Subgroups — C₅×#12649 #11935369

Order 1280.

428 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#2943	C₂	1	1	yes	maximal, characteristic
2	C₅×#2145	C₂	1	1	yes	maximal, characteristic
3	C₅×#2143	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅×#347	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#1773	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#1389	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#1384	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅×#135	C₂²	1	1	yes	characteristic
9	C₄×C₅×#41	—	2	1	no	—
10	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
11	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
12	C₅×#442	C₄	1	2	yes	complemented
13	C₅×#441	C₄	1	2	yes	complemented
14	C₅×#440	—	2	2	no	—
15	C₂×C₅×#108	C₂²	1	1	yes	characteristic
16	C₅×#427	C₂²	1	1	yes	characteristic
17	C₅×#427	C₂²	1	1	yes	characteristic
18	C₅×#416	—	2	1	no	—
19	C₅×#405	—	2	1	no	—
20	C₂×C₅×#97	C₂²	1	1	yes	characteristic
21	C₅×#360	C₄	1	4	yes	complemented
22	C₅×#359	C₄	1	4	yes	complemented
23	C₅×#347	—	2	2	no	—
24	C₅×#339	—	2	1	no	—
25	#12649	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
26	C₂×C₅×#41	—	2	1	no	—
27	C₅×#135	—	2	1	no	—
28	C₅×#135	—	2	1	no	—
29	C₅×#135	C₂×C₄	1	2	yes	—
30	C₅×#135	C₂×C₄	1	2	yes	—
31	C₂×C₅×Q₁₆	C₂×C₄	1	1	yes	characteristic
32	C₂×C₅×SD₁₆	—	2	1	no	—
33	C₂×C₅×D₈	C₂×C₄	1	1	yes	characteristic
34	C₂×C₅×M₁₆	—	2	1	no	—
35	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
36	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
37	C₄×C₅×Q₈	—	2	2	no	—
38	C₄×C₅×D₄	—	2	2	no	—
39	C₄×C₅×D₄	—	4	2	no	—
40	C₅×#117	—	2	1	no	—
41	C₅×#117	—	2	1	no	—
42	C₅×#117	—	4	1	no	—
43	C₂×C₄²×C₅	—	4	1	no	—
44	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
45	C₅×M₃₂	—	2	1	no	—
46	C₅×M₃₂	—	2	1	no	—
47	C₅×M₃₂	—	2	1	no	—
48	C₅×M₃₂	C₂×C₄	1	2	yes	complemented
49	C₂×C₅×C₁₆	—	2	1	no	—
50	C₂×C₅×C₁₆	C₂×C₄	1	2	yes	complemented
51	C₅×#108	—	2	1	no	—
52	C₅×#108	C₂×C₄	1	2	yes	complemented
53	C₅×#107	—	2	1	no	—
54	C₅×#107	—	2	1	no	—
55	C₅×#106	C₂×C₄	1	2	yes	—
56	C₅×#106	C₂×C₄	1	2	yes	complemented
57	C₅×#103	—	2	2	no	—
58	C₅×#102	—	2	2	no	—
59	C₅×#97	—	4	1	no	—
60	C₅×#97	C₂×C₄	1	2	yes	—
61	C₅×#97	D₄	1	2	yes	—
62	C₄×C₅×C₈	—	4	1	no	—
63	#2943	C₂×C₅	1	1	yes	characteristic
64	#2145	C₂×C₅	1	1	yes	characteristic
65	#2143	C₂×C₅	1	1	yes	characteristic
66	C₂×#347	C₂×C₅	1	1	yes	characteristic, complemented
67	#1773	C₂×C₅	1	1	yes	characteristic, complemented
68	#1389	C₂×C₅	1	1	yes	characteristic, complemented
69	#1384	C₂×C₅	1	1	yes	characteristic, complemented
70	C₅×#41	—	2	2	no	—
71	C₅×#41	—	2	2	no	—
72	C₅×#41	—	4	1	no	—
73	C₅×#41	—	4	1	no	—
74	C₂×C₅×Q₈	—	2	1	no	—
75	C₂×C₅×D₄	—	2	1	no	—
76	C₂×C₅×D₄	—	4	1	no	—
77	C₂²×C₄×C₅	—	2	1	no	—
78	C₂²×C₄×C₅	—	4	2	no	—
79	C₂²×C₄×C₅	—	4	1	no	—
80	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
81	C₅×Q₁₆	—	2	1	no	—
82	C₅×Q₁₆	C₄²	1	2	yes	—
83	C₅×SD₁₆	—	2	2	no	—
84	C₅×SD₁₆	—	2	2	no	—
85	C₅×D₈	—	2	1	no	—
86	C₅×D₈	C₄²	1	2	yes	—
87	C₅×M₁₆	—	2	2	no	—
88	C₅×M₁₆	—	4	1	no	—
89	C₂×C₅×C₈	—	2	1	no	—
90	C₂×C₅×C₈	—	2	1	no	—
91	C₂×C₅×C₈	—	4	1	no	—
92	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
93	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
97	C₂×C₅×C₈	#41	1	1	yes	characteristic
98	C₂×C₅×C₈	#41	1	1	yes	characteristic
99	C₅×C₄⋊C₄	—	2	2	no	—
100	C₅×C₄⋊C₄	—	2	2	no	—
101	C₅×C₄⋊C₄	—	4	2	no	—
102	C₅×C₂²⋊C₄	—	4	2	no	—
103	C₅×C₂²⋊C₄	—	4	1	no	—
104	C₅×C₂²⋊C₄	—	4	1	no	—
105	C₄²×C₅	—	2	1	no	—
106	C₄²×C₅	—	4	2	no	—
107	C₄²×C₅	—	4	1	no	—
108	C₄²×C₅	—	4	1	no	—
109	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
110	C₅×C₁₆	—	2	2	no	—
111	C₅×C₁₆	—	2	2	no	—
112	C₂×#135	C₂²×C₅	1	1	yes	characteristic
113	C₄×#41	—	2	1	no	—
114	C₂×M₃₂	C₂²×C₅	1	1	yes	characteristic
115	C₂×M₃₂	C₂²×C₅	1	1	yes	characteristic
116	#442	C₄×C₅	1	2	yes	complemented
117	#441	C₄×C₅	1	2	yes	complemented
118	#440	—	2	2	no	—
119	C₂×#108	C₂²×C₅	1	1	yes	characteristic
120	#427	C₂²×C₅	1	1	yes	characteristic
121	#427	C₂²×C₅	1	1	yes	characteristic
122	#416	—	2	1	no	—
123	#405	—	2	1	no	—
124	C₂×#97	C₂²×C₅	1	1	yes	characteristic
125	#360	C₄×C₅	1	4	yes	complemented
126	#359	C₄×C₅	1	4	yes	complemented
127	#347	—	2	2	no	—
128	#339	—	2	1	no	—
129	C₂³×C₅	—	4	1	no	—
130	C₂³×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
131	C₅×Q₈	—	2	2	no	—
132	C₅×Q₈	—	4	1	no	—
133	C₅×D₄	—	2	2	no	—
134	C₅×D₄	—	4	2	no	—
135	C₅×D₄	—	4	2	no	—
136	C₅×D₄	—	4	1	no	—
137	C₂×C₄×C₅	—	2	2	no	—
138	C₂×C₄×C₅	—	4	1	no	—
139	C₂×C₄×C₅	—	4	2	no	—
140	C₂×C₄×C₅	—	4	2	no	—
141	C₂×C₄×C₅	—	4	2	no	—
142	C₂×C₄×C₅	—	4	2	no	—
143	C₂×C₄×C₅	—	8	1	no	—
144	C₂×C₄×C₅	—	8	4	no	—
145	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
146	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
147	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
148	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
149	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
150	C₂×C₄×C₅	#137	1	1	yes	characteristic
151	C₅×C₈	—	2	2	no	—
152	C₅×C₈	—	4	2	no	—
153	C₅×C₈	C₂×C₄²	1	1	yes	characteristic
154	C₅×C₈	#117	1	1	yes	characteristic
155	C₅×C₈	C₄×D₄	1	1	yes	characteristic
156	C₅×C₈	C₄×D₄	1	1	yes	characteristic
157	C₂×#41	—	2	1	no	—
158	#135	—	2	1	no	—
159	#135	—	2	1	no	—
160	#135	C₂×C₄×C₅	1	2	yes	—
161	#135	C₂×C₄×C₅	1	2	yes	—
162	C₂×Q₁₆	C₂×C₄×C₅	1	1	yes	characteristic
163	C₂×SD₁₆	—	2	1	no	—
164	C₂×D₈	C₂×C₄×C₅	1	1	yes	characteristic
165	C₂×M₁₆	—	2	1	no	—
166	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
167	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
168	C₄×Q₈	—	2	2	no	—
169	C₄×D₄	—	2	2	no	—
170	C₄×D₄	—	4	2	no	—
171	#117	—	2	1	no	—
172	#117	—	2	1	no	—
173	#117	—	4	1	no	—
174	C₂×C₄²	—	4	1	no	—
175	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
176	M₃₂	—	2	1	no	—
177	M₃₂	—	2	1	no	—
178	M₃₂	—	2	1	no	—
179	M₃₂	C₂×C₄×C₅	1	2	yes	complemented
180	C₂×C₁₆	—	2	1	no	—
181	C₂×C₁₆	C₂×C₄×C₅	1	2	yes	complemented
182	#108	—	2	1	no	—
183	#108	C₂×C₄×C₅	1	2	yes	complemented
184	#107	—	2	1	no	—
185	#107	—	2	1	no	—
186	#106	C₂×C₄×C₅	1	2	yes	—
187	#106	C₂×C₄×C₅	1	2	yes	complemented
188	#103	—	2	2	no	—
189	#102	—	2	2	no	—
190	#97	—	4	1	no	—
191	#97	C₂×C₄×C₅	1	2	yes	—
192	#97	C₅×D₄	1	2	yes	—
193	C₄×C₈	—	4	1	no	—
194	C₂²×C₅	—	4	1	no	—
195	C₂²×C₅	—	4	2	no	—
196	C₂²×C₅	—	8	2	no	—
197	C₂²×C₅	#418	1	1	yes	characteristic
198	C₂²×C₅	#440	1	1	yes	characteristic
199	C₂²×C₅	#442	1	1	yes	characteristic
200	C₄×C₅	—	4	2	no	—
201	C₄×C₅	—	4	2	no	—
202	C₄×C₅	—	8	2	no	—
203	C₄×C₅	—	8	2	no	—
204	C₄×C₅	C₄×C₂²⋊C₄	1	1	yes	characteristic
205	C₄×C₅	#417	1	1	yes	characteristic
206	C₄×C₅	#440	1	1	yes	characteristic
207	C₄×C₅	#442	1	1	yes	characteristic
208	#41	—	2	2	no	—
209	#41	—	2	2	no	—
210	#41	—	4	1	no	—
211	#41	—	4	1	no	—
212	C₂×Q₈	—	2	1	no	—
213	C₂×D₄	—	2	1	no	—
214	C₂×D₄	—	4	1	no	—
215	C₂²×C₄	—	2	1	no	—
216	C₂²×C₄	—	4	2	no	—
217	C₂²×C₄	—	4	1	no	—
218	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
219	Q₁₆	—	2	1	no	—
220	Q₁₆	C₄²×C₅	1	2	yes	—
221	SD₁₆	—	2	2	no	—
222	SD₁₆	—	2	2	no	—
223	D₈	—	2	1	no	—
224	D₈	C₄²×C₅	1	2	yes	—
225	M₁₆	—	2	2	no	—
226	M₁₆	—	4	1	no	—
227	C₂×C₈	—	2	1	no	—
228	C₂×C₈	—	2	1	no	—
229	C₂×C₈	—	4	1	no	—
230	C₂×C₈	C₅×C₂²⋊C₄	1	2	yes	—
231	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
232	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
233	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
234	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
235	C₂×C₈	C₅×#41	1	1	yes	characteristic
236	C₂×C₈	C₅×#41	1	1	yes	characteristic
237	C₄⋊C₄	—	2	2	no	—
238	C₄⋊C₄	—	2	2	no	—
239	C₄⋊C₄	—	4	2	no	—
240	C₂²⋊C₄	—	4	2	no	—
241	C₂²⋊C₄	—	4	1	no	—
242	C₂²⋊C₄	—	4	1	no	—
243	C₄²	—	2	1	no	—
244	C₄²	—	4	2	no	—
245	C₄²	—	4	1	no	—
246	C₄²	—	4	1	no	—
247	C₄²	C₅×C₂²⋊C₄	1	2	yes	—
248	C₁₆	—	2	2	no	—
249	C₁₆	—	2	2	no	—
250	C₂×C₅	—	2	1	no	—
251	C₂×C₅	—	2	1	no	—
252	C₂×C₅	—	2	1	no	—
253	C₂×C₅	—	8	2	no	—
254	C₂×C₅	#1762	1	1	yes	characteristic
255	C₂³	—	4	1	no	—
256	C₂³	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
257	Q₈	—	2	2	no	—
258	Q₈	—	4	1	no	—
259	D₄	—	2	2	no	—
260	D₄	—	4	2	no	—
261	D₄	—	4	2	no	—
262	D₄	—	4	1	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	4	1	no	—
265	C₂×C₄	—	4	2	no	—
266	C₂×C₄	—	4	2	no	—
267	C₂×C₄	—	4	2	no	—
268	C₂×C₄	—	4	2	no	—
269	C₂×C₄	—	8	1	no	—
270	C₂×C₄	—	8	4	no	—
271	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
272	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
273	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
274	C₂×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
275	C₂×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
276	C₂×C₄	C₅×#137	1	1	yes	characteristic
277	C₈	—	2	2	no	—
278	C₈	—	4	2	no	—
279	C₈	C₂×C₄²×C₅	1	1	yes	characteristic
280	C₈	C₅×#117	1	1	yes	characteristic
281	C₈	C₄×C₅×D₄	1	1	yes	characteristic
282	C₈	C₄×C₅×D₄	1	1	yes	characteristic
283	C₅	#12649	1	1	yes	characteristic, direct product factor, Sylow
284	C₂²	—	4	1	no	—
285	C₂²	—	4	2	no	—
286	C₂²	—	8	2	no	—
287	C₂²	C₅×#418	1	1	yes	characteristic
288	C₂²	C₅×#440	1	1	yes	characteristic
289	C₂²	C₅×#442	1	1	yes	characteristic
290	C₄	—	4	2	no	—
291	C₄	—	4	2	no	—
292	C₄	—	8	2	no	—
293	C₄	—	8	2	no	—
294	C₄	C₄×C₅×C₂²⋊C₄	1	1	yes	characteristic
295	C₄	C₅×#417	1	1	yes	characteristic
296	C₄	C₅×#440	1	1	yes	characteristic
297	C₄	C₅×#442	1	1	yes	characteristic
298	C₂	—	2	1	no	—
299	C₂	—	2	1	no	—
300	C₂	—	2	1	no	—
301	C₂	—	8	2	no	—
302	C₂	C₅×#1762	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₅	#442
commutator	C₈	C₂×C₄²×C₅
Frattini, cosocle	C₂²×C₈	C₂³×C₅
socle	C₂×C₅	#1762
prime radical	C₂×C₅×#135	C₂²
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅ ◁ C₂²×C₄×C₅ ◁ C₂×C₅×#97 ◁ G

Complements

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

N	H ≅ G/N	type
4. C₂×C₅×#347	C₂	semidirect
5. C₅×#1773	C₂	semidirect
6. C₅×#1389	C₂	semidirect
7. C₅×#1384	C₂	semidirect
12. C₅×#442	C₄	semidirect
13. C₅×#441	C₄	semidirect
21. C₅×#360	C₄	semidirect
22. C₅×#359	C₄	semidirect
25. #12649	C₅	direct (N × H)
48. C₅×M₃₂	C₂×C₄	semidirect
50. C₂×C₅×C₁₆	C₂×C₄	semidirect
52. C₅×#108	C₂×C₄	semidirect
56. C₅×#106	C₂×C₄	semidirect
66. C₂×#347	C₂×C₅	semidirect
67. #1773	C₂×C₅	semidirect
68. #1389	C₂×C₅	semidirect
69. #1384	C₂×C₅	semidirect
116. #442	C₄×C₅	semidirect
117. #441	C₄×C₅	semidirect
125. #360	C₄×C₅	semidirect
126. #359	C₄×C₅	semidirect
179. M₃₂	C₂×C₄×C₅	semidirect
181. C₂×C₁₆	C₂×C₄×C₅	semidirect
183. #108	C₂×C₄×C₅	semidirect
187. #106	C₂×C₄×C₅	semidirect
283. C₅	#12649	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×#2943	2
C₅×#2145	2
C₅×#2143	2
C₂×C₅×#347	2
C₅×#1773	2
C₅×#1389	2
C₅×#1384	2
#12649	5

Subgroups — C5×#12649 #11935369

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#12649 #11935369