Subgroups — C₃×#10615070 #14439835

Order 1920.

712 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×C₅×#533	C₂	1	1	yes	maximal, characteristic
2	C₃×#65804	C₂	1	1	yes	maximal, characteristic
3	C₃×#65368	C₂	1	2	yes	maximal, complemented
4	C₃×#65367	C₂	1	2	yes	maximal, complemented
5	C₃×#65353	C₂	1	4	yes	maximal, complemented
6	C₃×#65287	C₂	1	4	yes	maximal
7	C₂×C₃×C₄×#168	C₂	1	1	yes	maximal, characteristic
8	#10615070	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₃×C₅×C₄⋊Q₈	C₂²	1	2	yes	—
10	C₃×C₅×#125	C₂²	1	2	yes	—
11	C₃×C₅×C₂²⋊Q₈	—	2	2	no	—
12	C₃×C₄×C₅×Q₈	—	2	2	no	—
13	C₃×C₅×#117	C₂²	1	2	yes	—
14	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
15	C₃×#4072	C₂²	1	2	yes	—
16	C₃×#4070	—	2	2	no	—
17	C₂²×C₃×#168	C₂²	1	2	yes	—
18	C₃×#4016	C₂²	1	2	yes	—
19	C₃×#4015	C₂²	1	2	yes	—
20	C₃×C₄×Dic₁₀	—	2	2	no	—
21	C₃×#3940	C₂²	1	8	yes	—
22	C₃×#3939	C₂²	1	8	yes	—
23	C₃×C₄×#168	C₂²	1	2	yes	—
24	C₃×C₄×#168	C₂²	1	1	yes	characteristic
25	C₃×C₄×#168	C₂²	1	1	yes	characteristic
26	C₃×#3160	—	5	1	no	maximal, Hall
27	C₅×#533	C₂×C₃	1	1	yes	characteristic
28	#65804	C₂×C₃	1	1	yes	characteristic
29	#65368	C₂×C₃	1	2	yes	complemented
30	#65367	C₂×C₃	1	2	yes	complemented
31	#65353	C₂×C₃	1	4	yes	complemented
32	#65287	C₂×C₃	1	4	yes	—
33	C₂×C₄×#168	C₂×C₃	1	1	yes	characteristic
34	C₂×C₃×C₅×Q₈	—	2	2	no	—
35	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
36	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
37	C₂×C₃×Dic₁₀	—	2	2	no	—
38	C₃×C₅×C₄⋊C₄	—	2	4	no	—
39	C₃×C₅×C₄⋊C₄	—	2	2	no	—
40	C₃×C₅×C₄⋊C₄	C₂³	1	4	yes	—
41	C₃×C₅×C₂²⋊C₄	—	2	2	no	—
42	C₃×C₄²×C₅	—	2	2	no	—
43	C₃×C₄²×C₅	D₄	1	2	yes	—
44	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
45	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
46	C₃×C₂²⋊Dic₅	—	2	2	no	—
47	C₃×C₄⋊Dic₅	—	2	2	no	—
48	C₃×C₄⋊Dic₅	C₂³	1	4	yes	—
49	C₃×#683	—	2	4	no	—
50	C₃×C₄×Dic₅	—	2	2	no	—
51	C₂×C₃×#168	Q₈	1	8	yes	complemented
52	C₂×C₃×#168	C₂³	1	2	yes	—
53	C₂×C₃×#168	C₂³	1	2	yes	—
54	C₃×#533	—	5	1	no	—
55	C₃×#533	D₅	1	1	yes	characteristic
56	C₃×#500	—	5	2	no	—
57	C₃×#499	—	5	4	no	—
58	C₃×#498	—	5	2	no	—
59	C₃×#427	—	5	4	no	—
60	C₂×C₃×C₄×C₈	—	5	1	no	—
61	C₅×C₄⋊Q₈	C₂²×C₃	1	2	yes	—
62	C₅×#125	C₂²×C₃	1	2	yes	—
63	C₅×C₂²⋊Q₈	—	2	2	no	—
64	C₄×C₅×Q₈	—	2	2	no	—
65	C₅×#117	C₂²×C₃	1	2	yes	—
66	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
67	#4072	C₂²×C₃	1	2	yes	—
68	#4070	—	2	2	no	—
69	C₂²×#168	C₂²×C₃	1	2	yes	—
70	#4016	C₂²×C₃	1	2	yes	—
71	#4015	C₂²×C₃	1	2	yes	—
72	C₄×Dic₁₀	—	2	2	no	—
73	#3940	C₂²×C₃	1	8	yes	—
74	#3939	C₂²×C₃	1	8	yes	—
75	C₄×#168	C₂²×C₃	1	2	yes	—
76	C₄×#168	C₂²×C₃	1	1	yes	characteristic
77	C₄×#168	C₂²×C₃	1	1	yes	characteristic
78	#3160	—	5	1	no	Sylow
79	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
80	C₃×C₅×Q₈	—	4	4	no	—
81	C₂×C₃×C₄×C₅	—	2	4	no	—
82	C₂×C₃×C₄×C₅	—	2	4	no	—
83	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
84	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
85	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
86	C₂×C₃×C₄×C₅	C₂×Q₈	1	4	yes	—
87	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
88	C₂×C₃×Dic₅	—	2	4	no	—
89	C₃×Dic₁₀	—	4	4	no	—
90	C₃×#168	C₂×Q₈	1	4	yes	—
91	C₃×#168	C₂×Q₈	1	4	yes	—
92	C₂²×C₃×C₈	—	5	2	no	—
93	C₃×C₄⋊Q₈	—	5	2	no	—
94	C₃×C₄⋊Q₈	C₂×D₅	1	2	yes	—
95	C₃×#125	—	5	2	no	—
96	C₃×#125	C₂×D₅	1	2	yes	—
97	C₃×C₂²⋊Q₈	—	2	2	no	—
98	C₃×C₂²⋊Q₈	—	10	2	no	—
99	C₃×C₄×Q₈	—	2	2	no	—
100	C₃×C₄×Q₈	—	10	2	no	—
101	C₃×#117	—	5	2	no	—
102	C₃×#117	C₂×D₅	1	2	yes	—
103	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
104	C₃×#107	—	5	8	no	—
105	C₃×#106	—	5	8	no	—
106	C₃×C₄×C₈	—	5	1	no	—
107	C₃×C₄×C₈	—	5	2	no	—
108	C₃×C₄×C₈	—	5	1	no	—
109	C₂×C₅×Q₈	—	2	2	no	—
110	C₂²×C₄×C₅	C₃×D₄	1	2	yes	—
111	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
112	C₂×Dic₁₀	—	2	2	no	—
113	C₅×C₄⋊C₄	—	2	4	no	—
114	C₅×C₄⋊C₄	—	2	2	no	—
115	C₅×C₄⋊C₄	C₂³×C₃	1	4	yes	—
116	C₅×C₂²⋊C₄	—	2	2	no	—
117	C₄²×C₅	—	2	2	no	—
118	C₄²×C₅	C₃×D₄	1	2	yes	—
119	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
120	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
121	C₂²⋊Dic₅	—	2	2	no	—
122	C₄⋊Dic₅	—	2	2	no	—
123	C₄⋊Dic₅	C₂³×C₃	1	4	yes	—
124	#683	—	2	4	no	—
125	C₄×Dic₅	—	2	2	no	—
126	C₂×#168	C₃×Q₈	1	8	yes	complemented
127	C₂×#168	C₂³×C₃	1	2	yes	—
128	C₂×#168	C₂³×C₃	1	2	yes	—
129	#533	—	5	1	no	—
130	#533	C₃×D₅	1	1	yes	characteristic
131	#500	—	5	2	no	—
132	#499	—	5	4	no	—
133	#498	—	5	2	no	—
134	#427	—	5	4	no	—
135	C₂×C₄×C₈	—	5	1	no	—
136	C₂²×C₃×C₅	—	2	2	no	—
137	C₂²×C₃×C₅	C₄⋊Q₈	1	2	yes	—
138	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
139	C₃×C₄×C₅	—	2	2	no	—
140	C₃×C₄×C₅	—	4	4	no	—
141	C₃×C₄×C₅	C₄⋊Q₈	1	2	yes	—
142	C₃×C₄×C₅	#135	1	4	yes	—
143	C₃×C₄×C₅	C₂²×Q₈	1	2	yes	—
144	C₃×Dic₅	—	4	4	no	—
145	C₂×C₃×Q₈	—	2	2	no	—
146	C₂×C₃×Q₈	—	10	2	no	—
147	C₂²×C₃×C₄	C₂²⋊D₅	1	2	yes	—
148	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
149	C₂×C₃×C₈	—	5	8	no	—
150	C₂×C₃×C₈	—	5	2	no	—
151	C₂×C₃×C₈	—	5	2	no	—
152	C₃×C₄⋊C₄	—	2	4	no	—
153	C₃×C₄⋊C₄	—	2	2	no	—
154	C₃×C₄⋊C₄	—	5	4	no	—
155	C₃×C₄⋊C₄	—	10	4	no	—
156	C₃×C₄⋊C₄	—	10	2	no	—
157	C₃×C₄⋊C₄	C₂²×D₅	1	4	yes	—
158	C₃×C₂²⋊C₄	—	2	2	no	—
159	C₃×C₂²⋊C₄	—	10	2	no	—
160	C₃×C₄²	—	2	2	no	—
161	C₃×C₄²	—	10	2	no	—
162	C₃×C₄²	C₂²⋊D₅	1	2	yes	—
163	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
164	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
165	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
166	C₅×Q₈	—	4	4	no	—
167	C₂×C₄×C₅	—	2	4	no	—
168	C₂×C₄×C₅	—	2	4	no	—
169	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
170	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
171	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
172	C₂×C₄×C₅	C₂×C₃×Q₈	1	4	yes	—
173	C₂×C₄×C₅	C₂⁴×C₃	1	1	yes	characteristic
174	C₂×Dic₅	—	2	4	no	—
175	Dic₁₀	—	4	4	no	—
176	#168	C₂×C₃×Q₈	1	4	yes	—
177	#168	C₂×C₃×Q₈	1	4	yes	—
178	C₂²×C₈	—	5	2	no	—
179	C₄⋊Q₈	—	5	2	no	—
180	C₄⋊Q₈	C₂×C₃×D₅	1	2	yes	—
181	#125	—	5	2	no	—
182	#125	C₂×C₃×D₅	1	2	yes	—
183	C₂²⋊Q₈	—	2	2	no	—
184	C₂²⋊Q₈	—	10	2	no	—
185	C₄×Q₈	—	2	2	no	—
186	C₄×Q₈	—	10	2	no	—
187	#117	—	5	2	no	—
188	#117	C₂×C₃×D₅	1	2	yes	—
189	C₂×C₄²	C₂×C₃×D₅	1	1	yes	characteristic
190	#107	—	5	8	no	—
191	#106	—	5	8	no	—
192	C₄×C₈	—	5	1	no	—
193	C₄×C₈	—	5	2	no	—
194	C₄×C₈	—	5	1	no	—
195	C₂×C₃×C₅	—	2	2	no	—
196	C₂×C₃×C₅	C₂×C₄⋊Q₈	1	1	yes	characteristic
197	C₂×C₃×C₅	C₂×#135	1	2	yes	—
198	C₂³×C₃	C₂×C₂²⋊D₅	1	1	yes	characteristic
199	C₃×Q₈	—	4	4	no	—
200	C₃×Q₈	—	20	4	no	—
201	C₂×C₃×C₄	—	2	4	no	—
202	C₂×C₃×C₄	—	2	4	no	—
203	C₂×C₃×C₄	—	10	4	no	—
204	C₂×C₃×C₄	Q₈×D₅	1	4	yes	—
205	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	2	yes	—
206	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	2	yes	—
207	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
208	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
209	C₃×C₈	—	5	4	no	—
210	C₃×C₈	—	5	4	no	—
211	C₂²×C₅	—	2	2	no	—
212	C₂²×C₅	C₃×C₄⋊Q₈	1	2	yes	—
213	C₂²×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
214	C₄×C₅	—	2	2	no	—
215	C₄×C₅	—	4	4	no	—
216	C₄×C₅	C₃×C₄⋊Q₈	1	2	yes	—
217	C₄×C₅	C₃×#135	1	4	yes	—
218	C₄×C₅	C₂²×C₃×Q₈	1	2	yes	—
219	Dic₅	—	4	4	no	—
220	C₂×Q₈	—	2	2	no	—
221	C₂×Q₈	—	10	2	no	—
222	C₂²×C₄	C₃×C₂²⋊D₅	1	2	yes	—
223	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
224	C₂×C₈	—	5	8	no	—
225	C₂×C₈	—	5	2	no	—
226	C₂×C₈	—	5	2	no	—
227	C₄⋊C₄	—	2	4	no	—
228	C₄⋊C₄	—	2	2	no	—
229	C₄⋊C₄	—	5	4	no	—
230	C₄⋊C₄	—	10	4	no	—
231	C₄⋊C₄	—	10	2	no	—
232	C₄⋊C₄	C₂²×C₃×D₅	1	4	yes	—
233	C₂²⋊C₄	—	2	2	no	—
234	C₂²⋊C₄	—	10	2	no	—
235	C₄²	—	2	2	no	—
236	C₄²	—	10	2	no	—
237	C₄²	C₃×C₂²⋊D₅	1	2	yes	—
238	C₄²	C₂²×C₃×D₅	1	1	yes	characteristic
239	C₄²	C₂²×C₃×D₅	1	1	yes	characteristic
240	C₃×C₅	#3160	1	1	yes	characteristic, complemented, Hall
241	C₂²×C₃	—	2	2	no	—
242	C₂²×C₃	#4090	1	2	yes	—
243	C₂²×C₃	C₂²×C₂²⋊D₅	1	1	yes	characteristic
244	C₃×C₄	—	2	2	no	—
245	C₃×C₄	—	4	4	no	—
246	C₃×C₄	—	20	4	no	—
247	C₃×C₄	#4090	1	2	yes	—
248	C₃×C₄	#4096	1	4	yes	—
249	C₃×C₄	C₂×Q₈×D₅	1	2	yes	—
250	C₂×C₅	—	2	2	no	—
251	C₂×C₅	C₂×C₃×C₄⋊Q₈	1	1	yes	characteristic
252	C₂×C₅	C₂×C₃×#135	1	2	yes	—
253	C₂³	C₂×C₃×C₂²⋊D₅	1	1	yes	characteristic
254	Q₈	—	4	4	no	—
255	Q₈	—	20	4	no	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	10	4	no	—
259	C₂×C₄	C₃×Q₈×D₅	1	4	yes	—
260	C₂×C₄	C₂×C₃×C₂²⋊D₅	1	2	yes	—
261	C₂×C₄	C₂×C₃×C₂²⋊D₅	1	1	yes	characteristic
262	C₂×C₄	C₂×C₃×C₂²⋊D₅	1	2	yes	—
263	C₂×C₄	C₂³×C₃×D₅	1	1	yes	characteristic
264	C₈	—	5	4	no	—
265	C₈	—	5	4	no	—
266	C₂×C₃	—	2	2	no	—
267	C₂×C₃	C₂×#4090	1	1	yes	characteristic
268	C₂×C₃	C₂×#4096	1	2	yes	—
269	C₅	C₃×#3160	1	1	yes	characteristic, complemented, Sylow
270	C₂²	—	2	2	no	—
271	C₂²	C₃×#4090	1	2	yes	—
272	C₂²	C₂²×C₃×C₂²⋊D₅	1	1	yes	characteristic
273	C₄	—	2	2	no	—
274	C₄	—	4	4	no	—
275	C₄	—	20	4	no	—
276	C₄	C₃×#4090	1	2	yes	—
277	C₄	C₃×#4096	1	4	yes	—
278	C₄	C₂×C₃×Q₈×D₅	1	2	yes	—
279	C₃	#10615070	1	1	yes	characteristic, direct product factor, Sylow
280	C₂	—	2	2	no	—
281	C₂	C₂×C₃×#4090	1	1	yes	characteristic
282	C₂	C₂×C₃×#4096	1	2	yes	—

	subgroup	G/H
center	C₂×C₃×C₄	C₂×C₂²⋊D₅
commutator, cosocle	C₂×C₄×C₅	C₂⁴×C₃
Frattini	C₂×C₄	C₂³×C₃×D₅
Fitting	C₃×C₅×#533	C₂
socle	C₂²×C₃×C₅	C₂²×D₄
prime radical	C₂³×C₃×C₅	C₂×D₄
hypercenter	C₃×#533	D₅
hypocenter	C₅	C₃×#3160
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂×C₃×C₄² ◁ C₃×#533

Complements

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

N	H ≅ G/N	type
3. C₃×#65368	C₂	semidirect
4. C₃×#65367	C₂	semidirect
5. C₃×#65353	C₂	semidirect
8. #10615070	C₃	direct (N × H)
29. #65368	C₂×C₃	semidirect
30. #65367	C₂×C₃	semidirect
31. #65353	C₂×C₃	semidirect
51. C₂×C₃×#168	Q₈	semidirect
126. C₂×#168	C₃×Q₈	semidirect
240. C₃×C₅	#3160	semidirect
269. C₅	C₃×#3160	semidirect
279. C₃	#10615070	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×#533	2
C₃×#65804	2
C₃×#65368	2
C₃×#65367	2
C₃×#65353	2
C₃×#65287	2
C₂×C₃×C₄×#168	2
#10615070	3
C₃×#3160	5

Subgroups — C3×#10615070 #14439835

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10615070 #14439835