Subgroups — C₅×#21690 #11944410

Order 1280.

1398 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#493	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#442	C₂	1	4	yes	maximal, complemented
3	C₄×C₅×#127	C₂	1	2	yes	maximal, complemented
4	C₅×#1972	C₂	1	4	yes	maximal, complemented
5	C₅×#1957	C₂	1	2	yes	maximal, complemented
6	C₅×#1844	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×C₅×#97	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₅×D₈	C₂²	1	4	yes	—
9	C₂×C₅×#127	C₂²	1	2	yes	—
10	C₂×C₄×C₅×D₄	—	2	4	no	—
11	C₂×C₄×C₅×D₄	C₂²	1	4	yes	—
12	C₅×#493	C₄	1	8	yes	complemented
13	C₅×#442	—	2	16	no	—
14	C₂×C₅×#106	C₂²	1	4	yes	—
15	C₂×C₅×#102	C₂²	1	8	yes	—
16	C₂×C₅×#97	C₂²	1	2	yes	—
17	C₂×C₅×#97	C₂²	1	2	yes	—
18	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
20	C₅×#390	—	2	4	no	—
21	C₅×#390	C₂²	1	4	yes	—
22	C₅×#384	C₂²	1	2	yes	—
23	C₄³×C₅	C₂²	1	1	yes	characteristic
24	#21690	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
25	C₂²×C₅×D₄	—	2	4	no	—
26	C₂²×C₅×D₄	C₂³	1	4	yes	—
27	C₂³×C₄×C₅	—	2	4	no	—
28	C₂×C₅×D₈	—	2	8	no	—
29	C₂×C₅×D₈	—	2	8	no	—
30	C₂×C₅×D₈	C₂×C₄	1	16	yes	complemented
31	C₂²×C₅×C₈	C₂³	1	2	yes	—
32	C₂²×C₅×C₈	C₂³	1	2	yes	—
33	C₅×#127	—	2	4	no	—
34	C₅×#127	C₂×C₄	1	8	yes	complemented
35	C₄×C₅×D₄	—	2	16	no	—
36	C₄×C₅×D₄	—	4	16	no	—
37	C₂×C₅×C₄⋊C₄	—	2	4	no	—
38	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
39	C₂×C₅×C₂²⋊C₄	—	2	4	no	—
40	C₂×C₅×C₂²⋊C₄	—	2	8	no	—
41	C₂×C₄²×C₅	D₄	1	4	yes	—
42	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
43	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
44	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
45	C₅×#106	—	2	8	no	—
46	C₅×#102	—	2	16	no	—
47	C₅×#97	—	2	4	no	—
48	C₅×#97	D₄	1	8	yes	complemented
49	C₄×C₅×C₈	—	2	2	no	—
50	C₄×C₅×C₈	C₂×C₄	1	4	yes	complemented
51	C₂×#493	C₂×C₅	1	1	yes	characteristic
52	C₂×#442	C₂×C₅	1	4	yes	complemented
53	C₄×#127	C₂×C₅	1	2	yes	complemented
54	#1972	C₂×C₅	1	4	yes	complemented
55	#1957	C₂×C₅	1	2	yes	complemented
56	#1844	C₂×C₅	1	1	yes	characteristic, complemented
57	C₄×#97	C₂×C₅	1	1	yes	characteristic, complemented
58	C₂⁴×C₅	—	2	4	no	—
59	C₂×C₅×D₄	—	2	16	no	—
60	C₂×C₅×D₄	—	2	8	no	—
61	C₂×C₅×D₄	—	2	8	no	—
62	C₂×C₅×D₄	—	2	4	no	—
63	C₂×C₅×D₄	—	4	8	no	—
64	C₂×C₅×D₄	—	4	8	no	—
65	C₂×C₅×D₄	C₂²×C₄	1	8	yes	—
66	C₂²×C₄×C₅	—	2	4	no	—
67	C₂²×C₄×C₅	—	4	16	no	—
68	C₂²×C₄×C₅	—	4	8	no	—
69	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
70	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
71	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
72	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
73	C₅×D₈	—	2	32	no	—
74	C₂×C₅×C₈	—	2	2	no	—
75	C₂×C₅×C₈	—	2	2	no	—
76	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
77	C₂×C₅×C₈	C₂×D₄	1	4	yes	—
78	C₂×C₅×C₈	C₂×D₄	1	4	yes	—
79	C₂×C₅×C₈	#41	1	4	yes	—
80	C₅×C₄⋊C₄	—	2	8	no	—
81	C₅×C₄⋊C₄	—	4	8	no	—
82	C₅×C₂²⋊C₄	—	4	16	no	—
83	C₅×C₂²⋊C₄	—	4	8	no	—
84	C₄²×C₅	—	2	8	no	—
85	C₄²×C₅	—	2	2	no	—
86	C₄²×C₅	C₂²×C₄	1	2	yes	—
87	C₄²×C₅	C₂×D₄	1	4	yes	—
88	C₄²×C₅	#41	1	2	yes	—
89	C₂²×D₈	C₂²×C₅	1	4	yes	—
90	C₂×#127	C₂²×C₅	1	2	yes	—
91	C₂×C₄×D₄	—	2	4	no	—
92	C₂×C₄×D₄	C₂²×C₅	1	4	yes	—
93	#493	C₄×C₅	1	8	yes	complemented
94	#442	—	2	16	no	—
95	C₂×#106	C₂²×C₅	1	4	yes	—
96	C₂×#102	C₂²×C₅	1	8	yes	—
97	C₂×#97	C₂²×C₅	1	2	yes	—
98	C₂×#97	C₂²×C₅	1	2	yes	—
99	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
100	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
101	#390	—	2	4	no	—
102	#390	C₂²×C₅	1	4	yes	—
103	#384	C₂²×C₅	1	2	yes	—
104	C₄³	C₂²×C₅	1	1	yes	characteristic
105	C₂³×C₅	—	2	8	no	—
106	C₂³×C₅	—	4	4	no	—
107	C₂³×C₅	—	4	4	no	—
108	C₂³×C₅	—	4	8	no	—
109	C₂³×C₅	—	4	8	no	—
110	C₂³×C₅	C₂²×D₄	1	1	yes	characteristic
111	C₅×D₄	—	2	16	no	—
112	C₅×D₄	—	4	8	no	—
113	C₅×D₄	—	4	32	no	—
114	C₂×C₄×C₅	—	2	4	no	—
115	C₂×C₄×C₅	—	2	2	no	—
116	C₂×C₄×C₅	—	2	4	no	—
117	C₂×C₄×C₅	—	4	4	no	—
118	C₂×C₄×C₅	—	4	8	no	—
119	C₂×C₄×C₅	—	8	16	no	—
120	C₂×C₄×C₅	C₄×D₄	1	4	yes	—
121	C₂×C₄×C₅	#127	1	4	yes	—
122	C₂×C₄×C₅	AGL(1,ℤ/8)	1	8	yes	—
123	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
124	C₂×C₄×C₅	C₂²×D₄	1	2	yes	—
125	C₂×C₄×C₅	C₂×#41	1	2	yes	—
126	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
127	C₅×C₈	—	2	4	no	—
128	C₅×C₈	C₄×D₄	1	8	yes	complemented
129	C₂²×D₄	—	2	4	no	—
130	C₂²×D₄	C₂³×C₅	1	4	yes	—
131	C₂³×C₄	—	2	4	no	—
132	C₂×D₈	—	2	8	no	—
133	C₂×D₈	—	2	8	no	—
134	C₂×D₈	C₂×C₄×C₅	1	16	yes	complemented
135	C₂²×C₈	C₂³×C₅	1	2	yes	—
136	C₂²×C₈	C₂³×C₅	1	2	yes	—
137	#127	—	2	4	no	—
138	#127	C₂×C₄×C₅	1	8	yes	complemented
139	C₄×D₄	—	2	16	no	—
140	C₄×D₄	—	4	16	no	—
141	C₂×C₄⋊C₄	—	2	4	no	—
142	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
143	C₂×C₂²⋊C₄	—	2	8	no	—
144	C₂×C₂²⋊C₄	—	2	4	no	—
145	C₂×C₄²	C₅×D₄	1	4	yes	—
146	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
147	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
148	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
149	#106	—	2	8	no	—
150	#102	—	2	16	no	—
151	#97	—	2	4	no	—
152	#97	C₅×D₄	1	8	yes	complemented
153	C₄×C₈	—	2	2	no	—
154	C₄×C₈	C₂×C₄×C₅	1	4	yes	complemented
155	C₂²×C₅	—	4	8	no	—
156	C₂²×C₅	—	4	16	no	—
157	C₂²×C₅	—	8	4	no	—
158	C₂²×C₅	—	8	4	no	—
159	C₂²×C₅	—	8	8	no	—
160	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
161	C₂²×C₅	C₂×#127	1	1	yes	characteristic
162	C₂²×C₅	#532	1	1	yes	characteristic
163	C₂²×C₅	C₂×AGL(1,ℤ/8)	1	4	yes	—
164	C₄×C₅	—	2	4	no	—
165	C₄×C₅	—	4	2	no	—
166	C₄×C₅	—	8	4	no	—
167	C₄×C₅	#442	1	8	yes	complemented
168	C₄×C₅	C₂×C₄×D₄	1	2	yes	—
169	C₄×C₅	#532	1	2	yes	—
170	C₂⁴	—	2	4	no	—
171	C₂×D₄	—	2	16	no	—
172	C₂×D₄	—	2	8	no	—
173	C₂×D₄	—	2	8	no	—
174	C₂×D₄	—	2	4	no	—
175	C₂×D₄	—	4	8	no	—
176	C₂×D₄	—	4	8	no	—
177	C₂×D₄	C₂²×C₄×C₅	1	8	yes	—
178	C₂²×C₄	—	2	4	no	—
179	C₂²×C₄	—	4	16	no	—
180	C₂²×C₄	—	4	8	no	—
181	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
182	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
183	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
184	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
185	D₈	—	2	32	no	—
186	C₂×C₈	—	2	2	no	—
187	C₂×C₈	—	2	2	no	—
188	C₂×C₈	C₂²×C₄×C₅	1	4	yes	—
189	C₂×C₈	C₂×C₅×D₄	1	4	yes	—
190	C₂×C₈	C₂×C₅×D₄	1	4	yes	—
191	C₂×C₈	C₅×#41	1	4	yes	—
192	C₄⋊C₄	—	2	8	no	—
193	C₄⋊C₄	—	4	8	no	—
194	C₂²⋊C₄	—	4	16	no	—
195	C₂²⋊C₄	—	4	8	no	—
196	C₄²	—	2	8	no	—
197	C₄²	—	2	2	no	—
198	C₄²	C₂²×C₄×C₅	1	2	yes	—
199	C₄²	C₂×C₅×D₄	1	4	yes	—
200	C₄²	C₅×#41	1	2	yes	—
201	C₂×C₅	—	8	8	no	—
202	C₂×C₅	C₄×#127	1	1	yes	characteristic
203	C₂×C₅	C₂×#442	1	2	yes	—
204	C₂×C₅	#3151	1	2	yes	—
205	C₂×C₅	#3184	1	2	yes	—
206	C₂³	—	2	8	no	—
207	C₂³	—	4	4	no	—
208	C₂³	—	4	4	no	—
209	C₂³	—	4	8	no	—
210	C₂³	—	4	8	no	—
211	C₂³	C₂²×C₅×D₄	1	1	yes	characteristic
212	D₄	—	2	16	no	—
213	D₄	—	4	32	no	—
214	D₄	—	4	8	no	—
215	C₂×C₄	—	2	4	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	4	no	—
218	C₂×C₄	—	4	4	no	—
219	C₂×C₄	—	4	8	no	—
220	C₂×C₄	—	8	16	no	—
221	C₂×C₄	C₄×C₅×D₄	1	4	yes	—
222	C₂×C₄	C₅×#127	1	4	yes	—
223	C₂×C₄	C₅×AGL(1,ℤ/8)	1	8	yes	—
224	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
225	C₂×C₄	C₂²×C₅×D₄	1	2	yes	—
226	C₂×C₄	C₂×C₅×#41	1	2	yes	—
227	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
228	C₈	—	2	4	no	—
229	C₈	C₄×C₅×D₄	1	8	yes	complemented
230	C₅	#21690	1	1	yes	characteristic, direct product factor, Sylow
231	C₂²	—	4	8	no	—
232	C₂²	—	4	16	no	—
233	C₂²	—	8	4	no	—
234	C₂²	—	8	4	no	—
235	C₂²	—	8	8	no	—
236	C₂²	C₂×C₄×C₅×D₄	1	1	yes	characteristic
237	C₂²	C₂×C₅×#127	1	1	yes	characteristic
238	C₂²	C₅×#532	1	1	yes	characteristic
239	C₂²	C₂×C₅×AGL(1,ℤ/8)	1	4	yes	—
240	C₄	—	2	4	no	—
241	C₄	—	4	2	no	—
242	C₄	—	8	4	no	—
243	C₄	C₅×#442	1	8	yes	complemented
244	C₄	C₂×C₄×C₅×D₄	1	2	yes	—
245	C₄	C₅×#532	1	2	yes	—
246	C₂	—	8	8	no	—
247	C₂	C₄×C₅×#127	1	1	yes	characteristic
248	C₂	C₂×C₅×#442	1	2	yes	—
249	C₂	C₅×#3151	1	2	yes	—
250	C₂	C₅×#3184	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₅	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
prime radical	C₂×C₅×#493	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₄³×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×#442	C₂	semidirect
3. C₄×C₅×#127	C₂	semidirect
4. C₅×#1972	C₂	semidirect
5. C₅×#1957	C₂	semidirect
6. C₅×#1844	C₂	semidirect
7. C₄×C₅×#97	C₂	semidirect
12. C₅×#493	C₄	semidirect
24. #21690	C₅	direct (N × H)
30. C₂×C₅×D₈	C₂×C₄	semidirect
34. C₅×#127	C₂×C₄	semidirect
48. C₅×#97	D₄	semidirect
50. C₄×C₅×C₈	C₂×C₄	semidirect
52. C₂×#442	C₂×C₅	semidirect
53. C₄×#127	C₂×C₅	semidirect
54. #1972	C₂×C₅	semidirect
55. #1957	C₂×C₅	semidirect
56. #1844	C₂×C₅	semidirect
57. C₄×#97	C₂×C₅	semidirect
93. #493	C₄×C₅	semidirect
128. C₅×C₈	C₄×D₄	semidirect
134. C₂×D₈	C₂×C₄×C₅	semidirect
138. #127	C₂×C₄×C₅	semidirect
152. #97	C₅×D₄	semidirect
154. C₄×C₈	C₂×C₄×C₅	semidirect
167. C₄×C₅	#442	semidirect
229. C₈	C₄×C₅×D₄	semidirect
230. C₅	#21690	direct (N × H)
243. C₄	C₅×#442	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#493	2
C₂×C₅×#442	2
C₄×C₅×#127	2
C₅×#1972	2
C₅×#1957	2
C₅×#1844	2
C₄×C₅×#97	2
#21690	5

Subgroups — C5×#21690 #11944410

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#21690 #11944410