Subgroups — C₅×#22264 #11944984

Order 1280.

1018 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#436	C₂	1	2	yes	maximal, complemented
2	C₄×C₅×#127	C₂	1	1	yes	maximal, characteristic
3	C₅×#1954	C₂	1	2	yes	maximal, complemented
4	C₅×#1870	C₂	1	8	yes	maximal, complemented
5	C₅×#1831	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#1744	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₅×M₁₆	C₂²	1	4	yes	—
8	C₂×C₅×#127	C₄	1	1	yes	characteristic
9	C₂×C₄×C₅×D₄	C₂²	1	2	yes	—
10	C₂×C₄×C₅×D₄	C₂²	1	4	yes	—
11	C₅×#436	—	2	8	no	—
12	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
13	C₂×C₅×C₂²⋊C₈	C₂²	1	4	yes	—
14	C₂×C₅×C₂²⋊C₈	C₂²	1	8	yes	—
15	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
17	C₅×#390	C₄	1	2	yes	—
18	C₅×#390	C₄	1	4	yes	—
19	C₅×#384	C₄	1	1	yes	characteristic
20	C₄³×C₅	C₂²	1	1	yes	characteristic
21	C₅×#336	C₂²	1	4	yes	—
22	C₅×#336	C₂²	1	4	yes	—
23	#22264	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
24	C₂²×C₅×D₄	C₂×C₄	1	2	yes	—
25	C₂²×C₅×D₄	C₂×C₄	1	4	yes	—
26	C₂³×C₄×C₅	C₂³	1	4	yes	—
27	C₂×C₅×M₁₆	—	2	8	no	—
28	C₂×C₅×M₁₆	—	4	8	no	—
29	C₂²×C₅×C₈	C₂³	1	2	yes	—
30	C₂²×C₅×C₈	C₂³	1	4	yes	—
31	C₂²×C₅×C₈	C₂³	1	2	yes	—
32	C₅×#127	—	2	4	no	—
33	C₄×C₅×D₄	—	2	4	no	—
34	C₄×C₅×D₄	—	2	4	no	—
35	C₄×C₅×D₄	—	4	8	no	—
36	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
37	C₂×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
38	C₂×C₅×C₂²⋊C₄	C₂×C₄	1	4	yes	—
39	C₂×C₅×C₂²⋊C₄	C₂×C₄	1	8	yes	—
40	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₅	C₂³	1	2	yes	—
44	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
45	C₅×C₄⋊C₈	—	2	4	no	—
46	C₅×C₂²⋊C₈	—	2	8	no	—
47	C₅×C₂²⋊C₈	—	4	8	no	—
48	C₄×C₅×C₈	—	2	2	no	—
49	C₄×C₅×C₈	D₄	1	4	yes	complemented
50	C₂×#436	C₂×C₅	1	2	yes	complemented
51	C₄×#127	C₂×C₅	1	1	yes	characteristic
52	#1954	C₂×C₅	1	2	yes	complemented
53	#1870	C₂×C₅	1	8	yes	complemented
54	#1831	C₂×C₅	1	1	yes	characteristic, complemented
55	#1744	C₂×C₅	1	1	yes	characteristic, complemented
56	C₂⁴×C₅	C₂²×C₄	1	4	yes	—
57	C₂×C₅×D₄	—	2	4	no	—
58	C₂×C₅×D₄	—	2	8	no	—
59	C₂×C₅×D₄	—	2	4	no	—
60	C₂×C₅×D₄	—	2	4	no	—
61	C₂×C₅×D₄	—	4	4	no	—
62	C₂×C₅×D₄	—	4	4	no	—
63	C₂²×C₄×C₅	—	2	4	no	—
64	C₂²×C₄×C₅	—	2	4	no	—
65	C₂²×C₄×C₅	—	2	8	no	—
66	C₂²×C₄×C₅	—	2	8	no	—
67	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
68	C₂²×C₄×C₅	C₂²×C₄	1	4	yes	—
69	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
71	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
73	C₅×M₁₆	—	4	16	no	—
74	C₂×C₅×C₈	—	2	4	no	—
75	C₂×C₅×C₈	—	2	4	no	—
76	C₂×C₅×C₈	—	2	2	no	—
77	C₂×C₅×C₈	—	2	8	no	—
78	C₂×C₅×C₈	—	2	4	no	—
79	C₂×C₅×C₈	C₂×D₄	1	4	yes	—
80	C₅×C₄⋊C₄	—	2	4	no	—
81	C₅×C₄⋊C₄	—	4	4	no	—
82	C₅×C₂²⋊C₄	—	2	8	no	—
83	C₅×C₂²⋊C₄	—	4	8	no	—
84	C₄²×C₅	—	2	2	no	—
85	C₄²×C₅	—	2	4	no	—
86	C₄²×C₅	—	4	2	no	—
87	C₄²×C₅	M₁₆	1	4	yes	complemented
88	C₄²×C₅	C₂×D₄	1	2	yes	—
89	C₄²×C₅	#41	1	2	yes	—
90	C₂²×M₁₆	C₂²×C₅	1	4	yes	—
91	C₂×#127	C₄×C₅	1	1	yes	characteristic
92	C₂×C₄×D₄	C₂²×C₅	1	2	yes	—
93	C₂×C₄×D₄	C₂²×C₅	1	4	yes	—
94	#436	—	2	8	no	—
95	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
96	C₂×C₂²⋊C₈	C₂²×C₅	1	4	yes	—
97	C₂×C₂²⋊C₈	C₂²×C₅	1	8	yes	—
98	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
99	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
100	#390	C₄×C₅	1	2	yes	—
101	#390	C₄×C₅	1	4	yes	—
102	#384	C₄×C₅	1	1	yes	characteristic
103	C₄³	C₂²×C₅	1	1	yes	characteristic
104	#336	C₂²×C₅	1	4	yes	—
105	#336	C₂²×C₅	1	4	yes	—
106	C₂³×C₅	—	2	4	no	—
107	C₂³×C₅	—	2	8	no	—
108	C₂³×C₅	—	2	4	no	—
109	C₂³×C₅	—	2	4	no	—
110	C₂³×C₅	—	2	8	no	—
111	C₂³×C₅	C₂³×C₄	1	1	yes	characteristic
112	C₅×D₄	—	4	16	no	—
113	C₅×D₄	—	4	8	no	—
114	C₂×C₄×C₅	—	2	2	no	—
115	C₂×C₄×C₅	—	2	2	no	—
116	C₂×C₄×C₅	—	2	2	no	—
117	C₂×C₄×C₅	—	2	4	no	—
118	C₂×C₄×C₅	—	2	2	no	—
119	C₂×C₄×C₅	—	2	8	no	—
120	C₂×C₄×C₅	—	2	2	no	—
121	C₂×C₄×C₅	—	2	2	no	—
122	C₂×C₄×C₅	—	4	4	no	—
123	C₂×C₄×C₅	—	4	4	no	—
124	C₂×C₄×C₅	—	4	8	no	—
125	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
126	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
127	C₂×C₄×C₅	C₂×M₁₆	1	4	yes	—
128	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
129	C₂×C₄×C₅	#131	1	2	yes	—
130	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
131	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
132	C₂×C₄×C₅	H(5,𝔽₂)	1	4	yes	—
133	C₅×C₈	—	2	4	no	—
134	C₅×C₈	—	4	4	no	—
135	C₅×C₈	—	4	2	no	—
136	C₂²×D₄	C₂×C₄×C₅	1	2	yes	—
137	C₂²×D₄	C₂×C₄×C₅	1	4	yes	—
138	C₂³×C₄	C₂³×C₅	1	4	yes	—
139	C₂×M₁₆	—	2	8	no	—
140	C₂×M₁₆	—	4	8	no	—
141	C₂²×C₈	C₂³×C₅	1	4	yes	—
142	C₂²×C₈	C₂³×C₅	1	2	yes	—
143	C₂²×C₈	C₂³×C₅	1	2	yes	—
144	#127	—	2	4	no	—
145	C₄×D₄	—	2	4	no	—
146	C₄×D₄	—	2	4	no	—
147	C₄×D₄	—	4	8	no	—
148	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
149	C₂×C₄⋊C₄	C₂×C₄×C₅	1	4	yes	—
150	C₂×C₂²⋊C₄	C₂×C₄×C₅	1	4	yes	—
151	C₂×C₂²⋊C₄	C₂×C₄×C₅	1	8	yes	—
152	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
153	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
154	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
155	C₂×C₄²	C₂³×C₅	1	2	yes	—
156	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
157	C₄⋊C₈	—	2	4	no	—
158	C₂²⋊C₈	—	2	8	no	—
159	C₂²⋊C₈	—	4	8	no	—
160	C₄×C₈	—	2	2	no	—
161	C₄×C₈	C₅×D₄	1	4	yes	complemented
162	C₂²×C₅	—	4	4	no	—
163	C₂²×C₅	—	4	4	no	—
164	C₂²×C₅	—	4	4	no	—
165	C₂²×C₅	—	4	8	no	—
166	C₂²×C₅	—	4	8	no	—
167	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
168	C₂²×C₅	#518	1	2	yes	—
169	C₂²×C₅	C₂²×M₁₆	1	1	yes	characteristic
170	C₂²×C₅	C₂×#131	1	1	yes	characteristic
171	C₂²×C₅	#568	1	2	yes	—
172	C₄×C₅	—	2	2	no	—
173	C₄×C₅	—	2	4	no	—
174	C₄×C₅	—	4	4	no	—
175	C₄×C₅	—	4	2	no	—
176	C₄×C₅	#436	1	4	yes	—
177	C₄×C₅	#535	1	2	yes	—
178	C₄×C₅	#540	1	2	yes	—
179	C₂⁴	C₂²×C₄×C₅	1	4	yes	—
180	C₂×D₄	—	2	4	no	—
181	C₂×D₄	—	2	8	no	—
182	C₂×D₄	—	2	4	no	—
183	C₂×D₄	—	2	4	no	—
184	C₂×D₄	—	4	4	no	—
185	C₂×D₄	—	4	4	no	—
186	C₂²×C₄	—	2	4	no	—
187	C₂²×C₄	—	2	4	no	—
188	C₂²×C₄	—	2	8	no	—
189	C₂²×C₄	—	2	8	no	—
190	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
191	C₂²×C₄	C₂²×C₄×C₅	1	4	yes	—
192	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
193	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
194	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
195	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
196	M₁₆	—	4	16	no	—
197	C₂×C₈	—	2	4	no	—
198	C₂×C₈	—	2	4	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	2	4	no	—
201	C₂×C₈	—	2	8	no	—
202	C₂×C₈	C₂×C₅×D₄	1	4	yes	—
203	C₄⋊C₄	—	2	4	no	—
204	C₄⋊C₄	—	4	4	no	—
205	C₂²⋊C₄	—	2	8	no	—
206	C₂²⋊C₄	—	4	8	no	—
207	C₄²	—	2	2	no	—
208	C₄²	—	2	4	no	—
209	C₄²	—	4	2	no	—
210	C₄²	C₅×M₁₆	1	4	yes	complemented
211	C₄²	C₂×C₅×D₄	1	2	yes	—
212	C₄²	C₅×#41	1	2	yes	—
213	C₂×C₅	—	8	4	no	—
214	C₂×C₅	#2328	1	1	yes	characteristic
215	C₂×C₅	C₂×#436	1	1	yes	characteristic
216	C₂×C₅	#2932	1	1	yes	characteristic
217	C₂×C₅	#2975	1	2	yes	—
218	C₂×C₅	#2981	1	2	yes	—
219	C₂³	—	2	4	no	—
220	C₂³	—	2	8	no	—
221	C₂³	—	2	4	no	—
222	C₂³	—	2	4	no	—
223	C₂³	—	2	8	no	—
224	C₂³	C₂³×C₄×C₅	1	1	yes	characteristic
225	D₄	—	4	16	no	—
226	D₄	—	4	8	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	8	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	4	4	no	—
236	C₂×C₄	—	4	4	no	—
237	C₂×C₄	—	4	8	no	—
238	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
239	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
240	C₂×C₄	C₂×C₅×M₁₆	1	4	yes	—
241	C₂×C₄	C₂×C₅×M₁₆	1	2	yes	—
242	C₂×C₄	C₅×#131	1	2	yes	—
243	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
244	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
245	C₂×C₄	C₅×H(5,𝔽₂)	1	4	yes	—
246	C₈	—	2	4	no	—
247	C₈	—	4	4	no	—
248	C₈	—	4	2	no	—
249	C₅	#22264	1	1	yes	characteristic, direct product factor, Sylow
250	C₂²	—	4	8	no	—
251	C₂²	—	4	4	no	—
252	C₂²	—	4	4	no	—
253	C₂²	—	4	4	no	—
254	C₂²	—	4	8	no	—
255	C₂²	C₂×C₄×C₅×D₄	1	1	yes	characteristic
256	C₂²	C₅×#518	1	2	yes	—
257	C₂²	C₂²×C₅×M₁₆	1	1	yes	characteristic
258	C₂²	C₂×C₅×#131	1	1	yes	characteristic
259	C₂²	C₅×#568	1	2	yes	—
260	C₄	—	2	2	no	—
261	C₄	—	2	4	no	—
262	C₄	—	4	2	no	—
263	C₄	—	4	4	no	—
264	C₄	C₅×#436	1	4	yes	—
265	C₄	C₅×#535	1	2	yes	—
266	C₄	C₅×#540	1	2	yes	—
267	C₂	—	8	4	no	—
268	C₂	C₅×#2328	1	1	yes	characteristic
269	C₂	C₂×C₅×#436	1	1	yes	characteristic
270	C₂	C₅×#2932	1	1	yes	characteristic
271	C₂	C₅×#2975	1	2	yes	—
272	C₂	C₅×#2981	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₅	C₂⁴
commutator	C₂³	C₂³×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
socle	C₂³×C₅	C₂³×C₄
prime radical	C₂×C₅×#127	C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₅×#436	C₂	semidirect
3. C₅×#1954	C₂	semidirect
4. C₅×#1870	C₂	semidirect
5. C₅×#1831	C₂	semidirect
6. C₅×#1744	C₂	semidirect
23. #22264	C₅	direct (N × H)
49. C₄×C₅×C₈	D₄	semidirect
50. C₂×#436	C₂×C₅	semidirect
52. #1954	C₂×C₅	semidirect
53. #1870	C₂×C₅	semidirect
54. #1831	C₂×C₅	semidirect
55. #1744	C₂×C₅	semidirect
87. C₄²×C₅	M₁₆	semidirect
161. C₄×C₈	C₅×D₄	semidirect
210. C₄²	C₅×M₁₆	semidirect
249. C₅	#22264	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×#436	2
C₄×C₅×#127	2
C₅×#1954	2
C₅×#1870	2
C₅×#1831	2
C₅×#1744	2
#22264	5

Subgroups — C5×#22264 #11944984

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#22264 #11944984