Subgroups — D₅×#1846 #12099703

Order 1280.

1182 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	C₂×D₅×#107	C₂	1	2	yes	maximal
2	C₂×D₅×#106	C₂	1	2	yes	maximal
3	C₂×C₄×C₈×D₅	C₂	1	1	yes	maximal, characteristic
4	D₅×#382	C₂	1	2	yes	maximal
5	C₅×#1846	C₂	1	1	yes	maximal, characteristic, complemented
6	#10601400	C₂	1	2	yes	maximal, complemented
7	#10601348	C₂	1	2	yes	maximal, complemented
8	#10600559	C₂	1	1	yes	maximal, characteristic, complemented
9	#10600342	C₂	1	2	yes	maximal, complemented
10	C₂²×C₈×D₅	C₂²	1	1	yes	characteristic
11	C₂²×C₈×D₅	C₂²	1	1	yes	characteristic
12	C₂×D₅×C₄⋊C₄	C₂²	1	4	yes	—
13	C₂×C₄²×D₅	C₂²	1	1	yes	characteristic
14	C₂×C₅×#107	C₂²	1	2	yes	—
15	C₂×C₅×#106	C₂²	1	2	yes	—
16	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
17	C₅×#382	C₂²	1	2	yes	—
18	C₂×#3950	C₂²	1	2	yes	—
19	C₂×#3949	C₂²	1	2	yes	—
20	C₂×C₈×Dic₅	C₂²	1	1	yes	characteristic
21	C₂×C₈×Dic₅	C₂²	1	1	yes	characteristic
22	#65266	C₂²	1	4	yes	—
23	C₂×#3940	C₂²	1	4	yes	—
24	C₂×#3939	C₂²	1	4	yes	—
25	#65225	C₂²	1	2	yes	—
26	C₂×C₄×#168	C₂²	1	1	yes	characteristic
27	D₅×#107	—	2	8	no	—
28	D₅×#106	—	2	8	no	—
29	C₄×C₈×D₅	C₄	1	8	yes	complemented
30	D₅×#95	—	2	4	no	—
31	#64939	—	2	4	no	—
32	C₂×#1846	—	5	1	no	maximal, Sylow
33	C₂²×C₄×D₅	—	2	4	no	—
34	C₂²×C₄×D₅	D₄	1	2	yes	—
35	C₂²×C₄×D₅	C₂³	1	1	yes	characteristic
36	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
37	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
38	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
39	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
40	C₂×C₄⋊Dic₅	C₂³	1	4	yes	—
41	C₂×#683	—	2	4	no	—
42	C₂×C₄×Dic₅	D₄	1	2	yes	—
43	C₂×C₄×Dic₅	C₂³	1	1	yes	characteristic
44	C₂²×#168	C₂³	1	1	yes	characteristic
45	C₂²×#168	C₂³	1	1	yes	characteristic
46	C₂×C₈×D₅	C₂×C₄	1	4	yes	—
47	C₂×C₈×D₅	C₂×C₄	1	4	yes	—
48	C₂×C₈×D₅	D₄	1	4	yes	—
49	C₂×C₈×D₅	D₄	1	4	yes	—
50	C₂×C₈×D₅	Q₈	1	4	yes	—
51	C₂×C₈×D₅	Q₈	1	4	yes	—
52	D₅×C₄⋊C₄	—	2	8	no	—
53	D₅×C₄⋊C₄	—	2	8	no	—
54	C₄²×D₅	C₂×C₄	1	4	yes	—
55	C₄²×D₅	Q₈	1	4	yes	—
56	C₅×#107	—	2	4	no	—
57	C₅×#106	—	2	4	no	—
58	C₄×C₅×C₈	C₂×C₄	1	4	yes	complemented
59	C₅×#95	—	2	4	no	—
60	#3963	—	2	4	no	—
61	#3963	—	2	8	no	—
62	#3950	—	2	4	no	—
63	#3949	—	2	4	no	—
64	C₈×Dic₅	C₂×C₄	1	4	yes	complemented
65	C₈×Dic₅	C₂×C₄	1	4	yes	complemented
66	#3940	—	2	8	no	—
67	#3939	—	2	8	no	—
68	C₄×#168	C₂×C₄	1	4	yes	complemented
69	C₂²×#107	—	5	2	no	—
70	C₂²×#106	—	5	2	no	—
71	C₂²×C₄×C₈	—	5	1	no	—
72	C₂×#382	—	5	2	no	—
73	#1846	—	5	2	no	—
74	#1846	—	5	2	no	—
75	#1846	—	5	1	no	—
76	#1846	—	5	2	no	—
77	#1846	D₅	1	1	yes	characteristic, direct product factor
78	C₂³×D₅	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₅	—	2	4	no	—
80	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
81	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
82	C₂²×Dic₅	—	2	4	no	—
83	C₂²×Dic₅	C₂×D₄	1	2	yes	—
84	C₂²×Dic₅	C₂×D₄	1	1	yes	characteristic
85	C₂×C₄×D₅	—	2	8	no	—
86	C₂×C₄×D₅	—	4	8	no	—
87	C₂×C₄×D₅	—	4	16	no	—
88	C₂×C₄×D₅	C₄⋊C₄	1	8	yes	—
89	C₂×C₄×D₅	C₂²×C₄	1	2	yes	—
90	C₂×C₄×D₅	C₂×D₄	1	2	yes	—
91	C₂×C₄×D₅	C₂×D₄	1	2	yes	—
92	C₂×C₄×D₅	C₂×Q₈	1	2	yes	—
93	C₂×C₄×D₅	C₂×Q₈	1	2	yes	—
94	C₂×C₄×D₅	C₂×Q₈	1	2	yes	—
95	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
96	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
97	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
98	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
99	C₂×C₅×C₈	C₂×Q₈	1	2	yes	—
100	C₂×C₅×C₈	C₂×Q₈	1	2	yes	—
101	C₅×C₄⋊C₄	—	2	4	no	—
102	C₅×C₄⋊C₄	—	2	4	no	—
103	C₄²×C₅	C₂²×C₄	1	2	yes	—
104	C₄²×C₅	C₂×Q₈	1	2	yes	—
105	C₄⋊Dic₅	—	2	4	no	—
106	C₄⋊Dic₅	—	2	4	no	—
107	#683	—	4	8	no	—
108	C₄×Dic₅	C₄⋊C₄	1	8	yes	—
109	C₄×Dic₅	C₂²×C₄	1	2	yes	—
110	C₄×Dic₅	C₂×Q₈	1	2	yes	—
111	C₂×#168	C₂²×C₄	1	2	yes	—
112	C₂×#168	C₂²×C₄	1	2	yes	—
113	C₂×#168	C₂×D₄	1	2	yes	—
114	C₂×#168	C₂×D₄	1	2	yes	—
115	C₂×#168	C₂×Q₈	1	2	yes	—
116	C₂×#168	C₂×Q₈	1	2	yes	—
117	C₈×D₅	C₄⋊C₄	1	16	yes	—
118	C₈×D₅	C₄⋊C₄	1	16	yes	—
119	C₂³×C₈	—	5	1	no	—
120	C₂³×C₈	—	5	1	no	—
121	C₂²×C₄⋊C₄	—	5	4	no	—
122	C₂²×C₄²	—	5	1	no	—
123	C₂×#107	—	5	2	no	—
124	C₂×#107	—	5	4	no	—
125	C₂×#107	—	10	8	no	—
126	C₂×#107	C₂×D₅	1	2	yes	—
127	C₂×#106	—	5	2	no	—
128	C₂×#106	—	5	4	no	—
129	C₂×#106	—	10	8	no	—
130	C₂×#106	C₂×D₅	1	2	yes	—
131	C₂×C₄×C₈	—	5	1	no	—
132	C₂×C₄×C₈	—	5	1	no	—
133	C₂×C₄×C₈	—	5	1	no	—
134	C₂×C₄×C₈	—	5	8	no	—
135	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
136	#382	—	5	4	no	—
137	#382	—	5	2	no	—
138	#382	—	10	4	no	—
139	#382	C₂×D₅	1	2	yes	—
140	C₂×#95	—	10	4	no	—
141	C₂³×C₅	C₂²×D₄	1	1	yes	characteristic
142	C₂²×D₅	C₂×C₄⋊C₄	1	2	yes	—
143	C₂²×D₅	#127	1	2	yes	—
144	C₂²×D₅	C₄⋊Q₈	1	2	yes	—
145	C₂²×D₅	#135	1	8	yes	—
146	C₂×C₄×C₅	—	2	4	no	—
147	C₂×C₄×C₅	—	4	4	no	—
148	C₂×C₄×C₅	—	4	8	no	—
149	C₂×C₄×C₅	C₂×C₄⋊C₄	1	4	yes	—
150	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
151	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
152	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
153	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
154	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
155	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
156	C₂×Dic₅	—	2	4	no	—
157	C₂×Dic₅	—	4	4	no	—
158	C₂×Dic₅	—	4	8	no	—
159	C₂×Dic₅	C₂×C₄⋊C₄	1	4	yes	—
160	C₂×Dic₅	C₂×C₄⋊C₄	1	2	yes	—
161	C₂×Dic₅	#127	1	2	yes	—
162	C₂×Dic₅	C₄⋊Q₈	1	2	yes	—
163	C₄×D₅	—	2	16	no	—
164	C₄×D₅	—	8	16	no	—
165	C₄×D₅	C₂×C₄⋊C₄	1	4	yes	—
166	C₄×D₅	C₂×C₄⋊C₄	1	4	yes	—
167	C₄×D₅	C₄⋊Q₈	1	4	yes	—
168	C₄×D₅	C₄⋊Q₈	1	4	yes	—
169	C₅×C₈	C₂×C₄⋊C₄	1	4	yes	—
170	C₅×C₈	C₂×C₄⋊C₄	1	4	yes	—
171	#168	C₂×C₄⋊C₄	1	4	yes	—
172	#168	C₂×C₄⋊C₄	1	4	yes	—
173	C₂³×C₄	—	5	1	no	—
174	C₂³×C₄	—	5	2	no	—
175	C₂³×C₄	—	10	4	no	—
176	C₂²×C₈	—	5	1	no	—
177	C₂²×C₈	—	5	1	no	—
178	C₂²×C₈	—	5	4	no	—
179	C₂²×C₈	—	5	4	no	—
180	C₂²×C₈	—	5	4	no	—
181	C₂²×C₈	—	5	4	no	—
182	C₂²×C₈	—	5	4	no	—
183	C₂²×C₈	—	5	4	no	—
184	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
185	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
186	C₂×C₄⋊C₄	—	5	4	no	—
187	C₂×C₄⋊C₄	—	10	8	no	—
188	C₂×C₄⋊C₄	—	10	8	no	—
189	C₂×C₄⋊C₄	—	10	4	no	—
190	C₂×C₄⋊C₄	C₂²×D₅	1	4	yes	—
191	C₂×C₄²	—	5	2	no	—
192	C₂×C₄²	—	5	1	no	—
193	C₂×C₄²	—	5	4	no	—
194	C₂×C₄²	—	5	4	no	—
195	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
196	#107	—	2	4	no	—
197	#107	—	10	8	no	—
198	#107	—	10	4	no	—
199	#106	—	2	4	no	—
200	#106	—	10	8	no	—
201	#106	—	10	4	no	—
202	C₄×C₈	—	5	4	no	—
203	C₄×C₈	—	5	4	no	—
204	C₄×C₈	—	5	4	no	—
205	C₄×C₈	C₄×D₅	1	4	yes	complemented
206	#95	—	2	4	no	—
207	#95	—	10	8	no	—
208	#95	—	10	4	no	—
209	C₂²×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
210	C₂²×C₅	C₂×#127	1	1	yes	characteristic
211	C₂²×C₅	C₂×C₄⋊Q₈	1	1	yes	characteristic
212	C₂²×C₅	C₂×#135	1	4	yes	—
213	C₂×D₅	#384	1	4	yes	—
214	C₂×D₅	#427	1	8	yes	—
215	C₂×D₅	#495	1	8	yes	—
216	C₂×D₅	#499	1	8	yes	—
217	C₄×C₅	—	2	4	no	—
218	C₄×C₅	—	8	4	no	—
219	C₄×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
220	C₄×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
221	C₄×C₅	C₂×C₄⋊Q₈	1	1	yes	characteristic
222	C₄×C₅	C₂×C₄⋊Q₈	1	1	yes	characteristic
223	Dic₅	—	2	4	no	—
224	Dic₅	—	8	4	no	—
225	Dic₅	#384	1	4	yes	—
226	C₂⁴	—	5	1	no	—
227	C₂²×C₄	—	2	4	no	—
228	C₂²×C₄	—	5	1	no	—
229	C₂²×C₄	—	5	2	no	—
230	C₂²×C₄	—	5	2	no	—
231	C₂²×C₄	—	5	2	no	—
232	C₂²×C₄	—	5	2	no	—
233	C₂²×C₄	—	5	2	no	—
234	C₂²×C₄	—	5	2	no	—
235	C₂²×C₄	—	5	2	no	—
236	C₂²×C₄	—	5	8	no	—
237	C₂²×C₄	—	10	8	no	—
238	C₂²×C₄	—	10	4	no	—
239	C₂²×C₄	—	20	8	no	—
240	C₂²×C₄	—	20	16	no	—
241	C₂²×C₄	D₄×D₅	1	2	yes	—
242	C₂²×C₄	C₂³×D₅	1	1	yes	characteristic
243	C₂×C₈	—	5	2	no	—
244	C₂×C₈	—	5	2	no	—
245	C₂×C₈	—	5	2	no	—
246	C₂×C₈	—	5	2	no	—
247	C₂×C₈	—	5	2	no	—
248	C₂×C₈	—	5	2	no	—
249	C₂×C₈	—	5	16	no	—
250	C₂×C₈	—	5	16	no	—
251	C₂×C₈	C₂×C₄×D₅	1	2	yes	—
252	C₂×C₈	C₂×C₄×D₅	1	2	yes	—
253	C₂×C₈	D₄×D₅	1	2	yes	—
254	C₂×C₈	D₄×D₅	1	2	yes	—
255	C₂×C₈	Q₈×D₅	1	2	yes	—
256	C₂×C₈	Q₈×D₅	1	2	yes	—
257	C₄⋊C₄	—	2	4	no	—
258	C₄⋊C₄	—	2	4	no	—
259	C₄⋊C₄	—	10	4	no	—
260	C₄⋊C₄	—	10	4	no	—
261	C₄⋊C₄	—	20	8	no	—
262	C₄²	—	5	2	no	—
263	C₄²	—	5	2	no	—
264	C₄²	—	5	8	no	—
265	C₄²	C₂×C₄×D₅	1	2	yes	—
266	C₄²	Q₈×D₅	1	2	yes	—
267	C₂×C₅	C₂×#384	1	1	yes	characteristic
268	C₂×C₅	C₂×#427	1	2	yes	—
269	C₂×C₅	C₂×#495	1	2	yes	—
270	C₂×C₅	C₂×#499	1	2	yes	—
271	D₅	#1846	1	8	yes	direct product factor
272	C₂³	—	5	2	no	—
273	C₂³	—	5	2	no	—
274	C₂³	—	5	2	no	—
275	C₂³	—	5	8	no	—
276	C₂³	C₂×D₄×D₅	1	1	yes	characteristic
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	4	4	no	—
279	C₂×C₄	—	4	8	no	—
280	C₂×C₄	—	5	2	no	—
281	C₂×C₄	—	5	2	no	—
282	C₂×C₄	—	5	2	no	—
283	C₂×C₄	—	5	4	no	—
284	C₂×C₄	—	5	4	no	—
285	C₂×C₄	—	5	4	no	—
286	C₂×C₄	—	5	4	no	—
287	C₂×C₄	—	5	4	no	—
288	C₂×C₄	—	10	16	no	—
289	C₂×C₄	—	10	4	no	—
290	C₂×C₄	—	20	4	no	—
291	C₂×C₄	—	20	8	no	—
292	C₂×C₄	—	40	16	no	—
293	C₂×C₄	D₅×C₄⋊C₄	1	4	yes	—
294	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
295	C₂×C₄	C₂×D₄×D₅	1	1	yes	characteristic
296	C₂×C₄	C₂×D₄×D₅	1	1	yes	characteristic
297	C₂×C₄	C₂×Q₈×D₅	1	1	yes	characteristic
298	C₂×C₄	C₂×Q₈×D₅	1	1	yes	characteristic
299	C₂×C₄	C₂×Q₈×D₅	1	1	yes	characteristic
300	C₈	—	5	4	no	—
301	C₈	—	5	4	no	—
302	C₈	D₅×C₄⋊C₄	1	4	yes	—
303	C₈	D₅×C₄⋊C₄	1	4	yes	—
304	C₅	C₂×#1846	1	1	yes	characteristic, complemented, Sylow
305	C₂²	—	5	4	no	—
306	C₂²	—	5	8	no	—
307	C₂²	—	5	8	no	—
308	C₂²	—	5	8	no	—
309	C₂²	C₂×D₅×C₄⋊C₄	1	1	yes	characteristic
310	C₂²	D₅×#127	1	1	yes	characteristic
311	C₂²	D₅×C₄⋊Q₈	1	1	yes	characteristic
312	C₂²	D₅×#135	1	4	yes	—
313	C₄	—	2	4	no	—
314	C₄	—	5	4	no	—
315	C₄	—	8	4	no	—
316	C₄	—	10	4	no	—
317	C₄	—	40	4	no	—
318	C₄	C₂×D₅×C₄⋊C₄	1	1	yes	characteristic
319	C₄	C₂×D₅×C₄⋊C₄	1	1	yes	characteristic
320	C₄	D₅×C₄⋊Q₈	1	1	yes	characteristic
321	C₄	D₅×C₄⋊Q₈	1	1	yes	characteristic
322	C₂	—	5	8	no	—
323	C₂	D₅×#384	1	1	yes	characteristic
324	C₂	D₅×#427	1	2	yes	—
325	C₂	D₅×#495	1	2	yes	—
326	C₂	D₅×#499	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₅
commutator	C₂×C₄×C₅	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅
Fitting	C₅×#1846	C₂
socle	C₂³×C₅	C₂²×D₄
prime radical	C₂³×D₅	C₂×D₄
cosocle	C₂²×C₄×C₅	C₂⁴
hypercenter	#1846	D₅
hypocenter	C₅	C₂×#1846
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1846

Complements

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

N	H ≅ G/N	type
5. C₅×#1846	C₂	semidirect
6. #10601400	C₂	semidirect
7. #10601348	C₂	semidirect
8. #10600559	C₂	semidirect
9. #10600342	C₂	semidirect
29. C₄×C₈×D₅	C₄	semidirect
58. C₄×C₅×C₈	C₂×C₄	semidirect
64. C₈×Dic₅	C₂×C₄	semidirect
65. C₈×Dic₅	C₂×C₄	semidirect
68. C₄×#168	C₂×C₄	semidirect
77. #1846	D₅	direct (N × H)
205. C₄×C₈	C₄×D₅	semidirect
271. D₅	#1846	direct (N × H)
304. C₅	C₂×#1846	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₅×#107	2
C₂×D₅×#106	2
C₂×C₄×C₈×D₅	2
D₅×#382	2
C₅×#1846	2
#10601400	2
#10601348	2
#10600559	2
#10600342	2
C₂×#1846	5

Subgroups — D5×#1846 #12099703

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₅×#1846 #12099703