Subgroups — C₃×#65902 #11755579

Order 960.

424 conjugacy classes of subgroups.

The subgroup lattice has 280 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×#4093	C₂	1	1	yes	maximal, characteristic
3	C₃×#4090	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#4044	C₂	1	2	yes	maximal, complemented
5	C₃×#4042	C₂	1	2	yes	maximal, complemented
6	C₃×#4039	C₂	1	1	yes	maximal, characteristic
7	C₃×#4033	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#4022	C₂	1	2	yes	maximal, complemented
9	C₃×#4021	C₂	1	1	yes	maximal, characteristic
10	C₃×#4018	C₂	1	2	yes	maximal, complemented
11	C₃×C₄×Dic₁₀	C₂	1	1	yes	maximal, characteristic, complemented
12	#65902	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
13	C₂×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic
14	C₂×C₃×D₂₀	C₂²	1	1	yes	characteristic
15	C₂×C₃×C₄×D₅	C₂²	1	2	yes	—
16	C₂×C₃×Dic₁₀	C₂²	1	1	yes	characteristic
17	C₂×C₃×Dic₁₀	C₂²	1	1	yes	characteristic
18	C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
19	C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
20	C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
21	C₃×C₄²×C₅	C₂²	1	2	yes	—
22	C₃×#685	C₂²	1	2	yes	—
23	C₃×#685	C₂²	1	2	yes	—
24	C₃×#685	C₂²	1	2	yes	—
25	C₃×#685	C₂²	1	2	yes	—
26	C₃×#685	C₂²	1	2	yes	—
27	C₃×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
28	C₃×#683	C₂²	1	2	yes	—
29	C₃×#683	C₂²	1	2	yes	—
30	C₃×#683	C₂²	1	1	yes	characteristic
31	C₃×#683	C₂²	1	1	yes	characteristic
32	C₃×#683	C₂²	1	2	yes	—
33	C₃×C₄×Dic₅	C₂²	1	1	yes	characteristic
34	C₃×C₄×Dic₅	C₂²	1	1	yes	characteristic
35	C₃×C₄×Dic₅	C₂²	1	2	yes	—
36	C₃×#556	—	5	1	no	maximal, Hall
37	C₄×C₅×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
38	#4093	C₂×C₃	1	1	yes	characteristic
39	#4090	C₂×C₃	1	1	yes	characteristic, complemented
40	#4044	C₂×C₃	1	2	yes	complemented
41	#4042	C₂×C₃	1	2	yes	complemented
42	#4039	C₂×C₃	1	1	yes	characteristic
43	#4033	C₂×C₃	1	1	yes	characteristic, complemented
44	#4022	C₂×C₃	1	2	yes	complemented
45	#4021	C₂×C₃	1	1	yes	characteristic
46	#4018	C₂×C₃	1	2	yes	complemented
47	C₄×Dic₁₀	C₂×C₃	1	1	yes	characteristic, complemented
48	C₂²×C₃×D₅	C₂³	1	2	yes	—
49	C₃×C₅×Q₈	—	2	2	no	—
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	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	1	yes	characteristic
57	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
58	C₂×C₃×Dic₅	C₂³	1	2	yes	—
59	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
60	C₃×D₂₀	—	2	2	no	—
61	C₃×C₄×D₅	—	2	4	no	—
62	C₃×Dic₁₀	—	2	2	no	—
63	C₃×Dic₁₀	—	2	2	no	—
64	C₃×C₄⋊Q₈	—	5	1	no	—
65	C₃×#126	—	5	2	no	—
66	C₃×#126	—	5	2	no	—
67	C₃×#124	—	5	1	no	—
68	C₃×#124	—	5	1	no	—
69	C₃×C₂²⋊Q₈	—	5	2	no	—
70	C₃×C₄×Q₈	—	5	1	no	—
71	C₃×C₄×Q₈	—	5	1	no	—
72	C₃×C₄×Q₈	D₅	1	1	yes	characteristic, complemented
73	C₃×C₄×D₄	—	5	1	no	—
74	C₃×#117	—	5	2	no	—
75	C₂×C₅×Q₈	C₂²×C₃	1	1	yes	characteristic
76	C₂×D₂₀	C₂²×C₃	1	1	yes	characteristic
77	C₂×C₄×D₅	C₂²×C₃	1	2	yes	—
78	C₂×Dic₁₀	C₂²×C₃	1	1	yes	characteristic
79	C₂×Dic₁₀	C₂²×C₃	1	1	yes	characteristic
80	C₅×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
81	C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
82	C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
83	C₄²×C₅	C₂²×C₃	1	2	yes	—
84	#685	C₂²×C₃	1	2	yes	—
85	#685	C₂²×C₃	1	2	yes	—
86	#685	C₂²×C₃	1	2	yes	—
87	#685	C₂²×C₃	1	2	yes	—
88	#685	C₂²×C₃	1	2	yes	—
89	C₄⋊Dic₅	C₂²×C₃	1	1	yes	characteristic
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	1	yes	characteristic
97	C₄×Dic₅	C₂²×C₃	1	2	yes	—
98	#556	—	5	1	no	Sylow
99	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
100	C₂×C₃×D₅	—	2	2	no	—
101	C₂×C₃×D₅	—	2	2	no	—
102	C₂×C₃×D₅	—	2	2	no	—
103	C₃×C₄×C₅	—	2	1	no	—
104	C₃×C₄×C₅	—	2	2	no	—
105	C₃×C₄×C₅	—	2	2	no	—
106	C₃×C₄×C₅	—	2	1	no	—
107	C₃×C₄×C₅	#41	1	2	yes	—
108	C₃×Dic₅	—	2	2	no	—
109	C₃×Dic₅	—	2	1	no	—
110	C₃×Dic₅	—	2	1	no	—
111	C₃×Dic₅	—	2	1	no	—
112	C₃×Dic₅	#41	1	2	yes	—
113	C₂×C₃×Q₈	—	5	1	no	—
114	C₂×C₃×Q₈	—	5	1	no	—
115	C₂×C₃×Q₈	C₂×D₅	1	1	yes	characteristic
116	C₂×C₃×D₄	—	5	1	no	—
117	C₂²×C₃×C₄	—	5	2	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	2	no	—
123	C₃×C₄⋊C₄	—	5	1	no	—
124	C₃×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
125	C₃×C₄⋊C₄	C₂×D₅	1	2	yes	—
126	C₃×C₂²⋊C₄	—	5	2	no	—
127	C₃×C₂²⋊C₄	—	5	2	no	—
128	C₃×C₂²⋊C₄	—	5	2	no	—
129	C₃×C₂²⋊C₄	—	5	2	no	—
130	C₃×C₂²⋊C₄	—	5	2	no	—
131	C₃×C₄²	—	5	2	no	—
132	C₃×C₄²	—	5	1	no	—
133	C₃×C₄²	—	5	1	no	—
134	C₃×C₄²	C₂×D₅	1	2	yes	—
135	C₃×C₄²	C₂×D₅	1	1	yes	characteristic
136	C₂²×D₅	C₂³×C₃	1	2	yes	—
137	C₅×Q₈	—	2	2	no	—
138	C₂×C₄×C₅	C₂³×C₃	1	2	yes	—
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	1	yes	characteristic
142	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
143	C₂×Dic₅	C₂³×C₃	1	1	yes	characteristic
144	C₂×Dic₅	C₂³×C₃	1	1	yes	characteristic
145	C₂×Dic₅	C₂³×C₃	1	1	yes	characteristic
146	C₂×Dic₅	C₂³×C₃	1	2	yes	—
147	C₂×Dic₅	C₂³×C₃	1	1	yes	characteristic
148	D₂₀	—	2	2	no	—
149	C₄×D₅	—	2	4	no	—
150	Dic₁₀	—	2	2	no	—
151	Dic₁₀	—	2	2	no	—
152	C₄⋊Q₈	—	5	1	no	—
153	#126	—	5	2	no	—
154	#126	—	5	2	no	—
155	#124	—	5	1	no	—
156	#124	—	5	1	no	—
157	C₂²⋊Q₈	—	5	2	no	—
158	C₄×Q₈	—	5	1	no	—
159	C₄×Q₈	—	5	1	no	—
160	C₄×Q₈	C₃×D₅	1	1	yes	characteristic, complemented
161	C₄×D₄	—	5	1	no	—
162	#117	—	5	2	no	—
163	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
164	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
165	C₂×C₃×C₅	2₋¹⁺⁴	1	1	yes	characteristic
166	C₃×D₅	—	4	2	no	—
167	C₂³×C₃	—	5	2	no	—
168	C₃×Q₈	—	2	2	no	—
169	C₃×Q₈	—	10	2	no	—
170	C₃×Q₈	—	10	2	no	—
171	C₃×D₄	—	10	2	no	—
172	C₂×C₃×C₄	—	5	1	no	—
173	C₂×C₃×C₄	—	5	1	no	—
174	C₂×C₃×C₄	—	5	1	no	—
175	C₂×C₃×C₄	—	5	2	no	—
176	C₂×C₃×C₄	—	5	1	no	—
177	C₂×C₃×C₄	—	10	4	no	—
178	C₂×C₃×C₄	C₂²×D₅	1	2	yes	—
179	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
180	C₂×C₃×C₄	C₂²×D₅	1	2	yes	—
181	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
182	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
183	C₂²×C₅	C₂⁴×C₃	1	1	yes	characteristic
184	C₂×D₅	—	2	2	no	—
185	C₂×D₅	—	2	2	no	—
186	C₂×D₅	—	2	2	no	—
187	C₄×C₅	—	2	1	no	—
188	C₄×C₅	—	2	2	no	—
189	C₄×C₅	—	2	2	no	—
190	C₄×C₅	—	2	1	no	—
191	C₄×C₅	C₃×#41	1	2	yes	—
192	Dic₅	—	2	2	no	—
193	Dic₅	—	2	1	no	—
194	Dic₅	—	2	1	no	—
195	Dic₅	—	2	1	no	—
196	Dic₅	C₃×#41	1	2	yes	—
197	C₂×Q₈	—	5	1	no	—
198	C₂×Q₈	—	5	1	no	—
199	C₂×Q₈	C₂×C₃×D₅	1	1	yes	characteristic
200	C₂×D₄	—	5	1	no	—
201	C₂²×C₄	—	5	2	no	—
202	C₄⋊C₄	—	5	1	no	—
203	C₄⋊C₄	—	5	2	no	—
204	C₄⋊C₄	—	5	2	no	—
205	C₄⋊C₄	—	5	1	no	—
206	C₄⋊C₄	—	5	1	no	—
207	C₄⋊C₄	—	5	2	no	—
208	C₄⋊C₄	C₂×C₃×D₅	1	1	yes	characteristic
209	C₄⋊C₄	C₂×C₃×D₅	1	2	yes	—
210	C₂²⋊C₄	—	5	2	no	—
211	C₂²⋊C₄	—	5	2	no	—
212	C₂²⋊C₄	—	5	2	no	—
213	C₂²⋊C₄	—	5	2	no	—
214	C₂²⋊C₄	—	5	2	no	—
215	C₄²	—	5	2	no	—
216	C₄²	—	5	1	no	—
217	C₄²	—	5	1	no	—
218	C₄²	C₂×C₃×D₅	1	2	yes	—
219	C₄²	C₂×C₃×D₅	1	1	yes	characteristic
220	C₃×C₅	#556	1	1	yes	characteristic, complemented, Hall
221	C₂²×C₃	—	10	2	no	—
222	C₂²×C₃	—	10	2	no	—
223	C₂²×C₃	—	10	2	no	—
224	C₂²×C₃	C₂³×D₅	1	1	yes	characteristic
225	C₃×C₄	—	2	2	no	—
226	C₃×C₄	—	2	1	no	—
227	C₃×C₄	—	2	2	no	—
228	C₃×C₄	—	2	1	no	—
229	C₃×C₄	—	5	2	no	—
230	C₃×C₄	—	10	2	no	—
231	C₃×C₄	—	10	1	no	—
232	C₃×C₄	—	10	1	no	—
233	C₃×C₄	—	10	1	no	—
234	C₃×C₄	#709	1	2	yes	—
235	C₂×C₅	C₂×C₃×#41	1	1	yes	characteristic
236	C₂×C₅	C₂×C₃×#41	1	1	yes	characteristic
237	C₂×C₅	C₃×2₋¹⁺⁴	1	1	yes	characteristic
238	D₅	—	4	2	no	—
239	C₂³	—	5	2	no	—
240	Q₈	—	2	2	no	—
241	Q₈	—	10	2	no	—
242	Q₈	—	10	2	no	—
243	D₄	—	10	2	no	—
244	C₂×C₄	—	5	1	no	—
245	C₂×C₄	—	5	1	no	—
246	C₂×C₄	—	5	1	no	—
247	C₂×C₄	—	5	2	no	—
248	C₂×C₄	—	5	1	no	—
249	C₂×C₄	—	10	4	no	—
250	C₂×C₄	C₂²×C₃×D₅	1	2	yes	—
251	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
252	C₂×C₄	C₂²×C₃×D₅	1	2	yes	—
253	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
254	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
255	C₂×C₃	—	20	2	no	—
256	C₂×C₃	C₂×#709	1	1	yes	characteristic
257	C₂×C₃	#4147	1	1	yes	characteristic
258	C₂×C₃	D₅×#41	1	1	yes	characteristic
259	C₅	C₃×#556	1	1	yes	characteristic, complemented, Sylow
260	C₂²	—	10	2	no	—
261	C₂²	—	10	2	no	—
262	C₂²	—	10	2	no	—
263	C₂²	C₂³×C₃×D₅	1	1	yes	characteristic
264	C₄	—	2	2	no	—
265	C₄	—	2	1	no	—
266	C₄	—	2	2	no	—
267	C₄	—	2	1	no	—
268	C₄	—	5	2	no	—
269	C₄	—	10	2	no	—
270	C₄	—	10	1	no	—
271	C₄	—	10	1	no	—
272	C₄	—	10	1	no	—
273	C₄	C₃×#709	1	2	yes	—
274	C₃	#65902	1	1	yes	characteristic, direct product factor, Sylow
275	C₂	—	20	2	no	—
276	C₂	C₂×C₃×#709	1	1	yes	characteristic
277	C₂	C₃×#4147	1	1	yes	characteristic
278	C₂	C₃×D₅×#41	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×C₄×C₅×Q₈	C₂
socle	C₂²×C₃×C₅	C₂⁴
prime radical	C₂×C₃×D₂₀	C₂²
hypercenter	C₃×C₄×Q₈	D₅
hypocenter	C₅	C₃×#556
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ 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
4. C₃×#4044	C₂	semidirect
5. C₃×#4042	C₂	semidirect
7. C₃×#4033	C₂	semidirect
8. C₃×#4022	C₂	semidirect
10. C₃×#4018	C₂	semidirect
11. C₃×C₄×Dic₁₀	C₂	semidirect
12. #65902	C₃	direct (N × H)
37. C₄×C₅×Q₈	C₂×C₃	semidirect
39. #4090	C₂×C₃	semidirect
40. #4044	C₂×C₃	semidirect
41. #4042	C₂×C₃	semidirect
43. #4033	C₂×C₃	semidirect
44. #4022	C₂×C₃	semidirect
46. #4018	C₂×C₃	semidirect
47. C₄×Dic₁₀	C₂×C₃	semidirect
72. C₃×C₄×Q₈	D₅	semidirect
160. C₄×Q₈	C₃×D₅	semidirect
220. C₃×C₅	#556	semidirect
259. C₅	C₃×#556	semidirect
274. C₃	#65902	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×#4093	2
C₃×#4090	2
C₃×#4044	2
C₃×#4042	2
C₃×#4039	2
C₃×#4033	2
C₃×#4022	2
C₃×#4021	2
C₃×#4018	2
C₃×C₄×Dic₁₀	2
#65902	3
C₃×#556	5

Subgroups — C3×#65902 #11755579

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#65902 #11755579