Subgroups — C₃×#10604053 #14296469

Order 1920.

464 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#3995	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₈×AGL(1,𝔽₅)	C₂	1	1	yes	maximal, characteristic
3	C₃×#65701	C₂	1	2	yes	maximal
4	C₃×#65149	C₂	1	1	yes	maximal, characteristic
5	C₃×#64875	C₂	1	2	yes	maximal
6	#10604053	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×C₄×AGL(1,𝔽₅)	C₂²	1	1	yes	characteristic
8	C₂×C₃×#700	C₂²	1	2	yes	—
9	C₂×C₃×D₅⋊C₈	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₈×D₅	C₂²	1	1	yes	characteristic
11	C₃×#4038	C₂²	1	2	yes	—
12	C₃×#4002	—	2	2	no	—
13	C₃×C₄×#170	—	2	2	no	—
14	C₃×#3995	C₄	1	4	yes	complemented
15	C₃×C₈×AGL(1,𝔽₅)	C₄	1	4	yes	complemented
16	C₃×C₅×#106	C₄	1	1	yes	characteristic, complemented
17	C₃×#3949	C₄	1	1	yes	characteristic, complemented
18	C₃×#3939	C₄	1	2	yes	complemented
19	C₃×#1772	—	5	1	no	maximal, Hall
20	C₂×#3995	C₂×C₃	1	1	yes	characteristic
21	C₂×C₈×AGL(1,𝔽₅)	C₂×C₃	1	1	yes	characteristic
22	#65701	C₂×C₃	1	2	yes	—
23	#65149	C₂×C₃	1	1	yes	characteristic
24	#64875	C₂×C₃	1	2	yes	—
25	C₂²×C₃×AGL(1,𝔽₅)	—	2	1	no	—
26	C₂×C₃×C₄×D₅	C₂³	1	1	yes	characteristic
27	C₂×C₃×#170	—	2	2	no	—
28	C₂×C₃×#170	D₄	1	1	yes	characteristic
29	C₂×C₃×#170	D₄	1	1	yes	characteristic
30	C₃×C₄×AGL(1,𝔽₅)	—	2	1	no	—
31	C₃×C₄×AGL(1,𝔽₅)	C₂×C₄	1	2	yes	—
32	C₃×#700	—	2	2	no	—
33	C₃×#700	C₂×C₄	1	4	yes	—
34	C₃×D₅⋊C₈	C₂×C₄	1	2	yes	—
35	C₃×D₅⋊C₈	Q₈	1	2	yes	—
36	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
37	C₃×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
38	C₃×#685	—	2	2	no	—
39	C₃×C₄⋊Dic₅	C₂×C₄	1	2	yes	—
40	C₃×C₄×Dic₅	—	2	2	no	—
41	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
42	C₃×C₈×D₅	C₂×C₄	1	2	yes	—
43	C₃×C₈×D₅	C₂×C₄	1	2	yes	—
44	C₂×C₃×#108	—	5	1	no	—
45	C₃×#427	—	5	1	no	—
46	C₃×#405	—	5	2	no	—
47	C₂×C₃×C₄×C₈	—	5	1	no	—
48	C₃×#339	—	5	2	no	—
49	C₂×C₄×AGL(1,𝔽₅)	C₂²×C₃	1	1	yes	characteristic
50	C₂×#700	C₂²×C₃	1	2	yes	—
51	C₂×D₅⋊C₈	C₂²×C₃	1	1	yes	characteristic
52	C₂×C₈×D₅	C₂²×C₃	1	1	yes	characteristic
53	#4038	C₂²×C₃	1	2	yes	—
54	#4002	—	2	2	no	—
55	C₄×#170	—	2	2	no	—
56	#3995	C₃×C₄	1	4	yes	complemented
57	C₈×AGL(1,𝔽₅)	C₃×C₄	1	4	yes	complemented
58	C₅×#106	C₃×C₄	1	1	yes	characteristic, complemented
59	#3949	C₃×C₄	1	1	yes	characteristic, complemented
60	#3939	C₃×C₄	1	2	yes	complemented
61	#1772	—	5	1	no	Sylow
62	C₂²×C₃×D₅	#41	1	1	yes	characteristic
63	C₂×C₃×AGL(1,𝔽₅)	—	2	4	no	—
64	C₂×C₃×AGL(1,𝔽₅)	—	2	2	no	—
65	C₂×C₃×C₄×C₅	—	2	2	no	—
66	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₃×Dic₅	—	2	2	no	—
68	C₂×C₃×Dic₅	C₂×D₄	1	1	yes	characteristic
69	C₃×C₄×D₅	C₂²×C₄	1	1	yes	characteristic
70	C₃×C₄×D₅	C₂²×C₄	1	1	yes	characteristic
71	C₃×C₄×D₅	C₂×Q₈	1	1	yes	characteristic
72	C₃×C₄×D₅	#41	1	1	yes	characteristic
73	C₃×#170	—	2	4	no	—
74	C₃×#170	C₄⋊C₄	1	2	yes	—
75	C₃×#170	C₄⋊C₄	1	2	yes	—
76	C₃×C₅×C₈	C₄²	1	2	yes	—
77	C₃×#168	C₄²	1	2	yes	—
78	C₂×C₃×M₁₆	—	5	2	no	—
79	C₂²×C₃×C₈	—	5	1	no	—
80	C₂²×C₃×C₈	—	5	1	no	—
81	C₃×#117	—	5	2	no	—
82	C₂×C₃×C₄²	—	5	1	no	—
83	C₃×#108	—	5	4	no	—
84	C₃×#107	—	5	2	no	—
85	C₃×#106	—	5	1	no	—
86	C₃×#106	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
87	C₃×#97	—	10	2	no	—
88	C₃×C₄×C₈	—	5	4	no	—
89	C₃×C₄×C₈	—	10	2	no	—
90	C₂²×AGL(1,𝔽₅)	—	2	1	no	—
91	C₂×C₄×D₅	C₂³×C₃	1	1	yes	characteristic
92	C₂×#170	—	2	2	no	—
93	C₂×#170	C₃×D₄	1	1	yes	characteristic
94	C₂×#170	C₃×D₄	1	1	yes	characteristic
95	C₄×AGL(1,𝔽₅)	—	2	1	no	—
96	C₄×AGL(1,𝔽₅)	C₂×C₃×C₄	1	2	yes	—
97	#700	—	2	2	no	—
98	#700	C₂×C₃×C₄	1	4	yes	—
99	D₅⋊C₈	C₂×C₃×C₄	1	2	yes	—
100	D₅⋊C₈	C₃×Q₈	1	2	yes	—
101	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
102	C₅×C₄⋊C₄	C₂×C₃×C₄	1	2	yes	—
103	#685	—	2	2	no	—
104	C₄⋊Dic₅	C₂×C₃×C₄	1	2	yes	—
105	C₄×Dic₅	—	2	2	no	—
106	C₂×#168	C₂×C₃×C₄	1	1	yes	characteristic
107	C₈×D₅	C₂×C₃×C₄	1	2	yes	—
108	C₈×D₅	C₂×C₃×C₄	1	2	yes	—
109	C₂×#108	—	5	1	no	—
110	#427	—	5	1	no	—
111	#405	—	5	2	no	—
112	C₂×C₄×C₈	—	5	1	no	—
113	#339	—	5	2	no	—
114	C₂²×C₃×C₅	C₄×D₄	1	1	yes	characteristic
115	C₂×C₃×D₅	—	2	2	no	—
116	C₂×C₃×D₅	#117	1	1	yes	characteristic
117	C₂×C₃×D₅	C₄×Q₈	1	1	yes	characteristic
118	C₃×AGL(1,𝔽₅)	—	2	4	no	—
119	C₃×C₄×C₅	—	4	2	no	—
120	C₃×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
121	C₃×C₄×C₅	C₄×Q₈	1	1	yes	characteristic
122	C₃×Dic₅	—	4	2	no	—
123	C₃×Dic₅	C₂×C₄⋊C₄	1	1	yes	characteristic
124	C₃×Dic₅	C₄×D₄	1	1	yes	characteristic
125	C₂²×C₃×C₄	—	5	1	no	—
126	C₂²×C₃×C₄	—	10	1	no	—
127	C₃×M₁₆	—	5	4	no	—
128	C₃×M₁₆	—	10	2	no	—
129	C₂×C₃×C₈	—	5	2	no	—
130	C₂×C₃×C₈	—	5	1	no	—
131	C₂×C₃×C₈	—	5	1	no	—
132	C₂×C₃×C₈	—	5	2	no	—
133	C₂×C₃×C₈	—	5	1	no	—
134	C₂×C₃×C₈	—	5	2	no	—
135	C₂×C₃×C₈	—	5	2	no	—
136	C₂×C₃×C₈	—	10	2	no	—
137	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
138	C₃×C₄⋊C₄	—	5	2	no	—
139	C₃×C₄⋊C₄	C₂×AGL(1,𝔽₅)	1	2	yes	—
140	C₃×C₂²⋊C₄	—	10	2	no	—
141	C₃×C₄²	—	5	2	no	—
142	C₃×C₄²	—	10	2	no	—
143	C₃×C₄²	—	10	1	no	—
144	C₂²×D₅	C₃×#41	1	1	yes	characteristic
145	C₂×AGL(1,𝔽₅)	—	2	4	no	—
146	C₂×AGL(1,𝔽₅)	—	2	2	no	—
147	C₂×C₄×C₅	—	2	2	no	—
148	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
149	C₂×Dic₅	—	2	2	no	—
150	C₂×Dic₅	C₂×C₃×D₄	1	1	yes	characteristic
151	C₄×D₅	C₂²×C₃×C₄	1	1	yes	characteristic
152	C₄×D₅	C₂²×C₃×C₄	1	1	yes	characteristic
153	C₄×D₅	C₂×C₃×Q₈	1	1	yes	characteristic
154	C₄×D₅	C₃×#41	1	1	yes	characteristic
155	#170	—	2	4	no	—
156	#170	C₃×C₄⋊C₄	1	2	yes	—
157	#170	C₃×C₄⋊C₄	1	2	yes	—
158	C₅×C₈	C₃×C₄²	1	2	yes	—
159	#168	C₃×C₄²	1	2	yes	—
160	C₂×M₁₆	—	5	2	no	—
161	C₂²×C₈	—	5	1	no	—
162	C₂²×C₈	—	5	1	no	—
163	#117	—	5	2	no	—
164	C₂×C₄²	—	5	1	no	—
165	#108	—	5	4	no	—
166	#107	—	5	2	no	—
167	#106	—	5	1	no	—
168	#106	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
169	#97	—	10	2	no	—
170	C₄×C₈	—	5	4	no	—
171	C₄×C₈	—	10	2	no	—
172	C₂×C₃×C₅	C₄×C₄⋊C₄	1	1	yes	characteristic
173	C₂×C₃×C₅	#443	1	2	yes	—
174	C₃×D₅	—	2	1	no	—
175	C₃×D₅	—	2	1	no	—
176	C₂³×C₃	—	5	1	no	—
177	C₂×C₃×C₄	—	2	2	no	—
178	C₂×C₃×C₄	—	5	1	no	—
179	C₂×C₃×C₄	—	5	1	no	—
180	C₂×C₃×C₄	—	5	1	no	—
181	C₂×C₃×C₄	—	5	1	no	—
182	C₂×C₃×C₄	—	5	1	no	—
183	C₂×C₃×C₄	—	10	2	no	—
184	C₂×C₃×C₄	—	10	4	no	—
185	C₂×C₃×C₄	—	10	2	no	—
186	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
187	C₃×C₈	—	5	2	no	—
188	C₃×C₈	—	5	2	no	—
189	C₃×C₈	—	5	2	no	—
190	C₃×C₈	—	10	4	no	—
191	C₃×C₈	C₄×AGL(1,𝔽₅)	1	2	yes	—
192	C₂²×C₅	C₃×C₄×D₄	1	1	yes	characteristic
193	C₂×D₅	—	2	2	no	—
194	C₂×D₅	C₃×#117	1	1	yes	characteristic
195	C₂×D₅	C₃×C₄×Q₈	1	1	yes	characteristic
196	AGL(1,𝔽₅)	—	2	4	no	—
197	C₄×C₅	—	4	2	no	—
198	C₄×C₅	C₂×C₃×C₄²	1	1	yes	characteristic
199	C₄×C₅	C₃×C₄×Q₈	1	1	yes	characteristic
200	Dic₅	—	4	2	no	—
201	Dic₅	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
202	Dic₅	C₃×C₄×D₄	1	1	yes	characteristic
203	C₂²×C₄	—	5	1	no	—
204	C₂²×C₄	—	10	1	no	—
205	M₁₆	—	5	4	no	—
206	M₁₆	—	10	2	no	—
207	C₂×C₈	—	5	2	no	—
208	C₂×C₈	—	5	1	no	—
209	C₂×C₈	—	5	1	no	—
210	C₂×C₈	—	5	2	no	—
211	C₂×C₈	—	5	1	no	—
212	C₂×C₈	—	5	2	no	—
213	C₂×C₈	—	5	2	no	—
214	C₂×C₈	—	10	2	no	—
215	C₂×C₈	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
216	C₄⋊C₄	—	5	2	no	—
217	C₄⋊C₄	C₂×C₃×AGL(1,𝔽₅)	1	2	yes	—
218	C₂²⋊C₄	—	10	2	no	—
219	C₄²	—	5	2	no	—
220	C₄²	—	10	2	no	—
221	C₄²	—	10	1	no	—
222	C₃×C₅	#1772	1	1	yes	characteristic, complemented, Hall
223	C₂²×C₃	—	5	1	no	—
224	C₂²×C₃	—	5	1	no	—
225	C₂²×C₃	—	10	2	no	—
226	C₂²×C₃	D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
227	C₃×C₄	—	4	2	no	—
228	C₃×C₄	—	5	1	no	—
229	C₃×C₄	—	5	1	no	—
230	C₃×C₄	—	10	4	no	—
231	C₃×C₄	—	20	2	no	—
232	C₃×C₄	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
233	C₃×C₄	Q₈×AGL(1,𝔽₅)	1	1	yes	characteristic
234	C₂×C₅	C₃×C₄×C₄⋊C₄	1	1	yes	characteristic
235	C₂×C₅	C₃×#443	1	2	yes	—
236	D₅	—	2	1	no	—
237	D₅	—	2	1	no	—
238	C₂³	—	5	1	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	5	1	no	—
241	C₂×C₄	—	5	1	no	—
242	C₂×C₄	—	5	1	no	—
243	C₂×C₄	—	5	1	no	—
244	C₂×C₄	—	5	1	no	—
245	C₂×C₄	—	10	2	no	—
246	C₂×C₄	—	10	4	no	—
247	C₂×C₄	—	10	2	no	—
248	C₂×C₄	C₂²×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
249	C₈	—	5	2	no	—
250	C₈	—	5	2	no	—
251	C₈	—	5	2	no	—
252	C₈	—	10	4	no	—
253	C₈	C₃×C₄×AGL(1,𝔽₅)	1	2	yes	—
254	C₂×C₃	—	10	1	no	—
255	C₂×C₃	—	10	1	no	—
256	C₂×C₃	C₄⋊C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
257	C₂×C₃	#65736	1	2	yes	—
258	C₅	C₃×#1772	1	1	yes	characteristic, complemented, Sylow
259	C₂²	—	5	1	no	—
260	C₂²	—	5	1	no	—
261	C₂²	—	10	2	no	—
262	C₂²	C₃×D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
263	C₄	—	4	2	no	—
264	C₄	—	5	1	no	—
265	C₄	—	5	1	no	—
266	C₄	—	10	4	no	—
267	C₄	—	20	2	no	—
268	C₄	C₂×C₃×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
269	C₄	C₃×Q₈×AGL(1,𝔽₅)	1	1	yes	characteristic
270	C₃	#10604053	1	1	yes	characteristic, direct product factor, Sylow
271	C₂	—	10	1	no	—
272	C₂	—	10	1	no	—
273	C₂	C₃×C₄⋊C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
274	C₂	C₃×#65736	1	2	yes	—

	subgroup	G/H
center	C₂²×C₃	D₄×AGL(1,𝔽₅)
commutator	C₄×C₅	C₂×C₃×C₄²
Frattini	C₂×C₄	C₂²×C₃×AGL(1,𝔽₅)
Fitting	C₃×C₅×#106	C₄
socle	C₂²×C₃×C₅	C₄×D₄
prime radical	C₂²×C₃×D₅	#41
cosocle	C₂×C₄×D₅	C₂³×C₃
hypercenter	C₃×#106	AGL(1,𝔽₅)
hypocenter	C₅	C₃×#1772
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄ ◁ C₃×#106

Complements

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

N	H ≅ G/N	type
6. #10604053	C₃	direct (N × H)
14. C₃×#3995	C₄	semidirect
15. C₃×C₈×AGL(1,𝔽₅)	C₄	semidirect
16. C₃×C₅×#106	C₄	semidirect
17. C₃×#3949	C₄	semidirect
18. C₃×#3939	C₄	semidirect
56. #3995	C₃×C₄	semidirect
57. C₈×AGL(1,𝔽₅)	C₃×C₄	semidirect
58. C₅×#106	C₃×C₄	semidirect
59. #3949	C₃×C₄	semidirect
60. #3939	C₃×C₄	semidirect
86. C₃×#106	AGL(1,𝔽₅)	semidirect
168. #106	C₃×AGL(1,𝔽₅)	semidirect
222. C₃×C₅	#1772	semidirect
258. C₅	C₃×#1772	semidirect
270. C₃	#10604053	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×#3995	2
C₂×C₃×C₈×AGL(1,𝔽₅)	2
C₃×#65701	2
C₃×#65149	2
C₃×#64875	2
#10604053	3
C₃×#1772	5

Subgroups — C3×#10604053 #14296469

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10604053 #14296469