Subgroups — C₅×#10545 #11933265

Order 1280.

454 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#500	C₂	1	1	yes	maximal, characteristic
2	C₅×#1981	C₂	1	2	yes	maximal
3	C₅×#1939	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅×#332	C₂	1	2	yes	maximal
5	C₅×#1277	C₂	1	1	yes	maximal, characteristic
6	C₂×C₅×C₄⋊Q₈	C₂²	1	2	yes	—
7	C₅×#500	—	2	4	no	—
8	C₂×C₅×#107	—	2	1	no	—
9	C₂×C₅×#107	—	2	1	no	—
10	C₂×C₅×C₄⋊C₈	—	2	1	no	—
11	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
13	C₂×C₅×#103	—	2	2	no	—
14	C₂×C₅×#103	—	2	2	no	—
15	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
16	C₄×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
17	C₅×#336	—	2	1	no	—
18	C₅×#332	—	2	4	no	—
19	#10545	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
20	C₂²×C₅×Q₈	—	2	2	no	—
21	C₂²×C₅×C₈	—	2	1	no	—
22	C₂²×C₅×C₈	—	2	2	no	—
23	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
24	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
25	C₅×C₄⋊Q₈	—	2	2	no	—
26	C₅×C₄⋊Q₈	—	2	2	no	—
27	C₅×C₄⋊Q₈	—	2	2	no	—
28	C₅×C₄⋊Q₈	D₄	1	4	yes	—
29	C₂×C₅×C₄⋊C₄	—	2	2	no	—
30	C₂×C₅×C₄⋊C₄	—	2	2	no	—
31	C₂×C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
32	C₂×C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
33	C₂×C₄²×C₅	—	2	1	no	—
34	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
35	C₅×#107	—	2	4	no	—
36	C₅×#107	—	2	4	no	—
37	C₅×C₄⋊C₈	—	2	2	no	—
38	C₅×C₄⋊C₈	—	2	2	no	—
39	C₅×C₄⋊C₈	—	2	2	no	—
40	C₅×C₄⋊C₈	—	4	2	no	—
41	C₅×#103	—	4	4	no	—
42	C₅×#103	—	4	4	no	—
43	C₄×C₅×C₈	—	2	2	no	—
44	C₅×#95	—	2	1	no	—
45	C₂×#500	C₂×C₅	1	1	yes	characteristic
46	#1981	C₂×C₅	1	2	yes	—
47	#1939	C₂×C₅	1	1	yes	characteristic
48	C₂×#332	C₂×C₅	1	2	yes	—
49	#1277	C₂×C₅	1	1	yes	characteristic
50	C₂×C₅×Q₈	—	2	4	no	—
51	C₂×C₅×Q₈	—	4	2	no	—
52	C₂×C₅×Q₈	—	4	2	no	—
53	C₂×C₅×Q₈	—	4	2	no	—
54	C₂×C₅×Q₈	—	4	2	no	—
55	C₂×C₅×Q₈	—	4	2	no	—
56	C₂²×C₄×C₅	—	2	1	no	—
57	C₂²×C₄×C₅	—	2	1	no	—
58	C₂²×C₄×C₅	—	4	2	no	—
59	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₅	#41	1	1	yes	characteristic
62	C₂×C₅×C₈	—	2	1	no	—
63	C₂×C₅×C₈	—	2	1	no	—
64	C₂×C₅×C₈	—	2	1	no	—
65	C₂×C₅×C₈	—	2	1	no	—
66	C₂×C₅×C₈	—	4	1	no	—
67	C₂×C₅×C₈	—	4	2	no	—
68	C₂×C₅×C₈	—	4	2	no	—
69	C₂×C₅×C₈	—	4	1	no	—
70	C₂×C₅×C₈	—	4	2	no	—
71	C₂×C₅×C₈	—	4	1	no	—
72	C₂×C₅×C₈	SD₁₆	1	2	yes	—
73	C₂×C₅×C₈	Q₁₆	1	2	yes	—
74	C₅×C₄⋊C₄	—	2	1	no	—
75	C₅×C₄⋊C₄	—	2	4	no	—
76	C₅×C₄⋊C₄	—	2	4	no	—
77	C₅×C₄⋊C₄	—	2	2	no	—
78	C₅×C₄⋊C₄	—	4	2	no	—
79	C₅×C₄⋊C₄	—	4	2	no	—
80	C₅×C₄⋊C₄	D₈	1	2	yes	—
81	C₄²×C₅	—	4	1	no	—
82	C₄²×C₅	—	4	1	no	—
83	C₄²×C₅	C₂×D₄	1	2	yes	—
84	C₄²×C₅	#41	1	2	yes	—
85	C₂×C₄⋊Q₈	C₂²×C₅	1	2	yes	—
86	#500	—	2	4	no	—
87	C₂×#107	—	2	1	no	—
88	C₂×#107	—	2	1	no	—
89	C₂×C₄⋊C₈	—	2	1	no	—
90	C₂×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
91	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
92	C₂×#103	—	2	2	no	—
93	C₂×#103	—	2	2	no	—
94	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
95	C₄×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
96	#336	—	2	1	no	—
97	#332	—	2	4	no	—
98	C₂³×C₅	#121	1	1	yes	characteristic
99	C₅×Q₈	—	4	4	no	—
100	C₅×Q₈	—	4	4	no	—
101	C₅×Q₈	—	8	2	no	—
102	C₅×Q₈	—	8	2	no	—
103	C₂×C₄×C₅	—	2	1	no	—
104	C₂×C₄×C₅	—	2	1	no	—
105	C₂×C₄×C₅	—	2	2	no	—
106	C₂×C₄×C₅	—	4	1	no	—
107	C₂×C₄×C₅	—	4	1	no	—
108	C₂×C₄×C₅	—	4	4	no	—
109	C₂×C₄×C₅	—	4	1	no	—
110	C₂×C₄×C₅	—	4	1	no	—
111	C₂×C₄×C₅	—	4	1	no	—
112	C₂×C₄×C₅	—	8	2	no	—
113	C₂×C₄×C₅	—	8	2	no	—
114	C₂×C₄×C₅	#120	1	1	yes	characteristic
115	C₂×C₄×C₅	#121	1	2	yes	—
116	C₂×C₄×C₅	#123	1	2	yes	—
117	C₂×C₄×C₅	#124	1	1	yes	characteristic
118	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
119	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
120	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
121	C₂×C₄×C₅	#135	1	1	yes	characteristic
122	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
123	C₂×C₄×C₅	#137	1	1	yes	characteristic
124	C₂×C₄×C₅	#137	1	1	yes	characteristic
125	C₅×C₈	—	2	2	no	—
126	C₅×C₈	—	2	2	no	—
127	C₅×C₈	—	8	1	no	—
128	C₅×C₈	—	8	2	no	—
129	C₂²×Q₈	—	2	2	no	—
130	C₂²×C₈	—	2	1	no	—
131	C₂²×C₈	—	2	2	no	—
132	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
133	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
134	C₄⋊Q₈	—	2	2	no	—
135	C₄⋊Q₈	—	2	2	no	—
136	C₄⋊Q₈	—	2	2	no	—
137	C₄⋊Q₈	C₅×D₄	1	4	yes	—
138	C₂×C₄⋊C₄	—	2	2	no	—
139	C₂×C₄⋊C₄	—	2	2	no	—
140	C₂×C₄⋊C₄	C₅×D₄	1	1	yes	characteristic
141	C₂×C₄⋊C₄	C₅×D₄	1	1	yes	characteristic
142	C₂×C₄²	—	2	1	no	—
143	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
144	#107	—	2	4	no	—
145	#107	—	2	4	no	—
146	C₄⋊C₈	—	2	2	no	—
147	C₄⋊C₈	—	2	2	no	—
148	C₄⋊C₈	—	2	2	no	—
149	C₄⋊C₈	—	4	2	no	—
150	#103	—	4	4	no	—
151	#103	—	4	4	no	—
152	C₄×C₈	—	2	2	no	—
153	#95	—	2	1	no	—
154	C₂²×C₅	#394	1	1	yes	characteristic
155	C₂²×C₅	#461	1	1	yes	characteristic
156	C₂²×C₅	#462	1	1	yes	characteristic
157	C₂²×C₅	#466	1	1	yes	characteristic
158	C₂²×C₅	#469	1	1	yes	characteristic
159	C₂²×C₅	#473	1	1	yes	characteristic
160	C₂²×C₅	#473	1	1	yes	characteristic
161	C₄×C₅	—	2	2	no	—
162	C₄×C₅	—	4	2	no	—
163	C₄×C₅	—	8	4	no	—
164	C₄×C₅	—	8	1	no	—
165	C₄×C₅	#448	1	1	yes	characteristic
166	C₄×C₅	#451	1	1	yes	characteristic
167	C₄×C₅	#459	1	2	yes	—
168	C₄×C₅	#463	1	2	yes	—
169	C₄×C₅	#487	1	1	yes	characteristic
170	C₄×C₅	#491	1	1	yes	characteristic
171	C₂×Q₈	—	2	4	no	—
172	C₂×Q₈	—	4	2	no	—
173	C₂×Q₈	—	4	2	no	—
174	C₂×Q₈	—	4	2	no	—
175	C₂×Q₈	—	4	2	no	—
176	C₂×Q₈	—	4	2	no	—
177	C₂²×C₄	—	2	1	no	—
178	C₂²×C₄	—	2	1	no	—
179	C₂²×C₄	—	4	2	no	—
180	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
181	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
182	C₂²×C₄	C₅×#41	1	1	yes	characteristic
183	C₂×C₈	—	2	1	no	—
184	C₂×C₈	—	2	1	no	—
185	C₂×C₈	—	2	1	no	—
186	C₂×C₈	—	2	1	no	—
187	C₂×C₈	—	4	1	no	—
188	C₂×C₈	—	4	2	no	—
189	C₂×C₈	—	4	2	no	—
190	C₂×C₈	—	4	1	no	—
191	C₂×C₈	—	4	2	no	—
192	C₂×C₈	—	4	1	no	—
193	C₂×C₈	C₅×SD₁₆	1	2	yes	—
194	C₂×C₈	C₅×Q₁₆	1	2	yes	—
195	C₄⋊C₄	—	2	1	no	—
196	C₄⋊C₄	—	2	4	no	—
197	C₄⋊C₄	—	2	4	no	—
198	C₄⋊C₄	—	2	2	no	—
199	C₄⋊C₄	—	4	2	no	—
200	C₄⋊C₄	—	4	2	no	—
201	C₄⋊C₄	C₅×D₈	1	2	yes	—
202	C₄²	—	4	1	no	—
203	C₄²	—	4	1	no	—
204	C₄²	C₂×C₅×D₄	1	2	yes	—
205	C₄²	C₅×#41	1	2	yes	—
206	C₂×C₅	#1673	1	1	yes	characteristic
207	C₂×C₅	#1674	1	1	yes	characteristic
208	C₂×C₅	#1693	1	1	yes	characteristic
209	C₂×C₅	#1694	1	1	yes	characteristic
210	C₂×C₅	#2043	1	1	yes	characteristic
211	C₂×C₅	#2047	1	1	yes	characteristic
212	C₂×C₅	#2054	1	1	yes	characteristic
213	C₂³	C₅×#121	1	1	yes	characteristic
214	Q₈	—	4	4	no	—
215	Q₈	—	4	4	no	—
216	Q₈	—	8	2	no	—
217	Q₈	—	8	2	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	4	1	no	—
222	C₂×C₄	—	4	1	no	—
223	C₂×C₄	—	4	4	no	—
224	C₂×C₄	—	4	1	no	—
225	C₂×C₄	—	4	1	no	—
226	C₂×C₄	—	4	1	no	—
227	C₂×C₄	—	8	2	no	—
228	C₂×C₄	—	8	2	no	—
229	C₂×C₄	C₅×#120	1	1	yes	characteristic
230	C₂×C₄	C₅×#121	1	2	yes	—
231	C₂×C₄	C₅×#123	1	2	yes	—
232	C₂×C₄	C₅×#124	1	1	yes	characteristic
233	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
234	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
235	C₂×C₄	C₂×C₅×Q₁₆	1	1	yes	characteristic
236	C₂×C₄	C₅×#135	1	1	yes	characteristic
237	C₂×C₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
238	C₂×C₄	C₅×#137	1	1	yes	characteristic
239	C₂×C₄	C₅×#137	1	1	yes	characteristic
240	C₈	—	2	2	no	—
241	C₈	—	2	2	no	—
242	C₈	—	8	1	no	—
243	C₈	—	8	2	no	—
244	C₅	#10545	1	1	yes	characteristic, direct product factor, Sylow
245	C₂²	C₅×#394	1	1	yes	characteristic
246	C₂²	C₅×#461	1	1	yes	characteristic
247	C₂²	C₅×#462	1	1	yes	characteristic
248	C₂²	C₅×#466	1	1	yes	characteristic
249	C₂²	C₅×#469	1	1	yes	characteristic
250	C₂²	C₅×#473	1	1	yes	characteristic
251	C₂²	C₅×#473	1	1	yes	characteristic
252	C₄	—	2	2	no	—
253	C₄	—	4	2	no	—
254	C₄	—	8	4	no	—
255	C₄	—	8	1	no	—
256	C₄	C₅×#448	1	1	yes	characteristic
257	C₄	C₅×#451	1	1	yes	characteristic
258	C₄	C₅×#459	1	2	yes	—
259	C₄	C₅×#463	1	2	yes	—
260	C₄	C₅×#487	1	1	yes	characteristic
261	C₄	C₅×#491	1	1	yes	characteristic
262	C₂	C₅×#1673	1	1	yes	characteristic
263	C₂	C₅×#1674	1	1	yes	characteristic
264	C₂	C₅×#1693	1	1	yes	characteristic
265	C₂	C₅×#1694	1	1	yes	characteristic
266	C₂	C₅×#2043	1	1	yes	characteristic
267	C₂	C₅×#2047	1	1	yes	characteristic
268	C₂	C₅×#2054	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₅	#121
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₅
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₂×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
19. #10545	C₅	direct (N × H)
244. C₅	#10545	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×#500	2
C₅×#1981	2
C₅×#1939	2
C₂×C₅×#332	2
C₅×#1277	2
#10545	5

Subgroups — C5×#10545 #11933265

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#10545 #11933265