Subgroups — C₃×#10604591 #14307348

Order 1920.

520 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	C₃×D₅×#125	C₂	1	1	yes	maximal, characteristic
2	C₃×#65688	C₂	1	1	yes	maximal, characteristic
3	C₃×#65676	C₂	1	1	yes	maximal, characteristic
4	C₃×#64874	C₂	1	4	yes	maximal
5	#10604591	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂×C₃×C₄⋊AGL(1,𝔽₅)	—	2	2	no	—
7	C₂×C₃×C₄⋊AGL(1,𝔽₅)	C₂²	1	2	yes	—
8	C₂×C₃×D₅⋊C₈	C₂²	1	2	yes	—
9	C₃×C₅×#125	C₄	1	1	yes	characteristic, complemented
10	C₃×D₅×C₄⋊C₄	—	2	2	no	—
11	C₃×D₅×C₄⋊C₄	C₂²	1	2	yes	—
12	C₃×#4036	C₄	1	2	yes	complemented
13	C₃×#4035	—	2	2	no	—
14	C₃×C₄²×D₅	C₂²	1	1	yes	characteristic
15	C₃×#4016	C₄	1	1	yes	characteristic, complemented
16	C₃×#4002	—	2	2	no	—
17	C₃×#1992	—	5	1	no	maximal, Hall
18	D₅×#125	C₂×C₃	1	1	yes	characteristic
19	#65688	C₂×C₃	1	1	yes	characteristic
20	#65676	C₂×C₃	1	1	yes	characteristic
21	#64874	C₂×C₃	1	4	yes	—
22	C₂²×C₃×AGL(1,𝔽₅)	—	2	2	no	—
23	C₂×C₃×C₄×D₅	—	2	2	no	—
24	C₂×C₃×C₄×D₅	D₄	1	2	yes	—
25	C₂×C₃×C₄×D₅	C₂³	1	1	yes	characteristic
26	C₂×C₃×#170	—	2	2	no	—
27	C₃×C₄⋊AGL(1,𝔽₅)	—	2	4	no	—
28	C₃×C₄⋊AGL(1,𝔽₅)	—	4	4	no	—
29	C₃×D₅⋊C₈	—	2	2	no	—
30	C₃×D₅⋊C₈	Q₈	1	4	yes	—
31	C₃×C₅×C₄⋊C₄	—	2	2	no	—
32	C₃×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
33	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
34	C₃×C₄⋊Dic₅	—	2	2	no	—
35	C₃×C₄⋊Dic₅	C₂×C₄	1	2	yes	—
36	C₃×#683	—	2	2	no	—
37	C₃×#683	—	2	2	no	—
38	C₃×#683	—	2	2	no	—
39	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
40	C₃×C₄×Dic₅	D₄	1	2	yes	—
41	C₂×C₃×#125	—	5	1	no	—
42	C₂×C₃×#97	—	5	1	no	—
43	C₃×#384	—	5	1	no	—
44	C₃×#340	—	5	4	no	—
45	C₂×C₄⋊AGL(1,𝔽₅)	—	2	2	no	—
46	C₂×C₄⋊AGL(1,𝔽₅)	C₂²×C₃	1	2	yes	—
47	C₂×D₅⋊C₈	C₂²×C₃	1	2	yes	—
48	C₅×#125	C₃×C₄	1	1	yes	characteristic, complemented
49	D₅×C₄⋊C₄	—	2	2	no	—
50	D₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
51	#4036	C₃×C₄	1	2	yes	complemented
52	#4035	—	2	2	no	—
53	C₄²×D₅	C₂²×C₃	1	1	yes	characteristic
54	#4016	C₃×C₄	1	1	yes	characteristic, complemented
55	#4002	—	2	2	no	—
56	#1992	—	5	1	no	Sylow
57	C₂²×C₃×D₅	C₂×D₄	1	1	yes	characteristic
58	C₂×C₃×AGL(1,𝔽₅)	—	4	2	no	—
59	C₂×C₃×AGL(1,𝔽₅)	—	4	4	no	—
60	C₂×C₃×C₄×C₅	—	2	2	no	—
61	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
62	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₃×Dic₅	—	2	2	no	—
64	C₂×C₃×Dic₅	C₂²⋊C₄	1	2	yes	—
65	C₂×C₃×Dic₅	C₂×D₄	1	1	yes	characteristic
66	C₃×C₄×D₅	—	2	2	no	—
67	C₃×C₄×D₅	—	2	2	no	—
68	C₃×C₄×D₅	—	4	2	no	—
69	C₃×C₄×D₅	—	4	2	no	—
70	C₃×C₄×D₅	C₂×Q₈	1	2	yes	—
71	C₃×C₄×D₅	#41	1	2	yes	—
72	C₃×#170	—	2	4	no	—
73	C₂²×C₃×C₈	—	5	2	no	—
74	C₃×#125	—	5	2	no	—
75	C₃×#125	—	5	1	no	—
76	C₃×#125	—	10	2	no	—
77	C₃×#125	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
78	C₂×C₃×C₄⋊C₄	—	5	2	no	—
79	C₂×C₃×C₄⋊C₄	—	5	2	no	—
80	C₂×C₃×C₄⋊C₄	—	10	2	no	—
81	C₂×C₃×C₄⋊C₄	—	10	2	no	—
82	C₂×C₃×C₄²	—	5	1	no	—
83	C₃×#97	—	10	2	no	—
84	C₂²×AGL(1,𝔽₅)	—	2	2	no	—
85	C₂×C₄×D₅	—	2	2	no	—
86	C₂×C₄×D₅	C₃×D₄	1	2	yes	—
87	C₂×C₄×D₅	C₂³×C₃	1	1	yes	characteristic
88	C₂×#170	—	2	2	no	—
89	C₄⋊AGL(1,𝔽₅)	—	2	4	no	—
90	C₄⋊AGL(1,𝔽₅)	—	4	4	no	—
91	D₅⋊C₈	—	2	2	no	—
92	D₅⋊C₈	C₃×Q₈	1	4	yes	—
93	C₅×C₄⋊C₄	—	2	2	no	—
94	C₅×C₄⋊C₄	C₂×C₃×C₄	1	2	yes	—
95	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
96	C₄⋊Dic₅	—	2	2	no	—
97	C₄⋊Dic₅	C₂×C₃×C₄	1	2	yes	—
98	#683	—	2	2	no	—
99	#683	—	2	2	no	—
100	#683	—	2	2	no	—
101	C₄×Dic₅	C₂×C₃×C₄	1	1	yes	characteristic
102	C₄×Dic₅	C₃×D₄	1	2	yes	—
103	C₂×#125	—	5	1	no	—
104	C₂×#97	—	5	1	no	—
105	#384	—	5	1	no	—
106	#340	—	5	4	no	—
107	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	C₂×C₃×D₅	#124	1	1	yes	characteristic
109	C₂×C₃×D₅	C₄⋊Q₈	1	1	yes	characteristic
110	C₂×C₃×D₅	AGL(1,ℤ/8)	1	2	yes	—
111	C₂×C₃×D₅	#137	1	2	yes	—
112	C₃×AGL(1,𝔽₅)	—	8	2	no	—
113	C₃×C₄×C₅	—	2	2	no	—
114	C₃×C₄×C₅	—	4	2	no	—
115	C₃×C₄×C₅	C₄×Q₈	1	2	yes	—
116	C₃×Dic₅	—	2	2	no	—
117	C₃×Dic₅	—	4	2	no	—
118	C₃×Dic₅	C₂²⋊Q₈	1	2	yes	—
119	C₂²×C₃×C₄	—	5	1	no	—
120	C₂²×C₃×C₄	—	5	2	no	—
121	C₂²×C₃×C₄	—	10	2	no	—
122	C₂²×C₃×C₄	—	10	2	no	—
123	C₂×C₃×C₈	—	5	4	no	—
124	C₂×C₃×C₈	—	10	2	no	—
125	C₂×C₃×C₈	—	10	2	no	—
126	C₃×C₄⋊C₄	—	2	2	no	—
127	C₃×C₄⋊C₄	—	5	2	no	—
128	C₃×C₄⋊C₄	—	10	2	no	—
129	C₃×C₄⋊C₄	—	10	2	no	—
130	C₃×C₄⋊C₄	—	10	2	no	—
131	C₃×C₄⋊C₄	—	10	4	no	—
132	C₃×C₄⋊C₄	—	10	2	no	—
133	C₃×C₄⋊C₄	—	20	4	no	—
134	C₃×C₄⋊C₄	C₂×AGL(1,𝔽₅)	1	2	yes	—
135	C₃×C₄²	—	5	2	no	—
136	C₃×C₄²	—	5	1	no	—
137	C₃×C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
138	C₂²×D₅	C₂×C₃×D₄	1	1	yes	characteristic
139	C₂×AGL(1,𝔽₅)	—	4	2	no	—
140	C₂×AGL(1,𝔽₅)	—	4	4	no	—
141	C₂×C₄×C₅	—	2	2	no	—
142	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	2	yes	—
143	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
144	C₂×Dic₅	—	2	2	no	—
145	C₂×Dic₅	C₃×C₂²⋊C₄	1	2	yes	—
146	C₂×Dic₅	C₂×C₃×D₄	1	1	yes	characteristic
147	C₄×D₅	—	2	2	no	—
148	C₄×D₅	—	2	2	no	—
149	C₄×D₅	—	4	2	no	—
150	C₄×D₅	—	4	2	no	—
151	C₄×D₅	C₂×C₃×Q₈	1	2	yes	—
152	C₄×D₅	C₃×#41	1	2	yes	—
153	#170	—	2	4	no	—
154	C₂²×C₈	—	5	2	no	—
155	#125	—	5	2	no	—
156	#125	—	5	1	no	—
157	#125	—	10	2	no	—
158	#125	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
159	C₂×C₄⋊C₄	—	5	2	no	—
160	C₂×C₄⋊C₄	—	5	2	no	—
161	C₂×C₄⋊C₄	—	10	2	no	—
162	C₂×C₄⋊C₄	—	10	2	no	—
163	C₂×C₄²	—	5	1	no	—
164	#97	—	10	2	no	—
165	C₂×C₃×C₅	#391	1	1	yes	characteristic
166	C₂×C₃×C₅	#417	1	2	yes	—
167	C₃×D₅	#490	1	1	yes	characteristic
168	C₃×D₅	#491	1	1	yes	characteristic
169	C₃×D₅	#501	1	2	yes	—
170	C₂³×C₃	—	5	1	no	—
171	C₂×C₃×C₄	—	2	2	no	—
172	C₂×C₃×C₄	—	5	1	no	—
173	C₂×C₃×C₄	—	5	2	no	—
174	C₂×C₃×C₄	—	5	2	no	—
175	C₂×C₃×C₄	—	5	2	no	—
176	C₂×C₃×C₄	—	10	2	no	—
177	C₂×C₃×C₄	—	10	2	no	—
178	C₂×C₃×C₄	—	10	2	no	—
179	C₂×C₃×C₄	—	20	2	no	—
180	C₂×C₃×C₄	—	20	2	no	—
181	C₂×C₃×C₄	—	20	2	no	—
182	C₂×C₃×C₄	—	20	4	no	—
183	C₂×C₃×C₄	C₂²⋊AGL(1,𝔽₅)	1	2	yes	—
184	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
185	C₃×C₈	—	10	4	no	—
186	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
187	C₂×D₅	C₃×#124	1	1	yes	characteristic
188	C₂×D₅	C₃×C₄⋊Q₈	1	1	yes	characteristic
189	C₂×D₅	C₃×AGL(1,ℤ/8)	1	2	yes	—
190	C₂×D₅	C₃×#137	1	2	yes	—
191	AGL(1,𝔽₅)	—	8	2	no	—
192	C₄×C₅	—	2	2	no	—
193	C₄×C₅	—	4	2	no	—
194	C₄×C₅	C₃×C₄×Q₈	1	2	yes	—
195	Dic₅	—	2	2	no	—
196	Dic₅	—	4	2	no	—
197	Dic₅	C₃×C₂²⋊Q₈	1	2	yes	—
198	C₂²×C₄	—	5	1	no	—
199	C₂²×C₄	—	5	2	no	—
200	C₂²×C₄	—	10	2	no	—
201	C₂²×C₄	—	10	2	no	—
202	C₂×C₈	—	5	4	no	—
203	C₂×C₈	—	10	2	no	—
204	C₂×C₈	—	10	2	no	—
205	C₄⋊C₄	—	2	2	no	—
206	C₄⋊C₄	—	5	2	no	—
207	C₄⋊C₄	—	10	2	no	—
208	C₄⋊C₄	—	10	2	no	—
209	C₄⋊C₄	—	10	2	no	—
210	C₄⋊C₄	—	10	4	no	—
211	C₄⋊C₄	—	10	2	no	—
212	C₄⋊C₄	—	20	4	no	—
213	C₄⋊C₄	C₂×C₃×AGL(1,𝔽₅)	1	2	yes	—
214	C₄²	—	5	2	no	—
215	C₄²	—	5	1	no	—
216	C₄²	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
217	C₃×C₅	#1992	1	1	yes	characteristic, complemented, Hall
218	C₂²×C₃	—	5	2	no	—
219	C₂²×C₃	—	5	2	no	—
220	C₂²×C₃	—	5	1	no	—
221	C₂²×C₃	—	5	1	no	—
222	C₂²×C₃	C₂×C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
223	C₃×C₄	—	2	2	no	—
224	C₃×C₄	—	4	2	no	—
225	C₃×C₄	—	5	2	no	—
226	C₃×C₄	—	10	2	no	—
227	C₃×C₄	—	20	2	no	—
228	C₃×C₄	—	40	2	no	—
229	C₃×C₄	Q₈×AGL(1,𝔽₅)	1	2	yes	—
230	C₂×C₅	C₃×#391	1	1	yes	characteristic
231	C₂×C₅	C₃×#417	1	2	yes	—
232	D₅	C₃×#490	1	1	yes	characteristic
233	D₅	C₃×#491	1	1	yes	characteristic
234	D₅	C₃×#501	1	2	yes	—
235	C₂³	—	5	1	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	5	1	no	—
238	C₂×C₄	—	5	2	no	—
239	C₂×C₄	—	5	2	no	—
240	C₂×C₄	—	5	2	no	—
241	C₂×C₄	—	10	2	no	—
242	C₂×C₄	—	10	2	no	—
243	C₂×C₄	—	10	2	no	—
244	C₂×C₄	—	20	2	no	—
245	C₂×C₄	—	20	2	no	—
246	C₂×C₄	—	20	2	no	—
247	C₂×C₄	—	20	4	no	—
248	C₂×C₄	C₃×C₂²⋊AGL(1,𝔽₅)	1	2	yes	—
249	C₂×C₄	C₂²×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
250	C₈	—	10	4	no	—
251	C₂×C₃	—	5	1	no	—
252	C₂×C₃	—	5	1	no	—
253	C₂×C₃	—	5	2	no	—
254	C₂×C₃	(C₂×AGL(1,𝔽₅))⋊Q₈	1	1	yes	characteristic
255	C₂×C₃	#65793	1	2	yes	—
256	C₅	C₃×#1992	1	1	yes	characteristic, complemented, Sylow
257	C₂²	—	5	1	no	—
258	C₂²	—	5	1	no	—
259	C₂²	—	5	2	no	—
260	C₂²	—	5	2	no	—
261	C₂²	C₂×C₃×C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
262	C₄	—	2	2	no	—
263	C₄	—	4	2	no	—
264	C₄	—	5	2	no	—
265	C₄	—	10	2	no	—
266	C₄	—	20	2	no	—
267	C₄	—	40	2	no	—
268	C₄	C₃×Q₈×AGL(1,𝔽₅)	1	2	yes	—
269	C₃	#10604591	1	1	yes	characteristic, direct product factor, Sylow
270	C₂	—	5	1	no	—
271	C₂	—	5	1	no	—
272	C₂	—	5	2	no	—
273	C₂	C₃×(C₂×AGL(1,𝔽₅))⋊Q₈	1	1	yes	characteristic
274	C₂	C₃×#65793	1	2	yes	—

	subgroup	G/H
center	C₂²×C₃	C₂×C₂²⋊AGL(1,𝔽₅)
commutator	C₂×C₄×C₅	C₂²×C₃×C₄
Frattini	C₂×C₄	C₂²×C₃×AGL(1,𝔽₅)
Fitting	C₃×C₅×#125	C₄
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₂²×C₃×D₅	C₂×D₄
cosocle	C₂×C₄×D₅	C₂³×C₃
hypercenter	C₃×#125	AGL(1,𝔽₅)
hypocenter	C₅	C₃×#1992
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ C₃×#125

Complements

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

N	H ≅ G/N	type
5. #10604591	C₃	direct (N × H)
9. C₃×C₅×#125	C₄	semidirect
12. C₃×#4036	C₄	semidirect
15. C₃×#4016	C₄	semidirect
48. C₅×#125	C₃×C₄	semidirect
51. #4036	C₃×C₄	semidirect
54. #4016	C₃×C₄	semidirect
77. C₃×#125	AGL(1,𝔽₅)	semidirect
158. #125	C₃×AGL(1,𝔽₅)	semidirect
217. C₃×C₅	#1992	semidirect
256. C₅	C₃×#1992	semidirect
269. C₃	#10604591	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×D₅×#125	2
C₃×#65688	2
C₃×#65676	2
C₃×#64874	2
#10604591	3
C₃×#1992	5

Subgroups — C3×#10604591 #14307348

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10604591 #14307348