Subgroups — C₃×C₅×#2315 #14275349

Order 1920.

1632 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₅×C₂²⋊C₄	C₂	1	2	yes	maximal
2	C₂²×C₃×C₅×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#385	C₂	1	8	yes	maximal, complemented
4	C₃×C₅×#380	C₂	1	4	yes	maximal, complemented
5	C₅×#2315	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂⁵×C₃×C₅	C₂²	1	1	yes	characteristic
7	C₂³×C₃×C₄×C₅	C₂²	1	4	yes	—
8	C₂³×C₃×C₄×C₅	C₂²	1	2	yes	—
9	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	8	yes	complemented
10	C₂×C₃×C₅×C₂²⋊C₄	—	2	8	no	—
11	C₂×C₃×C₅×C₂²⋊C₄	C₄	1	8	yes	complemented
12	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	4	yes	—
13	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	4	yes	—
14	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	4	yes	—
15	C₃×C₅×#95	C₂²	1	8	yes	complemented
16	C₃×#2315	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
17	C₂²×C₅×C₂²⋊C₄	C₂×C₃	1	2	yes	—
18	C₂²×C₅×C₂²⋊C₄	C₂×C₃	1	1	yes	characteristic
19	C₅×#385	C₂×C₃	1	8	yes	complemented
20	C₅×#380	C₂×C₃	1	4	yes	complemented
21	C₂⁴×C₃×C₅	—	4	4	no	—
22	C₂⁴×C₃×C₅	C₂×C₄	1	4	yes	—
23	C₂⁴×C₃×C₅	D₄	1	4	yes	—
24	C₂⁴×C₃×C₅	Q₈	1	4	yes	—
25	C₂⁴×C₃×C₅	C₂³	1	2	yes	—
26	C₂⁴×C₃×C₅	C₂³	1	1	yes	characteristic
27	C₂²×C₃×C₄×C₅	—	2	8	no	—
28	C₂²×C₃×C₄×C₅	—	4	4	no	—
29	C₂²×C₃×C₄×C₅	—	4	8	no	—
30	C₂²×C₃×C₄×C₅	C₂×C₄	1	8	yes	complemented
31	C₂²×C₃×C₄×C₅	C₂³	1	8	yes	—
32	C₂²×C₃×C₄×C₅	C₂³	1	4	yes	—
33	C₃×C₅×C₄⋊C₄	—	4	8	no	—
34	C₃×C₅×C₂²⋊C₄	—	2	8	no	—
35	C₃×C₅×C₂²⋊C₄	—	2	8	no	—
36	C₃×C₅×C₂²⋊C₄	C₂×C₄	1	16	yes	complemented
37	C₂²×C₃×C₂²⋊C₄	C₂×C₅	1	2	yes	—
38	C₂²×C₃×C₂²⋊C₄	C₂×C₅	1	1	yes	characteristic
39	C₃×#385	C₂×C₅	1	8	yes	complemented
40	C₃×#380	C₂×C₅	1	4	yes	complemented
41	C₂⁵×C₅	C₂²×C₃	1	1	yes	characteristic
42	C₂³×C₄×C₅	C₂²×C₃	1	4	yes	—
43	C₂³×C₄×C₅	C₂²×C₃	1	2	yes	—
44	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	8	yes	complemented
45	C₂×C₅×C₂²⋊C₄	—	2	8	no	—
46	C₂×C₅×C₂²⋊C₄	C₃×C₄	1	8	yes	complemented
47	C₂×C₅×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
48	C₂×C₅×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
49	C₂×C₅×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
50	C₅×#95	C₂²×C₃	1	8	yes	complemented
51	#2315	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
52	C₂³×C₃×C₅	—	2	4	no	—
53	C₂³×C₃×C₅	—	2	8	no	—
54	C₂³×C₃×C₅	—	4	8	no	—
55	C₂³×C₃×C₅	—	4	8	no	—
56	C₂³×C₃×C₅	—	4	8	no	—
57	C₂³×C₃×C₅	C₄⋊C₄	1	16	yes	complemented
58	C₂³×C₃×C₅	C₂²×C₄	1	4	yes	—
59	C₂³×C₃×C₅	C₂²×C₄	1	2	yes	—
60	C₂³×C₃×C₅	C₂×D₄	1	2	yes	—
61	C₂³×C₃×C₅	C₂×D₄	1	4	yes	—
62	C₂³×C₃×C₅	C₂×Q₈	1	4	yes	—
63	C₂³×C₃×C₅	C₂×Q₈	1	2	yes	—
64	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
65	C₂×C₃×C₄×C₅	—	2	8	no	—
66	C₂×C₃×C₄×C₅	—	2	8	no	—
67	C₂×C₃×C₄×C₅	—	2	16	no	—
68	C₂×C₃×C₄×C₅	—	4	8	no	—
69	C₂×C₃×C₄×C₅	—	4	16	no	—
70	C₂×C₃×C₄×C₅	C₂²×C₄	1	8	yes	—
71	C₂⁵×C₃	C₂²×C₅	1	1	yes	characteristic
72	C₂³×C₃×C₄	C₂²×C₅	1	4	yes	—
73	C₂³×C₃×C₄	C₂²×C₅	1	2	yes	—
74	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	8	yes	complemented
75	C₂×C₃×C₂²⋊C₄	—	2	8	no	—
76	C₂×C₃×C₂²⋊C₄	C₄×C₅	1	8	yes	complemented
77	C₂×C₃×C₂²⋊C₄	C₂²×C₅	1	4	yes	—
78	C₂×C₃×C₂²⋊C₄	C₂²×C₅	1	4	yes	—
79	C₂×C₃×C₂²⋊C₄	C₂²×C₅	1	4	yes	—
80	C₃×#95	C₂²×C₅	1	8	yes	complemented
81	C₂⁴×C₅	—	4	4	no	—
82	C₂⁴×C₅	C₂×C₃×C₄	1	4	yes	—
83	C₂⁴×C₅	C₃×D₄	1	4	yes	—
84	C₂⁴×C₅	C₃×Q₈	1	4	yes	—
85	C₂⁴×C₅	C₂³×C₃	1	2	yes	—
86	C₂⁴×C₅	C₂³×C₃	1	1	yes	characteristic
87	C₂²×C₄×C₅	—	2	8	no	—
88	C₂²×C₄×C₅	—	4	4	no	—
89	C₂²×C₄×C₅	—	4	8	no	—
90	C₂²×C₄×C₅	C₂×C₃×C₄	1	8	yes	complemented
91	C₂²×C₄×C₅	C₂³×C₃	1	8	yes	—
92	C₂²×C₄×C₅	C₂³×C₃	1	4	yes	—
93	C₅×C₄⋊C₄	—	4	8	no	—
94	C₅×C₂²⋊C₄	—	2	8	no	—
95	C₅×C₂²⋊C₄	—	2	8	no	—
96	C₅×C₂²⋊C₄	C₂×C₃×C₄	1	16	yes	complemented
97	C₂²×C₂²⋊C₄	C₂×C₃×C₅	1	2	yes	—
98	C₂²×C₂²⋊C₄	C₂×C₃×C₅	1	1	yes	characteristic
99	#385	C₂×C₃×C₅	1	8	yes	complemented
100	#380	C₂×C₃×C₅	1	4	yes	complemented
101	C₂²×C₃×C₅	—	2	4	no	—
102	C₂²×C₃×C₅	—	2	8	no	—
103	C₂²×C₃×C₅	—	2	8	no	—
104	C₂²×C₃×C₅	—	2	4	no	—
105	C₂²×C₃×C₅	—	2	4	no	—
106	C₂²×C₃×C₅	—	2	8	no	—
107	C₂²×C₃×C₅	—	4	16	no	—
108	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	8	yes	—
109	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	4	yes	—
110	C₂²×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
111	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
112	C₂²×C₃×C₅	C₂²×Q₈	1	1	yes	characteristic
113	C₂²×C₃×C₅	H(5,𝔽₂)	1	4	yes	—
114	C₃×C₄×C₅	—	2	8	no	—
115	C₃×C₄×C₅	—	4	8	no	—
116	C₂⁴×C₃	—	4	4	no	—
117	C₂⁴×C₃	C₂×C₄×C₅	1	4	yes	—
118	C₂⁴×C₃	C₅×D₄	1	4	yes	—
119	C₂⁴×C₃	C₅×Q₈	1	4	yes	—
120	C₂⁴×C₃	C₂³×C₅	1	2	yes	—
121	C₂⁴×C₃	C₂³×C₅	1	1	yes	characteristic
122	C₂²×C₃×C₄	—	2	8	no	—
123	C₂²×C₃×C₄	—	4	4	no	—
124	C₂²×C₃×C₄	—	4	8	no	—
125	C₂²×C₃×C₄	C₂×C₄×C₅	1	8	yes	complemented
126	C₂²×C₃×C₄	C₂³×C₅	1	8	yes	—
127	C₂²×C₃×C₄	C₂³×C₅	1	4	yes	—
128	C₃×C₄⋊C₄	—	4	8	no	—
129	C₃×C₂²⋊C₄	—	2	8	no	—
130	C₃×C₂²⋊C₄	—	2	8	no	—
131	C₃×C₂²⋊C₄	C₂×C₄×C₅	1	16	yes	complemented
132	C₂³×C₅	—	2	4	no	—
133	C₂³×C₅	—	2	8	no	—
134	C₂³×C₅	—	4	8	no	—
135	C₂³×C₅	—	4	8	no	—
136	C₂³×C₅	—	4	8	no	—
137	C₂³×C₅	C₃×C₄⋊C₄	1	16	yes	complemented
138	C₂³×C₅	C₂²×C₃×C₄	1	4	yes	—
139	C₂³×C₅	C₂²×C₃×C₄	1	2	yes	—
140	C₂³×C₅	C₂×C₃×D₄	1	2	yes	—
141	C₂³×C₅	C₂×C₃×D₄	1	4	yes	—
142	C₂³×C₅	C₂×C₃×Q₈	1	4	yes	—
143	C₂³×C₅	C₂×C₃×Q₈	1	2	yes	—
144	C₂³×C₅	C₂⁴×C₃	1	1	yes	characteristic
145	C₂×C₄×C₅	—	2	8	no	—
146	C₂×C₄×C₅	—	2	8	no	—
147	C₂×C₄×C₅	—	2	16	no	—
148	C₂×C₄×C₅	—	4	8	no	—
149	C₂×C₄×C₅	—	4	16	no	—
150	C₂×C₄×C₅	C₂²×C₃×C₄	1	8	yes	—
151	C₂⁵	C₂²×C₃×C₅	1	1	yes	characteristic
152	C₂³×C₄	C₂²×C₃×C₅	1	4	yes	—
153	C₂³×C₄	C₂²×C₃×C₅	1	2	yes	—
154	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	8	yes	complemented
155	C₂×C₂²⋊C₄	—	2	8	no	—
156	C₂×C₂²⋊C₄	C₃×C₄×C₅	1	8	yes	complemented
157	C₂×C₂²⋊C₄	C₂²×C₃×C₅	1	4	yes	—
158	C₂×C₂²⋊C₄	C₂²×C₃×C₅	1	4	yes	—
159	C₂×C₂²⋊C₄	C₂²×C₃×C₅	1	4	yes	—
160	#95	C₂²×C₃×C₅	1	8	yes	complemented
161	C₂×C₃×C₅	—	2	4	no	—
162	C₂×C₃×C₅	—	2	8	no	—
163	C₂×C₃×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
164	C₂×C₃×C₅	#518	1	2	yes	—
165	C₂×C₃×C₅	#534	1	2	yes	—
166	C₂×C₃×C₅	#543	1	2	yes	—
167	C₂³×C₃	—	2	4	no	—
168	C₂³×C₃	—	2	8	no	—
169	C₂³×C₃	—	4	8	no	—
170	C₂³×C₃	—	4	8	no	—
171	C₂³×C₃	—	4	8	no	—
172	C₂³×C₃	C₅×C₄⋊C₄	1	16	yes	complemented
173	C₂³×C₃	C₂²×C₄×C₅	1	4	yes	—
174	C₂³×C₃	C₂²×C₄×C₅	1	2	yes	—
175	C₂³×C₃	C₂×C₅×D₄	1	2	yes	—
176	C₂³×C₃	C₂×C₅×D₄	1	4	yes	—
177	C₂³×C₃	C₂×C₅×Q₈	1	4	yes	—
178	C₂³×C₃	C₂×C₅×Q₈	1	2	yes	—
179	C₂³×C₃	C₂⁴×C₅	1	1	yes	characteristic
180	C₂×C₃×C₄	—	2	8	no	—
181	C₂×C₃×C₄	—	2	8	no	—
182	C₂×C₃×C₄	—	2	16	no	—
183	C₂×C₃×C₄	—	4	8	no	—
184	C₂×C₃×C₄	—	4	16	no	—
185	C₂×C₃×C₄	C₂²×C₄×C₅	1	8	yes	—
186	C₂²×C₅	—	2	4	no	—
187	C₂²×C₅	—	2	8	no	—
188	C₂²×C₅	—	2	8	no	—
189	C₂²×C₅	—	2	4	no	—
190	C₂²×C₅	—	2	4	no	—
191	C₂²×C₅	—	2	8	no	—
192	C₂²×C₅	—	4	16	no	—
193	C₂²×C₅	C₂×C₃×C₄⋊C₄	1	8	yes	—
194	C₂²×C₅	C₂×C₃×C₄⋊C₄	1	4	yes	—
195	C₂²×C₅	C₂³×C₃×C₄	1	1	yes	characteristic
196	C₂²×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
197	C₂²×C₅	C₂²×C₃×Q₈	1	1	yes	characteristic
198	C₂²×C₅	C₃×H(5,𝔽₂)	1	4	yes	—
199	C₄×C₅	—	2	8	no	—
200	C₄×C₅	—	4	8	no	—
201	C₂⁴	—	4	4	no	—
202	C₂⁴	C₂×C₃×C₄×C₅	1	4	yes	—
203	C₂⁴	C₃×C₅×D₄	1	4	yes	—
204	C₂⁴	C₃×C₅×Q₈	1	4	yes	—
205	C₂⁴	C₂³×C₃×C₅	1	2	yes	—
206	C₂⁴	C₂³×C₃×C₅	1	1	yes	characteristic
207	C₂²×C₄	—	2	8	no	—
208	C₂²×C₄	—	4	4	no	—
209	C₂²×C₄	—	4	8	no	—
210	C₂²×C₄	C₂×C₃×C₄×C₅	1	8	yes	complemented
211	C₂²×C₄	C₂³×C₃×C₅	1	8	yes	—
212	C₂²×C₄	C₂³×C₃×C₅	1	4	yes	—
213	C₄⋊C₄	—	4	8	no	—
214	C₂²⋊C₄	—	2	8	no	—
215	C₂²⋊C₄	—	2	8	no	—
216	C₂²⋊C₄	C₂×C₃×C₄×C₅	1	16	yes	complemented
217	C₃×C₅	#2315	1	1	yes	characteristic, direct product factor, Hall
218	C₂²×C₃	—	2	4	no	—
219	C₂²×C₃	—	2	8	no	—
220	C₂²×C₃	—	2	8	no	—
221	C₂²×C₃	—	2	4	no	—
222	C₂²×C₃	—	2	4	no	—
223	C₂²×C₃	—	2	8	no	—
224	C₂²×C₃	—	4	16	no	—
225	C₂²×C₃	C₂×C₅×C₄⋊C₄	1	8	yes	—
226	C₂²×C₃	C₂×C₅×C₄⋊C₄	1	4	yes	—
227	C₂²×C₃	C₂³×C₄×C₅	1	1	yes	characteristic
228	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
229	C₂²×C₃	C₂²×C₅×Q₈	1	1	yes	characteristic
230	C₂²×C₃	C₅×H(5,𝔽₂)	1	4	yes	—
231	C₃×C₄	—	2	8	no	—
232	C₃×C₄	—	4	8	no	—
233	C₂×C₅	—	2	4	no	—
234	C₂×C₅	—	2	8	no	—
235	C₂×C₅	C₂²×C₃×C₄⋊C₄	1	1	yes	characteristic
236	C₂×C₅	C₃×#518	1	2	yes	—
237	C₂×C₅	C₃×#534	1	2	yes	—
238	C₂×C₅	C₃×#543	1	2	yes	—
239	C₂³	—	2	4	no	—
240	C₂³	—	2	8	no	—
241	C₂³	—	4	8	no	—
242	C₂³	—	4	8	no	—
243	C₂³	—	4	8	no	—
244	C₂³	C₃×C₅×C₄⋊C₄	1	16	yes	complemented
245	C₂³	C₂²×C₃×C₄×C₅	1	4	yes	—
246	C₂³	C₂²×C₃×C₄×C₅	1	2	yes	—
247	C₂³	C₂×C₃×C₅×D₄	1	2	yes	—
248	C₂³	C₂×C₃×C₅×D₄	1	4	yes	—
249	C₂³	C₂×C₃×C₅×Q₈	1	4	yes	—
250	C₂³	C₂×C₃×C₅×Q₈	1	2	yes	—
251	C₂³	C₂⁴×C₃×C₅	1	1	yes	characteristic
252	C₂×C₄	—	2	8	no	—
253	C₂×C₄	—	2	8	no	—
254	C₂×C₄	—	2	16	no	—
255	C₂×C₄	—	4	8	no	—
256	C₂×C₄	—	4	16	no	—
257	C₂×C₄	C₂²×C₃×C₄×C₅	1	8	yes	—
258	C₂×C₃	—	2	4	no	—
259	C₂×C₃	—	2	8	no	—
260	C₂×C₃	C₂²×C₅×C₄⋊C₄	1	1	yes	characteristic
261	C₂×C₃	C₅×#518	1	2	yes	—
262	C₂×C₃	C₅×#534	1	2	yes	—
263	C₂×C₃	C₅×#543	1	2	yes	—
264	C₅	C₃×#2315	1	1	yes	characteristic, direct product factor, Sylow
265	C₂²	—	2	4	no	—
266	C₂²	—	2	8	no	—
267	C₂²	—	2	8	no	—
268	C₂²	—	2	4	no	—
269	C₂²	—	2	4	no	—
270	C₂²	—	2	8	no	—
271	C₂²	—	4	16	no	—
272	C₂²	C₂×C₃×C₅×C₄⋊C₄	1	8	yes	—
273	C₂²	C₂×C₃×C₅×C₄⋊C₄	1	4	yes	—
274	C₂²	C₂³×C₃×C₄×C₅	1	1	yes	characteristic
275	C₂²	C₂²×C₃×C₅×D₄	1	1	yes	characteristic
276	C₂²	C₂²×C₃×C₅×Q₈	1	1	yes	characteristic
277	C₂²	C₃×C₅×H(5,𝔽₂)	1	4	yes	—
278	C₄	—	2	8	no	—
279	C₄	—	4	8	no	—
280	C₃	C₅×#2315	1	1	yes	characteristic, direct product factor, Sylow
281	C₂	—	2	4	no	—
282	C₂	—	2	8	no	—
283	C₂	C₂²×C₃×C₅×C₄⋊C₄	1	1	yes	characteristic
284	C₂	C₃×C₅×#518	1	2	yes	—
285	C₂	C₃×C₅×#534	1	2	yes	—
286	C₂	C₃×C₅×#543	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃×C₅	C₂⁴
commutator	C₂²	C₂³×C₃×C₄×C₅
Frattini, cosocle	C₂³	C₂⁴×C₃×C₅
prime radical	C₂⁵×C₃×C₅	C₂²
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
3. C₃×C₅×#385	C₂	semidirect
4. C₃×C₅×#380	C₂	semidirect
5. C₅×#2315	C₃	direct (N × H)
9. C₂×C₃×C₅×C₄⋊C₄	C₂²	semidirect
11. C₂×C₃×C₅×C₂²⋊C₄	C₄	semidirect
15. C₃×C₅×#95	C₂²	semidirect
16. C₃×#2315	C₅	direct (N × H)
19. C₅×#385	C₂×C₃	semidirect
20. C₅×#380	C₂×C₃	semidirect
30. C₂²×C₃×C₄×C₅	C₂×C₄	semidirect
36. C₃×C₅×C₂²⋊C₄	C₂×C₄	semidirect
39. C₃×#385	C₂×C₅	semidirect
40. C₃×#380	C₂×C₅	semidirect
44. C₂×C₅×C₄⋊C₄	C₂²×C₃	semidirect
46. C₂×C₅×C₂²⋊C₄	C₃×C₄	semidirect
50. C₅×#95	C₂²×C₃	semidirect
51. #2315	C₃×C₅	direct (N × H)
57. C₂³×C₃×C₅	C₄⋊C₄	semidirect
74. C₂×C₃×C₄⋊C₄	C₂²×C₅	semidirect
76. C₂×C₃×C₂²⋊C₄	C₄×C₅	semidirect
80. C₃×#95	C₂²×C₅	semidirect
90. C₂²×C₄×C₅	C₂×C₃×C₄	semidirect
96. C₅×C₂²⋊C₄	C₂×C₃×C₄	semidirect
99. #385	C₂×C₃×C₅	semidirect
100. #380	C₂×C₃×C₅	semidirect
125. C₂²×C₃×C₄	C₂×C₄×C₅	semidirect
131. C₃×C₂²⋊C₄	C₂×C₄×C₅	semidirect
137. C₂³×C₅	C₃×C₄⋊C₄	semidirect
154. C₂×C₄⋊C₄	C₂²×C₃×C₅	semidirect
156. C₂×C₂²⋊C₄	C₃×C₄×C₅	semidirect
160. #95	C₂²×C₃×C₅	semidirect
172. C₂³×C₃	C₅×C₄⋊C₄	semidirect
210. C₂²×C₄	C₂×C₃×C₄×C₅	semidirect
216. C₂²⋊C₄	C₂×C₃×C₄×C₅	semidirect
217. C₃×C₅	#2315	direct (N × H)
244. C₂³	C₃×C₅×C₄⋊C₄	semidirect
264. C₅	C₃×#2315	direct (N × H)
280. C₃	C₅×#2315	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃×C₅×C₂²⋊C₄	2
C₃×C₅×#385	2
C₃×C₅×#380	2
C₅×#2315	3
C₃×#2315	5

Subgroups — C3×C5×#2315 #14275349

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#2315 #14275349