Subgroups — C₃×C₅×#2298 #14275332

Order 1920.

1048 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×#117	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×#388	C₂	1	4	yes	maximal
3	C₃×C₅×#387	C₂	1	4	yes	maximal, complemented
4	C₃×C₄×C₅×C₄⋊C₄	C₂	1	2	yes	maximal, complemented
5	C₃×C₄×C₅×C₂²⋊C₄	C₂	1	2	yes	maximal
6	C₃×C₅×#376	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×C₄³×C₅	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#2298	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂³×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
10	C₃×C₅×#117	C₄	1	8	yes	complemented
11	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	4	yes	—
12	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
13	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	4	yes	—
14	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
15	C₂×C₃×C₄²×C₅	C₂²	1	4	yes	—
16	C₂×C₃×C₄²×C₅	C₂²	1	2	yes	—
17	C₂×C₃×C₄²×C₅	C₂²	1	2	yes	—
18	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
19	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
20	C₃×C₅×#95	C₂²	1	4	yes	—
21	C₃×C₅×#95	C₂²	1	4	yes	—
22	C₃×C₅×#95	C₂²	1	4	yes	—
23	C₃×#2298	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
24	C₂×C₅×#117	C₂×C₃	1	1	yes	characteristic
25	C₅×#388	C₂×C₃	1	4	yes	—
26	C₅×#387	C₂×C₃	1	4	yes	complemented
27	C₄×C₅×C₄⋊C₄	C₂×C₃	1	2	yes	complemented
28	C₄×C₅×C₂²⋊C₄	C₂×C₃	1	2	yes	—
29	C₅×#376	C₂×C₃	1	1	yes	characteristic, complemented
30	C₄³×C₅	C₂×C₃	1	1	yes	characteristic, complemented
31	C₂⁴×C₃×C₅	C₂³	1	1	yes	characteristic
32	C₂²×C₃×C₄×C₅	—	2	4	no	—
33	C₂²×C₃×C₄×C₅	C₂×C₄	1	4	yes	—
34	C₂²×C₃×C₄×C₅	C₂³	1	4	yes	—
35	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
36	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
37	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
38	C₂²×C₃×C₄×C₅	C₂³	1	4	yes	—
39	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
40	C₃×C₅×C₄⋊C₄	—	2	8	no	—
41	C₃×C₅×C₄⋊C₄	C₂×C₄	1	8	yes	—
42	C₃×C₅×C₂²⋊C₄	—	2	8	no	—
43	C₃×C₅×C₂²⋊C₄	C₂×C₄	1	8	yes	—
44	C₃×C₄²×C₅	—	2	8	no	—
45	C₃×C₄²×C₅	—	2	4	no	—
46	C₃×C₄²×C₅	—	2	4	no	—
47	C₃×C₄²×C₅	C₂×C₄	1	4	yes	—
48	C₃×C₄²×C₅	C₂×C₄	1	4	yes	—
49	C₂×C₃×#117	C₂×C₅	1	1	yes	characteristic
50	C₃×#388	C₂×C₅	1	4	yes	—
51	C₃×#387	C₂×C₅	1	4	yes	complemented
52	C₃×C₄×C₄⋊C₄	C₂×C₅	1	2	yes	complemented
53	C₃×C₄×C₂²⋊C₄	C₂×C₅	1	2	yes	—
54	C₃×#376	C₂×C₅	1	1	yes	characteristic, complemented
55	C₃×C₄³	C₂×C₅	1	1	yes	characteristic, complemented
56	C₂³×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
57	C₅×#117	C₃×C₄	1	8	yes	complemented
58	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	4	yes	—
59	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
60	C₂×C₅×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
61	C₂×C₅×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
62	C₂×C₄²×C₅	C₂²×C₃	1	2	yes	—
63	C₂×C₄²×C₅	C₂²×C₃	1	4	yes	—
64	C₂×C₄²×C₅	C₂²×C₃	1	2	yes	—
65	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
66	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
67	C₅×#95	C₂²×C₃	1	4	yes	—
68	C₅×#95	C₂²×C₃	1	4	yes	—
69	C₅×#95	C₂²×C₃	1	4	yes	—
70	#2298	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
71	C₂³×C₃×C₅	—	2	4	no	—
72	C₂³×C₃×C₅	—	2	2	no	—
73	C₂³×C₃×C₅	C₂²×C₄	1	2	yes	—
74	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
75	C₂×C₃×C₄×C₅	—	2	4	no	—
76	C₂×C₃×C₄×C₅	—	2	4	no	—
77	C₂×C₃×C₄×C₅	—	2	4	no	—
78	C₂×C₃×C₄×C₅	—	2	4	no	—
79	C₂×C₃×C₄×C₅	—	2	2	no	—
80	C₂×C₃×C₄×C₅	—	2	8	no	—
81	C₂×C₃×C₄×C₅	—	2	4	no	—
82	C₂×C₃×C₄×C₅	—	2	4	no	—
83	C₂×C₃×C₄×C₅	—	2	4	no	—
84	C₂×C₃×C₄×C₅	C₂²×C₄	1	4	yes	—
85	C₂×C₃×C₄×C₅	C₂²×C₄	1	4	yes	—
86	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
87	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
88	C₂×C₃×C₄×C₅	#41	1	8	yes	—
89	C₂×C₃×C₄×C₅	#41	1	4	yes	—
90	C₂³×C₃×C₄	C₂²×C₅	1	1	yes	characteristic
91	C₃×#117	C₄×C₅	1	8	yes	complemented
92	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	4	yes	—
93	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
94	C₂×C₃×C₂²⋊C₄	C₂²×C₅	1	4	yes	—
95	C₂×C₃×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
96	C₂×C₃×C₄²	C₂²×C₅	1	2	yes	—
97	C₂×C₃×C₄²	C₂²×C₅	1	4	yes	—
98	C₂×C₃×C₄²	C₂²×C₅	1	2	yes	—
99	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
100	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
101	C₃×#95	C₂²×C₅	1	4	yes	—
102	C₃×#95	C₂²×C₅	1	4	yes	—
103	C₃×#95	C₂²×C₅	1	4	yes	—
104	C₂⁴×C₅	C₂³×C₃	1	1	yes	characteristic
105	C₂²×C₄×C₅	—	2	4	no	—
106	C₂²×C₄×C₅	C₂×C₃×C₄	1	4	yes	—
107	C₂²×C₄×C₅	C₂³×C₃	1	4	yes	—
108	C₂²×C₄×C₅	C₂³×C₃	1	2	yes	—
109	C₂²×C₄×C₅	C₂³×C₃	1	2	yes	—
110	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
111	C₂²×C₄×C₅	C₂³×C₃	1	4	yes	—
112	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
113	C₅×C₄⋊C₄	—	2	8	no	—
114	C₅×C₄⋊C₄	C₂×C₃×C₄	1	8	yes	—
115	C₅×C₂²⋊C₄	—	2	8	no	—
116	C₅×C₂²⋊C₄	C₂×C₃×C₄	1	8	yes	—
117	C₄²×C₅	—	2	8	no	—
118	C₄²×C₅	—	2	4	no	—
119	C₄²×C₅	—	2	4	no	—
120	C₄²×C₅	C₂×C₃×C₄	1	4	yes	—
121	C₄²×C₅	C₂×C₃×C₄	1	4	yes	—
122	C₂×#117	C₂×C₃×C₅	1	1	yes	characteristic
123	#388	C₂×C₃×C₅	1	4	yes	—
124	#387	C₂×C₃×C₅	1	4	yes	complemented
125	C₄×C₄⋊C₄	C₂×C₃×C₅	1	2	yes	complemented
126	C₄×C₂²⋊C₄	C₂×C₃×C₅	1	2	yes	—
127	#376	C₂×C₃×C₅	1	1	yes	characteristic, complemented
128	C₄³	C₂×C₃×C₅	1	1	yes	characteristic, complemented
129	C₂²×C₃×C₅	—	2	2	no	—
130	C₂²×C₃×C₅	—	2	4	no	—
131	C₂²×C₃×C₅	—	4	4	no	—
132	C₂²×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
133	C₂²×C₃×C₅	C₂×#41	1	2	yes	—
134	C₂²×C₃×C₅	C₂×#41	1	4	yes	—
135	C₃×C₄×C₅	—	2	2	no	—
136	C₃×C₄×C₅	—	2	4	no	—
137	C₃×C₄×C₅	—	2	4	no	—
138	C₃×C₄×C₅	—	2	8	no	—
139	C₃×C₄×C₅	—	4	4	no	—
140	C₃×C₄×C₅	#117	1	4	yes	—
141	C₂⁴×C₃	C₂³×C₅	1	1	yes	characteristic
142	C₂²×C₃×C₄	—	2	4	no	—
143	C₂²×C₃×C₄	C₂×C₄×C₅	1	4	yes	—
144	C₂²×C₃×C₄	C₂³×C₅	1	4	yes	—
145	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
146	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
147	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
148	C₂²×C₃×C₄	C₂³×C₅	1	4	yes	—
149	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
150	C₃×C₄⋊C₄	—	2	8	no	—
151	C₃×C₄⋊C₄	C₂×C₄×C₅	1	8	yes	—
152	C₃×C₂²⋊C₄	—	2	8	no	—
153	C₃×C₂²⋊C₄	C₂×C₄×C₅	1	8	yes	—
154	C₃×C₄²	—	2	8	no	—
155	C₃×C₄²	—	2	4	no	—
156	C₃×C₄²	—	2	4	no	—
157	C₃×C₄²	C₂×C₄×C₅	1	4	yes	—
158	C₃×C₄²	C₂×C₄×C₅	1	4	yes	—
159	C₂³×C₅	—	2	4	no	—
160	C₂³×C₅	—	2	2	no	—
161	C₂³×C₅	C₂²×C₃×C₄	1	2	yes	—
162	C₂³×C₅	C₂⁴×C₃	1	1	yes	characteristic
163	C₂×C₄×C₅	—	2	4	no	—
164	C₂×C₄×C₅	—	2	4	no	—
165	C₂×C₄×C₅	—	2	4	no	—
166	C₂×C₄×C₅	—	2	4	no	—
167	C₂×C₄×C₅	—	2	2	no	—
168	C₂×C₄×C₅	—	2	8	no	—
169	C₂×C₄×C₅	—	2	4	no	—
170	C₂×C₄×C₅	—	2	4	no	—
171	C₂×C₄×C₅	—	2	4	no	—
172	C₂×C₄×C₅	C₂²×C₃×C₄	1	4	yes	—
173	C₂×C₄×C₅	C₂²×C₃×C₄	1	4	yes	—
174	C₂×C₄×C₅	C₂²×C₃×C₄	1	2	yes	—
175	C₂×C₄×C₅	C₂²×C₃×C₄	1	2	yes	—
176	C₂×C₄×C₅	C₃×#41	1	8	yes	—
177	C₂×C₄×C₅	C₃×#41	1	4	yes	—
178	C₂³×C₄	C₂²×C₃×C₅	1	1	yes	characteristic
179	#117	C₃×C₄×C₅	1	8	yes	complemented
180	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	4	yes	—
181	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	2	yes	—
182	C₂×C₂²⋊C₄	C₂²×C₃×C₅	1	4	yes	—
183	C₂×C₂²⋊C₄	C₂²×C₃×C₅	1	2	yes	—
184	C₂×C₄²	C₂²×C₃×C₅	1	2	yes	—
185	C₂×C₄²	C₂²×C₃×C₅	1	4	yes	—
186	C₂×C₄²	C₂²×C₃×C₅	1	2	yes	—
187	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
188	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
189	#95	C₂²×C₃×C₅	1	4	yes	—
190	#95	C₂²×C₃×C₅	1	4	yes	—
191	#95	C₂²×C₃×C₅	1	4	yes	—
192	C₂×C₃×C₅	—	4	2	no	—
193	C₂×C₃×C₅	C₂×#117	1	1	yes	characteristic
194	C₂×C₃×C₅	C₄×#41	1	2	yes	—
195	C₂×C₃×C₅	#529	1	4	yes	—
196	C₂³×C₃	—	2	4	no	—
197	C₂³×C₃	—	2	2	no	—
198	C₂³×C₃	C₂²×C₄×C₅	1	2	yes	—
199	C₂³×C₃	C₂⁴×C₅	1	1	yes	characteristic
200	C₂×C₃×C₄	—	2	4	no	—
201	C₂×C₃×C₄	—	2	4	no	—
202	C₂×C₃×C₄	—	2	4	no	—
203	C₂×C₃×C₄	—	2	4	no	—
204	C₂×C₃×C₄	—	2	2	no	—
205	C₂×C₃×C₄	—	2	8	no	—
206	C₂×C₃×C₄	—	2	4	no	—
207	C₂×C₃×C₄	—	2	4	no	—
208	C₂×C₃×C₄	—	2	4	no	—
209	C₂×C₃×C₄	C₂²×C₄×C₅	1	4	yes	—
210	C₂×C₃×C₄	C₂²×C₄×C₅	1	4	yes	—
211	C₂×C₃×C₄	C₂²×C₄×C₅	1	2	yes	—
212	C₂×C₃×C₄	C₂²×C₄×C₅	1	2	yes	—
213	C₂×C₃×C₄	C₅×#41	1	8	yes	—
214	C₂×C₃×C₄	C₅×#41	1	4	yes	—
215	C₂²×C₅	—	2	2	no	—
216	C₂²×C₅	—	2	4	no	—
217	C₂²×C₅	—	4	4	no	—
218	C₂²×C₅	C₂³×C₃×C₄	1	1	yes	characteristic
219	C₂²×C₅	C₂×C₃×#41	1	2	yes	—
220	C₂²×C₅	C₂×C₃×#41	1	4	yes	—
221	C₄×C₅	—	2	2	no	—
222	C₄×C₅	—	2	4	no	—
223	C₄×C₅	—	2	4	no	—
224	C₄×C₅	—	2	8	no	—
225	C₄×C₅	—	4	4	no	—
226	C₄×C₅	C₃×#117	1	4	yes	—
227	C₂⁴	C₂³×C₃×C₅	1	1	yes	characteristic
228	C₂²×C₄	—	2	4	no	—
229	C₂²×C₄	C₂×C₃×C₄×C₅	1	4	yes	—
230	C₂²×C₄	C₂³×C₃×C₅	1	4	yes	—
231	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
232	C₂²×C₄	C₂³×C₃×C₅	1	2	yes	—
233	C₂²×C₄	C₂³×C₃×C₅	1	2	yes	—
234	C₂²×C₄	C₂³×C₃×C₅	1	4	yes	—
235	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
236	C₄⋊C₄	—	2	8	no	—
237	C₄⋊C₄	C₂×C₃×C₄×C₅	1	8	yes	—
238	C₂²⋊C₄	—	2	8	no	—
239	C₂²⋊C₄	C₂×C₃×C₄×C₅	1	8	yes	—
240	C₄²	—	2	8	no	—
241	C₄²	—	2	4	no	—
242	C₄²	—	2	4	no	—
243	C₄²	C₂×C₃×C₄×C₅	1	4	yes	—
244	C₄²	C₂×C₃×C₄×C₅	1	4	yes	—
245	C₃×C₅	#2298	1	1	yes	characteristic, direct product factor, Hall
246	C₂²×C₃	—	2	2	no	—
247	C₂²×C₃	—	2	4	no	—
248	C₂²×C₃	—	4	4	no	—
249	C₂²×C₃	C₂³×C₄×C₅	1	1	yes	characteristic
250	C₂²×C₃	C₂×C₅×#41	1	2	yes	—
251	C₂²×C₃	C₂×C₅×#41	1	4	yes	—
252	C₃×C₄	—	2	2	no	—
253	C₃×C₄	—	2	4	no	—
254	C₃×C₄	—	2	4	no	—
255	C₃×C₄	—	2	8	no	—
256	C₃×C₄	—	4	4	no	—
257	C₃×C₄	C₅×#117	1	4	yes	—
258	C₂×C₅	—	4	2	no	—
259	C₂×C₅	C₂×C₃×#117	1	1	yes	characteristic
260	C₂×C₅	C₃×C₄×#41	1	2	yes	—
261	C₂×C₅	C₃×#529	1	4	yes	—
262	C₂³	—	2	4	no	—
263	C₂³	—	2	2	no	—
264	C₂³	C₂²×C₃×C₄×C₅	1	2	yes	—
265	C₂³	C₂⁴×C₃×C₅	1	1	yes	characteristic
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	8	no	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	2	4	no	—
274	C₂×C₄	—	2	4	no	—
275	C₂×C₄	C₂²×C₃×C₄×C₅	1	4	yes	—
276	C₂×C₄	C₂²×C₃×C₄×C₅	1	4	yes	—
277	C₂×C₄	C₂²×C₃×C₄×C₅	1	2	yes	—
278	C₂×C₄	C₂²×C₃×C₄×C₅	1	2	yes	—
279	C₂×C₄	C₃×C₅×#41	1	8	yes	—
280	C₂×C₄	C₃×C₅×#41	1	4	yes	—
281	C₂×C₃	—	4	2	no	—
282	C₂×C₃	C₂×C₅×#117	1	1	yes	characteristic
283	C₂×C₃	C₄×C₅×#41	1	2	yes	—
284	C₂×C₃	C₅×#529	1	4	yes	—
285	C₅	C₃×#2298	1	1	yes	characteristic, direct product factor, Sylow
286	C₂²	—	2	2	no	—
287	C₂²	—	2	4	no	—
288	C₂²	—	4	4	no	—
289	C₂²	C₂³×C₃×C₄×C₅	1	1	yes	characteristic
290	C₂²	C₂×C₃×C₅×#41	1	2	yes	—
291	C₂²	C₂×C₃×C₅×#41	1	4	yes	—
292	C₄	—	2	2	no	—
293	C₄	—	2	4	no	—
294	C₄	—	2	4	no	—
295	C₄	—	2	8	no	—
296	C₄	—	4	4	no	—
297	C₄	C₃×C₅×#117	1	4	yes	—
298	C₃	C₅×#2298	1	1	yes	characteristic, direct product factor, Sylow
299	C₂	—	4	2	no	—
300	C₂	C₂×C₃×C₅×#117	1	1	yes	characteristic
301	C₂	C₃×C₄×C₅×#41	1	2	yes	—
302	C₂	C₃×C₅×#529	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₃×C₅×#387	C₂	semidirect
4. C₃×C₄×C₅×C₄⋊C₄	C₂	semidirect
6. C₃×C₅×#376	C₂	semidirect
7. C₃×C₄³×C₅	C₂	semidirect
8. C₅×#2298	C₃	direct (N × H)
10. C₃×C₅×#117	C₄	semidirect
23. C₃×#2298	C₅	direct (N × H)
26. C₅×#387	C₂×C₃	semidirect
27. C₄×C₅×C₄⋊C₄	C₂×C₃	semidirect
29. C₅×#376	C₂×C₃	semidirect
30. C₄³×C₅	C₂×C₃	semidirect
51. C₃×#387	C₂×C₅	semidirect
52. C₃×C₄×C₄⋊C₄	C₂×C₅	semidirect
54. C₃×#376	C₂×C₅	semidirect
55. C₃×C₄³	C₂×C₅	semidirect
57. C₅×#117	C₃×C₄	semidirect
70. #2298	C₃×C₅	direct (N × H)
91. C₃×#117	C₄×C₅	semidirect
124. #387	C₂×C₃×C₅	semidirect
125. C₄×C₄⋊C₄	C₂×C₃×C₅	semidirect
127. #376	C₂×C₃×C₅	semidirect
128. C₄³	C₂×C₃×C₅	semidirect
179. #117	C₃×C₄×C₅	semidirect
245. C₃×C₅	#2298	direct (N × H)
285. C₅	C₃×#2298	direct (N × H)
298. C₃	C₅×#2298	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×#117	2
C₃×C₅×#388	2
C₃×C₅×#387	2
C₃×C₄×C₅×C₄⋊C₄	2
C₃×C₄×C₅×C₂²⋊C₄	2
C₃×C₅×#376	2
C₃×C₄³×C₅	2
C₅×#2298	3
C₃×#2298	5

Subgroups — C3×C5×#2298 #14275332

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#2298 #14275332