Subgroups — C₅×#13210 #11935930

Order 1280.

510 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3147	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#360	C₂	1	4	yes	maximal, complemented
3	C₄×C₅×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#1844	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₅×#135	—	2	1	no	—
6	C₂×C₅×#135	C₂²	1	2	yes	—
7	C₅×#532	—	2	1	no	—
8	C₂×C₅×M₃₂	C₂²	1	2	yes	—
9	C₅×#495	—	2	1	no	—
10	C₅×#495	C₄	1	2	yes	complemented
11	C₅×#494	C₄	1	1	yes	characteristic, complemented
12	C₅×#493	C₄	1	1	yes	characteristic, complemented
13	C₅×#492	—	2	1	no	—
14	C₂×C₅×#106	—	2	1	no	—
15	C₂×C₅×#106	C₂²	1	2	yes	—
16	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
17	C₅×#384	—	2	1	no	—
18	C₅×#360	—	2	8	no	—
19	C₅×#346	—	2	1	no	—
20	C₄×C₅×C₁₆	—	2	1	no	—
21	#13210	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
22	C₂×C₅×#41	—	2	2	no	—
23	C₅×#135	—	2	4	no	—
24	C₅×#135	—	2	4	no	—
25	C₅×#135	—	4	2	no	—
26	C₅×#135	—	4	2	no	—
27	C₂×C₅×Q₁₆	—	2	1	no	—
28	C₂×C₅×Q₁₆	C₂×C₄	1	2	yes	—
29	C₂×C₅×SD₁₆	—	2	2	no	—
30	C₂×C₅×SD₁₆	—	2	2	no	—
31	C₂×C₅×D₈	—	2	1	no	—
32	C₂×C₅×D₈	C₂×C₄	1	2	yes	—
33	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
34	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
35	C₅×C₄⋊Q₈	—	2	1	no	—
36	C₅×#127	—	2	1	no	—
37	C₅×#124	—	2	2	no	—
38	C₅×#121	—	4	2	no	—
39	C₄×C₅×D₄	—	4	2	no	—
40	C₂×C₅×C₄⋊C₄	—	2	2	no	—
41	C₂×C₅×C₄⋊C₄	—	4	2	no	—
42	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
43	C₅×M₃₂	—	2	2	no	—
44	C₅×M₃₂	D₄	1	4	yes	complemented
45	C₂×C₅×C₁₆	—	2	2	no	—
46	C₅×#106	—	2	4	no	—
47	C₅×#106	—	4	2	no	—
48	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
49	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
50	C₄×C₅×C₈	D₄	1	1	yes	characteristic
51	C₄×C₅×C₈	D₄	1	1	yes	characteristic
52	#3147	C₂×C₅	1	1	yes	characteristic
53	C₂×#360	C₂×C₅	1	4	yes	complemented
54	C₄×M₃₂	C₂×C₅	1	1	yes	characteristic, complemented
55	#1844	C₂×C₅	1	1	yes	characteristic, complemented
56	C₅×#41	—	4	2	no	—
57	C₅×#41	—	4	2	no	—
58	C₅×#41	—	4	2	no	—
59	C₅×#41	—	4	2	no	—
60	C₂×C₅×Q₈	—	2	2	no	—
61	C₂×C₅×D₄	—	2	2	no	—
62	C₂×C₅×D₄	—	4	2	no	—
63	C₂×C₅×D₄	—	4	2	no	—
64	C₂²×C₄×C₅	—	4	2	no	—
65	C₂²×C₄×C₅	—	4	2	no	—
66	C₂²×C₄×C₅	D₈	1	2	yes	—
67	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
68	C₅×Q₁₆	—	2	4	no	—
69	C₅×Q₁₆	—	4	2	no	—
70	C₅×SD₁₆	—	4	4	no	—
71	C₅×SD₁₆	—	4	4	no	—
72	C₅×D₈	—	2	4	no	—
73	C₅×D₈	—	4	2	no	—
74	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₅×C₈	D₈	1	2	yes	—
77	C₂×C₅×C₈	SD₁₆	1	2	yes	—
78	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
80	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
81	C₂×C₅×C₈	#41	1	2	yes	—
82	C₅×C₄⋊C₄	—	4	2	no	—
83	C₅×C₄⋊C₄	—	4	2	no	—
84	C₅×C₄⋊C₄	—	4	2	no	—
85	C₅×C₄⋊C₄	—	8	4	no	—
86	C₅×C₂²⋊C₄	—	4	4	no	—
87	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
88	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
89	C₄²×C₅	SD₁₆	1	2	yes	—
90	C₅×C₁₆	—	2	4	no	—
91	C₂×#135	—	2	1	no	—
92	C₂×#135	C₂²×C₅	1	2	yes	—
93	#532	—	2	1	no	—
94	C₂×M₃₂	C₂²×C₅	1	2	yes	—
95	#495	—	2	1	no	—
96	#495	C₄×C₅	1	2	yes	complemented
97	#494	C₄×C₅	1	1	yes	characteristic, complemented
98	#493	C₄×C₅	1	1	yes	characteristic, complemented
99	#492	—	2	1	no	—
100	C₂×#106	—	2	1	no	—
101	C₂×#106	C₂²×C₅	1	2	yes	—
102	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
103	#384	—	2	1	no	—
104	#360	—	2	8	no	—
105	#346	—	2	1	no	—
106	C₄×C₁₆	—	2	1	no	—
107	C₂³×C₅	—	4	2	no	—
108	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
109	C₅×Q₈	—	4	2	no	—
110	C₅×Q₈	—	4	2	no	—
111	C₅×D₄	—	4	2	no	—
112	C₅×D₄	—	4	2	no	—
113	C₅×D₄	—	8	2	no	—
114	C₅×D₄	—	8	2	no	—
115	C₅×D₄	—	8	4	no	—
116	C₂×C₄×C₅	—	2	4	no	—
117	C₂×C₄×C₅	—	4	2	no	—
118	C₂×C₄×C₅	—	8	2	no	—
119	C₂×C₄×C₅	—	8	2	no	—
120	C₂×C₄×C₅	—	8	4	no	—
121	C₂×C₄×C₅	—	8	2	no	—
122	C₂×C₄×C₅	#102	1	2	yes	—
123	C₂×C₄×C₅	#102	1	2	yes	—
124	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₅	#124	1	1	yes	characteristic
126	C₂×C₄×C₅	#127	1	1	yes	characteristic
127	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
128	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
129	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
130	C₅×C₈	#102	1	2	yes	—
131	C₅×C₈	#102	1	2	yes	—
132	C₅×C₈	C₄×D₄	1	2	yes	—
133	C₅×C₈	#121	1	2	yes	—
134	C₂×#41	—	2	2	no	—
135	#135	—	2	4	no	—
136	#135	—	2	4	no	—
137	#135	—	4	2	no	—
138	#135	—	4	2	no	—
139	C₂×Q₁₆	—	2	1	no	—
140	C₂×Q₁₆	C₂×C₄×C₅	1	2	yes	—
141	C₂×SD₁₆	—	2	2	no	—
142	C₂×SD₁₆	—	2	2	no	—
143	C₂×D₈	—	2	1	no	—
144	C₂×D₈	C₂×C₄×C₅	1	2	yes	—
145	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
146	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
147	C₄⋊Q₈	—	2	1	no	—
148	#127	—	2	1	no	—
149	#124	—	2	2	no	—
150	#121	—	4	2	no	—
151	C₄×D₄	—	4	2	no	—
152	C₂×C₄⋊C₄	—	2	2	no	—
153	C₂×C₄⋊C₄	—	4	2	no	—
154	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
155	M₃₂	—	2	2	no	—
156	M₃₂	C₅×D₄	1	4	yes	complemented
157	C₂×C₁₆	—	2	2	no	—
158	#106	—	2	4	no	—
159	#106	—	4	2	no	—
160	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
161	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
162	C₄×C₈	C₅×D₄	1	1	yes	characteristic
163	C₄×C₈	C₅×D₄	1	1	yes	characteristic
164	C₂²×C₅	—	2	2	no	—
165	C₂²×C₅	—	8	4	no	—
166	C₂²×C₅	—	8	2	no	—
167	C₂²×C₅	C₂×#102	1	1	yes	characteristic
168	C₂²×C₅	#486	1	1	yes	characteristic
169	C₂²×C₅	#493	1	1	yes	characteristic
170	C₄×C₅	—	2	2	no	—
171	C₄×C₅	—	8	2	no	—
172	C₄×C₅	—	16	2	no	—
173	C₄×C₅	#360	1	4	yes	complemented
174	C₄×C₅	#390	1	1	yes	characteristic
175	C₄×C₅	C₂×#102	1	1	yes	characteristic
176	C₄×C₅	#486	1	1	yes	characteristic
177	C₄×C₅	#492	1	1	yes	characteristic
178	#41	—	4	2	no	—
179	#41	—	4	2	no	—
180	#41	—	4	2	no	—
181	#41	—	4	2	no	—
182	C₂×Q₈	—	2	2	no	—
183	C₂×D₄	—	2	2	no	—
184	C₂×D₄	—	4	2	no	—
185	C₂×D₄	—	4	2	no	—
186	C₂²×C₄	—	4	2	no	—
187	C₂²×C₄	—	4	2	no	—
188	C₂²×C₄	C₅×D₈	1	2	yes	—
189	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
190	Q₁₆	—	2	4	no	—
191	Q₁₆	—	4	2	no	—
192	SD₁₆	—	4	4	no	—
193	SD₁₆	—	4	4	no	—
194	D₈	—	2	4	no	—
195	D₈	—	4	2	no	—
196	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
197	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
198	C₂×C₈	C₅×D₈	1	2	yes	—
199	C₂×C₈	C₅×SD₁₆	1	2	yes	—
200	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
201	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
202	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
203	C₂×C₈	C₅×#41	1	2	yes	—
204	C₄⋊C₄	—	4	2	no	—
205	C₄⋊C₄	—	4	2	no	—
206	C₄⋊C₄	—	4	2	no	—
207	C₄⋊C₄	—	8	4	no	—
208	C₂²⋊C₄	—	4	4	no	—
209	C₄²	C₅×C₂²⋊C₄	1	1	yes	characteristic
210	C₄²	C₅×C₂²⋊C₄	1	1	yes	characteristic
211	C₄²	C₅×SD₁₆	1	2	yes	—
212	C₁₆	—	2	4	no	—
213	C₂×C₅	—	2	1	no	—
214	C₂×C₅	—	2	1	no	—
215	C₂×C₅	—	16	2	no	—
216	C₂×C₅	#1957	1	1	yes	characteristic
217	C₂×C₅	C₂×#360	1	2	yes	—
218	C₂³	—	4	2	no	—
219	C₂³	C₂×C₅×D₈	1	1	yes	characteristic
220	Q₈	—	4	2	no	—
221	Q₈	—	4	2	no	—
222	D₄	—	4	2	no	—
223	D₄	—	4	2	no	—
224	D₄	—	8	2	no	—
225	D₄	—	8	2	no	—
226	D₄	—	8	4	no	—
227	C₂×C₄	—	2	4	no	—
228	C₂×C₄	—	4	2	no	—
229	C₂×C₄	—	8	2	no	—
230	C₂×C₄	—	8	2	no	—
231	C₂×C₄	—	8	4	no	—
232	C₂×C₄	—	8	2	no	—
233	C₂×C₄	C₅×#102	1	2	yes	—
234	C₂×C₄	C₅×#102	1	2	yes	—
235	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
236	C₂×C₄	C₅×#124	1	1	yes	characteristic
237	C₂×C₄	C₅×#127	1	1	yes	characteristic
238	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
239	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
240	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
241	C₈	C₅×#102	1	2	yes	—
242	C₈	C₅×#102	1	2	yes	—
243	C₈	C₄×C₅×D₄	1	2	yes	—
244	C₈	C₅×#121	1	2	yes	—
245	C₅	#13210	1	1	yes	characteristic, direct product factor, Sylow
246	C₂²	—	2	2	no	—
247	C₂²	—	8	4	no	—
248	C₂²	—	8	2	no	—
249	C₂²	C₂×C₅×#102	1	1	yes	characteristic
250	C₂²	C₅×#486	1	1	yes	characteristic
251	C₂²	C₅×#493	1	1	yes	characteristic
252	C₄	—	2	2	no	—
253	C₄	—	8	2	no	—
254	C₄	—	16	2	no	—
255	C₄	C₅×#360	1	4	yes	complemented
256	C₄	C₅×#390	1	1	yes	characteristic
257	C₄	C₂×C₅×#102	1	1	yes	characteristic
258	C₄	C₅×#486	1	1	yes	characteristic
259	C₄	C₅×#492	1	1	yes	characteristic
260	C₂	—	2	1	no	—
261	C₂	—	2	1	no	—
262	C₂	—	16	2	no	—
263	C₂	C₅×#1957	1	1	yes	characteristic
264	C₂	C₂×C₅×#360	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×#3147	C₂
LCS	G ▷ C₂×C₈ ▷ 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. C₂×C₅×#360	C₂	semidirect
3. C₄×C₅×M₃₂	C₂	semidirect
4. C₅×#1844	C₂	semidirect
10. C₅×#495	C₄	semidirect
11. C₅×#494	C₄	semidirect
12. C₅×#493	C₄	semidirect
21. #13210	C₅	direct (N × H)
44. C₅×M₃₂	D₄	semidirect
53. C₂×#360	C₂×C₅	semidirect
54. C₄×M₃₂	C₂×C₅	semidirect
55. #1844	C₂×C₅	semidirect
96. #495	C₄×C₅	semidirect
97. #494	C₄×C₅	semidirect
98. #493	C₄×C₅	semidirect
156. M₃₂	C₅×D₄	semidirect
173. C₄×C₅	#360	semidirect
245. C₅	#13210	direct (N × H)
255. C₄	C₅×#360	semidirect

Maximal subgroups

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

subgroup	index
C₅×#3147	2
C₂×C₅×#360	2
C₄×C₅×M₃₂	2
C₅×#1844	2
#13210	5

Subgroups — C5×#13210 #11935930

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#13210 #11935930