Subgroups — C₅×#17981 #11940701

Order 1280.

726 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3169	C₂	1	1	yes	maximal, characteristic
2	C₅×#2931	C₂	1	2	yes	maximal, complemented
3	C₅×#1586	C₂	1	8	yes	maximal, complemented
4	C₅×#1566	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#1492	C₂	1	2	yes	maximal, complemented
6	C₂×C₅×#322	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₅×#131	—	2	2	no	—
8	C₅×#532	C₂²	1	2	yes	—
9	C₅×#500	C₄	1	1	yes	characteristic
10	C₅×#493	C₄	1	1	yes	characteristic
11	C₅×#486	C₄	1	2	yes	—
12	C₅×#466	C₄	1	4	yes	—
13	C₄×C₅×D₈	C₂²	1	4	yes	—
14	C₅×#436	—	2	2	no	—
15	C₅×#436	C₂²	1	4	yes	—
16	C₅×C₈×D₄	—	2	2	no	—
17	C₅×C₈×D₄	C₂²	1	4	yes	—
18	C₅×#423	C₂²	1	2	yes	—
19	C₅×#408	—	2	2	no	—
20	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
23	C₅×#334	C₂²	1	4	yes	—
24	C₅×D₄⋊C₈	C₂²	1	8	yes	—
25	C₅×#322	C₂²	1	4	yes	—
26	#17981	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
27	C₂×C₅×#41	—	2	2	no	—
28	C₂×C₅×D₈	C₂×C₄	1	4	yes	—
29	C₅×#131	—	4	8	no	—
30	C₂×C₅×M₁₆	—	2	2	no	—
31	C₂×C₅×M₁₆	—	2	4	no	—
32	C₂²×C₅×C₈	—	2	1	no	—
33	C₂²×C₅×C₈	—	2	4	no	—
34	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
35	C₂²×C₅×C₈	D₄	1	2	yes	—
36	C₅×C₄⋊Q₈	C₂×C₄	1	2	yes	—
37	C₅×#127	C₂×C₄	1	2	yes	—
38	C₅×#124	—	2	2	no	—
39	C₅×#121	—	2	2	no	—
40	C₅×#121	C₂×C₄	1	4	yes	—
41	C₄×C₅×D₄	—	2	2	no	—
42	C₄×C₅×D₄	C₂³	1	4	yes	—
43	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
44	C₅×#107	C₂×C₄	1	4	yes	—
45	C₅×C₄⋊C₈	—	2	2	no	—
46	C₅×C₄⋊C₈	C₂³	1	4	yes	—
47	C₅×#102	C₂×C₄	1	8	yes	—
48	C₅×C₂²⋊C₈	—	2	4	no	—
49	C₅×C₂²⋊C₈	—	2	4	no	—
50	C₄×C₅×C₈	—	2	1	no	—
51	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
52	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
53	C₄×C₅×C₈	D₄	1	1	yes	characteristic
54	C₄×C₅×C₈	D₄	1	1	yes	characteristic
55	C₄×C₅×C₈	C₂³	1	2	yes	—
56	C₄×C₅×C₈	C₂³	1	2	yes	—
57	C₄×C₅×C₈	C₂³	1	2	yes	—
58	#3169	C₂×C₅	1	1	yes	characteristic
59	#2931	C₂×C₅	1	2	yes	complemented
60	#1586	C₂×C₅	1	8	yes	complemented
61	#1566	C₂×C₅	1	1	yes	characteristic, complemented
62	#1492	C₂×C₅	1	2	yes	complemented
63	C₂×#322	C₂×C₅	1	1	yes	characteristic, complemented
64	C₅×#41	—	4	4	no	—
65	C₅×#41	—	4	4	no	—
66	C₂×C₅×Q₈	—	2	2	no	—
67	C₂×C₅×D₄	—	2	4	no	—
68	C₂×C₅×D₄	—	2	2	no	—
69	C₂×C₅×D₄	—	2	2	no	—
70	C₂×C₅×D₄	C₂²×C₄	1	4	yes	—
71	C₂²×C₄×C₅	—	2	4	no	—
72	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₅	#41	1	1	yes	characteristic
75	C₅×D₈	—	4	4	no	—
76	C₅×M₁₆	—	4	8	no	—
77	C₅×M₁₆	—	4	4	no	—
78	C₂×C₅×C₈	—	2	4	no	—
79	C₂×C₅×C₈	—	2	4	no	—
80	C₂×C₅×C₈	—	2	2	no	—
81	C₂×C₅×C₈	—	2	4	no	—
82	C₂×C₅×C₈	—	4	2	no	—
83	C₂×C₅×C₈	—	4	8	no	—
84	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
85	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
86	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
87	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
88	C₅×C₄⋊C₄	—	2	2	no	—
89	C₅×C₄⋊C₄	C₂²×C₄	1	4	yes	—
90	C₅×C₂²⋊C₄	—	2	4	no	—
91	C₅×C₂²⋊C₄	—	2	4	no	—
92	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
93	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
94	C₄²×C₅	#41	1	1	yes	characteristic
95	C₄²×C₅	C₂⁴	1	1	yes	characteristic
96	C₂×#131	—	2	2	no	—
97	#532	C₂²×C₅	1	2	yes	—
98	#500	C₄×C₅	1	1	yes	characteristic
99	#493	C₄×C₅	1	1	yes	characteristic
100	#486	C₄×C₅	1	2	yes	—
101	#466	C₄×C₅	1	4	yes	—
102	C₄×D₈	C₂²×C₅	1	4	yes	—
103	#436	—	2	2	no	—
104	#436	C₂²×C₅	1	4	yes	—
105	C₈×D₄	—	2	2	no	—
106	C₈×D₄	C₂²×C₅	1	4	yes	—
107	#423	C₂²×C₅	1	2	yes	—
108	#408	—	2	2	no	—
109	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
110	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
111	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
112	#334	C₂²×C₅	1	4	yes	—
113	D₄⋊C₈	C₂²×C₅	1	8	yes	—
114	#322	C₂²×C₅	1	4	yes	—
115	C₂³×C₅	—	2	4	no	—
116	C₂³×C₅	C₄×D₄	1	1	yes	characteristic
117	C₅×Q₈	—	4	4	no	—
118	C₅×D₄	—	2	8	no	—
119	C₅×D₄	—	4	8	no	—
120	C₅×D₄	—	4	4	no	—
121	C₅×D₄	—	4	4	no	—
122	C₂×C₄×C₅	—	2	2	no	—
123	C₂×C₄×C₅	—	2	2	no	—
124	C₂×C₄×C₅	—	2	4	no	—
125	C₂×C₄×C₅	—	4	4	no	—
126	C₂×C₄×C₅	—	4	4	no	—
127	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
128	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
129	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
130	C₂×C₄×C₅	AGL(1,ℤ/8)	1	4	yes	—
131	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
132	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
133	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
134	C₅×C₈	—	2	4	no	—
135	C₅×C₈	—	2	2	no	—
136	C₅×C₈	—	2	2	no	—
137	C₅×C₈	—	2	2	no	—
138	C₅×C₈	—	2	2	no	—
139	C₅×C₈	—	4	4	no	—
140	C₂×#41	—	2	2	no	—
141	C₂×D₈	C₂×C₄×C₅	1	4	yes	—
142	#131	—	4	8	no	—
143	C₂×M₁₆	—	2	2	no	—
144	C₂×M₁₆	—	2	4	no	—
145	C₂²×C₈	—	2	1	no	—
146	C₂²×C₈	—	2	4	no	—
147	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
148	C₂²×C₈	C₅×D₄	1	2	yes	—
149	C₄⋊Q₈	C₂×C₄×C₅	1	2	yes	—
150	#127	C₂×C₄×C₅	1	2	yes	—
151	#124	—	2	2	no	—
152	#121	—	2	2	no	—
153	#121	C₂×C₄×C₅	1	4	yes	—
154	C₄×D₄	—	2	2	no	—
155	C₄×D₄	C₂³×C₅	1	4	yes	—
156	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
157	#107	C₂×C₄×C₅	1	4	yes	—
158	C₄⋊C₈	—	2	2	no	—
159	C₄⋊C₈	C₂³×C₅	1	4	yes	—
160	#102	C₂×C₄×C₅	1	8	yes	—
161	C₂²⋊C₈	—	2	4	no	—
162	C₂²⋊C₈	—	2	4	no	—
163	C₄×C₈	—	2	1	no	—
164	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
165	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
166	C₄×C₈	C₅×D₄	1	1	yes	characteristic
167	C₄×C₈	C₅×D₄	1	1	yes	characteristic
168	C₄×C₈	C₂³×C₅	1	2	yes	—
169	C₄×C₈	C₂³×C₅	1	2	yes	—
170	C₄×C₈	C₂³×C₅	1	2	yes	—
171	C₂²×C₅	—	2	1	no	—
172	C₂²×C₅	—	2	1	no	—
173	C₂²×C₅	—	4	4	no	—
174	C₂²×C₅	—	4	4	no	—
175	C₂²×C₅	—	4	4	no	—
176	C₂²×C₅	#442	1	2	yes	—
177	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
178	C₄×C₅	—	2	1	no	—
179	C₄×C₅	—	2	1	no	—
180	C₄×C₅	—	4	4	no	—
181	C₄×C₅	#442	1	2	yes	—
182	C₄×C₅	#443	1	2	yes	—
183	C₄×C₅	#568	1	2	yes	—
184	C₄×C₅	C₂×AGL(1,ℤ/8)	1	2	yes	—
185	#41	—	4	4	no	—
186	#41	—	4	4	no	—
187	C₂×Q₈	—	2	2	no	—
188	C₂×D₄	—	2	4	no	—
189	C₂×D₄	—	2	2	no	—
190	C₂×D₄	—	2	2	no	—
191	C₂×D₄	C₂²×C₄×C₅	1	4	yes	—
192	C₂²×C₄	—	2	4	no	—
193	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
194	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
195	C₂²×C₄	C₅×#41	1	1	yes	characteristic
196	D₈	—	4	4	no	—
197	M₁₆	—	4	8	no	—
198	M₁₆	—	4	4	no	—
199	C₂×C₈	—	2	4	no	—
200	C₂×C₈	—	2	4	no	—
201	C₂×C₈	—	2	2	no	—
202	C₂×C₈	—	2	4	no	—
203	C₂×C₈	—	4	2	no	—
204	C₂×C₈	—	4	8	no	—
205	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
206	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
207	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
208	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
209	C₄⋊C₄	—	2	2	no	—
210	C₄⋊C₄	C₂²×C₄×C₅	1	4	yes	—
211	C₂²⋊C₄	—	2	4	no	—
212	C₂²⋊C₄	—	2	4	no	—
213	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
214	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
215	C₄²	C₅×#41	1	1	yes	characteristic
216	C₄²	C₂⁴×C₅	1	1	yes	characteristic
217	C₂×C₅	—	2	2	no	—
218	C₂×C₅	—	8	4	no	—
219	C₂×C₅	#2932	1	1	yes	characteristic
220	C₂×C₅	C₂×#442	1	1	yes	characteristic
221	C₂×C₅	C₂×#443	1	1	yes	characteristic
222	C₂³	—	2	4	no	—
223	C₂³	C₄×C₅×D₄	1	1	yes	characteristic
224	Q₈	—	4	4	no	—
225	D₄	—	2	8	no	—
226	D₄	—	4	8	no	—
227	D₄	—	4	4	no	—
228	D₄	—	4	4	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	4	4	no	—
233	C₂×C₄	—	4	4	no	—
234	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
235	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
236	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
237	C₂×C₄	C₅×AGL(1,ℤ/8)	1	4	yes	—
238	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
239	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
240	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
241	C₈	—	2	4	no	—
242	C₈	—	2	2	no	—
243	C₈	—	2	2	no	—
244	C₈	—	2	2	no	—
245	C₈	—	2	2	no	—
246	C₈	—	4	4	no	—
247	C₅	#17981	1	1	yes	characteristic, direct product factor, Sylow
248	C₂²	—	2	1	no	—
249	C₂²	—	2	1	no	—
250	C₂²	—	4	4	no	—
251	C₂²	—	4	4	no	—
252	C₂²	—	4	4	no	—
253	C₂²	C₅×#442	1	2	yes	—
254	C₂²	C₂×C₄×C₅×D₄	1	1	yes	characteristic
255	C₄	—	2	1	no	—
256	C₄	—	2	1	no	—
257	C₄	—	4	4	no	—
258	C₄	C₅×#442	1	2	yes	—
259	C₄	C₅×#443	1	2	yes	—
260	C₄	C₅×#568	1	2	yes	—
261	C₄	C₂×C₅×AGL(1,ℤ/8)	1	2	yes	—
262	C₂	—	2	2	no	—
263	C₂	—	8	4	no	—
264	C₂	C₅×#2932	1	1	yes	characteristic
265	C₂	C₂×C₅×#442	1	1	yes	characteristic
266	C₂	C₂×C₅×#443	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅
Frattini, cosocle	C₄²	C₂⁴×C₅
socle	C₂²×C₅	C₂×C₄×D₄
prime radical	C₅×#3169	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×#1586	C₂	semidirect
4. C₅×#1566	C₂	semidirect
5. C₅×#1492	C₂	semidirect
6. C₂×C₅×#322	C₂	semidirect
26. #17981	C₅	direct (N × H)
59. #2931	C₂×C₅	semidirect
60. #1586	C₂×C₅	semidirect
61. #1566	C₂×C₅	semidirect
62. #1492	C₂×C₅	semidirect
63. C₂×#322	C₂×C₅	semidirect
247. C₅	#17981	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×#3169	2
C₅×#2931	2
C₅×#1586	2
C₅×#1566	2
C₅×#1492	2
C₂×C₅×#322	2
#17981	5

Subgroups — C5×#17981 #11940701

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#17981 #11940701