Subgroups — C₅×#22269 #11944989

Order 1280.

1222 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#493	C₂	1	1	yes	maximal, characteristic
2	C₄×C₅×#127	C₂	1	2	yes	maximal, complemented
3	C₅×#1957	C₂	1	2	yes	maximal, complemented
4	C₅×#1879	C₂	1	8	yes	maximal, complemented
5	C₅×#1834	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#1744	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₅×D₈	C₂²	1	4	yes	—
8	C₂×C₅×#127	C₂²	1	2	yes	—
9	C₂×C₄×C₅×D₄	—	2	2	no	—
10	C₂×C₄×C₅×D₄	—	2	2	no	—
11	C₂×C₄×C₅×D₄	C₂²	1	4	yes	—
12	C₅×#493	C₄	1	8	yes	complemented
13	C₂×C₅×#102	C₂²	1	8	yes	—
14	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
16	C₅×#390	—	2	2	no	—
17	C₅×#390	—	2	2	no	—
18	C₅×#390	C₂²	1	4	yes	—
19	C₅×#384	C₂²	1	2	yes	—
20	C₄³×C₅	C₂²	1	1	yes	characteristic
21	C₅×#340	C₂²	1	4	yes	—
22	C₅×#336	C₂²	1	4	yes	—
23	#22269	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
24	C₂²×C₅×D₄	—	2	2	no	—
25	C₂²×C₅×D₄	—	2	2	no	—
26	C₂²×C₅×D₄	C₂³	1	4	yes	—
27	C₂³×C₄×C₅	—	2	4	no	—
28	C₂×C₅×D₈	—	4	8	no	—
29	C₂×C₅×D₈	C₂×C₄	1	16	yes	complemented
30	C₂²×C₅×C₈	C₂³	1	2	yes	—
31	C₂²×C₅×C₈	C₂³	1	2	yes	—
32	C₅×#127	—	2	4	no	—
33	C₅×#127	C₂×C₄	1	8	yes	complemented
34	C₄×C₅×D₄	—	4	8	no	—
35	C₄×C₅×D₄	—	4	8	no	—
36	C₄×C₅×D₄	—	4	8	no	—
37	C₂×C₅×C₄⋊C₄	—	2	2	no	—
38	C₂×C₅×C₄⋊C₄	—	2	2	no	—
39	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
40	C₂×C₅×C₂²⋊C₄	—	2	4	no	—
41	C₂×C₅×C₂²⋊C₄	—	2	4	no	—
42	C₂×C₅×C₂²⋊C₄	—	2	4	no	—
43	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₅	D₄	1	2	yes	—
45	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₅	C₂³	1	2	yes	—
47	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
48	C₅×#102	—	4	8	no	—
49	C₄×C₅×C₈	—	2	2	no	—
50	C₄×C₅×C₈	C₂×C₄	1	4	yes	complemented
51	C₂×#493	C₂×C₅	1	1	yes	characteristic
52	C₄×#127	C₂×C₅	1	2	yes	complemented
53	#1957	C₂×C₅	1	2	yes	complemented
54	#1879	C₂×C₅	1	8	yes	complemented
55	#1834	C₂×C₅	1	1	yes	characteristic, complemented
56	#1744	C₂×C₅	1	1	yes	characteristic, complemented
57	C₂⁴×C₅	—	2	4	no	—
58	C₂×C₅×D₄	—	2	8	no	—
59	C₂×C₅×D₄	—	2	8	no	—
60	C₂×C₅×D₄	—	2	4	no	—
61	C₂×C₅×D₄	—	2	8	no	—
62	C₂×C₅×D₄	—	2	8	no	—
63	C₂×C₅×D₄	—	4	4	no	—
64	C₂×C₅×D₄	—	4	4	no	—
65	C₂×C₅×D₄	—	4	4	no	—
66	C₂×C₅×D₄	—	4	4	no	—
67	C₂×C₅×D₄	C₂²×C₄	1	8	yes	—
68	C₂²×C₄×C₅	—	2	4	no	—
69	C₂²×C₄×C₅	—	4	8	no	—
70	C₂²×C₄×C₅	—	4	8	no	—
71	C₂²×C₄×C₅	—	4	8	no	—
72	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
73	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
77	C₅×D₈	—	4	16	no	—
78	C₂×C₅×C₈	—	2	4	no	—
79	C₂×C₅×C₈	—	2	2	no	—
80	C₂×C₅×C₈	—	2	4	no	—
81	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
82	C₅×C₄⋊C₄	—	2	8	no	—
83	C₅×C₄⋊C₄	—	4	4	no	—
84	C₅×C₄⋊C₄	—	4	4	no	—
85	C₅×C₂²⋊C₄	—	4	8	no	—
86	C₅×C₂²⋊C₄	—	4	8	no	—
87	C₅×C₂²⋊C₄	—	4	8	no	—
88	C₄²×C₅	—	2	2	no	—
89	C₄²×C₅	—	2	4	no	—
90	C₄²×C₅	—	2	4	no	—
91	C₄²×C₅	D₈	1	4	yes	complemented
92	C₄²×C₅	C₂²×C₄	1	2	yes	—
93	C₄²×C₅	#41	1	2	yes	—
94	C₂²×D₈	C₂²×C₅	1	4	yes	—
95	C₂×#127	C₂²×C₅	1	2	yes	—
96	C₂×C₄×D₄	—	2	2	no	—
97	C₂×C₄×D₄	—	2	2	no	—
98	C₂×C₄×D₄	C₂²×C₅	1	4	yes	—
99	#493	C₄×C₅	1	8	yes	complemented
100	C₂×#102	C₂²×C₅	1	8	yes	—
101	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
102	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
103	#390	—	2	2	no	—
104	#390	—	2	2	no	—
105	#390	C₂²×C₅	1	4	yes	—
106	#384	C₂²×C₅	1	2	yes	—
107	C₄³	C₂²×C₅	1	1	yes	characteristic
108	#340	C₂²×C₅	1	4	yes	—
109	#336	C₂²×C₅	1	4	yes	—
110	C₂³×C₅	—	2	8	no	—
111	C₂³×C₅	—	4	4	no	—
112	C₂³×C₅	—	4	4	no	—
113	C₂³×C₅	—	4	4	no	—
114	C₂³×C₅	—	4	4	no	—
115	C₂³×C₅	—	4	4	no	—
116	C₂³×C₅	—	4	4	no	—
117	C₂³×C₅	C₂²×D₄	1	1	yes	characteristic
118	C₅×D₄	—	2	16	no	—
119	C₅×D₄	—	4	8	no	—
120	C₅×D₄	—	4	16	no	—
121	C₅×D₄	—	4	16	no	—
122	C₂×C₄×C₅	—	2	2	no	—
123	C₂×C₄×C₅	—	2	2	no	—
124	C₂×C₄×C₅	—	2	2	no	—
125	C₂×C₄×C₅	—	2	2	no	—
126	C₂×C₄×C₅	—	2	2	no	—
127	C₂×C₄×C₅	—	4	4	no	—
128	C₂×C₄×C₅	—	4	4	no	—
129	C₂×C₄×C₅	—	4	4	no	—
130	C₂×C₄×C₅	—	8	16	no	—
131	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
132	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
133	C₂×C₄×C₅	C₂×D₈	1	4	yes	—
134	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
135	C₂×C₄×C₅	#135	1	2	yes	—
136	C₂×C₄×C₅	AGL(1,ℤ/8)	1	4	yes	—
137	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
138	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
139	C₂×C₄×C₅	H(5,𝔽₂)	1	4	yes	—
140	C₅×C₈	—	2	4	no	—
141	C₅×C₈	—	4	2	no	—
142	C₂²×D₄	—	2	2	no	—
143	C₂²×D₄	—	2	2	no	—
144	C₂²×D₄	C₂³×C₅	1	4	yes	—
145	C₂³×C₄	—	2	4	no	—
146	C₂×D₈	—	4	8	no	—
147	C₂×D₈	C₂×C₄×C₅	1	16	yes	complemented
148	C₂²×C₈	C₂³×C₅	1	2	yes	—
149	C₂²×C₈	C₂³×C₅	1	2	yes	—
150	#127	—	2	4	no	—
151	#127	C₂×C₄×C₅	1	8	yes	complemented
152	C₄×D₄	—	4	8	no	—
153	C₄×D₄	—	4	8	no	—
154	C₄×D₄	—	4	8	no	—
155	C₂×C₄⋊C₄	—	2	2	no	—
156	C₂×C₄⋊C₄	—	2	2	no	—
157	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
158	C₂×C₂²⋊C₄	—	2	4	no	—
159	C₂×C₂²⋊C₄	—	2	4	no	—
160	C₂×C₂²⋊C₄	—	2	4	no	—
161	C₂×C₄²	C₅×D₄	1	2	yes	—
162	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
163	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
164	C₂×C₄²	C₂³×C₅	1	2	yes	—
165	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
166	#102	—	4	8	no	—
167	C₄×C₈	—	2	2	no	—
168	C₄×C₈	C₂×C₄×C₅	1	4	yes	complemented
169	C₂²×C₅	—	4	8	no	—
170	C₂²×C₅	—	4	8	no	—
171	C₂²×C₅	—	4	8	no	—
172	C₂²×C₅	—	8	4	no	—
173	C₂²×C₅	—	8	4	no	—
174	C₂²×C₅	—	8	4	no	—
175	C₂²×C₅	—	8	4	no	—
176	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
177	C₂²×C₅	#535	1	2	yes	—
178	C₂²×C₅	C₂²×D₈	1	1	yes	characteristic
179	C₂²×C₅	C₂×#135	1	1	yes	characteristic
180	C₂²×C₅	C₂×AGL(1,ℤ/8)	1	2	yes	—
181	C₄×C₅	—	2	4	no	—
182	C₄×C₅	—	4	2	no	—
183	C₄×C₅	—	8	4	no	—
184	C₄×C₅	C₄×D₈	1	4	yes	—
185	C₄×C₅	#442	1	4	yes	—
186	C₄×C₅	#518	1	2	yes	—
187	C₄×C₅	#540	1	2	yes	—
188	C₂⁴	—	2	4	no	—
189	C₂×D₄	—	2	8	no	—
190	C₂×D₄	—	2	8	no	—
191	C₂×D₄	—	2	4	no	—
192	C₂×D₄	—	2	8	no	—
193	C₂×D₄	—	2	8	no	—
194	C₂×D₄	—	4	4	no	—
195	C₂×D₄	—	4	4	no	—
196	C₂×D₄	—	4	4	no	—
197	C₂×D₄	—	4	4	no	—
198	C₂×D₄	C₂²×C₄×C₅	1	8	yes	—
199	C₂²×C₄	—	2	4	no	—
200	C₂²×C₄	—	4	8	no	—
201	C₂²×C₄	—	4	8	no	—
202	C₂²×C₄	—	4	8	no	—
203	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
204	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
205	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
206	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
207	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
208	D₈	—	4	16	no	—
209	C₂×C₈	—	2	4	no	—
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	—	2	4	no	—
212	C₂×C₈	C₂²×C₄×C₅	1	4	yes	—
213	C₄⋊C₄	—	2	8	no	—
214	C₄⋊C₄	—	4	4	no	—
215	C₄⋊C₄	—	4	4	no	—
216	C₂²⋊C₄	—	4	8	no	—
217	C₂²⋊C₄	—	4	8	no	—
218	C₂²⋊C₄	—	4	8	no	—
219	C₄²	—	2	2	no	—
220	C₄²	—	2	4	no	—
221	C₄²	—	2	4	no	—
222	C₄²	C₅×D₈	1	4	yes	complemented
223	C₄²	C₂²×C₄×C₅	1	2	yes	—
224	C₄²	C₅×#41	1	2	yes	—
225	C₂×C₅	—	8	8	no	—
226	C₂×C₅	#2328	1	1	yes	characteristic
227	C₂×C₅	C₂×C₄×D₈	1	1	yes	characteristic
228	C₂×C₅	C₂×#442	1	1	yes	characteristic
229	C₂×C₅	#3205	1	2	yes	—
230	C₂×C₅	#3207	1	2	yes	—
231	C₂³	—	2	8	no	—
232	C₂³	—	4	4	no	—
233	C₂³	—	4	4	no	—
234	C₂³	—	4	4	no	—
235	C₂³	—	4	4	no	—
236	C₂³	—	4	4	no	—
237	C₂³	—	4	4	no	—
238	C₂³	C₂²×C₅×D₄	1	1	yes	characteristic
239	D₄	—	2	16	no	—
240	D₄	—	4	16	no	—
241	D₄	—	4	16	no	—
242	D₄	—	4	8	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	4	4	no	—
249	C₂×C₄	—	4	4	no	—
250	C₂×C₄	—	4	4	no	—
251	C₂×C₄	—	8	16	no	—
252	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
253	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
254	C₂×C₄	C₂×C₅×D₈	1	4	yes	—
255	C₂×C₄	C₂×C₅×D₈	1	2	yes	—
256	C₂×C₄	C₅×#135	1	2	yes	—
257	C₂×C₄	C₅×AGL(1,ℤ/8)	1	4	yes	—
258	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
259	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
260	C₂×C₄	C₅×H(5,𝔽₂)	1	4	yes	—
261	C₈	—	2	4	no	—
262	C₈	—	4	2	no	—
263	C₅	#22269	1	1	yes	characteristic, direct product factor, Sylow
264	C₂²	—	4	8	no	—
265	C₂²	—	4	8	no	—
266	C₂²	—	4	8	no	—
267	C₂²	—	8	4	no	—
268	C₂²	—	8	4	no	—
269	C₂²	—	8	4	no	—
270	C₂²	—	8	4	no	—
271	C₂²	C₂×C₄×C₅×D₄	1	1	yes	characteristic
272	C₂²	C₅×#535	1	2	yes	—
273	C₂²	C₂²×C₅×D₈	1	1	yes	characteristic
274	C₂²	C₂×C₅×#135	1	1	yes	characteristic
275	C₂²	C₂×C₅×AGL(1,ℤ/8)	1	2	yes	—
276	C₄	—	2	4	no	—
277	C₄	—	4	2	no	—
278	C₄	—	8	4	no	—
279	C₄	C₄×C₅×D₈	1	4	yes	—
280	C₄	C₅×#442	1	4	yes	—
281	C₄	C₅×#518	1	2	yes	—
282	C₄	C₅×#540	1	2	yes	—
283	C₂	—	8	8	no	—
284	C₂	C₅×#2328	1	1	yes	characteristic
285	C₂	C₂×C₄×C₅×D₈	1	1	yes	characteristic
286	C₂	C₂×C₅×#442	1	1	yes	characteristic
287	C₂	C₅×#3205	1	2	yes	—
288	C₂	C₅×#3207	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₅	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
prime radical	C₂×C₅×#493	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₄³×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄×C₅×#127	C₂	semidirect
3. C₅×#1957	C₂	semidirect
4. C₅×#1879	C₂	semidirect
5. C₅×#1834	C₂	semidirect
6. C₅×#1744	C₂	semidirect
12. C₅×#493	C₄	semidirect
23. #22269	C₅	direct (N × H)
29. C₂×C₅×D₈	C₂×C₄	semidirect
33. C₅×#127	C₂×C₄	semidirect
50. C₄×C₅×C₈	C₂×C₄	semidirect
52. C₄×#127	C₂×C₅	semidirect
53. #1957	C₂×C₅	semidirect
54. #1879	C₂×C₅	semidirect
55. #1834	C₂×C₅	semidirect
56. #1744	C₂×C₅	semidirect
91. C₄²×C₅	D₈	semidirect
99. #493	C₄×C₅	semidirect
147. C₂×D₈	C₂×C₄×C₅	semidirect
151. #127	C₂×C₄×C₅	semidirect
168. C₄×C₈	C₂×C₄×C₅	semidirect
222. C₄²	C₅×D₈	semidirect
263. C₅	#22269	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×#493	2
C₄×C₅×#127	2
C₅×#1957	2
C₅×#1879	2
C₅×#1834	2
C₅×#1744	2
#22269	5

Subgroups — C5×#22269 #11944989

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#22269 #11944989