Subgroups — C₃×#10602658 #14302786

Order 1920.

624 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#536	C₂	1	1	yes	maximal, characteristic
2	C₃×#66153	C₂	1	1	yes	maximal, characteristic
3	C₃×#65529	C₂	1	1	yes	maximal, characteristic
4	C₃×#65514	C₂	1	2	yes	maximal, complemented
5	C₃×#64751	C₂	1	2	yes	maximal, complemented
6	#10602658	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×C₅×#41	C₂²	1	1	yes	characteristic
8	C₂²×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic
9	C₃×C₅×C₄⋊Q₈	C₄	1	2	yes	complemented
10	C₃×C₅×#124	C₄	1	2	yes	complemented
11	C₃×C₅×#123	—	2	2	no	—
12	C₃×C₅×C₂²⋊Q₈	—	2	2	no	—
13	C₃×C₅×#117	C₂²	1	1	yes	characteristic
14	C₃×#4094	—	2	4	no	—
15	C₃×#4072	C₂²	1	2	yes	—
16	C₂×C₃×#681	—	2	2	no	—
17	C₂×C₃×#681	C₂²	1	2	yes	—
18	C₃×#3967	—	2	4	no	—
19	C₃×#3966	—	2	2	no	—
20	C₃×#1909	—	5	1	no	maximal, Hall
21	C₅×#536	C₂×C₃	1	1	yes	characteristic
22	#66153	C₂×C₃	1	1	yes	characteristic
23	#65529	C₂×C₃	1	1	yes	characteristic
24	#65514	C₂×C₃	1	2	yes	complemented
25	#64751	C₂×C₃	1	2	yes	complemented
26	C₃×C₅×#41	—	2	2	no	—
27	C₃×C₅×#41	—	2	2	no	—
28	C₂×C₃×C₅×Q₈	—	2	1	no	—
29	C₂×C₃×C₅×Q₈	—	2	2	no	—
30	C₂×C₃×C₅×Q₈	—	2	2	no	—
31	C₂×C₃×C₅×Q₈	C₂×C₄	1	2	yes	—
32	C₂×C₃×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
33	C₂×C₃×C₅×D₄	C₂×C₄	1	1	yes	characteristic
34	C₂×C₃×C₅×D₄	D₄	1	2	yes	—
35	C₂²×C₃×C₄×C₅	—	2	1	no	—
36	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
37	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
38	C₃×C₅×C₄⋊C₄	—	2	4	no	—
39	C₃×C₅×C₄⋊C₄	—	2	1	no	—
40	C₃×C₅×C₂²⋊C₄	—	2	4	no	—
41	C₃×C₅×C₂²⋊C₄	—	2	1	no	—
42	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
43	C₃×C₂²⋊Dic₅	—	2	2	no	—
44	C₃×C₄⋊Dic₅	—	2	2	no	—
45	C₃×C₄×Dic₅	—	2	2	no	—
46	C₃×#681	—	2	4	no	—
47	C₃×#681	—	2	2	no	—
48	C₃×#681	—	2	2	no	—
49	C₂×C₃×#168	—	2	4	no	—
50	C₂×C₃×#168	D₄	1	4	yes	—
51	C₃×#568	—	5	1	no	—
52	C₃×#536	D₅	1	1	yes	characteristic
53	C₃×#419	—	5	2	no	—
54	C₃×#410	—	5	1	no	—
55	C₃×H(3,ℤ/4)	—	5	2	no	—
56	C₂×C₅×#41	C₂²×C₃	1	1	yes	characteristic
57	C₂²×C₅×Q₈	C₂²×C₃	1	1	yes	characteristic
58	C₅×C₄⋊Q₈	C₃×C₄	1	2	yes	complemented
59	C₅×#124	C₃×C₄	1	2	yes	complemented
60	C₅×#123	—	2	2	no	—
61	C₅×C₂²⋊Q₈	—	2	2	no	—
62	C₅×#117	C₂²×C₃	1	1	yes	characteristic
63	#4094	—	2	4	no	—
64	#4072	C₂²×C₃	1	2	yes	—
65	C₂×#681	—	2	2	no	—
66	C₂×#681	C₂²×C₃	1	2	yes	—
67	#3967	—	2	4	no	—
68	#3966	—	2	2	no	—
69	#1909	—	5	1	no	Sylow
70	C₂³×C₃×C₅	C₂²⋊C₄	1	2	yes	—
71	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
72	C₃×C₅×Q₈	—	2	2	no	—
73	C₃×C₅×Q₈	—	2	4	no	—
74	C₃×C₅×Q₈	—	4	1	no	—
75	C₃×C₅×Q₈	—	4	1	no	—
76	C₃×C₅×D₄	—	2	4	no	—
77	C₃×C₅×D₄	—	2	2	no	—
78	C₂×C₃×C₄×C₅	—	2	2	no	—
79	C₂×C₃×C₄×C₅	—	2	2	no	—
80	C₂×C₃×C₄×C₅	—	2	2	no	—
81	C₂×C₃×C₄×C₅	—	2	2	no	—
82	C₂×C₃×C₄×C₅	—	4	2	no	—
83	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
84	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
86	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
87	C₂×C₃×C₄×C₅	#41	1	2	yes	—
88	C₂×C₃×Dic₅	—	4	2	no	—
89	C₃×#168	—	2	4	no	—
90	C₂×C₃×#41	C₂×D₅	1	1	yes	characteristic
91	C₂²×C₃×Q₈	C₂×D₅	1	1	yes	characteristic
92	C₃×#131	—	10	4	no	—
93	C₂×C₃×M₁₆	—	5	2	no	—
94	C₂×C₃×M₁₆	—	10	2	no	—
95	C₃×C₄⋊Q₈	Dic₅	1	2	yes	complemented
96	C₃×#124	Dic₅	1	2	yes	complemented
97	C₃×#123	—	2	2	no	—
98	C₃×C₂²⋊Q₈	—	2	2	no	—
99	C₃×#117	—	5	2	no	—
100	C₃×#117	C₂×D₅	1	1	yes	characteristic
101	C₃×#103	—	10	4	no	—
102	C₃×#99	—	10	2	no	—
103	C₅×#41	—	2	2	no	—
104	C₅×#41	—	2	2	no	—
105	C₂×C₅×Q₈	—	2	1	no	—
106	C₂×C₅×Q₈	—	2	2	no	—
107	C₂×C₅×Q₈	—	2	2	no	—
108	C₂×C₅×Q₈	C₂×C₃×C₄	1	2	yes	—
109	C₂×C₅×Q₈	C₂×C₃×C₄	1	1	yes	characteristic
110	C₂×C₅×D₄	C₂×C₃×C₄	1	1	yes	characteristic
111	C₂×C₅×D₄	C₃×D₄	1	2	yes	—
112	C₂²×C₄×C₅	—	2	1	no	—
113	C₂²×C₄×C₅	C₃×D₄	1	2	yes	—
114	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
115	C₅×C₄⋊C₄	—	2	4	no	—
116	C₅×C₄⋊C₄	—	2	1	no	—
117	C₅×C₂²⋊C₄	—	2	4	no	—
118	C₅×C₂²⋊C₄	—	2	1	no	—
119	C₄²×C₅	C₂×C₃×C₄	1	2	yes	—
120	C₂²⋊Dic₅	—	2	2	no	—
121	C₄⋊Dic₅	—	2	2	no	—
122	C₄×Dic₅	—	2	2	no	—
123	#681	—	2	4	no	—
124	#681	—	2	2	no	—
125	#681	—	2	2	no	—
126	C₂×#168	—	2	4	no	—
127	C₂×#168	C₃×D₄	1	4	yes	—
128	#568	—	5	1	no	—
129	#536	C₃×D₅	1	1	yes	characteristic
130	#419	—	5	2	no	—
131	#410	—	5	1	no	—
132	H(3,ℤ/4)	—	5	2	no	—
133	C₂²×C₃×C₅	—	2	2	no	—
134	C₂²×C₃×C₅	—	4	1	no	—
135	C₂²×C₃×C₅	—	4	2	no	—
136	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
137	C₂²×C₃×C₅	#121	1	2	yes	—
138	C₃×C₄×C₅	—	2	2	no	—
139	C₃×C₄×C₅	—	4	2	no	—
140	C₃×C₄×C₅	—	4	2	no	—
141	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
142	C₃×C₄×C₅	#120	1	1	yes	characteristic
143	C₃×C₄×C₅	#123	1	1	yes	characteristic
144	C₃×Dic₅	—	8	2	no	—
145	C₃×#41	—	2	2	no	—
146	C₃×#41	—	2	2	no	—
147	C₂×C₃×Q₈	—	2	1	no	—
148	C₂×C₃×Q₈	—	2	2	no	—
149	C₂×C₃×Q₈	—	2	2	no	—
150	C₂×C₃×Q₈	C₂×Dic₅	1	2	yes	—
151	C₂×C₃×Q₈	C₂×Dic₅	1	1	yes	characteristic
152	C₂×C₃×D₄	C₂×Dic₅	1	1	yes	characteristic
153	C₂×C₃×D₄	C₂²⋊D₅	1	2	yes	—
154	C₂²×C₃×C₄	—	2	1	no	—
155	C₂²×C₃×C₄	C₂²⋊D₅	1	2	yes	—
156	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
157	C₃×M₁₆	—	10	4	no	—
158	C₃×M₁₆	—	10	2	no	—
159	C₃×M₁₆	—	10	2	no	—
160	C₂×C₃×C₈	—	5	4	no	—
161	C₂×C₃×C₈	—	10	4	no	—
162	C₃×C₄⋊C₄	—	2	4	no	—
163	C₃×C₄⋊C₄	—	2	1	no	—
164	C₃×C₄⋊C₄	—	10	2	no	—
165	C₃×C₂²⋊C₄	—	2	4	no	—
166	C₃×C₂²⋊C₄	—	2	1	no	—
167	C₃×C₂²⋊C₄	—	10	2	no	—
168	C₃×C₄²	—	10	2	no	—
169	C₃×C₄²	C₂×Dic₅	1	2	yes	—
170	C₂³×C₅	C₃×C₂²⋊C₄	1	2	yes	—
171	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
172	C₅×Q₈	—	2	2	no	—
173	C₅×Q₈	—	2	4	no	—
174	C₅×Q₈	—	4	1	no	—
175	C₅×Q₈	—	4	1	no	—
176	C₅×D₄	—	2	4	no	—
177	C₅×D₄	—	2	2	no	—
178	C₂×C₄×C₅	—	2	2	no	—
179	C₂×C₄×C₅	—	2	2	no	—
180	C₂×C₄×C₅	—	2	2	no	—
181	C₂×C₄×C₅	—	2	2	no	—
182	C₂×C₄×C₅	—	4	2	no	—
183	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	2	yes	—
184	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
185	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
186	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
187	C₂×C₄×C₅	C₃×#41	1	2	yes	—
188	C₂×Dic₅	—	4	2	no	—
189	#168	—	2	4	no	—
190	C₂×#41	C₂×C₃×D₅	1	1	yes	characteristic
191	C₂²×Q₈	C₂×C₃×D₅	1	1	yes	characteristic
192	#131	—	10	4	no	—
193	C₂×M₁₆	—	5	2	no	—
194	C₂×M₁₆	—	10	2	no	—
195	C₄⋊Q₈	C₃×Dic₅	1	2	yes	complemented
196	#124	C₃×Dic₅	1	2	yes	complemented
197	#123	—	2	2	no	—
198	C₂²⋊Q₈	—	2	2	no	—
199	#117	—	5	2	no	—
200	#117	C₂×C₃×D₅	1	1	yes	characteristic
201	#103	—	10	4	no	—
202	#99	—	10	2	no	—
203	C₂×C₃×C₅	—	2	2	no	—
204	C₂×C₃×C₅	—	2	1	no	—
205	C₂×C₃×C₅	—	4	2	no	—
206	C₂×C₃×C₅	#386	1	1	yes	characteristic
207	C₂³×C₃	C₂²⋊Dic₅	1	2	yes	—
208	C₂³×C₃	C₂×C₂²⋊D₅	1	1	yes	characteristic
209	C₃×Q₈	—	2	2	no	—
210	C₃×Q₈	—	2	4	no	—
211	C₃×Q₈	—	4	1	no	—
212	C₃×Q₈	—	4	1	no	—
213	C₃×D₄	—	2	4	no	—
214	C₃×D₄	—	2	2	no	—
215	C₂×C₃×C₄	—	2	2	no	—
216	C₂×C₃×C₄	—	2	2	no	—
217	C₂×C₃×C₄	—	2	2	no	—
218	C₂×C₃×C₄	—	2	2	no	—
219	C₂×C₃×C₄	—	4	2	no	—
220	C₂×C₃×C₄	—	20	2	no	—
221	C₂×C₃×C₄	C₂²⋊Dic₅	1	2	yes	—
222	C₂×C₃×C₄	D₄×D₅	1	2	yes	—
223	C₂×C₃×C₄	#711	1	2	yes	—
224	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
225	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
226	C₃×C₈	—	10	4	no	—
227	C₂²×C₅	—	2	2	no	—
228	C₂²×C₅	—	4	1	no	—
229	C₂²×C₅	—	4	2	no	—
230	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
231	C₂²×C₅	C₃×#121	1	2	yes	—
232	C₄×C₅	—	2	2	no	—
233	C₄×C₅	—	4	2	no	—
234	C₄×C₅	—	4	2	no	—
235	C₄×C₅	C₃×C₄×D₄	1	2	yes	—
236	C₄×C₅	C₃×#120	1	1	yes	characteristic
237	C₄×C₅	C₃×#123	1	1	yes	characteristic
238	Dic₅	—	8	2	no	—
239	#41	—	2	2	no	—
240	#41	—	2	2	no	—
241	C₂×Q₈	—	2	1	no	—
242	C₂×Q₈	—	2	2	no	—
243	C₂×Q₈	—	2	2	no	—
244	C₂×Q₈	C₂×C₃×Dic₅	1	2	yes	—
245	C₂×Q₈	C₂×C₃×Dic₅	1	1	yes	characteristic
246	C₂×D₄	C₂×C₃×Dic₅	1	1	yes	characteristic
247	C₂×D₄	C₃×C₂²⋊D₅	1	2	yes	—
248	C₂²×C₄	—	2	1	no	—
249	C₂²×C₄	C₃×C₂²⋊D₅	1	2	yes	—
250	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
251	M₁₆	—	10	4	no	—
252	M₁₆	—	10	2	no	—
253	M₁₆	—	10	2	no	—
254	C₂×C₈	—	5	4	no	—
255	C₂×C₈	—	10	4	no	—
256	C₄⋊C₄	—	2	4	no	—
257	C₄⋊C₄	—	2	1	no	—
258	C₄⋊C₄	—	10	2	no	—
259	C₂²⋊C₄	—	2	4	no	—
260	C₂²⋊C₄	—	2	1	no	—
261	C₂²⋊C₄	—	10	2	no	—
262	C₄²	—	10	2	no	—
263	C₄²	C₂×C₃×Dic₅	1	2	yes	—
264	C₃×C₅	#1909	1	1	yes	characteristic, complemented, Hall
265	C₂²×C₃	—	2	2	no	—
266	C₂²×C₃	—	4	1	no	—
267	C₂²×C₃	—	4	2	no	—
268	C₂²×C₃	#4085	1	2	yes	—
269	C₂²×C₃	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
270	C₃×C₄	—	2	2	no	—
271	C₃×C₄	—	4	2	no	—
272	C₃×C₄	—	4	2	no	—
273	C₃×C₄	—	40	2	no	—
274	C₃×C₄	D₄×Dic₅	1	2	yes	—
275	C₃×C₄	#4081	1	1	yes	characteristic
276	C₃×C₄	#4083	1	1	yes	characteristic
277	C₂×C₅	—	2	2	no	—
278	C₂×C₅	—	2	1	no	—
279	C₂×C₅	—	4	2	no	—
280	C₂×C₅	C₃×#386	1	1	yes	characteristic
281	C₂³	C₃×C₂²⋊Dic₅	1	2	yes	—
282	C₂³	C₂×C₃×C₂²⋊D₅	1	1	yes	characteristic
283	Q₈	—	2	2	no	—
284	Q₈	—	2	4	no	—
285	Q₈	—	4	1	no	—
286	Q₈	—	4	1	no	—
287	D₄	—	2	4	no	—
288	D₄	—	2	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	20	2	no	—
295	C₂×C₄	C₃×C₂²⋊Dic₅	1	2	yes	—
296	C₂×C₄	C₃×D₄×D₅	1	2	yes	—
297	C₂×C₄	C₃×#711	1	2	yes	—
298	C₂×C₄	C₂²×C₃×Dic₅	1	1	yes	characteristic
299	C₂×C₄	C₂×C₃×C₂²⋊D₅	1	1	yes	characteristic
300	C₈	—	10	4	no	—
301	C₂×C₃	—	2	1	no	—
302	C₂×C₃	—	2	2	no	—
303	C₂×C₃	—	4	2	no	—
304	C₂×C₃	#65509	1	1	yes	characteristic
305	C₅	C₃×#1909	1	1	yes	characteristic, complemented, Sylow
306	C₂²	—	2	2	no	—
307	C₂²	—	4	1	no	—
308	C₂²	—	4	2	no	—
309	C₂²	C₃×#4085	1	2	yes	—
310	C₂²	C₂×C₃×C₂²⋊Dic₅	1	1	yes	characteristic
311	C₄	—	2	2	no	—
312	C₄	—	4	2	no	—
313	C₄	—	4	2	no	—
314	C₄	—	40	2	no	—
315	C₄	C₃×D₄×Dic₅	1	2	yes	—
316	C₄	C₃×#4081	1	1	yes	characteristic
317	C₄	C₃×#4083	1	1	yes	characteristic
318	C₃	#10602658	1	1	yes	characteristic, direct product factor, Sylow
319	C₂	—	2	1	no	—
320	C₂	—	2	2	no	—
321	C₂	—	4	2	no	—
322	C₂	C₃×#65509	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃	#65509
commutator	C₂×C₄×C₅	C₂²×C₃×C₄
Frattini	C₂²×C₄	C₂²×C₃×D₅
Fitting	C₃×C₅×#536	C₂
socle	C₂×C₃×C₅	#386
prime radical	C₂×C₃×C₅×#41	C₂²
cosocle	C₂²×C₄×C₅	C₂³×C₃
hypercenter	C₃×#536	D₅
hypocenter	C₅	C₃×#1909
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃ ◁ C₂²×C₃×C₄ ◁ C₃×#536

Complements

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

N	H ≅ G/N	type
4. C₃×#65514	C₂	semidirect
5. C₃×#64751	C₂	semidirect
6. #10602658	C₃	direct (N × H)
9. C₃×C₅×C₄⋊Q₈	C₄	semidirect
10. C₃×C₅×#124	C₄	semidirect
24. #65514	C₂×C₃	semidirect
25. #64751	C₂×C₃	semidirect
58. C₅×C₄⋊Q₈	C₃×C₄	semidirect
59. C₅×#124	C₃×C₄	semidirect
95. C₃×C₄⋊Q₈	Dic₅	semidirect
96. C₃×#124	Dic₅	semidirect
195. C₄⋊Q₈	C₃×Dic₅	semidirect
196. #124	C₃×Dic₅	semidirect
264. C₃×C₅	#1909	semidirect
305. C₅	C₃×#1909	semidirect
318. C₃	#10602658	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×#536	2
C₃×#66153	2
C₃×#65529	2
C₃×#65514	2
C₃×#64751	2
#10602658	3
C₃×#1909	5

Subgroups — C3×#10602658 #14302786

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10602658 #14302786