Subgroups — C₅×#20573 #11943293

Order 1280.

898 conjugacy classes of subgroups.

The subgroup lattice has 272 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	C₄²×C₅×Q₈	C₂	1	1	yes	maximal, characteristic
3	C₅×#1771	C₂	1	4	yes	maximal
4	C₅×C₈×C₄⋊C₄	C₂	1	2	yes	maximal
5	C₅×#1767	C₂	1	1	yes	maximal, characteristic
6	C₄×C₅×C₄⋊C₈	C₂	1	2	yes	maximal
7	C₅×#1745	C₂	1	2	yes	maximal
8	C₅×#1744	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×C₅×Q₈	C₄	1	2	yes	—
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	4	yes	—
13	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
14	C₂×C₅×#97	C₂²	1	2	yes	—
15	C₂×C₅×#97	C₂²	1	4	yes	—
16	C₂×C₄×C₅×C₈	C₂²	1	4	yes	—
17	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
18	C₄×C₅×C₄⋊C₄	C₄	1	4	yes	—
19	C₄×C₅×C₄⋊C₄	C₄	1	2	yes	—
20	C₄×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
21	C₄×C₅×C₄⋊C₄	C₂²	1	2	yes	—
22	C₄³×C₅	C₂²	1	2	yes	—
23	C₄³×C₅	C₂²	1	1	yes	characteristic
24	C₅×#336	C₂²	1	1	yes	characteristic
25	C₅×#336	C₂²	1	2	yes	—
26	C₅×#336	C₂²	1	4	yes	—
27	C₅×#336	C₂²	1	2	yes	—
28	C₅×#336	C₂²	1	1	yes	characteristic
29	#20573	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
30	C₂²×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₅×C₈	C₂³	1	2	yes	—
32	C₂²×C₅×C₈	C₂³	1	2	yes	—
33	C₂²×C₅×C₈	C₂³	1	4	yes	—
34	C₄×C₅×Q₈	—	2	8	no	—
35	C₄×C₅×Q₈	C₂×C₄	1	8	yes	—
36	C₂×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
37	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
38	C₂×C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
39	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₅	C₂×C₄	1	4	yes	—
42	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
43	C₂×C₄²×C₅	C₂×C₄	1	4	yes	—
44	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
45	C₂×C₄²×C₅	C₂³	1	2	yes	—
46	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
47	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₅	C₂³	1	2	yes	—
49	C₅×C₄⋊C₈	C₂×C₄	1	8	yes	—
50	C₅×C₄⋊C₈	C₂×C₄	1	16	yes	—
51	C₅×#97	—	4	2	no	—
52	C₅×#97	C₂×C₄	1	16	yes	—
53	C₄×C₅×C₈	—	4	4	no	—
54	C₄×C₅×C₈	C₂×C₄	1	8	yes	—
55	C₅×#95	C₂×C₄	1	4	yes	—
56	C₅×#95	C₂×C₄	1	2	yes	—
57	C₂×#446	C₂×C₅	1	2	yes	—
58	C₄²×Q₈	C₂×C₅	1	1	yes	characteristic
59	#1771	C₂×C₅	1	4	yes	—
60	C₈×C₄⋊C₄	C₂×C₅	1	2	yes	—
61	#1767	C₂×C₅	1	1	yes	characteristic
62	C₄×C₄⋊C₈	C₂×C₅	1	2	yes	—
63	#1745	C₂×C₅	1	2	yes	—
64	#1744	C₂×C₅	1	1	yes	characteristic
65	C₂×C₅×Q₈	—	2	4	no	—
66	C₂×C₅×Q₈	C₄²	1	4	yes	—
67	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
68	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
69	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
71	C₂²×C₄×C₅	C₂²×C₄	1	4	yes	—
72	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
73	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
75	C₂×C₅×C₈	—	2	2	no	—
76	C₂×C₅×C₈	—	2	4	no	—
77	C₂×C₅×C₈	—	2	2	no	—
78	C₂×C₅×C₈	—	2	2	no	—
79	C₂×C₅×C₈	—	2	2	no	—
80	C₂×C₅×C₈	—	2	4	no	—
81	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
82	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
83	C₂×C₅×C₈	C₂²×C₄	1	8	yes	—
84	C₅×C₄⋊C₄	—	2	8	no	—
85	C₅×C₄⋊C₄	C₄²	1	8	yes	—
86	C₅×C₄⋊C₄	C₄²	1	4	yes	—
87	C₅×C₄⋊C₄	M₁₆	1	8	yes	—
88	C₄²×C₅	—	2	8	no	—
89	C₄²×C₅	—	2	4	no	—
90	C₄²×C₅	—	2	8	no	—
91	C₄²×C₅	—	2	4	no	—
92	C₄²×C₅	—	2	2	no	—
93	C₄²×C₅	—	2	1	no	—
94	C₄²×C₅	—	2	2	no	—
95	C₄²×C₅	—	2	2	no	—
96	C₄²×C₅	—	2	1	no	—
97	C₄²×C₅	C₂²×C₄	1	4	yes	—
98	C₄²×C₅	C₂²×C₄	1	8	yes	—
99	C₂×C₄×Q₈	C₄×C₅	1	2	yes	—
100	C₂×C₄×Q₈	C₂²×C₅	1	1	yes	characteristic
101	#446	C₄×C₅	1	16	yes	complemented
102	C₂×C₄⋊C₈	C₂²×C₅	1	4	yes	—
103	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
104	C₂×#97	C₂²×C₅	1	2	yes	—
105	C₂×#97	C₂²×C₅	1	4	yes	—
106	C₂×C₄×C₈	C₂²×C₅	1	4	yes	—
107	C₂×C₄×C₈	C₂²×C₅	1	2	yes	—
108	C₄×C₄⋊C₄	C₄×C₅	1	4	yes	—
109	C₄×C₄⋊C₄	C₄×C₅	1	2	yes	—
110	C₄×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
111	C₄×C₄⋊C₄	C₂²×C₅	1	2	yes	—
112	C₄³	C₂²×C₅	1	1	yes	characteristic
113	C₄³	C₂²×C₅	1	2	yes	—
114	#336	C₂²×C₅	1	1	yes	characteristic
115	#336	C₂²×C₅	1	1	yes	characteristic
116	#336	C₂²×C₅	1	2	yes	—
117	#336	C₂²×C₅	1	2	yes	—
118	#336	C₂²×C₅	1	4	yes	—
119	C₂³×C₅	C₂³×C₄	1	1	yes	characteristic
120	C₅×Q₈	—	2	8	no	—
121	C₂×C₄×C₅	—	2	2	no	—
122	C₂×C₄×C₅	—	2	4	no	—
123	C₂×C₄×C₅	—	2	2	no	—
124	C₂×C₄×C₅	—	2	4	no	—
125	C₂×C₄×C₅	—	2	4	no	—
126	C₂×C₄×C₅	—	2	2	no	—
127	C₂×C₄×C₅	—	2	4	no	—
128	C₂×C₄×C₅	—	2	2	no	—
129	C₂×C₄×C₅	—	2	2	no	—
130	C₂×C₄×C₅	—	2	2	no	—
131	C₂×C₄×C₅	C₂×C₄²	1	2	yes	—
132	C₂×C₄×C₅	C₂×C₄²	1	4	yes	—
133	C₂×C₄×C₅	C₂×C₄²	1	4	yes	—
134	C₂×C₄×C₅	C₂×C₄²	1	2	yes	—
135	C₂×C₄×C₅	C₂×M₁₆	1	8	yes	—
136	C₂×C₄×C₅	C₂×M₁₆	1	4	yes	—
137	C₂×C₄×C₅	#131	1	4	yes	—
138	C₂×C₄×C₅	C₂³×C₄	1	2	yes	—
139	C₂×C₄×C₅	H(5,𝔽₂)	1	1	yes	characteristic
140	C₂×C₄×C₅	2₋¹⁺⁴	1	1	yes	characteristic
141	C₂×C₄×C₅	2₋¹⁺⁴	1	2	yes	—
142	C₅×C₈	—	2	4	no	—
143	C₅×C₈	—	2	4	no	—
144	C₅×C₈	—	2	8	no	—
145	C₂²×Q₈	C₂×C₄×C₅	1	1	yes	characteristic
146	C₂²×C₈	C₂³×C₅	1	2	yes	—
147	C₂²×C₈	C₂³×C₅	1	2	yes	—
148	C₂²×C₈	C₂³×C₅	1	4	yes	—
149	C₄×Q₈	—	2	8	no	—
150	C₄×Q₈	C₂×C₄×C₅	1	8	yes	—
151	C₂×C₄⋊C₄	C₂×C₄×C₅	1	4	yes	—
152	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
153	C₂×C₄⋊C₄	C₂×C₄×C₅	1	1	yes	characteristic
154	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
155	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
156	C₂×C₄²	C₂×C₄×C₅	1	4	yes	—
157	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
158	C₂×C₄²	C₂×C₄×C₅	1	4	yes	—
159	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
160	C₂×C₄²	C₂³×C₅	1	2	yes	—
161	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
162	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
163	C₂×C₄²	C₂³×C₅	1	2	yes	—
164	C₄⋊C₈	C₂×C₄×C₅	1	8	yes	—
165	C₄⋊C₈	C₂×C₄×C₅	1	16	yes	—
166	#97	—	4	2	no	—
167	#97	C₂×C₄×C₅	1	16	yes	—
168	C₄×C₈	—	4	4	no	—
169	C₄×C₈	C₂×C₄×C₅	1	8	yes	—
170	#95	C₂×C₄×C₅	1	4	yes	—
171	#95	C₂×C₄×C₅	1	2	yes	—
172	C₂²×C₅	C₂²×C₄²	1	1	yes	characteristic
173	C₂²×C₅	#519	1	1	yes	characteristic
174	C₂²×C₅	#520	1	1	yes	characteristic
175	C₂²×C₅	C₂²×M₁₆	1	2	yes	—
176	C₂²×C₅	C₂×#131	1	2	yes	—
177	C₄×C₅	—	2	4	no	—
178	C₄×C₅	—	2	2	no	—
179	C₄×C₅	—	2	4	no	—
180	C₄×C₅	—	2	4	no	—
181	C₄×C₅	—	2	8	no	—
182	C₄×C₅	—	4	2	no	—
183	C₄×C₅	C₄×M₁₆	1	4	yes	—
184	C₄×C₅	#519	1	2	yes	—
185	C₄×C₅	#520	1	2	yes	—
186	C₂×Q₈	—	2	4	no	—
187	C₂×Q₈	C₄²×C₅	1	4	yes	—
188	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
189	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
190	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
191	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
192	C₂²×C₄	C₂²×C₄×C₅	1	4	yes	—
193	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
194	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
195	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
196	C₂×C₈	—	2	2	no	—
197	C₂×C₈	—	2	4	no	—
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	—	2	4	no	—
202	C₂×C₈	C₂²×C₄×C₅	1	4	yes	—
203	C₂×C₈	C₂²×C₄×C₅	1	4	yes	—
204	C₂×C₈	C₂²×C₄×C₅	1	8	yes	—
205	C₄⋊C₄	—	2	8	no	—
206	C₄⋊C₄	C₄²×C₅	1	8	yes	—
207	C₄⋊C₄	C₄²×C₅	1	4	yes	—
208	C₄⋊C₄	C₅×M₁₆	1	8	yes	—
209	C₄²	—	2	8	no	—
210	C₄²	—	2	4	no	—
211	C₄²	—	2	8	no	—
212	C₄²	—	2	4	no	—
213	C₄²	—	2	2	no	—
214	C₄²	—	2	1	no	—
215	C₄²	—	2	2	no	—
216	C₄²	—	2	2	no	—
217	C₄²	—	2	1	no	—
218	C₄²	C₂²×C₄×C₅	1	4	yes	—
219	C₄²	C₂²×C₄×C₅	1	8	yes	—
220	C₂×C₅	#2276	1	1	yes	characteristic
221	C₂×C₅	C₂×C₄×M₁₆	1	1	yes	characteristic
222	C₂×C₅	C₄×#131	1	1	yes	characteristic
223	C₂×C₅	#2984	1	2	yes	—
224	C₂×C₅	#2990	1	2	yes	—
225	C₂³	C₂³×C₄×C₅	1	1	yes	characteristic
226	Q₈	—	2	8	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	4	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	C₂×C₄²×C₅	1	2	yes	—
238	C₂×C₄	C₂×C₄²×C₅	1	4	yes	—
239	C₂×C₄	C₂×C₄²×C₅	1	4	yes	—
240	C₂×C₄	C₂×C₄²×C₅	1	2	yes	—
241	C₂×C₄	C₂×C₅×M₁₆	1	8	yes	—
242	C₂×C₄	C₂×C₅×M₁₆	1	4	yes	—
243	C₂×C₄	C₅×#131	1	4	yes	—
244	C₂×C₄	C₂³×C₄×C₅	1	2	yes	—
245	C₂×C₄	C₅×H(5,𝔽₂)	1	1	yes	characteristic
246	C₂×C₄	C₅×2₋¹⁺⁴	1	1	yes	characteristic
247	C₂×C₄	C₅×2₋¹⁺⁴	1	2	yes	—
248	C₈	—	2	4	no	—
249	C₈	—	2	4	no	—
250	C₈	—	2	8	no	—
251	C₅	#20573	1	1	yes	characteristic, direct product factor, Sylow
252	C₂²	C₂²×C₄²×C₅	1	1	yes	characteristic
253	C₂²	C₅×#519	1	1	yes	characteristic
254	C₂²	C₅×#520	1	1	yes	characteristic
255	C₂²	C₂²×C₅×M₁₆	1	2	yes	—
256	C₂²	C₂×C₅×#131	1	2	yes	—
257	C₄	—	2	4	no	—
258	C₄	—	2	8	no	—
259	C₄	—	2	4	no	—
260	C₄	—	2	2	no	—
261	C₄	—	2	4	no	—
262	C₄	—	4	2	no	—
263	C₄	C₄×C₅×M₁₆	1	4	yes	—
264	C₄	C₅×#519	1	2	yes	—
265	C₄	C₅×#520	1	2	yes	—
266	C₂	C₅×#2276	1	1	yes	characteristic
267	C₂	C₂×C₄×C₅×M₁₆	1	1	yes	characteristic
268	C₂	C₄×C₅×#131	1	1	yes	characteristic
269	C₂	C₅×#2984	1	2	yes	—
270	C₂	C₅×#2990	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
29. #20573	C₅	direct (N × H)
101. #446	C₄×C₅	semidirect
251. C₅	#20573	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
C₄²×C₅×Q₈	2
C₅×#1771	2
C₅×C₈×C₄⋊C₄	2
C₅×#1767	2
C₄×C₅×C₄⋊C₈	2
C₅×#1745	2
C₅×#1744	2
#20573	5

Subgroups — C5×#20573 #11943293

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#20573 #11943293