Subgroups — C₃×#10601496 #14296526

Order 1920.

448 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×#108	C₂	1	1	yes	maximal, characteristic
2	C₃×#65411	C₂	1	2	yes	maximal
3	C₃×#65395	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃×#3947	C₂	1	1	yes	maximal, characteristic
5	C₃×#64775	C₂	1	2	yes	maximal
6	#10601496	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×C₅×M₁₆	C₂²	1	2	yes	—
8	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
9	C₃×#4072	C₂²	1	2	yes	—
10	C₂×C₃×C₄×Dic₅	C₂²	1	1	yes	characteristic
11	C₂²×C₃×#168	C₂²	1	1	yes	characteristic
12	C₃×C₅×#108	C₄	1	4	yes	complemented
13	C₃×#3950	C₄	1	2	yes	complemented
14	C₃×#3949	C₄	1	2	yes	complemented
15	C₃×#3947	—	2	2	no	—
16	C₃×#3947	C₄	1	4	yes	complemented
17	C₃×C₈×Dic₅	—	2	2	no	—
18	C₃×#1773	—	5	1	no	maximal, Hall
19	C₂×C₅×#108	C₂×C₃	1	1	yes	characteristic
20	#65411	C₂×C₃	1	2	yes	—
21	#65395	C₂×C₃	1	1	yes	characteristic
22	C₂×#3947	C₂×C₃	1	1	yes	characteristic
23	#64775	C₂×C₃	1	2	yes	—
24	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
25	C₂²×C₃×Dic₅	—	2	1	no	—
26	C₃×C₅×M₁₆	—	2	2	no	—
27	C₃×C₅×M₁₆	C₂×C₄	1	4	yes	—
28	C₂×C₃×C₅×C₈	—	2	2	no	—
29	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
30	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
31	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
32	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
33	C₃×C₂²⋊Dic₅	—	2	2	no	—
34	C₃×C₄⋊Dic₅	C₂×C₄	1	4	yes	—
35	C₃×C₄×Dic₅	—	2	2	no	—
36	C₃×C₄×Dic₅	—	2	1	no	—
37	C₃×C₄×Dic₅	C₂×C₄	1	2	yes	—
38	C₂×C₃×#168	C₂×C₄	1	2	yes	—
39	C₂×C₃×#168	D₄	1	2	yes	—
40	C₂×C₃×#168	Q₈	1	2	yes	—
41	C₂×C₃×#108	D₅	1	1	yes	characteristic
42	C₃×#427	—	5	1	no	—
43	C₃×#405	—	5	2	no	—
44	C₂×C₃×#97	—	5	1	no	—
45	C₃×#339	—	5	2	no	—
46	C₂×C₅×M₁₆	C₂²×C₃	1	2	yes	—
47	C₂²×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
48	#4072	C₂²×C₃	1	2	yes	—
49	C₂×C₄×Dic₅	C₂²×C₃	1	1	yes	characteristic
50	C₂²×#168	C₂²×C₃	1	1	yes	characteristic
51	C₅×#108	C₃×C₄	1	4	yes	complemented
52	#3950	C₃×C₄	1	2	yes	complemented
53	#3949	C₃×C₄	1	2	yes	complemented
54	#3947	—	2	2	no	—
55	#3947	C₃×C₄	1	4	yes	complemented
56	C₈×Dic₅	—	2	2	no	—
57	#1773	—	5	1	no	Sylow
58	C₂³×C₃×C₅	#41	1	1	yes	characteristic
59	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
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	C₂×D₄	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×Dic₅	—	2	2	no	—
66	C₂×C₃×Dic₅	—	2	4	no	—
67	C₂×C₃×Dic₅	—	4	2	no	—
68	C₃×C₅×C₈	—	2	4	no	—
69	C₃×C₅×C₈	C₄²	1	2	yes	—
70	C₃×C₅×C₈	C₄²	1	2	yes	—
71	C₃×#168	C₄⋊C₄	1	4	yes	—
72	C₂×C₃×M₁₆	C₂×D₅	1	2	yes	—
73	C₂²×C₃×C₈	—	5	1	no	—
74	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
75	C₃×#117	—	5	2	no	—
76	C₂×C₃×C₄²	—	5	1	no	—
77	C₃×#108	Dic₅	1	4	yes	complemented
78	C₃×#107	—	5	2	no	—
79	C₃×#106	—	5	2	no	—
80	C₃×#97	—	5	4	no	—
81	C₃×#97	—	10	2	no	—
82	C₃×C₄×C₈	—	10	2	no	—
83	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
84	C₂²×Dic₅	—	2	1	no	—
85	C₅×M₁₆	—	2	2	no	—
86	C₅×M₁₆	C₂×C₃×C₄	1	4	yes	—
87	C₂×C₅×C₈	—	2	2	no	—
88	C₂×C₅×C₈	C₂×C₃×C₄	1	2	yes	—
89	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
90	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
91	C₂×C₅×C₈	C₂×C₃×C₄	1	2	yes	—
92	C₂²⋊Dic₅	—	2	2	no	—
93	C₄⋊Dic₅	C₂×C₃×C₄	1	4	yes	—
94	C₄×Dic₅	—	2	2	no	—
95	C₄×Dic₅	—	2	1	no	—
96	C₄×Dic₅	C₂×C₃×C₄	1	2	yes	—
97	C₂×#168	C₂×C₃×C₄	1	2	yes	—
98	C₂×#168	C₃×D₄	1	2	yes	—
99	C₂×#168	C₃×Q₈	1	2	yes	—
100	C₂×#108	C₃×D₅	1	1	yes	characteristic
101	#427	—	5	1	no	—
102	#405	—	5	2	no	—
103	C₂×#97	—	5	1	no	—
104	#339	—	5	2	no	—
105	C₂²×C₃×C₅	—	2	2	no	—
106	C₂²×C₃×C₅	#117	1	1	yes	characteristic
107	C₂²×C₃×C₅	C₄×D₄	1	1	yes	characteristic
108	C₂²×C₃×C₅	C₄×Q₈	1	1	yes	characteristic
109	C₃×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
110	C₃×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
111	C₃×C₄×C₅	C₄×D₄	1	1	yes	characteristic
112	C₃×C₄×C₅	C₄×Q₈	1	1	yes	characteristic
113	C₃×Dic₅	—	4	2	no	—
114	C₃×Dic₅	—	4	4	no	—
115	C₂²×C₃×C₄	—	10	1	no	—
116	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
117	C₃×M₁₆	—	2	2	no	—
118	C₃×M₁₆	C₂×Dic₅	1	4	yes	—
119	C₂×C₃×C₈	—	2	2	no	—
120	C₂×C₃×C₈	—	5	2	no	—
121	C₂×C₃×C₈	—	5	2	no	—
122	C₂×C₃×C₈	—	5	2	no	—
123	C₂×C₃×C₈	C₄×D₅	1	2	yes	—
124	C₂×C₃×C₈	C₄×D₅	1	1	yes	characteristic
125	C₂×C₃×C₈	C₄×D₅	1	1	yes	characteristic
126	C₂×C₃×C₈	C₂×Dic₅	1	2	yes	—
127	C₃×C₄⋊C₄	—	5	4	no	—
128	C₃×C₂²⋊C₄	—	10	2	no	—
129	C₃×C₄²	—	5	2	no	—
130	C₃×C₄²	—	10	2	no	—
131	C₃×C₄²	—	10	1	no	—
132	C₂³×C₅	C₃×#41	1	1	yes	characteristic
133	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
134	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
135	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
136	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
137	C₂×C₄×C₅	C₂×C₃×Q₈	1	1	yes	characteristic
138	C₂×C₄×C₅	C₃×#41	1	1	yes	characteristic
139	C₂×Dic₅	—	2	2	no	—
140	C₂×Dic₅	—	2	4	no	—
141	C₂×Dic₅	—	4	2	no	—
142	C₅×C₈	—	2	4	no	—
143	C₅×C₈	C₃×C₄²	1	2	yes	—
144	C₅×C₈	C₃×C₄²	1	2	yes	—
145	#168	C₃×C₄⋊C₄	1	4	yes	—
146	C₂×M₁₆	C₂×C₃×D₅	1	2	yes	—
147	C₂²×C₈	—	5	1	no	—
148	C₂²×C₈	C₂×C₃×D₅	1	1	yes	characteristic
149	#117	—	5	2	no	—
150	C₂×C₄²	—	5	1	no	—
151	#108	C₃×Dic₅	1	4	yes	complemented
152	#107	—	5	2	no	—
153	#106	—	5	2	no	—
154	#97	—	5	4	no	—
155	#97	—	10	2	no	—
156	C₄×C₈	—	10	2	no	—
157	C₂×C₃×C₅	—	2	1	no	—
158	C₂×C₃×C₅	—	2	1	no	—
159	C₂×C₃×C₅	C₄×C₄⋊C₄	1	1	yes	characteristic
160	C₂×C₃×C₅	#444	1	2	yes	—
161	C₂³×C₃	#713	1	1	yes	characteristic
162	C₂×C₃×C₄	—	10	2	no	—
163	C₂×C₃×C₄	—	10	4	no	—
164	C₂×C₃×C₄	—	20	2	no	—
165	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
166	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
167	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
168	C₂×C₃×C₄	#711	1	1	yes	characteristic
169	C₂×C₃×C₄	Q₈×D₅	1	1	yes	characteristic
170	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
171	C₃×C₈	—	2	4	no	—
172	C₃×C₈	—	5	4	no	—
173	C₃×C₈	C₄×Dic₅	1	2	yes	—
174	C₃×C₈	C₄×Dic₅	1	2	yes	—
175	C₂²×C₅	—	2	2	no	—
176	C₂²×C₅	C₃×#117	1	1	yes	characteristic
177	C₂²×C₅	C₃×C₄×D₄	1	1	yes	characteristic
178	C₂²×C₅	C₃×C₄×Q₈	1	1	yes	characteristic
179	C₄×C₅	C₂×C₃×C₄²	1	1	yes	characteristic
180	C₄×C₅	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
181	C₄×C₅	C₃×C₄×D₄	1	1	yes	characteristic
182	C₄×C₅	C₃×C₄×Q₈	1	1	yes	characteristic
183	Dic₅	—	4	2	no	—
184	Dic₅	—	4	4	no	—
185	C₂²×C₄	—	10	1	no	—
186	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
187	M₁₆	—	2	2	no	—
188	M₁₆	C₂×C₃×Dic₅	1	4	yes	—
189	C₂×C₈	—	2	2	no	—
190	C₂×C₈	—	5	2	no	—
191	C₂×C₈	—	5	2	no	—
192	C₂×C₈	—	5	2	no	—
193	C₂×C₈	C₃×C₄×D₅	1	2	yes	—
194	C₂×C₈	C₃×C₄×D₅	1	1	yes	characteristic
195	C₂×C₈	C₃×C₄×D₅	1	1	yes	characteristic
196	C₂×C₈	C₂×C₃×Dic₅	1	2	yes	—
197	C₄⋊C₄	—	5	4	no	—
198	C₂²⋊C₄	—	10	2	no	—
199	C₄²	—	5	2	no	—
200	C₄²	—	10	2	no	—
201	C₄²	—	10	1	no	—
202	C₃×C₅	#1773	1	1	yes	characteristic, complemented, Hall
203	C₂²×C₃	—	2	2	no	—
204	C₂²×C₃	#4038	1	1	yes	characteristic
205	C₂²×C₃	#4039	1	1	yes	characteristic
206	C₂²×C₃	Q₈×Dic₅	1	1	yes	characteristic
207	C₃×C₄	—	20	2	no	—
208	C₃×C₄	—	20	4	no	—
209	C₃×C₄	#4033	1	1	yes	characteristic
210	C₃×C₄	D₅×C₄⋊C₄	1	1	yes	characteristic
211	C₃×C₄	C₂×C₄×Dic₅	1	1	yes	characteristic
212	C₃×C₄	D₄×Dic₅	1	1	yes	characteristic
213	C₂×C₅	—	2	1	no	—
214	C₂×C₅	—	2	1	no	—
215	C₂×C₅	C₃×C₄×C₄⋊C₄	1	1	yes	characteristic
216	C₂×C₅	C₃×#444	1	2	yes	—
217	C₂³	C₃×#713	1	1	yes	characteristic
218	C₂×C₄	—	10	2	no	—
219	C₂×C₄	—	10	4	no	—
220	C₂×C₄	—	20	2	no	—
221	C₂×C₄	C₂×C₃×C₄×D₅	1	1	yes	characteristic
222	C₂×C₄	C₂×C₃×C₄×D₅	1	1	yes	characteristic
223	C₂×C₄	C₃×D₄×D₅	1	1	yes	characteristic
224	C₂×C₄	C₃×#711	1	1	yes	characteristic
225	C₂×C₄	C₃×Q₈×D₅	1	1	yes	characteristic
226	C₂×C₄	C₂²×C₃×Dic₅	1	1	yes	characteristic
227	C₈	—	2	4	no	—
228	C₈	—	5	4	no	—
229	C₈	C₃×C₄×Dic₅	1	2	yes	—
230	C₈	C₃×C₄×Dic₅	1	2	yes	—
231	C₂×C₃	—	2	1	no	—
232	C₂×C₃	—	2	1	no	—
233	C₂×C₃	#65183	1	2	yes	—
234	C₂×C₃	C₄⋊C₄×Dic₅	1	1	yes	characteristic
235	C₅	C₃×#1773	1	1	yes	characteristic, complemented, Sylow
236	C₂²	—	2	2	no	—
237	C₂²	C₃×#4038	1	1	yes	characteristic
238	C₂²	C₃×#4039	1	1	yes	characteristic
239	C₂²	C₃×Q₈×Dic₅	1	1	yes	characteristic
240	C₄	—	20	2	no	—
241	C₄	—	20	4	no	—
242	C₄	C₃×#4033	1	1	yes	characteristic
243	C₄	C₃×D₅×C₄⋊C₄	1	1	yes	characteristic
244	C₄	C₂×C₃×C₄×Dic₅	1	1	yes	characteristic
245	C₄	C₃×D₄×Dic₅	1	1	yes	characteristic
246	C₃	#10601496	1	1	yes	characteristic, direct product factor, Sylow
247	C₂	—	2	1	no	—
248	C₂	—	2	1	no	—
249	C₂	C₃×#65183	1	2	yes	—
250	C₂	C₃×C₄⋊C₄×Dic₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	D₄×D₅
commutator	C₄×C₅	C₂×C₃×C₄²
Frattini	C₂²×C₄	C₂²×C₃×D₅
Fitting	C₂×C₃×C₅×#108	C₂
socle	C₂²×C₃×C₅	C₄×D₄
prime radical	C₂³×C₃×C₅	#41
cosocle	C₂²×C₄×C₅	C₂³×C₃
hypercenter	C₂×C₃×#108	D₅
hypocenter	C₅	C₃×#1773
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂²×C₃×C₄ ◁ C₂×C₃×#108

Complements

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

N	H ≅ G/N	type
6. #10601496	C₃	direct (N × H)
12. C₃×C₅×#108	C₄	semidirect
13. C₃×#3950	C₄	semidirect
14. C₃×#3949	C₄	semidirect
16. C₃×#3947	C₄	semidirect
51. C₅×#108	C₃×C₄	semidirect
52. #3950	C₃×C₄	semidirect
53. #3949	C₃×C₄	semidirect
55. #3947	C₃×C₄	semidirect
77. C₃×#108	Dic₅	semidirect
151. #108	C₃×Dic₅	semidirect
202. C₃×C₅	#1773	semidirect
235. C₅	C₃×#1773	semidirect
246. C₃	#10601496	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×#108	2
C₃×#65411	2
C₃×#65395	2
C₂×C₃×#3947	2
C₃×#64775	2
#10601496	3
C₃×#1773	5

Subgroups — C3×#10601496 #14296526

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10601496 #14296526