Subgroups — D₅×#1384 #12014207

Order 1280.

586 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₅×M₃₂	C₂	1	1	yes	maximal, characteristic
2	D₅×#427	C₂	1	2	yes	maximal
3	C₅×#1384	C₂	1	1	yes	maximal, characteristic, complemented
4	#10598309	C₂	1	1	yes	maximal, characteristic, complemented
5	#10598216	C₂	1	2	yes	maximal, complemented
6	C₂²×C₈×D₅	C₂²	1	1	yes	characteristic
7	D₅×#117	—	2	2	no	—
8	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
9	C₅×#427	C₂²	1	2	yes	—
10	#65395	C₂²	1	2	yes	—
11	C₂×#3944	C₂²	1	1	yes	characteristic
12	#65287	—	2	2	no	—
13	D₅×M₃₂	—	2	2	no	—
14	D₅×M₃₂	—	2	2	no	—
15	C₂×#3930	C₄	1	2	yes	complemented
16	C₂×D₅×C₁₆	C₄	1	2	yes	complemented
17	#65169	—	2	2	no	—
18	D₅×#107	—	2	2	no	—
19	#65149	C₄	1	4	yes	complemented
20	D₅×#106	C₄	1	4	yes	complemented
21	C₂×#1384	—	5	1	no	maximal, Sylow
22	C₂²×C₄×D₅	D₄	1	1	yes	characteristic
23	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
24	C₅×#117	—	2	2	no	—
25	#4072	—	2	2	no	—
26	C₂²×#168	D₄	1	1	yes	characteristic
27	C₂×C₈×D₅	C₂×C₄	1	1	yes	characteristic
28	C₂×C₈×D₅	C₂×C₄	1	1	yes	characteristic
29	C₂×C₈×D₅	C₂×C₄	1	2	yes	—
30	C₂×C₈×D₅	C₂×C₄	1	2	yes	—
31	C₂×C₈×D₅	D₄	1	2	yes	—
32	C₂×C₈×D₅	D₄	1	2	yes	—
33	C₂×C₈×D₅	Q₈	1	1	yes	characteristic
34	C₂×C₈×D₅	Q₈	1	1	yes	characteristic
35	#4038	—	2	4	no	—
36	D₅×C₄⋊C₄	—	2	4	no	—
37	D₅×C₂²⋊C₄	—	4	2	no	—
38	#4023	—	4	2	no	—
39	#4018	—	4	2	no	—
40	C₄²×D₅	—	4	2	no	—
41	C₅×M₃₂	—	2	2	no	—
42	C₂×C₅×C₁₆	C₂×C₄	1	2	yes	complemented
43	C₅×#107	—	2	2	no	—
44	C₅×#106	C₂×C₄	1	4	yes	complemented
45	#3950	—	2	2	no	—
46	#3949	C₂×C₄	1	4	yes	complemented
47	#3944	—	2	2	no	—
48	C₂×#672	C₂×C₄	1	2	yes	complemented
49	#3940	—	2	4	no	—
50	#3939	—	2	4	no	—
51	#3930	—	2	4	no	—
52	D₅×C₁₆	—	2	2	no	—
53	D₅×C₁₆	—	2	2	no	—
54	C₂²×M₃₂	—	5	1	no	—
55	C₂×#427	—	5	2	no	—
56	#1384	—	5	1	no	—
57	#1384	—	5	2	no	—
58	#1384	D₅	1	1	yes	characteristic, direct product factor
59	C₂³×D₅	D₈	1	1	yes	characteristic
60	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
61	C₂²×Dic₅	D₈	1	1	yes	characteristic
62	C₂×C₄×D₅	—	4	4	no	—
63	C₂×C₄×D₅	C₂²⋊C₄	1	2	yes	—
64	C₂×C₄×D₅	C₂²⋊C₄	1	2	yes	—
65	C₂×C₄×D₅	C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₄×D₅	C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₄×D₅	SD₁₆	1	2	yes	—
68	C₂×C₄×D₅	SD₁₆	1	2	yes	—
69	C₂×C₄×D₅	Q₁₆	1	1	yes	characteristic
70	C₂×C₄×D₅	Q₁₆	1	1	yes	characteristic
71	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
73	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
74	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
75	C₅×C₄⋊C₄	—	2	4	no	—
76	C₅×C₂²⋊C₄	—	4	2	no	—
77	C₄²×C₅	—	4	2	no	—
78	C₂²⋊Dic₅	—	4	2	no	—
79	#685	—	4	4	no	—
80	C₄⋊Dic₅	—	2	4	no	—
81	#683	—	4	4	no	—
82	C₄×Dic₅	—	4	2	no	—
83	C₄×Dic₅	—	4	4	no	—
84	C₂×#168	C₂²⋊C₄	1	2	yes	—
85	C₂×#168	C₂²⋊C₄	1	2	yes	—
86	C₂×#168	C₄⋊C₄	1	1	yes	characteristic
87	C₂×#168	C₄⋊C₄	1	1	yes	characteristic
88	C₈×D₅	C₄²	1	1	yes	characteristic
89	C₈×D₅	C₄²	1	1	yes	characteristic
90	C₈×D₅	C₄²	1	2	yes	—
91	C₈×D₅	C₂²⋊C₄	1	1	yes	characteristic
92	C₈×D₅	C₂²⋊C₄	1	1	yes	characteristic
93	C₈×D₅	C₂²⋊C₄	1	2	yes	—
94	C₈×D₅	C₄⋊C₄	1	2	yes	—
95	C₈×D₅	C₄⋊C₄	1	2	yes	—
96	C₈×D₅	C₄⋊C₄	1	2	yes	—
97	C₈×D₅	C₄⋊C₄	1	2	yes	—
98	C₅×C₁₆	—	2	2	no	—
99	#672	—	2	2	no	—
100	C₂³×C₈	—	5	1	no	—
101	C₂×#117	—	10	2	no	—
102	C₂×M₃₂	—	5	1	no	—
103	C₂×M₃₂	—	5	2	no	—
104	C₂×M₃₂	—	10	2	no	—
105	C₂×M₃₂	—	10	2	no	—
106	C₂×M₃₂	C₂×D₅	1	1	yes	characteristic
107	C₂²×C₁₆	—	5	2	no	—
108	#427	—	5	2	no	—
109	#427	—	5	4	no	—
110	#427	—	10	2	no	—
111	#427	—	10	2	no	—
112	#427	C₂×D₅	1	2	yes	—
113	C₂×#107	—	10	2	no	—
114	C₂×#106	—	5	4	no	—
115	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
116	C₂²×D₅	—	4	2	no	—
117	C₂²×D₅	#102	1	2	yes	—
118	C₂²×D₅	#102	1	2	yes	—
119	C₂²×D₅	#107	1	1	yes	characteristic
120	C₂²×D₅	#107	1	1	yes	characteristic
121	C₂×C₄×C₅	—	4	4	no	—
122	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	2	yes	—
123	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₅	C₂×SD₁₆	1	2	yes	—
125	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
126	C₂×Dic₅	—	4	4	no	—
127	C₂×Dic₅	#102	1	2	yes	—
128	C₂×Dic₅	#102	1	2	yes	—
129	C₂×Dic₅	#107	1	1	yes	characteristic
130	C₂×Dic₅	#107	1	1	yes	characteristic
131	C₄×D₅	—	8	8	no	—
132	C₄×D₅	#95	1	1	yes	characteristic
133	C₄×D₅	#95	1	2	yes	—
134	C₄×D₅	#95	1	1	yes	characteristic
135	C₄×D₅	#103	1	2	yes	—
136	C₄×D₅	#103	1	2	yes	—
137	C₄×D₅	#103	1	2	yes	—
138	C₄×D₅	#103	1	2	yes	—
139	C₄×D₅	#106	1	2	yes	—
140	C₄×D₅	#106	1	1	yes	characteristic
141	C₄×D₅	#106	1	1	yes	characteristic
142	C₅×C₈	C₂×C₄²	1	1	yes	characteristic
143	C₅×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
144	C₅×C₈	C₂×C₄⋊C₄	1	2	yes	—
145	#168	#95	1	1	yes	characteristic
146	#168	#95	1	1	yes	characteristic
147	#168	#95	1	2	yes	—
148	C₂³×C₄	—	5	1	no	—
149	C₂²×C₈	—	5	1	no	—
150	C₂²×C₈	—	5	2	no	—
151	C₂²×C₈	—	5	2	no	—
152	C₂²×C₈	—	5	1	no	—
153	C₂²×C₈	—	5	1	no	—
154	C₂²×C₈	—	5	2	no	—
155	C₂²×C₈	—	5	2	no	—
156	C₂²×C₈	—	5	1	no	—
157	C₂²×C₈	—	5	1	no	—
158	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
159	#117	—	2	2	no	—
160	#117	—	10	4	no	—
161	#117	—	10	2	no	—
162	#117	—	20	2	no	—
163	#117	—	20	2	no	—
164	C₂×C₄⋊C₄	—	10	4	no	—
165	C₂×C₂²⋊C₄	—	20	2	no	—
166	C₂×C₄²	—	20	2	no	—
167	M₃₂	—	2	2	no	—
168	M₃₂	—	10	4	no	—
169	M₃₂	—	10	2	no	—
170	C₂×C₁₆	—	5	2	no	—
171	C₂×C₁₆	—	10	2	no	—
172	C₂×C₁₆	—	10	2	no	—
173	C₂×C₁₆	C₄×D₅	1	2	yes	complemented
174	#107	—	2	2	no	—
175	#107	—	10	4	no	—
176	#107	—	10	2	no	—
177	#106	—	5	4	no	—
178	#106	—	10	4	no	—
179	#106	C₄×D₅	1	4	yes	complemented
180	C₂²×C₅	—	4	1	no	—
181	C₂²×C₅	C₂×#102	1	2	yes	—
182	C₂²×C₅	C₂×#107	1	1	yes	characteristic
183	C₂×D₅	—	2	4	no	—
184	C₂×D₅	—	2	2	no	—
185	C₂×D₅	—	4	2	no	—
186	C₂×D₅	—	4	1	no	—
187	C₂×D₅	#340	1	2	yes	—
188	C₂×D₅	#340	1	1	yes	characteristic
189	C₂×D₅	#340	1	1	yes	characteristic
190	C₄×C₅	—	8	4	no	—
191	C₄×C₅	C₂×#95	1	1	yes	characteristic
192	C₄×C₅	C₂×#103	1	2	yes	—
193	C₄×C₅	C₂×#106	1	1	yes	characteristic
194	Dic₅	—	8	4	no	—
195	Dic₅	#340	1	1	yes	characteristic
196	Dic₅	#340	1	2	yes	—
197	Dic₅	#340	1	1	yes	characteristic
198	C₂⁴	—	5	1	no	—
199	C₂²×C₄	—	5	1	no	—
200	C₂²×C₄	—	5	2	no	—
201	C₂²×C₄	—	5	2	no	—
202	C₂²×C₄	—	5	1	no	—
203	C₂²×C₄	—	5	1	no	—
204	C₂²×C₄	—	5	2	no	—
205	C₂²×C₄	—	5	2	no	—
206	C₂²×C₄	—	5	1	no	—
207	C₂²×C₄	—	5	1	no	—
208	C₂²×C₄	—	20	4	no	—
209	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
210	C₂×C₈	—	5	2	no	—
211	C₂×C₈	—	5	2	no	—
212	C₂×C₈	—	5	1	no	—
213	C₂×C₈	—	5	1	no	—
214	C₂×C₈	—	5	1	no	—
215	C₂×C₈	—	5	2	no	—
216	C₂×C₈	—	5	1	no	—
217	C₂×C₈	—	5	1	no	—
218	C₂×C₈	—	5	1	no	—
219	C₂×C₈	—	5	2	no	—
220	C₂×C₈	—	5	2	no	—
221	C₂×C₈	—	5	2	no	—
222	C₂×C₈	—	5	2	no	—
223	C₂×C₈	—	5	2	no	—
224	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
225	C₂×C₈	C₂×C₄×D₅	1	2	yes	—
226	C₂×C₈	D₄×D₅	1	2	yes	—
227	C₂×C₈	Q₈×D₅	1	1	yes	characteristic
228	C₄⋊C₄	—	2	4	no	—
229	C₄⋊C₄	—	10	4	no	—
230	C₄⋊C₄	—	20	4	no	—
231	C₂²⋊C₄	—	4	2	no	—
232	C₂²⋊C₄	—	20	4	no	—
233	C₂²⋊C₄	—	20	2	no	—
234	C₄²	—	4	2	no	—
235	C₄²	—	20	4	no	—
236	C₄²	—	20	2	no	—
237	C₁₆	—	2	2	no	—
238	C₁₆	—	10	2	no	—
239	C₂×C₅	—	2	2	no	—
240	C₂×C₅	—	2	1	no	—
241	C₂×C₅	C₂×#340	1	1	yes	characteristic
242	D₅	—	2	2	no	—
243	D₅	—	2	1	no	—
244	D₅	#1384	1	2	yes	direct product factor
245	C₂³	—	5	2	no	—
246	C₂³	—	5	1	no	—
247	C₂³	—	5	2	no	—
248	C₂³	—	5	1	no	—
249	C₂³	—	20	2	no	—
250	C₂³	D₅×D₈	1	1	yes	characteristic
251	C₂×C₄	—	4	4	no	—
252	C₂×C₄	—	5	2	no	—
253	C₂×C₄	—	5	2	no	—
254	C₂×C₄	—	5	1	no	—
255	C₂×C₄	—	5	1	no	—
256	C₂×C₄	—	5	2	no	—
257	C₂×C₄	—	5	2	no	—
258	C₂×C₄	—	5	2	no	—
259	C₂×C₄	—	5	2	no	—
260	C₂×C₄	—	5	1	no	—
261	C₂×C₄	—	5	2	no	—
262	C₂×C₄	—	5	1	no	—
263	C₂×C₄	—	5	1	no	—
264	C₂×C₄	—	5	1	no	—
265	C₂×C₄	—	5	2	no	—
266	C₂×C₄	—	20	4	no	—
267	C₂×C₄	—	40	8	no	—
268	C₂×C₄	D₅×C₂²⋊C₄	1	2	yes	—
269	C₂×C₄	D₅×C₄⋊C₄	1	1	yes	characteristic
270	C₂×C₄	D₅×SD₁₆	1	2	yes	—
271	C₂×C₄	D₅×Q₁₆	1	1	yes	characteristic
272	C₈	—	5	1	no	—
273	C₈	—	5	2	no	—
274	C₈	—	5	1	no	—
275	C₈	C₄²×D₅	1	1	yes	characteristic
276	C₈	D₅×C₂²⋊C₄	1	1	yes	characteristic
277	C₈	D₅×C₄⋊C₄	1	2	yes	—
278	C₅	C₂×#1384	1	1	yes	characteristic, complemented, Sylow
279	C₂²	—	4	1	no	—
280	C₂²	—	5	1	no	—
281	C₂²	—	5	2	no	—
282	C₂²	—	5	1	no	—
283	C₂²	—	10	4	no	—
284	C₂²	—	10	2	no	—
285	C₂²	—	20	2	no	—
286	C₂²	—	20	1	no	—
287	C₂²	D₅×#102	1	2	yes	—
288	C₂²	D₅×#107	1	1	yes	characteristic
289	C₄	—	5	1	no	—
290	C₄	—	5	2	no	—
291	C₄	—	5	1	no	—
292	C₄	—	8	4	no	—
293	C₄	—	40	4	no	—
294	C₄	D₅×#95	1	1	yes	characteristic
295	C₄	D₅×#103	1	2	yes	—
296	C₄	D₅×#106	1	1	yes	characteristic
297	C₂	—	2	2	no	—
298	C₂	—	2	1	no	—
299	C₂	—	5	2	no	—
300	C₂	—	10	2	no	—
301	C₂	—	10	1	no	—
302	C₂	D₅×#340	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₅×#106
commutator	C₅×C₈	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅
Fitting	C₅×#1384	C₂
socle	C₂×C₅	C₂×#340
prime radical	C₂³×D₅	D₈
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	#1384	D₅
hypocenter	C₅	C₂×#1384
LCS	G ▷ C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1384

Complements

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

N	H ≅ G/N	type
3. C₅×#1384	C₂	semidirect
4. #10598309	C₂	semidirect
5. #10598216	C₂	semidirect
15. C₂×#3930	C₄	semidirect
16. C₂×D₅×C₁₆	C₄	semidirect
19. #65149	C₄	semidirect
20. D₅×#106	C₄	semidirect
42. C₂×C₅×C₁₆	C₂×C₄	semidirect
44. C₅×#106	C₂×C₄	semidirect
46. #3949	C₂×C₄	semidirect
48. C₂×#672	C₂×C₄	semidirect
58. #1384	D₅	direct (N × H)
173. C₂×C₁₆	C₄×D₅	semidirect
179. #106	C₄×D₅	semidirect
244. D₅	#1384	direct (N × H)
278. C₅	C₂×#1384	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₅×M₃₂	2
D₅×#427	2
C₅×#1384	2
#10598309	2
#10598216	2
C₂×#1384	5

Subgroups — D5×#1384 #12014207

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₅×#1384 #12014207