Subgroups — C₅×#13243 #11935963

Order 1280.

410 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#508	C₂	1	1	yes	maximal, characteristic
2	C₅×#2158	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#2145	C₂	1	2	yes	maximal, complemented
4	C₅×#2108	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#1976	C₂	1	2	yes	maximal, complemented
6	C₂×C₅×#135	C₂²	1	2	yes	—
7	C₅×#508	—	2	1	no	—
8	C₅×#508	—	2	1	no	—
9	C₅×#508	—	2	1	no	—
10	C₅×#508	—	2	1	no	—
11	C₂×C₅×Q₃₂	C₄	1	1	yes	characteristic, complemented
12	C₂×C₅×SD₃₂	C₄	1	2	yes	complemented
13	C₂×C₅×D₁₆	C₄	1	1	yes	characteristic, complemented
14	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
15	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
16	C₅×#427	C₂²	1	2	yes	—
17	C₅×#421	—	2	2	no	—
18	C₅×#410	—	2	2	no	—
19	C₅×#405	C₂²	1	1	yes	characteristic
20	C₅×#367	—	2	1	no	—
21	C₅×#366	—	2	1	no	—
22	C₅×#360	—	2	4	no	—
23	C₅×#346	—	2	1	no	—
24	C₄×C₅×C₁₆	—	2	1	no	—
25	#13243	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
26	C₂×C₅×#41	—	2	2	no	—
27	C₅×#135	—	2	2	no	—
28	C₅×#135	—	2	2	no	—
29	C₅×#135	—	2	2	no	—
30	C₅×#135	—	2	2	no	—
31	C₂×C₅×Q₁₆	C₂×C₄	1	2	yes	—
32	C₂×C₅×SD₁₆	—	2	2	no	—
33	C₂×C₅×D₈	C₂×C₄	1	2	yes	—
34	C₂×C₅×M₁₆	D₄	1	1	yes	characteristic
35	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
36	C₅×#117	—	2	2	no	—
37	C₅×#117	D₄	1	1	yes	characteristic
38	C₅×Q₃₂	—	2	1	no	—
39	C₅×Q₃₂	—	2	1	no	—
40	C₅×SD₃₂	—	2	2	no	—
41	C₅×SD₃₂	—	2	2	no	—
42	C₅×D₁₆	—	2	1	no	—
43	C₅×D₁₆	—	2	1	no	—
44	C₅×M₃₂	—	2	1	no	—
45	C₅×M₃₂	D₄	1	2	yes	—
46	C₂×C₅×C₁₆	—	2	1	no	—
47	C₂×C₅×C₁₆	C₂×C₄	1	1	yes	characteristic
48	C₂×C₅×C₁₆	C₂×C₄	1	1	yes	characteristic
49	C₂×C₅×C₁₆	D₄	1	1	yes	characteristic
50	C₂×C₅×C₁₆	D₄	1	1	yes	characteristic
51	C₂×C₅×C₁₆	D₄	1	1	yes	characteristic
52	C₂×C₅×C₁₆	D₄	1	1	yes	characteristic
53	C₅×#107	—	2	2	no	—
54	C₅×#106	—	2	2	no	—
55	C₅×#104	—	4	4	no	—
56	C₅×#99	—	4	4	no	—
57	C₅×#97	—	2	1	no	—
58	C₄×C₅×C₈	—	2	1	no	—
59	C₂×#508	C₂×C₅	1	1	yes	characteristic
60	#2158	C₂×C₅	1	1	yes	characteristic, complemented
61	#2145	C₂×C₅	1	2	yes	complemented
62	#2108	C₂×C₅	1	1	yes	characteristic, complemented
63	#1976	C₂×C₅	1	2	yes	complemented
64	C₅×#41	—	4	2	no	—
65	C₅×#41	—	4	2	no	—
66	C₅×#41	—	4	2	no	—
67	C₅×#41	—	4	2	no	—
68	C₂×C₅×Q₈	—	2	2	no	—
69	C₂×C₅×D₄	—	2	2	no	—
70	C₂×C₅×D₄	—	4	2	no	—
71	C₂²×C₄×C₅	—	4	2	no	—
72	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
73	C₅×Q₁₆	—	2	2	no	—
74	C₅×Q₁₆	—	2	2	no	—
75	C₅×SD₁₆	—	4	2	no	—
76	C₅×SD₁₆	—	4	2	no	—
77	C₅×D₈	—	2	2	no	—
78	C₅×D₈	—	2	2	no	—
79	C₅×M₁₆	—	2	1	no	—
80	C₅×M₁₆	D₈	1	2	yes	—
81	C₂×C₅×C₈	—	2	1	no	—
82	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
84	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
85	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
86	C₂×C₅×C₈	#41	1	1	yes	characteristic
87	C₂×C₅×C₈	#41	1	1	yes	characteristic
88	C₅×C₄⋊C₄	—	2	1	no	—
89	C₅×C₄⋊C₄	—	4	2	no	—
90	C₅×C₂²⋊C₄	—	4	2	no	—
91	C₅×C₂²⋊C₄	D₈	1	2	yes	—
92	C₄²×C₅	—	2	1	no	—
93	C₄²×C₅	—	4	2	no	—
94	C₅×C₁₆	—	2	2	no	—
95	C₅×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
96	C₅×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
97	C₅×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
98	C₅×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
99	C₂×#135	C₂²×C₅	1	2	yes	—
100	#508	—	2	1	no	—
101	#508	—	2	1	no	—
102	#508	—	2	1	no	—
103	#508	—	2	1	no	—
104	C₂×Q₃₂	C₄×C₅	1	1	yes	characteristic, complemented
105	C₂×SD₃₂	C₄×C₅	1	2	yes	complemented
106	C₂×D₁₆	C₄×C₅	1	1	yes	characteristic, complemented
107	C₂×M₃₂	C₂²×C₅	1	1	yes	characteristic
108	C₂²×C₁₆	C₂²×C₅	1	1	yes	characteristic
109	#427	C₂²×C₅	1	2	yes	—
110	#421	—	2	2	no	—
111	#410	—	2	2	no	—
112	#405	C₂²×C₅	1	1	yes	characteristic
113	#367	—	2	1	no	—
114	#366	—	2	1	no	—
115	#360	—	2	4	no	—
116	#346	—	2	1	no	—
117	C₄×C₁₆	—	2	1	no	—
118	C₂³×C₅	—	4	2	no	—
119	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
120	C₅×Q₈	—	4	2	no	—
121	C₅×Q₈	—	4	2	no	—
122	C₅×D₄	—	4	2	no	—
123	C₅×D₄	—	4	2	no	—
124	C₅×D₄	—	8	2	no	—
125	C₅×D₄	—	8	2	no	—
126	C₂×C₄×C₅	—	2	2	no	—
127	C₂×C₄×C₅	—	4	2	no	—
128	C₂×C₄×C₅	—	8	2	no	—
129	C₂×C₄×C₅	—	8	2	no	—
130	C₂×C₄×C₅	—	8	2	no	—
131	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
132	C₂×C₄×C₅	#121	1	1	yes	characteristic
133	C₂×C₄×C₅	#127	1	1	yes	characteristic
134	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
135	C₂×C₄×C₅	#135	1	1	yes	characteristic
136	C₂×C₄×C₅	#135	1	1	yes	characteristic
137	C₅×C₈	—	2	2	no	—
138	C₅×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
139	C₅×C₈	C₄×D₄	1	1	yes	characteristic
140	C₅×C₈	#121	1	1	yes	characteristic
141	C₅×C₈	#124	1	1	yes	characteristic
142	C₂×#41	—	2	2	no	—
143	#135	—	2	2	no	—
144	#135	—	2	2	no	—
145	#135	—	2	2	no	—
146	#135	—	2	2	no	—
147	C₂×Q₁₆	C₂×C₄×C₅	1	2	yes	—
148	C₂×SD₁₆	—	2	2	no	—
149	C₂×D₈	C₂×C₄×C₅	1	2	yes	—
150	C₂×M₁₆	C₅×D₄	1	1	yes	characteristic
151	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
152	#117	—	2	2	no	—
153	#117	C₅×D₄	1	1	yes	characteristic
154	Q₃₂	—	2	1	no	—
155	Q₃₂	—	2	1	no	—
156	SD₃₂	—	2	2	no	—
157	SD₃₂	—	2	2	no	—
158	D₁₆	—	2	1	no	—
159	D₁₆	—	2	1	no	—
160	M₃₂	—	2	1	no	—
161	M₃₂	C₅×D₄	1	2	yes	—
162	C₂×C₁₆	—	2	1	no	—
163	C₂×C₁₆	C₂×C₄×C₅	1	1	yes	characteristic
164	C₂×C₁₆	C₂×C₄×C₅	1	1	yes	characteristic
165	C₂×C₁₆	C₅×D₄	1	1	yes	characteristic
166	C₂×C₁₆	C₅×D₄	1	1	yes	characteristic
167	C₂×C₁₆	C₅×D₄	1	1	yes	characteristic
168	C₂×C₁₆	C₅×D₄	1	1	yes	characteristic
169	#107	—	2	2	no	—
170	#106	—	2	2	no	—
171	#104	—	4	4	no	—
172	#99	—	4	4	no	—
173	#97	—	2	1	no	—
174	C₄×C₈	—	2	1	no	—
175	C₂²×C₅	—	4	1	no	—
176	C₂²×C₅	—	8	2	no	—
177	C₂²×C₅	—	16	2	no	—
178	C₂²×C₅	C₄×D₈	1	1	yes	characteristic
179	C₂²×C₅	#466	1	1	yes	characteristic
180	C₂²×C₅	#493	1	1	yes	characteristic
181	C₄×C₅	—	4	2	no	—
182	C₄×C₅	—	8	2	no	—
183	C₄×C₅	—	16	2	no	—
184	C₄×C₅	#390	1	1	yes	characteristic
185	C₄×C₅	C₄×D₈	1	1	yes	characteristic
186	C₄×C₅	#466	1	1	yes	characteristic
187	C₄×C₅	#495	1	1	yes	characteristic
188	#41	—	4	2	no	—
189	#41	—	4	2	no	—
190	#41	—	4	2	no	—
191	#41	—	4	2	no	—
192	C₂×Q₈	—	2	2	no	—
193	C₂×D₄	—	2	2	no	—
194	C₂×D₄	—	4	2	no	—
195	C₂²×C₄	—	4	2	no	—
196	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
197	Q₁₆	—	2	2	no	—
198	Q₁₆	—	2	2	no	—
199	SD₁₆	—	4	2	no	—
200	SD₁₆	—	4	2	no	—
201	D₈	—	2	2	no	—
202	D₈	—	2	2	no	—
203	M₁₆	—	2	1	no	—
204	M₁₆	C₅×D₈	1	2	yes	—
205	C₂×C₈	—	2	1	no	—
206	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
207	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
208	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
209	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
210	C₂×C₈	C₅×#41	1	1	yes	characteristic
211	C₂×C₈	C₅×#41	1	1	yes	characteristic
212	C₄⋊C₄	—	2	1	no	—
213	C₄⋊C₄	—	4	2	no	—
214	C₂²⋊C₄	—	4	2	no	—
215	C₂²⋊C₄	C₅×D₈	1	2	yes	—
216	C₄²	—	2	1	no	—
217	C₄²	—	4	2	no	—
218	C₁₆	—	2	2	no	—
219	C₁₆	C₅×C₂²⋊C₄	1	1	yes	characteristic
220	C₁₆	C₅×C₂²⋊C₄	1	1	yes	characteristic
221	C₁₆	C₅×C₂²⋊C₄	1	1	yes	characteristic
222	C₁₆	C₅×C₂²⋊C₄	1	1	yes	characteristic
223	C₂×C₅	—	2	1	no	—
224	C₂×C₅	—	2	1	no	—
225	C₂×C₅	—	2	1	no	—
226	C₂×C₅	—	16	2	no	—
227	C₂×C₅	#1970	1	1	yes	characteristic
228	C₂³	—	4	2	no	—
229	C₂³	C₂×C₅×D₈	1	1	yes	characteristic
230	Q₈	—	4	2	no	—
231	Q₈	—	4	2	no	—
232	D₄	—	4	2	no	—
233	D₄	—	4	2	no	—
234	D₄	—	8	2	no	—
235	D₄	—	8	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	8	2	no	—
239	C₂×C₄	—	8	2	no	—
240	C₂×C₄	—	8	2	no	—
241	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
242	C₂×C₄	C₅×#121	1	1	yes	characteristic
243	C₂×C₄	C₅×#127	1	1	yes	characteristic
244	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
245	C₂×C₄	C₅×#135	1	1	yes	characteristic
246	C₂×C₄	C₅×#135	1	1	yes	characteristic
247	C₈	—	2	2	no	—
248	C₈	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
249	C₈	C₄×C₅×D₄	1	1	yes	characteristic
250	C₈	C₅×#121	1	1	yes	characteristic
251	C₈	C₅×#124	1	1	yes	characteristic
252	C₅	#13243	1	1	yes	characteristic, direct product factor, Sylow
253	C₂²	—	4	1	no	—
254	C₂²	—	8	2	no	—
255	C₂²	—	16	2	no	—
256	C₂²	C₄×C₅×D₈	1	1	yes	characteristic
257	C₂²	C₅×#466	1	1	yes	characteristic
258	C₂²	C₅×#493	1	1	yes	characteristic
259	C₄	—	4	2	no	—
260	C₄	—	8	2	no	—
261	C₄	—	16	2	no	—
262	C₄	C₅×#390	1	1	yes	characteristic
263	C₄	C₄×C₅×D₈	1	1	yes	characteristic
264	C₄	C₅×#466	1	1	yes	characteristic
265	C₄	C₅×#495	1	1	yes	characteristic
266	C₂	—	2	1	no	—
267	C₂	—	2	1	no	—
268	C₂	—	2	1	no	—
269	C₂	—	16	2	no	—
270	C₂	C₅×#1970	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₅	#466
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₂²×C₈	C₂³×C₅
socle	C₂×C₅	#1970
prime radical	C₂×C₅×#508	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅ ◁ C₂²×C₄×C₅ ◁ C₅×#405 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅×#2158	C₂	semidirect
3. C₅×#2145	C₂	semidirect
4. C₅×#2108	C₂	semidirect
5. C₅×#1976	C₂	semidirect
11. C₂×C₅×Q₃₂	C₄	semidirect
12. C₂×C₅×SD₃₂	C₄	semidirect
13. C₂×C₅×D₁₆	C₄	semidirect
25. #13243	C₅	direct (N × H)
60. #2158	C₂×C₅	semidirect
61. #2145	C₂×C₅	semidirect
62. #2108	C₂×C₅	semidirect
63. #1976	C₂×C₅	semidirect
104. C₂×Q₃₂	C₄×C₅	semidirect
105. C₂×SD₃₂	C₄×C₅	semidirect
106. C₂×D₁₆	C₄×C₅	semidirect
252. C₅	#13243	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×#508	2
C₅×#2158	2
C₅×#2145	2
C₅×#2108	2
C₅×#1976	2
#13243	5

Subgroups — C5×#13243 #11935963

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#13243 #11935963