Subgroups — C₅×#8932 #11931652

Order 1280.

426 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#2935	C₂	1	1	yes	maximal, characteristic
2	C₅×#1950	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#1570	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#1540	C₂	1	1	yes	maximal, characteristic
5	C₅×#1492	C₂	1	1	yes	maximal, characteristic
6	C₅×#1278	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#1277	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅×#131	—	2	1	no	—
9	C₅×#532	C₂²	1	1	yes	characteristic
10	C₅×#436	—	2	1	no	—
11	C₅×#435	—	2	2	no	—
12	C₅×C₈×D₄	—	2	1	no	—
13	C₅×#431	C₂²	1	1	yes	characteristic
14	C₂×C₅×#108	—	2	1	no	—
15	C₅×#423	C₂²	1	1	yes	characteristic
16	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
18	C₅×#408	—	2	1	no	—
19	C₄×C₅×M₁₆	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
21	C₅×#341	—	2	1	no	—
22	C₅×#334	—	2	1	no	—
23	C₅×#334	—	2	1	no	—
24	C₅×#332	C₄	1	1	yes	characteristic
25	C₅×#332	C₄	1	1	yes	characteristic
26	C₅×#331	C₄	1	1	yes	characteristic
27	C₅×#331	C₄	1	1	yes	characteristic
28	C₅×D₄⋊C₈	—	2	2	no	—
29	#8932	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
30	C₂×C₅×#41	—	2	1	no	—
31	C₅×#131	—	4	2	no	—
32	C₅×#131	—	4	2	no	—
33	C₂×C₅×M₁₆	—	2	2	no	—
34	C₂×C₅×M₁₆	—	2	1	no	—
35	C₂×C₅×M₁₆	D₄	1	1	yes	characteristic
36	C₂×C₅×M₁₆	D₄	1	1	yes	characteristic
37	C₂²×C₅×C₈	—	2	2	no	—
38	C₂²×C₅×C₈	—	2	1	no	—
39	C₂²×C₅×C₈	—	2	1	no	—
40	C₂²×C₅×C₈	—	2	1	no	—
41	C₅×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
42	C₅×#127	C₂×C₄	1	1	yes	characteristic
43	C₅×#124	—	2	1	no	—
44	C₅×#121	—	2	1	no	—
45	C₅×#121	Q₈	1	2	yes	—
46	C₄×C₅×D₄	—	2	1	no	—
47	C₄×C₅×D₄	D₄	1	2	yes	—
48	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
49	C₅×#108	—	4	2	no	—
50	C₅×C₄⋊C₈	—	2	1	no	—
51	C₅×C₄⋊C₈	—	2	1	no	—
52	C₅×C₄⋊C₈	—	2	1	no	—
53	C₅×C₄⋊C₈	—	2	1	no	—
54	C₅×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
55	C₅×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
56	C₅×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
57	C₅×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
58	C₅×C₄⋊C₈	D₄	1	1	yes	characteristic
59	C₅×C₄⋊C₈	D₄	1	1	yes	characteristic
60	C₅×C₂²⋊C₈	—	2	2	no	—
61	C₅×C₂²⋊C₈	—	2	2	no	—
62	C₅×C₂²⋊C₈	—	2	1	no	—
63	C₅×#97	—	2	1	no	—
64	C₄×C₅×C₈	—	2	1	no	—
65	C₄×C₅×C₈	—	2	1	no	—
66	C₄×C₅×C₈	—	2	1	no	—
67	C₄×C₅×C₈	—	2	1	no	—
68	#2935	C₂×C₅	1	1	yes	characteristic
69	#1950	C₂×C₅	1	1	yes	characteristic, complemented
70	#1570	C₂×C₅	1	1	yes	characteristic, complemented
71	#1540	C₂×C₅	1	1	yes	characteristic
72	#1492	C₂×C₅	1	1	yes	characteristic
73	#1278	C₂×C₅	1	1	yes	characteristic, complemented
74	#1277	C₂×C₅	1	1	yes	characteristic, complemented
75	C₅×#41	—	4	1	no	—
76	C₅×#41	—	4	1	no	—
77	C₅×#41	—	4	1	no	—
78	C₅×#41	—	4	1	no	—
79	C₂×C₅×Q₈	—	2	1	no	—
80	C₂×C₅×D₄	—	2	2	no	—
81	C₂×C₅×D₄	—	2	1	no	—
82	C₂×C₅×D₄	—	2	1	no	—
83	C₂×C₅×D₄	C₄⋊C₄	1	2	yes	—
84	C₂²×C₄×C₅	—	2	2	no	—
85	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
87	C₂²×C₄×C₅	#41	1	1	yes	characteristic
88	C₅×M₁₆	—	2	1	no	—
89	C₅×M₁₆	—	2	1	no	—
90	C₅×M₁₆	—	2	1	no	—
91	C₅×M₁₆	—	4	4	no	—
92	C₅×M₁₆	—	4	1	no	—
93	C₅×M₁₆	—	4	1	no	—
94	C₅×M₁₆	SD₁₆	1	2	yes	—
95	C₂×C₅×C₈	—	2	1	no	—
96	C₂×C₅×C₈	—	2	2	no	—
97	C₂×C₅×C₈	—	2	1	no	—
98	C₂×C₅×C₈	—	2	1	no	—
99	C₂×C₅×C₈	—	2	1	no	—
100	C₂×C₅×C₈	—	2	1	no	—
101	C₂×C₅×C₈	—	2	1	no	—
102	C₂×C₅×C₈	—	2	1	no	—
103	C₂×C₅×C₈	—	2	1	no	—
104	C₂×C₅×C₈	—	4	4	no	—
105	C₂×C₅×C₈	—	4	1	no	—
106	C₂×C₅×C₈	—	4	1	no	—
107	C₂×C₅×C₈	—	4	1	no	—
108	C₂×C₅×C₈	—	4	1	no	—
109	C₂×C₅×C₈	—	4	1	no	—
110	C₂×C₅×C₈	—	4	1	no	—
111	C₂×C₅×C₈	—	4	1	no	—
112	C₅×C₄⋊C₄	—	2	1	no	—
113	C₅×C₄⋊C₄	C₄⋊C₄	1	2	yes	—
114	C₅×C₂²⋊C₄	—	2	2	no	—
115	C₅×C₂²⋊C₄	—	2	2	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	#41	1	1	yes	characteristic
120	C₂×#131	—	2	1	no	—
121	#532	C₂²×C₅	1	1	yes	characteristic
122	#436	—	2	1	no	—
123	#435	—	2	2	no	—
124	C₈×D₄	—	2	1	no	—
125	#431	C₂²×C₅	1	1	yes	characteristic
126	C₂×#108	—	2	1	no	—
127	#423	C₂²×C₅	1	1	yes	characteristic
128	C₂×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
129	C₂×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
130	#408	—	2	1	no	—
131	C₄×M₁₆	C₂²×C₅	1	1	yes	characteristic
132	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
133	#341	—	2	1	no	—
134	#334	—	2	1	no	—
135	#334	—	2	1	no	—
136	#332	C₄×C₅	1	1	yes	characteristic
137	#332	C₄×C₅	1	1	yes	characteristic
138	#331	C₄×C₅	1	1	yes	characteristic
139	#331	C₄×C₅	1	1	yes	characteristic
140	D₄⋊C₈	—	2	2	no	—
141	C₂³×C₅	—	2	2	no	—
142	C₂³×C₅	C₂²⋊Q₈	1	1	yes	characteristic
143	C₅×Q₈	—	4	1	no	—
144	C₅×Q₈	—	4	1	no	—
145	C₅×D₄	—	4	2	no	—
146	C₅×D₄	—	4	2	no	—
147	C₅×D₄	—	4	2	no	—
148	C₅×D₄	—	4	1	no	—
149	C₅×D₄	—	4	1	no	—
150	C₅×D₄	—	4	1	no	—
151	C₅×D₄	—	4	1	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	2	no	—
157	C₂×C₄×C₅	—	4	2	no	—
158	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
159	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
160	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
161	C₂×C₄×C₅	#120	1	1	yes	characteristic
162	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
163	C₂×C₄×C₅	#123	1	1	yes	characteristic
164	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
165	C₂×C₄×C₅	#135	1	1	yes	characteristic
166	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
167	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
168	C₅×C₈	—	2	2	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	—	4	1	no	—
174	C₅×C₈	—	4	1	no	—
175	C₅×C₈	—	4	1	no	—
176	C₅×C₈	—	4	1	no	—
177	C₅×C₈	—	8	1	no	—
178	C₂×#41	—	2	1	no	—
179	#131	—	4	2	no	—
180	#131	—	4	2	no	—
181	C₂×M₁₆	—	2	2	no	—
182	C₂×M₁₆	—	2	1	no	—
183	C₂×M₁₆	C₅×D₄	1	1	yes	characteristic
184	C₂×M₁₆	C₅×D₄	1	1	yes	characteristic
185	C₂²×C₈	—	2	2	no	—
186	C₂²×C₈	—	2	1	no	—
187	C₂²×C₈	—	2	1	no	—
188	C₂²×C₈	—	2	1	no	—
189	C₄⋊Q₈	C₂×C₄×C₅	1	1	yes	characteristic
190	#127	C₂×C₄×C₅	1	1	yes	characteristic
191	#124	—	2	1	no	—
192	#121	—	2	1	no	—
193	#121	C₅×Q₈	1	2	yes	—
194	C₄×D₄	—	2	1	no	—
195	C₄×D₄	C₅×D₄	1	2	yes	—
196	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
197	#108	—	4	2	no	—
198	C₄⋊C₈	—	2	1	no	—
199	C₄⋊C₈	—	2	1	no	—
200	C₄⋊C₈	—	2	1	no	—
201	C₄⋊C₈	—	2	1	no	—
202	C₄⋊C₈	C₂×C₄×C₅	1	1	yes	characteristic
203	C₄⋊C₈	C₂×C₄×C₅	1	1	yes	characteristic
204	C₄⋊C₈	C₂×C₄×C₅	1	1	yes	characteristic
205	C₄⋊C₈	C₂×C₄×C₅	1	1	yes	characteristic
206	C₄⋊C₈	C₅×D₄	1	1	yes	characteristic
207	C₄⋊C₈	C₅×D₄	1	1	yes	characteristic
208	C₂²⋊C₈	—	2	2	no	—
209	C₂²⋊C₈	—	2	2	no	—
210	C₂²⋊C₈	—	2	1	no	—
211	#97	—	2	1	no	—
212	C₄×C₈	—	2	1	no	—
213	C₄×C₈	—	2	1	no	—
214	C₄×C₈	—	2	1	no	—
215	C₄×C₈	—	2	1	no	—
216	C₂²×C₅	—	2	1	no	—
217	C₂²×C₅	—	2	1	no	—
218	C₂²×C₅	—	4	2	no	—
219	C₂²×C₅	—	4	2	no	—
220	C₂²×C₅	—	4	2	no	—
221	C₂²×C₅	#385	1	1	yes	characteristic
222	C₂²×C₅	#476	1	1	yes	characteristic
223	C₂²×C₅	#478	1	1	yes	characteristic
224	C₄×C₅	—	2	1	no	—
225	C₄×C₅	—	2	1	no	—
226	C₄×C₅	—	2	1	no	—
227	C₄×C₅	—	4	2	no	—
228	C₄×C₅	C₄×SD₁₆	1	1	yes	characteristic
229	C₄×C₅	#442	1	1	yes	characteristic
230	C₄×C₅	#443	1	1	yes	characteristic
231	C₄×C₅	#444	1	1	yes	characteristic
232	C₄×C₅	#449	1	1	yes	characteristic
233	C₄×C₅	#450	1	1	yes	characteristic
234	#41	—	4	1	no	—
235	#41	—	4	1	no	—
236	#41	—	4	1	no	—
237	#41	—	4	1	no	—
238	C₂×Q₈	—	2	1	no	—
239	C₂×D₄	—	2	2	no	—
240	C₂×D₄	—	2	1	no	—
241	C₂×D₄	—	2	1	no	—
242	C₂×D₄	C₅×C₄⋊C₄	1	2	yes	—
243	C₂²×C₄	—	2	2	no	—
244	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
245	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
246	C₂²×C₄	C₅×#41	1	1	yes	characteristic
247	M₁₆	—	2	1	no	—
248	M₁₆	—	2	1	no	—
249	M₁₆	—	2	1	no	—
250	M₁₆	—	4	4	no	—
251	M₁₆	—	4	1	no	—
252	M₁₆	—	4	1	no	—
253	M₁₆	C₅×SD₁₆	1	2	yes	—
254	C₂×C₈	—	2	1	no	—
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	—	2	1	no	—
257	C₂×C₈	—	2	1	no	—
258	C₂×C₈	—	2	1	no	—
259	C₂×C₈	—	2	1	no	—
260	C₂×C₈	—	2	1	no	—
261	C₂×C₈	—	2	1	no	—
262	C₂×C₈	—	2	1	no	—
263	C₂×C₈	—	4	4	no	—
264	C₂×C₈	—	4	1	no	—
265	C₂×C₈	—	4	1	no	—
266	C₂×C₈	—	4	1	no	—
267	C₂×C₈	—	4	1	no	—
268	C₂×C₈	—	4	1	no	—
269	C₂×C₈	—	4	1	no	—
270	C₂×C₈	—	4	1	no	—
271	C₄⋊C₄	—	2	1	no	—
272	C₄⋊C₄	C₅×C₄⋊C₄	1	2	yes	—
273	C₂²⋊C₄	—	2	2	no	—
274	C₂²⋊C₄	—	2	2	no	—
275	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
276	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
277	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
278	C₄²	C₅×#41	1	1	yes	characteristic
279	C₂×C₅	—	2	1	no	—
280	C₂×C₅	—	2	1	no	—
281	C₂×C₅	—	8	2	no	—
282	C₂×C₅	#1858	1	1	yes	characteristic
283	C₂×C₅	#1864	1	1	yes	characteristic
284	C₂×C₅	#1866	1	1	yes	characteristic
285	C₂³	—	2	2	no	—
286	C₂³	C₅×C₂²⋊Q₈	1	1	yes	characteristic
287	Q₈	—	4	1	no	—
288	Q₈	—	4	1	no	—
289	D₄	—	4	2	no	—
290	D₄	—	4	2	no	—
291	D₄	—	4	2	no	—
292	D₄	—	4	1	no	—
293	D₄	—	4	1	no	—
294	D₄	—	4	1	no	—
295	D₄	—	4	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	4	1	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
303	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
304	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
305	C₂×C₄	C₅×#120	1	1	yes	characteristic
306	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
307	C₂×C₄	C₅×#123	1	1	yes	characteristic
308	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
309	C₂×C₄	C₅×#135	1	1	yes	characteristic
310	C₂×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
311	C₂×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
312	C₈	—	2	2	no	—
313	C₈	—	4	2	no	—
314	C₈	—	4	1	no	—
315	C₈	—	4	1	no	—
316	C₈	—	4	1	no	—
317	C₈	—	4	1	no	—
318	C₈	—	4	1	no	—
319	C₈	—	4	1	no	—
320	C₈	—	4	1	no	—
321	C₈	—	8	1	no	—
322	C₅	#8932	1	1	yes	characteristic, direct product factor, Sylow
323	C₂²	—	2	1	no	—
324	C₂²	—	2	1	no	—
325	C₂²	—	4	2	no	—
326	C₂²	—	4	2	no	—
327	C₂²	—	4	2	no	—
328	C₂²	C₅×#385	1	1	yes	characteristic
329	C₂²	C₅×#476	1	1	yes	characteristic
330	C₂²	C₅×#478	1	1	yes	characteristic
331	C₄	—	2	1	no	—
332	C₄	—	2	1	no	—
333	C₄	—	2	1	no	—
334	C₄	—	4	2	no	—
335	C₄	C₄×C₅×SD₁₆	1	1	yes	characteristic
336	C₄	C₅×#442	1	1	yes	characteristic
337	C₄	C₅×#443	1	1	yes	characteristic
338	C₄	C₅×#444	1	1	yes	characteristic
339	C₄	C₅×#449	1	1	yes	characteristic
340	C₄	C₅×#450	1	1	yes	characteristic
341	C₂	—	2	1	no	—
342	C₂	—	2	1	no	—
343	C₂	—	8	2	no	—
344	C₂	C₅×#1858	1	1	yes	characteristic
345	C₂	C₅×#1864	1	1	yes	characteristic
346	C₂	C₅×#1866	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	#120
commutator	C₄²	C₂²×C₄×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
socle	C₂²×C₅	#385
prime radical	C₅×#532	C₂²
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
2. C₅×#1950	C₂	semidirect
3. C₅×#1570	C₂	semidirect
6. C₅×#1278	C₂	semidirect
7. C₅×#1277	C₂	semidirect
29. #8932	C₅	direct (N × H)
69. #1950	C₂×C₅	semidirect
70. #1570	C₂×C₅	semidirect
73. #1278	C₂×C₅	semidirect
74. #1277	C₂×C₅	semidirect
322. C₅	#8932	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×#2935	2
C₅×#1950	2
C₅×#1570	2
C₅×#1540	2
C₅×#1492	2
C₅×#1278	2
C₅×#1277	2
#8932	5

Subgroups — C5×#8932 #11931652

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#8932 #11931652