Subgroups — C₅×#10166 #11932886

Order 1280.

514 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3226	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#498	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#3141	C₂	1	1	yes	maximal, characteristic
4	C₅×#1966	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#1548	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#1492	C₂	1	1	yes	maximal, characteristic
7	C₅×#1278	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#532	C₂²	1	1	yes	characteristic
9	C₂×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
10	C₂×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	C₅×#500	—	2	1	no	—
12	C₅×#498	—	2	2	no	—
13	C₅×#498	—	2	2	no	—
14	C₅×#489	—	2	2	no	—
15	C₅×#487	—	2	1	no	—
16	C₅×#486	—	2	1	no	—
17	C₅×#486	—	2	1	no	—
18	C₅×#480	—	2	2	no	—
19	C₅×#474	—	2	2	no	—
20	C₅×#431	C₂²	1	1	yes	characteristic
21	C₂×C₅×#106	—	2	2	no	—
22	C₅×#423	C₂²	1	1	yes	characteristic
23	C₅×#417	—	2	2	no	—
24	C₂×C₅×#101	—	2	2	no	—
25	C₄×C₅×M₁₆	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
27	C₅×#333	—	2	2	no	—
28	C₅×D₄⋊C₈	—	2	2	no	—
29	#10166	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
30	C₂×C₅×#41	—	2	1	no	—
31	C₂²×C₅×Q₈	—	2	1	no	—
32	C₂²×C₅×Q₈	—	2	1	no	—
33	C₂×C₅×M₁₆	—	2	1	no	—
34	C₂×C₅×M₁₆	D₄	1	2	yes	—
35	C₂²×C₅×C₈	—	2	1	no	—
36	C₅×C₄⋊Q₈	—	2	1	no	—
37	C₅×C₄⋊Q₈	—	2	1	no	—
38	C₅×C₄⋊Q₈	—	2	1	no	—
39	C₅×C₄⋊Q₈	—	2	1	no	—
40	C₅×C₄⋊Q₈	—	2	1	no	—
41	C₅×C₄⋊Q₈	—	2	1	no	—
42	C₅×C₄⋊Q₈	—	2	1	no	—
43	C₅×C₄⋊Q₈	D₄	1	1	yes	characteristic
44	C₅×C₄⋊Q₈	D₄	1	2	yes	—
45	C₅×#127	D₄	1	1	yes	characteristic
46	C₅×#124	—	2	1	no	—
47	C₅×#121	—	2	1	no	—
48	C₅×#121	—	2	1	no	—
49	C₄×C₅×D₄	—	2	1	no	—
50	C₄×C₅×D₄	—	2	1	no	—
51	C₂×C₅×C₄⋊C₄	—	2	1	no	—
52	C₂×C₅×C₄⋊C₄	—	2	1	no	—
53	C₂×C₅×C₄⋊C₄	—	2	1	no	—
54	C₂×C₅×C₄⋊C₄	D₄	1	2	yes	—
55	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
56	C₅×#107	—	2	4	no	—
57	C₅×#106	—	2	8	no	—
58	C₅×C₄⋊C₈	—	2	1	no	—
59	C₅×C₄⋊C₈	—	2	1	no	—
60	C₅×C₄⋊C₈	D₄	1	2	yes	—
61	C₅×#103	—	2	4	no	—
62	C₅×#102	—	2	4	no	—
63	C₅×#102	—	2	4	no	—
64	C₅×#101	—	4	4	no	—
65	C₅×C₂²⋊C₈	—	2	1	no	—
66	C₅×#97	—	2	1	no	—
67	C₄×C₅×C₈	—	2	1	no	—
68	C₄×C₅×C₈	—	2	1	no	—
69	C₄×C₅×C₈	—	2	1	no	—
70	C₄×C₅×C₈	D₄	1	2	yes	—
71	#3226	C₂×C₅	1	1	yes	characteristic
72	C₂×#498	C₂×C₅	1	1	yes	characteristic, complemented
73	#3141	C₂×C₅	1	1	yes	characteristic
74	#1966	C₂×C₅	1	1	yes	characteristic, complemented
75	#1548	C₂×C₅	1	1	yes	characteristic, complemented
76	#1492	C₂×C₅	1	1	yes	characteristic
77	#1278	C₂×C₅	1	1	yes	characteristic, complemented
78	C₅×#41	—	4	2	no	—
79	C₅×#41	—	4	2	no	—
80	C₂×C₅×Q₈	—	2	1	no	—
81	C₂×C₅×Q₈	—	2	2	no	—
82	C₂×C₅×Q₈	—	4	1	no	—
83	C₂×C₅×Q₈	—	4	1	no	—
84	C₂×C₅×Q₈	—	4	1	no	—
85	C₂×C₅×Q₈	—	4	2	no	—
86	C₂×C₅×Q₈	—	4	2	no	—
87	C₂×C₅×Q₈	—	8	2	no	—
88	C₂×C₅×D₄	—	2	1	no	—
89	C₂×C₅×D₄	—	2	1	no	—
90	C₂×C₅×D₄	—	2	1	no	—
91	C₂×C₅×D₄	—	4	1	no	—
92	C₂²×C₄×C₅	—	2	2	no	—
93	C₂²×C₄×C₅	—	4	1	no	—
94	C₂²×C₄×C₅	—	4	1	no	—
95	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
97	C₂²×C₄×C₅	#41	1	1	yes	characteristic
98	C₅×M₁₆	—	2	2	no	—
99	C₅×M₁₆	—	2	2	no	—
100	C₅×M₁₆	—	4	2	no	—
101	C₂×C₅×C₈	—	2	4	no	—
102	C₂×C₅×C₈	—	2	2	no	—
103	C₂×C₅×C₈	—	2	2	no	—
104	C₂×C₅×C₈	—	2	2	no	—
105	C₂×C₅×C₈	—	4	1	no	—
106	C₅×C₄⋊C₄	—	2	2	no	—
107	C₅×C₄⋊C₄	—	2	2	no	—
108	C₅×C₄⋊C₄	—	2	2	no	—
109	C₅×C₄⋊C₄	—	2	2	no	—
110	C₅×C₄⋊C₄	—	2	1	no	—
111	C₅×C₄⋊C₄	—	2	1	no	—
112	C₅×C₄⋊C₄	—	4	1	no	—
113	C₅×C₄⋊C₄	—	4	1	no	—
114	C₅×C₄⋊C₄	—	4	1	no	—
115	C₅×C₄⋊C₄	—	4	1	no	—
116	C₅×C₂²⋊C₄	—	4	1	no	—
117	C₅×C₂²⋊C₄	—	4	1	no	—
118	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
119	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
120	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
121	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
122	#532	C₂²×C₅	1	1	yes	characteristic
123	C₂×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
124	C₂×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
125	#500	—	2	1	no	—
126	#498	—	2	2	no	—
127	#498	—	2	2	no	—
128	#489	—	2	2	no	—
129	#487	—	2	1	no	—
130	#486	—	2	1	no	—
131	#486	—	2	1	no	—
132	#480	—	2	2	no	—
133	#474	—	2	2	no	—
134	#431	C₂²×C₅	1	1	yes	characteristic
135	C₂×#106	—	2	2	no	—
136	#423	C₂²×C₅	1	1	yes	characteristic
137	#417	—	2	2	no	—
138	C₂×#101	—	2	2	no	—
139	C₄×M₁₆	C₂²×C₅	1	1	yes	characteristic
140	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
141	#333	—	2	2	no	—
142	D₄⋊C₈	—	2	2	no	—
143	C₂³×C₅	—	4	1	no	—
144	C₂³×C₅	#121	1	1	yes	characteristic
145	C₅×Q₈	—	4	2	no	—
146	C₅×Q₈	—	8	2	no	—
147	C₅×Q₈	—	8	2	no	—
148	C₅×Q₈	—	8	2	no	—
149	C₅×Q₈	—	8	1	no	—
150	C₅×Q₈	—	8	1	no	—
151	C₅×D₄	—	4	2	no	—
152	C₅×D₄	—	4	2	no	—
153	C₅×D₄	—	4	2	no	—
154	C₅×D₄	—	8	2	no	—
155	C₂×C₄×C₅	—	2	2	no	—
156	C₂×C₄×C₅	—	2	2	no	—
157	C₂×C₄×C₅	—	4	1	no	—
158	C₂×C₄×C₅	—	4	2	no	—
159	C₂×C₄×C₅	—	4	2	no	—
160	C₂×C₄×C₅	—	4	4	no	—
161	C₂×C₄×C₅	—	8	2	no	—
162	C₂×C₄×C₅	—	8	1	no	—
163	C₂×C₄×C₅	—	8	1	no	—
164	C₂×C₄×C₅	#120	1	1	yes	characteristic
165	C₂×C₄×C₅	#120	1	1	yes	characteristic
166	C₂×C₄×C₅	#120	1	1	yes	characteristic
167	C₂×C₄×C₅	#121	1	1	yes	characteristic
168	C₂×C₄×C₅	#121	1	1	yes	characteristic
169	C₂×C₄×C₅	#127	1	1	yes	characteristic
170	C₂×C₄×C₅	#135	1	2	yes	—
171	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
172	C₅×C₈	—	2	4	no	—
173	C₅×C₈	—	4	2	no	—
174	C₅×C₈	—	4	2	no	—
175	C₅×C₈	—	8	1	no	—
176	C₂×#41	—	2	1	no	—
177	C₂²×Q₈	—	2	1	no	—
178	C₂²×Q₈	—	2	1	no	—
179	C₂×M₁₆	—	2	1	no	—
180	C₂×M₁₆	C₅×D₄	1	2	yes	—
181	C₂²×C₈	—	2	1	no	—
182	C₄⋊Q₈	—	2	1	no	—
183	C₄⋊Q₈	—	2	1	no	—
184	C₄⋊Q₈	—	2	1	no	—
185	C₄⋊Q₈	—	2	1	no	—
186	C₄⋊Q₈	—	2	1	no	—
187	C₄⋊Q₈	—	2	1	no	—
188	C₄⋊Q₈	—	2	1	no	—
189	C₄⋊Q₈	C₅×D₄	1	2	yes	—
190	C₄⋊Q₈	C₅×D₄	1	1	yes	characteristic
191	#127	C₅×D₄	1	1	yes	characteristic
192	#124	—	2	1	no	—
193	#121	—	2	1	no	—
194	#121	—	2	1	no	—
195	C₄×D₄	—	2	1	no	—
196	C₄×D₄	—	2	1	no	—
197	C₂×C₄⋊C₄	—	2	1	no	—
198	C₂×C₄⋊C₄	—	2	1	no	—
199	C₂×C₄⋊C₄	—	2	1	no	—
200	C₂×C₄⋊C₄	C₅×D₄	1	2	yes	—
201	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
202	#107	—	2	4	no	—
203	#106	—	2	8	no	—
204	C₄⋊C₈	—	2	1	no	—
205	C₄⋊C₈	—	2	1	no	—
206	C₄⋊C₈	C₅×D₄	1	2	yes	—
207	#103	—	2	4	no	—
208	#102	—	2	4	no	—
209	#102	—	2	4	no	—
210	#101	—	4	4	no	—
211	C₂²⋊C₈	—	2	1	no	—
212	#97	—	2	1	no	—
213	C₄×C₈	—	2	1	no	—
214	C₄×C₈	—	2	1	no	—
215	C₄×C₈	—	2	1	no	—
216	C₄×C₈	C₅×D₄	1	2	yes	—
217	C₂²×C₅	—	2	1	no	—
218	C₂²×C₅	—	2	1	no	—
219	C₂²×C₅	—	8	1	no	—
220	C₂²×C₅	—	8	1	no	—
221	C₂²×C₅	—	8	1	no	—
222	C₂²×C₅	#392	1	1	yes	characteristic
223	C₂²×C₅	#463	1	2	yes	—
224	C₄×C₅	—	2	1	no	—
225	C₄×C₅	—	2	1	no	—
226	C₄×C₅	—	8	2	no	—
227	C₄×C₅	—	8	1	no	—
228	C₄×C₅	—	8	2	no	—
229	C₄×C₅	#449	1	2	yes	—
230	C₄×C₅	#456	1	2	yes	—
231	C₄×C₅	#471	1	2	yes	—
232	C₄×C₅	#495	1	1	yes	characteristic
233	C₄×C₅	#496	1	1	yes	characteristic
234	#41	—	4	2	no	—
235	#41	—	4	2	no	—
236	C₂×Q₈	—	2	1	no	—
237	C₂×Q₈	—	2	2	no	—
238	C₂×Q₈	—	4	1	no	—
239	C₂×Q₈	—	4	1	no	—
240	C₂×Q₈	—	4	1	no	—
241	C₂×Q₈	—	4	2	no	—
242	C₂×Q₈	—	4	2	no	—
243	C₂×Q₈	—	8	2	no	—
244	C₂×D₄	—	2	1	no	—
245	C₂×D₄	—	2	1	no	—
246	C₂×D₄	—	2	1	no	—
247	C₂×D₄	—	4	1	no	—
248	C₂²×C₄	—	2	2	no	—
249	C₂²×C₄	—	4	1	no	—
250	C₂²×C₄	—	4	1	no	—
251	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
252	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
253	C₂²×C₄	C₅×#41	1	1	yes	characteristic
254	M₁₆	—	2	2	no	—
255	M₁₆	—	2	2	no	—
256	M₁₆	—	4	2	no	—
257	C₂×C₈	—	2	4	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	4	1	no	—
262	C₄⋊C₄	—	2	2	no	—
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	2	1	no	—
267	C₄⋊C₄	—	2	1	no	—
268	C₄⋊C₄	—	4	1	no	—
269	C₄⋊C₄	—	4	1	no	—
270	C₄⋊C₄	—	4	1	no	—
271	C₄⋊C₄	—	4	1	no	—
272	C₂²⋊C₄	—	4	1	no	—
273	C₂²⋊C₄	—	4	1	no	—
274	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
275	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
276	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
277	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
278	C₂×C₅	—	2	2	no	—
279	C₂×C₅	—	16	1	no	—
280	C₂×C₅	#2005	1	1	yes	characteristic
281	C₂×C₅	#2012	1	1	yes	characteristic
282	C₂×C₅	#2019	1	1	yes	characteristic
283	C₂³	—	4	1	no	—
284	C₂³	C₅×#121	1	1	yes	characteristic
285	Q₈	—	4	2	no	—
286	Q₈	—	8	2	no	—
287	Q₈	—	8	2	no	—
288	Q₈	—	8	2	no	—
289	Q₈	—	8	1	no	—
290	Q₈	—	8	1	no	—
291	D₄	—	4	2	no	—
292	D₄	—	4	2	no	—
293	D₄	—	4	2	no	—
294	D₄	—	8	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	4	1	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	4	no	—
301	C₂×C₄	—	8	2	no	—
302	C₂×C₄	—	8	1	no	—
303	C₂×C₄	—	8	1	no	—
304	C₂×C₄	C₅×#120	1	1	yes	characteristic
305	C₂×C₄	C₅×#120	1	1	yes	characteristic
306	C₂×C₄	C₅×#120	1	1	yes	characteristic
307	C₂×C₄	C₅×#121	1	1	yes	characteristic
308	C₂×C₄	C₅×#121	1	1	yes	characteristic
309	C₂×C₄	C₅×#127	1	1	yes	characteristic
310	C₂×C₄	C₅×#135	1	2	yes	—
311	C₂×C₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
312	C₈	—	2	4	no	—
313	C₈	—	4	2	no	—
314	C₈	—	4	2	no	—
315	C₈	—	8	1	no	—
316	C₅	#10166	1	1	yes	characteristic, direct product factor, Sylow
317	C₂²	—	2	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	—	8	1	no	—
320	C₂²	—	8	1	no	—
321	C₂²	—	8	1	no	—
322	C₂²	C₅×#392	1	1	yes	characteristic
323	C₂²	C₅×#463	1	2	yes	—
324	C₄	—	2	1	no	—
325	C₄	—	2	1	no	—
326	C₄	—	8	2	no	—
327	C₄	—	8	1	no	—
328	C₄	—	8	2	no	—
329	C₄	C₅×#449	1	2	yes	—
330	C₄	C₅×#456	1	2	yes	—
331	C₄	C₅×#471	1	2	yes	—
332	C₄	C₅×#495	1	1	yes	characteristic
333	C₄	C₅×#496	1	1	yes	characteristic
334	C₂	—	2	2	no	—
335	C₂	—	16	1	no	—
336	C₂	C₅×#2005	1	1	yes	characteristic
337	C₂	C₅×#2012	1	1	yes	characteristic
338	C₂	C₅×#2019	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₅
prime radical	C₅×#532	C₂²
LCS	G ▷ C₂×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
2. C₂×C₅×#498	C₂	semidirect
4. C₅×#1966	C₂	semidirect
5. C₅×#1548	C₂	semidirect
7. C₅×#1278	C₂	semidirect
29. #10166	C₅	direct (N × H)
72. C₂×#498	C₂×C₅	semidirect
74. #1966	C₂×C₅	semidirect
75. #1548	C₂×C₅	semidirect
77. #1278	C₂×C₅	semidirect
316. C₅	#10166	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×#3226	2
C₂×C₅×#498	2
C₅×#3141	2
C₅×#1966	2
C₅×#1548	2
C₅×#1492	2
C₅×#1278	2
#10166	5

Subgroups — C5×#10166 #11932886

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#10166 #11932886