Subgroups — C₃×#10603885 #14296528

Order 1920.

448 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×D₅⋊#108	C₂	1	1	yes	maximal, characteristic
2	C₃×#65721	C₂	1	1	yes	maximal, characteristic
3	C₃×#65718	C₂	1	2	yes	maximal
4	C₃×D₅×#97	C₂	1	1	yes	maximal, characteristic
5	C₃×#64862	C₂	1	2	yes	maximal
6	#10603885	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₃×#4130	C₂²	1	2	yes	—
8	C₃×D₅⋊M₁₆	C₂²	1	2	yes	—
9	C₂×C₃×C₈×D₅	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₈×D₅	C₂²	1	1	yes	characteristic
11	C₃×C₄²×D₅	C₂²	1	1	yes	characteristic
12	C₃×#3996	C₄	1	2	yes	complemented
13	C₃×#3995	C₄	1	2	yes	complemented
14	C₃×#3994	C₄	1	2	yes	complemented
15	C₃×#3993	C₄	1	2	yes	complemented
16	C₃×#3992	—	2	2	no	—
17	C₃×C₈×AGL(1,𝔽₅)	—	2	2	no	—
18	C₃×C₅×#97	C₄	1	1	yes	characteristic, complemented
19	C₃×#3947	C₄	1	1	yes	characteristic, complemented
20	C₃×#3947	C₄	1	1	yes	characteristic, complemented
21	C₃×#3935	C₄	1	1	yes	characteristic, complemented
22	C₃×#1773	—	5	1	no	maximal, Hall
23	D₅⋊#108	C₂×C₃	1	1	yes	characteristic
24	#65721	C₂×C₃	1	1	yes	characteristic
25	#65718	C₂×C₃	1	2	yes	—
26	D₅×#97	C₂×C₃	1	1	yes	characteristic
27	#64862	C₂×C₃	1	2	yes	—
28	C₂×C₃×C₄×D₅	—	2	1	no	—
29	C₂×C₃×C₄×D₅	C₂³	1	1	yes	characteristic
30	C₃×C₂²⋊AGL(1,𝔽₅)	—	2	2	no	—
31	C₃×#704	—	2	2	no	—
32	C₃×C₄⋊AGL(1,𝔽₅)	C₂×C₄	1	4	yes	—
33	C₃×C₄×AGL(1,𝔽₅)	—	2	2	no	—
34	C₃×#700	C₂×C₄	1	4	yes	—
35	C₃×D₅⋊C₈	—	2	2	no	—
36	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
37	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
38	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
39	C₃×C₄×Dic₅	—	2	1	no	—
40	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
41	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
42	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
43	C₃×C₈×D₅	C₂×C₄	1	1	yes	characteristic
44	C₃×C₈×D₅	C₂×C₄	1	1	yes	characteristic
45	C₃×C₈×D₅	C₂×C₄	1	1	yes	characteristic
46	C₃×C₈×D₅	C₂×C₄	1	1	yes	characteristic
47	C₃×C₈×D₅	D₄	1	2	yes	—
48	C₃×C₈×D₅	Q₈	1	2	yes	—
49	C₂×C₃×#108	—	5	1	no	—
50	C₃×#427	—	5	1	no	—
51	C₃×#405	—	5	2	no	—
52	C₂×C₃×#97	—	5	1	no	—
53	C₃×#339	—	5	2	no	—
54	#4130	C₂²×C₃	1	2	yes	—
55	D₅⋊M₁₆	C₂²×C₃	1	2	yes	—
56	C₂×C₈×D₅	C₂²×C₃	1	1	yes	characteristic
57	C₂×C₈×D₅	C₂²×C₃	1	1	yes	characteristic
58	C₄²×D₅	C₂²×C₃	1	1	yes	characteristic
59	#3996	C₃×C₄	1	2	yes	complemented
60	#3995	C₃×C₄	1	2	yes	complemented
61	#3994	C₃×C₄	1	2	yes	complemented
62	#3993	C₃×C₄	1	2	yes	complemented
63	#3992	—	2	2	no	—
64	C₈×AGL(1,𝔽₅)	—	2	2	no	—
65	C₅×#97	C₃×C₄	1	1	yes	characteristic, complemented
66	#3947	C₃×C₄	1	1	yes	characteristic, complemented
67	#3947	C₃×C₄	1	1	yes	characteristic, complemented
68	#3935	C₃×C₄	1	1	yes	characteristic, complemented
69	#1773	—	5	1	no	Sylow
70	C₂²×C₃×D₅	#41	1	1	yes	characteristic
71	C₂×C₃×AGL(1,𝔽₅)	—	2	4	no	—
72	C₂×C₃×C₄×C₅	—	2	1	no	—
73	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₃×Dic₅	—	2	1	no	—
75	C₂×C₃×Dic₅	#41	1	1	yes	characteristic
76	C₃×C₄×D₅	—	4	2	no	—
77	C₃×C₄×D₅	C₂²×C₄	1	1	yes	characteristic
78	C₃×C₄×D₅	C₂²×C₄	1	1	yes	characteristic
79	C₃×C₄×D₅	C₂×D₄	1	1	yes	characteristic
80	C₃×C₄×D₅	C₂×Q₈	1	1	yes	characteristic
81	C₃×#170	—	2	4	no	—
82	C₃×C₅×C₈	C₄²	1	1	yes	characteristic
83	C₃×C₅×C₈	C₄²	1	1	yes	characteristic
84	C₃×C₅×C₈	C₄⋊C₄	1	2	yes	—
85	C₃×#168	C₄²	1	1	yes	characteristic
86	C₃×#168	C₄²	1	1	yes	characteristic
87	C₃×#168	C₄⋊C₄	1	2	yes	—
88	C₂×C₃×M₁₆	—	5	2	no	—
89	C₂²×C₃×C₈	—	5	1	no	—
90	C₂²×C₃×C₈	—	5	1	no	—
91	C₃×#117	—	5	2	no	—
92	C₂×C₃×C₄²	—	5	1	no	—
93	C₃×#108	—	5	2	no	—
94	C₃×#108	—	5	2	no	—
95	C₃×#107	—	5	2	no	—
96	C₃×#106	—	5	2	no	—
97	C₃×#97	—	5	1	no	—
98	C₃×#97	—	5	1	no	—
99	C₃×#97	—	5	1	no	—
100	C₃×#97	—	10	2	no	—
101	C₃×#97	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
102	C₃×C₄×C₈	—	10	2	no	—
103	C₂×C₄×D₅	—	2	1	no	—
104	C₂×C₄×D₅	C₂³×C₃	1	1	yes	characteristic
105	C₂²⋊AGL(1,𝔽₅)	—	2	2	no	—
106	#704	—	2	2	no	—
107	C₄⋊AGL(1,𝔽₅)	C₂×C₃×C₄	1	4	yes	—
108	C₄×AGL(1,𝔽₅)	—	2	2	no	—
109	#700	C₂×C₃×C₄	1	4	yes	—
110	D₅⋊C₈	—	2	2	no	—
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	—	2	1	no	—
115	C₄×Dic₅	C₂×C₃×C₄	1	1	yes	characteristic
116	C₂×#168	C₂×C₃×C₄	1	1	yes	characteristic
117	C₂×#168	C₂×C₃×C₄	1	1	yes	characteristic
118	C₈×D₅	C₂×C₃×C₄	1	1	yes	characteristic
119	C₈×D₅	C₂×C₃×C₄	1	1	yes	characteristic
120	C₈×D₅	C₂×C₃×C₄	1	1	yes	characteristic
121	C₈×D₅	C₂×C₃×C₄	1	1	yes	characteristic
122	C₈×D₅	C₃×D₄	1	2	yes	—
123	C₈×D₅	C₃×Q₈	1	2	yes	—
124	C₂×#108	—	5	1	no	—
125	#427	—	5	1	no	—
126	#405	—	5	2	no	—
127	C₂×#97	—	5	1	no	—
128	#339	—	5	2	no	—
129	C₂²×C₃×C₅	#117	1	1	yes	characteristic
130	C₂×C₃×D₅	—	2	2	no	—
131	C₂×C₃×D₅	C₄×D₄	1	1	yes	characteristic
132	C₂×C₃×D₅	C₄×Q₈	1	1	yes	characteristic
133	C₃×AGL(1,𝔽₅)	—	4	4	no	—
134	C₃×C₄×C₅	—	4	1	no	—
135	C₃×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
136	C₃×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
137	C₃×Dic₅	—	4	1	no	—
138	C₃×Dic₅	C₄×D₄	1	1	yes	characteristic
139	C₃×Dic₅	C₄×Q₈	1	1	yes	characteristic
140	C₂²×C₃×C₄	—	5	1	no	—
141	C₂²×C₃×C₄	—	10	1	no	—
142	C₃×M₁₆	—	5	4	no	—
143	C₃×M₁₆	—	10	2	no	—
144	C₂×C₃×C₈	—	5	1	no	—
145	C₂×C₃×C₈	—	5	1	no	—
146	C₂×C₃×C₈	—	5	1	no	—
147	C₂×C₃×C₈	—	5	1	no	—
148	C₂×C₃×C₈	—	5	1	no	—
149	C₂×C₃×C₈	—	5	1	no	—
150	C₂×C₃×C₈	—	5	2	no	—
151	C₂×C₃×C₈	—	5	2	no	—
152	C₂×C₃×C₈	—	10	2	no	—
153	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
154	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
155	C₃×C₄⋊C₄	—	5	4	no	—
156	C₃×C₂²⋊C₄	—	10	2	no	—
157	C₃×C₄²	—	5	1	no	—
158	C₃×C₄²	—	10	1	no	—
159	C₃×C₄²	—	10	2	no	—
160	C₃×C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
161	C₂²×D₅	C₃×#41	1	1	yes	characteristic
162	C₂×AGL(1,𝔽₅)	—	2	4	no	—
163	C₂×C₄×C₅	—	2	1	no	—
164	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
165	C₂×Dic₅	—	2	1	no	—
166	C₂×Dic₅	C₃×#41	1	1	yes	characteristic
167	C₄×D₅	—	4	2	no	—
168	C₄×D₅	C₂²×C₃×C₄	1	1	yes	characteristic
169	C₄×D₅	C₂²×C₃×C₄	1	1	yes	characteristic
170	C₄×D₅	C₂×C₃×D₄	1	1	yes	characteristic
171	C₄×D₅	C₂×C₃×Q₈	1	1	yes	characteristic
172	#170	—	2	4	no	—
173	C₅×C₈	C₃×C₄²	1	1	yes	characteristic
174	C₅×C₈	C₃×C₄²	1	1	yes	characteristic
175	C₅×C₈	C₃×C₄⋊C₄	1	2	yes	—
176	#168	C₃×C₄²	1	1	yes	characteristic
177	#168	C₃×C₄²	1	1	yes	characteristic
178	#168	C₃×C₄⋊C₄	1	2	yes	—
179	C₂×M₁₆	—	5	2	no	—
180	C₂²×C₈	—	5	1	no	—
181	C₂²×C₈	—	5	1	no	—
182	#117	—	5	2	no	—
183	C₂×C₄²	—	5	1	no	—
184	#108	—	5	2	no	—
185	#108	—	5	2	no	—
186	#107	—	5	2	no	—
187	#106	—	5	2	no	—
188	#97	—	5	1	no	—
189	#97	—	5	1	no	—
190	#97	—	5	1	no	—
191	#97	—	10	2	no	—
192	#97	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
193	C₄×C₈	—	10	2	no	—
194	C₂×C₃×C₅	—	2	1	no	—
195	C₂×C₃×C₅	C₄×C₄⋊C₄	1	1	yes	characteristic
196	C₃×D₅	—	2	1	no	—
197	C₃×D₅	#444	1	2	yes	—
198	C₂³×C₃	—	5	1	no	—
199	C₂×C₃×C₄	—	2	1	no	—
200	C₂×C₃×C₄	—	5	1	no	—
201	C₂×C₃×C₄	—	5	1	no	—
202	C₂×C₃×C₄	—	5	1	no	—
203	C₂×C₃×C₄	—	5	1	no	—
204	C₂×C₃×C₄	—	5	1	no	—
205	C₂×C₃×C₄	—	10	4	no	—
206	C₂×C₃×C₄	—	10	1	no	—
207	C₂×C₃×C₄	—	20	2	no	—
208	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
209	C₃×C₈	—	5	1	no	—
210	C₃×C₈	—	5	1	no	—
211	C₃×C₈	—	5	2	no	—
212	C₃×C₈	—	10	4	no	—
213	C₃×C₈	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
214	C₃×C₈	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
215	C₃×C₈	C₄⋊AGL(1,𝔽₅)	1	2	yes	—
216	C₂²×C₅	C₃×#117	1	1	yes	characteristic
217	C₂×D₅	—	2	2	no	—
218	C₂×D₅	C₃×C₄×D₄	1	1	yes	characteristic
219	C₂×D₅	C₃×C₄×Q₈	1	1	yes	characteristic
220	AGL(1,𝔽₅)	—	4	4	no	—
221	C₄×C₅	—	4	1	no	—
222	C₄×C₅	C₂×C₃×C₄²	1	1	yes	characteristic
223	C₄×C₅	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
224	Dic₅	—	4	1	no	—
225	Dic₅	C₃×C₄×D₄	1	1	yes	characteristic
226	Dic₅	C₃×C₄×Q₈	1	1	yes	characteristic
227	C₂²×C₄	—	5	1	no	—
228	C₂²×C₄	—	10	1	no	—
229	M₁₆	—	5	4	no	—
230	M₁₆	—	10	2	no	—
231	C₂×C₈	—	5	1	no	—
232	C₂×C₈	—	5	1	no	—
233	C₂×C₈	—	5	1	no	—
234	C₂×C₈	—	5	1	no	—
235	C₂×C₈	—	5	1	no	—
236	C₂×C₈	—	5	1	no	—
237	C₂×C₈	—	5	2	no	—
238	C₂×C₈	—	5	2	no	—
239	C₂×C₈	—	10	2	no	—
240	C₂×C₈	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
241	C₂×C₈	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
242	C₄⋊C₄	—	5	4	no	—
243	C₂²⋊C₄	—	10	2	no	—
244	C₄²	—	5	1	no	—
245	C₄²	—	10	1	no	—
246	C₄²	—	10	2	no	—
247	C₄²	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
248	C₃×C₅	#1773	1	1	yes	characteristic, complemented, Hall
249	C₂²×C₃	—	5	1	no	—
250	C₂²×C₃	—	5	1	no	—
251	C₂²×C₃	—	10	2	no	—
252	C₂²×C₃	#4130	1	1	yes	characteristic
253	C₃×C₄	—	4	1	no	—
254	C₃×C₄	—	5	1	no	—
255	C₃×C₄	—	5	1	no	—
256	C₃×C₄	—	20	1	no	—
257	C₃×C₄	—	20	4	no	—
258	C₃×C₄	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
259	C₃×C₄	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
260	C₂×C₅	—	2	1	no	—
261	C₂×C₅	C₃×C₄×C₄⋊C₄	1	1	yes	characteristic
262	D₅	—	2	1	no	—
263	D₅	C₃×#444	1	2	yes	—
264	C₂³	—	5	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	5	1	no	—
267	C₂×C₄	—	5	1	no	—
268	C₂×C₄	—	5	1	no	—
269	C₂×C₄	—	5	1	no	—
270	C₂×C₄	—	5	1	no	—
271	C₂×C₄	—	10	4	no	—
272	C₂×C₄	—	10	1	no	—
273	C₂×C₄	—	20	2	no	—
274	C₂×C₄	C₂²×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
275	C₈	—	5	1	no	—
276	C₈	—	5	1	no	—
277	C₈	—	5	2	no	—
278	C₈	—	10	4	no	—
279	C₈	C₃×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
280	C₈	C₃×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
281	C₈	C₃×C₄⋊AGL(1,𝔽₅)	1	2	yes	—
282	C₂×C₃	—	2	1	no	—
283	C₂×C₃	—	5	2	no	—
284	C₂×C₃	—	10	1	no	—
285	C₂×C₃	C₄×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
286	C₅	C₃×#1773	1	1	yes	characteristic, complemented, Sylow
287	C₂²	—	5	1	no	—
288	C₂²	—	5	1	no	—
289	C₂²	—	10	2	no	—
290	C₂²	C₃×#4130	1	1	yes	characteristic
291	C₄	—	4	1	no	—
292	C₄	—	5	1	no	—
293	C₄	—	5	1	no	—
294	C₄	—	20	1	no	—
295	C₄	—	20	4	no	—
296	C₄	C₂×C₃×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
297	C₄	C₂×C₃×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
298	C₃	#10603885	1	1	yes	characteristic, direct product factor, Sylow
299	C₂	—	2	1	no	—
300	C₂	—	5	2	no	—
301	C₂	—	10	1	no	—
302	C₂	C₃×C₄×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₄	C₂×C₄⋊AGL(1,𝔽₅)
commutator	C₄×C₅	C₂×C₃×C₄²
Frattini	C₂×C₄	C₂²×C₃×AGL(1,𝔽₅)
Fitting	C₃×C₅×#97	C₄
socle	C₂×C₃×C₅	C₄×C₄⋊C₄
prime radical	C₂²×C₃×D₅	#41
cosocle	C₂×C₄×D₅	C₂³×C₃
hypercenter	C₃×#97	AGL(1,𝔽₅)
hypocenter	C₅	C₃×#1773
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₃×C₄ ◁ C₂×C₃×C₈ ◁ C₃×#97

Complements

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

N	H ≅ G/N	type
6. #10603885	C₃	direct (N × H)
12. C₃×#3996	C₄	semidirect
13. C₃×#3995	C₄	semidirect
14. C₃×#3994	C₄	semidirect
15. C₃×#3993	C₄	semidirect
18. C₃×C₅×#97	C₄	semidirect
19. C₃×#3947	C₄	semidirect
20. C₃×#3947	C₄	semidirect
21. C₃×#3935	C₄	semidirect
59. #3996	C₃×C₄	semidirect
60. #3995	C₃×C₄	semidirect
61. #3994	C₃×C₄	semidirect
62. #3993	C₃×C₄	semidirect
65. C₅×#97	C₃×C₄	semidirect
66. #3947	C₃×C₄	semidirect
67. #3947	C₃×C₄	semidirect
68. #3935	C₃×C₄	semidirect
101. C₃×#97	AGL(1,𝔽₅)	semidirect
192. #97	C₃×AGL(1,𝔽₅)	semidirect
248. C₃×C₅	#1773	semidirect
286. C₅	C₃×#1773	semidirect
298. C₃	#10603885	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×D₅⋊#108	2
C₃×#65721	2
C₃×#65718	2
C₃×D₅×#97	2
C₃×#64862	2
#10603885	3
C₃×#1773	5

Subgroups — C3×#10603885 #14296528

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10603885 #14296528