C₃×Q₈	1	1	yes	characteristic
199	C₂²×C₅	C₂×C₃×#41	1	2	yes	—
200	C₂²×C₅	C₃×H(5,𝔽₂)	1	1	yes	characteristic
201	C₂²×C₅	C₃×H(5,𝔽₂)	1	2	yes	—
202	C₂²×C₅	C₃×2₋¹⁺⁴	1	1	yes	characteristic
203	C₄×C₅	—	2	4	no	—
204	C₄×C₅	—	4	4	no	—
205	C₄×C₅	—	4	4	no	—
206	C₄×C₅	—	4	4	no	—
207	C₂⁴	C₂³×C₃×C₅	1	1	yes	characteristic
208	C₂²×C₄	—	2	4	no	—
209	C₂²×C₄	—	4	2	no	—
210	C₂²×C₄	—	4	2	no	—
211	C₂²×C₄	—	4	2	no	—
212	C₂²×C₄	C₃×C₅×Q₈	1	4	yes	—
213	C₂²×C₄	C₂³×C₃×C₅	1	4	yes	—
214	C₂²×C₄	C₂³×C₃×C₅	1	4	yes	—
215	C₂²×C₄	C₂³×C₃×C₅	1	4	yes	—
216	C₂²×C₄	C₂³×C₃×C₅	1	2	yes	—
217	C₄⋊C₄	—	4	2	no	—
218	C₄⋊C₄	—	4	2	no	—
219	C₄⋊C₄	—	4	4	no	—
220	C₄⋊C₄	—	4	4	no	—
221	C₂²⋊C₄	—	2	8	no	—
222	C₃×C₅	#2646	1	1	yes	characteristic, direct product factor, Hall
223	C₂²×C₃	—	2	4	no	—
224	C₂²×C₃	—	2	4	no	—
225	C₂²×C₃	—	2	2	no	—
226	C₂²×C₃	—	2	2	no	—
227	C₂²×C₃	C₅×#125	1	4	yes	—
228	C₂²×C₃	C₂²×C₅×Q₈	1	1	yes	characteristic
229	C₂²×C₃	C₂×C₅×#41	1	2	yes	—
230	C₂²×C₃	C₅×H(5,𝔽₂)	1	1	yes	characteristic
231	C₂²×C₃	C₅×H(5,𝔽₂)	1	2	yes	—
232	C₂²×C₃	C₅×2₋¹⁺⁴	1	1	yes	characteristic
233	C₃×C₄	—	2	4	no	—
234	C₃×C₄	—	4	4	no	—
235	C₃×C₄	—	4	4	no	—
236	C₃×C₄	—	4	4	no	—
237	C₂×C₅	—	2	4	no	—
238	C₂×C₅	C₂×C₃×#125	1	1	yes	characteristic
239	C₂×C₅	C₃×#538	1	2	yes	—
240	C₂×C₅	C₃×#539	1	2	yes	—
241	C₂×C₅	C₃×#543	1	1	yes	characteristic
242	C₂×C₅	C₃×#544	1	1	yes	characteristic
243	C₂³	—	2	1	no	—
244	C₂³	—	2	2	no	—
245	C₂³	—	2	1	no	—
246	C₂³	C₂×C₃×C₅×Q₈	1	2	yes	—
247	C₂³	C₃×C₅×#41	1	4	yes	—
248	C₂³	C₂⁴×C₃×C₅	1	1	yes	characteristic
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	2	4	no	—
260	C₂×C₄	—	4	4	no	—
261	C₂×C₄	—	4	8	no	—
262	C₂×C₄	C₂×C₃×C₅×Q₈	1	4	yes	—
263	C₂×C₃	—	2	4	no	—
264	C₂×C₃	C₂×C₅×#125	1	1	yes	characteristic
265	C₂×C₃	C₅×#538	1	2	yes	—
266	C₂×C₃	C₅×#539	1	2	yes	—
267	C₂×C₃	C₅×#543	1	1	yes	characteristic
268	C₂×C₃	C₅×#544	1	1	yes	characteristic
269	C₅	C₃×#2646	1	1	yes	characteristic, direct product factor, Sylow
270	C₂²	—	2	4	no	—
271	C₂²	—	2	4	no	—
272	C₂²	—	2	2	no	—
273	C₂²	—	2	2	no	—
274	C₂²	C₃×C₅×#125	1	4	yes	—
275	C₂²	C₂²×C₃×C₅×Q₈	1	1	yes	characteristic
276	C₂²	C₂×C₃×C₅×#41	1	2	yes	—
277	C₂²	C₃×C₅×H(5,𝔽₂)	1	1	yes	characteristic
278	C₂²	C₃×C₅×H(5,𝔽₂)	1	2	yes	—
279	C₂²	C₃×C₅×2₋¹⁺⁴	1	1	yes	characteristic
280	C₄	—	2	4	no	—
281	C₄	—	4	4	no	—
282	C₄	—	4	4	no	—
283	C₄	—	4	4	no	—
284	C₃	C₅×#2646	1	1	yes	characteristic, direct product factor, Sylow
285	C₂	—	2	4	no	—
286	C₂	C₂×C₃×C₅×#125	1	1	yes	characteristic
287	C₂	C₃×C₅×#538	1	2	yes	—
288	C₂	C₃×C₅×#539	1	2	yes	—
289	C₂	C₃×C₅×#543	1	1	yes	characteristic
290	C₂	C₃×C₅×#544	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃×C₅	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₃×C₅
prime radical	C₂⁴×C₃×C₅	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₃×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×#400	C₂	semidirect
2. C₃×C₅×#398	C₂	semidirect
6. C₅×#2646	C₃	direct (N × H)
18. C₃×#2646	C₅	direct (N × H)
19. C₅×#400	C₂×C₃	semidirect
20. C₅×#398	C₂×C₃	semidirect
39. C₃×#400	C₂×C₅	semidirect
40. C₃×#398	C₂×C₅	semidirect
55. #2646	C₃×C₅	direct (N × H)
102. #400	C₂×C₃×C₅	semidirect
103. #398	C₂×C₃×C₅	semidirect
222. C₃×C₅	#2646	direct (N × H)
269. C₅	C₃×#2646	direct (N × H)
284. C₃	C₅×#2646	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×#400	2
C₃×C₅×#398	2
C₃×C₅×#385	2
C₃×C₅×#380	2
C₂×C₃×C₅×#95	2
C₅×#2646	3
C₃×#2646	5

Subgroups — C3×C5×#2646 #14275680

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#2646 #14275680