Subgroups — C₃×#10602516 #14301442

Order 1920.

720 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	C₂×C₃×C₅×#124	C₂	1	1	yes	maximal, characteristic
2	C₃×#66132	C₂	1	1	yes	maximal, characteristic
3	C₃×#65514	C₂	1	2	yes	maximal, complemented
4	C₂×C₃×#3965	C₂	1	1	yes	maximal, characteristic
5	C₃×#64743	C₂	1	2	yes	maximal, complemented
6	#10602516	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂²×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic
8	C₂²×C₃×C₅×D₄	C₂²	1	1	yes	characteristic
9	C₃×C₅×#124	—	2	2	no	—
10	C₃×C₅×#124	C₄	1	4	yes	complemented
11	C₂×C₃×C₅×C₂²⋊C₄	—	2	2	no	—
12	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
13	C₂×C₃×C₂²⋊Dic₅	—	2	2	no	—
14	C₃×D₄×Dic₅	—	2	4	no	—
15	C₃×#4072	C₂²	1	2	yes	—
16	C₂×C₃×#681	C₂²	1	2	yes	—
17	C₃×#3967	—	2	4	no	—
18	C₃×#3965	—	2	2	no	—
19	C₃×#1885	—	5	1	no	maximal, Hall
20	C₂×C₅×#124	C₂×C₃	1	1	yes	characteristic
21	#66132	C₂×C₃	1	1	yes	characteristic
22	#65514	C₂×C₃	1	2	yes	complemented
23	C₂×#3965	C₂×C₃	1	1	yes	characteristic
24	#64743	C₂×C₃	1	2	yes	complemented
25	C₂⁴×C₃×C₅	D₄	1	2	yes	—
26	C₂×C₃×C₅×Q₈	—	2	1	no	—
27	C₂×C₃×C₅×Q₈	—	2	2	no	—
28	C₂×C₃×C₅×Q₈	—	2	2	no	—
29	C₂×C₃×C₅×Q₈	C₂×C₄	1	2	yes	complemented
30	C₂×C₃×C₅×D₄	—	2	2	no	—
31	C₂×C₃×C₅×D₄	—	2	2	no	—
32	C₂×C₃×C₅×D₄	C₂×C₄	1	2	yes	—
33	C₂×C₃×C₅×D₄	D₄	1	2	yes	—
34	C₂²×C₃×C₄×C₅	—	2	1	no	—
35	C₂²×C₃×C₄×C₅	—	2	1	no	—
36	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
37	C₂²×C₃×Dic₅	—	2	4	no	—
38	C₃×C₅×C₂²⋊C₄	—	2	8	no	—
39	C₃×C₄²×C₅	—	2	1	no	—
40	C₃×C₄²×C₅	C₂×C₄	1	2	yes	complemented
41	C₃×C₂²⋊Dic₅	—	2	4	no	—
42	C₃×C₂²⋊Dic₅	—	2	2	no	—
43	C₃×C₄⋊Dic₅	D₄	1	4	yes	—
44	C₃×C₄×Dic₅	—	2	2	no	—
45	C₃×#681	—	2	2	no	—
46	C₃×#681	—	2	2	no	—
47	C₂×C₃×#168	—	2	2	no	—
48	C₂×C₃×#124	D₅	1	1	yes	characteristic
49	C₃×#518	—	5	1	no	—
50	C₃×#419	—	5	2	no	—
51	C₂×C₃×#100	—	5	1	no	—
52	C₃×#339	—	5	2	no	—
53	C₂²×C₅×Q₈	C₂²×C₃	1	1	yes	characteristic
54	C₂²×C₅×D₄	C₂²×C₃	1	1	yes	characteristic
55	C₅×#124	—	2	2	no	—
56	C₅×#124	C₃×C₄	1	4	yes	complemented
57	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
58	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
59	C₂×C₂²⋊Dic₅	—	2	2	no	—
60	D₄×Dic₅	—	2	4	no	—
61	#4072	C₂²×C₃	1	2	yes	—
62	C₂×#681	C₂²×C₃	1	2	yes	—
63	#3967	—	2	4	no	—
64	#3965	—	2	2	no	—
65	#1885	—	5	1	no	Sylow
66	C₂³×C₃×C₅	—	2	2	no	—
67	C₂³×C₃×C₅	—	2	2	no	—
68	C₂³×C₃×C₅	—	4	4	no	—
69	C₂³×C₃×C₅	C₂²⋊C₄	1	4	yes	—
70	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
71	C₃×C₅×Q₈	—	2	4	no	—
72	C₃×C₅×Q₈	—	4	2	no	—
73	C₃×C₅×D₄	—	2	4	no	—
74	C₃×C₅×D₄	—	2	4	no	—
75	C₂×C₃×C₄×C₅	—	2	2	no	—
76	C₂×C₃×C₄×C₅	—	2	2	no	—
77	C₂×C₃×C₄×C₅	—	4	2	no	—
78	C₂×C₃×C₄×C₅	—	4	1	no	—
79	C₂×C₃×C₄×C₅	—	4	1	no	—
80	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
82	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
83	C₂×C₃×C₄×C₅	#41	1	2	yes	—
84	C₂×C₃×Dic₅	—	2	4	no	—
85	C₂×C₃×Dic₅	—	8	4	no	—
86	C₃×#168	—	4	2	no	—
87	C₂²×C₃×Q₈	C₂×D₅	1	1	yes	characteristic
88	C₂²×C₃×D₄	C₂×D₅	1	1	yes	characteristic
89	C₂×C₃×M₁₆	—	5	2	no	—
90	C₃×#124	—	2	2	no	—
91	C₃×#124	Dic₅	1	4	yes	complemented
92	C₃×C₄×D₄	—	10	4	no	—
93	C₃×#117	—	5	2	no	—
94	C₂×C₃×C₂²⋊C₄	—	2	2	no	—
95	C₂×C₃×C₂²⋊C₄	—	10	2	no	—
96	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
97	C₃×#103	—	10	4	no	—
98	C₃×#100	—	10	2	no	—
99	C₂⁴×C₅	C₃×D₄	1	2	yes	—
100	C₂×C₅×Q₈	—	2	1	no	—
101	C₂×C₅×Q₈	—	2	2	no	—
102	C₂×C₅×Q₈	—	2	2	no	—
103	C₂×C₅×Q₈	C₂×C₃×C₄	1	2	yes	complemented
104	C₂×C₅×D₄	—	2	2	no	—
105	C₂×C₅×D₄	—	2	2	no	—
106	C₂×C₅×D₄	C₂×C₃×C₄	1	2	yes	—
107	C₂×C₅×D₄	C₃×D₄	1	2	yes	—
108	C₂²×C₄×C₅	—	2	1	no	—
109	C₂²×C₄×C₅	—	2	1	no	—
110	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
111	C₂²×Dic₅	—	2	4	no	—
112	C₅×C₂²⋊C₄	—	2	8	no	—
113	C₄²×C₅	—	2	1	no	—
114	C₄²×C₅	C₂×C₃×C₄	1	2	yes	complemented
115	C₂²⋊Dic₅	—	2	4	no	—
116	C₂²⋊Dic₅	—	2	2	no	—
117	C₄⋊Dic₅	C₃×D₄	1	4	yes	—
118	C₄×Dic₅	—	2	2	no	—
119	#681	—	2	2	no	—
120	#681	—	2	2	no	—
121	C₂×#168	—	2	2	no	—
122	C₂×#124	C₃×D₅	1	1	yes	characteristic
123	#518	—	5	1	no	—
124	#419	—	5	2	no	—
125	C₂×#100	—	5	1	no	—
126	#339	—	5	2	no	—
127	C₂²×C₃×C₅	—	2	2	no	—
128	C₂²×C₃×C₅	—	2	4	no	—
129	C₂²×C₃×C₅	—	4	4	no	—
130	C₂²×C₃×C₅	—	4	4	no	—
131	C₂²×C₃×C₅	—	8	2	no	—
132	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
133	C₂²×C₃×C₅	#120	1	1	yes	characteristic
134	C₂²×C₃×C₅	#123	1	1	yes	characteristic
135	C₃×C₄×C₅	—	4	2	no	—
136	C₃×C₄×C₅	—	4	2	no	—
137	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
138	C₃×C₄×C₅	#121	1	2	yes	—
139	C₃×Dic₅	—	4	4	no	—
140	C₂⁴×C₃	C₂²⋊D₅	1	2	yes	—
141	C₂×C₃×Q₈	—	2	1	no	—
142	C₂×C₃×Q₈	—	2	2	no	—
143	C₂×C₃×Q₈	—	2	2	no	—
144	C₂×C₃×Q₈	C₂×Dic₅	1	2	yes	complemented
145	C₂×C₃×D₄	—	2	2	no	—
146	C₂×C₃×D₄	—	2	2	no	—
147	C₂×C₃×D₄	C₂×Dic₅	1	2	yes	—
148	C₂×C₃×D₄	C₂²⋊D₅	1	2	yes	—
149	C₂²×C₃×C₄	—	2	1	no	—
150	C₂²×C₃×C₄	—	2	1	no	—
151	C₂²×C₃×C₄	—	10	4	no	—
152	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
153	C₃×M₁₆	—	10	2	no	—
154	C₃×M₁₆	—	10	2	no	—
155	C₂×C₃×C₈	—	10	2	no	—
156	C₃×C₄⋊C₄	—	5	4	no	—
157	C₃×C₂²⋊C₄	—	2	8	no	—
158	C₃×C₂²⋊C₄	—	10	4	no	—
159	C₃×C₂²⋊C₄	—	10	2	no	—
160	C₃×C₄²	—	2	1	no	—
161	C₃×C₄²	—	10	2	no	—
162	C₃×C₄²	C₂×Dic₅	1	2	yes	complemented
163	C₂³×C₅	—	2	2	no	—
164	C₂³×C₅	—	2	2	no	—
165	C₂³×C₅	—	4	4	no	—
166	C₂³×C₅	C₃×C₂²⋊C₄	1	4	yes	—
167	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
168	C₅×Q₈	—	2	4	no	—
169	C₅×Q₈	—	4	2	no	—
170	C₅×D₄	—	2	4	no	—
171	C₅×D₄	—	2	4	no	—
172	C₂×C₄×C₅	—	2	2	no	—
173	C₂×C₄×C₅	—	2	2	no	—
174	C₂×C₄×C₅	—	4	2	no	—
175	C₂×C₄×C₅	—	4	1	no	—
176	C₂×C₄×C₅	—	4	1	no	—
177	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
178	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
179	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
180	C₂×C₄×C₅	C₃×#41	1	2	yes	—
181	C₂×Dic₅	—	2	4	no	—
182	C₂×Dic₅	—	8	4	no	—
183	#168	—	4	2	no	—
184	C₂²×Q₈	C₂×C₃×D₅	1	1	yes	characteristic
185	C₂²×D₄	C₂×C₃×D₅	1	1	yes	characteristic
186	C₂×M₁₆	—	5	2	no	—
187	#124	—	2	2	no	—
188	#124	C₃×Dic₅	1	4	yes	complemented
189	C₄×D₄	—	10	4	no	—
190	#117	—	5	2	no	—
191	C₂×C₂²⋊C₄	—	2	2	no	—
192	C₂×C₂²⋊C₄	—	10	2	no	—
193	C₂×C₄²	C₂×C₃×D₅	1	1	yes	characteristic
194	#103	—	10	4	no	—
195	#100	—	10	2	no	—
196	C₂×C₃×C₅	—	2	1	no	—
197	C₂×C₃×C₅	—	2	1	no	—
198	C₂×C₃×C₅	—	4	4	no	—
199	C₂×C₃×C₅	#386	1	1	yes	characteristic
200	C₂×C₃×C₅	#455	1	2	yes	—
201	C₂³×C₃	—	2	2	no	—
202	C₂³×C₃	—	2	2	no	—
203	C₂³×C₃	—	4	4	no	—
204	C₂³×C₃	C₂²⋊Dic₅	1	4	yes	—
205	C₂³×C₃	C₂×C₂²⋊D₅	1	1	yes	characteristic
206	C₃×Q₈	—	2	4	no	—
207	C₃×Q₈	—	4	2	no	—
208	C₃×D₄	—	2	4	no	—
209	C₃×D₄	—	2	4	no	—
210	C₂×C₃×C₄	—	2	2	no	—
211	C₂×C₃×C₄	—	2	2	no	—
212	C₂×C₃×C₄	—	4	2	no	—
213	C₂×C₃×C₄	—	4	1	no	—
214	C₂×C₃×C₄	—	4	1	no	—
215	C₂×C₃×C₄	—	10	4	no	—
216	C₂×C₃×C₄	—	40	4	no	—
217	C₂×C₃×C₄	D₄×D₅	1	2	yes	—
218	C₂×C₃×C₄	#711	1	2	yes	—
219	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
220	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
221	C₃×C₈	—	20	2	no	—
222	C₂²×C₅	—	2	2	no	—
223	C₂²×C₅	—	2	4	no	—
224	C₂²×C₅	—	4	4	no	—
225	C₂²×C₅	—	4	4	no	—
226	C₂²×C₅	—	8	2	no	—
227	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
228	C₂²×C₅	C₃×#120	1	1	yes	characteristic
229	C₂²×C₅	C₃×#123	1	1	yes	characteristic
230	C₄×C₅	—	4	2	no	—
231	C₄×C₅	—	4	2	no	—
232	C₄×C₅	C₃×C₄×D₄	1	2	yes	—
233	C₄×C₅	C₃×#121	1	2	yes	—
234	Dic₅	—	4	4	no	—
235	C₂⁴	C₃×C₂²⋊D₅	1	2	yes	—
236	C₂×Q₈	—	2	1	no	—
237	C₂×Q₈	—	2	2	no	—
238	C₂×Q₈	—	2	2	no	—
239	C₂×Q₈	C₂×C₃×Dic₅	1	2	yes	complemented
240	C₂×D₄	—	2	2	no	—
241	C₂×D₄	—	2	2	no	—
242	C₂×D₄	C₂×C₃×Dic₅	1	2	yes	—
243	C₂×D₄	C₃×C₂²⋊D₅	1	2	yes	—
244	C₂²×C₄	—	2	1	no	—
245	C₂²×C₄	—	2	1	no	—
246	C₂²×C₄	—	10	4	no	—
247	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
248	M₁₆	—	10	2	no	—
249	M₁₆	—	10	2	no	—
250	C₂×C₈	—	10	2	no	—
251	C₄⋊C₄	—	5	4	no	—
252	C₂²⋊C₄	—	2	8	no	—
253	C₂²⋊C₄	—	10	4	no	—
254	C₂²⋊C₄	—	10	2	no	—
255	C₄²	—	2	1	no	—
256	C₄²	—	10	2	no	—
257	C₄²	C₂×C₃×Dic₅	1	2	yes	complemented
258	C₃×C₅	#1885	1	1	yes	characteristic, complemented, Hall
259	C₂²×C₃	—	2	2	no	—
260	C₂²×C₃	—	2	4	no	—
261	C₂²×C₃	—	4	4	no	—
262	C₂²×C₃	—	4	4	no	—
263	C₂²×C₃	—	8	2	no	—
264	C₂²×C₃	#4081	1	1	yes	characteristic
265	C₂²×C₃	#4083	1	1	yes	characteristic
266	C₂²×C₃	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
267	C₃×C₄	—	4	2	no	—
268	C₃×C₄	—	4	2	no	—
269	C₃×C₄	—	20	4	no	—
270	C₃×C₄	D₄×Dic₅	1	2	yes	—
271	C₃×C₄	#4085	1	2	yes	—
272	C₂×C₅	—	2	1	no	—
273	C₂×C₅	—	2	1	no	—
274	C₂×C₅	—	4	4	no	—
275	C₂×C₅	C₃×#386	1	1	yes	characteristic
276	C₂×C₅	C₃×#455	1	2	yes	—
277	C₂³	—	2	2	no	—
278	C₂³	—	2	2	no	—
279	C₂³	—	4	4	no	—
280	C₂³	C₃×C₂²⋊Dic₅	1	4	yes	—
281	C₂³	C₂×C₃×C₂²⋊D₅	1	1	yes	characteristic
282	Q₈	—	2	4	no	—
283	Q₈	—	4	2	no	—
284	D₄	—	2	4	no	—
285	D₄	—	2	4	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	4	1	no	—
290	C₂×C₄	—	4	1	no	—
291	C₂×C₄	—	10	4	no	—
292	C₂×C₄	—	40	4	no	—
293	C₂×C₄	C₃×D₄×D₅	1	2	yes	—
294	C₂×C₄	C₃×#711	1	2	yes	—
295	C₂×C₄	C₂²×C₃×Dic₅	1	1	yes	characteristic
296	C₂×C₄	C₂×C₃×C₂²⋊D₅	1	1	yes	characteristic
297	C₈	—	20	2	no	—
298	C₂×C₃	—	2	1	no	—
299	C₂×C₃	—	2	1	no	—
300	C₂×C₃	—	4	4	no	—
301	C₂×C₃	#65350	1	2	yes	—
302	C₂×C₃	#65509	1	1	yes	characteristic
303	C₅	C₃×#1885	1	1	yes	characteristic, complemented, Sylow
304	C₂²	—	2	2	no	—
305	C₂²	—	2	4	no	—
306	C₂²	—	4	4	no	—
307	C₂²	—	4	4	no	—
308	C₂²	—	8	2	no	—
309	C₂²	C₃×#4081	1	1	yes	characteristic
310	C₂²	C₃×#4083	1	1	yes	characteristic
311	C₂²	C₂×C₃×C₂²⋊Dic₅	1	1	yes	characteristic
312	C₄	—	4	2	no	—
313	C₄	—	4	2	no	—
314	C₄	—	20	4	no	—
315	C₄	C₃×D₄×Dic₅	1	2	yes	—
316	C₄	C₃×#4085	1	2	yes	—
317	C₃	#10602516	1	1	yes	characteristic, direct product factor, Sylow
318	C₂	—	2	1	no	—
319	C₂	—	2	1	no	—
320	C₂	—	4	4	no	—
321	C₂	C₃×#65350	1	2	yes	—
322	C₂	C₃×#65509	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	#4083
commutator	C₂×C₄×C₅	C₂²×C₃×C₄
Frattini	C₂²×C₄	C₂²×C₃×D₅
Fitting	C₂×C₃×C₅×#124	C₂
socle	C₂²×C₃×C₅	#120
prime radical	C₂²×C₃×C₅×D₄	C₂²
cosocle	C₂²×C₄×C₅	C₂³×C₃
hypercenter	C₂×C₃×#124	D₅
hypocenter	C₅	C₃×#1885
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₃×C₄ ◁ C₂×C₃×#124

Complements

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

N	H ≅ G/N	type
3. C₃×#65514	C₂	semidirect
5. C₃×#64743	C₂	semidirect
6. #10602516	C₃	direct (N × H)
10. C₃×C₅×#124	C₄	semidirect
22. #65514	C₂×C₃	semidirect
24. #64743	C₂×C₃	semidirect
29. C₂×C₃×C₅×Q₈	C₂×C₄	semidirect
40. C₃×C₄²×C₅	C₂×C₄	semidirect
56. C₅×#124	C₃×C₄	semidirect
91. C₃×#124	Dic₅	semidirect
103. C₂×C₅×Q₈	C₂×C₃×C₄	semidirect
114. C₄²×C₅	C₂×C₃×C₄	semidirect
144. C₂×C₃×Q₈	C₂×Dic₅	semidirect
162. C₃×C₄²	C₂×Dic₅	semidirect
188. #124	C₃×Dic₅	semidirect
239. C₂×Q₈	C₂×C₃×Dic₅	semidirect
257. C₄²	C₂×C₃×Dic₅	semidirect
258. C₃×C₅	#1885	semidirect
303. C₅	C₃×#1885	semidirect
317. C₃	#10602516	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×#124	2
C₃×#66132	2
C₃×#65514	2
C₂×C₃×#3965	2
C₃×#64743	2
#10602516	3
C₃×#1885	5

Subgroups — C3×#10602516 #14301442

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10602516 #14301442