Subgroups — C₅×#9589 #11932309

Order 1280.

326 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#1986	C₂	1	1	yes	maximal, characteristic
2	C₅×#1950	C₂	1	1	yes	maximal, characteristic
3	C₅×#1542	C₂	1	1	yes	maximal, characteristic
4	C₅×#1493	C₂	1	1	yes	maximal, characteristic
5	C₅×C₈×M₁₆	C₂	1	1	yes	maximal, characteristic
6	C₅×#1303	C₂	1	1	yes	maximal, characteristic
7	C₅×#1279	C₂	1	1	yes	maximal, characteristic
8	C₅×#533	C₂²	1	1	yes	characteristic
9	C₅×#431	C₂²	1	1	yes	characteristic
10	C₂×C₅×#108	—	2	1	no	—
11	C₅×#423	C₂²	1	1	yes	characteristic
12	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
13	C₄×C₅×M₁₆	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
15	C₄×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
16	C₅×#341	—	2	1	no	—
17	C₅×#339	—	2	1	no	—
18	C₅×#339	—	2	1	no	—
19	C₅×#333	C₄	1	1	yes	characteristic, complemented
20	C₅×#333	C₄	1	1	yes	characteristic, complemented
21	C₅×#332	C₄	1	1	yes	characteristic, complemented
22	C₅×#332	C₄	1	1	yes	characteristic, complemented
23	C₅×Q₈⋊C₈	—	2	1	no	—
24	C₅×Q₈⋊C₈	—	2	1	no	—
25	C₅×#322	—	2	1	no	—
26	C₅×C₈²	—	2	1	no	—
27	#9589	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
28	C₂×C₅×M₁₆	—	2	1	no	—
29	C₂×C₅×M₁₆	D₄	1	1	yes	characteristic
30	C₂×C₅×M₁₆	D₄	1	1	yes	characteristic
31	C₂²×C₅×C₈	—	2	1	no	—
32	C₂²×C₅×C₈	—	2	1	no	—
33	C₅×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
34	C₅×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
35	C₅×#125	—	2	1	no	—
36	C₅×C₂²⋊Q₈	—	2	1	no	—
37	C₅×C₂²⋊Q₈	—	2	1	no	—
38	C₄×C₅×Q₈	—	2	1	no	—
39	C₄×C₅×Q₈	—	2	1	no	—
40	C₅×#117	—	2	1	no	—
41	C₂×C₅×C₄⋊C₄	—	2	1	no	—
42	C₂×C₄²×C₅	—	2	1	no	—
43	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
44	C₅×#108	—	4	2	no	—
45	C₅×C₄⋊C₈	—	2	1	no	—
46	C₅×C₄⋊C₈	—	2	1	no	—
47	C₅×C₄⋊C₈	—	2	1	no	—
48	C₅×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
49	C₅×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
50	C₅×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
51	C₅×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
52	C₅×C₂²⋊C₈	—	2	1	no	—
53	C₅×#97	—	2	1	no	—
54	C₄×C₅×C₈	—	2	1	no	—
55	C₄×C₅×C₈	—	2	1	no	—
56	C₄×C₅×C₈	—	2	1	no	—
57	C₄×C₅×C₈	D₄	1	1	yes	characteristic
58	C₄×C₅×C₈	D₄	1	1	yes	characteristic
59	C₅×#95	—	2	1	no	—
60	#1986	C₂×C₅	1	1	yes	characteristic
61	#1950	C₂×C₅	1	1	yes	characteristic
62	#1542	C₂×C₅	1	1	yes	characteristic
63	#1493	C₂×C₅	1	1	yes	characteristic
64	C₈×M₁₆	C₂×C₅	1	1	yes	characteristic
65	#1303	C₂×C₅	1	1	yes	characteristic
66	#1279	C₂×C₅	1	1	yes	characteristic
67	C₂×C₅×Q₈	—	2	1	no	—
68	C₂×C₅×Q₈	—	2	1	no	—
69	C₂²×C₄×C₅	—	4	1	no	—
70	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₅	#41	1	1	yes	characteristic
72	C₂²×C₄×C₅	#41	1	1	yes	characteristic
73	C₅×M₁₆	—	2	1	no	—
74	C₅×M₁₆	—	2	1	no	—
75	C₅×M₁₆	—	4	1	no	—
76	C₅×M₁₆	—	4	1	no	—
77	C₅×M₁₆	SD₁₆	1	2	yes	—
78	C₅×M₁₆	Q₁₆	1	2	yes	—
79	C₂×C₅×C₈	—	2	1	no	—
80	C₂×C₅×C₈	—	2	1	no	—
81	C₂×C₅×C₈	—	2	1	no	—
82	C₂×C₅×C₈	—	2	1	no	—
83	C₂×C₅×C₈	—	2	1	no	—
84	C₂×C₅×C₈	—	2	1	no	—
85	C₂×C₅×C₈	—	2	1	no	—
86	C₂×C₅×C₈	—	2	1	no	—
87	C₂×C₅×C₈	—	4	2	no	—
88	C₂×C₅×C₈	—	4	1	no	—
89	C₂×C₅×C₈	—	4	1	no	—
90	C₂×C₅×C₈	—	4	1	no	—
91	C₅×C₄⋊C₄	—	2	2	no	—
92	C₅×C₄⋊C₄	—	2	1	no	—
93	C₅×C₄⋊C₄	—	2	1	no	—
94	C₅×C₄⋊C₄	—	2	1	no	—
95	C₅×C₄⋊C₄	—	2	1	no	—
96	C₅×C₄⋊C₄	—	4	1	no	—
97	C₅×C₄⋊C₄	—	4	1	no	—
98	C₅×C₄⋊C₄	—	4	1	no	—
99	C₅×C₂²⋊C₄	—	4	1	no	—
100	C₄²×C₅	—	4	1	no	—
101	C₄²×C₅	—	4	1	no	—
102	C₄²×C₅	—	4	1	no	—
103	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
104	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
105	C₄²×C₅	#41	1	1	yes	characteristic
106	C₄²×C₅	#41	1	1	yes	characteristic
107	#533	C₂²×C₅	1	1	yes	characteristic
108	#431	C₂²×C₅	1	1	yes	characteristic
109	C₂×#108	—	2	1	no	—
110	#423	C₂²×C₅	1	1	yes	characteristic
111	C₂×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
112	C₄×M₁₆	C₂²×C₅	1	1	yes	characteristic
113	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
114	C₄×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
115	#341	—	2	1	no	—
116	#339	—	2	1	no	—
117	#339	—	2	1	no	—
118	#333	C₄×C₅	1	1	yes	characteristic, complemented
119	#333	C₄×C₅	1	1	yes	characteristic, complemented
120	#332	C₄×C₅	1	1	yes	characteristic, complemented
121	#332	C₄×C₅	1	1	yes	characteristic, complemented
122	Q₈⋊C₈	—	2	1	no	—
123	Q₈⋊C₈	—	2	1	no	—
124	#322	—	2	1	no	—
125	C₈²	—	2	1	no	—
126	C₂³×C₅	#124	1	1	yes	characteristic
127	C₅×Q₈	—	4	1	no	—
128	C₅×Q₈	—	4	1	no	—
129	C₅×Q₈	—	8	1	no	—
130	C₂×C₄×C₅	—	2	1	no	—
131	C₂×C₄×C₅	—	2	1	no	—
132	C₂×C₄×C₅	—	4	1	no	—
133	C₂×C₄×C₅	—	4	2	no	—
134	C₂×C₄×C₅	—	4	1	no	—
135	C₂×C₄×C₅	—	4	1	no	—
136	C₂×C₄×C₅	—	8	1	no	—
137	C₂×C₄×C₅	—	8	1	no	—
138	C₂×C₄×C₅	#117	1	1	yes	characteristic
139	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
140	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
141	C₂×C₄×C₅	#121	1	1	yes	characteristic
142	C₂×C₄×C₅	#123	1	1	yes	characteristic
143	C₂×C₄×C₅	#126	1	1	yes	characteristic
144	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
145	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
146	C₂×C₄×C₅	#135	1	1	yes	characteristic
147	C₂×C₄×C₅	#135	1	1	yes	characteristic
148	C₅×C₈	—	2	2	no	—
149	C₅×C₈	—	2	2	no	—
150	C₅×C₈	—	2	2	no	—
151	C₅×C₈	—	4	1	no	—
152	C₅×C₈	—	4	1	no	—
153	C₅×C₈	—	4	1	no	—
154	C₅×C₈	—	4	1	no	—
155	C₅×C₈	—	4	1	no	—
156	C₅×C₈	—	8	1	no	—
157	C₂×M₁₆	—	2	1	no	—
158	C₂×M₁₆	C₅×D₄	1	1	yes	characteristic
159	C₂×M₁₆	C₅×D₄	1	1	yes	characteristic
160	C₂²×C₈	—	2	1	no	—
161	C₂²×C₈	—	2	1	no	—
162	C₄⋊Q₈	C₂×C₄×C₅	1	1	yes	characteristic
163	C₄⋊Q₈	C₂×C₄×C₅	1	1	yes	characteristic
164	#125	—	2	1	no	—
165	C₂²⋊Q₈	—	2	1	no	—
166	C₂²⋊Q₈	—	2	1	no	—
167	C₄×Q₈	—	2	1	no	—
168	C₄×Q₈	—	2	1	no	—
169	#117	—	2	1	no	—
170	C₂×C₄⋊C₄	—	2	1	no	—
171	C₂×C₄²	—	2	1	no	—
172	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
173	#108	—	4	2	no	—
174	C₄⋊C₈	—	2	1	no	—
175	C₄⋊C₈	—	2	1	no	—
176	C₄⋊C₈	—	2	1	no	—
177	C₄⋊C₈	C₂×C₄×C₅	1	1	yes	characteristic
178	C₄⋊C₈	C₂×C₄×C₅	1	1	yes	characteristic
179	C₄⋊C₈	C₂×C₄×C₅	1	1	yes	characteristic
180	C₄⋊C₈	C₂×C₄×C₅	1	1	yes	characteristic
181	C₂²⋊C₈	—	2	1	no	—
182	#97	—	2	1	no	—
183	C₄×C₈	—	2	1	no	—
184	C₄×C₈	—	2	1	no	—
185	C₄×C₈	—	2	1	no	—
186	C₄×C₈	C₅×D₄	1	1	yes	characteristic
187	C₄×C₈	C₅×D₄	1	1	yes	characteristic
188	#95	—	2	1	no	—
189	C₂²×C₅	—	2	1	no	—
190	C₂²×C₅	—	2	1	no	—
191	C₂²×C₅	#388	1	1	yes	characteristic
192	C₂²×C₅	#487	1	1	yes	characteristic
193	C₂²×C₅	#488	1	1	yes	characteristic
194	C₄×C₅	—	2	1	no	—
195	C₄×C₅	—	2	1	no	—
196	C₄×C₅	—	2	1	no	—
197	C₄×C₅	—	8	2	no	—
198	C₄×C₅	—	8	1	no	—
199	C₄×C₅	—	8	1	no	—
200	C₄×C₅	C₄×SD₁₆	1	1	yes	characteristic
201	C₄×C₅	C₄×Q₁₆	1	1	yes	characteristic
202	C₄×C₅	#443	1	1	yes	characteristic
203	C₄×C₅	#444	1	1	yes	characteristic
204	C₄×C₅	#465	1	1	yes	characteristic
205	C₄×C₅	#467	1	1	yes	characteristic
206	C₂×Q₈	—	2	1	no	—
207	C₂×Q₈	—	2	1	no	—
208	C₂²×C₄	—	4	1	no	—
209	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
210	C₂²×C₄	C₅×#41	1	1	yes	characteristic
211	C₂²×C₄	C₅×#41	1	1	yes	characteristic
212	M₁₆	—	2	1	no	—
213	M₁₆	—	2	1	no	—
214	M₁₆	—	4	1	no	—
215	M₁₆	—	4	1	no	—
216	M₁₆	C₅×SD₁₆	1	2	yes	—
217	M₁₆	C₅×Q₁₆	1	2	yes	—
218	C₂×C₈	—	2	1	no	—
219	C₂×C₈	—	2	1	no	—
220	C₂×C₈	—	2	1	no	—
221	C₂×C₈	—	2	1	no	—
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	2	1	no	—
225	C₂×C₈	—	2	1	no	—
226	C₂×C₈	—	4	2	no	—
227	C₂×C₈	—	4	1	no	—
228	C₂×C₈	—	4	1	no	—
229	C₂×C₈	—	4	1	no	—
230	C₄⋊C₄	—	2	2	no	—
231	C₄⋊C₄	—	2	1	no	—
232	C₄⋊C₄	—	2	1	no	—
233	C₄⋊C₄	—	2	1	no	—
234	C₄⋊C₄	—	2	1	no	—
235	C₄⋊C₄	—	4	1	no	—
236	C₄⋊C₄	—	4	1	no	—
237	C₄⋊C₄	—	4	1	no	—
238	C₂²⋊C₄	—	4	1	no	—
239	C₄²	—	4	1	no	—
240	C₄²	—	4	1	no	—
241	C₄²	—	4	1	no	—
242	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
243	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
244	C₄²	C₅×#41	1	1	yes	characteristic
245	C₄²	C₅×#41	1	1	yes	characteristic
246	C₂×C₅	—	2	1	no	—
247	C₂×C₅	—	2	1	no	—
248	C₂×C₅	#1929	1	1	yes	characteristic
249	C₂×C₅	#1932	1	1	yes	characteristic
250	C₂×C₅	#1933	1	1	yes	characteristic
251	C₂³	C₅×#124	1	1	yes	characteristic
252	Q₈	—	4	1	no	—
253	Q₈	—	4	1	no	—
254	Q₈	—	8	1	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	4	1	no	—
258	C₂×C₄	—	4	2	no	—
259	C₂×C₄	—	4	1	no	—
260	C₂×C₄	—	4	1	no	—
261	C₂×C₄	—	8	1	no	—
262	C₂×C₄	—	8	1	no	—
263	C₂×C₄	C₅×#117	1	1	yes	characteristic
264	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
265	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
266	C₂×C₄	C₅×#121	1	1	yes	characteristic
267	C₂×C₄	C₅×#123	1	1	yes	characteristic
268	C₂×C₄	C₅×#126	1	1	yes	characteristic
269	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
270	C₂×C₄	C₂×C₅×Q₁₆	1	1	yes	characteristic
271	C₂×C₄	C₅×#135	1	1	yes	characteristic
272	C₂×C₄	C₅×#135	1	1	yes	characteristic
273	C₈	—	2	2	no	—
274	C₈	—	2	2	no	—
275	C₈	—	2	2	no	—
276	C₈	—	4	1	no	—
277	C₈	—	4	1	no	—
278	C₈	—	4	1	no	—
279	C₈	—	4	1	no	—
280	C₈	—	4	1	no	—
281	C₈	—	8	1	no	—
282	C₅	#9589	1	1	yes	characteristic, direct product factor, Sylow
283	C₂²	—	2	1	no	—
284	C₂²	—	2	1	no	—
285	C₂²	C₅×#388	1	1	yes	characteristic
286	C₂²	C₅×#487	1	1	yes	characteristic
287	C₂²	C₅×#488	1	1	yes	characteristic
288	C₄	—	2	1	no	—
289	C₄	—	2	1	no	—
290	C₄	—	2	1	no	—
291	C₄	—	8	2	no	—
292	C₄	—	8	1	no	—
293	C₄	—	8	1	no	—
294	C₄	C₄×C₅×SD₁₆	1	1	yes	characteristic
295	C₄	C₄×C₅×Q₁₆	1	1	yes	characteristic
296	C₄	C₅×#443	1	1	yes	characteristic
297	C₄	C₅×#444	1	1	yes	characteristic
298	C₄	C₅×#465	1	1	yes	characteristic
299	C₄	C₅×#467	1	1	yes	characteristic
300	C₂	—	2	1	no	—
301	C₂	—	2	1	no	—
302	C₂	C₅×#1929	1	1	yes	characteristic
303	C₂	C₅×#1932	1	1	yes	characteristic
304	C₂	C₅×#1933	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	#121
commutator	C₄²	C₂²×C₄×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
socle	C₂²×C₅	#388
prime radical	C₂³×C₅	#124
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
19. C₅×#333	C₄	semidirect
20. C₅×#333	C₄	semidirect
21. C₅×#332	C₄	semidirect
22. C₅×#332	C₄	semidirect
27. #9589	C₅	direct (N × H)
118. #333	C₄×C₅	semidirect
119. #333	C₄×C₅	semidirect
120. #332	C₄×C₅	semidirect
121. #332	C₄×C₅	semidirect
282. C₅	#9589	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅×#1986	2
C₅×#1950	2
C₅×#1542	2
C₅×#1493	2
C₅×C₈×M₁₆	2
C₅×#1303	2
C₅×#1279	2
#9589	5

Subgroups — C5×#9589 #11932309

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#9589 #11932309