Subgroups — C₅×#21705 #11944425

Order 1280.

1046 conjugacy classes of subgroups.

The subgroup lattice has 268 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×#443	C₂	1	4	yes	maximal, complemented
3	C₅×#2931	C₂	1	2	yes	maximal, complemented
4	C₅×#1974	C₂	1	4	yes	maximal, complemented
5	C₅×#1962	C₂	1	2	yes	maximal, complemented
6	C₅×#1848	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄²×C₅×C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅×#135	C₂²	1	4	yes	—
9	C₂×C₅×#131	C₂²	1	4	yes	—
10	C₅×#532	C₂²	1	2	yes	—
11	C₅×#495	C₄	1	4	yes	complemented
12	C₅×#494	C₄	1	1	yes	characteristic, complemented
13	C₅×#493	C₄	1	1	yes	characteristic, complemented
14	C₅×#492	C₄	1	2	yes	complemented
15	C₅×#443	—	2	16	no	—
16	C₅×#436	—	2	4	no	—
17	C₅×C₈×D₄	—	2	4	no	—
18	C₂×C₅×#108	C₂²	1	4	yes	—
19	C₅×#423	C₂²	1	2	yes	—
20	C₂×C₅×#104	C₂²	1	8	yes	—
21	C₅×#408	C₂²	1	4	yes	—
22	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
23	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
25	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
26	C₄³×C₅	C₂²	1	1	yes	characteristic
27	#21705	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
28	C₂×C₅×#41	C₂³	1	4	yes	—
29	C₅×#135	—	2	8	no	—
30	C₅×#135	—	2	8	no	—
31	C₂×C₅×Q₁₆	C₂×C₄	1	4	yes	—
32	C₂×C₅×SD₁₆	C₂×C₄	1	8	yes	—
33	C₂×C₅×D₈	C₂×C₄	1	4	yes	—
34	C₅×#131	—	2	16	no	—
35	C₂×C₅×M₁₆	—	2	4	no	—
36	C₂×C₅×M₁₆	C₂³	1	4	yes	—
37	C₂²×C₅×C₈	—	2	4	no	—
38	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
39	C₂²×C₅×C₈	C₂³	1	2	yes	—
40	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
41	C₅×C₄⋊Q₈	C₂×C₄	1	2	yes	—
42	C₅×#127	C₂×C₄	1	2	yes	—
43	C₅×#124	C₂×C₄	1	4	yes	—
44	C₅×#121	—	2	4	no	—
45	C₄×C₅×D₄	—	2	4	no	—
46	C₂×C₄²×C₅	—	2	2	no	—
47	C₂×C₄²×C₅	C₂³	1	2	yes	—
48	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
49	C₅×#108	—	2	8	no	—
50	C₅×C₄⋊C₈	—	2	4	no	—
51	C₅×#104	—	2	16	no	—
52	C₅×C₂²⋊C₈	—	2	8	no	—
53	C₄×C₅×C₈	—	2	4	no	—
54	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
55	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
56	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
57	C₄×C₅×C₈	D₄	1	2	yes	—
58	C₄×C₅×C₈	D₄	1	2	yes	—
59	C₄×C₅×C₈	D₄	1	8	yes	complemented
60	#3147	C₂×C₅	1	1	yes	characteristic
61	C₂×#443	C₂×C₅	1	4	yes	complemented
62	#2931	C₂×C₅	1	2	yes	complemented
63	#1974	C₂×C₅	1	4	yes	complemented
64	#1962	C₂×C₅	1	2	yes	complemented
65	#1848	C₂×C₅	1	1	yes	characteristic, complemented
66	C₄²×C₈	C₂×C₅	1	1	yes	characteristic, complemented
67	C₅×#41	—	2	8	no	—
68	C₅×#41	—	2	8	no	—
69	C₂×C₅×Q₈	C₂²×C₄	1	4	yes	—
70	C₂×C₅×D₄	—	2	4	no	—
71	C₂×C₅×D₄	—	2	4	no	—
72	C₂×C₅×D₄	C₂²×C₄	1	4	yes	—
73	C₂²×C₄×C₅	—	2	2	no	—
74	C₂²×C₄×C₅	—	2	4	no	—
75	C₂²×C₄×C₅	#41	1	2	yes	—
76	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
77	C₅×Q₁₆	—	2	8	no	—
78	C₅×SD₁₆	—	2	16	no	—
79	C₅×D₈	—	2	8	no	—
80	C₅×M₁₆	—	2	8	no	—
81	C₅×M₁₆	—	4	8	no	—
82	C₂×C₅×C₈	—	2	2	no	—
83	C₂×C₅×C₈	—	2	4	no	—
84	C₂×C₅×C₈	—	4	8	no	—
85	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
86	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
87	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
88	C₂×C₅×C₈	C₂×D₄	1	4	yes	—
89	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
90	C₂×C₅×C₈	C₂×D₄	1	4	yes	—
91	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
92	C₂×C₅×C₈	#41	1	4	yes	—
93	C₅×C₄⋊C₄	—	2	4	no	—
94	C₅×C₂²⋊C₄	—	2	8	no	—
95	C₄²×C₅	—	2	8	no	—
96	C₄²×C₅	—	2	2	no	—
97	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
98	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
99	C₄²×C₅	C₂×D₄	1	4	yes	—
100	C₄²×C₅	C₂×D₄	1	2	yes	—
101	C₂×#135	C₂²×C₅	1	4	yes	—
102	C₂×#131	C₂²×C₅	1	4	yes	—
103	#532	C₂²×C₅	1	2	yes	—
104	#495	C₄×C₅	1	4	yes	complemented
105	#494	C₄×C₅	1	1	yes	characteristic, complemented
106	#493	C₄×C₅	1	1	yes	characteristic, complemented
107	#492	C₄×C₅	1	2	yes	complemented
108	#443	—	2	16	no	—
109	#436	—	2	4	no	—
110	C₈×D₄	—	2	4	no	—
111	C₂×#108	C₂²×C₅	1	4	yes	—
112	#423	C₂²×C₅	1	2	yes	—
113	C₂×#104	C₂²×C₅	1	8	yes	—
114	#408	C₂²×C₅	1	4	yes	—
115	C₂×C₄×C₈	C₂²×C₅	1	2	yes	—
116	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
117	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
118	C₂×C₄×C₈	C₂²×C₅	1	2	yes	—
119	C₄³	C₂²×C₅	1	1	yes	characteristic
120	C₂³×C₅	—	2	4	no	—
121	C₂³×C₅	C₂×#41	1	1	yes	characteristic
122	C₅×Q₈	—	2	8	no	—
123	C₅×D₄	—	2	8	no	—
124	C₅×D₄	—	4	8	no	—
125	C₅×D₄	—	4	8	no	—
126	C₂×C₄×C₅	—	2	8	no	—
127	C₂×C₄×C₅	—	2	4	no	—
128	C₂×C₄×C₅	—	2	4	no	—
129	C₂×C₄×C₅	—	2	4	no	—
130	C₂×C₄×C₅	—	4	4	no	—
131	C₂×C₄×C₅	—	4	4	no	—
132	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
133	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
134	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
135	C₂×C₄×C₅	C₂²×D₄	1	2	yes	—
136	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
137	C₂×C₄×C₅	C₂×#41	1	2	yes	—
138	C₅×C₈	—	2	2	no	—
139	C₅×C₈	—	4	4	no	—
140	C₅×C₈	C₄×D₄	1	4	yes	—
141	C₅×C₈	C₄×D₄	1	4	yes	—
142	C₅×C₈	#127	1	2	yes	—
143	C₅×C₈	#127	1	2	yes	—
144	C₂×#41	C₂³×C₅	1	4	yes	—
145	#135	—	2	8	no	—
146	#135	—	2	8	no	—
147	C₂×Q₁₆	C₂×C₄×C₅	1	4	yes	—
148	C₂×SD₁₆	C₂×C₄×C₅	1	8	yes	—
149	C₂×D₈	C₂×C₄×C₅	1	4	yes	—
150	#131	—	2	16	no	—
151	C₂×M₁₆	—	2	4	no	—
152	C₂×M₁₆	C₂³×C₅	1	4	yes	—
153	C₂²×C₈	—	2	4	no	—
154	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
155	C₂²×C₈	C₂³×C₅	1	2	yes	—
156	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
157	C₄⋊Q₈	C₂×C₄×C₅	1	2	yes	—
158	#127	C₂×C₄×C₅	1	2	yes	—
159	#124	C₂×C₄×C₅	1	4	yes	—
160	#121	—	2	4	no	—
161	C₄×D₄	—	2	4	no	—
162	C₂×C₄²	—	2	2	no	—
163	C₂×C₄²	C₂³×C₅	1	2	yes	—
164	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
165	#108	—	2	8	no	—
166	C₄⋊C₈	—	2	4	no	—
167	#104	—	2	16	no	—
168	C₂²⋊C₈	—	2	8	no	—
169	C₄×C₈	—	2	4	no	—
170	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
171	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
172	C₄×C₈	C₂×C₄×C₅	1	2	yes	—
173	C₄×C₈	C₅×D₄	1	2	yes	—
174	C₄×C₈	C₅×D₄	1	2	yes	—
175	C₄×C₈	C₅×D₄	1	8	yes	complemented
176	C₂²×C₅	—	2	2	no	—
177	C₂²×C₅	—	4	4	no	—
178	C₂²×C₅	—	4	8	no	—
179	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
180	C₂²×C₅	#532	1	2	yes	—
181	C₄×C₅	—	2	8	no	—
182	C₄×C₅	—	2	2	no	—
183	C₄×C₅	—	4	4	no	—
184	C₄×C₅	#443	1	4	yes	complemented
185	C₄×C₅	C₂×C₄×D₄	1	2	yes	—
186	C₄×C₅	C₂×#127	1	1	yes	characteristic
187	C₄×C₅	#532	1	1	yes	characteristic
188	#41	—	2	8	no	—
189	#41	—	2	8	no	—
190	C₂×Q₈	C₂²×C₄×C₅	1	4	yes	—
191	C₂×D₄	—	2	4	no	—
192	C₂×D₄	—	2	4	no	—
193	C₂×D₄	C₂²×C₄×C₅	1	4	yes	—
194	C₂²×C₄	—	2	2	no	—
195	C₂²×C₄	—	2	4	no	—
196	C₂²×C₄	C₅×#41	1	2	yes	—
197	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
198	Q₁₆	—	2	8	no	—
199	SD₁₆	—	2	16	no	—
200	D₈	—	2	8	no	—
201	M₁₆	—	2	8	no	—
202	M₁₆	—	4	8	no	—
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	2	4	no	—
205	C₂×C₈	—	4	8	no	—
206	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
207	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
208	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
209	C₂×C₈	C₂×C₅×D₄	1	4	yes	—
210	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
211	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
212	C₂×C₈	C₂×C₅×D₄	1	4	yes	—
213	C₂×C₈	C₅×#41	1	4	yes	—
214	C₄⋊C₄	—	2	4	no	—
215	C₂²⋊C₄	—	2	8	no	—
216	C₄²	—	2	8	no	—
217	C₄²	—	2	2	no	—
218	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
219	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
220	C₄²	C₂×C₅×D₄	1	4	yes	—
221	C₄²	C₂×C₅×D₄	1	2	yes	—
222	C₂×C₅	—	2	2	no	—
223	C₂×C₅	—	8	4	no	—
224	C₂×C₅	C₄×#127	1	1	yes	characteristic
225	C₂×C₅	C₂×#443	1	2	yes	—
226	C₂³	—	2	4	no	—
227	C₂³	C₂×C₅×#41	1	1	yes	characteristic
228	Q₈	—	2	8	no	—
229	D₄	—	2	8	no	—
230	D₄	—	4	8	no	—
231	D₄	—	4	8	no	—
232	C₂×C₄	—	2	8	no	—
233	C₂×C₄	—	2	4	no	—
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	2	4	no	—
236	C₂×C₄	—	4	4	no	—
237	C₂×C₄	—	4	4	no	—
238	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
239	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
240	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
241	C₂×C₄	C₂²×C₅×D₄	1	2	yes	—
242	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
243	C₂×C₄	C₂×C₅×#41	1	2	yes	—
244	C₈	—	2	2	no	—
245	C₈	—	4	4	no	—
246	C₈	C₄×C₅×D₄	1	4	yes	—
247	C₈	C₄×C₅×D₄	1	4	yes	—
248	C₈	C₅×#127	1	2	yes	—
249	C₈	C₅×#127	1	2	yes	—
250	C₅	#21705	1	1	yes	characteristic, direct product factor, Sylow
251	C₂²	—	2	2	no	—
252	C₂²	—	4	4	no	—
253	C₂²	—	4	8	no	—
254	C₂²	C₂×C₄×C₅×D₄	1	1	yes	characteristic
255	C₂²	C₅×#532	1	2	yes	—
256	C₄	—	2	8	no	—
257	C₄	—	2	2	no	—
258	C₄	—	4	4	no	—
259	C₄	C₅×#443	1	4	yes	complemented
260	C₄	C₂×C₄×C₅×D₄	1	2	yes	—
261	C₄	C₂×C₅×#127	1	1	yes	characteristic
262	C₄	C₅×#532	1	1	yes	characteristic
263	C₂	—	2	2	no	—
264	C₂	—	8	4	no	—
265	C₂	C₄×C₅×#127	1	1	yes	characteristic
266	C₂	C₂×C₅×#443	1	2	yes	—

	subgroup	G/H
center	C₂×C₅×C₈	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
socle	C₂²×C₅	C₂×C₄×D₄
prime radical	C₅×#3147	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅×C₈ ◁ C₂×C₄×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×#443	C₂	semidirect
3. C₅×#2931	C₂	semidirect
4. C₅×#1974	C₂	semidirect
5. C₅×#1962	C₂	semidirect
6. C₅×#1848	C₂	semidirect
7. C₄²×C₅×C₈	C₂	semidirect
11. C₅×#495	C₄	semidirect
12. C₅×#494	C₄	semidirect
13. C₅×#493	C₄	semidirect
14. C₅×#492	C₄	semidirect
27. #21705	C₅	direct (N × H)
59. C₄×C₅×C₈	D₄	semidirect
61. C₂×#443	C₂×C₅	semidirect
62. #2931	C₂×C₅	semidirect
63. #1974	C₂×C₅	semidirect
64. #1962	C₂×C₅	semidirect
65. #1848	C₂×C₅	semidirect
66. C₄²×C₈	C₂×C₅	semidirect
104. #495	C₄×C₅	semidirect
105. #494	C₄×C₅	semidirect
106. #493	C₄×C₅	semidirect
107. #492	C₄×C₅	semidirect
175. C₄×C₈	C₅×D₄	semidirect
184. C₄×C₅	#443	semidirect
250. C₅	#21705	direct (N × H)
259. C₄	C₅×#443	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×#443	2
C₅×#2931	2
C₅×#1974	2
C₅×#1962	2
C₅×#1848	2
C₄²×C₅×C₈	2
#21705	5

Subgroups — C5×#21705 #11944425

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#21705 #11944425