Subgroups — C₃×#10603877 #14294244

Order 1920.

656 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#3992	C₂	1	2	yes	maximal
2	C₂×C₃×C₈×AGL(1,𝔽₅)	C₂	1	2	yes	maximal
3	C₃×#65686	C₂	1	1	yes	maximal, characteristic
4	C₃×C₄×D₅⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₃×D₅×#97	C₂	1	1	yes	maximal, characteristic
6	#10603877	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×C₄×AGL(1,𝔽₅)	C₂²	1	2	yes	—
8	C₂×C₃×D₅⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×C₃×D₅⋊C₈	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₈×D₅	C₂²	1	2	yes	—
11	C₃×C₄²×D₅	C₂²	1	1	yes	characteristic
12	C₃×(C₂×AGL(1,𝔽₅))⋊C₄	C₄	1	4	yes	—
13	C₃×C₄×#170	C₄	1	4	yes	complemented
14	C₃×#3992	C₄	1	8	yes	complemented
15	C₃×C₈×AGL(1,𝔽₅)	C₄	1	8	yes	complemented
16	C₃×C₅×#97	C₄	1	1	yes	characteristic, complemented
17	C₃×#3947	C₄	1	2	yes	complemented
18	C₃×#3935	C₄	1	1	yes	characteristic, complemented
19	C₃×#1726	—	5	1	no	maximal, Hall
20	C₂×#3992	C₂×C₃	1	2	yes	—
21	C₂×C₈×AGL(1,𝔽₅)	C₂×C₃	1	2	yes	—
22	#65686	C₂×C₃	1	1	yes	characteristic
23	C₄×D₅⋊C₈	C₂×C₃	1	1	yes	characteristic
24	D₅×#97	C₂×C₃	1	1	yes	characteristic
25	C₂²×C₃×AGL(1,𝔽₅)	C₂×C₄	1	4	yes	—
26	C₂×C₃×C₄×D₅	C₂×C₄	1	2	yes	—
27	C₂×C₃×C₄×D₅	C₂³	1	1	yes	characteristic
28	C₂×C₃×#170	C₂×C₄	1	2	yes	—
29	C₂×C₃×#170	C₂×C₄	1	2	yes	—
30	C₃×C₄×AGL(1,𝔽₅)	C₂×C₄	1	4	yes	—
31	C₃×C₄×AGL(1,𝔽₅)	C₂×C₄	1	4	yes	—
32	C₃×D₅⋊C₈	C₂×C₄	1	4	yes	—
33	C₃×D₅⋊C₈	C₂×C₄	1	4	yes	—
34	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
35	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
36	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
37	C₃×C₄×Dic₅	C₂×C₄	1	2	yes	—
38	C₂×C₃×#168	C₂×C₄	1	2	yes	—
39	C₃×C₈×D₅	C₂×C₄	1	4	yes	—
40	C₃×C₈×D₅	C₂×C₄	1	4	yes	—
41	C₂×C₃×#97	—	5	2	no	—
42	C₂×C₃×#97	—	5	1	no	—
43	C₂×C₃×C₄×C₈	—	5	1	no	—
44	C₂×C₃×C₄×C₈	—	5	2	no	—
45	C₃×#376	—	5	1	no	—
46	C₂×C₄×AGL(1,𝔽₅)	C₂²×C₃	1	2	yes	—
47	C₂×D₅⋊C₈	C₂²×C₃	1	1	yes	characteristic
48	C₂×D₅⋊C₈	C₂²×C₃	1	1	yes	characteristic
49	C₂×C₈×D₅	C₂²×C₃	1	2	yes	—
50	C₄²×D₅	C₂²×C₃	1	1	yes	characteristic
51	(C₂×AGL(1,𝔽₅))⋊C₄	C₃×C₄	1	4	yes	—
52	C₄×#170	C₃×C₄	1	4	yes	complemented
53	#3992	C₃×C₄	1	8	yes	complemented
54	C₈×AGL(1,𝔽₅)	C₃×C₄	1	8	yes	complemented
55	C₅×#97	C₃×C₄	1	1	yes	characteristic, complemented
56	#3947	C₃×C₄	1	2	yes	complemented
57	#3935	C₃×C₄	1	1	yes	characteristic, complemented
58	#1726	—	5	1	no	Sylow
59	C₂²×C₃×D₅	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₃×AGL(1,𝔽₅)	—	2	8	no	—
61	C₂×C₃×AGL(1,𝔽₅)	C₄²	1	8	yes	—
62	C₂×C₃×C₄×C₅	C₄²	1	2	yes	—
63	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₃×Dic₅	C₄²	1	2	yes	—
65	C₂×C₃×Dic₅	C₂²×C₄	1	1	yes	characteristic
66	C₃×C₄×D₅	—	2	4	no	—
67	C₃×C₄×D₅	C₂²×C₄	1	2	yes	—
68	C₃×C₄×D₅	C₂²×C₄	1	2	yes	—
69	C₃×#170	C₄²	1	4	yes	—
70	C₃×#170	C₄²	1	4	yes	—
71	C₃×C₅×C₈	C₄²	1	4	yes	—
72	C₃×#168	C₄²	1	4	yes	—
73	C₂²×C₃×C₈	—	5	1	no	—
74	C₂²×C₃×C₈	—	5	1	no	—
75	C₂²×C₃×C₈	—	5	2	no	—
76	C₂×C₃×C₄²	—	5	2	no	—
77	C₂×C₃×C₄²	—	5	1	no	—
78	C₃×#97	—	5	8	no	—
79	C₃×#97	—	5	2	no	—
80	C₃×#97	—	5	1	no	—
81	C₃×#97	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
82	C₃×C₄×C₈	—	5	8	no	—
83	C₃×C₄×C₈	—	5	4	no	—
84	C₃×#95	—	5	4	no	—
85	C₂²×AGL(1,𝔽₅)	C₂×C₃×C₄	1	4	yes	—
86	C₂×C₄×D₅	C₂×C₃×C₄	1	2	yes	—
87	C₂×C₄×D₅	C₂³×C₃	1	1	yes	characteristic
88	C₂×#170	C₂×C₃×C₄	1	2	yes	—
89	C₂×#170	C₂×C₃×C₄	1	2	yes	—
90	C₄×AGL(1,𝔽₅)	C₂×C₃×C₄	1	4	yes	—
91	C₄×AGL(1,𝔽₅)	C₂×C₃×C₄	1	4	yes	—
92	D₅⋊C₈	C₂×C₃×C₄	1	4	yes	—
93	D₅⋊C₈	C₂×C₃×C₄	1	4	yes	—
94	C₂×C₅×C₈	C₂×C₃×C₄	1	2	yes	—
95	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
96	C₄×Dic₅	C₂×C₃×C₄	1	1	yes	characteristic
97	C₄×Dic₅	C₂×C₃×C₄	1	2	yes	—
98	C₂×#168	C₂×C₃×C₄	1	2	yes	—
99	C₈×D₅	C₂×C₃×C₄	1	4	yes	—
100	C₈×D₅	C₂×C₃×C₄	1	4	yes	—
101	C₂×#97	—	5	2	no	—
102	C₂×#97	—	5	1	no	—
103	C₂×C₄×C₈	—	5	1	no	—
104	C₂×C₄×C₈	—	5	2	no	—
105	#376	—	5	1	no	—
106	C₂²×C₃×C₅	C₂×C₄²	1	1	yes	characteristic
107	C₂×C₃×D₅	C₂×C₄²	1	2	yes	—
108	C₂×C₃×D₅	#131	1	4	yes	—
109	C₃×AGL(1,𝔽₅)	—	2	8	no	—
110	C₃×C₄×C₅	—	2	2	no	—
111	C₃×C₄×C₅	C₂×C₄²	1	2	yes	—
112	C₃×Dic₅	—	2	2	no	—
113	C₃×Dic₅	C₂×C₄²	1	1	yes	characteristic
114	C₃×Dic₅	C₂×C₄²	1	1	yes	characteristic
115	C₂²×C₃×C₄	—	5	1	no	—
116	C₂²×C₃×C₄	—	5	4	no	—
117	C₂²×C₃×C₄	—	5	2	no	—
118	C₂×C₃×C₈	—	5	4	no	—
119	C₂×C₃×C₈	—	5	2	no	—
120	C₂×C₃×C₈	—	5	2	no	—
121	C₂×C₃×C₈	—	5	4	no	—
122	C₂×C₃×C₈	—	5	2	no	—
123	C₂×C₃×C₈	—	5	4	no	—
124	C₂×C₃×C₈	—	5	4	no	—
125	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	2	yes	—
126	C₃×C₄²	—	5	4	no	—
127	C₃×C₄²	—	5	4	no	—
128	C₃×C₄²	—	5	2	no	—
129	C₃×C₄²	—	5	1	no	—
130	C₃×C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
131	C₂²×D₅	C₂²×C₃×C₄	1	1	yes	characteristic
132	C₂×AGL(1,𝔽₅)	—	2	8	no	—
133	C₂×AGL(1,𝔽₅)	C₃×C₄²	1	8	yes	—
134	C₂×C₄×C₅	C₃×C₄²	1	2	yes	—
135	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
136	C₂×Dic₅	C₃×C₄²	1	2	yes	—
137	C₂×Dic₅	C₂²×C₃×C₄	1	1	yes	characteristic
138	C₄×D₅	—	2	4	no	—
139	C₄×D₅	C₂²×C₃×C₄	1	2	yes	—
140	C₄×D₅	C₂²×C₃×C₄	1	2	yes	—
141	#170	C₃×C₄²	1	4	yes	—
142	#170	C₃×C₄²	1	4	yes	—
143	C₅×C₈	C₃×C₄²	1	4	yes	—
144	#168	C₃×C₄²	1	4	yes	—
145	C₂²×C₈	—	5	1	no	—
146	C₂²×C₈	—	5	1	no	—
147	C₂²×C₈	—	5	2	no	—
148	C₂×C₄²	—	5	2	no	—
149	C₂×C₄²	—	5	1	no	—
150	#97	—	5	8	no	—
151	#97	—	5	2	no	—
152	#97	—	5	1	no	—
153	#97	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
154	C₄×C₈	—	5	8	no	—
155	C₄×C₈	—	5	4	no	—
156	#95	—	5	4	no	—
157	C₂×C₃×C₅	C₄³	1	1	yes	characteristic
158	C₂×C₃×C₅	#405	1	2	yes	—
159	C₃×D₅	#405	1	4	yes	—
160	C₂³×C₃	—	5	1	no	—
161	C₂×C₃×C₄	—	5	1	no	—
162	C₂×C₃×C₄	—	5	2	no	—
163	C₂×C₃×C₄	—	5	2	no	—
164	C₂×C₃×C₄	—	5	8	no	—
165	C₂×C₃×C₄	—	5	2	no	—
166	C₂×C₃×C₄	—	10	4	no	—
167	C₂×C₃×C₄	—	10	8	no	—
168	C₂×C₃×C₄	C₄×AGL(1,𝔽₅)	1	2	yes	—
169	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
170	C₃×C₈	—	5	4	no	—
171	C₃×C₈	—	5	4	no	—
172	C₃×C₈	—	5	4	no	—
173	C₃×C₈	C₄×AGL(1,𝔽₅)	1	4	yes	—
174	C₂²×C₅	C₂×C₃×C₄²	1	1	yes	characteristic
175	C₂×D₅	C₂×C₃×C₄²	1	2	yes	—
176	C₂×D₅	C₃×#131	1	4	yes	—
177	AGL(1,𝔽₅)	—	2	8	no	—
178	C₄×C₅	—	2	2	no	—
179	C₄×C₅	C₂×C₃×C₄²	1	2	yes	—
180	Dic₅	—	2	2	no	—
181	Dic₅	C₂×C₃×C₄²	1	1	yes	characteristic
182	Dic₅	C₂×C₃×C₄²	1	1	yes	characteristic
183	C₂²×C₄	—	5	1	no	—
184	C₂²×C₄	—	5	4	no	—
185	C₂²×C₄	—	5	2	no	—
186	C₂×C₈	—	5	4	no	—
187	C₂×C₈	—	5	2	no	—
188	C₂×C₈	—	5	2	no	—
189	C₂×C₈	—	5	4	no	—
190	C₂×C₈	—	5	2	no	—
191	C₂×C₈	—	5	4	no	—
192	C₂×C₈	—	5	4	no	—
193	C₂×C₈	C₂×C₃×AGL(1,𝔽₅)	1	2	yes	—
194	C₄²	—	5	4	no	—
195	C₄²	—	5	4	no	—
196	C₄²	—	5	2	no	—
197	C₄²	—	5	1	no	—
198	C₄²	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
199	C₃×C₅	#1726	1	1	yes	characteristic, complemented, Hall
200	C₂²×C₃	—	5	2	no	—
201	C₂²×C₃	—	5	4	no	—
202	C₂²×C₃	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
203	C₃×C₄	—	2	2	no	—
204	C₃×C₄	—	5	1	no	—
205	C₃×C₄	—	5	1	no	—
206	C₃×C₄	—	10	8	no	—
207	C₃×C₄	—	10	2	no	—
208	C₃×C₄	C₂×C₄×AGL(1,𝔽₅)	1	2	yes	—
209	C₂×C₅	C₃×C₄³	1	1	yes	characteristic
210	C₂×C₅	C₃×#405	1	2	yes	—
211	D₅	C₃×#405	1	4	yes	—
212	C₂³	—	5	1	no	—
213	C₂×C₄	—	5	1	no	—
214	C₂×C₄	—	5	2	no	—
215	C₂×C₄	—	5	2	no	—
216	C₂×C₄	—	5	8	no	—
217	C₂×C₄	—	5	2	no	—
218	C₂×C₄	—	10	4	no	—
219	C₂×C₄	—	10	8	no	—
220	C₂×C₄	C₃×C₄×AGL(1,𝔽₅)	1	2	yes	—
221	C₂×C₄	C₂²×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
222	C₈	—	5	4	no	—
223	C₈	—	5	4	no	—
224	C₈	—	5	4	no	—
225	C₈	C₃×C₄×AGL(1,𝔽₅)	1	4	yes	—
226	C₂×C₃	—	5	4	no	—
227	C₂×C₃	C₄²×AGL(1,𝔽₅)	1	1	yes	characteristic
228	C₂×C₃	#65728	1	2	yes	—
229	C₅	C₃×#1726	1	1	yes	characteristic, complemented, Sylow
230	C₂²	—	5	2	no	—
231	C₂²	—	5	4	no	—
232	C₂²	C₂×C₃×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
233	C₄	—	2	2	no	—
234	C₄	—	5	1	no	—
235	C₄	—	5	1	no	—
236	C₄	—	10	8	no	—
237	C₄	—	10	2	no	—
238	C₄	C₂×C₃×C₄×AGL(1,𝔽₅)	1	2	yes	—
239	C₃	#10603877	1	1	yes	characteristic, direct product factor, Sylow
240	C₂	—	5	4	no	—
241	C₂	C₃×C₄²×AGL(1,𝔽₅)	1	1	yes	characteristic
242	C₂	C₃×#65728	1	2	yes	—

	subgroup	G/H
center	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)
commutator	C₂×C₅	C₃×C₄³
Frattini	C₂×C₄	C₂²×C₃×AGL(1,𝔽₅)
Fitting	C₃×C₅×#97	C₄
socle	C₂²×C₃×C₅	C₂×C₄²
prime radical	C₂²×C₃×D₅	C₂²×C₄
cosocle	C₂×C₄×D₅	C₂³×C₃
hypercenter	C₃×#97	AGL(1,𝔽₅)
hypocenter	C₅	C₃×#1726
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₃×#97

Complements

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

N	H ≅ G/N	type
6. #10603877	C₃	direct (N × H)
13. C₃×C₄×#170	C₄	semidirect
14. C₃×#3992	C₄	semidirect
15. C₃×C₈×AGL(1,𝔽₅)	C₄	semidirect
16. C₃×C₅×#97	C₄	semidirect
17. C₃×#3947	C₄	semidirect
18. C₃×#3935	C₄	semidirect
52. C₄×#170	C₃×C₄	semidirect
53. #3992	C₃×C₄	semidirect
54. C₈×AGL(1,𝔽₅)	C₃×C₄	semidirect
55. C₅×#97	C₃×C₄	semidirect
56. #3947	C₃×C₄	semidirect
57. #3935	C₃×C₄	semidirect
81. C₃×#97	AGL(1,𝔽₅)	semidirect
153. #97	C₃×AGL(1,𝔽₅)	semidirect
199. C₃×C₅	#1726	semidirect
229. C₅	C₃×#1726	semidirect
239. C₃	#10603877	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×#3992	2
C₂×C₃×C₈×AGL(1,𝔽₅)	2
C₃×#65686	2
C₃×C₄×D₅⋊C₈	2
C₃×D₅×#97	2
#10603877	3
C₃×#1726	5

Subgroups — C3×#10603877 #14294244

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10603877 #14294244