Subgroups — C₃×#66166 #11755843

Order 960.

620 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×#41	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×#709	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#4093	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#4090	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#4086	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#4085	C₂	1	4	yes	maximal, complemented
7	C₃×#4082	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×C₄×C₂²⋊D₅	C₂	1	4	yes	maximal, complemented
9	C₂×C₃×C₄×Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
10	#66166	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
11	C₃×C₅×#41	—	2	4	no	—
12	C₂×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic
13	C₂×C₃×C₅×D₄	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₅×D₄	C₂²	1	2	yes	—
15	C₂²×C₃×C₄×C₅	C₂²	1	2	yes	—
16	C₂²×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
17	C₂×C₃×C₂²⋊D₅	C₂²	1	2	yes	—
18	C₂×C₃×C₂²⋊D₅	C₂²	1	4	yes	—
19	C₂²×C₃×Dic₅	C₂²	1	2	yes	—
20	C₃×#709	—	2	4	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	1	yes	characteristic
24	C₃×C₂²⋊Dic₅	C₂²	1	4	yes	—
25	C₃×#685	C₂²	1	4	yes	—
26	C₃×#683	C₂²	1	4	yes	—
27	C₃×C₄×Dic₅	C₂²	1	1	yes	characteristic
28	C₃×C₄×Dic₅	C₂²	1	1	yes	characteristic
29	C₃×C₄×Dic₅	C₂²	1	2	yes	—
30	C₃×#532	—	5	1	no	maximal, Hall
31	C₂×C₅×#41	C₂×C₃	1	1	yes	characteristic, complemented
32	C₂×#709	C₂×C₃	1	1	yes	characteristic, complemented
33	#4093	C₂×C₃	1	1	yes	characteristic, complemented
34	#4090	C₂×C₃	1	1	yes	characteristic, complemented
35	#4086	C₂×C₃	1	1	yes	characteristic, complemented
36	#4085	C₂×C₃	1	4	yes	complemented
37	#4082	C₂×C₃	1	1	yes	characteristic, complemented
38	C₄×C₂²⋊D₅	C₂×C₃	1	4	yes	complemented
39	C₂×C₄×Dic₅	C₂×C₃	1	1	yes	characteristic, complemented
40	C₂³×C₃×C₅	C₂³	1	2	yes	—
41	C₂³×C₃×C₅	C₂³	1	1	yes	characteristic
42	C₂²×C₃×D₅	C₂³	1	2	yes	—
43	C₃×C₅×Q₈	—	2	2	no	—
44	C₃×C₅×D₄	—	2	4	no	—
45	C₃×C₅×D₄	—	2	2	no	—
46	C₂×C₃×C₄×C₅	—	2	4	no	—
47	C₂×C₃×C₄×C₅	D₄	1	4	yes	complemented
48	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
49	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
50	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
51	C₃×C₂²⋊D₅	—	2	4	no	—
52	C₃×C₂²⋊D₅	—	2	8	no	—
53	C₂×C₃×Dic₅	—	2	4	no	—
54	C₂×C₃×Dic₅	C₂³	1	2	yes	—
55	C₂×C₃×Dic₅	C₂³	1	2	yes	—
56	C₂×C₃×Dic₅	C₂³	1	2	yes	—
57	C₃×D₂₀	—	2	2	no	—
58	C₃×C₄×D₅	—	2	4	no	—
59	C₃×Dic₁₀	—	2	2	no	—
60	C₂×C₃×#41	—	5	1	no	—
61	C₂×C₃×#41	D₅	1	1	yes	characteristic, complemented
62	C₃×C₄⋊Q₈	—	5	1	no	—
63	C₃×#127	—	5	1	no	—
64	C₃×#124	—	5	1	no	—
65	C₃×#124	—	5	1	no	—
66	C₃×#121	—	5	4	no	—
67	C₃×C₄×D₄	—	5	4	no	—
68	C₂×C₃×C₄²	—	5	1	no	—
69	C₅×#41	—	2	4	no	—
70	C₂×C₅×Q₈	C₂²×C₃	1	1	yes	characteristic
71	C₂×C₅×D₄	C₂²×C₃	1	1	yes	characteristic
72	C₂×C₅×D₄	C₂²×C₃	1	2	yes	—
73	C₂²×C₄×C₅	C₂²×C₃	1	2	yes	—
74	C₂²×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
75	C₂×C₂²⋊D₅	C₂²×C₃	1	2	yes	—
76	C₂×C₂²⋊D₅	C₂²×C₃	1	4	yes	—
77	C₂²×Dic₅	C₂²×C₃	1	2	yes	—
78	#709	—	2	4	no	—
79	C₂×D₂₀	C₂²×C₃	1	1	yes	characteristic
80	C₂×C₄×D₅	C₂²×C₃	1	2	yes	—
81	C₂×Dic₁₀	C₂²×C₃	1	1	yes	characteristic
82	C₂²⋊Dic₅	C₂²×C₃	1	4	yes	—
83	#685	C₂²×C₃	1	4	yes	—
84	#683	C₂²×C₃	1	4	yes	—
85	C₄×Dic₅	C₂²×C₃	1	2	yes	—
86	C₄×Dic₅	C₂²×C₃	1	1	yes	characteristic
87	C₄×Dic₅	C₂²×C₃	1	1	yes	characteristic
88	#532	—	5	1	no	Sylow
89	C₂²×C₃×C₅	—	2	2	no	—
90	C₂²×C₃×C₅	—	2	2	no	—
91	C₂²×C₃×C₅	—	2	4	no	—
92	C₂²×C₃×C₅	C₂×D₄	1	2	yes	—
93	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
94	C₂×C₃×D₅	—	2	2	no	—
95	C₂×C₃×D₅	—	2	4	no	—
96	C₃×C₄×C₅	—	2	2	no	—
97	C₃×C₄×C₅	C₂×D₄	1	2	yes	—
98	C₃×C₄×C₅	C₂×D₄	1	2	yes	—
99	C₃×Dic₅	—	2	2	no	—
100	C₃×Dic₅	—	2	2	no	—
101	C₃×Dic₅	#41	1	4	yes	complemented
102	C₃×#41	—	2	4	no	—
103	C₃×#41	—	10	4	no	—
104	C₂×C₃×Q₈	—	5	1	no	—
105	C₂×C₃×Q₈	C₂×D₅	1	1	yes	characteristic
106	C₂×C₃×D₄	—	5	4	no	—
107	C₂×C₃×D₄	—	5	2	no	—
108	C₂×C₃×D₄	—	5	1	no	—
109	C₂×C₃×D₄	C₂×D₅	1	2	yes	—
110	C₂×C₃×D₄	C₂×D₅	1	1	yes	characteristic
111	C₂²×C₃×C₄	—	5	2	no	—
112	C₂²×C₃×C₄	—	5	2	no	—
113	C₂²×C₃×C₄	C₂×D₅	1	2	yes	—
114	C₂²×C₃×C₄	C₂×D₅	1	1	yes	characteristic
115	C₃×C₄⋊C₄	—	5	4	no	—
116	C₃×C₂²⋊C₄	—	5	4	no	—
117	C₃×C₂²⋊C₄	—	5	4	no	—
118	C₃×C₄²	—	5	1	no	—
119	C₃×C₄²	—	5	1	no	—
120	C₃×C₄²	—	5	2	no	—
121	C₂³×C₅	C₂³×C₃	1	2	yes	—
122	C₂³×C₅	C₂³×C₃	1	1	yes	characteristic
123	C₂²×D₅	C₂³×C₃	1	2	yes	—
124	C₅×Q₈	—	2	2	no	—
125	C₅×D₄	—	2	4	no	—
126	C₅×D₄	—	2	2	no	—
127	C₂×C₄×C₅	—	2	4	no	—
128	C₂×C₄×C₅	C₃×D₄	1	4	yes	complemented
129	C₂×C₄×C₅	C₂³×C₃	1	2	yes	—
130	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
131	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
132	C₂²⋊D₅	—	2	4	no	—
133	C₂²⋊D₅	—	2	8	no	—
134	C₂×Dic₅	—	2	4	no	—
135	C₂×Dic₅	C₂³×C₃	1	2	yes	—
136	C₂×Dic₅	C₂³×C₃	1	2	yes	—
137	C₂×Dic₅	C₂³×C₃	1	2	yes	—
138	D₂₀	—	2	2	no	—
139	C₄×D₅	—	2	4	no	—
140	Dic₁₀	—	2	2	no	—
141	C₂×#41	—	5	1	no	—
142	C₂×#41	C₃×D₅	1	1	yes	characteristic, complemented
143	C₄⋊Q₈	—	5	1	no	—
144	#127	—	5	1	no	—
145	#124	—	5	1	no	—
146	#124	—	5	1	no	—
147	#121	—	5	4	no	—
148	C₄×D₄	—	5	4	no	—
149	C₂×C₄²	—	5	1	no	—
150	C₂×C₃×C₅	—	2	2	no	—
151	C₂×C₃×C₅	—	4	2	no	—
152	C₂×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
153	C₂×C₃×C₅	C₂×#41	1	2	yes	—
154	C₃×D₅	—	4	2	no	—
155	C₂³×C₃	—	5	2	no	—
156	C₂³×C₃	C₂²×D₅	1	2	yes	—
157	C₂³×C₃	C₂²×D₅	1	1	yes	characteristic
158	C₃×Q₈	—	2	2	no	—
159	C₃×Q₈	—	10	2	no	—
160	C₃×D₄	—	2	4	no	—
161	C₃×D₄	—	2	2	no	—
162	C₃×D₄	—	10	8	no	—
163	C₃×D₄	—	10	4	no	—
164	C₃×D₄	—	10	2	no	—
165	C₂×C₃×C₄	—	2	4	no	—
166	C₂×C₃×C₄	—	5	2	no	—
167	C₂×C₃×C₄	—	5	2	no	—
168	C₂×C₃×C₄	—	5	2	no	—
169	C₂×C₃×C₄	—	10	4	no	—
170	C₂×C₃×C₄	—	10	4	no	—
171	C₂×C₃×C₄	C₂²⋊D₅	1	4	yes	complemented
172	C₂×C₃×C₄	C₂²×D₅	1	2	yes	—
173	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
174	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
175	C₂²×C₅	—	2	2	no	—
176	C₂²×C₅	—	2	2	no	—
177	C₂²×C₅	—	2	4	no	—
178	C₂²×C₅	C₂×C₃×D₄	1	2	yes	—
179	C₂²×C₅	C₂⁴×C₃	1	1	yes	characteristic
180	C₂×D₅	—	2	2	no	—
181	C₂×D₅	—	2	4	no	—
182	C₄×C₅	—	2	2	no	—
183	C₄×C₅	C₂×C₃×D₄	1	2	yes	—
184	C₄×C₅	C₂×C₃×D₄	1	2	yes	—
185	Dic₅	—	2	2	no	—
186	Dic₅	—	2	2	no	—
187	Dic₅	C₃×#41	1	4	yes	complemented
188	#41	—	2	4	no	—
189	#41	—	10	4	no	—
190	C₂×Q₈	—	5	1	no	—
191	C₂×Q₈	C₂×C₃×D₅	1	1	yes	characteristic
192	C₂×D₄	—	5	4	no	—
193	C₂×D₄	—	5	2	no	—
194	C₂×D₄	—	5	1	no	—
195	C₂×D₄	C₂×C₃×D₅	1	2	yes	—
196	C₂×D₄	C₂×C₃×D₅	1	1	yes	characteristic
197	C₂²×C₄	—	5	2	no	—
198	C₂²×C₄	—	5	2	no	—
199	C₂²×C₄	C₂×C₃×D₅	1	2	yes	—
200	C₂²×C₄	C₂×C₃×D₅	1	1	yes	characteristic
201	C₄⋊C₄	—	5	4	no	—
202	C₂²⋊C₄	—	5	4	no	—
203	C₂²⋊C₄	—	5	4	no	—
204	C₄²	—	5	2	no	—
205	C₄²	—	5	1	no	—
206	C₄²	—	5	1	no	—
207	C₃×C₅	#532	1	1	yes	characteristic, complemented, Hall
208	C₂²×C₃	—	2	2	no	—
209	C₂²×C₃	—	2	4	no	—
210	C₂²×C₃	—	2	2	no	—
211	C₂²×C₃	—	10	4	no	—
212	C₂²×C₃	—	10	2	no	—
213	C₂²×C₃	C₂×C₂²⋊D₅	1	2	yes	—
214	C₂²×C₃	C₂³×D₅	1	1	yes	characteristic
215	C₃×C₄	—	2	2	no	—
216	C₃×C₄	—	5	4	no	—
217	C₃×C₄	—	10	2	no	—
218	C₃×C₄	—	10	2	no	—
219	C₃×C₄	C₂×C₂²⋊D₅	1	2	yes	—
220	C₃×C₄	C₂×C₂²⋊D₅	1	2	yes	—
221	C₂×C₅	—	2	2	no	—
222	C₂×C₅	—	4	2	no	—
223	C₂×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
224	C₂×C₅	C₂×C₃×#41	1	2	yes	—
225	D₅	—	4	2	no	—
226	C₂³	—	5	2	no	—
227	C₂³	C₂²×C₃×D₅	1	2	yes	—
228	C₂³	C₂²×C₃×D₅	1	1	yes	characteristic
229	Q₈	—	2	2	no	—
230	Q₈	—	10	2	no	—
231	D₄	—	2	4	no	—
232	D₄	—	2	2	no	—
233	D₄	—	10	8	no	—
234	D₄	—	10	4	no	—
235	D₄	—	10	2	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	5	2	no	—
238	C₂×C₄	—	5	2	no	—
239	C₂×C₄	—	5	2	no	—
240	C₂×C₄	—	10	4	no	—
241	C₂×C₄	—	10	4	no	—
242	C₂×C₄	C₃×C₂²⋊D₅	1	4	yes	complemented
243	C₂×C₄	C₂²×C₃×D₅	1	2	yes	—
244	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
245	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
246	C₂×C₃	—	2	2	no	—
247	C₂×C₃	—	4	2	no	—
248	C₂×C₃	—	20	2	no	—
249	C₂×C₃	D₅×#41	1	2	yes	—
250	C₂×C₃	C₂²×C₂²⋊D₅	1	1	yes	characteristic
251	C₅	C₃×#532	1	1	yes	characteristic, complemented, Sylow
252	C₂²	—	2	2	no	—
253	C₂²	—	2	4	no	—
254	C₂²	—	2	2	no	—
255	C₂²	—	10	4	no	—
256	C₂²	—	10	2	no	—
257	C₂²	C₂×C₃×C₂²⋊D₅	1	2	yes	—
258	C₂²	C₂³×C₃×D₅	1	1	yes	characteristic
259	C₄	—	2	2	no	—
260	C₄	—	5	4	no	—
261	C₄	—	10	2	no	—
262	C₄	—	10	2	no	—
263	C₄	C₂×C₃×C₂²⋊D₅	1	2	yes	—
264	C₄	C₂×C₃×C₂²⋊D₅	1	2	yes	—
265	C₃	#66166	1	1	yes	characteristic, direct product factor, Sylow
266	C₂	—	2	2	no	—
267	C₂	—	4	2	no	—
268	C₂	—	20	2	no	—
269	C₂	C₃×D₅×#41	1	2	yes	—
270	C₂	C₂²×C₃×C₂²⋊D₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	C₂²×D₅
commutator, cosocle	C₂²×C₅	C₂⁴×C₃
Frattini	C₂²	C₂³×C₃×D₅
Fitting	C₂×C₃×C₅×#41	C₂
socle	C₂²×C₃×C₅	C₂⁴
hypercenter	C₂×C₃×#41	D₅
hypocenter	C₅	C₃×#532
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂×C₃×#41

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₅×#41	C₂	semidirect
2. C₂×C₃×#709	C₂	semidirect
3. C₃×#4093	C₂	semidirect
4. C₃×#4090	C₂	semidirect
5. C₃×#4086	C₂	semidirect
6. C₃×#4085	C₂	semidirect
7. C₃×#4082	C₂	semidirect
8. C₃×C₄×C₂²⋊D₅	C₂	semidirect
9. C₂×C₃×C₄×Dic₅	C₂	semidirect
10. #66166	C₃	direct (N × H)
31. C₂×C₅×#41	C₂×C₃	semidirect
32. C₂×#709	C₂×C₃	semidirect
33. #4093	C₂×C₃	semidirect
34. #4090	C₂×C₃	semidirect
35. #4086	C₂×C₃	semidirect
36. #4085	C₂×C₃	semidirect
37. #4082	C₂×C₃	semidirect
38. C₄×C₂²⋊D₅	C₂×C₃	semidirect
39. C₂×C₄×Dic₅	C₂×C₃	semidirect
47. C₂×C₃×C₄×C₅	D₄	semidirect
61. C₂×C₃×#41	D₅	semidirect
101. C₃×Dic₅	#41	semidirect
128. C₂×C₄×C₅	C₃×D₄	semidirect
142. C₂×#41	C₃×D₅	semidirect
171. C₂×C₃×C₄	C₂²⋊D₅	semidirect
187. Dic₅	C₃×#41	semidirect
207. C₃×C₅	#532	semidirect
242. C₂×C₄	C₃×C₂²⋊D₅	semidirect
251. C₅	C₃×#532	semidirect
265. C₃	#66166	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×C₅×#41	2
C₂×C₃×#709	2
C₃×#4093	2
C₃×#4090	2
C₃×#4086	2
C₃×#4085	2
C₃×#4082	2
C₃×C₄×C₂²⋊D₅	2
C₂×C₃×C₄×Dic₅	2
#66166	3
C₃×#532	5

Subgroups — C3×#66166 #11755843

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#66166 #11755843