Subgroups — C₅×#8688 #11931408

Order 1280.

414 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#2931	C₂	1	1	yes	maximal, characteristic
2	C₅×#1837	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×#335	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#1547	C₂	1	2	yes	maximal
5	C₅×#1292	C₂	1	2	yes	maximal, complemented
6	C₂×C₅×#131	—	2	1	no	—
7	C₅×#532	C₂²	1	1	yes	characteristic
8	C₅×#436	—	2	1	no	—
9	C₅×#436	C₄	1	2	yes	—
10	C₅×C₈×D₄	—	2	1	no	—
11	C₅×C₈×D₄	C₄	1	2	yes	—
12	C₅×#423	C₂²	1	2	yes	—
13	C₅×#423	C₂²	1	1	yes	characteristic
14	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
15	C₅×#408	—	2	1	no	—
16	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
17	C₅×#341	—	2	1	no	—
18	C₅×#341	—	2	1	no	—
19	C₅×#335	—	2	1	no	—
20	C₅×#335	—	2	1	no	—
21	C₅×#332	—	2	2	no	—
22	C₅×#331	—	2	2	no	—
23	#8688	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
24	C₂×C₅×#41	—	2	1	no	—
25	C₅×#131	—	4	4	no	—
26	C₂×C₅×M₁₆	—	2	1	no	—
27	C₂×C₅×M₁₆	—	2	2	no	—
28	C₂×C₅×M₁₆	—	2	2	no	—
29	C₂²×C₅×C₈	—	2	2	no	—
30	C₂²×C₅×C₈	—	2	2	no	—
31	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
32	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
33	C₅×C₄⋊Q₈	D₄	1	1	yes	characteristic
34	C₅×#127	D₄	1	1	yes	characteristic
35	C₅×#124	—	2	1	no	—
36	C₅×#121	—	2	1	no	—
37	C₅×#121	Q₈	1	2	yes	—
38	C₄×C₅×D₄	—	2	1	no	—
39	C₄×C₅×D₄	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
41	C₅×C₄⋊C₈	—	2	1	no	—
42	C₅×C₄⋊C₈	—	2	2	no	—
43	C₅×C₄⋊C₈	—	2	2	no	—
44	C₅×C₄⋊C₈	—	2	2	no	—
45	C₅×C₄⋊C₈	—	2	2	no	—
46	C₅×C₄⋊C₈	C₂×C₄	1	2	yes	—
47	C₅×C₂²⋊C₈	—	2	2	no	—
48	C₅×C₂²⋊C₈	—	2	2	no	—
49	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
50	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
51	C₄×C₅×C₈	D₄	1	1	yes	characteristic
52	C₄×C₅×C₈	D₄	1	1	yes	characteristic
53	#2931	C₂×C₅	1	1	yes	characteristic
54	#1837	C₂×C₅	1	1	yes	characteristic, complemented
55	C₂×#335	C₂×C₅	1	1	yes	characteristic, complemented
56	#1547	C₂×C₅	1	2	yes	—
57	#1292	C₂×C₅	1	2	yes	complemented
58	C₅×#41	—	4	2	no	—
59	C₅×#41	—	4	2	no	—
60	C₂×C₅×Q₈	—	2	1	no	—
61	C₂×C₅×D₄	—	2	2	no	—
62	C₂×C₅×D₄	—	2	1	no	—
63	C₂×C₅×D₄	—	2	1	no	—
64	C₂×C₅×D₄	C₄⋊C₄	1	2	yes	—
65	C₂²×C₄×C₅	—	2	2	no	—
66	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
68	C₂²×C₄×C₅	#41	1	1	yes	characteristic
69	C₅×M₁₆	—	4	4	no	—
70	C₅×M₁₆	—	4	2	no	—
71	C₅×M₁₆	—	4	2	no	—
72	C₅×M₁₆	—	4	2	no	—
73	C₂×C₅×C₈	—	2	2	no	—
74	C₂×C₅×C₈	—	2	2	no	—
75	C₂×C₅×C₈	—	2	2	no	—
76	C₂×C₅×C₈	—	4	4	no	—
77	C₂×C₅×C₈	—	4	2	no	—
78	C₂×C₅×C₈	—	4	2	no	—
79	C₂×C₅×C₈	—	8	2	no	—
80	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
84	C₅×C₄⋊C₄	—	2	1	no	—
85	C₅×C₄⋊C₄	C₄⋊C₄	1	2	yes	—
86	C₅×C₂²⋊C₄	—	2	2	no	—
87	C₅×C₂²⋊C₄	—	2	2	no	—
88	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
89	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
90	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
91	C₄²×C₅	#41	1	1	yes	characteristic
92	C₂×#131	—	2	1	no	—
93	#532	C₂²×C₅	1	1	yes	characteristic
94	#436	—	2	1	no	—
95	#436	C₄×C₅	1	2	yes	—
96	C₈×D₄	—	2	1	no	—
97	C₈×D₄	C₄×C₅	1	2	yes	—
98	#423	C₂²×C₅	1	2	yes	—
99	#423	C₂²×C₅	1	1	yes	characteristic
100	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
101	#408	—	2	1	no	—
102	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
103	#341	—	2	1	no	—
104	#341	—	2	1	no	—
105	#335	—	2	1	no	—
106	#335	—	2	1	no	—
107	#332	—	2	2	no	—
108	#331	—	2	2	no	—
109	C₂³×C₅	—	2	2	no	—
110	C₂³×C₅	C₂²⋊Q₈	1	1	yes	characteristic
111	C₅×Q₈	—	4	2	no	—
112	C₅×D₄	—	4	2	no	—
113	C₅×D₄	—	4	4	no	—
114	C₅×D₄	—	4	2	no	—
115	C₅×D₄	—	4	1	no	—
116	C₅×D₄	—	4	1	no	—
117	C₂×C₄×C₅	—	2	1	no	—
118	C₂×C₄×C₅	—	2	1	no	—
119	C₂×C₄×C₅	—	2	2	no	—
120	C₂×C₄×C₅	—	4	1	no	—
121	C₂×C₄×C₅	—	4	2	no	—
122	C₂×C₄×C₅	—	4	2	no	—
123	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₅	#117	1	1	yes	characteristic
126	C₂×C₄×C₅	#121	1	1	yes	characteristic
127	C₂×C₄×C₅	#121	1	1	yes	characteristic
128	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
129	C₂×C₄×C₅	#135	1	2	yes	—
130	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
131	C₅×C₈	—	2	1	no	—
132	C₅×C₈	—	2	1	no	—
133	C₅×C₈	—	2	1	no	—
134	C₅×C₈	—	2	1	no	—
135	C₅×C₈	—	4	2	no	—
136	C₅×C₈	—	8	2	no	—
137	C₅×C₈	—	8	2	no	—
138	C₂×#41	—	2	1	no	—
139	#131	—	4	4	no	—
140	C₂×M₁₆	—	2	1	no	—
141	C₂×M₁₆	—	2	2	no	—
142	C₂×M₁₆	—	2	2	no	—
143	C₂²×C₈	—	2	2	no	—
144	C₂²×C₈	—	2	2	no	—
145	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
146	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
147	C₄⋊Q₈	C₅×D₄	1	1	yes	characteristic
148	#127	C₅×D₄	1	1	yes	characteristic
149	#124	—	2	1	no	—
150	#121	—	2	1	no	—
151	#121	C₅×Q₈	1	2	yes	—
152	C₄×D₄	—	2	1	no	—
153	C₄×D₄	C₂×C₄×C₅	1	2	yes	—
154	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
155	C₄⋊C₈	—	2	1	no	—
156	C₄⋊C₈	—	2	2	no	—
157	C₄⋊C₈	—	2	2	no	—
158	C₄⋊C₈	—	2	2	no	—
159	C₄⋊C₈	—	2	2	no	—
160	C₄⋊C₈	C₂×C₄×C₅	1	2	yes	—
161	C₂²⋊C₈	—	2	2	no	—
162	C₂²⋊C₈	—	2	2	no	—
163	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
164	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
165	C₄×C₈	C₅×D₄	1	1	yes	characteristic
166	C₄×C₈	C₅×D₄	1	1	yes	characteristic
167	C₂²×C₅	—	2	1	no	—
168	C₂²×C₅	—	2	1	no	—
169	C₂²×C₅	—	4	2	no	—
170	C₂²×C₅	—	4	2	no	—
171	C₂²×C₅	—	4	2	no	—
172	C₂²×C₅	#380	1	1	yes	characteristic
173	C₂²×C₅	#478	1	2	yes	—
174	C₄×C₅	—	2	1	no	—
175	C₄×C₅	—	2	1	no	—
176	C₄×C₅	—	2	1	no	—
177	C₄×C₅	—	4	2	no	—
178	C₄×C₅	#416	1	1	yes	characteristic
179	C₄×C₅	#418	1	1	yes	characteristic
180	C₄×C₅	#463	1	2	yes	—
181	C₄×C₅	#472	1	1	yes	characteristic
182	C₄×C₅	#473	1	1	yes	characteristic
183	#41	—	4	2	no	—
184	#41	—	4	2	no	—
185	C₂×Q₈	—	2	1	no	—
186	C₂×D₄	—	2	2	no	—
187	C₂×D₄	—	2	1	no	—
188	C₂×D₄	—	2	1	no	—
189	C₂×D₄	C₅×C₄⋊C₄	1	2	yes	—
190	C₂²×C₄	—	2	2	no	—
191	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
192	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
193	C₂²×C₄	C₅×#41	1	1	yes	characteristic
194	M₁₆	—	4	4	no	—
195	M₁₆	—	4	2	no	—
196	M₁₆	—	4	2	no	—
197	M₁₆	—	4	2	no	—
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	—	4	4	no	—
202	C₂×C₈	—	4	2	no	—
203	C₂×C₈	—	4	2	no	—
204	C₂×C₈	—	8	2	no	—
205	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
206	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
207	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
208	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
209	C₄⋊C₄	—	2	1	no	—
210	C₄⋊C₄	C₅×C₄⋊C₄	1	2	yes	—
211	C₂²⋊C₄	—	2	2	no	—
212	C₂²⋊C₄	—	2	2	no	—
213	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
214	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
215	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
216	C₄²	C₅×#41	1	1	yes	characteristic
217	C₂×C₅	—	2	2	no	—
218	C₂×C₅	—	8	2	no	—
219	C₂×C₅	#1804	1	1	yes	characteristic
220	C₂×C₅	#1813	1	1	yes	characteristic
221	C₂×C₅	#1816	1	1	yes	characteristic
222	C₂³	—	2	2	no	—
223	C₂³	C₅×C₂²⋊Q₈	1	1	yes	characteristic
224	Q₈	—	4	2	no	—
225	D₄	—	4	2	no	—
226	D₄	—	4	4	no	—
227	D₄	—	4	2	no	—
228	D₄	—	4	1	no	—
229	D₄	—	4	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	4	1	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
237	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
238	C₂×C₄	C₅×#117	1	1	yes	characteristic
239	C₂×C₄	C₅×#121	1	1	yes	characteristic
240	C₂×C₄	C₅×#121	1	1	yes	characteristic
241	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
242	C₂×C₄	C₅×#135	1	2	yes	—
243	C₂×C₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
244	C₈	—	2	1	no	—
245	C₈	—	2	1	no	—
246	C₈	—	2	1	no	—
247	C₈	—	2	1	no	—
248	C₈	—	4	2	no	—
249	C₈	—	8	2	no	—
250	C₈	—	8	2	no	—
251	C₅	#8688	1	1	yes	characteristic, direct product factor, Sylow
252	C₂²	—	2	1	no	—
253	C₂²	—	2	1	no	—
254	C₂²	—	4	2	no	—
255	C₂²	—	4	2	no	—
256	C₂²	—	4	2	no	—
257	C₂²	C₅×#380	1	1	yes	characteristic
258	C₂²	C₅×#478	1	2	yes	—
259	C₄	—	2	1	no	—
260	C₄	—	2	1	no	—
261	C₄	—	2	1	no	—
262	C₄	—	4	2	no	—
263	C₄	C₅×#416	1	1	yes	characteristic
264	C₄	C₅×#418	1	1	yes	characteristic
265	C₄	C₅×#463	1	2	yes	—
266	C₄	C₅×#472	1	1	yes	characteristic
267	C₄	C₅×#473	1	1	yes	characteristic
268	C₂	—	2	2	no	—
269	C₂	—	8	2	no	—
270	C₂	C₅×#1804	1	1	yes	characteristic
271	C₂	C₅×#1813	1	1	yes	characteristic
272	C₂	C₅×#1816	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	#380
commutator	C₄²	C₂²×C₄×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
prime radical	C₅×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅×#1837	C₂	semidirect
3. C₂×C₅×#335	C₂	semidirect
5. C₅×#1292	C₂	semidirect
23. #8688	C₅	direct (N × H)
54. #1837	C₂×C₅	semidirect
55. C₂×#335	C₂×C₅	semidirect
57. #1292	C₂×C₅	semidirect
251. C₅	#8688	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×#2931	2
C₅×#1837	2
C₂×C₅×#335	2
C₅×#1547	2
C₅×#1292	2
#8688	5

Subgroups — C5×#8688 #11931408

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#8688 #11931408