Subgroups — C₃×C₅×#892 #13087904

Order 1440.

330 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×C₅×D₃	C₂	1	1	yes	maximal, characteristic
2	C₃²×C₅×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₅×(C₂×D₃)⋊C₄	C₂	1	2	yes	maximal
4	C₃×C₅×C₄⋊Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₄×C₅×Dic₃	C₂	1	2	yes	maximal, complemented
6	C₃×C₅×#117	—	3	1	no	maximal
7	C₅×#892	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂²×C₃×C₅×D₃	C₂²	1	1	yes	characteristic
9	C₂×C₃²×C₄×C₅	C₂²	1	2	yes	—
10	C₂×C₃²×C₄×C₅	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₅×Dic₃	C₂²	1	2	yes	—
12	C₂×C₃×C₅×Dic₃	C₂²	1	1	yes	characteristic
13	C₃×C₄×C₅×D₃	C₄	1	4	yes	complemented
14	C₃×#892	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
15	C₂²×C₃×C₄×C₅	—	3	1	no	—
16	C₂×C₄×C₅×D₃	C₂×C₃	1	1	yes	characteristic
17	C₃×C₅×C₄⋊C₄	—	2	1	no	—
18	C₃×C₅×C₄⋊C₄	—	3	1	no	—
19	C₃×C₅×C₄⋊C₄	D₃	1	1	yes	characteristic, complemented
20	C₃×C₅×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic, complemented
21	C₃×C₅×C₂²⋊C₄	—	3	2	no	—
22	C₃×C₄²×C₅	—	3	2	no	—
23	C₅×(C₂×D₃)⋊C₄	C₂×C₃	1	2	yes	—
24	C₅×C₄⋊Dic₃	C₂×C₃	1	1	yes	characteristic, complemented
25	C₄×C₅×Dic₃	C₂×C₃	1	2	yes	complemented
26	C₂²×C₃²×C₅	C₂³	1	1	yes	characteristic
27	C₂×C₃×C₅×D₃	—	2	1	no	—
28	C₂×C₃×C₅×D₃	—	2	1	no	—
29	C₂×C₃×C₅×D₃	C₂×C₄	1	2	yes	—
30	C₃²×C₄×C₅	—	2	2	no	—
31	C₃²×C₄×C₅	C₂×C₄	1	2	yes	—
32	C₃×C₅×Dic₃	—	2	2	no	—
33	C₃×C₅×Dic₃	C₂×C₄	1	2	yes	—
34	C₅×#117	—	3	1	no	Hall
35	C₂×C₃×C₄×D₃	C₂×C₅	1	1	yes	characteristic
36	C₃²×C₄⋊C₄	C₂×C₅	1	1	yes	characteristic, complemented
37	C₃×(C₂×D₃)⋊C₄	C₂×C₅	1	2	yes	—
38	C₃×C₄⋊Dic₃	C₂×C₅	1	1	yes	characteristic, complemented
39	C₃×C₄×Dic₃	C₂×C₅	1	2	yes	complemented
40	C₂³×C₃×C₅	—	3	1	no	—
41	C₂²×C₅×D₃	C₂²×C₃	1	1	yes	characteristic
42	C₂×C₃×C₄×C₅	—	2	1	no	—
43	C₂×C₃×C₄×C₅	—	2	2	no	—
44	C₂×C₃×C₄×C₅	—	3	1	no	—
45	C₂×C₃×C₄×C₅	—	3	4	no	—
46	C₂×C₃×C₄×C₅	—	3	2	no	—
47	C₂×C₃×C₄×C₅	C₂×D₃	1	2	yes	—
48	C₂×C₃×C₄×C₅	C₂×D₃	1	1	yes	characteristic
49	C₂×C₃×C₄×C₅	C₂²×C₃	1	2	yes	—
50	C₂×C₃×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
51	C₂×C₅×Dic₃	C₂²×C₃	1	2	yes	—
52	C₂×C₅×Dic₃	C₂²×C₃	1	1	yes	characteristic
53	C₄×C₅×D₃	C₃×C₄	1	4	yes	complemented
54	C₃×#117	—	3	1	no	—
55	#892	C₃×C₅	1	1	yes	characteristic, direct product factor
56	C₂×C₃²×C₅	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₃²×C₅	#41	1	1	yes	characteristic
58	C₂×C₃²×C₅	#41	1	1	yes	characteristic
59	C₃×C₅×D₃	—	2	2	no	—
60	C₂²×C₄×C₅	—	3	1	no	—
61	C₅×C₄⋊C₄	—	3	1	no	—
62	C₅×C₄⋊C₄	C₃×D₃	1	1	yes	characteristic, complemented
63	C₅×C₂²⋊C₄	—	3	2	no	—
64	C₄²×C₅	—	3	2	no	—
65	C₂²×C₃×D₃	C₂²×C₅	1	1	yes	characteristic
66	C₂×C₃²×C₄	C₂²×C₅	1	2	yes	—
67	C₂×C₃²×C₄	C₂²×C₅	1	1	yes	characteristic
68	C₂×C₃×Dic₃	C₂²×C₅	1	2	yes	—
69	C₂×C₃×Dic₃	C₂²×C₅	1	1	yes	characteristic
70	C₃×C₄×D₃	C₄×C₅	1	4	yes	complemented
71	C₂²×C₃×C₅	—	2	1	no	—
72	C₂²×C₃×C₅	—	3	2	no	—
73	C₂²×C₃×C₅	—	6	1	no	—
74	C₂²×C₃×C₅	—	6	1	no	—
75	C₂²×C₃×C₅	C₂²×D₃	1	1	yes	characteristic
76	C₂²×C₃×C₅	C₂³×C₃	1	1	yes	characteristic
77	C₂×C₅×D₃	—	2	1	no	—
78	C₂×C₅×D₃	—	2	1	no	—
79	C₂×C₅×D₃	C₂×C₃×C₄	1	2	yes	—
80	C₃×C₄×C₅	—	2	2	no	—
81	C₃×C₄×C₅	—	2	2	no	—
82	C₃×C₄×C₅	—	2	2	no	—
83	C₃×C₄×C₅	—	3	2	no	—
84	C₃×C₄×C₅	—	4	2	no	—
85	C₃×C₄×C₅	—	6	2	no	—
86	C₃×C₄×C₅	C₄×D₃	1	2	yes	—
87	C₃×C₄×C₅	C₂×C₃×C₄	1	2	yes	—
88	C₅×Dic₃	—	2	2	no	—
89	C₅×Dic₃	C₂×C₃×C₄	1	2	yes	—
90	C₂²×C₃×C₄	—	3	1	no	—
91	C₂×C₄×D₃	C₂×C₃×C₅	1	1	yes	characteristic
92	C₃×C₄⋊C₄	—	2	1	no	—
93	C₃×C₄⋊C₄	—	3	1	no	—
94	C₃×C₄⋊C₄	C₅×D₃	1	1	yes	characteristic, complemented
95	C₃×C₄⋊C₄	C₂×C₃×C₅	1	1	yes	characteristic, complemented
96	C₃×C₂²⋊C₄	—	3	2	no	—
97	C₃×C₄²	—	3	2	no	—
98	(C₂×D₃)⋊C₄	C₂×C₃×C₅	1	2	yes	—
99	C₄⋊Dic₃	C₂×C₃×C₅	1	1	yes	characteristic, complemented
100	C₄×Dic₃	C₂×C₃×C₅	1	2	yes	complemented
101	C₃²×C₅	#117	1	1	yes	characteristic, complemented, Hall
102	C₂³×C₅	—	3	1	no	—
103	C₂×C₄×C₅	—	3	1	no	—
104	C₂×C₄×C₅	—	3	4	no	—
105	C₂×C₄×C₅	—	3	2	no	—
106	C₂×C₄×C₅	C₂×C₃×D₃	1	2	yes	—
107	C₂×C₄×C₅	C₂×C₃×D₃	1	1	yes	characteristic
108	C₂²×C₃²	C₂³×C₅	1	1	yes	characteristic
109	C₂×C₃×D₃	—	2	1	no	—
110	C₂×C₃×D₃	—	2	1	no	—
111	C₂×C₃×D₃	C₂×C₄×C₅	1	2	yes	—
112	C₃²×C₄	—	2	2	no	—
113	C₃²×C₄	C₂×C₄×C₅	1	2	yes	—
114	C₃×Dic₃	—	2	2	no	—
115	C₃×Dic₃	C₂×C₄×C₅	1	2	yes	—
116	#117	—	3	1	no	Sylow
117	C₂×C₃×C₅	—	2	1	no	—
118	C₂×C₃×C₅	—	2	1	no	—
119	C₂×C₃×C₅	—	2	1	no	—
120	C₂×C₃×C₅	—	6	2	no	—
121	C₂×C₃×C₅	C₂×C₄×D₃	1	1	yes	characteristic
122	C₂×C₃×C₅	#237	1	1	yes	characteristic
123	C₂×C₃×C₅	#239	1	1	yes	characteristic
124	C₂×C₃×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
125	C₂×C₃×C₅	C₃×#41	1	1	yes	characteristic
126	C₂×C₃×C₅	C₃×#41	1	1	yes	characteristic
127	C₅×D₃	—	2	2	no	—
128	C₂³×C₃	—	3	1	no	—
129	C₂²×D₃	C₂²×C₃×C₅	1	1	yes	characteristic
130	C₂×C₃×C₄	—	2	1	no	—
131	C₂×C₃×C₄	—	2	2	no	—
132	C₂×C₃×C₄	—	3	1	no	—
133	C₂×C₃×C₄	—	3	4	no	—
134	C₂×C₃×C₄	—	3	2	no	—
135	C₂×C₃×C₄	C₂×C₅×D₃	1	2	yes	—
136	C₂×C₃×C₄	C₂×C₅×D₃	1	1	yes	characteristic
137	C₂×C₃×C₄	C₂²×C₃×C₅	1	2	yes	—
138	C₂×C₃×C₄	C₂²×C₃×C₅	1	1	yes	characteristic
139	C₂×Dic₃	C₂²×C₃×C₅	1	2	yes	—
140	C₂×Dic₃	C₂²×C₃×C₅	1	1	yes	characteristic
141	C₄×D₃	C₃×C₄×C₅	1	4	yes	complemented
142	C₂²×C₅	—	3	2	no	—
143	C₂²×C₅	—	6	1	no	—
144	C₂²×C₅	—	6	1	no	—
145	C₂²×C₅	C₂²×C₃×D₃	1	1	yes	characteristic
146	C₄×C₅	—	2	2	no	—
147	C₄×C₅	—	3	2	no	—
148	C₄×C₅	—	6	2	no	—
149	C₄×C₅	C₃×C₄×D₃	1	2	yes	—
150	C₂×C₃²	C₂²×C₄×C₅	1	1	yes	characteristic
151	C₂×C₃²	C₅×#41	1	1	yes	characteristic
152	C₂×C₃²	C₅×#41	1	1	yes	characteristic
153	C₃×D₃	—	2	2	no	—
154	C₂²×C₄	—	3	1	no	—
155	C₄⋊C₄	—	3	1	no	—
156	C₄⋊C₄	C₃×C₅×D₃	1	1	yes	characteristic, complemented
157	C₂²⋊C₄	—	3	2	no	—
158	C₄²	—	3	2	no	—
159	C₃×C₅	—	2	1	no	—
160	C₃×C₅	#892	1	1	yes	characteristic, direct product factor
161	C₃×C₅	C₃×#117	1	1	yes	characteristic, complemented
162	C₂²×C₃	—	2	1	no	—
163	C₂²×C₃	—	3	2	no	—
164	C₂²×C₃	—	6	1	no	—
165	C₂²×C₃	—	6	1	no	—
166	C₂²×C₃	C₂²×C₅×D₃	1	1	yes	characteristic
167	C₂²×C₃	C₂³×C₃×C₅	1	1	yes	characteristic
168	C₂×D₃	—	2	1	no	—
169	C₂×D₃	—	2	1	no	—
170	C₂×D₃	C₂×C₃×C₄×C₅	1	2	yes	—
171	C₃×C₄	—	2	2	no	—
172	C₃×C₄	—	2	2	no	—
173	C₃×C₄	—	2	2	no	—
174	C₃×C₄	—	3	2	no	—
175	C₃×C₄	—	4	2	no	—
176	C₃×C₄	—	6	2	no	—
177	C₃×C₄	C₄×C₅×D₃	1	2	yes	—
178	C₃×C₄	C₂×C₃×C₄×C₅	1	2	yes	—
179	Dic₃	—	2	2	no	—
180	Dic₃	C₂×C₃×C₄×C₅	1	2	yes	—
181	C₂×C₅	—	6	2	no	—
182	C₂×C₅	C₂×C₃×C₄×D₃	1	1	yes	characteristic
183	C₂×C₅	C₃×#237	1	1	yes	characteristic
184	C₂×C₅	C₃×#239	1	1	yes	characteristic
185	C₃²	C₅×#117	1	1	yes	characteristic, complemented, Sylow
186	C₂³	—	3	1	no	—
187	C₂×C₄	—	3	1	no	—
188	C₂×C₄	—	3	4	no	—
189	C₂×C₄	—	3	2	no	—
190	C₂×C₄	C₂×C₃×C₅×D₃	1	2	yes	—
191	C₂×C₄	C₂×C₃×C₅×D₃	1	1	yes	characteristic
192	C₂×C₃	—	2	1	no	—
193	C₂×C₃	—	2	1	no	—
194	C₂×C₃	—	2	1	no	—
195	C₂×C₃	—	6	2	no	—
196	C₂×C₃	C₂×C₄×C₅×D₃	1	1	yes	characteristic
197	C₂×C₃	C₅×#237	1	1	yes	characteristic
198	C₂×C₃	C₅×#239	1	1	yes	characteristic
199	C₂×C₃	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
200	C₂×C₃	C₃×C₅×#41	1	1	yes	characteristic
201	C₂×C₃	C₃×C₅×#41	1	1	yes	characteristic
202	D₃	—	2	2	no	—
203	C₅	C₃×#892	1	1	yes	characteristic, direct product factor, Sylow
204	C₂²	—	3	2	no	—
205	C₂²	—	6	1	no	—
206	C₂²	—	6	1	no	—
207	C₂²	C₂²×C₃×C₅×D₃	1	1	yes	characteristic
208	C₄	—	2	2	no	—
209	C₄	—	3	2	no	—
210	C₄	—	6	2	no	—
211	C₄	C₃×C₄×C₅×D₃	1	2	yes	—
212	C₃	—	2	1	no	—
213	C₃	C₅×#892	1	1	yes	characteristic, direct product factor
214	C₃	C₃×C₅×#117	1	1	yes	characteristic, complemented
215	C₂	—	6	2	no	—
216	C₂	C₂×C₃×C₄×C₅×D₃	1	1	yes	characteristic
217	C₂	C₃×C₅×#237	1	1	yes	characteristic
218	C₂	C₃×C₅×#239	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃×C₅	C₂²×D₃
commutator	C₂×C₃	C₂²×C₃×C₄×C₅
Frattini	C₂²	C₂²×C₃×C₅×D₃
Fitting	C₃²×C₅×C₄⋊C₄	C₂
socle	C₂²×C₃²×C₅	C₂³
prime radical	C₂²×C₃×C₅×D₃	C₂²
cosocle	C₂²×C₃	C₂³×C₃×C₅
hypercenter	C₃×C₅×C₄⋊C₄	D₃
hypocenter	C₃	C₃×C₅×#117
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃×C₅ ◁ C₃×C₅×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
2. C₃²×C₅×C₄⋊C₄	C₂	semidirect
4. C₃×C₅×C₄⋊Dic₃	C₂	semidirect
5. C₃×C₄×C₅×Dic₃	C₂	semidirect
7. C₅×#892	C₃	direct (N × H)
13. C₃×C₄×C₅×D₃	C₄	semidirect
14. C₃×#892	C₅	direct (N × H)
19. C₃×C₅×C₄⋊C₄	D₃	semidirect
20. C₃×C₅×C₄⋊C₄	C₂×C₃	semidirect
24. C₅×C₄⋊Dic₃	C₂×C₃	semidirect
25. C₄×C₅×Dic₃	C₂×C₃	semidirect
36. C₃²×C₄⋊C₄	C₂×C₅	semidirect
38. C₃×C₄⋊Dic₃	C₂×C₅	semidirect
39. C₃×C₄×Dic₃	C₂×C₅	semidirect
53. C₄×C₅×D₃	C₃×C₄	semidirect
55. #892	C₃×C₅	direct (N × H)
62. C₅×C₄⋊C₄	C₃×D₃	semidirect
70. C₃×C₄×D₃	C₄×C₅	semidirect
94. C₃×C₄⋊C₄	C₅×D₃	semidirect
95. C₃×C₄⋊C₄	C₂×C₃×C₅	semidirect
99. C₄⋊Dic₃	C₂×C₃×C₅	semidirect
100. C₄×Dic₃	C₂×C₃×C₅	semidirect
101. C₃²×C₅	#117	semidirect
141. C₄×D₃	C₃×C₄×C₅	semidirect
156. C₄⋊C₄	C₃×C₅×D₃	semidirect
160. C₃×C₅	#892	direct (N × H)
161. C₃×C₅	C₃×#117	semidirect
185. C₃²	C₅×#117	semidirect
203. C₅	C₃×#892	direct (N × H)
213. C₃	C₅×#892	direct (N × H)
214. C₃	C₃×C₅×#117	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₄×C₅×D₃	2
C₃²×C₅×C₄⋊C₄	2
C₃×C₅×(C₂×D₃)⋊C₄	2
C₃×C₅×C₄⋊Dic₃	2
C₃×C₄×C₅×Dic₃	2
C₃×C₅×#117	3
C₅×#892	3
C₃×#892	5

Subgroups — C3×C5×#892 #13087904

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#892 #13087904