Subgroups — C₃×#10613669 #14439838

Order 1920.

712 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#66062	C₂	1	2	yes	maximal
2	C₃×C₅×#500	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#65368	C₂	1	2	yes	maximal, complemented
4	C₃×#65169	C₂	1	4	yes	maximal
5	C₃×#65167	C₂	1	4	yes	maximal, complemented
6	C₃×C₄×C₈×D₅	C₂	1	1	yes	maximal, characteristic
7	C₃×#64971	C₂	1	1	yes	maximal, characteristic, complemented
8	#10613669	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₃×C₅×C₄⋊Q₈	C₂²	1	2	yes	—
10	C₃×#4092	—	2	2	no	—
11	C₃×Q₈×Dic₅	—	2	2	no	—
12	C₂×C₃×C₈×D₅	C₂²	1	2	yes	—
13	C₃×#4043	—	2	2	no	—
14	C₃×#4038	C₂²	1	4	yes	—
15	C₃×#4036	C₂²	1	4	yes	—
16	C₃×#4033	—	2	2	no	—
17	C₃×C₄²×D₅	C₂²	1	1	yes	characteristic
18	C₃×#4015	C₂²	1	2	yes	—
19	C₃×C₅×#107	C₂²	1	4	yes	—
20	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
21	C₃×#3950	C₂²	1	4	yes	—
22	C₃×C₈×Dic₅	C₂²	1	2	yes	—
23	C₃×#3940	C₂²	1	8	yes	—
24	C₃×C₄×#168	C₂²	1	1	yes	characteristic
25	C₃×#3160	—	5	1	no	maximal, Hall
26	#66062	C₂×C₃	1	2	yes	—
27	C₅×#500	C₂×C₃	1	1	yes	characteristic, complemented
28	#65368	C₂×C₃	1	2	yes	complemented
29	#65169	C₂×C₃	1	4	yes	—
30	#65167	C₂×C₃	1	4	yes	complemented
31	C₄×C₈×D₅	C₂×C₃	1	1	yes	characteristic
32	#64971	C₂×C₃	1	1	yes	characteristic, complemented
33	C₂×C₃×C₅×Q₈	—	2	2	no	—
34	C₂×C₃×C₄×D₅	D₄	1	1	yes	characteristic
35	C₂×C₃×C₄×D₅	D₄	1	1	yes	characteristic
36	C₂×C₃×C₄×D₅	C₂³	1	1	yes	characteristic
37	C₂×C₃×Dic₁₀	—	2	2	no	—
38	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
39	C₃×C₅×C₄⋊C₄	—	2	2	no	—
40	C₃×C₅×C₄⋊C₄	C₂³	1	4	yes	—
41	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
42	C₃×#685	—	2	4	no	—
43	C₃×C₄⋊Dic₅	—	2	2	no	—
44	C₃×C₄⋊Dic₅	—	2	4	no	—
45	C₃×C₄⋊Dic₅	C₂³	1	4	yes	—
46	C₃×#683	—	2	4	no	—
47	C₃×C₄×Dic₅	—	2	4	no	—
48	C₃×C₄×Dic₅	D₄	1	1	yes	characteristic
49	C₃×C₄×Dic₅	D₄	1	1	yes	characteristic
50	C₃×C₄×Dic₅	C₂³	1	1	yes	characteristic
51	C₂×C₃×#168	C₂³	1	2	yes	—
52	C₃×C₈×D₅	Q₈	1	8	yes	complemented
53	C₃×#533	—	5	2	no	—
54	C₃×#500	—	5	1	no	—
55	C₃×#500	D₅	1	1	yes	characteristic, complemented
56	C₃×#499	—	5	4	no	—
57	C₃×#498	—	5	2	no	—
58	C₃×#427	—	5	4	no	—
59	C₂×C₃×C₄×C₈	—	5	1	no	—
60	C₅×C₄⋊Q₈	C₂²×C₃	1	2	yes	—
61	#4092	—	2	2	no	—
62	Q₈×Dic₅	—	2	2	no	—
63	C₂×C₈×D₅	C₂²×C₃	1	2	yes	—
64	#4043	—	2	2	no	—
65	#4038	C₂²×C₃	1	4	yes	—
66	#4036	C₂²×C₃	1	4	yes	—
67	#4033	—	2	2	no	—
68	C₄²×D₅	C₂²×C₃	1	1	yes	characteristic
69	#4015	C₂²×C₃	1	2	yes	—
70	C₅×#107	C₂²×C₃	1	4	yes	—
71	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
72	#3950	C₂²×C₃	1	4	yes	—
73	C₈×Dic₅	C₂²×C₃	1	2	yes	—
74	#3940	C₂²×C₃	1	8	yes	—
75	C₄×#168	C₂²×C₃	1	1	yes	characteristic
76	#3160	—	5	1	no	Sylow
77	C₂²×C₃×D₅	C₂×D₄	1	1	yes	characteristic
78	C₃×C₅×Q₈	—	4	4	no	—
79	C₂×C₃×C₄×C₅	—	2	4	no	—
80	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
81	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
82	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
83	C₂×C₃×Dic₅	—	2	4	no	—
84	C₂×C₃×Dic₅	C₂×D₄	1	1	yes	characteristic
85	C₂×C₃×Dic₅	C₂×D₄	1	1	yes	characteristic
86	C₂×C₃×Dic₅	C₂×D₄	1	1	yes	characteristic
87	C₃×C₄×D₅	—	2	2	no	—
88	C₃×C₄×D₅	—	2	2	no	—
89	C₃×C₄×D₅	C₂×Q₈	1	4	yes	—
90	C₃×Dic₁₀	—	4	4	no	—
91	C₃×C₅×C₈	C₂×Q₈	1	4	yes	—
92	C₃×#168	C₂×Q₈	1	4	yes	—
93	C₂²×C₃×C₈	—	5	2	no	—
94	C₃×C₄⋊Q₈	—	5	2	no	—
95	C₃×C₄⋊Q₈	C₂×D₅	1	2	yes	—
96	C₃×#125	—	5	4	no	—
97	C₃×C₂²⋊Q₈	—	10	2	no	—
98	C₃×C₂²⋊Q₈	—	10	2	no	—
99	C₃×C₄×Q₈	—	10	2	no	—
100	C₃×C₄×Q₈	—	10	2	no	—
101	C₃×#117	—	5	4	no	—
102	C₂×C₃×C₄²	—	5	1	no	—
103	C₃×#107	—	5	4	no	—
104	C₃×#107	C₂×D₅	1	4	yes	—
105	C₃×#106	—	5	8	no	—
106	C₃×C₄×C₈	—	5	2	no	—
107	C₃×C₄×C₈	—	5	1	no	—
108	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic
109	C₂×C₅×Q₈	—	2	2	no	—
110	C₂×C₄×D₅	C₃×D₄	1	1	yes	characteristic
111	C₂×C₄×D₅	C₃×D₄	1	1	yes	characteristic
112	C₂×C₄×D₅	C₂³×C₃	1	1	yes	characteristic
113	C₂×Dic₁₀	—	2	2	no	—
114	C₂×C₅×C₈	C₂³×C₃	1	2	yes	—
115	C₅×C₄⋊C₄	—	2	2	no	—
116	C₅×C₄⋊C₄	C₂³×C₃	1	4	yes	—
117	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
118	#685	—	2	4	no	—
119	C₄⋊Dic₅	—	2	2	no	—
120	C₄⋊Dic₅	—	2	4	no	—
121	C₄⋊Dic₅	C₂³×C₃	1	4	yes	—
122	#683	—	2	4	no	—
123	C₄×Dic₅	—	2	4	no	—
124	C₄×Dic₅	C₃×D₄	1	1	yes	characteristic
125	C₄×Dic₅	C₃×D₄	1	1	yes	characteristic
126	C₄×Dic₅	C₂³×C₃	1	1	yes	characteristic
127	C₂×#168	C₂³×C₃	1	2	yes	—
128	C₈×D₅	C₃×Q₈	1	8	yes	complemented
129	#533	—	5	2	no	—
130	#500	—	5	1	no	—
131	#500	C₃×D₅	1	1	yes	characteristic, complemented
132	#499	—	5	4	no	—
133	#498	—	5	2	no	—
134	#427	—	5	4	no	—
135	C₂×C₄×C₈	—	5	1	no	—
136	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
137	C₂×C₃×D₅	—	2	1	no	—
138	C₂×C₃×D₅	—	2	1	no	—
139	C₂×C₃×D₅	C₄⋊Q₈	1	2	yes	—
140	C₃×C₄×C₅	—	4	4	no	—
141	C₃×C₄×C₅	#135	1	2	yes	—
142	C₃×C₄×C₅	#135	1	2	yes	—
143	C₃×C₄×C₅	C₂²×Q₈	1	2	yes	—
144	C₃×Dic₅	—	2	1	no	—
145	C₃×Dic₅	—	2	1	no	—
146	C₃×Dic₅	—	4	4	no	—
147	C₃×Dic₅	C₄⋊Q₈	1	2	yes	—
148	C₂×C₃×Q₈	—	2	2	no	—
149	C₂×C₃×Q₈	—	10	2	no	—
150	C₂²×C₃×C₄	—	5	1	no	—
151	C₂²×C₃×C₄	—	5	1	no	—
152	C₂²×C₃×C₄	—	5	1	no	—
153	C₂×C₃×C₈	—	5	2	no	—
154	C₂×C₃×C₈	—	5	8	no	—
155	C₂×C₃×C₈	C₂²×D₅	1	2	yes	—
156	C₃×C₄⋊C₄	—	2	2	no	—
157	C₃×C₄⋊C₄	—	5	4	no	—
158	C₃×C₄⋊C₄	—	10	4	no	—
159	C₃×C₄⋊C₄	—	10	4	no	—
160	C₃×C₄⋊C₄	—	10	2	no	—
161	C₃×C₄⋊C₄	C₂²×D₅	1	4	yes	—
162	C₃×C₂²⋊C₄	—	10	4	no	—
163	C₃×C₄²	—	5	1	no	—
164	C₃×C₄²	—	5	1	no	—
165	C₃×C₄²	—	5	1	no	—
166	C₃×C₄²	—	10	4	no	—
167	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
168	C₂²×D₅	C₂×C₃×D₄	1	1	yes	characteristic
169	C₅×Q₈	—	4	4	no	—
170	C₂×C₄×C₅	—	2	4	no	—
171	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
172	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
173	C₂×C₄×C₅	C₂⁴×C₃	1	1	yes	characteristic
174	C₂×Dic₅	—	2	4	no	—
175	C₂×Dic₅	C₂×C₃×D₄	1	1	yes	characteristic
176	C₂×Dic₅	C₂×C₃×D₄	1	1	yes	characteristic
177	C₂×Dic₅	C₂×C₃×D₄	1	1	yes	characteristic
178	C₄×D₅	—	2	2	no	—
179	C₄×D₅	—	2	2	no	—
180	C₄×D₅	C₂×C₃×Q₈	1	4	yes	—
181	Dic₁₀	—	4	4	no	—
182	C₅×C₈	C₂×C₃×Q₈	1	4	yes	—
183	#168	C₂×C₃×Q₈	1	4	yes	—
184	C₂²×C₈	—	5	2	no	—
185	C₄⋊Q₈	—	5	2	no	—
186	C₄⋊Q₈	C₂×C₃×D₅	1	2	yes	—
187	#125	—	5	4	no	—
188	C₂²⋊Q₈	—	10	2	no	—
189	C₂²⋊Q₈	—	10	2	no	—
190	C₄×Q₈	—	10	2	no	—
191	C₄×Q₈	—	10	2	no	—
192	#117	—	5	4	no	—
193	C₂×C₄²	—	5	1	no	—
194	#107	—	5	4	no	—
195	#107	C₂×C₃×D₅	1	4	yes	—
196	#106	—	5	8	no	—
197	C₄×C₈	—	5	2	no	—
198	C₄×C₈	—	5	1	no	—
199	C₄×C₈	C₂×C₃×D₅	1	1	yes	characteristic
200	C₂×C₃×C₅	C₂×C₄⋊Q₈	1	1	yes	characteristic
201	C₂×C₃×C₅	C₂×#135	1	1	yes	characteristic
202	C₂×C₃×C₅	C₂×#135	1	1	yes	characteristic
203	C₃×D₅	—	2	2	no	—
204	C₂³×C₃	—	5	1	no	—
205	C₃×Q₈	—	4	4	no	—
206	C₃×Q₈	—	20	4	no	—
207	C₂×C₃×C₄	—	2	4	no	—
208	C₂×C₃×C₄	—	5	1	no	—
209	C₂×C₃×C₄	—	5	4	no	—
210	C₂×C₃×C₄	—	5	1	no	—
211	C₂×C₃×C₄	—	5	1	no	—
212	C₂×C₃×C₄	—	10	2	no	—
213	C₂×C₃×C₄	—	10	2	no	—
214	C₂×C₃×C₄	—	10	4	no	—
215	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
216	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
217	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
218	C₃×C₈	—	5	4	no	—
219	C₃×C₈	Q₈×D₅	1	4	yes	—
220	C₂²×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
221	C₂×D₅	—	2	1	no	—
222	C₂×D₅	—	2	1	no	—
223	C₂×D₅	C₃×C₄⋊Q₈	1	2	yes	—
224	C₄×C₅	—	4	4	no	—
225	C₄×C₅	C₃×#135	1	2	yes	—
226	C₄×C₅	C₃×#135	1	2	yes	—
227	C₄×C₅	C₂²×C₃×Q₈	1	2	yes	—
228	Dic₅	—	2	1	no	—
229	Dic₅	—	2	1	no	—
230	Dic₅	—	4	4	no	—
231	Dic₅	C₃×C₄⋊Q₈	1	2	yes	—
232	C₂×Q₈	—	2	2	no	—
233	C₂×Q₈	—	10	2	no	—
234	C₂²×C₄	—	5	1	no	—
235	C₂²×C₄	—	5	1	no	—
236	C₂²×C₄	—	5	1	no	—
237	C₂×C₈	—	5	2	no	—
238	C₂×C₈	—	5	8	no	—
239	C₂×C₈	C₂²×C₃×D₅	1	2	yes	—
240	C₄⋊C₄	—	2	2	no	—
241	C₄⋊C₄	—	5	4	no	—
242	C₄⋊C₄	—	10	4	no	—
243	C₄⋊C₄	—	10	4	no	—
244	C₄⋊C₄	—	10	2	no	—
245	C₄⋊C₄	C₂²×C₃×D₅	1	4	yes	—
246	C₂²⋊C₄	—	10	4	no	—
247	C₄²	—	5	1	no	—
248	C₄²	—	5	1	no	—
249	C₄²	—	5	1	no	—
250	C₄²	—	10	4	no	—
251	C₄²	C₂²×C₃×D₅	1	1	yes	characteristic
252	C₃×C₅	#3160	1	1	yes	characteristic, complemented, Hall
253	C₂²×C₃	—	5	2	no	—
254	C₂²×C₃	—	10	1	no	—
255	C₂²×C₃	—	10	1	no	—
256	C₂²×C₃	C₂×D₄×D₅	1	1	yes	characteristic
257	C₃×C₄	—	4	4	no	—
258	C₃×C₄	—	5	2	no	—
259	C₃×C₄	—	10	1	no	—
260	C₃×C₄	—	10	1	no	—
261	C₃×C₄	—	20	4	no	—
262	C₃×C₄	#4058	1	2	yes	—
263	C₃×C₄	#4065	1	2	yes	—
264	C₃×C₄	C₂×Q₈×D₅	1	2	yes	—
265	C₂×C₅	C₂×C₃×C₄⋊Q₈	1	1	yes	characteristic
266	C₂×C₅	C₂×C₃×#135	1	1	yes	characteristic
267	C₂×C₅	C₂×C₃×#135	1	1	yes	characteristic
268	D₅	—	2	2	no	—
269	C₂³	—	5	1	no	—
270	Q₈	—	4	4	no	—
271	Q₈	—	20	4	no	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	5	1	no	—
274	C₂×C₄	—	5	4	no	—
275	C₂×C₄	—	5	1	no	—
276	C₂×C₄	—	5	1	no	—
277	C₂×C₄	—	10	2	no	—
278	C₂×C₄	—	10	2	no	—
279	C₂×C₄	—	10	4	no	—
280	C₂×C₄	C₃×D₄×D₅	1	1	yes	characteristic
281	C₂×C₄	C₃×D₄×D₅	1	1	yes	characteristic
282	C₂×C₄	C₂³×C₃×D₅	1	1	yes	characteristic
283	C₈	—	5	4	no	—
284	C₈	C₃×Q₈×D₅	1	4	yes	—
285	C₂×C₃	—	10	2	no	—
286	C₂×C₃	D₅×C₄⋊Q₈	1	1	yes	characteristic
287	C₂×C₃	C₂×#4058	1	1	yes	characteristic
288	C₂×C₃	C₂×#4065	1	1	yes	characteristic
289	C₅	C₃×#3160	1	1	yes	characteristic, complemented, Sylow
290	C₂²	—	5	2	no	—
291	C₂²	—	10	1	no	—
292	C₂²	—	10	1	no	—
293	C₂²	C₂×C₃×D₄×D₅	1	1	yes	characteristic
294	C₄	—	4	4	no	—
295	C₄	—	5	2	no	—
296	C₄	—	10	1	no	—
297	C₄	—	10	1	no	—
298	C₄	—	20	4	no	—
299	C₄	C₃×#4058	1	2	yes	—
300	C₄	C₃×#4065	1	2	yes	—
301	C₄	C₂×C₃×Q₈×D₅	1	2	yes	—
302	C₃	#10613669	1	1	yes	characteristic, direct product factor, Sylow
303	C₂	—	10	2	no	—
304	C₂	C₃×D₅×C₄⋊Q₈	1	1	yes	characteristic
305	C₂	C₂×C₃×#4058	1	1	yes	characteristic
306	C₂	C₂×C₃×#4065	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂×D₄×D₅
commutator, cosocle	C₂×C₄×C₅	C₂⁴×C₃
Frattini	C₂×C₄	C₂³×C₃×D₅
Fitting	C₃×C₅×#500	C₂
socle	C₂²×C₃×C₅	C₂²×D₄
prime radical	C₂²×C₃×D₅	C₂×D₄
hypercenter	C₃×#500	D₅
hypocenter	C₅	C₃×#3160
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ C₃×#500

Complements

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

N	H ≅ G/N	type
2. C₃×C₅×#500	C₂	semidirect
3. C₃×#65368	C₂	semidirect
5. C₃×#65167	C₂	semidirect
7. C₃×#64971	C₂	semidirect
8. #10613669	C₃	direct (N × H)
27. C₅×#500	C₂×C₃	semidirect
28. #65368	C₂×C₃	semidirect
30. #65167	C₂×C₃	semidirect
32. #64971	C₂×C₃	semidirect
52. C₃×C₈×D₅	Q₈	semidirect
55. C₃×#500	D₅	semidirect
128. C₈×D₅	C₃×Q₈	semidirect
131. #500	C₃×D₅	semidirect
252. C₃×C₅	#3160	semidirect
289. C₅	C₃×#3160	semidirect
302. C₃	#10613669	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×#66062	2
C₃×C₅×#500	2
C₃×#65368	2
C₃×#65169	2
C₃×#65167	2
C₃×C₄×C₈×D₅	2
C₃×#64971	2
#10613669	3
C₃×#3160	5

Subgroups — C3×#10613669 #14439838

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10613669 #14439838