Subgroups — C₂×C₅×#1990 #11944036

Order 1280.

1550 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#384	C₂	1	1	yes	maximal, characteristic
4	C₅×#1990	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₅×#340	C₂	1	4	yes	maximal
6	C₂³×C₅×Q₈	—	2	1	no	—
7	C₂³×C₅×C₈	C₂²	1	2	yes	—
8	C₂×C₅×C₄⋊Q₈	—	2	2	no	—
9	C₂×C₅×C₄⋊Q₈	—	2	4	no	—
10	C₂×C₅×C₄⋊Q₈	C₄	1	8	yes	complemented
11	C₂×C₅×C₄⋊Q₈	C₂²	1	4	yes	—
12	C₂²×C₅×C₄⋊C₄	—	2	1	no	—
13	C₂²×C₅×C₄⋊C₄	—	2	1	no	—
14	C₂²×C₅×C₄⋊C₄	—	2	1	no	—
15	C₂²×C₅×C₄⋊C₄	C₂²	1	2	yes	—
16	C₂²×C₅×C₄⋊C₄	C₂²	1	2	yes	—
17	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₅×C₈	—	2	4	no	—
19	C₂×C₄×C₅×C₈	C₂²	1	4	yes	—
20	C₅×#384	—	2	2	no	—
21	C₅×#384	C₂²	1	4	yes	—
22	C₅×#340	C₂²	1	16	yes	—
23	C₂×#1990	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
24	C₂²×C₅×Q₈	—	2	8	no	—
25	C₂²×C₅×Q₈	—	2	4	no	—
26	C₂²×C₅×Q₈	—	4	4	no	—
27	C₂²×C₅×Q₈	—	4	4	no	—
28	C₂³×C₄×C₅	—	2	2	no	—
29	C₂³×C₄×C₅	—	2	2	no	—
30	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
31	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
32	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
33	C₂²×C₅×C₈	—	2	4	no	—
34	C₂²×C₅×C₈	—	2	4	no	—
35	C₂²×C₅×C₈	Q₈	1	8	yes	complemented
36	C₂²×C₅×C₈	C₂³	1	4	yes	—
37	C₅×C₄⋊Q₈	—	2	16	no	—
38	C₅×C₄⋊Q₈	—	2	8	no	—
39	C₅×C₄⋊Q₈	C₂×C₄	1	16	yes	complemented
40	C₂×C₅×C₄⋊C₄	—	2	8	no	—
41	C₂×C₅×C₄⋊C₄	—	2	4	no	—
42	C₂×C₅×C₄⋊C₄	—	2	2	no	—
43	C₂×C₅×C₄⋊C₄	—	2	4	no	—
44	C₂×C₅×C₄⋊C₄	—	2	8	no	—
45	C₂×C₅×C₄⋊C₄	—	2	2	no	—
46	C₂×C₅×C₄⋊C₄	—	2	4	no	—
47	C₂×C₅×C₄⋊C₄	—	2	4	no	—
48	C₂×C₅×C₄⋊C₄	—	4	4	no	—
49	C₂×C₅×C₄⋊C₄	—	4	4	no	—
50	C₂×C₅×C₄⋊C₄	C₂×C₄	1	8	yes	—
51	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
52	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
53	C₂×C₄²×C₅	C₂×C₄	1	4	yes	—
54	C₂×C₄²×C₅	D₄	1	2	yes	—
55	C₂×C₄²×C₅	D₄	1	4	yes	—
56	C₂×C₄²×C₅	C₂³	1	2	yes	—
57	C₄×C₅×C₈	—	2	8	no	—
58	C₂²×C₄⋊Q₈	C₂×C₅	1	1	yes	characteristic
59	C₂²×C₄×C₈	C₂×C₅	1	1	yes	characteristic
60	C₂×#384	C₂×C₅	1	1	yes	characteristic
61	#1990	C₂×C₅	1	8	yes	direct product factor
62	C₂×#340	C₂×C₅	1	4	yes	—
63	C₂⁴×C₅	C₂×D₄	1	1	yes	characteristic
64	C₂×C₅×Q₈	—	2	16	no	—
65	C₂×C₅×Q₈	—	4	32	no	—
66	C₂×C₅×Q₈	—	4	8	no	—
67	C₂×C₅×Q₈	—	4	8	no	—
68	C₂²×C₄×C₅	—	2	8	no	—
69	C₂²×C₄×C₅	—	2	4	no	—
70	C₂²×C₄×C₅	—	2	4	no	—
71	C₂²×C₄×C₅	—	4	4	no	—
72	C₂²×C₄×C₅	—	4	4	no	—
73	C₂²×C₄×C₅	—	4	4	no	—
74	C₂²×C₄×C₅	—	4	4	no	—
75	C₂²×C₄×C₅	—	4	4	no	—
76	C₂²×C₄×C₅	C₂²⋊C₄	1	4	yes	—
77	C₂²×C₄×C₅	C₂²⋊C₄	1	4	yes	—
78	C₂²×C₄×C₅	D₈	1	4	yes	—
79	C₂²×C₄×C₅	SD₁₆	1	4	yes	—
80	C₂²×C₄×C₅	SD₁₆	1	4	yes	—
81	C₂²×C₄×C₅	Q₁₆	1	4	yes	—
82	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
83	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
85	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
86	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
87	C₂²×C₄×C₅	C₂×Q₈	1	4	yes	—
88	C₂²×C₄×C₅	#41	1	4	yes	—
89	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
90	C₂×C₅×C₈	—	2	16	no	—
91	C₂×C₅×C₈	—	2	4	no	—
92	C₂×C₅×C₈	—	2	4	no	—
93	C₂×C₅×C₈	C₂×Q₈	1	8	yes	complemented
94	C₅×C₄⋊C₄	—	2	16	no	—
95	C₅×C₄⋊C₄	—	2	8	no	—
96	C₅×C₄⋊C₄	—	2	4	no	—
97	C₅×C₄⋊C₄	—	2	8	no	—
98	C₅×C₄⋊C₄	—	4	4	no	—
99	C₅×C₄⋊C₄	—	4	8	no	—
100	C₅×C₄⋊C₄	—	4	8	no	—
101	C₅×C₄⋊C₄	C₂²×C₄	1	8	yes	—
102	C₄²×C₅	C₂²⋊C₄	1	8	yes	—
103	C₄²×C₅	C₂²×C₄	1	4	yes	—
104	C₄²×C₅	C₂×D₄	1	4	yes	—
105	C₂³×Q₈	—	2	1	no	—
106	C₂³×C₈	C₂²×C₅	1	2	yes	—
107	C₂×C₄⋊Q₈	—	2	2	no	—
108	C₂×C₄⋊Q₈	—	2	4	no	—
109	C₂×C₄⋊Q₈	C₄×C₅	1	8	yes	complemented
110	C₂×C₄⋊Q₈	C₂²×C₅	1	4	yes	—
111	C₂²×C₄⋊C₄	—	2	1	no	—
112	C₂²×C₄⋊C₄	—	2	1	no	—
113	C₂²×C₄⋊C₄	—	2	1	no	—
114	C₂²×C₄⋊C₄	C₂²×C₅	1	2	yes	—
115	C₂²×C₄⋊C₄	C₂²×C₅	1	2	yes	—
116	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
117	C₂×C₄×C₈	—	2	4	no	—
118	C₂×C₄×C₈	C₂²×C₅	1	4	yes	—
119	#384	—	2	2	no	—
120	#384	C₂²×C₅	1	4	yes	—
121	#340	C₂²×C₅	1	16	yes	—
122	C₂³×C₅	C₂×C₂²⋊C₄	1	2	yes	—
123	C₂³×C₅	#124	1	2	yes	—
124	C₂³×C₅	C₄⋊Q₈	1	2	yes	—
125	C₂³×C₅	C₂×D₈	1	2	yes	—
126	C₂³×C₅	C₂×SD₁₆	1	2	yes	—
127	C₂³×C₅	C₂×SD₁₆	1	2	yes	—
128	C₂³×C₅	C₂×Q₁₆	1	2	yes	—
129	C₂³×C₅	C₂²×D₄	1	1	yes	characteristic
130	C₅×Q₈	—	4	32	no	—
131	C₂×C₄×C₅	—	2	8	no	—
132	C₂×C₄×C₅	—	4	4	no	—
133	C₂×C₄×C₅	—	4	4	no	—
134	C₂×C₄×C₅	—	4	16	no	—
135	C₂×C₄×C₅	—	4	4	no	—
136	C₂×C₄×C₅	—	4	4	no	—
137	C₂×C₄×C₅	—	4	4	no	—
138	C₂×C₄×C₅	—	8	8	no	—
139	C₂×C₄×C₅	#102	1	16	yes	—
140	C₂×C₄×C₅	#103	1	16	yes	—
141	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	2	yes	—
142	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	4	yes	—
143	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	4	yes	—
144	C₂×C₄×C₅	C₄×Q₈	1	8	yes	—
145	C₂×C₄×C₅	C₂²⋊Q₈	1	8	yes	—
146	C₂×C₄×C₅	#124	1	2	yes	—
147	C₂×C₄×C₅	C₄⋊Q₈	1	2	yes	—
148	C₂×C₄×C₅	C₂×D₈	1	4	yes	—
149	C₂×C₄×C₅	C₂×SD₁₆	1	4	yes	—
150	C₂×C₄×C₅	C₂×SD₁₆	1	4	yes	—
151	C₂×C₄×C₅	C₂×Q₁₆	1	4	yes	—
152	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
153	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
154	C₂×C₄×C₅	C₂²×Q₈	1	2	yes	—
155	C₂×C₄×C₅	C₂×#41	1	2	yes	—
156	C₅×C₈	—	2	8	no	—
157	C₂²×Q₈	—	2	8	no	—
158	C₂²×Q₈	—	2	4	no	—
159	C₂²×Q₈	—	4	4	no	—
160	C₂²×Q₈	—	4	4	no	—
161	C₂³×C₄	—	2	2	no	—
162	C₂³×C₄	—	2	2	no	—
163	C₂³×C₄	C₅×D₄	1	1	yes	characteristic
164	C₂³×C₄	C₅×D₄	1	1	yes	characteristic
165	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
166	C₂²×C₈	—	2	4	no	—
167	C₂²×C₈	—	2	4	no	—
168	C₂²×C₈	C₅×Q₈	1	8	yes	complemented
169	C₂²×C₈	C₂³×C₅	1	4	yes	—
170	C₄⋊Q₈	—	2	16	no	—
171	C₄⋊Q₈	—	2	8	no	—
172	C₄⋊Q₈	C₂×C₄×C₅	1	16	yes	complemented
173	C₂×C₄⋊C₄	—	2	8	no	—
174	C₂×C₄⋊C₄	—	2	4	no	—
175	C₂×C₄⋊C₄	—	2	2	no	—
176	C₂×C₄⋊C₄	—	2	4	no	—
177	C₂×C₄⋊C₄	—	2	8	no	—
178	C₂×C₄⋊C₄	—	2	4	no	—
179	C₂×C₄⋊C₄	—	2	4	no	—
180	C₂×C₄⋊C₄	—	2	2	no	—
181	C₂×C₄⋊C₄	—	4	4	no	—
182	C₂×C₄⋊C₄	—	4	4	no	—
183	C₂×C₄⋊C₄	C₂×C₄×C₅	1	8	yes	—
184	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
185	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
186	C₂×C₄²	C₂×C₄×C₅	1	4	yes	—
187	C₂×C₄²	C₅×D₄	1	2	yes	—
188	C₂×C₄²	C₅×D₄	1	4	yes	—
189	C₂×C₄²	C₂³×C₅	1	2	yes	—
190	C₄×C₈	—	2	8	no	—
191	C₂²×C₅	#391	1	4	yes	—
192	C₂²×C₅	C₂×#102	1	4	yes	—
193	C₂²×C₅	C₂×#103	1	4	yes	—
194	C₂²×C₅	#486	1	4	yes	—
195	C₂²×C₅	#487	1	4	yes	—
196	C₂²×C₅	#498	1	4	yes	—
197	C₂²×C₅	#500	1	4	yes	—
198	C₂²×C₅	C₂²×C₂²⋊C₄	1	1	yes	characteristic
199	C₂²×C₅	C₂×#124	1	1	yes	characteristic
200	C₂²×C₅	C₂×C₄⋊Q₈	1	1	yes	characteristic
201	C₂²×C₅	C₂²×D₈	1	1	yes	characteristic
202	C₂²×C₅	C₂²×SD₁₆	1	1	yes	characteristic
203	C₂²×C₅	C₂²×SD₁₆	1	1	yes	characteristic
204	C₂²×C₅	C₂²×Q₁₆	1	1	yes	characteristic
205	C₄×C₅	—	4	8	no	—
206	C₄×C₅	—	8	4	no	—
207	C₄×C₅	#391	1	4	yes	—
208	C₄×C₅	C₂×#102	1	8	yes	—
209	C₄×C₅	C₂×#103	1	8	yes	—
210	C₄×C₅	C₂×C₄×Q₈	1	2	yes	—
211	C₄×C₅	C₂×C₂²⋊Q₈	1	2	yes	—
212	C₂⁴	C₂×C₅×D₄	1	1	yes	characteristic
213	C₂×Q₈	—	2	16	no	—
214	C₂×Q₈	—	4	8	no	—
215	C₂×Q₈	—	4	8	no	—
216	C₂×Q₈	—	4	32	no	—
217	C₂²×C₄	—	2	8	no	—
218	C₂²×C₄	—	2	4	no	—
219	C₂²×C₄	—	2	4	no	—
220	C₂²×C₄	—	4	4	no	—
221	C₂²×C₄	—	4	4	no	—
222	C₂²×C₄	—	4	4	no	—
223	C₂²×C₄	—	4	4	no	—
224	C₂²×C₄	—	4	4	no	—
225	C₂²×C₄	C₅×C₂²⋊C₄	1	4	yes	—
226	C₂²×C₄	C₅×C₂²⋊C₄	1	4	yes	—
227	C₂²×C₄	C₅×D₈	1	4	yes	—
228	C₂²×C₄	C₅×SD₁₆	1	4	yes	—
229	C₂²×C₄	C₅×SD₁₆	1	4	yes	—
230	C₂²×C₄	C₅×Q₁₆	1	4	yes	—
231	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
232	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
233	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
234	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
235	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
236	C₂²×C₄	C₂×C₅×Q₈	1	4	yes	—
237	C₂²×C₄	C₅×#41	1	4	yes	—
238	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
239	C₂×C₈	—	2	16	no	—
240	C₂×C₈	—	2	4	no	—
241	C₂×C₈	—	2	4	no	—
242	C₂×C₈	C₂×C₅×Q₈	1	8	yes	complemented
243	C₄⋊C₄	—	2	16	no	—
244	C₄⋊C₄	—	2	8	no	—
245	C₄⋊C₄	—	2	4	no	—
246	C₄⋊C₄	—	2	8	no	—
247	C₄⋊C₄	—	4	4	no	—
248	C₄⋊C₄	—	4	8	no	—
249	C₄⋊C₄	—	4	8	no	—
250	C₄⋊C₄	C₂²×C₄×C₅	1	8	yes	—
251	C₄²	C₅×C₂²⋊C₄	1	8	yes	—
252	C₄²	C₂²×C₄×C₅	1	4	yes	—
253	C₄²	C₂×C₅×D₄	1	4	yes	—
254	C₂×C₅	#1990	1	8	yes	direct product factor
255	C₂×C₅	C₂×#391	1	1	yes	characteristic
256	C₂×C₅	C₂²×#102	1	1	yes	characteristic
257	C₂×C₅	C₂²×#103	1	1	yes	characteristic
258	C₂×C₅	C₂×#486	1	1	yes	characteristic
259	C₂×C₅	C₂×#487	1	1	yes	characteristic
260	C₂×C₅	C₂×#498	1	1	yes	characteristic
261	C₂×C₅	C₂×#500	1	1	yes	characteristic
262	C₂³	C₂×C₅×C₂²⋊C₄	1	2	yes	—
263	C₂³	C₅×#124	1	2	yes	—
264	C₂³	C₅×C₄⋊Q₈	1	2	yes	—
265	C₂³	C₂×C₅×D₈	1	2	yes	—
266	C₂³	C₂×C₅×SD₁₆	1	2	yes	—
267	C₂³	C₂×C₅×SD₁₆	1	2	yes	—
268	C₂³	C₂×C₅×Q₁₆	1	2	yes	—
269	C₂³	C₂²×C₅×D₄	1	1	yes	characteristic
270	Q₈	—	4	32	no	—
271	C₂×C₄	—	2	8	no	—
272	C₂×C₄	—	4	16	no	—
273	C₂×C₄	—	4	4	no	—
274	C₂×C₄	—	4	4	no	—
275	C₂×C₄	—	4	4	no	—
276	C₂×C₄	—	4	4	no	—
277	C₂×C₄	—	4	4	no	—
278	C₂×C₄	—	8	8	no	—
279	C₂×C₄	C₅×#102	1	16	yes	—
280	C₂×C₄	C₅×#103	1	16	yes	—
281	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	2	yes	—
282	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	4	yes	—
283	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	4	yes	—
284	C₂×C₄	C₄×C₅×Q₈	1	8	yes	—
285	C₂×C₄	C₅×C₂²⋊Q₈	1	8	yes	—
286	C₂×C₄	C₅×#124	1	2	yes	—
287	C₂×C₄	C₅×C₄⋊Q₈	1	2	yes	—
288	C₂×C₄	C₂×C₅×D₈	1	4	yes	—
289	C₂×C₄	C₂×C₅×SD₁₆	1	4	yes	—
290	C₂×C₄	C₂×C₅×SD₁₆	1	4	yes	—
291	C₂×C₄	C₂×C₅×Q₁₆	1	4	yes	—
292	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
293	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
294	C₂×C₄	C₂²×C₅×Q₈	1	2	yes	—
295	C₂×C₄	C₂×C₅×#41	1	2	yes	—
296	C₈	—	2	8	no	—
297	C₅	C₂×#1990	1	1	yes	characteristic, direct product factor, Sylow
298	C₂²	C₅×#391	1	4	yes	—
299	C₂²	C₂×C₅×#102	1	4	yes	—
300	C₂²	C₂×C₅×#103	1	4	yes	—
301	C₂²	C₅×#486	1	4	yes	—
302	C₂²	C₅×#487	1	4	yes	—
303	C₂²	C₅×#498	1	4	yes	—
304	C₂²	C₅×#500	1	4	yes	—
305	C₂²	C₂²×C₅×C₂²⋊C₄	1	1	yes	characteristic
306	C₂²	C₂×C₅×#124	1	1	yes	characteristic
307	C₂²	C₂×C₅×C₄⋊Q₈	1	1	yes	characteristic
308	C₂²	C₂²×C₅×D₈	1	1	yes	characteristic
309	C₂²	C₂²×C₅×SD₁₆	1	1	yes	characteristic
310	C₂²	C₂²×C₅×SD₁₆	1	1	yes	characteristic
311	C₂²	C₂²×C₅×Q₁₆	1	1	yes	characteristic
312	C₄	—	4	8	no	—
313	C₄	—	8	4	no	—
314	C₄	C₅×#391	1	4	yes	—
315	C₄	C₂×C₅×#102	1	8	yes	—
316	C₄	C₂×C₅×#103	1	8	yes	—
317	C₄	C₂×C₄×C₅×Q₈	1	2	yes	—
318	C₄	C₂×C₅×C₂²⋊Q₈	1	2	yes	—
319	C₂	C₅×#1990	1	8	yes	direct product factor
320	C₂	C₂×C₅×#391	1	1	yes	characteristic
321	C₂	C₂²×C₅×#102	1	1	yes	characteristic
322	C₂	C₂²×C₅×#103	1	1	yes	characteristic
323	C₂	C₂×C₅×#486	1	1	yes	characteristic
324	C₂	C₂×C₅×#487	1	1	yes	characteristic
325	C₂	C₂×C₅×#498	1	1	yes	characteristic
326	C₂	C₂×C₅×#500	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₅	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₅ ◁ C₂²×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₅×#1990	C₂	direct (N × H)
10. C₂×C₅×C₄⋊Q₈	C₄	semidirect
23. C₂×#1990	C₅	direct (N × H)
35. C₂²×C₅×C₈	Q₈	semidirect
39. C₅×C₄⋊Q₈	C₂×C₄	semidirect
61. #1990	C₂×C₅	direct (N × H)
93. C₂×C₅×C₈	C₂×Q₈	semidirect
109. C₂×C₄⋊Q₈	C₄×C₅	semidirect
168. C₂²×C₈	C₅×Q₈	semidirect
172. C₄⋊Q₈	C₂×C₄×C₅	semidirect
242. C₂×C₈	C₂×C₅×Q₈	semidirect
254. C₂×C₅	#1990	direct (N × H)
297. C₅	C₂×#1990	direct (N × H)
319. C₂	C₅×#1990	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×C₅×C₄⋊Q₈	2
C₂²×C₄×C₅×C₈	2
C₂×C₅×#384	2
C₅×#1990	2
C₂×C₅×#340	2
C₂×#1990	5

Subgroups — C2×C5×#1990 #11944036

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#1990 #11944036