Subgroups — C₅×#13255 #11935975

Order 1280.

498 conjugacy classes of subgroups.

The subgroup lattice has 282 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	C₅×#2150	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×#360	C₂	1	4	yes	maximal, complemented
4	C₄×C₅×#106	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₅×#135	—	2	1	no	—
6	C₂×C₅×#135	C₂²	1	2	yes	—
7	C₅×#532	—	2	1	no	—
8	C₂×C₅×M₃₂	C₂²	1	2	yes	—
9	C₅×#495	—	2	1	no	—
10	C₅×#495	C₄	1	2	yes	complemented
11	C₅×#494	C₄	1	1	yes	characteristic, complemented
12	C₅×#493	C₄	1	1	yes	characteristic, complemented
13	C₅×#492	—	2	1	no	—
14	C₂×C₅×#106	C₂²	1	2	yes	—
15	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
16	C₄×C₅×C₄⋊C₄	—	2	1	no	—
17	C₅×C₄⋊C₁₆	—	2	1	no	—
18	C₅×C₄⋊C₁₆	—	2	1	no	—
19	C₅×#360	—	2	8	no	—
20	C₅×#340	—	2	1	no	—
21	#13255	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
22	C₂×C₅×#41	—	2	2	no	—
23	C₅×#135	—	2	4	no	—
24	C₅×#135	—	2	4	no	—
25	C₅×#135	—	4	4	no	—
26	C₂×C₅×Q₁₆	—	2	1	no	—
27	C₂×C₅×Q₁₆	C₂×C₄	1	2	yes	—
28	C₂×C₅×SD₁₆	—	2	2	no	—
29	C₂×C₅×SD₁₆	—	2	2	no	—
30	C₂×C₅×D₈	—	2	1	no	—
31	C₂×C₅×D₈	C₂×C₄	1	2	yes	—
32	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
33	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
34	C₅×C₄⋊Q₈	—	2	1	no	—
35	C₅×#127	—	2	1	no	—
36	C₅×#124	—	2	2	no	—
37	C₅×#121	—	4	1	no	—
38	C₅×#121	—	4	1	no	—
39	C₄×C₅×D₄	—	4	1	no	—
40	C₄×C₅×D₄	—	4	1	no	—
41	C₂×C₅×C₄⋊C₄	—	2	2	no	—
42	C₂×C₄²×C₅	—	4	1	no	—
43	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
44	C₅×M₃₂	—	2	4	no	—
45	C₂×C₅×C₁₆	—	2	2	no	—
46	C₅×#106	—	2	2	no	—
47	C₅×#106	D₄	1	4	yes	complemented
48	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
49	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
50	C₄×C₅×C₈	D₄	1	1	yes	characteristic
51	C₄×C₅×C₈	D₄	1	1	yes	characteristic
52	C₅×#95	—	4	1	no	—
53	#3147	C₂×C₅	1	1	yes	characteristic
54	#2150	C₂×C₅	1	1	yes	characteristic, complemented
55	C₂×#360	C₂×C₅	1	4	yes	complemented
56	C₄×#106	C₂×C₅	1	1	yes	characteristic, complemented
57	C₅×#41	—	4	4	no	—
58	C₅×#41	—	4	4	no	—
59	C₂×C₅×Q₈	—	2	2	no	—
60	C₂×C₅×D₄	—	2	2	no	—
61	C₂×C₅×D₄	—	4	2	no	—
62	C₂×C₅×D₄	—	4	1	no	—
63	C₂×C₅×D₄	—	4	1	no	—
64	C₂²×C₄×C₅	—	4	2	no	—
65	C₂²×C₄×C₅	—	4	2	no	—
66	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
67	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
68	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
69	C₅×Q₁₆	—	2	4	no	—
70	C₅×Q₁₆	—	4	2	no	—
71	C₅×SD₁₆	—	4	4	no	—
72	C₅×SD₁₆	—	4	2	no	—
73	C₅×SD₁₆	—	4	2	no	—
74	C₅×D₈	—	2	4	no	—
75	C₅×D₈	—	4	2	no	—
76	C₂×C₅×C₈	—	2	2	no	—
77	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
81	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
82	C₂×C₅×C₈	#41	1	2	yes	—
83	C₅×C₄⋊C₄	—	2	4	no	—
84	C₅×C₄⋊C₄	—	4	1	no	—
85	C₅×C₄⋊C₄	—	4	1	no	—
86	C₅×C₄⋊C₄	—	4	2	no	—
87	C₅×C₂²⋊C₄	—	4	2	no	—
88	C₅×C₂²⋊C₄	—	4	2	no	—
89	C₄²×C₅	—	8	2	no	—
90	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
91	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
92	C₄²×C₅	SD₁₆	1	1	yes	characteristic
93	C₄²×C₅	SD₁₆	1	1	yes	characteristic
94	C₅×C₁₆	—	4	2	no	—
95	C₂×#135	—	2	1	no	—
96	C₂×#135	C₂²×C₅	1	2	yes	—
97	#532	—	2	1	no	—
98	C₂×M₃₂	C₂²×C₅	1	2	yes	—
99	#495	—	2	1	no	—
100	#495	C₄×C₅	1	2	yes	complemented
101	#494	C₄×C₅	1	1	yes	characteristic, complemented
102	#493	C₄×C₅	1	1	yes	characteristic, complemented
103	#492	—	2	1	no	—
104	C₂×#106	C₂²×C₅	1	2	yes	—
105	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
106	C₄×C₄⋊C₄	—	2	1	no	—
107	C₄⋊C₁₆	—	2	1	no	—
108	C₄⋊C₁₆	—	2	1	no	—
109	#360	—	2	8	no	—
110	#340	—	2	1	no	—
111	C₂³×C₅	—	4	2	no	—
112	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
113	C₅×Q₈	—	4	2	no	—
114	C₅×Q₈	—	4	2	no	—
115	C₅×D₄	—	4	2	no	—
116	C₅×D₄	—	4	2	no	—
117	C₅×D₄	—	8	2	no	—
118	C₅×D₄	—	8	4	no	—
119	C₅×D₄	—	8	2	no	—
120	C₂×C₄×C₅	—	2	2	no	—
121	C₂×C₄×C₅	—	4	1	no	—
122	C₂×C₄×C₅	—	4	2	no	—
123	C₂×C₄×C₅	—	4	4	no	—
124	C₂×C₄×C₅	—	8	2	no	—
125	C₂×C₄×C₅	—	8	2	no	—
126	C₂×C₄×C₅	—	8	4	no	—
127	C₂×C₄×C₅	#102	1	1	yes	characteristic
128	C₂×C₄×C₅	#102	1	1	yes	characteristic
129	C₂×C₄×C₅	#102	1	1	yes	characteristic
130	C₂×C₄×C₅	#102	1	1	yes	characteristic
131	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
132	C₂×C₄×C₅	#121	1	2	yes	—
133	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
134	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
135	C₅×C₈	—	2	1	no	—
136	C₅×C₈	—	2	1	no	—
137	C₅×C₈	C₄×D₄	1	2	yes	—
138	C₅×C₈	#124	1	1	yes	characteristic
139	C₅×C₈	#127	1	1	yes	characteristic
140	C₂×#41	—	2	2	no	—
141	#135	—	2	4	no	—
142	#135	—	2	4	no	—
143	#135	—	4	4	no	—
144	C₂×Q₁₆	—	2	1	no	—
145	C₂×Q₁₆	C₂×C₄×C₅	1	2	yes	—
146	C₂×SD₁₆	—	2	2	no	—
147	C₂×SD₁₆	—	2	2	no	—
148	C₂×D₈	—	2	1	no	—
149	C₂×D₈	C₂×C₄×C₅	1	2	yes	—
150	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
151	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
152	C₄⋊Q₈	—	2	1	no	—
153	#127	—	2	1	no	—
154	#124	—	2	2	no	—
155	#121	—	4	1	no	—
156	#121	—	4	1	no	—
157	C₄×D₄	—	4	1	no	—
158	C₄×D₄	—	4	1	no	—
159	C₂×C₄⋊C₄	—	2	2	no	—
160	C₂×C₄²	—	4	1	no	—
161	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
162	M₃₂	—	2	4	no	—
163	C₂×C₁₆	—	2	2	no	—
164	#106	—	2	2	no	—
165	#106	C₅×D₄	1	4	yes	complemented
166	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
167	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
168	C₄×C₈	C₅×D₄	1	1	yes	characteristic
169	C₄×C₈	C₅×D₄	1	1	yes	characteristic
170	#95	—	4	1	no	—
171	C₂²×C₅	—	2	1	no	—
172	C₂²×C₅	—	2	1	no	—
173	C₂²×C₅	—	8	2	no	—
174	C₂²×C₅	—	8	2	no	—
175	C₂²×C₅	—	8	2	no	—
176	C₂²×C₅	C₂×#102	1	1	yes	characteristic
177	C₂²×C₅	#459	1	2	yes	—
178	C₄×C₅	—	2	1	no	—
179	C₄×C₅	—	2	1	no	—
180	C₄×C₅	—	2	1	no	—
181	C₄×C₅	—	8	4	no	—
182	C₄×C₅	—	8	2	no	—
183	C₄×C₅	#360	1	2	yes	complemented
184	C₄×C₅	#390	1	1	yes	characteristic
185	C₄×C₅	#418	1	1	yes	characteristic
186	C₄×C₅	#460	1	2	yes	—
187	#41	—	4	4	no	—
188	#41	—	4	4	no	—
189	C₂×Q₈	—	2	2	no	—
190	C₂×D₄	—	2	2	no	—
191	C₂×D₄	—	4	2	no	—
192	C₂×D₄	—	4	1	no	—
193	C₂×D₄	—	4	1	no	—
194	C₂²×C₄	—	4	2	no	—
195	C₂²×C₄	—	4	2	no	—
196	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
197	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
198	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
199	Q₁₆	—	2	4	no	—
200	Q₁₆	—	4	2	no	—
201	SD₁₆	—	4	4	no	—
202	SD₁₆	—	4	2	no	—
203	SD₁₆	—	4	2	no	—
204	D₈	—	2	4	no	—
205	D₈	—	4	2	no	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
208	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
209	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
210	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
211	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
212	C₂×C₈	C₅×#41	1	2	yes	—
213	C₄⋊C₄	—	2	4	no	—
214	C₄⋊C₄	—	4	2	no	—
215	C₄⋊C₄	—	4	1	no	—
216	C₄⋊C₄	—	4	1	no	—
217	C₂²⋊C₄	—	4	2	no	—
218	C₂²⋊C₄	—	4	2	no	—
219	C₄²	—	8	2	no	—
220	C₄²	C₅×C₂²⋊C₄	1	1	yes	characteristic
221	C₄²	C₅×C₂²⋊C₄	1	1	yes	characteristic
222	C₄²	C₅×SD₁₆	1	1	yes	characteristic
223	C₄²	C₅×SD₁₆	1	1	yes	characteristic
224	C₁₆	—	4	2	no	—
225	C₂×C₅	—	2	2	no	—
226	C₂×C₅	—	16	2	no	—
227	C₂×C₅	#1982	1	1	yes	characteristic
228	C₂×C₅	C₂×#360	1	1	yes	characteristic
229	C₂×C₅	#2145	1	1	yes	characteristic
230	C₂³	—	4	2	no	—
231	C₂³	C₂×C₅×D₈	1	1	yes	characteristic
232	Q₈	—	4	2	no	—
233	Q₈	—	4	2	no	—
234	D₄	—	4	2	no	—
235	D₄	—	4	2	no	—
236	D₄	—	8	2	no	—
237	D₄	—	8	4	no	—
238	D₄	—	8	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	4	1	no	—
241	C₂×C₄	—	4	2	no	—
242	C₂×C₄	—	4	4	no	—
243	C₂×C₄	—	8	2	no	—
244	C₂×C₄	—	8	2	no	—
245	C₂×C₄	—	8	4	no	—
246	C₂×C₄	C₅×#102	1	1	yes	characteristic
247	C₂×C₄	C₅×#102	1	1	yes	characteristic
248	C₂×C₄	C₅×#102	1	1	yes	characteristic
249	C₂×C₄	C₅×#102	1	1	yes	characteristic
250	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
251	C₂×C₄	C₅×#121	1	2	yes	—
252	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
253	C₂×C₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
254	C₈	—	2	1	no	—
255	C₈	—	2	1	no	—
256	C₈	C₄×C₅×D₄	1	2	yes	—
257	C₈	C₅×#124	1	1	yes	characteristic
258	C₈	C₅×#127	1	1	yes	characteristic
259	C₅	#13255	1	1	yes	characteristic, direct product factor, Sylow
260	C₂²	—	2	1	no	—
261	C₂²	—	2	1	no	—
262	C₂²	—	8	2	no	—
263	C₂²	—	8	2	no	—
264	C₂²	—	8	2	no	—
265	C₂²	C₂×C₅×#102	1	1	yes	characteristic
266	C₂²	C₅×#459	1	2	yes	—
267	C₄	—	2	1	no	—
268	C₄	—	2	1	no	—
269	C₄	—	2	1	no	—
270	C₄	—	8	4	no	—
271	C₄	—	8	2	no	—
272	C₄	C₅×#360	1	2	yes	complemented
273	C₄	C₅×#390	1	1	yes	characteristic
274	C₄	C₅×#418	1	1	yes	characteristic
275	C₄	C₅×#460	1	2	yes	—
276	C₂	—	2	2	no	—
277	C₂	—	16	2	no	—
278	C₂	C₅×#1982	1	1	yes	characteristic
279	C₂	C₂×C₅×#360	1	1	yes	characteristic
280	C₂	C₅×#2145	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₂²×C₈	C₂³×C₅
prime radical	C₅×#3147	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ C₂×C₄×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅×#2150	C₂	semidirect
3. C₂×C₅×#360	C₂	semidirect
4. C₄×C₅×#106	C₂	semidirect
10. C₅×#495	C₄	semidirect
11. C₅×#494	C₄	semidirect
12. C₅×#493	C₄	semidirect
21. #13255	C₅	direct (N × H)
47. C₅×#106	D₄	semidirect
54. #2150	C₂×C₅	semidirect
55. C₂×#360	C₂×C₅	semidirect
56. C₄×#106	C₂×C₅	semidirect
100. #495	C₄×C₅	semidirect
101. #494	C₄×C₅	semidirect
102. #493	C₄×C₅	semidirect
165. #106	C₅×D₄	semidirect
183. C₄×C₅	#360	semidirect
259. C₅	#13255	direct (N × H)
272. C₄	C₅×#360	semidirect

Maximal subgroups

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

subgroup	index
C₅×#3147	2
C₅×#2150	2
C₂×C₅×#360	2
C₄×C₅×#106	2
#13255	5

Subgroups — C5×#13255 #11935975

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#13255 #11935975