Subgroups — C₅×#13158 #11935878

Order 1280.

316 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#361	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#361	C₂	1	1	yes	maximal, characteristic
3	C₅×#2110	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#1978	C₂	1	1	yes	maximal, characteristic
5	C₅×#1953	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#1388	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#1388	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₅×D₈	C₂²	1	1	yes	characteristic
9	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
10	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
11	C₅×#430	—	2	1	no	—
12	C₂×C₅×#108	C₂²	1	1	yes	characteristic
13	C₂×C₅×#108	C₂²	1	1	yes	characteristic
14	C₂×C₅×#108	C₂²	1	1	yes	characteristic
15	C₅×#418	—	2	1	no	—
16	C₂×C₅×#100	—	2	1	no	—
17	C₅×#405	C₂²	1	1	yes	characteristic
18	C₅×#361	—	2	2	no	—
19	C₅×#361	C₄	1	4	yes	complemented
20	C₅×#347	—	2	2	no	—
21	C₅×#344	—	2	1	no	—
22	#13158	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
23	C₂²×C₅×D₄	—	2	1	no	—
24	C₂×C₅×D₈	—	2	1	no	—
25	C₂×C₅×D₈	—	2	1	no	—
26	C₂×C₅×D₈	—	2	1	no	—
27	C₂×C₅×D₈	—	2	1	no	—
28	C₂×C₅×D₈	—	2	1	no	—
29	C₂×C₅×D₈	C₂×C₄	1	2	yes	—
30	C₂×C₅×M₁₆	—	2	1	no	—
31	C₂×C₅×M₁₆	—	2	1	no	—
32	C₂×C₅×M₁₆	—	2	1	no	—
33	C₂×C₅×M₁₆	D₄	1	1	yes	characteristic
34	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
35	C₅×#117	D₄	1	1	yes	characteristic
36	C₅×M₃₂	—	2	1	no	—
37	C₅×M₃₂	—	2	1	no	—
38	C₅×M₃₂	C₂×C₄	1	2	yes	—
39	C₅×M₃₂	D₄	1	2	yes	—
40	C₂×C₅×C₁₆	—	2	1	no	—
41	C₂×C₅×C₁₆	—	2	1	no	—
42	C₅×#108	—	2	1	no	—
43	C₅×#108	—	2	1	no	—
44	C₅×#108	—	2	1	no	—
45	C₅×#108	—	2	1	no	—
46	C₅×#108	—	2	1	no	—
47	C₅×#108	—	4	2	no	—
48	C₅×#108	C₂×C₄	1	2	yes	—
49	C₅×#102	—	4	2	no	—
50	C₅×#100	—	4	2	no	—
51	C₅×#97	—	2	1	no	—
52	C₄×C₅×C₈	—	2	1	no	—
53	C₂×#361	C₂×C₅	1	1	yes	characteristic
54	C₂×#361	C₂×C₅	1	1	yes	characteristic
55	#2110	C₂×C₅	1	1	yes	characteristic, complemented
56	#1978	C₂×C₅	1	1	yes	characteristic
57	#1953	C₂×C₅	1	1	yes	characteristic, complemented
58	#1388	C₂×C₅	1	1	yes	characteristic, complemented
59	#1388	C₂×C₅	1	1	yes	characteristic, complemented
60	C₂⁴×C₅	—	4	1	no	—
61	C₂×C₅×D₄	—	2	2	no	—
62	C₂×C₅×D₄	—	4	1	no	—
63	C₂×C₅×D₄	—	4	1	no	—
64	C₂×C₅×D₄	—	4	1	no	—
65	C₂×C₅×D₄	—	4	1	no	—
66	C₂×C₅×D₄	—	4	1	no	—
67	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
68	C₅×D₈	—	2	2	no	—
69	C₅×D₈	—	2	2	no	—
70	C₅×D₈	—	4	1	no	—
71	C₅×D₈	—	4	1	no	—
72	C₅×M₁₆	—	2	2	no	—
73	C₅×M₁₆	—	2	1	no	—
74	C₅×M₁₆	—	2	1	no	—
75	C₅×M₁₆	—	4	1	no	—
76	C₅×M₁₆	—	4	1	no	—
77	C₅×M₁₆	—	4	1	no	—
78	C₅×M₁₆	—	4	1	no	—
79	C₅×M₁₆	—	4	1	no	—
80	C₂×C₅×C₈	—	2	1	no	—
81	C₂×C₅×C₈	—	4	1	no	—
82	C₂×C₅×C₈	—	4	1	no	—
83	C₂×C₅×C₈	—	4	1	no	—
84	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
86	C₂×C₅×C₈	#41	1	1	yes	characteristic
87	C₂×C₅×C₈	#41	1	1	yes	characteristic
88	C₂×C₅×C₈	#41	1	1	yes	characteristic
89	C₂×C₅×C₈	#41	1	1	yes	characteristic
90	C₅×C₄⋊C₄	—	2	1	no	—
91	C₅×C₂²⋊C₄	—	2	1	no	—
92	C₄²×C₅	—	2	1	no	—
93	C₅×C₁₆	—	2	2	no	—
94	C₅×C₁₆	—	2	2	no	—
95	C₂²×D₈	C₂²×C₅	1	1	yes	characteristic
96	C₂×M₃₂	C₂²×C₅	1	1	yes	characteristic
97	C₂×M₃₂	C₂²×C₅	1	1	yes	characteristic
98	#430	—	2	1	no	—
99	C₂×#108	C₂²×C₅	1	1	yes	characteristic
100	C₂×#108	C₂²×C₅	1	1	yes	characteristic
101	C₂×#108	C₂²×C₅	1	1	yes	characteristic
102	#418	—	2	1	no	—
103	C₂×#100	—	2	1	no	—
104	#405	C₂²×C₅	1	1	yes	characteristic
105	#361	—	2	2	no	—
106	#361	C₄×C₅	1	4	yes	complemented
107	#347	—	2	2	no	—
108	#344	—	2	1	no	—
109	C₂³×C₅	—	4	2	no	—
110	C₂³×C₅	—	8	1	no	—
111	C₂³×C₅	—	8	1	no	—
112	C₂³×C₅	—	16	2	no	—
113	C₂³×C₅	#137	1	1	yes	characteristic
114	C₅×D₄	—	4	2	no	—
115	C₅×D₄	—	4	2	no	—
116	C₅×D₄	—	8	1	no	—
117	C₅×D₄	—	8	1	no	—
118	C₂×C₄×C₅	—	4	1	no	—
119	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
120	C₂×C₄×C₅	#121	1	1	yes	characteristic
121	C₂×C₄×C₅	#124	1	1	yes	characteristic
122	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
123	C₂×C₄×C₅	AGL(1,ℤ/8)	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	—	4	1	no	—
127	C₅×C₈	—	8	1	no	—
128	C₅×C₈	—	8	1	no	—
129	C₅×C₈	#117	1	1	yes	characteristic
130	C₅×C₈	C₄×D₄	1	1	yes	characteristic
131	C₅×C₈	#123	1	1	yes	characteristic
132	C₅×C₈	#126	1	1	yes	characteristic
133	C₂²×D₄	—	2	1	no	—
134	C₂×D₈	—	2	1	no	—
135	C₂×D₈	—	2	1	no	—
136	C₂×D₈	—	2	1	no	—
137	C₂×D₈	—	2	1	no	—
138	C₂×D₈	—	2	1	no	—
139	C₂×D₈	C₂×C₄×C₅	1	2	yes	—
140	C₂×M₁₆	—	2	1	no	—
141	C₂×M₁₆	—	2	1	no	—
142	C₂×M₁₆	—	2	1	no	—
143	C₂×M₁₆	C₅×D₄	1	1	yes	characteristic
144	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
145	#117	C₅×D₄	1	1	yes	characteristic
146	M₃₂	—	2	1	no	—
147	M₃₂	—	2	1	no	—
148	M₃₂	C₂×C₄×C₅	1	2	yes	—
149	M₃₂	C₅×D₄	1	2	yes	—
150	C₂×C₁₆	—	2	1	no	—
151	C₂×C₁₆	—	2	1	no	—
152	#108	—	2	1	no	—
153	#108	—	2	1	no	—
154	#108	—	2	1	no	—
155	#108	—	2	1	no	—
156	#108	—	2	1	no	—
157	#108	—	4	2	no	—
158	#108	C₂×C₄×C₅	1	2	yes	—
159	#102	—	4	2	no	—
160	#100	—	4	2	no	—
161	#97	—	2	1	no	—
162	C₄×C₈	—	2	1	no	—
163	C₂²×C₅	—	4	1	no	—
164	C₂²×C₅	—	8	2	no	—
165	C₂²×C₅	—	16	2	no	—
166	C₂²×C₅	—	16	2	no	—
167	C₂²×C₅	—	16	2	no	—
168	C₂²×C₅	#440	1	1	yes	characteristic
169	C₂²×C₅	#468	1	1	yes	characteristic
170	C₂²×C₅	#489	1	1	yes	characteristic
171	C₄×C₅	—	8	1	no	—
172	C₄×C₅	#388	1	1	yes	characteristic
173	C₄×C₅	#442	1	1	yes	characteristic
174	C₄×C₅	#469	1	1	yes	characteristic
175	C₄×C₅	#490	1	1	yes	characteristic
176	C₂⁴	—	4	1	no	—
177	C₂×D₄	—	2	2	no	—
178	C₂×D₄	—	4	1	no	—
179	C₂×D₄	—	4	1	no	—
180	C₂×D₄	—	4	1	no	—
181	C₂×D₄	—	4	1	no	—
182	C₂×D₄	—	4	1	no	—
183	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
184	D₈	—	2	2	no	—
185	D₈	—	2	2	no	—
186	D₈	—	4	1	no	—
187	D₈	—	4	1	no	—
188	M₁₆	—	2	2	no	—
189	M₁₆	—	2	1	no	—
190	M₁₆	—	2	1	no	—
191	M₁₆	—	4	1	no	—
192	M₁₆	—	4	1	no	—
193	M₁₆	—	4	1	no	—
194	M₁₆	—	4	1	no	—
195	M₁₆	—	4	1	no	—
196	C₂×C₈	—	2	1	no	—
197	C₂×C₈	—	4	1	no	—
198	C₂×C₈	—	4	1	no	—
199	C₂×C₈	—	4	1	no	—
200	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
201	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
202	C₂×C₈	C₅×#41	1	1	yes	characteristic
203	C₂×C₈	C₅×#41	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	1	no	—
207	C₂²⋊C₄	—	2	1	no	—
208	C₄²	—	2	1	no	—
209	C₁₆	—	2	2	no	—
210	C₁₆	—	2	2	no	—
211	C₂×C₅	—	2	1	no	—
212	C₂×C₅	—	2	1	no	—
213	C₂×C₅	—	2	1	no	—
214	C₂×C₅	—	16	2	no	—
215	C₂×C₅	#1936	1	1	yes	characteristic
216	C₂³	—	4	2	no	—
217	C₂³	—	8	1	no	—
218	C₂³	—	8	1	no	—
219	C₂³	—	16	2	no	—
220	C₂³	C₅×#137	1	1	yes	characteristic
221	D₄	—	4	2	no	—
222	D₄	—	4	2	no	—
223	D₄	—	8	1	no	—
224	D₄	—	8	1	no	—
225	C₂×C₄	—	4	1	no	—
226	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
227	C₂×C₄	C₅×#121	1	1	yes	characteristic
228	C₂×C₄	C₅×#124	1	1	yes	characteristic
229	C₂×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
230	C₂×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
231	C₂×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
232	C₈	—	4	2	no	—
233	C₈	—	4	1	no	—
234	C₈	—	8	1	no	—
235	C₈	—	8	1	no	—
236	C₈	C₅×#117	1	1	yes	characteristic
237	C₈	C₄×C₅×D₄	1	1	yes	characteristic
238	C₈	C₅×#123	1	1	yes	characteristic
239	C₈	C₅×#126	1	1	yes	characteristic
240	C₅	#13158	1	1	yes	characteristic, direct product factor, Sylow
241	C₂²	—	4	1	no	—
242	C₂²	—	8	2	no	—
243	C₂²	—	16	2	no	—
244	C₂²	—	16	2	no	—
245	C₂²	—	16	2	no	—
246	C₂²	C₅×#440	1	1	yes	characteristic
247	C₂²	C₅×#468	1	1	yes	characteristic
248	C₂²	C₅×#489	1	1	yes	characteristic
249	C₄	—	8	1	no	—
250	C₄	C₅×#388	1	1	yes	characteristic
251	C₄	C₅×#442	1	1	yes	characteristic
252	C₄	C₅×#469	1	1	yes	characteristic
253	C₄	C₅×#490	1	1	yes	characteristic
254	C₂	—	2	1	no	—
255	C₂	—	2	1	no	—
256	C₂	—	2	1	no	—
257	C₂	—	16	2	no	—
258	C₂	C₅×#1936	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₅	#1936
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₂²×C₈	C₂³×C₅
prime radical	C₂²×C₅×D₈	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂²×C₄×C₅ ◁ C₅×#405 ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅×#2110	C₂	semidirect
5. C₅×#1953	C₂	semidirect
6. C₅×#1388	C₂	semidirect
7. C₅×#1388	C₂	semidirect
19. C₅×#361	C₄	semidirect
22. #13158	C₅	direct (N × H)
55. #2110	C₂×C₅	semidirect
57. #1953	C₂×C₅	semidirect
58. #1388	C₂×C₅	semidirect
59. #1388	C₂×C₅	semidirect
106. #361	C₄×C₅	semidirect
240. C₅	#13158	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×#361	2
C₅×#2110	2
C₅×#1978	2
C₅×#1953	2
C₅×#1388	2
#13158	5

Subgroups — C5×#13158 #11935878

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#13158 #11935878