Subgroups — C₅×#13002 #11935722

Order 1280.

528 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	C₅×#3050	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#427	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×#360	C₂	1	2	yes	maximal, complemented
4	C₅×#2112	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#1386	C₂	1	2	yes	maximal, complemented
6	C₂×C₅×#135	C₂²	1	1	yes	characteristic
7	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₅×#525	—	2	1	no	—
9	C₂×C₅×#117	—	2	1	no	—
10	C₂×C₅×M₃₂	C₂²	1	2	yes	—
11	C₅×#467	C₄	1	2	yes	complemented
12	C₅×#466	C₄	1	2	yes	complemented
13	C₅×#465	—	2	2	no	—
14	C₅×#427	—	2	4	no	—
15	C₅×#427	C₂²	1	1	yes	characteristic
16	C₂×C₅×#107	—	2	1	no	—
17	C₂×C₅×#106	C₂²	1	2	yes	—
18	C₅×#416	—	2	1	no	—
19	C₅×#360	—	2	4	no	—
20	C₅×C₂²⋊C₁₆	—	2	2	no	—
21	#13002	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
22	C₂×C₅×#41	—	2	1	no	—
23	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
24	C₅×#135	—	2	2	no	—
25	C₅×#135	—	2	2	no	—
26	C₂×C₅×Q₁₆	C₂×C₄	1	1	yes	characteristic
27	C₂×C₅×SD₁₆	—	2	1	no	—
28	C₂×C₅×D₈	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₅×C₈	—	2	2	no	—
30	C₂²×C₅×C₈	—	2	2	no	—
31	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
32	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
33	C₂²×C₅×C₈	D₄	1	2	yes	—
34	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
35	C₅×#123	—	4	1	no	—
36	C₅×C₂²⋊Q₈	—	2	2	no	—
37	C₅×#121	—	2	2	no	—
38	C₅×#120	—	4	1	no	—
39	C₄×C₅×D₄	—	4	2	no	—
40	C₅×#117	—	2	4	no	—
41	C₅×#117	—	2	1	no	—
42	C₅×#117	—	4	2	no	—
43	C₂×C₅×C₄⋊C₄	—	2	2	no	—
44	C₂×C₅×C₂²⋊C₄	—	4	1	no	—
45	C₂×C₄²×C₅	—	4	1	no	—
46	C₅×M₃₂	—	2	2	no	—
47	C₅×M₃₂	—	2	2	no	—
48	C₂×C₅×C₁₆	—	2	2	no	—
49	C₅×#107	—	2	4	no	—
50	C₅×#107	C₂×C₄	1	2	yes	—
51	C₅×#106	—	2	2	no	—
52	C₅×#106	—	2	1	no	—
53	C₅×#106	D₄	1	4	yes	complemented
54	C₅×#103	—	2	2	no	—
55	C₅×#102	—	2	2	no	—
56	#3050	C₂×C₅	1	1	yes	characteristic
57	C₂×#427	C₂×C₅	1	1	yes	characteristic, complemented
58	C₂×#360	C₂×C₅	1	2	yes	complemented
59	#2112	C₂×C₅	1	1	yes	characteristic, complemented
60	#1386	C₂×C₅	1	2	yes	complemented
61	C₂⁴×C₅	D₈	1	1	yes	characteristic
62	C₅×#41	—	4	1	no	—
63	C₅×#41	—	4	1	no	—
64	C₅×#41	—	4	1	no	—
65	C₅×#41	—	4	1	no	—
66	C₂×C₅×Q₈	—	2	1	no	—
67	C₂×C₅×D₄	—	2	1	no	—
68	C₂×C₅×D₄	—	4	2	no	—
69	C₂×C₅×D₄	—	4	1	no	—
70	C₂²×C₄×C₅	—	2	1	no	—
71	C₂²×C₄×C₅	—	2	1	no	—
72	C₂²×C₄×C₅	—	2	1	no	—
73	C₂²×C₄×C₅	—	2	1	no	—
74	C₂²×C₄×C₅	—	4	2	no	—
75	C₂²×C₄×C₅	—	4	1	no	—
76	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
77	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
78	C₂²×C₄×C₅	SD₁₆	1	2	yes	—
79	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
80	C₅×Q₁₆	—	2	2	no	—
81	C₅×SD₁₆	—	4	2	no	—
82	C₅×D₈	—	2	2	no	—
83	C₂×C₅×C₈	—	2	1	no	—
84	C₂×C₅×C₈	—	2	1	no	—
85	C₂×C₅×C₈	—	2	4	no	—
86	C₂×C₅×C₈	—	2	4	no	—
87	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
88	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
90	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
91	C₂×C₅×C₈	#41	1	2	yes	—
92	C₅×C₄⋊C₄	—	2	2	no	—
93	C₅×C₄⋊C₄	—	2	4	no	—
94	C₅×C₄⋊C₄	—	4	2	no	—
95	C₅×C₄⋊C₄	—	4	2	no	—
96	C₅×C₂²⋊C₄	—	4	4	no	—
97	C₅×C₂²⋊C₄	—	4	2	no	—
98	C₅×C₂²⋊C₄	—	4	1	no	—
99	C₅×C₂²⋊C₄	—	4	2	no	—
100	C₄²×C₅	—	4	4	no	—
101	C₄²×C₅	—	4	1	no	—
102	C₅×C₁₆	—	4	2	no	—
103	C₂×#135	C₂²×C₅	1	1	yes	characteristic
104	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
105	#525	—	2	1	no	—
106	C₂×#117	—	2	1	no	—
107	C₂×M₃₂	C₂²×C₅	1	2	yes	—
108	#467	C₄×C₅	1	2	yes	complemented
109	#466	C₄×C₅	1	2	yes	complemented
110	#465	—	2	2	no	—
111	#427	—	2	4	no	—
112	#427	C₂²×C₅	1	1	yes	characteristic
113	C₂×#107	—	2	1	no	—
114	C₂×#106	C₂²×C₅	1	2	yes	—
115	#416	—	2	1	no	—
116	#360	—	2	4	no	—
117	C₂²⋊C₁₆	—	2	2	no	—
118	C₂³×C₅	—	2	1	no	—
119	C₂³×C₅	—	2	1	no	—
120	C₂³×C₅	—	4	2	no	—
121	C₂³×C₅	—	4	1	no	—
122	C₂³×C₅	#102	1	2	yes	—
123	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
124	C₅×Q₈	—	4	1	no	—
125	C₅×Q₈	—	4	1	no	—
126	C₅×D₄	—	4	1	no	—
127	C₅×D₄	—	4	1	no	—
128	C₅×D₄	—	8	4	no	—
129	C₅×D₄	—	8	1	no	—
130	C₅×D₄	—	8	1	no	—
131	C₂×C₄×C₅	—	2	1	no	—
132	C₂×C₄×C₅	—	2	1	no	—
133	C₂×C₄×C₅	—	2	2	no	—
134	C₂×C₄×C₅	—	2	2	no	—
135	C₂×C₄×C₅	—	2	2	no	—
136	C₂×C₄×C₅	—	2	2	no	—
137	C₂×C₄×C₅	—	4	2	no	—
138	C₂×C₄×C₅	—	4	1	no	—
139	C₂×C₄×C₅	—	4	4	no	—
140	C₂×C₄×C₅	—	8	4	no	—
141	C₂×C₄×C₅	—	8	1	no	—
142	C₂×C₄×C₅	—	8	1	no	—
143	C₂×C₄×C₅	#102	1	2	yes	—
144	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
145	C₂×C₄×C₅	#120	1	1	yes	characteristic
146	C₂×C₄×C₅	#123	1	1	yes	characteristic
147	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
148	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
149	C₂×C₄×C₅	#137	1	1	yes	characteristic
150	C₅×C₈	—	2	2	no	—
151	C₅×C₈	C₄×D₄	1	2	yes	—
152	C₅×C₈	#121	1	2	yes	—
153	C₂×#41	—	2	1	no	—
154	C₂³×C₄	C₅×D₄	1	1	yes	characteristic
155	#135	—	2	2	no	—
156	#135	—	2	2	no	—
157	C₂×Q₁₆	C₂×C₄×C₅	1	1	yes	characteristic
158	C₂×SD₁₆	—	2	1	no	—
159	C₂×D₈	C₂×C₄×C₅	1	1	yes	characteristic
160	C₂²×C₈	—	2	2	no	—
161	C₂²×C₈	—	2	2	no	—
162	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
163	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
164	C₂²×C₈	C₅×D₄	1	2	yes	—
165	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
166	#123	—	4	1	no	—
167	C₂²⋊Q₈	—	2	2	no	—
168	#121	—	2	2	no	—
169	#120	—	4	1	no	—
170	C₄×D₄	—	4	2	no	—
171	#117	—	2	4	no	—
172	#117	—	2	1	no	—
173	#117	—	4	2	no	—
174	C₂×C₄⋊C₄	—	2	2	no	—
175	C₂×C₂²⋊C₄	—	4	1	no	—
176	C₂×C₄²	—	4	1	no	—
177	M₃₂	—	2	2	no	—
178	M₃₂	—	2	2	no	—
179	C₂×C₁₆	—	2	2	no	—
180	#107	—	2	4	no	—
181	#107	C₂×C₄×C₅	1	2	yes	—
182	#106	—	2	2	no	—
183	#106	—	2	1	no	—
184	#106	C₅×D₄	1	4	yes	complemented
185	#103	—	2	2	no	—
186	#102	—	2	2	no	—
187	C₂²×C₅	—	2	2	no	—
188	C₂²×C₅	—	2	2	no	—
189	C₂²×C₅	—	2	2	no	—
190	C₂²×C₅	—	2	2	no	—
191	C₂²×C₅	—	4	2	no	—
192	C₂²×C₅	—	4	2	no	—
193	C₂²×C₅	—	8	1	no	—
194	C₂²×C₅	—	8	2	no	—
195	C₂²×C₅	C₂×#102	1	1	yes	characteristic
196	C₂²×C₅	#447	1	1	yes	characteristic
197	C₂²×C₅	#480	1	1	yes	characteristic
198	C₄×C₅	—	2	2	no	—
199	C₄×C₅	—	8	4	no	—
200	C₄×C₅	—	8	1	no	—
201	C₄×C₅	—	8	2	no	—
202	C₄×C₅	#386	1	1	yes	characteristic
203	C₄×C₅	#417	1	1	yes	characteristic
204	C₄×C₅	#448	1	1	yes	characteristic
205	C₄×C₅	#481	1	1	yes	characteristic
206	C₂⁴	C₅×D₈	1	1	yes	characteristic
207	#41	—	4	1	no	—
208	#41	—	4	1	no	—
209	#41	—	4	1	no	—
210	#41	—	4	1	no	—
211	C₂×Q₈	—	2	1	no	—
212	C₂×D₄	—	2	1	no	—
213	C₂×D₄	—	4	1	no	—
214	C₂×D₄	—	4	2	no	—
215	C₂²×C₄	—	2	1	no	—
216	C₂²×C₄	—	2	1	no	—
217	C₂²×C₄	—	2	1	no	—
218	C₂²×C₄	—	2	1	no	—
219	C₂²×C₄	—	4	2	no	—
220	C₂²×C₄	—	4	1	no	—
221	C₂²×C₄	C₅×C₂²⋊C₄	1	2	yes	—
222	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
223	C₂²×C₄	C₅×SD₁₆	1	2	yes	—
224	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
225	Q₁₆	—	2	2	no	—
226	SD₁₆	—	4	2	no	—
227	D₈	—	2	2	no	—
228	C₂×C₈	—	2	1	no	—
229	C₂×C₈	—	2	1	no	—
230	C₂×C₈	—	2	4	no	—
231	C₂×C₈	—	2	4	no	—
232	C₂×C₈	C₅×C₂²⋊C₄	1	2	yes	—
233	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
234	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
235	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
236	C₂×C₈	C₅×#41	1	2	yes	—
237	C₄⋊C₄	—	2	2	no	—
238	C₄⋊C₄	—	2	4	no	—
239	C₄⋊C₄	—	4	2	no	—
240	C₄⋊C₄	—	4	2	no	—
241	C₂²⋊C₄	—	4	4	no	—
242	C₂²⋊C₄	—	4	1	no	—
243	C₂²⋊C₄	—	4	2	no	—
244	C₂²⋊C₄	—	4	2	no	—
245	C₄²	—	4	4	no	—
246	C₄²	—	4	1	no	—
247	C₁₆	—	4	2	no	—
248	C₂×C₅	—	2	1	no	—
249	C₂×C₅	—	2	1	no	—
250	C₂×C₅	—	4	2	no	—
251	C₂×C₅	—	16	1	no	—
252	C₂×C₅	#1874	1	1	yes	characteristic
253	C₂×C₅	#2145	1	2	yes	—
254	C₂³	—	2	1	no	—
255	C₂³	—	2	1	no	—
256	C₂³	—	4	2	no	—
257	C₂³	—	4	1	no	—
258	C₂³	C₅×#102	1	2	yes	—
259	C₂³	C₂×C₅×D₈	1	1	yes	characteristic
260	Q₈	—	4	1	no	—
261	Q₈	—	4	1	no	—
262	D₄	—	4	1	no	—
263	D₄	—	4	1	no	—
264	D₄	—	8	4	no	—
265	D₄	—	8	1	no	—
266	D₄	—	8	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	—	4	1	no	—
275	C₂×C₄	—	4	4	no	—
276	C₂×C₄	—	8	4	no	—
277	C₂×C₄	—	8	1	no	—
278	C₂×C₄	—	8	1	no	—
279	C₂×C₄	C₅×#102	1	2	yes	—
280	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
281	C₂×C₄	C₅×#120	1	1	yes	characteristic
282	C₂×C₄	C₅×#123	1	1	yes	characteristic
283	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
284	C₂×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
285	C₂×C₄	C₅×#137	1	1	yes	characteristic
286	C₈	—	2	2	no	—
287	C₈	C₄×C₅×D₄	1	2	yes	—
288	C₈	C₅×#121	1	2	yes	—
289	C₅	#13002	1	1	yes	characteristic, direct product factor, Sylow
290	C₂²	—	2	2	no	—
291	C₂²	—	2	2	no	—
292	C₂²	—	2	2	no	—
293	C₂²	—	2	2	no	—
294	C₂²	—	4	2	no	—
295	C₂²	—	4	2	no	—
296	C₂²	—	8	1	no	—
297	C₂²	—	8	2	no	—
298	C₂²	C₂×C₅×#102	1	1	yes	characteristic
299	C₂²	C₅×#447	1	1	yes	characteristic
300	C₂²	C₅×#480	1	1	yes	characteristic
301	C₄	—	2	2	no	—
302	C₄	—	8	4	no	—
303	C₄	—	8	1	no	—
304	C₄	—	8	2	no	—
305	C₄	C₅×#386	1	1	yes	characteristic
306	C₄	C₅×#417	1	1	yes	characteristic
307	C₄	C₅×#448	1	1	yes	characteristic
308	C₄	C₅×#481	1	1	yes	characteristic
309	C₂	—	2	1	no	—
310	C₂	—	2	1	no	—
311	C₂	—	4	2	no	—
312	C₂	—	16	1	no	—
313	C₂	C₅×#1874	1	1	yes	characteristic
314	C₂	C₅×#2145	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₅	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₂²×C₈	C₂³×C₅
prime radical	C₅×#3050	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂³×C₄×C₅ ◁ C₂³×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×#427	C₂	semidirect
3. C₂×C₅×#360	C₂	semidirect
4. C₅×#2112	C₂	semidirect
5. C₅×#1386	C₂	semidirect
11. C₅×#467	C₄	semidirect
12. C₅×#466	C₄	semidirect
21. #13002	C₅	direct (N × H)
53. C₅×#106	D₄	semidirect
57. C₂×#427	C₂×C₅	semidirect
58. C₂×#360	C₂×C₅	semidirect
59. #2112	C₂×C₅	semidirect
60. #1386	C₂×C₅	semidirect
108. #467	C₄×C₅	semidirect
109. #466	C₄×C₅	semidirect
184. #106	C₅×D₄	semidirect
289. C₅	#13002	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×#3050	2
C₂×C₅×#427	2
C₂×C₅×#360	2
C₅×#2112	2
C₅×#1386	2
#13002	5

Subgroups — C5×#13002 #11935722

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#13002 #11935722