Subgroups — C₃×#10604014 #14280675

Order 1920.

356 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#65680	C₂	1	1	yes	maximal, characteristic
2	C₃×#65678	C₂	1	1	yes	maximal, characteristic
3	C₃×#65122	C₂	1	1	yes	maximal, characteristic
4	C₃×#170⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#170⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#64879	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×C₈×#170	C₂	1	1	yes	maximal, characteristic, complemented
8	#10604014	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₃×C₄²×D₅	C₂²	1	1	yes	characteristic
10	C₃×#4004	C₄	1	2	yes	—
11	C₃×#4003	C₄	1	2	yes	—
12	C₃×#4003	C₄	1	2	yes	—
13	C₃×C₄⋊#170	C₄	1	2	yes	—
14	C₃×C₄×#170	C₂²	1	1	yes	characteristic
15	C₃×C₄×#170	C₂²	1	1	yes	characteristic
16	C₃×C₄×#170	C₂²	1	1	yes	characteristic
17	C₃×C₄×#170	C₂²	1	1	yes	characteristic
18	C₃×C₅×C₄⋊C₈	C₄	1	1	yes	characteristic
19	C₃×#3952	C₄	1	1	yes	characteristic
20	C₃×#3952	C₄	1	1	yes	characteristic
21	C₃×C₈×Dic₅	C₂²	1	1	yes	characteristic
22	C₃×C₈×Dic₅	C₂²	1	1	yes	characteristic
23	C₃×C₄⋊#168	C₄	1	1	yes	characteristic
24	C₃×#1454	—	5	1	no	maximal, Hall
25	#65680	C₂×C₃	1	1	yes	characteristic
26	#65678	C₂×C₃	1	1	yes	characteristic
27	#65122	C₂×C₃	1	1	yes	characteristic
28	#170⋊C₈	C₂×C₃	1	1	yes	characteristic, complemented
29	#170⋊C₈	C₂×C₃	1	1	yes	characteristic, complemented
30	#64879	C₂×C₃	1	1	yes	characteristic, complemented
31	C₈×#170	C₂×C₃	1	1	yes	characteristic, complemented
32	C₂×C₃×C₄×D₅	C₂×C₄	1	1	yes	characteristic
33	C₂×C₃×C₄×D₅	C₂×C₄	1	1	yes	characteristic
34	C₂×C₃×C₄×D₅	C₂×C₄	1	1	yes	characteristic
35	C₂×C₃×#170	C₂×C₄	1	2	yes	—
36	C₂×C₃×#170	C₂×C₄	1	2	yes	—
37	C₂×C₃×#170	C₂×C₄	1	2	yes	—
38	C₂×C₃×#170	C₂×C₄	1	2	yes	—
39	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
40	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
41	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
42	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
43	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
44	C₃×C₄×Dic₅	C₂³	1	1	yes	characteristic
45	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
46	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
47	C₃×#431	—	5	1	no	—
48	C₃×#431	—	5	1	no	—
49	C₃×#431	—	5	1	no	—
50	C₃×#322	—	5	1	no	—
51	C₃×#322	—	5	1	no	—
52	C₃×#322	—	5	1	no	—
53	C₃×C₈²	—	5	1	no	—
54	C₄²×D₅	C₂²×C₃	1	1	yes	characteristic
55	#4004	C₃×C₄	1	2	yes	—
56	#4003	C₃×C₄	1	2	yes	—
57	#4003	C₃×C₄	1	2	yes	—
58	C₄⋊#170	C₃×C₄	1	2	yes	—
59	C₄×#170	C₂²×C₃	1	1	yes	characteristic
60	C₄×#170	C₂²×C₃	1	1	yes	characteristic
61	C₄×#170	C₂²×C₃	1	1	yes	characteristic
62	C₄×#170	C₂²×C₃	1	1	yes	characteristic
63	C₅×C₄⋊C₈	C₃×C₄	1	1	yes	characteristic
64	#3952	C₃×C₄	1	1	yes	characteristic
65	#3952	C₃×C₄	1	1	yes	characteristic
66	C₈×Dic₅	C₂²×C₃	1	1	yes	characteristic
67	C₈×Dic₅	C₂²×C₃	1	1	yes	characteristic
68	C₄⋊#168	C₃×C₄	1	1	yes	characteristic
69	#1454	—	5	1	no	Sylow
70	C₂²×C₃×D₅	C₄²	1	1	yes	characteristic
71	C₂×C₃×C₄×C₅	C₄²	1	1	yes	characteristic
72	C₂×C₃×C₄×C₅	C₄²	1	1	yes	characteristic
73	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₃×Dic₅	C₄²	1	1	yes	characteristic
75	C₂×C₃×Dic₅	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₃×Dic₅	C₂²×C₄	1	1	yes	characteristic
77	C₃×C₄×D₅	—	2	1	no	—
78	C₃×C₄×D₅	—	2	1	no	—
79	C₃×C₄×D₅	—	2	2	no	—
80	C₃×C₄×D₅	—	2	2	no	—
81	C₃×#170	—	2	2	no	—
82	C₃×#170	—	2	2	no	—
83	C₃×#170	—	2	2	no	—
84	C₃×#170	—	2	2	no	—
85	C₃×C₅×C₈	—	2	1	no	—
86	C₃×C₅×C₈	—	2	1	no	—
87	C₃×#168	—	2	1	no	—
88	C₃×#168	—	2	1	no	—
89	C₂×C₃×C₄²	—	5	1	no	—
90	C₃×C₄⋊C₈	—	5	2	no	—
91	C₃×C₄⋊C₈	—	5	2	no	—
92	C₃×C₄⋊C₈	—	5	1	no	—
93	C₃×C₄⋊C₈	AGL(1,𝔽₅)	1	1	yes	characteristic
94	C₃×C₂²⋊C₈	—	5	2	no	—
95	C₃×C₂²⋊C₈	—	5	2	no	—
96	C₃×C₂²⋊C₈	—	5	1	no	—
97	C₃×C₂²⋊C₈	—	5	1	no	—
98	C₃×C₄×C₈	—	5	1	no	—
99	C₃×C₄×C₈	—	5	1	no	—
100	C₃×C₄×C₈	—	5	1	no	—
101	C₃×C₄×C₈	—	5	1	no	—
102	C₃×C₄×C₈	—	5	1	no	—
103	C₃×C₄×C₈	—	5	1	no	—
104	C₂×C₄×D₅	C₂×C₃×C₄	1	1	yes	characteristic
105	C₂×C₄×D₅	C₂×C₃×C₄	1	1	yes	characteristic
106	C₂×C₄×D₅	C₂×C₃×C₄	1	1	yes	characteristic
107	C₂×#170	C₂×C₃×C₄	1	2	yes	—
108	C₂×#170	C₂×C₃×C₄	1	2	yes	—
109	C₂×#170	C₂×C₃×C₄	1	2	yes	—
110	C₂×#170	C₂×C₃×C₄	1	2	yes	—
111	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
112	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
113	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
114	C₄×Dic₅	C₂×C₃×C₄	1	1	yes	characteristic
115	C₄×Dic₅	C₂×C₃×C₄	1	1	yes	characteristic
116	C₄×Dic₅	C₂³×C₃	1	1	yes	characteristic
117	C₂×#168	C₂×C₃×C₄	1	1	yes	characteristic
118	C₂×#168	C₂×C₃×C₄	1	1	yes	characteristic
119	#431	—	5	1	no	—
120	#431	—	5	1	no	—
121	#431	—	5	1	no	—
122	#322	—	5	1	no	—
123	#322	—	5	1	no	—
124	#322	—	5	1	no	—
125	C₈²	—	5	1	no	—
126	C₂²×C₃×C₅	C₂×C₄²	1	1	yes	characteristic
127	C₂×C₃×D₅	—	2	1	no	—
128	C₂×C₃×D₅	—	2	1	no	—
129	C₂×C₃×D₅	—	2	1	no	—
130	C₃×C₄×C₅	—	2	1	no	—
131	C₃×C₄×C₅	—	2	1	no	—
132	C₃×C₄×C₅	#131	1	1	yes	characteristic
133	C₃×C₄×C₅	#131	1	1	yes	characteristic
134	C₃×Dic₅	—	2	1	no	—
135	C₃×Dic₅	#131	1	2	yes	—
136	C₃×Dic₅	#131	1	2	yes	—
137	C₂²×C₃×C₄	—	5	1	no	—
138	C₂²×C₃×C₄	—	5	1	no	—
139	C₂²×C₃×C₄	—	5	1	no	—
140	C₂×C₃×C₈	—	5	2	no	—
141	C₂×C₃×C₈	—	5	2	no	—
142	C₂×C₃×C₈	—	5	2	no	—
143	C₂×C₃×C₈	—	5	2	no	—
144	C₂×C₃×C₈	—	5	1	no	—
145	C₂×C₃×C₈	—	5	1	no	—
146	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
147	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
148	C₃×C₄²	—	5	1	no	—
149	C₃×C₄²	—	5	1	no	—
150	C₃×C₄²	—	5	1	no	—
151	C₃×C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
152	C₂²×D₅	C₃×C₄²	1	1	yes	characteristic
153	C₂×C₄×C₅	C₃×C₄²	1	1	yes	characteristic
154	C₂×C₄×C₅	C₃×C₄²	1	1	yes	characteristic
155	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
156	C₂×Dic₅	C₃×C₄²	1	1	yes	characteristic
157	C₂×Dic₅	C₂²×C₃×C₄	1	1	yes	characteristic
158	C₂×Dic₅	C₂²×C₃×C₄	1	1	yes	characteristic
159	C₄×D₅	—	2	1	no	—
160	C₄×D₅	—	2	1	no	—
161	C₄×D₅	—	2	2	no	—
162	C₄×D₅	—	2	2	no	—
163	#170	—	2	2	no	—
164	#170	—	2	2	no	—
165	#170	—	2	2	no	—
166	#170	—	2	2	no	—
167	C₅×C₈	—	2	1	no	—
168	C₅×C₈	—	2	1	no	—
169	#168	—	2	1	no	—
170	#168	—	2	1	no	—
171	C₂×C₄²	—	5	1	no	—
172	C₄⋊C₈	—	5	2	no	—
173	C₄⋊C₈	—	5	2	no	—
174	C₄⋊C₈	—	5	1	no	—
175	C₄⋊C₈	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
176	C₂²⋊C₈	—	5	2	no	—
177	C₂²⋊C₈	—	5	2	no	—
178	C₂²⋊C₈	—	5	1	no	—
179	C₂²⋊C₈	—	5	1	no	—
180	C₄×C₈	—	5	1	no	—
181	C₄×C₈	—	5	1	no	—
182	C₄×C₈	—	5	1	no	—
183	C₄×C₈	—	5	1	no	—
184	C₄×C₈	—	5	1	no	—
185	C₄×C₈	—	5	1	no	—
186	C₂×C₃×C₅	#405	1	1	yes	characteristic
187	C₂×C₃×C₅	#405	1	1	yes	characteristic
188	C₂×C₃×C₅	#405	1	1	yes	characteristic
189	C₃×D₅	—	4	1	no	—
190	C₂³×C₃	—	5	1	no	—
191	C₂×C₃×C₄	—	5	1	no	—
192	C₂×C₃×C₄	—	5	1	no	—
193	C₂×C₃×C₄	—	5	1	no	—
194	C₂×C₃×C₄	—	10	2	no	—
195	C₂×C₃×C₄	—	10	2	no	—
196	C₂×C₃×C₄	—	10	1	no	—
197	C₂×C₃×C₄	—	10	1	no	—
198	C₂×C₃×C₄	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
199	C₂×C₃×C₄	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
200	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
201	C₃×C₈	—	2	1	no	—
202	C₃×C₈	—	2	1	no	—
203	C₃×C₈	—	10	2	no	—
204	C₃×C₈	—	10	2	no	—
205	C₃×C₈	—	10	2	no	—
206	C₃×C₈	—	10	2	no	—
207	C₃×C₈	—	10	1	no	—
208	C₃×C₈	—	10	1	no	—
209	C₂²×C₅	C₂×C₃×C₄²	1	1	yes	characteristic
210	C₂×D₅	—	2	1	no	—
211	C₂×D₅	—	2	1	no	—
212	C₂×D₅	—	2	1	no	—
213	C₄×C₅	—	2	1	no	—
214	C₄×C₅	—	2	1	no	—
215	C₄×C₅	C₃×#131	1	1	yes	characteristic
216	C₄×C₅	C₃×#131	1	1	yes	characteristic
217	Dic₅	—	2	1	no	—
218	Dic₅	C₃×#131	1	2	yes	—
219	Dic₅	C₃×#131	1	2	yes	—
220	C₂²×C₄	—	5	1	no	—
221	C₂²×C₄	—	5	1	no	—
222	C₂²×C₄	—	5	1	no	—
223	C₂×C₈	—	5	2	no	—
224	C₂×C₈	—	5	2	no	—
225	C₂×C₈	—	5	2	no	—
226	C₂×C₈	—	5	2	no	—
227	C₂×C₈	—	5	1	no	—
228	C₂×C₈	—	5	1	no	—
229	C₂×C₈	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
230	C₂×C₈	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
231	C₄²	—	5	1	no	—
232	C₄²	—	5	1	no	—
233	C₄²	—	5	1	no	—
234	C₄²	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
235	C₃×C₅	#1454	1	1	yes	characteristic, complemented, Hall
236	C₂²×C₃	—	10	1	no	—
237	C₂²×C₃	—	10	1	no	—
238	C₂²×C₃	—	10	1	no	—
239	C₂²×C₃	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
240	C₃×C₄	—	2	1	no	—
241	C₃×C₄	—	2	1	no	—
242	C₃×C₄	—	5	2	no	—
243	C₃×C₄	—	5	2	no	—
244	C₃×C₄	—	10	1	no	—
245	C₃×C₄	#4131	1	1	yes	characteristic
246	C₃×C₄	#4133	1	1	yes	characteristic
247	C₂×C₅	C₃×#405	1	1	yes	characteristic
248	C₂×C₅	C₃×#405	1	1	yes	characteristic
249	C₂×C₅	C₃×#405	1	1	yes	characteristic
250	D₅	—	4	1	no	—
251	C₂³	—	5	1	no	—
252	C₂×C₄	—	5	1	no	—
253	C₂×C₄	—	5	1	no	—
254	C₂×C₄	—	5	1	no	—
255	C₂×C₄	—	10	2	no	—
256	C₂×C₄	—	10	2	no	—
257	C₂×C₄	—	10	1	no	—
258	C₂×C₄	—	10	1	no	—
259	C₂×C₄	C₃×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
260	C₂×C₄	C₃×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
261	C₂×C₄	C₂²×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
262	C₈	—	2	1	no	—
263	C₈	—	2	1	no	—
264	C₈	—	10	2	no	—
265	C₈	—	10	2	no	—
266	C₈	—	10	2	no	—
267	C₈	—	10	2	no	—
268	C₈	—	10	1	no	—
269	C₈	—	10	1	no	—
270	C₂×C₃	—	20	1	no	—
271	C₂×C₃	#65701	1	1	yes	characteristic
272	C₂×C₃	#65718	1	1	yes	characteristic
273	C₂×C₃	#65728	1	1	yes	characteristic
274	C₅	C₃×#1454	1	1	yes	characteristic, complemented, Sylow
275	C₂²	—	10	1	no	—
276	C₂²	—	10	1	no	—
277	C₂²	—	10	1	no	—
278	C₂²	C₂×C₃×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
279	C₄	—	2	1	no	—
280	C₄	—	2	1	no	—
281	C₄	—	5	2	no	—
282	C₄	—	5	2	no	—
283	C₄	—	10	1	no	—
284	C₄	C₃×#4131	1	1	yes	characteristic
285	C₄	C₃×#4133	1	1	yes	characteristic
286	C₃	#10604014	1	1	yes	characteristic, direct product factor, Sylow
287	C₂	—	20	1	no	—
288	C₂	C₃×#65701	1	1	yes	characteristic
289	C₂	C₃×#65718	1	1	yes	characteristic
290	C₂	C₃×#65728	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)
commutator	C₂²×C₅	C₂×C₃×C₄²
Frattini	C₂×C₄	C₂²×C₃×AGL(1,𝔽₅)
Fitting	C₃×C₅×C₄⋊C₈	C₄
socle	C₂²×C₃×C₅	C₂×C₄²
prime radical	C₂²×C₃×D₅	C₄²
cosocle	C₄×Dic₅	C₂³×C₃
hypercenter	C₃×C₄⋊C₈	AGL(1,𝔽₅)
hypocenter	C₅	C₃×#1454
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₃×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
4. C₃×#170⋊C₈	C₂	semidirect
5. C₃×#170⋊C₈	C₂	semidirect
6. C₃×#64879	C₂	semidirect
7. C₃×C₈×#170	C₂	semidirect
8. #10604014	C₃	direct (N × H)
28. #170⋊C₈	C₂×C₃	semidirect
29. #170⋊C₈	C₂×C₃	semidirect
30. #64879	C₂×C₃	semidirect
31. C₈×#170	C₂×C₃	semidirect
235. C₃×C₅	#1454	semidirect
274. C₅	C₃×#1454	semidirect
286. C₃	#10604014	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×#65680	2
C₃×#65678	2
C₃×#65122	2
C₃×#170⋊C₈	2
C₃×#64879	2
C₃×C₈×#170	2
#10604014	3
C₃×#1454	5

Subgroups — C3×#10604014 #14280675

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10604014 #14280675