Subgroups — C₅×#13129 #11935849

Order 1280.

306 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#2150	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#2117	C₂	1	1	yes	maximal, characteristic
3	C₅×#2115	C₂	1	1	yes	maximal, characteristic
4	C₅×#2110	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#1764	C₂	1	1	yes	maximal, characteristic
6	C₅×#1378	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#1376	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅×#131	C₂²	1	1	yes	characteristic
9	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
10	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
11	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
12	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
13	C₂×C₅×#104	C₄	1	1	yes	characteristic
14	C₅×#416	C₄	1	1	yes	characteristic
15	C₅×#405	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
17	C₅×C₄⋊C₁₆	—	2	2	no	—
18	C₅×#350	—	2	2	no	—
19	C₅×#349	—	2	2	no	—
20	C₅×#347	—	2	2	no	—
21	C₅×#341	C₄	1	1	yes	characteristic
22	C₅×#339	C₄	1	1	yes	characteristic
23	#13129	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
24	C₂×C₅×#41	C₂×C₄	1	1	yes	characteristic
25	C₅×#131	—	2	1	no	—
26	C₅×#131	—	2	1	no	—
27	C₅×#131	—	2	1	no	—
28	C₅×#131	—	2	1	no	—
29	C₂×C₅×M₁₆	—	2	1	no	—
30	C₂×C₅×M₁₆	C₂×C₄	1	1	yes	characteristic
31	C₂×C₅×M₁₆	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₅×C₈	—	2	1	no	—
33	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
35	C₅×#117	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
37	C₅×M₃₂	—	2	1	no	—
38	C₅×M₃₂	—	2	1	no	—
39	C₅×M₃₂	—	2	1	no	—
40	C₅×M₃₂	—	2	1	no	—
41	C₅×M₃₂	—	2	1	no	—
42	C₅×M₃₂	—	2	1	no	—
43	C₅×M₃₂	—	2	1	no	—
44	C₅×M₃₂	D₄	1	2	yes	—
45	C₂×C₅×C₁₆	—	2	1	no	—
46	C₂×C₅×C₁₆	—	2	1	no	—
47	C₂×C₅×C₁₆	—	2	1	no	—
48	C₂×C₅×C₁₆	D₄	1	2	yes	—
49	C₅×#104	—	2	1	no	—
50	C₅×#104	—	2	1	no	—
51	C₅×#103	—	2	1	no	—
52	C₅×#102	—	2	1	no	—
53	C₅×#97	—	2	1	no	—
54	C₄×C₅×C₈	—	2	1	no	—
55	C₄×C₅×C₈	—	2	1	no	—
56	C₄×C₅×C₈	—	2	1	no	—
57	#2150	C₂×C₅	1	1	yes	characteristic, complemented
58	#2117	C₂×C₅	1	1	yes	characteristic
59	#2115	C₂×C₅	1	1	yes	characteristic
60	#2110	C₂×C₅	1	1	yes	characteristic, complemented
61	#1764	C₂×C₅	1	1	yes	characteristic
62	#1378	C₂×C₅	1	1	yes	characteristic, complemented
63	#1376	C₂×C₅	1	1	yes	characteristic, complemented
64	C₅×#41	—	2	1	no	—
65	C₅×#41	—	2	1	no	—
66	C₅×#41	—	2	1	no	—
67	C₅×#41	—	2	1	no	—
68	C₂×C₅×Q₈	M₁₆	1	1	yes	characteristic
69	C₂×C₅×D₄	—	2	1	no	—
70	C₂×C₅×D₄	M₁₆	1	1	yes	characteristic
71	C₂²×C₄×C₅	—	2	1	no	—
72	C₂²×C₄×C₅	—	2	1	no	—
73	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
74	C₅×M₁₆	—	2	1	no	—
75	C₅×M₁₆	—	2	1	no	—
76	C₅×M₁₆	—	2	1	no	—
77	C₅×M₁₆	—	2	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	—	2	1	no	—
82	C₂×C₅×C₈	—	2	1	no	—
83	C₂×C₅×C₈	—	2	1	no	—
84	C₂×C₅×C₈	—	2	1	no	—
85	C₂×C₅×C₈	—	4	1	no	—
86	C₂×C₅×C₈	—	4	1	no	—
87	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
88	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
89	C₂×C₅×C₈	#41	1	1	yes	characteristic
90	C₂×C₅×C₈	#41	1	1	yes	characteristic
91	C₂×C₅×C₈	#41	1	1	yes	characteristic
92	C₂×C₅×C₈	#41	1	1	yes	characteristic
93	C₅×C₄⋊C₄	—	2	1	no	—
94	C₅×C₂²⋊C₄	—	2	1	no	—
95	C₄²×C₅	—	2	1	no	—
96	C₄²×C₅	—	2	1	no	—
97	C₄²×C₅	M₁₆	1	1	yes	characteristic
98	C₄²×C₅	M₁₆	1	1	yes	characteristic
99	C₅×C₁₆	—	2	2	no	—
100	C₅×C₁₆	—	2	2	no	—
101	C₅×C₁₆	—	4	1	no	—
102	C₅×C₁₆	—	4	1	no	—
103	C₂×#131	C₂²×C₅	1	1	yes	characteristic
104	C₂×M₃₂	C₂²×C₅	1	1	yes	characteristic
105	C₂×M₃₂	C₂²×C₅	1	1	yes	characteristic
106	C₂×M₃₂	C₂²×C₅	1	1	yes	characteristic
107	C₂×M₃₂	C₂²×C₅	1	1	yes	characteristic
108	C₂×#104	C₄×C₅	1	1	yes	characteristic
109	#416	C₄×C₅	1	1	yes	characteristic
110	#405	C₂²×C₅	1	1	yes	characteristic
111	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
112	C₄⋊C₁₆	—	2	2	no	—
113	#350	—	2	2	no	—
114	#349	—	2	2	no	—
115	#347	—	2	2	no	—
116	#341	C₄×C₅	1	1	yes	characteristic
117	#339	C₄×C₅	1	1	yes	characteristic
118	C₂³×C₅	—	2	1	no	—
119	C₂³×C₅	#131	1	1	yes	characteristic
120	C₅×Q₈	—	2	1	no	—
121	C₅×Q₈	—	2	1	no	—
122	C₅×D₄	—	2	1	no	—
123	C₅×D₄	—	2	1	no	—
124	C₅×D₄	—	4	1	no	—
125	C₅×D₄	—	4	1	no	—
126	C₂×C₄×C₅	—	2	1	no	—
127	C₂×C₄×C₅	—	2	1	no	—
128	C₂×C₄×C₅	—	2	1	no	—
129	C₂×C₄×C₅	—	4	2	no	—
130	C₂×C₄×C₅	—	4	1	no	—
131	C₂×C₄×C₅	—	4	1	no	—
132	C₂×C₄×C₅	—	4	1	no	—
133	C₂×C₄×C₅	#117	1	1	yes	characteristic
134	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
135	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
136	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
137	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
138	C₂×C₄×C₅	#131	1	1	yes	characteristic
139	C₅×C₈	—	4	1	no	—
140	C₅×C₈	—	4	1	no	—
141	C₅×C₈	—	4	1	no	—
142	C₅×C₈	#121	1	1	yes	characteristic
143	C₅×C₈	#123	1	1	yes	characteristic
144	C₅×C₈	#124	1	1	yes	characteristic
145	C₅×C₈	#126	1	1	yes	characteristic
146	C₂×#41	C₂×C₄×C₅	1	1	yes	characteristic
147	#131	—	2	1	no	—
148	#131	—	2	1	no	—
149	#131	—	2	1	no	—
150	#131	—	2	1	no	—
151	C₂×M₁₆	—	2	1	no	—
152	C₂×M₁₆	C₂×C₄×C₅	1	1	yes	characteristic
153	C₂×M₁₆	C₂×C₄×C₅	1	1	yes	characteristic
154	C₂²×C₈	—	2	1	no	—
155	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
156	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
157	#117	C₂×C₄×C₅	1	1	yes	characteristic
158	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
159	M₃₂	—	2	1	no	—
160	M₃₂	—	2	1	no	—
161	M₃₂	—	2	1	no	—
162	M₃₂	—	2	1	no	—
163	M₃₂	—	2	1	no	—
164	M₃₂	—	2	1	no	—
165	M₃₂	—	2	1	no	—
166	M₃₂	C₅×D₄	1	2	yes	—
167	C₂×C₁₆	—	2	1	no	—
168	C₂×C₁₆	—	2	1	no	—
169	C₂×C₁₆	—	2	1	no	—
170	C₂×C₁₆	C₅×D₄	1	2	yes	—
171	#104	—	2	1	no	—
172	#104	—	2	1	no	—
173	#103	—	2	1	no	—
174	#102	—	2	1	no	—
175	#97	—	2	1	no	—
176	C₄×C₈	—	2	1	no	—
177	C₄×C₈	—	2	1	no	—
178	C₄×C₈	—	2	1	no	—
179	C₂²×C₅	—	4	1	no	—
180	C₂²×C₅	—	4	1	no	—
181	C₂²×C₅	—	8	1	no	—
182	C₂²×C₅	#433	1	1	yes	characteristic
183	C₂²×C₅	#435	1	1	yes	characteristic
184	C₂²×C₅	#435	1	1	yes	characteristic
185	C₄×C₅	—	2	2	no	—
186	C₄×C₅	—	4	1	no	—
187	C₄×C₅	—	4	1	no	—
188	C₄×C₅	—	8	1	no	—
189	C₄×C₅	#388	1	1	yes	characteristic
190	C₄×C₅	#432	1	1	yes	characteristic
191	C₄×C₅	#435	1	1	yes	characteristic
192	C₄×C₅	#436	1	1	yes	characteristic
193	#41	—	2	1	no	—
194	#41	—	2	1	no	—
195	#41	—	2	1	no	—
196	#41	—	2	1	no	—
197	C₂×Q₈	C₅×M₁₆	1	1	yes	characteristic
198	C₂×D₄	—	2	1	no	—
199	C₂×D₄	C₅×M₁₆	1	1	yes	characteristic
200	C₂²×C₄	—	2	1	no	—
201	C₂²×C₄	—	2	1	no	—
202	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
203	M₁₆	—	2	1	no	—
204	M₁₆	—	2	1	no	—
205	M₁₆	—	2	1	no	—
206	M₁₆	—	2	1	no	—
207	M₁₆	—	4	1	no	—
208	M₁₆	—	4	1	no	—
209	C₂×C₈	—	2	1	no	—
210	C₂×C₈	—	2	1	no	—
211	C₂×C₈	—	2	1	no	—
212	C₂×C₈	—	2	1	no	—
213	C₂×C₈	—	2	1	no	—
214	C₂×C₈	—	4	1	no	—
215	C₂×C₈	—	4	1	no	—
216	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
217	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
218	C₂×C₈	C₅×#41	1	1	yes	characteristic
219	C₂×C₈	C₅×#41	1	1	yes	characteristic
220	C₂×C₈	C₅×#41	1	1	yes	characteristic
221	C₂×C₈	C₅×#41	1	1	yes	characteristic
222	C₄⋊C₄	—	2	1	no	—
223	C₂²⋊C₄	—	2	1	no	—
224	C₄²	—	2	1	no	—
225	C₄²	—	2	1	no	—
226	C₄²	C₅×M₁₆	1	1	yes	characteristic
227	C₄²	C₅×M₁₆	1	1	yes	characteristic
228	C₁₆	—	2	2	no	—
229	C₁₆	—	2	2	no	—
230	C₁₆	—	4	1	no	—
231	C₁₆	—	4	1	no	—
232	C₂×C₅	—	2	1	no	—
233	C₂×C₅	—	2	1	no	—
234	C₂×C₅	—	2	1	no	—
235	C₂×C₅	—	8	1	no	—
236	C₂×C₅	#1924	1	1	yes	characteristic
237	C₂³	—	2	1	no	—
238	C₂³	C₅×#131	1	1	yes	characteristic
239	Q₈	—	2	1	no	—
240	Q₈	—	2	1	no	—
241	D₄	—	2	1	no	—
242	D₄	—	2	1	no	—
243	D₄	—	4	1	no	—
244	D₄	—	4	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	4	2	no	—
249	C₂×C₄	—	4	1	no	—
250	C₂×C₄	—	4	1	no	—
251	C₂×C₄	—	4	1	no	—
252	C₂×C₄	C₅×#117	1	1	yes	characteristic
253	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
254	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
255	C₂×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
256	C₂×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
257	C₂×C₄	C₅×#131	1	1	yes	characteristic
258	C₈	—	4	1	no	—
259	C₈	—	4	1	no	—
260	C₈	—	4	1	no	—
261	C₈	C₅×#121	1	1	yes	characteristic
262	C₈	C₅×#123	1	1	yes	characteristic
263	C₈	C₅×#124	1	1	yes	characteristic
264	C₈	C₅×#126	1	1	yes	characteristic
265	C₅	#13129	1	1	yes	characteristic, direct product factor, Sylow
266	C₂²	—	4	1	no	—
267	C₂²	—	4	1	no	—
268	C₂²	—	8	1	no	—
269	C₂²	C₅×#433	1	1	yes	characteristic
270	C₂²	C₅×#435	1	1	yes	characteristic
271	C₂²	C₅×#435	1	1	yes	characteristic
272	C₄	—	2	2	no	—
273	C₄	—	4	1	no	—
274	C₄	—	4	1	no	—
275	C₄	—	8	1	no	—
276	C₄	C₅×#388	1	1	yes	characteristic
277	C₄	C₅×#432	1	1	yes	characteristic
278	C₄	C₅×#435	1	1	yes	characteristic
279	C₄	C₅×#436	1	1	yes	characteristic
280	C₂	—	2	1	no	—
281	C₂	—	2	1	no	—
282	C₂	—	2	1	no	—
283	C₂	—	8	1	no	—
284	C₂	C₅×#1924	1	1	yes	characteristic

	subgroup	G/H
center	C₅×C₈	#124
commutator	C₂²×C₄	C₂²×C₄×C₅
Frattini, cosocle	C₂²×C₈	C₂³×C₅
socle	C₂×C₅	#1924
prime radical	C₂×C₅×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₅×C₈ ◁ C₂²×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅×#2150	C₂	semidirect
4. C₅×#2110	C₂	semidirect
6. C₅×#1378	C₂	semidirect
7. C₅×#1376	C₂	semidirect
23. #13129	C₅	direct (N × H)
57. #2150	C₂×C₅	semidirect
60. #2110	C₂×C₅	semidirect
62. #1378	C₂×C₅	semidirect
63. #1376	C₂×C₅	semidirect
265. C₅	#13129	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×#2150	2
C₅×#2117	2
C₅×#2115	2
C₅×#2110	2
C₅×#1764	2
C₅×#1378	2
C₅×#1376	2
#13129	5

Subgroups — C5×#13129 #11935849

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#13129 #11935849