Subgroups — C₃×#10600295 #14298456

Order 1920.

688 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₄⋊Dic₅	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×#380	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₂²⋊#168	C₂	1	1	yes	maximal, characteristic
4	C₃×#64749	C₂	1	4	yes	maximal, complemented
5	#10600295	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂³×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
7	C₂³×C₃×Dic₅	—	2	1	no	—
8	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
9	C₂×C₃×C₅×C₂²⋊C₄	—	2	1	no	—
10	C₂×C₃×C₄⋊Dic₅	—	2	1	no	—
11	C₂×C₃×C₄⋊Dic₅	—	2	2	no	—
12	C₂×C₃×C₄⋊Dic₅	C₄	1	4	yes	complemented
13	C₂×C₃×C₄⋊Dic₅	C₂²	1	2	yes	—
14	C₂²×C₃×#168	C₂²	1	2	yes	—
15	C₃×C₅×#95	—	2	1	no	—
16	C₃×C₂²⋊#168	—	2	2	no	—
17	C₃×#1824	—	5	1	no	maximal, Hall
18	C₂²×C₄⋊Dic₅	C₂×C₃	1	1	yes	characteristic
19	C₅×#380	C₂×C₃	1	1	yes	characteristic
20	C₂×C₂²⋊#168	C₂×C₃	1	1	yes	characteristic
21	#64749	C₂×C₃	1	4	yes	complemented
22	C₂⁴×C₃×C₅	D₄	1	1	yes	characteristic
23	C₂²×C₃×C₄×C₅	—	2	4	no	—
24	C₂²×C₃×C₄×C₅	—	2	2	no	—
25	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
26	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
27	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
28	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
29	C₂²×C₃×Dic₅	—	2	4	no	—
30	C₂²×C₃×Dic₅	—	2	2	no	—
31	C₂²×C₃×Dic₅	—	4	4	no	—
32	C₃×C₅×C₄⋊C₄	—	2	4	no	—
33	C₃×C₅×C₂²⋊C₄	—	4	2	no	—
34	C₃×C₄⋊Dic₅	—	2	4	no	—
35	C₃×C₄⋊Dic₅	—	2	2	no	—
36	C₃×C₄⋊Dic₅	C₂×C₄	1	4	yes	—
37	C₂×C₃×#168	—	2	2	no	—
38	C₂×C₃×#168	—	2	4	no	—
39	C₂²×C₃×C₄⋊C₄	—	5	1	no	—
40	C₂×C₃×C₂²⋊C₈	—	5	1	no	—
41	C₃×#380	D₅	1	1	yes	characteristic
42	C₃×#340	—	5	4	no	—
43	C₂³×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
44	C₂³×Dic₅	—	2	1	no	—
45	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
46	C₂×C₅×C₂²⋊C₄	—	2	1	no	—
47	C₂×C₄⋊Dic₅	—	2	2	no	—
48	C₂×C₄⋊Dic₅	—	2	1	no	—
49	C₂×C₄⋊Dic₅	C₃×C₄	1	4	yes	complemented
50	C₂×C₄⋊Dic₅	C₂²×C₃	1	2	yes	—
51	C₂²×#168	C₂²×C₃	1	2	yes	—
52	C₅×#95	—	2	1	no	—
53	C₂²⋊#168	—	2	2	no	—
54	#1824	—	5	1	no	Sylow
55	C₂³×C₃×C₅	—	2	2	no	—
56	C₂³×C₃×C₅	—	2	2	no	—
57	C₂³×C₃×C₅	C₂²⋊C₄	1	2	yes	—
58	C₂³×C₃×C₅	SD₁₆	1	2	yes	—
59	C₂³×C₃×C₅	Q₁₆	1	2	yes	—
60	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
61	C₂×C₃×C₄×C₅	—	2	2	no	—
62	C₂×C₃×C₄×C₅	—	2	4	no	—
63	C₂×C₃×C₄×C₅	—	2	4	no	—
64	C₂×C₃×C₄×C₅	—	4	2	no	—
65	C₂×C₃×C₄×C₅	—	4	2	no	—
66	C₂×C₃×C₄×C₅	—	4	2	no	—
67	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
68	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
70	C₂×C₃×C₄×C₅	#41	1	4	yes	—
71	C₂×C₃×Dic₅	—	2	4	no	—
72	C₂×C₃×Dic₅	—	4	8	no	—
73	C₂×C₃×Dic₅	—	4	4	no	—
74	C₃×#168	—	4	2	no	—
75	C₂³×C₃×C₄	—	10	1	no	—
76	C₂³×C₃×C₄	C₂×D₅	1	1	yes	characteristic
77	C₂²×C₃×C₈	—	5	2	no	—
78	C₂×C₃×C₄⋊C₄	—	5	4	no	—
79	C₂×C₃×C₄⋊C₄	—	5	2	no	—
80	C₂×C₃×C₄⋊C₄	—	10	2	no	—
81	C₂×C₃×C₄⋊C₄	—	10	1	no	—
82	C₂×C₃×C₄⋊C₄	C₂×D₅	1	2	yes	—
83	C₂×C₃×C₂²⋊C₄	—	2	1	no	—
84	C₃×C₂²⋊C₈	—	10	2	no	—
85	C₃×#95	—	2	1	no	—
86	C₂⁴×C₅	C₃×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₅	—	2	4	no	—
88	C₂²×C₄×C₅	—	2	2	no	—
89	C₂²×C₄×C₅	C₂×C₃×C₄	1	2	yes	—
90	C₂²×C₄×C₅	C₃×D₄	1	2	yes	—
91	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
93	C₂²×Dic₅	—	2	4	no	—
94	C₂²×Dic₅	—	2	2	no	—
95	C₂²×Dic₅	—	4	4	no	—
96	C₅×C₄⋊C₄	—	2	4	no	—
97	C₅×C₂²⋊C₄	—	4	2	no	—
98	C₄⋊Dic₅	—	2	4	no	—
99	C₄⋊Dic₅	—	2	2	no	—
100	C₄⋊Dic₅	C₂×C₃×C₄	1	4	yes	—
101	C₂×#168	—	2	2	no	—
102	C₂×#168	—	2	4	no	—
103	C₂²×C₄⋊C₄	—	5	1	no	—
104	C₂×C₂²⋊C₈	—	5	1	no	—
105	#380	C₃×D₅	1	1	yes	characteristic
106	#340	—	5	4	no	—
107	C₂²×C₃×C₅	—	2	2	no	—
108	C₂²×C₃×C₅	—	2	2	no	—
109	C₂²×C₃×C₅	—	2	4	no	—
110	C₂²×C₃×C₅	—	2	4	no	—
111	C₂²×C₃×C₅	#103	1	4	yes	—
112	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₂²×C₃×C₅	#123	1	2	yes	—
114	C₂²×C₃×C₅	C₂×SD₁₆	1	1	yes	characteristic
115	C₂²×C₃×C₅	C₂×Q₁₆	1	1	yes	characteristic
116	C₂²×C₃×C₅	AGL(1,ℤ/8)	1	2	yes	—
117	C₃×C₄×C₅	—	2	2	no	—
118	C₃×C₄×C₅	—	8	2	no	—
119	C₃×C₄×C₅	#117	1	2	yes	—
120	C₃×C₄×C₅	#123	1	2	yes	—
121	C₃×Dic₅	—	4	4	no	—
122	C₂⁴×C₃	C₂²⋊D₅	1	1	yes	characteristic
123	C₂²×C₃×C₄	—	2	4	no	—
124	C₂²×C₃×C₄	—	2	2	no	—
125	C₂²×C₃×C₄	—	10	4	no	—
126	C₂²×C₃×C₄	—	10	2	no	—
127	C₂²×C₃×C₄	—	20	4	no	—
128	C₂²×C₃×C₄	C₄×D₅	1	2	yes	—
129	C₂²×C₃×C₄	D₂₀	1	2	yes	—
130	C₂²×C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
131	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
132	C₂×C₃×C₈	—	10	2	no	—
133	C₂×C₃×C₈	—	10	4	no	—
134	C₃×C₄⋊C₄	—	2	4	no	—
135	C₃×C₄⋊C₄	—	5	4	no	—
136	C₃×C₄⋊C₄	—	10	4	no	—
137	C₃×C₄⋊C₄	—	10	2	no	—
138	C₃×C₂²⋊C₄	—	4	2	no	—
139	C₂³×C₅	—	2	2	no	—
140	C₂³×C₅	—	2	2	no	—
141	C₂³×C₅	C₃×C₂²⋊C₄	1	2	yes	—
142	C₂³×C₅	C₃×SD₁₆	1	2	yes	—
143	C₂³×C₅	C₃×Q₁₆	1	2	yes	—
144	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
145	C₂×C₄×C₅	—	2	2	no	—
146	C₂×C₄×C₅	—	2	4	no	—
147	C₂×C₄×C₅	—	2	4	no	—
148	C₂×C₄×C₅	—	4	2	no	—
149	C₂×C₄×C₅	—	4	2	no	—
150	C₂×C₄×C₅	—	4	2	no	—
151	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	2	yes	—
152	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
153	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
154	C₂×C₄×C₅	C₃×#41	1	4	yes	—
155	C₂×Dic₅	—	2	4	no	—
156	C₂×Dic₅	—	4	8	no	—
157	C₂×Dic₅	—	4	4	no	—
158	#168	—	4	2	no	—
159	C₂³×C₄	—	10	1	no	—
160	C₂³×C₄	C₂×C₃×D₅	1	1	yes	characteristic
161	C₂²×C₈	—	5	2	no	—
162	C₂×C₄⋊C₄	—	5	4	no	—
163	C₂×C₄⋊C₄	—	5	2	no	—
164	C₂×C₄⋊C₄	—	10	2	no	—
165	C₂×C₄⋊C₄	—	10	1	no	—
166	C₂×C₄⋊C₄	C₂×C₃×D₅	1	2	yes	—
167	C₂×C₂²⋊C₄	—	2	1	no	—
168	C₂²⋊C₈	—	10	2	no	—
169	#95	—	2	1	no	—
170	C₂×C₃×C₅	—	2	4	no	—
171	C₂×C₃×C₅	#381	1	1	yes	characteristic
172	C₂×C₃×C₅	C₂×#103	1	1	yes	characteristic
173	C₂×C₃×C₅	#418	1	1	yes	characteristic
174	C₂×C₃×C₅	#481	1	2	yes	—
175	C₂×C₃×C₅	#484	1	2	yes	—
176	C₂³×C₃	—	2	2	no	—
177	C₂³×C₃	—	2	2	no	—
178	C₂³×C₃	#685	1	2	yes	—
179	C₂³×C₃	#687	1	2	yes	—
180	C₂³×C₃	#689	1	2	yes	—
181	C₂³×C₃	C₂×C₂²⋊D₅	1	1	yes	characteristic
182	C₂×C₃×C₄	—	2	2	no	—
183	C₂×C₃×C₄	—	2	4	no	—
184	C₂×C₃×C₄	—	2	4	no	—
185	C₂×C₃×C₄	—	4	2	no	—
186	C₂×C₃×C₄	—	4	2	no	—
187	C₂×C₃×C₄	—	4	2	no	—
188	C₂×C₃×C₄	—	10	4	no	—
189	C₂×C₃×C₄	—	20	8	no	—
190	C₂×C₃×C₄	—	20	4	no	—
191	C₂×C₃×C₄	#685	1	2	yes	—
192	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
193	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
194	C₂×C₃×C₄	#711	1	4	yes	—
195	C₃×C₈	—	20	2	no	—
196	C₂²×C₅	—	2	2	no	—
197	C₂²×C₅	—	2	2	no	—
198	C₂²×C₅	—	2	4	no	—
199	C₂²×C₅	—	2	4	no	—
200	C₂²×C₅	C₃×#103	1	4	yes	—
201	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
202	C₂²×C₅	C₃×#123	1	2	yes	—
203	C₂²×C₅	C₂×C₃×SD₁₆	1	1	yes	characteristic
204	C₂²×C₅	C₂×C₃×Q₁₆	1	1	yes	characteristic
205	C₂²×C₅	C₃×AGL(1,ℤ/8)	1	2	yes	—
206	C₄×C₅	—	2	2	no	—
207	C₄×C₅	—	8	2	no	—
208	C₄×C₅	C₃×#117	1	2	yes	—
209	C₄×C₅	C₃×#123	1	2	yes	—
210	Dic₅	—	4	4	no	—
211	C₂⁴	C₃×C₂²⋊D₅	1	1	yes	characteristic
212	C₂²×C₄	—	2	4	no	—
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	—	10	4	no	—
215	C₂²×C₄	—	10	2	no	—
216	C₂²×C₄	—	20	4	no	—
217	C₂²×C₄	C₃×C₄×D₅	1	2	yes	—
218	C₂²×C₄	C₃×D₂₀	1	2	yes	—
219	C₂²×C₄	C₃×C₂²⋊D₅	1	1	yes	characteristic
220	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
221	C₂×C₈	—	10	2	no	—
222	C₂×C₈	—	10	4	no	—
223	C₄⋊C₄	—	2	4	no	—
224	C₄⋊C₄	—	5	4	no	—
225	C₄⋊C₄	—	10	4	no	—
226	C₄⋊C₄	—	10	2	no	—
227	C₂²⋊C₄	—	4	2	no	—
228	C₃×C₅	#1824	1	1	yes	characteristic, complemented, Hall
229	C₂²×C₃	—	2	2	no	—
230	C₂²×C₃	—	2	4	no	—
231	C₂²×C₃	—	2	4	no	—
232	C₂²×C₃	—	2	2	no	—
233	C₂²×C₃	#3942	1	4	yes	—
234	C₂²×C₃	C₂×#685	1	1	yes	characteristic
235	C₂²×C₃	C₂×#687	1	1	yes	characteristic
236	C₂²×C₃	#4081	1	2	yes	—
237	C₂²×C₃	C₂×#689	1	1	yes	characteristic
238	C₂²×C₃	#4095	1	2	yes	—
239	C₃×C₄	—	2	2	no	—
240	C₃×C₄	—	8	2	no	—
241	C₃×C₄	—	20	4	no	—
242	C₃×C₄	#4023	1	2	yes	—
243	C₃×C₄	#4032	1	2	yes	—
244	C₂×C₅	—	2	4	no	—
245	C₂×C₅	C₃×#381	1	1	yes	characteristic
246	C₂×C₅	C₂×C₃×#103	1	1	yes	characteristic
247	C₂×C₅	C₃×#418	1	1	yes	characteristic
248	C₂×C₅	C₃×#481	1	2	yes	—
249	C₂×C₅	C₃×#484	1	2	yes	—
250	C₂³	—	2	2	no	—
251	C₂³	—	2	2	no	—
252	C₂³	C₃×#685	1	2	yes	—
253	C₂³	C₃×#687	1	2	yes	—
254	C₂³	C₃×#689	1	2	yes	—
255	C₂³	C₂×C₃×C₂²⋊D₅	1	1	yes	characteristic
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	10	4	no	—
263	C₂×C₄	—	20	8	no	—
264	C₂×C₄	—	20	4	no	—
265	C₂×C₄	C₃×#685	1	2	yes	—
266	C₂×C₄	C₂×C₃×C₄×D₅	1	1	yes	characteristic
267	C₂×C₄	C₂×C₃×D₂₀	1	1	yes	characteristic
268	C₂×C₄	C₃×#711	1	4	yes	—
269	C₈	—	20	2	no	—
270	C₂×C₃	—	2	4	no	—
271	C₂×C₃	#65232	1	1	yes	characteristic
272	C₂×C₃	C₂×#3942	1	1	yes	characteristic
273	C₂×C₃	#65290	1	1	yes	characteristic
274	C₂×C₃	#65323	1	2	yes	—
275	C₂×C₃	#65333	1	2	yes	—
276	C₅	C₃×#1824	1	1	yes	characteristic, complemented, Sylow
277	C₂²	—	2	2	no	—
278	C₂²	—	2	4	no	—
279	C₂²	—	2	4	no	—
280	C₂²	—	2	2	no	—
281	C₂²	C₃×#3942	1	4	yes	—
282	C₂²	C₂×C₃×#685	1	1	yes	characteristic
283	C₂²	C₂×C₃×#687	1	1	yes	characteristic
284	C₂²	C₃×#4081	1	2	yes	—
285	C₂²	C₂×C₃×#689	1	1	yes	characteristic
286	C₂²	C₃×#4095	1	2	yes	—
287	C₄	—	2	2	no	—
288	C₄	—	8	2	no	—
289	C₄	—	20	4	no	—
290	C₄	C₃×#4023	1	2	yes	—
291	C₄	C₃×#4032	1	2	yes	—
292	C₃	#10600295	1	1	yes	characteristic, direct product factor, Sylow
293	C₂	—	2	4	no	—
294	C₂	C₃×#65232	1	1	yes	characteristic
295	C₂	C₂×C₃×#3942	1	1	yes	characteristic
296	C₂	C₃×#65290	1	1	yes	characteristic
297	C₂	C₃×#65323	1	2	yes	—
298	C₂	C₃×#65333	1	2	yes	—

	subgroup	G/H
center	C₂³×C₃	C₂×C₂²⋊D₅
commutator	C₂×C₄×C₅	C₂²×C₃×C₄
Frattini	C₂²×C₄	C₂²×C₃×D₅
Fitting	C₃×C₅×#380	C₂
socle	C₂³×C₃×C₅	C₂×D₄
prime radical	C₂⁴×C₃×C₅	D₄
cosocle	C₂²×C₄×C₅	C₂³×C₃
hypercenter	C₃×#380	D₅
hypocenter	C₅	C₃×#1824
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³×C₃ ◁ C₂³×C₃×C₄ ◁ C₃×#380

Complements

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

N	H ≅ G/N	type
4. C₃×#64749	C₂	semidirect
5. #10600295	C₃	direct (N × H)
12. C₂×C₃×C₄⋊Dic₅	C₄	semidirect
21. #64749	C₂×C₃	semidirect
49. C₂×C₄⋊Dic₅	C₃×C₄	semidirect
228. C₃×C₅	#1824	semidirect
276. C₅	C₃×#1824	semidirect
292. C₃	#10600295	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₃×C₄⋊Dic₅	2
C₃×C₅×#380	2
C₂×C₃×C₂²⋊#168	2
C₃×#64749	2
#10600295	3
C₃×#1824	5

Subgroups — C3×#10600295 #14298456

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10600295 #14298456