Subgroups — C₃×#10604039 #14286147

Order 1920.

360 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#65703	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#65702	C₂	1	1	yes	maximal, characteristic
3	C₃×#65141	C₂	1	1	yes	maximal, characteristic
4	C₃×#64881	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₈×#170	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#64872	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#64871	C₂	1	1	yes	maximal, characteristic
8	#10604039	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂×C₃×D₅⋊C₈	—	2	1	no	—
10	C₂×C₃×#677	C₄	1	1	yes	characteristic
11	C₃×#4039	C₂²	1	1	yes	characteristic
12	C₃×#4033	C₂²	1	1	yes	characteristic
13	C₃×#4004	—	2	1	no	—
14	C₃×#4003	—	2	1	no	—
15	C₃×C₄⋊#170	C₂²	1	1	yes	characteristic
16	C₃×C₄⋊#170	C₂²	1	1	yes	characteristic
17	C₃×C₄×#170	—	2	1	no	—
18	C₃×C₄×#170	C₂²	1	1	yes	characteristic
19	C₃×C₄×#170	C₂²	1	1	yes	characteristic
20	C₃×C₅×#106	C₄	1	1	yes	characteristic
21	C₃×C₈×Dic₅	C₂²	1	1	yes	characteristic
22	C₃×#3942	C₄	1	1	yes	characteristic
23	C₃×#3941	C₄	1	1	yes	characteristic
24	C₃×C₈×SD₁₆	—	5	1	no	maximal, Hall
25	#65703	C₂×C₃	1	1	yes	characteristic, complemented
26	#65702	C₂×C₃	1	1	yes	characteristic
27	#65141	C₂×C₃	1	1	yes	characteristic
28	#64881	C₂×C₃	1	1	yes	characteristic, complemented
29	C₈×#170	C₂×C₃	1	1	yes	characteristic, complemented
30	#64872	C₂×C₃	1	1	yes	characteristic, complemented
31	#64871	C₂×C₃	1	1	yes	characteristic
32	C₂×C₃×D₂₀	C₂×C₄	1	1	yes	characteristic
33	C₂×C₃×C₄×D₅	—	2	1	no	—
34	C₂×C₃×Dic₁₀	C₂×C₄	1	1	yes	characteristic
35	C₂×C₃×#170	—	2	1	no	—
36	C₂×C₃×#170	—	2	1	no	—
37	C₂×C₃×#170	—	2	1	no	—
38	C₂×C₃×#170	D₄	1	1	yes	characteristic
39	C₂×C₃×#170	D₄	1	1	yes	characteristic
40	C₃×D₅⋊C₈	—	4	2	no	—
41	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
42	C₃×C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
43	C₃×C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
44	C₃×#685	—	2	1	no	—
45	C₃×#683	—	2	1	no	—
46	C₃×C₄×Dic₅	—	2	1	no	—
47	C₃×C₄×Dic₅	C₂³	1	1	yes	characteristic
48	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
49	C₃×#677	C₈	1	4	yes	complemented
50	C₃×C₈×Q₈	—	5	1	no	—
51	C₃×C₄×SD₁₆	—	5	1	no	—
52	C₃×C₈×D₄	—	5	1	no	—
53	C₃×#334	—	5	1	no	—
54	C₃×Q₈⋊C₈	—	5	1	no	—
55	C₃×D₄⋊C₈	—	5	1	no	—
56	C₃×C₈²	—	5	1	no	—
57	C₂×D₅⋊C₈	—	2	1	no	—
58	C₂×#677	C₃×C₄	1	1	yes	characteristic
59	#4039	C₂²×C₃	1	1	yes	characteristic
60	#4033	C₂²×C₃	1	1	yes	characteristic
61	#4004	—	2	1	no	—
62	#4003	—	2	1	no	—
63	C₄⋊#170	C₂²×C₃	1	1	yes	characteristic
64	C₄⋊#170	C₂²×C₃	1	1	yes	characteristic
65	C₄×#170	—	2	1	no	—
66	C₄×#170	C₂²×C₃	1	1	yes	characteristic
67	C₄×#170	C₂²×C₃	1	1	yes	characteristic
68	C₅×#106	C₃×C₄	1	1	yes	characteristic
69	C₈×Dic₅	C₂²×C₃	1	1	yes	characteristic
70	#3942	C₃×C₄	1	1	yes	characteristic
71	#3941	C₃×C₄	1	1	yes	characteristic
72	C₈×SD₁₆	—	5	1	no	Sylow
73	C₂²×C₃×D₅	—	2	1	no	—
74	C₂×C₃×C₄×C₅	—	2	1	no	—
75	C₂×C₃×C₄×C₅	—	2	1	no	—
76	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₃×Dic₅	—	2	1	no	—
78	C₂×C₃×Dic₅	C₂×D₄	1	1	yes	characteristic
79	C₂×C₃×Dic₅	#41	1	1	yes	characteristic
80	C₃×D₂₀	—	2	1	no	—
81	C₃×D₂₀	C₂×C₈	1	2	yes	—
82	C₃×C₄×D₅	—	4	1	no	—
83	C₃×C₄×D₅	—	4	1	no	—
84	C₃×Dic₁₀	—	2	1	no	—
85	C₃×Dic₁₀	C₂×C₈	1	2	yes	complemented
86	C₃×#170	—	2	2	no	—
87	C₃×#170	—	2	1	no	—
88	C₃×#170	—	4	1	no	—
89	C₃×#170	—	4	1	no	—
90	C₃×#170	SD₁₆	1	2	yes	complemented
91	C₃×C₅×C₈	C₂×C₈	1	2	yes	complemented
92	C₃×#168	—	2	1	no	—
93	C₂×C₃×SD₁₆	—	5	1	no	—
94	C₂²×C₃×C₈	—	10	1	no	—
95	C₃×C₄×Q₈	—	5	1	no	—
96	C₃×C₄×D₄	—	5	1	no	—
97	C₃×#106	AGL(1,𝔽₅)	1	1	yes	characteristic
98	C₃×C₄⋊C₈	—	5	1	no	—
99	C₃×C₄⋊C₈	—	5	1	no	—
100	C₃×C₄⋊C₈	—	10	1	no	—
101	C₃×#103	—	5	1	no	—
102	C₃×#102	—	5	1	no	—
103	C₃×C₂²⋊C₈	—	10	1	no	—
104	C₃×C₄×C₈	—	5	1	no	—
105	C₃×C₄×C₈	—	5	1	no	—
106	C₃×C₄×C₈	—	5	1	no	—
107	C₃×C₄×C₈	—	10	1	no	—
108	C₂×D₂₀	C₂×C₃×C₄	1	1	yes	characteristic
109	C₂×C₄×D₅	—	2	1	no	—
110	C₂×Dic₁₀	C₂×C₃×C₄	1	1	yes	characteristic
111	C₂×#170	—	2	1	no	—
112	C₂×#170	—	2	1	no	—
113	C₂×#170	—	2	1	no	—
114	C₂×#170	C₃×D₄	1	1	yes	characteristic
115	C₂×#170	C₃×D₄	1	1	yes	characteristic
116	D₅⋊C₈	—	4	2	no	—
117	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
118	C₅×C₄⋊C₄	C₂×C₃×C₄	1	1	yes	characteristic
119	C₅×C₄⋊C₄	C₂×C₃×C₄	1	1	yes	characteristic
120	#685	—	2	1	no	—
121	#683	—	2	1	no	—
122	C₄×Dic₅	—	2	1	no	—
123	C₄×Dic₅	C₂³×C₃	1	1	yes	characteristic
124	C₂×#168	C₂×C₃×C₄	1	1	yes	characteristic
125	#677	C₃×C₈	1	4	yes	complemented
126	C₈×Q₈	—	5	1	no	—
127	C₄×SD₁₆	—	5	1	no	—
128	C₈×D₄	—	5	1	no	—
129	#334	—	5	1	no	—
130	Q₈⋊C₈	—	5	1	no	—
131	D₄⋊C₈	—	5	1	no	—
132	C₈²	—	5	1	no	—
133	C₂²×C₃×C₅	C₄×D₄	1	1	yes	characteristic
134	C₂×C₃×D₅	—	2	2	no	—
135	C₂×C₃×D₅	—	4	1	no	—
136	C₂×C₃×D₅	—	4	1	no	—
137	C₃×C₄×C₅	—	4	1	no	—
138	C₃×C₄×C₅	—	4	1	no	—
139	C₃×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
140	C₃×C₄×C₅	#131	1	1	yes	characteristic
141	C₃×Dic₅	—	2	2	no	—
142	C₃×Dic₅	—	2	1	no	—
143	C₃×Dic₅	C₂×SD₁₆	1	1	yes	characteristic
144	C₃×Dic₅	#135	1	1	yes	characteristic
145	C₂×C₃×Q₈	—	5	1	no	—
146	C₂×C₃×D₄	—	5	1	no	—
147	C₂²×C₃×C₄	—	10	1	no	—
148	C₃×SD₁₆	—	5	4	no	—
149	C₂×C₃×C₈	—	5	1	no	—
150	C₂×C₃×C₈	—	5	1	no	—
151	C₂×C₃×C₈	—	5	1	no	—
152	C₂×C₃×C₈	—	10	1	no	—
153	C₂×C₃×C₈	—	10	1	no	—
154	C₂×C₃×C₈	—	10	1	no	—
155	C₂×C₃×C₈	—	20	2	no	—
156	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
157	C₃×C₄⋊C₄	—	10	1	no	—
158	C₃×C₄⋊C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
159	C₃×C₄⋊C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
160	C₃×C₂²⋊C₄	—	10	1	no	—
161	C₃×C₄²	—	5	1	no	—
162	C₃×C₄²	—	10	1	no	—
163	C₂²×D₅	—	2	1	no	—
164	C₂×C₄×C₅	—	2	1	no	—
165	C₂×C₄×C₅	—	2	1	no	—
166	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
167	C₂×Dic₅	—	2	1	no	—
168	C₂×Dic₅	C₂×C₃×D₄	1	1	yes	characteristic
169	C₂×Dic₅	C₃×#41	1	1	yes	characteristic
170	D₂₀	—	2	1	no	—
171	D₂₀	C₂×C₃×C₈	1	2	yes	—
172	C₄×D₅	—	4	1	no	—
173	C₄×D₅	—	4	1	no	—
174	Dic₁₀	—	2	1	no	—
175	Dic₁₀	C₂×C₃×C₈	1	2	yes	complemented
176	#170	—	2	2	no	—
177	#170	—	2	1	no	—
178	#170	—	4	1	no	—
179	#170	—	4	1	no	—
180	#170	C₃×SD₁₆	1	2	yes	complemented
181	C₅×C₈	C₂×C₃×C₈	1	2	yes	complemented
182	#168	—	2	1	no	—
183	C₂×SD₁₆	—	5	1	no	—
184	C₂²×C₈	—	10	1	no	—
185	C₄×Q₈	—	5	1	no	—
186	C₄×D₄	—	5	1	no	—
187	#106	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
188	C₄⋊C₈	—	5	1	no	—
189	C₄⋊C₈	—	5	1	no	—
190	C₄⋊C₈	—	10	1	no	—
191	#103	—	5	1	no	—
192	#102	—	5	1	no	—
193	C₂²⋊C₈	—	10	1	no	—
194	C₄×C₈	—	5	1	no	—
195	C₄×C₈	—	5	1	no	—
196	C₄×C₈	—	5	1	no	—
197	C₄×C₈	—	10	1	no	—
198	C₂×C₃×C₅	C₈×D₄	1	1	yes	characteristic
199	C₂×C₃×C₅	C₄×SD₁₆	1	1	yes	characteristic
200	C₂×C₃×C₅	#443	1	1	yes	characteristic
201	C₃×D₅	—	4	2	no	—
202	C₂³×C₃	—	10	1	no	—
203	C₃×Q₈	—	5	2	no	—
204	C₃×Q₈	—	10	1	no	—
205	C₃×D₄	—	5	2	no	—
206	C₃×D₄	—	10	1	no	—
207	C₂×C₃×C₄	—	2	1	no	—
208	C₂×C₃×C₄	—	2	1	no	—
209	C₂×C₃×C₄	—	5	1	no	—
210	C₂×C₃×C₄	—	5	1	no	—
211	C₂×C₃×C₄	—	10	1	no	—
212	C₂×C₃×C₄	—	20	1	no	—
213	C₂×C₃×C₄	—	20	1	no	—
214	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
215	C₃×C₈	—	5	2	no	—
216	C₃×C₈	—	10	1	no	—
217	C₃×C₈	—	10	2	no	—
218	C₃×C₈	—	10	1	no	—
219	C₃×C₈	—	20	1	no	—
220	C₃×C₈	—	20	1	no	—
221	C₃×C₈	D₅⋊C₈	1	2	yes	complemented
222	C₂²×C₅	C₃×C₄×D₄	1	1	yes	characteristic
223	C₂×D₅	—	2	2	no	—
224	C₂×D₅	—	4	1	no	—
225	C₂×D₅	—	4	1	no	—
226	C₄×C₅	—	4	1	no	—
227	C₄×C₅	—	4	1	no	—
228	C₄×C₅	C₂²×C₃×C₈	1	1	yes	characteristic
229	C₄×C₅	C₃×#131	1	1	yes	characteristic
230	Dic₅	—	2	2	no	—
231	Dic₅	—	2	1	no	—
232	Dic₅	C₂×C₃×SD₁₆	1	1	yes	characteristic
233	Dic₅	C₃×#135	1	1	yes	characteristic
234	C₂×Q₈	—	5	1	no	—
235	C₂×D₄	—	5	1	no	—
236	C₂²×C₄	—	10	1	no	—
237	SD₁₆	—	5	4	no	—
238	C₂×C₈	—	5	1	no	—
239	C₂×C₈	—	5	1	no	—
240	C₂×C₈	—	5	1	no	—
241	C₂×C₈	—	10	1	no	—
242	C₂×C₈	—	10	1	no	—
243	C₂×C₈	—	10	1	no	—
244	C₂×C₈	—	20	2	no	—
245	C₂×C₈	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
246	C₄⋊C₄	—	10	1	no	—
247	C₄⋊C₄	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
248	C₄⋊C₄	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
249	C₂²⋊C₄	—	10	1	no	—
250	C₄²	—	5	1	no	—
251	C₄²	—	10	1	no	—
252	C₃×C₅	C₈×SD₁₆	1	1	yes	characteristic, complemented, Hall
253	C₂²×C₃	—	10	2	no	—
254	C₂²×C₃	—	20	1	no	—
255	C₂²×C₃	—	20	1	no	—
256	C₂²×C₃	D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
257	C₃×C₄	—	4	1	no	—
258	C₃×C₄	—	4	1	no	—
259	C₃×C₄	—	5	1	no	—
260	C₃×C₄	—	5	1	no	—
261	C₃×C₄	—	10	2	no	—
262	C₃×C₄	—	10	1	no	—
263	C₃×C₄	C₂×D₅⋊C₈	1	1	yes	characteristic
264	C₃×C₄	#4133	1	1	yes	characteristic
265	C₂×C₅	C₃×C₈×D₄	1	1	yes	characteristic
266	C₂×C₅	C₃×C₄×SD₁₆	1	1	yes	characteristic
267	C₂×C₅	C₃×#443	1	1	yes	characteristic
268	D₅	—	4	2	no	—
269	C₂³	—	10	1	no	—
270	Q₈	—	5	2	no	—
271	Q₈	—	10	1	no	—
272	D₄	—	5	2	no	—
273	D₄	—	10	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	5	1	no	—
277	C₂×C₄	—	5	1	no	—
278	C₂×C₄	—	10	1	no	—
279	C₂×C₄	—	20	1	no	—
280	C₂×C₄	—	20	1	no	—
281	C₂×C₄	C₂²×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
282	C₈	—	5	2	no	—
283	C₈	—	10	1	no	—
284	C₈	—	10	2	no	—
285	C₈	—	10	1	no	—
286	C₈	—	20	1	no	—
287	C₈	—	20	1	no	—
288	C₈	C₃×D₅⋊C₈	1	2	yes	complemented
289	C₂×C₃	—	20	2	no	—
290	C₂×C₃	#65702	1	1	yes	characteristic
291	C₂×C₃	SD₁₆×AGL(1,𝔽₅)	1	1	yes	characteristic
292	C₂×C₃	#65736	1	1	yes	characteristic
293	C₅	C₃×C₈×SD₁₆	1	1	yes	characteristic, complemented, Sylow
294	C₂²	—	10	2	no	—
295	C₂²	—	20	1	no	—
296	C₂²	—	20	1	no	—
297	C₂²	C₃×D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
298	C₄	—	4	1	no	—
299	C₄	—	4	1	no	—
300	C₄	—	5	1	no	—
301	C₄	—	5	1	no	—
302	C₄	—	10	2	no	—
303	C₄	—	10	1	no	—
304	C₄	C₂×C₃×D₅⋊C₈	1	1	yes	characteristic
305	C₄	C₃×#4133	1	1	yes	characteristic
306	C₃	#10604039	1	1	yes	characteristic, direct product factor, Sylow
307	C₂	—	20	2	no	—
308	C₂	C₃×#65702	1	1	yes	characteristic
309	C₂	C₃×SD₁₆×AGL(1,𝔽₅)	1	1	yes	characteristic
310	C₂	C₃×#65736	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	D₄×AGL(1,𝔽₅)
commutator	C₄×C₅	C₂²×C₃×C₈
Frattini	C₂×C₄	C₂²×C₃×AGL(1,𝔽₅)
Fitting	C₃×C₅×#106	C₄
socle	C₂²×C₃×C₅	C₄×D₄
prime radical	C₂×C₃×D₂₀	C₂×C₄
cosocle	C₄×Dic₅	C₂³×C₃
hypercenter	C₃×#106	AGL(1,𝔽₅)
hypocenter	C₅	C₃×C₈×SD₁₆
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄ ◁ C₃×#106

Complements

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

N	H ≅ G/N	type
1. C₃×#65703	C₂	semidirect
4. C₃×#64881	C₂	semidirect
5. C₃×C₈×#170	C₂	semidirect
6. C₃×#64872	C₂	semidirect
8. #10604039	C₃	direct (N × H)
25. #65703	C₂×C₃	semidirect
28. #64881	C₂×C₃	semidirect
29. C₈×#170	C₂×C₃	semidirect
30. #64872	C₂×C₃	semidirect
49. C₃×#677	C₈	semidirect
85. C₃×Dic₁₀	C₂×C₈	semidirect
90. C₃×#170	SD₁₆	semidirect
91. C₃×C₅×C₈	C₂×C₈	semidirect
125. #677	C₃×C₈	semidirect
175. Dic₁₀	C₂×C₃×C₈	semidirect
180. #170	C₃×SD₁₆	semidirect
181. C₅×C₈	C₂×C₃×C₈	semidirect
221. C₃×C₈	D₅⋊C₈	semidirect
252. C₃×C₅	C₈×SD₁₆	semidirect
288. C₈	C₃×D₅⋊C₈	semidirect
293. C₅	C₃×C₈×SD₁₆	semidirect
306. C₃	#10604039	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×#65703	2
C₃×#65702	2
C₃×#65141	2
C₃×#64881	2
C₃×C₈×#170	2
C₃×#64872	2
C₃×#64871	2
#10604039	3
C₃×C₈×SD₁₆	5

Subgroups — C3×#10604039 #14286147

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10604039 #14286147