Subgroups — C₃×#10604018 #14285964

Order 1920.

320 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#65683	C₂	1	1	yes	maximal, characteristic
2	C₃×D₅⋊(C₄⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₃×#65122	C₂	1	1	yes	maximal, characteristic
4	C₃×#64884	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#64883	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#64882	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#64881	C₂	1	1	yes	maximal, characteristic, complemented
8	#10604018	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂×C₃×D₅⋊C₈	—	2	1	no	—
10	C₃×C₄²×D₅	C₂²	1	1	yes	characteristic
11	C₃×#4004	—	2	1	no	—
12	C₃×#4003	—	2	1	no	—
13	C₃×#4002	—	2	1	no	—
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×#1573	—	5	1	no	maximal, Hall
25	#65683	C₂×C₃	1	1	yes	characteristic
26	D₅⋊(C₄⋊C₈)	C₂×C₃	1	1	yes	characteristic
27	#65122	C₂×C₃	1	1	yes	characteristic
28	#64884	C₂×C₃	1	1	yes	characteristic, complemented
29	#64883	C₂×C₃	1	1	yes	characteristic, complemented
30	#64882	C₂×C₃	1	1	yes	characteristic, complemented
31	#64881	C₂×C₃	1	1	yes	characteristic, complemented
32	C₂×C₃×C₄×D₅	C₂×C₄	1	1	yes	characteristic
33	C₂×C₃×C₄×D₅	D₄	1	1	yes	characteristic
34	C₂×C₃×C₄×D₅	Q₈	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	—	2	1	no	—
39	C₃×D₅⋊C₈	—	4	2	no	—
40	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
41	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
42	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
43	C₃×C₄×Dic₅	D₄	1	1	yes	characteristic
44	C₃×C₄×Dic₅	Q₈	1	1	yes	characteristic
45	C₃×C₄×Dic₅	C₂³	1	1	yes	characteristic
46	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
47	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
48	C₃×#432	—	5	1	no	—
49	C₃×#431	—	5	1	no	—
50	C₂×C₃×C₄⋊C₈	—	5	1	no	—
51	C₃×#335	—	5	1	no	—
52	C₃×#335	—	5	1	no	—
53	C₃×#334	—	5	1	no	—
54	C₃×#334	—	5	1	no	—
55	C₂×D₅⋊C₈	—	2	1	no	—
56	C₄²×D₅	C₂²×C₃	1	1	yes	characteristic
57	#4004	—	2	1	no	—
58	#4003	—	2	1	no	—
59	#4002	—	2	1	no	—
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	C₄⋊#170	C₂²×C₃	1	1	yes	characteristic
64	C₅×C₄⋊C₈	C₃×C₄	1	1	yes	characteristic
65	#3952	C₃×C₄	1	1	yes	characteristic
66	#3952	C₃×C₄	1	1	yes	characteristic
67	C₈×Dic₅	C₂²×C₃	1	1	yes	characteristic
68	C₈×Dic₅	C₂²×C₃	1	1	yes	characteristic
69	C₄⋊#168	C₃×C₄	1	1	yes	characteristic
70	#1573	—	5	1	no	Sylow
71	C₂²×C₃×D₅	C₄⋊C₄	1	1	yes	characteristic
72	C₂×C₃×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
73	C₂×C₃×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₃×Dic₅	C₄⋊C₄	1	1	yes	characteristic
76	C₂×C₃×Dic₅	C₂×D₄	1	1	yes	characteristic
77	C₂×C₃×Dic₅	C₂×Q₈	1	1	yes	characteristic
78	C₃×C₄×D₅	—	2	2	no	—
79	C₃×C₄×D₅	—	2	1	no	—
80	C₃×C₄×D₅	—	2	1	no	—
81	C₃×C₄×D₅	—	2	1	no	—
82	C₃×C₄×D₅	—	2	1	no	—
83	C₃×#170	—	4	1	no	—
84	C₃×#170	—	4	1	no	—
85	C₃×#170	—	4	1	no	—
86	C₃×#170	—	4	1	no	—
87	C₃×C₅×C₈	—	2	1	no	—
88	C₃×C₅×C₈	—	2	1	no	—
89	C₃×#168	—	2	1	no	—
90	C₃×#168	—	2	1	no	—
91	C₂²×C₃×C₈	—	10	1	no	—
92	C₂×C₃×C₄²	—	5	1	no	—
93	C₃×C₄⋊C₈	—	5	1	no	—
94	C₃×C₄⋊C₈	—	5	1	no	—
95	C₃×C₄⋊C₈	—	5	1	no	—
96	C₃×C₄⋊C₈	—	5	1	no	—
97	C₃×C₄⋊C₈	—	5	1	no	—
98	C₃×C₄⋊C₈	—	10	1	no	—
99	C₃×C₄⋊C₈	AGL(1,𝔽₅)	1	1	yes	characteristic
100	C₃×C₂²⋊C₈	—	5	1	no	—
101	C₃×C₂²⋊C₈	—	5	1	no	—
102	C₃×C₂²⋊C₈	—	10	1	no	—
103	C₃×#97	—	10	1	no	—
104	C₃×C₄×C₈	—	5	1	no	—
105	C₃×C₄×C₈	—	5	1	no	—
106	C₂×C₄×D₅	C₂×C₃×C₄	1	1	yes	characteristic
107	C₂×C₄×D₅	C₃×D₄	1	1	yes	characteristic
108	C₂×C₄×D₅	C₃×Q₈	1	1	yes	characteristic
109	C₂×#170	—	2	1	no	—
110	C₂×#170	—	2	1	no	—
111	C₂×#170	—	2	1	no	—
112	C₂×#170	—	2	1	no	—
113	D₅⋊C₈	—	4	2	no	—
114	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
115	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
116	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
117	C₄×Dic₅	C₃×D₄	1	1	yes	characteristic
118	C₄×Dic₅	C₃×Q₈	1	1	yes	characteristic
119	C₄×Dic₅	C₂³×C₃	1	1	yes	characteristic
120	C₂×#168	C₂×C₃×C₄	1	1	yes	characteristic
121	C₂×#168	C₂×C₃×C₄	1	1	yes	characteristic
122	#432	—	5	1	no	—
123	#431	—	5	1	no	—
124	C₂×C₄⋊C₈	—	5	1	no	—
125	#335	—	5	1	no	—
126	#335	—	5	1	no	—
127	#334	—	5	1	no	—
128	#334	—	5	1	no	—
129	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
130	C₂×C₃×D₅	—	2	1	no	—
131	C₂×C₃×D₅	—	2	1	no	—
132	C₂×C₃×D₅	#108	1	2	yes	—
133	C₃×C₄×C₅	—	2	1	no	—
134	C₃×C₄×C₅	—	2	1	no	—
135	C₃×C₄×C₅	#131	1	1	yes	characteristic
136	C₃×C₄×C₅	#131	1	1	yes	characteristic
137	C₃×Dic₅	—	2	1	no	—
138	C₃×Dic₅	—	2	1	no	—
139	C₃×Dic₅	AGL(1,ℤ/8)	1	1	yes	characteristic
140	C₃×Dic₅	#137	1	1	yes	characteristic
141	C₂²×C₃×C₄	—	5	1	no	—
142	C₂²×C₃×C₄	—	5	1	no	—
143	C₂²×C₃×C₄	—	5	1	no	—
144	C₂×C₃×C₈	—	5	1	no	—
145	C₂×C₃×C₈	—	5	1	no	—
146	C₂×C₃×C₈	—	10	1	no	—
147	C₂×C₃×C₈	—	10	1	no	—
148	C₂×C₃×C₈	—	10	1	no	—
149	C₂×C₃×C₈	—	10	1	no	—
150	C₂×C₃×C₈	—	20	2	no	—
151	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
152	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
153	C₃×C₄²	—	5	1	no	—
154	C₃×C₄²	—	5	1	no	—
155	C₃×C₄²	—	5	1	no	—
156	C₃×C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
157	C₂²×D₅	C₃×C₄⋊C₄	1	1	yes	characteristic
158	C₂×C₄×C₅	C₃×C₄⋊C₄	1	1	yes	characteristic
159	C₂×C₄×C₅	C₃×C₄⋊C₄	1	1	yes	characteristic
160	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
161	C₂×Dic₅	C₃×C₄⋊C₄	1	1	yes	characteristic
162	C₂×Dic₅	C₂×C₃×D₄	1	1	yes	characteristic
163	C₂×Dic₅	C₂×C₃×Q₈	1	1	yes	characteristic
164	C₄×D₅	—	2	2	no	—
165	C₄×D₅	—	2	1	no	—
166	C₄×D₅	—	2	1	no	—
167	C₄×D₅	—	2	1	no	—
168	C₄×D₅	—	2	1	no	—
169	#170	—	4	1	no	—
170	#170	—	4	1	no	—
171	#170	—	4	1	no	—
172	#170	—	4	1	no	—
173	C₅×C₈	—	2	1	no	—
174	C₅×C₈	—	2	1	no	—
175	#168	—	2	1	no	—
176	#168	—	2	1	no	—
177	C₂²×C₈	—	10	1	no	—
178	C₂×C₄²	—	5	1	no	—
179	C₄⋊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	—	10	1	no	—
185	C₄⋊C₈	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
186	C₂²⋊C₈	—	5	1	no	—
187	C₂²⋊C₈	—	5	1	no	—
188	C₂²⋊C₈	—	10	1	no	—
189	#97	—	10	1	no	—
190	C₄×C₈	—	5	1	no	—
191	C₄×C₈	—	5	1	no	—
192	C₂×C₃×C₅	#424	1	1	yes	characteristic
193	C₂×C₃×C₅	#428	1	1	yes	characteristic
194	C₂×C₃×C₅	C₂×#108	1	1	yes	characteristic
195	C₃×D₅	—	2	2	no	—
196	C₂³×C₃	—	5	1	no	—
197	C₂×C₃×C₄	—	5	1	no	—
198	C₂×C₃×C₄	—	5	1	no	—
199	C₂×C₃×C₄	—	5	1	no	—
200	C₂×C₃×C₄	—	10	1	no	—
201	C₂×C₃×C₄	—	10	1	no	—
202	C₂×C₃×C₄	—	10	2	no	—
203	C₂×C₃×C₄	—	10	1	no	—
204	C₂×C₃×C₄	—	10	1	no	—
205	C₂×C₃×C₄	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
206	C₂×C₃×C₄	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
207	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
208	C₃×C₈	—	2	1	no	—
209	C₃×C₈	—	2	1	no	—
210	C₃×C₈	—	10	1	no	—
211	C₃×C₈	—	10	1	no	—
212	C₃×C₈	—	20	1	no	—
213	C₃×C₈	—	20	1	no	—
214	C₃×C₈	—	20	1	no	—
215	C₃×C₈	—	20	1	no	—
216	C₂²×C₅	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
217	C₂×D₅	—	2	1	no	—
218	C₂×D₅	—	2	1	no	—
219	C₂×D₅	C₃×#108	1	2	yes	—
220	C₄×C₅	—	2	1	no	—
221	C₄×C₅	—	2	1	no	—
222	C₄×C₅	C₃×#131	1	1	yes	characteristic
223	C₄×C₅	C₃×#131	1	1	yes	characteristic
224	Dic₅	—	2	1	no	—
225	Dic₅	—	2	1	no	—
226	Dic₅	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
227	Dic₅	C₃×#137	1	1	yes	characteristic
228	C₂²×C₄	—	5	1	no	—
229	C₂²×C₄	—	5	1	no	—
230	C₂²×C₄	—	5	1	no	—
231	C₂×C₈	—	5	1	no	—
232	C₂×C₈	—	5	1	no	—
233	C₂×C₈	—	10	1	no	—
234	C₂×C₈	—	10	1	no	—
235	C₂×C₈	—	10	1	no	—
236	C₂×C₈	—	10	1	no	—
237	C₂×C₈	—	20	2	no	—
238	C₂×C₈	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
239	C₂×C₈	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
240	C₄²	—	5	1	no	—
241	C₄²	—	5	1	no	—
242	C₄²	—	5	1	no	—
243	C₄²	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
244	C₃×C₅	#1573	1	1	yes	characteristic, complemented, Hall
245	C₂²×C₃	—	5	2	no	—
246	C₂²×C₃	—	10	1	no	—
247	C₂²×C₃	—	10	1	no	—
248	C₂²×C₃	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
249	C₃×C₄	—	2	1	no	—
250	C₃×C₄	—	2	1	no	—
251	C₃×C₄	—	5	1	no	—
252	C₃×C₄	—	5	1	no	—
253	C₃×C₄	—	10	1	no	—
254	C₃×C₄	—	10	1	no	—
255	C₃×C₄	#4131	1	1	yes	characteristic
256	C₃×C₄	#4133	1	1	yes	characteristic
257	C₂×C₅	C₃×#424	1	1	yes	characteristic
258	C₂×C₅	C₃×#428	1	1	yes	characteristic
259	C₂×C₅	C₂×C₃×#108	1	1	yes	characteristic
260	D₅	—	2	2	no	—
261	C₂³	—	5	1	no	—
262	C₂×C₄	—	5	1	no	—
263	C₂×C₄	—	5	1	no	—
264	C₂×C₄	—	5	1	no	—
265	C₂×C₄	—	10	1	no	—
266	C₂×C₄	—	10	1	no	—
267	C₂×C₄	—	10	2	no	—
268	C₂×C₄	—	10	1	no	—
269	C₂×C₄	—	10	1	no	—
270	C₂×C₄	C₃×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
271	C₂×C₄	C₃×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
272	C₂×C₄	C₂²×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
273	C₈	—	2	1	no	—
274	C₈	—	2	1	no	—
275	C₈	—	10	1	no	—
276	C₈	—	10	1	no	—
277	C₈	—	20	1	no	—
278	C₈	—	20	1	no	—
279	C₈	—	20	1	no	—
280	C₈	—	20	1	no	—
281	C₂×C₃	—	10	2	no	—
282	C₂×C₃	#65708	1	1	yes	characteristic
283	C₂×C₃	D₅⋊#108	1	1	yes	characteristic
284	C₂×C₃	#65727	1	1	yes	characteristic
285	C₅	C₃×#1573	1	1	yes	characteristic, complemented, Sylow
286	C₂²	—	5	2	no	—
287	C₂²	—	10	1	no	—
288	C₂²	—	10	1	no	—
289	C₂²	C₂×C₃×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
290	C₄	—	2	1	no	—
291	C₄	—	2	1	no	—
292	C₄	—	5	1	no	—
293	C₄	—	5	1	no	—
294	C₄	—	10	1	no	—
295	C₄	—	10	1	no	—
296	C₄	C₃×#4131	1	1	yes	characteristic
297	C₄	C₃×#4133	1	1	yes	characteristic
298	C₃	#10604018	1	1	yes	characteristic, direct product factor, Sylow
299	C₂	—	10	2	no	—
300	C₂	C₃×#65708	1	1	yes	characteristic
301	C₂	C₃×D₅⋊#108	1	1	yes	characteristic
302	C₂	C₃×#65727	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂×C₄⋊AGL(1,𝔽₅)
commutator	C₂×C₄×C₅	C₂²×C₃×C₄
Frattini	C₂×C₄	C₂²×C₃×AGL(1,𝔽₅)
Fitting	C₃×C₅×C₄⋊C₈	C₄
socle	C₂²×C₃×C₅	C₂×C₄⋊C₄
prime radical	C₂²×C₃×D₅	C₄⋊C₄
cosocle	C₄×Dic₅	C₂³×C₃
hypercenter	C₃×C₄⋊C₈	AGL(1,𝔽₅)
hypocenter	C₅	C₃×#1573
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ C₃×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
4. C₃×#64884	C₂	semidirect
5. C₃×#64883	C₂	semidirect
6. C₃×#64882	C₂	semidirect
7. C₃×#64881	C₂	semidirect
8. #10604018	C₃	direct (N × H)
28. #64884	C₂×C₃	semidirect
29. #64883	C₂×C₃	semidirect
30. #64882	C₂×C₃	semidirect
31. #64881	C₂×C₃	semidirect
244. C₃×C₅	#1573	semidirect
285. C₅	C₃×#1573	semidirect
298. C₃	#10604018	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×#65683	2
C₃×D₅⋊(C₄⋊C₈)	2
C₃×#65122	2
C₃×#64884	2
C₃×#64883	2
C₃×#64882	2
C₃×#64881	2
#10604018	3
C₃×#1573	5

Subgroups — C3×#10604018 #14285964

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10604018 #14285964