Subgroups — C₅×#9285 #11932005

Order 1280.

426 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3210	C₂	1	1	yes	maximal, characteristic
2	C₅×#2981	C₂	1	1	yes	maximal, characteristic
3	C₅×#1531	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#1530	C₂	1	1	yes	maximal, characteristic
5	C₅×#1497	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#1287	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#1275	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#533	C₂²	1	1	yes	characteristic
9	C₅×#532	C₂²	1	1	yes	characteristic
10	C₅×#494	C₄	1	1	yes	characteristic
11	C₅×#492	C₄	1	1	yes	characteristic
12	C₅×#487	C₄	1	1	yes	characteristic
13	C₅×#486	C₄	1	1	yes	characteristic
14	C₅×#464	—	2	2	no	—
15	C₅×#452	—	2	2	no	—
16	C₅×#436	—	2	1	no	—
17	C₅×#435	—	2	2	no	—
18	C₅×C₈×D₄	—	2	1	no	—
19	C₅×#432	C₂²	1	1	yes	characteristic
20	C₅×#432	C₂²	1	1	yes	characteristic
21	C₅×#431	C₂²	1	1	yes	characteristic
22	C₅×#431	C₂²	1	1	yes	characteristic
23	C₅×#408	—	2	2	no	—
24	C₂×C₅×#97	C₂²	1	1	yes	characteristic
25	C₅×#330	—	2	2	no	—
26	C₅×#329	—	2	2	no	—
27	C₅×Q₈⋊C₈	—	2	1	no	—
28	C₅×Q₈⋊C₈	—	2	1	no	—
29	#9285	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
30	C₂×C₅×#41	D₄	1	2	yes	—
31	C₂×C₅×Q₁₆	—	2	2	no	—
32	C₂×C₅×SD₁₆	—	2	2	no	—
33	C₂×C₅×M₁₆	—	2	2	no	—
34	C₂²×C₅×C₈	—	2	1	no	—
35	C₂²×C₅×C₈	—	2	2	no	—
36	C₅×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
37	C₅×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
38	C₅×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
39	C₅×#127	C₂×C₄	1	1	yes	characteristic
40	C₅×#125	—	2	1	no	—
41	C₅×#124	D₄	1	2	yes	—
42	C₅×C₂²⋊Q₈	—	2	1	no	—
43	C₅×C₂²⋊Q₈	—	2	1	no	—
44	C₅×#121	—	2	1	no	—
45	C₅×#121	—	2	1	no	—
46	C₄×C₅×Q₈	—	2	1	no	—
47	C₄×C₅×Q₈	—	2	1	no	—
48	C₄×C₅×D₄	—	2	1	no	—
49	C₄×C₅×D₄	—	2	1	no	—
50	C₅×#117	—	2	1	no	—
51	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
52	C₅×C₄⋊C₈	—	2	1	no	—
53	C₅×C₄⋊C₈	—	2	1	no	—
54	C₅×C₄⋊C₈	—	2	1	no	—
55	C₅×C₄⋊C₈	—	2	1	no	—
56	C₅×#103	—	2	2	no	—
57	C₅×#102	—	2	2	no	—
58	C₅×C₂²⋊C₈	—	2	2	no	—
59	C₅×C₂²⋊C₈	—	2	2	no	—
60	C₅×C₂²⋊C₈	—	2	1	no	—
61	C₅×C₂²⋊C₈	—	2	1	no	—
62	C₅×C₂²⋊C₈	—	2	1	no	—
63	C₅×C₂²⋊C₈	—	2	1	no	—
64	C₅×#97	—	2	1	no	—
65	C₅×#97	—	2	1	no	—
66	C₅×#97	—	2	1	no	—
67	C₅×#97	D₄	1	2	yes	—
68	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
69	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
70	C₄×C₅×C₈	D₄	1	1	yes	characteristic
71	C₄×C₅×C₈	D₄	1	1	yes	characteristic
72	#3210	C₂×C₅	1	1	yes	characteristic
73	#2981	C₂×C₅	1	1	yes	characteristic
74	#1531	C₂×C₅	1	1	yes	characteristic, complemented
75	#1530	C₂×C₅	1	1	yes	characteristic
76	#1497	C₂×C₅	1	1	yes	characteristic, complemented
77	#1287	C₂×C₅	1	1	yes	characteristic, complemented
78	#1275	C₂×C₅	1	1	yes	characteristic, complemented
79	C₅×#41	—	8	2	no	—
80	C₂×C₅×Q₈	—	2	1	no	—
81	C₂×C₅×Q₈	—	2	1	no	—
82	C₂×C₅×Q₈	C₂²⋊C₄	1	2	yes	—
83	C₂×C₅×D₄	—	2	2	no	—
84	C₂×C₅×D₄	—	2	1	no	—
85	C₂×C₅×D₄	—	2	1	no	—
86	C₂×C₅×D₄	C₂²⋊C₄	1	2	yes	—
87	C₂²×C₄×C₅	—	2	2	no	—
88	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₅	#41	1	1	yes	characteristic
90	C₂²×C₄×C₅	#41	1	1	yes	characteristic
91	C₅×Q₁₆	—	8	2	no	—
92	C₅×SD₁₆	—	8	2	no	—
93	C₅×M₁₆	—	8	2	no	—
94	C₂×C₅×C₈	—	2	2	no	—
95	C₂×C₅×C₈	—	2	1	no	—
96	C₂×C₅×C₈	—	2	1	no	—
97	C₂×C₅×C₈	—	2	1	no	—
98	C₂×C₅×C₈	—	2	2	no	—
99	C₂×C₅×C₈	—	2	1	no	—
100	C₂×C₅×C₈	—	4	1	no	—
101	C₂×C₅×C₈	—	4	1	no	—
102	C₂×C₅×C₈	—	4	1	no	—
103	C₂×C₅×C₈	—	8	2	no	—
104	C₅×C₄⋊C₄	—	2	1	no	—
105	C₅×C₄⋊C₄	—	2	1	no	—
106	C₅×C₄⋊C₄	—	2	1	no	—
107	C₅×C₄⋊C₄	—	2	1	no	—
108	C₅×C₄⋊C₄	—	2	1	no	—
109	C₅×C₄⋊C₄	—	2	1	no	—
110	C₅×C₄⋊C₄	—	4	1	no	—
111	C₅×C₄⋊C₄	—	4	1	no	—
112	C₅×C₂²⋊C₄	—	2	2	no	—
113	C₅×C₂²⋊C₄	—	2	2	no	—
114	C₅×C₂²⋊C₄	—	4	1	no	—
115	C₄²×C₅	—	4	1	no	—
116	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
117	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
118	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
119	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
120	#533	C₂²×C₅	1	1	yes	characteristic
121	#532	C₂²×C₅	1	1	yes	characteristic
122	#494	C₄×C₅	1	1	yes	characteristic
123	#492	C₄×C₅	1	1	yes	characteristic
124	#487	C₄×C₅	1	1	yes	characteristic
125	#486	C₄×C₅	1	1	yes	characteristic
126	#464	—	2	2	no	—
127	#452	—	2	2	no	—
128	#436	—	2	1	no	—
129	#435	—	2	2	no	—
130	C₈×D₄	—	2	1	no	—
131	#432	C₂²×C₅	1	1	yes	characteristic
132	#432	C₂²×C₅	1	1	yes	characteristic
133	#431	C₂²×C₅	1	1	yes	characteristic
134	#431	C₂²×C₅	1	1	yes	characteristic
135	#408	—	2	2	no	—
136	C₂×#97	C₂²×C₅	1	1	yes	characteristic
137	#330	—	2	2	no	—
138	#329	—	2	2	no	—
139	Q₈⋊C₈	—	2	1	no	—
140	Q₈⋊C₈	—	2	1	no	—
141	C₂³×C₅	—	2	2	no	—
142	C₂³×C₅	#123	1	1	yes	characteristic
143	C₅×Q₈	—	4	2	no	—
144	C₅×Q₈	—	4	2	no	—
145	C₅×Q₈	—	4	1	no	—
146	C₅×Q₈	—	4	1	no	—
147	C₅×D₄	—	4	2	no	—
148	C₅×D₄	—	4	1	no	—
149	C₅×D₄	—	4	1	no	—
150	C₅×D₄	—	8	1	no	—
151	C₅×D₄	—	8	2	no	—
152	C₂×C₄×C₅	—	2	1	no	—
153	C₂×C₄×C₅	—	2	1	no	—
154	C₂×C₄×C₅	—	2	2	no	—
155	C₂×C₄×C₅	—	4	1	no	—
156	C₂×C₄×C₅	—	4	1	no	—
157	C₂×C₄×C₅	—	4	2	no	—
158	C₂×C₄×C₅	—	4	2	no	—
159	C₂×C₄×C₅	—	4	1	no	—
160	C₂×C₄×C₅	—	4	1	no	—
161	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
162	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
163	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
164	C₂×C₄×C₅	#120	1	1	yes	characteristic
165	C₂×C₄×C₅	#121	1	1	yes	characteristic
166	C₂×C₄×C₅	#121	1	1	yes	characteristic
167	C₅×C₈	—	4	2	no	—
168	C₅×C₈	—	4	1	no	—
169	C₅×C₈	—	4	2	no	—
170	C₅×C₈	—	4	1	no	—
171	C₅×C₈	—	4	1	no	—
172	C₅×C₈	—	4	1	no	—
173	C₅×C₈	—	8	1	no	—
174	C₂×#41	C₅×D₄	1	2	yes	—
175	C₂×Q₁₆	—	2	2	no	—
176	C₂×SD₁₆	—	2	2	no	—
177	C₂×M₁₆	—	2	2	no	—
178	C₂²×C₈	—	2	1	no	—
179	C₂²×C₈	—	2	2	no	—
180	C₄⋊Q₈	C₂×C₄×C₅	1	1	yes	characteristic
181	C₄⋊Q₈	C₂×C₄×C₅	1	1	yes	characteristic
182	C₄⋊Q₈	C₂×C₄×C₅	1	1	yes	characteristic
183	#127	C₂×C₄×C₅	1	1	yes	characteristic
184	#125	—	2	1	no	—
185	#124	C₅×D₄	1	2	yes	—
186	C₂²⋊Q₈	—	2	1	no	—
187	C₂²⋊Q₈	—	2	1	no	—
188	#121	—	2	1	no	—
189	#121	—	2	1	no	—
190	C₄×Q₈	—	2	1	no	—
191	C₄×Q₈	—	2	1	no	—
192	C₄×D₄	—	2	1	no	—
193	C₄×D₄	—	2	1	no	—
194	#117	—	2	1	no	—
195	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
196	C₄⋊C₈	—	2	1	no	—
197	C₄⋊C₈	—	2	1	no	—
198	C₄⋊C₈	—	2	1	no	—
199	C₄⋊C₈	—	2	1	no	—
200	#103	—	2	2	no	—
201	#102	—	2	2	no	—
202	C₂²⋊C₈	—	2	2	no	—
203	C₂²⋊C₈	—	2	2	no	—
204	C₂²⋊C₈	—	2	1	no	—
205	C₂²⋊C₈	—	2	1	no	—
206	C₂²⋊C₈	—	2	1	no	—
207	C₂²⋊C₈	—	2	1	no	—
208	#97	—	2	1	no	—
209	#97	—	2	1	no	—
210	#97	—	2	1	no	—
211	#97	C₅×D₄	1	2	yes	—
212	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
213	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
214	C₄×C₈	C₅×D₄	1	1	yes	characteristic
215	C₄×C₈	C₅×D₄	1	1	yes	characteristic
216	C₂²×C₅	—	2	1	no	—
217	C₂²×C₅	—	2	1	no	—
218	C₂²×C₅	—	2	1	no	—
219	C₂²×C₅	—	4	2	no	—
220	C₂²×C₅	—	4	2	no	—
221	C₂²×C₅	—	4	2	no	—
222	C₂²×C₅	#386	1	1	yes	characteristic
223	C₄×C₅	—	2	1	no	—
224	C₄×C₅	—	2	1	no	—
225	C₄×C₅	—	2	1	no	—
226	C₄×C₅	—	2	1	no	—
227	C₄×C₅	—	2	1	no	—
228	C₄×C₅	—	4	2	no	—
229	C₄×C₅	—	8	1	no	—
230	C₄×C₅	—	8	1	no	—
231	C₄×C₅	#443	1	1	yes	characteristic
232	C₄×C₅	#444	1	1	yes	characteristic
233	#41	—	8	2	no	—
234	C₂×Q₈	—	2	1	no	—
235	C₂×Q₈	—	2	1	no	—
236	C₂×Q₈	C₅×C₂²⋊C₄	1	2	yes	—
237	C₂×D₄	—	2	2	no	—
238	C₂×D₄	—	2	1	no	—
239	C₂×D₄	—	2	1	no	—
240	C₂×D₄	C₅×C₂²⋊C₄	1	2	yes	—
241	C₂²×C₄	—	2	2	no	—
242	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
243	C₂²×C₄	C₅×#41	1	1	yes	characteristic
244	C₂²×C₄	C₅×#41	1	1	yes	characteristic
245	Q₁₆	—	8	2	no	—
246	SD₁₆	—	8	2	no	—
247	M₁₆	—	8	2	no	—
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	—	2	1	no	—
250	C₂×C₈	—	2	1	no	—
251	C₂×C₈	—	2	1	no	—
252	C₂×C₈	—	2	1	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	4	1	no	—
255	C₂×C₈	—	4	1	no	—
256	C₂×C₈	—	4	1	no	—
257	C₂×C₈	—	8	2	no	—
258	C₄⋊C₄	—	2	1	no	—
259	C₄⋊C₄	—	2	1	no	—
260	C₄⋊C₄	—	2	1	no	—
261	C₄⋊C₄	—	2	1	no	—
262	C₄⋊C₄	—	2	1	no	—
263	C₄⋊C₄	—	2	1	no	—
264	C₄⋊C₄	—	4	1	no	—
265	C₄⋊C₄	—	4	1	no	—
266	C₂²⋊C₄	—	2	2	no	—
267	C₂²⋊C₄	—	2	2	no	—
268	C₂²⋊C₄	—	4	1	no	—
269	C₄²	—	4	1	no	—
270	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
271	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
272	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
273	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
274	C₂×C₅	—	4	1	no	—
275	C₂×C₅	—	8	2	no	—
276	C₂×C₅	#1876	1	1	yes	characteristic
277	C₂×C₅	#1902	1	1	yes	characteristic
278	C₂×C₅	#1911	1	1	yes	characteristic
279	C₂³	—	2	2	no	—
280	C₂³	C₅×#123	1	1	yes	characteristic
281	Q₈	—	4	2	no	—
282	Q₈	—	4	2	no	—
283	Q₈	—	4	1	no	—
284	Q₈	—	4	1	no	—
285	D₄	—	4	2	no	—
286	D₄	—	4	1	no	—
287	D₄	—	4	1	no	—
288	D₄	—	8	1	no	—
289	D₄	—	8	2	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	4	1	no	—
294	C₂×C₄	—	4	1	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	1	no	—
298	C₂×C₄	—	4	1	no	—
299	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
300	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
301	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
302	C₂×C₄	C₅×#120	1	1	yes	characteristic
303	C₂×C₄	C₅×#121	1	1	yes	characteristic
304	C₂×C₄	C₅×#121	1	1	yes	characteristic
305	C₈	—	4	2	no	—
306	C₈	—	4	1	no	—
307	C₈	—	4	2	no	—
308	C₈	—	4	1	no	—
309	C₈	—	4	1	no	—
310	C₈	—	4	1	no	—
311	C₈	—	8	1	no	—
312	C₅	#9285	1	1	yes	characteristic, direct product factor, Sylow
313	C₂²	—	2	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	2	1	no	—
316	C₂²	—	4	2	no	—
317	C₂²	—	4	2	no	—
318	C₂²	—	4	2	no	—
319	C₂²	C₅×#386	1	1	yes	characteristic
320	C₄	—	2	1	no	—
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	2	1	no	—
325	C₄	—	4	2	no	—
326	C₄	—	8	1	no	—
327	C₄	—	8	1	no	—
328	C₄	C₅×#443	1	1	yes	characteristic
329	C₄	C₅×#444	1	1	yes	characteristic
330	C₂	—	4	1	no	—
331	C₂	—	8	2	no	—
332	C₂	C₅×#1876	1	1	yes	characteristic
333	C₂	C₅×#1902	1	1	yes	characteristic
334	C₂	C₅×#1911	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	#386
commutator	C₄²	C₂²×C₄×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
prime radical	C₅×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅×#1531	C₂	semidirect
5. C₅×#1497	C₂	semidirect
6. C₅×#1287	C₂	semidirect
7. C₅×#1275	C₂	semidirect
29. #9285	C₅	direct (N × H)
74. #1531	C₂×C₅	semidirect
76. #1497	C₂×C₅	semidirect
77. #1287	C₂×C₅	semidirect
78. #1275	C₂×C₅	semidirect
312. C₅	#9285	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×#3210	2
C₅×#2981	2
C₅×#1531	2
C₅×#1530	2
C₅×#1497	2
C₅×#1287	2
C₅×#1275	2
#9285	5

Subgroups — C5×#9285 #11932005

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#9285 #11932005