Subgroups — C₅×#22280 #11945000

Order 1280.

846 conjugacy classes of subgroups.

The subgroup lattice has 286 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×#2931	C₂	1	2	yes	maximal, complemented
3	C₅×#1962	C₂	1	2	yes	maximal, complemented
4	C₅×#1876	C₂	1	8	yes	maximal, complemented
5	C₅×#1837	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#1744	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₅×#135	C₂²	1	4	yes	—
8	C₂×C₅×#131	C₂²	1	4	yes	—
9	C₅×#532	C₂²	1	2	yes	—
10	C₅×#495	C₄	1	4	yes	complemented
11	C₅×#494	C₄	1	1	yes	characteristic, complemented
12	C₅×#493	C₄	1	1	yes	characteristic, complemented
13	C₅×#492	C₄	1	2	yes	complemented
14	C₅×#436	—	2	2	no	—
15	C₅×#436	—	2	2	no	—
16	C₅×C₈×D₄	—	2	2	no	—
17	C₅×C₈×D₄	—	2	2	no	—
18	C₅×#423	C₂²	1	2	yes	—
19	C₂×C₅×#104	C₂²	1	8	yes	—
20	C₅×#408	C₂²	1	4	yes	—
21	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
23	C₄³×C₅	C₂²	1	1	yes	characteristic
24	C₅×#341	C₂²	1	4	yes	—
25	C₅×#336	C₂²	1	4	yes	—
26	#22280	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
27	C₂×C₅×#41	C₂³	1	4	yes	—
28	C₅×#135	—	4	8	no	—
29	C₂×C₅×Q₁₆	C₂×C₄	1	4	yes	—
30	C₂×C₅×SD₁₆	C₂×C₄	1	8	yes	—
31	C₂×C₅×D₈	C₂×C₄	1	4	yes	—
32	C₅×#131	—	4	8	no	—
33	C₂×C₅×M₁₆	—	2	4	no	—
34	C₂×C₅×M₁₆	C₂³	1	4	yes	—
35	C₂²×C₅×C₈	—	2	4	no	—
36	C₂²×C₅×C₈	C₂³	1	2	yes	—
37	C₂²×C₅×C₈	C₂³	1	2	yes	—
38	C₅×C₄⋊Q₈	C₂×C₄	1	2	yes	—
39	C₅×#127	C₂×C₄	1	2	yes	—
40	C₅×#124	C₂×C₄	1	4	yes	—
41	C₅×#121	—	2	2	no	—
42	C₅×#121	—	2	2	no	—
43	C₄×C₅×D₄	—	2	2	no	—
44	C₄×C₅×D₄	—	2	2	no	—
45	C₂×C₄²×C₅	—	2	1	no	—
46	C₂×C₄²×C₅	—	2	1	no	—
47	C₂×C₄²×C₅	C₂³	1	2	yes	—
48	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
49	C₅×C₄⋊C₈	—	2	2	no	—
50	C₅×C₄⋊C₈	—	2	2	no	—
51	C₅×#104	—	4	8	no	—
52	C₅×C₂²⋊C₈	—	2	4	no	—
53	C₅×C₂²⋊C₈	—	2	4	no	—
54	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
55	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
56	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
57	C₄×C₅×C₈	D₄	1	1	yes	characteristic
58	C₄×C₅×C₈	D₄	1	2	yes	—
59	C₄×C₅×C₈	D₄	1	1	yes	characteristic
60	#3147	C₂×C₅	1	1	yes	characteristic
61	#2931	C₂×C₅	1	2	yes	complemented
62	#1962	C₂×C₅	1	2	yes	complemented
63	#1876	C₂×C₅	1	8	yes	complemented
64	#1837	C₂×C₅	1	1	yes	characteristic, complemented
65	#1744	C₂×C₅	1	1	yes	characteristic, complemented
66	C₅×#41	—	2	8	no	—
67	C₅×#41	—	2	8	no	—
68	C₂×C₅×Q₈	C₂²×C₄	1	4	yes	—
69	C₂×C₅×D₄	—	2	4	no	—
70	C₂×C₅×D₄	—	2	2	no	—
71	C₂×C₅×D₄	—	2	2	no	—
72	C₂×C₅×D₄	C₂²×C₄	1	4	yes	—
73	C₂²×C₄×C₅	—	2	2	no	—
74	C₂²×C₄×C₅	—	2	4	no	—
75	C₂²×C₄×C₅	#41	1	1	yes	characteristic
76	C₂²×C₄×C₅	#41	1	1	yes	characteristic
77	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
78	C₅×Q₁₆	—	4	4	no	—
79	C₅×SD₁₆	—	4	8	no	—
80	C₅×D₈	—	4	4	no	—
81	C₅×M₁₆	—	2	8	no	—
82	C₅×M₁₆	—	4	8	no	—
83	C₂×C₅×C₈	—	2	4	no	—
84	C₂×C₅×C₈	—	2	4	no	—
85	C₂×C₅×C₈	—	2	4	no	—
86	C₂×C₅×C₈	—	4	8	no	—
87	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
88	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
89	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
90	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
91	C₅×C₄⋊C₄	—	2	2	no	—
92	C₅×C₄⋊C₄	—	2	2	no	—
93	C₅×C₂²⋊C₄	—	2	4	no	—
94	C₅×C₂²⋊C₄	—	2	4	no	—
95	C₄²×C₅	—	2	4	no	—
96	C₄²×C₅	—	2	4	no	—
97	C₄²×C₅	—	4	2	no	—
98	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
99	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
100	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
101	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
102	C₂×#135	C₂²×C₅	1	4	yes	—
103	C₂×#131	C₂²×C₅	1	4	yes	—
104	#532	C₂²×C₅	1	2	yes	—
105	#495	C₄×C₅	1	4	yes	complemented
106	#494	C₄×C₅	1	1	yes	characteristic, complemented
107	#493	C₄×C₅	1	1	yes	characteristic, complemented
108	#492	C₄×C₅	1	2	yes	complemented
109	#436	—	2	2	no	—
110	#436	—	2	2	no	—
111	C₈×D₄	—	2	2	no	—
112	C₈×D₄	—	2	2	no	—
113	#423	C₂²×C₅	1	2	yes	—
114	C₂×#104	C₂²×C₅	1	8	yes	—
115	#408	C₂²×C₅	1	4	yes	—
116	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
117	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
118	C₄³	C₂²×C₅	1	1	yes	characteristic
119	#341	C₂²×C₅	1	4	yes	—
120	#336	C₂²×C₅	1	4	yes	—
121	C₂³×C₅	—	2	4	no	—
122	C₂³×C₅	C₂×#41	1	1	yes	characteristic
123	C₅×Q₈	—	2	8	no	—
124	C₅×D₄	—	2	8	no	—
125	C₅×D₄	—	4	8	no	—
126	C₅×D₄	—	4	4	no	—
127	C₅×D₄	—	4	4	no	—
128	C₂×C₄×C₅	—	2	2	no	—
129	C₂×C₄×C₅	—	2	2	no	—
130	C₂×C₄×C₅	—	2	4	no	—
131	C₂×C₄×C₅	—	2	4	no	—
132	C₂×C₄×C₅	—	4	2	no	—
133	C₂×C₄×C₅	—	4	2	no	—
134	C₂×C₄×C₅	—	4	4	no	—
135	C₂×C₄×C₅	—	4	4	no	—
136	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
137	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
138	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
139	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
140	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
141	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
142	C₂×C₄×C₅	H(5,𝔽₂)	1	4	yes	—
143	C₅×C₈	—	2	2	no	—
144	C₅×C₈	—	2	2	no	—
145	C₅×C₈	—	2	2	no	—
146	C₅×C₈	—	2	2	no	—
147	C₅×C₈	—	4	4	no	—
148	C₂×#41	C₂³×C₅	1	4	yes	—
149	#135	—	4	8	no	—
150	C₂×Q₁₆	C₂×C₄×C₅	1	4	yes	—
151	C₂×SD₁₆	C₂×C₄×C₅	1	8	yes	—
152	C₂×D₈	C₂×C₄×C₅	1	4	yes	—
153	#131	—	4	8	no	—
154	C₂×M₁₆	—	2	4	no	—
155	C₂×M₁₆	C₂³×C₅	1	4	yes	—
156	C₂²×C₈	—	2	4	no	—
157	C₂²×C₈	C₂³×C₅	1	2	yes	—
158	C₂²×C₈	C₂³×C₅	1	2	yes	—
159	C₄⋊Q₈	C₂×C₄×C₅	1	2	yes	—
160	#127	C₂×C₄×C₅	1	2	yes	—
161	#124	C₂×C₄×C₅	1	4	yes	—
162	#121	—	2	2	no	—
163	#121	—	2	2	no	—
164	C₄×D₄	—	2	2	no	—
165	C₄×D₄	—	2	2	no	—
166	C₂×C₄²	—	2	1	no	—
167	C₂×C₄²	—	2	1	no	—
168	C₂×C₄²	C₂³×C₅	1	2	yes	—
169	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
170	C₄⋊C₈	—	2	2	no	—
171	C₄⋊C₈	—	2	2	no	—
172	#104	—	4	8	no	—
173	C₂²⋊C₈	—	2	4	no	—
174	C₂²⋊C₈	—	2	4	no	—
175	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
176	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
177	C₄×C₈	C₂×C₄×C₅	1	2	yes	—
178	C₄×C₈	C₅×D₄	1	1	yes	characteristic
179	C₄×C₈	C₅×D₄	1	2	yes	—
180	C₄×C₈	C₅×D₄	1	1	yes	characteristic
181	C₂²×C₅	—	2	1	no	—
182	C₂²×C₅	—	2	1	no	—
183	C₂²×C₅	—	4	4	no	—
184	C₂²×C₅	—	4	4	no	—
185	C₂²×C₅	—	4	4	no	—
186	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
187	C₂²×C₅	#540	1	2	yes	—
188	C₄×C₅	—	2	1	no	—
189	C₄×C₅	—	2	1	no	—
190	C₄×C₅	—	4	4	no	—
191	C₄×C₅	—	4	4	no	—
192	C₄×C₅	#443	1	2	yes	—
193	C₄×C₅	#444	1	2	yes	—
194	C₄×C₅	#518	1	2	yes	—
195	C₄×C₅	#535	1	2	yes	—
196	#41	—	2	8	no	—
197	#41	—	2	8	no	—
198	C₂×Q₈	C₂²×C₄×C₅	1	4	yes	—
199	C₂×D₄	—	2	4	no	—
200	C₂×D₄	—	2	2	no	—
201	C₂×D₄	—	2	2	no	—
202	C₂×D₄	C₂²×C₄×C₅	1	4	yes	—
203	C₂²×C₄	—	2	2	no	—
204	C₂²×C₄	—	2	4	no	—
205	C₂²×C₄	C₅×#41	1	1	yes	characteristic
206	C₂²×C₄	C₅×#41	1	1	yes	characteristic
207	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
208	Q₁₆	—	4	4	no	—
209	SD₁₆	—	4	8	no	—
210	D₈	—	4	4	no	—
211	M₁₆	—	2	8	no	—
212	M₁₆	—	4	8	no	—
213	C₂×C₈	—	2	4	no	—
214	C₂×C₈	—	2	4	no	—
215	C₂×C₈	—	2	4	no	—
216	C₂×C₈	—	4	8	no	—
217	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
218	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
219	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
220	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
221	C₄⋊C₄	—	2	2	no	—
222	C₄⋊C₄	—	2	2	no	—
223	C₂²⋊C₄	—	2	4	no	—
224	C₂²⋊C₄	—	2	4	no	—
225	C₄²	—	2	4	no	—
226	C₄²	—	2	4	no	—
227	C₄²	—	4	2	no	—
228	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
229	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
230	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
231	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
232	C₂×C₅	—	2	2	no	—
233	C₂×C₅	—	8	4	no	—
234	C₂×C₅	#2328	1	1	yes	characteristic
235	C₂×C₅	C₂×#443	1	1	yes	characteristic
236	C₂×C₅	C₂×#444	1	1	yes	characteristic
237	C₂³	—	2	4	no	—
238	C₂³	C₂×C₅×#41	1	1	yes	characteristic
239	Q₈	—	2	8	no	—
240	D₄	—	2	8	no	—
241	D₄	—	4	8	no	—
242	D₄	—	4	4	no	—
243	D₄	—	4	4	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	4	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	4	2	no	—
249	C₂×C₄	—	4	2	no	—
250	C₂×C₄	—	4	4	no	—
251	C₂×C₄	—	4	4	no	—
252	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
253	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
254	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
255	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
256	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
257	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
258	C₂×C₄	C₅×H(5,𝔽₂)	1	4	yes	—
259	C₈	—	2	2	no	—
260	C₈	—	2	2	no	—
261	C₈	—	2	2	no	—
262	C₈	—	2	2	no	—
263	C₈	—	4	4	no	—
264	C₅	#22280	1	1	yes	characteristic, direct product factor, Sylow
265	C₂²	—	2	1	no	—
266	C₂²	—	2	1	no	—
267	C₂²	—	4	4	no	—
268	C₂²	—	4	4	no	—
269	C₂²	—	4	4	no	—
270	C₂²	C₂×C₄×C₅×D₄	1	1	yes	characteristic
271	C₂²	C₅×#540	1	2	yes	—
272	C₄	—	2	1	no	—
273	C₄	—	2	1	no	—
274	C₄	—	4	4	no	—
275	C₄	—	4	4	no	—
276	C₄	C₅×#443	1	2	yes	—
277	C₄	C₅×#444	1	2	yes	—
278	C₄	C₅×#518	1	2	yes	—
279	C₄	C₅×#535	1	2	yes	—
280	C₂	—	2	2	no	—
281	C₂	—	8	4	no	—
282	C₂	C₅×#2328	1	1	yes	characteristic
283	C₂	C₂×C₅×#443	1	1	yes	characteristic
284	C₂	C₂×C₅×#444	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
socle	C₂²×C₅	C₂×C₄×D₄
prime radical	C₅×#3147	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂×C₄×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅×#2931	C₂	semidirect
3. C₅×#1962	C₂	semidirect
4. C₅×#1876	C₂	semidirect
5. C₅×#1837	C₂	semidirect
6. C₅×#1744	C₂	semidirect
10. C₅×#495	C₄	semidirect
11. C₅×#494	C₄	semidirect
12. C₅×#493	C₄	semidirect
13. C₅×#492	C₄	semidirect
26. #22280	C₅	direct (N × H)
61. #2931	C₂×C₅	semidirect
62. #1962	C₂×C₅	semidirect
63. #1876	C₂×C₅	semidirect
64. #1837	C₂×C₅	semidirect
65. #1744	C₂×C₅	semidirect
105. #495	C₄×C₅	semidirect
106. #494	C₄×C₅	semidirect
107. #493	C₄×C₅	semidirect
108. #492	C₄×C₅	semidirect
264. C₅	#22280	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
C₅×#2931	2
C₅×#1962	2
C₅×#1876	2
C₅×#1837	2
C₅×#1744	2
#22280	5

Subgroups — C5×#22280 #11945000

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#22280 #11945000