Subgroups — C₃×#65868 #11755545

Order 960.

432 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#117	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#4076	C₂	1	1	yes	maximal, characteristic
3	C₃×#4070	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#4044	C₂	1	2	yes	maximal, complemented
5	C₃×#4042	C₂	1	2	yes	maximal, complemented
6	C₃×#4031	C₂	1	2	yes	maximal, complemented
7	C₃×#4029	C₂	1	2	yes	maximal, complemented
8	C₃×#4021	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃×C₄×D₂₀	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃×#4015	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃×C₄×Dic₁₀	C₂	1	1	yes	maximal, characteristic, complemented
12	#65868	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
13	C₂²×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₂²⋊D₅	C₂²	1	2	yes	—
15	C₂×C₃×D₂₀	C₂²	1	1	yes	characteristic
16	C₂×C₃×C₄×D₅	C₂²	1	2	yes	—
17	C₂×C₃×Dic₁₀	C₂²	1	1	yes	characteristic
18	C₂×C₃×Dic₁₀	C₂²	1	2	yes	—
19	C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
20	C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
21	C₃×C₅×C₂²⋊C₄	C₂²	1	1	yes	characteristic
22	C₃×C₅×C₂²⋊C₄	C₂²	1	1	yes	characteristic
23	C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
24	C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
25	C₃×C₂²⋊Dic₅	C₂²	1	2	yes	—
26	C₃×#685	C₂²	1	2	yes	—
27	C₃×#685	C₂²	1	2	yes	—
28	C₃×#685	C₂²	1	2	yes	—
29	C₃×#685	C₂²	1	2	yes	—
30	C₃×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
31	C₃×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
32	C₃×C₄⋊Dic₅	C₂²	1	2	yes	—
33	C₃×#683	C₂²	1	2	yes	—
34	C₃×#683	C₂²	1	2	yes	—
35	C₃×C₄×Dic₅	C₂²	1	2	yes	—
36	C₃×#542	—	5	1	no	maximal, Hall
37	C₅×#117	C₂×C₃	1	1	yes	characteristic, complemented
38	#4076	C₂×C₃	1	1	yes	characteristic
39	#4070	C₂×C₃	1	1	yes	characteristic, complemented
40	#4044	C₂×C₃	1	2	yes	complemented
41	#4042	C₂×C₃	1	2	yes	complemented
42	#4031	C₂×C₃	1	2	yes	complemented
43	#4029	C₂×C₃	1	2	yes	complemented
44	#4021	C₂×C₃	1	1	yes	characteristic, complemented
45	C₄×D₂₀	C₂×C₃	1	1	yes	characteristic, complemented
46	#4015	C₂×C₃	1	1	yes	characteristic, complemented
47	C₄×Dic₁₀	C₂×C₃	1	1	yes	characteristic, complemented
48	C₂³×C₃×C₅	C₂³	1	1	yes	characteristic
49	C₂²×C₃×D₅	C₂³	1	2	yes	—
50	C₂×C₃×C₄×C₅	—	2	2	no	—
51	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
52	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
53	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
54	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
55	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
56	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
57	C₃×C₂²⋊D₅	—	4	2	no	—
58	C₂×C₃×Dic₅	C₂³	1	2	yes	—
59	C₂×C₃×Dic₅	C₂³	1	2	yes	—
60	C₂×C₃×Dic₅	C₂³	1	2	yes	—
61	C₃×D₂₀	—	2	2	no	—
62	C₃×C₄×D₅	—	4	2	no	—
63	C₃×Dic₁₀	—	2	2	no	—
64	C₃×Dic₁₀	—	4	2	no	—
65	C₃×C₄⋊Q₈	—	5	1	no	—
66	C₃×#126	—	5	2	no	—
67	C₃×#124	—	5	2	no	—
68	C₃×#124	—	5	1	no	—
69	C₃×#123	—	5	2	no	—
70	C₃×C₂²⋊Q₈	—	5	1	no	—
71	C₃×C₂²⋊Q₈	—	5	2	no	—
72	C₃×#121	—	5	1	no	—
73	C₃×C₄×Q₈	—	5	1	no	—
74	C₃×C₄×D₄	—	5	1	no	—
75	C₃×#117	D₅	1	1	yes	characteristic, complemented
76	C₂²×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
77	C₂×C₂²⋊D₅	C₂²×C₃	1	2	yes	—
78	C₂×D₂₀	C₂²×C₃	1	1	yes	characteristic
79	C₂×C₄×D₅	C₂²×C₃	1	2	yes	—
80	C₂×Dic₁₀	C₂²×C₃	1	1	yes	characteristic
81	C₂×Dic₁₀	C₂²×C₃	1	2	yes	—
82	C₅×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
83	C₅×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
84	C₅×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
85	C₅×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
86	C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
87	C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
88	C₂²⋊Dic₅	C₂²×C₃	1	2	yes	—
89	#685	C₂²×C₃	1	2	yes	—
90	#685	C₂²×C₃	1	2	yes	—
91	#685	C₂²×C₃	1	2	yes	—
92	#685	C₂²×C₃	1	2	yes	—
93	C₄⋊Dic₅	C₂²×C₃	1	1	yes	characteristic
94	C₄⋊Dic₅	C₂²×C₃	1	1	yes	characteristic
95	C₄⋊Dic₅	C₂²×C₃	1	2	yes	—
96	#683	C₂²×C₃	1	2	yes	—
97	#683	C₂²×C₃	1	2	yes	—
98	C₄×Dic₅	C₂²×C₃	1	2	yes	—
99	#542	—	5	1	no	Sylow
100	C₂²×C₃×C₅	—	2	1	no	—
101	C₂²×C₃×C₅	—	2	1	no	—
102	C₂²×C₃×C₅	—	2	1	no	—
103	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
104	C₂×C₃×D₅	—	2	2	no	—
105	C₂×C₃×D₅	—	2	2	no	—
106	C₂×C₃×D₅	—	2	2	no	—
107	C₃×C₄×C₅	—	2	1	no	—
108	C₃×C₄×C₅	—	2	1	no	—
109	C₃×C₄×C₅	—	2	1	no	—
110	C₃×C₄×C₅	—	2	1	no	—
111	C₃×C₄×C₅	—	2	1	no	—
112	C₃×C₄×C₅	#41	1	2	yes	—
113	C₃×Dic₅	—	2	2	no	—
114	C₃×Dic₅	—	2	2	no	—
115	C₃×Dic₅	—	2	2	no	—
116	C₂×C₃×Q₈	—	5	1	no	—
117	C₂×C₃×Q₈	—	5	2	no	—
118	C₂×C₃×D₄	—	5	2	no	—
119	C₂×C₃×D₄	—	5	1	no	—
120	C₂²×C₃×C₄	—	5	2	no	—
121	C₂²×C₃×C₄	C₂×D₅	1	1	yes	characteristic
122	C₃×C₄⋊C₄	—	5	2	no	—
123	C₃×C₄⋊C₄	—	5	2	no	—
124	C₃×C₄⋊C₄	—	5	1	no	—
125	C₃×C₄⋊C₄	—	5	1	no	—
126	C₃×C₄⋊C₄	—	5	2	no	—
127	C₃×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
128	C₃×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
129	C₃×C₂²⋊C₄	—	5	2	no	—
130	C₃×C₂²⋊C₄	—	5	2	no	—
131	C₃×C₂²⋊C₄	—	5	2	no	—
132	C₃×C₂²⋊C₄	—	5	2	no	—
133	C₃×C₂²⋊C₄	—	5	2	no	—
134	C₃×C₂²⋊C₄	C₂×D₅	1	1	yes	characteristic
135	C₃×C₂²⋊C₄	C₂×D₅	1	1	yes	characteristic
136	C₃×C₄²	—	5	2	no	—
137	C₃×C₄²	C₂×D₅	1	1	yes	characteristic
138	C₃×C₄²	C₂×D₅	1	1	yes	characteristic
139	C₂³×C₅	C₂³×C₃	1	1	yes	characteristic
140	C₂²×D₅	C₂³×C₃	1	2	yes	—
141	C₂×C₄×C₅	—	2	2	no	—
142	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
143	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
144	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
145	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
146	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
147	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
148	C₂²⋊D₅	—	4	2	no	—
149	C₂×Dic₅	C₂³×C₃	1	2	yes	—
150	C₂×Dic₅	C₂³×C₃	1	2	yes	—
151	C₂×Dic₅	C₂³×C₃	1	2	yes	—
152	D₂₀	—	2	2	no	—
153	C₄×D₅	—	4	2	no	—
154	Dic₁₀	—	2	2	no	—
155	Dic₁₀	—	4	2	no	—
156	C₄⋊Q₈	—	5	1	no	—
157	#126	—	5	2	no	—
158	#124	—	5	2	no	—
159	#124	—	5	1	no	—
160	#123	—	5	2	no	—
161	C₂²⋊Q₈	—	5	1	no	—
162	C₂²⋊Q₈	—	5	2	no	—
163	#121	—	5	1	no	—
164	C₄×Q₈	—	5	1	no	—
165	C₄×D₄	—	5	1	no	—
166	#117	C₃×D₅	1	1	yes	characteristic, complemented
167	C₂×C₃×C₅	—	4	1	no	—
168	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
169	C₂×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
170	C₂×C₃×C₅	2₋¹⁺⁴	1	1	yes	characteristic
171	C₃×D₅	—	4	2	no	—
172	C₂³×C₃	—	5	2	no	—
173	C₂³×C₃	C₂²×D₅	1	1	yes	characteristic
174	C₃×Q₈	—	10	2	no	—
175	C₃×Q₈	—	20	2	no	—
176	C₃×D₄	—	10	2	no	—
177	C₃×D₄	—	20	2	no	—
178	C₂×C₃×C₄	—	2	2	no	—
179	C₂×C₃×C₄	—	5	2	no	—
180	C₂×C₃×C₄	—	5	2	no	—
181	C₂×C₃×C₄	—	5	2	no	—
182	C₂×C₃×C₄	—	20	2	no	—
183	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
184	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
185	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
186	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
187	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
188	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
189	C₂²×C₅	—	2	1	no	—
190	C₂²×C₅	—	2	1	no	—
191	C₂²×C₅	—	2	1	no	—
192	C₂²×C₅	C₂⁴×C₃	1	1	yes	characteristic
193	C₂×D₅	—	2	2	no	—
194	C₂×D₅	—	2	2	no	—
195	C₂×D₅	—	2	2	no	—
196	C₄×C₅	—	2	1	no	—
197	C₄×C₅	—	2	1	no	—
198	C₄×C₅	—	2	1	no	—
199	C₄×C₅	—	2	1	no	—
200	C₄×C₅	—	2	1	no	—
201	C₄×C₅	C₃×#41	1	2	yes	—
202	Dic₅	—	2	2	no	—
203	Dic₅	—	2	2	no	—
204	Dic₅	—	2	2	no	—
205	C₂×Q₈	—	5	1	no	—
206	C₂×Q₈	—	5	2	no	—
207	C₂×D₄	—	5	2	no	—
208	C₂×D₄	—	5	1	no	—
209	C₂²×C₄	—	5	2	no	—
210	C₂²×C₄	C₂×C₃×D₅	1	1	yes	characteristic
211	C₄⋊C₄	—	5	1	no	—
212	C₄⋊C₄	—	5	1	no	—
213	C₄⋊C₄	—	5	2	no	—
214	C₄⋊C₄	—	5	2	no	—
215	C₄⋊C₄	—	5	2	no	—
216	C₄⋊C₄	C₂×C₃×D₅	1	1	yes	characteristic
217	C₄⋊C₄	C₂×C₃×D₅	1	1	yes	characteristic
218	C₂²⋊C₄	—	5	2	no	—
219	C₂²⋊C₄	—	5	2	no	—
220	C₂²⋊C₄	—	5	2	no	—
221	C₂²⋊C₄	—	5	2	no	—
222	C₂²⋊C₄	—	5	2	no	—
223	C₂²⋊C₄	C₂×C₃×D₅	1	1	yes	characteristic
224	C₂²⋊C₄	C₂×C₃×D₅	1	1	yes	characteristic
225	C₄²	—	5	2	no	—
226	C₄²	C₂×C₃×D₅	1	1	yes	characteristic
227	C₄²	C₂×C₃×D₅	1	1	yes	characteristic
228	C₃×C₅	#542	1	1	yes	characteristic, complemented, Hall
229	C₂²×C₃	—	2	1	no	—
230	C₂²×C₃	—	2	1	no	—
231	C₂²×C₃	—	2	1	no	—
232	C₂²×C₃	—	10	2	no	—
233	C₂²×C₃	—	10	2	no	—
234	C₂²×C₃	—	10	2	no	—
235	C₂²×C₃	C₂³×D₅	1	1	yes	characteristic
236	C₃×C₄	—	2	1	no	—
237	C₃×C₄	—	2	1	no	—
238	C₃×C₄	—	2	1	no	—
239	C₃×C₄	—	2	1	no	—
240	C₃×C₄	—	2	1	no	—
241	C₃×C₄	—	10	2	no	—
242	C₃×C₄	—	10	2	no	—
243	C₃×C₄	—	10	2	no	—
244	C₃×C₄	#709	1	2	yes	—
245	C₂×C₅	—	4	1	no	—
246	C₂×C₅	C₂×C₃×#41	1	1	yes	characteristic
247	C₂×C₅	C₃×H(5,𝔽₂)	1	1	yes	characteristic
248	C₂×C₅	C₃×2₋¹⁺⁴	1	1	yes	characteristic
249	D₅	—	4	2	no	—
250	C₂³	—	5	2	no	—
251	C₂³	C₂²×C₃×D₅	1	1	yes	characteristic
252	Q₈	—	10	2	no	—
253	Q₈	—	20	2	no	—
254	D₄	—	10	2	no	—
255	D₄	—	20	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	5	2	no	—
258	C₂×C₄	—	5	2	no	—
259	C₂×C₄	—	5	2	no	—
260	C₂×C₄	—	20	2	no	—
261	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
262	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
263	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
264	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
265	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
266	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
267	C₂×C₃	—	4	1	no	—
268	C₂×C₃	—	20	2	no	—
269	C₂×C₃	C₂×#709	1	1	yes	characteristic
270	C₂×C₃	#4149	1	1	yes	characteristic
271	C₂×C₃	#4150	1	1	yes	characteristic
272	C₅	C₃×#542	1	1	yes	characteristic, complemented, Sylow
273	C₂²	—	2	1	no	—
274	C₂²	—	2	1	no	—
275	C₂²	—	2	1	no	—
276	C₂²	—	10	2	no	—
277	C₂²	—	10	2	no	—
278	C₂²	—	10	2	no	—
279	C₂²	C₂³×C₃×D₅	1	1	yes	characteristic
280	C₄	—	2	1	no	—
281	C₄	—	2	1	no	—
282	C₄	—	2	1	no	—
283	C₄	—	2	1	no	—
284	C₄	—	2	1	no	—
285	C₄	—	10	2	no	—
286	C₄	—	10	2	no	—
287	C₄	—	10	2	no	—
288	C₄	C₃×#709	1	2	yes	—
289	C₃	#65868	1	1	yes	characteristic, direct product factor, Sylow
290	C₂	—	4	1	no	—
291	C₂	—	20	2	no	—
292	C₂	C₂×C₃×#709	1	1	yes	characteristic
293	C₂	C₃×#4149	1	1	yes	characteristic
294	C₂	C₃×#4150	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂³×D₅
commutator, cosocle	C₂²×C₅	C₂⁴×C₃
Frattini	C₂²	C₂³×C₃×D₅
Fitting	C₃×C₅×#117	C₂
socle	C₂²×C₃×C₅	C₂⁴
prime radical	C₃×#4076	C₂
hypercenter	C₃×#117	D₅
hypocenter	C₅	C₃×#542
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×#117

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×#117	C₂	semidirect
3. C₃×#4070	C₂	semidirect
4. C₃×#4044	C₂	semidirect
5. C₃×#4042	C₂	semidirect
6. C₃×#4031	C₂	semidirect
7. C₃×#4029	C₂	semidirect
8. C₃×#4021	C₂	semidirect
9. C₃×C₄×D₂₀	C₂	semidirect
10. C₃×#4015	C₂	semidirect
11. C₃×C₄×Dic₁₀	C₂	semidirect
12. #65868	C₃	direct (N × H)
37. C₅×#117	C₂×C₃	semidirect
39. #4070	C₂×C₃	semidirect
40. #4044	C₂×C₃	semidirect
41. #4042	C₂×C₃	semidirect
42. #4031	C₂×C₃	semidirect
43. #4029	C₂×C₃	semidirect
44. #4021	C₂×C₃	semidirect
45. C₄×D₂₀	C₂×C₃	semidirect
46. #4015	C₂×C₃	semidirect
47. C₄×Dic₁₀	C₂×C₃	semidirect
75. C₃×#117	D₅	semidirect
166. #117	C₃×D₅	semidirect
228. C₃×C₅	#542	semidirect
272. C₅	C₃×#542	semidirect
289. C₃	#65868	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×C₅×#117	2
C₃×#4076	2
C₃×#4070	2
C₃×#4044	2
C₃×#4042	2
C₃×#4031	2
C₃×#4029	2
C₃×#4021	2
C₃×C₄×D₂₀	2
C₃×#4015	2
C₃×C₄×Dic₁₀	2
#65868	3
C₃×#542	5

Subgroups — C3×#65868 #11755545

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#65868 #11755545