Subgroups — C₅×#13260 #11935980

Order 1280.

482 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#495	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#364	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×#362	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅×#361	C₂	1	1	yes	maximal, characteristic
5	C₅×#2141	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#2140	C₂	1	1	yes	maximal, characteristic
7	C₅×#1772	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₅×Q₁₆	C₂²	1	1	yes	characteristic
9	C₂²×C₅×SD₁₆	—	2	1	no	—
10	C₂²×C₅×D₈	C₂²	1	1	yes	characteristic
11	C₂×C₅×#124	—	2	1	no	—
12	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
13	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
14	C₅×#495	—	2	2	no	—
15	C₅×#495	C₄	1	4	yes	complemented
16	C₂×C₅×#108	C₂²	1	1	yes	characteristic
17	C₅×#427	C₂²	1	1	yes	characteristic
18	C₅×#405	—	2	1	no	—
19	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
20	C₅×#364	—	2	2	no	—
21	C₅×#362	—	2	2	no	—
22	C₅×#361	—	2	2	no	—
23	C₅×#358	—	2	2	no	—
24	C₅×#357	—	2	2	no	—
25	C₅×#339	—	2	1	no	—
26	#13260	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
27	C₂²×C₅×Q₈	—	2	1	no	—
28	C₂²×C₅×D₄	—	2	1	no	—
29	C₂×C₅×Q₁₆	—	2	1	no	—
30	C₂×C₅×Q₁₆	—	2	1	no	—
31	C₂×C₅×Q₁₆	—	2	1	no	—
32	C₂×C₅×Q₁₆	—	2	1	no	—
33	C₂×C₅×Q₁₆	—	2	1	no	—
34	C₂×C₅×Q₁₆	C₂×C₄	1	2	yes	—
35	C₂×C₅×SD₁₆	—	2	4	no	—
36	C₂×C₅×SD₁₆	—	4	1	no	—
37	C₂×C₅×SD₁₆	—	4	1	no	—
38	C₂×C₅×SD₁₆	—	4	2	no	—
39	C₂×C₅×D₈	—	2	1	no	—
40	C₂×C₅×D₈	—	2	1	no	—
41	C₂×C₅×D₈	—	2	1	no	—
42	C₂×C₅×D₈	—	2	1	no	—
43	C₂×C₅×D₈	—	2	1	no	—
44	C₂×C₅×D₈	C₂×C₄	1	2	yes	—
45	C₂×C₅×M₁₆	—	2	1	no	—
46	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
47	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
48	C₅×#124	—	2	4	no	—
49	C₅×#124	—	4	2	no	—
50	C₅×#117	—	2	1	no	—
51	C₂×C₅×C₂²⋊C₄	—	4	2	no	—
52	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
53	C₅×M₃₂	—	2	1	no	—
54	C₅×M₃₂	—	2	1	no	—
55	C₅×M₃₂	—	2	1	no	—
56	C₅×M₃₂	—	2	1	no	—
57	C₂×C₅×C₁₆	—	2	1	no	—
58	C₂×C₅×C₁₆	—	2	1	no	—
59	C₅×#108	—	2	1	no	—
60	C₅×#108	D₄	1	2	yes	—
61	C₅×#107	D₄	1	2	yes	—
62	C₅×#106	—	2	1	no	—
63	C₅×#97	—	4	1	no	—
64	C₄×C₅×C₈	—	4	1	no	—
65	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
66	C₄×C₅×C₈	D₄	1	2	yes	—
67	C₂×#495	C₂×C₅	1	1	yes	characteristic
68	C₂×#364	C₂×C₅	1	1	yes	characteristic, complemented
69	C₂×#362	C₂×C₅	1	1	yes	characteristic, complemented
70	C₂×#361	C₂×C₅	1	1	yes	characteristic
71	#2141	C₂×C₅	1	1	yes	characteristic, complemented
72	#2140	C₂×C₅	1	1	yes	characteristic
73	#1772	C₂×C₅	1	1	yes	characteristic, complemented
74	C₂⁴×C₅	—	4	1	no	—
75	C₂×C₅×Q₈	—	2	2	no	—
76	C₂×C₅×Q₈	—	4	1	no	—
77	C₂×C₅×Q₈	—	4	1	no	—
78	C₂×C₅×Q₈	—	4	1	no	—
79	C₂×C₅×Q₈	—	4	1	no	—
80	C₂×C₅×Q₈	—	4	1	no	—
81	C₂×C₅×D₄	—	2	2	no	—
82	C₂×C₅×D₄	—	4	1	no	—
83	C₂×C₅×D₄	—	4	1	no	—
84	C₂×C₅×D₄	—	4	1	no	—
85	C₂×C₅×D₄	—	4	1	no	—
86	C₂×C₅×D₄	—	4	1	no	—
87	C₂²×C₄×C₅	—	2	1	no	—
88	C₂²×C₄×C₅	—	4	1	no	—
89	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
90	C₅×Q₁₆	—	2	2	no	—
91	C₅×Q₁₆	—	2	2	no	—
92	C₅×Q₁₆	—	4	1	no	—
93	C₅×Q₁₆	—	4	1	no	—
94	C₅×SD₁₆	—	4	4	no	—
95	C₅×SD₁₆	—	4	4	no	—
96	C₅×D₈	—	2	2	no	—
97	C₅×D₈	—	2	2	no	—
98	C₅×D₈	—	4	1	no	—
99	C₅×D₈	—	4	1	no	—
100	C₅×M₁₆	—	2	2	no	—
101	C₅×M₁₆	—	4	1	no	—
102	C₂×C₅×C₈	—	4	1	no	—
103	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
104	C₂×C₅×C₈	D₈	1	1	yes	characteristic
105	C₂×C₅×C₈	D₈	1	1	yes	characteristic
106	C₂×C₅×C₈	SD₁₆	1	2	yes	—
107	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
109	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
110	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
111	C₂×C₅×C₈	#41	1	1	yes	characteristic
112	C₂×C₅×C₈	#41	1	1	yes	characteristic
113	C₅×C₄⋊C₄	—	2	2	no	—
114	C₅×C₂²⋊C₄	—	4	8	no	—
115	C₅×C₂²⋊C₄	—	4	1	no	—
116	C₄²×C₅	—	2	1	no	—
117	C₄²×C₅	—	4	1	no	—
118	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
119	C₅×C₁₆	—	4	1	no	—
120	C₅×C₁₆	—	4	1	no	—
121	C₂²×Q₁₆	C₂²×C₅	1	1	yes	characteristic
122	C₂²×SD₁₆	—	2	1	no	—
123	C₂²×D₈	C₂²×C₅	1	1	yes	characteristic
124	C₂×#124	—	2	1	no	—
125	C₂×M₃₂	C₂²×C₅	1	1	yes	characteristic
126	C₂×M₃₂	C₂²×C₅	1	1	yes	characteristic
127	#495	—	2	2	no	—
128	#495	C₄×C₅	1	4	yes	complemented
129	C₂×#108	C₂²×C₅	1	1	yes	characteristic
130	#427	C₂²×C₅	1	1	yes	characteristic
131	#405	—	2	1	no	—
132	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
133	#364	—	2	2	no	—
134	#362	—	2	2	no	—
135	#361	—	2	2	no	—
136	#358	—	2	2	no	—
137	#357	—	2	2	no	—
138	#339	—	2	1	no	—
139	C₂³×C₅	—	4	2	no	—
140	C₂³×C₅	—	8	4	no	—
141	C₂³×C₅	—	8	1	no	—
142	C₂³×C₅	—	8	1	no	—
143	C₂³×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
144	C₅×Q₈	—	4	2	no	—
145	C₅×Q₈	—	4	2	no	—
146	C₅×Q₈	—	8	1	no	—
147	C₅×Q₈	—	8	1	no	—
148	C₅×D₄	—	4	2	no	—
149	C₅×D₄	—	4	2	no	—
150	C₅×D₄	—	8	1	no	—
151	C₅×D₄	—	8	1	no	—
152	C₂×C₄×C₅	—	2	2	no	—
153	C₂×C₄×C₅	—	4	2	no	—
154	C₂×C₄×C₅	—	4	2	no	—
155	C₂×C₄×C₅	—	4	2	no	—
156	C₂×C₄×C₅	—	8	1	no	—
157	C₂×C₄×C₅	—	8	1	no	—
158	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
159	C₂×C₄×C₅	#121	1	1	yes	characteristic
160	C₂×C₄×C₅	#121	1	1	yes	characteristic
161	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
162	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
163	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
164	C₅×C₈	—	4	2	no	—
165	C₅×C₈	#102	1	2	yes	—
166	C₅×C₈	#102	1	2	yes	—
167	C₅×C₈	C₄×D₄	1	1	yes	characteristic
168	C₅×C₈	C₄×D₄	1	1	yes	characteristic
169	C₅×C₈	#124	1	1	yes	characteristic
170	C₅×C₈	#127	1	1	yes	characteristic
171	C₂²×Q₈	—	2	1	no	—
172	C₂²×D₄	—	2	1	no	—
173	C₂×Q₁₆	—	2	1	no	—
174	C₂×Q₁₆	—	2	1	no	—
175	C₂×Q₁₆	—	2	1	no	—
176	C₂×Q₁₆	—	2	1	no	—
177	C₂×Q₁₆	—	2	1	no	—
178	C₂×Q₁₆	C₂×C₄×C₅	1	2	yes	—
179	C₂×SD₁₆	—	2	4	no	—
180	C₂×SD₁₆	—	4	1	no	—
181	C₂×SD₁₆	—	4	1	no	—
182	C₂×SD₁₆	—	4	2	no	—
183	C₂×D₈	—	2	1	no	—
184	C₂×D₈	—	2	1	no	—
185	C₂×D₈	—	2	1	no	—
186	C₂×D₈	—	2	1	no	—
187	C₂×D₈	—	2	1	no	—
188	C₂×D₈	C₂×C₄×C₅	1	2	yes	—
189	C₂×M₁₆	—	2	1	no	—
190	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
191	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
192	#124	—	2	4	no	—
193	#124	—	4	2	no	—
194	#117	—	2	1	no	—
195	C₂×C₂²⋊C₄	—	4	2	no	—
196	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
197	M₃₂	—	2	1	no	—
198	M₃₂	—	2	1	no	—
199	M₃₂	—	2	1	no	—
200	M₃₂	—	2	1	no	—
201	C₂×C₁₆	—	2	1	no	—
202	C₂×C₁₆	—	2	1	no	—
203	#108	—	2	1	no	—
204	#108	C₅×D₄	1	2	yes	—
205	#107	C₅×D₄	1	2	yes	—
206	#106	—	2	1	no	—
207	#97	—	4	1	no	—
208	C₄×C₈	—	4	1	no	—
209	C₄×C₈	C₂×C₄×C₅	1	2	yes	—
210	C₄×C₈	C₅×D₄	1	2	yes	—
211	C₂²×C₅	—	2	2	no	—
212	C₂²×C₅	—	8	4	no	—
213	C₂²×C₅	—	8	2	no	—
214	C₂²×C₅	—	16	2	no	—
215	C₂²×C₅	—	16	2	no	—
216	C₂²×C₅	#418	1	1	yes	characteristic
217	C₂²×C₅	#459	1	1	yes	characteristic
218	C₂²×C₅	#460	1	1	yes	characteristic
219	C₄×C₅	—	4	2	no	—
220	C₄×C₅	—	8	2	no	—
221	C₄×C₅	—	8	2	no	—
222	C₄×C₅	#390	1	1	yes	characteristic
223	C₄×C₅	C₂×#102	1	1	yes	characteristic
224	C₄×C₅	#459	1	1	yes	characteristic
225	C₄×C₅	#460	1	1	yes	characteristic
226	C₂⁴	—	4	1	no	—
227	C₂×Q₈	—	2	2	no	—
228	C₂×Q₈	—	4	1	no	—
229	C₂×Q₈	—	4	1	no	—
230	C₂×Q₈	—	4	1	no	—
231	C₂×Q₈	—	4	1	no	—
232	C₂×Q₈	—	4	1	no	—
233	C₂×D₄	—	2	2	no	—
234	C₂×D₄	—	4	1	no	—
235	C₂×D₄	—	4	1	no	—
236	C₂×D₄	—	4	1	no	—
237	C₂×D₄	—	4	1	no	—
238	C₂×D₄	—	4	1	no	—
239	C₂²×C₄	—	2	1	no	—
240	C₂²×C₄	—	4	1	no	—
241	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
242	Q₁₆	—	2	2	no	—
243	Q₁₆	—	2	2	no	—
244	Q₁₆	—	4	1	no	—
245	Q₁₆	—	4	1	no	—
246	SD₁₆	—	4	4	no	—
247	SD₁₆	—	4	4	no	—
248	D₈	—	2	2	no	—
249	D₈	—	2	2	no	—
250	D₈	—	4	1	no	—
251	D₈	—	4	1	no	—
252	M₁₆	—	2	2	no	—
253	M₁₆	—	4	1	no	—
254	C₂×C₈	—	4	1	no	—
255	C₂×C₈	C₅×C₂²⋊C₄	1	2	yes	—
256	C₂×C₈	C₅×D₈	1	1	yes	characteristic
257	C₂×C₈	C₅×D₈	1	1	yes	characteristic
258	C₂×C₈	C₅×SD₁₆	1	2	yes	—
259	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
260	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
261	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
262	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
263	C₂×C₈	C₅×#41	1	1	yes	characteristic
264	C₂×C₈	C₅×#41	1	1	yes	characteristic
265	C₄⋊C₄	—	2	2	no	—
266	C₂²⋊C₄	—	4	8	no	—
267	C₂²⋊C₄	—	4	1	no	—
268	C₄²	—	2	1	no	—
269	C₄²	—	4	1	no	—
270	C₄²	C₅×C₂²⋊C₄	1	2	yes	—
271	C₁₆	—	4	1	no	—
272	C₁₆	—	4	1	no	—
273	C₂×C₅	—	2	1	no	—
274	C₂×C₅	—	2	1	no	—
275	C₂×C₅	—	16	2	no	—
276	C₂×C₅	#1982	1	1	yes	characteristic
277	C₂×C₅	#2212	1	2	yes	—
278	C₂³	—	4	2	no	—
279	C₂³	—	8	4	no	—
280	C₂³	—	8	1	no	—
281	C₂³	—	8	1	no	—
282	C₂³	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
283	Q₈	—	4	2	no	—
284	Q₈	—	4	2	no	—
285	Q₈	—	8	1	no	—
286	Q₈	—	8	1	no	—
287	D₄	—	4	2	no	—
288	D₄	—	4	2	no	—
289	D₄	—	8	1	no	—
290	D₄	—	8	1	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	8	1	no	—
296	C₂×C₄	—	8	1	no	—
297	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
298	C₂×C₄	C₅×#121	1	1	yes	characteristic
299	C₂×C₄	C₅×#121	1	1	yes	characteristic
300	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
301	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
302	C₂×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
303	C₈	—	4	2	no	—
304	C₈	C₅×#102	1	2	yes	—
305	C₈	C₅×#102	1	2	yes	—
306	C₈	C₄×C₅×D₄	1	1	yes	characteristic
307	C₈	C₄×C₅×D₄	1	1	yes	characteristic
308	C₈	C₅×#124	1	1	yes	characteristic
309	C₈	C₅×#127	1	1	yes	characteristic
310	C₅	#13260	1	1	yes	characteristic, direct product factor, Sylow
311	C₂²	—	2	2	no	—
312	C₂²	—	8	4	no	—
313	C₂²	—	8	2	no	—
314	C₂²	—	16	2	no	—
315	C₂²	—	16	2	no	—
316	C₂²	C₅×#418	1	1	yes	characteristic
317	C₂²	C₅×#459	1	1	yes	characteristic
318	C₂²	C₅×#460	1	1	yes	characteristic
319	C₄	—	4	2	no	—
320	C₄	—	8	2	no	—
321	C₄	—	8	2	no	—
322	C₄	C₅×#390	1	1	yes	characteristic
323	C₄	C₂×C₅×#102	1	1	yes	characteristic
324	C₄	C₅×#459	1	1	yes	characteristic
325	C₄	C₅×#460	1	1	yes	characteristic
326	C₂	—	2	1	no	—
327	C₂	—	2	1	no	—
328	C₂	—	16	2	no	—
329	C₂	C₅×#1982	1	1	yes	characteristic
330	C₂	C₅×#2212	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₅	#459
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₂²×C₈	C₂³×C₅
prime radical	C₂²×C₅×D₈	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ C₂×C₄×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×#364	C₂	semidirect
3. C₂×C₅×#362	C₂	semidirect
5. C₅×#2141	C₂	semidirect
7. C₅×#1772	C₂	semidirect
15. C₅×#495	C₄	semidirect
26. #13260	C₅	direct (N × H)
68. C₂×#364	C₂×C₅	semidirect
69. C₂×#362	C₂×C₅	semidirect
71. #2141	C₂×C₅	semidirect
73. #1772	C₂×C₅	semidirect
128. #495	C₄×C₅	semidirect
310. C₅	#13260	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅×#495	2
C₂×C₅×#364	2
C₂×C₅×#362	2
C₂×C₅×#361	2
C₅×#2141	2
C₅×#2140	2
C₅×#1772	2
#13260	5

Subgroups — C5×#13260 #11935980

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#13260 #11935980