Subgroups — C₃×#65969 #11755646

Order 960.

456 conjugacy classes of subgroups.

The subgroup lattice has 300 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	C₂×C₃×C₄⋊Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#4043	C₂	1	2	yes	maximal, complemented
7	C₃×D₅×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#4035	C₂	1	2	yes	maximal, complemented
9	C₃×#4034	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃×#4027	C₂	1	2	yes	maximal
11	C₃×#4024	C₂	1	2	yes	maximal, complemented
12	#65969	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	C₂²×C₃×Dic₅	C₂²	1	2	yes	—
17	C₂×C₃×C₄×D₅	C₂²	1	1	yes	characteristic
18	C₂×C₃×C₄×D₅	C₂²	1	2	yes	—
19	C₂×C₃×Dic₁₀	C₂²	1	2	yes	—
20	C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
21	C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
22	C₃×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
23	C₃×C₂²⋊Dic₅	C₂²	1	1	yes	characteristic
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×C₄⋊Dic₅	C₂²	1	2	yes	—
30	C₃×#683	C₂²	1	2	yes	—
31	C₃×#683	C₂²	1	2	yes	—
32	C₃×#683	C₂²	1	2	yes	—
33	C₃×#683	C₂²	1	1	yes	characteristic
34	C₃×C₄×Dic₅	C₂²	1	2	yes	—
35	C₃×C₄×Dic₅	C₂²	1	1	yes	characteristic
36	C₃×#554	—	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	C₂×C₄⋊Dic₅	C₂×C₃	1	1	yes	characteristic, complemented
42	#4043	C₂×C₃	1	2	yes	complemented
43	D₅×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic, complemented
44	#4035	C₂×C₃	1	2	yes	complemented
45	#4034	C₂×C₃	1	1	yes	characteristic, complemented
46	#4027	C₂×C₃	1	2	yes	—
47	#4024	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	2	yes	—
53	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
54	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
55	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
56	C₃×C₂²⋊D₅	Q₈	1	4	yes	complemented
57	C₂×C₃×Dic₅	—	4	2	no	—
58	C₂×C₃×Dic₅	C₂³	1	2	yes	—
59	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
60	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
61	C₂×C₃×Dic₅	C₂³	1	2	yes	—
62	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
63	C₃×C₄×D₅	—	2	2	no	—
64	C₃×C₄×D₅	—	4	2	no	—
65	C₃×Dic₁₀	—	4	2	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	2	no	—
71	C₃×C₂²⋊Q₈	D₅	1	1	yes	characteristic, complemented
72	C₃×C₄×Q₈	—	5	1	no	—
73	C₃×C₄×D₄	—	5	1	no	—
74	C₃×C₄×D₄	—	5	2	no	—
75	C₂×C₃×C₄⋊C₄	—	5	1	no	—
76	C₂×C₃×C₄⋊C₄	—	5	1	no	—
77	C₂×C₅×Q₈	C₂²×C₃	1	1	yes	characteristic
78	C₂²×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
79	C₂×C₂²⋊D₅	C₂²×C₃	1	1	yes	characteristic
80	C₂²×Dic₅	C₂²×C₃	1	2	yes	—
81	C₂×C₄×D₅	C₂²×C₃	1	1	yes	characteristic
82	C₂×C₄×D₅	C₂²×C₃	1	2	yes	—
83	C₂×Dic₁₀	C₂²×C₃	1	2	yes	—
84	C₅×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
85	C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
86	C₅×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
87	C₂²⋊Dic₅	C₂²×C₃	1	1	yes	characteristic
88	#685	C₂²×C₃	1	2	yes	—
89	#685	C₂²×C₃	1	1	yes	characteristic
90	C₄⋊Dic₅	C₂²×C₃	1	2	yes	—
91	C₄⋊Dic₅	C₂²×C₃	1	1	yes	characteristic
92	C₄⋊Dic₅	C₂²×C₃	1	2	yes	—
93	C₄⋊Dic₅	C₂²×C₃	1	1	yes	characteristic
94	#683	C₂²×C₃	1	2	yes	—
95	#683	C₂²×C₃	1	2	yes	—
96	#683	C₂²×C₃	1	2	yes	—
97	#683	C₂²×C₃	1	1	yes	characteristic
98	C₄×Dic₅	C₂²×C₃	1	2	yes	—
99	C₄×Dic₅	C₂²×C₃	1	1	yes	characteristic
100	#554	—	5	1	no	Sylow
101	C₂²×C₃×C₅	—	2	1	no	—
102	C₂²×C₃×C₅	—	2	1	no	—
103	C₂²×C₃×C₅	C₂×Q₈	1	2	yes	—
104	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
105	C₂×C₃×D₅	—	2	1	no	—
106	C₂×C₃×D₅	—	2	1	no	—
107	C₂×C₃×D₅	C₂×Q₈	1	2	yes	—
108	C₃×C₄×C₅	—	2	1	no	—
109	C₃×C₄×C₅	—	2	2	no	—
110	C₃×C₄×C₅	—	2	2	no	—
111	C₃×C₄×C₅	#41	1	2	yes	—
112	C₃×Dic₅	—	2	1	no	—
113	C₃×Dic₅	—	2	1	no	—
114	C₃×Dic₅	—	2	2	no	—
115	C₃×Dic₅	—	2	2	no	—
116	C₃×Dic₅	C₂×Q₈	1	2	yes	—
117	C₂×C₃×Q₈	—	5	2	no	—
118	C₂×C₃×Q₈	C₂×D₅	1	1	yes	characteristic
119	C₂×C₃×D₄	—	5	1	no	—
120	C₂²×C₃×C₄	—	5	2	no	—
121	C₂²×C₃×C₄	—	5	1	no	—
122	C₂²×C₃×C₄	—	5	2	no	—
123	C₂²×C₃×C₄	C₂×D₅	1	1	yes	characteristic
124	C₃×C₄⋊C₄	—	5	2	no	—
125	C₃×C₄⋊C₄	—	5	1	no	—
126	C₃×C₄⋊C₄	—	5	2	no	—
127	C₃×C₄⋊C₄	—	5	2	no	—
128	C₃×C₄⋊C₄	—	5	1	no	—
129	C₃×C₄⋊C₄	—	5	2	no	—
130	C₃×C₄⋊C₄	—	5	1	no	—
131	C₃×C₄⋊C₄	—	5	2	no	—
132	C₃×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
133	C₃×C₄⋊C₄	C₂×D₅	1	2	yes	—
134	C₃×C₂²⋊C₄	—	5	1	no	—
135	C₃×C₂²⋊C₄	—	5	1	no	—
136	C₃×C₂²⋊C₄	—	5	2	no	—
137	C₃×C₂²⋊C₄	C₂×D₅	1	2	yes	—
138	C₃×C₄²	—	5	2	no	—
139	C₃×C₄²	—	5	1	no	—
140	C₂³×C₅	C₂³×C₃	1	1	yes	characteristic
141	C₂²×D₅	C₂³×C₃	1	1	yes	characteristic
142	C₅×Q₈	—	2	2	no	—
143	C₂×C₄×C₅	—	2	2	no	—
144	C₂×C₄×C₅	C₂³×C₃	1	2	yes	—
145	C₂×C₄×C₅	C₂³×C₃	1	2	yes	—
146	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
147	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
148	C₂²⋊D₅	C₃×Q₈	1	4	yes	complemented
149	C₂×Dic₅	—	4	2	no	—
150	C₂×Dic₅	C₂³×C₃	1	2	yes	—
151	C₂×Dic₅	C₂³×C₃	1	1	yes	characteristic
152	C₂×Dic₅	C₂³×C₃	1	1	yes	characteristic
153	C₂×Dic₅	C₂³×C₃	1	2	yes	—
154	C₂×Dic₅	C₂³×C₃	1	1	yes	characteristic
155	C₄×D₅	—	2	2	no	—
156	C₄×D₅	—	4	2	no	—
157	Dic₁₀	—	4	2	no	—
158	C₄⋊Q₈	—	5	1	no	—
159	#125	—	5	2	no	—
160	C₂²⋊Q₈	—	5	1	no	—
161	C₂²⋊Q₈	—	5	2	no	—
162	C₂²⋊Q₈	—	5	2	no	—
163	C₂²⋊Q₈	C₃×D₅	1	1	yes	characteristic, complemented
164	C₄×Q₈	—	5	1	no	—
165	C₄×D₄	—	5	1	no	—
166	C₄×D₄	—	5	2	no	—
167	C₂×C₄⋊C₄	—	5	1	no	—
168	C₂×C₄⋊C₄	—	5	1	no	—
169	C₂×C₃×C₅	—	2	2	no	—
170	C₂×C₃×C₅	C₂²×Q₈	1	1	yes	characteristic
171	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
172	C₂×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
173	C₃×D₅	—	2	2	no	—
174	C₂³×C₃	—	5	1	no	—
175	C₂³×C₃	C₂²×D₅	1	1	yes	characteristic
176	C₃×Q₈	—	2	2	no	—
177	C₃×Q₈	—	20	2	no	—
178	C₃×D₄	—	5	4	no	—
179	C₂×C₃×C₄	—	2	2	no	—
180	C₂×C₃×C₄	—	5	1	no	—
181	C₂×C₃×C₄	—	5	1	no	—
182	C₂×C₃×C₄	—	5	2	no	—
183	C₂×C₃×C₄	—	5	1	no	—
184	C₂×C₃×C₄	—	5	2	no	—
185	C₂×C₃×C₄	—	10	2	no	—
186	C₂×C₃×C₄	—	20	2	no	—
187	C₂×C₃×C₄	—	20	2	no	—
188	C₂×C₃×C₄	C₂²×D₅	1	2	yes	—
189	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
190	C₂×C₃×C₄	C₂²×D₅	1	2	yes	—
191	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
192	C₂²×C₅	—	2	1	no	—
193	C₂²×C₅	—	2	1	no	—
194	C₂²×C₅	C₂×C₃×Q₈	1	2	yes	—
195	C₂²×C₅	C₂⁴×C₃	1	1	yes	characteristic
196	C₂×D₅	—	2	1	no	—
197	C₂×D₅	—	2	1	no	—
198	C₂×D₅	C₂×C₃×Q₈	1	2	yes	—
199	C₄×C₅	—	2	1	no	—
200	C₄×C₅	—	2	2	no	—
201	C₄×C₅	—	2	2	no	—
202	C₄×C₅	C₃×#41	1	2	yes	—
203	Dic₅	—	2	1	no	—
204	Dic₅	—	2	1	no	—
205	Dic₅	—	2	2	no	—
206	Dic₅	—	2	2	no	—
207	Dic₅	C₂×C₃×Q₈	1	2	yes	—
208	C₂×Q₈	—	5	2	no	—
209	C₂×Q₈	C₂×C₃×D₅	1	1	yes	characteristic
210	C₂×D₄	—	5	1	no	—
211	C₂²×C₄	—	5	2	no	—
212	C₂²×C₄	—	5	1	no	—
213	C₂²×C₄	—	5	2	no	—
214	C₂²×C₄	C₂×C₃×D₅	1	1	yes	characteristic
215	C₄⋊C₄	—	5	1	no	—
216	C₄⋊C₄	—	5	2	no	—
217	C₄⋊C₄	—	5	2	no	—
218	C₄⋊C₄	—	5	1	no	—
219	C₄⋊C₄	—	5	1	no	—
220	C₄⋊C₄	—	5	2	no	—
221	C₄⋊C₄	—	5	2	no	—
222	C₄⋊C₄	—	5	2	no	—
223	C₄⋊C₄	C₂×C₃×D₅	1	1	yes	characteristic
224	C₄⋊C₄	C₂×C₃×D₅	1	2	yes	—
225	C₂²⋊C₄	—	5	1	no	—
226	C₂²⋊C₄	—	5	1	no	—
227	C₂²⋊C₄	—	5	2	no	—
228	C₂²⋊C₄	C₂×C₃×D₅	1	2	yes	—
229	C₄²	—	5	2	no	—
230	C₄²	—	5	1	no	—
231	C₃×C₅	#554	1	1	yes	characteristic, complemented, Hall
232	C₂²×C₃	—	2	1	no	—
233	C₂²×C₃	—	2	1	no	—
234	C₂²×C₃	—	5	2	no	—
235	C₂²×C₃	—	10	1	no	—
236	C₂²×C₃	—	10	1	no	—
237	C₂²×C₃	Q₈×D₅	1	2	yes	—
238	C₂²×C₃	C₂³×D₅	1	1	yes	characteristic
239	C₃×C₄	—	2	1	no	—
240	C₃×C₄	—	2	2	no	—
241	C₃×C₄	—	2	2	no	—
242	C₃×C₄	—	5	2	no	—
243	C₃×C₄	—	10	1	no	—
244	C₃×C₄	—	10	2	no	—
245	C₃×C₄	—	10	2	no	—
246	C₃×C₄	—	10	1	no	—
247	C₃×C₄	#711	1	2	yes	—
248	C₂×C₅	—	2	2	no	—
249	C₂×C₅	C₂²×C₃×Q₈	1	1	yes	characteristic
250	C₂×C₅	C₂×C₃×#41	1	1	yes	characteristic
251	C₂×C₅	C₃×H(5,𝔽₂)	1	1	yes	characteristic
252	D₅	—	2	2	no	—
253	C₂³	—	5	1	no	—
254	C₂³	C₂²×C₃×D₅	1	1	yes	characteristic
255	Q₈	—	2	2	no	—
256	Q₈	—	20	2	no	—
257	D₄	—	5	4	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	5	1	no	—
260	C₂×C₄	—	5	1	no	—
261	C₂×C₄	—	5	2	no	—
262	C₂×C₄	—	5	1	no	—
263	C₂×C₄	—	5	2	no	—
264	C₂×C₄	—	10	2	no	—
265	C₂×C₄	—	20	2	no	—
266	C₂×C₄	—	20	2	no	—
267	C₂×C₄	C₂²×C₃×D₅	1	2	yes	—
268	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
269	C₂×C₄	C₂²×C₃×D₅	1	2	yes	—
270	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
271	C₂×C₃	—	2	2	no	—
272	C₂×C₃	—	10	2	no	—
273	C₂×C₃	C₂×#711	1	1	yes	characteristic
274	C₂×C₃	C₂×Q₈×D₅	1	1	yes	characteristic
275	C₂×C₃	#4149	1	1	yes	characteristic
276	C₅	C₃×#554	1	1	yes	characteristic, complemented, Sylow
277	C₂²	—	2	1	no	—
278	C₂²	—	2	1	no	—
279	C₂²	—	5	2	no	—
280	C₂²	—	10	1	no	—
281	C₂²	—	10	1	no	—
282	C₂²	C₃×Q₈×D₅	1	2	yes	—
283	C₂²	C₂³×C₃×D₅	1	1	yes	characteristic
284	C₄	—	2	1	no	—
285	C₄	—	2	2	no	—
286	C₄	—	2	2	no	—
287	C₄	—	5	2	no	—
288	C₄	—	10	1	no	—
289	C₄	—	10	2	no	—
290	C₄	—	10	2	no	—
291	C₄	—	10	1	no	—
292	C₄	C₃×#711	1	2	yes	—
293	C₃	#65969	1	1	yes	characteristic, direct product factor, Sylow
294	C₂	—	2	2	no	—
295	C₂	—	10	2	no	—
296	C₂	C₂×C₃×#711	1	1	yes	characteristic
297	C₂	C₂×C₃×Q₈×D₅	1	1	yes	characteristic
298	C₂	C₃×#4149	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×#554
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. C₂×C₃×C₄⋊Dic₅	C₂	semidirect
6. C₃×#4043	C₂	semidirect
7. C₃×D₅×C₄⋊C₄	C₂	semidirect
8. C₃×#4035	C₂	semidirect
9. C₃×#4034	C₂	semidirect
11. C₃×#4024	C₂	semidirect
12. #65969	C₃	direct (N × H)
37. C₅×C₂²⋊Q₈	C₂×C₃	semidirect
38. #4092	C₂×C₃	semidirect
39. Q₈×Dic₅	C₂×C₃	semidirect
41. C₂×C₄⋊Dic₅	C₂×C₃	semidirect
42. #4043	C₂×C₃	semidirect
43. D₅×C₄⋊C₄	C₂×C₃	semidirect
44. #4035	C₂×C₃	semidirect
45. #4034	C₂×C₃	semidirect
47. #4024	C₂×C₃	semidirect
56. C₃×C₂²⋊D₅	Q₈	semidirect
71. C₃×C₂²⋊Q₈	D₅	semidirect
148. C₂²⋊D₅	C₃×Q₈	semidirect
163. C₂²⋊Q₈	C₃×D₅	semidirect
231. C₃×C₅	#554	semidirect
276. C₅	C₃×#554	semidirect
293. C₃	#65969	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
C₂×C₃×C₄⋊Dic₅	2
C₃×#4043	2
C₃×D₅×C₄⋊C₄	2
C₃×#4035	2
C₃×#4034	2
C₃×#4027	2
C₃×#4024	2
#65969	3
C₃×#554	5

Subgroups — C3×#65969 #11755646

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#65969 #11755646