Subgroups — C₃×#10617371 #14415244

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×#4133	C₂	1	2	yes	maximal, complemented
2	C₃×#66059	C₂	1	1	yes	maximal, characteristic
3	C₃×#65787	C₂	1	2	yes	maximal, complemented
4	C₃×#65705	C₂	1	4	yes	maximal, complemented
5	C₃×#65702	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	#10617371	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂×C₃×#713	C₂²	1	2	yes	—
10	C₃×#4133	—	2	8	no	—
11	C₂×C₃×#700	C₂²	1	2	yes	—
12	C₂×C₃×#700	C₂²	1	4	yes	—
13	C₂×C₃×D₅⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₃×D₅⋊C₈	C₂²	1	4	yes	—
15	C₂×C₃×D₅⋊C₈	C₂²	1	1	yes	characteristic
16	C₃×C₅×C₄⋊Q₈	C₄	1	1	yes	characteristic
17	C₃×#4093	C₄	1	2	yes	—
18	C₃×#4041	C₄	1	4	yes	—
19	C₃×#4039	C₂²	1	4	yes	—
20	C₃×#4020	C₄	1	1	yes	characteristic
21	C₃×C₄²×D₅	C₂²	1	1	yes	characteristic
22	C₃×#4003	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×#4133	C₂×C₃	1	2	yes	complemented
28	#66059	C₂×C₃	1	1	yes	characteristic
29	#65787	C₂×C₃	1	2	yes	complemented
30	#65705	C₂×C₃	1	4	yes	complemented
31	#65702	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×Q₈	C₂×C₄	1	2	yes	—
35	C₃×#713	—	2	4	no	—
36	C₃×#713	—	2	4	no	—
37	C₂×C₃×D₂₀	C₂×C₄	1	2	yes	—
38	C₂×C₃×D₂₀	C₂×C₄	1	4	yes	—
39	C₂×C₃×D₂₀	C₂×C₄	1	4	yes	—
40	C₂×C₃×C₄×D₅	C₂×C₄	1	2	yes	—
41	C₂×C₃×C₄×D₅	C₂³	1	4	yes	—
42	C₂×C₃×C₄×D₅	C₂³	1	1	yes	characteristic
43	C₂×C₃×#170	C₂³	1	2	yes	—
44	C₂×C₃×#170	C₂³	1	4	yes	—
45	C₂×C₃×#170	C₂³	1	1	yes	characteristic
46	C₂×C₃×#170	C₂³	1	1	yes	characteristic
47	C₃×#700	—	2	4	no	—
48	C₃×#700	—	2	8	no	—
49	C₃×D₅⋊C₈	—	2	2	no	—
50	C₃×D₅⋊C₈	—	2	8	no	—
51	C₃×D₅⋊C₈	D₄	1	4	yes	complemented
52	C₃×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
53	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
54	C₃×#685	C₂×C₄	1	8	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×#713	C₂²×C₃	1	2	yes	—
65	#4133	—	2	8	no	—
66	C₂×#700	C₂²×C₃	1	2	yes	—
67	C₂×#700	C₂²×C₃	1	4	yes	—
68	C₂×D₅⋊C₈	C₂²×C₃	1	1	yes	characteristic
69	C₂×D₅⋊C₈	C₂²×C₃	1	4	yes	—
70	C₂×D₅⋊C₈	C₂²×C₃	1	1	yes	characteristic
71	C₅×C₄⋊Q₈	C₃×C₄	1	1	yes	characteristic
72	#4093	C₃×C₄	1	2	yes	—
73	#4041	C₃×C₄	1	4	yes	—
74	#4039	C₂²×C₃	1	4	yes	—
75	#4020	C₃×C₄	1	1	yes	characteristic
76	C₄²×D₅	C₂²×C₃	1	1	yes	characteristic
77	#4003	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	C₂²×C₃×D₅	C₂²×C₄	1	4	yes	—
83	C₂²×C₃×D₅	C₂²×C₄	1	1	yes	characteristic
84	C₃×C₅×Q₈	—	2	4	no	—
85	C₂×C₃×C₄×C₅	C₂²×C₄	1	4	yes	—
86	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
88	C₂×C₃×Dic₅	C₂²×C₄	1	2	yes	—
89	C₂×C₃×Dic₅	C₂⁴	1	1	yes	characteristic
90	C₃×D₂₀	—	2	8	no	—
91	C₃×D₂₀	—	2	4	no	—
92	C₃×D₂₀	—	2	8	no	—
93	C₃×C₄×D₅	—	2	4	no	—
94	C₃×C₄×D₅	—	2	4	no	—
95	C₃×C₄×D₅	—	2	4	no	—
96	C₃×C₄×D₅	C₂×D₄	1	2	yes	—
97	C₃×C₄×D₅	#41	1	2	yes	—
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	AGL(1,𝔽₅)	1	1	yes	characteristic
110	C₃×#127	—	5	1	no	—
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×Q₈	C₂×C₃×C₄	1	2	yes	—
120	#713	—	2	4	no	—
121	#713	—	2	4	no	—
122	C₂×D₂₀	C₂×C₃×C₄	1	2	yes	—
123	C₂×D₂₀	C₂×C₃×C₄	1	4	yes	—
124	C₂×D₂₀	C₂×C₃×C₄	1	4	yes	—
125	C₂×C₄×D₅	C₂×C₃×C₄	1	2	yes	—
126	C₂×C₄×D₅	C₂³×C₃	1	4	yes	—
127	C₂×C₄×D₅	C₂³×C₃	1	1	yes	characteristic
128	C₂×#170	C₂³×C₃	1	4	yes	—
129	C₂×#170	C₂³×C₃	1	2	yes	—
130	C₂×#170	C₂³×C₃	1	1	yes	characteristic
131	C₂×#170	C₂³×C₃	1	1	yes	characteristic
132	#700	—	2	4	no	—
133	#700	—	2	8	no	—
134	D₅⋊C₈	—	2	2	no	—
135	D₅⋊C₈	—	2	8	no	—
136	D₅⋊C₈	C₃×D₄	1	4	yes	complemented
137	C₅×C₄⋊C₄	C₂×C₃×C₄	1	4	yes	—
138	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
139	#685	C₂×C₃×C₄	1	8	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	C₂³×C₄	1	1	yes	characteristic
150	C₂×C₃×D₅	—	2	2	no	—
151	C₂×C₃×D₅	—	2	4	no	—
152	C₂×C₃×D₅	—	2	8	no	—
153	C₂×C₃×D₅	C₄×D₄	1	2	yes	—
154	C₃×C₄×C₅	—	2	4	no	—
155	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
156	C₃×C₄×C₅	#131	1	4	yes	complemented
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	C₂×AGL(1,𝔽₅)	1	2	yes	—
163	C₂×C₃×D₄	—	5	4	no	—
164	C₂×C₃×D₄	—	5	4	no	—
165	C₂×C₃×D₄	—	5	2	no	—
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	C₂×AGL(1,𝔽₅)	1	4	yes	—
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	C₂²×D₅	C₂²×C₃×C₄	1	4	yes	—
184	C₂²×D₅	C₂²×C₃×C₄	1	1	yes	characteristic
185	C₅×Q₈	—	2	4	no	—
186	C₂×C₄×C₅	C₂²×C₃×C₄	1	4	yes	—
187	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
188	C₂×C₄×C₅	C₂²×C₃×C₄	1	2	yes	—
189	C₂×Dic₅	C₂²×C₃×C₄	1	2	yes	—
190	C₂×Dic₅	C₂⁴×C₃	1	1	yes	characteristic
191	D₂₀	—	2	8	no	—
192	D₂₀	—	2	4	no	—
193	D₂₀	—	2	8	no	—
194	C₄×D₅	—	2	4	no	—
195	C₄×D₅	—	2	4	no	—
196	C₄×D₅	—	2	4	no	—
197	C₄×D₅	C₂×C₃×D₄	1	2	yes	—
198	C₄×D₅	C₃×#41	1	2	yes	—
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	4	no	—
209	C₂²×C₈	—	5	1	no	—
210	C₄⋊Q₈	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
211	#127	—	5	1	no	—
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	C₂×C₄×D₄	1	1	yes	characteristic
221	C₂×C₃×C₅	C₂×#131	1	2	yes	—
222	C₃×D₅	—	2	2	no	—
223	C₃×D₅	—	4	4	no	—
224	C₂³×C₃	—	5	1	no	—
225	C₂³×C₃	—	5	4	no	—
226	C₃×Q₈	—	2	4	no	—
227	C₃×D₄	—	10	8	no	—
228	C₃×D₄	—	10	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	—	10	4	no	—
235	C₂×C₃×C₄	—	10	4	no	—
236	C₂×C₃×C₄	—	10	4	no	—
237	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	4	yes	—
238	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
239	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	2	yes	—
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	C₂³×C₃×C₄	1	1	yes	characteristic
245	C₂×D₅	—	2	2	no	—
246	C₂×D₅	—	2	8	no	—
247	C₂×D₅	—	2	4	no	—
248	C₂×D₅	C₃×C₄×D₄	1	2	yes	—
249	C₄×C₅	—	2	4	no	—
250	C₄×C₅	C₃×C₄×D₄	1	2	yes	—
251	C₄×C₅	C₃×#131	1	4	yes	complemented
252	Dic₅	—	2	2	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	C₂×C₃×AGL(1,𝔽₅)	1	2	yes	—
258	C₂×D₄	—	5	4	no	—
259	C₂×D₄	—	5	4	no	—
260	C₂×D₄	—	5	2	no	—
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	C₂×C₃×AGL(1,𝔽₅)	1	4	yes	—
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	—	5	2	no	—
280	C₂²×C₃	—	10	2	no	—
281	C₂²×C₃	—	10	4	no	—
282	C₂²×C₃	—	10	8	no	—
283	C₂²×C₃	C₂³×AGL(1,𝔽₅)	1	1	yes	characteristic
284	C₃×C₄	—	2	4	no	—
285	C₃×C₄	—	5	1	no	—
286	C₃×C₄	—	5	1	no	—
287	C₃×C₄	—	10	2	no	—
288	C₃×C₄	D₄×AGL(1,𝔽₅)	1	2	yes	—
289	C₃×C₄	#4133	1	4	yes	complemented
290	C₂×C₅	C₂×C₃×C₄×D₄	1	1	yes	characteristic
291	C₂×C₅	C₂×C₃×#131	1	2	yes	—
292	D₅	—	2	2	no	—
293	D₅	—	4	4	no	—
294	C₂³	—	5	1	no	—
295	C₂³	—	5	4	no	—
296	Q₈	—	2	4	no	—
297	D₄	—	10	8	no	—
298	D₄	—	10	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	—	10	4	no	—
305	C₂×C₄	—	10	4	no	—
306	C₂×C₄	—	10	4	no	—
307	C₂×C₄	C₂²×C₃×AGL(1,𝔽₅)	1	4	yes	—
308	C₂×C₄	C₂²×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
309	C₂×C₄	C₂²×C₃×AGL(1,𝔽₅)	1	2	yes	—
310	C₈	—	5	2	no	—
311	C₈	—	5	2	no	—
312	C₈	—	10	4	no	—
313	C₈	—	10	2	no	—
314	C₂×C₃	—	10	2	no	—
315	C₂×C₃	—	20	4	no	—
316	C₂×C₃	C₂×D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
317	C₂×C₃	C₂×#4133	1	2	yes	—
318	C₅	C₃×#2931	1	1	yes	characteristic, complemented, Sylow
319	C₂²	—	5	2	no	—
320	C₂²	—	10	2	no	—
321	C₂²	—	10	4	no	—
322	C₂²	—	10	8	no	—
323	C₂²	C₂³×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
324	C₄	—	2	4	no	—
325	C₄	—	5	1	no	—
326	C₄	—	5	1	no	—
327	C₄	—	10	2	no	—
328	C₄	C₃×D₄×AGL(1,𝔽₅)	1	2	yes	—
329	C₄	C₃×#4133	1	4	yes	complemented
330	C₃	#10617371	1	1	yes	characteristic, direct product factor, Sylow
331	C₂	—	10	2	no	—
332	C₂	—	20	4	no	—
333	C₂	C₂×C₃×D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
334	C₂	C₂×C₃×#4133	1	2	yes	—

	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×C₄⋊Q₈	C₄
socle	C₂²×C₃×C₅	C₂³×C₄
prime radical	C₃×#66059	C₂
cosocle	C₂×Dic₅	C₂⁴×C₃
hypercenter	C₃×C₄⋊Q₈	AGL(1,𝔽₅)
hypocenter	C₅	C₃×#2931
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×#4133	C₂	semidirect
3. C₃×#65787	C₂	semidirect
4. C₃×#65705	C₂	semidirect
5. C₃×#65702	C₂	semidirect
6. C₃×#65681	C₂	semidirect
7. C₃×C₄×D₅⋊C₈	C₂	semidirect
8. #10617371	C₃	direct (N × H)
27. C₂×#4133	C₂×C₃	semidirect
29. #65787	C₂×C₃	semidirect
30. #65705	C₂×C₃	semidirect
31. #65702	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
156. C₃×C₄×C₅	#131	semidirect
251. C₄×C₅	C₃×#131	semidirect
278. C₃×C₅	#2931	semidirect
289. C₃×C₄	#4133	semidirect
318. C₅	C₃×#2931	semidirect
329. C₄	C₃×#4133	semidirect
330. C₃	#10617371	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×#4133	2
C₃×#66059	2
C₃×#65787	2
C₃×#65705	2
C₃×#65702	2
C₃×#65681	2
C₃×C₄×D₅⋊C₈	2
#10617371	3
C₃×#2931	5

Subgroups — C3×#10617371 #14415244

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10617371 #14415244