Subgroups — C₃×#10602380 #14294780

Order 1920.

776 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×#117	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃×#681	C₂	1	2	yes	maximal
3	C₃×#64750	C₂	1	4	yes	maximal, complemented
4	#10602380	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₂³×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
6	C₃×C₅×#117	—	2	2	no	—
7	C₃×C₅×#117	—	2	2	no	—
8	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
9	C₂×C₃×C₅×C₂²⋊C₄	C₄	1	2	yes	—
10	C₂×C₃×C₄²×C₅	C₄	1	2	yes	—
11	C₂×C₃×#681	—	2	4	no	—
12	C₂×C₃×#681	—	2	2	no	—
13	C₂×C₃×#681	C₄	1	8	yes	complemented
14	C₂×C₃×#681	C₂²	1	4	yes	—
15	C₂²×C₃×#168	—	2	2	no	—
16	C₃×#1742	—	5	1	no	maximal, Hall
17	C₂×C₅×#117	C₂×C₃	1	1	yes	characteristic
18	C₂²×#681	C₂×C₃	1	2	yes	—
19	#64750	C₂×C₃	1	4	yes	complemented
20	C₂⁴×C₃×C₅	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₃×C₄×C₅	C₂×C₄	1	4	yes	—
22	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₃×C₄×C₅	C₂×C₄	1	4	yes	—
24	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
25	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
26	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
27	C₂²×C₃×C₄×C₅	Q₈	1	2	yes	—
28	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
29	C₃×C₅×C₄⋊C₄	—	2	4	no	—
30	C₃×C₅×C₂²⋊C₄	—	2	4	no	—
31	C₃×C₄²×C₅	—	2	2	no	—
32	C₃×C₄²×C₅	—	2	2	no	—
33	C₃×#681	—	2	8	no	—
34	C₃×#681	—	2	4	no	—
35	C₃×#681	C₂×C₄	1	8	yes	—
36	C₂×C₃×#168	—	2	8	no	—
37	C₂×C₃×#168	—	2	4	no	—
38	C₂²×C₃×M₁₆	—	5	2	no	—
39	C₂×C₃×#117	D₅	1	1	yes	characteristic
40	C₃×#343	—	5	4	no	—
41	C₂³×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
42	C₅×#117	—	2	2	no	—
43	C₅×#117	—	2	2	no	—
44	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
45	C₂×C₅×C₂²⋊C₄	C₃×C₄	1	2	yes	—
46	C₂×C₄²×C₅	C₃×C₄	1	2	yes	—
47	C₂×#681	—	2	4	no	—
48	C₂×#681	—	2	2	no	—
49	C₂×#681	C₃×C₄	1	8	yes	complemented
50	C₂×#681	C₂²×C₃	1	4	yes	—
51	C₂²×#168	—	2	2	no	—
52	#1742	—	5	1	no	Sylow
53	C₂³×C₃×C₅	—	4	2	no	—
54	C₂³×C₃×C₅	C₄²	1	2	yes	—
55	C₂³×C₃×C₅	C₂²⋊C₄	1	2	yes	—
56	C₂³×C₃×C₅	C₄⋊C₄	1	2	yes	—
57	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₃×C₄×C₅	—	2	4	no	—
59	C₂×C₃×C₄×C₅	—	4	4	no	—
60	C₂×C₃×C₄×C₅	C₄²	1	2	yes	—
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	C₂²⋊C₄	1	4	yes	—
64	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	4	yes	—
65	C₂×C₃×C₄×C₅	C₄⋊C₄	1	2	yes	—
66	C₂×C₃×C₄×C₅	C₄⋊C₄	1	4	yes	—
67	C₂×C₃×C₄×C₅	C₄⋊C₄	1	4	yes	—
68	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
69	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
70	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
71	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
72	C₃×#168	—	2	8	no	—
73	C₂³×C₃×C₄	C₂×D₅	1	1	yes	characteristic
74	C₂×C₃×M₁₆	—	5	8	no	—
75	C₂×C₃×M₁₆	—	5	4	no	—
76	C₂×C₃×M₁₆	—	10	4	no	—
77	C₂×C₃×M₁₆	—	10	2	no	—
78	C₂²×C₃×C₈	—	10	2	no	—
79	C₃×#117	—	2	2	no	—
80	C₃×#117	—	2	2	no	—
81	C₂×C₃×C₄⋊C₄	C₂×D₅	1	2	yes	—
82	C₂×C₃×C₂²⋊C₄	Dic₅	1	2	yes	—
83	C₂×C₃×C₄²	Dic₅	1	2	yes	—
84	C₂⁴×C₅	C₂×C₃×C₄	1	1	yes	characteristic
85	C₂²×C₄×C₅	C₂×C₃×C₄	1	4	yes	—
86	C₂²×C₄×C₅	C₂×C₃×C₄	1	4	yes	—
87	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
88	C₂²×C₄×C₅	C₃×D₄	1	2	yes	—
89	C₂²×C₄×C₅	C₃×D₄	1	2	yes	—
90	C₂²×C₄×C₅	C₃×D₄	1	2	yes	—
91	C₂²×C₄×C₅	C₃×Q₈	1	2	yes	—
92	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
93	C₅×C₄⋊C₄	—	2	4	no	—
94	C₅×C₂²⋊C₄	—	2	4	no	—
95	C₄²×C₅	—	2	2	no	—
96	C₄²×C₅	—	2	2	no	—
97	#681	—	2	8	no	—
98	#681	—	2	4	no	—
99	#681	C₂×C₃×C₄	1	8	yes	—
100	C₂×#168	—	2	8	no	—
101	C₂×#168	—	2	4	no	—
102	C₂²×M₁₆	—	5	2	no	—
103	C₂×#117	C₃×D₅	1	1	yes	characteristic
104	#343	—	5	4	no	—
105	C₂²×C₃×C₅	—	2	2	no	—
106	C₂²×C₃×C₅	—	2	4	no	—
107	C₂²×C₃×C₅	—	4	2	no	—
108	C₂²×C₃×C₅	—	4	2	no	—
109	C₂²×C₃×C₅	#95	1	4	yes	—
110	C₂²×C₃×C₅	C₂×C₄²	1	1	yes	characteristic
111	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₃×C₄×C₅	—	4	4	no	—
114	C₃×C₄×C₅	#95	1	2	yes	—
115	C₃×C₄×C₅	#95	1	2	yes	—
116	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	2	yes	—
117	C₃×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
118	C₂⁴×C₃	C₂×Dic₅	1	1	yes	characteristic
119	C₂²×C₃×C₄	Dic₁₀	1	2	yes	—
120	C₂²×C₃×C₄	C₄×D₅	1	4	yes	—
121	C₂²×C₃×C₄	D₂₀	1	2	yes	—
122	C₂²×C₃×C₄	C₂×Dic₅	1	4	yes	—
123	C₂²×C₃×C₄	C₂×Dic₅	1	1	yes	characteristic
124	C₂²×C₃×C₄	C₂²⋊D₅	1	2	yes	—
125	C₂²×C₃×C₄	C₂²⋊D₅	1	2	yes	—
126	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
127	C₃×M₁₆	—	5	8	no	—
128	C₃×M₁₆	—	10	8	no	—
129	C₃×M₁₆	—	10	4	no	—
130	C₂×C₃×C₈	—	10	8	no	—
131	C₂×C₃×C₈	—	10	4	no	—
132	C₃×C₄⋊C₄	—	2	4	no	—
133	C₃×C₂²⋊C₄	—	2	4	no	—
134	C₃×C₄²	—	2	2	no	—
135	C₃×C₄²	—	2	2	no	—
136	C₂³×C₅	—	4	2	no	—
137	C₂³×C₅	C₃×C₄²	1	2	yes	—
138	C₂³×C₅	C₃×C₂²⋊C₄	1	2	yes	—
139	C₂³×C₅	C₃×C₄⋊C₄	1	2	yes	—
140	C₂³×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
141	C₂×C₄×C₅	—	2	4	no	—
142	C₂×C₄×C₅	—	4	4	no	—
143	C₂×C₄×C₅	C₃×C₄²	1	2	yes	—
144	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	4	yes	—
145	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	4	yes	—
146	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
147	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
148	C₂×C₄×C₅	C₃×C₄⋊C₄	1	4	yes	—
149	C₂×C₄×C₅	C₃×C₄⋊C₄	1	4	yes	—
150	C₂×C₄×C₅	C₃×C₄⋊C₄	1	2	yes	—
151	C₂×C₄×C₅	C₂²×C₃×C₄	1	2	yes	—
152	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
153	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
154	C₂×C₄×C₅	C₂×C₃×Q₈	1	1	yes	characteristic
155	#168	—	2	8	no	—
156	C₂³×C₄	C₂×C₃×D₅	1	1	yes	characteristic
157	C₂×M₁₆	—	5	8	no	—
158	C₂×M₁₆	—	5	4	no	—
159	C₂×M₁₆	—	10	4	no	—
160	C₂×M₁₆	—	10	2	no	—
161	C₂²×C₈	—	10	2	no	—
162	#117	—	2	2	no	—
163	#117	—	2	2	no	—
164	C₂×C₄⋊C₄	C₂×C₃×D₅	1	2	yes	—
165	C₂×C₂²⋊C₄	C₃×Dic₅	1	2	yes	—
166	C₂×C₄²	C₃×Dic₅	1	2	yes	—
167	C₂×C₃×C₅	—	2	4	no	—
168	C₂×C₃×C₅	—	2	1	no	—
169	C₂×C₃×C₅	—	2	1	no	—
170	C₂×C₃×C₅	C₂×#95	1	1	yes	characteristic
171	C₂×C₃×C₅	#413	1	2	yes	—
172	C₂³×C₃	—	4	2	no	—
173	C₂³×C₃	C₄×Dic₅	1	2	yes	—
174	C₂³×C₃	C₄⋊Dic₅	1	2	yes	—
175	C₂³×C₃	C₂²⋊Dic₅	1	2	yes	—
176	C₂³×C₃	C₂²×Dic₅	1	1	yes	characteristic
177	C₂×C₃×C₄	—	2	4	no	—
178	C₂×C₃×C₄	—	4	4	no	—
179	C₂×C₃×C₄	C₄×Dic₅	1	2	yes	—
180	C₂×C₃×C₄	#683	1	4	yes	—
181	C₂×C₃×C₄	#683	1	4	yes	—
182	C₂×C₃×C₄	C₄⋊Dic₅	1	2	yes	—
183	C₂×C₃×C₄	#685	1	4	yes	—
184	C₂×C₃×C₄	#685	1	4	yes	—
185	C₂×C₃×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
186	C₂×C₃×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
187	C₂×C₃×C₄	C₂×Dic₁₀	1	1	yes	characteristic
188	C₂×C₃×C₄	C₂×C₄×D₅	1	2	yes	—
189	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
190	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	2	yes	—
191	C₃×C₈	—	10	8	no	—
192	C₂²×C₅	—	2	2	no	—
193	C₂²×C₅	—	2	4	no	—
194	C₂²×C₅	—	4	2	no	—
195	C₂²×C₅	—	4	2	no	—
196	C₂²×C₅	C₃×#95	1	4	yes	—
197	C₂²×C₅	C₂×C₃×C₄²	1	1	yes	characteristic
198	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
199	C₂²×C₅	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
200	C₄×C₅	—	4	4	no	—
201	C₄×C₅	C₃×#95	1	2	yes	—
202	C₄×C₅	C₃×#95	1	2	yes	—
203	C₄×C₅	C₂×C₃×C₂²⋊C₄	1	2	yes	—
204	C₄×C₅	C₂×C₃×C₄⋊C₄	1	2	yes	—
205	C₂⁴	C₂×C₃×Dic₅	1	1	yes	characteristic
206	C₂²×C₄	C₃×Dic₁₀	1	2	yes	—
207	C₂²×C₄	C₃×C₄×D₅	1	4	yes	—
208	C₂²×C₄	C₃×D₂₀	1	2	yes	—
209	C₂²×C₄	C₂×C₃×Dic₅	1	4	yes	—
210	C₂²×C₄	C₂×C₃×Dic₅	1	1	yes	characteristic
211	C₂²×C₄	C₃×C₂²⋊D₅	1	2	yes	—
212	C₂²×C₄	C₃×C₂²⋊D₅	1	2	yes	—
213	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
214	M₁₆	—	5	8	no	—
215	M₁₆	—	10	8	no	—
216	M₁₆	—	10	4	no	—
217	C₂×C₈	—	10	8	no	—
218	C₂×C₈	—	10	4	no	—
219	C₄⋊C₄	—	2	4	no	—
220	C₂²⋊C₄	—	2	4	no	—
221	C₄²	—	2	2	no	—
222	C₄²	—	2	2	no	—
223	C₃×C₅	#1742	1	1	yes	characteristic, complemented, Hall
224	C₂²×C₃	—	2	2	no	—
225	C₂²×C₃	—	2	4	no	—
226	C₂²×C₃	—	4	2	no	—
227	C₂²×C₃	—	4	2	no	—
228	C₂²×C₃	#3963	1	4	yes	—
229	C₂²×C₃	C₂×C₄×Dic₅	1	1	yes	characteristic
230	C₂²×C₃	C₂×C₄⋊Dic₅	1	1	yes	characteristic
231	C₂²×C₃	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
232	C₃×C₄	—	4	4	no	—
233	C₃×C₄	#3963	1	2	yes	—
234	C₃×C₄	#3963	1	2	yes	—
235	C₃×C₄	C₂×#683	1	2	yes	—
236	C₃×C₄	C₂×#685	1	2	yes	—
237	C₂×C₅	—	2	4	no	—
238	C₂×C₅	—	2	1	no	—
239	C₂×C₅	—	2	1	no	—
240	C₂×C₅	C₂×C₃×#95	1	1	yes	characteristic
241	C₂×C₅	C₃×#413	1	2	yes	—
242	C₂³	—	4	2	no	—
243	C₂³	C₃×C₄×Dic₅	1	2	yes	—
244	C₂³	C₃×C₄⋊Dic₅	1	2	yes	—
245	C₂³	C₃×C₂²⋊Dic₅	1	2	yes	—
246	C₂³	C₂²×C₃×Dic₅	1	1	yes	characteristic
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	4	4	no	—
249	C₂×C₄	C₃×C₄×Dic₅	1	2	yes	—
250	C₂×C₄	C₃×#683	1	4	yes	—
251	C₂×C₄	C₃×#683	1	4	yes	—
252	C₂×C₄	C₃×C₄⋊Dic₅	1	2	yes	—
253	C₂×C₄	C₃×#685	1	4	yes	—
254	C₂×C₄	C₃×#685	1	4	yes	—
255	C₂×C₄	C₃×C₂²⋊Dic₅	1	1	yes	characteristic
256	C₂×C₄	C₃×C₂²⋊Dic₅	1	1	yes	characteristic
257	C₂×C₄	C₂×C₃×Dic₁₀	1	1	yes	characteristic
258	C₂×C₄	C₂×C₃×C₄×D₅	1	2	yes	—
259	C₂×C₄	C₂×C₃×D₂₀	1	1	yes	characteristic
260	C₂×C₄	C₂×C₃×C₂²⋊D₅	1	2	yes	—
261	C₈	—	10	8	no	—
262	C₂×C₃	—	2	4	no	—
263	C₂×C₃	—	2	1	no	—
264	C₂×C₃	—	2	1	no	—
265	C₂×C₃	C₂×#3963	1	1	yes	characteristic
266	C₂×C₃	#65526	1	2	yes	—
267	C₅	C₃×#1742	1	1	yes	characteristic, complemented, Sylow
268	C₂²	—	2	2	no	—
269	C₂²	—	2	4	no	—
270	C₂²	—	4	2	no	—
271	C₂²	—	4	2	no	—
272	C₂²	C₃×#3963	1	4	yes	—
273	C₂²	C₂×C₃×C₄×Dic₅	1	1	yes	characteristic
274	C₂²	C₂×C₃×C₄⋊Dic₅	1	1	yes	characteristic
275	C₂²	C₂×C₃×C₂²⋊Dic₅	1	1	yes	characteristic
276	C₄	—	4	4	no	—
277	C₄	C₃×#3963	1	2	yes	—
278	C₄	C₃×#3963	1	2	yes	—
279	C₄	C₂×C₃×#683	1	2	yes	—
280	C₄	C₂×C₃×#685	1	2	yes	—
281	C₃	#10602380	1	1	yes	characteristic, direct product factor, Sylow
282	C₂	—	2	4	no	—
283	C₂	—	2	1	no	—
284	C₂	—	2	1	no	—
285	C₂	C₂×C₃×#3963	1	1	yes	characteristic
286	C₂	C₃×#65526	1	2	yes	—

	subgroup	G/H
center	C₂×C₃×C₄	C₂²⋊Dic₅
commutator	C₂²×C₅	C₂×C₃×C₄²
Frattini	C₂²×C₄	C₂²×C₃×D₅
Fitting	C₂×C₃×C₅×#117	C₂
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₃×C₅	C₂×C₄
cosocle	C₂²×C₄×C₅	C₂³×C₃
hypercenter	C₂×C₃×#117	D₅
hypocenter	C₅	C₃×#1742
LCS	G ▷ C₂²×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂³×C₃×C₄ ◁ C₂×C₃×#117

Complements

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

N	H ≅ G/N	type
3. C₃×#64750	C₂	semidirect
4. #10602380	C₃	direct (N × H)
13. C₂×C₃×#681	C₄	semidirect
19. #64750	C₂×C₃	semidirect
49. C₂×#681	C₃×C₄	semidirect
223. C₃×C₅	#1742	semidirect
267. C₅	C₃×#1742	semidirect
281. C₃	#10602380	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×#117	2
C₂²×C₃×#681	2
C₃×#64750	2
#10602380	3
C₃×#1742	5

Subgroups — C3×#10602380 #14294780

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10602380 #14294780