Subgroups — C₅×#13123 #11935843

Order 1280.

312 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#2153	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#2117	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#350	C₂	1	1	yes	maximal, characteristic
4	C₅×#2108	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#1757	C₂	1	1	yes	maximal, characteristic
6	C₅×#1379	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#1378	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	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
13	C₅×#413	C₄	1	1	yes	characteristic
14	C₅×#410	C₄	1	1	yes	characteristic
15	C₅×#405	C₂²	1	1	yes	characteristic
16	C₅×#405	C₂²	1	1	yes	characteristic
17	C₅×#364	—	2	2	no	—
18	C₅×#350	—	2	2	no	—
19	C₅×#350	—	2	1	no	—
20	C₅×#350	—	2	1	no	—
21	C₅×#346	—	2	1	no	—
22	C₄×C₅×C₁₆	—	2	1	no	—
23	C₅×#344	C₄	1	1	yes	characteristic
24	C₅×#344	C₄	1	1	yes	characteristic
25	#13123	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
26	C₂×C₅×#41	C₂×C₄	1	1	yes	characteristic
27	C₅×#131	—	2	1	no	—
28	C₅×#131	—	2	1	no	—
29	C₅×#131	—	2	1	no	—
30	C₅×#131	—	2	1	no	—
31	C₂×C₅×M₁₆	—	2	1	no	—
32	C₂×C₅×M₁₆	C₂×C₄	1	1	yes	characteristic
33	C₂×C₅×M₁₆	C₂×C₄	1	1	yes	characteristic
34	C₂×C₅×M₁₆	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₅×C₈	—	2	1	no	—
36	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
37	C₅×#117	C₂×C₄	1	1	yes	characteristic
38	C₅×#117	C₂×C₄	1	1	yes	characteristic
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	—	2	1	no	—
49	C₂×C₅×C₁₆	—	2	1	no	—
50	C₂×C₅×C₁₆	D₄	1	1	yes	characteristic
51	C₂×C₅×C₁₆	D₄	1	1	yes	characteristic
52	C₅×#101	C₈	1	2	yes	—
53	C₅×#100	C₈	1	2	yes	—
54	C₅×#99	—	2	2	no	—
55	C₅×#97	—	2	1	no	—
56	C₅×#97	—	2	1	no	—
57	C₄×C₅×C₈	—	2	1	no	—
58	C₄×C₅×C₈	—	2	1	no	—
59	#2153	C₂×C₅	1	1	yes	characteristic, complemented
60	#2117	C₂×C₅	1	1	yes	characteristic
61	C₂×#350	C₂×C₅	1	1	yes	characteristic
62	#2108	C₂×C₅	1	1	yes	characteristic, complemented
63	#1757	C₂×C₅	1	1	yes	characteristic
64	#1379	C₂×C₅	1	1	yes	characteristic, complemented
65	#1378	C₂×C₅	1	1	yes	characteristic, complemented
66	C₅×#41	—	2	1	no	—
67	C₅×#41	—	2	1	no	—
68	C₅×#41	—	2	1	no	—
69	C₅×#41	—	2	1	no	—
70	C₂×C₅×Q₈	C₂×C₈	1	1	yes	characteristic
71	C₂×C₅×D₄	—	2	1	no	—
72	C₂×C₅×D₄	C₂×C₈	1	1	yes	characteristic
73	C₂²×C₄×C₅	—	2	1	no	—
74	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
75	C₅×M₁₆	—	2	1	no	—
76	C₅×M₁₆	—	2	1	no	—
77	C₅×M₁₆	—	2	1	no	—
78	C₅×M₁₆	—	2	1	no	—
79	C₅×M₁₆	—	4	1	no	—
80	C₅×M₁₆	—	4	1	no	—
81	C₅×M₁₆	C₂×C₈	1	2	yes	—
82	C₅×M₁₆	C₂×C₈	1	2	yes	—
83	C₂×C₅×C₈	—	2	1	no	—
84	C₂×C₅×C₈	—	2	1	no	—
85	C₂×C₅×C₈	—	2	1	no	—
86	C₂×C₅×C₈	—	4	1	no	—
87	C₂×C₅×C₈	—	4	1	no	—
88	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
89	C₂×C₅×C₈	C₂×D₄	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	C₂×C₅×C₈	#41	1	1	yes	characteristic
94	C₅×C₄⋊C₄	—	2	1	no	—
95	C₅×C₄⋊C₄	—	2	1	no	—
96	C₅×C₂²⋊C₄	—	2	1	no	—
97	C₅×C₂²⋊C₄	M₁₆	1	2	yes	—
98	C₄²×C₅	—	2	1	no	—
99	C₄²×C₅	—	2	1	no	—
100	C₅×C₁₆	—	2	2	no	—
101	C₅×C₁₆	—	2	1	no	—
102	C₅×C₁₆	—	2	1	no	—
103	C₅×C₁₆	—	4	1	no	—
104	C₅×C₁₆	—	4	1	no	—
105	C₂×#131	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×M₃₂	C₂²×C₅	1	1	yes	characteristic
109	C₂²×C₁₆	C₂²×C₅	1	1	yes	characteristic
110	#413	C₄×C₅	1	1	yes	characteristic
111	#410	C₄×C₅	1	1	yes	characteristic
112	#405	C₂²×C₅	1	1	yes	characteristic
113	#405	C₂²×C₅	1	1	yes	characteristic
114	#364	—	2	2	no	—
115	#350	—	2	2	no	—
116	#350	—	2	1	no	—
117	#350	—	2	1	no	—
118	#346	—	2	1	no	—
119	C₄×C₁₆	—	2	1	no	—
120	#344	C₄×C₅	1	1	yes	characteristic
121	#344	C₄×C₅	1	1	yes	characteristic
122	C₂³×C₅	—	2	1	no	—
123	C₂³×C₅	C₂×M₁₆	1	1	yes	characteristic
124	C₅×Q₈	—	2	1	no	—
125	C₅×Q₈	—	2	1	no	—
126	C₅×D₄	—	2	1	no	—
127	C₅×D₄	—	2	1	no	—
128	C₅×D₄	—	4	1	no	—
129	C₅×D₄	—	4	1	no	—
130	C₂×C₄×C₅	—	2	1	no	—
131	C₂×C₄×C₅	—	2	2	no	—
132	C₂×C₄×C₅	—	4	1	no	—
133	C₂×C₄×C₅	—	4	1	no	—
134	C₂×C₄×C₅	—	4	1	no	—
135	C₂×C₄×C₅	#117	1	1	yes	characteristic
136	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
137	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
138	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
139	C₂×C₄×C₅	#131	1	1	yes	characteristic
140	C₂×C₄×C₅	#131	1	1	yes	characteristic
141	C₅×C₈	—	2	2	no	—
142	C₅×C₈	—	2	2	no	—
143	C₅×C₈	—	4	1	no	—
144	C₅×C₈	#121	1	1	yes	characteristic
145	C₅×C₈	#123	1	1	yes	characteristic
146	C₅×C₈	#124	1	1	yes	characteristic
147	C₅×C₈	#126	1	1	yes	characteristic
148	C₂×#41	C₂×C₄×C₅	1	1	yes	characteristic
149	#131	—	2	1	no	—
150	#131	—	2	1	no	—
151	#131	—	2	1	no	—
152	#131	—	2	1	no	—
153	C₂×M₁₆	—	2	1	no	—
154	C₂×M₁₆	C₂×C₄×C₅	1	1	yes	characteristic
155	C₂×M₁₆	C₂×C₄×C₅	1	1	yes	characteristic
156	C₂×M₁₆	C₂×C₄×C₅	1	1	yes	characteristic
157	C₂²×C₈	—	2	1	no	—
158	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
159	#117	C₂×C₄×C₅	1	1	yes	characteristic
160	#117	C₂×C₄×C₅	1	1	yes	characteristic
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	—	2	1	no	—
171	C₂×C₁₆	—	2	1	no	—
172	C₂×C₁₆	C₅×D₄	1	1	yes	characteristic
173	C₂×C₁₆	C₅×D₄	1	1	yes	characteristic
174	#101	C₅×C₈	1	2	yes	—
175	#100	C₅×C₈	1	2	yes	—
176	#99	—	2	2	no	—
177	#97	—	2	1	no	—
178	#97	—	2	1	no	—
179	C₄×C₈	—	2	1	no	—
180	C₄×C₈	—	2	1	no	—
181	C₂²×C₅	—	4	1	no	—
182	C₂²×C₅	—	4	1	no	—
183	C₂²×C₅	—	8	1	no	—
184	C₂²×C₅	#431	1	1	yes	characteristic
185	C₂²×C₅	#435	1	1	yes	characteristic
186	C₂²×C₅	#436	1	1	yes	characteristic
187	C₄×C₅	—	4	2	no	—
188	C₄×C₅	—	4	1	no	—
189	C₄×C₅	—	8	1	no	—
190	C₄×C₅	#388	1	1	yes	characteristic
191	C₄×C₅	#433	1	1	yes	characteristic
192	C₄×C₅	C₈×D₄	1	1	yes	characteristic
193	C₄×C₅	C₈×D₄	1	1	yes	characteristic
194	#41	—	2	1	no	—
195	#41	—	2	1	no	—
196	#41	—	2	1	no	—
197	#41	—	2	1	no	—
198	C₂×Q₈	C₂×C₅×C₈	1	1	yes	characteristic
199	C₂×D₄	—	2	1	no	—
200	C₂×D₄	C₂×C₅×C₈	1	1	yes	characteristic
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	M₁₆	C₂×C₅×C₈	1	2	yes	—
210	M₁₆	C₂×C₅×C₈	1	2	yes	—
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	C₄⋊C₄	—	2	1	no	—
224	C₂²⋊C₄	—	2	1	no	—
225	C₂²⋊C₄	C₅×M₁₆	1	2	yes	—
226	C₄²	—	2	1	no	—
227	C₄²	—	2	1	no	—
228	C₁₆	—	2	2	no	—
229	C₁₆	—	2	1	no	—
230	C₁₆	—	2	1	no	—
231	C₁₆	—	4	1	no	—
232	C₁₆	—	4	1	no	—
233	C₂×C₅	—	2	1	no	—
234	C₂×C₅	—	2	1	no	—
235	C₂×C₅	—	2	1	no	—
236	C₂×C₅	—	8	1	no	—
237	C₂×C₅	#1923	1	1	yes	characteristic
238	C₂³	—	2	1	no	—
239	C₂³	C₂×C₅×M₁₆	1	1	yes	characteristic
240	Q₈	—	2	1	no	—
241	Q₈	—	2	1	no	—
242	D₄	—	2	1	no	—
243	D₄	—	2	1	no	—
244	D₄	—	4	1	no	—
245	D₄	—	4	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	4	1	no	—
249	C₂×C₄	—	4	1	no	—
250	C₂×C₄	—	4	1	no	—
251	C₂×C₄	C₅×#117	1	1	yes	characteristic
252	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
253	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
254	C₂×C₄	C₂²×C₅×C₈	1	1	yes	characteristic
255	C₂×C₄	C₅×#131	1	1	yes	characteristic
256	C₂×C₄	C₅×#131	1	1	yes	characteristic
257	C₈	—	2	2	no	—
258	C₈	—	2	2	no	—
259	C₈	—	4	1	no	—
260	C₈	C₅×#121	1	1	yes	characteristic
261	C₈	C₅×#123	1	1	yes	characteristic
262	C₈	C₅×#124	1	1	yes	characteristic
263	C₈	C₅×#126	1	1	yes	characteristic
264	C₅	#13123	1	1	yes	characteristic, direct product factor, Sylow
265	C₂²	—	4	1	no	—
266	C₂²	—	4	1	no	—
267	C₂²	—	8	1	no	—
268	C₂²	C₅×#431	1	1	yes	characteristic
269	C₂²	C₅×#435	1	1	yes	characteristic
270	C₂²	C₅×#436	1	1	yes	characteristic
271	C₄	—	4	2	no	—
272	C₄	—	4	1	no	—
273	C₄	—	8	1	no	—
274	C₄	C₅×#388	1	1	yes	characteristic
275	C₄	C₅×#433	1	1	yes	characteristic
276	C₄	C₅×C₈×D₄	1	1	yes	characteristic
277	C₄	C₅×C₈×D₄	1	1	yes	characteristic
278	C₂	—	2	1	no	—
279	C₂	—	2	1	no	—
280	C₂	—	2	1	no	—
281	C₂	—	8	1	no	—
282	C₂	C₅×#1923	1	1	yes	characteristic

	subgroup	G/H
center	C₅×C₈	#121
commutator	C₂×C₄	C₂²×C₅×C₈
Frattini, cosocle	C₂²×C₈	C₂³×C₅
socle	C₂×C₅	#1923
prime radical	C₂×C₅×#41	C₂×C₄
LCS	G ▷ C₂×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×#2153	C₂	semidirect
4. C₅×#2108	C₂	semidirect
6. C₅×#1379	C₂	semidirect
7. C₅×#1378	C₂	semidirect
25. #13123	C₅	direct (N × H)
59. #2153	C₂×C₅	semidirect
62. #2108	C₂×C₅	semidirect
64. #1379	C₂×C₅	semidirect
65. #1378	C₂×C₅	semidirect
264. C₅	#13123	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×#2153	2
C₅×#2117	2
C₂×C₅×#350	2
C₅×#2108	2
C₅×#1757	2
C₅×#1379	2
C₅×#1378	2
#13123	5

Subgroups — C5×#13123 #11935843

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#13123 #11935843