Subgroups — C₃×#10599365 #14285852

Order 1920.

336 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#65126	C₂	1	1	yes	maximal, characteristic
2	C₃×D₅×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₃×#64974	C₂	1	1	yes	maximal, characteristic
4	C₃×C₅×#335	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#64699	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#64699	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#64678	C₂	1	1	yes	maximal, characteristic, complemented
8	#10599365	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂×C₃×#676	—	2	1	no	—
10	C₂×C₃×C₈×D₅	—	2	1	no	—
11	C₃×C₄²×D₅	C₂²	1	1	yes	characteristic
12	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
13	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
15	C₃×#3952	C₄	1	1	yes	characteristic
16	C₃×#3952	C₄	1	1	yes	characteristic
17	C₃×#3946	—	2	1	no	—
18	C₃×#3946	—	2	1	no	—
19	C₃×#3946	C₄	1	1	yes	characteristic
20	C₃×#3946	C₄	1	1	yes	characteristic
21	C₃×C₄⋊#168	C₂²	1	1	yes	characteristic
22	C₃×C₄⋊#168	C₂²	1	1	yes	characteristic
23	C₃×C₄×#168	C₂²	1	1	yes	characteristic
24	C₃×#1571	—	5	1	no	maximal, Hall
25	#65126	C₂×C₃	1	1	yes	characteristic
26	D₅×C₄⋊C₈	C₂×C₃	1	1	yes	characteristic
27	#64974	C₂×C₃	1	1	yes	characteristic
28	C₅×#335	C₂×C₃	1	1	yes	characteristic, complemented
29	#64699	C₂×C₃	1	1	yes	characteristic, complemented
30	#64699	C₂×C₃	1	1	yes	characteristic, complemented
31	#64678	C₂×C₃	1	1	yes	characteristic, complemented
32	C₂×C₃×C₄×D₅	C₂×C₄	1	1	yes	characteristic
33	C₂×C₃×C₄×D₅	D₄	1	1	yes	characteristic
34	C₂×C₃×C₄×D₅	Q₈	1	1	yes	characteristic
35	C₂×C₃×C₅×C₈	—	2	1	no	—
36	C₂×C₃×C₅×C₈	—	2	1	no	—
37	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
38	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
39	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
40	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
41	C₃×C₄×Dic₅	D₄	1	1	yes	characteristic
42	C₃×C₄×Dic₅	Q₈	1	1	yes	characteristic
43	C₂×C₃×#168	—	2	1	no	—
44	C₂×C₃×#168	—	2	1	no	—
45	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
46	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
47	C₃×#676	—	4	2	no	—
48	C₃×C₈×D₅	—	4	2	no	—
49	C₃×#431	—	5	1	no	—
50	C₃×#423	—	5	1	no	—
51	C₂×C₃×C₄⋊C₈	—	5	1	no	—
52	C₃×#335	—	5	1	no	—
53	C₃×#335	—	5	1	no	—
54	C₃×#335	—	5	1	no	—
55	C₃×#335	D₅	1	1	yes	characteristic, complemented
56	C₂×#676	—	2	1	no	—
57	C₂×C₈×D₅	—	2	1	no	—
58	C₄²×D₅	C₂²×C₃	1	1	yes	characteristic
59	C₅×C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
60	C₅×C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
61	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
62	#3952	C₃×C₄	1	1	yes	characteristic
63	#3952	C₃×C₄	1	1	yes	characteristic
64	#3946	—	2	1	no	—
65	#3946	—	2	1	no	—
66	#3946	C₃×C₄	1	1	yes	characteristic
67	#3946	C₃×C₄	1	1	yes	characteristic
68	C₄⋊#168	C₂²×C₃	1	1	yes	characteristic
69	C₄⋊#168	C₂²×C₃	1	1	yes	characteristic
70	C₄×#168	C₂²×C₃	1	1	yes	characteristic
71	#1571	—	5	1	no	Sylow
72	C₂²×C₃×D₅	C₄⋊C₄	1	1	yes	characteristic
73	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
75	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
76	C₂×C₃×Dic₅	C₄⋊C₄	1	1	yes	characteristic
77	C₂×C₃×Dic₅	C₄⋊C₄	1	1	yes	characteristic
78	C₂×C₃×Dic₅	C₄⋊C₄	1	1	yes	characteristic
79	C₃×C₄×D₅	—	2	1	no	—
80	C₃×C₄×D₅	—	2	1	no	—
81	C₃×C₄×D₅	—	4	1	no	—
82	C₃×C₄×D₅	D₈	1	2	yes	—
83	C₃×C₄×D₅	Q₁₆	1	2	yes	—
84	C₃×C₅×C₈	—	2	1	no	—
85	C₃×C₅×C₈	—	2	1	no	—
86	C₃×C₅×C₈	—	4	1	no	—
87	C₃×C₅×C₈	—	4	1	no	—
88	C₃×#168	—	2	1	no	—
89	C₃×#168	—	2	1	no	—
90	C₃×#168	—	4	1	no	—
91	C₃×#168	—	4	1	no	—
92	C₂×C₃×M₁₆	—	10	1	no	—
93	C₂²×C₃×C₈	—	10	1	no	—
94	C₂×C₃×C₄²	—	5	1	no	—
95	C₃×C₄⋊C₈	—	5	1	no	—
96	C₃×C₄⋊C₈	—	5	1	no	—
97	C₃×C₄⋊C₈	—	5	1	no	—
98	C₃×C₄⋊C₈	—	5	1	no	—
99	C₃×C₄⋊C₈	—	10	1	no	—
100	C₃×C₄⋊C₈	—	10	1	no	—
101	C₃×C₄⋊C₈	C₂×D₅	1	1	yes	characteristic
102	C₃×C₄⋊C₈	C₂×D₅	1	1	yes	characteristic
103	C₃×C₂²⋊C₈	—	5	1	no	—
104	C₃×C₂²⋊C₈	—	5	1	no	—
105	C₃×C₄×C₈	—	5	1	no	—
106	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic
107	C₂×C₄×D₅	C₂×C₃×C₄	1	1	yes	characteristic
108	C₂×C₄×D₅	C₃×D₄	1	1	yes	characteristic
109	C₂×C₄×D₅	C₃×Q₈	1	1	yes	characteristic
110	C₂×C₅×C₈	—	2	1	no	—
111	C₂×C₅×C₈	—	2	1	no	—
112	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
113	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
114	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
115	C₄×Dic₅	C₂×C₃×C₄	1	1	yes	characteristic
116	C₄×Dic₅	C₃×D₄	1	1	yes	characteristic
117	C₄×Dic₅	C₃×Q₈	1	1	yes	characteristic
118	C₂×#168	—	2	1	no	—
119	C₂×#168	—	2	1	no	—
120	C₂×#168	C₂×C₃×C₄	1	1	yes	characteristic
121	C₂×#168	C₂×C₃×C₄	1	1	yes	characteristic
122	#676	—	4	2	no	—
123	C₈×D₅	—	4	2	no	—
124	#431	—	5	1	no	—
125	#423	—	5	1	no	—
126	C₂×C₄⋊C₈	—	5	1	no	—
127	#335	—	5	1	no	—
128	#335	—	5	1	no	—
129	#335	—	5	1	no	—
130	#335	C₃×D₅	1	1	yes	characteristic, complemented
131	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
132	C₂×C₃×D₅	—	2	1	no	—
133	C₂×C₃×D₅	—	2	1	no	—
134	C₂×C₃×D₅	#107	1	2	yes	—
135	C₃×C₄×C₅	—	2	1	no	—
136	C₃×C₄×C₅	#131	1	1	yes	characteristic
137	C₃×C₄×C₅	#131	1	1	yes	characteristic
138	C₃×C₄×C₅	C₂×D₈	1	1	yes	characteristic
139	C₃×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
140	C₃×Dic₅	—	2	1	no	—
141	C₃×Dic₅	—	2	1	no	—
142	C₃×Dic₅	#107	1	2	yes	—
143	C₂²×C₃×C₄	—	5	1	no	—
144	C₂²×C₃×C₄	—	5	1	no	—
145	C₂²×C₃×C₄	—	5	1	no	—
146	C₃×M₁₆	—	20	2	no	—
147	C₂×C₃×C₈	—	2	1	no	—
148	C₂×C₃×C₈	—	2	1	no	—
149	C₂×C₃×C₈	—	5	1	no	—
150	C₂×C₃×C₈	—	5	1	no	—
151	C₂×C₃×C₈	—	10	1	no	—
152	C₂×C₃×C₈	—	10	1	no	—
153	C₂×C₃×C₈	—	20	2	no	—
154	C₂×C₃×C₈	C₄×D₅	1	1	yes	characteristic
155	C₂×C₃×C₈	C₄×D₅	1	1	yes	characteristic
156	C₃×C₄²	—	5	1	no	—
157	C₃×C₄²	—	5	1	no	—
158	C₃×C₄²	—	5	1	no	—
159	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
160	C₂²×D₅	C₃×C₄⋊C₄	1	1	yes	characteristic
161	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
162	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
163	C₂×C₄×C₅	C₂×C₃×Q₈	1	1	yes	characteristic
164	C₂×Dic₅	C₃×C₄⋊C₄	1	1	yes	characteristic
165	C₂×Dic₅	C₃×C₄⋊C₄	1	1	yes	characteristic
166	C₂×Dic₅	C₃×C₄⋊C₄	1	1	yes	characteristic
167	C₄×D₅	—	2	1	no	—
168	C₄×D₅	—	2	1	no	—
169	C₄×D₅	—	4	1	no	—
170	C₄×D₅	C₃×D₈	1	2	yes	—
171	C₄×D₅	C₃×Q₁₆	1	2	yes	—
172	C₅×C₈	—	2	1	no	—
173	C₅×C₈	—	2	1	no	—
174	C₅×C₈	—	4	1	no	—
175	C₅×C₈	—	4	1	no	—
176	#168	—	2	1	no	—
177	#168	—	2	1	no	—
178	#168	—	4	1	no	—
179	#168	—	4	1	no	—
180	C₂×M₁₆	—	10	1	no	—
181	C₂²×C₈	—	10	1	no	—
182	C₂×C₄²	—	5	1	no	—
183	C₄⋊C₈	—	5	1	no	—
184	C₄⋊C₈	—	5	1	no	—
185	C₄⋊C₈	—	5	1	no	—
186	C₄⋊C₈	—	5	1	no	—
187	C₄⋊C₈	—	10	1	no	—
188	C₄⋊C₈	—	10	1	no	—
189	C₄⋊C₈	C₂×C₃×D₅	1	1	yes	characteristic
190	C₄⋊C₈	C₂×C₃×D₅	1	1	yes	characteristic
191	C₂²⋊C₈	—	5	1	no	—
192	C₂²⋊C₈	—	5	1	no	—
193	C₄×C₈	—	5	1	no	—
194	C₄×C₈	C₂×C₃×D₅	1	1	yes	characteristic
195	C₂×C₃×C₅	#424	1	1	yes	characteristic
196	C₂×C₃×C₅	C₂×#107	1	1	yes	characteristic
197	C₂×C₃×C₅	#430	1	1	yes	characteristic
198	C₃×D₅	—	2	2	no	—
199	C₂³×C₃	—	5	1	no	—
200	C₂×C₃×C₄	—	5	1	no	—
201	C₂×C₃×C₄	—	5	2	no	—
202	C₂×C₃×C₄	—	5	2	no	—
203	C₂×C₃×C₄	—	5	1	no	—
204	C₂×C₃×C₄	—	5	1	no	—
205	C₂×C₃×C₄	—	10	1	no	—
206	C₂×C₃×C₄	—	10	1	no	—
207	C₂×C₃×C₄	—	20	1	no	—
208	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
209	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
210	C₂×C₃×C₄	Q₈×D₅	1	1	yes	characteristic
211	C₃×C₈	—	2	1	no	—
212	C₃×C₈	—	2	1	no	—
213	C₃×C₈	—	4	1	no	—
214	C₃×C₈	—	4	1	no	—
215	C₃×C₈	—	10	1	no	—
216	C₃×C₈	—	10	1	no	—
217	C₃×C₈	—	20	1	no	—
218	C₃×C₈	—	20	1	no	—
219	C₂²×C₅	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
220	C₂×D₅	—	2	1	no	—
221	C₂×D₅	—	2	1	no	—
222	C₂×D₅	C₃×#107	1	2	yes	—
223	C₄×C₅	—	2	1	no	—
224	C₄×C₅	C₃×#131	1	1	yes	characteristic
225	C₄×C₅	C₃×#131	1	1	yes	characteristic
226	C₄×C₅	C₂×C₃×D₈	1	1	yes	characteristic
227	C₄×C₅	C₂×C₃×Q₁₆	1	1	yes	characteristic
228	Dic₅	—	2	1	no	—
229	Dic₅	—	2	1	no	—
230	Dic₅	C₃×#107	1	2	yes	—
231	C₂²×C₄	—	5	1	no	—
232	C₂²×C₄	—	5	1	no	—
233	C₂²×C₄	—	5	1	no	—
234	M₁₆	—	20	2	no	—
235	C₂×C₈	—	2	1	no	—
236	C₂×C₈	—	2	1	no	—
237	C₂×C₈	—	5	1	no	—
238	C₂×C₈	—	5	1	no	—
239	C₂×C₈	—	10	1	no	—
240	C₂×C₈	—	10	1	no	—
241	C₂×C₈	—	20	2	no	—
242	C₂×C₈	C₃×C₄×D₅	1	1	yes	characteristic
243	C₂×C₈	C₃×C₄×D₅	1	1	yes	characteristic
244	C₄²	—	5	1	no	—
245	C₄²	—	5	1	no	—
246	C₄²	—	5	1	no	—
247	C₄²	C₂²×C₃×D₅	1	1	yes	characteristic
248	C₃×C₅	#1571	1	1	yes	characteristic, complemented, Hall
249	C₂²×C₃	—	5	2	no	—
250	C₂²×C₃	—	10	1	no	—
251	C₂²×C₃	—	10	1	no	—
252	C₂²×C₃	D₅×C₄⋊C₄	1	1	yes	characteristic
253	C₃×C₄	—	2	1	no	—
254	C₃×C₄	—	5	2	no	—
255	C₃×C₄	—	10	1	no	—
256	C₃×C₄	—	10	1	no	—
257	C₃×C₄	#4047	1	1	yes	characteristic
258	C₃×C₄	#4053	1	1	yes	characteristic
259	C₃×C₄	D₅×D₈	1	1	yes	characteristic
260	C₃×C₄	D₅×Q₁₆	1	1	yes	characteristic
261	C₂×C₅	C₃×#424	1	1	yes	characteristic
262	C₂×C₅	C₂×C₃×#107	1	1	yes	characteristic
263	C₂×C₅	C₃×#430	1	1	yes	characteristic
264	D₅	—	2	2	no	—
265	C₂³	—	5	1	no	—
266	C₂×C₄	—	5	1	no	—
267	C₂×C₄	—	5	2	no	—
268	C₂×C₄	—	5	2	no	—
269	C₂×C₄	—	5	1	no	—
270	C₂×C₄	—	5	1	no	—
271	C₂×C₄	—	10	1	no	—
272	C₂×C₄	—	10	1	no	—
273	C₂×C₄	—	20	1	no	—
274	C₂×C₄	C₂×C₃×C₄×D₅	1	1	yes	characteristic
275	C₂×C₄	C₃×D₄×D₅	1	1	yes	characteristic
276	C₂×C₄	C₃×Q₈×D₅	1	1	yes	characteristic
277	C₈	—	2	1	no	—
278	C₈	—	2	1	no	—
279	C₈	—	4	1	no	—
280	C₈	—	4	1	no	—
281	C₈	—	10	1	no	—
282	C₈	—	10	1	no	—
283	C₈	—	20	1	no	—
284	C₈	—	20	1	no	—
285	C₂×C₃	—	10	2	no	—
286	C₂×C₃	#65128	1	1	yes	characteristic
287	C₂×C₃	D₅×#107	1	1	yes	characteristic
288	C₂×C₃	#65182	1	1	yes	characteristic
289	C₅	C₃×#1571	1	1	yes	characteristic, complemented, Sylow
290	C₂²	—	5	2	no	—
291	C₂²	—	10	1	no	—
292	C₂²	—	10	1	no	—
293	C₂²	C₃×D₅×C₄⋊C₄	1	1	yes	characteristic
294	C₄	—	2	1	no	—
295	C₄	—	5	2	no	—
296	C₄	—	10	1	no	—
297	C₄	—	10	1	no	—
298	C₄	C₃×#4047	1	1	yes	characteristic
299	C₄	C₃×#4053	1	1	yes	characteristic
300	C₄	C₃×D₅×D₈	1	1	yes	characteristic
301	C₄	C₃×D₅×Q₁₆	1	1	yes	characteristic
302	C₃	#10599365	1	1	yes	characteristic, direct product factor, Sylow
303	C₂	—	10	2	no	—
304	C₂	C₃×#65128	1	1	yes	characteristic
305	C₂	C₃×D₅×#107	1	1	yes	characteristic
306	C₂	C₃×#65182	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	D₄×D₅
commutator	C₂×C₄×C₅	C₂²×C₃×C₄
Frattini	C₄²	C₂²×C₃×D₅
Fitting	C₃×C₅×#335	C₂
socle	C₂²×C₃×C₅	C₂×C₄⋊C₄
prime radical	C₂²×C₃×D₅	C₄⋊C₄
cosocle	C₄²×C₅	C₂³×C₃
hypercenter	C₃×#335	D₅
hypocenter	C₅	C₃×#1571
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₃×C₄² ◁ C₃×#335

Complements

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

N	H ≅ G/N	type
4. C₃×C₅×#335	C₂	semidirect
5. C₃×#64699	C₂	semidirect
6. C₃×#64699	C₂	semidirect
7. C₃×#64678	C₂	semidirect
8. #10599365	C₃	direct (N × H)
28. C₅×#335	C₂×C₃	semidirect
29. #64699	C₂×C₃	semidirect
30. #64699	C₂×C₃	semidirect
31. #64678	C₂×C₃	semidirect
55. C₃×#335	D₅	semidirect
130. #335	C₃×D₅	semidirect
248. C₃×C₅	#1571	semidirect
289. C₅	C₃×#1571	semidirect
302. C₃	#10599365	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×#65126	2
C₃×D₅×C₄⋊C₈	2
C₃×#64974	2
C₃×C₅×#335	2
C₃×#64699	2
C₃×#64678	2
#10599365	3
C₃×#1571	5

Subgroups — C3×#10599365 #14285852

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10599365 #14285852