Subgroups — C₃×#10602322 #14295041

Order 1920.

824 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	C₂²×C₃×C₄²×C₅	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₂²⋊#168	C₂	1	2	yes	maximal
3	C₂×C₃×#3935	C₂	1	2	yes	maximal, complemented
4	C₃×#64744	C₂	1	2	yes	maximal, complemented
5	#10602322	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂³×C₃×C₄×C₅	C₄	1	2	yes	—
7	C₂³×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
8	C₂×C₃×C₄²×C₅	—	2	4	no	—
9	C₂×C₃×C₄²×C₅	C₄	1	2	yes	—
10	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
12	C₂²×C₃×#168	C₂²	1	4	yes	—
13	C₃×C₂²⋊#168	C₄	1	8	yes	complemented
14	C₃×#3935	—	2	4	no	—
15	C₃×#1749	—	5	1	no	maximal, Hall
16	C₂²×C₄²×C₅	C₂×C₃	1	1	yes	characteristic
17	C₂×C₂²⋊#168	C₂×C₃	1	2	yes	—
18	C₂×#3935	C₂×C₃	1	2	yes	complemented
19	#64744	C₂×C₃	1	2	yes	complemented
20	C₂⁴×C₃×C₅	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₃×C₄×C₅	—	2	4	no	—
22	C₂²×C₃×C₄×C₅	—	2	4	no	—
23	C₂²×C₃×C₄×C₅	—	2	4	no	—
24	C₂²×C₃×C₄×C₅	—	2	2	no	—
25	C₂²×C₃×C₄×C₅	—	2	2	no	—
26	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₃×C₄×C₅	C₂×C₄	1	4	yes	—
28	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
29	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
30	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
31	C₃×C₄²×C₅	—	2	4	no	—
32	C₃×C₄²×C₅	—	2	2	no	—
33	C₃×C₄²×C₅	D₄	1	4	yes	—
34	C₂×C₃×#168	—	2	4	no	—
35	C₂×C₃×#168	—	2	4	no	—
36	C₂×C₃×#168	C₂×C₄	1	8	yes	complemented
37	C₂²×C₃×C₄²	D₅	1	1	yes	characteristic
38	C₂×C₃×C₂²⋊C₈	—	5	2	no	—
39	C₂×C₃×#97	—	5	2	no	—
40	C₃×#336	—	5	2	no	—
41	C₂³×C₄×C₅	C₃×C₄	1	2	yes	—
42	C₂³×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
43	C₂×C₄²×C₅	—	2	4	no	—
44	C₂×C₄²×C₅	C₃×C₄	1	2	yes	—
45	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
46	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
47	C₂²×#168	C₂²×C₃	1	4	yes	—
48	C₂²⋊#168	C₃×C₄	1	8	yes	complemented
49	#3935	—	2	4	no	—
50	#1749	—	5	1	no	Sylow
51	C₂³×C₃×C₅	—	2	1	no	—
52	C₂³×C₃×C₅	—	2	1	no	—
53	C₂³×C₃×C₅	—	2	2	no	—
54	C₂³×C₃×C₅	C₄²	1	2	yes	—
55	C₂³×C₃×C₅	M₁₆	1	4	yes	—
56	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₃×C₄×C₅	—	2	16	no	—
58	C₂×C₃×C₄×C₅	—	2	1	no	—
59	C₂×C₃×C₄×C₅	—	2	2	no	—
60	C₂×C₃×C₄×C₅	—	2	1	no	—
61	C₂×C₃×C₄×C₅	—	2	2	no	—
62	C₂×C₃×C₄×C₅	—	2	2	no	—
63	C₂×C₃×C₄×C₅	—	2	2	no	—
64	C₂×C₃×C₄×C₅	—	2	8	no	—
65	C₂×C₃×C₄×C₅	C₄²	1	2	yes	—
66	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	4	yes	—
67	C₂×C₃×C₄×C₅	M₁₆	1	4	yes	—
68	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
69	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
70	C₂×C₃×C₄×C₅	#41	1	2	yes	—
71	C₃×#168	—	2	8	no	—
72	C₂³×C₃×C₄	Dic₅	1	2	yes	—
73	C₂³×C₃×C₄	C₂×D₅	1	1	yes	characteristic
74	C₂²×C₃×C₈	—	5	4	no	—
75	C₂×C₃×C₄²	—	2	4	no	—
76	C₂×C₃×C₄²	Dic₅	1	2	yes	—
77	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
78	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
79	C₃×C₂²⋊C₈	—	5	8	no	—
80	C₃×#97	—	10	4	no	—
81	C₂⁴×C₅	C₂×C₃×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₅	—	2	4	no	—
83	C₂²×C₄×C₅	—	2	4	no	—
84	C₂²×C₄×C₅	—	2	4	no	—
85	C₂²×C₄×C₅	—	2	2	no	—
86	C₂²×C₄×C₅	—	2	2	no	—
87	C₂²×C₄×C₅	C₂×C₃×C₄	1	4	yes	—
88	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
89	C₂²×C₄×C₅	C₂×C₃×C₄	1	2	yes	—
90	C₂²×C₄×C₅	C₂×C₃×C₄	1	2	yes	—
91	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
92	C₄²×C₅	—	2	4	no	—
93	C₄²×C₅	—	2	2	no	—
94	C₄²×C₅	C₃×D₄	1	4	yes	—
95	C₂×#168	—	2	4	no	—
96	C₂×#168	—	2	4	no	—
97	C₂×#168	C₂×C₃×C₄	1	8	yes	complemented
98	C₂²×C₄²	C₃×D₅	1	1	yes	characteristic
99	C₂×C₂²⋊C₈	—	5	2	no	—
100	C₂×#97	—	5	2	no	—
101	#336	—	5	2	no	—
102	C₂²×C₃×C₅	—	2	2	no	—
103	C₂²×C₃×C₅	—	2	2	no	—
104	C₂²×C₃×C₅	—	2	4	no	—
105	C₂²×C₃×C₅	—	2	4	no	—
106	C₂²×C₃×C₅	#97	1	4	yes	complemented
107	C₂²×C₃×C₅	C₂×C₄²	1	1	yes	characteristic
108	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂²×C₃×C₅	#117	1	1	yes	characteristic
110	C₂²×C₃×C₅	C₂×M₁₆	1	2	yes	—
111	C₂²×C₃×C₅	C₂×M₁₆	1	2	yes	—
112	C₃×C₄×C₅	—	2	2	no	—
113	C₃×C₄×C₅	—	2	4	no	—
114	C₃×C₄×C₅	—	2	4	no	—
115	C₃×C₄×C₅	C₄×D₄	1	4	yes	—
116	C₂⁴×C₃	C₂×Dic₅	1	1	yes	characteristic
117	C₂²×C₃×C₄	—	2	4	no	—
118	C₂²×C₃×C₄	—	2	4	no	—
119	C₂²×C₃×C₄	—	2	4	no	—
120	C₂²×C₃×C₄	—	2	2	no	—
121	C₂²×C₃×C₄	—	2	2	no	—
122	C₂²×C₃×C₄	C₄×D₅	1	4	yes	—
123	C₂²×C₃×C₄	C₂×Dic₅	1	2	yes	—
124	C₂²×C₃×C₄	C₂×Dic₅	1	1	yes	characteristic
125	C₂²×C₃×C₄	C₂×Dic₅	1	2	yes	—
126	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
127	C₂×C₃×C₈	—	5	8	no	—
128	C₂×C₃×C₈	—	10	4	no	—
129	C₂×C₃×C₈	—	10	4	no	—
130	C₃×C₄²	—	2	4	no	—
131	C₃×C₄²	—	2	2	no	—
132	C₃×C₄²	C₂²⋊D₅	1	4	yes	—
133	C₂³×C₅	—	2	1	no	—
134	C₂³×C₅	—	2	1	no	—
135	C₂³×C₅	—	2	2	no	—
136	C₂³×C₅	C₃×C₄²	1	2	yes	—
137	C₂³×C₅	C₃×M₁₆	1	4	yes	—
138	C₂³×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
139	C₂×C₄×C₅	—	2	16	no	—
140	C₂×C₄×C₅	—	2	1	no	—
141	C₂×C₄×C₅	—	2	2	no	—
142	C₂×C₄×C₅	—	2	1	no	—
143	C₂×C₄×C₅	—	2	2	no	—
144	C₂×C₄×C₅	—	2	2	no	—
145	C₂×C₄×C₅	—	2	2	no	—
146	C₂×C₄×C₅	—	2	8	no	—
147	C₂×C₄×C₅	C₃×C₄²	1	2	yes	—
148	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	4	yes	—
149	C₂×C₄×C₅	C₃×M₁₆	1	4	yes	—
150	C₂×C₄×C₅	C₂²×C₃×C₄	1	2	yes	—
151	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
152	C₂×C₄×C₅	C₃×#41	1	2	yes	—
153	#168	—	2	8	no	—
154	C₂³×C₄	C₃×Dic₅	1	2	yes	—
155	C₂³×C₄	C₂×C₃×D₅	1	1	yes	characteristic
156	C₂²×C₈	—	5	4	no	—
157	C₂×C₄²	—	2	4	no	—
158	C₂×C₄²	C₃×Dic₅	1	2	yes	—
159	C₂×C₄²	C₂×C₃×D₅	1	1	yes	characteristic
160	C₂×C₄²	C₂×C₃×D₅	1	1	yes	characteristic
161	C₂²⋊C₈	—	5	8	no	—
162	#97	—	10	4	no	—
163	C₂×C₃×C₅	—	2	4	no	—
164	C₂×C₃×C₅	C₄×C₂²⋊C₄	1	1	yes	characteristic
165	C₂×C₃×C₅	C₂×#97	1	1	yes	characteristic
166	C₂×C₃×C₅	C₄×M₁₆	1	1	yes	characteristic
167	C₂×C₃×C₅	#407	1	2	yes	—
168	C₂×C₃×C₅	#432	1	2	yes	—
169	C₂³×C₃	—	2	1	no	—
170	C₂³×C₃	—	2	2	no	—
171	C₂³×C₃	—	2	1	no	—
172	C₂³×C₃	#681	1	4	yes	—
173	C₂³×C₃	C₄×Dic₅	1	2	yes	—
174	C₂³×C₃	C₂²×Dic₅	1	1	yes	characteristic
175	C₂×C₃×C₄	—	2	16	no	—
176	C₂×C₃×C₄	—	2	2	no	—
177	C₂×C₃×C₄	—	2	1	no	—
178	C₂×C₃×C₄	—	2	2	no	—
179	C₂×C₃×C₄	—	2	2	no	—
180	C₂×C₃×C₄	—	2	2	no	—
181	C₂×C₃×C₄	—	2	1	no	—
182	C₂×C₃×C₄	—	2	8	no	—
183	C₂×C₃×C₄	#681	1	4	yes	—
184	C₂×C₃×C₄	C₄×Dic₅	1	2	yes	—
185	C₂×C₃×C₄	C₂²⋊Dic₅	1	4	yes	—
186	C₂×C₃×C₄	C₂×C₄×D₅	1	2	yes	—
187	C₂×C₃×C₄	#709	1	2	yes	—
188	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	2	yes	—
189	C₃×C₈	—	10	8	no	—
190	C₂²×C₅	—	2	2	no	—
191	C₂²×C₅	—	2	2	no	—
192	C₂²×C₅	—	2	4	no	—
193	C₂²×C₅	—	2	4	no	—
194	C₂²×C₅	C₃×#97	1	4	yes	complemented
195	C₂²×C₅	C₂×C₃×C₄²	1	1	yes	characteristic
196	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
197	C₂²×C₅	C₃×#117	1	1	yes	characteristic
198	C₂²×C₅	C₂×C₃×M₁₆	1	2	yes	—
199	C₂²×C₅	C₂×C₃×M₁₆	1	2	yes	—
200	C₄×C₅	—	2	2	no	—
201	C₄×C₅	—	2	4	no	—
202	C₄×C₅	—	2	4	no	—
203	C₄×C₅	C₃×C₄×D₄	1	4	yes	—
204	C₂⁴	C₂×C₃×Dic₅	1	1	yes	characteristic
205	C₂²×C₄	—	2	4	no	—
206	C₂²×C₄	—	2	4	no	—
207	C₂²×C₄	—	2	4	no	—
208	C₂²×C₄	—	2	2	no	—
209	C₂²×C₄	—	2	2	no	—
210	C₂²×C₄	C₃×C₄×D₅	1	4	yes	—
211	C₂²×C₄	C₂×C₃×Dic₅	1	2	yes	—
212	C₂²×C₄	C₂×C₃×Dic₅	1	1	yes	characteristic
213	C₂²×C₄	C₂×C₃×Dic₅	1	2	yes	—
214	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
215	C₂×C₈	—	5	8	no	—
216	C₂×C₈	—	10	4	no	—
217	C₂×C₈	—	10	4	no	—
218	C₄²	—	2	4	no	—
219	C₄²	—	2	2	no	—
220	C₄²	C₃×C₂²⋊D₅	1	4	yes	—
221	C₃×C₅	#1749	1	1	yes	characteristic, complemented, Hall
222	C₂²×C₃	—	2	2	no	—
223	C₂²×C₃	—	2	4	no	—
224	C₂²×C₃	—	2	4	no	—
225	C₂²×C₃	—	2	2	no	—
226	C₂²×C₃	#3935	1	4	yes	complemented
227	C₂²×C₃	C₂×#681	1	2	yes	—
228	C₂²×C₃	C₂×#681	1	2	yes	—
229	C₂²×C₃	C₂×C₄×Dic₅	1	1	yes	characteristic
230	C₂²×C₃	#4072	1	1	yes	characteristic
231	C₂²×C₃	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
232	C₃×C₄	—	2	4	no	—
233	C₃×C₄	—	2	4	no	—
234	C₃×C₄	—	2	2	no	—
235	C₃×C₄	C₄×C₂²⋊D₅	1	4	yes	—
236	C₂×C₅	—	2	4	no	—
237	C₂×C₅	C₃×C₄×C₂²⋊C₄	1	1	yes	characteristic
238	C₂×C₅	C₂×C₃×#97	1	1	yes	characteristic
239	C₂×C₅	C₃×C₄×M₁₆	1	1	yes	characteristic
240	C₂×C₅	C₃×#407	1	2	yes	—
241	C₂×C₅	C₃×#432	1	2	yes	—
242	C₂³	—	2	1	no	—
243	C₂³	—	2	2	no	—
244	C₂³	—	2	1	no	—
245	C₂³	C₃×#681	1	4	yes	—
246	C₂³	C₃×C₄×Dic₅	1	2	yes	—
247	C₂³	C₂²×C₃×Dic₅	1	1	yes	characteristic
248	C₂×C₄	—	2	16	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	8	no	—
256	C₂×C₄	C₃×#681	1	4	yes	—
257	C₂×C₄	C₃×C₄×Dic₅	1	2	yes	—
258	C₂×C₄	C₃×C₂²⋊Dic₅	1	4	yes	—
259	C₂×C₄	C₂×C₃×C₄×D₅	1	2	yes	—
260	C₂×C₄	C₃×#709	1	2	yes	—
261	C₂×C₄	C₂×C₃×C₂²⋊D₅	1	2	yes	—
262	C₈	—	10	8	no	—
263	C₂×C₃	—	2	4	no	—
264	C₂×C₃	C₂×#3935	1	1	yes	characteristic
265	C₂×C₃	C₄×#681	1	1	yes	characteristic
266	C₂×C₃	#65215	1	2	yes	—
267	C₂×C₃	#65496	1	2	yes	—
268	C₂×C₃	C₄×C₂²⋊Dic₅	1	1	yes	characteristic
269	C₅	C₃×#1749	1	1	yes	characteristic, complemented, Sylow
270	C₂²	—	2	2	no	—
271	C₂²	—	2	4	no	—
272	C₂²	—	2	4	no	—
273	C₂²	—	2	2	no	—
274	C₂²	C₃×#3935	1	4	yes	complemented
275	C₂²	C₂×C₃×#681	1	2	yes	—
276	C₂²	C₂×C₃×#681	1	2	yes	—
277	C₂²	C₂×C₃×C₄×Dic₅	1	1	yes	characteristic
278	C₂²	C₃×#4072	1	1	yes	characteristic
279	C₂²	C₂×C₃×C₂²⋊Dic₅	1	1	yes	characteristic
280	C₄	—	2	4	no	—
281	C₄	—	2	4	no	—
282	C₄	—	2	2	no	—
283	C₄	C₃×C₄×C₂²⋊D₅	1	4	yes	—
284	C₃	#10602322	1	1	yes	characteristic, direct product factor, Sylow
285	C₂	—	2	4	no	—
286	C₂	C₂×C₃×#3935	1	1	yes	characteristic
287	C₂	C₃×C₄×#681	1	1	yes	characteristic
288	C₂	C₃×#65215	1	2	yes	—
289	C₂	C₃×#65496	1	2	yes	—
290	C₂	C₃×C₄×C₂²⋊Dic₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃×C₄	C₂²×D₅
commutator	C₂²×C₅	C₂×C₃×C₄²
Frattini	C₂²×C₄	C₂²×C₃×D₅
Fitting	C₂²×C₃×C₄²×C₅	C₂
socle	C₂³×C₃×C₅	C₂²×C₄
prime radical	C₂⁴×C₃×C₅	C₂×C₄
cosocle	C₂²×C₄×C₅	C₂³×C₃
hypercenter	C₂²×C₃×C₄²	D₅
hypocenter	C₅	C₃×#1749
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃×C₄ ◁ C₂²×C₃×C₄²

Complements

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

N	H ≅ G/N	type
3. C₂×C₃×#3935	C₂	semidirect
4. C₃×#64744	C₂	semidirect
5. #10602322	C₃	direct (N × H)
13. C₃×C₂²⋊#168	C₄	semidirect
18. C₂×#3935	C₂×C₃	semidirect
19. #64744	C₂×C₃	semidirect
36. C₂×C₃×#168	C₂×C₄	semidirect
48. C₂²⋊#168	C₃×C₄	semidirect
97. C₂×#168	C₂×C₃×C₄	semidirect
106. C₂²×C₃×C₅	#97	semidirect
194. C₂²×C₅	C₃×#97	semidirect
221. C₃×C₅	#1749	semidirect
226. C₂²×C₃	#3935	semidirect
269. C₅	C₃×#1749	semidirect
274. C₂²	C₃×#3935	semidirect
284. C₃	#10602322	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃×C₄²×C₅	2
C₂×C₃×C₂²⋊#168	2
C₂×C₃×#3935	2
C₃×#64744	2
#10602322	3
C₃×#1749	5

Subgroups — C3×#10602322 #14295041

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10602322 #14295041