Subgroups — C₃×C₅×#938 #13087951

Order 1440.

430 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×C₅×D₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×C₅×#67	C₂	1	2	yes	maximal, complemented
3	C₂×C₃×C₄×C₅×D₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₅×#217	C₂	1	2	yes	maximal, complemented
5	C₃×C₅×C₄⋊Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×C₅×#121	—	3	1	no	maximal
7	C₅×#938	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂³×C₃²×C₅	C₂²	1	2	yes	—
9	C₂²×C₃×C₅×D₃	C₂²	1	1	yes	characteristic
10	C₃²×C₅×D₄	—	2	2	no	—
11	C₂×C₃²×C₄×C₅	C₂²	1	1	yes	characteristic
12	C₃×C₅×#67	—	2	4	no	—
13	C₂×C₃×C₅×Dic₃	C₂²	1	2	yes	—
14	C₂×C₃×C₅×Dic₃	C₂²	1	1	yes	characteristic
15	C₃×C₄×C₅×D₃	—	2	2	no	—
16	C₃×#938	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
17	C₂×C₃×C₅×D₄	—	2	1	no	—
18	C₂×C₃×C₅×D₄	—	3	2	no	—
19	C₂×C₃×C₅×D₄	D₃	1	1	yes	characteristic, complemented
20	C₂×C₃×C₅×D₄	C₂×C₃	1	1	yes	characteristic, complemented
21	C₂²×C₃×C₄×C₅	—	3	1	no	—
22	C₂×C₅×#67	C₂×C₃	1	2	yes	complemented
23	C₂×C₄×C₅×D₃	C₂×C₃	1	1	yes	characteristic, complemented
24	C₃×C₅×C₄⋊C₄	—	3	1	no	—
25	C₃×C₅×C₂²⋊C₄	—	3	2	no	—
26	C₅×#217	C₂×C₃	1	2	yes	complemented
27	C₅×C₄⋊Dic₃	C₂×C₃	1	1	yes	characteristic, complemented
28	C₂²×C₃²×C₅	—	2	2	no	—
29	C₂²×C₃²×C₅	—	2	2	no	—
30	C₂²×C₃²×C₅	—	2	2	no	—
31	C₂²×C₃²×C₅	C₂³	1	1	yes	characteristic
32	C₂×C₃×C₅×D₃	—	2	1	no	—
33	C₂×C₃×C₅×D₃	—	2	1	no	—
34	C₂×C₃×C₅×D₃	D₄	1	2	yes	complemented
35	C₃²×C₄×C₅	D₄	1	2	yes	complemented
36	C₃×C₅×Dic₃	—	2	1	no	—
37	C₃×C₅×Dic₃	—	2	2	no	—
38	C₅×#121	—	3	1	no	Hall
39	C₂×C₃²×D₄	C₂×C₅	1	1	yes	characteristic, complemented
40	C₂×C₃×#67	C₂×C₅	1	2	yes	complemented
41	C₂×C₃×C₄×D₃	C₂×C₅	1	1	yes	characteristic, complemented
42	C₃×#217	C₂×C₅	1	2	yes	complemented
43	C₃×C₄⋊Dic₃	C₂×C₅	1	1	yes	characteristic, complemented
44	C₂³×C₃×C₅	—	2	2	no	—
45	C₂³×C₃×C₅	—	3	1	no	—
46	C₂³×C₃×C₅	C₂×D₃	1	2	yes	—
47	C₂³×C₃×C₅	C₂²×C₃	1	2	yes	—
48	C₂²×C₅×D₃	C₂²×C₃	1	1	yes	characteristic
49	C₃×C₅×D₄	—	2	4	no	—
50	C₃×C₅×D₄	—	2	2	no	—
51	C₃×C₅×D₄	—	2	2	no	—
52	C₃×C₅×D₄	—	6	4	no	—
53	C₂×C₃×C₄×C₅	—	2	1	no	—
54	C₂×C₃×C₄×C₅	—	3	1	no	—
55	C₂×C₃×C₄×C₅	—	3	2	no	—
56	C₂×C₃×C₄×C₅	—	6	2	no	—
57	C₂×C₃×C₄×C₅	C₂×D₃	1	1	yes	characteristic
58	C₂×C₃×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
59	C₅×#67	—	2	4	no	—
60	C₂×C₅×Dic₃	C₂²×C₃	1	2	yes	—
61	C₂×C₅×Dic₃	C₂²×C₃	1	1	yes	characteristic
62	C₄×C₅×D₃	—	2	2	no	—
63	C₃×#121	—	3	1	no	—
64	#938	C₃×C₅	1	1	yes	characteristic, direct product factor
65	C₂×C₃²×C₅	—	4	2	no	—
66	C₂×C₃²×C₅	C₂×D₄	1	1	yes	characteristic
67	C₂×C₃²×C₅	C₂×D₄	1	1	yes	characteristic
68	C₂×C₃²×C₅	#41	1	1	yes	characteristic
69	C₃×C₅×D₃	—	2	2	no	—
70	C₂×C₅×D₄	—	3	2	no	—
71	C₂×C₅×D₄	C₃×D₃	1	1	yes	characteristic, complemented
72	C₂²×C₄×C₅	—	3	1	no	—
73	C₅×C₄⋊C₄	—	3	1	no	—
74	C₅×C₂²⋊C₄	—	3	2	no	—
75	C₂³×C₃²	C₂²×C₅	1	2	yes	—
76	C₂²×C₃×D₃	C₂²×C₅	1	1	yes	characteristic
77	C₃²×D₄	—	2	2	no	—
78	C₂×C₃²×C₄	C₂²×C₅	1	1	yes	characteristic
79	C₃×#67	—	2	4	no	—
80	C₂×C₃×Dic₃	C₂²×C₅	1	2	yes	—
81	C₂×C₃×Dic₃	C₂²×C₅	1	1	yes	characteristic
82	C₃×C₄×D₃	—	2	2	no	—
83	C₂²×C₃×C₅	—	2	4	no	—
84	C₂²×C₃×C₅	—	2	2	no	—
85	C₂²×C₃×C₅	—	2	2	no	—
86	C₂²×C₃×C₅	—	2	2	no	—
87	C₂²×C₃×C₅	—	2	2	no	—
88	C₂²×C₃×C₅	—	2	2	no	—
89	C₂²×C₃×C₅	—	2	2	no	—
90	C₂²×C₃×C₅	—	2	1	no	—
91	C₂²×C₃×C₅	—	3	2	no	—
92	C₂²×C₃×C₅	—	4	2	no	—
93	C₂²×C₃×C₅	—	4	2	no	—
94	C₂²×C₃×C₅	—	6	1	no	—
95	C₂²×C₃×C₅	—	6	1	no	—
96	C₂²×C₃×C₅	C₂²×D₃	1	1	yes	characteristic
97	C₂²×C₃×C₅	C₂³×C₃	1	1	yes	characteristic
98	C₂×C₅×D₃	—	2	1	no	—
99	C₂×C₅×D₃	—	2	1	no	—
100	C₂×C₅×D₃	C₃×D₄	1	2	yes	complemented
101	C₃×C₄×C₅	—	2	2	no	—
102	C₃×C₄×C₅	—	6	2	no	—
103	C₃×C₄×C₅	—	6	1	no	—
104	C₃×C₄×C₅	#67	1	2	yes	complemented
105	C₃×C₄×C₅	C₃×D₄	1	2	yes	complemented
106	C₅×Dic₃	—	2	1	no	—
107	C₅×Dic₃	—	2	2	no	—
108	C₂×C₃×D₄	—	2	1	no	—
109	C₂×C₃×D₄	—	3	2	no	—
110	C₂×C₃×D₄	C₅×D₃	1	1	yes	characteristic, complemented
111	C₂×C₃×D₄	C₂×C₃×C₅	1	1	yes	characteristic, complemented
112	C₂²×C₃×C₄	—	3	1	no	—
113	C₂×#67	C₂×C₃×C₅	1	2	yes	complemented
114	C₂×C₄×D₃	C₂×C₃×C₅	1	1	yes	characteristic, complemented
115	C₃×C₄⋊C₄	—	3	1	no	—
116	C₃×C₂²⋊C₄	—	3	2	no	—
117	#217	C₂×C₃×C₅	1	2	yes	complemented
118	C₄⋊Dic₃	C₂×C₃×C₅	1	1	yes	characteristic, complemented
119	C₃²×C₅	#121	1	1	yes	characteristic, complemented, Hall
120	C₂³×C₅	—	3	1	no	—
121	C₂³×C₅	C₂×C₃×D₃	1	2	yes	—
122	C₅×D₄	—	2	2	no	—
123	C₅×D₄	—	6	4	no	—
124	C₂×C₄×C₅	—	3	1	no	—
125	C₂×C₄×C₅	—	3	2	no	—
126	C₂×C₄×C₅	—	6	2	no	—
127	C₂×C₄×C₅	C₂×C₃×D₃	1	1	yes	characteristic
128	C₂²×C₃²	—	2	2	no	—
129	C₂²×C₃²	—	2	2	no	—
130	C₂²×C₃²	—	2	2	no	—
131	C₂²×C₃²	C₂³×C₅	1	1	yes	characteristic
132	C₂×C₃×D₃	—	2	1	no	—
133	C₂×C₃×D₃	—	2	1	no	—
134	C₂×C₃×D₃	C₅×D₄	1	2	yes	complemented
135	C₃²×C₄	C₅×D₄	1	2	yes	complemented
136	C₃×Dic₃	—	2	1	no	—
137	C₃×Dic₃	—	2	2	no	—
138	#121	—	3	1	no	Sylow
139	C₂×C₃×C₅	—	2	1	no	—
140	C₂×C₃×C₅	—	2	1	no	—
141	C₂×C₃×C₅	—	2	1	no	—
142	C₂×C₃×C₅	—	4	4	no	—
143	C₂×C₃×C₅	—	4	2	no	—
144	C₂×C₃×C₅	—	4	2	no	—
145	C₂×C₃×C₅	—	6	2	no	—
146	C₂×C₃×C₅	D₃×D₄	1	1	yes	characteristic
147	C₂×C₃×C₅	#237	1	1	yes	characteristic
148	C₂×C₃×C₅	C₂×#67	1	1	yes	characteristic
149	C₂×C₃×C₅	C₂×C₃×D₄	1	1	yes	characteristic
150	C₂×C₃×C₅	C₂×C₃×D₄	1	1	yes	characteristic
151	C₂×C₃×C₅	C₃×#41	1	1	yes	characteristic
152	C₅×D₃	—	2	2	no	—
153	C₂³×C₃	—	2	2	no	—
154	C₂³×C₃	—	3	1	no	—
155	C₂³×C₃	C₂×C₅×D₃	1	2	yes	—
156	C₂³×C₃	C₂²×C₃×C₅	1	2	yes	—
157	C₂²×D₃	C₂²×C₃×C₅	1	1	yes	characteristic
158	C₃×D₄	—	2	4	no	—
159	C₃×D₄	—	2	2	no	—
160	C₃×D₄	—	2	2	no	—
161	C₃×D₄	—	6	4	no	—
162	C₂×C₃×C₄	—	2	1	no	—
163	C₂×C₃×C₄	—	3	1	no	—
164	C₂×C₃×C₄	—	3	2	no	—
165	C₂×C₃×C₄	—	6	2	no	—
166	C₂×C₃×C₄	C₂×C₅×D₃	1	1	yes	characteristic
167	C₂×C₃×C₄	C₂²×C₃×C₅	1	1	yes	characteristic
168	#67	—	2	4	no	—
169	C₂×Dic₃	C₂²×C₃×C₅	1	2	yes	—
170	C₂×Dic₃	C₂²×C₃×C₅	1	1	yes	characteristic
171	C₄×D₃	—	2	2	no	—
172	C₂²×C₅	—	2	2	no	—
173	C₂²×C₅	—	2	2	no	—
174	C₂²×C₅	—	2	2	no	—
175	C₂²×C₅	—	3	2	no	—
176	C₂²×C₅	—	6	1	no	—
177	C₂²×C₅	—	6	1	no	—
178	C₂²×C₅	C₂²×C₃×D₃	1	1	yes	characteristic
179	C₄×C₅	—	6	2	no	—
180	C₄×C₅	—	6	1	no	—
181	C₄×C₅	C₃×#67	1	2	yes	complemented
182	C₂×C₃²	—	4	2	no	—
183	C₂×C₃²	C₂×C₅×D₄	1	1	yes	characteristic
184	C₂×C₃²	C₂×C₅×D₄	1	1	yes	characteristic
185	C₂×C₃²	C₅×#41	1	1	yes	characteristic
186	C₃×D₃	—	2	2	no	—
187	C₂×D₄	—	3	2	no	—
188	C₂×D₄	C₃×C₅×D₃	1	1	yes	characteristic, complemented
189	C₂²×C₄	—	3	1	no	—
190	C₄⋊C₄	—	3	1	no	—
191	C₂²⋊C₄	—	3	2	no	—
192	C₃×C₅	—	2	1	no	—
193	C₃×C₅	#938	1	1	yes	characteristic, direct product factor
194	C₃×C₅	C₃×#121	1	1	yes	characteristic, complemented
195	C₂²×C₃	—	2	4	no	—
196	C₂²×C₃	—	2	2	no	—
197	C₂²×C₃	—	2	2	no	—
198	C₂²×C₃	—	2	2	no	—
199	C₂²×C₃	—	2	2	no	—
200	C₂²×C₃	—	2	2	no	—
201	C₂²×C₃	—	2	2	no	—
202	C₂²×C₃	—	2	1	no	—
203	C₂²×C₃	—	3	2	no	—
204	C₂²×C₃	—	4	2	no	—
205	C₂²×C₃	—	4	2	no	—
206	C₂²×C₃	—	6	1	no	—
207	C₂²×C₃	—	6	1	no	—
208	C₂²×C₃	C₂²×C₅×D₃	1	1	yes	characteristic
209	C₂²×C₃	C₂³×C₃×C₅	1	1	yes	characteristic
210	C₂×D₃	—	2	1	no	—
211	C₂×D₃	—	2	1	no	—
212	C₂×D₃	C₃×C₅×D₄	1	2	yes	complemented
213	C₃×C₄	—	2	2	no	—
214	C₃×C₄	—	6	2	no	—
215	C₃×C₄	—	6	1	no	—
216	C₃×C₄	C₅×#67	1	2	yes	complemented
217	C₃×C₄	C₃×C₅×D₄	1	2	yes	complemented
218	Dic₃	—	2	1	no	—
219	Dic₃	—	2	2	no	—
220	C₂×C₅	—	4	2	no	—
221	C₂×C₅	—	6	2	no	—
222	C₂×C₅	C₃×D₃×D₄	1	1	yes	characteristic
223	C₂×C₅	C₃×#237	1	1	yes	characteristic
224	C₂×C₅	C₂×C₃×#67	1	1	yes	characteristic
225	C₃²	C₅×#121	1	1	yes	characteristic, complemented, Sylow
226	C₂³	—	3	1	no	—
227	C₂³	C₂×C₃×C₅×D₃	1	2	yes	—
228	D₄	—	2	2	no	—
229	D₄	—	6	4	no	—
230	C₂×C₄	—	3	1	no	—
231	C₂×C₄	—	3	2	no	—
232	C₂×C₄	—	6	2	no	—
233	C₂×C₄	C₂×C₃×C₅×D₃	1	1	yes	characteristic
234	C₂×C₃	—	2	1	no	—
235	C₂×C₃	—	2	1	no	—
236	C₂×C₃	—	2	1	no	—
237	C₂×C₃	—	4	4	no	—
238	C₂×C₃	—	4	2	no	—
239	C₂×C₃	—	4	2	no	—
240	C₂×C₃	—	6	2	no	—
241	C₂×C₃	C₅×D₃×D₄	1	1	yes	characteristic
242	C₂×C₃	C₅×#237	1	1	yes	characteristic
243	C₂×C₃	C₂×C₅×#67	1	1	yes	characteristic
244	C₂×C₃	C₂×C₃×C₅×D₄	1	1	yes	characteristic
245	C₂×C₃	C₂×C₃×C₅×D₄	1	1	yes	characteristic
246	C₂×C₃	C₃×C₅×#41	1	1	yes	characteristic
247	D₃	—	2	2	no	—
248	C₅	C₃×#938	1	1	yes	characteristic, direct product factor, Sylow
249	C₂²	—	2	2	no	—
250	C₂²	—	2	2	no	—
251	C₂²	—	2	2	no	—
252	C₂²	—	3	2	no	—
253	C₂²	—	6	1	no	—
254	C₂²	—	6	1	no	—
255	C₂²	C₂²×C₃×C₅×D₃	1	1	yes	characteristic
256	C₄	—	6	1	no	—
257	C₄	—	6	2	no	—
258	C₄	C₃×C₅×#67	1	2	yes	complemented
259	C₃	—	2	1	no	—
260	C₃	C₅×#938	1	1	yes	characteristic, direct product factor
261	C₃	C₃×C₅×#121	1	1	yes	characteristic, complemented
262	C₂	—	4	2	no	—
263	C₂	—	6	2	no	—
264	C₂	C₃×C₅×D₃×D₄	1	1	yes	characteristic
265	C₂	C₃×C₅×#237	1	1	yes	characteristic
266	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×#121
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. C₂×C₃²×C₅×D₄	C₂	semidirect
2. C₂×C₃×C₅×#67	C₂	semidirect
3. C₂×C₃×C₄×C₅×D₃	C₂	semidirect
4. C₃×C₅×#217	C₂	semidirect
5. C₃×C₅×C₄⋊Dic₃	C₂	semidirect
7. C₅×#938	C₃	direct (N × H)
16. C₃×#938	C₅	direct (N × H)
19. C₂×C₃×C₅×D₄	D₃	semidirect
20. C₂×C₃×C₅×D₄	C₂×C₃	semidirect
22. C₂×C₅×#67	C₂×C₃	semidirect
23. C₂×C₄×C₅×D₃	C₂×C₃	semidirect
26. C₅×#217	C₂×C₃	semidirect
27. C₅×C₄⋊Dic₃	C₂×C₃	semidirect
34. C₂×C₃×C₅×D₃	D₄	semidirect
35. C₃²×C₄×C₅	D₄	semidirect
39. C₂×C₃²×D₄	C₂×C₅	semidirect
40. C₂×C₃×#67	C₂×C₅	semidirect
41. C₂×C₃×C₄×D₃	C₂×C₅	semidirect
42. C₃×#217	C₂×C₅	semidirect
43. C₃×C₄⋊Dic₃	C₂×C₅	semidirect
64. #938	C₃×C₅	direct (N × H)
71. C₂×C₅×D₄	C₃×D₃	semidirect
100. C₂×C₅×D₃	C₃×D₄	semidirect
104. C₃×C₄×C₅	#67	semidirect
105. C₃×C₄×C₅	C₃×D₄	semidirect
110. C₂×C₃×D₄	C₅×D₃	semidirect
111. C₂×C₃×D₄	C₂×C₃×C₅	semidirect
113. C₂×#67	C₂×C₃×C₅	semidirect
114. C₂×C₄×D₃	C₂×C₃×C₅	semidirect
117. #217	C₂×C₃×C₅	semidirect
118. C₄⋊Dic₃	C₂×C₃×C₅	semidirect
119. C₃²×C₅	#121	semidirect
134. C₂×C₃×D₃	C₅×D₄	semidirect
135. C₃²×C₄	C₅×D₄	semidirect
181. C₄×C₅	C₃×#67	semidirect
188. C₂×D₄	C₃×C₅×D₃	semidirect
193. C₃×C₅	#938	direct (N × H)
194. C₃×C₅	C₃×#121	semidirect
212. C₂×D₃	C₃×C₅×D₄	semidirect
216. C₃×C₄	C₅×#67	semidirect
217. C₃×C₄	C₃×C₅×D₄	semidirect
225. C₃²	C₅×#121	semidirect
248. C₅	C₃×#938	direct (N × H)
258. C₄	C₃×C₅×#67	semidirect
260. C₃	C₅×#938	direct (N × H)
261. C₃	C₃×C₅×#121	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃²×C₅×D₄	2
C₂×C₃×C₅×#67	2
C₂×C₃×C₄×C₅×D₃	2
C₃×C₅×#217	2
C₃×C₅×C₄⋊Dic₃	2
C₃×C₅×#121	3
C₅×#938	3
C₃×#938	5

Subgroups — C3×C5×#938 #13087951

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#938 #13087951