Subgroups — C₅×#11666 #11934386

Order 1280.

368 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₅×#135	C₂	1	1	yes	maximal, characteristic
2	C₅×#2163	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#1566	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	C₂×C₅×#135	C₄	1	1	yes	characteristic
7	C₄×C₅×#41	—	2	1	no	—
8	C₄×C₅×Q₁₆	C₄	1	1	yes	characteristic
9	C₄×C₅×Q₁₆	C₂²	1	1	yes	characteristic
10	C₄×C₅×SD₁₆	—	2	1	no	—
11	C₄×C₅×SD₁₆	—	2	1	no	—
12	C₄×C₅×D₈	C₄	1	1	yes	characteristic
13	C₄×C₅×D₈	C₂²	1	1	yes	characteristic
14	C₅×#427	C₄	1	1	yes	characteristic
15	C₅×#423	—	2	1	no	—
16	C₅×#416	—	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×#334	—	2	1	no	—
22	C₅×#334	C₂²	1	2	yes	—
23	#11666	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
24	C₂×C₅×#41	—	2	1	no	—
25	C₅×#135	—	2	2	no	—
26	C₅×#135	—	2	2	no	—
27	C₂×C₅×Q₁₆	C₂×C₄	1	1	yes	characteristic
28	C₂×C₅×SD₁₆	—	2	1	no	—
29	C₂×C₅×D₈	C₂×C₄	1	1	yes	characteristic
30	C₂×C₅×M₁₆	—	4	1	no	—
31	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
33	C₄×C₅×Q₈	—	2	1	no	—
34	C₄×C₅×Q₈	—	2	1	no	—
35	C₄×C₅×D₄	—	2	1	no	—
36	C₄×C₅×D₄	—	2	1	no	—
37	C₄×C₅×D₄	—	4	1	no	—
38	C₄×C₅×D₄	—	4	1	no	—
39	C₅×#117	—	2	1	no	—
40	C₅×#117	—	4	1	no	—
41	C₂×C₄²×C₅	—	4	1	no	—
42	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
43	C₂×C₅×C₁₆	—	2	2	no	—
44	C₅×#107	C₂×C₄	1	1	yes	characteristic
45	C₅×#107	C₂×C₄	1	1	yes	characteristic
46	C₅×#106	—	2	1	no	—
47	C₅×C₄⋊C₈	—	2	2	no	—
48	C₅×C₄⋊C₈	—	4	1	no	—
49	C₅×#103	—	2	1	no	—
50	C₅×#103	—	2	1	no	—
51	C₅×#102	—	2	1	no	—
52	C₅×#102	—	2	1	no	—
53	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
54	C₄×C₅×C₈	D₄	1	1	yes	characteristic
55	C₄×C₅×C₈	D₄	1	1	yes	characteristic
56	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
57	C₄×#135	C₂×C₅	1	1	yes	characteristic
58	#2163	C₂×C₅	1	1	yes	characteristic, complemented
59	#1566	C₂×C₅	1	1	yes	characteristic, complemented
60	#1334	C₂×C₅	1	2	yes	complemented
61	#1333	C₂×C₅	1	2	yes	complemented
62	C₅×#41	—	2	4	no	—
63	C₅×#41	—	4	2	no	—
64	C₂×C₅×Q₈	—	2	1	no	—
65	C₂×C₅×D₄	—	2	1	no	—
66	C₂×C₅×D₄	—	4	1	no	—
67	C₂²×C₄×C₅	—	4	1	no	—
68	C₂²×C₄×C₅	—	4	1	no	—
69	C₂²×C₄×C₅	—	4	1	no	—
70	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
71	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
72	C₂²×C₄×C₅	SD₁₆	1	1	yes	characteristic
73	C₅×Q₁₆	—	2	1	no	—
74	C₅×Q₁₆	M₁₆	1	2	yes	complemented
75	C₅×SD₁₆	—	2	4	no	—
76	C₅×D₈	—	2	1	no	—
77	C₅×D₈	M₁₆	1	2	yes	complemented
78	C₅×M₁₆	—	8	2	no	—
79	C₂×C₅×C₈	—	2	1	no	—
80	C₂×C₅×C₈	—	2	1	no	—
81	C₂×C₅×C₈	—	4	1	no	—
82	C₂×C₅×C₈	—	4	2	no	—
83	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
87	C₅×C₄⋊C₄	—	2	1	no	—
88	C₅×C₄⋊C₄	—	2	1	no	—
89	C₅×C₄⋊C₄	—	4	1	no	—
90	C₅×C₄⋊C₄	—	4	1	no	—
91	C₅×C₂²⋊C₄	—	4	1	no	—
92	C₅×C₂²⋊C₄	—	4	1	no	—
93	C₅×C₂²⋊C₄	—	4	1	no	—
94	C₄²×C₅	—	4	1	no	—
95	C₄²×C₅	—	4	1	no	—
96	C₄²×C₅	—	4	1	no	—
97	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
98	C₄²×C₅	D₈	1	1	yes	characteristic
99	C₄²×C₅	SD₁₆	1	1	yes	characteristic
100	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
101	C₅×C₁₆	—	4	2	no	—
102	C₂×#135	C₄×C₅	1	1	yes	characteristic
103	C₄×#41	—	2	1	no	—
104	C₄×Q₁₆	C₄×C₅	1	1	yes	characteristic
105	C₄×Q₁₆	C₂²×C₅	1	1	yes	characteristic
106	C₄×SD₁₆	—	2	1	no	—
107	C₄×SD₁₆	—	2	1	no	—
108	C₄×D₈	C₄×C₅	1	1	yes	characteristic
109	C₄×D₈	C₂²×C₅	1	1	yes	characteristic
110	#427	C₄×C₅	1	1	yes	characteristic
111	#423	—	2	1	no	—
112	#416	—	2	1	no	—
113	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
114	C₄⋊C₁₆	—	2	1	no	—
115	C₂²⋊C₁₆	—	2	1	no	—
116	#346	C₂²×C₅	1	2	yes	—
117	#334	—	2	1	no	—
118	#334	C₂²×C₅	1	2	yes	—
119	C₂³×C₅	—	4	1	no	—
120	C₂³×C₅	#102	1	1	yes	characteristic
121	C₅×Q₈	—	2	2	no	—
122	C₅×Q₈	—	4	1	no	—
123	C₅×D₄	—	2	2	no	—
124	C₅×D₄	—	4	4	no	—
125	C₅×D₄	—	4	1	no	—
126	C₂×C₄×C₅	—	2	2	no	—
127	C₂×C₄×C₅	—	2	1	no	—
128	C₂×C₄×C₅	—	2	1	no	—
129	C₂×C₄×C₅	—	4	1	no	—
130	C₂×C₄×C₅	—	4	4	no	—
131	C₂×C₄×C₅	—	4	1	no	—
132	C₂×C₄×C₅	—	4	1	no	—
133	C₂×C₄×C₅	—	4	1	no	—
134	C₂×C₄×C₅	—	8	2	no	—
135	C₂×C₄×C₅	—	8	2	no	—
136	C₂×C₄×C₅	#102	1	1	yes	characteristic
137	C₂×C₄×C₅	#102	1	1	yes	characteristic
138	C₂×C₄×C₅	#102	1	1	yes	characteristic
139	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
140	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
141	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
142	C₅×C₈	—	2	1	no	—
143	C₅×C₈	—	2	1	no	—
144	C₅×C₈	—	2	1	no	—
145	C₅×C₈	—	8	2	no	—
146	C₅×C₈	C₂×M₁₆	1	1	yes	characteristic
147	C₅×C₈	C₂×M₁₆	1	1	yes	characteristic
148	C₂×#41	—	2	1	no	—
149	#135	—	2	2	no	—
150	#135	—	2	2	no	—
151	C₂×Q₁₆	C₂×C₄×C₅	1	1	yes	characteristic
152	C₂×SD₁₆	—	2	1	no	—
153	C₂×D₈	C₂×C₄×C₅	1	1	yes	characteristic
154	C₂×M₁₆	—	4	1	no	—
155	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
156	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
157	C₄×Q₈	—	2	1	no	—
158	C₄×Q₈	—	2	1	no	—
159	C₄×D₄	—	2	1	no	—
160	C₄×D₄	—	2	1	no	—
161	C₄×D₄	—	4	1	no	—
162	C₄×D₄	—	4	1	no	—
163	#117	—	2	1	no	—
164	#117	—	4	1	no	—
165	C₂×C₄²	—	4	1	no	—
166	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
167	C₂×C₁₆	—	2	2	no	—
168	#107	C₂×C₄×C₅	1	1	yes	characteristic
169	#107	C₂×C₄×C₅	1	1	yes	characteristic
170	#106	—	2	1	no	—
171	C₄⋊C₈	—	2	2	no	—
172	C₄⋊C₈	—	4	1	no	—
173	#103	—	2	1	no	—
174	#103	—	2	1	no	—
175	#102	—	2	1	no	—
176	#102	—	2	1	no	—
177	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
178	C₄×C₈	C₅×D₄	1	1	yes	characteristic
179	C₄×C₈	C₅×D₄	1	1	yes	characteristic
180	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
181	C₂²×C₅	—	2	1	no	—
182	C₂²×C₅	—	2	1	no	—
183	C₂²×C₅	—	2	1	no	—
184	C₂²×C₅	—	4	2	no	—
185	C₂²×C₅	—	8	1	no	—
186	C₂²×C₅	—	8	1	no	—
187	C₂²×C₅	C₂×#102	1	1	yes	characteristic
188	C₄×C₅	—	2	1	no	—
189	C₄×C₅	—	2	1	no	—
190	C₄×C₅	—	2	1	no	—
191	C₄×C₅	—	2	1	no	—
192	C₄×C₅	—	4	2	no	—
193	C₄×C₅	—	8	1	no	—
194	C₄×C₅	—	8	1	no	—
195	C₄×C₅	#360	1	2	yes	—
196	C₄×C₅	#407	1	1	yes	characteristic
197	C₄×C₅	#421	1	1	yes	characteristic
198	#41	—	2	4	no	—
199	#41	—	4	2	no	—
200	C₂×Q₈	—	2	1	no	—
201	C₂×D₄	—	2	1	no	—
202	C₂×D₄	—	4	1	no	—
203	C₂²×C₄	—	4	1	no	—
204	C₂²×C₄	—	4	1	no	—
205	C₂²×C₄	—	4	1	no	—
206	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
207	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
208	C₂²×C₄	C₅×SD₁₆	1	1	yes	characteristic
209	Q₁₆	—	2	1	no	—
210	Q₁₆	C₅×M₁₆	1	2	yes	complemented
211	SD₁₆	—	2	4	no	—
212	D₈	—	2	1	no	—
213	D₈	C₅×M₁₆	1	2	yes	complemented
214	M₁₆	—	8	2	no	—
215	C₂×C₈	—	2	1	no	—
216	C₂×C₈	—	2	1	no	—
217	C₂×C₈	—	4	1	no	—
218	C₂×C₈	—	4	2	no	—
219	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
220	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
221	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
222	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
223	C₄⋊C₄	—	2	1	no	—
224	C₄⋊C₄	—	2	1	no	—
225	C₄⋊C₄	—	4	1	no	—
226	C₄⋊C₄	—	4	1	no	—
227	C₂²⋊C₄	—	4	1	no	—
228	C₂²⋊C₄	—	4	1	no	—
229	C₂²⋊C₄	—	4	1	no	—
230	C₄²	—	4	1	no	—
231	C₄²	—	4	1	no	—
232	C₄²	—	4	1	no	—
233	C₄²	C₅×C₂²⋊C₄	1	1	yes	characteristic
234	C₄²	C₅×D₈	1	1	yes	characteristic
235	C₄²	C₅×SD₁₆	1	1	yes	characteristic
236	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
237	C₁₆	—	4	2	no	—
238	C₂×C₅	—	4	1	no	—
239	C₂×C₅	—	8	2	no	—
240	C₂×C₅	#1486	1	1	yes	characteristic
241	C₂×C₅	C₂×#360	1	1	yes	characteristic
242	C₂×C₅	#2148	1	1	yes	characteristic
243	C₂³	—	4	1	no	—
244	C₂³	C₅×#102	1	1	yes	characteristic
245	Q₈	—	2	2	no	—
246	Q₈	—	4	1	no	—
247	D₄	—	2	2	no	—
248	D₄	—	4	4	no	—
249	D₄	—	4	1	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	4	1	no	—
254	C₂×C₄	—	4	4	no	—
255	C₂×C₄	—	4	1	no	—
256	C₂×C₄	—	4	1	no	—
257	C₂×C₄	—	4	1	no	—
258	C₂×C₄	—	8	2	no	—
259	C₂×C₄	—	8	2	no	—
260	C₂×C₄	C₅×#102	1	1	yes	characteristic
261	C₂×C₄	C₅×#102	1	1	yes	characteristic
262	C₂×C₄	C₅×#102	1	1	yes	characteristic
263	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
264	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
265	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
266	C₈	—	2	1	no	—
267	C₈	—	2	1	no	—
268	C₈	—	2	1	no	—
269	C₈	—	8	2	no	—
270	C₈	C₂×C₅×M₁₆	1	1	yes	characteristic
271	C₈	C₂×C₅×M₁₆	1	1	yes	characteristic
272	C₅	#11666	1	1	yes	characteristic, direct product factor, Sylow
273	C₂²	—	2	1	no	—
274	C₂²	—	2	1	no	—
275	C₂²	—	2	1	no	—
276	C₂²	—	4	2	no	—
277	C₂²	—	8	1	no	—
278	C₂²	—	8	1	no	—
279	C₂²	C₂×C₅×#102	1	1	yes	characteristic
280	C₄	—	2	1	no	—
281	C₄	—	2	1	no	—
282	C₄	—	2	1	no	—
283	C₄	—	2	1	no	—
284	C₄	—	4	2	no	—
285	C₄	—	8	1	no	—
286	C₄	—	8	1	no	—
287	C₄	C₅×#360	1	2	yes	—
288	C₄	C₅×#407	1	1	yes	characteristic
289	C₄	C₅×#421	1	1	yes	characteristic
290	C₂	—	4	1	no	—
291	C₂	—	8	2	no	—
292	C₂	C₅×#1486	1	1	yes	characteristic
293	C₂	C₂×C₅×#360	1	1	yes	characteristic
294	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	C₂×C₅×#135	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×#1566	C₂	semidirect
4. C₅×#1334	C₂	semidirect
5. C₅×#1333	C₂	semidirect
23. #11666	C₅	direct (N × H)
58. #2163	C₂×C₅	semidirect
59. #1566	C₂×C₅	semidirect
60. #1334	C₂×C₅	semidirect
61. #1333	C₂×C₅	semidirect
74. C₅×Q₁₆	M₁₆	semidirect
77. C₅×D₈	M₁₆	semidirect
210. Q₁₆	C₅×M₁₆	semidirect
213. D₈	C₅×M₁₆	semidirect
272. C₅	#11666	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₅×#135	2
C₅×#2163	2
C₅×#1566	2
C₅×#1334	2
C₅×#1333	2
#11666	5

Subgroups — C5×#11666 #11934386

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#11666 #11934386