Subgroups — C₃×#10610901 #14439841

Order 1920.

712 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#65851	C₂	1	2	yes	maximal
2	C₃×C₅×#427	C₂	1	1	yes	maximal, characteristic
3	C₃×#65395	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃×C₈×Dic₅	C₂	1	1	yes	maximal, characteristic
5	C₃×#65287	C₂	1	2	yes	maximal
6	C₃×#65167	C₂	1	2	yes	maximal, complemented
7	C₃×#65163	C₂	1	2	yes	maximal, complemented
8	C₃×#65147	C₂	1	2	yes	maximal, complemented
9	C₃×#65144	C₂	1	2	yes	maximal, complemented
10	#10610901	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
11	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₃×C₅×#117	C₂²	1	2	yes	—
13	C₃×#4072	C₂²	1	2	yes	—
14	C₂×C₃×C₄×Dic₅	C₂²	1	1	yes	characteristic
15	C₂²×C₃×#168	C₂²	1	1	yes	characteristic
16	C₃×#4036	C₂²	1	4	yes	—
17	C₃×#4034	C₂²	1	4	yes	—
18	C₃×#4024	—	2	4	no	—
19	C₃×C₄×Dic₁₀	—	2	4	no	—
20	C₃×C₅×#107	C₂²	1	2	yes	—
21	C₃×C₅×#106	C₂²	1	2	yes	—
22	C₃×#3950	C₂²	1	2	yes	—
23	C₃×#3949	C₂²	1	2	yes	—
24	C₃×C₈×Dic₅	C₂²	1	1	yes	characteristic
25	C₃×C₈×Dic₅	C₂²	1	1	yes	characteristic
26	C₃×C₈×Dic₅	C₂²	1	1	yes	characteristic
27	C₃×C₈×Dic₅	C₂²	1	1	yes	characteristic
28	C₃×#3940	C₂²	1	4	yes	—
29	C₃×#3939	C₂²	1	4	yes	—
30	C₃×#3160	—	5	1	no	maximal, Hall
31	#65851	C₂×C₃	1	2	yes	—
32	C₅×#427	C₂×C₃	1	1	yes	characteristic
33	#65395	C₂×C₃	1	1	yes	characteristic
34	C₂×C₈×Dic₅	C₂×C₃	1	1	yes	characteristic
35	#65287	C₂×C₃	1	2	yes	—
36	#65167	C₂×C₃	1	2	yes	complemented
37	#65163	C₂×C₃	1	2	yes	complemented
38	#65147	C₂×C₃	1	2	yes	complemented
39	#65144	C₂×C₃	1	2	yes	complemented
40	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
41	C₂²×C₃×Dic₅	D₄	1	2	yes	—
42	C₂×C₃×Dic₁₀	—	2	4	no	—
43	C₂×C₃×C₅×C₈	Q₈	1	4	yes	complemented
44	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
45	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
46	C₃×C₅×C₄⋊C₄	C₂³	1	4	yes	—
47	C₃×C₅×C₂²⋊C₄	—	2	2	no	—
48	C₃×C₄²×C₅	—	2	2	no	—
49	C₃×C₂²⋊Dic₅	—	2	2	no	—
50	C₃×C₄⋊Dic₅	—	2	4	no	—
51	C₃×C₄⋊Dic₅	C₂³	1	4	yes	—
52	C₃×#683	—	2	4	no	—
53	C₃×#683	—	2	4	no	—
54	C₃×C₄×Dic₅	—	2	2	no	—
55	C₃×C₄×Dic₅	D₄	1	2	yes	—
56	C₃×C₄×Dic₅	C₂³	1	1	yes	characteristic
57	C₃×C₄×Dic₅	C₂³	1	1	yes	characteristic
58	C₂×C₃×#168	Q₈	1	4	yes	complemented
59	C₂×C₃×#168	C₂³	1	1	yes	characteristic
60	C₂×C₃×#168	C₂³	1	1	yes	characteristic
61	C₃×#533	—	5	2	no	—
62	C₃×#500	—	5	2	no	—
63	C₃×#499	—	5	2	no	—
64	C₃×#499	—	5	2	no	—
65	C₃×#498	—	5	2	no	—
66	C₃×#427	—	5	1	no	—
67	C₃×#427	—	5	2	no	—
68	C₃×#427	D₅	1	1	yes	characteristic
69	C₂×C₃×C₄×C₈	—	5	1	no	—
70	C₂²×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
71	C₅×#117	C₂²×C₃	1	2	yes	—
72	#4072	C₂²×C₃	1	2	yes	—
73	C₂×C₄×Dic₅	C₂²×C₃	1	1	yes	characteristic
74	C₂²×#168	C₂²×C₃	1	1	yes	characteristic
75	#4036	C₂²×C₃	1	4	yes	—
76	#4034	C₂²×C₃	1	4	yes	—
77	#4024	—	2	4	no	—
78	C₄×Dic₁₀	—	2	4	no	—
79	C₅×#107	C₂²×C₃	1	2	yes	—
80	C₅×#106	C₂²×C₃	1	2	yes	—
81	#3950	C₂²×C₃	1	2	yes	—
82	#3949	C₂²×C₃	1	2	yes	—
83	C₈×Dic₅	C₂²×C₃	1	1	yes	characteristic
84	C₈×Dic₅	C₂²×C₃	1	1	yes	characteristic
85	C₈×Dic₅	C₂²×C₃	1	1	yes	characteristic
86	C₈×Dic₅	C₂²×C₃	1	1	yes	characteristic
87	#3940	C₂²×C₃	1	4	yes	—
88	#3939	C₂²×C₃	1	4	yes	—
89	#3160	—	5	1	no	Sylow
90	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
91	C₂×C₃×C₄×C₅	—	2	4	no	—
92	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
93	C₂×C₃×C₄×C₅	C₂×Q₈	1	2	yes	—
94	C₂×C₃×C₄×C₅	C₂×Q₈	1	2	yes	—
95	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
96	C₂×C₃×Dic₅	—	2	4	no	—
97	C₂×C₃×Dic₅	—	2	4	no	—
98	C₂×C₃×Dic₅	C₂×D₄	1	2	yes	—
99	C₂×C₃×Dic₅	C₂×D₄	1	2	yes	—
100	C₃×Dic₁₀	—	4	8	no	—
101	C₃×C₅×C₈	C₂×Q₈	1	2	yes	—
102	C₃×C₅×C₈	C₂×Q₈	1	2	yes	—
103	C₃×#168	C₂×Q₈	1	2	yes	—
104	C₃×#168	C₂×Q₈	1	2	yes	—
105	C₂²×C₃×C₈	—	5	1	no	—
106	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
107	C₃×C₄⋊Q₈	—	5	4	no	—
108	C₃×#125	—	5	4	no	—
109	C₃×C₂²⋊Q₈	—	10	4	no	—
110	C₃×C₄×Q₈	—	10	4	no	—
111	C₃×#117	—	5	2	no	—
112	C₃×#117	C₂×D₅	1	2	yes	—
113	C₂×C₃×C₄²	—	5	1	no	—
114	C₃×#107	—	5	2	no	—
115	C₃×#107	—	5	4	no	—
116	C₃×#107	C₂×D₅	1	2	yes	—
117	C₃×#106	—	5	2	no	—
118	C₃×#106	—	5	4	no	—
119	C₃×#106	C₂×D₅	1	2	yes	—
120	C₃×C₄×C₈	—	5	1	no	—
121	C₃×C₄×C₈	—	5	1	no	—
122	C₃×C₄×C₈	—	5	1	no	—
123	C₃×C₄×C₈	—	5	1	no	—
124	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
125	C₂²×Dic₅	C₃×D₄	1	2	yes	—
126	C₂×Dic₁₀	—	2	4	no	—
127	C₂×C₅×C₈	C₃×Q₈	1	4	yes	complemented
128	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
129	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
130	C₅×C₄⋊C₄	C₂³×C₃	1	4	yes	—
131	C₅×C₂²⋊C₄	—	2	2	no	—
132	C₄²×C₅	—	2	2	no	—
133	C₂²⋊Dic₅	—	2	2	no	—
134	C₄⋊Dic₅	—	2	4	no	—
135	C₄⋊Dic₅	C₂³×C₃	1	4	yes	—
136	#683	—	2	4	no	—
137	#683	—	2	4	no	—
138	C₄×Dic₅	—	2	2	no	—
139	C₄×Dic₅	C₃×D₄	1	2	yes	—
140	C₄×Dic₅	C₂³×C₃	1	1	yes	characteristic
141	C₄×Dic₅	C₂³×C₃	1	1	yes	characteristic
142	C₂×#168	C₃×Q₈	1	4	yes	complemented
143	C₂×#168	C₂³×C₃	1	1	yes	characteristic
144	C₂×#168	C₂³×C₃	1	1	yes	characteristic
145	#533	—	5	2	no	—
146	#500	—	5	2	no	—
147	#499	—	5	2	no	—
148	#499	—	5	2	no	—
149	#498	—	5	2	no	—
150	#427	—	5	1	no	—
151	#427	—	5	2	no	—
152	#427	C₃×D₅	1	1	yes	characteristic
153	C₂×C₄×C₈	—	5	1	no	—
154	C₂²×C₃×C₅	—	2	2	no	—
155	C₂²×C₃×C₅	C₄⋊Q₈	1	2	yes	—
156	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
157	C₃×C₄×C₅	—	4	4	no	—
158	C₃×C₄×C₅	C₄⋊Q₈	1	2	yes	—
159	C₃×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
160	C₃×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
161	C₃×Dic₅	—	2	2	no	—
162	C₃×Dic₅	—	4	4	no	—
163	C₃×Dic₅	#135	1	4	yes	—
164	C₂×C₃×Q₈	—	10	4	no	—
165	C₂²×C₃×C₄	—	5	2	no	—
166	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
167	C₂×C₃×C₈	—	5	4	no	—
168	C₂×C₃×C₈	—	5	1	no	—
169	C₂×C₃×C₈	—	5	1	no	—
170	C₂×C₃×C₈	Dic₁₀	1	4	yes	complemented
171	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
172	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
173	C₃×C₄⋊C₄	—	5	4	no	—
174	C₃×C₄⋊C₄	—	10	4	no	—
175	C₃×C₄⋊C₄	—	10	4	no	—
176	C₃×C₄⋊C₄	—	10	4	no	—
177	C₃×C₄⋊C₄	C₂²×D₅	1	4	yes	—
178	C₃×C₂²⋊C₄	—	2	2	no	—
179	C₃×C₂²⋊C₄	—	10	2	no	—
180	C₃×C₄²	—	2	2	no	—
181	C₃×C₄²	—	5	2	no	—
182	C₃×C₄²	—	5	1	no	—
183	C₃×C₄²	—	5	1	no	—
184	C₃×C₄²	—	10	2	no	—
185	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
186	C₂×C₄×C₅	—	2	4	no	—
187	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
188	C₂×C₄×C₅	C₂×C₃×Q₈	1	2	yes	—
189	C₂×C₄×C₅	C₂×C₃×Q₈	1	2	yes	—
190	C₂×C₄×C₅	C₂⁴×C₃	1	1	yes	characteristic
191	C₂×Dic₅	—	2	4	no	—
192	C₂×Dic₅	—	2	4	no	—
193	C₂×Dic₅	C₂×C₃×D₄	1	2	yes	—
194	C₂×Dic₅	C₂×C₃×D₄	1	2	yes	—
195	Dic₁₀	—	4	8	no	—
196	C₅×C₈	C₂×C₃×Q₈	1	2	yes	—
197	C₅×C₈	C₂×C₃×Q₈	1	2	yes	—
198	#168	C₂×C₃×Q₈	1	2	yes	—
199	#168	C₂×C₃×Q₈	1	2	yes	—
200	C₂²×C₈	—	5	1	no	—
201	C₂²×C₈	C₂×C₃×D₅	1	1	yes	characteristic
202	C₄⋊Q₈	—	5	4	no	—
203	#125	—	5	4	no	—
204	C₂²⋊Q₈	—	10	4	no	—
205	C₄×Q₈	—	10	4	no	—
206	#117	—	5	2	no	—
207	#117	C₂×C₃×D₅	1	2	yes	—
208	C₂×C₄²	—	5	1	no	—
209	#107	—	5	2	no	—
210	#107	—	5	4	no	—
211	#107	C₂×C₃×D₅	1	2	yes	—
212	#106	—	5	2	no	—
213	#106	—	5	4	no	—
214	#106	C₂×C₃×D₅	1	2	yes	—
215	C₄×C₈	—	5	1	no	—
216	C₄×C₈	—	5	1	no	—
217	C₄×C₈	—	5	1	no	—
218	C₄×C₈	—	5	1	no	—
219	C₂×C₃×C₅	—	2	2	no	—
220	C₂×C₃×C₅	C₂×C₄⋊Q₈	1	1	yes	characteristic
221	C₂×C₃×C₅	C₂×#135	1	2	yes	—
222	C₂³×C₃	D₄×D₅	1	1	yes	characteristic
223	C₃×Q₈	—	20	8	no	—
224	C₂×C₃×C₄	—	2	4	no	—
225	C₂×C₃×C₄	—	5	2	no	—
226	C₂×C₃×C₄	—	5	2	no	—
227	C₂×C₃×C₄	—	10	4	no	—
228	C₂×C₃×C₄	—	10	4	no	—
229	C₂×C₃×C₄	C₂×Dic₁₀	1	2	yes	—
230	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
231	C₂×C₃×C₄	Q₈×D₅	1	2	yes	—
232	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
233	C₃×C₈	—	5	2	no	—
234	C₃×C₈	—	5	2	no	—
235	C₃×C₈	C₂×Dic₁₀	1	2	yes	—
236	C₃×C₈	C₂×Dic₁₀	1	2	yes	—
237	C₂²×C₅	—	2	2	no	—
238	C₂²×C₅	C₃×C₄⋊Q₈	1	2	yes	—
239	C₂²×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
240	C₄×C₅	—	4	4	no	—
241	C₄×C₅	C₃×C₄⋊Q₈	1	2	yes	—
242	C₄×C₅	C₂²×C₃×Q₈	1	1	yes	characteristic
243	C₄×C₅	C₂²×C₃×Q₈	1	1	yes	characteristic
244	Dic₅	—	2	2	no	—
245	Dic₅	—	4	4	no	—
246	Dic₅	C₃×#135	1	4	yes	—
247	C₂×Q₈	—	10	4	no	—
248	C₂²×C₄	—	5	2	no	—
249	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
250	C₂×C₈	—	5	4	no	—
251	C₂×C₈	—	5	1	no	—
252	C₂×C₈	—	5	1	no	—
253	C₂×C₈	C₃×Dic₁₀	1	4	yes	complemented
254	C₂×C₈	C₂²×C₃×D₅	1	1	yes	characteristic
255	C₂×C₈	C₂²×C₃×D₅	1	1	yes	characteristic
256	C₄⋊C₄	—	5	4	no	—
257	C₄⋊C₄	—	10	4	no	—
258	C₄⋊C₄	—	10	4	no	—
259	C₄⋊C₄	—	10	4	no	—
260	C₄⋊C₄	C₂²×C₃×D₅	1	4	yes	—
261	C₂²⋊C₄	—	2	2	no	—
262	C₂²⋊C₄	—	10	2	no	—
263	C₄²	—	2	2	no	—
264	C₄²	—	5	2	no	—
265	C₄²	—	5	1	no	—
266	C₄²	—	5	1	no	—
267	C₄²	—	10	2	no	—
268	C₃×C₅	#3160	1	1	yes	characteristic, complemented, Hall
269	C₂²×C₃	—	2	2	no	—
270	C₂²×C₃	#4034	1	2	yes	—
271	C₂²×C₃	C₂×D₄×D₅	1	1	yes	characteristic
272	C₃×C₄	—	4	4	no	—
273	C₃×C₄	—	5	4	no	—
274	C₃×C₄	—	10	2	no	—
275	C₃×C₄	—	20	4	no	—
276	C₃×C₄	#4034	1	2	yes	—
277	C₃×C₄	C₂²×Dic₁₀	1	1	yes	characteristic
278	C₃×C₄	C₂×Q₈×D₅	1	1	yes	characteristic
279	C₂×C₅	—	2	2	no	—
280	C₂×C₅	C₂×C₃×C₄⋊Q₈	1	1	yes	characteristic
281	C₂×C₅	C₂×C₃×#135	1	2	yes	—
282	C₂³	C₃×D₄×D₅	1	1	yes	characteristic
283	Q₈	—	20	8	no	—
284	C₂×C₄	—	2	4	no	—
285	C₂×C₄	—	5	2	no	—
286	C₂×C₄	—	5	2	no	—
287	C₂×C₄	—	10	4	no	—
288	C₂×C₄	—	10	4	no	—
289	C₂×C₄	C₂×C₃×Dic₁₀	1	2	yes	—
290	C₂×C₄	C₃×D₄×D₅	1	1	yes	characteristic
291	C₂×C₄	C₃×Q₈×D₅	1	2	yes	—
292	C₂×C₄	C₂³×C₃×D₅	1	1	yes	characteristic
293	C₈	—	5	2	no	—
294	C₈	—	5	2	no	—
295	C₈	C₂×C₃×Dic₁₀	1	2	yes	—
296	C₈	C₂×C₃×Dic₁₀	1	2	yes	—
297	C₂×C₃	—	2	2	no	—
298	C₂×C₃	C₂×#4034	1	1	yes	characteristic
299	C₂×C₃	D₅×#135	1	2	yes	—
300	C₅	C₃×#3160	1	1	yes	characteristic, complemented, Sylow
301	C₂²	—	2	2	no	—
302	C₂²	C₃×#4034	1	2	yes	—
303	C₂²	C₂×C₃×D₄×D₅	1	1	yes	characteristic
304	C₄	—	4	4	no	—
305	C₄	—	5	4	no	—
306	C₄	—	10	2	no	—
307	C₄	—	20	4	no	—
308	C₄	C₃×#4034	1	2	yes	—
309	C₄	C₂²×C₃×Dic₁₀	1	1	yes	characteristic
310	C₄	C₂×C₃×Q₈×D₅	1	1	yes	characteristic
311	C₃	#10610901	1	1	yes	characteristic, direct product factor, Sylow
312	C₂	—	2	2	no	—
313	C₂	C₂×C₃×#4034	1	1	yes	characteristic
314	C₂	C₃×D₅×#135	1	2	yes	—

	subgroup	G/H
center	C₂×C₃×C₄	D₄×D₅
commutator, cosocle	C₂×C₄×C₅	C₂⁴×C₃
Frattini	C₂×C₄	C₂³×C₃×D₅
Fitting	C₃×C₅×#427	C₂
socle	C₂²×C₃×C₅	C₂²×D₄
prime radical	C₂³×C₃×C₅	C₂×D₄
hypercenter	C₃×#427	D₅
hypocenter	C₅	C₃×#3160
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂²×C₃×C₄ ◁ C₃×#427

Complements

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

N	H ≅ G/N	type
6. C₃×#65167	C₂	semidirect
7. C₃×#65163	C₂	semidirect
8. C₃×#65147	C₂	semidirect
9. C₃×#65144	C₂	semidirect
10. #10610901	C₃	direct (N × H)
36. #65167	C₂×C₃	semidirect
37. #65163	C₂×C₃	semidirect
38. #65147	C₂×C₃	semidirect
39. #65144	C₂×C₃	semidirect
43. C₂×C₃×C₅×C₈	Q₈	semidirect
58. C₂×C₃×#168	Q₈	semidirect
127. C₂×C₅×C₈	C₃×Q₈	semidirect
142. C₂×#168	C₃×Q₈	semidirect
170. C₂×C₃×C₈	Dic₁₀	semidirect
253. C₂×C₈	C₃×Dic₁₀	semidirect
268. C₃×C₅	#3160	semidirect
300. C₅	C₃×#3160	semidirect
311. C₃	#10610901	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×#65851	2
C₃×C₅×#427	2
C₃×#65395	2
C₂×C₃×C₈×Dic₅	2
C₃×#65287	2
C₃×#65167	2
C₃×#65163	2
C₃×#65147	2
C₃×#65144	2
#10610901	3
C₃×#3160	5

Subgroups — C3×#10610901 #14439841

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10610901 #14439841