Subgroups — C₃×#10600607 #14296656

Order 1920.

544 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	C₂×C₃×C₅×#97	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#3950	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×#3949	C₂	1	1	yes	maximal, characteristic
4	C₃×C₄×C₄⋊Dic₅	C₂	1	2	yes	maximal
5	C₃×#64771	C₂	1	2	yes	maximal
6	#10600607	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
9	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₄⋊Dic₅	C₂²	1	2	yes	—
11	C₂×C₃×C₄⋊Dic₅	C₂²	1	2	yes	—
12	C₂×C₃×C₄×Dic₅	—	2	2	no	—
13	C₃×C₅×#97	C₄	1	4	yes	complemented
14	C₃×#3963	—	2	2	no	—
15	C₃×#3950	C₄	1	4	yes	complemented
16	C₃×#3949	C₄	1	4	yes	complemented
17	C₃×#1776	—	5	1	no	maximal, Hall
18	C₂×C₅×#97	C₂×C₃	1	1	yes	characteristic
19	C₂×#3950	C₂×C₃	1	1	yes	characteristic
20	C₂×#3949	C₂×C₃	1	1	yes	characteristic
21	C₄×C₄⋊Dic₅	C₂×C₃	1	2	yes	—
22	#64771	C₂×C₃	1	2	yes	—
23	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
24	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
25	C₂²×C₃×Dic₅	—	2	2	no	—
26	C₂²×C₃×Dic₅	—	2	2	no	—
27	C₂×C₃×C₅×C₈	—	2	1	no	—
28	C₂×C₃×C₅×C₈	—	2	1	no	—
29	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
30	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
31	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
32	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
33	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
34	C₃×C₄²×C₅	Q₈	1	2	yes	—
35	C₃×C₄⋊Dic₅	—	2	2	no	—
36	C₃×C₄⋊Dic₅	—	2	2	no	—
37	C₃×C₄⋊Dic₅	C₂×C₄	1	4	yes	—
38	C₃×C₄⋊Dic₅	C₂×C₄	1	4	yes	—
39	C₃×C₄×Dic₅	—	4	4	no	—
40	C₂×C₃×#107	—	5	1	no	—
41	C₂×C₃×#106	—	5	1	no	—
42	C₂×C₃×#97	D₅	1	1	yes	characteristic
43	C₃×C₄×C₄⋊C₄	—	5	2	no	—
44	C₃×#340	—	5	2	no	—
45	C₂²×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
46	C₂²×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
47	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
48	C₂×C₄⋊Dic₅	C₂²×C₃	1	2	yes	—
49	C₂×C₄⋊Dic₅	C₂²×C₃	1	2	yes	—
50	C₂×C₄×Dic₅	—	2	2	no	—
51	C₅×#97	C₃×C₄	1	4	yes	complemented
52	#3963	—	2	2	no	—
53	#3950	C₃×C₄	1	4	yes	complemented
54	#3949	C₃×C₄	1	4	yes	complemented
55	#1776	—	5	1	no	Sylow
56	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
57	C₂×C₃×C₄×C₅	—	2	2	no	—
58	C₂×C₃×C₄×C₅	—	2	2	no	—
59	C₂×C₃×C₄×C₅	C₄⋊C₄	1	4	yes	—
60	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
64	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
65	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
66	C₂×C₃×Dic₅	—	2	4	no	—
67	C₂×C₃×Dic₅	—	2	4	no	—
68	C₂×C₃×Dic₅	—	4	2	no	—
69	C₂×C₃×Dic₅	—	4	2	no	—
70	C₂×C₃×Dic₅	—	4	2	no	—
71	C₂×C₃×Dic₅	—	4	2	no	—
72	C₃×C₅×C₈	—	2	2	no	—
73	C₃×C₅×C₈	C₄²	1	4	yes	complemented
74	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
75	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
76	C₂×C₃×C₄⋊C₄	—	5	2	no	—
77	C₂×C₃×C₄⋊C₄	—	5	2	no	—
78	C₂×C₃×C₄²	—	10	2	no	—
79	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
80	C₃×#107	—	5	4	no	—
81	C₃×#106	—	5	4	no	—
82	C₃×#97	Dic₅	1	4	yes	complemented
83	C₃×#95	—	10	2	no	—
84	C₂²×C₄×C₅	C₃×D₄	1	2	yes	—
85	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
86	C₂²×Dic₅	—	2	2	no	—
87	C₂²×Dic₅	—	2	2	no	—
88	C₂×C₅×C₈	—	2	1	no	—
89	C₂×C₅×C₈	—	2	1	no	—
90	C₂×C₅×C₈	C₂×C₃×C₄	1	2	yes	—
91	C₂×C₅×C₈	C₂×C₃×C₄	1	2	yes	—
92	C₂×C₅×C₈	C₂×C₃×C₄	1	2	yes	—
93	C₂×C₅×C₈	C₂×C₃×C₄	1	2	yes	—
94	C₄²×C₅	C₂×C₃×C₄	1	2	yes	—
95	C₄²×C₅	C₃×Q₈	1	2	yes	—
96	C₄⋊Dic₅	—	2	2	no	—
97	C₄⋊Dic₅	—	2	2	no	—
98	C₄⋊Dic₅	C₂×C₃×C₄	1	4	yes	—
99	C₄⋊Dic₅	C₂×C₃×C₄	1	4	yes	—
100	C₄×Dic₅	—	4	4	no	—
101	C₂×#107	—	5	1	no	—
102	C₂×#106	—	5	1	no	—
103	C₂×#97	C₃×D₅	1	1	yes	characteristic
104	C₄×C₄⋊C₄	—	5	2	no	—
105	#340	—	5	2	no	—
106	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂²×C₃×C₅	C₄×D₄	1	1	yes	characteristic
108	C₂²×C₃×C₅	C₄×D₄	1	1	yes	characteristic
109	C₂²×C₃×C₅	AGL(1,ℤ/8)	1	2	yes	—
110	C₂²×C₃×C₅	#137	1	2	yes	—
111	C₃×C₄×C₅	—	2	4	no	—
112	C₃×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
113	C₃×C₄×C₅	#117	1	1	yes	characteristic
114	C₃×C₄×C₅	C₄×Q₈	1	1	yes	characteristic
115	C₃×C₄×C₅	C₄×Q₈	1	1	yes	characteristic
116	C₃×Dic₅	—	4	4	no	—
117	C₃×Dic₅	—	4	4	no	—
118	C₂²×C₃×C₄	—	10	2	no	—
119	C₂²×C₃×C₄	—	10	2	no	—
120	C₂²×C₃×C₄	D₂₀	1	2	yes	—
121	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
122	C₂×C₃×C₈	—	2	1	no	—
123	C₂×C₃×C₈	—	2	1	no	—
124	C₂×C₃×C₈	C₄×D₅	1	2	yes	—
125	C₂×C₃×C₈	C₄×D₅	1	2	yes	—
126	C₂×C₃×C₈	C₂×Dic₅	1	2	yes	—
127	C₂×C₃×C₈	C₂×Dic₅	1	2	yes	—
128	C₃×C₄⋊C₄	—	5	4	no	—
129	C₃×C₄⋊C₄	—	5	4	no	—
130	C₃×C₄⋊C₄	—	10	2	no	—
131	C₃×C₄⋊C₄	—	10	2	no	—
132	C₃×C₄²	—	20	4	no	—
133	C₃×C₄²	Dic₁₀	1	2	yes	—
134	C₃×C₄²	C₂×Dic₅	1	2	yes	—
135	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
136	C₂×C₄×C₅	—	2	2	no	—
137	C₂×C₄×C₅	—	2	2	no	—
138	C₂×C₄×C₅	C₃×C₄⋊C₄	1	4	yes	—
139	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
140	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
141	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
142	C₂×C₄×C₅	C₂×C₃×Q₈	1	1	yes	characteristic
143	C₂×C₄×C₅	C₃×#41	1	1	yes	characteristic
144	C₂×C₄×C₅	C₃×#41	1	1	yes	characteristic
145	C₂×Dic₅	—	2	4	no	—
146	C₂×Dic₅	—	2	4	no	—
147	C₂×Dic₅	—	4	2	no	—
148	C₂×Dic₅	—	4	2	no	—
149	C₂×Dic₅	—	4	2	no	—
150	C₂×Dic₅	—	4	2	no	—
151	C₅×C₈	—	2	2	no	—
152	C₅×C₈	C₃×C₄²	1	4	yes	complemented
153	C₂²×C₈	C₂×C₃×D₅	1	1	yes	characteristic
154	C₂²×C₈	C₂×C₃×D₅	1	1	yes	characteristic
155	C₂×C₄⋊C₄	—	5	2	no	—
156	C₂×C₄⋊C₄	—	5	2	no	—
157	C₂×C₄²	—	10	2	no	—
158	C₂×C₄²	C₂×C₃×D₅	1	1	yes	characteristic
159	#107	—	5	4	no	—
160	#106	—	5	4	no	—
161	#97	C₃×Dic₅	1	4	yes	complemented
162	#95	—	10	2	no	—
163	C₂×C₃×C₅	C₄×C₄⋊C₄	1	1	yes	characteristic
164	C₂×C₃×C₅	#428	1	2	yes	—
165	C₂×C₃×C₅	#440	1	2	yes	—
166	C₂×C₃×C₅	#441	1	1	yes	characteristic
167	C₂×C₃×C₅	#442	1	1	yes	characteristic
168	C₂³×C₃	C₂×D₂₀	1	1	yes	characteristic
169	C₂×C₃×C₄	—	2	2	no	—
170	C₂×C₃×C₄	—	2	2	no	—
171	C₂×C₃×C₄	—	10	4	no	—
172	C₂×C₃×C₄	—	10	4	no	—
173	C₂×C₃×C₄	—	20	2	no	—
174	C₂×C₃×C₄	—	20	2	no	—
175	C₂×C₃×C₄	—	20	2	no	—
176	C₂×C₃×C₄	—	20	2	no	—
177	C₂×C₃×C₄	C₄⋊Dic₅	1	4	yes	—
178	C₂×C₃×C₄	C₂×Dic₁₀	1	1	yes	characteristic
179	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
180	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
181	C₂×C₃×C₄	#709	1	1	yes	characteristic
182	C₂×C₃×C₄	#709	1	1	yes	characteristic
183	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
184	C₃×C₈	—	2	2	no	—
185	C₃×C₈	C₄×Dic₅	1	4	yes	complemented
186	C₂²×C₅	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
187	C₂²×C₅	C₃×C₄×D₄	1	1	yes	characteristic
188	C₂²×C₅	C₃×C₄×D₄	1	1	yes	characteristic
189	C₂²×C₅	C₃×AGL(1,ℤ/8)	1	2	yes	—
190	C₂²×C₅	C₃×#137	1	2	yes	—
191	C₄×C₅	—	2	4	no	—
192	C₄×C₅	C₂×C₃×C₄²	1	1	yes	characteristic
193	C₄×C₅	C₃×#117	1	1	yes	characteristic
194	C₄×C₅	C₃×C₄×Q₈	1	1	yes	characteristic
195	C₄×C₅	C₃×C₄×Q₈	1	1	yes	characteristic
196	Dic₅	—	4	4	no	—
197	Dic₅	—	4	4	no	—
198	C₂²×C₄	—	10	2	no	—
199	C₂²×C₄	—	10	2	no	—
200	C₂²×C₄	C₃×D₂₀	1	2	yes	—
201	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
202	C₂×C₈	—	2	1	no	—
203	C₂×C₈	—	2	1	no	—
204	C₂×C₈	C₃×C₄×D₅	1	2	yes	—
205	C₂×C₈	C₃×C₄×D₅	1	2	yes	—
206	C₂×C₈	C₂×C₃×Dic₅	1	2	yes	—
207	C₂×C₈	C₂×C₃×Dic₅	1	2	yes	—
208	C₄⋊C₄	—	5	4	no	—
209	C₄⋊C₄	—	5	4	no	—
210	C₄⋊C₄	—	10	2	no	—
211	C₄⋊C₄	—	10	2	no	—
212	C₄²	—	20	4	no	—
213	C₄²	C₃×Dic₁₀	1	2	yes	—
214	C₄²	C₂×C₃×Dic₅	1	2	yes	—
215	C₃×C₅	#1776	1	1	yes	characteristic, complemented, Hall
216	C₂²×C₃	C₄×D₂₀	1	1	yes	characteristic
217	C₂²×C₃	C₄×D₂₀	1	1	yes	characteristic
218	C₂²×C₃	#4054	1	2	yes	—
219	C₂²×C₃	#4055	1	2	yes	—
220	C₂²×C₃	C₂×C₄⋊Dic₅	1	1	yes	characteristic
221	C₃×C₄	—	2	4	no	—
222	C₃×C₄	—	20	4	no	—
223	C₃×C₄	—	20	4	no	—
224	C₃×C₄	C₄×Dic₁₀	1	1	yes	characteristic
225	C₃×C₄	C₄×Dic₁₀	1	1	yes	characteristic
226	C₃×C₄	C₂×C₄×Dic₅	1	1	yes	characteristic
227	C₃×C₄	#4072	1	1	yes	characteristic
228	C₂×C₅	C₃×C₄×C₄⋊C₄	1	1	yes	characteristic
229	C₂×C₅	C₃×#428	1	2	yes	—
230	C₂×C₅	C₃×#440	1	2	yes	—
231	C₂×C₅	C₃×#441	1	1	yes	characteristic
232	C₂×C₅	C₃×#442	1	1	yes	characteristic
233	C₂³	C₂×C₃×D₂₀	1	1	yes	characteristic
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	10	4	no	—
237	C₂×C₄	—	10	4	no	—
238	C₂×C₄	—	20	2	no	—
239	C₂×C₄	—	20	2	no	—
240	C₂×C₄	—	20	2	no	—
241	C₂×C₄	—	20	2	no	—
242	C₂×C₄	C₃×C₄⋊Dic₅	1	4	yes	—
243	C₂×C₄	C₂×C₃×Dic₁₀	1	1	yes	characteristic
244	C₂×C₄	C₂×C₃×C₄×D₅	1	1	yes	characteristic
245	C₂×C₄	C₂×C₃×C₄×D₅	1	1	yes	characteristic
246	C₂×C₄	C₃×#709	1	1	yes	characteristic
247	C₂×C₄	C₃×#709	1	1	yes	characteristic
248	C₂×C₄	C₂²×C₃×Dic₅	1	1	yes	characteristic
249	C₈	—	2	2	no	—
250	C₈	C₃×C₄×Dic₅	1	4	yes	complemented
251	C₂×C₃	#64999	1	2	yes	—
252	C₂×C₃	#65000	1	1	yes	characteristic
253	C₂×C₃	#65005	1	1	yes	characteristic
254	C₂×C₃	C₄×C₄⋊Dic₅	1	1	yes	characteristic
255	C₂×C₃	#65410	1	2	yes	—
256	C₅	C₃×#1776	1	1	yes	characteristic, complemented, Sylow
257	C₂²	C₃×C₄×D₂₀	1	1	yes	characteristic
258	C₂²	C₃×C₄×D₂₀	1	1	yes	characteristic
259	C₂²	C₃×#4054	1	2	yes	—
260	C₂²	C₃×#4055	1	2	yes	—
261	C₂²	C₂×C₃×C₄⋊Dic₅	1	1	yes	characteristic
262	C₄	—	2	4	no	—
263	C₄	—	20	4	no	—
264	C₄	—	20	4	no	—
265	C₄	C₃×C₄×Dic₁₀	1	1	yes	characteristic
266	C₄	C₃×C₄×Dic₁₀	1	1	yes	characteristic
267	C₄	C₂×C₃×C₄×Dic₅	1	1	yes	characteristic
268	C₄	C₃×#4072	1	1	yes	characteristic
269	C₃	#10600607	1	1	yes	characteristic, direct product factor, Sylow
270	C₂	C₃×#64999	1	2	yes	—
271	C₂	C₃×#65000	1	1	yes	characteristic
272	C₂	C₃×#65005	1	1	yes	characteristic
273	C₂	C₃×C₄×C₄⋊Dic₅	1	1	yes	characteristic
274	C₂	C₃×#65410	1	2	yes	—

	subgroup	G/H
center	C₂³×C₃	C₂×D₂₀
commutator	C₄×C₅	C₂×C₃×C₄²
Frattini	C₂²×C₄	C₂²×C₃×D₅
Fitting	C₂×C₃×C₅×#97	C₂
socle, prime radical	C₂³×C₃×C₅	C₂×D₄
cosocle	C₂²×C₄×C₅	C₂³×C₃
hypercenter	C₂×C₃×#97	D₅
hypocenter	C₅	C₃×#1776
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×C₄² ◁ C₂×C₃×#97

Complements

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

N	H ≅ G/N	type
6. #10600607	C₃	direct (N × H)
13. C₃×C₅×#97	C₄	semidirect
15. C₃×#3950	C₄	semidirect
16. C₃×#3949	C₄	semidirect
51. C₅×#97	C₃×C₄	semidirect
53. #3950	C₃×C₄	semidirect
54. #3949	C₃×C₄	semidirect
73. C₃×C₅×C₈	C₄²	semidirect
82. C₃×#97	Dic₅	semidirect
152. C₅×C₈	C₃×C₄²	semidirect
161. #97	C₃×Dic₅	semidirect
185. C₃×C₈	C₄×Dic₅	semidirect
215. C₃×C₅	#1776	semidirect
250. C₈	C₃×C₄×Dic₅	semidirect
256. C₅	C₃×#1776	semidirect
269. C₃	#10600607	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×C₅×#97	2
C₂×C₃×#3950	2
C₂×C₃×#3949	2
C₃×C₄×C₄⋊Dic₅	2
C₃×#64771	2
#10600607	3
C₃×#1776	5

Subgroups — C3×#10600607 #14296656

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10600607 #14296656