Subgroups — C₃×#65975 #11755652

Order 960.

424 conjugacy classes of subgroups.

The subgroup lattice has 292 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, complemented
2	C₃×#4092	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×Q₈×Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₄×C₂²⋊D₅	C₂	1	1	yes	maximal, characteristic
5	C₃×#4072	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#4044	C₂	1	2	yes	maximal, complemented
7	C₃×#4038	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#4034	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃×#4033	C₂	1	2	yes	maximal, complemented
10	C₃×#4031	C₂	1	2	yes	maximal
11	C₃×#4025	C₂	1	2	yes	maximal, complemented
12	#65975	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×C₂²⋊D₅	C₂²	1	1	yes	characteristic
16	C₂×C₃×C₄×D₅	C₂²	1	1	yes	characteristic
17	C₂×C₃×Dic₁₀	C₂²	1	2	yes	—
18	C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
19	C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
20	C₃×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
21	C₃×C₂²⋊Dic₅	C₂²	1	1	yes	characteristic
22	C₃×C₂²⋊Dic₅	C₂²	1	2	yes	—
23	C₃×#685	C₂²	1	2	yes	—
24	C₃×#685	C₂²	1	2	yes	—
25	C₃×#685	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	1	yes	characteristic
29	C₃×#683	C₂²	1	2	yes	—
30	C₃×#683	C₂²	1	2	yes	—
31	C₃×#683	C₂²	1	1	yes	characteristic
32	C₃×C₄×Dic₅	C₂²	1	2	yes	—
33	C₃×C₄×Dic₅	C₂²	1	1	yes	characteristic
34	C₃×C₄×Dic₅	C₂²	1	2	yes	—
35	C₃×C₄×Dic₅	C₂²	1	2	yes	—
36	C₃×#556	—	5	1	no	maximal, Hall
37	C₅×C₂²⋊Q₈	C₂×C₃	1	1	yes	characteristic, complemented
38	#4092	C₂×C₃	1	1	yes	characteristic, complemented
39	Q₈×Dic₅	C₂×C₃	1	1	yes	characteristic, complemented
40	C₄×C₂²⋊D₅	C₂×C₃	1	1	yes	characteristic
41	#4072	C₂×C₃	1	1	yes	characteristic, complemented
42	#4044	C₂×C₃	1	2	yes	complemented
43	#4038	C₂×C₃	1	1	yes	characteristic, complemented
44	#4034	C₂×C₃	1	1	yes	characteristic, complemented
45	#4033	C₂×C₃	1	2	yes	complemented
46	#4031	C₂×C₃	1	2	yes	—
47	#4025	C₂×C₃	1	2	yes	complemented
48	C₂³×C₃×C₅	C₂³	1	1	yes	characteristic
49	C₂²×C₃×D₅	C₂³	1	1	yes	characteristic
50	C₃×C₅×Q₈	—	2	2	no	—
51	C₂×C₃×C₄×C₅	—	2	2	no	—
52	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
53	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
54	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
55	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
56	C₃×C₂²⋊D₅	—	2	2	no	—
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	C₂×C₃×Dic₅	C₂³	1	2	yes	—
61	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
62	C₃×C₄×D₅	—	2	2	no	—
63	C₃×Dic₁₀	—	2	4	no	—
64	C₃×C₄⋊Q₈	—	5	1	no	—
65	C₃×#126	—	5	2	no	—
66	C₃×#126	—	5	2	no	—
67	C₃×#124	—	5	2	no	—
68	C₃×C₂²⋊Q₈	—	5	1	no	—
69	C₃×C₂²⋊Q₈	D₅	1	1	yes	characteristic, complemented
70	C₃×C₄×Q₈	—	5	1	no	—
71	C₃×C₄×Q₈	—	5	2	no	—
72	C₃×C₄×D₄	—	5	1	no	—
73	C₃×#117	—	5	1	no	—
74	C₃×#117	—	5	1	no	—
75	C₂×C₅×Q₈	C₂²×C₃	1	1	yes	characteristic
76	C₂²×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
77	C₂×C₂²⋊D₅	C₂²×C₃	1	1	yes	characteristic
78	C₂×C₄×D₅	C₂²×C₃	1	1	yes	characteristic
79	C₂×Dic₁₀	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	C₅×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
83	C₂²⋊Dic₅	C₂²×C₃	1	1	yes	characteristic
84	C₂²⋊Dic₅	C₂²×C₃	1	2	yes	—
85	#685	C₂²×C₃	1	2	yes	—
86	#685	C₂²×C₃	1	2	yes	—
87	#685	C₂²×C₃	1	1	yes	characteristic
88	C₄⋊Dic₅	C₂²×C₃	1	2	yes	—
89	C₄⋊Dic₅	C₂²×C₃	1	1	yes	characteristic
90	C₄⋊Dic₅	C₂²×C₃	1	1	yes	characteristic
91	#683	C₂²×C₃	1	2	yes	—
92	#683	C₂²×C₃	1	2	yes	—
93	#683	C₂²×C₃	1	1	yes	characteristic
94	C₄×Dic₅	C₂²×C₃	1	2	yes	—
95	C₄×Dic₅	C₂²×C₃	1	2	yes	—
96	C₄×Dic₅	C₂²×C₃	1	2	yes	—
97	C₄×Dic₅	C₂²×C₃	1	1	yes	characteristic
98	#556	—	5	1	no	Sylow
99	C₂²×C₃×C₅	—	2	1	no	—
100	C₂²×C₃×C₅	—	2	1	no	—
101	C₂²×C₃×C₅	—	2	1	no	—
102	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
103	C₂×C₃×D₅	—	2	1	no	—
104	C₂×C₃×D₅	—	2	1	no	—
105	C₂×C₃×D₅	—	2	1	no	—
106	C₃×C₄×C₅	—	2	2	no	—
107	C₃×C₄×C₅	—	2	1	no	—
108	C₃×C₄×C₅	—	2	2	no	—
109	C₃×C₄×C₅	#41	1	2	yes	—
110	C₃×Dic₅	—	2	2	no	—
111	C₃×Dic₅	—	2	1	no	—
112	C₃×Dic₅	—	2	2	no	—
113	C₃×Dic₅	—	2	1	no	—
114	C₃×Dic₅	#41	1	2	yes	—
115	C₂×C₃×Q₈	—	5	2	no	—
116	C₂×C₃×Q₈	C₂×D₅	1	1	yes	characteristic
117	C₂×C₃×D₄	—	5	1	no	—
118	C₂²×C₃×C₄	—	5	1	no	—
119	C₂²×C₃×C₄	C₂×D₅	1	1	yes	characteristic
120	C₃×C₄⋊C₄	—	5	1	no	—
121	C₃×C₄⋊C₄	—	5	2	no	—
122	C₃×C₄⋊C₄	—	5	2	no	—
123	C₃×C₄⋊C₄	—	5	1	no	—
124	C₃×C₄⋊C₄	—	5	1	no	—
125	C₃×C₄⋊C₄	—	5	2	no	—
126	C₃×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
127	C₃×C₄⋊C₄	C₂×D₅	1	2	yes	—
128	C₃×C₂²⋊C₄	—	5	2	no	—
129	C₃×C₂²⋊C₄	—	5	1	no	—
130	C₃×C₂²⋊C₄	—	5	2	no	—
131	C₃×C₂²⋊C₄	—	5	1	no	—
132	C₃×C₂²⋊C₄	—	5	2	no	—
133	C₃×C₂²⋊C₄	C₂×D₅	1	2	yes	—
134	C₃×C₄²	—	5	2	no	—
135	C₃×C₄²	—	5	2	no	—
136	C₃×C₄²	—	5	1	no	—
137	C₃×C₄²	—	5	2	no	—
138	C₂³×C₅	C₂³×C₃	1	1	yes	characteristic
139	C₂²×D₅	C₂³×C₃	1	1	yes	characteristic
140	C₅×Q₈	—	2	2	no	—
141	C₂×C₄×C₅	—	2	2	no	—
142	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
143	C₂×C₄×C₅	C₂³×C₃	1	2	yes	—
144	C₂×C₄×C₅	C₂³×C₃	1	2	yes	—
145	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
146	C₂²⋊D₅	—	2	2	no	—
147	C₂×Dic₅	C₂³×C₃	1	2	yes	—
148	C₂×Dic₅	C₂³×C₃	1	1	yes	characteristic
149	C₂×Dic₅	C₂³×C₃	1	1	yes	characteristic
150	C₂×Dic₅	C₂³×C₃	1	2	yes	—
151	C₂×Dic₅	C₂³×C₃	1	1	yes	characteristic
152	C₄×D₅	—	2	2	no	—
153	Dic₁₀	—	2	4	no	—
154	C₄⋊Q₈	—	5	1	no	—
155	#126	—	5	2	no	—
156	#126	—	5	2	no	—
157	#124	—	5	2	no	—
158	C₂²⋊Q₈	—	5	1	no	—
159	C₂²⋊Q₈	C₃×D₅	1	1	yes	characteristic, complemented
160	C₄×Q₈	—	5	1	no	—
161	C₄×Q₈	—	5	2	no	—
162	C₄×D₄	—	5	1	no	—
163	#117	—	5	1	no	—
164	#117	—	5	1	no	—
165	C₂×C₃×C₅	—	4	1	no	—
166	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
167	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
168	C₂×C₃×C₅	2₋¹⁺⁴	1	1	yes	characteristic
169	C₃×D₅	—	4	1	no	—
170	C₂³×C₃	—	5	1	no	—
171	C₂³×C₃	C₂²×D₅	1	1	yes	characteristic
172	C₃×Q₈	—	2	2	no	—
173	C₃×Q₈	—	10	4	no	—
174	C₃×D₄	—	10	2	no	—
175	C₂×C₃×C₄	—	2	2	no	—
176	C₂×C₃×C₄	—	5	1	no	—
177	C₂×C₃×C₄	—	5	2	no	—
178	C₂×C₃×C₄	—	5	1	no	—
179	C₂×C₃×C₄	—	5	1	no	—
180	C₂×C₃×C₄	—	5	2	no	—
181	C₂×C₃×C₄	—	10	2	no	—
182	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
183	C₂×C₃×C₄	C₂²×D₅	1	2	yes	—
184	C₂×C₃×C₄	C₂²×D₅	1	2	yes	—
185	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
186	C₂²×C₅	—	2	1	no	—
187	C₂²×C₅	—	2	1	no	—
188	C₂²×C₅	—	2	1	no	—
189	C₂²×C₅	C₂⁴×C₃	1	1	yes	characteristic
190	C₂×D₅	—	2	1	no	—
191	C₂×D₅	—	2	1	no	—
192	C₂×D₅	—	2	1	no	—
193	C₄×C₅	—	2	2	no	—
194	C₄×C₅	—	2	1	no	—
195	C₄×C₅	—	2	2	no	—
196	C₄×C₅	C₃×#41	1	2	yes	—
197	Dic₅	—	2	2	no	—
198	Dic₅	—	2	1	no	—
199	Dic₅	—	2	2	no	—
200	Dic₅	—	2	1	no	—
201	Dic₅	C₃×#41	1	2	yes	—
202	C₂×Q₈	—	5	2	no	—
203	C₂×Q₈	C₂×C₃×D₅	1	1	yes	characteristic
204	C₂×D₄	—	5	1	no	—
205	C₂²×C₄	—	5	1	no	—
206	C₂²×C₄	C₂×C₃×D₅	1	1	yes	characteristic
207	C₄⋊C₄	—	5	1	no	—
208	C₄⋊C₄	—	5	1	no	—
209	C₄⋊C₄	—	5	1	no	—
210	C₄⋊C₄	—	5	2	no	—
211	C₄⋊C₄	—	5	2	no	—
212	C₄⋊C₄	—	5	2	no	—
213	C₄⋊C₄	C₂×C₃×D₅	1	1	yes	characteristic
214	C₄⋊C₄	C₂×C₃×D₅	1	2	yes	—
215	C₂²⋊C₄	—	5	2	no	—
216	C₂²⋊C₄	—	5	1	no	—
217	C₂²⋊C₄	—	5	2	no	—
218	C₂²⋊C₄	—	5	1	no	—
219	C₂²⋊C₄	—	5	2	no	—
220	C₂²⋊C₄	C₂×C₃×D₅	1	2	yes	—
221	C₄²	—	5	2	no	—
222	C₄²	—	5	2	no	—
223	C₄²	—	5	1	no	—
224	C₄²	—	5	2	no	—
225	C₃×C₅	#556	1	1	yes	characteristic, complemented, Hall
226	C₂²×C₃	—	2	1	no	—
227	C₂²×C₃	—	2	1	no	—
228	C₂²×C₃	—	2	1	no	—
229	C₂²×C₃	—	10	1	no	—
230	C₂²×C₃	—	10	1	no	—
231	C₂²×C₃	—	10	1	no	—
232	C₂²×C₃	C₂³×D₅	1	1	yes	characteristic
233	C₃×C₄	—	2	1	no	—
234	C₃×C₄	—	2	2	no	—
235	C₃×C₄	—	2	2	no	—
236	C₃×C₄	—	5	2	no	—
237	C₃×C₄	—	10	1	no	—
238	C₃×C₄	—	10	2	no	—
239	C₃×C₄	—	10	1	no	—
240	C₃×C₄	—	10	2	no	—
241	C₃×C₄	#711	1	2	yes	—
242	C₂×C₅	—	4	1	no	—
243	C₂×C₅	C₂×C₃×#41	1	1	yes	characteristic
244	C₂×C₅	C₂×C₃×#41	1	1	yes	characteristic
245	C₂×C₅	C₃×2₋¹⁺⁴	1	1	yes	characteristic
246	D₅	—	4	1	no	—
247	C₂³	—	5	1	no	—
248	C₂³	C₂²×C₃×D₅	1	1	yes	characteristic
249	Q₈	—	2	2	no	—
250	Q₈	—	10	4	no	—
251	D₄	—	10	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	5	1	no	—
254	C₂×C₄	—	5	2	no	—
255	C₂×C₄	—	5	1	no	—
256	C₂×C₄	—	5	1	no	—
257	C₂×C₄	—	5	2	no	—
258	C₂×C₄	—	10	2	no	—
259	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
260	C₂×C₄	C₂²×C₃×D₅	1	2	yes	—
261	C₂×C₄	C₂²×C₃×D₅	1	2	yes	—
262	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
263	C₂×C₃	—	4	1	no	—
264	C₂×C₃	—	20	1	no	—
265	C₂×C₃	C₂×#711	1	1	yes	characteristic
266	C₂×C₃	#4147	1	1	yes	characteristic
267	C₂×C₃	D₅×#41	1	1	yes	characteristic
268	C₅	C₃×#556	1	1	yes	characteristic, complemented, Sylow
269	C₂²	—	2	1	no	—
270	C₂²	—	2	1	no	—
271	C₂²	—	2	1	no	—
272	C₂²	—	10	1	no	—
273	C₂²	—	10	1	no	—
274	C₂²	—	10	1	no	—
275	C₂²	C₂³×C₃×D₅	1	1	yes	characteristic
276	C₄	—	2	1	no	—
277	C₄	—	2	2	no	—
278	C₄	—	2	2	no	—
279	C₄	—	5	2	no	—
280	C₄	—	10	1	no	—
281	C₄	—	10	2	no	—
282	C₄	—	10	1	no	—
283	C₄	—	10	2	no	—
284	C₄	C₃×#711	1	2	yes	—
285	C₃	#65975	1	1	yes	characteristic, direct product factor, Sylow
286	C₂	—	4	1	no	—
287	C₂	—	20	1	no	—
288	C₂	C₂×C₃×#711	1	1	yes	characteristic
289	C₂	C₃×#4147	1	1	yes	characteristic
290	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×C₅×C₂²⋊Q₈	C₂
socle	C₂²×C₃×C₅	C₂⁴
prime radical	C₂×C₃×C₂²⋊D₅	C₂²
hypercenter	C₃×C₂²⋊Q₈	D₅
hypocenter	C₅	C₃×#556
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
1. C₃×C₅×C₂²⋊Q₈	C₂	semidirect
2. C₃×#4092	C₂	semidirect
3. C₃×Q₈×Dic₅	C₂	semidirect
5. C₃×#4072	C₂	semidirect
6. C₃×#4044	C₂	semidirect
7. C₃×#4038	C₂	semidirect
8. C₃×#4034	C₂	semidirect
9. C₃×#4033	C₂	semidirect
11. C₃×#4025	C₂	semidirect
12. #65975	C₃	direct (N × H)
37. C₅×C₂²⋊Q₈	C₂×C₃	semidirect
38. #4092	C₂×C₃	semidirect
39. Q₈×Dic₅	C₂×C₃	semidirect
41. #4072	C₂×C₃	semidirect
42. #4044	C₂×C₃	semidirect
43. #4038	C₂×C₃	semidirect
44. #4034	C₂×C₃	semidirect
45. #4033	C₂×C₃	semidirect
47. #4025	C₂×C₃	semidirect
69. C₃×C₂²⋊Q₈	D₅	semidirect
159. C₂²⋊Q₈	C₃×D₅	semidirect
225. C₃×C₅	#556	semidirect
268. C₅	C₃×#556	semidirect
285. C₃	#65975	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×#4092	2
C₃×Q₈×Dic₅	2
C₃×C₄×C₂²⋊D₅	2
C₃×#4072	2
C₃×#4044	2
C₃×#4038	2
C₃×#4034	2
C₃×#4033	2
C₃×#4031	2
C₃×#4025	2
#65975	3
C₃×#556	5

Subgroups — C3×#65975 #11755652

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#65975 #11755652