Subgroups — C₅×#10861 #11933581

Order 1280.

982 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×#106	C₂	1	2	yes	maximal
2	C₂²×C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#340	C₂	1	2	yes	maximal
4	C₅×C₄⋊#95	C₂	1	2	yes	maximal
5	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
6	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
7	C₂²×C₅×C₄⋊C₄	C₂²	1	4	yes	—
8	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
9	C₂×C₅×#106	—	2	4	no	—
10	C₂×C₅×#106	—	2	4	no	—
11	C₂×C₅×#106	C₄	1	8	yes	complemented
12	C₂×C₄×C₅×C₈	—	2	4	no	—
13	C₂×C₄×C₅×C₈	C₄	1	4	yes	complemented
14	C₂×C₅×#95	—	2	4	no	—
15	C₅×#340	—	2	4	no	—
16	#10861	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
17	C₂³×C₄×C₅	—	2	4	no	—
18	C₂³×C₄×C₅	D₄	1	2	yes	—
19	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
20	C₂²×C₅×C₈	—	2	2	no	—
21	C₂²×C₅×C₈	—	2	2	no	—
22	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
23	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
24	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
25	C₂²×C₅×C₈	D₄	1	2	yes	—
26	C₂²×C₅×C₈	D₄	1	4	yes	—
27	C₂²×C₅×C₈	D₄	1	2	yes	—
28	C₂²×C₅×C₈	Q₈	1	2	yes	—
29	C₂²×C₅×C₈	Q₈	1	2	yes	—
30	C₂×C₅×C₄⋊C₄	—	2	4	no	—
31	C₂×C₅×C₄⋊C₄	—	2	4	no	—
32	C₂×C₅×C₄⋊C₄	—	2	4	no	—
33	C₂×C₅×C₄⋊C₄	—	2	4	no	—
34	C₂×C₅×C₄⋊C₄	—	2	4	no	—
35	C₂×C₅×C₄⋊C₄	C₂×C₄	1	8	yes	—
36	C₂×C₄²×C₅	—	2	2	no	—
37	C₂×C₄²×C₅	—	2	2	no	—
38	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₅	Q₈	1	2	yes	—
40	C₅×#106	—	2	16	no	—
41	C₄×C₅×C₈	—	2	8	no	—
42	C₅×#95	—	4	8	no	—
43	C₂²×#106	C₂×C₅	1	2	yes	—
44	C₂²×C₄×C₈	C₂×C₅	1	1	yes	characteristic
45	C₂×#340	C₂×C₅	1	2	yes	—
46	C₄⋊#95	C₂×C₅	1	2	yes	—
47	C₂⁴×C₅	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₅	—	2	4	no	—
49	C₂²×C₄×C₅	—	2	4	no	—
50	C₂²×C₄×C₅	—	2	8	no	—
51	C₂²×C₄×C₅	—	4	4	no	—
52	C₂²×C₄×C₅	—	4	4	no	—
53	C₂²×C₄×C₅	—	4	8	no	—
54	C₂²×C₄×C₅	—	4	8	no	—
55	C₂²×C₄×C₅	C₄⋊C₄	1	4	yes	—
56	C₂²×C₄×C₅	SD₁₆	1	8	yes	—
57	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
59	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
60	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
63	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
64	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
65	C₂²×C₄×C₅	#41	1	2	yes	—
66	C₂²×C₄×C₅	#41	1	2	yes	—
67	C₂×C₅×C₈	—	2	2	no	—
68	C₂×C₅×C₈	—	2	2	no	—
69	C₂×C₅×C₈	—	2	4	no	—
70	C₂×C₅×C₈	—	2	4	no	—
71	C₂×C₅×C₈	C₄²	1	4	yes	—
72	C₂×C₅×C₈	C₂²⋊C₄	1	4	yes	—
73	C₂×C₅×C₈	C₂²⋊C₄	1	8	yes	—
74	C₂×C₅×C₈	C₄⋊C₄	1	4	yes	—
75	C₂×C₅×C₈	C₄⋊C₄	1	4	yes	—
76	C₂×C₅×C₈	C₄⋊C₄	1	8	yes	—
77	C₅×C₄⋊C₄	—	2	8	no	—
78	C₅×C₄⋊C₄	—	2	8	no	—
79	C₅×C₄⋊C₄	—	4	4	no	—
80	C₅×C₄⋊C₄	—	4	4	no	—
81	C₄²×C₅	—	2	2	no	—
82	C₄²×C₅	—	2	2	no	—
83	C₄²×C₅	—	2	4	no	—
84	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
85	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
86	C₂²×C₄⋊C₄	C₂²×C₅	1	4	yes	—
87	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
88	C₂×#106	—	2	4	no	—
89	C₂×#106	—	2	4	no	—
90	C₂×#106	C₄×C₅	1	8	yes	complemented
91	C₂×C₄×C₈	—	2	4	no	—
92	C₂×C₄×C₈	C₄×C₅	1	4	yes	complemented
93	C₂×#95	—	2	4	no	—
94	#340	—	2	4	no	—
95	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
96	C₂³×C₅	C₄×D₄	1	2	yes	—
97	C₂³×C₅	#121	1	2	yes	—
98	C₂³×C₅	#127	1	1	yes	characteristic
99	C₂³×C₅	C₄⋊Q₈	1	1	yes	characteristic
100	C₂³×C₅	C₂×SD₁₆	1	4	yes	—
101	C₂³×C₅	#135	1	4	yes	—
102	C₂×C₄×C₅	—	2	4	no	—
103	C₂×C₄×C₅	—	2	4	no	—
104	C₂×C₄×C₅	—	2	16	no	—
105	C₂×C₄×C₅	—	4	8	no	—
106	C₂×C₄×C₅	—	4	16	no	—
107	C₂×C₄×C₅	—	8	8	no	—
108	C₂×C₄×C₅	—	8	4	no	—
109	C₂×C₄×C₅	—	8	4	no	—
110	C₂×C₄×C₅	#106	1	8	yes	complemented
111	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
112	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	2	yes	—
114	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
116	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₅	#117	1	1	yes	characteristic
118	C₂×C₄×C₅	#117	1	1	yes	characteristic
119	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
120	C₂×C₄×C₅	C₄×Q₈	1	2	yes	—
121	C₂×C₄×C₅	C₄×Q₈	1	2	yes	—
122	C₂×C₄×C₅	#121	1	2	yes	—
123	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
124	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
125	C₂×C₄×C₅	#124	1	2	yes	—
126	C₂×C₄×C₅	#125	1	2	yes	—
127	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
128	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
129	C₅×C₈	—	2	4	no	—
130	C₅×C₈	#95	1	8	yes	complemented
131	C₂³×C₄	—	2	4	no	—
132	C₂³×C₄	C₅×D₄	1	2	yes	—
133	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
134	C₂²×C₈	—	2	2	no	—
135	C₂²×C₈	—	2	2	no	—
136	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
137	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
138	C₂²×C₈	C₂×C₄×C₅	1	4	yes	—
139	C₂²×C₈	C₅×D₄	1	2	yes	—
140	C₂²×C₈	C₅×D₄	1	4	yes	—
141	C₂²×C₈	C₅×D₄	1	2	yes	—
142	C₂²×C₈	C₅×Q₈	1	2	yes	—
143	C₂²×C₈	C₅×Q₈	1	2	yes	—
144	C₂×C₄⋊C₄	—	2	4	no	—
145	C₂×C₄⋊C₄	—	2	4	no	—
146	C₂×C₄⋊C₄	—	2	4	no	—
147	C₂×C₄⋊C₄	—	2	4	no	—
148	C₂×C₄⋊C₄	—	2	4	no	—
149	C₂×C₄⋊C₄	C₂×C₄×C₅	1	8	yes	—
150	C₂×C₄²	—	2	2	no	—
151	C₂×C₄²	—	2	2	no	—
152	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
153	C₂×C₄²	C₅×Q₈	1	2	yes	—
154	#106	—	2	16	no	—
155	C₄×C₈	—	2	8	no	—
156	#95	—	4	8	no	—
157	C₂²×C₅	C₄×C₄⋊C₄	1	1	yes	characteristic
158	C₂²×C₅	#380	1	1	yes	characteristic
159	C₂²×C₅	#384	1	1	yes	characteristic
160	C₂²×C₅	#389	1	2	yes	—
161	C₂²×C₅	#390	1	2	yes	—
162	C₂²×C₅	C₂×#106	1	2	yes	—
163	C₂²×C₅	#427	1	2	yes	—
164	C₂²×C₅	C₄×SD₁₆	1	8	yes	—
165	C₂²×C₅	#465	1	8	yes	—
166	C₂²×C₅	#492	1	2	yes	—
167	C₂²×C₅	#495	1	2	yes	—
168	C₂²×C₅	#498	1	2	yes	—
169	C₂²×C₅	#499	1	2	yes	—
170	C₄×C₅	—	2	8	no	—
171	C₄×C₅	—	8	8	no	—
172	C₄×C₅	C₂×#95	1	1	yes	characteristic
173	C₄×C₅	C₄×C₄⋊C₄	1	1	yes	characteristic
174	C₄×C₅	#380	1	1	yes	characteristic
175	C₄×C₅	#382	1	1	yes	characteristic
176	C₄×C₅	#389	1	2	yes	—
177	C₄×C₅	#391	1	2	yes	—
178	C₂⁴	C₂×C₅×D₄	1	1	yes	characteristic
179	C₂²×C₄	—	2	4	no	—
180	C₂²×C₄	—	2	4	no	—
181	C₂²×C₄	—	2	8	no	—
182	C₂²×C₄	—	4	4	no	—
183	C₂²×C₄	—	4	4	no	—
184	C₂²×C₄	—	4	8	no	—
185	C₂²×C₄	—	4	8	no	—
186	C₂²×C₄	C₅×C₄⋊C₄	1	4	yes	—
187	C₂²×C₄	C₅×SD₁₆	1	8	yes	—
188	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
189	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
190	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
191	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
192	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
193	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
194	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
195	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
196	C₂²×C₄	C₅×#41	1	2	yes	—
197	C₂²×C₄	C₅×#41	1	2	yes	—
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	2	4	no	—
201	C₂×C₈	—	2	4	no	—
202	C₂×C₈	C₄²×C₅	1	4	yes	—
203	C₂×C₈	C₅×C₂²⋊C₄	1	4	yes	—
204	C₂×C₈	C₅×C₂²⋊C₄	1	8	yes	—
205	C₂×C₈	C₅×C₄⋊C₄	1	4	yes	—
206	C₂×C₈	C₅×C₄⋊C₄	1	4	yes	—
207	C₂×C₈	C₅×C₄⋊C₄	1	8	yes	—
208	C₄⋊C₄	—	2	8	no	—
209	C₄⋊C₄	—	2	8	no	—
210	C₄⋊C₄	—	4	4	no	—
211	C₄⋊C₄	—	4	4	no	—
212	C₄²	—	2	2	no	—
213	C₄²	—	2	2	no	—
214	C₄²	—	2	4	no	—
215	C₂×C₅	C₄⋊#95	1	1	yes	characteristic
216	C₂×C₅	C₄×#106	1	2	yes	—
217	C₂×C₅	#1807	1	2	yes	—
218	C₂×C₅	#1844	1	1	yes	characteristic
219	C₂×C₅	#1846	1	1	yes	characteristic
220	C₂×C₅	#1941	1	4	yes	—
221	C₂×C₅	#1968	1	4	yes	—
222	C₂³	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
223	C₂³	C₄×C₅×D₄	1	2	yes	—
224	C₂³	C₅×#121	1	2	yes	—
225	C₂³	C₅×#127	1	1	yes	characteristic
226	C₂³	C₅×C₄⋊Q₈	1	1	yes	characteristic
227	C₂³	C₂×C₅×SD₁₆	1	4	yes	—
228	C₂³	C₅×#135	1	4	yes	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	2	16	no	—
232	C₂×C₄	—	4	8	no	—
233	C₂×C₄	—	4	16	no	—
234	C₂×C₄	—	8	8	no	—
235	C₂×C₄	—	8	4	no	—
236	C₂×C₄	—	8	4	no	—
237	C₂×C₄	C₅×#106	1	8	yes	complemented
238	C₂×C₄	C₂×C₄²×C₅	1	1	yes	characteristic
239	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
240	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	2	yes	—
241	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
242	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
243	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
244	C₂×C₄	C₅×#117	1	1	yes	characteristic
245	C₂×C₄	C₅×#117	1	1	yes	characteristic
246	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
247	C₂×C₄	C₄×C₅×Q₈	1	2	yes	—
248	C₂×C₄	C₄×C₅×Q₈	1	2	yes	—
249	C₂×C₄	C₅×#121	1	2	yes	—
250	C₂×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
251	C₂×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
252	C₂×C₄	C₅×#124	1	2	yes	—
253	C₂×C₄	C₅×#125	1	2	yes	—
254	C₂×C₄	C₅×C₄⋊Q₈	1	1	yes	characteristic
255	C₂×C₄	C₅×C₄⋊Q₈	1	1	yes	characteristic
256	C₈	—	2	4	no	—
257	C₈	C₅×#95	1	8	yes	complemented
258	C₅	#10861	1	1	yes	characteristic, direct product factor, Sylow
259	C₂²	C₄×C₅×C₄⋊C₄	1	1	yes	characteristic
260	C₂²	C₅×#380	1	1	yes	characteristic
261	C₂²	C₅×#384	1	1	yes	characteristic
262	C₂²	C₅×#389	1	2	yes	—
263	C₂²	C₅×#390	1	2	yes	—
264	C₂²	C₂×C₅×#106	1	2	yes	—
265	C₂²	C₅×#427	1	2	yes	—
266	C₂²	C₄×C₅×SD₁₆	1	8	yes	—
267	C₂²	C₅×#465	1	8	yes	—
268	C₂²	C₅×#492	1	2	yes	—
269	C₂²	C₅×#495	1	2	yes	—
270	C₂²	C₅×#498	1	2	yes	—
271	C₂²	C₅×#499	1	2	yes	—
272	C₄	—	2	8	no	—
273	C₄	—	8	8	no	—
274	C₄	C₂×C₅×#95	1	1	yes	characteristic
275	C₄	C₄×C₅×C₄⋊C₄	1	1	yes	characteristic
276	C₄	C₅×#380	1	1	yes	characteristic
277	C₄	C₅×#382	1	1	yes	characteristic
278	C₄	C₅×#389	1	2	yes	—
279	C₄	C₅×#391	1	2	yes	—
280	C₂	C₅×C₄⋊#95	1	1	yes	characteristic
281	C₂	C₄×C₅×#106	1	2	yes	—
282	C₂	C₅×#1807	1	2	yes	—
283	C₂	C₅×#1844	1	1	yes	characteristic
284	C₂	C₅×#1846	1	1	yes	characteristic
285	C₂	C₅×#1941	1	4	yes	—
286	C₂	C₅×#1968	1	4	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₅	C₂×D₄
commutator	C₂×C₄	C₂×C₄²×C₅
Frattini, cosocle	C₂³×C₄	C₂³×C₅
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₅ ◁ C₂²×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
11. C₂×C₅×#106	C₄	semidirect
13. C₂×C₄×C₅×C₈	C₄	semidirect
16. #10861	C₅	direct (N × H)
90. C₂×#106	C₄×C₅	semidirect
92. C₂×C₄×C₈	C₄×C₅	semidirect
110. C₂×C₄×C₅	#106	semidirect
130. C₅×C₈	#95	semidirect
237. C₂×C₄	C₅×#106	semidirect
257. C₈	C₅×#95	semidirect
258. C₅	#10861	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₅×#106	2
C₂²×C₄×C₅×C₈	2
C₂×C₅×#340	2
C₅×C₄⋊#95	2
#10861	5

Subgroups — C5×#10861 #11933581

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#10861 #11933581