Subgroups — C₃×C₅×#2330 #14275364

Order 1920.

888 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×#125	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×#389	C₂	1	2	yes	maximal
3	C₃×C₅×#389	C₂	1	4	yes	maximal
4	C₃×C₅×#387	C₂	1	4	yes	maximal
5	C₃×C₅×#384	C₂	1	1	yes	maximal, characteristic
6	C₃×C₄×C₅×C₄⋊C₄	C₂	1	2	yes	maximal
7	C₃×C₅×#376	C₂	1	1	yes	maximal, characteristic
8	C₅×#2330	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₃×C₅×#125	C₄	1	8	yes	complemented
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×C₄⋊C₄	C₂²	1	2	yes	—
14	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	4	yes	—
15	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
16	C₂×C₃×C₄²×C₅	C₂²	1	4	yes	—
17	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
18	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
19	C₃×C₅×#95	C₂²	1	2	yes	—
20	C₃×C₅×#95	C₂²	1	2	yes	—
21	C₃×C₅×#95	C₂²	1	4	yes	—
22	C₃×C₅×#95	C₂²	1	4	yes	—
23	C₃×#2330	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
24	C₂×C₅×#125	C₂×C₃	1	1	yes	characteristic
25	C₅×#389	C₂×C₃	1	2	yes	—
26	C₅×#389	C₂×C₃	1	4	yes	—
27	C₅×#387	C₂×C₃	1	4	yes	—
28	C₅×#384	C₂×C₃	1	1	yes	characteristic
29	C₄×C₅×C₄⋊C₄	C₂×C₃	1	2	yes	—
30	C₅×#376	C₂×C₃	1	1	yes	characteristic
31	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
32	C₂²×C₃×C₄×C₅	C₂³	1	4	yes	—
33	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
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	C₃×C₅×C₄⋊C₄	—	2	4	no	—
39	C₃×C₅×C₄⋊C₄	—	4	4	no	—
40	C₃×C₅×C₄⋊C₄	—	4	4	no	—
41	C₃×C₅×C₄⋊C₄	C₂×C₄	1	16	yes	—
42	C₃×C₅×C₄⋊C₄	C₂×C₄	1	8	yes	—
43	C₃×C₄²×C₅	—	2	2	no	—
44	C₃×C₄²×C₅	—	4	4	no	—
45	C₃×C₄²×C₅	C₂×C₄	1	4	yes	—
46	C₃×C₄²×C₅	Q₈	1	4	yes	—
47	C₂×C₃×#125	C₂×C₅	1	1	yes	characteristic
48	C₃×#389	C₂×C₅	1	4	yes	—
49	C₃×#389	C₂×C₅	1	2	yes	—
50	C₃×#387	C₂×C₅	1	4	yes	—
51	C₃×#384	C₂×C₅	1	1	yes	characteristic
52	C₃×C₄×C₄⋊C₄	C₂×C₅	1	2	yes	—
53	C₃×#376	C₂×C₅	1	1	yes	characteristic
54	C₅×#125	C₃×C₄	1	8	yes	complemented
55	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
56	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	4	yes	—
57	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	4	yes	—
58	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
59	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	4	yes	—
60	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
61	C₂×C₄²×C₅	C₂²×C₃	1	4	yes	—
62	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
63	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
64	C₅×#95	C₂²×C₃	1	2	yes	—
65	C₅×#95	C₂²×C₃	1	2	yes	—
66	C₅×#95	C₂²×C₃	1	4	yes	—
67	C₅×#95	C₂²×C₃	1	4	yes	—
68	#2330	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
69	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
70	C₂×C₃×C₄×C₅	—	2	2	no	—
71	C₂×C₃×C₄×C₅	—	2	4	no	—
72	C₂×C₃×C₄×C₅	—	2	2	no	—
73	C₂×C₃×C₄×C₅	—	2	4	no	—
74	C₂×C₃×C₄×C₅	—	2	2	no	—
75	C₂×C₃×C₄×C₅	—	2	2	no	—
76	C₂×C₃×C₄×C₅	—	2	1	no	—
77	C₂×C₃×C₄×C₅	—	2	1	no	—
78	C₂×C₃×C₄×C₅	—	2	4	no	—
79	C₂×C₃×C₄×C₅	—	2	4	no	—
80	C₂×C₃×C₄×C₅	—	2	2	no	—
81	C₂×C₃×C₄×C₅	—	2	4	no	—
82	C₂×C₃×C₄×C₅	—	2	2	no	—
83	C₂×C₃×C₄×C₅	C₂²×C₄	1	4	yes	—
84	C₂×C₃×C₄×C₅	C₂²×C₄	1	8	yes	—
85	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
86	C₂×C₃×C₄×C₅	C₂×Q₈	1	4	yes	—
87	C₂×C₃×C₄×C₅	C₂×Q₈	1	2	yes	—
88	C₂×C₃×C₄×C₅	#41	1	2	yes	—
89	C₃×#125	C₄×C₅	1	8	yes	complemented
90	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
91	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	4	yes	—
92	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	4	yes	—
93	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
94	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	4	yes	—
95	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
96	C₂×C₃×C₄²	C₂²×C₅	1	4	yes	—
97	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
98	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
99	C₃×#95	C₂²×C₅	1	2	yes	—
100	C₃×#95	C₂²×C₅	1	2	yes	—
101	C₃×#95	C₂²×C₅	1	4	yes	—
102	C₃×#95	C₂²×C₅	1	4	yes	—
103	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
104	C₂²×C₄×C₅	C₂³×C₃	1	4	yes	—
105	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
106	C₂²×C₄×C₅	C₂³×C₃	1	4	yes	—
107	C₂²×C₄×C₅	C₂³×C₃	1	2	yes	—
108	C₂²×C₄×C₅	C₂³×C₃	1	2	yes	—
109	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
110	C₅×C₄⋊C₄	—	2	4	no	—
111	C₅×C₄⋊C₄	—	4	4	no	—
112	C₅×C₄⋊C₄	—	4	4	no	—
113	C₅×C₄⋊C₄	C₂×C₃×C₄	1	16	yes	—
114	C₅×C₄⋊C₄	C₂×C₃×C₄	1	8	yes	—
115	C₄²×C₅	—	2	2	no	—
116	C₄²×C₅	—	4	4	no	—
117	C₄²×C₅	C₂×C₃×C₄	1	4	yes	—
118	C₄²×C₅	C₃×Q₈	1	4	yes	—
119	C₂×#125	C₂×C₃×C₅	1	1	yes	characteristic
120	#389	C₂×C₃×C₅	1	4	yes	—
121	#389	C₂×C₃×C₅	1	2	yes	—
122	#387	C₂×C₃×C₅	1	4	yes	—
123	#384	C₂×C₃×C₅	1	1	yes	characteristic
124	C₄×C₄⋊C₄	C₂×C₃×C₅	1	2	yes	—
125	#376	C₂×C₃×C₅	1	1	yes	characteristic
126	C₂²×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
127	C₂²×C₃×C₅	C₂²×Q₈	1	1	yes	characteristic
128	C₂²×C₃×C₅	C₂×#41	1	1	yes	characteristic
129	C₂²×C₃×C₅	H(5,𝔽₂)	1	2	yes	—
130	C₂²×C₃×C₅	2₋¹⁺⁴	1	2	yes	—
131	C₃×C₄×C₅	—	2	8	no	—
132	C₃×C₄×C₅	—	2	4	no	—
133	C₃×C₄×C₅	—	2	4	no	—
134	C₃×C₄×C₅	—	4	1	no	—
135	C₃×C₄×C₅	—	4	4	no	—
136	C₃×C₄×C₅	—	4	1	no	—
137	C₃×C₄×C₅	C₄×Q₈	1	4	yes	—
138	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
139	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
140	C₂²×C₃×C₄	C₂³×C₅	1	4	yes	—
141	C₂²×C₃×C₄	C₂³×C₅	1	4	yes	—
142	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
143	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
144	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
145	C₃×C₄⋊C₄	—	2	4	no	—
146	C₃×C₄⋊C₄	—	4	4	no	—
147	C₃×C₄⋊C₄	—	4	4	no	—
148	C₃×C₄⋊C₄	C₂×C₄×C₅	1	16	yes	—
149	C₃×C₄⋊C₄	C₂×C₄×C₅	1	8	yes	—
150	C₃×C₄²	—	2	2	no	—
151	C₃×C₄²	—	4	4	no	—
152	C₃×C₄²	C₂×C₄×C₅	1	4	yes	—
153	C₃×C₄²	C₅×Q₈	1	4	yes	—
154	C₂³×C₅	C₂⁴×C₃	1	1	yes	characteristic
155	C₂×C₄×C₅	—	2	2	no	—
156	C₂×C₄×C₅	—	2	4	no	—
157	C₂×C₄×C₅	—	2	2	no	—
158	C₂×C₄×C₅	—	2	4	no	—
159	C₂×C₄×C₅	—	2	2	no	—
160	C₂×C₄×C₅	—	2	2	no	—
161	C₂×C₄×C₅	—	2	1	no	—
162	C₂×C₄×C₅	—	2	1	no	—
163	C₂×C₄×C₅	—	2	4	no	—
164	C₂×C₄×C₅	—	2	4	no	—
165	C₂×C₄×C₅	—	2	2	no	—
166	C₂×C₄×C₅	—	2	4	no	—
167	C₂×C₄×C₅	—	2	2	no	—
168	C₂×C₄×C₅	C₂²×C₃×C₄	1	4	yes	—
169	C₂×C₄×C₅	C₂²×C₃×C₄	1	8	yes	—
170	C₂×C₄×C₅	C₂²×C₃×C₄	1	2	yes	—
171	C₂×C₄×C₅	C₂×C₃×Q₈	1	4	yes	—
172	C₂×C₄×C₅	C₂×C₃×Q₈	1	2	yes	—
173	C₂×C₄×C₅	C₃×#41	1	2	yes	—
174	#125	C₃×C₄×C₅	1	8	yes	complemented
175	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	2	yes	—
176	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	4	yes	—
177	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	4	yes	—
178	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	2	yes	—
179	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	4	yes	—
180	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
181	C₂×C₄²	C₂²×C₃×C₅	1	4	yes	—
182	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
183	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
184	#95	C₂²×C₃×C₅	1	2	yes	—
185	#95	C₂²×C₃×C₅	1	2	yes	—
186	#95	C₂²×C₃×C₅	1	4	yes	—
187	#95	C₂²×C₃×C₅	1	4	yes	—
188	C₂×C₃×C₅	C₂×C₄×Q₈	1	1	yes	characteristic
189	C₂×C₃×C₅	#520	1	2	yes	—
190	C₂×C₃×C₅	#540	1	1	yes	characteristic
191	C₂×C₃×C₅	#541	1	1	yes	characteristic
192	C₂×C₃×C₅	#544	1	2	yes	—
193	C₂³×C₃	C₂⁴×C₅	1	1	yes	characteristic
194	C₂×C₃×C₄	—	2	2	no	—
195	C₂×C₃×C₄	—	2	4	no	—
196	C₂×C₃×C₄	—	2	2	no	—
197	C₂×C₃×C₄	—	2	4	no	—
198	C₂×C₃×C₄	—	2	2	no	—
199	C₂×C₃×C₄	—	2	2	no	—
200	C₂×C₃×C₄	—	2	1	no	—
201	C₂×C₃×C₄	—	2	1	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	4	no	—
206	C₂×C₃×C₄	—	2	2	no	—
207	C₂×C₃×C₄	C₂²×C₄×C₅	1	4	yes	—
208	C₂×C₃×C₄	C₂²×C₄×C₅	1	8	yes	—
209	C₂×C₃×C₄	C₂²×C₄×C₅	1	2	yes	—
210	C₂×C₃×C₄	C₂×C₅×Q₈	1	4	yes	—
211	C₂×C₃×C₄	C₂×C₅×Q₈	1	2	yes	—
212	C₂×C₃×C₄	C₅×#41	1	2	yes	—
213	C₂²×C₅	C₂³×C₃×C₄	1	1	yes	characteristic
214	C₂²×C₅	C₂²×C₃×Q₈	1	1	yes	characteristic
215	C₂²×C₅	C₂×C₃×#41	1	1	yes	characteristic
216	C₂²×C₅	C₃×H(5,𝔽₂)	1	2	yes	—
217	C₂²×C₅	C₃×2₋¹⁺⁴	1	2	yes	—
218	C₄×C₅	—	2	8	no	—
219	C₄×C₅	—	2	4	no	—
220	C₄×C₅	—	2	4	no	—
221	C₄×C₅	—	4	1	no	—
222	C₄×C₅	—	4	4	no	—
223	C₄×C₅	—	4	1	no	—
224	C₄×C₅	C₃×C₄×Q₈	1	4	yes	—
225	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
226	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
227	C₂²×C₄	C₂³×C₃×C₅	1	4	yes	—
228	C₂²×C₄	C₂³×C₃×C₅	1	4	yes	—
229	C₂²×C₄	C₂³×C₃×C₅	1	2	yes	—
230	C₂²×C₄	C₂³×C₃×C₅	1	2	yes	—
231	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
232	C₄⋊C₄	—	2	4	no	—
233	C₄⋊C₄	—	4	4	no	—
234	C₄⋊C₄	—	4	4	no	—
235	C₄⋊C₄	C₂×C₃×C₄×C₅	1	16	yes	—
236	C₄⋊C₄	C₂×C₃×C₄×C₅	1	8	yes	—
237	C₄²	—	2	2	no	—
238	C₄²	—	4	4	no	—
239	C₄²	C₂×C₃×C₄×C₅	1	4	yes	—
240	C₄²	C₃×C₅×Q₈	1	4	yes	—
241	C₃×C₅	#2330	1	1	yes	characteristic, direct product factor, Hall
242	C₂²×C₃	C₂³×C₄×C₅	1	1	yes	characteristic
243	C₂²×C₃	C₂²×C₅×Q₈	1	1	yes	characteristic
244	C₂²×C₃	C₂×C₅×#41	1	1	yes	characteristic
245	C₂²×C₃	C₅×H(5,𝔽₂)	1	2	yes	—
246	C₂²×C₃	C₅×2₋¹⁺⁴	1	2	yes	—
247	C₃×C₄	—	2	8	no	—
248	C₃×C₄	—	2	4	no	—
249	C₃×C₄	—	2	4	no	—
250	C₃×C₄	—	4	1	no	—
251	C₃×C₄	—	4	4	no	—
252	C₃×C₄	—	4	1	no	—
253	C₃×C₄	C₄×C₅×Q₈	1	4	yes	—
254	C₂×C₅	C₂×C₃×C₄×Q₈	1	1	yes	characteristic
255	C₂×C₅	C₃×#520	1	2	yes	—
256	C₂×C₅	C₃×#540	1	1	yes	characteristic
257	C₂×C₅	C₃×#541	1	1	yes	characteristic
258	C₂×C₅	C₃×#544	1	2	yes	—
259	C₂³	C₂⁴×C₃×C₅	1	1	yes	characteristic
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	4	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	C₂²×C₃×C₄×C₅	1	4	yes	—
274	C₂×C₄	C₂²×C₃×C₄×C₅	1	8	yes	—
275	C₂×C₄	C₂²×C₃×C₄×C₅	1	2	yes	—
276	C₂×C₄	C₂×C₃×C₅×Q₈	1	4	yes	—
277	C₂×C₄	C₂×C₃×C₅×Q₈	1	2	yes	—
278	C₂×C₄	C₃×C₅×#41	1	2	yes	—
279	C₂×C₃	C₂×C₄×C₅×Q₈	1	1	yes	characteristic
280	C₂×C₃	C₅×#520	1	2	yes	—
281	C₂×C₃	C₅×#540	1	1	yes	characteristic
282	C₂×C₃	C₅×#541	1	1	yes	characteristic
283	C₂×C₃	C₅×#544	1	2	yes	—
284	C₅	C₃×#2330	1	1	yes	characteristic, direct product factor, Sylow
285	C₂²	C₂³×C₃×C₄×C₅	1	1	yes	characteristic
286	C₂²	C₂²×C₃×C₅×Q₈	1	1	yes	characteristic
287	C₂²	C₂×C₃×C₅×#41	1	1	yes	characteristic
288	C₂²	C₃×C₅×H(5,𝔽₂)	1	2	yes	—
289	C₂²	C₃×C₅×2₋¹⁺⁴	1	2	yes	—
290	C₄	—	2	8	no	—
291	C₄	—	2	4	no	—
292	C₄	—	2	4	no	—
293	C₄	—	4	1	no	—
294	C₄	—	4	4	no	—
295	C₄	—	4	1	no	—
296	C₄	C₃×C₄×C₅×Q₈	1	4	yes	—
297	C₃	C₅×#2330	1	1	yes	characteristic, direct product factor, Sylow
298	C₂	C₂×C₃×C₄×C₅×Q₈	1	1	yes	characteristic
299	C₂	C₃×C₅×#520	1	2	yes	—
300	C₂	C₃×C₅×#540	1	1	yes	characteristic
301	C₂	C₃×C₅×#541	1	1	yes	characteristic
302	C₂	C₃×C₅×#544	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₃×C₅	C₂⁴
commutator	C₂²	C₂³×C₃×C₄×C₅
Frattini, cosocle	C₂³	C₂⁴×C₃×C₅
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
8. C₅×#2330	C₃	direct (N × H)
9. C₃×C₅×#125	C₄	semidirect
23. C₃×#2330	C₅	direct (N × H)
54. C₅×#125	C₃×C₄	semidirect
68. #2330	C₃×C₅	direct (N × H)
89. C₃×#125	C₄×C₅	semidirect
174. #125	C₃×C₄×C₅	semidirect
241. C₃×C₅	#2330	direct (N × H)
284. C₅	C₃×#2330	direct (N × H)
297. C₃	C₅×#2330	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×#125	2
C₃×C₅×#389	2
C₃×C₅×#387	2
C₃×C₅×#384	2
C₃×C₄×C₅×C₄⋊C₄	2
C₃×C₅×#376	2
C₅×#2330	3
C₃×#2330	5

Subgroups — C3×C5×#2330 #14275364

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#2330 #14275364