Subgroups — C₅×#11980 #11934700

Order 1280.

314 conjugacy classes of subgroups.

The subgroup lattice has 284 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×#2150	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#1566	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#1363	C₂	1	2	yes	maximal, complemented
5	C₅×#1360	C₂	1	2	yes	maximal, complemented
6	C₅×#529	—	2	1	no	—
7	C₂×C₅×M₃₂	—	2	1	no	—
8	C₅×#499	C₂²	1	1	yes	characteristic
9	C₅×#495	C₂²	1	1	yes	characteristic
10	C₅×#488	—	2	1	no	—
11	C₅×#467	C₄	1	1	yes	characteristic
12	C₅×#466	C₄	1	1	yes	characteristic
13	C₅×#465	—	2	1	no	—
14	C₄×C₅×Q₁₆	C₄	1	1	yes	characteristic
15	C₄×C₅×SD₁₆	—	2	1	no	—
16	C₄×C₅×D₈	C₄	1	1	yes	characteristic
17	C₅×#423	—	2	1	no	—
18	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
19	C₅×C₄⋊C₁₆	—	2	1	no	—
20	C₅×C₄⋊C₁₆	C₂²	1	2	yes	—
21	C₅×#334	—	2	1	no	—
22	C₅×#334	C₂²	1	2	yes	—
23	#11980	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	C₂×C₅×M₁₆	—	4	1	no	—
28	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
30	C₅×#126	—	4	1	no	—
31	C₅×#125	—	2	1	no	—
32	C₅×#124	—	2	1	no	—
33	C₅×#123	—	4	1	no	—
34	C₅×C₂²⋊Q₈	—	2	1	no	—
35	C₅×#121	—	2	1	no	—
36	C₄×C₅×Q₈	—	2	1	no	—
37	C₄×C₅×D₄	—	2	1	no	—
38	C₄×C₅×D₄	—	4	1	no	—
39	C₅×#117	—	4	1	no	—
40	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
41	C₅×M₃₂	—	4	2	no	—
42	C₂×C₅×C₁₆	—	2	2	no	—
43	C₅×#107	C₂×C₄	1	1	yes	characteristic
44	C₅×#107	C₂×C₄	1	1	yes	characteristic
45	C₅×#106	—	2	1	no	—
46	C₅×C₄⋊C₈	—	2	2	no	—
47	C₅×C₄⋊C₈	—	4	1	no	—
48	C₅×#103	—	2	1	no	—
49	C₅×#103	—	2	1	no	—
50	C₅×#102	—	2	1	no	—
51	C₅×#102	—	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	#3168	C₂×C₅	1	1	yes	characteristic
57	#2150	C₂×C₅	1	1	yes	characteristic, complemented
58	#1566	C₂×C₅	1	1	yes	characteristic, complemented
59	#1363	C₂×C₅	1	2	yes	complemented
60	#1360	C₂×C₅	1	2	yes	complemented
61	C₂×C₅×Q₈	—	2	1	no	—
62	C₂×C₅×D₄	—	2	1	no	—
63	C₂×C₅×D₄	—	4	1	no	—
64	C₂²×C₄×C₅	—	4	1	no	—
65	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
67	C₂²×C₄×C₅	SD₁₆	1	1	yes	characteristic
68	C₅×Q₁₆	—	4	1	no	—
69	C₅×SD₁₆	—	4	1	no	—
70	C₅×SD₁₆	—	4	1	no	—
71	C₅×D₈	—	4	1	no	—
72	C₅×M₁₆	—	8	2	no	—
73	C₂×C₅×C₈	—	2	1	no	—
74	C₂×C₅×C₈	—	2	1	no	—
75	C₂×C₅×C₈	—	4	1	no	—
76	C₂×C₅×C₈	—	4	2	no	—
77	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
81	C₅×C₄⋊C₄	—	2	1	no	—
82	C₅×C₄⋊C₄	—	2	1	no	—
83	C₅×C₄⋊C₄	—	4	1	no	—
84	C₅×C₄⋊C₄	—	4	1	no	—
85	C₅×C₄⋊C₄	—	4	1	no	—
86	C₅×C₄⋊C₄	—	4	1	no	—
87	C₅×C₂²⋊C₄	—	4	1	no	—
88	C₅×C₂²⋊C₄	—	4	1	no	—
89	C₅×C₂²⋊C₄	—	4	1	no	—
90	C₅×C₂²⋊C₄	—	4	1	no	—
91	C₅×C₂²⋊C₄	—	4	1	no	—
92	C₄²×C₅	—	4	1	no	—
93	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
94	C₄²×C₅	D₈	1	1	yes	characteristic
95	C₄²×C₅	SD₁₆	1	1	yes	characteristic
96	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
97	C₅×C₁₆	—	4	2	no	—
98	#529	—	2	1	no	—
99	C₂×M₃₂	—	2	1	no	—
100	#499	C₂²×C₅	1	1	yes	characteristic
101	#495	C₂²×C₅	1	1	yes	characteristic
102	#488	—	2	1	no	—
103	#467	C₄×C₅	1	1	yes	characteristic
104	#466	C₄×C₅	1	1	yes	characteristic
105	#465	—	2	1	no	—
106	C₄×Q₁₆	C₄×C₅	1	1	yes	characteristic
107	C₄×SD₁₆	—	2	1	no	—
108	C₄×D₈	C₄×C₅	1	1	yes	characteristic
109	#423	—	2	1	no	—
110	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
111	C₄⋊C₁₆	—	2	1	no	—
112	C₄⋊C₁₆	C₂²×C₅	1	2	yes	—
113	#334	—	2	1	no	—
114	#334	C₂²×C₅	1	2	yes	—
115	C₂³×C₅	—	4	1	no	—
116	C₂³×C₅	#102	1	1	yes	characteristic
117	C₅×Q₈	—	4	1	no	—
118	C₅×Q₈	—	4	1	no	—
119	C₅×D₄	—	4	1	no	—
120	C₅×D₄	—	4	1	no	—
121	C₅×D₄	—	8	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	—	4	1	no	—
126	C₂×C₄×C₅	—	4	1	no	—
127	C₂×C₄×C₅	—	4	1	no	—
128	C₂×C₄×C₅	—	4	1	no	—
129	C₂×C₄×C₅	—	8	2	no	—
130	C₂×C₄×C₅	#102	1	1	yes	characteristic
131	C₂×C₄×C₅	#102	1	1	yes	characteristic
132	C₂×C₄×C₅	#102	1	1	yes	characteristic
133	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
134	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
135	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
136	C₅×C₈	—	2	1	no	—
137	C₅×C₈	—	2	1	no	—
138	C₅×C₈	—	2	1	no	—
139	C₅×C₈	—	8	2	no	—
140	C₅×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
141	C₅×C₈	#137	1	1	yes	characteristic
142	C₂×Q₁₆	C₂×C₄×C₅	1	1	yes	characteristic
143	C₂×SD₁₆	—	2	1	no	—
144	C₂×D₈	C₂×C₄×C₅	1	1	yes	characteristic
145	C₂×M₁₆	—	4	1	no	—
146	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
147	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
148	#126	—	4	1	no	—
149	#125	—	2	1	no	—
150	#124	—	2	1	no	—
151	#123	—	4	1	no	—
152	C₂²⋊Q₈	—	2	1	no	—
153	#121	—	2	1	no	—
154	C₄×Q₈	—	2	1	no	—
155	C₄×D₄	—	2	1	no	—
156	C₄×D₄	—	4	1	no	—
157	#117	—	4	1	no	—
158	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
159	M₃₂	—	4	2	no	—
160	C₂×C₁₆	—	2	2	no	—
161	#107	C₂×C₄×C₅	1	1	yes	characteristic
162	#107	C₂×C₄×C₅	1	1	yes	characteristic
163	#106	—	2	1	no	—
164	C₄⋊C₈	—	2	2	no	—
165	C₄⋊C₈	—	4	1	no	—
166	#103	—	2	1	no	—
167	#103	—	2	1	no	—
168	#102	—	2	1	no	—
169	#102	—	2	1	no	—
170	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
171	C₄×C₈	C₅×D₄	1	1	yes	characteristic
172	C₄×C₈	C₅×D₄	1	1	yes	characteristic
173	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
174	C₂²×C₅	—	2	1	no	—
175	C₂²×C₅	—	2	1	no	—
176	C₂²×C₅	—	2	1	no	—
177	C₂²×C₅	—	8	1	no	—
178	C₂²×C₅	—	8	1	no	—
179	C₂²×C₅	—	8	1	no	—
180	C₂²×C₅	C₂×#102	1	1	yes	characteristic
181	C₄×C₅	—	2	1	no	—
182	C₄×C₅	—	2	1	no	—
183	C₄×C₅	—	2	1	no	—
184	C₄×C₅	—	2	1	no	—
185	C₄×C₅	—	8	1	no	—
186	C₄×C₅	—	8	1	no	—
187	C₄×C₅	—	8	1	no	—
188	C₄×C₅	#360	1	2	yes	—
189	C₄×C₅	#413	1	1	yes	characteristic
190	C₄×C₅	#417	1	1	yes	characteristic
191	C₂×Q₈	—	2	1	no	—
192	C₂×D₄	—	2	1	no	—
193	C₂×D₄	—	4	1	no	—
194	C₂²×C₄	—	4	1	no	—
195	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
196	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
197	C₂²×C₄	C₅×SD₁₆	1	1	yes	characteristic
198	Q₁₆	—	4	1	no	—
199	SD₁₆	—	4	1	no	—
200	SD₁₆	—	4	1	no	—
201	D₈	—	4	1	no	—
202	M₁₆	—	8	2	no	—
203	C₂×C₈	—	2	1	no	—
204	C₂×C₈	—	2	1	no	—
205	C₂×C₈	—	4	1	no	—
206	C₂×C₈	—	4	2	no	—
207	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
208	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
209	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
210	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
211	C₄⋊C₄	—	2	1	no	—
212	C₄⋊C₄	—	2	1	no	—
213	C₄⋊C₄	—	4	1	no	—
214	C₄⋊C₄	—	4	1	no	—
215	C₄⋊C₄	—	4	1	no	—
216	C₄⋊C₄	—	4	1	no	—
217	C₂²⋊C₄	—	4	1	no	—
218	C₂²⋊C₄	—	4	1	no	—
219	C₂²⋊C₄	—	4	1	no	—
220	C₂²⋊C₄	—	4	1	no	—
221	C₂²⋊C₄	—	4	1	no	—
222	C₄²	—	4	1	no	—
223	C₄²	C₅×C₂²⋊C₄	1	1	yes	characteristic
224	C₄²	C₅×D₈	1	1	yes	characteristic
225	C₄²	C₅×SD₁₆	1	1	yes	characteristic
226	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
227	C₁₆	—	4	2	no	—
228	C₂×C₅	—	4	1	no	—
229	C₂×C₅	—	16	1	no	—
230	C₂×C₅	#1547	1	1	yes	characteristic
231	C₂×C₅	#2143	1	1	yes	characteristic
232	C₂×C₅	C₂×#360	1	1	yes	characteristic
233	C₂³	—	4	1	no	—
234	C₂³	C₅×#102	1	1	yes	characteristic
235	Q₈	—	4	1	no	—
236	Q₈	—	4	1	no	—
237	D₄	—	4	1	no	—
238	D₄	—	4	1	no	—
239	D₄	—	8	2	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	4	1	no	—
244	C₂×C₄	—	4	1	no	—
245	C₂×C₄	—	4	1	no	—
246	C₂×C₄	—	4	1	no	—
247	C₂×C₄	—	8	2	no	—
248	C₂×C₄	C₅×#102	1	1	yes	characteristic
249	C₂×C₄	C₅×#102	1	1	yes	characteristic
250	C₂×C₄	C₅×#102	1	1	yes	characteristic
251	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
252	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
253	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
254	C₈	—	2	1	no	—
255	C₈	—	2	1	no	—
256	C₈	—	2	1	no	—
257	C₈	—	8	2	no	—
258	C₈	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
259	C₈	C₅×#137	1	1	yes	characteristic
260	C₅	#11980	1	1	yes	characteristic, direct product factor, Sylow
261	C₂²	—	2	1	no	—
262	C₂²	—	2	1	no	—
263	C₂²	—	2	1	no	—
264	C₂²	—	8	1	no	—
265	C₂²	—	8	1	no	—
266	C₂²	—	8	1	no	—
267	C₂²	C₂×C₅×#102	1	1	yes	characteristic
268	C₄	—	2	1	no	—
269	C₄	—	2	1	no	—
270	C₄	—	2	1	no	—
271	C₄	—	2	1	no	—
272	C₄	—	8	1	no	—
273	C₄	—	8	1	no	—
274	C₄	—	8	1	no	—
275	C₄	C₅×#360	1	2	yes	—
276	C₄	C₅×#413	1	1	yes	characteristic
277	C₄	C₅×#417	1	1	yes	characteristic
278	C₂	—	4	1	no	—
279	C₂	—	16	1	no	—
280	C₂	C₅×#1547	1	1	yes	characteristic
281	C₂	C₅×#2143	1	1	yes	characteristic
282	C₂	C₂×C₅×#360	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 ▷ C₂×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×#2150	C₂	semidirect
3. C₅×#1566	C₂	semidirect
4. C₅×#1363	C₂	semidirect
5. C₅×#1360	C₂	semidirect
23. #11980	C₅	direct (N × H)
57. #2150	C₂×C₅	semidirect
58. #1566	C₂×C₅	semidirect
59. #1363	C₂×C₅	semidirect
60. #1360	C₂×C₅	semidirect
260. C₅	#11980	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×#2150	2
C₅×#1566	2
C₅×#1363	2
C₅×#1360	2
#11980	5

Subgroups — C5×#11980 #11934700

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#11980 #11934700