Subgroups — C₃×#10617361 #14415242

Order 1920.

1000 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#4131	C₂	1	2	yes	maximal, complemented
2	C₃×#66050	C₂	1	1	yes	maximal, characteristic
3	C₃×#65776	C₂	1	2	yes	maximal, complemented
4	C₃×#65774	C₂	1	4	yes	maximal, complemented
5	C₃×D₄×#170	C₂	1	4	yes	maximal, complemented
6	C₃×#65681	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×C₄×D₅⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	#10617361	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂×C₃×#711	C₂²	1	2	yes	—
10	C₂×C₃×#704	C₂²	1	4	yes	—
11	C₂²×C₃×#170	C₂²	1	4	yes	—
12	C₃×#4131	—	2	8	no	—
13	C₂×C₃×#700	C₂²	1	2	yes	—
14	C₂×C₃×D₅⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₃×D₅⋊C₈	C₂²	1	1	yes	characteristic
16	C₃×C₅×#127	C₄	1	1	yes	characteristic
17	C₃×#4084	C₄	1	4	yes	—
18	C₃×#4082	C₄	1	2	yes	—
19	C₃×D₄×Dic₅	C₂²	1	4	yes	—
20	C₃×C₄²×D₅	C₂²	1	1	yes	characteristic
21	C₃×#4015	C₄	1	1	yes	characteristic
22	C₃×C₂²⋊#170	C₂²	1	8	yes	—
23	C₃×C₄⋊#170	C₂²	1	4	yes	—
24	C₃×C₄×#170	C₂²	1	2	yes	—
25	C₃×C₄×#170	C₂²	1	2	yes	—
26	C₃×#2931	—	5	1	no	maximal, Hall
27	C₂×#4131	C₂×C₃	1	2	yes	complemented
28	#66050	C₂×C₃	1	1	yes	characteristic
29	#65776	C₂×C₃	1	2	yes	complemented
30	#65774	C₂×C₃	1	4	yes	complemented
31	D₄×#170	C₂×C₃	1	4	yes	complemented
32	#65681	C₂×C₃	1	1	yes	characteristic, complemented
33	C₄×D₅⋊C₈	C₂×C₃	1	1	yes	characteristic, complemented
34	C₂×C₃×C₅×D₄	C₂×C₄	1	4	yes	—
35	C₂×C₃×C₅×D₄	C₂×C₄	1	2	yes	—
36	C₂×C₃×C₂²⋊D₅	C₂×C₄	1	4	yes	—
37	C₂²×C₃×Dic₅	C₂³	1	4	yes	—
38	C₃×#711	—	2	4	no	—
39	C₃×#711	—	2	4	no	—
40	C₂×C₃×C₄×D₅	C₂×C₄	1	2	yes	—
41	C₂×C₃×C₄×D₅	C₂³	1	1	yes	characteristic
42	C₂×C₃×Dic₁₀	C₂×C₄	1	2	yes	—
43	C₃×#704	—	2	8	no	—
44	C₂×C₃×#170	—	2	8	no	—
45	C₂×C₃×#170	C₂³	1	2	yes	—
46	C₂×C₃×#170	C₂³	1	4	yes	—
47	C₂×C₃×#170	C₂³	1	1	yes	characteristic
48	C₂×C₃×#170	C₂³	1	1	yes	characteristic
49	C₃×#700	—	2	4	no	—
50	C₃×D₅⋊C₈	—	2	2	no	—
51	C₃×D₅⋊C₈	D₄	1	4	yes	complemented
52	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
53	C₃×C₂²⋊Dic₅	C₂×C₄	1	8	yes	—
54	C₃×C₄⋊Dic₅	C₂×C₄	1	4	yes	—
55	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
56	C₃×C₄×Dic₅	C₂³	1	2	yes	—
57	C₂×C₃×#131	—	5	2	no	—
58	C₃×#532	—	5	1	no	—
59	C₃×#436	—	5	4	no	—
60	C₃×C₈×D₄	—	5	4	no	—
61	C₃×#423	—	5	1	no	—
62	C₃×#408	—	5	2	no	—
63	C₂×C₃×C₄×C₈	—	5	1	no	—
64	C₂×#711	C₂²×C₃	1	2	yes	—
65	C₂×#704	C₂²×C₃	1	4	yes	—
66	C₂²×#170	C₂²×C₃	1	4	yes	—
67	#4131	—	2	8	no	—
68	C₂×#700	C₂²×C₃	1	2	yes	—
69	C₂×D₅⋊C₈	C₂²×C₃	1	1	yes	characteristic
70	C₂×D₅⋊C₈	C₂²×C₃	1	1	yes	characteristic
71	C₅×#127	C₃×C₄	1	1	yes	characteristic
72	#4084	C₃×C₄	1	4	yes	—
73	#4082	C₃×C₄	1	2	yes	—
74	D₄×Dic₅	C₂²×C₃	1	4	yes	—
75	C₄²×D₅	C₂²×C₃	1	1	yes	characteristic
76	#4015	C₃×C₄	1	1	yes	characteristic
77	C₂²⋊#170	C₂²×C₃	1	8	yes	—
78	C₄⋊#170	C₂²×C₃	1	4	yes	—
79	C₄×#170	C₂²×C₃	1	2	yes	—
80	C₄×#170	C₂²×C₃	1	2	yes	—
81	#2931	—	5	1	no	Sylow
82	C₂³×C₃×C₅	C₂²×C₄	1	4	yes	—
83	C₂²×C₃×D₅	C₂²×C₄	1	1	yes	characteristic
84	C₃×C₅×D₄	—	2	4	no	—
85	C₃×C₅×D₄	—	2	8	no	—
86	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
88	C₃×C₂²⋊D₅	—	2	8	no	—
89	C₂×C₃×Dic₅	—	2	4	no	—
90	C₂×C₃×Dic₅	—	2	4	no	—
91	C₂×C₃×Dic₅	C₂²×C₄	1	2	yes	—
92	C₂×C₃×Dic₅	C₂²×C₄	1	4	yes	—
93	C₂×C₃×Dic₅	C₂⁴	1	1	yes	characteristic
94	C₃×C₄×D₅	—	2	4	no	—
95	C₃×C₄×D₅	C₂×D₄	1	2	yes	—
96	C₃×C₄×D₅	#41	1	2	yes	—
97	C₃×Dic₁₀	—	2	4	no	—
98	C₃×#170	—	2	4	no	—
99	C₃×#170	—	2	2	no	—
100	C₃×#170	C₂×D₄	1	2	yes	—
101	C₃×#170	C₂×D₄	1	2	yes	—
102	C₂×C₃×#41	—	5	2	no	—
103	C₃×#131	—	10	8	no	—
104	C₂×C₃×M₁₆	—	5	2	no	—
105	C₂×C₃×M₁₆	—	5	4	no	—
106	C₂²×C₃×C₈	—	5	1	no	—
107	C₂²×C₃×C₈	—	5	4	no	—
108	C₂²×C₃×C₈	—	5	1	no	—
109	C₃×C₄⋊Q₈	—	5	1	no	—
110	C₃×#127	AGL(1,𝔽₅)	1	1	yes	characteristic
111	C₃×#124	—	5	2	no	—
112	C₃×#121	—	5	4	no	—
113	C₃×C₄×D₄	—	5	4	no	—
114	C₂×C₃×C₄²	—	5	1	no	—
115	C₃×C₄⋊C₈	—	5	4	no	—
116	C₃×C₂²⋊C₈	—	5	8	no	—
117	C₃×C₄×C₈	—	5	2	no	—
118	C₃×C₄×C₈	—	5	2	no	—
119	C₂×C₅×D₄	C₂×C₃×C₄	1	2	yes	—
120	C₂×C₅×D₄	C₂×C₃×C₄	1	4	yes	—
121	C₂×C₂²⋊D₅	C₂×C₃×C₄	1	4	yes	—
122	C₂²×Dic₅	C₂³×C₃	1	4	yes	—
123	#711	—	2	4	no	—
124	#711	—	2	4	no	—
125	C₂×C₄×D₅	C₂×C₃×C₄	1	2	yes	—
126	C₂×C₄×D₅	C₂³×C₃	1	1	yes	characteristic
127	C₂×Dic₁₀	C₂×C₃×C₄	1	2	yes	—
128	#704	—	2	8	no	—
129	C₂×#170	—	2	8	no	—
130	C₂×#170	C₂³×C₃	1	2	yes	—
131	C₂×#170	C₂³×C₃	1	4	yes	—
132	C₂×#170	C₂³×C₃	1	1	yes	characteristic
133	C₂×#170	C₂³×C₃	1	1	yes	characteristic
134	#700	—	2	4	no	—
135	D₅⋊C₈	—	2	2	no	—
136	D₅⋊C₈	C₃×D₄	1	4	yes	complemented
137	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
138	C₂²⋊Dic₅	C₂×C₃×C₄	1	8	yes	—
139	C₄⋊Dic₅	C₂×C₃×C₄	1	4	yes	—
140	C₄×Dic₅	C₂×C₃×C₄	1	1	yes	characteristic
141	C₄×Dic₅	C₂³×C₃	1	2	yes	—
142	C₂×#131	—	5	2	no	—
143	#532	—	5	1	no	—
144	#436	—	5	4	no	—
145	C₈×D₄	—	5	4	no	—
146	#423	—	5	1	no	—
147	#408	—	5	2	no	—
148	C₂×C₄×C₈	—	5	1	no	—
149	C₂²×C₃×C₅	—	2	4	no	—
150	C₂²×C₃×C₅	—	2	8	no	—
151	C₂²×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
152	C₂×C₃×D₅	—	2	2	no	—
153	C₂×C₃×D₅	C₄×D₄	1	2	yes	—
154	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
155	C₃×C₄×C₅	#131	1	4	yes	complemented
156	C₃×Dic₅	—	2	4	no	—
157	C₃×Dic₅	—	2	2	no	—
158	C₃×Dic₅	C₂²×D₄	1	1	yes	characteristic
159	C₃×Dic₅	C₂×#41	1	1	yes	characteristic
160	C₃×#41	—	10	4	no	—
161	C₃×#41	—	10	4	no	—
162	C₂×C₃×Q₈	—	5	2	no	—
163	C₂×C₃×D₄	—	5	4	no	—
164	C₂×C₃×D₄	C₂×AGL(1,𝔽₅)	1	4	yes	—
165	C₂×C₃×D₄	C₂×AGL(1,𝔽₅)	1	2	yes	—
166	C₂²×C₃×C₄	—	5	1	no	—
167	C₂²×C₃×C₄	—	5	4	no	—
168	C₂²×C₃×C₄	—	5	2	no	—
169	C₃×M₁₆	—	10	8	no	—
170	C₃×M₁₆	—	10	4	no	—
171	C₂×C₃×C₈	—	5	1	no	—
172	C₂×C₃×C₈	—	5	1	no	—
173	C₂×C₃×C₈	—	5	4	no	—
174	C₂×C₃×C₈	—	5	2	no	—
175	C₂×C₃×C₈	—	5	4	no	—
176	C₂×C₃×C₈	—	10	8	no	—
177	C₂×C₃×C₈	—	10	2	no	—
178	C₃×C₄⋊C₄	—	5	4	no	—
179	C₃×C₂²⋊C₄	—	5	8	no	—
180	C₃×C₄²	—	5	1	no	—
181	C₃×C₄²	—	5	2	no	—
182	C₃×C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
183	C₂³×C₅	C₂²×C₃×C₄	1	4	yes	—
184	C₂²×D₅	C₂²×C₃×C₄	1	1	yes	characteristic
185	C₅×D₄	—	2	4	no	—
186	C₅×D₄	—	2	8	no	—
187	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
188	C₂×C₄×C₅	C₂²×C₃×C₄	1	2	yes	—
189	C₂²⋊D₅	—	2	8	no	—
190	C₂×Dic₅	—	2	4	no	—
191	C₂×Dic₅	—	2	4	no	—
192	C₂×Dic₅	C₂²×C₃×C₄	1	2	yes	—
193	C₂×Dic₅	C₂²×C₃×C₄	1	4	yes	—
194	C₂×Dic₅	C₂⁴×C₃	1	1	yes	characteristic
195	C₄×D₅	—	2	4	no	—
196	C₄×D₅	C₂×C₃×D₄	1	2	yes	—
197	C₄×D₅	C₃×#41	1	2	yes	—
198	Dic₁₀	—	2	4	no	—
199	#170	—	2	4	no	—
200	#170	—	2	2	no	—
201	#170	C₂×C₃×D₄	1	2	yes	—
202	#170	C₂×C₃×D₄	1	2	yes	—
203	C₂×#41	—	5	2	no	—
204	#131	—	10	8	no	—
205	C₂×M₁₆	—	5	2	no	—
206	C₂×M₁₆	—	5	4	no	—
207	C₂²×C₈	—	5	1	no	—
208	C₂²×C₈	—	5	1	no	—
209	C₂²×C₈	—	5	4	no	—
210	C₄⋊Q₈	—	5	1	no	—
211	#127	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
212	#124	—	5	2	no	—
213	#121	—	5	4	no	—
214	C₄×D₄	—	5	4	no	—
215	C₂×C₄²	—	5	1	no	—
216	C₄⋊C₈	—	5	4	no	—
217	C₂²⋊C₈	—	5	8	no	—
218	C₄×C₈	—	5	2	no	—
219	C₄×C₈	—	5	2	no	—
220	C₂×C₃×C₅	—	4	4	no	—
221	C₂×C₃×C₅	C₂×C₄×D₄	1	1	yes	characteristic
222	C₂×C₃×C₅	C₂×#131	1	2	yes	—
223	C₃×D₅	—	2	2	no	—
224	C₂³×C₃	—	5	1	no	—
225	C₂³×C₃	C₂²×AGL(1,𝔽₅)	1	4	yes	—
226	C₃×Q₈	—	10	4	no	—
227	C₃×D₄	—	2	8	no	—
228	C₃×D₄	—	2	4	no	—
229	C₃×D₄	—	10	8	no	—
230	C₂×C₃×C₄	—	5	1	no	—
231	C₂×C₃×C₄	—	5	2	no	—
232	C₂×C₃×C₄	—	5	2	no	—
233	C₂×C₃×C₄	—	5	2	no	—
234	C₂×C₃×C₄	—	5	4	no	—
235	C₂×C₃×C₄	—	10	4	no	—
236	C₂×C₃×C₄	—	10	4	no	—
237	C₂×C₃×C₄	—	10	4	no	—
238	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	2	yes	—
239	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
240	C₃×C₈	—	5	2	no	—
241	C₃×C₈	—	5	2	no	—
242	C₃×C₈	—	10	2	no	—
243	C₃×C₈	—	10	4	no	—
244	C₂²×C₅	—	2	8	no	—
245	C₂²×C₅	—	2	4	no	—
246	C₂²×C₅	C₂³×C₃×C₄	1	1	yes	characteristic
247	C₂×D₅	—	2	2	no	—
248	C₂×D₅	C₃×C₄×D₄	1	2	yes	—
249	C₄×C₅	C₃×C₄×D₄	1	2	yes	—
250	C₄×C₅	C₃×#131	1	4	yes	complemented
251	Dic₅	—	2	2	no	—
252	Dic₅	—	2	4	no	—
253	Dic₅	C₂²×C₃×D₄	1	1	yes	characteristic
254	Dic₅	C₂×C₃×#41	1	1	yes	characteristic
255	#41	—	10	4	no	—
256	#41	—	10	4	no	—
257	C₂×Q₈	—	5	2	no	—
258	C₂×D₄	—	5	4	no	—
259	C₂×D₄	C₂×C₃×AGL(1,𝔽₅)	1	4	yes	—
260	C₂×D₄	C₂×C₃×AGL(1,𝔽₅)	1	2	yes	—
261	C₂²×C₄	—	5	1	no	—
262	C₂²×C₄	—	5	4	no	—
263	C₂²×C₄	—	5	2	no	—
264	M₁₆	—	10	8	no	—
265	M₁₆	—	10	4	no	—
266	C₂×C₈	—	5	1	no	—
267	C₂×C₈	—	5	1	no	—
268	C₂×C₈	—	5	4	no	—
269	C₂×C₈	—	5	2	no	—
270	C₂×C₈	—	5	4	no	—
271	C₂×C₈	—	10	8	no	—
272	C₂×C₈	—	10	2	no	—
273	C₄⋊C₄	—	5	4	no	—
274	C₂²⋊C₄	—	5	8	no	—
275	C₄²	—	5	2	no	—
276	C₄²	—	5	1	no	—
277	C₄²	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
278	C₃×C₅	#2931	1	1	yes	characteristic, complemented, Hall
279	C₂²×C₃	—	2	8	no	—
280	C₂²×C₃	—	2	4	no	—
281	C₂²×C₃	—	5	2	no	—
282	C₂²×C₃	—	10	2	no	—
283	C₂²×C₃	C₂³×AGL(1,𝔽₅)	1	1	yes	characteristic
284	C₃×C₄	—	5	1	no	—
285	C₃×C₄	—	5	1	no	—
286	C₃×C₄	—	10	2	no	—
287	C₃×C₄	—	10	4	no	—
288	C₃×C₄	#4131	1	4	yes	complemented
289	C₃×C₄	D₄×AGL(1,𝔽₅)	1	2	yes	—
290	C₂×C₅	—	4	4	no	—
291	C₂×C₅	C₂×C₃×C₄×D₄	1	1	yes	characteristic
292	C₂×C₅	C₂×C₃×#131	1	2	yes	—
293	D₅	—	2	2	no	—
294	C₂³	—	5	1	no	—
295	C₂³	C₂²×C₃×AGL(1,𝔽₅)	1	4	yes	—
296	Q₈	—	10	4	no	—
297	D₄	—	2	8	no	—
298	D₄	—	2	4	no	—
299	D₄	—	10	8	no	—
300	C₂×C₄	—	5	1	no	—
301	C₂×C₄	—	5	2	no	—
302	C₂×C₄	—	5	2	no	—
303	C₂×C₄	—	5	2	no	—
304	C₂×C₄	—	5	4	no	—
305	C₂×C₄	—	10	4	no	—
306	C₂×C₄	—	10	4	no	—
307	C₂×C₄	—	10	4	no	—
308	C₂×C₄	C₂²×C₃×AGL(1,𝔽₅)	1	2	yes	—
309	C₂×C₄	C₂²×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
310	C₈	—	5	2	no	—
311	C₈	—	5	2	no	—
312	C₈	—	10	4	no	—
313	C₈	—	10	2	no	—
314	C₂×C₃	—	4	4	no	—
315	C₂×C₃	—	10	2	no	—
316	C₂×C₃	C₂×#4131	1	2	yes	—
317	C₂×C₃	C₂×D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
318	C₅	C₃×#2931	1	1	yes	characteristic, complemented, Sylow
319	C₂²	—	2	8	no	—
320	C₂²	—	2	4	no	—
321	C₂²	—	5	2	no	—
322	C₂²	—	10	2	no	—
323	C₂²	C₂³×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
324	C₄	—	5	1	no	—
325	C₄	—	5	1	no	—
326	C₄	—	10	2	no	—
327	C₄	—	10	4	no	—
328	C₄	C₃×#4131	1	4	yes	complemented
329	C₄	C₃×D₄×AGL(1,𝔽₅)	1	2	yes	—
330	C₃	#10617361	1	1	yes	characteristic, direct product factor, Sylow
331	C₂	—	4	4	no	—
332	C₂	—	10	2	no	—
333	C₂	C₂×C₃×#4131	1	2	yes	—
334	C₂	C₂×C₃×D₄×AGL(1,𝔽₅)	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂³×AGL(1,𝔽₅)
commutator	C₂²×C₅	C₂³×C₃×C₄
Frattini	C₂²	C₂³×C₃×AGL(1,𝔽₅)
Fitting	C₃×C₅×#127	C₄
socle	C₂²×C₃×C₅	C₂³×C₄
prime radical	C₃×#66050	C₂
cosocle	C₂×Dic₅	C₂⁴×C₃
hypercenter	C₃×#127	AGL(1,𝔽₅)
hypocenter	C₅	C₃×#2931
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×#127

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×#4131	C₂	semidirect
3. C₃×#65776	C₂	semidirect
4. C₃×#65774	C₂	semidirect
5. C₃×D₄×#170	C₂	semidirect
6. C₃×#65681	C₂	semidirect
7. C₃×C₄×D₅⋊C₈	C₂	semidirect
8. #10617361	C₃	direct (N × H)
27. C₂×#4131	C₂×C₃	semidirect
29. #65776	C₂×C₃	semidirect
30. #65774	C₂×C₃	semidirect
31. D₄×#170	C₂×C₃	semidirect
32. #65681	C₂×C₃	semidirect
33. C₄×D₅⋊C₈	C₂×C₃	semidirect
51. C₃×D₅⋊C₈	D₄	semidirect
136. D₅⋊C₈	C₃×D₄	semidirect
155. C₃×C₄×C₅	#131	semidirect
250. C₄×C₅	C₃×#131	semidirect
278. C₃×C₅	#2931	semidirect
288. C₃×C₄	#4131	semidirect
318. C₅	C₃×#2931	semidirect
328. C₄	C₃×#4131	semidirect
330. C₃	#10617361	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×#4131	2
C₃×#66050	2
C₃×#65776	2
C₃×#65774	2
C₃×D₄×#170	2
C₃×#65681	2
C₃×C₄×D₅⋊C₈	2
#10617361	3
C₃×#2931	5

Subgroups — C3×#10617361 #14415242

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10617361 #14415242