Subgroups — C₃×#10603839 #14294870

Order 1920.

592 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄²×AGL(1,𝔽₅)	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄×D₅⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₃×C₄×C₈×D₅	C₂	1	1	yes	maximal, characteristic
4	C₃×(C₂×AGL(1,𝔽₅))⋊C₈	C₂	1	4	yes	maximal
5	#10603839	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂×C₃×C₄×AGL(1,𝔽₅)	C₄	1	2	yes	—
7	C₂×C₃×C₄×AGL(1,𝔽₅)	C₄	1	2	yes	—
8	C₂×C₃×C₄×AGL(1,𝔽₅)	C₂²	1	2	yes	—
9	C₂×C₃×D₅⋊C₈	C₂²	1	2	yes	—
10	C₂×C₃×C₈×D₅	C₂²	1	2	yes	—
11	C₃×C₄²×D₅	C₂²	1	1	yes	characteristic
12	C₃×C₄×#170	C₄	1	2	yes	complemented
13	C₃×C₄×#170	C₄	1	2	yes	complemented
14	C₃×C₄×C₅×C₈	C₄	1	1	yes	characteristic, complemented
15	C₃×C₈×Dic₅	C₄	1	2	yes	complemented
16	C₃×C₄×#168	C₄	1	1	yes	characteristic, complemented
17	C₃×#1744	—	5	1	no	maximal, Hall
18	C₄²×AGL(1,𝔽₅)	C₂×C₃	1	1	yes	characteristic
19	C₄×D₅⋊C₈	C₂×C₃	1	1	yes	characteristic
20	C₄×C₈×D₅	C₂×C₃	1	1	yes	characteristic
21	(C₂×AGL(1,𝔽₅))⋊C₈	C₂×C₃	1	4	yes	—
22	C₂²×C₃×AGL(1,𝔽₅)	C₂×C₄	1	4	yes	—
23	C₂×C₃×C₄×D₅	C₂×C₄	1	1	yes	characteristic
24	C₂×C₃×C₄×D₅	C₂×C₄	1	1	yes	characteristic
25	C₂×C₃×C₄×D₅	C₂³	1	1	yes	characteristic
26	C₂×C₃×#170	C₂×C₄	1	4	yes	—
27	C₃×C₄×AGL(1,𝔽₅)	—	2	4	no	—
28	C₃×C₄×AGL(1,𝔽₅)	—	2	4	no	—
29	C₃×C₄×AGL(1,𝔽₅)	C₈	1	8	yes	complemented
30	C₃×D₅⋊C₈	—	2	4	no	—
31	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
32	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
33	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
34	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
35	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
36	C₂×C₃×#168	C₂×C₄	1	2	yes	—
37	C₃×C₈×D₅	—	2	4	no	—
38	C₂×C₃×C₄×C₈	—	5	1	no	—
39	C₂×C₃×C₄×C₈	—	5	1	no	—
40	C₃×C₄³	—	5	1	no	—
41	C₃×#336	—	5	4	no	—
42	C₂×C₄×AGL(1,𝔽₅)	C₃×C₄	1	2	yes	—
43	C₂×C₄×AGL(1,𝔽₅)	C₃×C₄	1	2	yes	—
44	C₂×C₄×AGL(1,𝔽₅)	C₂²×C₃	1	2	yes	—
45	C₂×D₅⋊C₈	C₂²×C₃	1	2	yes	—
46	C₂×C₈×D₅	C₂²×C₃	1	2	yes	—
47	C₄²×D₅	C₂²×C₃	1	1	yes	characteristic
48	C₄×#170	C₃×C₄	1	2	yes	complemented
49	C₄×#170	C₃×C₄	1	2	yes	complemented
50	C₄×C₅×C₈	C₃×C₄	1	1	yes	characteristic, complemented
51	C₈×Dic₅	C₃×C₄	1	2	yes	complemented
52	C₄×#168	C₃×C₄	1	1	yes	characteristic, complemented
53	#1744	—	5	1	no	Sylow
54	C₂²×C₃×D₅	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₃×AGL(1,𝔽₅)	—	2	4	no	—
56	C₂×C₃×AGL(1,𝔽₅)	—	2	4	no	—
57	C₂×C₃×AGL(1,𝔽₅)	C₂×C₈	1	8	yes	—
58	C₂×C₃×C₄×C₅	C₄²	1	1	yes	characteristic
59	C₂×C₃×C₄×C₅	C₄²	1	1	yes	characteristic
60	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₃×Dic₅	C₄²	1	1	yes	characteristic
62	C₂×C₃×Dic₅	C₄²	1	1	yes	characteristic
63	C₂×C₃×Dic₅	C₂²×C₄	1	1	yes	characteristic
64	C₃×C₄×D₅	C₂×C₈	1	4	yes	—
65	C₃×C₄×D₅	M₁₆	1	4	yes	—
66	C₃×C₄×D₅	#41	1	4	yes	—
67	C₃×#170	—	2	4	no	—
68	C₃×C₅×C₈	—	2	2	no	—
69	C₃×#168	—	2	2	no	—
70	C₂²×C₃×C₈	—	5	2	no	—
71	C₂²×C₃×C₈	—	5	2	no	—
72	C₂×C₃×C₄²	—	5	2	no	—
73	C₂×C₃×C₄²	—	5	1	no	—
74	C₂×C₃×C₄²	—	5	2	no	—
75	C₂×C₃×C₄²	—	5	2	no	—
76	C₃×C₄×C₈	—	5	2	no	—
77	C₃×C₄×C₈	—	5	2	no	—
78	C₃×C₄×C₈	—	5	2	no	—
79	C₃×C₄×C₈	—	5	1	no	—
80	C₃×C₄×C₈	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
81	C₂²×AGL(1,𝔽₅)	C₂×C₃×C₄	1	4	yes	—
82	C₂×C₄×D₅	C₂×C₃×C₄	1	1	yes	characteristic
83	C₂×C₄×D₅	C₂×C₃×C₄	1	1	yes	characteristic
84	C₂×C₄×D₅	C₂³×C₃	1	1	yes	characteristic
85	C₂×#170	C₂×C₃×C₄	1	4	yes	—
86	C₄×AGL(1,𝔽₅)	—	2	4	no	—
87	C₄×AGL(1,𝔽₅)	—	2	4	no	—
88	C₄×AGL(1,𝔽₅)	C₃×C₈	1	8	yes	complemented
89	D₅⋊C₈	—	2	4	no	—
90	C₂×C₅×C₈	C₂×C₃×C₄	1	2	yes	—
91	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
92	C₄×Dic₅	C₂×C₃×C₄	1	1	yes	characteristic
93	C₄×Dic₅	C₂×C₃×C₄	1	1	yes	characteristic
94	C₄×Dic₅	C₂×C₃×C₄	1	1	yes	characteristic
95	C₂×#168	C₂×C₃×C₄	1	2	yes	—
96	C₈×D₅	—	2	4	no	—
97	C₂×C₄×C₈	—	5	1	no	—
98	C₂×C₄×C₈	—	5	1	no	—
99	C₄³	—	5	1	no	—
100	#336	—	5	4	no	—
101	C₂²×C₃×C₅	C₂×C₄²	1	1	yes	characteristic
102	C₂×C₃×D₅	#117	1	2	yes	—
103	C₂×C₃×D₅	C₂²×C₈	1	2	yes	—
104	C₂×C₃×D₅	C₂×M₁₆	1	2	yes	—
105	C₃×AGL(1,𝔽₅)	—	2	8	no	—
106	C₃×C₄×C₅	C₄×C₈	1	2	yes	—
107	C₃×C₄×C₅	#97	1	2	yes	—
108	C₃×C₄×C₅	#117	1	2	yes	—
109	C₃×Dic₅	C₄×C₈	1	2	yes	—
110	C₃×Dic₅	#97	1	2	yes	—
111	C₃×Dic₅	#117	1	2	yes	—
112	C₂²×C₃×C₄	—	5	1	no	—
113	C₂²×C₃×C₄	—	5	1	no	—
114	C₂²×C₃×C₄	—	5	1	no	—
115	C₂²×C₃×C₄	—	5	4	no	—
116	C₂×C₃×C₈	—	5	4	no	—
117	C₂×C₃×C₈	—	5	2	no	—
118	C₂×C₃×C₈	—	10	4	no	—
119	C₂×C₃×C₈	—	10	4	no	—
120	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	2	yes	—
121	C₃×C₄²	—	5	8	no	—
122	C₃×C₄²	—	5	1	no	—
123	C₃×C₄²	—	5	1	no	—
124	C₃×C₄²	—	5	1	no	—
125	C₃×C₄²	—	10	4	no	—
126	C₃×C₄²	—	10	4	no	—
127	C₃×C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
128	C₂²×D₅	C₂²×C₃×C₄	1	1	yes	characteristic
129	C₂×AGL(1,𝔽₅)	—	2	4	no	—
130	C₂×AGL(1,𝔽₅)	—	2	4	no	—
131	C₂×AGL(1,𝔽₅)	C₂×C₃×C₈	1	8	yes	—
132	C₂×C₄×C₅	C₃×C₄²	1	1	yes	characteristic
133	C₂×C₄×C₅	C₃×C₄²	1	1	yes	characteristic
134	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
135	C₂×Dic₅	C₃×C₄²	1	1	yes	characteristic
136	C₂×Dic₅	C₃×C₄²	1	1	yes	characteristic
137	C₂×Dic₅	C₂²×C₃×C₄	1	1	yes	characteristic
138	C₄×D₅	C₂×C₃×C₈	1	4	yes	—
139	C₄×D₅	C₃×M₁₆	1	4	yes	—
140	C₄×D₅	C₃×#41	1	4	yes	—
141	#170	—	2	4	no	—
142	C₅×C₈	—	2	2	no	—
143	#168	—	2	2	no	—
144	C₂²×C₈	—	5	2	no	—
145	C₂²×C₈	—	5	2	no	—
146	C₂×C₄²	—	5	2	no	—
147	C₂×C₄²	—	5	1	no	—
148	C₂×C₄²	—	5	2	no	—
149	C₂×C₄²	—	5	2	no	—
150	C₄×C₈	—	5	2	no	—
151	C₄×C₈	—	5	2	no	—
152	C₄×C₈	—	5	2	no	—
153	C₄×C₈	—	5	1	no	—
154	C₄×C₈	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
155	C₂×C₃×C₅	#376	1	1	yes	characteristic
156	C₂×C₃×C₅	C₂×C₄×C₈	1	1	yes	characteristic
157	C₂×C₃×C₅	C₂×#97	1	1	yes	characteristic
158	C₃×D₅	#431	1	4	yes	—
159	C₂³×C₃	—	5	1	no	—
160	C₂×C₃×C₄	—	5	1	no	—
161	C₂×C₃×C₄	—	5	1	no	—
162	C₂×C₃×C₄	—	5	1	no	—
163	C₂×C₃×C₄	—	5	4	no	—
164	C₂×C₃×C₄	—	5	4	no	—
165	C₂×C₃×C₄	—	5	4	no	—
166	C₂×C₃×C₄	—	5	8	no	—
167	C₂×C₃×C₄	—	10	4	no	—
168	C₂×C₃×C₄	—	10	4	no	—
169	C₂×C₃×C₄	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
170	C₂×C₃×C₄	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
171	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
172	C₃×C₈	—	2	2	no	—
173	C₃×C₈	—	10	4	no	—
174	C₃×C₈	—	10	2	no	—
175	C₂²×C₅	C₂×C₃×C₄²	1	1	yes	characteristic
176	C₂×D₅	C₃×#117	1	2	yes	—
177	C₂×D₅	C₂²×C₃×C₈	1	2	yes	—
178	C₂×D₅	C₂×C₃×M₁₆	1	2	yes	—
179	AGL(1,𝔽₅)	—	2	8	no	—
180	C₄×C₅	C₃×C₄×C₈	1	2	yes	—
181	C₄×C₅	C₃×#97	1	2	yes	—
182	C₄×C₅	C₃×#117	1	2	yes	—
183	Dic₅	C₃×C₄×C₈	1	2	yes	—
184	Dic₅	C₃×#97	1	2	yes	—
185	Dic₅	C₃×#117	1	2	yes	—
186	C₂²×C₄	—	5	1	no	—
187	C₂²×C₄	—	5	1	no	—
188	C₂²×C₄	—	5	1	no	—
189	C₂²×C₄	—	5	4	no	—
190	C₂×C₈	—	5	4	no	—
191	C₂×C₈	—	5	2	no	—
192	C₂×C₈	—	10	4	no	—
193	C₂×C₈	—	10	4	no	—
194	C₂×C₈	C₂×C₃×AGL(1,𝔽₅)	1	2	yes	—
195	C₄²	—	5	8	no	—
196	C₄²	—	5	1	no	—
197	C₄²	—	5	1	no	—
198	C₄²	—	5	1	no	—
199	C₄²	—	10	4	no	—
200	C₄²	—	10	4	no	—
201	C₄²	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
202	C₃×C₅	#1744	1	1	yes	characteristic, complemented, Hall
203	C₂²×C₃	—	5	2	no	—
204	C₂²×C₃	—	5	2	no	—
205	C₂²×C₃	—	5	2	no	—
206	C₂²×C₃	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
207	C₃×C₄	—	5	2	no	—
208	C₃×C₄	—	5	2	no	—
209	C₃×C₄	—	5	2	no	—
210	C₃×C₄	—	10	8	no	—
211	C₃×C₄	C₈×AGL(1,𝔽₅)	1	2	yes	—
212	C₃×C₄	#3992	1	2	yes	—
213	C₃×C₄	#4130	1	2	yes	—
214	C₂×C₅	C₃×#376	1	1	yes	characteristic
215	C₂×C₅	C₂×C₃×C₄×C₈	1	1	yes	characteristic
216	C₂×C₅	C₂×C₃×#97	1	1	yes	characteristic
217	D₅	C₃×#431	1	4	yes	—
218	C₂³	—	5	1	no	—
219	C₂×C₄	—	5	1	no	—
220	C₂×C₄	—	5	1	no	—
221	C₂×C₄	—	5	1	no	—
222	C₂×C₄	—	5	4	no	—
223	C₂×C₄	—	5	4	no	—
224	C₂×C₄	—	5	4	no	—
225	C₂×C₄	—	5	8	no	—
226	C₂×C₄	—	10	4	no	—
227	C₂×C₄	—	10	4	no	—
228	C₂×C₄	C₃×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
229	C₂×C₄	C₃×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
230	C₂×C₄	C₂²×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
231	C₈	—	2	2	no	—
232	C₈	—	10	4	no	—
233	C₈	—	10	2	no	—
234	C₂×C₃	—	5	4	no	—
235	C₂×C₃	#65686	1	1	yes	characteristic
236	C₂×C₃	C₂×C₈×AGL(1,𝔽₅)	1	1	yes	characteristic
237	C₂×C₃	C₂×#3992	1	1	yes	characteristic
238	C₅	C₃×#1744	1	1	yes	characteristic, complemented, Sylow
239	C₂²	—	5	2	no	—
240	C₂²	—	5	2	no	—
241	C₂²	—	5	2	no	—
242	C₂²	C₂×C₃×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
243	C₄	—	5	2	no	—
244	C₄	—	5	2	no	—
245	C₄	—	5	2	no	—
246	C₄	—	10	8	no	—
247	C₄	C₃×C₈×AGL(1,𝔽₅)	1	2	yes	—
248	C₄	C₃×#3992	1	2	yes	—
249	C₄	C₃×#4130	1	2	yes	—
250	C₃	#10603839	1	1	yes	characteristic, direct product factor, Sylow
251	C₂	—	5	4	no	—
252	C₂	C₃×#65686	1	1	yes	characteristic
253	C₂	C₂×C₃×C₈×AGL(1,𝔽₅)	1	1	yes	characteristic
254	C₂	C₂×C₃×#3992	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₄²	C₂×AGL(1,𝔽₅)
commutator	C₂×C₅	C₂×C₃×C₄×C₈
Frattini	C₂×C₄	C₂²×C₃×AGL(1,𝔽₅)
Fitting	C₃×C₄×C₅×C₈	C₄
socle	C₂²×C₃×C₅	C₂×C₄²
prime radical	C₂²×C₃×D₅	C₂²×C₄
cosocle	C₂×C₄×D₅	C₂³×C₃
hypercenter	C₃×C₄×C₈	AGL(1,𝔽₅)
hypocenter	C₅	C₃×#1744
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₃×C₄² ◁ C₃×C₄×C₈

Complements

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

N	H ≅ G/N	type
5. #10603839	C₃	direct (N × H)
12. C₃×C₄×#170	C₄	semidirect
13. C₃×C₄×#170	C₄	semidirect
14. C₃×C₄×C₅×C₈	C₄	semidirect
15. C₃×C₈×Dic₅	C₄	semidirect
16. C₃×C₄×#168	C₄	semidirect
29. C₃×C₄×AGL(1,𝔽₅)	C₈	semidirect
48. C₄×#170	C₃×C₄	semidirect
49. C₄×#170	C₃×C₄	semidirect
50. C₄×C₅×C₈	C₃×C₄	semidirect
51. C₈×Dic₅	C₃×C₄	semidirect
52. C₄×#168	C₃×C₄	semidirect
80. C₃×C₄×C₈	AGL(1,𝔽₅)	semidirect
88. C₄×AGL(1,𝔽₅)	C₃×C₈	semidirect
154. C₄×C₈	C₃×AGL(1,𝔽₅)	semidirect
202. C₃×C₅	#1744	semidirect
238. C₅	C₃×#1744	semidirect
250. C₃	#10603839	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×C₄²×AGL(1,𝔽₅)	2
C₃×C₄×D₅⋊C₈	2
C₃×C₄×C₈×D₅	2
C₃×(C₂×AGL(1,𝔽₅))⋊C₈	2
#10603839	3
C₃×#1744	5

Subgroups — C3×#10603839 #14294870

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10603839 #14294870