Subgroups — C₃×#10603850 #14294972

Order 1920.

400 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#65718	C₂	1	2	yes	maximal
2	C₃×#64978	C₂	1	1	yes	maximal, characteristic
3	C₃×#64897	C₂	1	1	yes	maximal, characteristic
4	C₃×#64895	C₂	1	1	yes	maximal, characteristic
5	C₃×#64863	C₂	1	2	yes	maximal
6	#10603850	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₃×#4130	C₂²	1	2	yes	—
8	C₃×D₅⋊M₁₆	C₂²	1	2	yes	—
9	C₂×C₃×#676	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₈×D₅	C₂²	1	1	yes	characteristic
11	C₃×#4018	C₂²	1	1	yes	characteristic
12	C₃×#3992	C₄	1	4	yes	complemented
13	C₃×C₈×AGL(1,𝔽₅)	C₄	1	4	yes	complemented
14	C₃×C₄×C₅×C₈	C₄	1	1	yes	characteristic, complemented
15	C₃×#3947	C₄	1	1	yes	characteristic, complemented
16	C₃×C₈×Dic₅	C₄	1	1	yes	characteristic, complemented
17	C₃×#3935	C₄	1	1	yes	characteristic, complemented
18	C₃×#1747	—	5	1	no	maximal, Hall
19	#65718	C₂×C₃	1	2	yes	—
20	#64978	C₂×C₃	1	1	yes	characteristic
21	#64897	C₂×C₃	1	1	yes	characteristic
22	#64895	C₂×C₃	1	1	yes	characteristic
23	#64863	C₂×C₃	1	2	yes	—
24	C₂×C₃×C₄×D₅	C₂³	1	1	yes	characteristic
25	C₃×C₂²⋊AGL(1,𝔽₅)	—	2	2	no	—
26	C₃×#704	—	2	2	no	—
27	C₃×C₄⋊AGL(1,𝔽₅)	—	2	2	no	—
28	C₃×C₄×AGL(1,𝔽₅)	C₂×C₄	1	4	yes	—
29	C₃×#700	—	2	2	no	—
30	C₃×D₅⋊C₈	C₂×C₄	1	4	yes	—
31	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
32	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
33	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
34	C₃×#685	—	2	1	no	—
35	C₃×#683	—	2	1	no	—
36	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
37	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
38	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
39	C₃×#676	—	2	2	no	—
40	C₃×C₈×D₅	C₂×C₄	1	2	yes	—
41	C₃×C₈×D₅	C₂×C₄	1	2	yes	—
42	C₃×#405	—	5	2	no	—
43	C₃×#405	—	5	1	no	—
44	C₃×#338	—	5	1	no	—
45	C₃×H(3,ℤ/4)	—	5	2	no	—
46	C₃×H(3,ℤ/4)	—	5	1	no	—
47	#4130	C₂²×C₃	1	2	yes	—
48	D₅⋊M₁₆	C₂²×C₃	1	2	yes	—
49	C₂×#676	C₂²×C₃	1	1	yes	characteristic
50	C₂×C₈×D₅	C₂²×C₃	1	1	yes	characteristic
51	#4018	C₂²×C₃	1	1	yes	characteristic
52	#3992	C₃×C₄	1	4	yes	complemented
53	C₈×AGL(1,𝔽₅)	C₃×C₄	1	4	yes	complemented
54	C₄×C₅×C₈	C₃×C₄	1	1	yes	characteristic, complemented
55	#3947	C₃×C₄	1	1	yes	characteristic, complemented
56	C₈×Dic₅	C₃×C₄	1	1	yes	characteristic, complemented
57	#3935	C₃×C₄	1	1	yes	characteristic, complemented
58	#1747	—	5	1	no	Sylow
59	C₂²×C₃×D₅	#41	1	1	yes	characteristic
60	C₂×C₃×AGL(1,𝔽₅)	—	2	4	no	—
61	C₂×C₃×C₄×C₅	—	2	1	no	—
62	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₃×Dic₅	—	2	1	no	—
64	C₂×C₃×Dic₅	#41	1	1	yes	characteristic
65	C₃×C₄×D₅	C₂²×C₄	1	2	yes	—
66	C₃×C₄×D₅	#41	1	2	yes	—
67	C₃×#170	—	2	4	no	—
68	C₃×C₅×C₈	—	2	1	no	—
69	C₃×C₅×C₈	C₄²	1	1	yes	characteristic
70	C₃×C₅×C₈	C₄²	1	1	yes	characteristic
71	C₃×#168	—	2	1	no	—
72	C₃×#168	C₄²	1	2	yes	—
73	C₂×C₃×M₁₆	—	5	2	no	—
74	C₂×C₃×M₁₆	—	5	1	no	—
75	C₂²×C₃×C₈	—	5	1	no	—
76	C₃×#117	—	5	2	no	—
77	C₃×#117	—	5	1	no	—
78	C₃×#97	—	5	4	no	—
79	C₃×#97	—	5	1	no	—
80	C₃×#97	—	5	1	no	—
81	C₃×C₄×C₈	—	5	4	no	—
82	C₃×C₄×C₈	—	5	1	no	—
83	C₃×C₄×C₈	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
84	C₂×C₄×D₅	C₂³×C₃	1	1	yes	characteristic
85	C₂²⋊AGL(1,𝔽₅)	—	2	2	no	—
86	#704	—	2	2	no	—
87	C₄⋊AGL(1,𝔽₅)	—	2	2	no	—
88	C₄×AGL(1,𝔽₅)	C₂×C₃×C₄	1	4	yes	—
89	#700	—	2	2	no	—
90	D₅⋊C₈	C₂×C₃×C₄	1	4	yes	—
91	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
92	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
93	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
94	#685	—	2	1	no	—
95	#683	—	2	1	no	—
96	C₄×Dic₅	C₂×C₃×C₄	1	1	yes	characteristic
97	C₂×#168	C₂×C₃×C₄	1	1	yes	characteristic
98	C₂×#168	C₂×C₃×C₄	1	1	yes	characteristic
99	#676	—	2	2	no	—
100	C₈×D₅	C₂×C₃×C₄	1	2	yes	—
101	C₈×D₅	C₂×C₃×C₄	1	2	yes	—
102	#405	—	5	2	no	—
103	#405	—	5	1	no	—
104	#338	—	5	1	no	—
105	H(3,ℤ/4)	—	5	2	no	—
106	H(3,ℤ/4)	—	5	1	no	—
107	C₂²×C₃×C₅	#117	1	1	yes	characteristic
108	C₂×C₃×D₅	—	4	1	no	—
109	C₂×C₃×D₅	#117	1	2	yes	—
110	C₃×AGL(1,𝔽₅)	—	4	4	no	—
111	C₃×C₄×C₅	—	4	1	no	—
112	C₃×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
113	C₃×C₄×C₅	#117	1	1	yes	characteristic
114	C₃×Dic₅	—	4	1	no	—
115	C₃×Dic₅	#117	1	2	yes	—
116	C₂²×C₃×C₄	—	5	1	no	—
117	C₃×M₁₆	—	10	2	no	—
118	C₃×M₁₆	—	10	2	no	—
119	C₃×M₁₆	—	10	2	no	—
120	C₂×C₃×C₈	—	5	1	no	—
121	C₂×C₃×C₈	—	5	2	no	—
122	C₂×C₃×C₈	—	5	2	no	—
123	C₂×C₃×C₈	—	5	4	no	—
124	C₂×C₃×C₈	—	5	1	no	—
125	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
126	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
127	C₃×C₄⋊C₄	—	10	1	no	—
128	C₃×C₄⋊C₄	—	10	2	no	—
129	C₃×C₂²⋊C₄	—	10	1	no	—
130	C₃×C₂²⋊C₄	—	10	2	no	—
131	C₃×C₄²	—	5	4	no	—
132	C₃×C₄²	—	5	1	no	—
133	C₃×C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
134	C₂²×D₅	C₃×#41	1	1	yes	characteristic
135	C₂×AGL(1,𝔽₅)	—	2	4	no	—
136	C₂×C₄×C₅	—	2	1	no	—
137	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
138	C₂×Dic₅	—	2	1	no	—
139	C₂×Dic₅	C₃×#41	1	1	yes	characteristic
140	C₄×D₅	C₂²×C₃×C₄	1	2	yes	—
141	C₄×D₅	C₃×#41	1	2	yes	—
142	#170	—	2	4	no	—
143	C₅×C₈	—	2	1	no	—
144	C₅×C₈	C₃×C₄²	1	1	yes	characteristic
145	C₅×C₈	C₃×C₄²	1	1	yes	characteristic
146	#168	—	2	1	no	—
147	#168	C₃×C₄²	1	2	yes	—
148	C₂×M₁₆	—	5	2	no	—
149	C₂×M₁₆	—	5	1	no	—
150	C₂²×C₈	—	5	1	no	—
151	#117	—	5	2	no	—
152	#117	—	5	1	no	—
153	#97	—	5	4	no	—
154	#97	—	5	1	no	—
155	#97	—	5	1	no	—
156	C₄×C₈	—	5	4	no	—
157	C₄×C₈	—	5	1	no	—
158	C₄×C₈	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
159	C₂×C₃×C₅	—	2	1	no	—
160	C₂×C₃×C₅	#376	1	1	yes	characteristic
161	C₃×D₅	—	2	2	no	—
162	C₂³×C₃	—	5	1	no	—
163	C₂×C₃×C₄	—	2	1	no	—
164	C₂×C₃×C₄	—	5	1	no	—
165	C₂×C₃×C₄	—	5	2	no	—
166	C₂×C₃×C₄	—	5	2	no	—
167	C₂×C₃×C₄	—	10	4	no	—
168	C₂×C₃×C₄	—	10	1	no	—
169	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
170	C₃×C₈	—	2	1	no	—
171	C₃×C₈	—	5	2	no	—
172	C₃×C₈	—	10	4	no	—
173	C₃×C₈	—	10	1	no	—
174	C₃×C₈	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
175	C₃×C₈	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
176	C₂²×C₅	C₃×#117	1	1	yes	characteristic
177	C₂×D₅	—	4	1	no	—
178	C₂×D₅	C₃×#117	1	2	yes	—
179	AGL(1,𝔽₅)	—	4	4	no	—
180	C₄×C₅	—	4	1	no	—
181	C₄×C₅	C₂×C₃×C₄²	1	1	yes	characteristic
182	C₄×C₅	C₃×#117	1	1	yes	characteristic
183	Dic₅	—	4	1	no	—
184	Dic₅	C₃×#117	1	2	yes	—
185	C₂²×C₄	—	5	1	no	—
186	M₁₆	—	10	2	no	—
187	M₁₆	—	10	2	no	—
188	M₁₆	—	10	2	no	—
189	C₂×C₈	—	5	1	no	—
190	C₂×C₈	—	5	2	no	—
191	C₂×C₈	—	5	2	no	—
192	C₂×C₈	—	5	4	no	—
193	C₂×C₈	—	5	1	no	—
194	C₂×C₈	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
195	C₂×C₈	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
196	C₄⋊C₄	—	10	1	no	—
197	C₄⋊C₄	—	10	2	no	—
198	C₂²⋊C₄	—	10	1	no	—
199	C₂²⋊C₄	—	10	2	no	—
200	C₄²	—	5	4	no	—
201	C₄²	—	5	1	no	—
202	C₄²	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
203	C₃×C₅	#1747	1	1	yes	characteristic, complemented, Hall
204	C₂²×C₃	—	5	2	no	—
205	C₂²×C₃	—	20	1	no	—
206	C₂²×C₃	#4130	1	1	yes	characteristic
207	C₃×C₄	—	4	1	no	—
208	C₃×C₄	—	5	2	no	—
209	C₃×C₄	—	20	1	no	—
210	C₃×C₄	—	20	4	no	—
211	C₃×C₄	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
212	C₃×C₄	#4130	1	1	yes	characteristic
213	C₂×C₅	—	2	1	no	—
214	C₂×C₅	C₃×#376	1	1	yes	characteristic
215	D₅	—	2	2	no	—
216	C₂³	—	5	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	5	1	no	—
219	C₂×C₄	—	5	2	no	—
220	C₂×C₄	—	5	2	no	—
221	C₂×C₄	—	10	4	no	—
222	C₂×C₄	—	10	1	no	—
223	C₂×C₄	C₂²×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
224	C₈	—	2	1	no	—
225	C₈	—	5	2	no	—
226	C₈	—	10	4	no	—
227	C₈	—	10	1	no	—
228	C₈	C₃×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
229	C₈	C₃×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
230	C₂×C₃	—	2	1	no	—
231	C₂×C₃	—	10	2	no	—
232	C₂×C₃	#65686	1	1	yes	characteristic
233	C₅	C₃×#1747	1	1	yes	characteristic, complemented, Sylow
234	C₂²	—	5	2	no	—
235	C₂²	—	20	1	no	—
236	C₂²	C₃×#4130	1	1	yes	characteristic
237	C₄	—	4	1	no	—
238	C₄	—	5	2	no	—
239	C₄	—	20	1	no	—
240	C₄	—	20	4	no	—
241	C₄	C₂×C₃×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
242	C₄	C₃×#4130	1	1	yes	characteristic
243	C₃	#10603850	1	1	yes	characteristic, direct product factor, Sylow
244	C₂	—	2	1	no	—
245	C₂	—	10	2	no	—
246	C₂	C₃×#65686	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₈	C₄×AGL(1,𝔽₅)
commutator	C₄×C₅	C₂×C₃×C₄²
Frattini	C₂×C₄	C₂²×C₃×AGL(1,𝔽₅)
Fitting	C₃×C₄×C₅×C₈	C₄
socle	C₂×C₃×C₅	#376
prime radical	C₂²×C₃×D₅	#41
cosocle	C₂×C₄×D₅	C₂³×C₃
hypercenter	C₃×C₄×C₈	AGL(1,𝔽₅)
hypocenter	C₅	C₃×#1747
LCS	G ▷ C₄×C₅ ▷ C₅
UCS	C₁ ◁ C₃×C₈ ◁ C₃×C₄×C₈

Complements

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

N	H ≅ G/N	type
6. #10603850	C₃	direct (N × H)
12. C₃×#3992	C₄	semidirect
13. C₃×C₈×AGL(1,𝔽₅)	C₄	semidirect
14. C₃×C₄×C₅×C₈	C₄	semidirect
15. C₃×#3947	C₄	semidirect
16. C₃×C₈×Dic₅	C₄	semidirect
17. C₃×#3935	C₄	semidirect
52. #3992	C₃×C₄	semidirect
53. C₈×AGL(1,𝔽₅)	C₃×C₄	semidirect
54. C₄×C₅×C₈	C₃×C₄	semidirect
55. #3947	C₃×C₄	semidirect
56. C₈×Dic₅	C₃×C₄	semidirect
57. #3935	C₃×C₄	semidirect
83. C₃×C₄×C₈	AGL(1,𝔽₅)	semidirect
158. C₄×C₈	C₃×AGL(1,𝔽₅)	semidirect
203. C₃×C₅	#1747	semidirect
233. C₅	C₃×#1747	semidirect
243. C₃	#10603850	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×#65718	2
C₃×#64978	2
C₃×#64897	2
C₃×#64895	2
C₃×#64863	2
#10603850	3
C₃×#1747	5

Subgroups — C3×#10603850 #14294972

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10603850 #14294972