Subgroups — C₃×#10602609 #14306037

Order 1920.

640 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#532	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#3969	C₂	1	4	yes	maximal, complemented
3	C₃×#65221	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#65213	C₂	1	1	yes	maximal, characteristic, complemented
5	#10602609	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂×C₃×C₅×#41	C₂²	1	2	yes	—
7	C₃×C₅×C₄⋊Q₈	C₄	1	1	yes	characteristic, complemented
8	C₃×C₅×#127	C₄	1	1	yes	characteristic, complemented
9	C₃×C₅×#124	C₄	1	2	yes	complemented
10	C₃×C₅×#121	—	2	2	no	—
11	C₃×C₄×C₅×D₄	—	2	2	no	—
12	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
13	C₂×C₃×C₄×Dic₅	C₂²	1	2	yes	—
14	C₂×C₃×#681	C₂²	1	2	yes	—
15	C₃×#3969	—	2	8	no	—
16	C₃×#3963	—	2	2	no	—
17	C₃×C₄⋊#168	—	2	2	no	—
18	C₃×#1963	—	5	1	no	maximal, Hall
19	C₅×#532	C₂×C₃	1	1	yes	characteristic
20	C₂×#3969	C₂×C₃	1	4	yes	complemented
21	#65221	C₂×C₃	1	1	yes	characteristic, complemented
22	#65213	C₂×C₃	1	1	yes	characteristic, complemented
23	C₃×C₅×#41	—	2	4	no	—
24	C₃×C₅×#41	—	2	4	no	—
25	C₂×C₃×C₅×Q₈	C₂×C₄	1	2	yes	—
26	C₂×C₃×C₅×D₄	—	2	2	no	—
27	C₂×C₃×C₅×D₄	—	2	2	no	—
28	C₂×C₃×C₅×D₄	C₂×C₄	1	2	yes	—
29	C₂²×C₃×C₄×C₅	—	2	2	no	—
30	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
31	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
32	C₂²×C₃×Dic₅	—	2	2	no	—
33	C₃×C₅×C₄⋊C₄	—	2	2	no	—
34	C₃×C₅×C₂²⋊C₄	—	2	4	no	—
35	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
36	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
37	C₃×C₄²×C₅	D₄	1	2	yes	—
38	C₃×C₄×Dic₅	—	2	2	no	—
39	C₃×C₄×Dic₅	D₄	1	4	yes	complemented
40	C₃×#681	—	2	4	no	—
41	C₂×C₃×#168	—	2	2	no	—
42	C₃×#532	D₅	1	1	yes	characteristic
43	C₃×#423	—	5	1	no	—
44	C₂×C₃×#104	—	5	4	no	—
45	C₃×#376	—	5	1	no	—
46	C₂×C₅×#41	C₂²×C₃	1	2	yes	—
47	C₅×C₄⋊Q₈	C₃×C₄	1	1	yes	characteristic, complemented
48	C₅×#127	C₃×C₄	1	1	yes	characteristic, complemented
49	C₅×#124	C₃×C₄	1	2	yes	complemented
50	C₅×#121	—	2	2	no	—
51	C₄×C₅×D₄	—	2	2	no	—
52	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
53	C₂×C₄×Dic₅	C₂²×C₃	1	2	yes	—
54	C₂×#681	C₂²×C₃	1	2	yes	—
55	#3969	—	2	8	no	—
56	#3963	—	2	2	no	—
57	C₄⋊#168	—	2	2	no	—
58	#1963	—	5	1	no	Sylow
59	C₂³×C₃×C₅	—	2	2	no	—
60	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
61	C₃×C₅×Q₈	—	2	4	no	—
62	C₃×C₅×D₄	—	2	4	no	—
63	C₃×C₅×D₄	—	4	4	no	—
64	C₃×C₅×D₄	—	4	4	no	—
65	C₂×C₃×C₄×C₅	—	2	2	no	—
66	C₂×C₃×C₄×C₅	—	4	2	no	—
67	C₂×C₃×C₄×C₅	—	4	2	no	—
68	C₂×C₃×C₄×C₅	—	4	2	no	—
69	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
70	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
71	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
73	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
74	C₂×C₃×C₄×C₅	#41	1	2	yes	—
75	C₂×C₃×Dic₅	—	2	4	no	—
76	C₂×C₃×Dic₅	—	4	4	no	—
77	C₃×#168	—	4	2	no	—
78	C₂×C₃×#41	C₂×D₅	1	2	yes	—
79	C₂×C₃×M₁₆	—	5	2	no	—
80	C₃×C₄⋊Q₈	Dic₅	1	1	yes	characteristic, complemented
81	C₃×#127	Dic₅	1	1	yes	characteristic, complemented
82	C₃×#124	Dic₅	1	2	yes	complemented
83	C₃×#121	—	2	2	no	—
84	C₃×C₄×D₄	—	2	2	no	—
85	C₂×C₃×C₄²	—	5	2	no	—
86	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
87	C₃×C₄⋊C₈	—	10	2	no	—
88	C₃×#104	—	10	8	no	—
89	C₃×#95	—	10	2	no	—
90	C₅×#41	—	2	4	no	—
91	C₅×#41	—	2	4	no	—
92	C₂×C₅×Q₈	C₂×C₃×C₄	1	2	yes	—
93	C₂×C₅×D₄	—	2	2	no	—
94	C₂×C₅×D₄	—	2	2	no	—
95	C₂×C₅×D₄	C₂×C₃×C₄	1	2	yes	—
96	C₂²×C₄×C₅	—	2	2	no	—
97	C₂²×C₄×C₅	C₃×D₄	1	2	yes	—
98	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
99	C₂²×Dic₅	—	2	2	no	—
100	C₅×C₄⋊C₄	—	2	2	no	—
101	C₅×C₂²⋊C₄	—	2	4	no	—
102	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
103	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
104	C₄²×C₅	C₃×D₄	1	2	yes	—
105	C₄×Dic₅	—	2	2	no	—
106	C₄×Dic₅	C₃×D₄	1	4	yes	complemented
107	#681	—	2	4	no	—
108	C₂×#168	—	2	2	no	—
109	#532	C₃×D₅	1	1	yes	characteristic
110	#423	—	5	1	no	—
111	C₂×#104	—	5	4	no	—
112	#376	—	5	1	no	—
113	C₂²×C₃×C₅	—	2	2	no	—
114	C₂²×C₃×C₅	—	4	2	no	—
115	C₂²×C₃×C₅	—	4	4	no	—
116	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	C₂²×C₃×C₅	#121	1	2	yes	—
118	C₃×C₄×C₅	—	2	2	no	—
119	C₃×C₄×C₅	—	2	2	no	—
120	C₃×C₄×C₅	—	4	2	no	—
121	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
122	C₃×C₄×C₅	#124	1	1	yes	characteristic
123	C₃×C₄×C₅	#127	1	1	yes	characteristic
124	C₃×Dic₅	—	4	4	no	—
125	C₃×#41	—	2	4	no	—
126	C₃×#41	—	2	4	no	—
127	C₂×C₃×Q₈	C₂×Dic₅	1	2	yes	—
128	C₂×C₃×D₄	—	2	2	no	—
129	C₂×C₃×D₄	—	2	2	no	—
130	C₂×C₃×D₄	C₂×Dic₅	1	2	yes	—
131	C₂²×C₃×C₄	—	2	2	no	—
132	C₂²×C₃×C₄	—	10	2	no	—
133	C₂²×C₃×C₄	C₂²⋊D₅	1	2	yes	—
134	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
135	C₃×M₁₆	—	10	4	no	—
136	C₂×C₃×C₈	—	10	2	no	—
137	C₃×C₄⋊C₄	—	2	2	no	—
138	C₃×C₂²⋊C₄	—	2	4	no	—
139	C₃×C₄²	—	5	4	no	—
140	C₃×C₄²	—	10	2	no	—
141	C₃×C₄²	C₂×Dic₅	1	1	yes	characteristic
142	C₃×C₄²	C₂×Dic₅	1	1	yes	characteristic
143	C₃×C₄²	C₂²⋊D₅	1	2	yes	—
144	C₂³×C₅	—	2	2	no	—
145	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
146	C₅×Q₈	—	2	4	no	—
147	C₅×D₄	—	2	4	no	—
148	C₅×D₄	—	4	4	no	—
149	C₅×D₄	—	4	4	no	—
150	C₂×C₄×C₅	—	2	2	no	—
151	C₂×C₄×C₅	—	4	2	no	—
152	C₂×C₄×C₅	—	4	2	no	—
153	C₂×C₄×C₅	—	4	2	no	—
154	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	2	yes	—
155	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	2	yes	—
156	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
157	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
158	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
159	C₂×C₄×C₅	C₃×#41	1	2	yes	—
160	C₂×Dic₅	—	2	4	no	—
161	C₂×Dic₅	—	4	4	no	—
162	#168	—	4	2	no	—
163	C₂×#41	C₂×C₃×D₅	1	2	yes	—
164	C₂×M₁₆	—	5	2	no	—
165	C₄⋊Q₈	C₃×Dic₅	1	1	yes	characteristic, complemented
166	#127	C₃×Dic₅	1	1	yes	characteristic, complemented
167	#124	C₃×Dic₅	1	2	yes	complemented
168	#121	—	2	2	no	—
169	C₄×D₄	—	2	2	no	—
170	C₂×C₄²	—	5	2	no	—
171	C₂×C₄²	C₂×C₃×D₅	1	1	yes	characteristic
172	C₄⋊C₈	—	10	2	no	—
173	#104	—	10	8	no	—
174	#95	—	10	2	no	—
175	C₂×C₃×C₅	—	2	2	no	—
176	C₂×C₃×C₅	—	8	2	no	—
177	C₂×C₃×C₅	#390	1	1	yes	characteristic
178	C₂×C₃×C₅	#421	1	2	yes	—
179	C₂³×C₃	—	2	2	no	—
180	C₂³×C₃	C₂×C₂²⋊D₅	1	1	yes	characteristic
181	C₃×Q₈	—	2	4	no	—
182	C₃×D₄	—	2	4	no	—
183	C₃×D₄	—	4	4	no	—
184	C₃×D₄	—	4	4	no	—
185	C₂×C₃×C₄	—	2	2	no	—
186	C₂×C₃×C₄	—	4	2	no	—
187	C₂×C₃×C₄	—	4	2	no	—
188	C₂×C₃×C₄	—	4	2	no	—
189	C₂×C₃×C₄	—	10	4	no	—
190	C₂×C₃×C₄	—	20	4	no	—
191	C₂×C₃×C₄	C₂²⋊Dic₅	1	2	yes	—
192	C₂×C₃×C₄	C₂²⋊Dic₅	1	2	yes	—
193	C₂×C₃×C₄	D₄×D₅	1	2	yes	—
194	C₂×C₃×C₄	#711	1	2	yes	—
195	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
196	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
197	C₃×C₈	—	20	2	no	—
198	C₂²×C₅	—	2	2	no	—
199	C₂²×C₅	—	4	2	no	—
200	C₂²×C₅	—	4	4	no	—
201	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
202	C₂²×C₅	C₃×#121	1	2	yes	—
203	C₄×C₅	—	2	2	no	—
204	C₄×C₅	—	2	2	no	—
205	C₄×C₅	—	4	2	no	—
206	C₄×C₅	C₃×C₄×D₄	1	2	yes	—
207	C₄×C₅	C₃×#124	1	1	yes	characteristic
208	C₄×C₅	C₃×#127	1	1	yes	characteristic
209	Dic₅	—	4	4	no	—
210	#41	—	2	4	no	—
211	#41	—	2	4	no	—
212	C₂×Q₈	C₂×C₃×Dic₅	1	2	yes	—
213	C₂×D₄	—	2	2	no	—
214	C₂×D₄	—	2	2	no	—
215	C₂×D₄	C₂×C₃×Dic₅	1	2	yes	—
216	C₂²×C₄	—	2	2	no	—
217	C₂²×C₄	—	10	2	no	—
218	C₂²×C₄	C₃×C₂²⋊D₅	1	2	yes	—
219	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
220	M₁₆	—	10	4	no	—
221	C₂×C₈	—	10	2	no	—
222	C₄⋊C₄	—	2	2	no	—
223	C₂²⋊C₄	—	2	4	no	—
224	C₄²	—	5	4	no	—
225	C₄²	—	10	2	no	—
226	C₄²	C₂×C₃×Dic₅	1	1	yes	characteristic
227	C₄²	C₂×C₃×Dic₅	1	1	yes	characteristic
228	C₄²	C₃×C₂²⋊D₅	1	2	yes	—
229	C₃×C₅	#1963	1	1	yes	characteristic, complemented, Hall
230	C₂²×C₃	—	2	2	no	—
231	C₂²×C₃	—	4	2	no	—
232	C₂²×C₃	—	4	4	no	—
233	C₂²×C₃	#4084	1	2	yes	—
234	C₂²×C₃	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
235	C₃×C₄	—	2	2	no	—
236	C₃×C₄	—	2	2	no	—
237	C₃×C₄	—	4	2	no	—
238	C₃×C₄	—	20	4	no	—
239	C₃×C₄	D₄×Dic₅	1	2	yes	—
240	C₃×C₄	#4082	1	1	yes	characteristic
241	C₃×C₄	#4086	1	1	yes	characteristic
242	C₂×C₅	—	2	2	no	—
243	C₂×C₅	—	8	2	no	—
244	C₂×C₅	C₃×#390	1	1	yes	characteristic
245	C₂×C₅	C₃×#421	1	2	yes	—
246	C₂³	—	2	2	no	—
247	C₂³	C₂×C₃×C₂²⋊D₅	1	1	yes	characteristic
248	Q₈	—	2	4	no	—
249	D₄	—	2	4	no	—
250	D₄	—	4	4	no	—
251	D₄	—	4	4	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	4	2	no	—
255	C₂×C₄	—	4	2	no	—
256	C₂×C₄	—	10	4	no	—
257	C₂×C₄	—	20	4	no	—
258	C₂×C₄	C₃×C₂²⋊Dic₅	1	2	yes	—
259	C₂×C₄	C₃×C₂²⋊Dic₅	1	2	yes	—
260	C₂×C₄	C₃×D₄×D₅	1	2	yes	—
261	C₂×C₄	C₃×#711	1	2	yes	—
262	C₂×C₄	C₂²×C₃×Dic₅	1	1	yes	characteristic
263	C₂×C₄	C₂×C₃×C₂²⋊D₅	1	1	yes	characteristic
264	C₈	—	20	2	no	—
265	C₂×C₃	—	2	2	no	—
266	C₂×C₃	—	8	2	no	—
267	C₂×C₃	#65510	1	1	yes	characteristic
268	C₂×C₃	#65525	1	2	yes	—
269	C₅	C₃×#1963	1	1	yes	characteristic, complemented, Sylow
270	C₂²	—	2	2	no	—
271	C₂²	—	4	2	no	—
272	C₂²	—	4	4	no	—
273	C₂²	C₃×#4084	1	2	yes	—
274	C₂²	C₂×C₃×C₂²⋊Dic₅	1	1	yes	characteristic
275	C₄	—	2	2	no	—
276	C₄	—	2	2	no	—
277	C₄	—	4	2	no	—
278	C₄	—	20	4	no	—
279	C₄	C₃×D₄×Dic₅	1	2	yes	—
280	C₄	C₃×#4082	1	1	yes	characteristic
281	C₄	C₃×#4086	1	1	yes	characteristic
282	C₃	#10602609	1	1	yes	characteristic, direct product factor, Sylow
283	C₂	—	2	2	no	—
284	C₂	—	8	2	no	—
285	C₂	C₃×#65510	1	1	yes	characteristic
286	C₂	C₃×#65525	1	2	yes	—

	subgroup	G/H
center	C₂×C₃×C₄	C₂×C₂²⋊D₅
commutator	C₂×C₄×C₅	C₂²×C₃×C₄
Frattini	C₂²×C₄	C₂²×C₃×D₅
Fitting, prime radical	C₃×C₅×#532	C₂
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
cosocle	C₂²×C₄×C₅	C₂³×C₃
hypercenter	C₃×#532	D₅
hypocenter	C₅	C₃×#1963
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂×C₃×C₄² ◁ C₃×#532

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×#3969	C₂	semidirect
3. C₃×#65221	C₂	semidirect
4. C₃×#65213	C₂	semidirect
5. #10602609	C₃	direct (N × H)
7. C₃×C₅×C₄⋊Q₈	C₄	semidirect
8. C₃×C₅×#127	C₄	semidirect
9. C₃×C₅×#124	C₄	semidirect
20. C₂×#3969	C₂×C₃	semidirect
21. #65221	C₂×C₃	semidirect
22. #65213	C₂×C₃	semidirect
39. C₃×C₄×Dic₅	D₄	semidirect
47. C₅×C₄⋊Q₈	C₃×C₄	semidirect
48. C₅×#127	C₃×C₄	semidirect
49. C₅×#124	C₃×C₄	semidirect
80. C₃×C₄⋊Q₈	Dic₅	semidirect
81. C₃×#127	Dic₅	semidirect
82. C₃×#124	Dic₅	semidirect
106. C₄×Dic₅	C₃×D₄	semidirect
165. C₄⋊Q₈	C₃×Dic₅	semidirect
166. #127	C₃×Dic₅	semidirect
167. #124	C₃×Dic₅	semidirect
229. C₃×C₅	#1963	semidirect
269. C₅	C₃×#1963	semidirect
282. C₃	#10602609	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×C₅×#532	2
C₂×C₃×#3969	2
C₃×#65221	2
C₃×#65213	2
#10602609	3
C₃×#1963	5

Subgroups — C3×#10602609 #14306037

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10602609 #14306037