Subgroups — C₅×#12917 #11935637

Order 1280.

384 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#504	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#364	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#2151	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅×#350	C₂	1	1	yes	maximal, characteristic
5	C₅×#1764	C₂	1	1	yes	maximal, characteristic
6	C₅×#1379	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#1374	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅×#131	C₂²	1	1	yes	characteristic
9	C₅×#504	—	2	2	no	—
10	C₅×#504	—	2	2	no	—
11	C₂×C₅×M₃₂	—	2	1	no	—
12	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
13	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
14	C₂²×C₅×C₁₆	—	2	1	no	—
15	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
16	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
17	C₂×C₅×#104	C₄	1	1	yes	characteristic
18	C₅×#416	C₄	1	1	yes	characteristic
19	C₅×#405	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
21	C₅×#364	—	2	2	no	—
22	C₅×C₄⋊C₁₆	—	2	1	no	—
23	C₅×C₄⋊C₁₆	—	2	1	no	—
24	C₅×#350	—	2	2	no	—
25	C₅×#341	C₄	1	1	yes	characteristic
26	C₅×#339	C₄	1	1	yes	characteristic
27	#12917	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
28	C₂×C₅×#41	C₂×C₄	1	1	yes	characteristic
29	C₅×#131	—	2	1	no	—
30	C₅×#131	—	2	1	no	—
31	C₅×#131	D₄	1	2	yes	—
32	C₅×#131	Q₈	1	2	yes	—
33	C₂×C₅×M₁₆	—	2	1	no	—
34	C₂×C₅×M₁₆	C₂×C₄	1	1	yes	characteristic
35	C₂×C₅×M₁₆	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₅×C₈	—	2	1	no	—
37	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
39	C₅×#117	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
41	C₅×M₃₂	—	2	4	no	—
42	C₅×M₃₂	—	2	1	no	—
43	C₅×M₃₂	—	2	1	no	—
44	C₅×M₃₂	—	2	1	no	—
45	C₅×M₃₂	D₄	1	2	yes	—
46	C₂×C₅×C₁₆	—	2	4	no	—
47	C₂×C₅×C₁₆	—	2	1	no	—
48	C₂×C₅×C₁₆	—	2	1	no	—
49	C₂×C₅×C₁₆	—	2	1	no	—
50	C₂×C₅×C₁₆	—	2	1	no	—
51	C₂×C₅×C₁₆	—	2	1	no	—
52	C₂×C₅×C₁₆	—	2	1	no	—
53	C₂×C₅×C₁₆	—	2	1	no	—
54	C₂×C₅×C₁₆	D₄	1	1	yes	characteristic
55	C₂×C₅×C₁₆	D₄	1	1	yes	characteristic
56	C₅×#104	—	2	2	no	—
57	C₅×#103	C₈	1	1	yes	characteristic
58	C₅×#103	C₈	1	1	yes	characteristic
59	C₅×#102	C₈	1	1	yes	characteristic
60	C₅×#102	C₈	1	1	yes	characteristic
61	C₅×#97	—	2	1	no	—
62	C₄×C₅×C₈	—	2	1	no	—
63	C₄×C₅×C₈	—	2	1	no	—
64	C₄×C₅×C₈	—	2	1	no	—
65	C₂×#504	C₂×C₅	1	1	yes	characteristic
66	C₂×#364	C₂×C₅	1	1	yes	characteristic, complemented
67	#2151	C₂×C₅	1	1	yes	characteristic, complemented
68	C₂×#350	C₂×C₅	1	1	yes	characteristic
69	#1764	C₂×C₅	1	1	yes	characteristic
70	#1379	C₂×C₅	1	1	yes	characteristic, complemented
71	#1374	C₂×C₅	1	1	yes	characteristic, complemented
72	C₅×#41	—	2	1	no	—
73	C₅×#41	—	2	1	no	—
74	C₅×#41	C₄⋊C₄	1	2	yes	—
75	C₅×#41	M₁₆	1	2	yes	—
76	C₂×C₅×Q₈	C₂×C₈	1	1	yes	characteristic
77	C₂×C₅×D₄	—	2	1	no	—
78	C₂×C₅×D₄	C₂×C₈	1	1	yes	characteristic
79	C₂²×C₄×C₅	—	2	1	no	—
80	C₂²×C₄×C₅	—	2	1	no	—
81	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
82	C₅×M₁₆	—	2	1	no	—
83	C₅×M₁₆	—	2	2	no	—
84	C₅×M₁₆	—	2	2	no	—
85	C₅×M₁₆	—	2	1	no	—
86	C₅×M₁₆	—	2	1	no	—
87	C₅×M₁₆	C₄⋊C₄	1	2	yes	—
88	C₂×C₅×C₈	—	2	1	no	—
89	C₂×C₅×C₈	—	2	1	no	—
90	C₂×C₅×C₈	—	2	1	no	—
91	C₂×C₅×C₈	—	2	2	no	—
92	C₂×C₅×C₈	—	2	2	no	—
93	C₂×C₅×C₈	—	2	1	no	—
94	C₂×C₅×C₈	C₂×C₈	1	1	yes	characteristic
95	C₂×C₅×C₈	C₂×C₈	1	1	yes	characteristic
96	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
97	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
98	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
99	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
100	C₂×C₅×C₈	#41	1	1	yes	characteristic
101	C₂×C₅×C₈	#41	1	1	yes	characteristic
102	C₅×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
103	C₅×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
104	C₅×C₂²⋊C₄	—	2	1	no	—
105	C₄²×C₅	—	2	1	no	—
106	C₄²×C₅	—	2	1	no	—
107	C₄²×C₅	—	2	1	no	—
108	C₅×C₁₆	—	2	1	no	—
109	C₅×C₁₆	—	2	2	no	—
110	C₅×C₁₆	—	2	1	no	—
111	C₅×C₁₆	—	4	1	no	—
112	C₅×C₁₆	—	4	1	no	—
113	C₂×#131	C₂²×C₅	1	1	yes	characteristic
114	#504	—	2	2	no	—
115	#504	—	2	2	no	—
116	C₂×M₃₂	—	2	1	no	—
117	C₂×M₃₂	C₂²×C₅	1	1	yes	characteristic
118	C₂×M₃₂	C₂²×C₅	1	1	yes	characteristic
119	C₂²×C₁₆	—	2	1	no	—
120	C₂²×C₁₆	C₂²×C₅	1	1	yes	characteristic
121	C₂²×C₁₆	C₂²×C₅	1	1	yes	characteristic
122	C₂×#104	C₄×C₅	1	1	yes	characteristic
123	#416	C₄×C₅	1	1	yes	characteristic
124	#405	C₂²×C₅	1	1	yes	characteristic
125	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
126	#364	—	2	2	no	—
127	C₄⋊C₁₆	—	2	1	no	—
128	C₄⋊C₁₆	—	2	1	no	—
129	#350	—	2	2	no	—
130	#341	C₄×C₅	1	1	yes	characteristic
131	#339	C₄×C₅	1	1	yes	characteristic
132	C₂³×C₅	—	2	1	no	—
133	C₂³×C₅	#131	1	1	yes	characteristic
134	C₅×Q₈	—	2	1	no	—
135	C₅×Q₈	C₄⋊C₈	1	2	yes	—
136	C₅×D₄	—	2	2	no	—
137	C₅×D₄	—	2	2	no	—
138	C₅×D₄	—	2	1	no	—
139	C₅×D₄	C₄⋊C₈	1	2	yes	—
140	C₂×C₄×C₅	—	2	1	no	—
141	C₂×C₄×C₅	—	2	2	no	—
142	C₂×C₄×C₅	—	2	2	no	—
143	C₂×C₄×C₅	—	2	2	no	—
144	C₂×C₄×C₅	—	2	1	no	—
145	C₂×C₄×C₅	—	2	1	no	—
146	C₂×C₄×C₅	—	4	1	no	—
147	C₂×C₄×C₅	—	4	1	no	—
148	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
149	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
150	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
151	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
152	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
153	C₂×C₄×C₅	#131	1	1	yes	characteristic
154	C₅×C₈	—	2	2	no	—
155	C₅×C₈	—	2	1	no	—
156	C₅×C₈	—	2	1	no	—
157	C₅×C₈	—	4	1	no	—
158	C₅×C₈	#120	1	1	yes	characteristic
159	C₅×C₈	C₂²⋊Q₈	1	1	yes	characteristic
160	C₅×C₈	C₂²⋊Q₈	1	1	yes	characteristic
161	C₅×C₈	#123	1	1	yes	characteristic
162	C₂×#41	C₂×C₄×C₅	1	1	yes	characteristic
163	#131	—	2	1	no	—
164	#131	—	2	1	no	—
165	#131	C₅×D₄	1	2	yes	—
166	#131	C₅×Q₈	1	2	yes	—
167	C₂×M₁₆	—	2	1	no	—
168	C₂×M₁₆	C₂×C₄×C₅	1	1	yes	characteristic
169	C₂×M₁₆	C₂×C₄×C₅	1	1	yes	characteristic
170	C₂²×C₈	—	2	1	no	—
171	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
172	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
173	#117	C₂×C₄×C₅	1	1	yes	characteristic
174	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
175	M₃₂	—	2	4	no	—
176	M₃₂	—	2	1	no	—
177	M₃₂	—	2	1	no	—
178	M₃₂	—	2	1	no	—
179	M₃₂	C₅×D₄	1	2	yes	—
180	C₂×C₁₆	—	2	4	no	—
181	C₂×C₁₆	—	2	1	no	—
182	C₂×C₁₆	—	2	1	no	—
183	C₂×C₁₆	—	2	1	no	—
184	C₂×C₁₆	—	2	1	no	—
185	C₂×C₁₆	—	2	1	no	—
186	C₂×C₁₆	—	2	1	no	—
187	C₂×C₁₆	—	2	1	no	—
188	C₂×C₁₆	C₅×D₄	1	1	yes	characteristic
189	C₂×C₁₆	C₅×D₄	1	1	yes	characteristic
190	#104	—	2	2	no	—
191	#103	C₅×C₈	1	1	yes	characteristic
192	#103	C₅×C₈	1	1	yes	characteristic
193	#102	C₅×C₈	1	1	yes	characteristic
194	#102	C₅×C₈	1	1	yes	characteristic
195	#97	—	2	1	no	—
196	C₄×C₈	—	2	1	no	—
197	C₄×C₈	—	2	1	no	—
198	C₄×C₈	—	2	1	no	—
199	C₂²×C₅	—	2	1	no	—
200	C₂²×C₅	—	2	1	no	—
201	C₂²×C₅	—	2	2	no	—
202	C₂²×C₅	—	4	1	no	—
203	C₂²×C₅	—	4	1	no	—
204	C₂²×C₅	#424	1	1	yes	characteristic
205	C₂²×C₅	C₈×D₄	1	1	yes	characteristic
206	C₂²×C₅	#435	1	1	yes	characteristic
207	C₄×C₅	—	2	2	no	—
208	C₄×C₅	—	4	2	no	—
209	C₄×C₅	—	4	1	no	—
210	C₄×C₅	—	4	1	no	—
211	C₄×C₅	#385	1	1	yes	characteristic
212	C₄×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
213	C₄×C₅	C₈×D₄	1	1	yes	characteristic
214	C₄×C₅	#435	1	1	yes	characteristic
215	#41	—	2	1	no	—
216	#41	—	2	1	no	—
217	#41	C₅×C₄⋊C₄	1	2	yes	—
218	#41	C₅×M₁₆	1	2	yes	—
219	C₂×Q₈	C₂×C₅×C₈	1	1	yes	characteristic
220	C₂×D₄	—	2	1	no	—
221	C₂×D₄	C₂×C₅×C₈	1	1	yes	characteristic
222	C₂²×C₄	—	2	1	no	—
223	C₂²×C₄	—	2	1	no	—
224	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
225	M₁₆	—	2	1	no	—
226	M₁₆	—	2	2	no	—
227	M₁₆	—	2	2	no	—
228	M₁₆	—	2	1	no	—
229	M₁₆	—	2	1	no	—
230	M₁₆	C₅×C₄⋊C₄	1	2	yes	—
231	C₂×C₈	—	2	1	no	—
232	C₂×C₈	—	2	1	no	—
233	C₂×C₈	—	2	1	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	2	1	no	—
237	C₂×C₈	C₂×C₅×C₈	1	1	yes	characteristic
238	C₂×C₈	C₂×C₅×C₈	1	1	yes	characteristic
239	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
240	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
241	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
242	C₂×C₈	C₂×C₅×Q₈	1	1	yes	characteristic
243	C₂×C₈	C₅×#41	1	1	yes	characteristic
244	C₂×C₈	C₅×#41	1	1	yes	characteristic
245	C₄⋊C₄	C₂×C₅×C₈	1	1	yes	characteristic
246	C₄⋊C₄	C₂×C₅×C₈	1	1	yes	characteristic
247	C₂²⋊C₄	—	2	1	no	—
248	C₄²	—	2	1	no	—
249	C₄²	—	2	1	no	—
250	C₄²	—	2	1	no	—
251	C₁₆	—	2	1	no	—
252	C₁₆	—	2	2	no	—
253	C₁₆	—	2	1	no	—
254	C₁₆	—	4	1	no	—
255	C₁₆	—	4	1	no	—
256	C₂×C₅	—	2	1	no	—
257	C₂×C₅	—	2	1	no	—
258	C₂×C₅	—	4	2	no	—
259	C₂×C₅	#1850	1	1	yes	characteristic
260	C₂×C₅	#2170	1	1	yes	characteristic
261	C₂×C₅	#2171	1	1	yes	characteristic
262	C₂³	—	2	1	no	—
263	C₂³	C₅×#131	1	1	yes	characteristic
264	Q₈	—	2	1	no	—
265	Q₈	C₅×C₄⋊C₈	1	2	yes	—
266	D₄	—	2	2	no	—
267	D₄	—	2	2	no	—
268	D₄	—	2	1	no	—
269	D₄	C₅×C₄⋊C₈	1	2	yes	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	4	1	no	—
277	C₂×C₄	—	4	1	no	—
278	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
279	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
280	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
281	C₂×C₄	C₂²×C₅×C₈	1	1	yes	characteristic
282	C₂×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
283	C₂×C₄	C₅×#131	1	1	yes	characteristic
284	C₈	—	2	2	no	—
285	C₈	—	2	1	no	—
286	C₈	—	2	1	no	—
287	C₈	—	4	1	no	—
288	C₈	C₅×#120	1	1	yes	characteristic
289	C₈	C₅×C₂²⋊Q₈	1	1	yes	characteristic
290	C₈	C₅×C₂²⋊Q₈	1	1	yes	characteristic
291	C₈	C₅×#123	1	1	yes	characteristic
292	C₅	#12917	1	1	yes	characteristic, direct product factor, Sylow
293	C₂²	—	2	1	no	—
294	C₂²	—	2	1	no	—
295	C₂²	—	2	2	no	—
296	C₂²	—	4	1	no	—
297	C₂²	—	4	1	no	—
298	C₂²	C₅×#424	1	1	yes	characteristic
299	C₂²	C₅×C₈×D₄	1	1	yes	characteristic
300	C₂²	C₅×#435	1	1	yes	characteristic
301	C₄	—	2	2	no	—
302	C₄	—	4	2	no	—
303	C₄	—	4	1	no	—
304	C₄	—	4	1	no	—
305	C₄	C₅×#385	1	1	yes	characteristic
306	C₄	C₂×C₅×C₄⋊C₈	1	1	yes	characteristic
307	C₄	C₅×C₈×D₄	1	1	yes	characteristic
308	C₄	C₅×#435	1	1	yes	characteristic
309	C₂	—	2	1	no	—
310	C₂	—	2	1	no	—
311	C₂	—	4	2	no	—
312	C₂	C₅×#1850	1	1	yes	characteristic
313	C₂	C₅×#2170	1	1	yes	characteristic
314	C₂	C₅×#2171	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₅×C₈	C₂×D₄
commutator	C₂×C₄	C₂²×C₅×C₈
Frattini, cosocle	C₂²×C₈	C₂³×C₅
socle	C₂²×C₅	C₈×D₄
prime radical	C₂×C₅×#41	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅×C₈ ◁ C₂²×C₅×C₁₆ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×#364	C₂	semidirect
3. C₅×#2151	C₂	semidirect
6. C₅×#1379	C₂	semidirect
7. C₅×#1374	C₂	semidirect
27. #12917	C₅	direct (N × H)
66. C₂×#364	C₂×C₅	semidirect
67. #2151	C₂×C₅	semidirect
70. #1379	C₂×C₅	semidirect
71. #1374	C₂×C₅	semidirect
292. C₅	#12917	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅×#504	2
C₂×C₅×#364	2
C₅×#2151	2
C₂×C₅×#350	2
C₅×#1764	2
C₅×#1379	2
C₅×#1374	2
#12917	5

Subgroups — C5×#12917 #11935637

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#12917 #11935637