Subgroups — C₃×C₅×#951 #13087962

Order 1440.

280 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₅×#41	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₅×Dic₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₅×#216	C₂	1	2	yes	maximal, complemented
4	C₃×C₅×#214	C₂	1	2	yes	maximal, complemented
5	C₃×C₅×#208	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×C₅×#137	—	3	1	no	maximal
7	C₅×#951	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₃²×C₅×Q₈	C₂²	1	1	yes	characteristic
9	C₃²×C₅×D₄	—	2	1	no	—
10	C₃²×C₅×D₄	C₂²	1	1	yes	characteristic
11	C₂×C₃²×C₄×C₅	—	2	1	no	—
12	C₂×C₃²×C₄×C₅	C₂²	1	1	yes	characteristic
13	C₂×C₃×C₅×Dic₃	—	2	1	no	—
14	C₃×C₅×Dic₆	—	2	1	no	—
15	C₃×C₅×Dic₆	C₂²	1	2	yes	—
16	C₃×C₅×C₃⋊C₈	C₂²	1	2	yes	—
17	C₃×#951	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
18	C₃×C₅×#41	—	2	1	no	—
19	C₃×C₅×#41	D₃	1	1	yes	characteristic
20	C₃×C₅×#41	C₂×C₃	1	1	yes	characteristic
21	C₂×C₃×C₅×Q₈	—	3	1	no	—
22	C₂×C₅×Dic₆	C₂×C₃	1	1	yes	characteristic, complemented
23	C₃×C₅×Q₁₆	—	3	2	no	—
24	C₃×C₅×SD₁₆	—	3	2	no	—
25	C₃×C₅×M₁₆	—	3	1	no	—
26	C₅×#216	C₂×C₃	1	2	yes	complemented
27	C₅×#214	C₂×C₃	1	2	yes	complemented
28	C₅×#208	C₂×C₃	1	1	yes	characteristic, complemented
29	C₂²×C₃²×C₅	—	2	1	no	—
30	C₂²×C₃²×C₅	D₄	1	1	yes	characteristic
31	C₃²×C₄×C₅	—	2	1	no	—
32	C₃²×C₄×C₅	D₄	1	1	yes	characteristic
33	C₃²×C₄×C₅	C₂³	1	1	yes	characteristic
34	C₃×C₅×Dic₃	—	2	2	no	—
35	C₅×#137	—	3	1	no	Hall
36	C₃²×#41	C₂×C₅	1	1	yes	characteristic
37	C₂×C₃×Dic₆	C₂×C₅	1	1	yes	characteristic, complemented
38	C₃×#216	C₂×C₅	1	2	yes	complemented
39	C₃×#214	C₂×C₅	1	2	yes	complemented
40	C₃×#208	C₂×C₅	1	1	yes	characteristic, complemented
41	C₃×C₅×Q₈	—	2	1	no	—
42	C₃×C₅×Q₈	—	3	2	no	—
43	C₃×C₅×Q₈	—	6	1	no	—
44	C₃×C₅×Q₈	C₂×D₃	1	1	yes	characteristic
45	C₃×C₅×Q₈	C₂²×C₃	1	1	yes	characteristic
46	C₃×C₅×D₄	—	2	2	no	—
47	C₃×C₅×D₄	—	2	1	no	—
48	C₃×C₅×D₄	—	2	1	no	—
49	C₃×C₅×D₄	—	2	1	no	—
50	C₃×C₅×D₄	C₂×D₃	1	1	yes	characteristic
51	C₃×C₅×D₄	C₂²×C₃	1	1	yes	characteristic
52	C₂×C₃×C₄×C₅	—	2	2	no	—
53	C₂×C₃×C₄×C₅	—	2	1	no	—
54	C₂×C₃×C₄×C₅	—	2	1	no	—
55	C₂×C₃×C₄×C₅	—	2	1	no	—
56	C₂×C₃×C₄×C₅	—	6	1	no	—
57	C₂×C₃×C₄×C₅	C₂×D₃	1	1	yes	characteristic
58	C₂×C₃×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
59	C₂×C₅×Dic₃	—	2	1	no	—
60	C₅×Dic₆	—	2	1	no	—
61	C₅×Dic₆	C₂²×C₃	1	2	yes	—
62	C₃×C₅×C₈	—	3	2	no	—
63	C₅×C₃⋊C₈	C₂²×C₃	1	2	yes	—
64	C₃×#137	—	3	1	no	—
65	#951	C₃×C₅	1	1	yes	characteristic, direct product factor
66	C₂×C₃²×C₅	—	2	1	no	—
67	C₂×C₃²×C₅	—	4	1	no	—
68	C₂×C₃²×C₅	C₂×D₄	1	1	yes	characteristic
69	C₅×#41	C₃×D₃	1	1	yes	characteristic
70	C₂×C₅×Q₈	—	3	1	no	—
71	C₅×Q₁₆	—	3	2	no	—
72	C₅×SD₁₆	—	3	2	no	—
73	C₅×M₁₆	—	3	1	no	—
74	C₃²×Q₈	C₂²×C₅	1	1	yes	characteristic
75	C₃²×D₄	—	2	1	no	—
76	C₃²×D₄	C₂²×C₅	1	1	yes	characteristic
77	C₂×C₃²×C₄	—	2	1	no	—
78	C₂×C₃²×C₄	C₂²×C₅	1	1	yes	characteristic
79	C₂×C₃×Dic₃	—	2	1	no	—
80	C₃×Dic₆	—	2	1	no	—
81	C₃×Dic₆	C₂²×C₅	1	2	yes	—
82	C₃×C₃⋊C₈	C₂²×C₅	1	2	yes	—
83	C₂²×C₃×C₅	—	2	2	no	—
84	C₂²×C₃×C₅	—	2	1	no	—
85	C₂²×C₃×C₅	—	2	1	no	—
86	C₂²×C₃×C₅	—	2	1	no	—
87	C₂²×C₃×C₅	#67	1	1	yes	characteristic
88	C₂²×C₃×C₅	C₃×D₄	1	1	yes	characteristic
89	C₃×C₄×C₅	—	2	2	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	—	6	2	no	—
95	C₃×C₄×C₅	#67	1	1	yes	characteristic
96	C₃×C₄×C₅	C₃×D₄	1	1	yes	characteristic
97	C₃×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
98	C₃×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
99	C₅×Dic₃	—	2	2	no	—
100	C₃×#41	—	2	1	no	—
101	C₃×#41	C₅×D₃	1	1	yes	characteristic
102	C₃×#41	C₂×C₃×C₅	1	1	yes	characteristic
103	C₂×C₃×Q₈	—	3	1	no	—
104	C₂×Dic₆	C₂×C₃×C₅	1	1	yes	characteristic, complemented
105	C₃×Q₁₆	—	3	2	no	—
106	C₃×SD₁₆	—	3	2	no	—
107	C₃×M₁₆	—	3	1	no	—
108	#216	C₂×C₃×C₅	1	2	yes	complemented
109	#214	C₂×C₃×C₅	1	2	yes	complemented
110	#208	C₂×C₃×C₅	1	1	yes	characteristic, complemented
111	C₃²×C₅	#137	1	1	yes	characteristic, complemented, Hall
112	C₅×Q₈	—	3	2	no	—
113	C₅×Q₈	—	6	1	no	—
114	C₅×Q₈	C₂×C₃×D₃	1	1	yes	characteristic
115	C₅×D₄	—	2	1	no	—
116	C₅×D₄	C₂×C₃×D₃	1	1	yes	characteristic
117	C₂×C₄×C₅	—	2	1	no	—
118	C₂×C₄×C₅	—	6	1	no	—
119	C₂×C₄×C₅	C₂×C₃×D₃	1	1	yes	characteristic
120	C₅×C₈	—	3	2	no	—
121	C₂²×C₃²	—	2	1	no	—
122	C₂²×C₃²	C₅×D₄	1	1	yes	characteristic
123	C₃²×C₄	—	2	1	no	—
124	C₃²×C₄	C₅×D₄	1	1	yes	characteristic
125	C₃²×C₄	C₂³×C₅	1	1	yes	characteristic
126	C₃×Dic₃	—	2	2	no	—
127	#137	—	3	1	no	Sylow
128	C₂×C₃×C₅	—	2	1	no	—
129	C₂×C₃×C₅	—	2	1	no	—
130	C₂×C₃×C₅	—	2	1	no	—
131	C₂×C₃×C₅	—	4	1	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	C₂×#67	1	1	yes	characteristic
136	C₂×C₃×C₅	C₂×C₃×D₄	1	1	yes	characteristic
137	C₃×Q₈	—	2	1	no	—
138	C₃×Q₈	—	3	2	no	—
139	C₃×Q₈	—	6	1	no	—
140	C₃×Q₈	C₂×C₅×D₃	1	1	yes	characteristic
141	C₃×Q₈	C₂²×C₃×C₅	1	1	yes	characteristic
142	C₃×D₄	—	2	2	no	—
143	C₃×D₄	—	2	1	no	—
144	C₃×D₄	—	2	1	no	—
145	C₃×D₄	—	2	1	no	—
146	C₃×D₄	C₂×C₅×D₃	1	1	yes	characteristic
147	C₃×D₄	C₂²×C₃×C₅	1	1	yes	characteristic
148	C₂×C₃×C₄	—	2	2	no	—
149	C₂×C₃×C₄	—	2	1	no	—
150	C₂×C₃×C₄	—	2	1	no	—
151	C₂×C₃×C₄	—	2	1	no	—
152	C₂×C₃×C₄	—	6	1	no	—
153	C₂×C₃×C₄	C₂×C₅×D₃	1	1	yes	characteristic
154	C₂×C₃×C₄	C₂²×C₃×C₅	1	1	yes	characteristic
155	C₂×Dic₃	—	2	1	no	—
156	Dic₆	—	2	1	no	—
157	Dic₆	C₂²×C₃×C₅	1	2	yes	—
158	C₃×C₈	—	3	2	no	—
159	C₃⋊C₈	C₂²×C₃×C₅	1	2	yes	—
160	C₂²×C₅	—	2	1	no	—
161	C₂²×C₅	C₃×#67	1	1	yes	characteristic
162	C₄×C₅	—	2	1	no	—
163	C₄×C₅	—	6	2	no	—
164	C₄×C₅	C₃×#67	1	1	yes	characteristic
165	C₄×C₅	C₂²×C₃×D₃	1	1	yes	characteristic
166	C₂×C₃²	—	2	1	no	—
167	C₂×C₃²	—	4	1	no	—
168	C₂×C₃²	C₂×C₅×D₄	1	1	yes	characteristic
169	#41	C₃×C₅×D₃	1	1	yes	characteristic
170	C₂×Q₈	—	3	1	no	—
171	Q₁₆	—	3	2	no	—
172	SD₁₆	—	3	2	no	—
173	M₁₆	—	3	1	no	—
174	C₃×C₅	—	2	1	no	—
175	C₃×C₅	#951	1	1	yes	characteristic, direct product factor
176	C₃×C₅	C₃×#137	1	1	yes	characteristic, complemented
177	C₂²×C₃	—	2	2	no	—
178	C₂²×C₃	—	2	1	no	—
179	C₂²×C₃	—	2	1	no	—
180	C₂²×C₃	—	2	1	no	—
181	C₂²×C₃	C₅×#67	1	1	yes	characteristic
182	C₂²×C₃	C₃×C₅×D₄	1	1	yes	characteristic
183	C₃×C₄	—	2	2	no	—
184	C₃×C₄	—	2	1	no	—
185	C₃×C₄	—	2	1	no	—
186	C₃×C₄	—	2	1	no	—
187	C₃×C₄	—	2	1	no	—
188	C₃×C₄	—	6	2	no	—
189	C₃×C₄	C₅×#67	1	1	yes	characteristic
190	C₃×C₄	C₃×C₅×D₄	1	1	yes	characteristic
191	C₃×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
192	C₃×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
193	Dic₃	—	2	2	no	—
194	C₂×C₅	—	2	1	no	—
195	C₂×C₅	—	4	1	no	—
196	C₂×C₅	C₂×C₃×#67	1	1	yes	characteristic
197	C₃²	C₅×#137	1	1	yes	characteristic, complemented, Sylow
198	Q₈	—	3	2	no	—
199	Q₈	—	6	1	no	—
200	Q₈	C₂×C₃×C₅×D₃	1	1	yes	characteristic
201	D₄	—	2	1	no	—
202	D₄	C₂×C₃×C₅×D₃	1	1	yes	characteristic
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	6	1	no	—
205	C₂×C₄	C₂×C₃×C₅×D₃	1	1	yes	characteristic
206	C₈	—	3	2	no	—
207	C₂×C₃	—	2	1	no	—
208	C₂×C₃	—	2	1	no	—
209	C₂×C₃	—	2	1	no	—
210	C₂×C₃	—	4	1	no	—
211	C₂×C₃	—	4	2	no	—
212	C₂×C₃	—	4	1	no	—
213	C₂×C₃	—	4	1	no	—
214	C₂×C₃	C₂×C₅×#67	1	1	yes	characteristic
215	C₂×C₃	C₂×C₃×C₅×D₄	1	1	yes	characteristic
216	C₅	C₃×#951	1	1	yes	characteristic, direct product factor, Sylow
217	C₂²	—	2	1	no	—
218	C₂²	C₃×C₅×#67	1	1	yes	characteristic
219	C₄	—	2	1	no	—
220	C₄	—	6	2	no	—
221	C₄	C₃×C₅×#67	1	1	yes	characteristic
222	C₄	C₂²×C₃×C₅×D₃	1	1	yes	characteristic
223	C₃	—	2	1	no	—
224	C₃	C₅×#951	1	1	yes	characteristic, direct product factor
225	C₃	C₃×C₅×#137	1	1	yes	characteristic, complemented
226	C₂	—	2	1	no	—
227	C₂	—	4	1	no	—
228	C₂	C₂×C₃×C₅×#67	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₅	C₂×#67
commutator, cosocle	C₃×C₄	C₂³×C₃×C₅
Frattini	C₄	C₂²×C₃×C₅×D₃
Fitting, prime radical	C₃²×C₅×#41	C₂
socle	C₂×C₃²×C₅	C₂×D₄
hypercenter	C₃×C₅×#41	D₃
hypocenter	C₃	C₃×C₅×#137
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₃×C₅ ◁ C₂×C₃×C₄×C₅ ◁ C₃×C₅×#41

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₅×Dic₆	C₂	semidirect
3. C₃×C₅×#216	C₂	semidirect
4. C₃×C₅×#214	C₂	semidirect
5. C₃×C₅×#208	C₂	semidirect
7. C₅×#951	C₃	direct (N × H)
17. C₃×#951	C₅	direct (N × H)
22. C₂×C₅×Dic₆	C₂×C₃	semidirect
26. C₅×#216	C₂×C₃	semidirect
27. C₅×#214	C₂×C₃	semidirect
28. C₅×#208	C₂×C₃	semidirect
37. C₂×C₃×Dic₆	C₂×C₅	semidirect
38. C₃×#216	C₂×C₅	semidirect
39. C₃×#214	C₂×C₅	semidirect
40. C₃×#208	C₂×C₅	semidirect
65. #951	C₃×C₅	direct (N × H)
104. C₂×Dic₆	C₂×C₃×C₅	semidirect
108. #216	C₂×C₃×C₅	semidirect
109. #214	C₂×C₃×C₅	semidirect
110. #208	C₂×C₃×C₅	semidirect
111. C₃²×C₅	#137	semidirect
175. C₃×C₅	#951	direct (N × H)
176. C₃×C₅	C₃×#137	semidirect
197. C₃²	C₅×#137	semidirect
216. C₅	C₃×#951	direct (N × H)
224. C₃	C₅×#951	direct (N × H)
225. C₃	C₃×C₅×#137	semidirect

Maximal subgroups

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

subgroup	index
C₃²×C₅×#41	2
C₂×C₃×C₅×Dic₆	2
C₃×C₅×#216	2
C₃×C₅×#214	2
C₃×C₅×#208	2
C₃×C₅×#137	3
C₅×#951	3
C₃×#951	5

Subgroups — C3×C5×#951 #13087962

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#951 #13087962