Subgroups — C₅×#13259 #11935979

Order 1280.

472 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3147	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#364	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#2145	C₂	1	2	yes	maximal, complemented
4	C₅×#2143	C₂	1	2	yes	maximal, complemented
5	C₅×#1772	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₅×#135	—	2	1	no	—
7	C₂×C₅×#135	C₂²	1	2	yes	—
8	C₅×#532	—	2	1	no	—
9	C₂×C₅×M₃₂	C₂²	1	2	yes	—
10	C₅×#495	—	2	1	no	—
11	C₅×#495	C₄	1	2	yes	complemented
12	C₅×#494	C₄	1	1	yes	characteristic, complemented
13	C₅×#493	C₄	1	1	yes	characteristic, complemented
14	C₅×#492	—	2	1	no	—
15	C₂×C₅×#108	C₂²	1	1	yes	characteristic
16	C₅×#427	C₂²	1	1	yes	characteristic
17	C₅×#405	—	2	1	no	—
18	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
19	C₅×#364	—	2	1	no	—
20	C₅×#364	—	2	1	no	—
21	C₅×#360	—	2	4	no	—
22	C₅×#359	—	2	4	no	—
23	C₅×#339	—	2	1	no	—
24	#13259	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
25	C₂×C₅×#41	—	2	2	no	—
26	C₅×#135	—	2	2	no	—
27	C₅×#135	—	2	2	no	—
28	C₅×#135	—	2	2	no	—
29	C₅×#135	—	2	2	no	—
30	C₅×#135	—	4	1	no	—
31	C₅×#135	—	4	1	no	—
32	C₅×#135	—	4	1	no	—
33	C₅×#135	—	4	1	no	—
34	C₂×C₅×Q₁₆	—	2	1	no	—
35	C₂×C₅×Q₁₆	C₂×C₄	1	2	yes	—
36	C₂×C₅×SD₁₆	—	2	2	no	—
37	C₂×C₅×SD₁₆	—	2	2	no	—
38	C₂×C₅×D₈	—	2	1	no	—
39	C₂×C₅×D₈	C₂×C₄	1	2	yes	—
40	C₂×C₅×M₁₆	—	2	1	no	—
41	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
42	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
43	C₅×C₄⋊Q₈	—	2	1	no	—
44	C₅×#127	—	2	1	no	—
45	C₅×#124	—	2	2	no	—
46	C₅×#121	—	4	2	no	—
47	C₄×C₅×D₄	—	4	2	no	—
48	C₅×#117	—	2	1	no	—
49	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
50	C₅×M₃₂	—	2	2	no	—
51	C₅×M₃₂	—	2	2	no	—
52	C₂×C₅×C₁₆	—	2	2	no	—
53	C₅×#108	—	2	1	no	—
54	C₅×#108	D₄	1	2	yes	complemented
55	C₅×#107	—	2	1	no	—
56	C₅×#106	D₄	1	2	yes	complemented
57	C₅×#97	—	4	1	no	—
58	C₄×C₅×C₈	—	4	1	no	—
59	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
60	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
61	C₄×C₅×C₈	D₄	1	1	yes	characteristic
62	C₄×C₅×C₈	D₄	1	1	yes	characteristic
63	#3147	C₂×C₅	1	1	yes	characteristic
64	C₂×#364	C₂×C₅	1	1	yes	characteristic, complemented
65	#2145	C₂×C₅	1	2	yes	complemented
66	#2143	C₂×C₅	1	2	yes	complemented
67	#1772	C₂×C₅	1	1	yes	characteristic, complemented
68	C₅×#41	—	4	2	no	—
69	C₅×#41	—	4	2	no	—
70	C₅×#41	—	4	2	no	—
71	C₅×#41	—	4	2	no	—
72	C₂×C₅×Q₈	—	2	2	no	—
73	C₂×C₅×D₄	—	2	2	no	—
74	C₂×C₅×D₄	—	4	2	no	—
75	C₂×C₅×D₄	—	4	2	no	—
76	C₂²×C₄×C₅	—	2	1	no	—
77	C₂²×C₄×C₅	—	4	2	no	—
78	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
79	C₅×Q₁₆	—	2	2	no	—
80	C₅×Q₁₆	—	2	2	no	—
81	C₅×Q₁₆	—	4	1	no	—
82	C₅×Q₁₆	—	4	1	no	—
83	C₅×SD₁₆	—	4	2	no	—
84	C₅×SD₁₆	—	4	2	no	—
85	C₅×SD₁₆	—	4	2	no	—
86	C₅×SD₁₆	—	4	2	no	—
87	C₅×D₈	—	2	2	no	—
88	C₅×D₈	—	2	2	no	—
89	C₅×D₈	—	4	1	no	—
90	C₅×D₈	—	4	1	no	—
91	C₅×M₁₆	—	2	2	no	—
92	C₅×M₁₆	—	4	1	no	—
93	C₂×C₅×C₈	—	4	1	no	—
94	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₅×C₈	D₈	1	1	yes	characteristic
97	C₂×C₅×C₈	D₈	1	1	yes	characteristic
98	C₂×C₅×C₈	SD₁₆	1	1	yes	characteristic
99	C₂×C₅×C₈	SD₁₆	1	1	yes	characteristic
100	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
102	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
103	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
104	C₂×C₅×C₈	#41	1	1	yes	characteristic
105	C₂×C₅×C₈	#41	1	1	yes	characteristic
106	C₅×C₄⋊C₄	—	2	2	no	—
107	C₅×C₄⋊C₄	—	4	2	no	—
108	C₅×C₂²⋊C₄	—	4	4	no	—
109	C₅×C₂²⋊C₄	—	4	1	no	—
110	C₄²×C₅	—	2	1	no	—
111	C₄²×C₅	—	4	1	no	—
112	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
113	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
114	C₅×C₁₆	—	4	2	no	—
115	C₂×#135	—	2	1	no	—
116	C₂×#135	C₂²×C₅	1	2	yes	—
117	#532	—	2	1	no	—
118	C₂×M₃₂	C₂²×C₅	1	2	yes	—
119	#495	—	2	1	no	—
120	#495	C₄×C₅	1	2	yes	complemented
121	#494	C₄×C₅	1	1	yes	characteristic, complemented
122	#493	C₄×C₅	1	1	yes	characteristic, complemented
123	#492	—	2	1	no	—
124	C₂×#108	C₂²×C₅	1	1	yes	characteristic
125	#427	C₂²×C₅	1	1	yes	characteristic
126	#405	—	2	1	no	—
127	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
128	#364	—	2	1	no	—
129	#364	—	2	1	no	—
130	#360	—	2	4	no	—
131	#359	—	2	4	no	—
132	#339	—	2	1	no	—
133	C₂³×C₅	—	4	2	no	—
134	C₂³×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
135	C₅×Q₈	—	4	2	no	—
136	C₅×Q₈	—	4	2	no	—
137	C₅×D₄	—	4	2	no	—
138	C₅×D₄	—	4	2	no	—
139	C₅×D₄	—	8	4	no	—
140	C₅×D₄	—	8	2	no	—
141	C₅×D₄	—	8	2	no	—
142	C₂×C₄×C₅	—	2	1	no	—
143	C₂×C₄×C₅	—	2	1	no	—
144	C₂×C₄×C₅	—	4	2	no	—
145	C₂×C₄×C₅	—	4	2	no	—
146	C₂×C₄×C₅	—	4	2	no	—
147	C₂×C₄×C₅	—	8	2	no	—
148	C₂×C₄×C₅	—	8	2	no	—
149	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
150	C₂×C₄×C₅	#121	1	1	yes	characteristic
151	C₂×C₄×C₅	#121	1	1	yes	characteristic
152	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
153	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
154	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
155	C₅×C₈	—	4	2	no	—
156	C₅×C₈	#102	1	1	yes	characteristic
157	C₅×C₈	#102	1	1	yes	characteristic
158	C₅×C₈	#102	1	1	yes	characteristic
159	C₅×C₈	#102	1	1	yes	characteristic
160	C₅×C₈	C₄×D₄	1	1	yes	characteristic
161	C₅×C₈	C₄×D₄	1	1	yes	characteristic
162	C₅×C₈	#124	1	1	yes	characteristic
163	C₅×C₈	#127	1	1	yes	characteristic
164	C₂×#41	—	2	2	no	—
165	#135	—	2	2	no	—
166	#135	—	2	2	no	—
167	#135	—	2	2	no	—
168	#135	—	2	2	no	—
169	#135	—	4	1	no	—
170	#135	—	4	1	no	—
171	#135	—	4	1	no	—
172	#135	—	4	1	no	—
173	C₂×Q₁₆	—	2	1	no	—
174	C₂×Q₁₆	C₂×C₄×C₅	1	2	yes	—
175	C₂×SD₁₆	—	2	2	no	—
176	C₂×SD₁₆	—	2	2	no	—
177	C₂×D₈	—	2	1	no	—
178	C₂×D₈	C₂×C₄×C₅	1	2	yes	—
179	C₂×M₁₆	—	2	1	no	—
180	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
181	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
182	C₄⋊Q₈	—	2	1	no	—
183	#127	—	2	1	no	—
184	#124	—	2	2	no	—
185	#121	—	4	2	no	—
186	C₄×D₄	—	4	2	no	—
187	#117	—	2	1	no	—
188	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
189	M₃₂	—	2	2	no	—
190	M₃₂	—	2	2	no	—
191	C₂×C₁₆	—	2	2	no	—
192	#108	—	2	1	no	—
193	#108	C₅×D₄	1	2	yes	complemented
194	#107	—	2	1	no	—
195	#106	C₅×D₄	1	2	yes	complemented
196	#97	—	4	1	no	—
197	C₄×C₈	—	4	1	no	—
198	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
199	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
200	C₄×C₈	C₅×D₄	1	1	yes	characteristic
201	C₄×C₈	C₅×D₄	1	1	yes	characteristic
202	C₂²×C₅	—	4	1	no	—
203	C₂²×C₅	—	8	4	no	—
204	C₂²×C₅	—	8	2	no	—
205	C₂²×C₅	#418	1	1	yes	characteristic
206	C₂²×C₅	#459	1	1	yes	characteristic
207	C₂²×C₅	#460	1	1	yes	characteristic
208	C₄×C₅	—	2	2	no	—
209	C₄×C₅	—	4	1	no	—
210	C₄×C₅	—	8	2	no	—
211	C₄×C₅	—	8	2	no	—
212	C₄×C₅	#390	1	1	yes	characteristic
213	C₄×C₅	C₂×#102	1	1	yes	characteristic
214	C₄×C₅	#459	1	1	yes	characteristic
215	C₄×C₅	#460	1	1	yes	characteristic
216	#41	—	4	2	no	—
217	#41	—	4	2	no	—
218	#41	—	4	2	no	—
219	#41	—	4	2	no	—
220	C₂×Q₈	—	2	2	no	—
221	C₂×D₄	—	2	2	no	—
222	C₂×D₄	—	4	2	no	—
223	C₂×D₄	—	4	2	no	—
224	C₂²×C₄	—	2	1	no	—
225	C₂²×C₄	—	4	2	no	—
226	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
227	Q₁₆	—	2	2	no	—
228	Q₁₆	—	2	2	no	—
229	Q₁₆	—	4	1	no	—
230	Q₁₆	—	4	1	no	—
231	SD₁₆	—	4	2	no	—
232	SD₁₆	—	4	2	no	—
233	SD₁₆	—	4	2	no	—
234	SD₁₆	—	4	2	no	—
235	D₈	—	2	2	no	—
236	D₈	—	2	2	no	—
237	D₈	—	4	1	no	—
238	D₈	—	4	1	no	—
239	M₁₆	—	2	2	no	—
240	M₁₆	—	4	1	no	—
241	C₂×C₈	—	4	1	no	—
242	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
243	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
244	C₂×C₈	C₅×D₈	1	1	yes	characteristic
245	C₂×C₈	C₅×D₈	1	1	yes	characteristic
246	C₂×C₈	C₅×SD₁₆	1	1	yes	characteristic
247	C₂×C₈	C₅×SD₁₆	1	1	yes	characteristic
248	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
249	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
250	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
251	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
252	C₂×C₈	C₅×#41	1	1	yes	characteristic
253	C₂×C₈	C₅×#41	1	1	yes	characteristic
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	—	4	2	no	—
256	C₂²⋊C₄	—	4	4	no	—
257	C₂²⋊C₄	—	4	1	no	—
258	C₄²	—	2	1	no	—
259	C₄²	—	4	1	no	—
260	C₄²	C₅×C₂²⋊C₄	1	1	yes	characteristic
261	C₄²	C₅×C₂²⋊C₄	1	1	yes	characteristic
262	C₁₆	—	4	2	no	—
263	C₂×C₅	—	2	1	no	—
264	C₂×C₅	—	2	1	no	—
265	C₂×C₅	—	2	1	no	—
266	C₂×C₅	—	16	2	no	—
267	C₂×C₅	#1982	1	1	yes	characteristic
268	C₂³	—	4	2	no	—
269	C₂³	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
270	Q₈	—	4	2	no	—
271	Q₈	—	4	2	no	—
272	D₄	—	4	2	no	—
273	D₄	—	4	2	no	—
274	D₄	—	8	4	no	—
275	D₄	—	8	2	no	—
276	D₄	—	8	2	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	8	2	no	—
283	C₂×C₄	—	8	2	no	—
284	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
285	C₂×C₄	C₅×#121	1	1	yes	characteristic
286	C₂×C₄	C₅×#121	1	1	yes	characteristic
287	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
288	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
289	C₂×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
290	C₈	—	4	2	no	—
291	C₈	C₅×#102	1	1	yes	characteristic
292	C₈	C₅×#102	1	1	yes	characteristic
293	C₈	C₅×#102	1	1	yes	characteristic
294	C₈	C₅×#102	1	1	yes	characteristic
295	C₈	C₄×C₅×D₄	1	1	yes	characteristic
296	C₈	C₄×C₅×D₄	1	1	yes	characteristic
297	C₈	C₅×#124	1	1	yes	characteristic
298	C₈	C₅×#127	1	1	yes	characteristic
299	C₅	#13259	1	1	yes	characteristic, direct product factor, Sylow
300	C₂²	—	4	1	no	—
301	C₂²	—	8	4	no	—
302	C₂²	—	8	2	no	—
303	C₂²	C₅×#418	1	1	yes	characteristic
304	C₂²	C₅×#459	1	1	yes	characteristic
305	C₂²	C₅×#460	1	1	yes	characteristic
306	C₄	—	2	2	no	—
307	C₄	—	4	1	no	—
308	C₄	—	8	2	no	—
309	C₄	—	8	2	no	—
310	C₄	C₅×#390	1	1	yes	characteristic
311	C₄	C₂×C₅×#102	1	1	yes	characteristic
312	C₄	C₅×#459	1	1	yes	characteristic
313	C₄	C₅×#460	1	1	yes	characteristic
314	C₂	—	2	1	no	—
315	C₂	—	2	1	no	—
316	C₂	—	2	1	no	—
317	C₂	—	16	2	no	—
318	C₂	C₅×#1982	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₅	#459
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₂²×C₈	C₂³×C₅
socle	C₂×C₅	#1982
prime radical	C₅×#3147	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×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. C₅×#2145	C₂	semidirect
4. C₅×#2143	C₂	semidirect
5. C₅×#1772	C₂	semidirect
11. C₅×#495	C₄	semidirect
12. C₅×#494	C₄	semidirect
13. C₅×#493	C₄	semidirect
24. #13259	C₅	direct (N × H)
54. C₅×#108	D₄	semidirect
56. C₅×#106	D₄	semidirect
64. C₂×#364	C₂×C₅	semidirect
65. #2145	C₂×C₅	semidirect
66. #2143	C₂×C₅	semidirect
67. #1772	C₂×C₅	semidirect
120. #495	C₄×C₅	semidirect
121. #494	C₄×C₅	semidirect
122. #493	C₄×C₅	semidirect
193. #108	C₅×D₄	semidirect
195. #106	C₅×D₄	semidirect
299. C₅	#13259	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×#3147	2
C₂×C₅×#364	2
C₅×#2145	2
C₅×#2143	2
C₅×#1772	2
#13259	5

Subgroups — C5×#13259 #11935979

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#13259 #11935979