Subgroups — C₃×C₅×#935 #13087945

Order 1440.

366 conjugacy classes of subgroups.

The subgroup lattice has 240 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
2	C₂²×C₃×C₅×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₅×#217	C₂	1	2	yes	maximal, complemented
4	C₃×C₅×#217	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₅×Dic₃⋊C₄	C₂	1	2	yes	maximal, complemented
6	C₃×C₅×#123	—	3	1	no	maximal
7	C₅×#935	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
10	C₃²×C₅×D₄	—	2	2	no	—
11	C₂×C₃²×C₄×C₅	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₅×Dic₃	—	2	2	no	—
13	C₂×C₃×C₅×Dic₃	C₂²	1	2	yes	—
14	C₂×C₃×C₅×Dic₃	C₂²	1	2	yes	—
15	C₃×#935	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
16	C₂×C₃×C₅×D₄	—	2	1	no	—
17	C₂×C₃×C₅×D₄	D₃	1	1	yes	characteristic
18	C₂×C₃×C₅×D₄	C₂×C₃	1	1	yes	characteristic
19	C₂²×C₃×C₄×C₅	—	3	1	no	—
20	C₂²×C₅×Dic₃	C₂×C₃	1	1	yes	characteristic, complemented
21	C₃×C₅×C₄⋊C₄	—	3	2	no	—
22	C₃×C₅×C₂²⋊C₄	—	3	1	no	—
23	C₃×C₅×C₂²⋊C₄	—	3	2	no	—
24	C₅×#217	C₂×C₃	1	1	yes	characteristic, complemented
25	C₅×#217	C₂×C₃	1	2	yes	complemented
26	C₅×Dic₃⋊C₄	C₂×C₃	1	2	yes	complemented
27	C₂²×C₃²×C₅	—	2	2	no	—
28	C₂²×C₃²×C₅	—	2	1	no	—
29	C₂²×C₃²×C₅	—	2	2	no	—
30	C₂²×C₃²×C₅	D₄	1	2	yes	—
31	C₂²×C₃²×C₅	C₂³	1	1	yes	characteristic
32	C₃²×C₄×C₅	—	2	1	no	—
33	C₃×C₅×Dic₃	—	2	2	no	—
34	C₃×C₅×Dic₃	—	2	2	no	—
35	C₅×#123	—	3	1	no	Hall
36	C₂×C₃²×D₄	C₂×C₅	1	1	yes	characteristic
37	C₂²×C₃×Dic₃	C₂×C₅	1	1	yes	characteristic, complemented
38	C₃×#217	C₂×C₅	1	1	yes	characteristic, complemented
39	C₃×#217	C₂×C₅	1	2	yes	complemented
40	C₃×Dic₃⋊C₄	C₂×C₅	1	2	yes	complemented
41	C₂³×C₃×C₅	—	2	1	no	—
42	C₂³×C₃×C₅	—	2	1	no	—
43	C₂³×C₃×C₅	C₂×D₃	1	1	yes	characteristic
44	C₂³×C₃×C₅	C₂×D₃	1	1	yes	characteristic
45	C₂³×C₃×C₅	C₂²×C₃	1	1	yes	characteristic
46	C₂³×C₃×C₅	C₂²×C₃	1	1	yes	characteristic
47	C₃×C₅×D₄	—	2	4	no	—
48	C₃×C₅×D₄	—	2	2	no	—
49	C₃×C₅×D₄	—	2	2	no	—
50	C₂×C₃×C₄×C₅	—	2	1	no	—
51	C₂×C₃×C₄×C₅	—	3	2	no	—
52	C₂×C₃×C₄×C₅	—	3	2	no	—
53	C₂×C₃×C₄×C₅	—	6	2	no	—
54	C₂×C₃×C₄×C₅	C₂×D₃	1	1	yes	characteristic
55	C₂×C₃×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
56	C₂×C₅×Dic₃	—	2	2	no	—
57	C₂×C₅×Dic₃	C₂²×C₃	1	2	yes	—
58	C₂×C₅×Dic₃	C₂²×C₃	1	2	yes	—
59	C₃×#123	—	3	1	no	—
60	#935	C₃×C₅	1	1	yes	characteristic, direct product factor
61	C₂×C₃²×C₅	—	2	2	no	—
62	C₂×C₃²×C₅	—	4	1	no	—
63	C₂×C₃²×C₅	C₂×D₄	1	1	yes	characteristic
64	C₂×C₃²×C₅	#41	1	2	yes	—
65	C₂×C₅×D₄	C₃×D₃	1	1	yes	characteristic
66	C₂²×C₄×C₅	—	3	1	no	—
67	C₅×C₄⋊C₄	—	3	2	no	—
68	C₅×C₂²⋊C₄	—	3	1	no	—
69	C₅×C₂²⋊C₄	—	3	2	no	—
70	C₂³×C₃²	C₂²×C₅	1	1	yes	characteristic
71	C₂³×C₃²	C₂²×C₅	1	1	yes	characteristic
72	C₃²×D₄	—	2	2	no	—
73	C₂×C₃²×C₄	C₂²×C₅	1	1	yes	characteristic
74	C₂×C₃×Dic₃	—	2	2	no	—
75	C₂×C₃×Dic₃	C₂²×C₅	1	2	yes	—
76	C₂×C₃×Dic₃	C₂²×C₅	1	2	yes	—
77	C₂²×C₃×C₅	—	2	2	no	—
78	C₂²×C₃×C₅	—	2	2	no	—
79	C₂²×C₃×C₅	—	2	2	no	—
80	C₂²×C₃×C₅	—	2	2	no	—
81	C₂²×C₃×C₅	—	2	1	no	—
82	C₂²×C₃×C₅	—	2	1	no	—
83	C₂²×C₃×C₅	—	2	2	no	—
84	C₂²×C₃×C₅	—	2	2	no	—
85	C₂²×C₃×C₅	—	2	1	no	—
86	C₂²×C₃×C₅	—	4	2	no	—
87	C₂²×C₃×C₅	—	4	2	no	—
88	C₂²×C₃×C₅	#67	1	2	yes	—
89	C₂²×C₃×C₅	C₃×D₄	1	2	yes	—
90	C₂²×C₃×C₅	C₂²×D₃	1	1	yes	characteristic
91	C₂²×C₃×C₅	C₂³×C₃	1	1	yes	characteristic
92	C₃×C₄×C₅	—	2	2	no	—
93	C₃×C₄×C₅	—	2	1	no	—
94	C₃×C₄×C₅	—	2	1	no	—
95	C₃×C₄×C₅	—	6	2	no	—
96	C₃×C₄×C₅	—	6	2	no	—
97	C₅×Dic₃	—	2	2	no	—
98	C₅×Dic₃	—	2	2	no	—
99	C₂×C₃×D₄	—	2	1	no	—
100	C₂×C₃×D₄	C₅×D₃	1	1	yes	characteristic
101	C₂×C₃×D₄	C₂×C₃×C₅	1	1	yes	characteristic
102	C₂²×C₃×C₄	—	3	1	no	—
103	C₂²×Dic₃	C₂×C₃×C₅	1	1	yes	characteristic, complemented
104	C₃×C₄⋊C₄	—	3	2	no	—
105	C₃×C₂²⋊C₄	—	3	1	no	—
106	C₃×C₂²⋊C₄	—	3	2	no	—
107	#217	C₂×C₃×C₅	1	1	yes	characteristic, complemented
108	#217	C₂×C₃×C₅	1	2	yes	complemented
109	Dic₃⋊C₄	C₂×C₃×C₅	1	2	yes	complemented
110	C₃²×C₅	#123	1	1	yes	characteristic, complemented, Hall
111	C₂³×C₅	C₂×C₃×D₃	1	1	yes	characteristic
112	C₂³×C₅	C₂×C₃×D₃	1	1	yes	characteristic
113	C₅×D₄	—	2	2	no	—
114	C₂×C₄×C₅	—	3	2	no	—
115	C₂×C₄×C₅	—	3	2	no	—
116	C₂×C₄×C₅	—	6	2	no	—
117	C₂×C₄×C₅	C₂×C₃×D₃	1	1	yes	characteristic
118	C₂²×C₃²	—	2	1	no	—
119	C₂²×C₃²	—	2	2	no	—
120	C₂²×C₃²	—	2	2	no	—
121	C₂²×C₃²	C₅×D₄	1	2	yes	—
122	C₂²×C₃²	C₂³×C₅	1	1	yes	characteristic
123	C₃²×C₄	—	2	1	no	—
124	C₃×Dic₃	—	2	2	no	—
125	C₃×Dic₃	—	2	2	no	—
126	#123	—	3	1	no	Sylow
127	C₂×C₃×C₅	—	2	2	no	—
128	C₂×C₃×C₅	—	2	2	no	—
129	C₂×C₃×C₅	—	2	2	no	—
130	C₂×C₃×C₅	—	2	1	no	—
131	C₂×C₃×C₅	—	4	2	no	—
132	C₂×C₃×C₅	—	4	2	no	—
133	C₂×C₃×C₅	—	4	1	no	—
134	C₂×C₃×C₅	—	4	1	no	—
135	C₂×C₃×C₅	#237	1	2	yes	—
136	C₂×C₃×C₅	C₂×#67	1	1	yes	characteristic
137	C₂×C₃×C₅	C₂×C₃×D₄	1	1	yes	characteristic
138	C₂×C₃×C₅	C₃×#41	1	2	yes	—
139	C₂³×C₃	—	2	1	no	—
140	C₂³×C₃	—	2	1	no	—
141	C₂³×C₃	C₂×C₅×D₃	1	1	yes	characteristic
142	C₂³×C₃	C₂×C₅×D₃	1	1	yes	characteristic
143	C₂³×C₃	C₂²×C₃×C₅	1	1	yes	characteristic
144	C₂³×C₃	C₂²×C₃×C₅	1	1	yes	characteristic
145	C₃×D₄	—	2	4	no	—
146	C₃×D₄	—	2	2	no	—
147	C₃×D₄	—	2	2	no	—
148	C₂×C₃×C₄	—	2	1	no	—
149	C₂×C₃×C₄	—	3	2	no	—
150	C₂×C₃×C₄	—	3	2	no	—
151	C₂×C₃×C₄	—	6	2	no	—
152	C₂×C₃×C₄	C₂×C₅×D₃	1	1	yes	characteristic
153	C₂×C₃×C₄	C₂²×C₃×C₅	1	1	yes	characteristic
154	C₂×Dic₃	—	2	2	no	—
155	C₂×Dic₃	C₂²×C₃×C₅	1	2	yes	—
156	C₂×Dic₃	C₂²×C₃×C₅	1	2	yes	—
157	C₂²×C₅	—	2	2	no	—
158	C₂²×C₅	—	2	2	no	—
159	C₂²×C₅	—	2	1	no	—
160	C₂²×C₅	C₃×#67	1	2	yes	—
161	C₂²×C₅	C₂²×C₃×D₃	1	1	yes	characteristic
162	C₄×C₅	—	2	1	no	—
163	C₄×C₅	—	6	2	no	—
164	C₄×C₅	—	6	2	no	—
165	C₂×C₃²	—	2	2	no	—
166	C₂×C₃²	—	4	1	no	—
167	C₂×C₃²	C₂×C₅×D₄	1	1	yes	characteristic
168	C₂×C₃²	C₅×#41	1	2	yes	—
169	C₂×D₄	C₃×C₅×D₃	1	1	yes	characteristic
170	C₂²×C₄	—	3	1	no	—
171	C₄⋊C₄	—	3	2	no	—
172	C₂²⋊C₄	—	3	1	no	—
173	C₂²⋊C₄	—	3	2	no	—
174	C₃×C₅	—	2	1	no	—
175	C₃×C₅	#935	1	1	yes	characteristic, direct product factor
176	C₃×C₅	C₃×#123	1	1	yes	characteristic, complemented
177	C₂²×C₃	—	2	2	no	—
178	C₂²×C₃	—	2	2	no	—
179	C₂²×C₃	—	2	2	no	—
180	C₂²×C₃	—	2	2	no	—
181	C₂²×C₃	—	2	1	no	—
182	C₂²×C₃	—	2	1	no	—
183	C₂²×C₃	—	2	2	no	—
184	C₂²×C₃	—	2	2	no	—
185	C₂²×C₃	—	2	1	no	—
186	C₂²×C₃	—	4	2	no	—
187	C₂²×C₃	—	4	2	no	—
188	C₂²×C₃	C₅×#67	1	2	yes	—
189	C₂²×C₃	C₃×C₅×D₄	1	2	yes	—
190	C₂²×C₃	C₂²×C₅×D₃	1	1	yes	characteristic
191	C₂²×C₃	C₂³×C₃×C₅	1	1	yes	characteristic
192	C₃×C₄	—	2	2	no	—
193	C₃×C₄	—	2	1	no	—
194	C₃×C₄	—	2	1	no	—
195	C₃×C₄	—	6	2	no	—
196	C₃×C₄	—	6	2	no	—
197	Dic₃	—	2	2	no	—
198	Dic₃	—	2	2	no	—
199	C₂×C₅	—	2	2	no	—
200	C₂×C₅	—	4	1	no	—
201	C₂×C₅	C₃×#237	1	2	yes	—
202	C₂×C₅	C₂×C₃×#67	1	1	yes	characteristic
203	C₃²	C₅×#123	1	1	yes	characteristic, complemented, Sylow
204	C₂³	C₂×C₃×C₅×D₃	1	1	yes	characteristic
205	C₂³	C₂×C₃×C₅×D₃	1	1	yes	characteristic
206	D₄	—	2	2	no	—
207	C₂×C₄	—	3	2	no	—
208	C₂×C₄	—	3	2	no	—
209	C₂×C₄	—	6	2	no	—
210	C₂×C₄	C₂×C₃×C₅×D₃	1	1	yes	characteristic
211	C₂×C₃	—	2	2	no	—
212	C₂×C₃	—	2	2	no	—
213	C₂×C₃	—	2	1	no	—
214	C₂×C₃	—	2	2	no	—
215	C₂×C₃	—	4	2	no	—
216	C₂×C₃	—	4	2	no	—
217	C₂×C₃	—	4	1	no	—
218	C₂×C₃	—	4	1	no	—
219	C₂×C₃	C₅×#237	1	2	yes	—
220	C₂×C₃	C₂×C₅×#67	1	1	yes	characteristic
221	C₂×C₃	C₂×C₃×C₅×D₄	1	1	yes	characteristic
222	C₂×C₃	C₃×C₅×#41	1	2	yes	—
223	C₅	C₃×#935	1	1	yes	characteristic, direct product factor, Sylow
224	C₂²	—	2	2	no	—
225	C₂²	—	2	1	no	—
226	C₂²	—	2	2	no	—
227	C₂²	C₃×C₅×#67	1	2	yes	—
228	C₂²	C₂²×C₃×C₅×D₃	1	1	yes	characteristic
229	C₄	—	2	1	no	—
230	C₄	—	6	2	no	—
231	C₄	—	6	2	no	—
232	C₃	—	2	1	no	—
233	C₃	C₅×#935	1	1	yes	characteristic, direct product factor
234	C₃	C₃×C₅×#123	1	1	yes	characteristic, complemented
235	C₂	—	2	2	no	—
236	C₂	—	4	1	no	—
237	C₂	C₃×C₅×#237	1	2	yes	—
238	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, prime radical	C₂×C₃²×C₅×D₄	C₂
socle	C₂²×C₃²×C₅	C₂³
hypercenter	C₂×C₃×C₅×D₄	D₃
hypocenter	C₃	C₃×C₅×#123
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
2. C₂²×C₃×C₅×Dic₃	C₂	semidirect
3. C₃×C₅×#217	C₂	semidirect
4. C₃×C₅×#217	C₂	semidirect
5. C₃×C₅×Dic₃⋊C₄	C₂	semidirect
7. C₅×#935	C₃	direct (N × H)
15. C₃×#935	C₅	direct (N × H)
20. C₂²×C₅×Dic₃	C₂×C₃	semidirect
24. C₅×#217	C₂×C₃	semidirect
25. C₅×#217	C₂×C₃	semidirect
26. C₅×Dic₃⋊C₄	C₂×C₃	semidirect
37. C₂²×C₃×Dic₃	C₂×C₅	semidirect
38. C₃×#217	C₂×C₅	semidirect
39. C₃×#217	C₂×C₅	semidirect
40. C₃×Dic₃⋊C₄	C₂×C₅	semidirect
60. #935	C₃×C₅	direct (N × H)
103. C₂²×Dic₃	C₂×C₃×C₅	semidirect
107. #217	C₂×C₃×C₅	semidirect
108. #217	C₂×C₃×C₅	semidirect
109. Dic₃⋊C₄	C₂×C₃×C₅	semidirect
110. C₃²×C₅	#123	semidirect
175. C₃×C₅	#935	direct (N × H)
176. C₃×C₅	C₃×#123	semidirect
203. C₃²	C₅×#123	semidirect
223. C₅	C₃×#935	direct (N × H)
233. C₃	C₅×#935	direct (N × H)
234. C₃	C₃×C₅×#123	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
C₂²×C₃×C₅×Dic₃	2
C₃×C₅×#217	2
C₃×C₅×Dic₃⋊C₄	2
C₃×C₅×#123	3
C₅×#935	3
C₃×#935	5

Subgroups — C3×C5×#935 #13087945

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#935 #13087945