Subgroups — C₅×#20764 #11943484

Order 1280.

1014 conjugacy classes of subgroups.

The subgroup lattice has 274 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	1	yes	maximal, characteristic
3	C₂×C₅×#384	C₂	1	1	yes	maximal, characteristic
4	C₅×#1836	C₂	1	4	yes	maximal, complemented
5	C₅×#1834	C₂	1	4	yes	maximal, complemented
6	C₅×#1736	C₂	1	4	yes	maximal
7	C₂²×C₅×M₁₆	C₂²	1	2	yes	—
8	C₂×C₅×#117	C₂²	1	2	yes	—
9	C₂²×C₅×C₄⋊C₄	—	2	2	no	—
10	C₂²×C₅×C₄⋊C₄	C₂²	1	2	yes	—
11	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
12	C₄×C₅×M₁₆	C₄	1	8	yes	complemented
13	C₂×C₅×#97	C₂²	1	2	yes	—
14	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
15	C₅×#384	—	2	2	no	—
16	C₅×#384	C₂²	1	2	yes	—
17	C₅×#384	C₂²	1	2	yes	—
18	C₅×#384	C₂²	1	2	yes	—
19	C₅×#382	C₂²	1	2	yes	—
20	C₅×#380	—	2	2	no	—
21	C₄×C₅×C₄⋊C₄	—	2	2	no	—
22	C₅×#340	C₂²	1	16	yes	—
23	#20764	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
24	C₂³×C₄×C₅	—	2	2	no	—
25	C₂³×C₄×C₅	D₄	1	2	yes	—
26	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
27	C₂×C₅×M₁₆	—	2	4	no	—
28	C₂×C₅×M₁₆	—	2	4	no	—
29	C₂×C₅×M₁₆	C₂×C₄	1	8	yes	—
30	C₂²×C₅×C₈	C₂³	1	4	yes	—
31	C₅×#117	—	2	8	no	—
32	C₂×C₅×C₄⋊C₄	—	2	8	no	—
33	C₂×C₅×C₄⋊C₄	—	2	2	no	—
34	C₂×C₅×C₄⋊C₄	—	2	2	no	—
35	C₂×C₅×C₄⋊C₄	—	2	2	no	—
36	C₂×C₅×C₄⋊C₄	—	2	4	no	—
37	C₂×C₅×C₄⋊C₄	—	2	4	no	—
38	C₂×C₅×C₄⋊C₄	—	4	8	no	—
39	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
40	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
41	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
42	C₂×C₄²×C₅	—	2	2	no	—
43	C₂×C₄²×C₅	C₂×C₄	1	4	yes	—
44	C₂×C₄²×C₅	D₄	1	2	yes	—
45	C₂×C₄²×C₅	Q₈	1	4	yes	—
46	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
47	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
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	C₂×C₄×M₁₆	C₂×C₅	1	1	yes	characteristic
52	#2285	C₂×C₅	1	1	yes	characteristic
53	C₂×#384	C₂×C₅	1	1	yes	characteristic
54	#1836	C₂×C₅	1	4	yes	complemented
55	#1834	C₂×C₅	1	4	yes	complemented
56	#1736	C₂×C₅	1	4	yes	—
57	C₂⁴×C₅	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₅	—	2	4	no	—
59	C₂²×C₄×C₅	—	2	4	no	—
60	C₂²×C₄×C₅	—	2	4	no	—
61	C₂²×C₄×C₅	—	2	4	no	—
62	C₂²×C₄×C₅	—	4	8	no	—
63	C₂²×C₄×C₅	—	4	4	no	—
64	C₂²×C₄×C₅	C₄⋊C₄	1	8	yes	—
65	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
66	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
67	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
69	C₂²×C₄×C₅	C₂×Q₈	1	2	yes	—
70	C₂²×C₄×C₅	#41	1	4	yes	—
71	C₂²×C₄×C₅	#41	1	4	yes	—
72	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
73	C₅×M₁₆	—	2	16	no	—
74	C₂×C₅×C₈	—	2	4	no	—
75	C₂×C₅×C₈	—	2	4	no	—
76	C₂×C₅×C₈	C₂²×C₄	1	8	yes	—
77	C₅×C₄⋊C₄	—	2	8	no	—
78	C₅×C₄⋊C₄	—	2	8	no	—
79	C₅×C₄⋊C₄	—	4	8	no	—
80	C₅×C₄⋊C₄	—	4	4	no	—
81	C₅×C₄⋊C₄	—	4	4	no	—
82	C₅×C₄⋊C₄	—	4	8	no	—
83	C₅×C₄⋊C₄	—	4	4	no	—
84	C₅×C₂²⋊C₄	—	4	4	no	—
85	C₄²×C₅	—	4	4	no	—
86	C₄²×C₅	C₄⋊C₄	1	8	yes	—
87	C₄²×C₅	C₂²×C₄	1	2	yes	—
88	C₄²×C₅	C₂²×C₄	1	2	yes	—
89	C₄²×C₅	C₂×Q₈	1	2	yes	—
90	C₄²×C₅	C₂×Q₈	1	2	yes	—
91	C₂²×M₁₆	C₂²×C₅	1	2	yes	—
92	C₂×#117	C₂²×C₅	1	2	yes	—
93	C₂²×C₄⋊C₄	—	2	2	no	—
94	C₂²×C₄⋊C₄	C₂²×C₅	1	2	yes	—
95	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
96	C₄×M₁₆	C₄×C₅	1	8	yes	complemented
97	C₂×#97	C₂²×C₅	1	2	yes	—
98	C₂×C₄×C₈	C₂²×C₅	1	2	yes	—
99	#384	—	2	2	no	—
100	#384	C₂²×C₅	1	2	yes	—
101	#384	C₂²×C₅	1	2	yes	—
102	#384	C₂²×C₅	1	2	yes	—
103	#382	C₂²×C₅	1	2	yes	—
104	#380	—	2	2	no	—
105	C₄×C₄⋊C₄	—	2	2	no	—
106	#340	C₂²×C₅	1	16	yes	—
107	C₂³×C₅	—	2	2	no	—
108	C₂³×C₅	—	2	2	no	—
109	C₂³×C₅	C₂×C₄⋊C₄	1	2	yes	—
110	C₂³×C₅	#124	1	2	yes	—
111	C₂³×C₅	#124	1	2	yes	—
112	C₂³×C₅	C₂²×D₄	1	1	yes	characteristic
113	C₂×C₄×C₅	—	2	2	no	—
114	C₂×C₄×C₅	—	2	2	no	—
115	C₂×C₄×C₅	—	2	16	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	4	no	—
120	C₂×C₄×C₅	—	4	4	no	—
121	C₂×C₄×C₅	—	8	8	no	—
122	C₂×C₄×C₅	C₂×C₄⋊C₄	1	4	yes	—
123	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
124	C₂×C₄×C₅	C₂×C₄⋊C₄	1	4	yes	—
125	C₂×C₄×C₅	#117	1	8	yes	—
126	C₂×C₄×C₅	#124	1	2	yes	—
127	C₂×C₄×C₅	#124	1	2	yes	—
128	C₂×C₄×C₅	#125	1	8	yes	—
129	C₂×C₄×C₅	AGL(1,ℤ/8)	1	4	yes	—
130	C₂×C₄×C₅	#137	1	4	yes	—
131	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
132	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
133	C₂×C₄×C₅	C₂×#41	1	2	yes	—
134	C₂×C₄×C₅	C₂×#41	1	2	yes	—
135	C₅×C₈	—	2	8	no	—
136	C₂³×C₄	—	2	2	no	—
137	C₂³×C₄	C₅×D₄	1	2	yes	—
138	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
139	C₂×M₁₆	—	2	4	no	—
140	C₂×M₁₆	—	2	4	no	—
141	C₂×M₁₆	C₂×C₄×C₅	1	8	yes	—
142	C₂²×C₈	C₂³×C₅	1	4	yes	—
143	#117	—	2	8	no	—
144	C₂×C₄⋊C₄	—	2	8	no	—
145	C₂×C₄⋊C₄	—	2	4	no	—
146	C₂×C₄⋊C₄	—	2	4	no	—
147	C₂×C₄⋊C₄	—	2	2	no	—
148	C₂×C₄⋊C₄	—	2	2	no	—
149	C₂×C₄⋊C₄	—	2	2	no	—
150	C₂×C₄⋊C₄	—	4	8	no	—
151	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
152	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
153	C₂×C₂²⋊C₄	—	2	2	no	—
154	C₂×C₄²	—	2	2	no	—
155	C₂×C₄²	C₂×C₄×C₅	1	4	yes	—
156	C₂×C₄²	C₅×D₄	1	2	yes	—
157	C₂×C₄²	C₅×Q₈	1	4	yes	—
158	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
159	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
160	#97	C₂×C₄×C₅	1	8	yes	complemented
161	C₄×C₈	C₂×C₄×C₅	1	8	yes	complemented
162	#95	—	2	4	no	—
163	C₂²×C₅	—	2	2	no	—
164	C₂²×C₅	—	2	2	no	—
165	C₂²×C₅	—	2	4	no	—
166	C₂²×C₅	—	2	4	no	—
167	C₂²×C₅	#382	1	4	yes	—
168	C₂²×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
169	C₂²×C₅	C₂×#124	1	1	yes	characteristic
170	C₂²×C₅	C₂×#124	1	1	yes	characteristic
171	C₂²×C₅	C₂×AGL(1,ℤ/8)	1	2	yes	—
172	C₂²×C₅	C₂×#137	1	2	yes	—
173	C₄×C₅	—	2	4	no	—
174	C₄×C₅	—	8	4	no	—
175	C₄×C₅	—	8	4	no	—
176	C₄×C₅	#382	1	4	yes	—
177	C₄×C₅	#428	1	8	yes	—
178	C₄×C₅	C₂×#117	1	2	yes	—
179	C₄×C₅	C₂×#125	1	2	yes	—
180	C₂⁴	C₂×C₅×D₄	1	1	yes	characteristic
181	C₂²×C₄	—	2	4	no	—
182	C₂²×C₄	—	2	4	no	—
183	C₂²×C₄	—	2	4	no	—
184	C₂²×C₄	—	2	4	no	—
185	C₂²×C₄	—	4	8	no	—
186	C₂²×C₄	—	4	4	no	—
187	C₂²×C₄	C₅×C₄⋊C₄	1	8	yes	—
188	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
189	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
190	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
191	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
192	C₂²×C₄	C₂×C₅×Q₈	1	2	yes	—
193	C₂²×C₄	C₅×#41	1	4	yes	—
194	C₂²×C₄	C₅×#41	1	4	yes	—
195	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
196	M₁₆	—	2	16	no	—
197	C₂×C₈	—	2	4	no	—
198	C₂×C₈	—	2	4	no	—
199	C₂×C₈	C₂²×C₄×C₅	1	8	yes	—
200	C₄⋊C₄	—	2	8	no	—
201	C₄⋊C₄	—	2	8	no	—
202	C₄⋊C₄	—	4	8	no	—
203	C₄⋊C₄	—	4	4	no	—
204	C₄⋊C₄	—	4	8	no	—
205	C₄⋊C₄	—	4	4	no	—
206	C₄⋊C₄	—	4	4	no	—
207	C₂²⋊C₄	—	4	4	no	—
208	C₄²	—	4	4	no	—
209	C₄²	C₅×C₄⋊C₄	1	8	yes	—
210	C₄²	C₂²×C₄×C₅	1	2	yes	—
211	C₄²	C₂²×C₄×C₅	1	2	yes	—
212	C₄²	C₂×C₅×Q₈	1	2	yes	—
213	C₄²	C₂×C₅×Q₈	1	2	yes	—
214	C₂×C₅	—	2	4	no	—
215	C₂×C₅	C₂×#382	1	1	yes	characteristic
216	C₂×C₅	C₂×#428	1	2	yes	—
217	C₂×C₅	#3138	1	1	yes	characteristic
218	C₂×C₅	#3139	1	1	yes	characteristic
219	C₂×C₅	#3141	1	2	yes	—
220	C₂³	—	2	2	no	—
221	C₂³	—	2	2	no	—
222	C₂³	C₂×C₅×C₄⋊C₄	1	2	yes	—
223	C₂³	C₅×#124	1	2	yes	—
224	C₂³	C₅×#124	1	2	yes	—
225	C₂³	C₂²×C₅×D₄	1	1	yes	characteristic
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	16	no	—
229	C₂×C₄	—	4	4	no	—
230	C₂×C₄	—	4	4	no	—
231	C₂×C₄	—	4	8	no	—
232	C₂×C₄	—	4	4	no	—
233	C₂×C₄	—	4	4	no	—
234	C₂×C₄	—	8	8	no	—
235	C₂×C₄	C₂×C₅×C₄⋊C₄	1	4	yes	—
236	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
237	C₂×C₄	C₂×C₅×C₄⋊C₄	1	4	yes	—
238	C₂×C₄	C₅×#117	1	8	yes	—
239	C₂×C₄	C₅×#124	1	2	yes	—
240	C₂×C₄	C₅×#124	1	2	yes	—
241	C₂×C₄	C₅×#125	1	8	yes	—
242	C₂×C₄	C₅×AGL(1,ℤ/8)	1	4	yes	—
243	C₂×C₄	C₅×#137	1	4	yes	—
244	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
245	C₂×C₄	C₂²×C₅×Q₈	1	1	yes	characteristic
246	C₂×C₄	C₂×C₅×#41	1	2	yes	—
247	C₂×C₄	C₂×C₅×#41	1	2	yes	—
248	C₈	—	2	8	no	—
249	C₅	#20764	1	1	yes	characteristic, direct product factor, Sylow
250	C₂²	—	2	2	no	—
251	C₂²	—	2	2	no	—
252	C₂²	—	2	4	no	—
253	C₂²	—	2	4	no	—
254	C₂²	C₅×#382	1	4	yes	—
255	C₂²	C₂²×C₅×C₄⋊C₄	1	1	yes	characteristic
256	C₂²	C₂×C₅×#124	1	1	yes	characteristic
257	C₂²	C₂×C₅×#124	1	1	yes	characteristic
258	C₂²	C₂×C₅×AGL(1,ℤ/8)	1	2	yes	—
259	C₂²	C₂×C₅×#137	1	2	yes	—
260	C₄	—	2	4	no	—
261	C₄	—	8	4	no	—
262	C₄	—	8	4	no	—
263	C₄	C₅×#382	1	4	yes	—
264	C₄	C₅×#428	1	8	yes	—
265	C₄	C₂×C₅×#117	1	2	yes	—
266	C₄	C₂×C₅×#125	1	2	yes	—
267	C₂	—	2	4	no	—
268	C₂	C₂×C₅×#382	1	1	yes	characteristic
269	C₂	C₂×C₅×#428	1	2	yes	—
270	C₂	C₅×#3138	1	1	yes	characteristic
271	C₂	C₅×#3139	1	1	yes	characteristic
272	C₂	C₅×#3141	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
4. C₅×#1836	C₂	semidirect
5. C₅×#1834	C₂	semidirect
12. C₄×C₅×M₁₆	C₄	semidirect
23. #20764	C₅	direct (N × H)
48. C₅×#97	C₂×C₄	semidirect
49. C₄×C₅×C₈	C₂×C₄	semidirect
54. #1836	C₂×C₅	semidirect
55. #1834	C₂×C₅	semidirect
96. C₄×M₁₆	C₄×C₅	semidirect
160. #97	C₂×C₄×C₅	semidirect
161. C₄×C₈	C₂×C₄×C₅	semidirect
249. C₅	#20764	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
C₂×C₅×#384	2
C₅×#1836	2
C₅×#1834	2
C₅×#1736	2
#20764	5

Subgroups — C5×#20764 #11943484

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#20764 #11943484