Subgroups — C₃×#10600188 #14305991

Order 1920.

720 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×#65811	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄³×C₅	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	#10600188	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂×C₃×#709	C₂²	1	2	yes	—
7	C₂×C₃×C₄²×C₅	—	2	2	no	—
8	C₂×C₃×C₄²×C₅	C₂²	1	2	yes	—
9	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
10	C₃×#4076	—	2	2	no	—
11	C₂×C₃×#681	C₂²	1	2	yes	—
12	C₃×#4021	C₄	1	2	yes	complemented
13	C₃×#4020	C₄	1	1	yes	characteristic, complemented
14	C₃×C₄×D₂₀	—	2	2	no	—
15	C₃×#4015	C₄	1	1	yes	characteristic, complemented
16	C₃×#3937	—	2	8	no	—
17	C₃×C₄⋊#168	—	2	2	no	—
18	C₃×#1962	—	5	1	no	maximal, Hall
19	#65811	C₂×C₃	1	1	yes	characteristic
20	C₄³×C₅	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	8	no	—
34	C₃×C₄²×C₅	—	2	2	no	—
35	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
36	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
37	C₃×C₄²×C₅	D₄	1	2	yes	—
38	C₃×C₄²×C₅	D₄	1	4	yes	complemented
39	C₃×#685	—	2	4	no	—
40	C₃×C₄⋊Dic₅	—	2	2	no	—
41	C₃×#681	—	2	4	no	—
42	C₂×C₃×#168	—	2	2	no	—
43	C₃×#532	—	5	1	no	—
44	C₃×#423	—	5	1	no	—
45	C₂×C₃×#104	—	5	4	no	—
46	C₃×C₄³	D₅	1	1	yes	characteristic, complemented
47	C₂×#709	C₂²×C₃	1	2	yes	—
48	C₂×C₄²×C₅	—	2	2	no	—
49	C₂×C₄²×C₅	C₂²×C₃	1	2	yes	—
50	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
51	#4076	—	2	2	no	—
52	C₂×#681	C₂²×C₃	1	2	yes	—
53	#4021	C₃×C₄	1	2	yes	complemented
54	#4020	C₃×C₄	1	1	yes	characteristic, complemented
55	C₄×D₂₀	—	2	2	no	—
56	#4015	C₃×C₄	1	1	yes	characteristic, complemented
57	#3937	—	2	8	no	—
58	C₄⋊#168	—	2	2	no	—
59	#1962	—	5	1	no	Sylow
60	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₃×D₅	—	2	2	no	—
62	C₂×C₃×C₄×C₅	—	2	8	no	—
63	C₂×C₃×C₄×C₅	—	2	4	no	—
64	C₂×C₃×C₄×C₅	—	2	4	no	—
65	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
66	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
67	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
69	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
70	C₂×C₃×C₄×C₅	#41	1	2	yes	—
71	C₃×C₂²⋊D₅	—	4	4	no	—
72	C₂×C₃×Dic₅	—	2	2	no	—
73	C₃×D₂₀	—	2	4	no	—
74	C₃×D₂₀	—	4	4	no	—
75	C₃×C₄×D₅	—	4	2	no	—
76	C₃×C₄×D₅	—	4	2	no	—
77	C₃×Dic₁₀	—	2	4	no	—
78	C₃×#168	—	4	2	no	—
79	C₂×C₃×#41	—	5	2	no	—
80	C₂×C₃×M₁₆	—	5	2	no	—
81	C₃×C₄⋊Q₈	—	5	1	no	—
82	C₃×#127	—	5	1	no	—
83	C₃×#124	—	5	2	no	—
84	C₃×#121	—	10	2	no	—
85	C₃×C₄×D₄	—	10	2	no	—
86	C₂×C₃×C₄²	—	2	2	no	—
87	C₂×C₃×C₄²	C₂×D₅	1	2	yes	—
88	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
89	C₃×C₄⋊C₈	—	10	2	no	—
90	C₃×#104	—	10	8	no	—
91	C₂²×C₄×C₅	—	2	2	no	—
92	C₂²×C₄×C₅	C₃×D₄	1	2	yes	—
93	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
94	C₂×C₂²⋊D₅	—	2	2	no	—
95	#709	—	2	4	no	—
96	#709	—	2	4	no	—
97	C₂×D₂₀	—	2	2	no	—
98	C₂×D₂₀	C₂×C₃×C₄	1	2	yes	—
99	C₂×C₄×D₅	—	2	2	no	—
100	C₂×Dic₁₀	C₂×C₃×C₄	1	2	yes	—
101	C₄²×C₅	—	2	8	no	—
102	C₄²×C₅	—	2	2	no	—
103	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
104	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
105	C₄²×C₅	C₃×D₄	1	2	yes	—
106	C₄²×C₅	C₃×D₄	1	4	yes	complemented
107	#685	—	2	4	no	—
108	C₄⋊Dic₅	—	2	2	no	—
109	#681	—	2	4	no	—
110	C₂×#168	—	2	2	no	—
111	#532	—	5	1	no	—
112	#423	—	5	1	no	—
113	C₂×#104	—	5	4	no	—
114	C₄³	C₃×D₅	1	1	yes	characteristic, complemented
115	C₂²×C₃×C₅	—	2	2	no	—
116	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	C₂²×C₃×C₅	#121	1	2	yes	—
118	C₂×C₃×D₅	—	4	2	no	—
119	C₂×C₃×D₅	—	4	4	no	—
120	C₃×C₄×C₅	—	2	8	no	—
121	C₃×C₄×C₅	—	2	2	no	—
122	C₃×C₄×C₅	#104	1	4	yes	complemented
123	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
124	C₃×C₄×C₅	#124	1	1	yes	characteristic
125	C₃×C₄×C₅	#127	1	1	yes	characteristic
126	C₃×Dic₅	—	4	2	no	—
127	C₃×#41	—	10	4	no	—
128	C₃×#41	—	10	4	no	—
129	C₂×C₃×Q₈	—	5	2	no	—
130	C₂×C₃×D₄	—	5	2	no	—
131	C₂×C₃×D₄	—	10	2	no	—
132	C₂×C₃×D₄	—	10	2	no	—
133	C₂²×C₃×C₄	—	2	2	no	—
134	C₂²×C₃×C₄	—	10	2	no	—
135	C₂²×C₃×C₄	C₂²⋊D₅	1	2	yes	—
136	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
137	C₃×M₁₆	—	10	4	no	—
138	C₂×C₃×C₈	—	10	2	no	—
139	C₃×C₄⋊C₄	—	10	2	no	—
140	C₃×C₂²⋊C₄	—	10	4	no	—
141	C₃×C₄²	—	2	8	no	—
142	C₃×C₄²	—	2	2	no	—
143	C₃×C₄²	C₄×D₅	1	1	yes	characteristic
144	C₃×C₄²	C₄×D₅	1	1	yes	characteristic
145	C₃×C₄²	D₂₀	1	2	yes	—
146	C₃×C₄²	D₂₀	1	4	yes	complemented
147	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
148	C₂²×D₅	—	2	2	no	—
149	C₂×C₄×C₅	—	2	8	no	—
150	C₂×C₄×C₅	—	2	4	no	—
151	C₂×C₄×C₅	—	2	4	no	—
152	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	2	yes	—
153	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	2	yes	—
154	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
155	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
156	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
157	C₂×C₄×C₅	C₃×#41	1	2	yes	—
158	C₂²⋊D₅	—	4	4	no	—
159	C₂×Dic₅	—	2	2	no	—
160	D₂₀	—	2	4	no	—
161	D₂₀	—	4	4	no	—
162	C₄×D₅	—	4	2	no	—
163	C₄×D₅	—	4	2	no	—
164	Dic₁₀	—	2	4	no	—
165	#168	—	4	2	no	—
166	C₂×#41	—	5	2	no	—
167	C₂×M₁₆	—	5	2	no	—
168	C₄⋊Q₈	—	5	1	no	—
169	#127	—	5	1	no	—
170	#124	—	5	2	no	—
171	#121	—	10	2	no	—
172	C₄×D₄	—	10	2	no	—
173	C₂×C₄²	—	2	2	no	—
174	C₂×C₄²	C₂×C₃×D₅	1	2	yes	—
175	C₂×C₄²	C₂×C₃×D₅	1	1	yes	characteristic
176	C₄⋊C₈	—	10	2	no	—
177	#104	—	10	8	no	—
178	C₂×C₃×C₅	—	2	2	no	—
179	C₂×C₃×C₅	#390	1	1	yes	characteristic
180	C₂×C₃×C₅	C₂×#104	1	2	yes	—
181	C₃×D₅	—	8	2	no	—
182	C₂³×C₃	—	10	2	no	—
183	C₂³×C₃	C₂×C₂²⋊D₅	1	1	yes	characteristic
184	C₃×Q₈	—	10	4	no	—
185	C₃×D₄	—	10	4	no	—
186	C₃×D₄	—	20	4	no	—
187	C₃×D₄	—	20	4	no	—
188	C₂×C₃×C₄	—	2	8	no	—
189	C₂×C₃×C₄	—	2	4	no	—
190	C₂×C₃×C₄	—	2	4	no	—
191	C₂×C₃×C₄	—	10	2	no	—
192	C₂×C₃×C₄	—	20	2	no	—
193	C₂×C₃×C₄	—	20	2	no	—
194	C₂×C₃×C₄	#685	1	2	yes	—
195	C₂×C₃×C₄	#685	1	2	yes	—
196	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
197	C₂×C₃×C₄	C₂×D₂₀	1	2	yes	—
198	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
199	C₂×C₃×C₄	#709	1	2	yes	—
200	C₃×C₈	—	20	2	no	—
201	C₂²×C₅	—	2	2	no	—
202	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
203	C₂²×C₅	C₃×#121	1	2	yes	—
204	C₂×D₅	—	4	2	no	—
205	C₂×D₅	—	4	4	no	—
206	C₄×C₅	—	2	8	no	—
207	C₄×C₅	—	2	2	no	—
208	C₄×C₅	C₃×#104	1	4	yes	complemented
209	C₄×C₅	C₃×C₄×D₄	1	2	yes	—
210	C₄×C₅	C₃×#124	1	1	yes	characteristic
211	C₄×C₅	C₃×#127	1	1	yes	characteristic
212	Dic₅	—	4	2	no	—
213	#41	—	10	4	no	—
214	#41	—	10	4	no	—
215	C₂×Q₈	—	5	2	no	—
216	C₂×D₄	—	5	2	no	—
217	C₂×D₄	—	10	2	no	—
218	C₂×D₄	—	10	2	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	10	2	no	—
221	C₂²×C₄	C₃×C₂²⋊D₅	1	2	yes	—
222	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
223	M₁₆	—	10	4	no	—
224	C₂×C₈	—	10	2	no	—
225	C₄⋊C₄	—	10	2	no	—
226	C₂²⋊C₄	—	10	4	no	—
227	C₄²	—	2	8	no	—
228	C₄²	—	2	2	no	—
229	C₄²	C₃×C₄×D₅	1	1	yes	characteristic
230	C₄²	C₃×C₄×D₅	1	1	yes	characteristic
231	C₄²	C₃×D₂₀	1	2	yes	—
232	C₄²	C₃×D₂₀	1	4	yes	complemented
233	C₃×C₅	#1962	1	1	yes	characteristic, complemented, Hall
234	C₂²×C₃	—	2	2	no	—
235	C₂²×C₃	—	20	4	no	—
236	C₂²×C₃	—	20	2	no	—
237	C₂²×C₃	C₂×#685	1	1	yes	characteristic
238	C₂²×C₃	#4076	1	2	yes	—
239	C₃×C₄	—	2	8	no	—
240	C₃×C₄	—	2	2	no	—
241	C₃×C₄	—	20	2	no	—
242	C₃×C₄	#3937	1	4	yes	complemented
243	C₃×C₄	C₄×D₂₀	1	2	yes	—
244	C₃×C₄	#4020	1	1	yes	characteristic
245	C₃×C₄	#4021	1	1	yes	characteristic
246	C₂×C₅	—	2	2	no	—
247	C₂×C₅	C₃×#390	1	1	yes	characteristic
248	C₂×C₅	C₂×C₃×#104	1	2	yes	—
249	D₅	—	8	2	no	—
250	C₂³	—	10	2	no	—
251	C₂³	C₂×C₃×C₂²⋊D₅	1	1	yes	characteristic
252	Q₈	—	10	4	no	—
253	D₄	—	10	4	no	—
254	D₄	—	20	4	no	—
255	D₄	—	20	4	no	—
256	C₂×C₄	—	2	8	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	10	2	no	—
260	C₂×C₄	—	20	2	no	—
261	C₂×C₄	—	20	2	no	—
262	C₂×C₄	C₃×#685	1	2	yes	—
263	C₂×C₄	C₃×#685	1	2	yes	—
264	C₂×C₄	C₂×C₃×C₄×D₅	1	1	yes	characteristic
265	C₂×C₄	C₂×C₃×D₂₀	1	2	yes	—
266	C₂×C₄	C₂×C₃×D₂₀	1	1	yes	characteristic
267	C₂×C₄	C₃×#709	1	2	yes	—
268	C₈	—	20	2	no	—
269	C₂×C₃	—	2	2	no	—
270	C₂×C₃	—	40	2	no	—
271	C₂×C₃	C₂×#3937	1	2	yes	—
272	C₂×C₃	#65228	1	1	yes	characteristic
273	C₅	C₃×#1962	1	1	yes	characteristic, complemented, Sylow
274	C₂²	—	2	2	no	—
275	C₂²	—	20	4	no	—
276	C₂²	—	20	2	no	—
277	C₂²	C₂×C₃×#685	1	1	yes	characteristic
278	C₂²	C₃×#4076	1	2	yes	—
279	C₄	—	2	8	no	—
280	C₄	—	2	2	no	—
281	C₄	—	20	2	no	—
282	C₄	C₃×#3937	1	4	yes	complemented
283	C₄	C₃×C₄×D₂₀	1	2	yes	—
284	C₄	C₃×#4020	1	1	yes	characteristic
285	C₄	C₃×#4021	1	1	yes	characteristic
286	C₃	#10600188	1	1	yes	characteristic, direct product factor, Sylow
287	C₂	—	2	2	no	—
288	C₂	—	40	2	no	—
289	C₂	C₂×C₃×#3937	1	2	yes	—
290	C₂	C₃×#65228	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	C₂×D₂₀
commutator	C₂×C₄×C₅	C₂²×C₃×C₄
Frattini	C₂²×C₄	C₂²×C₃×D₅
Fitting	C₃×C₄³×C₅	C₂
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₃×#65811	C₂
cosocle	C₂²×C₄×C₅	C₂³×C₃
hypercenter	C₃×C₄³	D₅
hypocenter	C₅	C₃×#1962
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂×C₃×C₄² ◁ C₃×C₄³

Complements

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

N	H ≅ G/N	type
2. C₃×C₄³×C₅	C₂	semidirect
3. C₂×C₃×#3937	C₂	semidirect
4. C₃×#65213	C₂	semidirect
5. #10600188	C₃	direct (N × H)
12. C₃×#4021	C₄	semidirect
13. C₃×#4020	C₄	semidirect
15. C₃×#4015	C₄	semidirect
20. C₄³×C₅	C₂×C₃	semidirect
21. C₂×#3937	C₂×C₃	semidirect
22. #65213	C₂×C₃	semidirect
38. C₃×C₄²×C₅	D₄	semidirect
46. C₃×C₄³	D₅	semidirect
53. #4021	C₃×C₄	semidirect
54. #4020	C₃×C₄	semidirect
56. #4015	C₃×C₄	semidirect
106. C₄²×C₅	C₃×D₄	semidirect
114. C₄³	C₃×D₅	semidirect
122. C₃×C₄×C₅	#104	semidirect
146. C₃×C₄²	D₂₀	semidirect
208. C₄×C₅	C₃×#104	semidirect
232. C₄²	C₃×D₂₀	semidirect
233. C₃×C₅	#1962	semidirect
242. C₃×C₄	#3937	semidirect
273. C₅	C₃×#1962	semidirect
282. C₄	C₃×#3937	semidirect
286. C₃	#10600188	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×C₄³×C₅	2
C₂×C₃×#3937	2
C₃×#65213	2
#10600188	3
C₃×#1962	5

Subgroups — C3×#10600188 #14305991

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10600188 #14305991