Subgroups — C₃×#65880 #11755557

Order 960.

588 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#711	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄×C₅×D₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×D₄×Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₄×C₂²⋊D₅	C₂	1	2	yes	maximal, complemented
5	C₂×C₃×#683	C₂	1	2	yes	maximal, complemented
6	C₃×D₅×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#4033	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#4027	C₂	1	2	yes	maximal, complemented
9	C₃×#4023	C₂	1	2	yes	maximal, complemented
10	C₃×#4018	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃×C₄×Dic₁₀	C₂	1	1	yes	maximal, characteristic, complemented
12	#65880	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
13	C₂×C₃×C₅×D₄	C₂²	1	1	yes	characteristic
14	C₂²×C₃×C₄×C₅	C₂²	1	2	yes	—
15	C₂×C₃×C₂²⋊D₅	C₂²	1	2	yes	—
16	C₂²×C₃×Dic₅	C₂²	1	2	yes	—
17	C₂²×C₃×Dic₅	C₂²	1	2	yes	—
18	C₃×#711	C₄	1	8	yes	complemented
19	C₂×C₃×C₄×D₅	C₂²	1	2	yes	—
20	C₂×C₃×C₄×D₅	C₂²	1	1	yes	characteristic
21	C₂×C₃×Dic₁₀	C₂²	1	1	yes	characteristic
22	C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
23	C₃×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
24	C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
25	C₃×C₂²⋊Dic₅	C₂²	1	2	yes	—
26	C₃×#685	C₂²	1	1	yes	characteristic
27	C₃×#685	C₂²	1	1	yes	characteristic
28	C₃×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
29	C₃×#683	C₂²	1	2	yes	—
30	C₃×#683	C₂²	1	1	yes	characteristic
31	C₃×#683	C₂²	1	1	yes	characteristic
32	C₃×#683	C₂²	1	2	yes	—
33	C₃×#683	C₂²	1	2	yes	—
34	C₃×C₄×Dic₅	C₂²	1	1	yes	characteristic
35	C₃×C₄×Dic₅	C₂²	1	2	yes	—
36	C₃×C₄×Dic₅	C₂²	1	2	yes	—
37	C₃×#520	—	5	1	no	maximal, Hall
38	C₂×#711	C₂×C₃	1	1	yes	characteristic
39	C₄×C₅×D₄	C₂×C₃	1	1	yes	characteristic, complemented
40	D₄×Dic₅	C₂×C₃	1	1	yes	characteristic, complemented
41	C₄×C₂²⋊D₅	C₂×C₃	1	2	yes	complemented
42	C₂×#683	C₂×C₃	1	2	yes	complemented
43	D₅×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic, complemented
44	#4033	C₂×C₃	1	1	yes	characteristic, complemented
45	#4027	C₂×C₃	1	2	yes	complemented
46	#4023	C₂×C₃	1	2	yes	complemented
47	#4018	C₂×C₃	1	1	yes	characteristic, complemented
48	C₄×Dic₁₀	C₂×C₃	1	1	yes	characteristic, complemented
49	C₂³×C₃×C₅	C₂³	1	2	yes	—
50	C₂²×C₃×D₅	C₂³	1	1	yes	characteristic
51	C₃×C₅×D₄	C₂×C₄	1	4	yes	complemented
52	C₂×C₃×C₄×C₅	—	4	2	no	—
53	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
54	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
55	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
56	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
57	C₃×C₂²⋊D₅	C₂×C₄	1	8	yes	complemented
58	C₂×C₃×Dic₅	—	4	2	no	—
59	C₂×C₃×Dic₅	C₂×C₄	1	8	yes	complemented
60	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
61	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
62	C₂×C₃×Dic₅	C₂³	1	2	yes	—
63	C₂×C₃×Dic₅	C₂³	1	2	yes	—
64	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
65	C₃×C₄×D₅	—	4	2	no	—
66	C₃×C₄×D₅	C₂×C₄	1	4	yes	complemented
67	C₃×Dic₁₀	C₂×C₄	1	4	yes	complemented
68	C₂×C₃×#41	—	5	1	no	—
69	C₃×C₄×Q₈	—	5	1	no	—
70	C₃×C₄×Q₈	—	5	1	no	—
71	C₃×C₄×D₄	—	5	1	no	—
72	C₃×C₄×D₄	—	5	2	no	—
73	C₃×C₄×D₄	—	5	2	no	—
74	C₃×C₄×D₄	D₅	1	1	yes	characteristic, complemented
75	C₃×#117	—	5	2	no	—
76	C₃×#117	—	5	1	no	—
77	C₂×C₃×C₄⋊C₄	—	5	2	no	—
78	C₂×C₃×C₄⋊C₄	—	5	1	no	—
79	C₂×C₅×D₄	C₂²×C₃	1	1	yes	characteristic
80	C₂²×C₄×C₅	C₂²×C₃	1	2	yes	—
81	C₂×C₂²⋊D₅	C₂²×C₃	1	2	yes	—
82	C₂²×Dic₅	C₂²×C₃	1	2	yes	—
83	C₂²×Dic₅	C₂²×C₃	1	2	yes	—
84	#711	C₃×C₄	1	8	yes	complemented
85	C₂×C₄×D₅	C₂²×C₃	1	2	yes	—
86	C₂×C₄×D₅	C₂²×C₃	1	1	yes	characteristic
87	C₂×Dic₁₀	C₂²×C₃	1	1	yes	characteristic
88	C₅×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
89	C₅×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
90	C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
91	C₂²⋊Dic₅	C₂²×C₃	1	2	yes	—
92	#685	C₂²×C₃	1	1	yes	characteristic
93	#685	C₂²×C₃	1	1	yes	characteristic
94	C₄⋊Dic₅	C₂²×C₃	1	1	yes	characteristic
95	#683	C₂²×C₃	1	2	yes	—
96	#683	C₂²×C₃	1	1	yes	characteristic
97	#683	C₂²×C₃	1	1	yes	characteristic
98	#683	C₂²×C₃	1	2	yes	—
99	#683	C₂²×C₃	1	2	yes	—
100	C₄×Dic₅	C₂²×C₃	1	2	yes	—
101	C₄×Dic₅	C₂²×C₃	1	1	yes	characteristic
102	C₄×Dic₅	C₂²×C₃	1	2	yes	—
103	#520	—	5	1	no	Sylow
104	C₂²×C₃×C₅	—	2	2	no	—
105	C₂²×C₃×C₅	—	2	2	no	—
106	C₂²×C₃×C₅	C₂²×C₄	1	4	yes	—
107	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
108	C₂×C₃×D₅	—	2	1	no	—
109	C₂×C₃×D₅	—	2	1	no	—
110	C₂×C₃×D₅	C₂²×C₄	1	2	yes	—
111	C₃×C₄×C₅	—	2	2	no	—
112	C₃×C₄×C₅	—	2	1	no	—
113	C₃×C₄×C₅	—	2	1	no	—
114	C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
115	C₃×Dic₅	—	2	2	no	—
116	C₃×Dic₅	—	2	1	no	—
117	C₃×Dic₅	—	2	1	no	—
118	C₃×Dic₅	C₂²×C₄	1	4	yes	—
119	C₃×Dic₅	C₂²×C₄	1	2	yes	—
120	C₃×#41	—	5	8	no	—
121	C₂×C₃×Q₈	—	5	1	no	—
122	C₂×C₃×D₄	—	5	2	no	—
123	C₂×C₃×D₄	C₂×D₅	1	1	yes	characteristic
124	C₂²×C₃×C₄	—	5	2	no	—
125	C₂²×C₃×C₄	—	5	1	no	—
126	C₂²×C₃×C₄	—	5	2	no	—
127	C₂²×C₃×C₄	—	5	2	no	—
128	C₂²×C₃×C₄	C₂×D₅	1	2	yes	—
129	C₃×C₄⋊C₄	—	5	2	no	—
130	C₃×C₄⋊C₄	—	5	1	no	—
131	C₃×C₄⋊C₄	—	5	2	no	—
132	C₃×C₄⋊C₄	—	5	2	no	—
133	C₃×C₄⋊C₄	—	5	1	no	—
134	C₃×C₄⋊C₄	—	5	1	no	—
135	C₃×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
136	C₃×C₂²⋊C₄	—	5	2	no	—
137	C₃×C₂²⋊C₄	—	5	1	no	—
138	C₃×C₂²⋊C₄	—	5	1	no	—
139	C₃×C₂²⋊C₄	C₂×D₅	1	2	yes	—
140	C₃×C₄²	—	5	2	no	—
141	C₃×C₄²	—	5	2	no	—
142	C₃×C₄²	—	5	1	no	—
143	C₃×C₄²	C₂×D₅	1	1	yes	characteristic
144	C₂³×C₅	C₂³×C₃	1	2	yes	—
145	C₂²×D₅	C₂³×C₃	1	1	yes	characteristic
146	C₅×D₄	C₂×C₃×C₄	1	4	yes	complemented
147	C₂×C₄×C₅	—	4	2	no	—
148	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
149	C₂×C₄×C₅	C₂³×C₃	1	2	yes	—
150	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
151	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
152	C₂²⋊D₅	C₂×C₃×C₄	1	8	yes	complemented
153	C₂×Dic₅	—	4	2	no	—
154	C₂×Dic₅	C₂×C₃×C₄	1	8	yes	complemented
155	C₂×Dic₅	C₂³×C₃	1	1	yes	characteristic
156	C₂×Dic₅	C₂³×C₃	1	1	yes	characteristic
157	C₂×Dic₅	C₂³×C₃	1	2	yes	—
158	C₂×Dic₅	C₂³×C₃	1	2	yes	—
159	C₂×Dic₅	C₂³×C₃	1	1	yes	characteristic
160	C₄×D₅	—	4	2	no	—
161	C₄×D₅	C₂×C₃×C₄	1	4	yes	complemented
162	Dic₁₀	C₂×C₃×C₄	1	4	yes	complemented
163	C₂×#41	—	5	1	no	—
164	C₄×Q₈	—	5	1	no	—
165	C₄×Q₈	—	5	1	no	—
166	C₄×D₄	—	5	1	no	—
167	C₄×D₄	—	5	2	no	—
168	C₄×D₄	—	5	2	no	—
169	C₄×D₄	C₃×D₅	1	1	yes	characteristic, complemented
170	#117	—	5	2	no	—
171	#117	—	5	1	no	—
172	C₂×C₄⋊C₄	—	5	2	no	—
173	C₂×C₄⋊C₄	—	5	1	no	—
174	C₂×C₃×C₅	—	2	4	no	—
175	C₂×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
176	C₂×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
177	C₂×C₃×C₅	2₋¹⁺⁴	1	1	yes	characteristic
178	C₃×D₅	—	2	2	no	—
179	C₂³×C₃	—	5	1	no	—
180	C₂³×C₃	C₂²×D₅	1	2	yes	—
181	C₃×Q₈	—	5	4	no	—
182	C₃×D₄	—	5	8	no	—
183	C₃×D₄	C₄×D₅	1	4	yes	complemented
184	C₂×C₃×C₄	—	4	2	no	—
185	C₂×C₃×C₄	—	5	8	no	—
186	C₂×C₃×C₄	—	5	2	no	—
187	C₂×C₃×C₄	—	5	1	no	—
188	C₂×C₃×C₄	—	5	4	no	—
189	C₂×C₃×C₄	—	5	1	no	—
190	C₂×C₃×C₄	—	5	1	no	—
191	C₂×C₃×C₄	—	5	2	no	—
192	C₂×C₃×C₄	—	20	2	no	—
193	C₂×C₃×C₄	—	20	2	no	—
194	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
195	C₂×C₃×C₄	C₂²×D₅	1	2	yes	—
196	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
197	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
198	C₂²×C₅	—	2	2	no	—
199	C₂²×C₅	—	2	2	no	—
200	C₂²×C₅	C₂²×C₃×C₄	1	4	yes	—
201	C₂²×C₅	C₂⁴×C₃	1	1	yes	characteristic
202	C₂×D₅	—	2	1	no	—
203	C₂×D₅	—	2	1	no	—
204	C₂×D₅	C₂²×C₃×C₄	1	2	yes	—
205	C₄×C₅	—	2	2	no	—
206	C₄×C₅	—	2	1	no	—
207	C₄×C₅	—	2	1	no	—
208	C₄×C₅	C₂²×C₃×C₄	1	2	yes	—
209	Dic₅	—	2	2	no	—
210	Dic₅	—	2	1	no	—
211	Dic₅	—	2	1	no	—
212	Dic₅	C₂²×C₃×C₄	1	4	yes	—
213	Dic₅	C₂²×C₃×C₄	1	2	yes	—
214	#41	—	5	8	no	—
215	C₂×Q₈	—	5	1	no	—
216	C₂×D₄	—	5	2	no	—
217	C₂×D₄	C₂×C₃×D₅	1	1	yes	characteristic
218	C₂²×C₄	—	5	2	no	—
219	C₂²×C₄	—	5	1	no	—
220	C₂²×C₄	—	5	2	no	—
221	C₂²×C₄	—	5	2	no	—
222	C₂²×C₄	C₂×C₃×D₅	1	2	yes	—
223	C₄⋊C₄	—	5	2	no	—
224	C₄⋊C₄	—	5	1	no	—
225	C₄⋊C₄	—	5	2	no	—
226	C₄⋊C₄	—	5	2	no	—
227	C₄⋊C₄	—	5	1	no	—
228	C₄⋊C₄	—	5	1	no	—
229	C₄⋊C₄	C₂×C₃×D₅	1	1	yes	characteristic
230	C₂²⋊C₄	—	5	2	no	—
231	C₂²⋊C₄	—	5	1	no	—
232	C₂²⋊C₄	—	5	1	no	—
233	C₂²⋊C₄	C₂×C₃×D₅	1	2	yes	—
234	C₄²	—	5	2	no	—
235	C₄²	—	5	2	no	—
236	C₄²	—	5	1	no	—
237	C₄²	C₂×C₃×D₅	1	1	yes	characteristic
238	C₃×C₅	#520	1	1	yes	characteristic, complemented, Hall
239	C₂²×C₃	—	2	2	no	—
240	C₂²×C₃	—	2	2	no	—
241	C₂²×C₃	—	5	2	no	—
242	C₂²×C₃	—	10	1	no	—
243	C₂²×C₃	—	10	1	no	—
244	C₂²×C₃	C₂×C₄×D₅	1	4	yes	—
245	C₂²×C₃	C₂³×D₅	1	1	yes	characteristic
246	C₃×C₄	—	2	1	no	—
247	C₃×C₄	—	2	2	no	—
248	C₃×C₄	—	2	1	no	—
249	C₃×C₄	—	5	2	no	—
250	C₃×C₄	—	5	4	no	—
251	C₃×C₄	—	10	2	no	—
252	C₃×C₄	—	10	1	no	—
253	C₃×C₄	—	10	1	no	—
254	C₃×C₄	C₂×C₄×D₅	1	2	yes	—
255	C₂×C₅	—	2	4	no	—
256	C₂×C₅	C₂³×C₃×C₄	1	1	yes	characteristic
257	C₂×C₅	C₃×H(5,𝔽₂)	1	1	yes	characteristic
258	C₂×C₅	C₃×2₋¹⁺⁴	1	1	yes	characteristic
259	D₅	—	2	2	no	—
260	C₂³	—	5	1	no	—
261	C₂³	C₂²×C₃×D₅	1	2	yes	—
262	Q₈	—	5	4	no	—
263	D₄	—	5	8	no	—
264	D₄	C₃×C₄×D₅	1	4	yes	complemented
265	C₂×C₄	—	4	2	no	—
266	C₂×C₄	—	5	8	no	—
267	C₂×C₄	—	5	2	no	—
268	C₂×C₄	—	5	1	no	—
269	C₂×C₄	—	5	4	no	—
270	C₂×C₄	—	5	1	no	—
271	C₂×C₄	—	5	2	no	—
272	C₂×C₄	—	5	1	no	—
273	C₂×C₄	—	20	2	no	—
274	C₂×C₄	—	20	2	no	—
275	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
276	C₂×C₄	C₂²×C₃×D₅	1	2	yes	—
277	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
278	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
279	C₂×C₃	—	2	4	no	—
280	C₂×C₃	—	10	2	no	—
281	C₂×C₃	C₂²×C₄×D₅	1	1	yes	characteristic
282	C₂×C₃	#4144	1	1	yes	characteristic
283	C₂×C₃	#4150	1	1	yes	characteristic
284	C₅	C₃×#520	1	1	yes	characteristic, complemented, Sylow
285	C₂²	—	2	2	no	—
286	C₂²	—	2	2	no	—
287	C₂²	—	5	2	no	—
288	C₂²	—	10	1	no	—
289	C₂²	—	10	1	no	—
290	C₂²	C₂×C₃×C₄×D₅	1	4	yes	—
291	C₂²	C₂³×C₃×D₅	1	1	yes	characteristic
292	C₄	—	2	1	no	—
293	C₄	—	2	2	no	—
294	C₄	—	2	1	no	—
295	C₄	—	5	2	no	—
296	C₄	—	5	4	no	—
297	C₄	—	10	2	no	—
298	C₄	—	10	1	no	—
299	C₄	—	10	1	no	—
300	C₄	C₂×C₃×C₄×D₅	1	2	yes	—
301	C₃	#65880	1	1	yes	characteristic, direct product factor, Sylow
302	C₂	—	2	4	no	—
303	C₂	—	10	2	no	—
304	C₂	C₂²×C₃×C₄×D₅	1	1	yes	characteristic
305	C₂	C₃×#4144	1	1	yes	characteristic
306	C₂	C₃×#4150	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂³×D₅
commutator	C₂×C₅	C₂³×C₃×C₄
Frattini	C₂²	C₂³×C₃×D₅
Fitting	C₃×C₄×C₅×D₄	C₂
socle	C₂²×C₃×C₅	C₂⁴
prime radical	C₂×C₃×#711	C₂
cosocle	C₂²×C₅	C₂⁴×C₃
hypercenter	C₃×C₄×D₄	D₅
hypocenter	C₅	C₃×#520
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄×D₄

Complements

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

N	H ≅ G/N	type
2. C₃×C₄×C₅×D₄	C₂	semidirect
3. C₃×D₄×Dic₅	C₂	semidirect
4. C₃×C₄×C₂²⋊D₅	C₂	semidirect
5. C₂×C₃×#683	C₂	semidirect
6. C₃×D₅×C₄⋊C₄	C₂	semidirect
7. C₃×#4033	C₂	semidirect
8. C₃×#4027	C₂	semidirect
9. C₃×#4023	C₂	semidirect
10. C₃×#4018	C₂	semidirect
11. C₃×C₄×Dic₁₀	C₂	semidirect
12. #65880	C₃	direct (N × H)
18. C₃×#711	C₄	semidirect
39. C₄×C₅×D₄	C₂×C₃	semidirect
40. D₄×Dic₅	C₂×C₃	semidirect
41. C₄×C₂²⋊D₅	C₂×C₃	semidirect
42. C₂×#683	C₂×C₃	semidirect
43. D₅×C₄⋊C₄	C₂×C₃	semidirect
44. #4033	C₂×C₃	semidirect
45. #4027	C₂×C₃	semidirect
46. #4023	C₂×C₃	semidirect
47. #4018	C₂×C₃	semidirect
48. C₄×Dic₁₀	C₂×C₃	semidirect
51. C₃×C₅×D₄	C₂×C₄	semidirect
57. C₃×C₂²⋊D₅	C₂×C₄	semidirect
59. C₂×C₃×Dic₅	C₂×C₄	semidirect
66. C₃×C₄×D₅	C₂×C₄	semidirect
67. C₃×Dic₁₀	C₂×C₄	semidirect
74. C₃×C₄×D₄	D₅	semidirect
84. #711	C₃×C₄	semidirect
146. C₅×D₄	C₂×C₃×C₄	semidirect
152. C₂²⋊D₅	C₂×C₃×C₄	semidirect
154. C₂×Dic₅	C₂×C₃×C₄	semidirect
161. C₄×D₅	C₂×C₃×C₄	semidirect
162. Dic₁₀	C₂×C₃×C₄	semidirect
169. C₄×D₄	C₃×D₅	semidirect
183. C₃×D₄	C₄×D₅	semidirect
238. C₃×C₅	#520	semidirect
264. D₄	C₃×C₄×D₅	semidirect
284. C₅	C₃×#520	semidirect
301. C₃	#65880	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×#711	2
C₃×C₄×C₅×D₄	2
C₃×D₄×Dic₅	2
C₃×C₄×C₂²⋊D₅	2
C₂×C₃×#683	2
C₃×D₅×C₄⋊C₄	2
C₃×#4033	2
C₃×#4027	2
C₃×#4023	2
C₃×#4018	2
C₃×C₄×Dic₁₀	2
#65880	3
C₃×#520	5

Subgroups — C3×#65880 #11755557

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#65880 #11755557