Subgroups — C₃×#10610989 #14418117

Order 1920.

916 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#66082	C₂	1	1	yes	maximal, characteristic
2	C₃×#66079	C₂	1	2	yes	maximal, complemented
3	C₃×C₅×#430	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃×#3957	C₂	1	2	yes	maximal, complemented
5	C₃×M₁₆×Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#65184	C₂	1	4	yes	maximal, complemented
7	C₃×#65183	C₂	1	4	yes	maximal, complemented
8	#10610989	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂×C₃×#709	C₂²	1	2	yes	—
10	C₂×C₃×C₅×M₁₆	C₂²	1	2	yes	—
11	C₂×C₃×C₅×M₁₆	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₄×Dic₅	C₂²	1	1	yes	characteristic
13	C₂×C₃×#681	C₂²	1	1	yes	characteristic
14	C₃×#4055	C₄	1	4	yes	complemented
15	C₃×#4054	C₄	1	4	yes	complemented
16	C₃×#4053	C₂²	1	8	yes	complemented
17	C₃×D₅×M₁₆	—	2	4	no	—
18	C₃×#4050	C₂²	1	4	yes	—
19	C₃×#4047	—	2	4	no	—
20	C₃×C₅×#108	C₂²	1	4	yes	complemented
21	C₃×#3957	C₂²	1	4	yes	complemented
22	C₃×#3957	C₂²	1	4	yes	complemented
23	C₃×#3947	C₂²	1	2	yes	complemented
24	C₃×C₈×Dic₅	C₂²	1	2	yes	complemented
25	C₃×#2956	—	5	1	no	maximal, Hall
26	#66082	C₂×C₃	1	1	yes	characteristic
27	#66079	C₂×C₃	1	2	yes	complemented
28	C₅×#430	C₂×C₃	1	1	yes	characteristic, complemented
29	C₂×#3957	C₂×C₃	1	2	yes	complemented
30	M₁₆×Dic₅	C₂×C₃	1	1	yes	characteristic, complemented
31	#65184	C₂×C₃	1	4	yes	complemented
32	#65183	C₂×C₃	1	4	yes	complemented
33	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
34	C₂×C₃×C₂²⋊D₅	—	2	2	no	—
35	C₂²×C₃×Dic₅	—	2	1	no	—
36	C₃×#709	—	2	2	no	—
37	C₃×#709	—	2	2	no	—
38	C₃×#709	C₂×C₄	1	4	yes	—
39	C₃×#709	C₂³	1	4	yes	—
40	C₂×C₃×D₂₀	C₂×C₄	1	2	yes	—
41	C₂×C₃×C₄×D₅	—	2	2	no	—
42	C₂×C₃×Dic₁₀	C₂×C₄	1	2	yes	—
43	C₃×C₅×M₁₆	—	2	2	no	—
44	C₃×C₅×M₁₆	C₂×C₄	1	4	yes	—
45	C₃×C₅×M₁₆	C₂³	1	4	yes	—
46	C₂×C₃×C₅×C₈	—	2	2	no	—
47	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
48	C₃×C₄×Dic₅	—	2	1	no	—
49	C₃×C₄×Dic₅	C₂³	1	1	yes	characteristic
50	C₃×C₄×Dic₅	C₂³	1	1	yes	characteristic
51	C₃×#681	D₄	1	4	yes	—
52	C₂×C₃×#168	C₂³	1	2	yes	—
53	C₃×Dic₂₀	C₂×C₄	1	4	yes	complemented
54	C₃×D₄₀	C₂×C₄	1	4	yes	complemented
55	C₃×#677	C₂×C₄	1	8	yes	complemented
56	C₃×#676	—	2	8	no	—
57	C₃×C₈×D₅	—	2	8	no	—
58	C₃×#575	—	5	1	no	—
59	C₃×#568	—	5	2	no	—
60	C₃×#444	—	5	4	no	—
61	C₃×#443	—	5	4	no	—
62	C₃×#430	D₅	1	1	yes	characteristic, complemented
63	C₂×C₃×#104	—	5	2	no	—
64	C₃×C₄×M₁₆	—	5	1	no	—
65	C₂×#709	C₂²×C₃	1	2	yes	—
66	C₂×C₅×M₁₆	C₂²×C₃	1	2	yes	—
67	C₂×C₅×M₁₆	C₂²×C₃	1	1	yes	characteristic
68	C₂×C₄×Dic₅	C₂²×C₃	1	1	yes	characteristic
69	C₂×#681	C₂²×C₃	1	1	yes	characteristic
70	#4055	C₃×C₄	1	4	yes	complemented
71	#4054	C₃×C₄	1	4	yes	complemented
72	#4053	C₂²×C₃	1	8	yes	complemented
73	D₅×M₁₆	—	2	4	no	—
74	#4050	C₂²×C₃	1	4	yes	—
75	#4047	—	2	4	no	—
76	C₅×#108	C₂²×C₃	1	4	yes	complemented
77	#3957	C₂²×C₃	1	4	yes	complemented
78	#3957	C₂²×C₃	1	4	yes	complemented
79	#3947	C₂²×C₃	1	2	yes	complemented
80	C₈×Dic₅	C₂²×C₃	1	2	yes	complemented
81	#2956	—	5	1	no	Sylow
82	C₂³×C₃×C₅	#41	1	1	yes	characteristic
83	C₂²×C₃×D₅	—	2	2	no	—
84	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
85	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
86	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
87	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
88	C₃×C₂²⋊D₅	—	2	4	no	—
89	C₃×C₂²⋊D₅	—	2	4	no	—
90	C₂×C₃×Dic₅	—	2	2	no	—
91	C₂×C₃×Dic₅	—	2	1	no	—
92	C₂×C₃×Dic₅	—	2	1	no	—
93	C₂×C₃×Dic₅	—	4	2	no	—
94	C₃×D₂₀	—	2	2	no	—
95	C₃×D₂₀	C₂²×C₄	1	4	yes	—
96	C₃×C₄×D₅	—	2	4	no	—
97	C₃×C₄×D₅	—	2	4	no	—
98	C₃×Dic₁₀	—	2	2	no	—
99	C₃×Dic₁₀	C₂²×C₄	1	4	yes	—
100	C₃×C₅×C₈	—	2	4	no	—
101	C₃×C₅×C₈	C₂²×C₄	1	4	yes	—
102	C₃×#168	C₂×D₄	1	4	yes	—
103	C₂×C₃×#41	—	5	2	no	—
104	C₃×#137	—	5	4	no	—
105	C₃×AGL(1,ℤ/8)	—	5	4	no	—
106	C₃×#135	—	5	4	no	—
107	C₃×#131	—	5	8	no	—
108	C₃×#131	—	10	4	no	—
109	C₂×C₃×M₁₆	—	5	1	no	—
110	C₂×C₃×M₁₆	—	10	4	no	—
111	C₂×C₃×M₁₆	C₂×D₅	1	2	yes	—
112	C₂×C₃×M₁₆	C₂×D₅	1	1	yes	characteristic
113	C₂×C₃×C₄²	—	5	1	no	—
114	C₃×#108	C₂×D₅	1	4	yes	complemented
115	C₃×#104	—	5	4	no	—
116	C₃×#104	—	5	4	no	—
117	C₃×#97	—	5	2	no	—
118	C₃×C₄×C₈	—	5	2	no	—
119	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
120	C₂×C₂²⋊D₅	—	2	2	no	—
121	C₂²×Dic₅	—	2	1	no	—
122	#709	—	2	2	no	—
123	#709	—	2	2	no	—
124	#709	C₂×C₃×C₄	1	4	yes	—
125	#709	C₂³×C₃	1	4	yes	—
126	C₂×D₂₀	C₂×C₃×C₄	1	2	yes	—
127	C₂×C₄×D₅	—	2	2	no	—
128	C₂×Dic₁₀	C₂×C₃×C₄	1	2	yes	—
129	C₅×M₁₆	—	2	2	no	—
130	C₅×M₁₆	C₂×C₃×C₄	1	4	yes	—
131	C₅×M₁₆	C₂³×C₃	1	4	yes	—
132	C₂×C₅×C₈	—	2	2	no	—
133	C₂×C₅×C₈	C₂³×C₃	1	2	yes	—
134	C₄×Dic₅	—	2	1	no	—
135	C₄×Dic₅	C₂³×C₃	1	1	yes	characteristic
136	C₄×Dic₅	C₂³×C₃	1	1	yes	characteristic
137	#681	C₃×D₄	1	4	yes	—
138	C₂×#168	C₂³×C₃	1	2	yes	—
139	Dic₂₀	C₂×C₃×C₄	1	4	yes	complemented
140	D₄₀	C₂×C₃×C₄	1	4	yes	complemented
141	#677	C₂×C₃×C₄	1	8	yes	complemented
142	#676	—	2	8	no	—
143	C₈×D₅	—	2	8	no	—
144	#575	—	5	1	no	—
145	#568	—	5	2	no	—
146	#444	—	5	4	no	—
147	#443	—	5	4	no	—
148	#430	C₃×D₅	1	1	yes	characteristic, complemented
149	C₂×#104	—	5	2	no	—
150	C₄×M₁₆	—	5	1	no	—
151	C₂²×C₃×C₅	—	4	1	no	—
152	C₂²×C₃×C₅	C₄×D₄	1	2	yes	—
153	C₂²×C₃×C₅	C₂×#41	1	1	yes	characteristic
154	C₂×C₃×D₅	—	2	4	no	—
155	C₂×C₃×D₅	—	8	2	no	—
156	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
157	C₃×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
158	C₃×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
159	C₃×Dic₅	—	2	2	no	—
160	C₃×Dic₅	—	2	4	no	—
161	C₃×Dic₅	—	4	1	no	—
162	C₃×#41	—	5	4	no	—
163	C₃×#41	—	5	4	no	—
164	C₃×#41	—	10	2	no	—
165	C₃×#41	—	10	2	no	—
166	C₂×C₃×Q₈	—	5	2	no	—
167	C₂×C₃×D₄	—	5	2	no	—
168	C₂×C₃×D₄	—	10	2	no	—
169	C₂²×C₃×C₄	—	10	2	no	—
170	C₂²×C₃×C₄	—	10	1	no	—
171	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
172	C₃×Q₁₆	—	5	4	no	—
173	C₃×SD₁₆	—	5	8	no	—
174	C₃×D₈	—	5	4	no	—
175	C₃×M₁₆	—	2	2	no	—
176	C₃×M₁₆	—	5	4	no	—
177	C₃×M₁₆	—	10	8	no	—
178	C₃×M₁₆	C₄×D₅	1	4	yes	—
179	C₃×M₁₆	C₂²×D₅	1	4	yes	—
180	C₂×C₃×C₈	—	2	2	no	—
181	C₂×C₃×C₈	—	5	2	no	—
182	C₂×C₃×C₈	—	10	8	no	—
183	C₂×C₃×C₈	C₂²×D₅	1	2	yes	—
184	C₃×C₄²	—	5	1	no	—
185	C₃×C₄²	—	5	1	no	—
186	C₃×C₄²	—	10	1	no	—
187	C₂³×C₅	C₃×#41	1	1	yes	characteristic
188	C₂²×D₅	—	2	2	no	—
189	C₂×C₄×C₅	C₂²×C₃×C₄	1	2	yes	—
190	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
191	C₂×C₄×C₅	C₃×#41	1	1	yes	characteristic
192	C₂×C₄×C₅	C₂⁴×C₃	1	1	yes	characteristic
193	C₂²⋊D₅	—	2	4	no	—
194	C₂²⋊D₅	—	2	4	no	—
195	C₂×Dic₅	—	2	2	no	—
196	C₂×Dic₅	—	2	1	no	—
197	C₂×Dic₅	—	2	1	no	—
198	C₂×Dic₅	—	4	2	no	—
199	D₂₀	—	2	2	no	—
200	D₂₀	C₂²×C₃×C₄	1	4	yes	—
201	C₄×D₅	—	2	4	no	—
202	C₄×D₅	—	2	4	no	—
203	Dic₁₀	—	2	2	no	—
204	Dic₁₀	C₂²×C₃×C₄	1	4	yes	—
205	C₅×C₈	—	2	4	no	—
206	C₅×C₈	C₂²×C₃×C₄	1	4	yes	—
207	#168	C₂×C₃×D₄	1	4	yes	—
208	C₂×#41	—	5	2	no	—
209	#137	—	5	4	no	—
210	AGL(1,ℤ/8)	—	5	4	no	—
211	#135	—	5	4	no	—
212	#131	—	5	8	no	—
213	#131	—	10	4	no	—
214	C₂×M₁₆	—	5	1	no	—
215	C₂×M₁₆	—	10	4	no	—
216	C₂×M₁₆	C₂×C₃×D₅	1	2	yes	—
217	C₂×M₁₆	C₂×C₃×D₅	1	1	yes	characteristic
218	C₂×C₄²	—	5	1	no	—
219	#108	C₂×C₃×D₅	1	4	yes	complemented
220	#104	—	5	4	no	—
221	#104	—	5	4	no	—
222	#97	—	5	2	no	—
223	C₄×C₈	—	5	2	no	—
224	C₂×C₃×C₅	—	2	1	no	—
225	C₂×C₃×C₅	—	2	2	no	—
226	C₂×C₃×C₅	C₂×C₄×D₄	1	1	yes	characteristic
227	C₃×D₅	—	4	4	no	—
228	C₂³×C₃	—	10	2	no	—
229	C₂³×C₃	#713	1	1	yes	characteristic
230	C₃×Q₈	—	5	4	no	—
231	C₃×Q₈	—	10	2	no	—
232	C₃×D₄	—	5	4	no	—
233	C₃×D₄	—	10	4	no	—
234	C₃×D₄	—	10	4	no	—
235	C₃×D₄	—	10	2	no	—
236	C₂×C₃×C₄	—	10	1	no	—
237	C₂×C₃×C₄	—	10	2	no	—
238	C₂×C₃×C₄	—	10	4	no	—
239	C₂×C₃×C₄	—	10	4	no	—
240	C₂×C₃×C₄	—	10	1	no	—
241	C₂×C₃×C₄	—	20	2	no	—
242	C₂×C₃×C₄	C₂×C₄×D₅	1	2	yes	—
243	C₂×C₃×C₄	D₄×D₅	1	2	yes	—
244	C₂×C₃×C₄	#713	1	1	yes	characteristic
245	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
246	C₃×C₈	—	2	4	no	—
247	C₃×C₈	—	5	4	no	—
248	C₃×C₈	C₂×C₄×D₅	1	4	yes	—
249	C₂²×C₅	—	4	1	no	—
250	C₂²×C₅	C₃×C₄×D₄	1	2	yes	—
251	C₂²×C₅	C₂×C₃×#41	1	1	yes	characteristic
252	C₂×D₅	—	2	4	no	—
253	C₂×D₅	—	8	2	no	—
254	C₄×C₅	C₃×C₄×D₄	1	2	yes	—
255	C₄×C₅	C₂³×C₃×C₄	1	1	yes	characteristic
256	C₄×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
257	Dic₅	—	2	2	no	—
258	Dic₅	—	2	4	no	—
259	Dic₅	—	4	1	no	—
260	#41	—	5	4	no	—
261	#41	—	5	4	no	—
262	#41	—	10	2	no	—
263	#41	—	10	2	no	—
264	C₂×Q₈	—	5	2	no	—
265	C₂×D₄	—	5	2	no	—
266	C₂×D₄	—	10	2	no	—
267	C₂²×C₄	—	10	2	no	—
268	C₂²×C₄	—	10	1	no	—
269	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
270	Q₁₆	—	5	4	no	—
271	SD₁₆	—	5	8	no	—
272	D₈	—	5	4	no	—
273	M₁₆	—	2	2	no	—
274	M₁₆	—	5	4	no	—
275	M₁₆	—	10	8	no	—
276	M₁₆	C₃×C₄×D₅	1	4	yes	—
277	M₁₆	C₂²×C₃×D₅	1	4	yes	—
278	C₂×C₈	—	2	2	no	—
279	C₂×C₈	—	5	2	no	—
280	C₂×C₈	—	10	8	no	—
281	C₂×C₈	C₂²×C₃×D₅	1	2	yes	—
282	C₄²	—	5	1	no	—
283	C₄²	—	5	1	no	—
284	C₄²	—	10	1	no	—
285	C₃×C₅	#2956	1	1	yes	characteristic, complemented, Hall
286	C₂²×C₃	—	4	1	no	—
287	C₂²×C₃	—	10	4	no	—
288	C₂²×C₃	—	40	2	no	—
289	C₂²×C₃	#4039	1	2	yes	—
290	C₂²×C₃	C₂×#713	1	1	yes	characteristic
291	C₃×C₄	—	10	2	no	—
292	C₃×C₄	—	10	4	no	—
293	C₃×C₄	—	20	1	no	—
294	C₃×C₄	#4039	1	2	yes	—
295	C₃×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
296	C₃×C₄	C₂×D₄×D₅	1	1	yes	characteristic
297	C₂×C₅	—	2	1	no	—
298	C₂×C₅	—	2	2	no	—
299	C₂×C₅	C₂×C₃×C₄×D₄	1	1	yes	characteristic
300	D₅	—	4	4	no	—
301	C₂³	—	10	2	no	—
302	C₂³	C₃×#713	1	1	yes	characteristic
303	Q₈	—	5	4	no	—
304	Q₈	—	10	2	no	—
305	D₄	—	5	4	no	—
306	D₄	—	10	4	no	—
307	D₄	—	10	4	no	—
308	D₄	—	10	2	no	—
309	C₂×C₄	—	10	1	no	—
310	C₂×C₄	—	10	2	no	—
311	C₂×C₄	—	10	4	no	—
312	C₂×C₄	—	10	4	no	—
313	C₂×C₄	—	10	1	no	—
314	C₂×C₄	—	20	2	no	—
315	C₂×C₄	C₂×C₃×C₄×D₅	1	2	yes	—
316	C₂×C₄	C₃×D₄×D₅	1	2	yes	—
317	C₂×C₄	C₃×#713	1	1	yes	characteristic
318	C₂×C₄	C₂³×C₃×D₅	1	1	yes	characteristic
319	C₈	—	2	4	no	—
320	C₈	—	5	4	no	—
321	C₈	C₂×C₃×C₄×D₅	1	4	yes	—
322	C₂×C₃	—	2	1	no	—
323	C₂×C₃	—	2	2	no	—
324	C₂×C₃	—	20	4	no	—
325	C₂×C₃	C₂×#4039	1	1	yes	characteristic
326	C₅	C₃×#2956	1	1	yes	characteristic, complemented, Sylow
327	C₂²	—	4	1	no	—
328	C₂²	—	10	4	no	—
329	C₂²	—	40	2	no	—
330	C₂²	C₃×#4039	1	2	yes	—
331	C₂²	C₂×C₃×#713	1	1	yes	characteristic
332	C₄	—	10	2	no	—
333	C₄	—	10	4	no	—
334	C₄	—	20	1	no	—
335	C₄	C₃×#4039	1	2	yes	—
336	C₄	C₂²×C₃×C₄×D₅	1	1	yes	characteristic
337	C₄	C₂×C₃×D₄×D₅	1	1	yes	characteristic
338	C₃	#10610989	1	1	yes	characteristic, direct product factor, Sylow
339	C₂	—	2	1	no	—
340	C₂	—	2	2	no	—
341	C₂	—	20	4	no	—
342	C₂	C₂×C₃×#4039	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₄	C₂×D₄×D₅
commutator	C₄×C₅	C₂³×C₃×C₄
Frattini	C₂×C₄	C₂³×C₃×D₅
Fitting	C₃×C₅×#430	C₂
socle	C₂×C₃×C₅	C₂×C₄×D₄
prime radical	C₃×#66082	C₂
cosocle	C₂×C₄×C₅	C₂⁴×C₃
hypercenter	C₃×#430	D₅
hypocenter	C₅	C₃×#2956
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₃×C₄ ◁ C₂²×C₃×C₄ ◁ C₃×#430

Complements

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

N	H ≅ G/N	type
2. C₃×#66079	C₂	semidirect
3. C₃×C₅×#430	C₂	semidirect
4. C₂×C₃×#3957	C₂	semidirect
5. C₃×M₁₆×Dic₅	C₂	semidirect
6. C₃×#65184	C₂	semidirect
7. C₃×#65183	C₂	semidirect
8. #10610989	C₃	direct (N × H)
14. C₃×#4055	C₄	semidirect
15. C₃×#4054	C₄	semidirect
16. C₃×#4053	C₂²	semidirect
20. C₃×C₅×#108	C₂²	semidirect
21. C₃×#3957	C₂²	semidirect
22. C₃×#3957	C₂²	semidirect
23. C₃×#3947	C₂²	semidirect
24. C₃×C₈×Dic₅	C₂²	semidirect
27. #66079	C₂×C₃	semidirect
28. C₅×#430	C₂×C₃	semidirect
29. C₂×#3957	C₂×C₃	semidirect
30. M₁₆×Dic₅	C₂×C₃	semidirect
31. #65184	C₂×C₃	semidirect
32. #65183	C₂×C₃	semidirect
53. C₃×Dic₂₀	C₂×C₄	semidirect
54. C₃×D₄₀	C₂×C₄	semidirect
55. C₃×#677	C₂×C₄	semidirect
62. C₃×#430	D₅	semidirect
70. #4055	C₃×C₄	semidirect
71. #4054	C₃×C₄	semidirect
72. #4053	C₂²×C₃	semidirect
76. C₅×#108	C₂²×C₃	semidirect
77. #3957	C₂²×C₃	semidirect
78. #3957	C₂²×C₃	semidirect
79. #3947	C₂²×C₃	semidirect
80. C₈×Dic₅	C₂²×C₃	semidirect
114. C₃×#108	C₂×D₅	semidirect
139. Dic₂₀	C₂×C₃×C₄	semidirect
140. D₄₀	C₂×C₃×C₄	semidirect
141. #677	C₂×C₃×C₄	semidirect
148. #430	C₃×D₅	semidirect
219. #108	C₂×C₃×D₅	semidirect
285. C₃×C₅	#2956	semidirect
326. C₅	C₃×#2956	semidirect
338. C₃	#10610989	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×#66082	2
C₃×#66079	2
C₃×C₅×#430	2
C₂×C₃×#3957	2
C₃×M₁₆×Dic₅	2
C₃×#65184	2
C₃×#65183	2
#10610989	3
C₃×#2956	5

Subgroups — C3×#10610989 #14418117

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10610989 #14418117