Subgroups — C₃×C₅×#2383 #14275417

Order 1920.

912 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	C₂×C₃×C₅×#126	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×#389	C₂	1	3	yes	maximal, complemented
3	C₃×C₅×#388	C₂	1	3	yes	maximal
4	C₃×C₅×#387	C₂	1	3	yes	maximal, complemented
5	C₃×C₅×#385	C₂	1	3	yes	maximal
6	C₃×C₅×#384	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×C₅×#383	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#2383	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂³×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
10	C₃×C₅×#126	C₄	1	8	yes	complemented
11	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	3	yes	—
12	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	3	yes	—
13	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	3	yes	—
14	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	3	yes	—
15	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	3	yes	—
16	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	3	yes	—
17	C₂×C₃×C₄²×C₅	C₂²	1	3	yes	—
18	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
19	C₃×C₅×#95	C₂²	1	3	yes	—
20	C₃×C₅×#95	C₂²	1	3	yes	—
21	C₃×C₅×#95	C₂²	1	3	yes	—
22	C₃×C₅×#95	C₂²	1	3	yes	—
23	C₃×#2383	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
24	C₂×C₅×#126	C₂×C₃	1	1	yes	characteristic
25	C₅×#389	C₂×C₃	1	3	yes	complemented
26	C₅×#388	C₂×C₃	1	3	yes	—
27	C₅×#387	C₂×C₃	1	3	yes	complemented
28	C₅×#385	C₂×C₃	1	3	yes	—
29	C₅×#384	C₂×C₃	1	1	yes	characteristic, complemented
30	C₅×#383	C₂×C₃	1	1	yes	characteristic, complemented
31	C₂⁴×C₃×C₅	C₂³	1	1	yes	characteristic
32	C₂²×C₃×C₄×C₅	—	4	3	no	—
33	C₂²×C₃×C₄×C₅	C₂³	1	3	yes	—
34	C₂²×C₃×C₄×C₅	C₂³	1	3	yes	—
35	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
36	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
37	C₂²×C₃×C₄×C₅	C₂³	1	3	yes	—
38	C₂²×C₃×C₄×C₅	C₂³	1	3	yes	—
39	C₃×C₅×C₄⋊C₄	—	4	3	no	—
40	C₃×C₅×C₄⋊C₄	—	4	3	no	—
41	C₃×C₅×C₄⋊C₄	—	4	3	no	—
42	C₃×C₅×C₄⋊C₄	C₂×C₄	1	12	yes	complemented
43	C₃×C₅×C₂²⋊C₄	—	4	3	no	—
44	C₃×C₅×C₂²⋊C₄	C₂×C₄	1	12	yes	—
45	C₃×C₄²×C₅	—	4	3	no	—
46	C₃×C₄²×C₅	C₂×C₄	1	4	yes	complemented
47	C₂×C₃×#126	C₂×C₅	1	1	yes	characteristic
48	C₃×#389	C₂×C₅	1	3	yes	complemented
49	C₃×#388	C₂×C₅	1	3	yes	—
50	C₃×#387	C₂×C₅	1	3	yes	complemented
51	C₃×#385	C₂×C₅	1	3	yes	—
52	C₃×#384	C₂×C₅	1	1	yes	characteristic, complemented
53	C₃×#383	C₂×C₅	1	1	yes	characteristic, complemented
54	C₂³×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
55	C₅×#126	C₃×C₄	1	8	yes	complemented
56	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	3	yes	—
57	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	3	yes	—
58	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	3	yes	—
59	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	3	yes	—
60	C₂×C₅×C₂²⋊C₄	C₂²×C₃	1	3	yes	—
61	C₂×C₅×C₂²⋊C₄	C₂²×C₃	1	3	yes	—
62	C₂×C₄²×C₅	C₂²×C₃	1	3	yes	—
63	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
64	C₅×#95	C₂²×C₃	1	3	yes	—
65	C₅×#95	C₂²×C₃	1	3	yes	—
66	C₅×#95	C₂²×C₃	1	3	yes	—
67	C₅×#95	C₂²×C₃	1	3	yes	—
68	#2383	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
69	C₂³×C₃×C₅	—	2	3	no	—
70	C₂³×C₃×C₅	—	2	3	no	—
71	C₂³×C₃×C₅	C₂²×C₄	1	2	yes	—
72	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
73	C₂×C₃×C₄×C₅	—	2	3	no	—
74	C₂×C₃×C₄×C₅	—	2	3	no	—
75	C₂×C₃×C₄×C₅	—	2	3	no	—
76	C₂×C₃×C₄×C₅	—	2	3	no	—
77	C₂×C₃×C₄×C₅	—	2	3	no	—
78	C₂×C₃×C₄×C₅	—	2	3	no	—
79	C₂×C₃×C₄×C₅	—	2	3	no	—
80	C₂×C₃×C₄×C₅	—	2	3	no	—
81	C₂×C₃×C₄×C₅	—	2	3	no	—
82	C₂×C₃×C₄×C₅	—	2	3	no	—
83	C₂×C₃×C₄×C₅	—	2	6	no	—
84	C₂×C₃×C₄×C₅	—	4	4	no	—
85	C₂×C₃×C₄×C₅	C₂²×C₄	1	6	yes	—
86	C₂×C₃×C₄×C₅	C₂²×C₄	1	6	yes	—
87	C₂³×C₃×C₄	C₂²×C₅	1	1	yes	characteristic
88	C₃×#126	C₄×C₅	1	8	yes	complemented
89	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	3	yes	—
90	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	3	yes	—
91	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	3	yes	—
92	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	3	yes	—
93	C₂×C₃×C₂²⋊C₄	C₂²×C₅	1	3	yes	—
94	C₂×C₃×C₂²⋊C₄	C₂²×C₅	1	3	yes	—
95	C₂×C₃×C₄²	C₂²×C₅	1	3	yes	—
96	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
97	C₃×#95	C₂²×C₅	1	3	yes	—
98	C₃×#95	C₂²×C₅	1	3	yes	—
99	C₃×#95	C₂²×C₅	1	3	yes	—
100	C₃×#95	C₂²×C₅	1	3	yes	—
101	C₂⁴×C₅	C₂³×C₃	1	1	yes	characteristic
102	C₂²×C₄×C₅	—	4	3	no	—
103	C₂²×C₄×C₅	C₂³×C₃	1	3	yes	—
104	C₂²×C₄×C₅	C₂³×C₃	1	3	yes	—
105	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
106	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
107	C₂²×C₄×C₅	C₂³×C₃	1	3	yes	—
108	C₂²×C₄×C₅	C₂³×C₃	1	3	yes	—
109	C₅×C₄⋊C₄	—	4	3	no	—
110	C₅×C₄⋊C₄	—	4	3	no	—
111	C₅×C₄⋊C₄	—	4	3	no	—
112	C₅×C₄⋊C₄	C₂×C₃×C₄	1	12	yes	complemented
113	C₅×C₂²⋊C₄	—	4	3	no	—
114	C₅×C₂²⋊C₄	C₂×C₃×C₄	1	12	yes	—
115	C₄²×C₅	—	4	3	no	—
116	C₄²×C₅	C₂×C₃×C₄	1	4	yes	complemented
117	C₂×#126	C₂×C₃×C₅	1	1	yes	characteristic
118	#389	C₂×C₃×C₅	1	3	yes	complemented
119	#388	C₂×C₃×C₅	1	3	yes	—
120	#387	C₂×C₃×C₅	1	3	yes	complemented
121	#385	C₂×C₃×C₅	1	3	yes	—
122	#384	C₂×C₃×C₅	1	1	yes	characteristic, complemented
123	#383	C₂×C₃×C₅	1	1	yes	characteristic, complemented
124	C₂²×C₃×C₅	—	2	6	no	—
125	C₂²×C₃×C₅	—	4	1	no	—
126	C₂²×C₃×C₅	—	4	3	no	—
127	C₂²×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
128	C₂²×C₃×C₅	H(5,𝔽₂)	1	3	yes	—
129	C₂²×C₃×C₅	2₋¹⁺⁴	1	3	yes	—
130	C₃×C₄×C₅	—	2	6	no	—
131	C₃×C₄×C₅	—	2	6	no	—
132	C₃×C₄×C₅	—	4	3	no	—
133	C₃×C₄×C₅	—	4	1	no	—
134	C₃×C₄×C₅	—	4	1	no	—
135	C₃×C₄×C₅	—	4	3	no	—
136	C₂⁴×C₃	C₂³×C₅	1	1	yes	characteristic
137	C₂²×C₃×C₄	—	4	3	no	—
138	C₂²×C₃×C₄	C₂³×C₅	1	3	yes	—
139	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
140	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
141	C₂²×C₃×C₄	C₂³×C₅	1	3	yes	—
142	C₂²×C₃×C₄	C₂³×C₅	1	3	yes	—
143	C₂²×C₃×C₄	C₂³×C₅	1	3	yes	—
144	C₃×C₄⋊C₄	—	4	3	no	—
145	C₃×C₄⋊C₄	—	4	3	no	—
146	C₃×C₄⋊C₄	—	4	3	no	—
147	C₃×C₄⋊C₄	C₂×C₄×C₅	1	12	yes	complemented
148	C₃×C₂²⋊C₄	—	4	3	no	—
149	C₃×C₂²⋊C₄	C₂×C₄×C₅	1	12	yes	—
150	C₃×C₄²	—	4	3	no	—
151	C₃×C₄²	C₂×C₄×C₅	1	4	yes	complemented
152	C₂³×C₅	—	2	3	no	—
153	C₂³×C₅	—	2	3	no	—
154	C₂³×C₅	C₂²×C₃×C₄	1	2	yes	—
155	C₂³×C₅	C₂⁴×C₃	1	1	yes	characteristic
156	C₂×C₄×C₅	—	2	3	no	—
157	C₂×C₄×C₅	—	2	3	no	—
158	C₂×C₄×C₅	—	2	3	no	—
159	C₂×C₄×C₅	—	2	3	no	—
160	C₂×C₄×C₅	—	2	3	no	—
161	C₂×C₄×C₅	—	2	3	no	—
162	C₂×C₄×C₅	—	2	3	no	—
163	C₂×C₄×C₅	—	2	3	no	—
164	C₂×C₄×C₅	—	2	3	no	—
165	C₂×C₄×C₅	—	2	3	no	—
166	C₂×C₄×C₅	—	2	6	no	—
167	C₂×C₄×C₅	—	4	4	no	—
168	C₂×C₄×C₅	C₂²×C₃×C₄	1	6	yes	—
169	C₂×C₄×C₅	C₂²×C₃×C₄	1	6	yes	—
170	C₂³×C₄	C₂²×C₃×C₅	1	1	yes	characteristic
171	#126	C₃×C₄×C₅	1	8	yes	complemented
172	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	3	yes	—
173	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	3	yes	—
174	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	3	yes	—
175	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	3	yes	—
176	C₂×C₂²⋊C₄	C₂²×C₃×C₅	1	3	yes	—
177	C₂×C₂²⋊C₄	C₂²×C₃×C₅	1	3	yes	—
178	C₂×C₄²	C₂²×C₃×C₅	1	3	yes	—
179	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
180	#95	C₂²×C₃×C₅	1	3	yes	—
181	#95	C₂²×C₃×C₅	1	3	yes	—
182	#95	C₂²×C₃×C₅	1	3	yes	—
183	#95	C₂²×C₃×C₅	1	3	yes	—
184	C₂×C₃×C₅	—	4	2	no	—
185	C₂×C₃×C₅	#520	1	3	yes	—
186	C₂×C₃×C₅	#560	1	1	yes	characteristic
187	C₂×C₃×C₅	#563	1	3	yes	—
188	C₂³×C₃	—	2	3	no	—
189	C₂³×C₃	—	2	3	no	—
190	C₂³×C₃	C₂²×C₄×C₅	1	2	yes	—
191	C₂³×C₃	C₂⁴×C₅	1	1	yes	characteristic
192	C₂×C₃×C₄	—	2	3	no	—
193	C₂×C₃×C₄	—	2	3	no	—
194	C₂×C₃×C₄	—	2	3	no	—
195	C₂×C₃×C₄	—	2	3	no	—
196	C₂×C₃×C₄	—	2	3	no	—
197	C₂×C₃×C₄	—	2	3	no	—
198	C₂×C₃×C₄	—	2	3	no	—
199	C₂×C₃×C₄	—	2	3	no	—
200	C₂×C₃×C₄	—	2	3	no	—
201	C₂×C₃×C₄	—	2	3	no	—
202	C₂×C₃×C₄	—	2	6	no	—
203	C₂×C₃×C₄	—	4	4	no	—
204	C₂×C₃×C₄	C₂²×C₄×C₅	1	6	yes	—
205	C₂×C₃×C₄	C₂²×C₄×C₅	1	6	yes	—
206	C₂²×C₅	—	2	6	no	—
207	C₂²×C₅	—	4	1	no	—
208	C₂²×C₅	—	4	3	no	—
209	C₂²×C₅	C₂³×C₃×C₄	1	1	yes	characteristic
210	C₂²×C₅	C₃×H(5,𝔽₂)	1	3	yes	—
211	C₂²×C₅	C₃×2₋¹⁺⁴	1	3	yes	—
212	C₄×C₅	—	2	6	no	—
213	C₄×C₅	—	2	6	no	—
214	C₄×C₅	—	4	3	no	—
215	C₄×C₅	—	4	1	no	—
216	C₄×C₅	—	4	1	no	—
217	C₄×C₅	—	4	3	no	—
218	C₂⁴	C₂³×C₃×C₅	1	1	yes	characteristic
219	C₂²×C₄	—	4	3	no	—
220	C₂²×C₄	C₂³×C₃×C₅	1	3	yes	—
221	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
222	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
223	C₂²×C₄	C₂³×C₃×C₅	1	3	yes	—
224	C₂²×C₄	C₂³×C₃×C₅	1	3	yes	—
225	C₂²×C₄	C₂³×C₃×C₅	1	3	yes	—
226	C₄⋊C₄	—	4	3	no	—
227	C₄⋊C₄	—	4	3	no	—
228	C₄⋊C₄	—	4	3	no	—
229	C₄⋊C₄	C₂×C₃×C₄×C₅	1	12	yes	complemented
230	C₂²⋊C₄	—	4	3	no	—
231	C₂²⋊C₄	C₂×C₃×C₄×C₅	1	12	yes	—
232	C₄²	—	4	3	no	—
233	C₄²	C₂×C₃×C₄×C₅	1	4	yes	complemented
234	C₃×C₅	#2383	1	1	yes	characteristic, direct product factor, Hall
235	C₂²×C₃	—	2	6	no	—
236	C₂²×C₃	—	4	1	no	—
237	C₂²×C₃	—	4	3	no	—
238	C₂²×C₃	C₂³×C₄×C₅	1	1	yes	characteristic
239	C₂²×C₃	C₅×H(5,𝔽₂)	1	3	yes	—
240	C₂²×C₃	C₅×2₋¹⁺⁴	1	3	yes	—
241	C₃×C₄	—	2	6	no	—
242	C₃×C₄	—	2	6	no	—
243	C₃×C₄	—	4	3	no	—
244	C₃×C₄	—	4	1	no	—
245	C₃×C₄	—	4	1	no	—
246	C₃×C₄	—	4	3	no	—
247	C₂×C₅	—	4	2	no	—
248	C₂×C₅	C₃×#520	1	3	yes	—
249	C₂×C₅	C₃×#560	1	1	yes	characteristic
250	C₂×C₅	C₃×#563	1	3	yes	—
251	C₂³	—	2	3	no	—
252	C₂³	—	2	3	no	—
253	C₂³	C₂²×C₃×C₄×C₅	1	2	yes	—
254	C₂³	C₂⁴×C₃×C₅	1	1	yes	characteristic
255	C₂×C₄	—	2	3	no	—
256	C₂×C₄	—	2	3	no	—
257	C₂×C₄	—	2	3	no	—
258	C₂×C₄	—	2	3	no	—
259	C₂×C₄	—	2	3	no	—
260	C₂×C₄	—	2	3	no	—
261	C₂×C₄	—	2	3	no	—
262	C₂×C₄	—	2	3	no	—
263	C₂×C₄	—	2	3	no	—
264	C₂×C₄	—	2	3	no	—
265	C₂×C₄	—	2	6	no	—
266	C₂×C₄	—	4	4	no	—
267	C₂×C₄	C₂²×C₃×C₄×C₅	1	6	yes	—
268	C₂×C₄	C₂²×C₃×C₄×C₅	1	6	yes	—
269	C₂×C₃	—	4	2	no	—
270	C₂×C₃	C₅×#520	1	3	yes	—
271	C₂×C₃	C₅×#560	1	1	yes	characteristic
272	C₂×C₃	C₅×#563	1	3	yes	—
273	C₅	C₃×#2383	1	1	yes	characteristic, direct product factor, Sylow
274	C₂²	—	2	6	no	—
275	C₂²	—	4	1	no	—
276	C₂²	—	4	3	no	—
277	C₂²	C₂³×C₃×C₄×C₅	1	1	yes	characteristic
278	C₂²	C₃×C₅×H(5,𝔽₂)	1	3	yes	—
279	C₂²	C₃×C₅×2₋¹⁺⁴	1	3	yes	—
280	C₄	—	2	6	no	—
281	C₄	—	2	6	no	—
282	C₄	—	4	1	no	—
283	C₄	—	4	1	no	—
284	C₄	—	4	3	no	—
285	C₄	—	4	3	no	—
286	C₃	C₅×#2383	1	1	yes	characteristic, direct product factor, Sylow
287	C₂	—	4	2	no	—
288	C₂	C₃×C₅×#520	1	3	yes	—
289	C₂	C₃×C₅×#560	1	1	yes	characteristic
290	C₂	C₃×C₅×#563	1	3	yes	—

	subgroup	G/H
center, socle	C₂³×C₃×C₅	C₂⁴
commutator	C₂²	C₂³×C₃×C₄×C₅
Frattini, cosocle	C₂³	C₂⁴×C₃×C₅
prime radical	C₂⁴×C₃×C₅	C₂³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×C₅×#389	C₂	semidirect
4. C₃×C₅×#387	C₂	semidirect
6. C₃×C₅×#384	C₂	semidirect
7. C₃×C₅×#383	C₂	semidirect
8. C₅×#2383	C₃	direct (N × H)
10. C₃×C₅×#126	C₄	semidirect
23. C₃×#2383	C₅	direct (N × H)
25. C₅×#389	C₂×C₃	semidirect
27. C₅×#387	C₂×C₃	semidirect
29. C₅×#384	C₂×C₃	semidirect
30. C₅×#383	C₂×C₃	semidirect
42. C₃×C₅×C₄⋊C₄	C₂×C₄	semidirect
46. C₃×C₄²×C₅	C₂×C₄	semidirect
48. C₃×#389	C₂×C₅	semidirect
50. C₃×#387	C₂×C₅	semidirect
52. C₃×#384	C₂×C₅	semidirect
53. C₃×#383	C₂×C₅	semidirect
55. C₅×#126	C₃×C₄	semidirect
68. #2383	C₃×C₅	direct (N × H)
88. C₃×#126	C₄×C₅	semidirect
112. C₅×C₄⋊C₄	C₂×C₃×C₄	semidirect
116. C₄²×C₅	C₂×C₃×C₄	semidirect
118. #389	C₂×C₃×C₅	semidirect
120. #387	C₂×C₃×C₅	semidirect
122. #384	C₂×C₃×C₅	semidirect
123. #383	C₂×C₃×C₅	semidirect
147. C₃×C₄⋊C₄	C₂×C₄×C₅	semidirect
151. C₃×C₄²	C₂×C₄×C₅	semidirect
171. #126	C₃×C₄×C₅	semidirect
229. C₄⋊C₄	C₂×C₃×C₄×C₅	semidirect
233. C₄²	C₂×C₃×C₄×C₅	semidirect
234. C₃×C₅	#2383	direct (N × H)
273. C₅	C₃×#2383	direct (N × H)
286. C₃	C₅×#2383	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₃×C₅×#126	2
C₃×C₅×#389	2
C₃×C₅×#388	2
C₃×C₅×#387	2
C₃×C₅×#385	2
C₃×C₅×#384	2
C₃×C₅×#383	2
C₅×#2383	3
C₃×#2383	5

Subgroups — C3×C5×#2383 #14275417

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#2383 #14275417