Subgroups — C₃×#10601640 #14291413

Order 1920.

408 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×C₈×D₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#65304	C₂	1	2	yes	maximal, complemented
3	C₃×#64984	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#64705	C₂	1	2	yes	maximal, complemented
5	C₃×#64678	C₂	1	1	yes	maximal, characteristic, complemented
6	#10601640	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂²×C₃×C₅×C₈	—	2	1	no	—
8	C₃×C₄×C₅×D₄	C₂²	1	1	yes	characteristic
9	C₂×C₃×#686	—	2	2	no	—
10	C₂×C₃×D₄₀	—	2	1	no	—
11	C₂×C₃×D₄₀	—	2	1	no	—
12	C₃×#4020	C₂²	1	2	yes	—
13	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₃×C₅×C₂²⋊C₈	—	2	1	no	—
15	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
16	C₃×#3941	—	2	2	no	—
17	C₃×C₄⋊#168	C₂²	1	2	yes	—
18	C₃×#1667	—	5	1	no	maximal, Hall
19	C₅×C₈×D₄	C₂×C₃	1	1	yes	characteristic, complemented
20	#65304	C₂×C₃	1	2	yes	complemented
21	#64984	C₂×C₃	1	1	yes	characteristic, complemented
22	#64705	C₂×C₃	1	2	yes	complemented
23	#64678	C₂×C₃	1	1	yes	characteristic, complemented
24	C₂×C₃×C₅×D₄	D₄	1	1	yes	characteristic
25	C₂²×C₃×C₄×C₅	—	2	1	no	—
26	C₂×C₃×D₂₀	—	2	2	no	—
27	C₂×C₃×D₂₀	—	2	2	no	—
28	C₂×C₃×D₂₀	—	2	2	no	—
29	C₂×C₃×C₅×C₈	—	2	1	no	—
30	C₂×C₃×C₅×C₈	—	4	2	no	—
31	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
32	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
33	C₃×C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
34	C₃×C₅×C₂²⋊C₄	—	2	1	no	—
35	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
36	C₃×#686	—	4	4	no	—
37	C₂×C₃×#168	—	2	2	no	—
38	C₃×D₄₀	—	4	2	no	—
39	C₃×D₄₀	—	4	2	no	—
40	C₃×#493	—	5	1	no	—
41	C₃×#459	—	5	2	no	—
42	C₃×C₈×D₄	D₅	1	1	yes	characteristic, complemented
43	C₃×#335	—	5	1	no	—
44	C₃×#331	—	5	2	no	—
45	C₂²×C₅×C₈	—	2	1	no	—
46	C₄×C₅×D₄	C₂²×C₃	1	1	yes	characteristic
47	C₂×#686	—	2	2	no	—
48	C₂×D₄₀	—	2	1	no	—
49	C₂×D₄₀	—	2	1	no	—
50	#4020	C₂²×C₃	1	2	yes	—
51	C₅×C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
52	C₅×C₂²⋊C₈	—	2	1	no	—
53	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
54	#3941	—	2	2	no	—
55	C₄⋊#168	C₂²×C₃	1	2	yes	—
56	#1667	—	5	1	no	Sylow
57	C₂³×C₃×C₅	—	2	1	no	—
58	C₂²×C₃×D₅	—	4	2	no	—
59	C₃×C₅×D₄	—	2	1	no	—
60	C₃×C₅×D₄	D₈	1	2	yes	complemented
61	C₂×C₃×C₄×C₅	—	2	1	no	—
62	C₂×C₃×C₄×C₅	—	4	1	no	—
63	C₂×C₃×C₄×C₅	—	4	1	no	—
64	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
65	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
66	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
67	C₃×D₂₀	—	4	2	no	—
68	C₃×D₂₀	—	4	2	no	—
69	C₃×D₂₀	—	4	2	no	—
70	C₃×D₂₀	—	4	2	no	—
71	C₃×D₂₀	—	8	2	no	—
72	C₃×C₅×C₈	—	2	1	no	—
73	C₃×C₅×C₈	—	4	1	no	—
74	C₃×C₅×C₈	D₈	1	2	yes	complemented
75	C₃×#168	—	4	2	no	—
76	C₂×C₃×D₈	—	10	2	no	—
77	C₂×C₃×D₈	—	10	1	no	—
78	C₂×C₃×D₈	—	10	1	no	—
79	C₂²×C₃×C₈	—	2	1	no	—
80	C₃×#127	—	5	2	no	—
81	C₃×C₄×D₄	C₂×D₅	1	1	yes	characteristic
82	C₃×C₄⋊C₈	—	5	2	no	—
83	C₃×C₄⋊C₈	C₂×D₅	1	1	yes	characteristic
84	C₃×#102	—	10	2	no	—
85	C₃×C₂²⋊C₈	—	2	1	no	—
86	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic
87	C₂×C₅×D₄	C₃×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₅	—	2	1	no	—
89	C₂×D₂₀	—	2	2	no	—
90	C₂×D₂₀	—	2	2	no	—
91	C₂×D₂₀	—	2	2	no	—
92	C₂×C₅×C₈	—	2	1	no	—
93	C₂×C₅×C₈	—	4	2	no	—
94	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
95	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
96	C₅×C₄⋊C₄	C₃×D₄	1	1	yes	characteristic
97	C₅×C₂²⋊C₄	—	2	1	no	—
98	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
99	#686	—	4	4	no	—
100	C₂×#168	—	2	2	no	—
101	D₄₀	—	4	2	no	—
102	D₄₀	—	4	2	no	—
103	#493	—	5	1	no	—
104	#459	—	5	2	no	—
105	C₈×D₄	C₃×D₅	1	1	yes	characteristic, complemented
106	#335	—	5	1	no	—
107	#331	—	5	2	no	—
108	C₂²×C₃×C₅	—	2	2	no	—
109	C₂²×C₃×C₅	—	4	1	no	—
110	C₂²×C₃×C₅	—	4	1	no	—
111	C₂²×C₃×C₅	#121	1	1	yes	characteristic
112	C₂×C₃×D₅	—	8	2	no	—
113	C₂×C₃×D₅	—	8	2	no	—
114	C₂×C₃×D₅	—	8	2	no	—
115	C₃×C₄×C₅	—	2	1	no	—
116	C₃×C₄×C₅	—	4	1	no	—
117	C₃×C₄×C₅	C₂×D₈	1	1	yes	characteristic
118	C₃×C₄×C₅	C₂×D₈	1	1	yes	characteristic
119	C₃×C₄×C₅	#135	1	1	yes	characteristic
120	C₃×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
121	C₂×C₃×D₄	—	10	2	no	—
122	C₂×C₃×D₄	—	10	2	no	—
123	C₂×C₃×D₄	—	10	2	no	—
124	C₂×C₃×D₄	D₂₀	1	1	yes	characteristic
125	C₂²×C₃×C₄	—	2	1	no	—
126	C₃×D₈	—	20	2	no	—
127	C₃×D₈	—	20	4	no	—
128	C₃×D₈	—	20	2	no	—
129	C₂×C₃×C₈	—	2	1	no	—
130	C₂×C₃×C₈	—	4	2	no	—
131	C₂×C₃×C₈	—	10	2	no	—
132	C₂×C₃×C₈	C₂²⋊D₅	1	1	yes	characteristic
133	C₂×C₃×C₈	C₂²⋊D₅	1	1	yes	characteristic
134	C₃×C₄⋊C₄	D₂₀	1	1	yes	characteristic
135	C₃×C₂²⋊C₄	—	2	1	no	—
136	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
137	C₂³×C₅	—	2	1	no	—
138	C₂²×D₅	—	4	2	no	—
139	C₅×D₄	—	2	1	no	—
140	C₅×D₄	C₃×D₈	1	2	yes	complemented
141	C₂×C₄×C₅	—	2	1	no	—
142	C₂×C₄×C₅	—	4	1	no	—
143	C₂×C₄×C₅	—	4	1	no	—
144	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
145	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
146	C₂×C₄×C₅	C₃×#41	1	1	yes	characteristic
147	D₂₀	—	4	2	no	—
148	D₂₀	—	4	2	no	—
149	D₂₀	—	4	2	no	—
150	D₂₀	—	4	2	no	—
151	D₂₀	—	8	2	no	—
152	C₅×C₈	—	2	1	no	—
153	C₅×C₈	—	4	1	no	—
154	C₅×C₈	C₃×D₈	1	2	yes	complemented
155	#168	—	4	2	no	—
156	C₂×D₈	—	10	2	no	—
157	C₂×D₈	—	10	1	no	—
158	C₂×D₈	—	10	1	no	—
159	C₂²×C₈	—	2	1	no	—
160	#127	—	5	2	no	—
161	C₄×D₄	C₂×C₃×D₅	1	1	yes	characteristic
162	C₄⋊C₈	—	5	2	no	—
163	C₄⋊C₈	C₂×C₃×D₅	1	1	yes	characteristic
164	#102	—	10	2	no	—
165	C₂²⋊C₈	—	2	1	no	—
166	C₄×C₈	C₂×C₃×D₅	1	1	yes	characteristic
167	C₂×C₃×C₅	—	4	2	no	—
168	C₂×C₃×C₅	#459	1	1	yes	characteristic
169	C₂×C₃×C₅	#466	1	1	yes	characteristic
170	C₂×C₃×C₅	#472	1	1	yes	characteristic
171	C₃×D₅	—	16	2	no	—
172	C₂³×C₃	—	2	1	no	—
173	C₂³×C₃	—	20	2	no	—
174	C₃×D₄	—	2	1	no	—
175	C₃×D₄	—	20	2	no	—
176	C₃×D₄	—	20	2	no	—
177	C₃×D₄	—	20	2	no	—
178	C₃×D₄	—	20	2	no	—
179	C₃×D₄	—	40	2	no	—
180	C₃×D₄	D₄₀	1	2	yes	complemented
181	C₂×C₃×C₄	—	2	1	no	—
182	C₂×C₃×C₄	—	4	1	no	—
183	C₂×C₃×C₄	—	4	1	no	—
184	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
185	C₂×C₃×C₄	#709	1	1	yes	characteristic
186	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
187	C₃×C₈	—	2	1	no	—
188	C₃×C₈	—	4	1	no	—
189	C₃×C₈	—	20	2	no	—
190	C₃×C₈	#686	1	2	yes	complemented
191	C₂²×C₅	—	2	2	no	—
192	C₂²×C₅	—	4	1	no	—
193	C₂²×C₅	—	4	1	no	—
194	C₂²×C₅	C₃×#121	1	1	yes	characteristic
195	C₂×D₅	—	8	2	no	—
196	C₂×D₅	—	8	2	no	—
197	C₂×D₅	—	8	2	no	—
198	C₄×C₅	—	2	1	no	—
199	C₄×C₅	—	4	1	no	—
200	C₄×C₅	C₂×C₃×D₈	1	1	yes	characteristic
201	C₄×C₅	C₂×C₃×D₈	1	1	yes	characteristic
202	C₄×C₅	C₃×#135	1	1	yes	characteristic
203	C₄×C₅	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
204	C₂×D₄	—	10	2	no	—
205	C₂×D₄	—	10	2	no	—
206	C₂×D₄	—	10	2	no	—
207	C₂×D₄	C₃×D₂₀	1	1	yes	characteristic
208	C₂²×C₄	—	2	1	no	—
209	D₈	—	20	2	no	—
210	D₈	—	20	4	no	—
211	D₈	—	20	2	no	—
212	C₂×C₈	—	2	1	no	—
213	C₂×C₈	—	4	2	no	—
214	C₂×C₈	—	10	2	no	—
215	C₂×C₈	C₃×C₂²⋊D₅	1	1	yes	characteristic
216	C₂×C₈	C₃×C₂²⋊D₅	1	1	yes	characteristic
217	C₄⋊C₄	C₃×D₂₀	1	1	yes	characteristic
218	C₂²⋊C₄	—	2	1	no	—
219	C₄²	C₂²×C₃×D₅	1	1	yes	characteristic
220	C₃×C₅	#1667	1	1	yes	characteristic, complemented, Hall
221	C₂²×C₃	—	2	2	no	—
222	C₂²×C₃	—	4	1	no	—
223	C₂²×C₃	—	4	1	no	—
224	C₂²×C₃	—	40	2	no	—
225	C₂²×C₃	—	40	2	no	—
226	C₂²×C₃	—	40	2	no	—
227	C₂²×C₃	#4076	1	1	yes	characteristic
228	C₃×C₄	—	2	1	no	—
229	C₃×C₄	—	4	1	no	—
230	C₃×C₄	C₂×D₄₀	1	1	yes	characteristic
231	C₃×C₄	#4050	1	1	yes	characteristic
232	C₃×C₄	C₂×#686	1	1	yes	characteristic
233	C₃×C₄	#4095	1	1	yes	characteristic
234	C₂×C₅	—	4	2	no	—
235	C₂×C₅	C₃×#459	1	1	yes	characteristic
236	C₂×C₅	C₃×#466	1	1	yes	characteristic
237	C₂×C₅	C₃×#472	1	1	yes	characteristic
238	D₅	—	16	2	no	—
239	C₂³	—	2	1	no	—
240	C₂³	—	20	2	no	—
241	D₄	—	2	1	no	—
242	D₄	—	20	2	no	—
243	D₄	—	20	2	no	—
244	D₄	—	20	2	no	—
245	D₄	—	20	2	no	—
246	D₄	—	40	2	no	—
247	D₄	C₃×D₄₀	1	2	yes	complemented
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	4	1	no	—
250	C₂×C₄	—	4	1	no	—
251	C₂×C₄	C₂×C₃×D₂₀	1	1	yes	characteristic
252	C₂×C₄	C₃×#709	1	1	yes	characteristic
253	C₂×C₄	C₂×C₃×C₂²⋊D₅	1	1	yes	characteristic
254	C₈	—	2	1	no	—
255	C₈	—	4	1	no	—
256	C₈	—	20	2	no	—
257	C₈	C₃×#686	1	2	yes	complemented
258	C₂×C₃	—	4	2	no	—
259	C₂×C₃	—	80	2	no	—
260	C₂×C₃	#65304	1	1	yes	characteristic
261	C₂×C₃	#65404	1	1	yes	characteristic
262	C₂×C₃	#65431	1	1	yes	characteristic
263	C₅	C₃×#1667	1	1	yes	characteristic, complemented, Sylow
264	C₂²	—	2	2	no	—
265	C₂²	—	4	1	no	—
266	C₂²	—	4	1	no	—
267	C₂²	—	40	2	no	—
268	C₂²	—	40	2	no	—
269	C₂²	—	40	2	no	—
270	C₂²	C₃×#4076	1	1	yes	characteristic
271	C₄	—	2	1	no	—
272	C₄	—	4	1	no	—
273	C₄	C₂×C₃×D₄₀	1	1	yes	characteristic
274	C₄	C₃×#4050	1	1	yes	characteristic
275	C₄	C₂×C₃×#686	1	1	yes	characteristic
276	C₄	C₃×#4095	1	1	yes	characteristic
277	C₃	#10601640	1	1	yes	characteristic, direct product factor, Sylow
278	C₂	—	4	2	no	—
279	C₂	—	80	2	no	—
280	C₂	C₃×#65304	1	1	yes	characteristic
281	C₂	C₃×#65404	1	1	yes	characteristic
282	C₂	C₃×#65431	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	#4076
commutator, cosocle	C₄²×C₅	C₂³×C₃
Frattini	C₄²	C₂²×C₃×D₅
Fitting	C₃×C₅×C₈×D₄	C₂
socle	C₂²×C₃×C₅	#121
hypercenter	C₃×C₈×D₄	D₅
hypocenter	C₅	C₃×#1667
LCS	G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ C₃×C₈×D₄

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×C₈×D₄	C₂	semidirect
2. C₃×#65304	C₂	semidirect
3. C₃×#64984	C₂	semidirect
4. C₃×#64705	C₂	semidirect
5. C₃×#64678	C₂	semidirect
6. #10601640	C₃	direct (N × H)
19. C₅×C₈×D₄	C₂×C₃	semidirect
20. #65304	C₂×C₃	semidirect
21. #64984	C₂×C₃	semidirect
22. #64705	C₂×C₃	semidirect
23. #64678	C₂×C₃	semidirect
42. C₃×C₈×D₄	D₅	semidirect
60. C₃×C₅×D₄	D₈	semidirect
74. C₃×C₅×C₈	D₈	semidirect
105. C₈×D₄	C₃×D₅	semidirect
140. C₅×D₄	C₃×D₈	semidirect
154. C₅×C₈	C₃×D₈	semidirect
180. C₃×D₄	D₄₀	semidirect
190. C₃×C₈	#686	semidirect
220. C₃×C₅	#1667	semidirect
247. D₄	C₃×D₄₀	semidirect
257. C₈	C₃×#686	semidirect
263. C₅	C₃×#1667	semidirect
277. C₃	#10601640	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×C₈×D₄	2
C₃×#65304	2
C₃×#64984	2
C₃×#64705	2
C₃×#64678	2
#10601640	3
C₃×#1667	5

Subgroups — C3×#10601640 #14291413

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10601640 #14291413