Subgroups — C₃×#10606092 #14331370

Order 1920.

912 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	C₂×C₃×#4022	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×#384	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#65279	C₂	1	3	yes	maximal
4	C₃×#65278	C₂	1	3	yes	maximal
5	C₃×#65267	C₂	1	3	yes	maximal, complemented
6	C₃×#65261	C₂	1	3	yes	maximal, complemented
7	C₃×#65226	C₂	1	1	yes	maximal, characteristic, complemented
8	#10606092	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂²×C₃×C₄×D₅	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	3	yes	—
11	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	3	yes	—
12	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
13	C₂×C₃×#685	C₂²	1	3	yes	—
14	C₂×C₃×#685	C₂²	1	3	yes	—
15	C₂×C₃×#683	C₂²	1	3	yes	—
16	C₂×C₃×#683	C₂²	1	3	yes	—
17	C₂×C₃×C₄×Dic₅	C₂²	1	3	yes	—
18	C₃×#4022	C₄	1	8	yes	complemented
19	C₃×#3963	C₂²	1	3	yes	—
20	C₃×#3963	C₂²	1	3	yes	—
21	C₃×#3963	C₂²	1	3	yes	—
22	C₃×#3963	C₂²	1	3	yes	—
23	C₃×#2383	—	5	1	no	maximal, Hall
24	C₂×#4022	C₂×C₃	1	1	yes	characteristic
25	C₅×#384	C₂×C₃	1	1	yes	characteristic, complemented
26	#65279	C₂×C₃	1	3	yes	—
27	#65278	C₂×C₃	1	3	yes	—
28	#65267	C₂×C₃	1	3	yes	complemented
29	#65261	C₂×C₃	1	3	yes	complemented
30	#65226	C₂×C₃	1	1	yes	characteristic, complemented
31	C₂³×C₃×D₅	C₂³	1	1	yes	characteristic
32	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
33	C₂²×C₃×C₄×C₅	C₂³	1	3	yes	—
34	C₂²×C₃×C₄×C₅	C₂³	1	3	yes	—
35	C₂²×C₃×Dic₅	C₂³	1	3	yes	—
36	C₂²×C₃×Dic₅	C₂³	1	1	yes	characteristic
37	C₂²×C₃×Dic₅	C₂³	1	3	yes	—
38	C₂×C₃×C₄×D₅	—	4	3	no	—
39	C₃×C₅×C₄⋊C₄	—	4	3	no	—
40	C₃×C₅×C₄⋊C₄	—	4	3	no	—
41	C₃×C₄²×C₅	C₂×C₄	1	4	yes	complemented
42	C₃×#685	—	4	3	no	—
43	C₃×#685	C₂×C₄	1	12	yes	—
44	C₃×#683	—	4	3	no	—
45	C₃×#683	C₂×C₄	1	12	yes	complemented
46	C₃×C₄×Dic₅	—	4	3	no	—
47	C₂×C₃×#126	—	5	1	no	—
48	C₃×#389	—	5	3	no	—
49	C₃×#388	—	5	3	no	—
50	C₃×#387	—	5	3	no	—
51	C₃×#385	—	5	3	no	—
52	C₃×#384	D₅	1	1	yes	characteristic, complemented
53	C₃×#383	—	5	1	no	—
54	C₂²×C₄×D₅	C₂²×C₃	1	1	yes	characteristic
55	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	3	yes	—
56	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	3	yes	—
57	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
58	C₂×#685	C₂²×C₃	1	3	yes	—
59	C₂×#685	C₂²×C₃	1	3	yes	—
60	C₂×#683	C₂²×C₃	1	3	yes	—
61	C₂×#683	C₂²×C₃	1	3	yes	—
62	C₂×C₄×Dic₅	C₂²×C₃	1	3	yes	—
63	#4022	C₃×C₄	1	8	yes	complemented
64	#3963	C₂²×C₃	1	3	yes	—
65	#3963	C₂²×C₃	1	3	yes	—
66	#3963	C₂²×C₃	1	3	yes	—
67	#3963	C₂²×C₃	1	3	yes	—
68	#2383	—	5	1	no	Sylow
69	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
70	C₂²×C₃×D₅	—	2	3	no	—
71	C₂²×C₃×D₅	—	2	3	no	—
72	C₂²×C₃×D₅	C₂²×C₄	1	2	yes	—
73	C₂×C₃×C₄×C₅	—	2	3	no	—
74	C₂×C₃×C₄×C₅	—	2	3	no	—
75	C₂×C₃×C₄×C₅	—	2	3	no	—
76	C₂×C₃×C₄×C₅	—	2	3	no	—
77	C₂×C₃×C₄×C₅	—	2	3	no	—
78	C₂×C₃×C₄×C₅	—	2	3	no	—
79	C₂×C₃×C₄×C₅	C₂²×C₄	1	6	yes	—
80	C₂×C₃×Dic₅	—	2	3	no	—
81	C₂×C₃×Dic₅	—	2	3	no	—
82	C₂×C₃×Dic₅	—	2	3	no	—
83	C₂×C₃×Dic₅	—	2	3	no	—
84	C₂×C₃×Dic₅	—	2	6	no	—
85	C₂×C₃×Dic₅	C₂²×C₄	1	6	yes	—
86	C₃×C₄×D₅	—	4	4	no	—
87	C₂³×C₃×C₄	—	5	1	no	—
88	C₃×#126	—	5	8	no	—
89	C₂×C₃×C₄⋊C₄	—	5	3	no	—
90	C₂×C₃×C₄⋊C₄	—	5	3	no	—
91	C₂×C₃×C₄⋊C₄	C₂×D₅	1	3	yes	—
92	C₂×C₃×C₄⋊C₄	C₂×D₅	1	3	yes	—
93	C₂×C₃×C₂²⋊C₄	—	5	3	no	—
94	C₂×C₃×C₂²⋊C₄	—	5	3	no	—
95	C₂×C₃×C₄²	—	5	3	no	—
96	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
97	C₃×#95	—	5	3	no	—
98	C₃×#95	—	5	3	no	—
99	C₃×#95	—	5	3	no	—
100	C₃×#95	—	5	3	no	—
101	C₂³×D₅	C₂³×C₃	1	1	yes	characteristic
102	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
103	C₂²×C₄×C₅	C₂³×C₃	1	3	yes	—
104	C₂²×C₄×C₅	C₂³×C₃	1	3	yes	—
105	C₂²×Dic₅	C₂³×C₃	1	1	yes	characteristic
106	C₂²×Dic₅	C₂³×C₃	1	3	yes	—
107	C₂²×Dic₅	C₂³×C₃	1	3	yes	—
108	C₂×C₄×D₅	—	4	3	no	—
109	C₅×C₄⋊C₄	—	4	3	no	—
110	C₅×C₄⋊C₄	—	4	3	no	—
111	C₄²×C₅	C₂×C₃×C₄	1	4	yes	complemented
112	#685	—	4	3	no	—
113	#685	C₂×C₃×C₄	1	12	yes	—
114	#683	—	4	3	no	—
115	#683	C₂×C₃×C₄	1	12	yes	complemented
116	C₄×Dic₅	—	4	3	no	—
117	C₂×#126	—	5	1	no	—
118	#389	—	5	3	no	—
119	#388	—	5	3	no	—
120	#387	—	5	3	no	—
121	#385	—	5	3	no	—
122	#384	C₃×D₅	1	1	yes	characteristic, complemented
123	#383	—	5	1	no	—
124	C₂²×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
125	C₂²×C₃×C₅	H(5,𝔽₂)	1	3	yes	—
126	C₂²×C₃×C₅	2₋¹⁺⁴	1	3	yes	—
127	C₂×C₃×D₅	—	2	6	no	—
128	C₂×C₃×D₅	—	4	1	no	—
129	C₂×C₃×D₅	—	4	3	no	—
130	C₃×C₄×C₅	—	2	6	no	—
131	C₃×C₄×C₅	—	4	1	no	—
132	C₃×C₄×C₅	—	4	3	no	—
133	C₃×Dic₅	—	2	6	no	—
134	C₃×Dic₅	—	4	1	no	—
135	C₃×Dic₅	—	4	3	no	—
136	C₂⁴×C₃	—	5	1	no	—
137	C₂²×C₃×C₄	—	5	3	no	—
138	C₂²×C₃×C₄	—	5	3	no	—
139	C₂²×C₃×C₄	—	5	1	no	—
140	C₂²×C₃×C₄	—	20	3	no	—
141	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
142	C₂²×C₃×C₄	C₂²×D₅	1	3	yes	—
143	C₂²×C₃×C₄	C₂²×D₅	1	3	yes	—
144	C₃×C₄⋊C₄	—	4	3	no	—
145	C₃×C₄⋊C₄	—	4	3	no	—
146	C₃×C₄⋊C₄	—	5	12	no	—
147	C₃×C₄⋊C₄	—	20	3	no	—
148	C₃×C₂²⋊C₄	—	5	12	no	—
149	C₃×C₂²⋊C₄	—	20	3	no	—
150	C₃×C₄²	—	20	3	no	—
151	C₃×C₄²	C₄×D₅	1	4	yes	complemented
152	C₂³×C₅	C₂⁴×C₃	1	1	yes	characteristic
153	C₂²×D₅	—	2	3	no	—
154	C₂²×D₅	—	2	3	no	—
155	C₂²×D₅	C₂²×C₃×C₄	1	2	yes	—
156	C₂×C₄×C₅	—	2	3	no	—
157	C₂×C₄×C₅	—	2	3	no	—
158	C₂×C₄×C₅	—	2	3	no	—
159	C₂×C₄×C₅	—	2	3	no	—
160	C₂×C₄×C₅	—	2	3	no	—
161	C₂×C₄×C₅	—	2	3	no	—
162	C₂×C₄×C₅	C₂²×C₃×C₄	1	6	yes	—
163	C₂×Dic₅	—	2	3	no	—
164	C₂×Dic₅	—	2	3	no	—
165	C₂×Dic₅	—	2	3	no	—
166	C₂×Dic₅	—	2	3	no	—
167	C₂×Dic₅	—	2	6	no	—
168	C₂×Dic₅	C₂²×C₃×C₄	1	6	yes	—
169	C₄×D₅	—	4	4	no	—
170	C₂³×C₄	—	5	1	no	—
171	#126	—	5	8	no	—
172	C₂×C₄⋊C₄	—	5	3	no	—
173	C₂×C₄⋊C₄	—	5	3	no	—
174	C₂×C₄⋊C₄	C₂×C₃×D₅	1	3	yes	—
175	C₂×C₄⋊C₄	C₂×C₃×D₅	1	3	yes	—
176	C₂×C₂²⋊C₄	—	5	3	no	—
177	C₂×C₂²⋊C₄	—	5	3	no	—
178	C₂×C₄²	—	5	3	no	—
179	C₂×C₄²	C₂×C₃×D₅	1	1	yes	characteristic
180	#95	—	5	3	no	—
181	#95	—	5	3	no	—
182	#95	—	5	3	no	—
183	#95	—	5	3	no	—
184	C₂×C₃×C₅	#520	1	3	yes	—
185	C₂×C₃×C₅	#560	1	1	yes	characteristic
186	C₂×C₃×C₅	#563	1	3	yes	—
187	C₃×D₅	—	4	2	no	—
188	C₂³×C₃	—	5	2	no	—
189	C₂³×C₃	—	10	3	no	—
190	C₂³×C₃	—	10	3	no	—
191	C₂³×C₃	C₂³×D₅	1	1	yes	characteristic
192	C₂×C₃×C₄	—	2	3	no	—
193	C₂×C₃×C₄	—	2	3	no	—
194	C₂×C₃×C₄	—	2	3	no	—
195	C₂×C₃×C₄	—	2	3	no	—
196	C₂×C₃×C₄	—	2	3	no	—
197	C₂×C₃×C₄	—	2	3	no	—
198	C₂×C₃×C₄	—	5	6	no	—
199	C₂×C₃×C₄	—	10	3	no	—
200	C₂×C₃×C₄	—	10	3	no	—
201	C₂×C₃×C₄	—	10	3	no	—
202	C₂×C₃×C₄	—	10	3	no	—
203	C₂×C₃×C₄	—	10	6	no	—
204	C₂×C₃×C₄	—	20	4	no	—
205	C₂×C₃×C₄	C₂×C₄×D₅	1	6	yes	—
206	C₂²×C₅	C₂³×C₃×C₄	1	1	yes	characteristic
207	C₂²×C₅	C₃×H(5,𝔽₂)	1	3	yes	—
208	C₂²×C₅	C₃×2₋¹⁺⁴	1	3	yes	—
209	C₂×D₅	—	2	6	no	—
210	C₂×D₅	—	4	1	no	—
211	C₂×D₅	—	4	3	no	—
212	C₄×C₅	—	2	6	no	—
213	C₄×C₅	—	4	1	no	—
214	C₄×C₅	—	4	3	no	—
215	Dic₅	—	2	6	no	—
216	Dic₅	—	4	1	no	—
217	Dic₅	—	4	3	no	—
218	C₂⁴	—	5	1	no	—
219	C₂²×C₄	—	5	3	no	—
220	C₂²×C₄	—	5	3	no	—
221	C₂²×C₄	—	5	1	no	—
222	C₂²×C₄	—	20	3	no	—
223	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
224	C₂²×C₄	C₂²×C₃×D₅	1	3	yes	—
225	C₂²×C₄	C₂²×C₃×D₅	1	3	yes	—
226	C₄⋊C₄	—	4	3	no	—
227	C₄⋊C₄	—	4	3	no	—
228	C₄⋊C₄	—	5	12	no	—
229	C₄⋊C₄	—	20	3	no	—
230	C₂²⋊C₄	—	5	12	no	—
231	C₂²⋊C₄	—	20	3	no	—
232	C₄²	—	20	3	no	—
233	C₄²	C₃×C₄×D₅	1	4	yes	complemented
234	C₃×C₅	#2383	1	1	yes	characteristic, complemented, Hall
235	C₂²×C₃	—	10	6	no	—
236	C₂²×C₃	—	20	1	no	—
237	C₂²×C₃	—	20	3	no	—
238	C₂²×C₃	C₂²×C₄×D₅	1	1	yes	characteristic
239	C₂²×C₃	#4144	1	3	yes	—
240	C₂²×C₃	#4147	1	3	yes	—
241	C₃×C₄	—	2	6	no	—
242	C₃×C₄	—	4	3	no	—
243	C₃×C₄	—	4	1	no	—
244	C₃×C₄	—	10	6	no	—
245	C₃×C₄	—	20	1	no	—
246	C₃×C₄	—	20	3	no	—
247	C₂×C₅	C₃×#520	1	3	yes	—
248	C₂×C₅	C₃×#560	1	1	yes	characteristic
249	C₂×C₅	C₃×#563	1	3	yes	—
250	D₅	—	4	2	no	—
251	C₂³	—	5	2	no	—
252	C₂³	—	10	3	no	—
253	C₂³	—	10	3	no	—
254	C₂³	C₂³×C₃×D₅	1	1	yes	characteristic
255	C₂×C₄	—	2	3	no	—
256	C₂×C₄	—	2	3	no	—
257	C₂×C₄	—	2	3	no	—
258	C₂×C₄	—	2	3	no	—
259	C₂×C₄	—	2	3	no	—
260	C₂×C₄	—	2	3	no	—
261	C₂×C₄	—	5	6	no	—
262	C₂×C₄	—	10	3	no	—
263	C₂×C₄	—	10	3	no	—
264	C₂×C₄	—	10	3	no	—
265	C₂×C₄	—	10	3	no	—
266	C₂×C₄	—	10	6	no	—
267	C₂×C₄	—	20	4	no	—
268	C₂×C₄	C₂×C₃×C₄×D₅	1	6	yes	—
269	C₂×C₃	—	20	2	no	—
270	C₂×C₃	#65838	1	3	yes	—
271	C₂×C₃	#66054	1	1	yes	characteristic
272	C₂×C₃	#66069	1	3	yes	—
273	C₅	C₃×#2383	1	1	yes	characteristic, complemented, Sylow
274	C₂²	—	10	6	no	—
275	C₂²	—	20	1	no	—
276	C₂²	—	20	3	no	—
277	C₂²	C₂²×C₃×C₄×D₅	1	1	yes	characteristic
278	C₂²	C₃×#4144	1	3	yes	—
279	C₂²	C₃×#4147	1	3	yes	—
280	C₄	—	2	6	no	—
281	C₄	—	4	1	no	—
282	C₄	—	4	3	no	—
283	C₄	—	10	6	no	—
284	C₄	—	20	1	no	—
285	C₄	—	20	3	no	—
286	C₃	#10606092	1	1	yes	characteristic, direct product factor, Sylow
287	C₂	—	20	2	no	—
288	C₂	C₃×#65838	1	3	yes	—
289	C₂	C₃×#66054	1	1	yes	characteristic
290	C₂	C₃×#66069	1	3	yes	—

	subgroup	G/H
center	C₂³×C₃	C₂³×D₅
commutator	C₂²×C₅	C₂³×C₃×C₄
Frattini	C₂³	C₂³×C₃×D₅
Fitting	C₃×C₅×#384	C₂
socle	C₂³×C₃×C₅	C₂⁴
prime radical	C₂³×C₃×D₅	C₂³
cosocle	C₂³×C₅	C₂⁴×C₃
hypercenter	C₃×#384	D₅
hypocenter	C₅	C₃×#2383
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³×C₃ ◁ C₃×#384

Complements

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

N	H ≅ G/N	type
2. C₃×C₅×#384	C₂	semidirect
5. C₃×#65267	C₂	semidirect
6. C₃×#65261	C₂	semidirect
7. C₃×#65226	C₂	semidirect
8. #10606092	C₃	direct (N × H)
18. C₃×#4022	C₄	semidirect
25. C₅×#384	C₂×C₃	semidirect
28. #65267	C₂×C₃	semidirect
29. #65261	C₂×C₃	semidirect
30. #65226	C₂×C₃	semidirect
41. C₃×C₄²×C₅	C₂×C₄	semidirect
45. C₃×#683	C₂×C₄	semidirect
52. C₃×#384	D₅	semidirect
63. #4022	C₃×C₄	semidirect
111. C₄²×C₅	C₂×C₃×C₄	semidirect
115. #683	C₂×C₃×C₄	semidirect
122. #384	C₃×D₅	semidirect
151. C₃×C₄²	C₄×D₅	semidirect
233. C₄²	C₃×C₄×D₅	semidirect
234. C₃×C₅	#2383	semidirect
273. C₅	C₃×#2383	semidirect
286. C₃	#10606092	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₃×#4022	2
C₃×C₅×#384	2
C₃×#65279	2
C₃×#65278	2
C₃×#65267	2
C₃×#65261	2
C₃×#65226	2
#10606092	3
C₃×#2383	5

Subgroups — C3×#10606092 #14331370

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10606092 #14331370