Subgroups — C₃×#10602804 #14295503

Order 1920.

568 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×#131	C₂	1	1	yes	maximal, characteristic
2	C₃×#65529	C₂	1	1	yes	maximal, characteristic
3	C₃×#65526	C₂	1	1	yes	maximal, characteristic
4	C₃×#65390	C₂	1	2	yes	maximal, complemented
5	C₃×#64776	C₂	1	2	yes	maximal, complemented
6	#10602804	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×C₅×#41	C₂²	1	1	yes	characteristic
8	C₃×C₅×#131	—	2	2	no	—
9	C₃×C₅×#131	—	2	2	no	—
10	C₂×C₃×C₅×M₁₆	C₄	1	2	yes	complemented
11	C₂×C₃×C₅×M₁₆	C₂²	1	1	yes	characteristic
12	C₂²×C₃×C₅×C₈	C₄	1	2	yes	complemented
13	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
14	C₃×#4072	C₂²	1	2	yes	—
15	C₂×C₃×#681	C₂²	1	2	yes	—
16	C₃×#3968	C₄	1	2	yes	complemented
17	C₃×#3966	C₄	1	4	yes	—
18	C₃×#3965	C₄	1	2	yes	complemented
19	C₃×#3947	—	2	2	no	—
20	C₃×C₈×Dic₅	—	2	2	no	—
21	C₃×#1757	—	5	1	no	maximal, Hall
22	C₂×C₅×#131	C₂×C₃	1	1	yes	characteristic
23	#65529	C₂×C₃	1	1	yes	characteristic
24	#65526	C₂×C₃	1	1	yes	characteristic
25	#65390	C₂×C₃	1	2	yes	complemented
26	#64776	C₂×C₃	1	2	yes	complemented
27	C₃×C₅×#41	—	2	2	no	—
28	C₃×C₅×#41	—	2	2	no	—
29	C₂×C₃×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
30	C₂×C₃×C₅×D₄	C₂×C₄	1	2	yes	—
31	C₂×C₃×C₅×D₄	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
33	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
34	C₃×C₅×M₁₆	—	2	2	no	—
35	C₃×C₅×M₁₆	—	2	4	no	—
36	C₂×C₃×C₅×C₈	—	2	2	no	—
37	C₂×C₃×C₅×C₈	—	2	2	no	—
38	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
39	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
40	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
41	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
42	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
43	C₃×C₂²⋊Dic₅	C₂×C₄	1	4	yes	—
44	C₃×C₄⋊Dic₅	—	2	2	no	—
45	C₃×C₄×Dic₅	—	2	2	no	—
46	C₃×#681	—	2	2	no	—
47	C₃×#681	C₂×C₄	1	4	yes	—
48	C₂×C₃×#168	—	2	2	no	—
49	C₂×C₃×#131	D₅	1	1	yes	characteristic
50	C₃×#413	—	5	1	no	—
51	C₃×#410	—	5	1	no	—
52	C₃×#405	—	5	2	no	—
53	C₃×#344	—	5	2	no	—
54	C₂×C₅×#41	C₂²×C₃	1	1	yes	characteristic
55	C₅×#131	—	2	2	no	—
56	C₅×#131	—	2	2	no	—
57	C₂×C₅×M₁₆	C₃×C₄	1	2	yes	complemented
58	C₂×C₅×M₁₆	C₂²×C₃	1	1	yes	characteristic
59	C₂²×C₅×C₈	C₃×C₄	1	2	yes	complemented
60	C₂²×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
61	#4072	C₂²×C₃	1	2	yes	—
62	C₂×#681	C₂²×C₃	1	2	yes	—
63	#3968	C₃×C₄	1	2	yes	complemented
64	#3966	C₃×C₄	1	4	yes	—
65	#3965	C₃×C₄	1	2	yes	complemented
66	#3947	—	2	2	no	—
67	C₈×Dic₅	—	2	2	no	—
68	#1757	—	5	1	no	Sylow
69	C₂³×C₃×C₅	C₄²	1	2	yes	—
70	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
71	C₃×C₅×Q₈	—	2	2	no	—
72	C₃×C₅×D₄	—	2	4	no	—
73	C₃×C₅×D₄	—	2	2	no	—
74	C₂×C₃×C₄×C₅	—	2	2	no	—
75	C₂×C₃×C₄×C₅	—	2	2	no	—
76	C₂×C₃×C₄×C₅	C₄²	1	2	yes	—
77	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
80	C₂×C₃×C₄×C₅	#41	1	2	yes	—
81	C₂×C₃×Dic₅	—	2	4	no	—
82	C₃×C₅×C₈	—	2	2	no	—
83	C₃×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
84	C₃×C₅×C₈	C₂²⋊C₄	1	2	yes	—
85	C₃×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
86	C₃×#168	—	2	4	no	—
87	C₂×C₃×#41	C₂×D₅	1	1	yes	characteristic
88	C₃×#131	—	2	2	no	—
89	C₃×#131	—	2	2	no	—
90	C₂×C₃×M₁₆	—	5	2	no	—
91	C₂×C₃×M₁₆	Dic₅	1	2	yes	complemented
92	C₂×C₃×M₁₆	C₂×D₅	1	1	yes	characteristic
93	C₂²×C₃×C₈	Dic₅	1	2	yes	complemented
94	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
95	C₃×#117	—	5	2	no	—
96	C₃×#101	—	5	2	no	—
97	C₃×#100	—	5	2	no	—
98	C₃×#99	—	5	4	no	—
99	C₃×#97	—	10	2	no	—
100	C₃×C₄×C₈	—	10	2	no	—
101	C₅×#41	—	2	2	no	—
102	C₅×#41	—	2	2	no	—
103	C₂×C₅×Q₈	C₂×C₃×C₄	1	1	yes	characteristic
104	C₂×C₅×D₄	C₂×C₃×C₄	1	2	yes	—
105	C₂×C₅×D₄	C₂×C₃×C₄	1	1	yes	characteristic
106	C₂²×C₄×C₅	C₂×C₃×C₄	1	2	yes	—
107	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
108	C₅×M₁₆	—	2	2	no	—
109	C₅×M₁₆	—	2	4	no	—
110	C₂×C₅×C₈	—	2	2	no	—
111	C₂×C₅×C₈	—	2	2	no	—
112	C₂×C₅×C₈	C₂×C₃×C₄	1	2	yes	—
113	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
114	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
115	C₂×C₅×C₈	C₃×D₄	1	2	yes	—
116	C₂×C₅×C₈	C₃×D₄	1	2	yes	—
117	C₂²⋊Dic₅	C₂×C₃×C₄	1	4	yes	—
118	C₄⋊Dic₅	—	2	2	no	—
119	C₄×Dic₅	—	2	2	no	—
120	#681	—	2	2	no	—
121	#681	C₂×C₃×C₄	1	4	yes	—
122	C₂×#168	—	2	2	no	—
123	C₂×#131	C₃×D₅	1	1	yes	characteristic
124	#413	—	5	1	no	—
125	#410	—	5	1	no	—
126	#405	—	5	2	no	—
127	#344	—	5	2	no	—
128	C₂²×C₃×C₅	—	2	2	no	—
129	C₂²×C₃×C₅	—	4	1	no	—
130	C₂²×C₃×C₅	—	4	2	no	—
131	C₂²×C₃×C₅	C₂×C₄²	1	1	yes	characteristic
132	C₂²×C₃×C₅	C₄×D₄	1	2	yes	—
133	C₃×C₄×C₅	—	2	2	no	—
134	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
135	C₃×C₄×C₅	#117	1	1	yes	characteristic
136	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
137	C₃×Dic₅	—	4	4	no	—
138	C₃×#41	—	2	2	no	—
139	C₃×#41	—	2	2	no	—
140	C₂×C₃×Q₈	C₄×D₅	1	1	yes	characteristic
141	C₂×C₃×D₄	C₄×D₅	1	2	yes	—
142	C₂×C₃×D₄	C₄×D₅	1	1	yes	characteristic
143	C₂²×C₃×C₄	C₂×Dic₅	1	2	yes	—
144	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
145	C₃×M₁₆	—	2	2	no	—
146	C₃×M₁₆	—	2	4	no	—
147	C₃×M₁₆	—	5	4	no	—
148	C₃×M₁₆	—	10	2	no	—
149	C₂×C₃×C₈	—	2	2	no	—
150	C₂×C₃×C₈	—	2	2	no	—
151	C₂×C₃×C₈	—	10	2	no	—
152	C₂×C₃×C₈	C₂×Dic₅	1	2	yes	—
153	C₂×C₃×C₈	C₂×Dic₅	1	1	yes	characteristic
154	C₂×C₃×C₈	C₂×Dic₅	1	1	yes	characteristic
155	C₂×C₃×C₈	C₂²⋊D₅	1	2	yes	—
156	C₂×C₃×C₈	C₂²⋊D₅	1	2	yes	—
157	C₃×C₄⋊C₄	—	10	2	no	—
158	C₃×C₂²⋊C₄	—	5	4	no	—
159	C₃×C₄²	—	10	2	no	—
160	C₂³×C₅	C₃×C₄²	1	2	yes	—
161	C₂³×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
162	C₅×Q₈	—	2	2	no	—
163	C₅×D₄	—	2	4	no	—
164	C₅×D₄	—	2	2	no	—
165	C₂×C₄×C₅	—	2	2	no	—
166	C₂×C₄×C₅	—	2	2	no	—
167	C₂×C₄×C₅	C₃×C₄²	1	2	yes	—
168	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
169	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
170	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
171	C₂×C₄×C₅	C₃×#41	1	2	yes	—
172	C₂×Dic₅	—	2	4	no	—
173	C₅×C₈	—	2	2	no	—
174	C₅×C₈	C₃×C₂²⋊C₄	1	1	yes	characteristic
175	C₅×C₈	C₃×C₂²⋊C₄	1	1	yes	characteristic
176	C₅×C₈	C₃×C₂²⋊C₄	1	2	yes	—
177	#168	—	2	4	no	—
178	C₂×#41	C₂×C₃×D₅	1	1	yes	characteristic
179	#131	—	2	2	no	—
180	#131	—	2	2	no	—
181	C₂×M₁₆	—	5	2	no	—
182	C₂×M₁₆	C₃×Dic₅	1	2	yes	complemented
183	C₂×M₁₆	C₂×C₃×D₅	1	1	yes	characteristic
184	C₂²×C₈	C₃×Dic₅	1	2	yes	complemented
185	C₂²×C₈	C₂×C₃×D₅	1	1	yes	characteristic
186	#117	—	5	2	no	—
187	#101	—	5	2	no	—
188	#100	—	5	2	no	—
189	#99	—	5	4	no	—
190	#97	—	10	2	no	—
191	C₄×C₈	—	10	2	no	—
192	C₂×C₃×C₅	—	2	2	no	—
193	C₂×C₃×C₅	—	2	1	no	—
194	C₂×C₃×C₅	—	4	2	no	—
195	C₂×C₃×C₅	C₄×C₂²⋊C₄	1	1	yes	characteristic
196	C₂³×C₃	C₄×Dic₅	1	2	yes	—
197	C₂³×C₃	C₂×C₄×D₅	1	1	yes	characteristic
198	C₃×Q₈	—	2	2	no	—
199	C₃×D₄	—	2	4	no	—
200	C₃×D₄	—	2	2	no	—
201	C₂×C₃×C₄	—	2	2	no	—
202	C₂×C₃×C₄	—	2	2	no	—
203	C₂×C₃×C₄	—	10	4	no	—
204	C₂×C₃×C₄	C₄×Dic₅	1	2	yes	—
205	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
206	C₂×C₃×C₄	#709	1	2	yes	—
207	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
208	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	2	yes	—
209	C₃×C₈	—	2	2	no	—
210	C₃×C₈	—	10	4	no	—
211	C₃×C₈	C₂²⋊Dic₅	1	1	yes	characteristic
212	C₃×C₈	C₂²⋊Dic₅	1	2	yes	—
213	C₃×C₈	C₂²⋊Dic₅	1	1	yes	characteristic
214	C₂²×C₅	—	2	2	no	—
215	C₂²×C₅	—	4	1	no	—
216	C₂²×C₅	—	4	2	no	—
217	C₂²×C₅	C₂×C₃×C₄²	1	1	yes	characteristic
218	C₂²×C₅	C₃×C₄×D₄	1	2	yes	—
219	C₄×C₅	—	2	2	no	—
220	C₄×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
221	C₄×C₅	C₃×#117	1	1	yes	characteristic
222	C₄×C₅	C₃×C₄×D₄	1	2	yes	—
223	Dic₅	—	4	4	no	—
224	#41	—	2	2	no	—
225	#41	—	2	2	no	—
226	C₂×Q₈	C₃×C₄×D₅	1	1	yes	characteristic
227	C₂×D₄	C₃×C₄×D₅	1	2	yes	—
228	C₂×D₄	C₃×C₄×D₅	1	1	yes	characteristic
229	C₂²×C₄	C₂×C₃×Dic₅	1	2	yes	—
230	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
231	M₁₆	—	2	2	no	—
232	M₁₆	—	2	4	no	—
233	M₁₆	—	5	4	no	—
234	M₁₆	—	10	2	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	10	2	no	—
238	C₂×C₈	C₂×C₃×Dic₅	1	2	yes	—
239	C₂×C₈	C₂×C₃×Dic₅	1	1	yes	characteristic
240	C₂×C₈	C₂×C₃×Dic₅	1	1	yes	characteristic
241	C₂×C₈	C₃×C₂²⋊D₅	1	2	yes	—
242	C₂×C₈	C₃×C₂²⋊D₅	1	2	yes	—
243	C₄⋊C₄	—	10	2	no	—
244	C₂²⋊C₄	—	5	4	no	—
245	C₄²	—	10	2	no	—
246	C₃×C₅	#1757	1	1	yes	characteristic, complemented, Hall
247	C₂²×C₃	—	2	2	no	—
248	C₂²×C₃	—	4	1	no	—
249	C₂²×C₃	—	4	2	no	—
250	C₂²×C₃	C₂×C₄×Dic₅	1	1	yes	characteristic
251	C₂²×C₃	C₄×C₂²⋊D₅	1	2	yes	—
252	C₃×C₄	—	2	2	no	—
253	C₃×C₄	—	20	4	no	—
254	C₃×C₄	#4072	1	1	yes	characteristic
255	C₃×C₄	C₄×C₂²⋊D₅	1	2	yes	—
256	C₃×C₄	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
257	C₂×C₅	—	2	2	no	—
258	C₂×C₅	—	2	1	no	—
259	C₂×C₅	—	4	2	no	—
260	C₂×C₅	C₃×C₄×C₂²⋊C₄	1	1	yes	characteristic
261	C₂³	C₃×C₄×Dic₅	1	2	yes	—
262	C₂³	C₂×C₃×C₄×D₅	1	1	yes	characteristic
263	Q₈	—	2	2	no	—
264	D₄	—	2	4	no	—
265	D₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	10	4	no	—
269	C₂×C₄	C₃×C₄×Dic₅	1	2	yes	—
270	C₂×C₄	C₂×C₃×C₄×D₅	1	1	yes	characteristic
271	C₂×C₄	C₃×#709	1	2	yes	—
272	C₂×C₄	C₂²×C₃×Dic₅	1	1	yes	characteristic
273	C₂×C₄	C₂×C₃×C₂²⋊D₅	1	2	yes	—
274	C₈	—	2	2	no	—
275	C₈	—	10	4	no	—
276	C₈	C₃×C₂²⋊Dic₅	1	1	yes	characteristic
277	C₈	C₃×C₂²⋊Dic₅	1	1	yes	characteristic
278	C₈	C₃×C₂²⋊Dic₅	1	2	yes	—
279	C₂×C₃	—	2	2	no	—
280	C₂×C₃	—	2	1	no	—
281	C₂×C₃	—	4	2	no	—
282	C₂×C₃	C₄×C₂²⋊Dic₅	1	1	yes	characteristic
283	C₅	C₃×#1757	1	1	yes	characteristic, complemented, Sylow
284	C₂²	—	2	2	no	—
285	C₂²	—	4	1	no	—
286	C₂²	—	4	2	no	—
287	C₂²	C₂×C₃×C₄×Dic₅	1	1	yes	characteristic
288	C₂²	C₃×C₄×C₂²⋊D₅	1	2	yes	—
289	C₄	—	2	2	no	—
290	C₄	—	20	4	no	—
291	C₄	C₃×#4072	1	1	yes	characteristic
292	C₄	C₃×C₄×C₂²⋊D₅	1	2	yes	—
293	C₄	C₂×C₃×C₂²⋊Dic₅	1	1	yes	characteristic
294	C₃	#10602804	1	1	yes	characteristic, direct product factor, Sylow
295	C₂	—	2	2	no	—
296	C₂	—	2	1	no	—
297	C₂	—	4	2	no	—
298	C₂	C₃×C₄×C₂²⋊Dic₅	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₈	C₂²⋊Dic₅
commutator	C₂²×C₅	C₂×C₃×C₄²
Frattini	C₂²×C₄	C₂²×C₃×D₅
Fitting	C₂×C₃×C₅×#131	C₂
socle	C₂×C₃×C₅	C₄×C₂²⋊C₄
prime radical	C₂×C₃×C₅×#41	C₂²
cosocle	C₂²×C₄×C₅	C₂³×C₃
hypercenter	C₂×C₃×#131	D₅
hypocenter	C₅	C₃×#1757
LCS	G ▷ C₂²×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₃×C₈ ◁ C₂²×C₃×C₈ ◁ C₂×C₃×#131

Complements

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

N	H ≅ G/N	type
4. C₃×#65390	C₂	semidirect
5. C₃×#64776	C₂	semidirect
6. #10602804	C₃	direct (N × H)
10. C₂×C₃×C₅×M₁₆	C₄	semidirect
12. C₂²×C₃×C₅×C₈	C₄	semidirect
16. C₃×#3968	C₄	semidirect
18. C₃×#3965	C₄	semidirect
25. #65390	C₂×C₃	semidirect
26. #64776	C₂×C₃	semidirect
57. C₂×C₅×M₁₆	C₃×C₄	semidirect
59. C₂²×C₅×C₈	C₃×C₄	semidirect
63. #3968	C₃×C₄	semidirect
65. #3965	C₃×C₄	semidirect
91. C₂×C₃×M₁₆	Dic₅	semidirect
93. C₂²×C₃×C₈	Dic₅	semidirect
182. C₂×M₁₆	C₃×Dic₅	semidirect
184. C₂²×C₈	C₃×Dic₅	semidirect
246. C₃×C₅	#1757	semidirect
283. C₅	C₃×#1757	semidirect
294. C₃	#10602804	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×C₅×#131	2
C₃×#65529	2
C₃×#65526	2
C₃×#65390	2
C₃×#64776	2
#10602804	3
C₃×#1757	5

Subgroups — C3×#10602804 #14295503

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10602804 #14295503