Subgroups — C₃×#65931 #11755608

Order 960.

588 conjugacy classes of subgroups.

The subgroup lattice has 304 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	2	yes	maximal, complemented
2	C₃×C₅×#121	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#4085	C₂	1	2	yes	maximal, complemented
4	C₃×#4085	C₂	1	2	yes	maximal, complemented
5	C₃×#4076	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₃×#683	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#4042	C₂	1	2	yes	maximal, complemented
8	C₃×#4030	C₂	1	2	yes	maximal, complemented
9	C₃×#4024	C₂	1	2	yes	maximal, complemented
10	#65931	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
11	C₂×C₃×C₅×D₄	C₂²	1	2	yes	—
12	C₂×C₃×C₅×D₄	C₂²	1	1	yes	characteristic
13	C₂²×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₂²⋊D₅	C₂²	1	2	yes	—
15	C₂×C₃×C₂²⋊D₅	C₂²	1	2	yes	—
16	C₂×C₃×C₂²⋊D₅	C₂²	1	2	yes	—
17	C₂²×C₃×Dic₅	C₂²	1	2	yes	—
18	C₂²×C₃×Dic₅	C₂²	1	1	yes	characteristic
19	C₂²×C₃×Dic₅	C₂²	1	1	yes	characteristic
20	C₃×#711	—	2	8	no	—
21	C₂×C₃×D₂₀	C₂²	1	1	yes	characteristic
22	C₂×C₃×C₄×D₅	C₂²	1	2	yes	—
23	C₂×C₃×Dic₁₀	C₂²	1	2	yes	—
24	C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
25	C₃×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
26	C₃×C₂²⋊Dic₅	C₂²	1	2	yes	—
27	C₃×#685	C₂²	1	2	yes	—
28	C₃×#685	C₂²	1	2	yes	—
29	C₃×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
30	C₃×#683	C₂²	1	2	yes	—
31	C₃×#683	C₂²	1	2	yes	—
32	C₃×#683	C₂²	1	2	yes	—
33	C₃×#537	—	5	1	no	maximal, Hall
34	C₂×#711	C₂×C₃	1	2	yes	complemented
35	C₅×#121	C₂×C₃	1	1	yes	characteristic, complemented
36	#4085	C₂×C₃	1	2	yes	complemented
37	#4085	C₂×C₃	1	2	yes	complemented
38	#4076	C₂×C₃	1	1	yes	characteristic, complemented
39	C₂×#683	C₂×C₃	1	1	yes	characteristic, complemented
40	#4042	C₂×C₃	1	2	yes	complemented
41	#4030	C₂×C₃	1	2	yes	complemented
42	#4024	C₂×C₃	1	2	yes	complemented
43	C₂³×C₃×C₅	C₂³	1	2	yes	—
44	C₂³×C₃×C₅	C₂³	1	1	yes	characteristic
45	C₂²×C₃×D₅	C₂³	1	2	yes	—
46	C₃×C₅×D₄	—	2	4	no	—
47	C₃×C₅×D₄	—	4	1	no	—
48	C₂×C₃×C₄×C₅	—	4	1	no	—
49	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
50	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
51	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
52	C₃×C₂²⋊D₅	—	2	4	no	—
53	C₃×C₂²⋊D₅	—	2	4	no	—
54	C₃×C₂²⋊D₅	—	4	2	no	—
55	C₂×C₃×Dic₅	—	2	4	no	—
56	C₂×C₃×Dic₅	—	4	1	no	—
57	C₂×C₃×Dic₅	D₄	1	4	yes	complemented
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	—	4	1	no	—
62	C₃×C₄×D₅	—	2	4	no	—
63	C₃×Dic₁₀	—	2	4	no	—
64	C₂×C₃×#41	—	5	2	no	—
65	C₃×C₂²⋊Q₈	—	5	2	no	—
66	C₃×C₂²⋊Q₈	—	5	2	no	—
67	C₃×#121	—	5	2	no	—
68	C₃×#121	—	5	2	no	—
69	C₃×#121	—	5	1	no	—
70	C₃×#121	—	5	2	no	—
71	C₃×#121	D₅	1	1	yes	characteristic, complemented
72	C₂×C₃×C₄⋊C₄	—	5	1	no	—
73	C₂×C₅×D₄	C₂²×C₃	1	2	yes	—
74	C₂×C₅×D₄	C₂²×C₃	1	1	yes	characteristic
75	C₂²×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
76	C₂×C₂²⋊D₅	C₂²×C₃	1	2	yes	—
77	C₂×C₂²⋊D₅	C₂²×C₃	1	2	yes	—
78	C₂×C₂²⋊D₅	C₂²×C₃	1	2	yes	—
79	C₂²×Dic₅	C₂²×C₃	1	2	yes	—
80	C₂²×Dic₅	C₂²×C₃	1	1	yes	characteristic
81	C₂²×Dic₅	C₂²×C₃	1	1	yes	characteristic
82	#711	—	2	8	no	—
83	C₂×D₂₀	C₂²×C₃	1	1	yes	characteristic
84	C₂×C₄×D₅	C₂²×C₃	1	2	yes	—
85	C₂×Dic₁₀	C₂²×C₃	1	2	yes	—
86	C₅×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
87	C₅×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
88	C₂²⋊Dic₅	C₂²×C₃	1	2	yes	—
89	#685	C₂²×C₃	1	2	yes	—
90	#685	C₂²×C₃	1	2	yes	—
91	C₄⋊Dic₅	C₂²×C₃	1	1	yes	characteristic
92	#683	C₂²×C₃	1	2	yes	—
93	#683	C₂²×C₃	1	2	yes	—
94	#683	C₂²×C₃	1	2	yes	—
95	#537	—	5	1	no	Sylow
96	C₂²×C₃×C₅	—	2	1	no	—
97	C₂²×C₃×C₅	—	2	1	no	—
98	C₂²×C₃×C₅	—	2	2	no	—
99	C₂²×C₃×C₅	—	2	2	no	—
100	C₂²×C₃×C₅	—	2	2	no	—
101	C₂²×C₃×C₅	C₂×D₄	1	2	yes	—
102	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
103	C₂×C₃×D₅	—	2	2	no	—
104	C₂×C₃×D₅	—	2	2	no	—
105	C₂×C₃×D₅	—	2	2	no	—
106	C₃×C₄×C₅	—	2	2	no	—
107	C₃×C₄×C₅	—	2	1	no	—
108	C₃×C₄×C₅	—	2	1	no	—
109	C₃×Dic₅	—	2	2	no	—
110	C₃×Dic₅	—	2	2	no	—
111	C₃×Dic₅	C₂×D₄	1	4	yes	—
112	C₃×#41	—	10	8	no	—
113	C₂×C₃×Q₈	—	5	2	no	—
114	C₂×C₃×D₄	—	5	2	no	—
115	C₂×C₃×D₄	—	5	2	no	—
116	C₂×C₃×D₄	—	5	2	no	—
117	C₂×C₃×D₄	—	5	1	no	—
118	C₂×C₃×D₄	C₂×D₅	1	2	yes	—
119	C₂×C₃×D₄	C₂×D₅	1	1	yes	characteristic
120	C₂²×C₃×C₄	—	5	1	no	—
121	C₂²×C₃×C₄	—	5	2	no	—
122	C₂²×C₃×C₄	—	5	1	no	—
123	C₂²×C₃×C₄	—	5	2	no	—
124	C₂²×C₃×C₄	C₂×D₅	1	1	yes	characteristic
125	C₃×C₄⋊C₄	—	5	2	no	—
126	C₃×C₄⋊C₄	—	5	2	no	—
127	C₃×C₄⋊C₄	—	5	2	no	—
128	C₃×C₄⋊C₄	—	5	1	no	—
129	C₃×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
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	C₂×D₅	1	2	yes	—
134	C₂³×C₅	C₂³×C₃	1	2	yes	—
135	C₂³×C₅	C₂³×C₃	1	1	yes	characteristic
136	C₂²×D₅	C₂³×C₃	1	2	yes	—
137	C₅×D₄	—	2	4	no	—
138	C₅×D₄	—	4	1	no	—
139	C₂×C₄×C₅	—	4	1	no	—
140	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
141	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
142	C₂×C₄×C₅	C₂³×C₃	1	2	yes	—
143	C₂²⋊D₅	—	2	4	no	—
144	C₂²⋊D₅	—	2	4	no	—
145	C₂²⋊D₅	—	4	2	no	—
146	C₂×Dic₅	—	2	4	no	—
147	C₂×Dic₅	—	4	1	no	—
148	C₂×Dic₅	C₃×D₄	1	4	yes	complemented
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	—	4	1	no	—
153	C₄×D₅	—	2	4	no	—
154	Dic₁₀	—	2	4	no	—
155	C₂×#41	—	5	2	no	—
156	C₂²⋊Q₈	—	5	2	no	—
157	C₂²⋊Q₈	—	5	2	no	—
158	#121	—	5	2	no	—
159	#121	—	5	2	no	—
160	#121	—	5	1	no	—
161	#121	—	5	2	no	—
162	#121	C₃×D₅	1	1	yes	characteristic, complemented
163	C₂×C₄⋊C₄	—	5	1	no	—
164	C₂×C₃×C₅	—	2	2	no	—
165	C₂×C₃×C₅	—	4	2	no	—
166	C₂×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
167	C₂×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
168	C₂×C₃×C₅	2₋¹⁺⁴	1	1	yes	characteristic
169	C₃×D₅	—	4	2	no	—
170	C₂³×C₃	—	5	2	no	—
171	C₂³×C₃	C₂²×D₅	1	2	yes	—
172	C₂³×C₃	C₂²×D₅	1	1	yes	characteristic
173	C₃×Q₈	—	10	4	no	—
174	C₃×D₄	—	2	4	no	—
175	C₃×D₄	—	4	1	no	—
176	C₃×D₄	—	10	4	no	—
177	C₃×D₄	—	10	4	no	—
178	C₃×D₄	—	20	1	no	—
179	C₃×D₄	—	20	2	no	—
180	C₂×C₃×C₄	—	4	1	no	—
181	C₂×C₃×C₄	—	5	4	no	—
182	C₂×C₃×C₄	—	5	2	no	—
183	C₂×C₃×C₄	—	5	2	no	—
184	C₂×C₃×C₄	—	5	2	no	—
185	C₂×C₃×C₄	—	10	4	no	—
186	C₂×C₃×C₄	—	10	4	no	—
187	C₂×C₃×C₄	—	20	1	no	—
188	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
189	C₂×C₃×C₄	C₂²×D₅	1	2	yes	—
190	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
191	C₂²×C₅	—	2	1	no	—
192	C₂²×C₅	—	2	1	no	—
193	C₂²×C₅	—	2	2	no	—
194	C₂²×C₅	—	2	2	no	—
195	C₂²×C₅	—	2	2	no	—
196	C₂²×C₅	C₂×C₃×D₄	1	2	yes	—
197	C₂²×C₅	C₂⁴×C₃	1	1	yes	characteristic
198	C₂×D₅	—	2	2	no	—
199	C₂×D₅	—	2	2	no	—
200	C₂×D₅	—	2	2	no	—
201	C₄×C₅	—	2	2	no	—
202	C₄×C₅	—	2	1	no	—
203	C₄×C₅	—	2	1	no	—
204	Dic₅	—	2	2	no	—
205	Dic₅	—	2	2	no	—
206	Dic₅	C₂×C₃×D₄	1	4	yes	—
207	#41	—	10	8	no	—
208	C₂×Q₈	—	5	2	no	—
209	C₂×D₄	—	5	2	no	—
210	C₂×D₄	—	5	2	no	—
211	C₂×D₄	—	5	2	no	—
212	C₂×D₄	—	5	1	no	—
213	C₂×D₄	C₂×C₃×D₅	1	2	yes	—
214	C₂×D₄	C₂×C₃×D₅	1	1	yes	characteristic
215	C₂²×C₄	—	5	1	no	—
216	C₂²×C₄	—	5	2	no	—
217	C₂²×C₄	—	5	1	no	—
218	C₂²×C₄	—	5	2	no	—
219	C₂²×C₄	C₂×C₃×D₅	1	1	yes	characteristic
220	C₄⋊C₄	—	5	2	no	—
221	C₄⋊C₄	—	5	2	no	—
222	C₄⋊C₄	—	5	2	no	—
223	C₄⋊C₄	—	5	1	no	—
224	C₄⋊C₄	C₂×C₃×D₅	1	1	yes	characteristic
225	C₂²⋊C₄	—	5	2	no	—
226	C₂²⋊C₄	—	5	2	no	—
227	C₂²⋊C₄	—	5	2	no	—
228	C₂²⋊C₄	C₂×C₃×D₅	1	2	yes	—
229	C₃×C₅	#537	1	1	yes	characteristic, complemented, Hall
230	C₂²×C₃	—	2	1	no	—
231	C₂²×C₃	—	2	1	no	—
232	C₂²×C₃	—	2	2	no	—
233	C₂²×C₃	—	2	2	no	—
234	C₂²×C₃	—	2	2	no	—
235	C₂²×C₃	—	10	2	no	—
236	C₂²×C₃	—	10	2	no	—
237	C₂²×C₃	—	10	2	no	—
238	C₂²×C₃	D₄×D₅	1	2	yes	—
239	C₂²×C₃	C₂³×D₅	1	1	yes	characteristic
240	C₃×C₄	—	2	2	no	—
241	C₃×C₄	—	2	1	no	—
242	C₃×C₄	—	2	1	no	—
243	C₃×C₄	—	5	4	no	—
244	C₃×C₄	—	10	2	no	—
245	C₃×C₄	—	10	2	no	—
246	C₂×C₅	—	2	2	no	—
247	C₂×C₅	—	4	2	no	—
248	C₂×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
249	C₂×C₅	C₃×H(5,𝔽₂)	1	1	yes	characteristic
250	C₂×C₅	C₃×2₋¹⁺⁴	1	1	yes	characteristic
251	D₅	—	4	2	no	—
252	C₂³	—	5	2	no	—
253	C₂³	C₂²×C₃×D₅	1	2	yes	—
254	C₂³	C₂²×C₃×D₅	1	1	yes	characteristic
255	Q₈	—	10	4	no	—
256	D₄	—	2	4	no	—
257	D₄	—	4	1	no	—
258	D₄	—	10	4	no	—
259	D₄	—	10	4	no	—
260	D₄	—	20	1	no	—
261	D₄	—	20	2	no	—
262	C₂×C₄	—	4	1	no	—
263	C₂×C₄	—	5	4	no	—
264	C₂×C₄	—	5	2	no	—
265	C₂×C₄	—	5	2	no	—
266	C₂×C₄	—	5	2	no	—
267	C₂×C₄	—	10	4	no	—
268	C₂×C₄	—	10	4	no	—
269	C₂×C₄	—	20	1	no	—
270	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
271	C₂×C₄	C₂²×C₃×D₅	1	2	yes	—
272	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
273	C₂×C₃	—	2	2	no	—
274	C₂×C₃	—	4	2	no	—
275	C₂×C₃	—	20	2	no	—
276	C₂×C₃	C₂×D₄×D₅	1	1	yes	characteristic
277	C₂×C₃	#4144	1	1	yes	characteristic
278	C₂×C₃	#4150	1	1	yes	characteristic
279	C₅	C₃×#537	1	1	yes	characteristic, complemented, Sylow
280	C₂²	—	2	1	no	—
281	C₂²	—	2	1	no	—
282	C₂²	—	2	2	no	—
283	C₂²	—	2	2	no	—
284	C₂²	—	2	2	no	—
285	C₂²	—	10	2	no	—
286	C₂²	—	10	2	no	—
287	C₂²	—	10	2	no	—
288	C₂²	C₃×D₄×D₅	1	2	yes	—
289	C₂²	C₂³×C₃×D₅	1	1	yes	characteristic
290	C₄	—	2	2	no	—
291	C₄	—	2	1	no	—
292	C₄	—	2	1	no	—
293	C₄	—	5	4	no	—
294	C₄	—	10	2	no	—
295	C₄	—	10	2	no	—
296	C₃	#65931	1	1	yes	characteristic, direct product factor, Sylow
297	C₂	—	2	2	no	—
298	C₂	—	4	2	no	—
299	C₂	—	20	2	no	—
300	C₂	C₂×C₃×D₄×D₅	1	1	yes	characteristic
301	C₂	C₃×#4144	1	1	yes	characteristic
302	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×#121	C₂
socle	C₂²×C₃×C₅	C₂⁴
hypercenter	C₃×#121	D₅
hypocenter	C₅	C₃×#537
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×#121

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×#711	C₂	semidirect
2. C₃×C₅×#121	C₂	semidirect
3. C₃×#4085	C₂	semidirect
4. C₃×#4085	C₂	semidirect
5. C₃×#4076	C₂	semidirect
6. C₂×C₃×#683	C₂	semidirect
7. C₃×#4042	C₂	semidirect
8. C₃×#4030	C₂	semidirect
9. C₃×#4024	C₂	semidirect
10. #65931	C₃	direct (N × H)
34. C₂×#711	C₂×C₃	semidirect
35. C₅×#121	C₂×C₃	semidirect
36. #4085	C₂×C₃	semidirect
37. #4085	C₂×C₃	semidirect
38. #4076	C₂×C₃	semidirect
39. C₂×#683	C₂×C₃	semidirect
40. #4042	C₂×C₃	semidirect
41. #4030	C₂×C₃	semidirect
42. #4024	C₂×C₃	semidirect
57. C₂×C₃×Dic₅	D₄	semidirect
71. C₃×#121	D₅	semidirect
148. C₂×Dic₅	C₃×D₄	semidirect
162. #121	C₃×D₅	semidirect
229. C₃×C₅	#537	semidirect
279. C₅	C₃×#537	semidirect
296. C₃	#65931	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×C₅×#121	2
C₃×#4085	2
C₃×#4076	2
C₂×C₃×#683	2
C₃×#4042	2
C₃×#4030	2
C₃×#4024	2
#65931	3
C₃×#537	5

Subgroups — C3×#65931 #11755608

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#65931 #11755608