Subgroups — C₂×C₃×#64982 #14438731

Order 1920.

1312 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×#677	C₂	1	4	yes	maximal, complemented
2	C₂×C₃×#4020	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×#4015	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃×C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#64982	C₂	1	8	yes	maximal, direct product factor
6	C₂×#64982	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂²×C₃×D₂₀	—	2	2	no	—
8	C₂²×C₃×D₂₀	C₂²	1	2	yes	—
9	C₂²×C₃×Dic₁₀	C₂²	1	2	yes	—
10	C₂²×C₃×C₅×C₈	C₂²	1	2	yes	—
11	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₄⋊Dic₅	—	2	2	no	—
13	C₂×C₃×#677	—	2	16	no	—
14	C₂×C₃×#677	C₂²	1	16	yes	complemented
15	C₃×#4020	—	2	2	no	—
16	C₃×#4020	C₂²	1	4	yes	—
17	C₃×#4015	—	2	2	no	—
18	C₃×#4015	C₂²	1	4	yes	complemented
19	C₃×C₄×C₅×C₈	C₂²	1	4	yes	complemented
20	C₂×C₃×#492	—	5	1	no	maximal, Hall
21	C₂²×#677	C₂×C₃	1	4	yes	complemented
22	C₂×#4020	C₂×C₃	1	1	yes	characteristic
23	C₂×#4015	C₂×C₃	1	1	yes	characteristic, complemented
24	C₂×C₄×C₅×C₈	C₂×C₃	1	1	yes	characteristic, complemented
25	#64982	C₂×C₃	1	8	yes	direct product factor
26	C₂³×C₃×D₅	—	2	2	no	—
27	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
28	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
29	C₂²×C₃×Dic₅	—	2	2	no	—
30	C₂×C₃×D₂₀	—	2	8	no	—
31	C₂×C₃×D₂₀	—	2	8	no	—
32	C₂×C₃×D₂₀	—	2	2	no	—
33	C₂×C₃×D₂₀	—	4	8	no	—
34	C₂×C₃×D₂₀	C₂³	1	4	yes	—
35	C₂×C₃×Dic₁₀	—	2	8	no	—
36	C₂×C₃×Dic₁₀	—	2	2	no	—
37	C₂×C₃×Dic₁₀	C₂³	1	4	yes	—
38	C₂×C₃×C₅×C₈	D₄	1	8	yes	complemented
39	C₂×C₃×C₅×C₈	C₂³	1	4	yes	—
40	C₃×C₄²×C₅	D₄	1	2	yes	—
41	C₃×C₄²×C₅	C₂³	1	2	yes	—
42	C₃×C₄⋊Dic₅	—	2	8	no	—
43	C₃×#677	—	2	32	no	—
44	C₂²×C₃×SD₁₆	—	5	4	no	—
45	C₂×C₃×C₄⋊Q₈	—	5	1	no	—
46	C₂×C₃×#127	—	5	1	no	—
47	C₃×#492	—	5	8	no	—
48	C₂×C₃×C₄×C₈	D₅	1	1	yes	characteristic, complemented
49	C₂²×D₂₀	—	2	2	no	—
50	C₂²×D₂₀	C₂²×C₃	1	2	yes	—
51	C₂²×Dic₁₀	C₂²×C₃	1	2	yes	—
52	C₂²×C₅×C₈	C₂²×C₃	1	2	yes	—
53	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
54	C₂×C₄⋊Dic₅	—	2	2	no	—
55	C₂×#677	—	2	16	no	—
56	C₂×#677	C₂²×C₃	1	16	yes	complemented
57	#4020	—	2	2	no	—
58	#4020	C₂²×C₃	1	4	yes	—
59	#4015	—	2	2	no	—
60	#4015	C₂²×C₃	1	4	yes	complemented
61	C₄×C₅×C₈	C₂²×C₃	1	4	yes	complemented
62	C₂×#492	—	5	1	no	Sylow
63	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₃×D₅	—	2	4	no	—
65	C₂²×C₃×D₅	—	4	4	no	—
66	C₂²×C₃×D₅	—	4	8	no	—
67	C₂×C₃×C₄×C₅	SD₁₆	1	8	yes	complemented
68	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
69	C₂×C₃×C₄×C₅	C₂×D₄	1	4	yes	—
70	C₂×C₃×C₄×C₅	C₂×D₄	1	4	yes	—
71	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
72	C₂×C₃×Dic₅	—	2	4	no	—
73	C₂×C₃×Dic₅	—	4	4	no	—
74	C₃×D₂₀	—	2	8	no	—
75	C₃×D₂₀	—	4	4	no	—
76	C₃×D₂₀	—	4	16	no	—
77	C₃×Dic₁₀	—	2	8	no	—
78	C₃×Dic₁₀	—	4	4	no	—
79	C₃×C₅×C₈	C₂×D₄	1	8	yes	complemented
80	C₂²×C₃×Q₈	—	5	2	no	—
81	C₂²×C₃×D₄	—	5	2	no	—
82	C₂²×C₃×D₄	—	10	2	no	—
83	C₂×C₃×SD₁₆	—	5	16	no	—
84	C₂×C₃×SD₁₆	—	10	16	no	—
85	C₂²×C₃×C₈	C₂×D₅	1	2	yes	—
86	C₃×C₄⋊Q₈	—	5	4	no	—
87	C₃×C₄⋊Q₈	—	10	2	no	—
88	C₃×#127	—	5	4	no	—
89	C₃×#127	—	10	2	no	—
90	C₂×C₃×C₄⋊C₄	—	10	2	no	—
91	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
92	C₃×C₄×C₈	C₂×D₅	1	4	yes	complemented
93	C₂³×D₅	—	2	2	no	—
94	C₂²×C₄×C₅	C₃×D₄	1	2	yes	—
95	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
96	C₂²×Dic₅	—	2	2	no	—
97	C₂×D₂₀	—	2	8	no	—
98	C₂×D₂₀	—	2	8	no	—
99	C₂×D₂₀	—	2	2	no	—
100	C₂×D₂₀	—	4	8	no	—
101	C₂×D₂₀	C₂³×C₃	1	4	yes	—
102	C₂×Dic₁₀	—	2	8	no	—
103	C₂×Dic₁₀	—	2	2	no	—
104	C₂×Dic₁₀	C₂³×C₃	1	4	yes	—
105	C₂×C₅×C₈	C₃×D₄	1	8	yes	complemented
106	C₂×C₅×C₈	C₂³×C₃	1	4	yes	—
107	C₄²×C₅	C₃×D₄	1	2	yes	—
108	C₄²×C₅	C₂³×C₃	1	2	yes	—
109	C₄⋊Dic₅	—	2	8	no	—
110	#677	—	2	32	no	—
111	C₂²×SD₁₆	—	5	4	no	—
112	C₂×C₄⋊Q₈	—	5	1	no	—
113	C₂×#127	—	5	1	no	—
114	#492	—	5	8	no	—
115	C₂×C₄×C₈	C₃×D₅	1	1	yes	characteristic, complemented
116	C₂²×C₃×C₅	#127	1	2	yes	—
117	C₂²×C₃×C₅	C₂×SD₁₆	1	4	yes	—
118	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
119	C₂×C₃×D₅	—	4	4	no	—
120	C₂×C₃×D₅	—	4	8	no	—
121	C₂×C₃×D₅	—	8	8	no	—
122	C₃×C₄×C₅	#127	1	2	yes	—
123	C₃×C₄×C₅	C₂×SD₁₆	1	8	yes	complemented
124	C₃×C₄×C₅	C₂²×D₄	1	2	yes	—
125	C₃×Dic₅	—	4	4	no	—
126	C₂⁴×C₃	—	10	2	no	—
127	C₂×C₃×Q₈	—	5	4	no	—
128	C₂×C₃×Q₈	—	10	2	no	—
129	C₂×C₃×Q₈	—	10	8	no	—
130	C₂×C₃×D₄	—	5	4	no	—
131	C₂×C₃×D₄	—	10	8	no	—
132	C₂×C₃×D₄	—	10	8	no	—
133	C₂×C₃×D₄	—	10	2	no	—
134	C₂×C₃×D₄	—	20	8	no	—
135	C₂²×C₃×C₄	—	10	2	no	—
136	C₂²×C₃×C₄	D₂₀	1	2	yes	—
137	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
138	C₃×SD₁₆	—	10	32	no	—
139	C₂×C₃×C₈	D₂₀	1	8	yes	complemented
140	C₂×C₃×C₈	C₂²×D₅	1	4	yes	—
141	C₃×C₄⋊C₄	—	10	8	no	—
142	C₃×C₄²	D₂₀	1	2	yes	—
143	C₃×C₄²	C₂²×D₅	1	2	yes	—
144	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
145	C₂²×D₅	—	2	4	no	—
146	C₂²×D₅	—	4	4	no	—
147	C₂²×D₅	—	4	8	no	—
148	C₂×C₄×C₅	C₃×SD₁₆	1	8	yes	complemented
149	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
150	C₂×C₄×C₅	C₂×C₃×D₄	1	4	yes	—
151	C₂×C₄×C₅	C₂×C₃×D₄	1	4	yes	—
152	C₂×C₄×C₅	C₂⁴×C₃	1	1	yes	characteristic
153	C₂×Dic₅	—	2	4	no	—
154	C₂×Dic₅	—	4	4	no	—
155	D₂₀	—	2	8	no	—
156	D₂₀	—	4	4	no	—
157	D₂₀	—	4	16	no	—
158	Dic₁₀	—	2	8	no	—
159	Dic₁₀	—	4	4	no	—
160	C₅×C₈	C₂×C₃×D₄	1	8	yes	complemented
161	C₂²×Q₈	—	5	2	no	—
162	C₂²×D₄	—	5	2	no	—
163	C₂²×D₄	—	10	2	no	—
164	C₂×SD₁₆	—	5	16	no	—
165	C₂×SD₁₆	—	10	16	no	—
166	C₂²×C₈	C₂×C₃×D₅	1	2	yes	—
167	C₄⋊Q₈	—	5	4	no	—
168	C₄⋊Q₈	—	10	2	no	—
169	#127	—	5	4	no	—
170	#127	—	10	2	no	—
171	C₂×C₄⋊C₄	—	10	2	no	—
172	C₂×C₄²	C₂×C₃×D₅	1	1	yes	characteristic
173	C₄×C₈	C₂×C₃×D₅	1	4	yes	complemented
174	C₂×C₃×C₅	#492	1	4	yes	complemented
175	C₂×C₃×C₅	C₂×#127	1	1	yes	characteristic
176	C₂×C₃×C₅	C₂²×SD₁₆	1	2	yes	—
177	C₃×D₅	—	8	4	no	—
178	C₂³×C₃	—	10	4	no	—
179	C₂³×C₃	—	20	4	no	—
180	C₂³×C₃	—	20	8	no	—
181	C₂³×C₃	C₂×D₂₀	1	1	yes	characteristic
182	C₃×Q₈	—	10	8	no	—
183	C₃×Q₈	—	20	4	no	—
184	C₃×D₄	—	10	8	no	—
185	C₃×D₄	—	20	4	no	—
186	C₃×D₄	—	20	16	no	—
187	C₂×C₃×C₄	—	10	4	no	—
188	C₂×C₃×C₄	—	20	4	no	—
189	C₂×C₃×C₄	#677	1	8	yes	complemented
190	C₂×C₃×C₄	C₂×D₂₀	1	4	yes	—
191	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
192	C₂×C₃×C₄	C₂×D₂₀	1	4	yes	—
193	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
194	C₃×C₈	C₂×D₂₀	1	8	yes	complemented
195	C₂²×C₅	C₃×#127	1	2	yes	—
196	C₂²×C₅	C₂×C₃×SD₁₆	1	4	yes	—
197	C₂²×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
198	C₂×D₅	—	4	8	no	—
199	C₂×D₅	—	4	4	no	—
200	C₂×D₅	—	8	8	no	—
201	C₄×C₅	C₃×#127	1	2	yes	—
202	C₄×C₅	C₂×C₃×SD₁₆	1	8	yes	complemented
203	C₄×C₅	C₂²×C₃×D₄	1	2	yes	—
204	Dic₅	—	4	4	no	—
205	C₂⁴	—	10	2	no	—
206	C₂×Q₈	—	5	4	no	—
207	C₂×Q₈	—	10	2	no	—
208	C₂×Q₈	—	10	8	no	—
209	C₂×D₄	—	5	4	no	—
210	C₂×D₄	—	10	8	no	—
211	C₂×D₄	—	10	8	no	—
212	C₂×D₄	—	10	2	no	—
213	C₂×D₄	—	20	8	no	—
214	C₂²×C₄	—	10	2	no	—
215	C₂²×C₄	C₃×D₂₀	1	2	yes	—
216	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
217	SD₁₆	—	10	32	no	—
218	C₂×C₈	C₃×D₂₀	1	8	yes	complemented
219	C₂×C₈	C₂²×C₃×D₅	1	4	yes	—
220	C₄⋊C₄	—	10	8	no	—
221	C₄²	C₃×D₂₀	1	2	yes	—
222	C₄²	C₂²×C₃×D₅	1	2	yes	—
223	C₃×C₅	C₂×#492	1	1	yes	characteristic, complemented, Hall
224	C₂²×C₃	—	20	4	no	—
225	C₂²×C₃	—	20	8	no	—
226	C₂²×C₃	—	40	8	no	—
227	C₂²×C₃	#4020	1	2	yes	—
228	C₂²×C₃	C₂×#677	1	4	yes	—
229	C₂²×C₃	C₂²×D₂₀	1	1	yes	characteristic
230	C₃×C₄	—	20	4	no	—
231	C₃×C₄	#4020	1	2	yes	—
232	C₃×C₄	C₂×#677	1	8	yes	complemented
233	C₃×C₄	C₂²×D₂₀	1	2	yes	—
234	C₂×C₅	C₃×#492	1	4	yes	complemented
235	C₂×C₅	C₂×C₃×#127	1	1	yes	characteristic
236	C₂×C₅	C₂²×C₃×SD₁₆	1	2	yes	—
237	D₅	—	8	4	no	—
238	C₂³	—	10	4	no	—
239	C₂³	—	20	4	no	—
240	C₂³	—	20	8	no	—
241	C₂³	C₂×C₃×D₂₀	1	1	yes	characteristic
242	Q₈	—	10	8	no	—
243	Q₈	—	20	4	no	—
244	D₄	—	10	8	no	—
245	D₄	—	20	4	no	—
246	D₄	—	20	16	no	—
247	C₂×C₄	—	10	4	no	—
248	C₂×C₄	—	20	4	no	—
249	C₂×C₄	C₃×#677	1	8	yes	complemented
250	C₂×C₄	C₂×C₃×D₂₀	1	4	yes	—
251	C₂×C₄	C₂×C₃×D₂₀	1	1	yes	characteristic
252	C₂×C₄	C₂×C₃×D₂₀	1	4	yes	—
253	C₂×C₄	C₂³×C₃×D₅	1	1	yes	characteristic
254	C₈	C₂×C₃×D₂₀	1	8	yes	complemented
255	C₂×C₃	—	40	4	no	—
256	C₂×C₃	#64982	1	4	yes	direct product factor
257	C₂×C₃	C₂×#4020	1	1	yes	characteristic
258	C₂×C₃	C₂²×#677	1	2	yes	—
259	C₅	C₂×C₃×#492	1	1	yes	characteristic, complemented, Sylow
260	C₂²	—	20	4	no	—
261	C₂²	—	20	8	no	—
262	C₂²	—	40	8	no	—
263	C₂²	C₃×#4020	1	2	yes	—
264	C₂²	C₂×C₃×#677	1	4	yes	—
265	C₂²	C₂²×C₃×D₂₀	1	1	yes	characteristic
266	C₄	—	20	4	no	—
267	C₄	C₃×#4020	1	2	yes	—
268	C₄	C₂×C₃×#677	1	8	yes	complemented
269	C₄	C₂²×C₃×D₂₀	1	2	yes	—
270	C₃	C₂×#64982	1	1	yes	characteristic, direct product factor, Sylow
271	C₂	—	40	4	no	—
272	C₂	C₃×#64982	1	4	yes	direct product factor
273	C₂	C₂×C₃×#4020	1	1	yes	characteristic
274	C₂	C₂²×C₃×#677	1	2	yes	—

	subgroup	G/H
center	C₂³×C₃	C₂×D₂₀
commutator, cosocle	C₂×C₄×C₅	C₂⁴×C₃
Frattini	C₂×C₄	C₂³×C₃×D₅
Fitting	C₂×C₃×C₄×C₅×C₈	C₂
socle	C₂³×C₃×C₅	C₂×D₄
prime radical	C₂×C₃×#4020	C₂
hypercenter	C₂×C₃×C₄×C₈	D₅
hypocenter	C₅	C₂×C₃×#492
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×C₄² ◁ C₂×C₃×C₄×C₈

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃×#677	C₂	semidirect
3. C₂×C₃×#4015	C₂	semidirect
4. C₂×C₃×C₄×C₅×C₈	C₂	semidirect
5. C₃×#64982	C₂	direct (N × H)
6. C₂×#64982	C₃	direct (N × H)
14. C₂×C₃×#677	C₂²	semidirect
18. C₃×#4015	C₂²	semidirect
19. C₃×C₄×C₅×C₈	C₂²	semidirect
21. C₂²×#677	C₂×C₃	semidirect
23. C₂×#4015	C₂×C₃	semidirect
24. C₂×C₄×C₅×C₈	C₂×C₃	semidirect
25. #64982	C₂×C₃	direct (N × H)
38. C₂×C₃×C₅×C₈	D₄	semidirect
48. C₂×C₃×C₄×C₈	D₅	semidirect
56. C₂×#677	C₂²×C₃	semidirect
60. #4015	C₂²×C₃	semidirect
61. C₄×C₅×C₈	C₂²×C₃	semidirect
67. C₂×C₃×C₄×C₅	SD₁₆	semidirect
79. C₃×C₅×C₈	C₂×D₄	semidirect
92. C₃×C₄×C₈	C₂×D₅	semidirect
105. C₂×C₅×C₈	C₃×D₄	semidirect
115. C₂×C₄×C₈	C₃×D₅	semidirect
123. C₃×C₄×C₅	C₂×SD₁₆	semidirect
139. C₂×C₃×C₈	D₂₀	semidirect
148. C₂×C₄×C₅	C₃×SD₁₆	semidirect
160. C₅×C₈	C₂×C₃×D₄	semidirect
173. C₄×C₈	C₂×C₃×D₅	semidirect
174. C₂×C₃×C₅	#492	semidirect
189. C₂×C₃×C₄	#677	semidirect
194. C₃×C₈	C₂×D₂₀	semidirect
202. C₄×C₅	C₂×C₃×SD₁₆	semidirect
218. C₂×C₈	C₃×D₂₀	semidirect
223. C₃×C₅	C₂×#492	semidirect
232. C₃×C₄	C₂×#677	semidirect
234. C₂×C₅	C₃×#492	semidirect
249. C₂×C₄	C₃×#677	semidirect
254. C₈	C₂×C₃×D₂₀	semidirect
256. C₂×C₃	#64982	direct (N × H)
259. C₅	C₂×C₃×#492	semidirect
268. C₄	C₂×C₃×#677	semidirect
270. C₃	C₂×#64982	direct (N × H)
272. C₂	C₃×#64982	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₃×#677	2
C₂×C₃×#4020	2
C₂×C₃×#4015	2
C₂×C₃×C₄×C₅×C₈	2
C₃×#64982	2
C₂×#64982	3
C₂×C₃×#492	5

Subgroups — C2×C3×#64982 #14438731

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#64982 #14438731