Subgroups — C₅×#13261 #11935981

Order 1280.

566 conjugacy classes of subgroups.

The subgroup lattice has 264 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	C₂×C₅×#364	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×#361	C₂	1	2	yes	maximal, complemented
4	C₅×#2140	C₂	1	2	yes	maximal, complemented
5	C₅×#1772	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₅×D₈	—	2	1	no	—
7	C₂²×C₅×D₈	C₂²	1	2	yes	—
8	C₂×C₅×#127	—	2	1	no	—
9	C₂×C₅×M₃₂	C₂²	1	2	yes	—
10	C₅×#493	—	2	2	no	—
11	C₅×#493	C₄	1	4	yes	complemented
12	C₂×C₅×#108	C₂²	1	1	yes	characteristic
13	C₅×#427	C₂²	1	1	yes	characteristic
14	C₅×#405	—	2	1	no	—
15	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
16	C₅×#364	—	2	2	no	—
17	C₅×#361	—	2	4	no	—
18	C₅×#357	—	2	4	no	—
19	C₅×#339	—	2	1	no	—
20	#13261	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₂²×C₅×D₄	—	2	2	no	—
22	C₂²×C₅×D₄	—	4	2	no	—
23	C₂×C₅×D₈	—	2	4	no	—
24	C₂×C₅×D₈	—	2	2	no	—
25	C₂×C₅×D₈	—	2	2	no	—
26	C₂×C₅×D₈	—	2	2	no	—
27	C₂×C₅×D₈	—	2	2	no	—
28	C₂×C₅×D₈	—	2	2	no	—
29	C₂×C₅×D₈	—	4	1	no	—
30	C₂×C₅×D₈	—	4	1	no	—
31	C₂×C₅×D₈	—	4	2	no	—
32	C₂×C₅×D₈	C₂×C₄	1	4	yes	—
33	C₂×C₅×M₁₆	—	2	1	no	—
34	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
35	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
36	C₅×#127	—	2	4	no	—
37	C₅×#127	—	4	2	no	—
38	C₅×#117	—	2	1	no	—
39	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
40	C₅×M₃₂	—	2	2	no	—
41	C₅×M₃₂	—	2	2	no	—
42	C₂×C₅×C₁₆	—	2	2	no	—
43	C₅×#108	—	2	1	no	—
44	C₅×#108	D₄	1	2	yes	complemented
45	C₅×#107	D₄	1	2	yes	complemented
46	C₅×#106	—	2	1	no	—
47	C₅×#97	—	4	1	no	—
48	C₄×C₅×C₈	—	4	1	no	—
49	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
50	C₄×C₅×C₈	D₄	1	2	yes	—
51	C₂×#493	C₂×C₅	1	1	yes	characteristic
52	C₂×#364	C₂×C₅	1	1	yes	characteristic, complemented
53	C₂×#361	C₂×C₅	1	2	yes	complemented
54	#2140	C₂×C₅	1	2	yes	complemented
55	#1772	C₂×C₅	1	1	yes	characteristic, complemented
56	C₂⁴×C₅	—	4	2	no	—
57	C₂×C₅×D₄	—	2	4	no	—
58	C₂×C₅×D₄	—	4	8	no	—
59	C₂×C₅×D₄	—	4	2	no	—
60	C₂×C₅×D₄	—	4	2	no	—
61	C₂×C₅×D₄	—	4	2	no	—
62	C₂×C₅×D₄	—	4	2	no	—
63	C₂×C₅×D₄	—	4	2	no	—
64	C₂×C₅×D₄	—	8	8	no	—
65	C₂²×C₄×C₅	—	2	1	no	—
66	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
67	C₅×D₈	—	2	4	no	—
68	C₅×D₈	—	2	4	no	—
69	C₅×D₈	—	4	4	no	—
70	C₅×D₈	—	4	4	no	—
71	C₅×D₈	—	4	2	no	—
72	C₅×D₈	—	4	2	no	—
73	C₅×M₁₆	—	2	2	no	—
74	C₅×M₁₆	—	4	1	no	—
75	C₂×C₅×C₈	—	4	1	no	—
76	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
77	C₂×C₅×C₈	D₈	1	1	yes	characteristic
78	C₂×C₅×C₈	D₈	1	1	yes	characteristic
79	C₂×C₅×C₈	SD₁₆	1	2	yes	—
80	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
82	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
83	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
84	C₂×C₅×C₈	#41	1	1	yes	characteristic
85	C₂×C₅×C₈	#41	1	1	yes	characteristic
86	C₅×C₄⋊C₄	—	2	2	no	—
87	C₅×C₂²⋊C₄	—	4	1	no	—
88	C₄²×C₅	—	2	1	no	—
89	C₄²×C₅	—	4	1	no	—
90	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
91	C₅×C₁₆	—	4	2	no	—
92	C₂²×D₈	—	2	1	no	—
93	C₂²×D₈	C₂²×C₅	1	2	yes	—
94	C₂×#127	—	2	1	no	—
95	C₂×M₃₂	C₂²×C₅	1	2	yes	—
96	#493	—	2	2	no	—
97	#493	C₄×C₅	1	4	yes	complemented
98	C₂×#108	C₂²×C₅	1	1	yes	characteristic
99	#427	C₂²×C₅	1	1	yes	characteristic
100	#405	—	2	1	no	—
101	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
102	#364	—	2	2	no	—
103	#361	—	2	4	no	—
104	#357	—	2	4	no	—
105	#339	—	2	1	no	—
106	C₂³×C₅	—	4	4	no	—
107	C₂³×C₅	—	8	8	no	—
108	C₂³×C₅	—	8	2	no	—
109	C₂³×C₅	—	8	2	no	—
110	C₂³×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
111	C₅×D₄	—	4	4	no	—
112	C₅×D₄	—	4	4	no	—
113	C₅×D₄	—	8	16	no	—
114	C₅×D₄	—	8	2	no	—
115	C₅×D₄	—	8	2	no	—
116	C₂×C₄×C₅	—	2	4	no	—
117	C₂×C₄×C₅	—	2	2	no	—
118	C₂×C₄×C₅	—	4	2	no	—
119	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₅	#121	1	1	yes	characteristic
121	C₂×C₄×C₅	#121	1	1	yes	characteristic
122	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
123	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
124	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
125	C₅×C₈	—	4	2	no	—
126	C₅×C₈	#102	1	2	yes	—
127	C₅×C₈	#102	1	2	yes	—
128	C₅×C₈	C₄×D₄	1	1	yes	characteristic
129	C₅×C₈	C₄×D₄	1	1	yes	characteristic
130	C₅×C₈	#124	1	1	yes	characteristic
131	C₅×C₈	#127	1	1	yes	characteristic
132	C₂²×D₄	—	2	2	no	—
133	C₂²×D₄	—	4	2	no	—
134	C₂×D₈	—	2	4	no	—
135	C₂×D₈	—	2	2	no	—
136	C₂×D₈	—	2	2	no	—
137	C₂×D₈	—	2	2	no	—
138	C₂×D₈	—	2	2	no	—
139	C₂×D₈	—	2	2	no	—
140	C₂×D₈	—	4	1	no	—
141	C₂×D₈	—	4	1	no	—
142	C₂×D₈	—	4	2	no	—
143	C₂×D₈	C₂×C₄×C₅	1	4	yes	—
144	C₂×M₁₆	—	2	1	no	—
145	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
146	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
147	#127	—	2	4	no	—
148	#127	—	4	2	no	—
149	#117	—	2	1	no	—
150	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
151	M₃₂	—	2	2	no	—
152	M₃₂	—	2	2	no	—
153	C₂×C₁₆	—	2	2	no	—
154	#108	—	2	1	no	—
155	#108	C₅×D₄	1	2	yes	complemented
156	#107	C₅×D₄	1	2	yes	complemented
157	#106	—	2	1	no	—
158	#97	—	4	1	no	—
159	C₄×C₈	—	4	1	no	—
160	C₄×C₈	C₂×C₄×C₅	1	2	yes	—
161	C₄×C₈	C₅×D₄	1	2	yes	—
162	C₂²×C₅	—	2	2	no	—
163	C₂²×C₅	—	8	8	no	—
164	C₂²×C₅	—	8	4	no	—
165	C₂²×C₅	—	16	4	no	—
166	C₂²×C₅	—	16	4	no	—
167	C₂²×C₅	#418	1	1	yes	characteristic
168	C₂²×C₅	#459	1	1	yes	characteristic
169	C₂²×C₅	#460	1	1	yes	characteristic
170	C₄×C₅	—	2	4	no	—
171	C₄×C₅	—	8	2	no	—
172	C₄×C₅	#390	1	1	yes	characteristic
173	C₄×C₅	C₂×#102	1	1	yes	characteristic
174	C₄×C₅	#459	1	1	yes	characteristic
175	C₄×C₅	#460	1	1	yes	characteristic
176	C₂⁴	—	4	2	no	—
177	C₂×D₄	—	2	4	no	—
178	C₂×D₄	—	4	8	no	—
179	C₂×D₄	—	4	2	no	—
180	C₂×D₄	—	4	2	no	—
181	C₂×D₄	—	4	2	no	—
182	C₂×D₄	—	4	2	no	—
183	C₂×D₄	—	4	2	no	—
184	C₂×D₄	—	8	8	no	—
185	C₂²×C₄	—	2	1	no	—
186	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
187	D₈	—	2	4	no	—
188	D₈	—	2	4	no	—
189	D₈	—	4	2	no	—
190	D₈	—	4	2	no	—
191	D₈	—	4	4	no	—
192	D₈	—	4	4	no	—
193	M₁₆	—	2	2	no	—
194	M₁₆	—	4	1	no	—
195	C₂×C₈	—	4	1	no	—
196	C₂×C₈	C₅×C₂²⋊C₄	1	2	yes	—
197	C₂×C₈	C₅×D₈	1	1	yes	characteristic
198	C₂×C₈	C₅×D₈	1	1	yes	characteristic
199	C₂×C₈	C₅×SD₁₆	1	2	yes	—
200	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
201	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
202	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
203	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
204	C₂×C₈	C₅×#41	1	1	yes	characteristic
205	C₂×C₈	C₅×#41	1	1	yes	characteristic
206	C₄⋊C₄	—	2	2	no	—
207	C₂²⋊C₄	—	4	1	no	—
208	C₄²	—	2	1	no	—
209	C₄²	—	4	1	no	—
210	C₄²	C₅×C₂²⋊C₄	1	2	yes	—
211	C₁₆	—	4	2	no	—
212	C₂×C₅	—	2	1	no	—
213	C₂×C₅	—	2	1	no	—
214	C₂×C₅	—	16	4	no	—
215	C₂×C₅	#1982	1	1	yes	characteristic
216	C₂×C₅	#2213	1	2	yes	—
217	C₂³	—	4	4	no	—
218	C₂³	—	8	8	no	—
219	C₂³	—	8	2	no	—
220	C₂³	—	8	2	no	—
221	C₂³	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
222	D₄	—	4	4	no	—
223	D₄	—	4	4	no	—
224	D₄	—	8	16	no	—
225	D₄	—	8	2	no	—
226	D₄	—	8	2	no	—
227	C₂×C₄	—	2	4	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	4	2	no	—
230	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
231	C₂×C₄	C₅×#121	1	1	yes	characteristic
232	C₂×C₄	C₅×#121	1	1	yes	characteristic
233	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
234	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
235	C₂×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
236	C₈	—	4	2	no	—
237	C₈	C₅×#102	1	2	yes	—
238	C₈	C₅×#102	1	2	yes	—
239	C₈	C₄×C₅×D₄	1	1	yes	characteristic
240	C₈	C₄×C₅×D₄	1	1	yes	characteristic
241	C₈	C₅×#124	1	1	yes	characteristic
242	C₈	C₅×#127	1	1	yes	characteristic
243	C₅	#13261	1	1	yes	characteristic, direct product factor, Sylow
244	C₂²	—	2	2	no	—
245	C₂²	—	8	8	no	—
246	C₂²	—	8	4	no	—
247	C₂²	—	16	4	no	—
248	C₂²	—	16	4	no	—
249	C₂²	C₅×#418	1	1	yes	characteristic
250	C₂²	C₅×#459	1	1	yes	characteristic
251	C₂²	C₅×#460	1	1	yes	characteristic
252	C₄	—	2	4	no	—
253	C₄	—	8	2	no	—
254	C₄	C₅×#390	1	1	yes	characteristic
255	C₄	C₂×C₅×#102	1	1	yes	characteristic
256	C₄	C₅×#459	1	1	yes	characteristic
257	C₄	C₅×#460	1	1	yes	characteristic
258	C₂	—	2	1	no	—
259	C₂	—	2	1	no	—
260	C₂	—	16	4	no	—
261	C₂	C₅×#1982	1	1	yes	characteristic
262	C₂	C₅×#2213	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₅	#459
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₂²×C₈	C₂³×C₅
prime radical	C₂×C₅×#493	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ C₂×C₄×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×#364	C₂	semidirect
3. C₂×C₅×#361	C₂	semidirect
4. C₅×#2140	C₂	semidirect
5. C₅×#1772	C₂	semidirect
11. C₅×#493	C₄	semidirect
20. #13261	C₅	direct (N × H)
44. C₅×#108	D₄	semidirect
45. C₅×#107	D₄	semidirect
52. C₂×#364	C₂×C₅	semidirect
53. C₂×#361	C₂×C₅	semidirect
54. #2140	C₂×C₅	semidirect
55. #1772	C₂×C₅	semidirect
97. #493	C₄×C₅	semidirect
155. #108	C₅×D₄	semidirect
156. #107	C₅×D₄	semidirect
243. C₅	#13261	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
C₂×C₅×#364	2
C₂×C₅×#361	2
C₅×#2140	2
C₅×#1772	2
#13261	5

Subgroups — C5×#13261 #11935981

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#13261 #11935981