Subgroups — C₃×#66023 #11755700

Order 960.

412 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#125	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#4043	C₂	1	2	yes	maximal
3	C₃×#4042	C₂	1	2	yes	maximal
4	C₃×#4038	C₂	1	1	yes	maximal, characteristic
5	C₃×D₅×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
6	C₃×#4036	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#4035	C₂	1	2	yes	maximal, complemented
8	C₃×#4034	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃×#4034	C₂	1	2	yes	maximal, complemented
10	C₃×#4018	C₂	1	1	yes	maximal, characteristic
11	C₃×#4015	C₂	1	1	yes	maximal, characteristic, complemented
12	#66023	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
13	C₂×C₃×C₄×D₅	C₂²	1	2	yes	—
14	C₂×C₃×C₄×D₅	C₂²	1	1	yes	characteristic
15	C₂×C₃×Dic₁₀	C₂²	1	2	yes	—
16	C₂×C₃×Dic₁₀	C₂²	1	2	yes	—
17	C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
18	C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
19	C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
20	C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
21	C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
22	C₃×#685	C₂²	1	1	yes	characteristic
23	C₃×#685	C₂²	1	1	yes	characteristic
24	C₃×#685	C₂²	1	2	yes	—
25	C₃×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
26	C₃×C₄⋊Dic₅	C₂²	1	2	yes	—
27	C₃×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
28	C₃×C₄⋊Dic₅	C₂²	1	2	yes	—
29	C₃×#683	C₂²	1	1	yes	characteristic
30	C₃×#683	C₂²	1	1	yes	characteristic
31	C₃×#683	C₂²	1	2	yes	—
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×#544	—	5	1	no	maximal, Hall
37	C₅×#125	C₂×C₃	1	1	yes	characteristic, complemented
38	#4043	C₂×C₃	1	2	yes	—
39	#4042	C₂×C₃	1	2	yes	—
40	#4038	C₂×C₃	1	1	yes	characteristic
41	D₅×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic
42	#4036	C₂×C₃	1	1	yes	characteristic, complemented
43	#4035	C₂×C₃	1	2	yes	complemented
44	#4034	C₂×C₃	1	1	yes	characteristic, complemented
45	#4034	C₂×C₃	1	2	yes	complemented
46	#4018	C₂×C₃	1	1	yes	characteristic
47	#4015	C₂×C₃	1	1	yes	characteristic, complemented
48	C₂²×C₃×D₅	C₂³	1	1	yes	characteristic
49	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
50	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
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×Dic₅	C₂³	1	1	yes	characteristic
55	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
56	C₂×C₃×Dic₅	C₂³	1	2	yes	—
57	C₂×C₃×Dic₅	C₂³	1	2	yes	—
58	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
59	C₃×C₄×D₅	—	4	2	no	—
60	C₃×C₄×D₅	Q₈	1	4	yes	complemented
61	C₃×Dic₁₀	—	4	2	no	—
62	C₃×Dic₁₀	—	4	2	no	—
63	C₃×C₄⋊Q₈	—	5	2	no	—
64	C₃×C₄⋊Q₈	—	5	1	no	—
65	C₃×C₄⋊Q₈	—	5	1	no	—
66	C₃×#125	—	5	2	no	—
67	C₃×#125	—	5	1	no	—
68	C₃×#125	D₅	1	1	yes	characteristic, complemented
69	C₃×C₂²⋊Q₈	—	5	2	no	—
70	C₃×C₂²⋊Q₈	—	5	2	no	—
71	C₃×#117	—	5	1	no	—
72	C₃×#117	—	5	1	no	—
73	C₂×C₃×C₄⋊C₄	—	5	1	no	—
74	C₂×C₄×D₅	C₂²×C₃	1	2	yes	—
75	C₂×C₄×D₅	C₂²×C₃	1	1	yes	characteristic
76	C₂×Dic₁₀	C₂²×C₃	1	2	yes	—
77	C₂×Dic₁₀	C₂²×C₃	1	2	yes	—
78	C₅×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
79	C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
80	C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
81	C₅×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
82	C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
83	#685	C₂²×C₃	1	1	yes	characteristic
84	#685	C₂²×C₃	1	1	yes	characteristic
85	#685	C₂²×C₃	1	2	yes	—
86	C₄⋊Dic₅	C₂²×C₃	1	1	yes	characteristic
87	C₄⋊Dic₅	C₂²×C₃	1	2	yes	—
88	C₄⋊Dic₅	C₂²×C₃	1	1	yes	characteristic
89	C₄⋊Dic₅	C₂²×C₃	1	2	yes	—
90	#683	C₂²×C₃	1	1	yes	characteristic
91	#683	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	C₄×Dic₅	C₂²×C₃	1	1	yes	characteristic
96	C₄×Dic₅	C₂²×C₃	1	2	yes	—
97	#544	—	5	1	no	Sylow
98	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
99	C₂×C₃×D₅	—	2	1	no	—
100	C₂×C₃×D₅	—	2	1	no	—
101	C₂×C₃×D₅	C₂×Q₈	1	2	yes	—
102	C₃×C₄×C₅	—	2	1	no	—
103	C₃×C₄×C₅	—	2	2	no	—
104	C₃×C₄×C₅	—	2	2	no	—
105	C₃×C₄×C₅	—	2	1	no	—
106	C₃×C₄×C₅	C₂×Q₈	1	2	yes	—
107	C₃×Dic₅	—	2	1	no	—
108	C₃×Dic₅	—	2	2	no	—
109	C₃×Dic₅	—	2	2	no	—
110	C₃×Dic₅	—	2	1	no	—
111	C₃×Dic₅	C₂×Q₈	1	2	yes	—
112	C₂×C₃×Q₈	—	5	2	no	—
113	C₂×C₃×Q₈	—	5	2	no	—
114	C₂²×C₃×C₄	—	5	1	no	—
115	C₂²×C₃×C₄	—	5	2	no	—
116	C₃×C₄⋊C₄	—	5	2	no	—
117	C₃×C₄⋊C₄	—	5	1	no	—
118	C₃×C₄⋊C₄	—	5	1	no	—
119	C₃×C₄⋊C₄	—	5	2	no	—
120	C₃×C₄⋊C₄	—	5	2	no	—
121	C₃×C₄⋊C₄	—	5	1	no	—
122	C₃×C₄⋊C₄	—	5	1	no	—
123	C₃×C₄⋊C₄	—	5	2	no	—
124	C₃×C₄⋊C₄	—	5	2	no	—
125	C₃×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
126	C₃×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
127	C₃×C₄⋊C₄	C₂×D₅	1	2	yes	—
128	C₃×C₄⋊C₄	C₂×D₅	1	2	yes	—
129	C₃×C₂²⋊C₄	—	5	1	no	—
130	C₃×C₂²⋊C₄	—	5	2	no	—
131	C₃×C₂²⋊C₄	—	5	1	no	—
132	C₃×C₄²	—	5	2	no	—
133	C₃×C₄²	—	5	1	no	—
134	C₃×C₄²	C₂×D₅	1	1	yes	characteristic
135	C₂²×D₅	C₂³×C₃	1	1	yes	characteristic
136	C₂×C₄×C₅	C₂³×C₃	1	2	yes	—
137	C₂×C₄×C₅	C₂³×C₃	1	2	yes	—
138	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
139	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
140	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
141	C₂×Dic₅	C₂³×C₃	1	1	yes	characteristic
142	C₂×Dic₅	C₂³×C₃	1	1	yes	characteristic
143	C₂×Dic₅	C₂³×C₃	1	2	yes	—
144	C₂×Dic₅	C₂³×C₃	1	2	yes	—
145	C₂×Dic₅	C₂³×C₃	1	1	yes	characteristic
146	C₄×D₅	—	4	2	no	—
147	C₄×D₅	C₃×Q₈	1	4	yes	complemented
148	Dic₁₀	—	4	2	no	—
149	Dic₁₀	—	4	2	no	—
150	C₄⋊Q₈	—	5	2	no	—
151	C₄⋊Q₈	—	5	1	no	—
152	C₄⋊Q₈	—	5	1	no	—
153	#125	—	5	2	no	—
154	#125	—	5	1	no	—
155	#125	C₃×D₅	1	1	yes	characteristic, complemented
156	C₂²⋊Q₈	—	5	2	no	—
157	C₂²⋊Q₈	—	5	2	no	—
158	#117	—	5	1	no	—
159	#117	—	5	1	no	—
160	C₂×C₄⋊C₄	—	5	1	no	—
161	C₂×C₃×C₅	C₂²×Q₈	1	1	yes	characteristic
162	C₂×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
163	C₂×C₃×C₅	2₋¹⁺⁴	1	1	yes	characteristic
164	C₃×D₅	—	2	2	no	—
165	C₂³×C₃	—	5	1	no	—
166	C₃×Q₈	—	20	2	no	—
167	C₃×Q₈	—	20	2	no	—
168	C₂×C₃×C₄	—	5	1	no	—
169	C₂×C₃×C₄	—	5	4	no	—
170	C₂×C₃×C₄	—	5	1	no	—
171	C₂×C₃×C₄	—	5	1	no	—
172	C₂×C₃×C₄	—	5	2	no	—
173	C₂×C₃×C₄	—	5	2	no	—
174	C₂×C₃×C₄	—	20	2	no	—
175	C₂×C₃×C₄	C₂²×D₅	1	2	yes	—
176	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
177	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
178	C₂×C₃×C₄	C₂²×D₅	1	2	yes	—
179	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
180	C₂²×C₅	C₂⁴×C₃	1	1	yes	characteristic
181	C₂×D₅	—	2	1	no	—
182	C₂×D₅	—	2	1	no	—
183	C₂×D₅	C₂×C₃×Q₈	1	2	yes	—
184	C₄×C₅	—	2	1	no	—
185	C₄×C₅	—	2	2	no	—
186	C₄×C₅	—	2	2	no	—
187	C₄×C₅	—	2	1	no	—
188	C₄×C₅	C₂×C₃×Q₈	1	2	yes	—
189	Dic₅	—	2	1	no	—
190	Dic₅	—	2	2	no	—
191	Dic₅	—	2	2	no	—
192	Dic₅	—	2	1	no	—
193	Dic₅	C₂×C₃×Q₈	1	2	yes	—
194	C₂×Q₈	—	5	2	no	—
195	C₂×Q₈	—	5	2	no	—
196	C₂²×C₄	—	5	1	no	—
197	C₂²×C₄	—	5	2	no	—
198	C₄⋊C₄	—	5	2	no	—
199	C₄⋊C₄	—	5	1	no	—
200	C₄⋊C₄	—	5	1	no	—
201	C₄⋊C₄	—	5	1	no	—
202	C₄⋊C₄	—	5	1	no	—
203	C₄⋊C₄	—	5	2	no	—
204	C₄⋊C₄	—	5	2	no	—
205	C₄⋊C₄	—	5	2	no	—
206	C₄⋊C₄	—	5	2	no	—
207	C₄⋊C₄	C₂×C₃×D₅	1	1	yes	characteristic
208	C₄⋊C₄	C₂×C₃×D₅	1	1	yes	characteristic
209	C₄⋊C₄	C₂×C₃×D₅	1	2	yes	—
210	C₄⋊C₄	C₂×C₃×D₅	1	2	yes	—
211	C₂²⋊C₄	—	5	1	no	—
212	C₂²⋊C₄	—	5	2	no	—
213	C₂²⋊C₄	—	5	1	no	—
214	C₄²	—	5	2	no	—
215	C₄²	—	5	1	no	—
216	C₄²	C₂×C₃×D₅	1	1	yes	characteristic
217	C₃×C₅	#544	1	1	yes	characteristic, complemented, Hall
218	C₂²×C₃	—	5	2	no	—
219	C₂²×C₃	—	10	1	no	—
220	C₂²×C₃	—	10	1	no	—
221	C₂²×C₃	C₂³×D₅	1	1	yes	characteristic
222	C₃×C₄	—	2	2	no	—
223	C₃×C₄	—	2	1	no	—
224	C₃×C₄	—	2	2	no	—
225	C₃×C₄	—	2	1	no	—
226	C₃×C₄	—	5	2	no	—
227	C₃×C₄	—	10	2	no	—
228	C₃×C₄	—	10	1	no	—
229	C₃×C₄	—	10	2	no	—
230	C₃×C₄	—	10	1	no	—
231	C₃×C₄	Q₈×D₅	1	2	yes	—
232	C₂×C₅	C₂²×C₃×Q₈	1	1	yes	characteristic
233	C₂×C₅	C₃×H(5,𝔽₂)	1	1	yes	characteristic
234	C₂×C₅	C₃×2₋¹⁺⁴	1	1	yes	characteristic
235	D₅	—	2	2	no	—
236	C₂³	—	5	1	no	—
237	Q₈	—	20	2	no	—
238	Q₈	—	20	2	no	—
239	C₂×C₄	—	5	1	no	—
240	C₂×C₄	—	5	4	no	—
241	C₂×C₄	—	5	1	no	—
242	C₂×C₄	—	5	1	no	—
243	C₂×C₄	—	5	2	no	—
244	C₂×C₄	—	5	2	no	—
245	C₂×C₄	—	20	2	no	—
246	C₂×C₄	C₂²×C₃×D₅	1	2	yes	—
247	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
248	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
249	C₂×C₄	C₂²×C₃×D₅	1	2	yes	—
250	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
251	C₂×C₃	—	10	2	no	—
252	C₂×C₃	C₂×Q₈×D₅	1	1	yes	characteristic
253	C₂×C₃	#4149	1	1	yes	characteristic
254	C₂×C₃	#4150	1	1	yes	characteristic
255	C₅	C₃×#544	1	1	yes	characteristic, complemented, Sylow
256	C₂²	—	5	2	no	—
257	C₂²	—	10	1	no	—
258	C₂²	—	10	1	no	—
259	C₂²	C₂³×C₃×D₅	1	1	yes	characteristic
260	C₄	—	2	2	no	—
261	C₄	—	2	1	no	—
262	C₄	—	2	2	no	—
263	C₄	—	2	1	no	—
264	C₄	—	5	2	no	—
265	C₄	—	10	2	no	—
266	C₄	—	10	1	no	—
267	C₄	—	10	2	no	—
268	C₄	—	10	1	no	—
269	C₄	C₃×Q₈×D₅	1	2	yes	—
270	C₃	#66023	1	1	yes	characteristic, direct product factor, Sylow
271	C₂	—	10	2	no	—
272	C₂	C₂×C₃×Q₈×D₅	1	1	yes	characteristic
273	C₂	C₃×#4149	1	1	yes	characteristic
274	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×#125	C₂
socle	C₂²×C₃×C₅	C₂⁴
prime radical	C₂²×C₃×D₅	C₂³
hypercenter	C₃×#125	D₅
hypocenter	C₅	C₃×#544
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×#125

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×#125	C₂	semidirect
6. C₃×#4036	C₂	semidirect
7. C₃×#4035	C₂	semidirect
8. C₃×#4034	C₂	semidirect
9. C₃×#4034	C₂	semidirect
11. C₃×#4015	C₂	semidirect
12. #66023	C₃	direct (N × H)
37. C₅×#125	C₂×C₃	semidirect
42. #4036	C₂×C₃	semidirect
43. #4035	C₂×C₃	semidirect
44. #4034	C₂×C₃	semidirect
45. #4034	C₂×C₃	semidirect
47. #4015	C₂×C₃	semidirect
60. C₃×C₄×D₅	Q₈	semidirect
68. C₃×#125	D₅	semidirect
147. C₄×D₅	C₃×Q₈	semidirect
155. #125	C₃×D₅	semidirect
217. C₃×C₅	#544	semidirect
255. C₅	C₃×#544	semidirect
270. C₃	#66023	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×#125	2
C₃×#4043	2
C₃×#4042	2
C₃×#4038	2
C₃×D₅×C₄⋊C₄	2
C₃×#4036	2
C₃×#4035	2
C₃×#4034	2
C₃×#4018	2
C₃×#4015	2
#66023	3
C₃×#544	5

Subgroups — C3×#66023 #11755700

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#66023 #11755700