Subgroups — C₅×#11625 #11934345

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×#2966	C₂	1	1	yes	maximal, characteristic
2	C₅×#2163	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#1338	C₂	1	2	yes	maximal, complemented
5	C₅×#1330	C₂	1	2	yes	maximal, complemented
6	C₅×#529	C₄	1	1	yes	characteristic
7	C₂²×C₅×C₁₆	—	2	1	no	—
8	C₅×#446	C₂²	1	1	yes	characteristic
9	C₅×C₈×Q₈	C₄	1	1	yes	characteristic
10	C₅×#436	C₂²	1	1	yes	characteristic
11	C₅×#435	—	2	1	no	—
12	C₅×C₈×D₄	—	2	1	no	—
13	C₅×C₈×D₄	C₄	1	1	yes	characteristic
14	C₅×#433	—	2	1	no	—
15	C₅×#432	C₄	1	1	yes	characteristic
16	C₅×#405	—	2	1	no	—
17	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
18	C₅×C₄⋊C₁₆	—	2	1	no	—
19	C₅×C₄⋊C₁₆	—	2	1	no	—
20	C₅×C₄⋊C₁₆	C₂²	1	2	yes	—
21	C₅×C₂²⋊C₁₆	—	2	1	no	—
22	C₅×#346	C₂²	1	2	yes	—
23	#11625	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
24	C₂×C₅×M₁₆	—	2	1	no	—
25	C₂²×C₅×C₈	—	2	1	no	—
26	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
27	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
28	C₅×#126	—	2	1	no	—
29	C₅×#125	C₈	1	1	yes	characteristic
30	C₅×#124	C₈	1	1	yes	characteristic
31	C₅×#123	—	2	1	no	—
32	C₅×C₂²⋊Q₈	C₈	1	1	yes	characteristic
33	C₅×#121	C₈	1	1	yes	characteristic
34	C₄×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
35	C₄×C₅×D₄	—	2	1	no	—
36	C₄×C₅×D₄	C₂×C₄	1	1	yes	characteristic
37	C₅×#117	—	2	1	no	—
38	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
39	C₂×C₅×C₁₆	—	2	2	no	—
40	C₂×C₅×C₁₆	—	2	2	no	—
41	C₂×C₅×C₁₆	—	4	2	no	—
42	C₅×C₄⋊C₈	—	2	1	no	—
43	C₅×C₄⋊C₈	—	2	1	no	—
44	C₅×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
45	C₅×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
46	C₅×C₂²⋊C₈	—	2	1	no	—
47	C₅×C₂²⋊C₈	—	2	1	no	—
48	C₅×C₂²⋊C₈	—	2	1	no	—
49	C₅×#97	—	2	1	no	—
50	C₅×#97	—	2	1	no	—
51	C₄×C₅×C₈	—	2	1	no	—
52	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
53	C₄×C₅×C₈	D₄	1	1	yes	characteristic
54	C₄×C₅×C₈	D₄	1	1	yes	characteristic
55	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
56	#2966	C₂×C₅	1	1	yes	characteristic
57	#2163	C₂×C₅	1	1	yes	characteristic, complemented
58	C₂×C₄⋊C₁₆	C₂×C₅	1	1	yes	characteristic, complemented
59	#1338	C₂×C₅	1	2	yes	complemented
60	#1330	C₂×C₅	1	2	yes	complemented
61	C₂×C₅×Q₈	C₂×C₈	1	1	yes	characteristic
62	C₂×C₅×D₄	—	2	1	no	—
63	C₂×C₅×D₄	C₂×C₈	1	1	yes	characteristic
64	C₂²×C₄×C₅	—	2	1	no	—
65	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
67	C₂²×C₄×C₅	M₁₆	1	1	yes	characteristic
68	C₅×M₁₆	—	4	1	no	—
69	C₅×M₁₆	—	4	1	no	—
70	C₂×C₅×C₈	—	2	1	no	—
71	C₂×C₅×C₈	—	2	1	no	—
72	C₂×C₅×C₈	—	2	1	no	—
73	C₂×C₅×C₈	—	2	1	no	—
74	C₂×C₅×C₈	—	4	1	no	—
75	C₂×C₅×C₈	—	4	2	no	—
76	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
79	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
80	C₅×C₄⋊C₄	—	2	1	no	—
81	C₅×C₄⋊C₄	—	2	1	no	—
82	C₅×C₄⋊C₄	—	2	1	no	—
83	C₅×C₄⋊C₄	—	2	1	no	—
84	C₅×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
85	C₅×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
86	C₅×C₂²⋊C₄	—	2	1	no	—
87	C₅×C₂²⋊C₄	—	2	1	no	—
88	C₅×C₂²⋊C₄	—	2	1	no	—
89	C₅×C₂²⋊C₄	—	2	1	no	—
90	C₅×C₂²⋊C₄	—	2	1	no	—
91	C₄²×C₅	—	2	1	no	—
92	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
93	C₄²×C₅	C₂×C₈	1	1	yes	characteristic
94	C₄²×C₅	M₁₆	1	1	yes	characteristic
95	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
96	C₅×C₁₆	—	4	2	no	—
97	C₅×C₁₆	—	4	2	no	—
98	#529	C₄×C₅	1	1	yes	characteristic
99	C₂²×C₁₆	—	2	1	no	—
100	#446	C₂²×C₅	1	1	yes	characteristic
101	C₈×Q₈	C₄×C₅	1	1	yes	characteristic
102	#436	C₂²×C₅	1	1	yes	characteristic
103	#435	—	2	1	no	—
104	C₈×D₄	—	2	1	no	—
105	C₈×D₄	C₄×C₅	1	1	yes	characteristic
106	#433	—	2	1	no	—
107	#432	C₄×C₅	1	1	yes	characteristic
108	#405	—	2	1	no	—
109	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
110	C₄⋊C₁₆	—	2	1	no	—
111	C₄⋊C₁₆	—	2	1	no	—
112	C₄⋊C₁₆	C₂²×C₅	1	2	yes	—
113	C₂²⋊C₁₆	—	2	1	no	—
114	#346	C₂²×C₅	1	2	yes	—
115	C₂³×C₅	—	2	1	no	—
116	C₂³×C₅	C₂²⋊C₈	1	1	yes	characteristic
117	C₅×Q₈	—	4	1	no	—
118	C₅×D₄	—	4	1	no	—
119	C₅×D₄	—	4	2	no	—
120	C₂×C₄×C₅	—	2	2	no	—
121	C₂×C₄×C₅	—	2	1	no	—
122	C₂×C₄×C₅	—	2	1	no	—
123	C₂×C₄×C₅	—	2	1	no	—
124	C₂×C₄×C₅	—	2	1	no	—
125	C₂×C₄×C₅	—	2	1	no	—
126	C₂×C₄×C₅	—	4	1	no	—
127	C₂×C₄×C₅	—	4	1	no	—
128	C₂×C₄×C₅	—	4	1	no	—
129	C₂×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
130	C₂×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
131	C₂×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
132	C₂×C₄×C₅	C₂×C₂²⋊C₄	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	C₅×C₈	—	2	1	no	—
136	C₅×C₈	—	2	1	no	—
137	C₅×C₈	—	2	1	no	—
138	C₅×C₈	—	4	1	no	—
139	C₅×C₈	—	4	1	no	—
140	C₅×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
141	C₅×C₈	#137	1	1	yes	characteristic
142	C₂×M₁₆	—	2	1	no	—
143	C₂²×C₈	—	2	1	no	—
144	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
145	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
146	#126	—	2	1	no	—
147	#125	C₅×C₈	1	1	yes	characteristic
148	#124	C₅×C₈	1	1	yes	characteristic
149	#123	—	2	1	no	—
150	C₂²⋊Q₈	C₅×C₈	1	1	yes	characteristic
151	#121	C₅×C₈	1	1	yes	characteristic
152	C₄×Q₈	C₂×C₄×C₅	1	1	yes	characteristic
153	C₄×D₄	—	2	1	no	—
154	C₄×D₄	C₂×C₄×C₅	1	1	yes	characteristic
155	#117	—	2	1	no	—
156	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
157	C₂×C₁₆	—	2	2	no	—
158	C₂×C₁₆	—	2	2	no	—
159	C₂×C₁₆	—	4	2	no	—
160	C₄⋊C₈	—	2	1	no	—
161	C₄⋊C₈	—	2	1	no	—
162	C₄⋊C₈	C₂×C₄×C₅	1	1	yes	characteristic
163	C₄⋊C₈	C₂×C₄×C₅	1	1	yes	characteristic
164	C₂²⋊C₈	—	2	1	no	—
165	C₂²⋊C₈	—	2	1	no	—
166	C₂²⋊C₈	—	2	1	no	—
167	#97	—	2	1	no	—
168	#97	—	2	1	no	—
169	C₄×C₈	—	2	1	no	—
170	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
171	C₄×C₈	C₅×D₄	1	1	yes	characteristic
172	C₄×C₈	C₅×D₄	1	1	yes	characteristic
173	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
174	C₂²×C₅	—	2	1	no	—
175	C₂²×C₅	—	2	1	no	—
176	C₂²×C₅	—	4	1	no	—
177	C₂²×C₅	—	4	1	no	—
178	C₂²×C₅	—	4	1	no	—
179	C₂²×C₅	#350	1	2	yes	—
180	C₂²×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
181	C₄×C₅	—	2	1	no	—
182	C₄×C₅	—	2	1	no	—
183	C₄×C₅	—	2	1	no	—
184	C₄×C₅	—	2	1	no	—
185	C₄×C₅	—	2	1	no	—
186	C₄×C₅	—	4	1	no	—
187	C₄×C₅	—	4	1	no	—
188	C₄×C₅	—	4	1	no	—
189	C₄×C₅	#417	1	1	yes	characteristic
190	C₄×C₅	#421	1	1	yes	characteristic
191	C₂×Q₈	C₂×C₅×C₈	1	1	yes	characteristic
192	C₂×D₄	—	2	1	no	—
193	C₂×D₄	C₂×C₅×C₈	1	1	yes	characteristic
194	C₂²×C₄	—	2	1	no	—
195	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
196	C₂²×C₄	C₂×C₅×C₈	1	1	yes	characteristic
197	C₂²×C₄	C₅×M₁₆	1	1	yes	characteristic
198	M₁₆	—	4	1	no	—
199	M₁₆	—	4	1	no	—
200	C₂×C₈	—	2	1	no	—
201	C₂×C₈	—	2	1	no	—
202	C₂×C₈	—	2	1	no	—
203	C₂×C₈	—	2	1	no	—
204	C₂×C₈	—	4	1	no	—
205	C₂×C₈	—	4	2	no	—
206	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
207	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
208	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
209	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
210	C₄⋊C₄	—	2	1	no	—
211	C₄⋊C₄	—	2	1	no	—
212	C₄⋊C₄	—	2	1	no	—
213	C₄⋊C₄	—	2	1	no	—
214	C₄⋊C₄	C₂×C₅×C₈	1	1	yes	characteristic
215	C₄⋊C₄	C₂×C₅×C₈	1	1	yes	characteristic
216	C₂²⋊C₄	—	2	1	no	—
217	C₂²⋊C₄	—	2	1	no	—
218	C₂²⋊C₄	—	2	1	no	—
219	C₂²⋊C₄	—	2	1	no	—
220	C₂²⋊C₄	—	2	1	no	—
221	C₄²	—	2	1	no	—
222	C₄²	C₅×C₂²⋊C₄	1	1	yes	characteristic
223	C₄²	C₂×C₅×C₈	1	1	yes	characteristic
224	C₄²	C₅×M₁₆	1	1	yes	characteristic
225	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
226	C₁₆	—	4	2	no	—
227	C₁₆	—	4	2	no	—
228	C₂×C₅	—	2	2	no	—
229	C₂×C₅	—	8	1	no	—
230	C₂×C₅	#1477	1	1	yes	characteristic
231	C₂×C₅	#2115	1	1	yes	characteristic
232	C₂×C₅	C₂×#350	1	1	yes	characteristic
233	C₂³	—	2	1	no	—
234	C₂³	C₅×C₂²⋊C₈	1	1	yes	characteristic
235	Q₈	—	4	1	no	—
236	D₄	—	4	1	no	—
237	D₄	—	4	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	4	1	no	—
245	C₂×C₄	—	4	1	no	—
246	C₂×C₄	—	4	1	no	—
247	C₂×C₄	C₅×C₂²⋊C₈	1	1	yes	characteristic
248	C₂×C₄	C₅×C₂²⋊C₈	1	1	yes	characteristic
249	C₂×C₄	C₅×C₂²⋊C₈	1	1	yes	characteristic
250	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
251	C₂×C₄	C₂²×C₅×C₈	1	1	yes	characteristic
252	C₂×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
253	C₈	—	2	1	no	—
254	C₈	—	2	1	no	—
255	C₈	—	2	1	no	—
256	C₈	—	4	1	no	—
257	C₈	—	4	1	no	—
258	C₈	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
259	C₈	C₅×#137	1	1	yes	characteristic
260	C₅	#11625	1	1	yes	characteristic, direct product factor, Sylow
261	C₂²	—	2	1	no	—
262	C₂²	—	2	1	no	—
263	C₂²	—	4	1	no	—
264	C₂²	—	4	1	no	—
265	C₂²	—	4	1	no	—
266	C₂²	C₅×#350	1	2	yes	—
267	C₂²	C₂×C₅×C₂²⋊C₈	1	1	yes	characteristic
268	C₄	—	2	1	no	—
269	C₄	—	2	1	no	—
270	C₄	—	2	1	no	—
271	C₄	—	2	1	no	—
272	C₄	—	2	1	no	—
273	C₄	—	4	1	no	—
274	C₄	—	4	1	no	—
275	C₄	—	4	1	no	—
276	C₄	C₅×#417	1	1	yes	characteristic
277	C₄	C₅×#421	1	1	yes	characteristic
278	C₂	—	2	2	no	—
279	C₂	—	8	1	no	—
280	C₂	C₅×#1477	1	1	yes	characteristic
281	C₂	C₅×#2115	1	1	yes	characteristic
282	C₂	C₂×C₅×#350	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₅×C₈
Frattini, cosocle	C₄×C₈	C₂³×C₅
socle	C₂²×C₅	C₂×C₂²⋊C₈
prime radical	C₅×#121	C₈
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂×C₄×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅×#2163	C₂	semidirect
3. C₂×C₅×C₄⋊C₁₆	C₂	semidirect
4. C₅×#1338	C₂	semidirect
5. C₅×#1330	C₂	semidirect
23. #11625	C₅	direct (N × H)
57. #2163	C₂×C₅	semidirect
58. C₂×C₄⋊C₁₆	C₂×C₅	semidirect
59. #1338	C₂×C₅	semidirect
60. #1330	C₂×C₅	semidirect
260. C₅	#11625	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×#2966	2
C₅×#2163	2
C₂×C₅×C₄⋊C₁₆	2
C₅×#1338	2
C₅×#1330	2
#11625	5

Subgroups — C5×#11625 #11934345

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#11625 #11934345