Subgroups — C₃×#10603895 #14280612

Order 1920.

380 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×#704	C₂	1	1	yes	maximal, characteristic
2	C₃×#65748	C₂	1	1	yes	maximal, characteristic
3	C₃×#65007	C₂	1	1	yes	maximal, characteristic
4	C₃×#170⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#170⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#64879	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×C₈×#170	C₂	1	1	yes	maximal, characteristic, complemented
8	#10603895	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂×C₃×#704	C₄	1	2	yes	—
10	C₂×C₃×C₄×Dic₅	C₂²	1	1	yes	characteristic
11	C₃×C₂²⋊#170	C₄	1	2	yes	—
12	C₃×#4002	C₄	1	2	yes	—
13	C₃×C₄⋊#170	C₄	1	2	yes	—
14	C₃×C₄×#170	C₂²	1	1	yes	characteristic
15	C₃×C₄×#170	C₂²	1	1	yes	characteristic
16	C₃×C₄×#170	C₂²	1	1	yes	characteristic
17	C₃×C₄×#170	C₂²	1	1	yes	characteristic
18	C₃×C₅×C₂²⋊C₈	C₄	1	1	yes	characteristic
19	C₃×C₂²⋊#168	C₄	1	1	yes	characteristic
20	C₃×#3946	C₄	1	1	yes	characteristic
21	C₃×#3946	C₄	1	1	yes	characteristic
22	C₃×C₈×Dic₅	C₂²	1	1	yes	characteristic
23	C₃×C₈×Dic₅	C₂²	1	1	yes	characteristic
24	C₃×#1453	—	5	1	no	maximal, Hall
25	C₄×#704	C₂×C₃	1	1	yes	characteristic
26	#65748	C₂×C₃	1	1	yes	characteristic
27	#65007	C₂×C₃	1	1	yes	characteristic
28	#170⋊C₈	C₂×C₃	1	1	yes	characteristic, complemented
29	#170⋊C₈	C₂×C₃	1	1	yes	characteristic, complemented
30	#64879	C₂×C₃	1	1	yes	characteristic, complemented
31	C₈×#170	C₂×C₃	1	1	yes	characteristic, complemented
32	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₃×Dic₅	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₃×Dic₅	C₂×C₄	1	1	yes	characteristic
35	C₃×#704	—	2	4	no	—
36	C₂×C₃×#170	C₂×C₄	1	2	yes	—
37	C₂×C₃×#170	C₂×C₄	1	2	yes	—
38	C₂×C₃×#170	C₂×C₄	1	2	yes	—
39	C₂×C₃×#170	C₂×C₄	1	2	yes	—
40	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
41	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
42	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
43	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
44	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
45	C₃×C₄×Dic₅	C₂³	1	1	yes	characteristic
46	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
47	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
48	C₃×#431	—	5	1	no	—
49	C₃×#431	—	5	1	no	—
50	C₃×C₄×M₁₆	—	5	1	no	—
51	C₃×#322	—	5	1	no	—
52	C₃×#322	—	5	1	no	—
53	C₃×#322	—	5	1	no	—
54	C₃×C₈²	—	5	1	no	—
55	C₂×#704	C₃×C₄	1	2	yes	—
56	C₂×C₄×Dic₅	C₂²×C₃	1	1	yes	characteristic
57	C₂²⋊#170	C₃×C₄	1	2	yes	—
58	#4002	C₃×C₄	1	2	yes	—
59	C₄⋊#170	C₃×C₄	1	2	yes	—
60	C₄×#170	C₂²×C₃	1	1	yes	characteristic
61	C₄×#170	C₂²×C₃	1	1	yes	characteristic
62	C₄×#170	C₂²×C₃	1	1	yes	characteristic
63	C₄×#170	C₂²×C₃	1	1	yes	characteristic
64	C₅×C₂²⋊C₈	C₃×C₄	1	1	yes	characteristic
65	C₂²⋊#168	C₃×C₄	1	1	yes	characteristic
66	#3946	C₃×C₄	1	1	yes	characteristic
67	#3946	C₃×C₄	1	1	yes	characteristic
68	C₈×Dic₅	C₂²×C₃	1	1	yes	characteristic
69	C₈×Dic₅	C₂²×C₃	1	1	yes	characteristic
70	#1453	—	5	1	no	Sylow
71	C₂³×C₃×C₅	C₄²	1	1	yes	characteristic
72	C₂×C₃×C₄×C₅	—	2	1	no	—
73	C₂×C₃×C₄×C₅	—	2	1	no	—
74	C₂×C₃×C₄×C₅	C₄²	1	1	yes	characteristic
75	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₃×Dic₅	—	2	2	no	—
77	C₂×C₃×Dic₅	—	2	2	no	—
78	C₂×C₃×Dic₅	C₄²	1	1	yes	characteristic
79	C₂×C₃×Dic₅	C₄²	1	1	yes	characteristic
80	C₂×C₃×Dic₅	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₃×Dic₅	C₂²×C₄	1	1	yes	characteristic
82	C₃×#170	—	2	2	no	—
83	C₃×#170	—	2	2	no	—
84	C₃×#170	—	2	2	no	—
85	C₃×#170	M₁₆	1	4	yes	—
86	C₃×C₅×C₈	—	2	1	no	—
87	C₃×C₅×C₈	—	2	1	no	—
88	C₃×#168	—	2	1	no	—
89	C₃×#168	—	2	1	no	—
90	C₂×C₃×M₁₆	—	5	2	no	—
91	C₂×C₃×C₄²	—	5	1	no	—
92	C₃×C₄⋊C₈	—	5	2	no	—
93	C₃×C₄⋊C₈	—	5	1	no	—
94	C₃×C₄⋊C₈	—	5	1	no	—
95	C₃×C₂²⋊C₈	—	5	2	no	—
96	C₃×C₂²⋊C₈	—	5	1	no	—
97	C₃×C₂²⋊C₈	AGL(1,𝔽₅)	1	1	yes	characteristic
98	C₃×#97	—	5	2	no	—
99	C₃×C₄×C₈	—	5	1	no	—
100	C₃×C₄×C₈	—	5	1	no	—
101	C₃×C₄×C₈	—	5	1	no	—
102	C₃×C₄×C₈	—	5	1	no	—
103	C₃×C₄×C₈	—	5	1	no	—
104	C₃×C₄×C₈	—	5	1	no	—
105	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
106	C₂²×Dic₅	C₂×C₃×C₄	1	1	yes	characteristic
107	C₂²×Dic₅	C₂×C₃×C₄	1	1	yes	characteristic
108	#704	—	2	4	no	—
109	C₂×#170	C₂×C₃×C₄	1	2	yes	—
110	C₂×#170	C₂×C₃×C₄	1	2	yes	—
111	C₂×#170	C₂×C₃×C₄	1	2	yes	—
112	C₂×#170	C₂×C₃×C₄	1	2	yes	—
113	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
114	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
115	C₄×Dic₅	C₂×C₃×C₄	1	1	yes	characteristic
116	C₄×Dic₅	C₂×C₃×C₄	1	1	yes	characteristic
117	C₄×Dic₅	C₂×C₃×C₄	1	1	yes	characteristic
118	C₄×Dic₅	C₂³×C₃	1	1	yes	characteristic
119	C₂×#168	C₂×C₃×C₄	1	1	yes	characteristic
120	C₂×#168	C₂×C₃×C₄	1	1	yes	characteristic
121	#431	—	5	1	no	—
122	#431	—	5	1	no	—
123	C₄×M₁₆	—	5	1	no	—
124	#322	—	5	1	no	—
125	#322	—	5	1	no	—
126	#322	—	5	1	no	—
127	C₈²	—	5	1	no	—
128	C₂²×C₃×C₅	—	2	1	no	—
129	C₂²×C₃×C₅	—	2	1	no	—
130	C₂²×C₃×C₅	—	2	1	no	—
131	C₂²×C₃×C₅	C₂×C₄²	1	1	yes	characteristic
132	C₃×C₄×C₅	—	2	1	no	—
133	C₃×C₄×C₅	#131	1	1	yes	characteristic
134	C₃×C₄×C₅	#131	1	1	yes	characteristic
135	C₃×Dic₅	—	2	1	no	—
136	C₃×Dic₅	—	2	1	no	—
137	C₃×Dic₅	C₂×M₁₆	1	2	yes	—
138	C₃×Dic₅	#131	1	2	yes	—
139	C₂²×C₃×C₄	—	5	1	no	—
140	C₂²×C₃×C₄	—	5	1	no	—
141	C₂²×C₃×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
142	C₃×M₁₆	—	10	4	no	—
143	C₂×C₃×C₈	—	5	2	no	—
144	C₂×C₃×C₈	—	5	2	no	—
145	C₂×C₃×C₈	—	5	2	no	—
146	C₂×C₃×C₈	—	5	2	no	—
147	C₂×C₃×C₈	—	5	1	no	—
148	C₂×C₃×C₈	—	5	1	no	—
149	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
150	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
151	C₃×C₄²	—	5	1	no	—
152	C₃×C₄²	—	5	1	no	—
153	C₃×C₄²	—	5	1	no	—
154	C₃×C₄²	—	5	1	no	—
155	C₂³×C₅	C₃×C₄²	1	1	yes	characteristic
156	C₂×C₄×C₅	—	2	1	no	—
157	C₂×C₄×C₅	—	2	1	no	—
158	C₂×C₄×C₅	C₃×C₄²	1	1	yes	characteristic
159	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
160	C₂×Dic₅	—	2	2	no	—
161	C₂×Dic₅	—	2	2	no	—
162	C₂×Dic₅	C₃×C₄²	1	1	yes	characteristic
163	C₂×Dic₅	C₃×C₄²	1	1	yes	characteristic
164	C₂×Dic₅	C₂²×C₃×C₄	1	1	yes	characteristic
165	C₂×Dic₅	C₂²×C₃×C₄	1	1	yes	characteristic
166	#170	—	2	2	no	—
167	#170	—	2	2	no	—
168	#170	—	2	2	no	—
169	#170	C₃×M₁₆	1	4	yes	—
170	C₅×C₈	—	2	1	no	—
171	C₅×C₈	—	2	1	no	—
172	#168	—	2	1	no	—
173	#168	—	2	1	no	—
174	C₂×M₁₆	—	5	2	no	—
175	C₂×C₄²	—	5	1	no	—
176	C₄⋊C₈	—	5	2	no	—
177	C₄⋊C₈	—	5	1	no	—
178	C₄⋊C₈	—	5	1	no	—
179	C₂²⋊C₈	—	5	2	no	—
180	C₂²⋊C₈	—	5	1	no	—
181	C₂²⋊C₈	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
182	#97	—	5	2	no	—
183	C₄×C₈	—	5	1	no	—
184	C₄×C₈	—	5	1	no	—
185	C₄×C₈	—	5	1	no	—
186	C₄×C₈	—	5	1	no	—
187	C₄×C₈	—	5	1	no	—
188	C₄×C₈	—	5	1	no	—
189	C₂×C₃×C₅	—	4	1	no	—
190	C₂×C₃×C₅	C₄×M₁₆	1	1	yes	characteristic
191	C₂×C₃×C₅	#405	1	1	yes	characteristic
192	C₂×C₃×C₅	#405	1	1	yes	characteristic
193	C₂³×C₃	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
194	C₂×C₃×C₄	—	2	1	no	—
195	C₂×C₃×C₄	—	2	1	no	—
196	C₂×C₃×C₄	—	5	1	no	—
197	C₂×C₃×C₄	—	5	1	no	—
198	C₂×C₃×C₄	—	5	1	no	—
199	C₂×C₃×C₄	—	5	1	no	—
200	C₂×C₃×C₄	—	10	2	no	—
201	C₂×C₃×C₄	—	10	2	no	—
202	C₂×C₃×C₄	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
203	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
204	C₃×C₈	—	2	1	no	—
205	C₃×C₈	—	2	1	no	—
206	C₃×C₈	—	5	4	no	—
207	C₃×C₈	—	10	2	no	—
208	C₃×C₈	—	10	2	no	—
209	C₃×C₈	—	10	2	no	—
210	C₃×C₈	—	10	1	no	—
211	C₃×C₈	—	10	1	no	—
212	C₂²×C₅	—	2	1	no	—
213	C₂²×C₅	—	2	1	no	—
214	C₂²×C₅	—	2	1	no	—
215	C₂²×C₅	C₂×C₃×C₄²	1	1	yes	characteristic
216	C₄×C₅	—	2	1	no	—
217	C₄×C₅	C₃×#131	1	1	yes	characteristic
218	C₄×C₅	C₃×#131	1	1	yes	characteristic
219	Dic₅	—	2	1	no	—
220	Dic₅	—	2	1	no	—
221	Dic₅	C₂×C₃×M₁₆	1	2	yes	—
222	Dic₅	C₃×#131	1	2	yes	—
223	C₂²×C₄	—	5	1	no	—
224	C₂²×C₄	—	5	1	no	—
225	C₂²×C₄	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
226	M₁₆	—	10	4	no	—
227	C₂×C₈	—	5	2	no	—
228	C₂×C₈	—	5	2	no	—
229	C₂×C₈	—	5	2	no	—
230	C₂×C₈	—	5	2	no	—
231	C₂×C₈	—	5	1	no	—
232	C₂×C₈	—	5	1	no	—
233	C₂×C₈	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
234	C₂×C₈	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
235	C₄²	—	5	1	no	—
236	C₄²	—	5	1	no	—
237	C₄²	—	5	1	no	—
238	C₄²	—	5	1	no	—
239	C₃×C₅	#1453	1	1	yes	characteristic, complemented, Hall
240	C₂²×C₃	—	2	1	no	—
241	C₂²×C₃	—	2	1	no	—
242	C₂²×C₃	—	2	1	no	—
243	C₂²×C₃	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
244	C₃×C₄	—	2	1	no	—
245	C₃×C₄	—	5	2	no	—
246	C₃×C₄	—	5	2	no	—
247	C₃×C₄	—	10	1	no	—
248	C₃×C₄	—	10	1	no	—
249	C₃×C₄	#4131	1	1	yes	characteristic
250	C₃×C₄	#4131	1	1	yes	characteristic
251	C₂×C₅	—	4	1	no	—
252	C₂×C₅	C₃×C₄×M₁₆	1	1	yes	characteristic
253	C₂×C₅	C₃×#405	1	1	yes	characteristic
254	C₂×C₅	C₃×#405	1	1	yes	characteristic
255	C₂³	C₃×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	5	1	no	—
259	C₂×C₄	—	5	1	no	—
260	C₂×C₄	—	5	1	no	—
261	C₂×C₄	—	5	1	no	—
262	C₂×C₄	—	10	2	no	—
263	C₂×C₄	—	10	2	no	—
264	C₂×C₄	C₃×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
265	C₂×C₄	C₂²×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
266	C₈	—	2	1	no	—
267	C₈	—	2	1	no	—
268	C₈	—	5	4	no	—
269	C₈	—	10	2	no	—
270	C₈	—	10	2	no	—
271	C₈	—	10	2	no	—
272	C₈	—	10	1	no	—
273	C₈	—	10	1	no	—
274	C₂×C₃	—	4	1	no	—
275	C₂×C₃	#65691	1	1	yes	characteristic
276	C₂×C₃	#65718	1	1	yes	characteristic
277	C₂×C₃	M₁₆×AGL(1,𝔽₅)	1	1	yes	characteristic
278	C₅	C₃×#1453	1	1	yes	characteristic, complemented, Sylow
279	C₂²	—	2	1	no	—
280	C₂²	—	2	1	no	—
281	C₂²	—	2	1	no	—
282	C₂²	C₂×C₃×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
283	C₄	—	2	1	no	—
284	C₄	—	5	2	no	—
285	C₄	—	5	2	no	—
286	C₄	—	10	1	no	—
287	C₄	—	10	1	no	—
288	C₄	C₃×#4131	1	1	yes	characteristic
289	C₄	C₃×#4131	1	1	yes	characteristic
290	C₃	#10603895	1	1	yes	characteristic, direct product factor, Sylow
291	C₂	—	4	1	no	—
292	C₂	C₃×#65691	1	1	yes	characteristic
293	C₂	C₃×#65718	1	1	yes	characteristic
294	C₂	C₃×M₁₆×AGL(1,𝔽₅)	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)
commutator	C₂²×C₅	C₂×C₃×C₄²
Frattini	C₂×C₄	C₂²×C₃×AGL(1,𝔽₅)
Fitting	C₃×C₅×C₂²⋊C₈	C₄
socle	C₂²×C₃×C₅	C₂×C₄²
prime radical	C₂³×C₃×C₅	C₄²
cosocle	C₄×Dic₅	C₂³×C₃
hypercenter	C₃×C₂²⋊C₈	AGL(1,𝔽₅)
hypocenter	C₅	C₃×#1453
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₃×C₂²⋊C₈

Complements

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

N	H ≅ G/N	type
4. C₃×#170⋊C₈	C₂	semidirect
5. C₃×#170⋊C₈	C₂	semidirect
6. C₃×#64879	C₂	semidirect
7. C₃×C₈×#170	C₂	semidirect
8. #10603895	C₃	direct (N × H)
28. #170⋊C₈	C₂×C₃	semidirect
29. #170⋊C₈	C₂×C₃	semidirect
30. #64879	C₂×C₃	semidirect
31. C₈×#170	C₂×C₃	semidirect
239. C₃×C₅	#1453	semidirect
278. C₅	C₃×#1453	semidirect
290. C₃	#10603895	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×C₄×#704	2
C₃×#65748	2
C₃×#65007	2
C₃×#170⋊C₈	2
C₃×#64879	2
C₃×C₈×#170	2
#10603895	3
C₃×#1453	5

Subgroups — C3×#10603895 #14280612

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10603895 #14280612