Subgroups — C₅×#13271 #11935991

Order 1280.

332 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3160	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#364	C₂	1	1	yes	maximal, characteristic
3	C₅×#1772	C₂	1	1	yes	maximal, characteristic
4	C₅×#1389	C₂	1	2	yes	maximal
5	C₅×#1384	C₂	1	2	yes	maximal
6	C₅×#533	—	2	1	no	—
7	C₂×C₅×M₃₂	C₂²	1	2	yes	—
8	C₅×#500	—	2	1	no	—
9	C₅×#499	—	2	1	no	—
10	C₅×#499	C₄	1	2	yes	complemented
11	C₅×#498	C₄	1	2	yes	complemented
12	C₂×C₅×#108	C₂²	1	1	yes	characteristic
13	C₅×#427	—	2	1	no	—
14	C₅×#427	C₂²	1	2	yes	—
15	C₅×#427	C₂²	1	1	yes	characteristic
16	C₅×#405	—	2	1	no	—
17	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
18	C₅×#364	—	2	1	no	—
19	C₅×#364	—	2	1	no	—
20	C₅×#339	—	2	1	no	—
21	#13271	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
22	C₂×C₅×M₁₆	—	2	1	no	—
23	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
24	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
25	C₅×C₄⋊Q₈	—	2	2	no	—
26	C₅×#125	—	2	2	no	—
27	C₅×C₂²⋊Q₈	—	4	2	no	—
28	C₄×C₅×Q₈	—	4	2	no	—
29	C₅×#117	—	2	1	no	—
30	C₅×#117	—	2	2	no	—
31	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
32	C₅×M₃₂	—	2	2	no	—
33	C₅×M₃₂	—	2	2	no	—
34	C₂×C₅×C₁₆	—	2	2	no	—
35	C₅×#108	—	2	1	no	—
36	C₅×#108	Q₈	1	2	yes	complemented
37	C₅×#107	—	2	2	no	—
38	C₅×#107	—	2	2	no	—
39	C₅×#107	—	2	1	no	—
40	C₅×#106	—	2	2	no	—
41	C₅×#106	C₂×C₄	1	4	yes	—
42	C₅×#106	Q₈	1	2	yes	complemented
43	C₅×#97	—	4	1	no	—
44	C₄×C₅×C₈	—	4	1	no	—
45	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
46	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
47	C₄×C₅×C₈	D₄	1	1	yes	characteristic
48	C₄×C₅×C₈	D₄	1	1	yes	characteristic
49	#3160	C₂×C₅	1	1	yes	characteristic
50	C₂×#364	C₂×C₅	1	1	yes	characteristic
51	#1772	C₂×C₅	1	1	yes	characteristic
52	#1389	C₂×C₅	1	2	yes	—
53	#1384	C₂×C₅	1	2	yes	—
54	C₂×C₅×Q₈	—	4	2	no	—
55	C₂²×C₄×C₅	—	2	1	no	—
56	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
57	C₅×M₁₆	—	2	2	no	—
58	C₅×M₁₆	—	4	1	no	—
59	C₂×C₅×C₈	—	4	1	no	—
60	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
61	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₅×C₈	SD₁₆	1	1	yes	characteristic
63	C₂×C₅×C₈	SD₁₆	1	1	yes	characteristic
64	C₂×C₅×C₈	Q₁₆	1	1	yes	characteristic
65	C₂×C₅×C₈	Q₁₆	1	1	yes	characteristic
66	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
68	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
69	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
70	C₂×C₅×C₈	#41	1	1	yes	characteristic
71	C₂×C₅×C₈	#41	1	1	yes	characteristic
72	C₅×C₄⋊C₄	—	2	4	no	—
73	C₅×C₄⋊C₄	—	2	2	no	—
74	C₅×C₄⋊C₄	—	4	4	no	—
75	C₅×C₄⋊C₄	—	4	2	no	—
76	C₅×C₂²⋊C₄	—	4	1	no	—
77	C₅×C₂²⋊C₄	—	4	2	no	—
78	C₄²×C₅	—	2	1	no	—
79	C₄²×C₅	—	4	1	no	—
80	C₄²×C₅	—	4	2	no	—
81	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
82	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
83	C₅×C₁₆	—	4	2	no	—
84	#533	—	2	1	no	—
85	C₂×M₃₂	C₂²×C₅	1	2	yes	—
86	#500	—	2	1	no	—
87	#499	—	2	1	no	—
88	#499	C₄×C₅	1	2	yes	complemented
89	#498	C₄×C₅	1	2	yes	complemented
90	C₂×#108	C₂²×C₅	1	1	yes	characteristic
91	#427	—	2	1	no	—
92	#427	C₂²×C₅	1	2	yes	—
93	#427	C₂²×C₅	1	1	yes	characteristic
94	#405	—	2	1	no	—
95	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
96	#364	—	2	1	no	—
97	#364	—	2	1	no	—
98	#339	—	2	1	no	—
99	C₂³×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
100	C₅×Q₈	—	8	4	no	—
101	C₂×C₄×C₅	—	2	1	no	—
102	C₂×C₄×C₅	—	2	1	no	—
103	C₂×C₄×C₅	—	4	2	no	—
104	C₂×C₄×C₅	—	4	4	no	—
105	C₂×C₄×C₅	—	4	2	no	—
106	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
108	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
109	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
110	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
111	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
112	C₅×C₈	—	4	2	no	—
113	C₅×C₈	#103	1	1	yes	characteristic
114	C₅×C₈	#103	1	1	yes	characteristic
115	C₅×C₈	#103	1	1	yes	characteristic
116	C₅×C₈	#103	1	1	yes	characteristic
117	C₅×C₈	C₄×Q₈	1	1	yes	characteristic
118	C₅×C₈	C₄×Q₈	1	1	yes	characteristic
119	C₅×C₈	#124	1	1	yes	characteristic
120	C₅×C₈	C₄⋊Q₈	1	1	yes	characteristic
121	C₂×M₁₆	—	2	1	no	—
122	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
123	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
124	C₄⋊Q₈	—	2	2	no	—
125	#125	—	2	2	no	—
126	C₂²⋊Q₈	—	4	2	no	—
127	C₄×Q₈	—	4	2	no	—
128	#117	—	2	2	no	—
129	#117	—	2	1	no	—
130	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
131	M₃₂	—	2	2	no	—
132	M₃₂	—	2	2	no	—
133	C₂×C₁₆	—	2	2	no	—
134	#108	—	2	1	no	—
135	#108	C₅×Q₈	1	2	yes	complemented
136	#107	—	2	2	no	—
137	#107	—	2	2	no	—
138	#107	—	2	1	no	—
139	#106	—	2	2	no	—
140	#106	C₂×C₄×C₅	1	4	yes	—
141	#106	C₅×Q₈	1	2	yes	complemented
142	#97	—	4	1	no	—
143	C₄×C₈	—	4	1	no	—
144	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
145	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
146	C₄×C₈	C₅×D₄	1	1	yes	characteristic
147	C₄×C₈	C₅×D₄	1	1	yes	characteristic
148	C₂²×C₅	—	4	1	no	—
149	C₂²×C₅	#418	1	1	yes	characteristic
150	C₂²×C₅	#476	1	1	yes	characteristic
151	C₂²×C₅	#477	1	1	yes	characteristic
152	C₄×C₅	—	2	2	no	—
153	C₄×C₅	—	4	1	no	—
154	C₄×C₅	—	8	2	no	—
155	C₄×C₅	—	8	4	no	—
156	C₄×C₅	#391	1	1	yes	characteristic
157	C₄×C₅	C₂×#103	1	1	yes	characteristic
158	C₄×C₅	#476	1	1	yes	characteristic
159	C₄×C₅	#477	1	1	yes	characteristic
160	C₂×Q₈	—	4	2	no	—
161	C₂²×C₄	—	2	1	no	—
162	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
163	M₁₆	—	2	2	no	—
164	M₁₆	—	4	1	no	—
165	C₂×C₈	—	4	1	no	—
166	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
167	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
168	C₂×C₈	C₅×SD₁₆	1	1	yes	characteristic
169	C₂×C₈	C₅×SD₁₆	1	1	yes	characteristic
170	C₂×C₈	C₅×Q₁₆	1	1	yes	characteristic
171	C₂×C₈	C₅×Q₁₆	1	1	yes	characteristic
172	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
173	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
174	C₂×C₈	C₂×C₅×Q₈	1	1	yes	characteristic
175	C₂×C₈	C₂×C₅×Q₈	1	1	yes	characteristic
176	C₂×C₈	C₅×#41	1	1	yes	characteristic
177	C₂×C₈	C₅×#41	1	1	yes	characteristic
178	C₄⋊C₄	—	2	4	no	—
179	C₄⋊C₄	—	2	2	no	—
180	C₄⋊C₄	—	4	4	no	—
181	C₄⋊C₄	—	4	2	no	—
182	C₂²⋊C₄	—	4	2	no	—
183	C₂²⋊C₄	—	4	1	no	—
184	C₄²	—	2	1	no	—
185	C₄²	—	4	1	no	—
186	C₄²	—	4	2	no	—
187	C₄²	C₅×C₂²⋊C₄	1	1	yes	characteristic
188	C₄²	C₅×C₂²⋊C₄	1	1	yes	characteristic
189	C₁₆	—	4	2	no	—
190	C₂×C₅	—	2	1	no	—
191	C₂×C₅	—	2	1	no	—
192	C₂×C₅	—	2	1	no	—
193	C₂×C₅	#1987	1	1	yes	characteristic
194	C₂³	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
195	Q₈	—	8	4	no	—
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	4	2	no	—
199	C₂×C₄	—	4	4	no	—
200	C₂×C₄	—	4	2	no	—
201	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
202	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
203	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
204	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
205	C₂×C₄	C₂×C₅×Q₁₆	1	1	yes	characteristic
206	C₂×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
207	C₈	—	4	2	no	—
208	C₈	C₅×#103	1	1	yes	characteristic
209	C₈	C₅×#103	1	1	yes	characteristic
210	C₈	C₅×#103	1	1	yes	characteristic
211	C₈	C₅×#103	1	1	yes	characteristic
212	C₈	C₄×C₅×Q₈	1	1	yes	characteristic
213	C₈	C₄×C₅×Q₈	1	1	yes	characteristic
214	C₈	C₅×#124	1	1	yes	characteristic
215	C₈	C₅×C₄⋊Q₈	1	1	yes	characteristic
216	C₅	#13271	1	1	yes	characteristic, direct product factor, Sylow
217	C₂²	—	4	1	no	—
218	C₂²	C₅×#418	1	1	yes	characteristic
219	C₂²	C₅×#476	1	1	yes	characteristic
220	C₂²	C₅×#477	1	1	yes	characteristic
221	C₄	—	2	2	no	—
222	C₄	—	4	1	no	—
223	C₄	—	8	4	no	—
224	C₄	—	8	2	no	—
225	C₄	C₅×#391	1	1	yes	characteristic
226	C₄	C₂×C₅×#103	1	1	yes	characteristic
227	C₄	C₅×#476	1	1	yes	characteristic
228	C₄	C₅×#477	1	1	yes	characteristic
229	C₂	—	2	1	no	—
230	C₂	—	2	1	no	—
231	C₂	—	2	1	no	—
232	C₂	C₅×#1987	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₅	#476
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₂²×C₈	C₂³×C₅
socle	C₂×C₅	#1987
prime radical	C₂³×C₅	AGL(1,ℤ/8)
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×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
10. C₅×#499	C₄	semidirect
11. C₅×#498	C₄	semidirect
21. #13271	C₅	direct (N × H)
36. C₅×#108	Q₈	semidirect
42. C₅×#106	Q₈	semidirect
88. #499	C₄×C₅	semidirect
89. #498	C₄×C₅	semidirect
135. #108	C₅×Q₈	semidirect
141. #106	C₅×Q₈	semidirect
216. C₅	#13271	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×#3160	2
C₂×C₅×#364	2
C₅×#1772	2
C₅×#1389	2
C₅×#1384	2
#13271	5

Subgroups — C5×#13271 #11935991

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#13271 #11935991