Subgroups — C₅×#11626 #11934346

Order 1280.

328 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#2965	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	C₄×C₅×#41	C₄	1	1	yes	characteristic
7	C₂²×C₅×C₁₆	—	2	1	no	—
8	C₅×#446	C₄	1	1	yes	characteristic
9	C₅×#446	C₂²	1	1	yes	characteristic
10	C₅×#436	C₄	1	1	yes	characteristic
11	C₅×#436	C₂²	1	1	yes	characteristic
12	C₅×#435	—	2	1	no	—
13	C₅×#435	—	2	1	no	—
14	C₅×#431	—	2	1	no	—
15	C₅×#431	C₄	1	1	yes	characteristic
16	C₄×C₅×M₁₆	—	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	#11626	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
24	C₂×C₅×#41	C₈	1	1	yes	characteristic
25	C₂×C₅×M₁₆	—	2	1	no	—
26	C₂×C₅×M₁₆	—	2	1	no	—
27	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
28	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
29	C₄×C₅×Q₈	C₈	1	1	yes	characteristic
30	C₄×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
31	C₄×C₅×D₄	—	2	1	no	—
32	C₄×C₅×D₄	—	2	1	no	—
33	C₄×C₅×D₄	C₈	1	1	yes	characteristic
34	C₄×C₅×D₄	C₂×C₄	1	1	yes	characteristic
35	C₅×#117	—	2	1	no	—
36	C₅×#117	C₈	1	1	yes	characteristic
37	C₂×C₄²×C₅	—	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	#2965	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	C₅×#41	—	4	2	no	—
62	C₂×C₅×Q₈	C₂×C₈	1	1	yes	characteristic
63	C₂×C₅×D₄	—	2	1	no	—
64	C₂×C₅×D₄	C₂×C₈	1	1	yes	characteristic
65	C₂²×C₄×C₅	—	2	1	no	—
66	C₂²×C₄×C₅	—	2	1	no	—
67	C₂²×C₄×C₅	—	2	1	no	—
68	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
70	C₂²×C₄×C₅	M₁₆	1	1	yes	characteristic
71	C₅×M₁₆	—	4	2	no	—
72	C₅×M₁₆	—	4	1	no	—
73	C₅×M₁₆	—	4	1	no	—
74	C₂×C₅×C₈	—	2	1	no	—
75	C₂×C₅×C₈	—	2	1	no	—
76	C₂×C₅×C₈	—	2	1	no	—
77	C₂×C₅×C₈	—	2	1	no	—
78	C₂×C₅×C₈	—	4	1	no	—
79	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
82	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
83	C₅×C₄⋊C₄	—	2	1	no	—
84	C₅×C₄⋊C₄	—	2	1	no	—
85	C₅×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
86	C₅×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
87	C₅×C₂²⋊C₄	—	2	1	no	—
88	C₅×C₂²⋊C₄	—	2	1	no	—
89	C₅×C₂²⋊C₄	—	2	1	no	—
90	C₄²×C₅	—	2	1	no	—
91	C₄²×C₅	—	2	1	no	—
92	C₄²×C₅	—	2	1	no	—
93	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
94	C₄²×C₅	C₂×C₈	1	1	yes	characteristic
95	C₄²×C₅	M₁₆	1	1	yes	characteristic
96	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
97	C₅×C₁₆	—	4	2	no	—
98	C₅×C₁₆	—	4	2	no	—
99	C₄×#41	C₄×C₅	1	1	yes	characteristic
100	C₂²×C₁₆	—	2	1	no	—
101	#446	C₄×C₅	1	1	yes	characteristic
102	#446	C₂²×C₅	1	1	yes	characteristic
103	#436	C₄×C₅	1	1	yes	characteristic
104	#436	C₂²×C₅	1	1	yes	characteristic
105	#435	—	2	1	no	—
106	#435	—	2	1	no	—
107	#431	—	2	1	no	—
108	#431	C₄×C₅	1	1	yes	characteristic
109	C₄×M₁₆	—	2	1	no	—
110	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
111	C₄⋊C₁₆	—	2	1	no	—
112	C₄⋊C₁₆	—	2	1	no	—
113	C₄⋊C₁₆	C₂²×C₅	1	2	yes	—
114	C₂²⋊C₁₆	—	2	1	no	—
115	#346	C₂²×C₅	1	2	yes	—
116	C₂³×C₅	—	2	1	no	—
117	C₂³×C₅	C₂²⋊C₈	1	1	yes	characteristic
118	C₅×Q₈	—	4	1	no	—
119	C₅×D₄	—	4	1	no	—
120	C₅×D₄	—	4	2	no	—
121	C₂×C₄×C₅	—	2	2	no	—
122	C₂×C₄×C₅	—	2	2	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	—	2	1	no	—
127	C₂×C₄×C₅	—	2	1	no	—
128	C₂×C₄×C₅	—	4	2	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	C₂²⋊C₈	1	1	yes	characteristic
133	C₂×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
134	C₂×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
135	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
136	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
137	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
138	C₅×C₈	—	2	1	no	—
139	C₅×C₈	—	2	1	no	—
140	C₅×C₈	—	2	1	no	—
141	C₅×C₈	—	4	1	no	—
142	C₅×C₈	—	4	1	no	—
143	C₅×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
144	C₅×C₈	#137	1	1	yes	characteristic
145	C₂×#41	C₅×C₈	1	1	yes	characteristic
146	C₂×M₁₆	—	2	1	no	—
147	C₂×M₁₆	—	2	1	no	—
148	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
149	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
150	C₄×Q₈	C₅×C₈	1	1	yes	characteristic
151	C₄×Q₈	C₂×C₄×C₅	1	1	yes	characteristic
152	C₄×D₄	—	2	1	no	—
153	C₄×D₄	—	2	1	no	—
154	C₄×D₄	C₅×C₈	1	1	yes	characteristic
155	C₄×D₄	C₂×C₄×C₅	1	1	yes	characteristic
156	#117	—	2	1	no	—
157	#117	C₅×C₈	1	1	yes	characteristic
158	C₂×C₄²	—	2	1	no	—
159	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
160	C₂×C₁₆	—	2	2	no	—
161	C₂×C₁₆	—	2	2	no	—
162	C₂×C₁₆	—	4	2	no	—
163	C₄⋊C₈	—	2	1	no	—
164	C₄⋊C₈	—	2	1	no	—
165	C₄⋊C₈	C₂×C₄×C₅	1	1	yes	characteristic
166	C₄⋊C₈	C₂×C₄×C₅	1	1	yes	characteristic
167	C₂²⋊C₈	—	2	1	no	—
168	C₂²⋊C₈	—	2	1	no	—
169	C₂²⋊C₈	—	2	1	no	—
170	#97	—	2	1	no	—
171	#97	—	2	1	no	—
172	C₄×C₈	—	2	1	no	—
173	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
174	C₄×C₈	C₅×D₄	1	1	yes	characteristic
175	C₄×C₈	C₅×D₄	1	1	yes	characteristic
176	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
177	C₂²×C₅	—	2	1	no	—
178	C₂²×C₅	—	2	1	no	—
179	C₂²×C₅	—	4	1	no	—
180	C₂²×C₅	—	4	1	no	—
181	C₂²×C₅	—	4	1	no	—
182	C₂²×C₅	#349	1	2	yes	—
183	C₂²×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
184	C₄×C₅	—	2	1	no	—
185	C₄×C₅	—	2	1	no	—
186	C₄×C₅	—	2	1	no	—
187	C₄×C₅	—	2	1	no	—
188	C₄×C₅	—	4	1	no	—
189	C₄×C₅	—	4	1	no	—
190	C₄×C₅	—	4	1	no	—
191	C₄×C₅	#350	1	2	yes	—
192	C₄×C₅	#417	1	1	yes	characteristic
193	C₄×C₅	#421	1	1	yes	characteristic
194	#41	—	4	2	no	—
195	C₂×Q₈	C₂×C₅×C₈	1	1	yes	characteristic
196	C₂×D₄	—	2	1	no	—
197	C₂×D₄	C₂×C₅×C₈	1	1	yes	characteristic
198	C₂²×C₄	—	2	1	no	—
199	C₂²×C₄	—	2	1	no	—
200	C₂²×C₄	—	2	1	no	—
201	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
202	C₂²×C₄	C₂×C₅×C₈	1	1	yes	characteristic
203	C₂²×C₄	C₅×M₁₆	1	1	yes	characteristic
204	M₁₆	—	4	2	no	—
205	M₁₆	—	4	1	no	—
206	M₁₆	—	4	1	no	—
207	C₂×C₈	—	2	1	no	—
208	C₂×C₈	—	2	1	no	—
209	C₂×C₈	—	2	1	no	—
210	C₂×C₈	—	2	1	no	—
211	C₂×C₈	—	4	1	no	—
212	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
213	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
214	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
215	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
216	C₄⋊C₄	—	2	1	no	—
217	C₄⋊C₄	—	2	1	no	—
218	C₄⋊C₄	C₂×C₅×C₈	1	1	yes	characteristic
219	C₄⋊C₄	C₂×C₅×C₈	1	1	yes	characteristic
220	C₂²⋊C₄	—	2	1	no	—
221	C₂²⋊C₄	—	2	1	no	—
222	C₂²⋊C₄	—	2	1	no	—
223	C₄²	—	2	1	no	—
224	C₄²	—	2	1	no	—
225	C₄²	—	2	1	no	—
226	C₄²	C₅×C₂²⋊C₄	1	1	yes	characteristic
227	C₄²	C₂×C₅×C₈	1	1	yes	characteristic
228	C₄²	C₅×M₁₆	1	1	yes	characteristic
229	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
230	C₁₆	—	4	2	no	—
231	C₁₆	—	4	2	no	—
232	C₂×C₅	—	2	2	no	—
233	C₂×C₅	—	8	1	no	—
234	C₂×C₅	#1477	1	1	yes	characteristic
235	C₂×C₅	C₂×#349	1	1	yes	characteristic
236	C₂×C₅	C₂×#350	1	1	yes	characteristic
237	C₂³	—	2	1	no	—
238	C₂³	C₅×C₂²⋊C₈	1	1	yes	characteristic
239	Q₈	—	4	1	no	—
240	D₄	—	4	1	no	—
241	D₄	—	4	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	4	2	no	—
250	C₂×C₄	—	4	1	no	—
251	C₂×C₄	—	4	1	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	C₅×C₂²⋊C₈	1	1	yes	characteristic
254	C₂×C₄	C₅×C₂²⋊C₈	1	1	yes	characteristic
255	C₂×C₄	C₅×C₂²⋊C₈	1	1	yes	characteristic
256	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
257	C₂×C₄	C₂²×C₅×C₈	1	1	yes	characteristic
258	C₂×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
259	C₈	—	2	1	no	—
260	C₈	—	2	1	no	—
261	C₈	—	2	1	no	—
262	C₈	—	4	1	no	—
263	C₈	—	4	1	no	—
264	C₈	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
265	C₈	C₅×#137	1	1	yes	characteristic
266	C₅	#11626	1	1	yes	characteristic, direct product factor, Sylow
267	C₂²	—	2	1	no	—
268	C₂²	—	2	1	no	—
269	C₂²	—	4	1	no	—
270	C₂²	—	4	1	no	—
271	C₂²	—	4	1	no	—
272	C₂²	C₅×#349	1	2	yes	—
273	C₂²	C₂×C₅×C₂²⋊C₈	1	1	yes	characteristic
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	2	1	no	—
277	C₄	—	2	1	no	—
278	C₄	—	4	1	no	—
279	C₄	—	4	1	no	—
280	C₄	—	4	1	no	—
281	C₄	C₅×#350	1	2	yes	—
282	C₄	C₅×#417	1	1	yes	characteristic
283	C₄	C₅×#421	1	1	yes	characteristic
284	C₂	—	2	2	no	—
285	C₂	—	8	1	no	—
286	C₂	C₅×#1477	1	1	yes	characteristic
287	C₂	C₂×C₅×#349	1	1	yes	characteristic
288	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	C₂×C₅×#41	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. #11626	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
266. C₅	#11626	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×#2965	2
C₅×#2163	2
C₂×C₅×C₄⋊C₁₆	2
C₅×#1338	2
C₅×#1330	2
#11626	5

Subgroups — C5×#11626 #11934346

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#11626 #11934346