Subgroups — C₃×#10600233 #14306034

Order 1920.

640 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#65811	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×#376	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×#3937	C₂	1	4	yes	maximal, complemented
4	C₃×#65213	C₂	1	1	yes	maximal, characteristic, complemented
5	#10600233	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂×C₃×#709	C₂²	1	2	yes	—
7	C₂×C₃×C₄²×C₅	C₂²	1	2	yes	—
8	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
9	C₃×#4076	—	2	2	no	—
10	C₂×C₃×#681	C₂²	1	2	yes	—
11	C₃×#4021	C₄	1	2	yes	complemented
12	C₃×#4020	C₄	1	1	yes	characteristic, complemented
13	C₃×C₄×D₂₀	—	2	2	no	—
14	C₃×#4015	C₄	1	1	yes	characteristic, complemented
15	C₃×C₅×#95	—	2	2	no	—
16	C₃×#3937	—	2	8	no	—
17	C₃×C₄⋊#168	—	2	2	no	—
18	C₃×#1963	—	5	1	no	maximal, Hall
19	#65811	C₂×C₃	1	1	yes	characteristic
20	C₅×#376	C₂×C₃	1	1	yes	characteristic, complemented
21	C₂×#3937	C₂×C₃	1	4	yes	complemented
22	#65213	C₂×C₃	1	1	yes	characteristic, complemented
23	C₂²×C₃×C₄×C₅	—	2	2	no	—
24	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
25	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
26	C₂×C₃×C₂²⋊D₅	—	2	2	no	—
27	C₃×#709	—	2	4	no	—
28	C₃×#709	—	2	4	no	—
29	C₂×C₃×D₂₀	—	2	2	no	—
30	C₂×C₃×D₂₀	C₂×C₄	1	2	yes	—
31	C₂×C₃×C₄×D₅	—	2	2	no	—
32	C₂×C₃×Dic₁₀	C₂×C₄	1	2	yes	—
33	C₃×C₄²×C₅	—	2	2	no	—
34	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
35	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
36	C₃×C₄²×C₅	D₄	1	2	yes	—
37	C₃×C₄²×C₅	D₄	1	4	yes	complemented
38	C₃×#685	—	2	4	no	—
39	C₃×C₄⋊Dic₅	—	2	2	no	—
40	C₃×#681	—	2	4	no	—
41	C₂×C₃×#168	—	2	2	no	—
42	C₃×#532	—	5	1	no	—
43	C₃×#423	—	5	1	no	—
44	C₂×C₃×#104	—	5	4	no	—
45	C₃×#376	D₅	1	1	yes	characteristic, complemented
46	C₂×#709	C₂²×C₃	1	2	yes	—
47	C₂×C₄²×C₅	C₂²×C₃	1	2	yes	—
48	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
49	#4076	—	2	2	no	—
50	C₂×#681	C₂²×C₃	1	2	yes	—
51	#4021	C₃×C₄	1	2	yes	complemented
52	#4020	C₃×C₄	1	1	yes	characteristic, complemented
53	C₄×D₂₀	—	2	2	no	—
54	#4015	C₃×C₄	1	1	yes	characteristic, complemented
55	C₅×#95	—	2	2	no	—
56	#3937	—	2	8	no	—
57	C₄⋊#168	—	2	2	no	—
58	#1963	—	5	1	no	Sylow
59	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₃×D₅	—	2	2	no	—
61	C₂×C₃×C₄×C₅	—	2	4	no	—
62	C₂×C₃×C₄×C₅	—	4	2	no	—
63	C₂×C₃×C₄×C₅	—	4	4	no	—
64	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
65	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
66	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
68	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
69	C₂×C₃×C₄×C₅	#41	1	2	yes	—
70	C₃×C₂²⋊D₅	—	4	4	no	—
71	C₂×C₃×Dic₅	—	2	2	no	—
72	C₃×D₂₀	—	2	4	no	—
73	C₃×D₂₀	—	4	4	no	—
74	C₃×C₄×D₅	—	4	2	no	—
75	C₃×C₄×D₅	—	4	2	no	—
76	C₃×Dic₁₀	—	2	4	no	—
77	C₃×#168	—	4	2	no	—
78	C₂×C₃×#41	—	5	2	no	—
79	C₂×C₃×M₁₆	—	5	2	no	—
80	C₃×C₄⋊Q₈	—	5	1	no	—
81	C₃×#127	—	5	1	no	—
82	C₃×#124	—	5	2	no	—
83	C₃×#121	—	10	2	no	—
84	C₃×C₄×D₄	—	10	2	no	—
85	C₂×C₃×C₄²	C₂×D₅	1	2	yes	—
86	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
87	C₃×C₄⋊C₈	—	10	2	no	—
88	C₃×#104	—	10	8	no	—
89	C₃×#95	—	2	2	no	—
90	C₂²×C₄×C₅	—	2	2	no	—
91	C₂²×C₄×C₅	C₃×D₄	1	2	yes	—
92	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
93	C₂×C₂²⋊D₅	—	2	2	no	—
94	#709	—	2	4	no	—
95	#709	—	2	4	no	—
96	C₂×D₂₀	—	2	2	no	—
97	C₂×D₂₀	C₂×C₃×C₄	1	2	yes	—
98	C₂×C₄×D₅	—	2	2	no	—
99	C₂×Dic₁₀	C₂×C₃×C₄	1	2	yes	—
100	C₄²×C₅	—	2	2	no	—
101	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
102	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
103	C₄²×C₅	C₃×D₄	1	2	yes	—
104	C₄²×C₅	C₃×D₄	1	4	yes	complemented
105	#685	—	2	4	no	—
106	C₄⋊Dic₅	—	2	2	no	—
107	#681	—	2	4	no	—
108	C₂×#168	—	2	2	no	—
109	#532	—	5	1	no	—
110	#423	—	5	1	no	—
111	C₂×#104	—	5	4	no	—
112	#376	C₃×D₅	1	1	yes	characteristic, complemented
113	C₂²×C₃×C₅	—	2	2	no	—
114	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₂²×C₃×C₅	#121	1	2	yes	—
116	C₂×C₃×D₅	—	4	2	no	—
117	C₂×C₃×D₅	—	4	4	no	—
118	C₃×C₄×C₅	—	2	2	no	—
119	C₃×C₄×C₅	—	2	2	no	—
120	C₃×C₄×C₅	—	4	4	no	—
121	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
122	C₃×C₄×C₅	#124	1	1	yes	characteristic
123	C₃×C₄×C₅	#127	1	1	yes	characteristic
124	C₃×Dic₅	—	4	2	no	—
125	C₃×#41	—	10	4	no	—
126	C₃×#41	—	10	4	no	—
127	C₂×C₃×Q₈	—	5	2	no	—
128	C₂×C₃×D₄	—	5	2	no	—
129	C₂×C₃×D₄	—	10	2	no	—
130	C₂×C₃×D₄	—	10	2	no	—
131	C₂²×C₃×C₄	—	2	2	no	—
132	C₂²×C₃×C₄	—	10	2	no	—
133	C₂²×C₃×C₄	C₂²⋊D₅	1	2	yes	—
134	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
135	C₃×M₁₆	—	10	4	no	—
136	C₂×C₃×C₈	—	10	2	no	—
137	C₃×C₄⋊C₄	—	10	2	no	—
138	C₃×C₂²⋊C₄	—	10	4	no	—
139	C₃×C₄²	—	2	2	no	—
140	C₃×C₄²	C₄×D₅	1	1	yes	characteristic
141	C₃×C₄²	C₄×D₅	1	1	yes	characteristic
142	C₃×C₄²	D₂₀	1	2	yes	—
143	C₃×C₄²	D₂₀	1	4	yes	complemented
144	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
145	C₂²×D₅	—	2	2	no	—
146	C₂×C₄×C₅	—	2	4	no	—
147	C₂×C₄×C₅	—	4	2	no	—
148	C₂×C₄×C₅	—	4	4	no	—
149	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	2	yes	—
150	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	2	yes	—
151	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
152	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
153	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
154	C₂×C₄×C₅	C₃×#41	1	2	yes	—
155	C₂²⋊D₅	—	4	4	no	—
156	C₂×Dic₅	—	2	2	no	—
157	D₂₀	—	2	4	no	—
158	D₂₀	—	4	4	no	—
159	C₄×D₅	—	4	2	no	—
160	C₄×D₅	—	4	2	no	—
161	Dic₁₀	—	2	4	no	—
162	#168	—	4	2	no	—
163	C₂×#41	—	5	2	no	—
164	C₂×M₁₆	—	5	2	no	—
165	C₄⋊Q₈	—	5	1	no	—
166	#127	—	5	1	no	—
167	#124	—	5	2	no	—
168	#121	—	10	2	no	—
169	C₄×D₄	—	10	2	no	—
170	C₂×C₄²	C₂×C₃×D₅	1	2	yes	—
171	C₂×C₄²	C₂×C₃×D₅	1	1	yes	characteristic
172	C₄⋊C₈	—	10	2	no	—
173	#104	—	10	8	no	—
174	#95	—	2	2	no	—
175	C₂×C₃×C₅	—	2	2	no	—
176	C₂×C₃×C₅	#390	1	1	yes	characteristic
177	C₂×C₃×C₅	#421	1	2	yes	—
178	C₃×D₅	—	8	2	no	—
179	C₂³×C₃	—	10	2	no	—
180	C₂³×C₃	C₂×C₂²⋊D₅	1	1	yes	characteristic
181	C₃×Q₈	—	10	4	no	—
182	C₃×D₄	—	10	4	no	—
183	C₃×D₄	—	20	4	no	—
184	C₃×D₄	—	20	4	no	—
185	C₂×C₃×C₄	—	2	4	no	—
186	C₂×C₃×C₄	—	4	2	no	—
187	C₂×C₃×C₄	—	4	4	no	—
188	C₂×C₃×C₄	—	10	2	no	—
189	C₂×C₃×C₄	—	20	2	no	—
190	C₂×C₃×C₄	—	20	2	no	—
191	C₂×C₃×C₄	#685	1	2	yes	—
192	C₂×C₃×C₄	#685	1	2	yes	—
193	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
194	C₂×C₃×C₄	C₂×D₂₀	1	2	yes	—
195	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
196	C₂×C₃×C₄	#709	1	2	yes	—
197	C₃×C₈	—	20	2	no	—
198	C₂²×C₅	—	2	2	no	—
199	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
200	C₂²×C₅	C₃×#121	1	2	yes	—
201	C₂×D₅	—	4	2	no	—
202	C₂×D₅	—	4	4	no	—
203	C₄×C₅	—	2	2	no	—
204	C₄×C₅	—	2	2	no	—
205	C₄×C₅	—	4	4	no	—
206	C₄×C₅	C₃×C₄×D₄	1	2	yes	—
207	C₄×C₅	C₃×#124	1	1	yes	characteristic
208	C₄×C₅	C₃×#127	1	1	yes	characteristic
209	Dic₅	—	4	2	no	—
210	#41	—	10	4	no	—
211	#41	—	10	4	no	—
212	C₂×Q₈	—	5	2	no	—
213	C₂×D₄	—	5	2	no	—
214	C₂×D₄	—	10	2	no	—
215	C₂×D₄	—	10	2	no	—
216	C₂²×C₄	—	2	2	no	—
217	C₂²×C₄	—	10	2	no	—
218	C₂²×C₄	C₃×C₂²⋊D₅	1	2	yes	—
219	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
220	M₁₆	—	10	4	no	—
221	C₂×C₈	—	10	2	no	—
222	C₄⋊C₄	—	10	2	no	—
223	C₂²⋊C₄	—	10	4	no	—
224	C₄²	—	2	2	no	—
225	C₄²	C₃×C₄×D₅	1	1	yes	characteristic
226	C₄²	C₃×C₄×D₅	1	1	yes	characteristic
227	C₄²	C₃×D₂₀	1	2	yes	—
228	C₄²	C₃×D₂₀	1	4	yes	complemented
229	C₃×C₅	#1963	1	1	yes	characteristic, complemented, Hall
230	C₂²×C₃	—	2	2	no	—
231	C₂²×C₃	—	20	4	no	—
232	C₂²×C₃	—	20	2	no	—
233	C₂²×C₃	C₂×#685	1	1	yes	characteristic
234	C₂²×C₃	#4076	1	2	yes	—
235	C₃×C₄	—	2	2	no	—
236	C₃×C₄	—	2	2	no	—
237	C₃×C₄	—	4	4	no	—
238	C₃×C₄	—	20	2	no	—
239	C₃×C₄	C₄×D₂₀	1	2	yes	—
240	C₃×C₄	#4020	1	1	yes	characteristic
241	C₃×C₄	#4021	1	1	yes	characteristic
242	C₂×C₅	—	2	2	no	—
243	C₂×C₅	C₃×#390	1	1	yes	characteristic
244	C₂×C₅	C₃×#421	1	2	yes	—
245	D₅	—	8	2	no	—
246	C₂³	—	10	2	no	—
247	C₂³	C₂×C₃×C₂²⋊D₅	1	1	yes	characteristic
248	Q₈	—	10	4	no	—
249	D₄	—	10	4	no	—
250	D₄	—	20	4	no	—
251	D₄	—	20	4	no	—
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	4	4	no	—
255	C₂×C₄	—	10	2	no	—
256	C₂×C₄	—	20	2	no	—
257	C₂×C₄	—	20	2	no	—
258	C₂×C₄	C₃×#685	1	2	yes	—
259	C₂×C₄	C₃×#685	1	2	yes	—
260	C₂×C₄	C₂×C₃×C₄×D₅	1	1	yes	characteristic
261	C₂×C₄	C₂×C₃×D₂₀	1	2	yes	—
262	C₂×C₄	C₂×C₃×D₂₀	1	1	yes	characteristic
263	C₂×C₄	C₃×#709	1	2	yes	—
264	C₈	—	20	2	no	—
265	C₂×C₃	—	2	2	no	—
266	C₂×C₃	—	40	2	no	—
267	C₂×C₃	#65228	1	1	yes	characteristic
268	C₂×C₃	#65294	1	2	yes	—
269	C₅	C₃×#1963	1	1	yes	characteristic, complemented, Sylow
270	C₂²	—	2	2	no	—
271	C₂²	—	20	4	no	—
272	C₂²	—	20	2	no	—
273	C₂²	C₂×C₃×#685	1	1	yes	characteristic
274	C₂²	C₃×#4076	1	2	yes	—
275	C₄	—	2	2	no	—
276	C₄	—	2	2	no	—
277	C₄	—	4	4	no	—
278	C₄	—	20	2	no	—
279	C₄	C₃×C₄×D₂₀	1	2	yes	—
280	C₄	C₃×#4020	1	1	yes	characteristic
281	C₄	C₃×#4021	1	1	yes	characteristic
282	C₃	#10600233	1	1	yes	characteristic, direct product factor, Sylow
283	C₂	—	2	2	no	—
284	C₂	—	40	2	no	—
285	C₂	C₃×#65228	1	1	yes	characteristic
286	C₂	C₃×#65294	1	2	yes	—

	subgroup	G/H
center	C₂×C₃×C₄	C₂×D₂₀
commutator	C₂×C₄×C₅	C₂²×C₃×C₄
Frattini	C₂²×C₄	C₂²×C₃×D₅
Fitting	C₃×C₅×#376	C₂
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₃×#65811	C₂
cosocle	C₂²×C₄×C₅	C₂³×C₃
hypercenter	C₃×#376	D₅
hypocenter	C₅	C₃×#1963
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂×C₃×C₄² ◁ C₃×#376

Complements

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

N	H ≅ G/N	type
2. C₃×C₅×#376	C₂	semidirect
3. C₂×C₃×#3937	C₂	semidirect
4. C₃×#65213	C₂	semidirect
5. #10600233	C₃	direct (N × H)
11. C₃×#4021	C₄	semidirect
12. C₃×#4020	C₄	semidirect
14. C₃×#4015	C₄	semidirect
20. C₅×#376	C₂×C₃	semidirect
21. C₂×#3937	C₂×C₃	semidirect
22. #65213	C₂×C₃	semidirect
37. C₃×C₄²×C₅	D₄	semidirect
45. C₃×#376	D₅	semidirect
51. #4021	C₃×C₄	semidirect
52. #4020	C₃×C₄	semidirect
54. #4015	C₃×C₄	semidirect
104. C₄²×C₅	C₃×D₄	semidirect
112. #376	C₃×D₅	semidirect
143. C₃×C₄²	D₂₀	semidirect
228. C₄²	C₃×D₂₀	semidirect
229. C₃×C₅	#1963	semidirect
269. C₅	C₃×#1963	semidirect
282. C₃	#10600233	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×#65811	2
C₃×C₅×#376	2
C₂×C₃×#3937	2
C₃×#65213	2
#10600233	3
C₃×#1963	5

Subgroups — C3×#10600233 #14306034

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10600233 #14306034