Subgroups — C₃×#10602619 #14282412

Order 1920.

464 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#533	C₂	1	1	yes	maximal, characteristic
2	C₃×#65213	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₄×#168	C₂	1	1	yes	maximal, characteristic
4	C₃×Q₈⋊#168	C₂	1	4	yes	maximal, complemented
5	#10602619	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₃×C₅×C₄⋊Q₈	C₄	1	1	yes	characteristic
7	C₃×C₅×C₄⋊Q₈	C₄	1	1	yes	characteristic
8	C₃×C₅×#125	—	2	1	no	—
9	C₃×C₅×C₂²⋊Q₈	—	2	1	no	—
10	C₃×C₅×C₂²⋊Q₈	C₄	1	2	yes	—
11	C₃×C₄×C₅×Q₈	—	2	1	no	—
12	C₃×C₄×C₅×Q₈	C₂²	1	2	yes	—
13	C₃×C₅×#117	—	2	1	no	—
14	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
15	C₂×C₃×#681	—	2	1	no	—
16	C₂²×C₃×#168	—	2	1	no	—
17	C₃×C₄⋊#168	—	2	1	no	—
18	C₃×C₄⋊#168	C₂²	1	2	yes	—
19	C₃×C₄×#168	—	2	1	no	—
20	C₃×C₄×#168	C₂²	1	2	yes	—
21	C₃×#1493	—	5	1	no	maximal, Hall
22	C₅×#533	C₂×C₃	1	1	yes	characteristic
23	#65213	C₂×C₃	1	1	yes	characteristic
24	C₂×C₄×#168	C₂×C₃	1	1	yes	characteristic
25	Q₈⋊#168	C₂×C₃	1	4	yes	complemented
26	C₂×C₃×C₅×Q₈	—	2	1	no	—
27	C₂×C₃×C₅×Q₈	C₂×C₄	1	2	yes	—
28	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
30	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
31	C₃×C₅×C₄⋊C₄	—	2	1	no	—
32	C₃×C₅×C₄⋊C₄	—	2	1	no	—
33	C₃×C₅×C₄⋊C₄	—	2	1	no	—
34	C₃×C₅×C₄⋊C₄	—	2	2	no	—
35	C₃×C₅×C₄⋊C₄	—	2	2	no	—
36	C₃×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
37	C₃×C₅×C₂²⋊C₄	—	2	2	no	—
38	C₃×C₄²×C₅	—	2	2	no	—
39	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
40	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
41	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
42	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
43	C₃×#681	—	4	2	no	—
44	C₂×C₃×#168	—	2	4	no	—
45	C₂×C₃×#168	—	2	2	no	—
46	C₂×C₃×#168	—	2	2	no	—
47	C₃×#533	D₅	1	1	yes	characteristic
48	C₃×#423	—	5	1	no	—
49	C₂×C₃×C₄×C₈	—	5	1	no	—
50	C₃×Q₈⋊C₈	—	5	4	no	—
51	C₅×C₄⋊Q₈	C₃×C₄	1	1	yes	characteristic
52	C₅×C₄⋊Q₈	C₃×C₄	1	1	yes	characteristic
53	C₅×#125	—	2	1	no	—
54	C₅×C₂²⋊Q₈	—	2	1	no	—
55	C₅×C₂²⋊Q₈	C₃×C₄	1	2	yes	—
56	C₄×C₅×Q₈	—	2	1	no	—
57	C₄×C₅×Q₈	C₂²×C₃	1	2	yes	—
58	C₅×#117	—	2	1	no	—
59	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
60	C₂×#681	—	2	1	no	—
61	C₂²×#168	—	2	1	no	—
62	C₄⋊#168	—	2	1	no	—
63	C₄⋊#168	C₂²×C₃	1	2	yes	—
64	C₄×#168	—	2	1	no	—
65	C₄×#168	C₂²×C₃	1	2	yes	—
66	#1493	—	5	1	no	Sylow
67	C₂³×C₃×C₅	C₂²⋊C₄	1	1	yes	characteristic
68	C₃×C₅×Q₈	—	2	4	no	—
69	C₃×C₅×Q₈	—	4	2	no	—
70	C₂×C₃×C₄×C₅	—	2	2	no	—
71	C₂×C₃×C₄×C₅	—	2	2	no	—
72	C₂×C₃×C₄×C₅	—	2	2	no	—
73	C₂×C₃×C₄×C₅	—	2	2	no	—
74	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₃×C₄×C₅	SD₁₆	1	2	yes	—
78	C₂×C₃×C₄×C₅	Q₁₆	1	2	yes	—
79	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
81	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
82	C₃×#168	—	2	4	no	—
83	C₃×#168	—	4	2	no	—
84	C₂×C₃×M₁₆	—	10	1	no	—
85	C₂²×C₃×C₈	—	10	1	no	—
86	C₃×C₄⋊Q₈	Dic₅	1	1	yes	characteristic
87	C₃×C₄⋊Q₈	Dic₅	1	1	yes	characteristic
88	C₃×#125	—	2	1	no	—
89	C₃×C₂²⋊Q₈	—	2	1	no	—
90	C₃×C₂²⋊Q₈	Dic₅	1	2	yes	—
91	C₃×C₄×Q₈	—	2	1	no	—
92	C₃×C₄×Q₈	C₂×D₅	1	2	yes	—
93	C₃×#117	—	2	1	no	—
94	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
95	C₃×C₄⋊C₈	—	5	2	no	—
96	C₃×C₄⋊C₈	—	10	1	no	—
97	C₃×C₄×C₈	—	5	2	no	—
98	C₃×C₄×C₈	—	10	1	no	—
99	C₂×C₅×Q₈	—	2	1	no	—
100	C₂×C₅×Q₈	C₂×C₃×C₄	1	2	yes	—
101	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
102	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
103	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
104	C₅×C₄⋊C₄	—	2	1	no	—
105	C₅×C₄⋊C₄	—	2	1	no	—
106	C₅×C₄⋊C₄	—	2	1	no	—
107	C₅×C₄⋊C₄	—	2	2	no	—
108	C₅×C₄⋊C₄	—	2	2	no	—
109	C₅×C₄⋊C₄	C₂×C₃×C₄	1	2	yes	—
110	C₅×C₂²⋊C₄	—	2	2	no	—
111	C₄²×C₅	—	2	2	no	—
112	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
113	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
114	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
115	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
116	#681	—	4	2	no	—
117	C₂×#168	—	2	4	no	—
118	C₂×#168	—	2	2	no	—
119	C₂×#168	—	2	2	no	—
120	#533	C₃×D₅	1	1	yes	characteristic
121	#423	—	5	1	no	—
122	C₂×C₄×C₈	—	5	1	no	—
123	Q₈⋊C₈	—	5	4	no	—
124	C₂²×C₃×C₅	—	2	1	no	—
125	C₂²×C₃×C₅	—	2	1	no	—
126	C₂²×C₃×C₅	#103	1	2	yes	—
127	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
128	C₃×C₄×C₅	—	2	1	no	—
129	C₃×C₄×C₅	—	2	1	no	—
130	C₃×C₄×C₅	—	4	2	no	—
131	C₃×C₄×C₅	—	4	2	no	—
132	C₃×C₄×C₅	#103	1	2	yes	—
133	C₃×C₄×C₅	#104	1	2	yes	—
134	C₃×C₄×C₅	#131	1	2	yes	—
135	C₃×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
136	C₃×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
137	C₂×C₃×Q₈	—	2	1	no	—
138	C₂×C₃×Q₈	C₂×Dic₅	1	2	yes	—
139	C₂²×C₃×C₄	C₂×Dic₅	1	1	yes	characteristic
140	C₂²×C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
141	C₂²×C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
142	C₃×M₁₆	—	20	2	no	—
143	C₂×C₃×C₈	—	10	4	no	—
144	C₂×C₃×C₈	—	10	2	no	—
145	C₂×C₃×C₈	—	10	2	no	—
146	C₃×C₄⋊C₄	—	2	1	no	—
147	C₃×C₄⋊C₄	—	2	1	no	—
148	C₃×C₄⋊C₄	—	2	1	no	—
149	C₃×C₄⋊C₄	—	2	2	no	—
150	C₃×C₄⋊C₄	—	2	2	no	—
151	C₃×C₄⋊C₄	C₂×Dic₅	1	2	yes	—
152	C₃×C₂²⋊C₄	—	2	2	no	—
153	C₃×C₄²	—	2	2	no	—
154	C₃×C₄²	C₂×Dic₅	1	1	yes	characteristic
155	C₃×C₄²	C₂²⋊D₅	1	1	yes	characteristic
156	C₃×C₄²	C₂²⋊D₅	1	1	yes	characteristic
157	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
158	C₂³×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
159	C₅×Q₈	—	2	4	no	—
160	C₅×Q₈	—	4	2	no	—
161	C₂×C₄×C₅	—	2	2	no	—
162	C₂×C₄×C₅	—	2	2	no	—
163	C₂×C₄×C₅	—	2	2	no	—
164	C₂×C₄×C₅	—	2	2	no	—
165	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
166	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
167	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
168	C₂×C₄×C₅	C₃×SD₁₆	1	2	yes	—
169	C₂×C₄×C₅	C₃×Q₁₆	1	2	yes	—
170	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
171	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
172	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
173	#168	—	2	4	no	—
174	#168	—	4	2	no	—
175	C₂×M₁₆	—	10	1	no	—
176	C₂²×C₈	—	10	1	no	—
177	C₄⋊Q₈	C₃×Dic₅	1	1	yes	characteristic
178	C₄⋊Q₈	C₃×Dic₅	1	1	yes	characteristic
179	#125	—	2	1	no	—
180	C₂²⋊Q₈	—	2	1	no	—
181	C₂²⋊Q₈	C₃×Dic₅	1	2	yes	—
182	C₄×Q₈	—	2	1	no	—
183	C₄×Q₈	C₂×C₃×D₅	1	2	yes	—
184	#117	—	2	1	no	—
185	C₂×C₄²	C₂×C₃×D₅	1	1	yes	characteristic
186	C₄⋊C₈	—	5	2	no	—
187	C₄⋊C₈	—	10	1	no	—
188	C₄×C₈	—	5	2	no	—
189	C₄×C₈	—	10	1	no	—
190	C₂×C₃×C₅	—	2	2	no	—
191	C₂×C₃×C₅	#408	1	1	yes	characteristic
192	C₂×C₃×C₅	C₂×#103	1	1	yes	characteristic
193	C₂×C₃×C₅	C₂×#104	1	1	yes	characteristic
194	C₂³×C₃	C₂²⋊Dic₅	1	1	yes	characteristic
195	C₃×Q₈	—	2	4	no	—
196	C₃×Q₈	—	4	2	no	—
197	C₂×C₃×C₄	—	2	2	no	—
198	C₂×C₃×C₄	—	2	2	no	—
199	C₂×C₃×C₄	—	2	2	no	—
200	C₂×C₃×C₄	—	2	2	no	—
201	C₂×C₃×C₄	#688	1	2	yes	—
202	C₂×C₃×C₄	#689	1	2	yes	—
203	C₂×C₃×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
204	C₂×C₃×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
205	C₂×C₃×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
206	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
207	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
208	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
209	C₃×C₈	—	10	4	no	—
210	C₃×C₈	—	20	2	no	—
211	C₂²×C₅	—	2	1	no	—
212	C₂²×C₅	—	2	1	no	—
213	C₂²×C₅	C₃×#103	1	2	yes	—
214	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
215	C₄×C₅	—	2	1	no	—
216	C₄×C₅	—	2	1	no	—
217	C₄×C₅	—	4	2	no	—
218	C₄×C₅	—	4	2	no	—
219	C₄×C₅	C₃×#103	1	2	yes	—
220	C₄×C₅	C₃×#104	1	2	yes	—
221	C₄×C₅	C₃×#131	1	2	yes	—
222	C₄×C₅	C₂×C₃×SD₁₆	1	1	yes	characteristic
223	C₄×C₅	C₂×C₃×Q₁₆	1	1	yes	characteristic
224	C₂×Q₈	—	2	1	no	—
225	C₂×Q₈	C₂×C₃×Dic₅	1	2	yes	—
226	C₂²×C₄	C₂×C₃×Dic₅	1	1	yes	characteristic
227	C₂²×C₄	C₃×C₂²⋊D₅	1	1	yes	characteristic
228	C₂²×C₄	C₃×C₂²⋊D₅	1	1	yes	characteristic
229	M₁₆	—	20	2	no	—
230	C₂×C₈	—	10	4	no	—
231	C₂×C₈	—	10	2	no	—
232	C₂×C₈	—	10	2	no	—
233	C₄⋊C₄	—	2	1	no	—
234	C₄⋊C₄	—	2	1	no	—
235	C₄⋊C₄	—	2	1	no	—
236	C₄⋊C₄	—	2	2	no	—
237	C₄⋊C₄	—	2	2	no	—
238	C₄⋊C₄	C₂×C₃×Dic₅	1	2	yes	—
239	C₂²⋊C₄	—	2	2	no	—
240	C₄²	—	2	2	no	—
241	C₄²	C₂×C₃×Dic₅	1	1	yes	characteristic
242	C₄²	C₃×C₂²⋊D₅	1	1	yes	characteristic
243	C₄²	C₃×C₂²⋊D₅	1	1	yes	characteristic
244	C₄²	C₂²×C₃×D₅	1	1	yes	characteristic
245	C₃×C₅	#1493	1	1	yes	characteristic, complemented, Hall
246	C₂²×C₃	—	2	1	no	—
247	C₂²×C₃	—	2	1	no	—
248	C₂²×C₃	#3967	1	2	yes	—
249	C₂²×C₃	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
250	C₃×C₄	—	2	1	no	—
251	C₃×C₄	—	2	1	no	—
252	C₃×C₄	—	4	2	no	—
253	C₃×C₄	—	4	2	no	—
254	C₃×C₄	#3967	1	2	yes	—
255	C₃×C₄	#3969	1	2	yes	—
256	C₃×C₄	C₂×#688	1	1	yes	characteristic
257	C₃×C₄	C₂×#689	1	1	yes	characteristic
258	C₃×C₄	#4094	1	2	yes	—
259	C₂×C₅	—	2	2	no	—
260	C₂×C₅	C₃×#408	1	1	yes	characteristic
261	C₂×C₅	C₂×C₃×#103	1	1	yes	characteristic
262	C₂×C₅	C₂×C₃×#104	1	1	yes	characteristic
263	C₂³	C₃×C₂²⋊Dic₅	1	1	yes	characteristic
264	Q₈	—	2	4	no	—
265	Q₈	—	4	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	C₃×#688	1	2	yes	—
271	C₂×C₄	C₃×#689	1	2	yes	—
272	C₂×C₄	C₃×C₂²⋊Dic₅	1	1	yes	characteristic
273	C₂×C₄	C₃×C₂²⋊Dic₅	1	1	yes	characteristic
274	C₂×C₄	C₃×C₂²⋊Dic₅	1	1	yes	characteristic
275	C₂×C₄	C₂²×C₃×Dic₅	1	1	yes	characteristic
276	C₂×C₄	C₂×C₃×C₂²⋊D₅	1	1	yes	characteristic
277	C₂×C₄	C₂×C₃×C₂²⋊D₅	1	1	yes	characteristic
278	C₈	—	10	4	no	—
279	C₈	—	20	2	no	—
280	C₂×C₃	—	2	2	no	—
281	C₂×C₃	C₂×#3967	1	1	yes	characteristic
282	C₂×C₃	#65523	1	1	yes	characteristic
283	C₂×C₃	C₂×#3969	1	1	yes	characteristic
284	C₅	C₃×#1493	1	1	yes	characteristic, complemented, Sylow
285	C₂²	—	2	1	no	—
286	C₂²	—	2	1	no	—
287	C₂²	C₃×#3967	1	2	yes	—
288	C₂²	C₂×C₃×C₂²⋊Dic₅	1	1	yes	characteristic
289	C₄	—	2	1	no	—
290	C₄	—	2	1	no	—
291	C₄	—	4	2	no	—
292	C₄	—	4	2	no	—
293	C₄	C₃×#3967	1	2	yes	—
294	C₄	C₃×#3969	1	2	yes	—
295	C₄	C₂×C₃×#688	1	1	yes	characteristic
296	C₄	C₂×C₃×#689	1	1	yes	characteristic
297	C₄	C₃×#4094	1	2	yes	—
298	C₃	#10602619	1	1	yes	characteristic, direct product factor, Sylow
299	C₂	—	2	2	no	—
300	C₂	C₂×C₃×#3967	1	1	yes	characteristic
301	C₂	C₃×#65523	1	1	yes	characteristic
302	C₂	C₂×C₃×#3969	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	C₂×C₂²⋊D₅
commutator	C₂×C₄×C₅	C₂²×C₃×C₄
Frattini	C₄²	C₂²×C₃×D₅
Fitting	C₃×C₅×#533	C₂
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₂³×C₃×C₅	C₂²⋊C₄
cosocle	C₄²×C₅	C₂³×C₃
hypercenter	C₃×#533	D₅
hypocenter	C₅	C₃×#1493
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂×C₃×C₄² ◁ C₃×#533

Complements

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

N	H ≅ G/N	type
4. C₃×Q₈⋊#168	C₂	semidirect
5. #10602619	C₃	direct (N × H)
25. Q₈⋊#168	C₂×C₃	semidirect
245. C₃×C₅	#1493	semidirect
284. C₅	C₃×#1493	semidirect
298. C₃	#10602619	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×C₅×#533	2
C₃×#65213	2
C₂×C₃×C₄×#168	2
C₃×Q₈⋊#168	2
#10602619	3
C₃×#1493	5

Subgroups — C3×#10602619 #14282412

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10602619 #14282412