Subgroups — C₅×#13124 #11935844

Order 1280.

332 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#2151	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅×#350	C₂	1	1	yes	maximal, characteristic
3	C₅×#2113	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×C₅×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#1764	C₂	1	1	yes	maximal, characteristic
6	C₅×#1379	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	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
12	C₂²×C₅×C₁₆	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×C₂²⋊C₁₆	—	2	2	no	—
20	C₄×C₅×C₁₆	—	2	2	no	—
21	C₅×#341	C₄	1	1	yes	characteristic
22	C₅×#339	C₄	1	1	yes	characteristic
23	#13124	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	C₂×C₅×C₁₆	—	2	1	no	—
42	C₂×C₅×C₁₆	—	2	1	no	—
43	C₂×C₅×C₁₆	—	2	1	no	—
44	C₂×C₅×C₁₆	—	2	1	no	—
45	C₂×C₅×C₁₆	—	2	1	no	—
46	C₂×C₅×C₁₆	—	2	1	no	—
47	C₂×C₅×C₁₆	D₄	1	2	yes	—
48	C₂×C₅×C₁₆	D₄	1	2	yes	—
49	C₅×#104	—	2	2	no	—
50	C₅×#103	C₈	1	2	yes	—
51	C₅×#102	C₈	1	2	yes	—
52	C₅×#97	—	2	1	no	—
53	C₄×C₅×C₈	—	2	1	no	—
54	C₄×C₅×C₈	—	2	1	no	—
55	C₄×C₅×C₈	—	2	1	no	—
56	#2151	C₂×C₅	1	1	yes	characteristic, complemented
57	C₂×#350	C₂×C₅	1	1	yes	characteristic
58	#2113	C₂×C₅	1	1	yes	characteristic
59	C₂×C₄×C₁₆	C₂×C₅	1	1	yes	characteristic, complemented
60	#1764	C₂×C₅	1	1	yes	characteristic
61	#1379	C₂×C₅	1	1	yes	characteristic, complemented
62	#1376	C₂×C₅	1	1	yes	characteristic, complemented
63	C₅×#41	—	2	1	no	—
64	C₅×#41	—	2	1	no	—
65	C₅×#41	—	2	1	no	—
66	C₅×#41	—	2	1	no	—
67	C₂×C₅×Q₈	C₂×C₈	1	1	yes	characteristic
68	C₂×C₅×D₄	—	2	1	no	—
69	C₂×C₅×D₄	C₂×C₈	1	1	yes	characteristic
70	C₂²×C₄×C₅	—	2	1	no	—
71	C₂²×C₄×C₅	—	2	1	no	—
72	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
73	C₅×M₁₆	—	2	1	no	—
74	C₅×M₁₆	—	2	1	no	—
75	C₅×M₁₆	—	2	1	no	—
76	C₅×M₁₆	—	2	1	no	—
77	C₅×M₁₆	—	4	1	no	—
78	C₅×M₁₆	—	4	1	no	—
79	C₂×C₅×C₈	—	2	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	—	4	1	no	—
84	C₂×C₅×C₈	—	4	1	no	—
85	C₂×C₅×C₈	C₂×C₈	1	2	yes	—
86	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
87	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
88	C₂×C₅×C₈	#41	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	C₅×C₄⋊C₄	C₂×C₈	1	2	yes	—
93	C₅×C₂²⋊C₄	—	2	1	no	—
94	C₄²×C₅	—	2	1	no	—
95	C₄²×C₅	—	2	1	no	—
96	C₄²×C₅	M₁₆	1	2	yes	—
97	C₅×C₁₆	—	2	2	no	—
98	C₅×C₁₆	—	2	2	no	—
99	C₅×C₁₆	—	4	1	no	—
100	C₅×C₁₆	—	4	1	no	—
101	C₂×#131	C₂²×C₅	1	1	yes	characteristic
102	C₂×M₃₂	C₂²×C₅	1	1	yes	characteristic
103	C₂×M₃₂	C₂²×C₅	1	1	yes	characteristic
104	C₂²×C₁₆	C₂²×C₅	1	1	yes	characteristic
105	C₂²×C₁₆	C₂²×C₅	1	1	yes	characteristic
106	C₂×#104	C₄×C₅	1	1	yes	characteristic
107	#416	C₄×C₅	1	1	yes	characteristic
108	#405	C₂²×C₅	1	1	yes	characteristic
109	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
110	C₄⋊C₁₆	—	2	2	no	—
111	#350	—	2	2	no	—
112	C₂²⋊C₁₆	—	2	2	no	—
113	C₄×C₁₆	—	2	2	no	—
114	#341	C₄×C₅	1	1	yes	characteristic
115	#339	C₄×C₅	1	1	yes	characteristic
116	C₂³×C₅	—	2	1	no	—
117	C₂³×C₅	#131	1	1	yes	characteristic
118	C₅×Q₈	—	2	1	no	—
119	C₅×Q₈	—	2	1	no	—
120	C₅×D₄	—	2	1	no	—
121	C₅×D₄	—	2	1	no	—
122	C₅×D₄	—	4	1	no	—
123	C₅×D₄	—	4	1	no	—
124	C₂×C₄×C₅	—	2	4	no	—
125	C₂×C₄×C₅	—	2	2	no	—
126	C₂×C₄×C₅	—	2	2	no	—
127	C₂×C₄×C₅	—	2	1	no	—
128	C₂×C₄×C₅	—	4	1	no	—
129	C₂×C₄×C₅	—	4	1	no	—
130	C₂×C₄×C₅	#117	1	1	yes	characteristic
131	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
132	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
133	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
134	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
135	C₂×C₄×C₅	#131	1	1	yes	characteristic
136	C₅×C₈	—	2	2	no	—
137	C₅×C₈	—	4	1	no	—
138	C₅×C₈	—	4	1	no	—
139	C₅×C₈	#121	1	1	yes	characteristic
140	C₅×C₈	#123	1	1	yes	characteristic
141	C₅×C₈	#124	1	1	yes	characteristic
142	C₅×C₈	#126	1	1	yes	characteristic
143	C₂×#41	C₂×C₄×C₅	1	1	yes	characteristic
144	#131	—	2	1	no	—
145	#131	—	2	1	no	—
146	#131	—	2	1	no	—
147	#131	—	2	1	no	—
148	C₂×M₁₆	—	2	1	no	—
149	C₂×M₁₆	C₂×C₄×C₅	1	1	yes	characteristic
150	C₂×M₁₆	C₂×C₄×C₅	1	1	yes	characteristic
151	C₂²×C₈	—	2	1	no	—
152	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
153	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
154	#117	C₂×C₄×C₅	1	1	yes	characteristic
155	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
156	M₃₂	—	2	1	no	—
157	M₃₂	—	2	1	no	—
158	M₃₂	—	2	1	no	—
159	M₃₂	—	2	1	no	—
160	C₂×C₁₆	—	2	1	no	—
161	C₂×C₁₆	—	2	1	no	—
162	C₂×C₁₆	—	2	1	no	—
163	C₂×C₁₆	—	2	1	no	—
164	C₂×C₁₆	—	2	1	no	—
165	C₂×C₁₆	—	2	1	no	—
166	C₂×C₁₆	C₅×D₄	1	2	yes	—
167	C₂×C₁₆	C₅×D₄	1	2	yes	—
168	#104	—	2	2	no	—
169	#103	C₅×C₈	1	2	yes	—
170	#102	C₅×C₈	1	2	yes	—
171	#97	—	2	1	no	—
172	C₄×C₈	—	2	1	no	—
173	C₄×C₈	—	2	1	no	—
174	C₄×C₈	—	2	1	no	—
175	C₂²×C₅	—	2	2	no	—
176	C₂²×C₅	—	4	2	no	—
177	C₂²×C₅	—	4	1	no	—
178	C₂²×C₅	#433	1	1	yes	characteristic
179	C₂²×C₅	C₈×D₄	1	1	yes	characteristic
180	C₂²×C₅	#435	1	1	yes	characteristic
181	C₄×C₅	—	2	4	no	—
182	C₄×C₅	—	4	2	no	—
183	C₄×C₅	—	4	1	no	—
184	C₄×C₅	#388	1	1	yes	characteristic
185	C₄×C₅	#431	1	1	yes	characteristic
186	C₄×C₅	C₈×D₄	1	1	yes	characteristic
187	C₄×C₅	#436	1	1	yes	characteristic
188	#41	—	2	1	no	—
189	#41	—	2	1	no	—
190	#41	—	2	1	no	—
191	#41	—	2	1	no	—
192	C₂×Q₈	C₂×C₅×C₈	1	1	yes	characteristic
193	C₂×D₄	—	2	1	no	—
194	C₂×D₄	C₂×C₅×C₈	1	1	yes	characteristic
195	C₂²×C₄	—	2	1	no	—
196	C₂²×C₄	—	2	1	no	—
197	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
198	M₁₆	—	2	1	no	—
199	M₁₆	—	2	1	no	—
200	M₁₆	—	2	1	no	—
201	M₁₆	—	2	1	no	—
202	M₁₆	—	4	1	no	—
203	M₁₆	—	4	1	no	—
204	C₂×C₈	—	2	1	no	—
205	C₂×C₈	—	2	1	no	—
206	C₂×C₈	—	2	1	no	—
207	C₂×C₈	—	2	1	no	—
208	C₂×C₈	—	4	1	no	—
209	C₂×C₈	—	4	1	no	—
210	C₂×C₈	C₂×C₅×C₈	1	2	yes	—
211	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
212	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
213	C₂×C₈	C₅×#41	1	1	yes	characteristic
214	C₂×C₈	C₅×#41	1	1	yes	characteristic
215	C₂×C₈	C₅×#41	1	1	yes	characteristic
216	C₂×C₈	C₅×#41	1	1	yes	characteristic
217	C₄⋊C₄	C₂×C₅×C₈	1	2	yes	—
218	C₂²⋊C₄	—	2	1	no	—
219	C₄²	—	2	1	no	—
220	C₄²	—	2	1	no	—
221	C₄²	C₅×M₁₆	1	2	yes	—
222	C₁₆	—	2	2	no	—
223	C₁₆	—	2	2	no	—
224	C₁₆	—	4	1	no	—
225	C₁₆	—	4	1	no	—
226	C₂×C₅	—	2	1	no	—
227	C₂×C₅	—	2	1	no	—
228	C₂×C₅	—	8	1	no	—
229	C₂×C₅	#1923	1	1	yes	characteristic
230	C₂×C₅	#2170	1	2	yes	—
231	C₂³	—	2	1	no	—
232	C₂³	C₅×#131	1	1	yes	characteristic
233	Q₈	—	2	1	no	—
234	Q₈	—	2	1	no	—
235	D₄	—	2	1	no	—
236	D₄	—	2	1	no	—
237	D₄	—	4	1	no	—
238	D₄	—	4	1	no	—
239	C₂×C₄	—	2	4	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	4	1	no	—
244	C₂×C₄	—	4	1	no	—
245	C₂×C₄	C₅×#117	1	1	yes	characteristic
246	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
247	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
248	C₂×C₄	C₂²×C₅×C₈	1	1	yes	characteristic
249	C₂×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
250	C₂×C₄	C₅×#131	1	1	yes	characteristic
251	C₈	—	2	2	no	—
252	C₈	—	4	1	no	—
253	C₈	—	4	1	no	—
254	C₈	C₅×#121	1	1	yes	characteristic
255	C₈	C₅×#123	1	1	yes	characteristic
256	C₈	C₅×#124	1	1	yes	characteristic
257	C₈	C₅×#126	1	1	yes	characteristic
258	C₅	#13124	1	1	yes	characteristic, direct product factor, Sylow
259	C₂²	—	2	2	no	—
260	C₂²	—	4	2	no	—
261	C₂²	—	4	1	no	—
262	C₂²	C₅×#433	1	1	yes	characteristic
263	C₂²	C₅×C₈×D₄	1	1	yes	characteristic
264	C₂²	C₅×#435	1	1	yes	characteristic
265	C₄	—	2	4	no	—
266	C₄	—	4	2	no	—
267	C₄	—	4	1	no	—
268	C₄	C₅×#388	1	1	yes	characteristic
269	C₄	C₅×#431	1	1	yes	characteristic
270	C₄	C₅×C₈×D₄	1	1	yes	characteristic
271	C₄	C₅×#436	1	1	yes	characteristic
272	C₂	—	2	1	no	—
273	C₂	—	2	1	no	—
274	C₂	—	8	1	no	—
275	C₂	C₅×#1923	1	1	yes	characteristic
276	C₂	C₅×#2170	1	2	yes	—

	subgroup	G/H
center	C₂×C₅×C₈	C₂×D₄
commutator	C₂×C₄	C₂²×C₅×C₈
Frattini, cosocle	C₂²×C₈	C₂³×C₅
socle	C₂²×C₅	C₈×D₄
prime radical	C₂×C₅×#41	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅×C₈ ◁ C₂²×C₅×C₁₆ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅×#2151	C₂	semidirect
4. C₂×C₄×C₅×C₁₆	C₂	semidirect
6. C₅×#1379	C₂	semidirect
7. C₅×#1376	C₂	semidirect
23. #13124	C₅	direct (N × H)
56. #2151	C₂×C₅	semidirect
59. C₂×C₄×C₁₆	C₂×C₅	semidirect
61. #1379	C₂×C₅	semidirect
62. #1376	C₂×C₅	semidirect
258. C₅	#13124	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×#2151	2
C₂×C₅×#350	2
C₅×#2113	2
C₂×C₄×C₅×C₁₆	2
C₅×#1764	2
C₅×#1379	2
C₅×#1376	2
#13124	5

Subgroups — C5×#13124 #11935844

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#13124 #11935844