Subgroups — C₃×C₅×#889 #13087901

Order 1440.

250 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	C₃²×C₅×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×C₄⋊Dic₃	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×Dic₃⋊C₄	C₂	1	1	yes	maximal, characteristic
4	C₃×C₅×Dic₃⋊C₄	C₂	1	1	yes	maximal, characteristic
5	C₃×C₅×Dic₃⋊C₄	C₂	1	1	yes	maximal, characteristic
6	C₃×C₅×Dic₃⋊C₄	C₂	1	1	yes	maximal, characteristic
7	C₃×C₄×C₅×Dic₃	C₂	1	1	yes	maximal, characteristic
8	C₃×C₅×#125	—	3	1	no	maximal
9	C₅×#889	C₃	1	1	yes	maximal, characteristic, direct product factor
10	C₂×C₃²×C₄×C₅	C₂²	1	1	yes	characteristic
11	C₂×C₃²×C₄×C₅	C₂²	1	1	yes	characteristic
12	C₂×C₃²×C₄×C₅	C₂²	1	1	yes	characteristic
13	C₂×C₃×C₅×Dic₃	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₅×Dic₃	C₂²	1	1	yes	characteristic
15	C₂×C₃×C₅×Dic₃	C₂²	1	1	yes	characteristic
16	C₂×C₃×C₅×Dic₃	C₂²	1	1	yes	characteristic
17	C₃×#889	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
18	C₃×C₅×C₄⋊C₄	—	2	1	no	—
19	C₃×C₅×C₄⋊C₄	—	3	1	no	—
20	C₃×C₅×C₄⋊C₄	—	3	1	no	—
21	C₃×C₅×C₄⋊C₄	—	3	1	no	—
22	C₃×C₅×C₄⋊C₄	—	3	1	no	—
23	C₃×C₅×C₄⋊C₄	—	3	1	no	—
24	C₃×C₅×C₄⋊C₄	D₃	1	1	yes	characteristic
25	C₃×C₅×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic
26	C₃×C₄²×C₅	—	3	1	no	—
27	C₅×C₄⋊Dic₃	C₂×C₃	1	1	yes	characteristic
28	C₅×Dic₃⋊C₄	C₂×C₃	1	1	yes	characteristic
29	C₅×Dic₃⋊C₄	C₂×C₃	1	1	yes	characteristic
30	C₅×Dic₃⋊C₄	C₂×C₃	1	1	yes	characteristic
31	C₅×Dic₃⋊C₄	C₂×C₃	1	1	yes	characteristic
32	C₄×C₅×Dic₃	C₂×C₃	1	1	yes	characteristic
33	C₂²×C₃²×C₅	C₂³	1	1	yes	characteristic
34	C₃²×C₄×C₅	—	2	1	no	—
35	C₃²×C₄×C₅	—	2	1	no	—
36	C₃²×C₄×C₅	—	2	1	no	—
37	C₃×C₅×Dic₃	—	2	1	no	—
38	C₃×C₅×Dic₃	—	2	1	no	—
39	C₃×C₅×Dic₃	—	2	1	no	—
40	C₃×C₅×Dic₃	Q₈	1	2	yes	—
41	C₅×#125	—	3	1	no	Hall
42	C₃²×C₄⋊C₄	C₂×C₅	1	1	yes	characteristic
43	C₃×C₄⋊Dic₃	C₂×C₅	1	1	yes	characteristic
44	C₃×Dic₃⋊C₄	C₂×C₅	1	1	yes	characteristic
45	C₃×Dic₃⋊C₄	C₂×C₅	1	1	yes	characteristic
46	C₃×Dic₃⋊C₄	C₂×C₅	1	1	yes	characteristic
47	C₃×Dic₃⋊C₄	C₂×C₅	1	1	yes	characteristic
48	C₃×C₄×Dic₃	C₂×C₅	1	1	yes	characteristic
49	C₂×C₃×C₄×C₅	—	2	1	no	—
50	C₂×C₃×C₄×C₅	—	2	1	no	—
51	C₂×C₃×C₄×C₅	—	2	1	no	—
52	C₂×C₃×C₄×C₅	—	3	1	no	—
53	C₂×C₃×C₄×C₅	—	3	1	no	—
54	C₂×C₃×C₄×C₅	—	3	1	no	—
55	C₂×C₃×C₄×C₅	—	3	1	no	—
56	C₂×C₃×C₄×C₅	C₂×D₃	1	1	yes	characteristic
57	C₂×C₃×C₄×C₅	C₂×D₃	1	1	yes	characteristic
58	C₂×C₃×C₄×C₅	C₂×D₃	1	1	yes	characteristic
59	C₂×C₃×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
60	C₂×C₃×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
61	C₂×C₃×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
62	C₂×C₅×Dic₃	C₂²×C₃	1	1	yes	characteristic
63	C₂×C₅×Dic₃	C₂²×C₃	1	1	yes	characteristic
64	C₂×C₅×Dic₃	C₂²×C₃	1	1	yes	characteristic
65	C₂×C₅×Dic₃	C₂²×C₃	1	1	yes	characteristic
66	C₃×#125	—	3	1	no	—
67	#889	C₃×C₅	1	1	yes	characteristic, direct product factor
68	C₂×C₃²×C₅	C₂×Q₈	1	1	yes	characteristic
69	C₂×C₃²×C₅	#41	1	1	yes	characteristic
70	C₂×C₃²×C₅	#41	1	1	yes	characteristic
71	C₅×C₄⋊C₄	—	3	1	no	—
72	C₅×C₄⋊C₄	—	3	1	no	—
73	C₅×C₄⋊C₄	—	3	1	no	—
74	C₅×C₄⋊C₄	—	3	1	no	—
75	C₅×C₄⋊C₄	—	3	1	no	—
76	C₅×C₄⋊C₄	C₃×D₃	1	1	yes	characteristic
77	C₄²×C₅	—	3	1	no	—
78	C₂×C₃²×C₄	C₂²×C₅	1	1	yes	characteristic
79	C₂×C₃²×C₄	C₂²×C₅	1	1	yes	characteristic
80	C₂×C₃²×C₄	C₂²×C₅	1	1	yes	characteristic
81	C₂×C₃×Dic₃	C₂²×C₅	1	1	yes	characteristic
82	C₂×C₃×Dic₃	C₂²×C₅	1	1	yes	characteristic
83	C₂×C₃×Dic₃	C₂²×C₅	1	1	yes	characteristic
84	C₂×C₃×Dic₃	C₂²×C₅	1	1	yes	characteristic
85	C₂²×C₃×C₅	—	2	1	no	—
86	C₂²×C₃×C₅	C₂²×D₃	1	1	yes	characteristic
87	C₂²×C₃×C₅	C₂³×C₃	1	1	yes	characteristic
88	C₃×C₄×C₅	—	2	2	no	—
89	C₃×C₄×C₅	—	2	1	no	—
90	C₃×C₄×C₅	—	2	1	no	—
91	C₃×C₄×C₅	—	2	1	no	—
92	C₃×C₄×C₅	—	2	1	no	—
93	C₃×C₄×C₅	—	2	1	no	—
94	C₃×C₄×C₅	—	2	1	no	—
95	C₃×C₄×C₅	—	3	2	no	—
96	C₃×C₄×C₅	—	4	1	no	—
97	C₃×C₄×C₅	—	4	1	no	—
98	C₃×C₄×C₅	—	6	1	no	—
99	C₃×C₄×C₅	—	6	1	no	—
100	C₃×C₄×C₅	—	6	1	no	—
101	C₅×Dic₃	—	2	1	no	—
102	C₅×Dic₃	—	2	1	no	—
103	C₅×Dic₃	—	2	1	no	—
104	C₅×Dic₃	C₃×Q₈	1	2	yes	—
105	C₃×C₄⋊C₄	—	2	1	no	—
106	C₃×C₄⋊C₄	—	3	1	no	—
107	C₃×C₄⋊C₄	—	3	1	no	—
108	C₃×C₄⋊C₄	—	3	1	no	—
109	C₃×C₄⋊C₄	—	3	1	no	—
110	C₃×C₄⋊C₄	—	3	1	no	—
111	C₃×C₄⋊C₄	C₅×D₃	1	1	yes	characteristic
112	C₃×C₄⋊C₄	C₂×C₃×C₅	1	1	yes	characteristic
113	C₃×C₄²	—	3	1	no	—
114	C₄⋊Dic₃	C₂×C₃×C₅	1	1	yes	characteristic
115	Dic₃⋊C₄	C₂×C₃×C₅	1	1	yes	characteristic
116	Dic₃⋊C₄	C₂×C₃×C₅	1	1	yes	characteristic
117	Dic₃⋊C₄	C₂×C₃×C₅	1	1	yes	characteristic
118	Dic₃⋊C₄	C₂×C₃×C₅	1	1	yes	characteristic
119	C₄×Dic₃	C₂×C₃×C₅	1	1	yes	characteristic
120	C₃²×C₅	#125	1	1	yes	characteristic, complemented, Hall
121	C₂×C₄×C₅	—	3	1	no	—
122	C₂×C₄×C₅	—	3	1	no	—
123	C₂×C₄×C₅	—	3	1	no	—
124	C₂×C₄×C₅	—	3	1	no	—
125	C₂×C₄×C₅	C₂×C₃×D₃	1	1	yes	characteristic
126	C₂×C₄×C₅	C₂×C₃×D₃	1	1	yes	characteristic
127	C₂×C₄×C₅	C₂×C₃×D₃	1	1	yes	characteristic
128	C₂²×C₃²	C₂³×C₅	1	1	yes	characteristic
129	C₃²×C₄	—	2	1	no	—
130	C₃²×C₄	—	2	1	no	—
131	C₃²×C₄	—	2	1	no	—
132	C₃×Dic₃	—	2	1	no	—
133	C₃×Dic₃	—	2	1	no	—
134	C₃×Dic₃	—	2	1	no	—
135	C₃×Dic₃	C₅×Q₈	1	2	yes	—
136	#125	—	3	1	no	Sylow
137	C₂×C₃×C₅	—	2	1	no	—
138	C₂×C₃×C₅	—	2	1	no	—
139	C₂×C₃×C₅	—	2	1	no	—
140	C₂×C₃×C₅	#235	1	1	yes	characteristic
141	C₂×C₃×C₅	#237	1	1	yes	characteristic
142	C₂×C₃×C₅	D₃×Q₈	1	1	yes	characteristic
143	C₂×C₃×C₅	C₂×C₃×Q₈	1	1	yes	characteristic
144	C₂×C₃×C₅	C₃×#41	1	1	yes	characteristic
145	C₂×C₃×C₅	C₃×#41	1	1	yes	characteristic
146	C₂×C₃×C₄	—	2	1	no	—
147	C₂×C₃×C₄	—	2	1	no	—
148	C₂×C₃×C₄	—	2	1	no	—
149	C₂×C₃×C₄	—	3	1	no	—
150	C₂×C₃×C₄	—	3	1	no	—
151	C₂×C₃×C₄	—	3	1	no	—
152	C₂×C₃×C₄	—	3	1	no	—
153	C₂×C₃×C₄	C₂×C₅×D₃	1	1	yes	characteristic
154	C₂×C₃×C₄	C₂×C₅×D₃	1	1	yes	characteristic
155	C₂×C₃×C₄	C₂×C₅×D₃	1	1	yes	characteristic
156	C₂×C₃×C₄	C₂²×C₃×C₅	1	1	yes	characteristic
157	C₂×C₃×C₄	C₂²×C₃×C₅	1	1	yes	characteristic
158	C₂×C₃×C₄	C₂²×C₃×C₅	1	1	yes	characteristic
159	C₂×Dic₃	C₂²×C₃×C₅	1	1	yes	characteristic
160	C₂×Dic₃	C₂²×C₃×C₅	1	1	yes	characteristic
161	C₂×Dic₃	C₂²×C₃×C₅	1	1	yes	characteristic
162	C₂×Dic₃	C₂²×C₃×C₅	1	1	yes	characteristic
163	C₂²×C₅	C₂²×C₃×D₃	1	1	yes	characteristic
164	C₄×C₅	—	2	1	no	—
165	C₄×C₅	—	2	1	no	—
166	C₄×C₅	—	2	1	no	—
167	C₄×C₅	—	3	2	no	—
168	C₄×C₅	—	6	1	no	—
169	C₄×C₅	—	6	1	no	—
170	C₄×C₅	—	6	1	no	—
171	C₂×C₃²	C₂×C₅×Q₈	1	1	yes	characteristic
172	C₂×C₃²	C₅×#41	1	1	yes	characteristic
173	C₂×C₃²	C₅×#41	1	1	yes	characteristic
174	C₄⋊C₄	—	3	1	no	—
175	C₄⋊C₄	—	3	1	no	—
176	C₄⋊C₄	—	3	1	no	—
177	C₄⋊C₄	—	3	1	no	—
178	C₄⋊C₄	—	3	1	no	—
179	C₄⋊C₄	C₃×C₅×D₃	1	1	yes	characteristic
180	C₄²	—	3	1	no	—
181	C₃×C₅	—	2	1	no	—
182	C₃×C₅	#889	1	1	yes	characteristic, direct product factor
183	C₃×C₅	C₃×#125	1	1	yes	characteristic, complemented
184	C₂²×C₃	—	2	1	no	—
185	C₂²×C₃	C₂²×C₅×D₃	1	1	yes	characteristic
186	C₂²×C₃	C₂³×C₃×C₅	1	1	yes	characteristic
187	C₃×C₄	—	2	2	no	—
188	C₃×C₄	—	2	1	no	—
189	C₃×C₄	—	2	1	no	—
190	C₃×C₄	—	2	1	no	—
191	C₃×C₄	—	2	1	no	—
192	C₃×C₄	—	2	1	no	—
193	C₃×C₄	—	2	1	no	—
194	C₃×C₄	—	3	2	no	—
195	C₃×C₄	—	4	1	no	—
196	C₃×C₄	—	4	1	no	—
197	C₃×C₄	—	6	1	no	—
198	C₃×C₄	—	6	1	no	—
199	C₃×C₄	—	6	1	no	—
200	Dic₃	—	2	1	no	—
201	Dic₃	—	2	1	no	—
202	Dic₃	—	2	1	no	—
203	Dic₃	C₃×C₅×Q₈	1	2	yes	—
204	C₂×C₅	C₃×#235	1	1	yes	characteristic
205	C₂×C₅	C₃×#237	1	1	yes	characteristic
206	C₂×C₅	C₃×D₃×Q₈	1	1	yes	characteristic
207	C₃²	C₅×#125	1	1	yes	characteristic, complemented, Sylow
208	C₂×C₄	—	3	1	no	—
209	C₂×C₄	—	3	1	no	—
210	C₂×C₄	—	3	1	no	—
211	C₂×C₄	—	3	1	no	—
212	C₂×C₄	C₂×C₃×C₅×D₃	1	1	yes	characteristic
213	C₂×C₄	C₂×C₃×C₅×D₃	1	1	yes	characteristic
214	C₂×C₄	C₂×C₃×C₅×D₃	1	1	yes	characteristic
215	C₂×C₃	—	2	1	no	—
216	C₂×C₃	—	2	1	no	—
217	C₂×C₃	—	2	1	no	—
218	C₂×C₃	C₅×#235	1	1	yes	characteristic
219	C₂×C₃	C₅×#237	1	1	yes	characteristic
220	C₂×C₃	C₅×D₃×Q₈	1	1	yes	characteristic
221	C₂×C₃	C₂×C₃×C₅×Q₈	1	1	yes	characteristic
222	C₂×C₃	C₃×C₅×#41	1	1	yes	characteristic
223	C₂×C₃	C₃×C₅×#41	1	1	yes	characteristic
224	C₅	C₃×#889	1	1	yes	characteristic, direct product factor, Sylow
225	C₂²	C₂²×C₃×C₅×D₃	1	1	yes	characteristic
226	C₄	—	2	1	no	—
227	C₄	—	2	1	no	—
228	C₄	—	2	1	no	—
229	C₄	—	3	2	no	—
230	C₄	—	6	1	no	—
231	C₄	—	6	1	no	—
232	C₄	—	6	1	no	—
233	C₃	—	2	1	no	—
234	C₃	C₅×#889	1	1	yes	characteristic, direct product factor
235	C₃	C₃×C₅×#125	1	1	yes	characteristic, complemented
236	C₂	C₃×C₅×#235	1	1	yes	characteristic
237	C₂	C₃×C₅×#237	1	1	yes	characteristic
238	C₂	C₃×C₅×D₃×Q₈	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	C₃²×C₅×C₄⋊C₄	C₂
socle, prime radical	C₂²×C₃²×C₅	C₂³
hypercenter	C₃×C₅×C₄⋊C₄	D₃
hypocenter	C₃	C₃×C₅×#125
LCS	G ▷ C₂²×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
9. C₅×#889	C₃	direct (N × H)
17. C₃×#889	C₅	direct (N × H)
67. #889	C₃×C₅	direct (N × H)
120. C₃²×C₅	#125	semidirect
182. C₃×C₅	#889	direct (N × H)
183. C₃×C₅	C₃×#125	semidirect
207. C₃²	C₅×#125	semidirect
224. C₅	C₃×#889	direct (N × H)
234. C₃	C₅×#889	direct (N × H)
235. C₃	C₃×C₅×#125	semidirect

Maximal subgroups

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

subgroup	index
C₃²×C₅×C₄⋊C₄	2
C₃×C₅×C₄⋊Dic₃	2
C₃×C₅×Dic₃⋊C₄	2
C₃×C₄×C₅×Dic₃	2
C₃×C₅×#125	3
C₅×#889	3
C₃×#889	5

Subgroups — C3×C5×#889 #13087901

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#889 #13087901