Subgroups — C₃×#10601495 #14296463

Order 1920.

464 conjugacy classes of subgroups.

The subgroup lattice has 260 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×C₈×Dic₅	C₂	1	1	yes	maximal, characteristic
5	C₃×#64775	C₂	1	2	yes	maximal
6	#10601495	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×C₈×Dic₅	—	2	2	no	—
17	C₃×C₈×Dic₅	C₄	1	4	yes	complemented
18	C₃×#1772	—	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×C₈×Dic₅	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	1	yes	characteristic
40	C₂×C₃×#168	D₄	1	1	yes	characteristic
41	C₂×C₃×#168	Q₈	1	2	yes	—
42	C₂×C₃×#108	D₅	1	1	yes	characteristic
43	C₃×#427	—	5	1	no	—
44	C₃×#405	—	5	2	no	—
45	C₂×C₃×C₄×C₈	—	5	1	no	—
46	C₃×#339	—	5	2	no	—
47	C₂×C₅×M₁₆	C₂²×C₃	1	2	yes	—
48	C₂²×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
49	#4072	C₂²×C₃	1	2	yes	—
50	C₂×C₄×Dic₅	C₂²×C₃	1	1	yes	characteristic
51	C₂²×#168	C₂²×C₃	1	1	yes	characteristic
52	C₅×#108	C₃×C₄	1	4	yes	complemented
53	#3950	C₃×C₄	1	2	yes	complemented
54	#3949	C₃×C₄	1	2	yes	complemented
55	#3947	—	2	2	no	—
56	C₈×Dic₅	—	2	2	no	—
57	C₈×Dic₅	C₃×C₄	1	4	yes	complemented
58	#1772	—	5	1	no	Sylow
59	C₂³×C₃×C₅	#41	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	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
64	C₂×C₃×C₄×C₅	C₂×Q₈	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	2	no	—
68	C₂×C₃×Dic₅	—	2	4	no	—
69	C₃×C₅×C₈	—	2	4	no	—
70	C₃×C₅×C₈	C₄²	1	2	yes	—
71	C₃×C₅×C₈	C₄²	1	2	yes	—
72	C₃×#168	C₄⋊C₄	1	2	yes	—
73	C₃×#168	C₄⋊C₄	1	2	yes	—
74	C₂×C₃×M₁₆	C₂×D₅	1	2	yes	—
75	C₂²×C₃×C₈	—	5	1	no	—
76	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
77	C₃×#117	—	5	2	no	—
78	C₂×C₃×C₄²	—	5	1	no	—
79	C₃×#108	Dic₅	1	4	yes	complemented
80	C₃×#107	—	5	2	no	—
81	C₃×#106	—	5	2	no	—
82	C₃×#97	—	10	2	no	—
83	C₃×C₄×C₈	—	5	4	no	—
84	C₃×C₄×C₈	—	10	2	no	—
85	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
86	C₂²×Dic₅	—	2	1	no	—
87	C₅×M₁₆	—	2	2	no	—
88	C₅×M₁₆	C₂×C₃×C₄	1	4	yes	—
89	C₂×C₅×C₈	—	2	2	no	—
90	C₂×C₅×C₈	C₂×C₃×C₄	1	2	yes	—
91	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
92	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
93	C₂×C₅×C₈	C₂×C₃×C₄	1	2	yes	—
94	C₂²⋊Dic₅	—	2	2	no	—
95	C₄⋊Dic₅	C₂×C₃×C₄	1	4	yes	—
96	C₄×Dic₅	—	2	2	no	—
97	C₄×Dic₅	—	2	1	no	—
98	C₄×Dic₅	C₂×C₃×C₄	1	2	yes	—
99	C₂×#168	C₂×C₃×C₄	1	2	yes	—
100	C₂×#168	C₃×D₄	1	1	yes	characteristic
101	C₂×#168	C₃×D₄	1	1	yes	characteristic
102	C₂×#168	C₃×Q₈	1	2	yes	—
103	C₂×#108	C₃×D₅	1	1	yes	characteristic
104	#427	—	5	1	no	—
105	#405	—	5	2	no	—
106	C₂×C₄×C₈	—	5	1	no	—
107	#339	—	5	2	no	—
108	C₂²×C₃×C₅	—	2	2	no	—
109	C₂²×C₃×C₅	#117	1	1	yes	characteristic
110	C₂²×C₃×C₅	C₄×D₄	1	1	yes	characteristic
111	C₂²×C₃×C₅	C₄×Q₈	1	1	yes	characteristic
112	C₃×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
113	C₃×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
114	C₃×C₄×C₅	C₄×D₄	1	1	yes	characteristic
115	C₃×C₄×C₅	C₄×Q₈	1	1	yes	characteristic
116	C₃×Dic₅	—	2	4	no	—
117	C₃×Dic₅	—	4	4	no	—
118	C₂²×C₃×C₄	—	10	1	no	—
119	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
120	C₃×M₁₆	—	2	2	no	—
121	C₃×M₁₆	C₂×Dic₅	1	4	yes	—
122	C₂×C₃×C₈	—	2	2	no	—
123	C₂×C₃×C₈	—	5	2	no	—
124	C₂×C₃×C₈	—	5	1	no	—
125	C₂×C₃×C₈	—	5	1	no	—
126	C₂×C₃×C₈	—	5	2	no	—
127	C₂×C₃×C₈	C₄×D₅	1	2	yes	—
128	C₂×C₃×C₈	C₄×D₅	1	1	yes	characteristic
129	C₂×C₃×C₈	C₄×D₅	1	1	yes	characteristic
130	C₂×C₃×C₈	C₂×Dic₅	1	2	yes	—
131	C₃×C₄⋊C₄	—	5	4	no	—
132	C₃×C₂²⋊C₄	—	10	2	no	—
133	C₃×C₄²	—	5	2	no	—
134	C₃×C₄²	—	10	2	no	—
135	C₃×C₄²	—	10	1	no	—
136	C₂³×C₅	C₃×#41	1	1	yes	characteristic
137	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
138	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
139	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
140	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
141	C₂×C₄×C₅	C₂×C₃×Q₈	1	1	yes	characteristic
142	C₂×C₄×C₅	C₃×#41	1	1	yes	characteristic
143	C₂×Dic₅	—	2	4	no	—
144	C₂×Dic₅	—	2	2	no	—
145	C₂×Dic₅	—	2	4	no	—
146	C₅×C₈	—	2	4	no	—
147	C₅×C₈	C₃×C₄²	1	2	yes	—
148	C₅×C₈	C₃×C₄²	1	2	yes	—
149	#168	C₃×C₄⋊C₄	1	2	yes	—
150	#168	C₃×C₄⋊C₄	1	2	yes	—
151	C₂×M₁₆	C₂×C₃×D₅	1	2	yes	—
152	C₂²×C₈	—	5	1	no	—
153	C₂²×C₈	C₂×C₃×D₅	1	1	yes	characteristic
154	#117	—	5	2	no	—
155	C₂×C₄²	—	5	1	no	—
156	#108	C₃×Dic₅	1	4	yes	complemented
157	#107	—	5	2	no	—
158	#106	—	5	2	no	—
159	#97	—	10	2	no	—
160	C₄×C₈	—	5	4	no	—
161	C₄×C₈	—	10	2	no	—
162	C₂×C₃×C₅	—	2	1	no	—
163	C₂×C₃×C₅	—	2	1	no	—
164	C₂×C₃×C₅	C₄×C₄⋊C₄	1	1	yes	characteristic
165	C₂×C₃×C₅	#443	1	2	yes	—
166	C₂³×C₃	#713	1	1	yes	characteristic
167	C₂×C₃×C₄	—	10	4	no	—
168	C₂×C₃×C₄	—	10	2	no	—
169	C₂×C₃×C₄	—	10	4	no	—
170	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
171	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
172	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
173	C₂×C₃×C₄	#711	1	1	yes	characteristic
174	C₂×C₃×C₄	Q₈×D₅	1	1	yes	characteristic
175	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
176	C₃×C₈	—	2	4	no	—
177	C₃×C₈	—	5	2	no	—
178	C₃×C₈	—	5	2	no	—
179	C₃×C₈	C₄×Dic₅	1	2	yes	—
180	C₃×C₈	C₄×Dic₅	1	2	yes	—
181	C₂²×C₅	—	2	2	no	—
182	C₂²×C₅	C₃×#117	1	1	yes	characteristic
183	C₂²×C₅	C₃×C₄×D₄	1	1	yes	characteristic
184	C₂²×C₅	C₃×C₄×Q₈	1	1	yes	characteristic
185	C₄×C₅	C₂×C₃×C₄²	1	1	yes	characteristic
186	C₄×C₅	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
187	C₄×C₅	C₃×C₄×D₄	1	1	yes	characteristic
188	C₄×C₅	C₃×C₄×Q₈	1	1	yes	characteristic
189	Dic₅	—	2	4	no	—
190	Dic₅	—	4	4	no	—
191	C₂²×C₄	—	10	1	no	—
192	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
193	M₁₆	—	2	2	no	—
194	M₁₆	C₂×C₃×Dic₅	1	4	yes	—
195	C₂×C₈	—	2	2	no	—
196	C₂×C₈	—	5	2	no	—
197	C₂×C₈	—	5	1	no	—
198	C₂×C₈	—	5	1	no	—
199	C₂×C₈	—	5	2	no	—
200	C₂×C₈	C₃×C₄×D₅	1	2	yes	—
201	C₂×C₈	C₃×C₄×D₅	1	1	yes	characteristic
202	C₂×C₈	C₃×C₄×D₅	1	1	yes	characteristic
203	C₂×C₈	C₂×C₃×Dic₅	1	2	yes	—
204	C₄⋊C₄	—	5	4	no	—
205	C₂²⋊C₄	—	10	2	no	—
206	C₄²	—	5	2	no	—
207	C₄²	—	10	2	no	—
208	C₄²	—	10	1	no	—
209	C₃×C₅	#1772	1	1	yes	characteristic, complemented, Hall
210	C₂²×C₃	—	2	2	no	—
211	C₂²×C₃	#4038	1	1	yes	characteristic
212	C₂²×C₃	#4039	1	1	yes	characteristic
213	C₂²×C₃	Q₈×Dic₅	1	1	yes	characteristic
214	C₃×C₄	—	10	4	no	—
215	C₃×C₄	—	20	4	no	—
216	C₃×C₄	#4033	1	1	yes	characteristic
217	C₃×C₄	D₅×C₄⋊C₄	1	1	yes	characteristic
218	C₃×C₄	C₂×C₄×Dic₅	1	1	yes	characteristic
219	C₃×C₄	D₄×Dic₅	1	1	yes	characteristic
220	C₂×C₅	—	2	1	no	—
221	C₂×C₅	—	2	1	no	—
222	C₂×C₅	C₃×C₄×C₄⋊C₄	1	1	yes	characteristic
223	C₂×C₅	C₃×#443	1	2	yes	—
224	C₂³	C₃×#713	1	1	yes	characteristic
225	C₂×C₄	—	10	4	no	—
226	C₂×C₄	—	10	2	no	—
227	C₂×C₄	—	10	4	no	—
228	C₂×C₄	C₂×C₃×C₄×D₅	1	1	yes	characteristic
229	C₂×C₄	C₂×C₃×C₄×D₅	1	1	yes	characteristic
230	C₂×C₄	C₃×D₄×D₅	1	1	yes	characteristic
231	C₂×C₄	C₃×#711	1	1	yes	characteristic
232	C₂×C₄	C₃×Q₈×D₅	1	1	yes	characteristic
233	C₂×C₄	C₂²×C₃×Dic₅	1	1	yes	characteristic
234	C₈	—	2	4	no	—
235	C₈	—	5	2	no	—
236	C₈	—	5	2	no	—
237	C₈	C₃×C₄×Dic₅	1	2	yes	—
238	C₈	C₃×C₄×Dic₅	1	2	yes	—
239	C₂×C₃	—	2	1	no	—
240	C₂×C₃	—	2	1	no	—
241	C₂×C₃	#65184	1	2	yes	—
242	C₂×C₃	C₄⋊C₄×Dic₅	1	1	yes	characteristic
243	C₅	C₃×#1772	1	1	yes	characteristic, complemented, Sylow
244	C₂²	—	2	2	no	—
245	C₂²	C₃×#4038	1	1	yes	characteristic
246	C₂²	C₃×#4039	1	1	yes	characteristic
247	C₂²	C₃×Q₈×Dic₅	1	1	yes	characteristic
248	C₄	—	10	4	no	—
249	C₄	—	20	4	no	—
250	C₄	C₃×#4033	1	1	yes	characteristic
251	C₄	C₃×D₅×C₄⋊C₄	1	1	yes	characteristic
252	C₄	C₂×C₃×C₄×Dic₅	1	1	yes	characteristic
253	C₄	C₃×D₄×Dic₅	1	1	yes	characteristic
254	C₃	#10601495	1	1	yes	characteristic, direct product factor, Sylow
255	C₂	—	2	1	no	—
256	C₂	—	2	1	no	—
257	C₂	C₃×#65184	1	2	yes	—
258	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×#1772
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. #10601495	C₃	direct (N × H)
12. C₃×C₅×#108	C₄	semidirect
13. C₃×#3950	C₄	semidirect
14. C₃×#3949	C₄	semidirect
17. C₃×C₈×Dic₅	C₄	semidirect
52. C₅×#108	C₃×C₄	semidirect
53. #3950	C₃×C₄	semidirect
54. #3949	C₃×C₄	semidirect
57. C₈×Dic₅	C₃×C₄	semidirect
79. C₃×#108	Dic₅	semidirect
156. #108	C₃×Dic₅	semidirect
209. C₃×C₅	#1772	semidirect
243. C₅	C₃×#1772	semidirect
254. C₃	#10601495	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×C₈×Dic₅	2
C₃×#64775	2
#10601495	3
C₃×#1772	5

Subgroups — C3×#10601495 #14296463

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10601495 #14296463