Subgroups — C₃×C₅×#937 #13087950

Order 1440.

574 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	C₂³×C₃×C₅×D₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃²×C₅×D₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×C₅×#67	C₂	1	2	yes	maximal, complemented
4	C₃×C₅×#217	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₅×(C₂×D₃)⋊C₄	C₂	1	2	yes	maximal, complemented
6	C₃×C₅×#120	—	3	1	no	maximal
7	C₅×#937	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂³×C₃²×C₅	C₂²	1	1	yes	characteristic
9	C₂³×C₃²×C₅	C₂²	1	1	yes	characteristic, complemented
10	C₂²×C₃×C₅×D₃	—	2	2	no	—
11	C₂²×C₃×C₅×D₃	—	2	4	no	—
12	C₂²×C₃×C₅×D₃	C₂²	1	2	yes	—
13	C₃²×C₅×D₄	—	2	2	no	—
14	C₂×C₃²×C₄×C₅	C₂²	1	1	yes	characteristic, complemented
15	C₃×C₅×#67	—	2	4	no	—
16	C₂×C₃×C₅×Dic₃	C₂²	1	2	yes	complemented
17	C₃×#937	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
18	C₂⁴×C₃×C₅	—	3	1	no	—
19	C₂³×C₅×D₃	C₂×C₃	1	1	yes	characteristic, complemented
20	C₂×C₃×C₅×D₄	—	2	1	no	—
21	C₂×C₃×C₅×D₄	—	3	2	no	—
22	C₂×C₃×C₅×D₄	D₃	1	1	yes	characteristic, complemented
23	C₂×C₃×C₅×D₄	C₂×C₃	1	1	yes	characteristic, complemented
24	C₂×C₅×#67	C₂×C₃	1	2	yes	complemented
25	C₃×C₅×C₂²⋊C₄	—	3	1	no	—
26	C₃×C₅×C₂²⋊C₄	—	3	2	no	—
27	C₅×#217	C₂×C₃	1	1	yes	characteristic, complemented
28	C₅×(C₂×D₃)⋊C₄	C₂×C₃	1	2	yes	complemented
29	C₂²×C₃²×C₅	—	2	2	no	—
30	C₂²×C₃²×C₅	—	2	1	no	—
31	C₂²×C₃²×C₅	—	2	2	no	—
32	C₂²×C₃²×C₅	D₄	1	2	yes	complemented
33	C₂²×C₃²×C₅	C₂³	1	1	yes	characteristic
34	C₂×C₃×C₅×D₃	—	2	8	no	—
35	C₂×C₃×C₅×D₃	—	2	2	no	—
36	C₂×C₃×C₅×D₃	—	2	2	no	—
37	C₂×C₃×C₅×D₃	D₄	1	4	yes	complemented
38	C₃²×C₄×C₅	—	2	1	no	—
39	C₃×C₅×Dic₃	—	2	2	no	—
40	C₅×#120	—	3	1	no	Hall
41	C₂³×C₃×D₃	C₂×C₅	1	1	yes	characteristic, complemented
42	C₂×C₃²×D₄	C₂×C₅	1	1	yes	characteristic, complemented
43	C₂×C₃×#67	C₂×C₅	1	2	yes	complemented
44	C₃×#217	C₂×C₅	1	1	yes	characteristic, complemented
45	C₃×(C₂×D₃)⋊C₄	C₂×C₅	1	2	yes	complemented
46	C₂³×C₃×C₅	—	2	1	no	—
47	C₂³×C₃×C₅	—	2	1	no	—
48	C₂³×C₃×C₅	—	3	2	no	—
49	C₂³×C₃×C₅	—	6	4	no	—
50	C₂³×C₃×C₅	—	6	2	no	—
51	C₂³×C₃×C₅	C₂×D₃	1	1	yes	characteristic
52	C₂³×C₃×C₅	C₂×D₃	1	1	yes	characteristic, complemented
53	C₂³×C₃×C₅	C₂²×C₃	1	1	yes	characteristic
54	C₂³×C₃×C₅	C₂²×C₃	1	1	yes	characteristic, complemented
55	C₂²×C₅×D₃	—	2	2	no	—
56	C₂²×C₅×D₃	—	2	4	no	—
57	C₂²×C₅×D₃	C₂²×C₃	1	2	yes	—
58	C₃×C₅×D₄	—	2	4	no	—
59	C₃×C₅×D₄	—	2	2	no	—
60	C₃×C₅×D₄	—	2	2	no	—
61	C₃×C₅×D₄	—	6	4	no	—
62	C₂×C₃×C₄×C₅	—	2	1	no	—
63	C₂×C₃×C₄×C₅	—	3	2	no	—
64	C₂×C₃×C₄×C₅	C₂×D₃	1	1	yes	characteristic, complemented
65	C₂×C₃×C₄×C₅	C₂²×C₃	1	1	yes	characteristic, complemented
66	C₅×#67	—	2	4	no	—
67	C₂×C₅×Dic₃	C₂²×C₃	1	2	yes	complemented
68	C₃×#120	—	3	1	no	—
69	#937	C₃×C₅	1	1	yes	characteristic, direct product factor
70	C₂×C₃²×C₅	—	2	2	no	—
71	C₂×C₃²×C₅	—	4	1	no	—
72	C₂×C₃²×C₅	C₂×D₄	1	2	yes	—
73	C₂×C₃²×C₅	C₂×D₄	1	1	yes	characteristic
74	C₃×C₅×D₃	—	2	4	no	—
75	C₂⁴×C₅	—	3	1	no	—
76	C₂×C₅×D₄	—	3	2	no	—
77	C₂×C₅×D₄	C₃×D₃	1	1	yes	characteristic, complemented
78	C₅×C₂²⋊C₄	—	3	1	no	—
79	C₅×C₂²⋊C₄	—	3	2	no	—
80	C₂³×C₃²	C₂²×C₅	1	1	yes	characteristic
81	C₂³×C₃²	C₂²×C₅	1	1	yes	characteristic, complemented
82	C₂²×C₃×D₃	—	2	2	no	—
83	C₂²×C₃×D₃	—	2	4	no	—
84	C₂²×C₃×D₃	C₂²×C₅	1	2	yes	—
85	C₃²×D₄	—	2	2	no	—
86	C₂×C₃²×C₄	C₂²×C₅	1	1	yes	characteristic, complemented
87	C₃×#67	—	2	4	no	—
88	C₂×C₃×Dic₃	C₂²×C₅	1	2	yes	complemented
89	C₂²×C₃×C₅	—	2	2	no	—
90	C₂²×C₃×C₅	—	2	2	no	—
91	C₂²×C₃×C₅	—	2	2	no	—
92	C₂²×C₃×C₅	—	2	2	no	—
93	C₂²×C₃×C₅	—	2	1	no	—
94	C₂²×C₃×C₅	—	2	2	no	—
95	C₂²×C₃×C₅	—	2	1	no	—
96	C₂²×C₃×C₅	—	2	2	no	—
97	C₂²×C₃×C₅	—	2	1	no	—
98	C₂²×C₃×C₅	—	3	4	no	—
99	C₂²×C₃×C₅	—	4	2	no	—
100	C₂²×C₃×C₅	—	4	2	no	—
101	C₂²×C₃×C₅	—	6	8	no	—
102	C₂²×C₃×C₅	—	6	2	no	—
103	C₂²×C₃×C₅	—	6	2	no	—
104	C₂²×C₃×C₅	#67	1	2	yes	complemented
105	C₂²×C₃×C₅	C₃×D₄	1	2	yes	complemented
106	C₂²×C₃×C₅	C₂²×D₃	1	1	yes	characteristic
107	C₂²×C₃×C₅	C₂³×C₃	1	1	yes	characteristic
108	C₂×C₅×D₃	—	2	8	no	—
109	C₂×C₅×D₃	—	2	2	no	—
110	C₂×C₅×D₃	—	2	2	no	—
111	C₂×C₅×D₃	C₃×D₄	1	4	yes	complemented
112	C₃×C₄×C₅	—	2	2	no	—
113	C₃×C₄×C₅	—	2	1	no	—
114	C₃×C₄×C₅	—	2	1	no	—
115	C₃×C₄×C₅	—	6	2	no	—
116	C₅×Dic₃	—	2	2	no	—
117	C₂⁴×C₃	—	3	1	no	—
118	C₂³×D₃	C₂×C₃×C₅	1	1	yes	characteristic, complemented
119	C₂×C₃×D₄	—	2	1	no	—
120	C₂×C₃×D₄	—	3	2	no	—
121	C₂×C₃×D₄	C₅×D₃	1	1	yes	characteristic, complemented
122	C₂×C₃×D₄	C₂×C₃×C₅	1	1	yes	characteristic, complemented
123	C₂×#67	C₂×C₃×C₅	1	2	yes	complemented
124	C₃×C₂²⋊C₄	—	3	1	no	—
125	C₃×C₂²⋊C₄	—	3	2	no	—
126	#217	C₂×C₃×C₅	1	1	yes	characteristic, complemented
127	(C₂×D₃)⋊C₄	C₂×C₃×C₅	1	2	yes	complemented
128	C₃²×C₅	#120	1	1	yes	characteristic, complemented, Hall
129	C₂³×C₅	—	3	2	no	—
130	C₂³×C₅	—	6	4	no	—
131	C₂³×C₅	—	6	2	no	—
132	C₂³×C₅	C₂×C₃×D₃	1	1	yes	characteristic
133	C₂³×C₅	C₂×C₃×D₃	1	1	yes	characteristic, complemented
134	C₅×D₄	—	2	2	no	—
135	C₅×D₄	—	6	4	no	—
136	C₂×C₄×C₅	—	3	2	no	—
137	C₂×C₄×C₅	C₂×C₃×D₃	1	1	yes	characteristic, complemented
138	C₂²×C₃²	—	2	1	no	—
139	C₂²×C₃²	—	2	2	no	—
140	C₂²×C₃²	—	2	2	no	—
141	C₂²×C₃²	C₅×D₄	1	2	yes	complemented
142	C₂²×C₃²	C₂³×C₅	1	1	yes	characteristic
143	C₂×C₃×D₃	—	2	8	no	—
144	C₂×C₃×D₃	—	2	2	no	—
145	C₂×C₃×D₃	—	2	2	no	—
146	C₂×C₃×D₃	C₅×D₄	1	4	yes	complemented
147	C₃²×C₄	—	2	1	no	—
148	C₃×Dic₃	—	2	2	no	—
149	#120	—	3	1	no	Sylow
150	C₂×C₃×C₅	—	2	2	no	—
151	C₂×C₃×C₅	—	2	2	no	—
152	C₂×C₃×C₅	—	2	2	no	—
153	C₂×C₃×C₅	—	2	1	no	—
154	C₂×C₃×C₅	—	4	2	no	—
155	C₂×C₃×C₅	—	4	2	no	—
156	C₂×C₃×C₅	—	4	1	no	—
157	C₂×C₃×C₅	—	4	1	no	—
158	C₂×C₃×C₅	—	6	4	no	—
159	C₂×C₃×C₅	D₃×D₄	1	2	yes	—
160	C₂×C₃×C₅	C₂×#67	1	1	yes	characteristic
161	C₂×C₃×C₅	C₂×C₃×D₄	1	2	yes	—
162	C₂×C₃×C₅	C₂×C₃×D₄	1	1	yes	characteristic
163	C₅×D₃	—	2	4	no	—
164	C₂³×C₃	—	2	1	no	—
165	C₂³×C₃	—	2	1	no	—
166	C₂³×C₃	—	3	2	no	—
167	C₂³×C₃	—	6	4	no	—
168	C₂³×C₃	—	6	2	no	—
169	C₂³×C₃	C₂×C₅×D₃	1	1	yes	characteristic
170	C₂³×C₃	C₂×C₅×D₃	1	1	yes	characteristic, complemented
171	C₂³×C₃	C₂²×C₃×C₅	1	1	yes	characteristic
172	C₂³×C₃	C₂²×C₃×C₅	1	1	yes	characteristic, complemented
173	C₂²×D₃	—	2	2	no	—
174	C₂²×D₃	—	2	4	no	—
175	C₂²×D₃	C₂²×C₃×C₅	1	2	yes	—
176	C₃×D₄	—	2	4	no	—
177	C₃×D₄	—	2	2	no	—
178	C₃×D₄	—	2	2	no	—
179	C₃×D₄	—	6	4	no	—
180	C₂×C₃×C₄	—	2	1	no	—
181	C₂×C₃×C₄	—	3	2	no	—
182	C₂×C₃×C₄	C₂×C₅×D₃	1	1	yes	characteristic, complemented
183	C₂×C₃×C₄	C₂²×C₃×C₅	1	1	yes	characteristic, complemented
184	#67	—	2	4	no	—
185	C₂×Dic₃	C₂²×C₃×C₅	1	2	yes	complemented
186	C₂²×C₅	—	2	2	no	—
187	C₂²×C₅	—	2	2	no	—
188	C₂²×C₅	—	2	1	no	—
189	C₂²×C₅	—	3	4	no	—
190	C₂²×C₅	—	6	8	no	—
191	C₂²×C₅	—	6	2	no	—
192	C₂²×C₅	—	6	2	no	—
193	C₂²×C₅	C₃×#67	1	2	yes	complemented
194	C₂²×C₅	C₂²×C₃×D₃	1	1	yes	characteristic
195	C₄×C₅	—	2	1	no	—
196	C₄×C₅	—	6	2	no	—
197	C₂×C₃²	—	2	2	no	—
198	C₂×C₃²	—	4	1	no	—
199	C₂×C₃²	C₂×C₅×D₄	1	2	yes	—
200	C₂×C₃²	C₂×C₅×D₄	1	1	yes	characteristic
201	C₃×D₃	—	2	4	no	—
202	C₂⁴	—	3	1	no	—
203	C₂×D₄	—	3	2	no	—
204	C₂×D₄	C₃×C₅×D₃	1	1	yes	characteristic, complemented
205	C₂²⋊C₄	—	3	1	no	—
206	C₂²⋊C₄	—	3	2	no	—
207	C₃×C₅	—	2	1	no	—
208	C₃×C₅	#937	1	1	yes	characteristic, direct product factor
209	C₃×C₅	C₃×#120	1	1	yes	characteristic, complemented
210	C₂²×C₃	—	2	2	no	—
211	C₂²×C₃	—	2	2	no	—
212	C₂²×C₃	—	2	2	no	—
213	C₂²×C₃	—	2	2	no	—
214	C₂²×C₃	—	2	1	no	—
215	C₂²×C₃	—	2	2	no	—
216	C₂²×C₃	—	2	1	no	—
217	C₂²×C₃	—	2	2	no	—
218	C₂²×C₃	—	2	1	no	—
219	C₂²×C₃	—	3	4	no	—
220	C₂²×C₃	—	4	2	no	—
221	C₂²×C₃	—	4	2	no	—
222	C₂²×C₃	—	6	8	no	—
223	C₂²×C₃	—	6	2	no	—
224	C₂²×C₃	—	6	2	no	—
225	C₂²×C₃	C₅×#67	1	2	yes	complemented
226	C₂²×C₃	C₃×C₅×D₄	1	2	yes	complemented
227	C₂²×C₃	C₂²×C₅×D₃	1	1	yes	characteristic
228	C₂²×C₃	C₂³×C₃×C₅	1	1	yes	characteristic
229	C₂×D₃	—	2	8	no	—
230	C₂×D₃	—	2	2	no	—
231	C₂×D₃	—	2	2	no	—
232	C₂×D₃	C₃×C₅×D₄	1	4	yes	complemented
233	C₃×C₄	—	2	2	no	—
234	C₃×C₄	—	2	1	no	—
235	C₃×C₄	—	2	1	no	—
236	C₃×C₄	—	6	2	no	—
237	Dic₃	—	2	2	no	—
238	C₂×C₅	—	2	2	no	—
239	C₂×C₅	—	4	1	no	—
240	C₂×C₅	—	6	4	no	—
241	C₂×C₅	C₃×D₃×D₄	1	2	yes	—
242	C₂×C₅	C₂×C₃×#67	1	1	yes	characteristic
243	C₃²	C₅×#120	1	1	yes	characteristic, complemented, Sylow
244	C₂³	—	3	2	no	—
245	C₂³	—	6	4	no	—
246	C₂³	—	6	2	no	—
247	C₂³	C₂×C₃×C₅×D₃	1	1	yes	characteristic
248	C₂³	C₂×C₃×C₅×D₃	1	1	yes	characteristic, complemented
249	D₄	—	2	2	no	—
250	D₄	—	6	4	no	—
251	C₂×C₄	—	3	2	no	—
252	C₂×C₄	C₂×C₃×C₅×D₃	1	1	yes	characteristic, complemented
253	C₂×C₃	—	2	2	no	—
254	C₂×C₃	—	2	2	no	—
255	C₂×C₃	—	2	1	no	—
256	C₂×C₃	—	2	2	no	—
257	C₂×C₃	—	4	2	no	—
258	C₂×C₃	—	4	2	no	—
259	C₂×C₃	—	4	1	no	—
260	C₂×C₃	—	4	1	no	—
261	C₂×C₃	—	6	4	no	—
262	C₂×C₃	C₅×D₃×D₄	1	2	yes	—
263	C₂×C₃	C₂×C₅×#67	1	1	yes	characteristic
264	C₂×C₃	C₂×C₃×C₅×D₄	1	1	yes	characteristic
265	C₂×C₃	C₂×C₃×C₅×D₄	1	2	yes	—
266	D₃	—	2	4	no	—
267	C₅	C₃×#937	1	1	yes	characteristic, direct product factor, Sylow
268	C₂²	—	2	2	no	—
269	C₂²	—	2	1	no	—
270	C₂²	—	2	2	no	—
271	C₂²	—	3	4	no	—
272	C₂²	—	6	8	no	—
273	C₂²	—	6	2	no	—
274	C₂²	—	6	2	no	—
275	C₂²	C₃×C₅×#67	1	2	yes	complemented
276	C₂²	C₂²×C₃×C₅×D₃	1	1	yes	characteristic
277	C₄	—	2	1	no	—
278	C₄	—	6	2	no	—
279	C₃	—	2	1	no	—
280	C₃	C₅×#937	1	1	yes	characteristic, direct product factor
281	C₃	C₃×C₅×#120	1	1	yes	characteristic, complemented
282	C₂	—	2	2	no	—
283	C₂	—	4	1	no	—
284	C₂	—	6	4	no	—
285	C₂	C₃×C₅×D₃×D₄	1	2	yes	—
286	C₂	C₂×C₃×C₅×#67	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃×C₅	C₂²×D₃
commutator, cosocle	C₂²×C₃	C₂³×C₃×C₅
Frattini	C₂²	C₂²×C₃×C₅×D₃
Fitting	C₂×C₃²×C₅×D₄	C₂
socle	C₂²×C₃²×C₅	C₂³
hypercenter	C₂×C₃×C₅×D₄	D₃
hypocenter	C₃	C₃×C₅×#120
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃×C₅ ◁ C₂×C₃×C₅×D₄

Complements

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

N	H ≅ G/N	type
1. C₂³×C₃×C₅×D₃	C₂	semidirect
2. C₂×C₃²×C₅×D₄	C₂	semidirect
3. C₂×C₃×C₅×#67	C₂	semidirect
4. C₃×C₅×#217	C₂	semidirect
5. C₃×C₅×(C₂×D₃)⋊C₄	C₂	semidirect
7. C₅×#937	C₃	direct (N × H)
9. C₂³×C₃²×C₅	C₂²	semidirect
14. C₂×C₃²×C₄×C₅	C₂²	semidirect
16. C₂×C₃×C₅×Dic₃	C₂²	semidirect
17. C₃×#937	C₅	direct (N × H)
19. C₂³×C₅×D₃	C₂×C₃	semidirect
22. C₂×C₃×C₅×D₄	D₃	semidirect
23. C₂×C₃×C₅×D₄	C₂×C₃	semidirect
24. C₂×C₅×#67	C₂×C₃	semidirect
27. C₅×#217	C₂×C₃	semidirect
28. C₅×(C₂×D₃)⋊C₄	C₂×C₃	semidirect
32. C₂²×C₃²×C₅	D₄	semidirect
37. C₂×C₃×C₅×D₃	D₄	semidirect
41. C₂³×C₃×D₃	C₂×C₅	semidirect
42. C₂×C₃²×D₄	C₂×C₅	semidirect
43. C₂×C₃×#67	C₂×C₅	semidirect
44. C₃×#217	C₂×C₅	semidirect
45. C₃×(C₂×D₃)⋊C₄	C₂×C₅	semidirect
52. C₂³×C₃×C₅	C₂×D₃	semidirect
54. C₂³×C₃×C₅	C₂²×C₃	semidirect
64. C₂×C₃×C₄×C₅	C₂×D₃	semidirect
65. C₂×C₃×C₄×C₅	C₂²×C₃	semidirect
67. C₂×C₅×Dic₃	C₂²×C₃	semidirect
69. #937	C₃×C₅	direct (N × H)
77. C₂×C₅×D₄	C₃×D₃	semidirect
81. C₂³×C₃²	C₂²×C₅	semidirect
86. C₂×C₃²×C₄	C₂²×C₅	semidirect
88. C₂×C₃×Dic₃	C₂²×C₅	semidirect
104. C₂²×C₃×C₅	#67	semidirect
105. C₂²×C₃×C₅	C₃×D₄	semidirect
111. C₂×C₅×D₃	C₃×D₄	semidirect
118. C₂³×D₃	C₂×C₃×C₅	semidirect
121. C₂×C₃×D₄	C₅×D₃	semidirect
122. C₂×C₃×D₄	C₂×C₃×C₅	semidirect
123. C₂×#67	C₂×C₃×C₅	semidirect
126. #217	C₂×C₃×C₅	semidirect
127. (C₂×D₃)⋊C₄	C₂×C₃×C₅	semidirect
128. C₃²×C₅	#120	semidirect
133. C₂³×C₅	C₂×C₃×D₃	semidirect
137. C₂×C₄×C₅	C₂×C₃×D₃	semidirect
141. C₂²×C₃²	C₅×D₄	semidirect
146. C₂×C₃×D₃	C₅×D₄	semidirect
170. C₂³×C₃	C₂×C₅×D₃	semidirect
172. C₂³×C₃	C₂²×C₃×C₅	semidirect
182. C₂×C₃×C₄	C₂×C₅×D₃	semidirect
183. C₂×C₃×C₄	C₂²×C₃×C₅	semidirect
185. C₂×Dic₃	C₂²×C₃×C₅	semidirect
193. C₂²×C₅	C₃×#67	semidirect
204. C₂×D₄	C₃×C₅×D₃	semidirect
208. C₃×C₅	#937	direct (N × H)
209. C₃×C₅	C₃×#120	semidirect
225. C₂²×C₃	C₅×#67	semidirect
226. C₂²×C₃	C₃×C₅×D₄	semidirect
232. C₂×D₃	C₃×C₅×D₄	semidirect
243. C₃²	C₅×#120	semidirect
248. C₂³	C₂×C₃×C₅×D₃	semidirect
252. C₂×C₄	C₂×C₃×C₅×D₃	semidirect
267. C₅	C₃×#937	direct (N × H)
275. C₂²	C₃×C₅×#67	semidirect
280. C₃	C₅×#937	direct (N × H)
281. C₃	C₃×C₅×#120	semidirect

Maximal subgroups

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

subgroup	index
C₂³×C₃×C₅×D₃	2
C₂×C₃²×C₅×D₄	2
C₂×C₃×C₅×#67	2
C₃×C₅×#217	2
C₃×C₅×(C₂×D₃)⋊C₄	2
C₃×C₅×#120	3
C₅×#937	3
C₃×#937	5

Subgroups — C3×C5×#937 #13087950

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#937 #13087950