Subgroups — C₅×#20801 #11943521

Order 1280.

1102 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#428	C₂	1	4	yes	maximal
2	C₂×C₄×C₅×M₁₆	C₂	1	1	yes	maximal, characteristic
3	C₅×#2285	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅×#382	C₂	1	1	yes	maximal, characteristic
5	C₅×#1847	C₂	1	4	yes	maximal, complemented
6	C₅×#1846	C₂	1	4	yes	maximal, complemented
7	C₂²×C₅×M₁₆	C₂²	1	2	yes	—
8	C₂×C₅×#117	C₂²	1	2	yes	—
9	C₂²×C₅×C₄⋊C₄	C₂²	1	2	yes	—
10	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
11	C₅×#428	—	2	16	no	—
12	C₂×C₅×#107	C₂²	1	8	yes	—
13	C₂×C₅×#106	C₂²	1	8	yes	—
14	C₄×C₅×M₁₆	C₄	1	8	yes	complemented
15	C₂×C₅×#97	C₂²	1	2	yes	—
16	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
17	C₅×#384	C₂²	1	2	yes	—
18	C₅×#382	—	2	2	no	—
19	C₅×#382	C₂²	1	2	yes	—
20	C₅×#382	C₂²	1	2	yes	—
21	C₅×#382	C₂²	1	2	yes	—
22	C₅×#380	—	2	2	no	—
23	C₄×C₅×C₄⋊C₄	—	2	2	no	—
24	C₂×C₅×#95	—	2	2	no	—
25	#20801	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
26	C₂³×C₄×C₅	—	2	2	no	—
27	C₂³×C₄×C₅	D₄	1	2	yes	—
28	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
29	C₂×C₅×M₁₆	C₂×C₄	1	8	yes	—
30	C₂×C₅×M₁₆	D₄	1	8	yes	—
31	C₂×C₅×M₁₆	Q₈	1	8	yes	—
32	C₂²×C₅×C₈	C₂³	1	4	yes	—
33	C₅×#117	—	2	8	no	—
34	C₂×C₅×C₄⋊C₄	—	2	8	no	—
35	C₂×C₅×C₄⋊C₄	—	2	4	no	—
36	C₂×C₅×C₄⋊C₄	—	2	2	no	—
37	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
38	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
39	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
40	C₂×C₄²×C₅	—	2	2	no	—
41	C₂×C₄²×C₅	C₂×C₄	1	4	yes	—
42	C₂×C₄²×C₅	D₄	1	2	yes	—
43	C₂×C₄²×C₅	Q₈	1	4	yes	—
44	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
45	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
46	C₅×#107	—	2	16	no	—
47	C₅×#106	—	2	16	no	—
48	C₅×#97	C₂×C₄	1	8	yes	complemented
49	C₄×C₅×C₈	C₂×C₄	1	8	yes	complemented
50	C₅×#95	—	2	4	no	—
51	C₅×#95	—	2	4	no	—
52	C₅×#95	—	2	4	no	—
53	C₂×#428	C₂×C₅	1	4	yes	—
54	C₂×C₄×M₁₆	C₂×C₅	1	1	yes	characteristic
55	#2285	C₂×C₅	1	1	yes	characteristic
56	C₂×#382	C₂×C₅	1	1	yes	characteristic
57	#1847	C₂×C₅	1	4	yes	complemented
58	#1846	C₂×C₅	1	4	yes	complemented
59	C₂⁴×C₅	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₅	—	2	4	no	—
61	C₂²×C₄×C₅	—	2	4	no	—
62	C₂²×C₄×C₅	—	4	4	no	—
63	C₂²×C₄×C₅	—	4	8	no	—
64	C₂²×C₄×C₅	—	4	4	no	—
65	C₂²×C₄×C₅	C₄⋊C₄	1	8	yes	—
66	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
67	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
68	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
69	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₅	C₂×D₄	1	4	yes	—
71	C₂²×C₄×C₅	C₂×Q₈	1	4	yes	—
72	C₂²×C₄×C₅	C₂×Q₈	1	2	yes	—
73	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
74	C₅×M₁₆	C₄⋊C₄	1	32	yes	complemented
75	C₂×C₅×C₈	C₂²×C₄	1	8	yes	—
76	C₂×C₅×C₈	C₂×D₄	1	8	yes	—
77	C₂×C₅×C₈	C₂×Q₈	1	8	yes	—
78	C₅×C₄⋊C₄	—	2	8	no	—
79	C₅×C₄⋊C₄	—	2	4	no	—
80	C₅×C₄⋊C₄	—	2	4	no	—
81	C₅×C₄⋊C₄	—	4	8	no	—
82	C₅×C₄⋊C₄	—	4	4	no	—
83	C₅×C₂²⋊C₄	—	4	4	no	—
84	C₄²×C₅	—	4	4	no	—
85	C₄²×C₅	C₄⋊C₄	1	8	yes	—
86	C₄²×C₅	C₂²×C₄	1	2	yes	—
87	C₄²×C₅	C₂²×C₄	1	2	yes	—
88	C₄²×C₅	C₂×Q₈	1	2	yes	—
89	C₄²×C₅	C₂×Q₈	1	2	yes	—
90	C₂²×M₁₆	C₂²×C₅	1	2	yes	—
91	C₂×#117	C₂²×C₅	1	2	yes	—
92	C₂²×C₄⋊C₄	C₂²×C₅	1	2	yes	—
93	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
94	#428	—	2	16	no	—
95	C₂×#107	C₂²×C₅	1	8	yes	—
96	C₂×#106	C₂²×C₅	1	8	yes	—
97	C₄×M₁₆	C₄×C₅	1	8	yes	complemented
98	C₂×#97	C₂²×C₅	1	2	yes	—
99	C₂×C₄×C₈	C₂²×C₅	1	2	yes	—
100	#384	C₂²×C₅	1	2	yes	—
101	#382	—	2	2	no	—
102	#382	C₂²×C₅	1	2	yes	—
103	#382	C₂²×C₅	1	2	yes	—
104	#382	C₂²×C₅	1	2	yes	—
105	#380	—	2	2	no	—
106	C₄×C₄⋊C₄	—	2	2	no	—
107	C₂×#95	—	2	2	no	—
108	C₂³×C₅	—	2	4	no	—
109	C₂³×C₅	C₂×C₄⋊C₄	1	2	yes	—
110	C₂³×C₅	#127	1	2	yes	—
111	C₂³×C₅	C₄⋊Q₈	1	2	yes	—
112	C₂³×C₅	C₂²×D₄	1	1	yes	characteristic
113	C₂×C₄×C₅	—	2	4	no	—
114	C₂×C₄×C₅	—	2	4	no	—
115	C₂×C₄×C₅	—	4	8	no	—
116	C₂×C₄×C₅	—	4	4	no	—
117	C₂×C₄×C₅	—	4	4	no	—
118	C₂×C₄×C₅	—	4	8	no	—
119	C₂×C₄×C₅	—	4	8	no	—
120	C₂×C₄×C₅	—	8	8	no	—
121	C₂×C₄×C₅	C₂×C₄⋊C₄	1	4	yes	—
122	C₂×C₄×C₅	C₂×C₄⋊C₄	1	4	yes	—
123	C₂×C₄×C₅	C₂×C₄⋊C₄	1	8	yes	—
124	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
125	C₂×C₄×C₅	#127	1	2	yes	—
126	C₂×C₄×C₅	C₄⋊Q₈	1	8	yes	—
127	C₂×C₄×C₅	C₄⋊Q₈	1	2	yes	—
128	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
129	C₂×C₄×C₅	C₂²×D₄	1	2	yes	—
130	C₂×C₄×C₅	C₂²×Q₈	1	2	yes	—
131	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
132	C₅×C₈	C₂×C₄⋊C₄	1	16	yes	—
133	C₂³×C₄	—	2	2	no	—
134	C₂³×C₄	C₅×D₄	1	2	yes	—
135	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
136	C₂×M₁₆	C₂×C₄×C₅	1	8	yes	—
137	C₂×M₁₆	C₅×D₄	1	8	yes	—
138	C₂×M₁₆	C₅×Q₈	1	8	yes	—
139	C₂²×C₈	C₂³×C₅	1	4	yes	—
140	#117	—	2	8	no	—
141	C₂×C₄⋊C₄	—	2	8	no	—
142	C₂×C₄⋊C₄	—	2	2	no	—
143	C₂×C₄⋊C₄	—	2	4	no	—
144	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
145	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
146	C₂×C₂²⋊C₄	—	2	2	no	—
147	C₂×C₄²	—	2	2	no	—
148	C₂×C₄²	C₂×C₄×C₅	1	4	yes	—
149	C₂×C₄²	C₅×D₄	1	2	yes	—
150	C₂×C₄²	C₅×Q₈	1	4	yes	—
151	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
152	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
153	#107	—	2	16	no	—
154	#106	—	2	16	no	—
155	#97	C₂×C₄×C₅	1	8	yes	complemented
156	C₄×C₈	C₂×C₄×C₅	1	8	yes	complemented
157	#95	—	2	4	no	—
158	#95	—	2	4	no	—
159	#95	—	2	4	no	—
160	C₂²×C₅	—	2	4	no	—
161	C₂²×C₅	—	2	4	no	—
162	C₂²×C₅	—	2	4	no	—
163	C₂²×C₅	#384	1	4	yes	—
164	C₂²×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
165	C₂²×C₅	C₂×#127	1	1	yes	characteristic
166	C₂²×C₅	C₂×C₄⋊Q₈	1	1	yes	characteristic
167	C₂²×C₅	#575	1	4	yes	—
168	C₄×C₅	—	2	4	no	—
169	C₄×C₅	—	2	4	no	—
170	C₄×C₅	—	8	4	no	—
171	C₄×C₅	—	8	4	no	—
172	C₄×C₅	#384	1	4	yes	—
173	C₄×C₅	C₂²×C₄⋊C₄	1	2	yes	—
174	C₄×C₅	C₂×C₄⋊Q₈	1	2	yes	—
175	C₂⁴	C₂×C₅×D₄	1	1	yes	characteristic
176	C₂²×C₄	—	2	4	no	—
177	C₂²×C₄	—	2	4	no	—
178	C₂²×C₄	—	4	4	no	—
179	C₂²×C₄	—	4	8	no	—
180	C₂²×C₄	—	4	4	no	—
181	C₂²×C₄	C₅×C₄⋊C₄	1	8	yes	—
182	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
183	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
184	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
185	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
186	C₂²×C₄	C₂×C₅×D₄	1	4	yes	—
187	C₂²×C₄	C₂×C₅×Q₈	1	4	yes	—
188	C₂²×C₄	C₂×C₅×Q₈	1	2	yes	—
189	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
190	M₁₆	C₅×C₄⋊C₄	1	32	yes	complemented
191	C₂×C₈	C₂²×C₄×C₅	1	8	yes	—
192	C₂×C₈	C₂×C₅×D₄	1	8	yes	—
193	C₂×C₈	C₂×C₅×Q₈	1	8	yes	—
194	C₄⋊C₄	—	2	4	no	—
195	C₄⋊C₄	—	2	8	no	—
196	C₄⋊C₄	—	2	4	no	—
197	C₄⋊C₄	—	4	8	no	—
198	C₄⋊C₄	—	4	4	no	—
199	C₂²⋊C₄	—	4	4	no	—
200	C₄²	—	4	4	no	—
201	C₄²	C₅×C₄⋊C₄	1	8	yes	—
202	C₄²	C₂²×C₄×C₅	1	2	yes	—
203	C₄²	C₂²×C₄×C₅	1	2	yes	—
204	C₄²	C₂×C₅×Q₈	1	2	yes	—
205	C₄²	C₂×C₅×Q₈	1	2	yes	—
206	C₂×C₅	—	2	4	no	—
207	C₂×C₅	C₂×#384	1	1	yes	characteristic
208	C₂×C₅	#2911	1	2	yes	—
209	C₂×C₅	#3153	1	2	yes	—
210	C₂×C₅	#3166	1	2	yes	—
211	C₂³	—	2	4	no	—
212	C₂³	C₂×C₅×C₄⋊C₄	1	2	yes	—
213	C₂³	C₅×#127	1	2	yes	—
214	C₂³	C₅×C₄⋊Q₈	1	2	yes	—
215	C₂³	C₂²×C₅×D₄	1	1	yes	characteristic
216	C₂×C₄	—	2	4	no	—
217	C₂×C₄	—	2	4	no	—
218	C₂×C₄	—	4	8	no	—
219	C₂×C₄	—	4	4	no	—
220	C₂×C₄	—	4	4	no	—
221	C₂×C₄	—	4	8	no	—
222	C₂×C₄	—	4	8	no	—
223	C₂×C₄	—	8	8	no	—
224	C₂×C₄	C₂×C₅×C₄⋊C₄	1	4	yes	—
225	C₂×C₄	C₂×C₅×C₄⋊C₄	1	4	yes	—
226	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
227	C₂×C₄	C₂×C₅×C₄⋊C₄	1	8	yes	—
228	C₂×C₄	C₅×#127	1	2	yes	—
229	C₂×C₄	C₅×C₄⋊Q₈	1	2	yes	—
230	C₂×C₄	C₅×C₄⋊Q₈	1	8	yes	—
231	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
232	C₂×C₄	C₂²×C₅×D₄	1	2	yes	—
233	C₂×C₄	C₂²×C₅×Q₈	1	2	yes	—
234	C₂×C₄	C₂²×C₅×Q₈	1	1	yes	characteristic
235	C₈	C₂×C₅×C₄⋊C₄	1	16	yes	—
236	C₅	#20801	1	1	yes	characteristic, direct product factor, Sylow
237	C₂²	—	2	4	no	—
238	C₂²	—	2	4	no	—
239	C₂²	—	2	4	no	—
240	C₂²	C₅×#384	1	4	yes	—
241	C₂²	C₂²×C₅×C₄⋊C₄	1	1	yes	characteristic
242	C₂²	C₂×C₅×#127	1	1	yes	characteristic
243	C₂²	C₂×C₅×C₄⋊Q₈	1	1	yes	characteristic
244	C₂²	C₅×#575	1	4	yes	—
245	C₄	—	2	4	no	—
246	C₄	—	2	4	no	—
247	C₄	—	8	4	no	—
248	C₄	—	8	4	no	—
249	C₄	C₅×#384	1	4	yes	—
250	C₄	C₂²×C₅×C₄⋊C₄	1	2	yes	—
251	C₄	C₂×C₅×C₄⋊Q₈	1	2	yes	—
252	C₂	—	2	4	no	—
253	C₂	C₂×C₅×#384	1	1	yes	characteristic
254	C₂	C₅×#2911	1	2	yes	—
255	C₂	C₅×#3153	1	2	yes	—
256	C₂	C₅×#3166	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	C₂⁴×C₅	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₂²×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
5. C₅×#1847	C₂	semidirect
6. C₅×#1846	C₂	semidirect
14. C₄×C₅×M₁₆	C₄	semidirect
25. #20801	C₅	direct (N × H)
48. C₅×#97	C₂×C₄	semidirect
49. C₄×C₅×C₈	C₂×C₄	semidirect
57. #1847	C₂×C₅	semidirect
58. #1846	C₂×C₅	semidirect
74. C₅×M₁₆	C₄⋊C₄	semidirect
97. C₄×M₁₆	C₄×C₅	semidirect
155. #97	C₂×C₄×C₅	semidirect
156. C₄×C₈	C₂×C₄×C₅	semidirect
190. M₁₆	C₅×C₄⋊C₄	semidirect
236. C₅	#20801	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×#428	2
C₂×C₄×C₅×M₁₆	2
C₅×#2285	2
C₂×C₅×#382	2
C₅×#1847	2
C₅×#1846	2
#20801	5

Subgroups — C5×#20801 #11943521

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#20801 #11943521