Subgroups — C₅×#20824 #11943544

Order 1280.

930 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₅×M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₅×#2285	C₂	1	2	yes	maximal
3	C₅×#1845	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#1844	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#1834	C₂	1	2	yes	maximal, complemented
6	C₅×#1809	C₂	1	4	yes	maximal
7	C₅×#1776	C₂	1	4	yes	maximal, complemented
8	C₂²×C₅×M₁₆	C₂²	1	2	yes	—
9	C₂×C₅×#117	—	2	2	no	—
10	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
11	C₂×C₅×#107	C₂²	1	4	yes	—
12	C₂×C₅×#106	C₂²	1	4	yes	—
13	C₄×C₅×M₁₆	C₄	1	8	yes	complemented
14	C₂×C₅×#97	C₂²	1	2	yes	—
15	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
17	C₅×#384	C₂²	1	2	yes	—
18	C₅×#384	C₂²	1	2	yes	—
19	C₅×#382	—	2	2	no	—
20	C₅×#380	—	2	2	no	—
21	C₅×#380	C₂²	1	4	yes	—
22	C₄×C₅×C₄⋊C₄	—	2	2	no	—
23	C₄×C₅×C₄⋊C₄	C₂²	1	4	yes	—
24	C₅×#340	C₂²	1	8	yes	—
25	#20824	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
26	C₂³×C₄×C₅	D₄	1	2	yes	—
27	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
28	C₂×C₅×M₁₆	—	2	4	no	—
29	C₂×C₅×M₁₆	—	2	4	no	—
30	C₂×C₅×M₁₆	C₂×C₄	1	8	yes	—
31	C₂²×C₅×C₈	C₂³	1	2	yes	—
32	C₂²×C₅×C₈	C₂³	1	2	yes	—
33	C₅×#117	—	4	8	no	—
34	C₂×C₅×C₄⋊C₄	—	2	2	no	—
35	C₂×C₅×C₄⋊C₄	—	2	2	no	—
36	C₂×C₅×C₄⋊C₄	—	2	2	no	—
37	C₂×C₅×C₄⋊C₄	—	2	2	no	—
38	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
39	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
40	C₂×C₅×C₂²⋊C₄	—	2	4	no	—
41	C₂×C₄²×C₅	—	2	4	no	—
42	C₂×C₄²×C₅	C₂×C₄	1	4	yes	—
43	C₂×C₄²×C₅	D₄	1	2	yes	—
44	C₂×C₄²×C₅	Q₈	1	4	yes	—
45	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
47	C₅×#107	—	2	8	no	—
48	C₅×#106	—	2	8	no	—
49	C₅×#97	C₂×C₄	1	8	yes	—
50	C₄×C₅×C₈	C₂×C₄	1	4	yes	—
51	C₄×C₅×C₈	C₂×C₄	1	4	yes	—
52	C₅×#95	—	2	4	no	—
53	C₅×#95	—	2	4	no	—
54	C₂×C₄×M₁₆	C₂×C₅	1	1	yes	characteristic
55	#2285	C₂×C₅	1	2	yes	—
56	#1845	C₂×C₅	1	1	yes	characteristic, complemented
57	#1844	C₂×C₅	1	1	yes	characteristic, complemented
58	#1834	C₂×C₅	1	2	yes	complemented
59	#1809	C₂×C₅	1	4	yes	—
60	#1776	C₂×C₅	1	4	yes	complemented
61	C₂⁴×C₅	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₅	—	2	4	no	—
63	C₂²×C₄×C₅	—	2	4	no	—
64	C₂²×C₄×C₅	—	2	2	no	—
65	C₂²×C₄×C₅	—	2	2	no	—
66	C₂²×C₄×C₅	—	2	4	no	—
67	C₂²×C₄×C₅	—	2	4	no	—
68	C₂²×C₄×C₅	C₄⋊C₄	1	8	yes	—
69	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
70	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
71	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
73	C₂²×C₄×C₅	C₂×Q₈	1	2	yes	—
74	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
75	C₅×M₁₆	—	2	16	no	—
76	C₂×C₅×C₈	—	2	2	no	—
77	C₂×C₅×C₈	—	2	2	no	—
78	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
79	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
80	C₂×C₅×C₈	#41	1	4	yes	—
81	C₂×C₅×C₈	#41	1	4	yes	—
82	C₅×C₄⋊C₄	—	2	8	no	—
83	C₅×C₄⋊C₄	—	2	8	no	—
84	C₅×C₄⋊C₄	—	4	4	no	—
85	C₅×C₄⋊C₄	—	4	4	no	—
86	C₅×C₄⋊C₄	—	4	4	no	—
87	C₅×C₄⋊C₄	—	4	4	no	—
88	C₅×C₂²⋊C₄	—	4	8	no	—
89	C₄²×C₅	—	4	8	no	—
90	C₄²×C₅	C₄⋊C₄	1	8	yes	—
91	C₄²×C₅	C₂²×C₄	1	2	yes	—
92	C₄²×C₅	C₂²×C₄	1	2	yes	—
93	C₄²×C₅	C₂×Q₈	1	2	yes	—
94	C₄²×C₅	C₂×Q₈	1	2	yes	—
95	C₂²×M₁₆	C₂²×C₅	1	2	yes	—
96	C₂×#117	—	2	2	no	—
97	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
98	C₂×#107	C₂²×C₅	1	4	yes	—
99	C₂×#106	C₂²×C₅	1	4	yes	—
100	C₄×M₁₆	C₄×C₅	1	8	yes	complemented
101	C₂×#97	C₂²×C₅	1	2	yes	—
102	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
103	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
104	#384	C₂²×C₅	1	2	yes	—
105	#384	C₂²×C₅	1	2	yes	—
106	#382	—	2	2	no	—
107	#380	—	2	2	no	—
108	#380	C₂²×C₅	1	4	yes	—
109	C₄×C₄⋊C₄	—	2	2	no	—
110	C₄×C₄⋊C₄	C₂²×C₅	1	4	yes	—
111	#340	C₂²×C₅	1	8	yes	—
112	C₂³×C₅	—	2	1	no	—
113	C₂³×C₅	—	2	1	no	—
114	C₂³×C₅	—	2	2	no	—
115	C₂³×C₅	—	2	2	no	—
116	C₂³×C₅	C₂×C₄⋊C₄	1	2	yes	—
117	C₂³×C₅	C₂²×D₄	1	1	yes	characteristic
118	C₂×C₄×C₅	—	2	1	no	—
119	C₂×C₄×C₅	—	2	1	no	—
120	C₂×C₄×C₅	—	2	2	no	—
121	C₂×C₄×C₅	—	2	2	no	—
122	C₂×C₄×C₅	—	2	4	no	—
123	C₂×C₄×C₅	—	2	4	no	—
124	C₂×C₄×C₅	—	2	16	no	—
125	C₂×C₄×C₅	—	4	4	no	—
126	C₂×C₄×C₅	—	4	4	no	—
127	C₂×C₄×C₅	—	4	4	no	—
128	C₂×C₄×C₅	—	4	4	no	—
129	C₂×C₄×C₅	—	4	4	no	—
130	C₂×C₄×C₅	—	4	4	no	—
131	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
132	C₂×C₄×C₅	C₂×C₄⋊C₄	1	4	yes	—
133	C₂×C₄×C₅	C₂×C₄⋊C₄	1	4	yes	—
134	C₂×C₄×C₅	AGL(1,ℤ/8)	1	4	yes	—
135	C₂×C₄×C₅	#137	1	4	yes	—
136	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
137	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
138	C₂×C₄×C₅	C₂×#41	1	2	yes	—
139	C₂×C₄×C₅	C₂×#41	1	2	yes	—
140	C₅×C₈	—	2	4	no	—
141	C₅×C₈	#117	1	8	yes	complemented
142	C₂³×C₄	C₅×D₄	1	2	yes	—
143	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
144	C₂×M₁₆	—	2	4	no	—
145	C₂×M₁₆	—	2	4	no	—
146	C₂×M₁₆	C₂×C₄×C₅	1	8	yes	—
147	C₂²×C₈	C₂³×C₅	1	2	yes	—
148	C₂²×C₈	C₂³×C₅	1	2	yes	—
149	#117	—	4	8	no	—
150	C₂×C₄⋊C₄	—	2	2	no	—
151	C₂×C₄⋊C₄	—	2	2	no	—
152	C₂×C₄⋊C₄	—	2	2	no	—
153	C₂×C₄⋊C₄	—	2	2	no	—
154	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
155	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
156	C₂×C₂²⋊C₄	—	2	4	no	—
157	C₂×C₄²	—	2	4	no	—
158	C₂×C₄²	C₂×C₄×C₅	1	4	yes	—
159	C₂×C₄²	C₅×D₄	1	2	yes	—
160	C₂×C₄²	C₅×Q₈	1	4	yes	—
161	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
162	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
163	#107	—	2	8	no	—
164	#106	—	2	8	no	—
165	#97	C₂×C₄×C₅	1	8	yes	—
166	C₄×C₈	C₂×C₄×C₅	1	4	yes	—
167	C₄×C₈	C₂×C₄×C₅	1	4	yes	—
168	#95	—	2	4	no	—
169	#95	—	2	4	no	—
170	C₂²×C₅	—	2	4	no	—
171	C₂²×C₅	—	2	2	no	—
172	C₂²×C₅	—	4	2	no	—
173	C₂²×C₅	—	4	1	no	—
174	C₂²×C₅	—	4	1	no	—
175	C₂²×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
176	C₂²×C₅	#532	1	1	yes	characteristic
177	C₂²×C₅	#532	1	1	yes	characteristic
178	C₂²×C₅	C₂×AGL(1,ℤ/8)	1	2	yes	—
179	C₂²×C₅	C₂×#137	1	2	yes	—
180	C₄×C₅	—	2	4	no	—
181	C₄×C₅	—	2	2	no	—
182	C₄×C₅	—	8	4	no	—
183	C₄×C₅	—	8	4	no	—
184	C₄×C₅	#428	1	8	yes	—
185	C₄×C₅	C₂×#117	1	2	yes	—
186	C₄×C₅	#533	1	2	yes	—
187	C₂⁴	C₂×C₅×D₄	1	1	yes	characteristic
188	C₂²×C₄	—	2	4	no	—
189	C₂²×C₄	—	2	4	no	—
190	C₂²×C₄	—	2	2	no	—
191	C₂²×C₄	—	2	2	no	—
192	C₂²×C₄	—	2	4	no	—
193	C₂²×C₄	—	2	4	no	—
194	C₂²×C₄	C₅×C₄⋊C₄	1	8	yes	—
195	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
196	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
197	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
198	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
199	C₂²×C₄	C₂×C₅×Q₈	1	2	yes	—
200	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
201	M₁₆	—	2	16	no	—
202	C₂×C₈	—	2	2	no	—
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	C₂²×C₄×C₅	1	4	yes	—
205	C₂×C₈	C₂²×C₄×C₅	1	4	yes	—
206	C₂×C₈	C₅×#41	1	4	yes	—
207	C₂×C₈	C₅×#41	1	4	yes	—
208	C₄⋊C₄	—	2	8	no	—
209	C₄⋊C₄	—	2	8	no	—
210	C₄⋊C₄	—	4	4	no	—
211	C₄⋊C₄	—	4	4	no	—
212	C₄⋊C₄	—	4	4	no	—
213	C₄⋊C₄	—	4	4	no	—
214	C₂²⋊C₄	—	4	8	no	—
215	C₄²	—	4	8	no	—
216	C₄²	C₅×C₄⋊C₄	1	8	yes	—
217	C₄²	C₂²×C₄×C₅	1	2	yes	—
218	C₄²	C₂²×C₄×C₅	1	2	yes	—
219	C₄²	C₂×C₅×Q₈	1	2	yes	—
220	C₄²	C₂×C₅×Q₈	1	2	yes	—
221	C₂×C₅	—	4	2	no	—
222	C₂×C₅	#2285	1	1	yes	characteristic
223	C₂×C₅	C₂×#428	1	2	yes	—
224	C₂×C₅	#3182	1	2	yes	—
225	C₂×C₅	#3184	1	1	yes	characteristic
226	C₂×C₅	#3185	1	1	yes	characteristic
227	C₂³	—	2	1	no	—
228	C₂³	—	2	1	no	—
229	C₂³	—	2	2	no	—
230	C₂³	—	2	2	no	—
231	C₂³	C₂×C₅×C₄⋊C₄	1	2	yes	—
232	C₂³	C₂²×C₅×D₄	1	1	yes	characteristic
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	16	no	—
238	C₂×C₄	—	2	4	no	—
239	C₂×C₄	—	2	4	no	—
240	C₂×C₄	—	4	4	no	—
241	C₂×C₄	—	4	4	no	—
242	C₂×C₄	—	4	4	no	—
243	C₂×C₄	—	4	4	no	—
244	C₂×C₄	—	4	4	no	—
245	C₂×C₄	—	4	4	no	—
246	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
247	C₂×C₄	C₂×C₅×C₄⋊C₄	1	4	yes	—
248	C₂×C₄	C₂×C₅×C₄⋊C₄	1	4	yes	—
249	C₂×C₄	C₅×AGL(1,ℤ/8)	1	4	yes	—
250	C₂×C₄	C₅×#137	1	4	yes	—
251	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
252	C₂×C₄	C₂²×C₅×Q₈	1	1	yes	characteristic
253	C₂×C₄	C₂×C₅×#41	1	2	yes	—
254	C₂×C₄	C₂×C₅×#41	1	2	yes	—
255	C₈	—	2	4	no	—
256	C₈	C₅×#117	1	8	yes	complemented
257	C₅	#20824	1	1	yes	characteristic, direct product factor, Sylow
258	C₂²	—	2	2	no	—
259	C₂²	—	2	4	no	—
260	C₂²	—	4	2	no	—
261	C₂²	—	4	1	no	—
262	C₂²	—	4	1	no	—
263	C₂²	C₂²×C₅×C₄⋊C₄	1	1	yes	characteristic
264	C₂²	C₅×#532	1	1	yes	characteristic
265	C₂²	C₅×#532	1	1	yes	characteristic
266	C₂²	C₂×C₅×AGL(1,ℤ/8)	1	2	yes	—
267	C₂²	C₂×C₅×#137	1	2	yes	—
268	C₄	—	2	2	no	—
269	C₄	—	2	4	no	—
270	C₄	—	8	4	no	—
271	C₄	—	8	4	no	—
272	C₄	C₅×#428	1	8	yes	—
273	C₄	C₂×C₅×#117	1	2	yes	—
274	C₄	C₅×#533	1	2	yes	—
275	C₂	—	4	2	no	—
276	C₂	C₅×#2285	1	1	yes	characteristic
277	C₂	C₂×C₅×#428	1	2	yes	—
278	C₂	C₅×#3182	1	2	yes	—
279	C₂	C₅×#3184	1	1	yes	characteristic
280	C₂	C₅×#3185	1	1	yes	characteristic

	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
3. C₅×#1845	C₂	semidirect
4. C₅×#1844	C₂	semidirect
5. C₅×#1834	C₂	semidirect
7. C₅×#1776	C₂	semidirect
13. C₄×C₅×M₁₆	C₄	semidirect
25. #20824	C₅	direct (N × H)
56. #1845	C₂×C₅	semidirect
57. #1844	C₂×C₅	semidirect
58. #1834	C₂×C₅	semidirect
60. #1776	C₂×C₅	semidirect
100. C₄×M₁₆	C₄×C₅	semidirect
141. C₅×C₈	#117	semidirect
256. C₈	C₅×#117	semidirect
257. C₅	#20824	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×C₄×C₅×M₁₆	2
C₅×#2285	2
C₅×#1845	2
C₅×#1844	2
C₅×#1834	2
C₅×#1809	2
C₅×#1776	2
#20824	5

Subgroups — C5×#20824 #11943544

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#20824 #11943544