Subgroups — C₃×#10604057 #14305232

Order 1920.

416 conjugacy classes of subgroups.

The subgroup lattice has 268 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×#65708	C₂	1	2	yes	maximal
4	C₃×#65149	C₂	1	1	yes	maximal, characteristic
5	C₃×#64875	C₂	1	2	yes	maximal
6	#10604057	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×#4004	—	2	2	no	—
13	C₃×C₄⋊#170	—	2	2	no	—
14	C₃×#3995	—	2	2	no	—
15	C₃×C₈×AGL(1,𝔽₅)	—	2	2	no	—
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×#1945	—	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	#65708	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	2	yes	—
29	C₃×C₄×AGL(1,𝔽₅)	—	2	1	no	—
30	C₃×C₄×AGL(1,𝔽₅)	Q₈	1	2	yes	—
31	C₃×#700	—	2	2	no	—
32	C₃×#700	—	2	2	no	—
33	C₃×D₅⋊C₈	—	2	1	no	—
34	C₃×D₅⋊C₈	—	2	1	no	—
35	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
36	C₃×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
37	C₃×#685	—	2	2	no	—
38	C₃×C₄⋊Dic₅	C₂×C₄	1	2	yes	—
39	C₃×C₄×Dic₅	—	2	2	no	—
40	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
41	C₃×C₈×D₅	D₄	1	2	yes	—
42	C₃×C₈×D₅	Q₈	1	2	yes	—
43	C₂×C₃×#108	—	5	1	no	—
44	C₃×#427	—	5	1	no	—
45	C₃×#424	—	5	2	no	—
46	C₂×C₃×C₄×C₈	—	5	1	no	—
47	C₃×#339	—	5	2	no	—
48	C₂×C₄×AGL(1,𝔽₅)	C₂²×C₃	1	1	yes	characteristic
49	C₂×#700	C₂²×C₃	1	2	yes	—
50	C₂×D₅⋊C₈	C₂²×C₃	1	1	yes	characteristic
51	C₂×C₈×D₅	C₂²×C₃	1	1	yes	characteristic
52	#4038	C₂²×C₃	1	2	yes	—
53	#4004	—	2	2	no	—
54	C₄⋊#170	—	2	2	no	—
55	#3995	—	2	2	no	—
56	C₈×AGL(1,𝔽₅)	—	2	2	no	—
57	C₅×#106	C₃×C₄	1	1	yes	characteristic, complemented
58	#3949	C₃×C₄	1	1	yes	characteristic, complemented
59	#3939	C₃×C₄	1	2	yes	complemented
60	#1945	—	5	1	no	Sylow
61	C₂²×C₃×D₅	#41	1	1	yes	characteristic
62	C₂×C₃×AGL(1,𝔽₅)	—	2	4	no	—
63	C₂×C₃×AGL(1,𝔽₅)	—	2	2	no	—
64	C₂×C₃×C₄×C₅	—	2	2	no	—
65	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₃×Dic₅	—	2	2	no	—
67	C₂×C₃×Dic₅	C₂×D₄	1	1	yes	characteristic
68	C₃×C₄×D₅	C₂×D₄	1	1	yes	characteristic
69	C₃×C₄×D₅	C₂×Q₈	1	1	yes	characteristic
70	C₃×C₄×D₅	C₂×Q₈	1	1	yes	characteristic
71	C₃×C₄×D₅	#41	1	1	yes	characteristic
72	C₃×#170	—	2	2	no	—
73	C₃×#170	—	4	2	no	—
74	C₃×C₅×C₈	C₄⋊C₄	1	2	yes	—
75	C₃×#168	C₄⋊C₄	1	2	yes	—
76	C₂×C₃×M₁₆	—	5	2	no	—
77	C₂²×C₃×C₈	—	5	1	no	—
78	C₂²×C₃×C₈	—	5	1	no	—
79	C₃×#117	—	5	2	no	—
80	C₂×C₃×C₄²	—	5	1	no	—
81	C₃×#108	—	10	2	no	—
82	C₃×#107	—	5	2	no	—
83	C₃×#106	—	5	1	no	—
84	C₃×#106	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
85	C₃×C₄⋊C₈	—	10	2	no	—
86	C₃×C₄⋊C₈	—	10	2	no	—
87	C₃×C₄×C₈	—	10	2	no	—
88	C₂²×AGL(1,𝔽₅)	—	2	1	no	—
89	C₂×C₄×D₅	C₂³×C₃	1	1	yes	characteristic
90	C₂×#170	—	2	2	no	—
91	C₂×#170	C₃×D₄	1	2	yes	—
92	C₄×AGL(1,𝔽₅)	—	2	1	no	—
93	C₄×AGL(1,𝔽₅)	C₃×Q₈	1	2	yes	—
94	#700	—	2	2	no	—
95	#700	—	2	2	no	—
96	D₅⋊C₈	—	2	1	no	—
97	D₅⋊C₈	—	2	1	no	—
98	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
99	C₅×C₄⋊C₄	C₂×C₃×C₄	1	2	yes	—
100	#685	—	2	2	no	—
101	C₄⋊Dic₅	C₂×C₃×C₄	1	2	yes	—
102	C₄×Dic₅	—	2	2	no	—
103	C₂×#168	C₂×C₃×C₄	1	1	yes	characteristic
104	C₈×D₅	C₃×D₄	1	2	yes	—
105	C₈×D₅	C₃×Q₈	1	2	yes	—
106	C₂×#108	—	5	1	no	—
107	#427	—	5	1	no	—
108	#424	—	5	2	no	—
109	C₂×C₄×C₈	—	5	1	no	—
110	#339	—	5	2	no	—
111	C₂²×C₃×C₅	C₄×D₄	1	1	yes	characteristic
112	C₂×C₃×D₅	—	2	2	no	—
113	C₂×C₃×D₅	C₂²⋊Q₈	1	1	yes	characteristic
114	C₂×C₃×D₅	#125	1	1	yes	characteristic
115	C₃×AGL(1,𝔽₅)	—	2	4	no	—
116	C₃×C₄×C₅	—	4	2	no	—
117	C₃×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
118	C₃×C₄×C₅	C₄×Q₈	1	1	yes	characteristic
119	C₃×Dic₅	—	4	2	no	—
120	C₃×Dic₅	#121	1	1	yes	characteristic
121	C₃×Dic₅	C₄⋊Q₈	1	1	yes	characteristic
122	C₂²×C₃×C₄	—	5	1	no	—
123	C₂²×C₃×C₄	—	10	1	no	—
124	C₃×M₁₆	—	10	2	no	—
125	C₃×M₁₆	—	10	2	no	—
126	C₂×C₃×C₈	—	5	1	no	—
127	C₂×C₃×C₈	—	5	2	no	—
128	C₂×C₃×C₈	—	5	2	no	—
129	C₂×C₃×C₈	—	5	2	no	—
130	C₂×C₃×C₈	—	10	1	no	—
131	C₂×C₃×C₈	—	10	1	no	—
132	C₂×C₃×C₈	—	10	2	no	—
133	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
134	C₃×C₄⋊C₄	—	5	2	no	—
135	C₃×C₄⋊C₄	C₂×AGL(1,𝔽₅)	1	2	yes	—
136	C₃×C₂²⋊C₄	—	10	2	no	—
137	C₃×C₄²	—	5	2	no	—
138	C₃×C₄²	—	10	2	no	—
139	C₃×C₄²	—	10	1	no	—
140	C₂²×D₅	C₃×#41	1	1	yes	characteristic
141	C₂×AGL(1,𝔽₅)	—	2	4	no	—
142	C₂×AGL(1,𝔽₅)	—	2	2	no	—
143	C₂×C₄×C₅	—	2	2	no	—
144	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
145	C₂×Dic₅	—	2	2	no	—
146	C₂×Dic₅	C₂×C₃×D₄	1	1	yes	characteristic
147	C₄×D₅	C₂×C₃×D₄	1	1	yes	characteristic
148	C₄×D₅	C₂×C₃×Q₈	1	1	yes	characteristic
149	C₄×D₅	C₂×C₃×Q₈	1	1	yes	characteristic
150	C₄×D₅	C₃×#41	1	1	yes	characteristic
151	#170	—	2	2	no	—
152	#170	—	4	2	no	—
153	C₅×C₈	C₃×C₄⋊C₄	1	2	yes	—
154	#168	C₃×C₄⋊C₄	1	2	yes	—
155	C₂×M₁₆	—	5	2	no	—
156	C₂²×C₈	—	5	1	no	—
157	C₂²×C₈	—	5	1	no	—
158	#117	—	5	2	no	—
159	C₂×C₄²	—	5	1	no	—
160	#108	—	10	2	no	—
161	#107	—	5	2	no	—
162	#106	—	5	1	no	—
163	#106	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
164	C₄⋊C₈	—	10	2	no	—
165	C₄⋊C₈	—	10	2	no	—
166	C₄×C₈	—	10	2	no	—
167	C₂×C₃×C₅	#389	1	1	yes	characteristic
168	C₂×C₃×C₅	#443	1	2	yes	—
169	C₃×D₅	—	2	1	no	—
170	C₃×D₅	—	2	1	no	—
171	C₂³×C₃	—	5	1	no	—
172	C₂×C₃×C₄	—	2	2	no	—
173	C₂×C₃×C₄	—	5	1	no	—
174	C₂×C₃×C₄	—	5	1	no	—
175	C₂×C₃×C₄	—	5	1	no	—
176	C₂×C₃×C₄	—	5	1	no	—
177	C₂×C₃×C₄	—	5	1	no	—
178	C₂×C₃×C₄	—	10	2	no	—
179	C₂×C₃×C₄	—	10	4	no	—
180	C₂×C₃×C₄	—	10	2	no	—
181	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
182	C₃×C₈	—	5	2	no	—
183	C₃×C₈	—	10	2	no	—
184	C₃×C₈	—	20	2	no	—
185	C₃×C₈	C₄⋊AGL(1,𝔽₅)	1	2	yes	—
186	C₂²×C₅	C₃×C₄×D₄	1	1	yes	characteristic
187	C₂×D₅	—	2	2	no	—
188	C₂×D₅	C₃×C₂²⋊Q₈	1	1	yes	characteristic
189	C₂×D₅	C₃×#125	1	1	yes	characteristic
190	AGL(1,𝔽₅)	—	2	4	no	—
191	C₄×C₅	—	4	2	no	—
192	C₄×C₅	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
193	C₄×C₅	C₃×C₄×Q₈	1	1	yes	characteristic
194	Dic₅	—	4	2	no	—
195	Dic₅	C₃×#121	1	1	yes	characteristic
196	Dic₅	C₃×C₄⋊Q₈	1	1	yes	characteristic
197	C₂²×C₄	—	5	1	no	—
198	C₂²×C₄	—	10	1	no	—
199	M₁₆	—	10	2	no	—
200	M₁₆	—	10	2	no	—
201	C₂×C₈	—	5	1	no	—
202	C₂×C₈	—	5	2	no	—
203	C₂×C₈	—	5	2	no	—
204	C₂×C₈	—	5	2	no	—
205	C₂×C₈	—	10	1	no	—
206	C₂×C₈	—	10	1	no	—
207	C₂×C₈	—	10	2	no	—
208	C₂×C₈	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
209	C₄⋊C₄	—	5	2	no	—
210	C₄⋊C₄	C₂×C₃×AGL(1,𝔽₅)	1	2	yes	—
211	C₂²⋊C₄	—	10	2	no	—
212	C₄²	—	5	2	no	—
213	C₄²	—	10	2	no	—
214	C₄²	—	10	1	no	—
215	C₃×C₅	#1945	1	1	yes	characteristic, complemented, Hall
216	C₂²×C₃	—	5	1	no	—
217	C₂²×C₃	—	5	1	no	—
218	C₂²×C₃	—	10	2	no	—
219	C₂²×C₃	D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
220	C₃×C₄	—	4	2	no	—
221	C₃×C₄	—	5	1	no	—
222	C₃×C₄	—	5	1	no	—
223	C₃×C₄	—	10	4	no	—
224	C₃×C₄	—	20	2	no	—
225	C₃×C₄	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
226	C₃×C₄	Q₈×AGL(1,𝔽₅)	1	1	yes	characteristic
227	C₂×C₅	C₃×#389	1	1	yes	characteristic
228	C₂×C₅	C₃×#443	1	2	yes	—
229	D₅	—	2	1	no	—
230	D₅	—	2	1	no	—
231	C₂³	—	5	1	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	5	1	no	—
234	C₂×C₄	—	5	1	no	—
235	C₂×C₄	—	5	1	no	—
236	C₂×C₄	—	5	1	no	—
237	C₂×C₄	—	5	1	no	—
238	C₂×C₄	—	10	2	no	—
239	C₂×C₄	—	10	4	no	—
240	C₂×C₄	—	10	2	no	—
241	C₂×C₄	C₂²×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
242	C₈	—	5	2	no	—
243	C₈	—	10	2	no	—
244	C₈	—	20	2	no	—
245	C₈	C₃×C₄⋊AGL(1,𝔽₅)	1	2	yes	—
246	C₂×C₃	—	10	1	no	—
247	C₂×C₃	—	10	1	no	—
248	C₂×C₃	#65712	1	1	yes	characteristic
249	C₂×C₃	#65736	1	2	yes	—
250	C₅	C₃×#1945	1	1	yes	characteristic, complemented, Sylow
251	C₂²	—	5	1	no	—
252	C₂²	—	5	1	no	—
253	C₂²	—	10	2	no	—
254	C₂²	C₃×D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
255	C₄	—	4	2	no	—
256	C₄	—	5	1	no	—
257	C₄	—	5	1	no	—
258	C₄	—	10	4	no	—
259	C₄	—	20	2	no	—
260	C₄	C₂×C₃×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
261	C₄	C₃×Q₈×AGL(1,𝔽₅)	1	1	yes	characteristic
262	C₃	#10604057	1	1	yes	characteristic, direct product factor, Sylow
263	C₂	—	10	1	no	—
264	C₂	—	10	1	no	—
265	C₂	C₃×#65712	1	1	yes	characteristic
266	C₂	C₃×#65736	1	2	yes	—

	subgroup	G/H
center	C₂²×C₃	D₄×AGL(1,𝔽₅)
commutator	C₂×C₄×C₅	C₂²×C₃×C₄
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×#1945
LCS	G ▷ C₂×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. #10604057	C₃	direct (N × H)
16. C₃×C₅×#106	C₄	semidirect
17. C₃×#3949	C₄	semidirect
18. C₃×#3939	C₄	semidirect
57. C₅×#106	C₃×C₄	semidirect
58. #3949	C₃×C₄	semidirect
59. #3939	C₃×C₄	semidirect
84. C₃×#106	AGL(1,𝔽₅)	semidirect
163. #106	C₃×AGL(1,𝔽₅)	semidirect
215. C₃×C₅	#1945	semidirect
250. C₅	C₃×#1945	semidirect
262. C₃	#10604057	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×#65708	2
C₃×#65149	2
C₃×#64875	2
#10604057	3
C₃×#1945	5

Subgroups — C3×#10604057 #14305232

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10604057 #14305232