Subgroups — C₃×#65959 #11755636

Order 960.

384 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	C₃×C₅×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₃×Q₈×Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#4090	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#4072	C₂	1	1	yes	maximal, characteristic
5	C₃×#4070	C₂	1	1	yes	maximal, characteristic
6	C₃×#4036	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#4036	C₂	1	2	yes	maximal, complemented
8	C₃×#4035	C₂	1	2	yes	maximal, complemented
9	C₃×#4033	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃×#4025	C₂	1	2	yes	maximal
11	C₃×#4025	C₂	1	2	yes	maximal
12	#65959	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
13	C₂×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic
14	C₂²×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
15	C₂×C₃×Dic₁₀	C₂²	1	1	yes	characteristic
16	C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
17	C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
18	C₃×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
19	C₃×C₂²⋊Dic₅	C₂²	1	2	yes	—
20	C₃×C₂²⋊Dic₅	C₂²	1	1	yes	characteristic
21	C₃×C₂²⋊Dic₅	C₂²	1	1	yes	characteristic
22	C₃×C₄⋊Dic₅	C₂²	1	2	yes	—
23	C₃×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
24	C₃×C₄⋊Dic₅	C₂²	1	2	yes	—
25	C₃×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
26	C₃×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
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	2	yes	—
34	C₃×C₄×Dic₅	C₂²	1	1	yes	characteristic
35	C₃×C₄×Dic₅	C₂²	1	2	yes	—
36	C₃×#541	—	5	1	no	maximal, Hall
37	C₅×C₂²⋊Q₈	C₂×C₃	1	1	yes	characteristic
38	Q₈×Dic₅	C₂×C₃	1	1	yes	characteristic, complemented
39	#4090	C₂×C₃	1	1	yes	characteristic, complemented
40	#4072	C₂×C₃	1	1	yes	characteristic
41	#4070	C₂×C₃	1	1	yes	characteristic
42	#4036	C₂×C₃	1	2	yes	complemented
43	#4036	C₂×C₃	1	1	yes	characteristic, complemented
44	#4035	C₂×C₃	1	2	yes	complemented
45	#4033	C₂×C₃	1	1	yes	characteristic, complemented
46	#4025	C₂×C₃	1	2	yes	—
47	#4025	C₂×C₃	1	2	yes	—
48	C₂³×C₃×C₅	C₂³	1	1	yes	characteristic
49	C₃×C₅×Q₈	—	2	2	no	—
50	C₂×C₃×C₄×C₅	—	2	2	no	—
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	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	C₂×C₃×Dic₅	C₂³	1	2	yes	—
61	C₃×Dic₁₀	—	2	2	no	—
62	C₃×C₄⋊Q₈	—	5	1	no	—
63	C₃×#126	—	5	2	no	—
64	C₃×#126	—	5	2	no	—
65	C₃×#125	—	5	2	no	—
66	C₃×#125	—	5	1	no	—
67	C₃×#125	—	5	2	no	—
68	C₃×C₂²⋊Q₈	—	5	1	no	—
69	C₃×C₂²⋊Q₈	D₅	1	1	yes	characteristic
70	C₃×C₄×Q₈	—	5	1	no	—
71	C₃×C₄×Q₈	—	5	1	no	—
72	C₃×#117	—	5	1	no	—
73	C₂×C₅×Q₈	C₂²×C₃	1	1	yes	characteristic
74	C₂²×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
75	C₂×Dic₁₀	C₂²×C₃	1	1	yes	characteristic
76	C₅×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
77	C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
78	C₅×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
79	C₂²⋊Dic₅	C₂²×C₃	1	2	yes	—
80	C₂²⋊Dic₅	C₂²×C₃	1	1	yes	characteristic
81	C₂²⋊Dic₅	C₂²×C₃	1	1	yes	characteristic
82	C₄⋊Dic₅	C₂²×C₃	1	2	yes	—
83	C₄⋊Dic₅	C₂²×C₃	1	2	yes	—
84	C₄⋊Dic₅	C₂²×C₃	1	1	yes	characteristic
85	C₄⋊Dic₅	C₂²×C₃	1	1	yes	characteristic
86	C₄⋊Dic₅	C₂²×C₃	1	1	yes	characteristic
87	#683	C₂²×C₃	1	2	yes	—
88	#683	C₂²×C₃	1	2	yes	—
89	#683	C₂²×C₃	1	2	yes	—
90	#683	C₂²×C₃	1	2	yes	—
91	#683	C₂²×C₃	1	2	yes	—
92	C₄×Dic₅	C₂²×C₃	1	2	yes	—
93	C₄×Dic₅	C₂²×C₃	1	2	yes	—
94	C₄×Dic₅	C₂²×C₃	1	1	yes	characteristic
95	C₄×Dic₅	C₂²×C₃	1	1	yes	characteristic
96	#541	—	5	1	no	Sylow
97	C₂²×C₃×C₅	—	2	1	no	—
98	C₂²×C₃×C₅	—	2	1	no	—
99	C₂²×C₃×C₅	—	2	1	no	—
100	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
101	C₃×C₄×C₅	—	2	1	no	—
102	C₃×C₄×C₅	—	2	2	no	—
103	C₃×C₄×C₅	—	2	2	no	—
104	C₃×C₄×C₅	#41	1	2	yes	—
105	C₃×Dic₅	—	2	2	no	—
106	C₃×Dic₅	—	2	1	no	—
107	C₃×Dic₅	—	2	1	no	—
108	C₃×Dic₅	—	2	2	no	—
109	C₃×Dic₅	—	2	1	no	—
110	C₃×Dic₅	—	2	1	no	—
111	C₂×C₃×Q₈	—	5	1	no	—
112	C₂×C₃×Q₈	C₂×D₅	1	1	yes	characteristic
113	C₂²×C₃×C₄	C₂×D₅	1	1	yes	characteristic
114	C₃×C₄⋊C₄	—	5	2	no	—
115	C₃×C₄⋊C₄	—	5	2	no	—
116	C₃×C₄⋊C₄	—	5	2	no	—
117	C₃×C₄⋊C₄	—	5	2	no	—
118	C₃×C₄⋊C₄	—	5	2	no	—
119	C₃×C₄⋊C₄	—	5	1	no	—
120	C₃×C₄⋊C₄	—	5	1	no	—
121	C₃×C₄⋊C₄	—	5	1	no	—
122	C₃×C₄⋊C₄	—	5	2	no	—
123	C₃×C₄⋊C₄	—	5	2	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	1	no	—
127	C₃×C₂²⋊C₄	—	5	1	no	—
128	C₃×C₂²⋊C₄	—	5	2	no	—
129	C₃×C₂²⋊C₄	C₂×D₅	1	2	yes	—
130	C₃×C₄²	—	5	1	no	—
131	C₃×C₄²	—	5	1	no	—
132	C₃×C₄²	—	5	2	no	—
133	C₃×C₄²	—	5	2	no	—
134	C₂³×C₅	C₂³×C₃	1	1	yes	characteristic
135	C₅×Q₈	—	2	2	no	—
136	C₂×C₄×C₅	—	2	2	no	—
137	C₂×C₄×C₅	C₂³×C₃	1	2	yes	—
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	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	1	yes	characteristic
144	C₂×Dic₅	C₂³×C₃	1	2	yes	—
145	C₂×Dic₅	C₂³×C₃	1	2	yes	—
146	C₂×Dic₅	C₂³×C₃	1	1	yes	characteristic
147	Dic₁₀	—	2	2	no	—
148	C₄⋊Q₈	—	5	1	no	—
149	#126	—	5	2	no	—
150	#126	—	5	2	no	—
151	#125	—	5	2	no	—
152	#125	—	5	1	no	—
153	#125	—	5	2	no	—
154	C₂²⋊Q₈	—	5	1	no	—
155	C₂²⋊Q₈	C₃×D₅	1	1	yes	characteristic
156	C₄×Q₈	—	5	1	no	—
157	C₄×Q₈	—	5	1	no	—
158	#117	—	5	1	no	—
159	C₂×C₃×C₅	—	4	1	no	—
160	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
161	C₂×C₃×C₅	2₋¹⁺⁴	1	1	yes	characteristic
162	C₂×C₃×C₅	2₋¹⁺⁴	1	1	yes	characteristic
163	C₂³×C₃	C₂²×D₅	1	1	yes	characteristic
164	C₃×Q₈	—	2	2	no	—
165	C₃×Q₈	—	10	2	no	—
166	C₂×C₃×C₄	—	2	2	no	—
167	C₂×C₃×C₄	—	5	1	no	—
168	C₂×C₃×C₄	—	5	1	no	—
169	C₂×C₃×C₄	—	5	1	no	—
170	C₂×C₃×C₄	—	5	2	no	—
171	C₂×C₃×C₄	—	5	1	no	—
172	C₂×C₃×C₄	—	5	2	no	—
173	C₂×C₃×C₄	C₂²×D₅	1	2	yes	—
174	C₂×C₃×C₄	C₂²×D₅	1	2	yes	—
175	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
176	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
177	C₂²×C₅	—	2	1	no	—
178	C₂²×C₅	—	2	1	no	—
179	C₂²×C₅	—	2	1	no	—
180	C₂²×C₅	C₂⁴×C₃	1	1	yes	characteristic
181	C₄×C₅	—	2	1	no	—
182	C₄×C₅	—	2	2	no	—
183	C₄×C₅	—	2	2	no	—
184	C₄×C₅	C₃×#41	1	2	yes	—
185	Dic₅	—	2	2	no	—
186	Dic₅	—	2	1	no	—
187	Dic₅	—	2	2	no	—
188	Dic₅	—	2	1	no	—
189	Dic₅	—	2	1	no	—
190	Dic₅	—	2	1	no	—
191	C₂×Q₈	—	5	1	no	—
192	C₂×Q₈	C₂×C₃×D₅	1	1	yes	characteristic
193	C₂²×C₄	C₂×C₃×D₅	1	1	yes	characteristic
194	C₄⋊C₄	—	5	1	no	—
195	C₄⋊C₄	—	5	1	no	—
196	C₄⋊C₄	—	5	1	no	—
197	C₄⋊C₄	—	5	2	no	—
198	C₄⋊C₄	—	5	2	no	—
199	C₄⋊C₄	—	5	2	no	—
200	C₄⋊C₄	—	5	2	no	—
201	C₄⋊C₄	—	5	2	no	—
202	C₄⋊C₄	—	5	2	no	—
203	C₄⋊C₄	—	5	2	no	—
204	C₄⋊C₄	C₂×C₃×D₅	1	1	yes	characteristic
205	C₄⋊C₄	C₂×C₃×D₅	1	2	yes	—
206	C₂²⋊C₄	—	5	1	no	—
207	C₂²⋊C₄	—	5	1	no	—
208	C₂²⋊C₄	—	5	2	no	—
209	C₂²⋊C₄	C₂×C₃×D₅	1	2	yes	—
210	C₄²	—	5	1	no	—
211	C₄²	—	5	1	no	—
212	C₄²	—	5	2	no	—
213	C₄²	—	5	2	no	—
214	C₃×C₅	#541	1	1	yes	characteristic, complemented, Hall
215	C₂²×C₃	—	2	1	no	—
216	C₂²×C₃	—	2	1	no	—
217	C₂²×C₃	—	2	1	no	—
218	C₂²×C₃	C₂³×D₅	1	1	yes	characteristic
219	C₃×C₄	—	2	1	no	—
220	C₃×C₄	—	2	2	no	—
221	C₃×C₄	—	2	2	no	—
222	C₃×C₄	—	10	2	no	—
223	C₃×C₄	—	10	1	no	—
224	C₃×C₄	—	10	1	no	—
225	C₃×C₄	—	10	2	no	—
226	C₃×C₄	—	10	1	no	—
227	C₃×C₄	—	10	1	no	—
228	C₃×C₄	#711	1	2	yes	—
229	C₂×C₅	—	4	1	no	—
230	C₂×C₅	C₂×C₃×#41	1	1	yes	characteristic
231	C₂×C₅	C₃×2₋¹⁺⁴	1	1	yes	characteristic
232	C₂×C₅	C₃×2₋¹⁺⁴	1	1	yes	characteristic
233	C₂³	C₂²×C₃×D₅	1	1	yes	characteristic
234	Q₈	—	2	2	no	—
235	Q₈	—	10	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	5	1	no	—
238	C₂×C₄	—	5	1	no	—
239	C₂×C₄	—	5	1	no	—
240	C₂×C₄	—	5	2	no	—
241	C₂×C₄	—	5	1	no	—
242	C₂×C₄	—	5	2	no	—
243	C₂×C₄	C₂²×C₃×D₅	1	2	yes	—
244	C₂×C₄	C₂²×C₃×D₅	1	2	yes	—
245	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
246	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
247	C₂×C₃	—	4	1	no	—
248	C₂×C₃	C₂×#711	1	1	yes	characteristic
249	C₂×C₃	#4147	1	1	yes	characteristic
250	C₂×C₃	#4150	1	1	yes	characteristic
251	C₅	C₃×#541	1	1	yes	characteristic, complemented, Sylow
252	C₂²	—	2	1	no	—
253	C₂²	—	2	1	no	—
254	C₂²	—	2	1	no	—
255	C₂²	C₂³×C₃×D₅	1	1	yes	characteristic
256	C₄	—	2	1	no	—
257	C₄	—	2	2	no	—
258	C₄	—	2	2	no	—
259	C₄	—	10	2	no	—
260	C₄	—	10	1	no	—
261	C₄	—	10	1	no	—
262	C₄	—	10	2	no	—
263	C₄	—	10	1	no	—
264	C₄	—	10	1	no	—
265	C₄	C₃×#711	1	2	yes	—
266	C₃	#65959	1	1	yes	characteristic, direct product factor, Sylow
267	C₂	—	4	1	no	—
268	C₂	C₂×C₃×#711	1	1	yes	characteristic
269	C₂	C₃×#4147	1	1	yes	characteristic
270	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×C₂²⋊Q₈	C₂
socle	C₂²×C₃×C₅	C₂⁴
prime radical	C₂³×C₃×C₅	C₂³
hypercenter	C₃×C₂²⋊Q₈	D₅
hypocenter	C₅	C₃×#541
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₂²⋊Q₈

Complements

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

N	H ≅ G/N	type
2. C₃×Q₈×Dic₅	C₂	semidirect
3. C₃×#4090	C₂	semidirect
6. C₃×#4036	C₂	semidirect
7. C₃×#4036	C₂	semidirect
8. C₃×#4035	C₂	semidirect
9. C₃×#4033	C₂	semidirect
12. #65959	C₃	direct (N × H)
38. Q₈×Dic₅	C₂×C₃	semidirect
39. #4090	C₂×C₃	semidirect
42. #4036	C₂×C₃	semidirect
43. #4036	C₂×C₃	semidirect
44. #4035	C₂×C₃	semidirect
45. #4033	C₂×C₃	semidirect
214. C₃×C₅	#541	semidirect
251. C₅	C₃×#541	semidirect
266. C₃	#65959	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×C₂²⋊Q₈	2
C₃×Q₈×Dic₅	2
C₃×#4090	2
C₃×#4072	2
C₃×#4070	2
C₃×#4036	2
C₃×#4035	2
C₃×#4033	2
C₃×#4025	2
#65959	3
C₃×#541	5

Subgroups — C3×#65959 #11755636

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#65959 #11755636