Subgroups — C₅×#22256 #11944976

Order 1280.

614 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3160	C₂	1	1	yes	maximal, characteristic
2	C₅×#2960	C₂	1	2	yes	maximal
3	C₅×#1994	C₂	1	2	yes	maximal
4	C₅×#1922	C₂	1	8	yes	maximal
5	C₅×#1837	C₂	1	1	yes	maximal, characteristic
6	C₅×#1744	C₂	1	1	yes	maximal, characteristic
7	C₅×#533	C₂²	1	2	yes	—
8	C₅×#500	C₄	1	1	yes	characteristic, complemented
9	C₅×#500	C₄	1	1	yes	characteristic, complemented
10	C₅×#499	C₄	1	4	yes	complemented
11	C₅×#498	C₄	1	2	yes	complemented
12	C₅×#446	—	2	2	no	—
13	C₅×#446	—	2	2	no	—
14	C₅×C₈×Q₈	—	2	2	no	—
15	C₅×C₈×Q₈	—	2	2	no	—
16	C₅×#427	C₂²	1	4	yes	—
17	C₅×#424	C₂²	1	4	yes	—
18	C₅×#423	C₂²	1	2	yes	—
19	C₅×#405	C₂²	1	4	yes	—
20	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
22	C₄³×C₅	C₂²	1	1	yes	characteristic
23	C₅×#341	C₂²	1	4	yes	—
24	C₅×#339	C₂²	1	8	yes	—
25	C₅×#336	C₂²	1	4	yes	—
26	#22256	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
27	C₂×C₅×M₁₆	C₂³	1	4	yes	—
28	C₂²×C₅×C₈	C₂³	1	2	yes	—
29	C₂²×C₅×C₈	C₂³	1	2	yes	—
30	C₅×C₄⋊Q₈	C₂×C₄	1	2	yes	—
31	C₅×C₄⋊Q₈	C₂×C₄	1	2	yes	—
32	C₅×#125	C₂×C₄	1	4	yes	—
33	C₅×C₂²⋊Q₈	—	2	2	no	—
34	C₅×C₂²⋊Q₈	—	2	2	no	—
35	C₄×C₅×Q₈	—	2	2	no	—
36	C₄×C₅×Q₈	—	2	2	no	—
37	C₅×#117	C₂³	1	4	yes	—
38	C₂×C₄²×C₅	—	2	1	no	—
39	C₂×C₄²×C₅	—	2	1	no	—
40	C₂×C₄²×C₅	C₂³	1	2	yes	—
41	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
42	C₅×#107	C₂×C₄	1	4	yes	—
43	C₅×#107	C₂×C₄	1	4	yes	—
44	C₅×#106	C₂×C₄	1	8	yes	—
45	C₅×C₄⋊C₈	—	2	4	no	—
46	C₅×C₄⋊C₈	—	2	4	no	—
47	C₅×C₄⋊C₈	—	2	2	no	—
48	C₅×C₄⋊C₈	—	2	2	no	—
49	C₅×#97	—	2	4	no	—
50	C₄×C₅×C₈	—	2	4	no	—
51	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
52	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
53	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
54	C₄×C₅×C₈	D₄	1	1	yes	characteristic
55	C₄×C₅×C₈	D₄	1	2	yes	—
56	C₄×C₅×C₈	D₄	1	1	yes	characteristic
57	#3160	C₂×C₅	1	1	yes	characteristic
58	#2960	C₂×C₅	1	2	yes	—
59	#1994	C₂×C₅	1	2	yes	—
60	#1922	C₂×C₅	1	8	yes	—
61	#1837	C₂×C₅	1	1	yes	characteristic
62	#1744	C₂×C₅	1	1	yes	characteristic
63	C₂×C₅×Q₈	—	2	2	no	—
64	C₂×C₅×Q₈	—	2	2	no	—
65	C₂²×C₄×C₅	—	2	2	no	—
66	C₂²×C₄×C₅	#41	1	1	yes	characteristic
67	C₂²×C₄×C₅	#41	1	1	yes	characteristic
68	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
69	C₅×M₁₆	—	2	8	no	—
70	C₂×C₅×C₈	—	2	4	no	—
71	C₂×C₅×C₈	—	2	4	no	—
72	C₂×C₅×C₈	—	2	4	no	—
73	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
74	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
75	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
76	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
77	C₅×C₄⋊C₄	—	2	4	no	—
78	C₅×C₄⋊C₄	—	2	4	no	—
79	C₅×C₄⋊C₄	—	2	2	no	—
80	C₅×C₄⋊C₄	—	2	2	no	—
81	C₅×C₄⋊C₄	C₂²×C₄	1	4	yes	—
82	C₅×C₄⋊C₄	C₂²×C₄	1	4	yes	—
83	C₅×C₂²⋊C₄	—	2	4	no	—
84	C₄²×C₅	—	2	4	no	—
85	C₄²×C₅	—	2	4	no	—
86	C₄²×C₅	—	2	4	no	—
87	C₄²×C₅	—	4	2	no	—
88	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
89	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
90	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
91	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
92	#533	C₂²×C₅	1	2	yes	—
93	#500	C₄×C₅	1	1	yes	characteristic, complemented
94	#500	C₄×C₅	1	1	yes	characteristic, complemented
95	#499	C₄×C₅	1	4	yes	complemented
96	#498	C₄×C₅	1	2	yes	complemented
97	#446	—	2	2	no	—
98	#446	—	2	2	no	—
99	C₈×Q₈	—	2	2	no	—
100	C₈×Q₈	—	2	2	no	—
101	#427	C₂²×C₅	1	4	yes	—
102	#424	C₂²×C₅	1	4	yes	—
103	#423	C₂²×C₅	1	2	yes	—
104	#405	C₂²×C₅	1	4	yes	—
105	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
106	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
107	C₄³	C₂²×C₅	1	1	yes	characteristic
108	#341	C₂²×C₅	1	4	yes	—
109	#339	C₂²×C₅	1	8	yes	—
110	#336	C₂²×C₅	1	4	yes	—
111	C₂³×C₅	C₂×#41	1	1	yes	characteristic
112	C₅×Q₈	—	4	4	no	—
113	C₅×Q₈	—	4	4	no	—
114	C₂×C₄×C₅	—	2	2	no	—
115	C₂×C₄×C₅	—	2	2	no	—
116	C₂×C₄×C₅	—	2	4	no	—
117	C₂×C₄×C₅	—	2	4	no	—
118	C₂×C₄×C₅	—	2	4	no	—
119	C₂×C₄×C₅	—	4	2	no	—
120	C₂×C₄×C₅	—	4	2	no	—
121	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
122	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
123	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
124	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
125	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
126	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
127	C₂×C₄×C₅	2₋¹⁺⁴	1	4	yes	—
128	C₅×C₈	—	2	2	no	—
129	C₅×C₈	—	2	2	no	—
130	C₅×C₈	—	2	2	no	—
131	C₅×C₈	—	2	2	no	—
132	C₅×C₈	—	4	4	no	—
133	C₂×M₁₆	C₂³×C₅	1	4	yes	—
134	C₂²×C₈	C₂³×C₅	1	2	yes	—
135	C₂²×C₈	C₂³×C₅	1	2	yes	—
136	C₄⋊Q₈	C₂×C₄×C₅	1	2	yes	—
137	C₄⋊Q₈	C₂×C₄×C₅	1	2	yes	—
138	#125	C₂×C₄×C₅	1	4	yes	—
139	C₂²⋊Q₈	—	2	2	no	—
140	C₂²⋊Q₈	—	2	2	no	—
141	C₄×Q₈	—	2	2	no	—
142	C₄×Q₈	—	2	2	no	—
143	#117	C₂³×C₅	1	4	yes	—
144	C₂×C₄²	—	2	1	no	—
145	C₂×C₄²	—	2	1	no	—
146	C₂×C₄²	C₂³×C₅	1	2	yes	—
147	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
148	#107	C₂×C₄×C₅	1	4	yes	—
149	#107	C₂×C₄×C₅	1	4	yes	—
150	#106	C₂×C₄×C₅	1	8	yes	—
151	C₄⋊C₈	—	2	4	no	—
152	C₄⋊C₈	—	2	4	no	—
153	C₄⋊C₈	—	2	2	no	—
154	C₄⋊C₈	—	2	2	no	—
155	#97	—	2	4	no	—
156	C₄×C₈	—	2	4	no	—
157	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
158	C₄×C₈	C₂×C₄×C₅	1	2	yes	—
159	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
160	C₄×C₈	C₅×D₄	1	1	yes	characteristic
161	C₄×C₈	C₅×D₄	1	2	yes	—
162	C₄×C₈	C₅×D₄	1	1	yes	characteristic
163	C₂²×C₅	—	2	1	no	—
164	C₂²×C₅	—	2	1	no	—
165	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
166	C₂²×C₅	#541	1	2	yes	—
167	C₄×C₅	—	2	1	no	—
168	C₄×C₅	—	2	1	no	—
169	C₄×C₅	—	4	4	no	—
170	C₄×C₅	—	4	4	no	—
171	C₄×C₅	—	4	4	no	—
172	C₄×C₅	#443	1	2	yes	—
173	C₄×C₅	#444	1	2	yes	—
174	C₄×C₅	#519	1	2	yes	—
175	C₄×C₅	#536	1	2	yes	—
176	C₂×Q₈	—	2	2	no	—
177	C₂×Q₈	—	2	2	no	—
178	C₂²×C₄	—	2	2	no	—
179	C₂²×C₄	C₅×#41	1	1	yes	characteristic
180	C₂²×C₄	C₅×#41	1	1	yes	characteristic
181	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
182	M₁₆	—	2	8	no	—
183	C₂×C₈	—	2	4	no	—
184	C₂×C₈	—	2	4	no	—
185	C₂×C₈	—	2	4	no	—
186	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
187	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
188	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
189	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
190	C₄⋊C₄	—	2	4	no	—
191	C₄⋊C₄	—	2	4	no	—
192	C₄⋊C₄	—	2	2	no	—
193	C₄⋊C₄	—	2	2	no	—
194	C₄⋊C₄	C₂²×C₄×C₅	1	4	yes	—
195	C₄⋊C₄	C₂²×C₄×C₅	1	4	yes	—
196	C₂²⋊C₄	—	2	4	no	—
197	C₄²	—	2	4	no	—
198	C₄²	—	2	4	no	—
199	C₄²	—	2	4	no	—
200	C₄²	—	4	2	no	—
201	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
202	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
203	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
204	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
205	C₂×C₅	—	2	2	no	—
206	C₂×C₅	#2327	1	1	yes	characteristic
207	C₂×C₅	C₂×#443	1	1	yes	characteristic
208	C₂×C₅	C₂×#444	1	1	yes	characteristic
209	C₂³	C₂×C₅×#41	1	1	yes	characteristic
210	Q₈	—	4	4	no	—
211	Q₈	—	4	4	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	4	no	—
215	C₂×C₄	—	2	4	no	—
216	C₂×C₄	—	2	4	no	—
217	C₂×C₄	—	4	2	no	—
218	C₂×C₄	—	4	2	no	—
219	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
220	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
221	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
222	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
223	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
224	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
225	C₂×C₄	C₅×2₋¹⁺⁴	1	4	yes	—
226	C₈	—	2	2	no	—
227	C₈	—	2	2	no	—
228	C₈	—	2	2	no	—
229	C₈	—	2	2	no	—
230	C₈	—	4	4	no	—
231	C₅	#22256	1	1	yes	characteristic, direct product factor, Sylow
232	C₂²	—	2	1	no	—
233	C₂²	—	2	1	no	—
234	C₂²	C₂×C₄×C₅×D₄	1	1	yes	characteristic
235	C₂²	C₅×#541	1	2	yes	—
236	C₄	—	2	1	no	—
237	C₄	—	2	1	no	—
238	C₄	—	4	4	no	—
239	C₄	—	4	4	no	—
240	C₄	—	4	4	no	—
241	C₄	C₅×#443	1	2	yes	—
242	C₄	C₅×#444	1	2	yes	—
243	C₄	C₅×#519	1	2	yes	—
244	C₄	C₅×#536	1	2	yes	—
245	C₂	—	2	2	no	—
246	C₂	C₅×#2327	1	1	yes	characteristic
247	C₂	C₂×C₅×#443	1	1	yes	characteristic
248	C₂	C₂×C₅×#444	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
socle	C₂²×C₅	C₂×C₄×D₄
prime radical	C₂³×C₅	C₂×#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂×C₄×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
8. C₅×#500	C₄	semidirect
9. C₅×#500	C₄	semidirect
10. C₅×#499	C₄	semidirect
11. C₅×#498	C₄	semidirect
26. #22256	C₅	direct (N × H)
93. #500	C₄×C₅	semidirect
94. #500	C₄×C₅	semidirect
95. #499	C₄×C₅	semidirect
96. #498	C₄×C₅	semidirect
231. C₅	#22256	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×#3160	2
C₅×#2960	2
C₅×#1994	2
C₅×#1922	2
C₅×#1837	2
C₅×#1744	2
#22256	5

Subgroups — C5×#22256 #11944976

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#22256 #11944976