Subgroups — C₃×#65247 #11749855

Order 960.

572 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₂²⋊D₅	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₃×Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×C₅×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃×#685	C₂	1	2	yes	maximal, complemented
5	C₃×#3963	C₂	1	2	yes	maximal, complemented
6	#65247	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂⁴×C₃×C₅	C₂²	1	1	yes	characteristic
8	C₂³×C₃×D₅	C₂²	1	1	yes	characteristic
9	C₂²×C₃×C₄×C₅	C₂²	1	2	yes	—
10	C₂×C₃×C₂²⋊D₅	—	2	2	no	—
11	C₂×C₃×C₂²⋊D₅	—	2	2	no	—
12	C₂×C₃×C₂²⋊D₅	C₄	1	4	yes	complemented
13	C₂²×C₃×Dic₅	—	2	2	no	—
14	C₂²×C₃×Dic₅	—	2	4	no	—
15	C₂²×C₃×Dic₅	C₂²	1	1	yes	characteristic
16	C₂²×C₃×Dic₅	C₂²	1	2	yes	—
17	C₃×C₅×C₂²⋊C₄	—	2	2	no	—
18	C₃×#685	—	2	4	no	—
19	C₃×#386	—	5	1	no	maximal, Hall
20	C₂²×C₂²⋊D₅	C₂×C₃	1	1	yes	characteristic
21	C₂³×Dic₅	C₂×C₃	1	1	yes	characteristic, complemented
22	C₂×C₅×C₂²⋊C₄	C₂×C₃	1	1	yes	characteristic, complemented
23	C₂×#685	C₂×C₃	1	2	yes	complemented
24	#3963	C₂×C₃	1	2	yes	complemented
25	C₂³×C₃×C₅	—	2	2	no	—
26	C₂³×C₃×C₅	—	2	2	no	—
27	C₂³×C₃×C₅	C₂×C₄	1	2	yes	—
28	C₂³×C₃×C₅	D₄	1	2	yes	—
29	C₂³×C₃×C₅	D₄	1	2	yes	—
30	C₂³×C₃×C₅	C₂³	1	1	yes	characteristic
31	C₂²×C₃×D₅	—	2	2	no	—
32	C₂²×C₃×D₅	—	2	1	no	—
33	C₂²×C₃×D₅	—	2	1	no	—
34	C₂²×C₃×D₅	—	2	2	no	—
35	C₂²×C₃×D₅	C₂×C₄	1	2	yes	complemented
36	C₂×C₃×C₄×C₅	—	2	2	no	—
37	C₂×C₃×C₄×C₅	—	2	2	no	—
38	C₂×C₃×C₄×C₅	—	2	2	no	—
39	C₃×C₂²⋊D₅	—	2	8	no	—
40	C₂×C₃×Dic₅	—	2	8	no	—
41	C₂×C₃×Dic₅	—	2	2	no	—
42	C₂×C₃×Dic₅	—	2	2	no	—
43	C₂×C₃×Dic₅	—	2	1	no	—
44	C₂×C₃×Dic₅	—	2	1	no	—
45	C₂×C₃×Dic₅	C₂×C₄	1	2	yes	complemented
46	C₂×C₃×Dic₅	D₄	1	4	yes	complemented
47	C₂²×C₃×D₄	—	5	1	no	—
48	C₂³×C₃×C₄	—	5	1	no	—
49	C₂×C₃×C₂²⋊C₄	—	5	2	no	—
50	C₂×C₃×C₂²⋊C₄	D₅	1	1	yes	characteristic, complemented
51	C₃×#95	—	5	2	no	—
52	C₂⁴×C₅	C₂²×C₃	1	1	yes	characteristic
53	C₂³×D₅	C₂²×C₃	1	1	yes	characteristic
54	C₂²×C₄×C₅	C₂²×C₃	1	2	yes	—
55	C₂×C₂²⋊D₅	—	2	2	no	—
56	C₂×C₂²⋊D₅	—	2	2	no	—
57	C₂×C₂²⋊D₅	C₃×C₄	1	4	yes	complemented
58	C₂²×Dic₅	—	2	2	no	—
59	C₂²×Dic₅	—	2	4	no	—
60	C₂²×Dic₅	C₂²×C₃	1	1	yes	characteristic
61	C₂²×Dic₅	C₂²×C₃	1	2	yes	—
62	C₅×C₂²⋊C₄	—	2	2	no	—
63	#685	—	2	4	no	—
64	#386	—	5	1	no	Sylow
65	C₂²×C₃×C₅	—	2	4	no	—
66	C₂²×C₃×C₅	—	2	2	no	—
67	C₂²×C₃×C₅	—	2	4	no	—
68	C₂²×C₃×C₅	—	2	2	no	—
69	C₂²×C₃×C₅	C₂²⋊C₄	1	4	yes	complemented
70	C₂²×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₃×C₅	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₃×C₅	C₂×D₄	1	2	yes	—
73	C₂²×C₃×C₅	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₃×C₅	#41	1	2	yes	—
75	C₂×C₃×D₅	—	2	2	no	—
76	C₂×C₃×D₅	—	2	4	no	—
77	C₂×C₃×D₅	—	4	1	no	—
78	C₂×C₃×D₅	—	4	2	no	—
79	C₂×C₃×D₅	—	4	1	no	—
80	C₃×C₄×C₅	—	4	2	no	—
81	C₃×Dic₅	—	2	2	no	—
82	C₃×Dic₅	—	2	4	no	—
83	C₂⁴×C₃	—	5	1	no	—
84	C₂⁴×C₃	C₂×D₅	1	1	yes	characteristic
85	C₂×C₃×D₄	—	5	4	no	—
86	C₂×C₃×D₄	—	10	2	no	—
87	C₂×C₃×D₄	—	10	2	no	—
88	C₂²×C₃×C₄	—	5	1	no	—
89	C₂²×C₃×C₄	—	5	2	no	—
90	C₂²×C₃×C₄	—	10	4	no	—
91	C₂²×C₃×C₄	—	10	2	no	—
92	C₂²×C₃×C₄	C₂×D₅	1	2	yes	—
93	C₃×C₂²⋊C₄	—	2	2	no	—
94	C₃×C₂²⋊C₄	—	10	4	no	—
95	C₂³×C₅	—	2	2	no	—
96	C₂³×C₅	—	2	2	no	—
97	C₂³×C₅	C₂×C₃×C₄	1	2	yes	—
98	C₂³×C₅	C₃×D₄	1	2	yes	—
99	C₂³×C₅	C₃×D₄	1	2	yes	—
100	C₂³×C₅	C₂³×C₃	1	1	yes	characteristic
101	C₂²×D₅	—	2	2	no	—
102	C₂²×D₅	—	2	1	no	—
103	C₂²×D₅	—	2	1	no	—
104	C₂²×D₅	—	2	2	no	—
105	C₂²×D₅	C₂×C₃×C₄	1	2	yes	complemented
106	C₂×C₄×C₅	—	2	2	no	—
107	C₂×C₄×C₅	—	2	2	no	—
108	C₂×C₄×C₅	—	2	2	no	—
109	C₂²⋊D₅	—	2	8	no	—
110	C₂×Dic₅	—	2	8	no	—
111	C₂×Dic₅	—	2	2	no	—
112	C₂×Dic₅	—	2	2	no	—
113	C₂×Dic₅	—	2	1	no	—
114	C₂×Dic₅	—	2	1	no	—
115	C₂×Dic₅	C₂×C₃×C₄	1	2	yes	complemented
116	C₂×Dic₅	C₃×D₄	1	4	yes	complemented
117	C₂²×D₄	—	5	1	no	—
118	C₂³×C₄	—	5	1	no	—
119	C₂×C₂²⋊C₄	—	5	2	no	—
120	C₂×C₂²⋊C₄	C₃×D₅	1	1	yes	characteristic, complemented
121	#95	—	5	2	no	—
122	C₂×C₃×C₅	—	2	4	no	—
123	C₂×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₂×C₃×C₅	C₄×D₄	1	2	yes	—
125	C₂×C₃×C₅	#120	1	1	yes	characteristic
126	C₂×C₃×C₅	#121	1	2	yes	—
127	C₂×C₃×C₅	#123	1	1	yes	characteristic
128	C₃×D₅	—	4	2	no	—
129	C₂³×C₃	—	2	2	no	—
130	C₂³×C₃	—	2	2	no	—
131	C₂³×C₃	—	5	2	no	—
132	C₂³×C₃	—	10	1	no	—
133	C₂³×C₃	—	10	2	no	—
134	C₂³×C₃	—	10	2	no	—
135	C₂³×C₃	—	10	1	no	—
136	C₂³×C₃	C₄×D₅	1	2	yes	—
137	C₂³×C₃	D₂₀	1	2	yes	—
138	C₂³×C₃	C₂²⋊D₅	1	2	yes	—
139	C₂³×C₃	C₂²×D₅	1	1	yes	characteristic
140	C₃×D₄	—	10	8	no	—
141	C₂×C₃×C₄	—	2	2	no	—
142	C₂×C₃×C₄	—	2	2	no	—
143	C₂×C₃×C₄	—	2	2	no	—
144	C₂×C₃×C₄	—	5	2	no	—
145	C₂×C₃×C₄	—	5	4	no	—
146	C₂×C₃×C₄	—	10	8	no	—
147	C₂×C₃×C₄	—	10	2	no	—
148	C₂×C₃×C₄	—	10	1	no	—
149	C₂×C₃×C₄	—	10	2	no	—
150	C₂×C₃×C₄	—	10	1	no	—
151	C₂²×C₅	—	2	4	no	—
152	C₂²×C₅	—	2	2	no	—
153	C₂²×C₅	—	2	4	no	—
154	C₂²×C₅	—	2	2	no	—
155	C₂²×C₅	C₃×C₂²⋊C₄	1	4	yes	complemented
156	C₂²×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
157	C₂²×C₅	C₂×C₃×D₄	1	1	yes	characteristic
158	C₂²×C₅	C₂×C₃×D₄	1	2	yes	—
159	C₂²×C₅	C₂×C₃×D₄	1	1	yes	characteristic
160	C₂²×C₅	C₃×#41	1	2	yes	—
161	C₂×D₅	—	2	2	no	—
162	C₂×D₅	—	2	4	no	—
163	C₂×D₅	—	4	1	no	—
164	C₂×D₅	—	4	2	no	—
165	C₂×D₅	—	4	1	no	—
166	C₄×C₅	—	4	2	no	—
167	Dic₅	—	2	2	no	—
168	Dic₅	—	2	4	no	—
169	C₂⁴	—	5	1	no	—
170	C₂⁴	C₂×C₃×D₅	1	1	yes	characteristic
171	C₂×D₄	—	5	4	no	—
172	C₂×D₄	—	10	2	no	—
173	C₂×D₄	—	10	2	no	—
174	C₂²×C₄	—	5	1	no	—
175	C₂²×C₄	—	5	2	no	—
176	C₂²×C₄	—	10	4	no	—
177	C₂²×C₄	—	10	2	no	—
178	C₂²×C₄	C₂×C₃×D₅	1	2	yes	—
179	C₂²⋊C₄	—	2	2	no	—
180	C₂²⋊C₄	—	10	4	no	—
181	C₃×C₅	#386	1	1	yes	characteristic, complemented, Hall
182	C₂²×C₃	—	2	4	no	—
183	C₂²×C₃	—	2	2	no	—
184	C₂²×C₃	—	2	2	no	—
185	C₂²×C₃	—	2	4	no	—
186	C₂²×C₃	—	10	4	no	—
187	C₂²×C₃	—	10	2	no	—
188	C₂²×C₃	—	20	1	no	—
189	C₂²×C₃	—	20	1	no	—
190	C₂²×C₃	—	20	2	no	—
191	C₂²×C₃	#685	1	4	yes	complemented
192	C₂²×C₃	C₂×C₄×D₅	1	1	yes	characteristic
193	C₂²×C₃	C₂×D₂₀	1	1	yes	characteristic
194	C₂²×C₃	D₄×D₅	1	2	yes	—
195	C₂²×C₃	#711	1	2	yes	—
196	C₂²×C₃	C₂×C₂²⋊D₅	1	1	yes	characteristic
197	C₃×C₄	—	4	2	no	—
198	C₃×C₄	—	10	4	no	—
199	C₃×C₄	—	10	2	no	—
200	C₂×C₅	—	2	4	no	—
201	C₂×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
202	C₂×C₅	C₃×C₄×D₄	1	2	yes	—
203	C₂×C₅	C₃×#120	1	1	yes	characteristic
204	C₂×C₅	C₃×#121	1	2	yes	—
205	C₂×C₅	C₃×#123	1	1	yes	characteristic
206	D₅	—	4	2	no	—
207	C₂³	—	2	2	no	—
208	C₂³	—	2	2	no	—
209	C₂³	—	5	2	no	—
210	C₂³	—	10	1	no	—
211	C₂³	—	10	2	no	—
212	C₂³	—	10	2	no	—
213	C₂³	—	10	1	no	—
214	C₂³	C₃×C₄×D₅	1	2	yes	—
215	C₂³	C₃×D₂₀	1	2	yes	—
216	C₂³	C₃×C₂²⋊D₅	1	2	yes	—
217	C₂³	C₂²×C₃×D₅	1	1	yes	characteristic
218	D₄	—	10	8	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	5	2	no	—
223	C₂×C₄	—	5	4	no	—
224	C₂×C₄	—	10	8	no	—
225	C₂×C₄	—	10	2	no	—
226	C₂×C₄	—	10	1	no	—
227	C₂×C₄	—	10	2	no	—
228	C₂×C₄	—	10	1	no	—
229	C₂×C₃	—	2	4	no	—
230	C₂×C₃	—	20	2	no	—
231	C₂×C₃	#4027	1	2	yes	—
232	C₂×C₃	#4028	1	1	yes	characteristic
233	C₂×C₃	#4032	1	1	yes	characteristic
234	C₂×C₃	C₂×#685	1	1	yes	characteristic
235	C₂×C₃	#4085	1	2	yes	—
236	C₅	C₃×#386	1	1	yes	characteristic, complemented, Sylow
237	C₂²	—	2	4	no	—
238	C₂²	—	2	4	no	—
239	C₂²	—	2	2	no	—
240	C₂²	—	2	2	no	—
241	C₂²	—	10	4	no	—
242	C₂²	—	10	2	no	—
243	C₂²	—	20	2	no	—
244	C₂²	—	20	1	no	—
245	C₂²	—	20	1	no	—
246	C₂²	C₃×#685	1	4	yes	complemented
247	C₂²	C₂×C₃×C₄×D₅	1	1	yes	characteristic
248	C₂²	C₂×C₃×D₂₀	1	1	yes	characteristic
249	C₂²	C₃×D₄×D₅	1	2	yes	—
250	C₂²	C₃×#711	1	2	yes	—
251	C₂²	C₂×C₃×C₂²⋊D₅	1	1	yes	characteristic
252	C₄	—	4	2	no	—
253	C₄	—	10	4	no	—
254	C₄	—	10	2	no	—
255	C₃	#65247	1	1	yes	characteristic, direct product factor, Sylow
256	C₂	—	2	4	no	—
257	C₂	—	20	2	no	—
258	C₂	C₃×#4027	1	2	yes	—
259	C₂	C₃×#4028	1	1	yes	characteristic
260	C₂	C₃×#4032	1	1	yes	characteristic
261	C₂	C₂×C₃×#685	1	1	yes	characteristic
262	C₂	C₃×#4085	1	2	yes	—

	subgroup	G/H
center	C₂³×C₃	C₂²×D₅
commutator	C₂²×C₅	C₂²×C₃×C₄
Frattini	C₂³	C₂²×C₃×D₅
Fitting	C₂×C₃×C₅×C₂²⋊C₄	C₂
socle	C₂³×C₃×C₅	C₂³
prime radical	C₂²×C₃×C₂²⋊D₅	C₂
cosocle	C₂³×C₅	C₂³×C₃
hypercenter	C₂×C₃×C₂²⋊C₄	D₅
hypocenter	C₅	C₃×#386
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
2. C₂³×C₃×Dic₅	C₂	semidirect
3. C₂×C₃×C₅×C₂²⋊C₄	C₂	semidirect
4. C₂×C₃×#685	C₂	semidirect
5. C₃×#3963	C₂	semidirect
6. #65247	C₃	direct (N × H)
12. C₂×C₃×C₂²⋊D₅	C₄	semidirect
21. C₂³×Dic₅	C₂×C₃	semidirect
22. C₂×C₅×C₂²⋊C₄	C₂×C₃	semidirect
23. C₂×#685	C₂×C₃	semidirect
24. #3963	C₂×C₃	semidirect
35. C₂²×C₃×D₅	C₂×C₄	semidirect
45. C₂×C₃×Dic₅	C₂×C₄	semidirect
46. C₂×C₃×Dic₅	D₄	semidirect
50. C₂×C₃×C₂²⋊C₄	D₅	semidirect
57. C₂×C₂²⋊D₅	C₃×C₄	semidirect
69. C₂²×C₃×C₅	C₂²⋊C₄	semidirect
105. C₂²×D₅	C₂×C₃×C₄	semidirect
115. C₂×Dic₅	C₂×C₃×C₄	semidirect
116. C₂×Dic₅	C₃×D₄	semidirect
120. C₂×C₂²⋊C₄	C₃×D₅	semidirect
155. C₂²×C₅	C₃×C₂²⋊C₄	semidirect
181. C₃×C₅	#386	semidirect
191. C₂²×C₃	#685	semidirect
236. C₅	C₃×#386	semidirect
246. C₂²	C₃×#685	semidirect
255. C₃	#65247	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×C₂²⋊D₅	2
C₂³×C₃×Dic₅	2
C₂×C₃×C₅×C₂²⋊C₄	2
C₂×C₃×#685	2
C₃×#3963	2
#65247	3
C₃×#386	5

Subgroups — C3×#65247 #11749855

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#65247 #11749855