Subgroups — C₅×#20576 #11943296

Order 1280.

778 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#446	C₂	1	2	yes	maximal
2	C₅×#2295	C₂	1	1	yes	maximal, characteristic
3	C₅×#1771	C₂	1	2	yes	maximal
4	C₅×#1771	C₂	1	2	yes	maximal
5	C₅×#1770	C₂	1	2	yes	maximal
6	C₅×#1768	C₂	1	2	yes	maximal
7	C₅×#1767	C₂	1	1	yes	maximal, characteristic
8	C₅×#1746	C₂	1	2	yes	maximal
9	C₅×#1745	C₂	1	1	yes	maximal, characteristic
10	C₂×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
11	C₅×#446	C₄	1	16	yes	complemented
12	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
13	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
14	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
15	C₂×C₅×#97	C₂²	1	1	yes	characteristic
16	C₂×C₅×#97	C₂²	1	1	yes	characteristic
17	C₂×C₅×#97	C₂²	1	2	yes	—
18	C₂×C₅×#97	C₂²	1	2	yes	—
19	C₂×C₅×#97	C₂²	1	2	yes	—
20	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
21	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
22	C₅×#391	C₄	1	2	yes	—
23	C₅×#389	C₄	1	2	yes	—
24	C₅×#387	C₄	1	4	yes	—
25	C₄×C₅×C₄⋊C₄	C₂²	1	2	yes	—
26	C₄×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
27	C₅×#376	C₂²	1	2	yes	—
28	C₄³×C₅	C₂²	1	1	yes	characteristic
29	C₅×#336	C₂²	1	2	yes	—
30	C₅×#336	C₂²	1	1	yes	characteristic
31	C₅×#336	C₂²	1	2	yes	—
32	C₅×#336	C₂²	1	2	yes	—
33	C₅×#336	C₂²	1	2	yes	—
34	C₅×#336	C₂²	1	1	yes	characteristic
35	#20576	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
36	C₂²×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₅×C₈	C₂³	1	2	yes	—
38	C₂²×C₅×C₈	C₂³	1	2	yes	—
39	C₂²×C₅×C₈	C₂³	1	2	yes	—
40	C₂²×C₅×C₈	C₂³	1	2	yes	—
41	C₄×C₅×Q₈	C₂×C₄	1	8	yes	—
42	C₂×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
43	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
44	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
45	C₂×C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
47	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
48	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
49	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
50	C₂×C₄²×C₅	C₂³	1	2	yes	—
51	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
52	C₂×C₄²×C₅	C₂³	1	2	yes	—
53	C₅×C₄⋊C₈	C₂×C₄	1	8	yes	—
54	C₅×C₄⋊C₈	C₂×C₄	1	8	yes	—
55	C₅×C₄⋊C₈	C₂×C₄	1	8	yes	—
56	C₅×#97	—	4	2	no	—
57	C₅×#97	—	4	1	no	—
58	C₅×#97	—	4	1	no	—
59	C₅×#97	C₂×C₄	1	8	yes	—
60	C₅×#97	C₂×C₄	1	8	yes	—
61	C₄×C₅×C₈	—	4	2	no	—
62	C₄×C₅×C₈	C₂×C₄	1	8	yes	—
63	C₅×#95	C₂×C₄	1	4	yes	—
64	C₅×#95	C₂×C₄	1	4	yes	—
65	C₅×#95	C₂×C₄	1	2	yes	—
66	C₅×#95	C₂×C₄	1	4	yes	—
67	C₂×#446	C₂×C₅	1	2	yes	—
68	#2295	C₂×C₅	1	1	yes	characteristic
69	#1771	C₂×C₅	1	2	yes	—
70	#1771	C₂×C₅	1	2	yes	—
71	#1770	C₂×C₅	1	2	yes	—
72	#1768	C₂×C₅	1	2	yes	—
73	#1767	C₂×C₅	1	1	yes	characteristic
74	#1746	C₂×C₅	1	2	yes	—
75	#1745	C₂×C₅	1	1	yes	characteristic
76	C₂×C₅×Q₈	—	4	2	no	—
77	C₂×C₅×Q₈	C₄²	1	4	yes	—
78	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
79	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
80	C₂²×C₄×C₅	C₂²×C₄	1	4	yes	—
81	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
84	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
85	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
86	C₂×C₅×C₈	—	2	2	no	—
87	C₂×C₅×C₈	—	2	2	no	—
88	C₂×C₅×C₈	—	2	2	no	—
89	C₂×C₅×C₈	—	2	2	no	—
90	C₂×C₅×C₈	—	2	2	no	—
91	C₂×C₅×C₈	—	2	2	no	—
92	C₂×C₅×C₈	—	2	2	no	—
93	C₂×C₅×C₈	—	2	2	no	—
94	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
95	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
96	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
97	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
98	C₅×C₄⋊C₄	—	4	2	no	—
99	C₅×C₄⋊C₄	—	4	4	no	—
100	C₅×C₄⋊C₄	C₄²	1	8	yes	—
101	C₅×C₄⋊C₄	C₄²	1	4	yes	—
102	C₄²×C₅	—	2	2	no	—
103	C₄²×C₅	—	2	1	no	—
104	C₄²×C₅	—	2	2	no	—
105	C₄²×C₅	—	2	1	no	—
106	C₄²×C₅	—	2	2	no	—
107	C₄²×C₅	—	4	2	no	—
108	C₄²×C₅	—	4	2	no	—
109	C₄²×C₅	C₂²×C₄	1	4	yes	—
110	C₄²×C₅	C₂²×C₄	1	4	yes	—
111	C₄²×C₅	C₂²×C₄	1	4	yes	—
112	C₂×C₄×Q₈	C₂²×C₅	1	1	yes	characteristic
113	#446	C₄×C₅	1	16	yes	complemented
114	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
115	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
116	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
117	C₂×#97	C₂²×C₅	1	1	yes	characteristic
118	C₂×#97	C₂²×C₅	1	1	yes	characteristic
119	C₂×#97	C₂²×C₅	1	2	yes	—
120	C₂×#97	C₂²×C₅	1	2	yes	—
121	C₂×#97	C₂²×C₅	1	2	yes	—
122	C₂×C₄×C₈	C₂²×C₅	1	2	yes	—
123	C₂×C₄×C₈	C₂²×C₅	1	2	yes	—
124	#391	C₄×C₅	1	2	yes	—
125	#389	C₄×C₅	1	2	yes	—
126	#387	C₄×C₅	1	4	yes	—
127	C₄×C₄⋊C₄	C₂²×C₅	1	2	yes	—
128	C₄×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
129	#376	C₂²×C₅	1	2	yes	—
130	C₄³	C₂²×C₅	1	1	yes	characteristic
131	#336	C₂²×C₅	1	1	yes	characteristic
132	#336	C₂²×C₅	1	1	yes	characteristic
133	#336	C₂²×C₅	1	2	yes	—
134	#336	C₂²×C₅	1	2	yes	—
135	#336	C₂²×C₅	1	2	yes	—
136	#336	C₂²×C₅	1	2	yes	—
137	C₂³×C₅	C₂³×C₄	1	1	yes	characteristic
138	C₅×Q₈	—	4	4	no	—
139	C₂×C₄×C₅	—	2	2	no	—
140	C₂×C₄×C₅	—	2	2	no	—
141	C₂×C₄×C₅	—	2	4	no	—
142	C₂×C₄×C₅	—	2	4	no	—
143	C₂×C₄×C₅	—	2	2	no	—
144	C₂×C₄×C₅	—	2	4	no	—
145	C₂×C₄×C₅	—	2	2	no	—
146	C₂×C₄×C₅	—	2	2	no	—
147	C₂×C₄×C₅	—	2	2	no	—
148	C₂×C₄×C₅	—	2	2	no	—
149	C₂×C₄×C₅	—	2	2	no	—
150	C₂×C₄×C₅	—	2	4	no	—
151	C₂×C₄×C₅	—	2	4	no	—
152	C₂×C₄×C₅	C₂×C₄²	1	4	yes	—
153	C₂×C₄×C₅	C₂×C₄²	1	2	yes	—
154	C₂×C₄×C₅	C₂×C₄²	1	4	yes	—
155	C₂×C₄×C₅	C₂×C₄²	1	2	yes	—
156	C₂×C₄×C₅	C₂³×C₄	1	2	yes	—
157	C₂×C₄×C₅	H(5,𝔽₂)	1	1	yes	characteristic
158	C₂×C₄×C₅	2₋¹⁺⁴	1	1	yes	characteristic
159	C₂×C₄×C₅	2₋¹⁺⁴	1	2	yes	—
160	C₅×C₈	—	2	4	no	—
161	C₅×C₈	—	2	4	no	—
162	C₅×C₈	—	2	4	no	—
163	C₅×C₈	—	2	4	no	—
164	C₂²×Q₈	C₂×C₄×C₅	1	1	yes	characteristic
165	C₂²×C₈	C₂³×C₅	1	2	yes	—
166	C₂²×C₈	C₂³×C₅	1	2	yes	—
167	C₂²×C₈	C₂³×C₅	1	2	yes	—
168	C₂²×C₈	C₂³×C₅	1	2	yes	—
169	C₄×Q₈	C₂×C₄×C₅	1	8	yes	—
170	C₂×C₄⋊C₄	C₂×C₄×C₅	1	4	yes	—
171	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
172	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
173	C₂×C₄⋊C₄	C₂×C₄×C₅	1	1	yes	characteristic
174	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
175	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
176	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
177	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
178	C₂×C₄²	C₂³×C₅	1	2	yes	—
179	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
180	C₂×C₄²	C₂³×C₅	1	2	yes	—
181	C₄⋊C₈	C₂×C₄×C₅	1	8	yes	—
182	C₄⋊C₈	C₂×C₄×C₅	1	8	yes	—
183	C₄⋊C₈	C₂×C₄×C₅	1	8	yes	—
184	#97	—	4	2	no	—
185	#97	—	4	1	no	—
186	#97	—	4	1	no	—
187	#97	C₂×C₄×C₅	1	8	yes	—
188	#97	C₂×C₄×C₅	1	8	yes	—
189	C₄×C₈	—	4	2	no	—
190	C₄×C₈	C₂×C₄×C₅	1	8	yes	—
191	#95	C₂×C₄×C₅	1	4	yes	—
192	#95	C₂×C₄×C₅	1	4	yes	—
193	#95	C₂×C₄×C₅	1	2	yes	—
194	#95	C₂×C₄×C₅	1	4	yes	—
195	C₂²×C₅	C₂²×C₄²	1	1	yes	characteristic
196	C₂²×C₅	#519	1	1	yes	characteristic
197	C₂²×C₅	#520	1	1	yes	characteristic
198	C₂²×C₅	#568	1	2	yes	—
199	C₂²×C₅	#568	1	2	yes	—
200	C₄×C₅	—	2	2	no	—
201	C₄×C₅	—	2	4	no	—
202	C₄×C₅	—	2	2	no	—
203	C₄×C₅	—	2	4	no	—
204	C₄×C₅	—	4	2	no	—
205	C₄×C₅	—	4	2	no	—
206	C₄×C₅	—	4	4	no	—
207	C₄×C₅	#519	1	2	yes	—
208	C₄×C₅	#520	1	2	yes	—
209	C₂×Q₈	—	4	2	no	—
210	C₂×Q₈	C₄²×C₅	1	4	yes	—
211	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
212	C₂²×C₄	C₂²×C₄×C₅	1	4	yes	—
213	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
214	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
215	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
216	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
217	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
218	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	C₂²×C₄×C₅	1	4	yes	—
228	C₂×C₈	C₂²×C₄×C₅	1	4	yes	—
229	C₂×C₈	C₂²×C₄×C₅	1	4	yes	—
230	C₂×C₈	C₂²×C₄×C₅	1	4	yes	—
231	C₄⋊C₄	—	4	2	no	—
232	C₄⋊C₄	—	4	4	no	—
233	C₄⋊C₄	C₄²×C₅	1	8	yes	—
234	C₄⋊C₄	C₄²×C₅	1	4	yes	—
235	C₄²	—	2	2	no	—
236	C₄²	—	2	1	no	—
237	C₄²	—	2	2	no	—
238	C₄²	—	2	1	no	—
239	C₄²	—	2	2	no	—
240	C₄²	—	4	2	no	—
241	C₄²	—	4	2	no	—
242	C₄²	C₂²×C₄×C₅	1	4	yes	—
243	C₄²	C₂²×C₄×C₅	1	4	yes	—
244	C₄²	C₂²×C₄×C₅	1	4	yes	—
245	C₂×C₅	#2276	1	1	yes	characteristic
246	C₂×C₅	#2873	1	1	yes	characteristic
247	C₂×C₅	#2875	1	1	yes	characteristic
248	C₂×C₅	#2987	1	2	yes	—
249	C₂×C₅	#2995	1	2	yes	—
250	C₂³	C₂³×C₄×C₅	1	1	yes	characteristic
251	Q₈	—	4	4	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	4	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	C₂×C₄²×C₅	1	4	yes	—
266	C₂×C₄	C₂×C₄²×C₅	1	2	yes	—
267	C₂×C₄	C₂×C₄²×C₅	1	4	yes	—
268	C₂×C₄	C₂×C₄²×C₅	1	2	yes	—
269	C₂×C₄	C₂³×C₄×C₅	1	2	yes	—
270	C₂×C₄	C₅×H(5,𝔽₂)	1	1	yes	characteristic
271	C₂×C₄	C₅×2₋¹⁺⁴	1	1	yes	characteristic
272	C₂×C₄	C₅×2₋¹⁺⁴	1	2	yes	—
273	C₈	—	2	4	no	—
274	C₈	—	2	4	no	—
275	C₈	—	2	4	no	—
276	C₈	—	2	4	no	—
277	C₅	#20576	1	1	yes	characteristic, direct product factor, Sylow
278	C₂²	C₂²×C₄²×C₅	1	1	yes	characteristic
279	C₂²	C₅×#519	1	1	yes	characteristic
280	C₂²	C₅×#520	1	1	yes	characteristic
281	C₂²	C₅×#568	1	2	yes	—
282	C₂²	C₅×#568	1	2	yes	—
283	C₄	—	2	2	no	—
284	C₄	—	2	4	no	—
285	C₄	—	2	2	no	—
286	C₄	—	2	4	no	—
287	C₄	—	4	2	no	—
288	C₄	—	4	2	no	—
289	C₄	—	4	4	no	—
290	C₄	C₅×#519	1	2	yes	—
291	C₄	C₅×#520	1	2	yes	—
292	C₂	C₅×#2276	1	1	yes	characteristic
293	C₂	C₅×#2873	1	1	yes	characteristic
294	C₂	C₅×#2875	1	1	yes	characteristic
295	C₂	C₅×#2987	1	2	yes	—
296	C₂	C₅×#2995	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₅	C₂⁴
commutator	C₂²	C₂²×C₄²×C₅
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
socle, prime radical	C₂³×C₅	C₂³×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
11. C₅×#446	C₄	semidirect
35. #20576	C₅	direct (N × H)
113. #446	C₄×C₅	semidirect
277. C₅	#20576	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×#446	2
C₅×#2295	2
C₅×#1771	2
C₅×#1770	2
C₅×#1768	2
C₅×#1767	2
C₅×#1746	2
C₅×#1745	2
#20576	5

Subgroups — C5×#20576 #11943296

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#20576 #11943296