Subgroups — C₂²×C₅×#940 #14479961

Order 1920.

6000 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×C₅×D₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×C₅×#67	C₂	1	4	yes	maximal, complemented
3	C₂³×C₅×D₁₂	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅×#940	C₂	1	24	yes	maximal, direct product factor
5	C₂²×C₄×C₅×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₅×#127	—	3	1	no	maximal, Hall
7	C₂⁵×C₃×C₅	C₂²	1	2	yes	—
8	C₂⁴×C₅×D₃	C₂²	1	2	yes	—
9	C₂²×C₃×C₅×D₄	—	2	8	no	—
10	C₂²×C₃×C₅×D₄	C₂²	1	12	yes	complemented
11	C₂³×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
12	C₂²×C₅×#67	—	2	32	no	—
13	C₂²×C₅×#67	C₂²	1	48	yes	complemented
14	C₂³×C₅×Dic₃	C₂²	1	2	yes	—
15	C₂²×C₅×D₁₂	—	2	8	no	—
16	C₂²×C₅×D₁₂	C₂²	1	12	yes	complemented
17	C₅×#940	C₂²	1	64	yes	direct product factor
18	C₂×C₄×C₅×Dic₃	C₂²	1	12	yes	complemented
19	C₂²×#940	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
20	C₂³×C₅×D₄	—	3	4	no	—
21	C₂³×C₅×D₄	—	3	1	no	—
22	C₂³×C₅×D₄	D₃	1	1	yes	characteristic, complemented
23	C₂×C₅×#127	—	3	24	no	—
24	C₂²×C₄²×C₅	—	3	1	no	—
25	C₂⁴×C₃×C₅	—	2	16	no	—
26	C₂⁴×C₃×C₅	—	2	8	no	—
27	C₂⁴×C₃×C₅	C₂³	1	12	yes	—
28	C₂⁴×C₃×C₅	C₂³	1	1	yes	characteristic
29	C₂³×C₅×D₃	—	2	8	no	—
30	C₂³×C₅×D₃	—	2	16	no	—
31	C₂³×C₅×D₃	C₂³	1	12	yes	—
32	C₂×C₃×C₅×D₄	—	2	48	no	—
33	C₂×C₃×C₅×D₄	C₂³	1	16	yes	complemented
34	C₂²×C₃×C₄×C₅	D₄	1	8	yes	complemented
35	C₂²×C₃×C₄×C₅	C₂³	1	6	yes	—
36	C₂×C₅×#67	—	2	192	no	—
37	C₂×C₅×#67	C₂³	1	64	yes	complemented
38	C₂²×C₅×Dic₃	D₄	1	16	yes	complemented
39	C₂²×C₅×Dic₃	C₂³	1	12	yes	—
40	C₂×C₅×D₁₂	—	2	48	no	—
41	C₂×C₅×D₁₂	C₂³	1	16	yes	complemented
42	C₄×C₅×Dic₃	C₂³	1	16	yes	complemented
43	C₂³×C₃×D₄	C₂×C₅	1	1	yes	characteristic, complemented
44	C₂³×#67	C₂×C₅	1	4	yes	complemented
45	C₂³×D₁₂	C₂×C₅	1	1	yes	characteristic, complemented
46	C₂×#940	C₂×C₅	1	24	yes	direct product factor
47	C₂²×C₄×Dic₃	C₂×C₅	1	1	yes	characteristic, complemented
48	C₂⁵×C₅	—	3	2	no	—
49	C₂⁵×C₅	C₂×D₃	1	2	yes	—
50	C₂²×C₅×D₄	—	2	8	no	—
51	C₂²×C₅×D₄	—	3	48	no	—
52	C₂²×C₅×D₄	—	3	12	no	—
53	C₂²×C₅×D₄	—	6	32	no	—
54	C₂²×C₅×D₄	—	6	8	no	—
55	C₂²×C₅×D₄	C₂×D₃	1	12	yes	complemented
56	C₂³×C₄×C₅	—	3	2	no	—
57	C₂³×C₄×C₅	C₂×D₃	1	1	yes	characteristic
58	C₅×#127	—	3	64	no	—
59	C₂×C₄²×C₅	—	3	12	no	—
60	C₂²×#127	—	3	1	no	Sylow
61	C₂³×C₃×C₅	—	2	48	no	—
62	C₂³×C₃×C₅	—	2	24	no	—
63	C₂³×C₃×C₅	—	4	32	no	—
64	C₂³×C₃×C₅	C₂×D₄	1	8	yes	—
65	C₂³×C₃×C₅	C₂×D₄	1	4	yes	—
66	C₂³×C₃×C₅	C₂⁴	1	8	yes	—
67	C₂³×C₃×C₅	C₂⁴	1	3	yes	—
68	C₂²×C₅×D₃	—	2	24	no	—
69	C₂²×C₅×D₃	—	2	48	no	—
70	C₂²×C₅×D₃	—	4	32	no	—
71	C₂²×C₅×D₃	C₂⁴	1	8	yes	—
72	C₃×C₅×D₄	—	2	32	no	—
73	C₂×C₃×C₄×C₅	C₂×D₄	1	24	yes	complemented
74	C₂×C₃×C₄×C₅	C₂⁴	1	4	yes	—
75	C₅×#67	—	2	128	no	—
76	C₂×C₅×Dic₃	C₂×D₄	1	48	yes	complemented
77	C₂×C₅×Dic₃	C₂⁴	1	8	yes	—
78	C₅×D₁₂	—	2	32	no	—
79	C₂⁵×C₃	C₂²×C₅	1	2	yes	—
80	C₂⁴×D₃	C₂²×C₅	1	2	yes	—
81	C₂²×C₃×D₄	—	2	8	no	—
82	C₂²×C₃×D₄	C₂²×C₅	1	12	yes	complemented
83	C₂³×C₃×C₄	C₂²×C₅	1	1	yes	characteristic
84	C₂²×#67	—	2	32	no	—
85	C₂²×#67	C₂²×C₅	1	48	yes	complemented
86	C₂³×Dic₃	C₂²×C₅	1	2	yes	—
87	C₂²×D₁₂	—	2	8	no	—
88	C₂²×D₁₂	C₂²×C₅	1	12	yes	complemented
89	#940	C₂²×C₅	1	64	yes	direct product factor
90	C₂×C₄×Dic₃	C₂²×C₅	1	12	yes	complemented
91	C₂⁴×C₅	—	2	16	no	—
92	C₂⁴×C₅	—	2	8	no	—
93	C₂⁴×C₅	—	3	12	no	—
94	C₂⁴×C₅	—	6	16	no	—
95	C₂⁴×C₅	—	6	8	no	—
96	C₂⁴×C₅	C₂²×D₃	1	12	yes	—
97	C₂⁴×C₅	C₂²×D₃	1	1	yes	characteristic
98	C₂×C₅×D₄	—	2	48	no	—
99	C₂×C₅×D₄	—	3	64	no	—
100	C₂×C₅×D₄	—	3	16	no	—
101	C₂×C₅×D₄	—	6	192	no	—
102	C₂×C₅×D₄	—	6	48	no	—
103	C₂×C₅×D₄	C₂²×D₃	1	16	yes	complemented
104	C₂²×C₄×C₅	—	3	16	no	—
105	C₂²×C₄×C₅	—	3	12	no	—
106	C₂²×C₄×C₅	#67	1	8	yes	complemented
107	C₂²×C₄×C₅	C₂²×D₃	1	6	yes	—
108	C₄²×C₅	—	3	16	no	—
109	C₂³×D₄	—	3	4	no	—
110	C₂³×D₄	—	3	1	no	—
111	C₂³×D₄	C₅×D₃	1	1	yes	characteristic, complemented
112	C₂×#127	—	3	24	no	—
113	C₂²×C₄²	—	3	1	no	—
114	C₂²×C₃×C₅	—	2	16	no	—
115	C₂²×C₃×C₅	—	2	8	no	—
116	C₂²×C₃×C₅	—	4	48	no	—
117	C₂²×C₃×C₅	#127	1	16	yes	complemented
118	C₂²×C₃×C₅	C₂²×D₄	1	12	yes	—
119	C₂²×C₃×C₅	C₂²×D₄	1	6	yes	—
120	C₂²×C₃×C₅	C₂⁵	1	1	yes	characteristic
121	C₂×C₅×D₃	—	2	8	no	—
122	C₂×C₅×D₃	—	2	16	no	—
123	C₂×C₅×D₃	—	4	48	no	—
124	C₃×C₄×C₅	C₂²×D₄	1	8	yes	complemented
125	C₅×Dic₃	C₂²×D₄	1	16	yes	complemented
126	C₂⁴×C₃	—	2	16	no	—
127	C₂⁴×C₃	—	2	8	no	—
128	C₂⁴×C₃	C₂³×C₅	1	12	yes	—
129	C₂⁴×C₃	C₂³×C₅	1	1	yes	characteristic
130	C₂³×D₃	—	2	8	no	—
131	C₂³×D₃	—	2	16	no	—
132	C₂³×D₃	C₂³×C₅	1	12	yes	—
133	C₂×C₃×D₄	—	2	48	no	—
134	C₂×C₃×D₄	C₂³×C₅	1	16	yes	complemented
135	C₂²×C₃×C₄	C₅×D₄	1	8	yes	complemented
136	C₂²×C₃×C₄	C₂³×C₅	1	6	yes	—
137	C₂×#67	—	2	192	no	—
138	C₂×#67	C₂³×C₅	1	64	yes	complemented
139	C₂²×Dic₃	C₅×D₄	1	16	yes	complemented
140	C₂²×Dic₃	C₂³×C₅	1	12	yes	—
141	C₂×D₁₂	—	2	48	no	—
142	C₂×D₁₂	C₂³×C₅	1	16	yes	complemented
143	C₄×Dic₃	C₂³×C₅	1	16	yes	complemented
144	C₂³×C₅	—	2	48	no	—
145	C₂³×C₅	—	2	24	no	—
146	C₂³×C₅	—	3	8	no	—
147	C₂³×C₅	—	4	32	no	—
148	C₂³×C₅	—	6	48	no	—
149	C₂³×C₅	—	6	24	no	—
150	C₂³×C₅	—	12	32	no	—
151	C₂³×C₅	D₃×D₄	1	8	yes	—
152	C₂³×C₅	C₂×#67	1	4	yes	—
153	C₂³×C₅	C₂³×D₃	1	8	yes	—
154	C₂³×C₅	C₂³×D₃	1	3	yes	—
155	C₅×D₄	—	2	32	no	—
156	C₅×D₄	—	6	128	no	—
157	C₅×D₄	—	6	32	no	—
158	C₂×C₄×C₅	—	3	48	no	—
159	C₂×C₄×C₅	—	3	8	no	—
160	C₂×C₄×C₅	C₂×#67	1	24	yes	complemented
161	C₂×C₄×C₅	C₂³×D₃	1	4	yes	—
162	C₂⁵	—	3	2	no	—
163	C₂⁵	C₂×C₅×D₃	1	2	yes	—
164	C₂²×D₄	—	2	8	no	—
165	C₂²×D₄	—	3	48	no	—
166	C₂²×D₄	—	3	12	no	—
167	C₂²×D₄	—	6	32	no	—
168	C₂²×D₄	—	6	8	no	—
169	C₂²×D₄	C₂×C₅×D₃	1	12	yes	complemented
170	C₂³×C₄	—	3	2	no	—
171	C₂³×C₄	C₂×C₅×D₃	1	1	yes	characteristic
172	#127	—	3	64	no	—
173	C₂×C₄²	—	3	12	no	—
174	C₂×C₃×C₅	—	4	8	no	—
175	C₂×C₃×C₅	C₂×#127	1	12	yes	complemented
176	C₂×C₃×C₅	C₂³×D₄	1	2	yes	—
177	C₂×C₃×C₅	C₂³×D₄	1	1	yes	characteristic
178	C₅×D₃	—	4	8	no	—
179	C₂³×C₃	—	2	48	no	—
180	C₂³×C₃	—	2	24	no	—
181	C₂³×C₃	—	4	32	no	—
182	C₂³×C₃	C₂×C₅×D₄	1	8	yes	—
183	C₂³×C₃	C₂×C₅×D₄	1	4	yes	—
184	C₂³×C₃	C₂⁴×C₅	1	8	yes	—
185	C₂³×C₃	C₂⁴×C₅	1	3	yes	—
186	C₂²×D₃	—	2	24	no	—
187	C₂²×D₃	—	2	48	no	—
188	C₂²×D₃	—	4	32	no	—
189	C₂²×D₃	C₂⁴×C₅	1	8	yes	—
190	C₃×D₄	—	2	32	no	—
191	C₂×C₃×C₄	C₂×C₅×D₄	1	24	yes	complemented
192	C₂×C₃×C₄	C₂⁴×C₅	1	4	yes	—
193	#67	—	2	128	no	—
194	C₂×Dic₃	C₂×C₅×D₄	1	48	yes	complemented
195	C₂×Dic₃	C₂⁴×C₅	1	8	yes	—
196	D₁₂	—	2	32	no	—
197	C₂²×C₅	—	2	16	no	—
198	C₂²×C₅	—	2	8	no	—
199	C₂²×C₅	—	4	48	no	—
200	C₂²×C₅	—	6	16	no	—
201	C₂²×C₅	—	6	8	no	—
202	C₂²×C₅	—	12	48	no	—
203	C₂²×C₅	#940	1	16	yes	direct product factor
204	C₂²×C₅	C₂×D₃×D₄	1	12	yes	—
205	C₂²×C₅	C₂²×#67	1	6	yes	—
206	C₂²×C₅	C₂⁴×D₃	1	1	yes	characteristic
207	C₄×C₅	—	3	16	no	—
208	C₄×C₅	C₂²×#67	1	8	yes	complemented
209	C₂⁴	—	2	16	no	—
210	C₂⁴	—	2	8	no	—
211	C₂⁴	—	3	12	no	—
212	C₂⁴	—	6	16	no	—
213	C₂⁴	—	6	8	no	—
214	C₂⁴	C₂²×C₅×D₃	1	12	yes	—
215	C₂⁴	C₂²×C₅×D₃	1	1	yes	characteristic
216	C₂×D₄	—	2	48	no	—
217	C₂×D₄	—	3	64	no	—
218	C₂×D₄	—	3	16	no	—
219	C₂×D₄	—	6	192	no	—
220	C₂×D₄	—	6	48	no	—
221	C₂×D₄	C₂²×C₅×D₃	1	16	yes	complemented
222	C₂²×C₄	—	3	16	no	—
223	C₂²×C₄	—	3	12	no	—
224	C₂²×C₄	C₅×#67	1	8	yes	complemented
225	C₂²×C₄	C₂²×C₅×D₃	1	6	yes	—
226	C₄²	—	3	16	no	—
227	C₃×C₅	C₂²×#127	1	1	yes	characteristic, complemented, Hall
228	C₂²×C₃	—	2	16	no	—
229	C₂²×C₃	—	2	8	no	—
230	C₂²×C₃	—	4	48	no	—
231	C₂²×C₃	C₅×#127	1	16	yes	complemented
232	C₂²×C₃	C₂²×C₅×D₄	1	12	yes	—
233	C₂²×C₃	C₂²×C₅×D₄	1	6	yes	—
234	C₂²×C₃	C₂⁵×C₅	1	1	yes	characteristic
235	C₂×D₃	—	2	8	no	—
236	C₂×D₃	—	2	16	no	—
237	C₂×D₃	—	4	48	no	—
238	C₃×C₄	C₂²×C₅×D₄	1	8	yes	complemented
239	Dic₃	C₂²×C₅×D₄	1	16	yes	complemented
240	C₂×C₅	—	4	8	no	—
241	C₂×C₅	—	12	8	no	—
242	C₂×C₅	C₂×#940	1	12	yes	direct product factor
243	C₂×C₅	C₂²×D₃×D₄	1	2	yes	—
244	C₂×C₅	C₂³×#67	1	1	yes	characteristic
245	C₂³	—	2	48	no	—
246	C₂³	—	2	24	no	—
247	C₂³	—	3	8	no	—
248	C₂³	—	4	32	no	—
249	C₂³	—	6	48	no	—
250	C₂³	—	6	24	no	—
251	C₂³	—	12	32	no	—
252	C₂³	C₅×D₃×D₄	1	8	yes	—
253	C₂³	C₂×C₅×#67	1	4	yes	—
254	C₂³	C₂³×C₅×D₃	1	8	yes	—
255	C₂³	C₂³×C₅×D₃	1	3	yes	—
256	D₄	—	2	32	no	—
257	D₄	—	6	128	no	—
258	D₄	—	6	32	no	—
259	C₂×C₄	—	3	48	no	—
260	C₂×C₄	—	3	8	no	—
261	C₂×C₄	C₂×C₅×#67	1	24	yes	complemented
262	C₂×C₄	C₂³×C₅×D₃	1	4	yes	—
263	C₂×C₃	—	4	8	no	—
264	C₂×C₃	C₂×C₅×#127	1	12	yes	complemented
265	C₂×C₃	C₂³×C₅×D₄	1	2	yes	—
266	C₂×C₃	C₂³×C₅×D₄	1	1	yes	characteristic
267	D₃	—	4	8	no	—
268	C₅	C₂²×#940	1	1	yes	characteristic, direct product factor, Sylow
269	C₂²	—	2	16	no	—
270	C₂²	—	2	8	no	—
271	C₂²	—	4	48	no	—
272	C₂²	—	6	16	no	—
273	C₂²	—	6	8	no	—
274	C₂²	—	12	48	no	—
275	C₂²	C₅×#940	1	16	yes	direct product factor
276	C₂²	C₂×C₅×D₃×D₄	1	12	yes	—
277	C₂²	C₂²×C₅×#67	1	6	yes	—
278	C₂²	C₂⁴×C₅×D₃	1	1	yes	characteristic
279	C₄	—	3	16	no	—
280	C₄	C₂²×C₅×#67	1	8	yes	complemented
281	C₃	C₂²×C₅×#127	1	1	yes	characteristic, complemented, Sylow
282	C₂	—	4	8	no	—
283	C₂	—	12	8	no	—
284	C₂	C₂×C₅×#940	1	12	yes	direct product factor
285	C₂	C₂²×C₅×D₃×D₄	1	2	yes	—
286	C₂	C₂³×C₅×#67	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴×C₅	C₂²×D₃
commutator, cosocle	C₂²×C₃	C₂⁵×C₅
Frattini	C₂²	C₂⁴×C₅×D₃
Fitting	C₂³×C₃×C₅×D₄	C₂
socle	C₂⁴×C₃×C₅	C₂³
hypercenter	C₂³×C₅×D₄	D₃
hypocenter	C₃	C₂²×C₅×#127
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂⁴×C₅ ◁ C₂³×C₅×D₄

Complements

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

N	H ≅ G/N	type
1. C₂³×C₃×C₅×D₄	C₂	semidirect
2. C₂³×C₅×#67	C₂	semidirect
3. C₂³×C₅×D₁₂	C₂	semidirect
4. C₂×C₅×#940	C₂	direct (N × H)
5. C₂²×C₄×C₅×Dic₃	C₂	semidirect
10. C₂²×C₃×C₅×D₄	C₂²	semidirect
13. C₂²×C₅×#67	C₂²	semidirect
16. C₂²×C₅×D₁₂	C₂²	semidirect
17. C₅×#940	C₂²	direct (N × H)
18. C₂×C₄×C₅×Dic₃	C₂²	semidirect
19. C₂²×#940	C₅	direct (N × H)
22. C₂³×C₅×D₄	D₃	semidirect
33. C₂×C₃×C₅×D₄	C₂³	semidirect
34. C₂²×C₃×C₄×C₅	D₄	semidirect
37. C₂×C₅×#67	C₂³	semidirect
38. C₂²×C₅×Dic₃	D₄	semidirect
41. C₂×C₅×D₁₂	C₂³	semidirect
42. C₄×C₅×Dic₃	C₂³	semidirect
43. C₂³×C₃×D₄	C₂×C₅	semidirect
44. C₂³×#67	C₂×C₅	semidirect
45. C₂³×D₁₂	C₂×C₅	semidirect
46. C₂×#940	C₂×C₅	direct (N × H)
47. C₂²×C₄×Dic₃	C₂×C₅	semidirect
55. C₂²×C₅×D₄	C₂×D₃	semidirect
73. C₂×C₃×C₄×C₅	C₂×D₄	semidirect
76. C₂×C₅×Dic₃	C₂×D₄	semidirect
82. C₂²×C₃×D₄	C₂²×C₅	semidirect
85. C₂²×#67	C₂²×C₅	semidirect
88. C₂²×D₁₂	C₂²×C₅	semidirect
89. #940	C₂²×C₅	direct (N × H)
90. C₂×C₄×Dic₃	C₂²×C₅	semidirect
103. C₂×C₅×D₄	C₂²×D₃	semidirect
106. C₂²×C₄×C₅	#67	semidirect
111. C₂³×D₄	C₅×D₃	semidirect
117. C₂²×C₃×C₅	#127	semidirect
124. C₃×C₄×C₅	C₂²×D₄	semidirect
125. C₅×Dic₃	C₂²×D₄	semidirect
134. C₂×C₃×D₄	C₂³×C₅	semidirect
135. C₂²×C₃×C₄	C₅×D₄	semidirect
138. C₂×#67	C₂³×C₅	semidirect
139. C₂²×Dic₃	C₅×D₄	semidirect
142. C₂×D₁₂	C₂³×C₅	semidirect
143. C₄×Dic₃	C₂³×C₅	semidirect
160. C₂×C₄×C₅	C₂×#67	semidirect
169. C₂²×D₄	C₂×C₅×D₃	semidirect
175. C₂×C₃×C₅	C₂×#127	semidirect
191. C₂×C₃×C₄	C₂×C₅×D₄	semidirect
194. C₂×Dic₃	C₂×C₅×D₄	semidirect
203. C₂²×C₅	#940	direct (N × H)
208. C₄×C₅	C₂²×#67	semidirect
221. C₂×D₄	C₂²×C₅×D₃	semidirect
224. C₂²×C₄	C₅×#67	semidirect
227. C₃×C₅	C₂²×#127	semidirect
231. C₂²×C₃	C₅×#127	semidirect
238. C₃×C₄	C₂²×C₅×D₄	semidirect
239. Dic₃	C₂²×C₅×D₄	semidirect
242. C₂×C₅	C₂×#940	direct (N × H)
261. C₂×C₄	C₂×C₅×#67	semidirect
264. C₂×C₃	C₂×C₅×#127	semidirect
268. C₅	C₂²×#940	direct (N × H)
275. C₂²	C₅×#940	direct (N × H)
280. C₄	C₂²×C₅×#67	semidirect
281. C₃	C₂²×C₅×#127	semidirect
284. C₂	C₂×C₅×#940	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₃×C₅×D₄	2
C₂³×C₅×#67	2
C₂³×C₅×D₁₂	2
C₂×C₅×#940	2
C₂²×C₄×C₅×Dic₃	2
C₂²×C₅×#127	3
C₂²×#940	5

Subgroups — C22×C5×#940 #14479961

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₅×#940 #14479961