Subgroups — C₅×#11656 #11934376

Order 1280.

306 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×#3168	C₂	1	1	yes	maximal, characteristic
2	C₅×#2163	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#1564	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#1334	C₂	1	2	yes	maximal, complemented
5	C₅×#1333	C₂	1	2	yes	maximal, complemented
6	C₅×#529	—	2	1	no	—
7	C₅×#499	C₄	1	1	yes	characteristic
8	C₅×#495	C₄	1	1	yes	characteristic
9	C₅×#488	—	2	1	no	—
10	C₅×#467	C₄	1	1	yes	characteristic
11	C₅×#466	C₄	1	1	yes	characteristic
12	C₅×#465	—	2	1	no	—
13	C₄×C₅×Q₁₆	C₂²	1	1	yes	characteristic
14	C₄×C₅×SD₁₆	—	2	1	no	—
15	C₄×C₅×D₈	C₂²	1	1	yes	characteristic
16	C₅×#431	—	2	1	no	—
17	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
18	C₅×C₄⋊C₁₆	—	2	1	no	—
19	C₅×C₂²⋊C₁₆	—	2	1	no	—
20	C₅×#346	C₂²	1	2	yes	—
21	C₅×#335	—	2	1	no	—
22	C₅×#334	C₂²	1	2	yes	—
23	#11656	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
24	C₂×C₅×Q₁₆	C₂×C₄	1	1	yes	characteristic
25	C₂×C₅×SD₁₆	—	2	1	no	—
26	C₂×C₅×D₈	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
29	C₅×#126	—	4	1	no	—
30	C₅×#125	—	2	1	no	—
31	C₅×#124	—	2	1	no	—
32	C₅×#123	—	4	1	no	—
33	C₅×C₂²⋊Q₈	—	2	1	no	—
34	C₅×#121	—	2	1	no	—
35	C₄×C₅×Q₈	—	2	1	no	—
36	C₄×C₅×D₄	—	2	1	no	—
37	C₄×C₅×D₄	—	4	1	no	—
38	C₅×#117	—	4	1	no	—
39	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
40	C₂×C₅×C₁₆	—	2	2	no	—
41	C₅×#107	C₂×C₄	1	1	yes	characteristic
42	C₅×#107	C₂×C₄	1	1	yes	characteristic
43	C₅×#106	—	2	1	no	—
44	C₅×C₄⋊C₈	—	2	2	no	—
45	C₅×#103	—	2	1	no	—
46	C₅×#103	—	2	1	no	—
47	C₅×#102	—	2	1	no	—
48	C₅×#102	—	2	1	no	—
49	C₅×C₂²⋊C₈	—	4	1	no	—
50	C₄×C₅×C₈	—	4	1	no	—
51	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
52	C₄×C₅×C₈	D₄	1	1	yes	characteristic
53	C₄×C₅×C₈	D₄	1	1	yes	characteristic
54	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
55	#3168	C₂×C₅	1	1	yes	characteristic
56	#2163	C₂×C₅	1	1	yes	characteristic, complemented
57	#1564	C₂×C₅	1	1	yes	characteristic, complemented
58	#1334	C₂×C₅	1	2	yes	complemented
59	#1333	C₂×C₅	1	2	yes	complemented
60	C₂×C₅×Q₈	—	2	1	no	—
61	C₂×C₅×D₄	—	2	1	no	—
62	C₂×C₅×D₄	—	4	1	no	—
63	C₂²×C₄×C₅	—	4	1	no	—
64	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
66	C₂²×C₄×C₅	SD₁₆	1	1	yes	characteristic
67	C₅×Q₁₆	—	2	2	no	—
68	C₅×SD₁₆	—	4	2	no	—
69	C₅×D₈	—	2	2	no	—
70	C₂×C₅×C₈	—	2	2	no	—
71	C₂×C₅×C₈	—	4	1	no	—
72	C₂×C₅×C₈	—	4	2	no	—
73	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
77	C₅×C₄⋊C₄	—	2	1	no	—
78	C₅×C₄⋊C₄	—	2	1	no	—
79	C₅×C₄⋊C₄	—	4	1	no	—
80	C₅×C₄⋊C₄	—	4	1	no	—
81	C₅×C₄⋊C₄	—	4	1	no	—
82	C₅×C₄⋊C₄	—	4	1	no	—
83	C₅×C₂²⋊C₄	—	4	1	no	—
84	C₅×C₂²⋊C₄	—	4	1	no	—
85	C₅×C₂²⋊C₄	—	4	1	no	—
86	C₅×C₂²⋊C₄	—	4	1	no	—
87	C₅×C₂²⋊C₄	—	4	1	no	—
88	C₄²×C₅	—	4	1	no	—
89	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
90	C₄²×C₅	D₈	1	1	yes	characteristic
91	C₄²×C₅	SD₁₆	1	1	yes	characteristic
92	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
93	C₅×C₁₆	—	4	2	no	—
94	#529	—	2	1	no	—
95	#499	C₄×C₅	1	1	yes	characteristic
96	#495	C₄×C₅	1	1	yes	characteristic
97	#488	—	2	1	no	—
98	#467	C₄×C₅	1	1	yes	characteristic
99	#466	C₄×C₅	1	1	yes	characteristic
100	#465	—	2	1	no	—
101	C₄×Q₁₆	C₂²×C₅	1	1	yes	characteristic
102	C₄×SD₁₆	—	2	1	no	—
103	C₄×D₈	C₂²×C₅	1	1	yes	characteristic
104	#431	—	2	1	no	—
105	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
106	C₄⋊C₁₆	—	2	1	no	—
107	C₂²⋊C₁₆	—	2	1	no	—
108	#346	C₂²×C₅	1	2	yes	—
109	#335	—	2	1	no	—
110	#334	C₂²×C₅	1	2	yes	—
111	C₂³×C₅	—	4	1	no	—
112	C₂³×C₅	#102	1	1	yes	characteristic
113	C₅×Q₈	—	4	1	no	—
114	C₅×Q₈	—	4	1	no	—
115	C₅×D₄	—	4	1	no	—
116	C₅×D₄	—	4	1	no	—
117	C₅×D₄	—	8	2	no	—
118	C₂×C₄×C₅	—	2	1	no	—
119	C₂×C₄×C₅	—	2	1	no	—
120	C₂×C₄×C₅	—	4	1	no	—
121	C₂×C₄×C₅	—	4	1	no	—
122	C₂×C₄×C₅	—	4	1	no	—
123	C₂×C₄×C₅	—	4	1	no	—
124	C₂×C₄×C₅	—	4	1	no	—
125	C₂×C₄×C₅	—	8	2	no	—
126	C₂×C₄×C₅	#102	1	1	yes	characteristic
127	C₂×C₄×C₅	#102	1	1	yes	characteristic
128	C₂×C₄×C₅	#102	1	1	yes	characteristic
129	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
130	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
131	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
132	C₅×C₈	—	2	1	no	—
133	C₅×C₈	—	2	1	no	—
134	C₅×C₈	—	2	1	no	—
135	C₅×C₈	—	8	2	no	—
136	C₅×C₈	#131	1	2	yes	—
137	C₂×Q₁₆	C₂×C₄×C₅	1	1	yes	characteristic
138	C₂×SD₁₆	—	2	1	no	—
139	C₂×D₈	C₂×C₄×C₅	1	1	yes	characteristic
140	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
141	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
142	#126	—	4	1	no	—
143	#125	—	2	1	no	—
144	#124	—	2	1	no	—
145	#123	—	4	1	no	—
146	C₂²⋊Q₈	—	2	1	no	—
147	#121	—	2	1	no	—
148	C₄×Q₈	—	2	1	no	—
149	C₄×D₄	—	2	1	no	—
150	C₄×D₄	—	4	1	no	—
151	#117	—	4	1	no	—
152	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
153	C₂×C₁₆	—	2	2	no	—
154	#107	C₂×C₄×C₅	1	1	yes	characteristic
155	#107	C₂×C₄×C₅	1	1	yes	characteristic
156	#106	—	2	1	no	—
157	C₄⋊C₈	—	2	2	no	—
158	#103	—	2	1	no	—
159	#103	—	2	1	no	—
160	#102	—	2	1	no	—
161	#102	—	2	1	no	—
162	C₂²⋊C₈	—	4	1	no	—
163	C₄×C₈	—	4	1	no	—
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	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
168	C₂²×C₅	—	2	1	no	—
169	C₂²×C₅	—	2	1	no	—
170	C₂²×C₅	—	2	1	no	—
171	C₂²×C₅	—	8	1	no	—
172	C₂²×C₅	—	8	1	no	—
173	C₂²×C₅	—	8	1	no	—
174	C₂²×C₅	C₂×#102	1	1	yes	characteristic
175	C₄×C₅	—	2	1	no	—
176	C₄×C₅	—	2	1	no	—
177	C₄×C₅	—	2	1	no	—
178	C₄×C₅	—	2	1	no	—
179	C₄×C₅	—	2	1	no	—
180	C₄×C₅	—	8	1	no	—
181	C₄×C₅	—	8	1	no	—
182	C₄×C₅	—	8	1	no	—
183	C₄×C₅	#408	1	1	yes	characteristic
184	C₄×C₅	#421	1	1	yes	characteristic
185	C₂×Q₈	—	2	1	no	—
186	C₂×D₄	—	2	1	no	—
187	C₂×D₄	—	4	1	no	—
188	C₂²×C₄	—	4	1	no	—
189	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
190	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
191	C₂²×C₄	C₅×SD₁₆	1	1	yes	characteristic
192	Q₁₆	—	2	2	no	—
193	SD₁₆	—	4	2	no	—
194	D₈	—	2	2	no	—
195	C₂×C₈	—	2	2	no	—
196	C₂×C₈	—	4	1	no	—
197	C₂×C₈	—	4	2	no	—
198	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
199	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
200	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
201	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
202	C₄⋊C₄	—	2	1	no	—
203	C₄⋊C₄	—	2	1	no	—
204	C₄⋊C₄	—	4	1	no	—
205	C₄⋊C₄	—	4	1	no	—
206	C₄⋊C₄	—	4	1	no	—
207	C₄⋊C₄	—	4	1	no	—
208	C₂²⋊C₄	—	4	1	no	—
209	C₂²⋊C₄	—	4	1	no	—
210	C₂²⋊C₄	—	4	1	no	—
211	C₂²⋊C₄	—	4	1	no	—
212	C₂²⋊C₄	—	4	1	no	—
213	C₄²	—	4	1	no	—
214	C₄²	C₅×C₂²⋊C₄	1	1	yes	characteristic
215	C₄²	C₅×D₈	1	1	yes	characteristic
216	C₄²	C₅×SD₁₆	1	1	yes	characteristic
217	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
218	C₁₆	—	4	2	no	—
219	C₂×C₅	—	4	1	no	—
220	C₂×C₅	—	16	1	no	—
221	C₂×C₅	#1480	1	1	yes	characteristic
222	C₂×C₅	#2145	1	1	yes	characteristic
223	C₂×C₅	#2148	1	1	yes	characteristic
224	C₂³	—	4	1	no	—
225	C₂³	C₅×#102	1	1	yes	characteristic
226	Q₈	—	4	1	no	—
227	Q₈	—	4	1	no	—
228	D₄	—	4	1	no	—
229	D₄	—	4	1	no	—
230	D₄	—	8	2	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	4	1	no	—
234	C₂×C₄	—	4	1	no	—
235	C₂×C₄	—	4	1	no	—
236	C₂×C₄	—	4	1	no	—
237	C₂×C₄	—	4	1	no	—
238	C₂×C₄	—	8	2	no	—
239	C₂×C₄	C₅×#102	1	1	yes	characteristic
240	C₂×C₄	C₅×#102	1	1	yes	characteristic
241	C₂×C₄	C₅×#102	1	1	yes	characteristic
242	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
243	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
244	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
245	C₈	—	2	1	no	—
246	C₈	—	2	1	no	—
247	C₈	—	2	1	no	—
248	C₈	—	8	2	no	—
249	C₈	C₅×#131	1	2	yes	—
250	C₅	#11656	1	1	yes	characteristic, direct product factor, Sylow
251	C₂²	—	2	1	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	2	1	no	—
254	C₂²	—	8	1	no	—
255	C₂²	—	8	1	no	—
256	C₂²	—	8	1	no	—
257	C₂²	C₂×C₅×#102	1	1	yes	characteristic
258	C₄	—	2	1	no	—
259	C₄	—	2	1	no	—
260	C₄	—	2	1	no	—
261	C₄	—	2	1	no	—
262	C₄	—	2	1	no	—
263	C₄	—	8	1	no	—
264	C₄	—	8	1	no	—
265	C₄	—	8	1	no	—
266	C₄	C₅×#408	1	1	yes	characteristic
267	C₄	C₅×#421	1	1	yes	characteristic
268	C₂	—	4	1	no	—
269	C₂	—	16	1	no	—
270	C₂	C₅×#1480	1	1	yes	characteristic
271	C₂	C₅×#2145	1	1	yes	characteristic
272	C₂	C₅×#2148	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₄×C₈	C₂³×C₅
prime radical	C₅×#466	C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×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. C₅×#1564	C₂	semidirect
4. C₅×#1334	C₂	semidirect
5. C₅×#1333	C₂	semidirect
23. #11656	C₅	direct (N × H)
56. #2163	C₂×C₅	semidirect
57. #1564	C₂×C₅	semidirect
58. #1334	C₂×C₅	semidirect
59. #1333	C₂×C₅	semidirect
250. C₅	#11656	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×#3168	2
C₅×#2163	2
C₅×#1564	2
C₅×#1334	2
C₅×#1333	2
#11656	5

Subgroups — C5×#11656 #11934376

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#11656 #11934376