Subgroups — C₃×#65912 #11755589

Order 960.

444 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×C₅×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#4092	C₂	1	1	yes	maximal, characteristic
3	C₃×#4090	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#4043	C₂	1	1	yes	maximal, characteristic
5	C₃×#4042	C₂	1	2	yes	maximal
6	C₃×#4035	C₂	1	2	yes	maximal, complemented
7	C₃×#4034	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#4018	C₂	1	2	yes	maximal
9	C₃×C₄²×D₅	C₂	1	1	yes	maximal, characteristic
10	C₃×C₄×Dic₁₀	C₂	1	2	yes	maximal, complemented
11	C₃×C₄×Dic₁₀	C₂	1	1	yes	maximal, characteristic, complemented
12	#65912	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
13	C₂×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₄×D₅	C₂²	1	1	yes	characteristic
15	C₂×C₃×C₄×D₅	C₂²	1	1	yes	characteristic
16	C₂×C₃×C₄×D₅	C₂²	1	1	yes	characteristic
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	2	yes	—
21	C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
22	C₃×C₄²×C₅	C₂²	1	2	yes	—
23	C₃×#685	C₂²	1	2	yes	—
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×#683	C₂²	1	2	yes	—
28	C₃×#683	C₂²	1	2	yes	—
29	C₃×#683	C₂²	1	2	yes	—
30	C₃×#683	C₂²	1	2	yes	—
31	C₃×#683	C₂²	1	2	yes	—
32	C₃×C₄×Dic₅	C₂²	1	1	yes	characteristic
33	C₃×C₄×Dic₅	C₂²	1	1	yes	characteristic
34	C₃×C₄×Dic₅	C₂²	1	2	yes	—
35	C₃×C₄×Dic₅	C₂²	1	1	yes	characteristic
36	C₃×#533	—	5	1	no	maximal, Hall
37	C₄×C₅×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
38	#4092	C₂×C₃	1	1	yes	characteristic
39	#4090	C₂×C₃	1	1	yes	characteristic, complemented
40	#4043	C₂×C₃	1	1	yes	characteristic
41	#4042	C₂×C₃	1	2	yes	—
42	#4035	C₂×C₃	1	2	yes	complemented
43	#4034	C₂×C₃	1	1	yes	characteristic, complemented
44	#4018	C₂×C₃	1	2	yes	—
45	C₄²×D₅	C₂×C₃	1	1	yes	characteristic
46	C₄×Dic₁₀	C₂×C₃	1	2	yes	complemented
47	C₄×Dic₁₀	C₂×C₃	1	1	yes	characteristic, complemented
48	C₂²×C₃×D₅	C₂³	1	1	yes	characteristic
49	C₃×C₅×Q₈	—	2	2	no	—
50	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
51	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
52	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
53	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
54	C₂×C₃×C₄×C₅	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	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
60	C₃×C₄×D₅	—	2	2	no	—
61	C₃×C₄×D₅	—	2	2	no	—
62	C₃×C₄×D₅	Q₈	1	4	yes	complemented
63	C₃×Dic₁₀	—	2	2	no	—
64	C₃×Dic₁₀	—	2	4	no	—
65	C₃×C₄⋊Q₈	—	5	1	no	—
66	C₃×C₄⋊Q₈	—	5	1	no	—
67	C₃×#125	—	5	2	no	—
68	C₃×C₂²⋊Q₈	—	5	1	no	—
69	C₃×C₂²⋊Q₈	—	5	2	no	—
70	C₃×C₂²⋊Q₈	—	5	1	no	—
71	C₃×C₄×Q₈	—	5	2	no	—
72	C₃×C₄×Q₈	—	5	1	no	—
73	C₃×C₄×Q₈	D₅	1	1	yes	characteristic, complemented
74	C₃×#117	—	5	2	no	—
75	C₂×C₃×C₄²	—	5	1	no	—
76	C₂×C₅×Q₈	C₂²×C₃	1	1	yes	characteristic
77	C₂×C₄×D₅	C₂²×C₃	1	1	yes	characteristic
78	C₂×C₄×D₅	C₂²×C₃	1	1	yes	characteristic
79	C₂×C₄×D₅	C₂²×C₃	1	1	yes	characteristic
80	C₂×Dic₁₀	C₂²×C₃	1	2	yes	—
81	C₂×Dic₁₀	C₂²×C₃	1	1	yes	characteristic
82	C₅×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
83	C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
84	C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
85	C₄²×C₅	C₂²×C₃	1	2	yes	—
86	#685	C₂²×C₃	1	2	yes	—
87	#685	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	2	yes	—
91	#683	C₂²×C₃	1	2	yes	—
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	C₄×Dic₅	C₂²×C₃	1	1	yes	characteristic
98	C₄×Dic₅	C₂²×C₃	1	1	yes	characteristic
99	#533	—	5	1	no	Sylow
100	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
101	C₂×C₃×D₅	—	2	1	no	—
102	C₂×C₃×D₅	—	2	1	no	—
103	C₂×C₃×D₅	C₂×Q₈	1	2	yes	—
104	C₃×C₄×C₅	—	2	2	no	—
105	C₃×C₄×C₅	—	2	1	no	—
106	C₃×C₄×C₅	—	2	2	no	—
107	C₃×C₄×C₅	C₂×Q₈	1	2	yes	—
108	C₃×C₄×C₅	#41	1	2	yes	—
109	C₃×Dic₅	—	2	2	no	—
110	C₃×Dic₅	—	2	2	no	—
111	C₃×Dic₅	—	2	1	no	—
112	C₃×Dic₅	C₂×Q₈	1	2	yes	—
113	C₃×Dic₅	#41	1	2	yes	—
114	C₂×C₃×Q₈	—	5	2	no	—
115	C₂×C₃×Q₈	—	5	1	no	—
116	C₂×C₃×Q₈	C₂×D₅	1	1	yes	characteristic
117	C₂²×C₃×C₄	—	5	1	no	—
118	C₂²×C₃×C₄	—	5	1	no	—
119	C₂²×C₃×C₄	—	5	1	no	—
120	C₃×C₄⋊C₄	—	5	2	no	—
121	C₃×C₄⋊C₄	—	5	2	no	—
122	C₃×C₄⋊C₄	—	5	2	no	—
123	C₃×C₄⋊C₄	—	5	2	no	—
124	C₃×C₄⋊C₄	—	5	2	no	—
125	C₃×C₄⋊C₄	—	5	2	no	—
126	C₃×C₄⋊C₄	—	5	1	no	—
127	C₃×C₄⋊C₄	C₂×D₅	1	2	yes	—
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×C₄²	—	5	1	no	—
132	C₃×C₄²	—	5	1	no	—
133	C₃×C₄²	—	5	1	no	—
134	C₃×C₄²	—	5	2	no	—
135	C₃×C₄²	C₂×D₅	1	1	yes	characteristic
136	C₃×C₄²	C₂×D₅	1	2	yes	—
137	C₂²×D₅	C₂³×C₃	1	1	yes	characteristic
138	C₅×Q₈	—	2	2	no	—
139	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
140	C₂×C₄×C₅	C₂³×C₃	1	2	yes	—
141	C₂×C₄×C₅	C₂³×C₃	1	2	yes	—
142	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
143	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
144	C₂×Dic₅	C₂³×C₃	1	1	yes	characteristic
145	C₂×Dic₅	C₂³×C₃	1	2	yes	—
146	C₂×Dic₅	C₂³×C₃	1	2	yes	—
147	C₂×Dic₅	C₂³×C₃	1	1	yes	characteristic
148	C₂×Dic₅	C₂³×C₃	1	1	yes	characteristic
149	C₄×D₅	—	2	2	no	—
150	C₄×D₅	—	2	2	no	—
151	C₄×D₅	C₃×Q₈	1	4	yes	complemented
152	Dic₁₀	—	2	2	no	—
153	Dic₁₀	—	2	4	no	—
154	C₄⋊Q₈	—	5	1	no	—
155	C₄⋊Q₈	—	5	1	no	—
156	#125	—	5	2	no	—
157	C₂²⋊Q₈	—	5	1	no	—
158	C₂²⋊Q₈	—	5	2	no	—
159	C₂²⋊Q₈	—	5	1	no	—
160	C₄×Q₈	—	5	2	no	—
161	C₄×Q₈	—	5	1	no	—
162	C₄×Q₈	C₃×D₅	1	1	yes	characteristic, complemented
163	#117	—	5	2	no	—
164	C₂×C₄²	—	5	1	no	—
165	C₂×C₃×C₅	C₂²×Q₈	1	1	yes	characteristic
166	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
167	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
168	C₃×D₅	—	2	2	no	—
169	C₂³×C₃	—	5	1	no	—
170	C₃×Q₈	—	2	2	no	—
171	C₃×Q₈	—	10	4	no	—
172	C₃×Q₈	—	10	2	no	—
173	C₂×C₃×C₄	—	5	1	no	—
174	C₂×C₃×C₄	—	5	1	no	—
175	C₂×C₃×C₄	—	5	4	no	—
176	C₂×C₃×C₄	—	5	1	no	—
177	C₂×C₃×C₄	—	5	2	no	—
178	C₂×C₃×C₄	—	5	2	no	—
179	C₂×C₃×C₄	—	10	2	no	—
180	C₂×C₃×C₄	—	10	2	no	—
181	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
182	C₂×C₃×C₄	C₂²×D₅	1	2	yes	—
183	C₂×C₃×C₄	C₂²×D₅	1	2	yes	—
184	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
185	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
186	C₂²×C₅	C₂⁴×C₃	1	1	yes	characteristic
187	C₂×D₅	—	2	1	no	—
188	C₂×D₅	—	2	1	no	—
189	C₂×D₅	C₂×C₃×Q₈	1	2	yes	—
190	C₄×C₅	—	2	2	no	—
191	C₄×C₅	—	2	1	no	—
192	C₄×C₅	—	2	2	no	—
193	C₄×C₅	C₂×C₃×Q₈	1	2	yes	—
194	C₄×C₅	C₃×#41	1	2	yes	—
195	Dic₅	—	2	2	no	—
196	Dic₅	—	2	2	no	—
197	Dic₅	—	2	1	no	—
198	Dic₅	C₂×C₃×Q₈	1	2	yes	—
199	Dic₅	C₃×#41	1	2	yes	—
200	C₂×Q₈	—	5	2	no	—
201	C₂×Q₈	—	5	1	no	—
202	C₂×Q₈	C₂×C₃×D₅	1	1	yes	characteristic
203	C₂²×C₄	—	5	1	no	—
204	C₂²×C₄	—	5	1	no	—
205	C₂²×C₄	—	5	1	no	—
206	C₄⋊C₄	—	5	2	no	—
207	C₄⋊C₄	—	5	2	no	—
208	C₄⋊C₄	—	5	2	no	—
209	C₄⋊C₄	—	5	2	no	—
210	C₄⋊C₄	—	5	2	no	—
211	C₄⋊C₄	—	5	2	no	—
212	C₄⋊C₄	—	5	1	no	—
213	C₄⋊C₄	C₂×C₃×D₅	1	2	yes	—
214	C₄⋊C₄	C₂×C₃×D₅	1	1	yes	characteristic
215	C₂²⋊C₄	—	5	2	no	—
216	C₂²⋊C₄	—	5	2	no	—
217	C₄²	—	5	1	no	—
218	C₄²	—	5	1	no	—
219	C₄²	—	5	1	no	—
220	C₄²	—	5	2	no	—
221	C₄²	C₂×C₃×D₅	1	1	yes	characteristic
222	C₄²	C₂×C₃×D₅	1	2	yes	—
223	C₃×C₅	#533	1	1	yes	characteristic, complemented, Hall
224	C₂²×C₃	—	5	2	no	—
225	C₂²×C₃	—	10	1	no	—
226	C₂²×C₃	—	10	1	no	—
227	C₂²×C₃	C₂³×D₅	1	1	yes	characteristic
228	C₃×C₄	—	2	2	no	—
229	C₃×C₄	—	2	1	no	—
230	C₃×C₄	—	2	2	no	—
231	C₃×C₄	—	5	2	no	—
232	C₃×C₄	—	5	2	no	—
233	C₃×C₄	—	10	1	no	—
234	C₃×C₄	—	10	2	no	—
235	C₃×C₄	—	10	2	no	—
236	C₃×C₄	#709	1	2	yes	—
237	C₃×C₄	Q₈×D₅	1	2	yes	—
238	C₂×C₅	C₂²×C₃×Q₈	1	1	yes	characteristic
239	C₂×C₅	C₂×C₃×#41	1	1	yes	characteristic
240	C₂×C₅	C₂×C₃×#41	1	1	yes	characteristic
241	D₅	—	2	2	no	—
242	C₂³	—	5	1	no	—
243	Q₈	—	2	2	no	—
244	Q₈	—	10	4	no	—
245	Q₈	—	10	2	no	—
246	C₂×C₄	—	5	1	no	—
247	C₂×C₄	—	5	1	no	—
248	C₂×C₄	—	5	4	no	—
249	C₂×C₄	—	5	1	no	—
250	C₂×C₄	—	5	2	no	—
251	C₂×C₄	—	5	2	no	—
252	C₂×C₄	—	10	2	no	—
253	C₂×C₄	—	10	2	no	—
254	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
255	C₂×C₄	C₂²×C₃×D₅	1	2	yes	—
256	C₂×C₄	C₂²×C₃×D₅	1	2	yes	—
257	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
258	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
259	C₂×C₃	—	10	2	no	—
260	C₂×C₃	C₂×#709	1	1	yes	characteristic
261	C₂×C₃	C₂×Q₈×D₅	1	1	yes	characteristic
262	C₂×C₃	D₅×#41	1	1	yes	characteristic
263	C₅	C₃×#533	1	1	yes	characteristic, complemented, Sylow
264	C₂²	—	5	2	no	—
265	C₂²	—	10	1	no	—
266	C₂²	—	10	1	no	—
267	C₂²	C₂³×C₃×D₅	1	1	yes	characteristic
268	C₄	—	2	2	no	—
269	C₄	—	2	1	no	—
270	C₄	—	2	2	no	—
271	C₄	—	5	2	no	—
272	C₄	—	5	2	no	—
273	C₄	—	10	1	no	—
274	C₄	—	10	2	no	—
275	C₄	—	10	2	no	—
276	C₄	C₃×#709	1	2	yes	—
277	C₄	C₃×Q₈×D₅	1	2	yes	—
278	C₃	#65912	1	1	yes	characteristic, direct product factor, Sylow
279	C₂	—	10	2	no	—
280	C₂	C₂×C₃×#709	1	1	yes	characteristic
281	C₂	C₂×C₃×Q₈×D₅	1	1	yes	characteristic
282	C₂	C₃×D₅×#41	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	C₃×C₄×C₅×Q₈	C₂
socle	C₂²×C₃×C₅	C₂⁴
prime radical	C₂²×C₃×D₅	C₂³
hypercenter	C₃×C₄×Q₈	D₅
hypocenter	C₅	C₃×#533
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₃×C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. C₃×C₄×C₅×Q₈	C₂	semidirect
3. C₃×#4090	C₂	semidirect
6. C₃×#4035	C₂	semidirect
7. C₃×#4034	C₂	semidirect
10. C₃×C₄×Dic₁₀	C₂	semidirect
11. C₃×C₄×Dic₁₀	C₂	semidirect
12. #65912	C₃	direct (N × H)
37. C₄×C₅×Q₈	C₂×C₃	semidirect
39. #4090	C₂×C₃	semidirect
42. #4035	C₂×C₃	semidirect
43. #4034	C₂×C₃	semidirect
46. C₄×Dic₁₀	C₂×C₃	semidirect
47. C₄×Dic₁₀	C₂×C₃	semidirect
62. C₃×C₄×D₅	Q₈	semidirect
73. C₃×C₄×Q₈	D₅	semidirect
151. C₄×D₅	C₃×Q₈	semidirect
162. C₄×Q₈	C₃×D₅	semidirect
223. C₃×C₅	#533	semidirect
263. C₅	C₃×#533	semidirect
278. C₃	#65912	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×C₄×C₅×Q₈	2
C₃×#4092	2
C₃×#4090	2
C₃×#4043	2
C₃×#4042	2
C₃×#4035	2
C₃×#4034	2
C₃×#4018	2
C₃×C₄²×D₅	2
C₃×C₄×Dic₁₀	2
#65912	3
C₃×#533	5

Subgroups — C3×#65912 #11755589

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#65912 #11755589