Subgroups — C₅×#8196 #11930916

Order 1280.

500 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#2965	C₂	1	1	yes	maximal, characteristic
2	C₄×C₅×#104	C₂	1	1	yes	maximal, characteristic
3	C₄×C₅×#97	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#1476	C₂	1	1	yes	maximal, characteristic
5	C₅×C₈×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#1278	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#1276	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×C₅×#41	C₂²	1	1	yes	characteristic
9	C₅×#446	C₄	1	2	yes	complemented
10	C₅×#436	C₄	1	2	yes	complemented
11	C₅×#435	—	2	2	no	—
12	C₅×#431	—	2	1	no	—
13	C₅×#431	C₂²	1	1	yes	characteristic
14	C₅×#431	C₂²	1	1	yes	characteristic
15	C₂×C₅×#104	C₄	1	2	yes	—
16	C₄×C₅×M₁₆	—	2	1	no	—
17	C₄×C₅×M₁₆	C₂²	1	1	yes	characteristic
18	C₂×C₅×#97	—	2	2	no	—
19	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
21	C₄³×C₅	C₂²	1	1	yes	characteristic
22	C₅×#339	C₄	1	2	yes	—
23	C₅×Q₈⋊C₈	C₄	1	2	yes	complemented
24	C₅×D₄⋊C₈	C₄	1	2	yes	complemented
25	C₅×#322	—	2	1	no	—
26	C₅×C₈²	—	2	1	no	—
27	#8196	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
28	C₂×C₅×#41	C₂×C₄	1	1	yes	characteristic
29	C₂×C₅×M₁₆	—	2	2	no	—
30	C₂×C₅×M₁₆	C₂×C₄	1	2	yes	—
31	C₂²×C₅×C₈	—	2	1	no	—
32	C₂²×C₅×C₈	D₄	1	2	yes	—
33	C₄×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
34	C₄×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
35	C₄×C₅×D₄	—	2	1	no	—
36	C₄×C₅×D₄	—	2	1	no	—
37	C₄×C₅×D₄	C₂×C₄	1	1	yes	characteristic
38	C₄×C₅×D₄	C₂×C₄	1	1	yes	characteristic
39	C₅×#117	—	2	1	no	—
40	C₅×#117	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₅	—	2	1	no	—
42	C₂×C₄²×C₅	—	2	1	no	—
43	C₂×C₄²×C₅	—	2	1	no	—
44	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
46	C₅×C₄⋊C₈	—	2	2	no	—
47	C₅×C₄⋊C₈	C₂×C₄	1	2	yes	—
48	C₅×C₄⋊C₈	C₂×C₄	1	2	yes	—
49	C₅×#104	—	2	4	no	—
50	C₅×C₂²⋊C₈	—	2	1	no	—
51	C₅×C₂²⋊C₈	—	2	1	no	—
52	C₅×C₂²⋊C₈	—	2	2	no	—
53	C₅×#97	—	2	8	no	—
54	C₅×#97	—	2	2	no	—
55	C₅×#97	—	2	1	no	—
56	C₄×C₅×C₈	—	2	1	no	—
57	C₄×C₅×C₈	—	2	1	no	—
58	C₄×C₅×C₈	—	2	1	no	—
59	C₄×C₅×C₈	—	2	1	no	—
60	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
61	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
62	C₄×C₅×C₈	D₄	1	1	yes	characteristic
63	C₄×C₅×C₈	D₄	1	1	yes	characteristic
64	#2965	C₂×C₅	1	1	yes	characteristic
65	C₄×#104	C₂×C₅	1	1	yes	characteristic
66	C₄×#97	C₂×C₅	1	1	yes	characteristic, complemented
67	#1476	C₂×C₅	1	1	yes	characteristic
68	C₈×M₁₆	C₂×C₅	1	1	yes	characteristic, complemented
69	#1278	C₂×C₅	1	1	yes	characteristic, complemented
70	#1276	C₂×C₅	1	1	yes	characteristic, complemented
71	C₅×#41	—	2	2	no	—
72	C₅×#41	—	2	2	no	—
73	C₂×C₅×Q₈	C₄²	1	1	yes	characteristic
74	C₂×C₅×D₄	—	2	1	no	—
75	C₂×C₅×D₄	C₄²	1	1	yes	characteristic
76	C₂²×C₄×C₅	—	2	2	no	—
77	C₂²×C₄×C₅	—	2	1	no	—
78	C₂²×C₄×C₅	—	2	1	no	—
79	C₂²×C₄×C₅	—	2	1	no	—
80	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
81	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₅	#41	1	1	yes	characteristic
83	C₅×M₁₆	—	2	2	no	—
84	C₅×M₁₆	—	4	4	no	—
85	C₅×M₁₆	M₁₆	1	4	yes	complemented
86	C₂×C₅×C₈	—	2	4	no	—
87	C₂×C₅×C₈	—	2	2	no	—
88	C₂×C₅×C₈	—	2	2	no	—
89	C₂×C₅×C₈	—	2	4	no	—
90	C₂×C₅×C₈	—	2	2	no	—
91	C₂×C₅×C₈	—	2	2	no	—
92	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
93	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
94	C₅×C₄⋊C₄	—	2	1	no	—
95	C₅×C₄⋊C₄	—	2	1	no	—
96	C₅×C₄⋊C₄	C₄²	1	1	yes	characteristic
97	C₅×C₄⋊C₄	C₄²	1	1	yes	characteristic
98	C₅×C₂²⋊C₄	—	2	1	no	—
99	C₅×C₂²⋊C₄	—	2	1	no	—
100	C₅×C₂²⋊C₄	—	2	1	no	—
101	C₄²×C₅	—	2	4	no	—
102	C₄²×C₅	—	2	1	no	—
103	C₄²×C₅	—	2	1	no	—
104	C₄²×C₅	—	2	1	no	—
105	C₄²×C₅	—	2	2	no	—
106	C₄²×C₅	—	2	2	no	—
107	C₄²×C₅	—	4	2	no	—
108	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
109	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
110	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
111	C₄²×C₅	#41	1	1	yes	characteristic
112	C₄×#41	C₂²×C₅	1	1	yes	characteristic
113	#446	C₄×C₅	1	2	yes	complemented
114	#436	C₄×C₅	1	2	yes	complemented
115	#435	—	2	2	no	—
116	#431	—	2	1	no	—
117	#431	C₂²×C₅	1	1	yes	characteristic
118	#431	C₂²×C₅	1	1	yes	characteristic
119	C₂×#104	C₄×C₅	1	2	yes	—
120	C₄×M₁₆	—	2	1	no	—
121	C₄×M₁₆	C₂²×C₅	1	1	yes	characteristic
122	C₂×#97	—	2	2	no	—
123	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
124	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
125	C₄³	C₂²×C₅	1	1	yes	characteristic
126	#339	C₄×C₅	1	2	yes	—
127	Q₈⋊C₈	C₄×C₅	1	2	yes	complemented
128	D₄⋊C₈	C₄×C₅	1	2	yes	complemented
129	#322	—	2	1	no	—
130	C₈²	—	2	1	no	—
131	C₂³×C₅	—	2	1	no	—
132	C₂³×C₅	C₄×D₄	1	1	yes	characteristic
133	C₅×Q₈	—	2	1	no	—
134	C₅×Q₈	—	2	1	no	—
135	C₅×D₄	—	2	1	no	—
136	C₅×D₄	—	2	1	no	—
137	C₅×D₄	—	4	2	no	—
138	C₂×C₄×C₅	—	2	2	no	—
139	C₂×C₄×C₅	—	2	2	no	—
140	C₂×C₄×C₅	—	2	4	no	—
141	C₂×C₄×C₅	—	2	1	no	—
142	C₂×C₄×C₅	—	2	1	no	—
143	C₂×C₄×C₅	—	2	1	no	—
144	C₂×C₄×C₅	—	4	2	no	—
145	C₂×C₄×C₅	—	4	2	no	—
146	C₂×C₄×C₅	—	4	2	no	—
147	C₂×C₄×C₅	—	4	2	no	—
148	C₂×C₄×C₅	—	4	2	no	—
149	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
150	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
151	C₂×C₄×C₅	#117	1	1	yes	characteristic
152	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
153	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
154	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
155	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
156	C₂×C₄×C₅	#131	1	2	yes	—
157	C₅×C₈	—	2	4	no	—
158	C₅×C₈	—	2	2	no	—
159	C₅×C₈	—	2	4	no	—
160	C₅×C₈	—	4	2	no	—
161	C₅×C₈	—	4	2	no	—
162	C₅×C₈	#104	1	4	yes	complemented
163	C₂×#41	C₂×C₄×C₅	1	1	yes	characteristic
164	C₂×M₁₆	—	2	2	no	—
165	C₂×M₁₆	C₂×C₄×C₅	1	2	yes	—
166	C₂²×C₈	—	2	1	no	—
167	C₂²×C₈	C₅×D₄	1	2	yes	—
168	C₄×Q₈	C₂×C₄×C₅	1	1	yes	characteristic
169	C₄×Q₈	C₂×C₄×C₅	1	1	yes	characteristic
170	C₄×D₄	—	2	1	no	—
171	C₄×D₄	—	2	1	no	—
172	C₄×D₄	C₂×C₄×C₅	1	1	yes	characteristic
173	C₄×D₄	C₂×C₄×C₅	1	1	yes	characteristic
174	#117	—	2	1	no	—
175	#117	C₂×C₄×C₅	1	1	yes	characteristic
176	C₂×C₄²	—	2	1	no	—
177	C₂×C₄²	—	2	1	no	—
178	C₂×C₄²	—	2	1	no	—
179	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
180	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
181	C₄⋊C₈	—	2	2	no	—
182	C₄⋊C₈	C₂×C₄×C₅	1	2	yes	—
183	C₄⋊C₈	C₂×C₄×C₅	1	2	yes	—
184	#104	—	2	4	no	—
185	C₂²⋊C₈	—	2	1	no	—
186	C₂²⋊C₈	—	2	1	no	—
187	C₂²⋊C₈	—	2	2	no	—
188	#97	—	2	8	no	—
189	#97	—	2	2	no	—
190	#97	—	2	1	no	—
191	C₄×C₈	—	2	1	no	—
192	C₄×C₈	—	2	1	no	—
193	C₄×C₈	—	2	1	no	—
194	C₄×C₈	—	2	1	no	—
195	C₄×C₈	C₂×C₄×C₅	1	2	yes	—
196	C₄×C₈	C₂×C₄×C₅	1	2	yes	—
197	C₄×C₈	C₅×D₄	1	1	yes	characteristic
198	C₄×C₈	C₅×D₄	1	1	yes	characteristic
199	C₂²×C₅	—	2	1	no	—
200	C₂²×C₅	—	2	1	no	—
201	C₂²×C₅	—	4	1	no	—
202	C₂²×C₅	—	4	1	no	—
203	C₂²×C₅	—	4	1	no	—
204	C₂²×C₅	C₄×C₂²⋊C₄	1	1	yes	characteristic
205	C₂²×C₅	#436	1	2	yes	—
206	C₄×C₅	—	2	1	no	—
207	C₄×C₅	—	2	1	no	—
208	C₄×C₅	—	4	4	no	—
209	C₄×C₅	—	4	1	no	—
210	C₄×C₅	—	4	1	no	—
211	C₄×C₅	—	4	1	no	—
212	C₄×C₅	C₄×M₁₆	1	1	yes	characteristic
213	C₄×C₅	#405	1	1	yes	characteristic
214	C₄×C₅	C₂×#104	1	2	yes	—
215	C₄×C₅	#435	1	2	yes	—
216	C₄×C₅	#444	1	2	yes	—
217	#41	—	2	2	no	—
218	#41	—	2	2	no	—
219	C₂×Q₈	C₄²×C₅	1	1	yes	characteristic
220	C₂×D₄	—	2	1	no	—
221	C₂×D₄	C₄²×C₅	1	1	yes	characteristic
222	C₂²×C₄	—	2	2	no	—
223	C₂²×C₄	—	2	1	no	—
224	C₂²×C₄	—	2	1	no	—
225	C₂²×C₄	—	2	1	no	—
226	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
227	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
228	C₂²×C₄	C₅×#41	1	1	yes	characteristic
229	M₁₆	—	2	2	no	—
230	M₁₆	—	4	4	no	—
231	M₁₆	C₅×M₁₆	1	4	yes	complemented
232	C₂×C₈	—	2	4	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	2	4	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	C₅×C₂²⋊C₄	1	2	yes	—
239	C₂×C₈	C₅×C₂²⋊C₄	1	2	yes	—
240	C₄⋊C₄	—	2	1	no	—
241	C₄⋊C₄	—	2	1	no	—
242	C₄⋊C₄	C₄²×C₅	1	1	yes	characteristic
243	C₄⋊C₄	C₄²×C₅	1	1	yes	characteristic
244	C₂²⋊C₄	—	2	1	no	—
245	C₂²⋊C₄	—	2	1	no	—
246	C₂²⋊C₄	—	2	1	no	—
247	C₄²	—	2	4	no	—
248	C₄²	—	2	1	no	—
249	C₄²	—	2	1	no	—
250	C₄²	—	2	1	no	—
251	C₄²	—	2	2	no	—
252	C₄²	—	2	2	no	—
253	C₄²	—	4	2	no	—
254	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
255	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
256	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
257	C₄²	C₅×#41	1	1	yes	characteristic
258	C₂×C₅	—	2	2	no	—
259	C₂×C₅	—	8	1	no	—
260	C₂×C₅	#1753	1	1	yes	characteristic
261	C₂×C₅	C₄×#104	1	1	yes	characteristic
262	C₂×C₅	#1765	1	1	yes	characteristic
263	C₂³	—	2	1	no	—
264	C₂³	C₄×C₅×D₄	1	1	yes	characteristic
265	Q₈	—	2	1	no	—
266	Q₈	—	2	1	no	—
267	D₄	—	2	1	no	—
268	D₄	—	2	1	no	—
269	D₄	—	4	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	C₂×C₄²×C₅	1	1	yes	characteristic
282	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
283	C₂×C₄	C₅×#117	1	1	yes	characteristic
284	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
285	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
286	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
287	C₂×C₄	C₂×C₅×M₁₆	1	2	yes	—
288	C₂×C₄	C₅×#131	1	2	yes	—
289	C₈	—	2	4	no	—
290	C₈	—	2	2	no	—
291	C₈	—	2	4	no	—
292	C₈	—	4	2	no	—
293	C₈	—	4	2	no	—
294	C₈	C₅×#104	1	4	yes	complemented
295	C₅	#8196	1	1	yes	characteristic, direct product factor, Sylow
296	C₂²	—	2	1	no	—
297	C₂²	—	2	1	no	—
298	C₂²	—	4	1	no	—
299	C₂²	—	4	1	no	—
300	C₂²	—	4	1	no	—
301	C₂²	C₄×C₅×C₂²⋊C₄	1	1	yes	characteristic
302	C₂²	C₅×#436	1	2	yes	—
303	C₄	—	2	1	no	—
304	C₄	—	2	1	no	—
305	C₄	—	4	4	no	—
306	C₄	—	4	1	no	—
307	C₄	—	4	1	no	—
308	C₄	—	4	1	no	—
309	C₄	C₄×C₅×M₁₆	1	1	yes	characteristic
310	C₄	C₅×#405	1	1	yes	characteristic
311	C₄	C₂×C₅×#104	1	2	yes	—
312	C₄	C₅×#435	1	2	yes	—
313	C₄	C₅×#444	1	2	yes	—
314	C₂	—	2	2	no	—
315	C₂	—	8	1	no	—
316	C₂	C₅×#1753	1	1	yes	characteristic
317	C₂	C₄×C₅×#104	1	1	yes	characteristic
318	C₂	C₅×#1765	1	1	yes	characteristic

	subgroup	G/H
center	C₄²×C₅	C₂×D₄
commutator	C₂×C₄	C₂×C₄²×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
socle	C₂²×C₅	C₄×C₂²⋊C₄
prime radical	C₂×C₅×#41	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄²×C₅ ◁ C₂×C₄×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₄×C₅×#97	C₂	semidirect
5. C₅×C₈×M₁₆	C₂	semidirect
6. C₅×#1278	C₂	semidirect
7. C₅×#1276	C₂	semidirect
9. C₅×#446	C₄	semidirect
10. C₅×#436	C₄	semidirect
23. C₅×Q₈⋊C₈	C₄	semidirect
24. C₅×D₄⋊C₈	C₄	semidirect
27. #8196	C₅	direct (N × H)
66. C₄×#97	C₂×C₅	semidirect
68. C₈×M₁₆	C₂×C₅	semidirect
69. #1278	C₂×C₅	semidirect
70. #1276	C₂×C₅	semidirect
85. C₅×M₁₆	M₁₆	semidirect
113. #446	C₄×C₅	semidirect
114. #436	C₄×C₅	semidirect
127. Q₈⋊C₈	C₄×C₅	semidirect
128. D₄⋊C₈	C₄×C₅	semidirect
162. C₅×C₈	#104	semidirect
231. M₁₆	C₅×M₁₆	semidirect
294. C₈	C₅×#104	semidirect
295. C₅	#8196	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×#2965	2
C₄×C₅×#104	2
C₄×C₅×#97	2
C₅×#1476	2
C₅×C₈×M₁₆	2
C₅×#1278	2
C₅×#1276	2
#8196	5

Subgroups — C5×#8196 #11930916

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#8196 #11930916