Subgroups — C₅×#9823 #11932543

Order 1280.

526 conjugacy classes of subgroups.

The subgroup lattice has 234 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×#1990	C₂	1	2	yes	maximal
3	C₅×#1956	C₂	1	2	yes	maximal
4	C₅×#1845	C₂	1	1	yes	maximal, characteristic
5	C₂×C₅×C₈²	C₂	1	1	yes	maximal, characteristic
6	C₂×C₅×C₄⋊Q₈	C₂²	1	2	yes	—
7	C₅×#500	—	2	2	no	—
8	C₅×#500	C₄	1	4	yes	complemented
9	C₂×C₅×#107	—	2	2	no	—
10	C₂×C₅×#107	—	2	2	no	—
11	C₂×C₅×#103	—	2	4	no	—
12	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
15	C₅×#384	C₂²	1	2	yes	—
16	C₅×#340	—	2	4	no	—
17	C₅×C₈²	—	2	2	no	—
18	#9823	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
19	C₂²×C₅×Q₈	—	2	2	no	—
20	C₂²×C₅×C₈	D₄	1	2	yes	—
21	C₂²×C₅×C₈	D₄	1	2	yes	—
22	C₂²×C₅×C₈	D₄	1	2	yes	—
23	C₅×C₄⋊Q₈	—	2	2	no	—
24	C₅×C₄⋊Q₈	—	2	2	no	—
25	C₅×C₄⋊Q₈	—	2	2	no	—
26	C₅×C₄⋊Q₈	C₂×C₄	1	4	yes	—
27	C₂×C₅×C₄⋊C₄	—	2	2	no	—
28	C₂×C₅×C₄⋊C₄	—	2	2	no	—
29	C₂×C₅×C₄⋊C₄	—	2	2	no	—
30	C₂×C₅×C₄⋊C₄	—	2	2	no	—
31	C₂×C₅×C₄⋊C₄	—	2	2	no	—
32	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
33	C₅×#107	—	2	8	no	—
34	C₅×#107	—	4	4	no	—
35	C₅×#103	—	4	8	no	—
36	C₄×C₅×C₈	—	2	1	no	—
37	C₄×C₅×C₈	—	2	1	no	—
38	C₄×C₅×C₈	—	2	1	no	—
39	C₄×C₅×C₈	—	2	1	no	—
40	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
41	C₄×C₅×C₈	D₄	1	2	yes	—
42	C₂×#500	C₂×C₅	1	1	yes	characteristic
43	#1990	C₂×C₅	1	2	yes	—
44	#1956	C₂×C₅	1	2	yes	—
45	#1845	C₂×C₅	1	1	yes	characteristic
46	C₂×C₈²	C₂×C₅	1	1	yes	characteristic
47	C₂×C₅×Q₈	—	2	4	no	—
48	C₂×C₅×Q₈	—	4	2	no	—
49	C₂×C₅×Q₈	—	4	4	no	—
50	C₂×C₅×Q₈	—	4	4	no	—
51	C₂²×C₄×C₅	—	4	2	no	—
52	C₂²×C₄×C₅	—	4	2	no	—
53	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
56	C₂×C₅×C₈	—	2	2	no	—
57	C₂×C₅×C₈	—	2	2	no	—
58	C₂×C₅×C₈	—	2	2	no	—
59	C₂×C₅×C₈	—	2	2	no	—
60	C₂×C₅×C₈	C₂²⋊C₄	1	4	yes	—
61	C₂×C₅×C₈	D₈	1	4	yes	—
62	C₂×C₅×C₈	SD₁₆	1	4	yes	—
63	C₂×C₅×C₈	Q₁₆	1	4	yes	—
64	C₂×C₅×C₈	Q₁₆	1	4	yes	—
65	C₅×C₄⋊C₄	—	2	4	no	—
66	C₅×C₄⋊C₄	—	2	4	no	—
67	C₅×C₄⋊C₄	—	4	2	no	—
68	C₅×C₄⋊C₄	—	4	2	no	—
69	C₅×C₄⋊C₄	—	4	4	no	—
70	C₅×C₄⋊C₄	—	4	4	no	—
71	C₅×C₄⋊C₄	—	4	4	no	—
72	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
73	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
74	C₄²×C₅	#41	1	1	yes	characteristic
75	C₄²×C₅	#41	1	1	yes	characteristic
76	C₂×C₄⋊Q₈	C₂²×C₅	1	2	yes	—
77	#500	—	2	2	no	—
78	#500	C₄×C₅	1	4	yes	complemented
79	C₂×#107	—	2	2	no	—
80	C₂×#107	—	2	2	no	—
81	C₂×#103	—	2	4	no	—
82	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
83	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
84	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
85	#384	C₂²×C₅	1	2	yes	—
86	#340	—	2	4	no	—
87	C₈²	—	2	2	no	—
88	C₂³×C₅	#127	1	1	yes	characteristic
89	C₅×Q₈	—	4	8	no	—
90	C₅×Q₈	—	8	4	no	—
91	C₂×C₄×C₅	—	4	4	no	—
92	C₂×C₄×C₅	—	8	2	no	—
93	C₂×C₄×C₅	—	8	2	no	—
94	C₂×C₄×C₅	—	8	2	no	—
95	C₂×C₄×C₅	—	8	2	no	—
96	C₂×C₄×C₅	—	8	2	no	—
97	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
99	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
100	C₂×C₄×C₅	#121	1	1	yes	characteristic
101	C₂×C₄×C₅	#121	1	1	yes	characteristic
102	C₂×C₄×C₅	#124	1	1	yes	characteristic
103	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
104	C₂×C₄×C₅	C₂×SD₁₆	1	2	yes	—
105	C₂×C₄×C₅	C₂×Q₁₆	1	2	yes	—
106	C₂×C₄×C₅	C₂×Q₁₆	1	2	yes	—
107	C₂×C₄×C₅	#135	1	2	yes	—
108	C₂×C₄×C₅	#135	1	2	yes	—
109	C₅×C₈	—	2	4	no	—
110	C₅×C₈	—	2	4	no	—
111	C₅×C₈	#103	1	8	yes	complemented
112	C₂²×Q₈	—	2	2	no	—
113	C₂²×C₈	C₅×D₄	1	2	yes	—
114	C₂²×C₈	C₅×D₄	1	2	yes	—
115	C₂²×C₈	C₅×D₄	1	2	yes	—
116	C₄⋊Q₈	—	2	2	no	—
117	C₄⋊Q₈	—	2	2	no	—
118	C₄⋊Q₈	—	2	2	no	—
119	C₄⋊Q₈	C₂×C₄×C₅	1	4	yes	—
120	C₂×C₄⋊C₄	—	2	2	no	—
121	C₂×C₄⋊C₄	—	2	2	no	—
122	C₂×C₄⋊C₄	—	2	2	no	—
123	C₂×C₄⋊C₄	—	2	2	no	—
124	C₂×C₄⋊C₄	—	2	2	no	—
125	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
126	#107	—	2	8	no	—
127	#107	—	4	4	no	—
128	#103	—	4	8	no	—
129	C₄×C₈	—	2	1	no	—
130	C₄×C₈	—	2	1	no	—
131	C₄×C₈	—	2	1	no	—
132	C₄×C₈	—	2	1	no	—
133	C₄×C₈	C₂×C₄×C₅	1	2	yes	—
134	C₄×C₈	C₅×D₄	1	2	yes	—
135	C₂²×C₅	#390	1	1	yes	characteristic
136	C₂²×C₅	#492	1	1	yes	characteristic
137	C₂²×C₅	#493	1	1	yes	characteristic
138	C₂²×C₅	#494	1	1	yes	characteristic
139	C₂²×C₅	#494	1	1	yes	characteristic
140	C₂²×C₅	#495	1	1	yes	characteristic
141	C₂²×C₅	#495	1	1	yes	characteristic
142	C₄×C₅	—	8	4	no	—
143	C₄×C₅	—	16	2	no	—
144	C₄×C₅	C₂×#103	1	2	yes	—
145	C₄×C₅	C₄×D₈	1	2	yes	—
146	C₄×C₅	C₄×Q₁₆	1	2	yes	—
147	C₄×C₅	#466	1	2	yes	—
148	C₄×C₅	#467	1	2	yes	—
149	C₄×C₅	#487	1	2	yes	—
150	C₂×Q₈	—	2	4	no	—
151	C₂×Q₈	—	4	2	no	—
152	C₂×Q₈	—	4	4	no	—
153	C₂×Q₈	—	4	4	no	—
154	C₂²×C₄	—	4	2	no	—
155	C₂²×C₄	—	4	2	no	—
156	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
157	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
158	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
159	C₂×C₈	—	2	2	no	—
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	—	2	2	no	—
162	C₂×C₈	—	2	2	no	—
163	C₂×C₈	C₅×C₂²⋊C₄	1	4	yes	—
164	C₂×C₈	C₅×D₈	1	4	yes	—
165	C₂×C₈	C₅×SD₁₆	1	4	yes	—
166	C₂×C₈	C₅×Q₁₆	1	4	yes	—
167	C₂×C₈	C₅×Q₁₆	1	4	yes	—
168	C₄⋊C₄	—	2	4	no	—
169	C₄⋊C₄	—	2	4	no	—
170	C₄⋊C₄	—	4	2	no	—
171	C₄⋊C₄	—	4	2	no	—
172	C₄⋊C₄	—	4	4	no	—
173	C₄⋊C₄	—	4	4	no	—
174	C₄⋊C₄	—	4	4	no	—
175	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
176	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
177	C₄²	C₅×#41	1	1	yes	characteristic
178	C₄²	C₅×#41	1	1	yes	characteristic
179	C₂×C₅	#1706	1	1	yes	characteristic
180	C₂×C₅	#1707	1	1	yes	characteristic
181	C₂×C₅	#1710	1	1	yes	characteristic
182	C₂×C₅	#1713	1	1	yes	characteristic
183	C₂×C₅	#1956	1	1	yes	characteristic
184	C₂×C₅	#1969	1	1	yes	characteristic
185	C₂×C₅	#1970	1	1	yes	characteristic
186	C₂³	C₅×#127	1	1	yes	characteristic
187	Q₈	—	4	8	no	—
188	Q₈	—	8	4	no	—
189	C₂×C₄	—	4	4	no	—
190	C₂×C₄	—	8	2	no	—
191	C₂×C₄	—	8	2	no	—
192	C₂×C₄	—	8	2	no	—
193	C₂×C₄	—	8	2	no	—
194	C₂×C₄	—	8	2	no	—
195	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
196	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
197	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
198	C₂×C₄	C₅×#121	1	1	yes	characteristic
199	C₂×C₄	C₅×#121	1	1	yes	characteristic
200	C₂×C₄	C₅×#124	1	1	yes	characteristic
201	C₂×C₄	C₂×C₅×D₈	1	2	yes	—
202	C₂×C₄	C₂×C₅×SD₁₆	1	2	yes	—
203	C₂×C₄	C₂×C₅×Q₁₆	1	2	yes	—
204	C₂×C₄	C₂×C₅×Q₁₆	1	2	yes	—
205	C₂×C₄	C₅×#135	1	2	yes	—
206	C₂×C₄	C₅×#135	1	2	yes	—
207	C₈	—	2	4	no	—
208	C₈	—	2	4	no	—
209	C₈	C₅×#103	1	8	yes	complemented
210	C₅	#9823	1	1	yes	characteristic, direct product factor, Sylow
211	C₂²	C₅×#390	1	1	yes	characteristic
212	C₂²	C₅×#492	1	1	yes	characteristic
213	C₂²	C₅×#493	1	1	yes	characteristic
214	C₂²	C₅×#494	1	1	yes	characteristic
215	C₂²	C₅×#494	1	1	yes	characteristic
216	C₂²	C₅×#495	1	1	yes	characteristic
217	C₂²	C₅×#495	1	1	yes	characteristic
218	C₄	—	8	4	no	—
219	C₄	—	16	2	no	—
220	C₄	C₂×C₅×#103	1	2	yes	—
221	C₄	C₄×C₅×D₈	1	2	yes	—
222	C₄	C₄×C₅×Q₁₆	1	2	yes	—
223	C₄	C₅×#466	1	2	yes	—
224	C₄	C₅×#467	1	2	yes	—
225	C₄	C₅×#487	1	2	yes	—
226	C₂	C₅×#1706	1	1	yes	characteristic
227	C₂	C₅×#1707	1	1	yes	characteristic
228	C₂	C₅×#1710	1	1	yes	characteristic
229	C₂	C₅×#1713	1	1	yes	characteristic
230	C₂	C₅×#1956	1	1	yes	characteristic
231	C₂	C₅×#1969	1	1	yes	characteristic
232	C₂	C₅×#1970	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₅	#127
commutator	C₄²	C₂²×C₄×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
LCS	G ▷ 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
8. C₅×#500	C₄	semidirect
18. #9823	C₅	direct (N × H)
78. #500	C₄×C₅	semidirect
111. C₅×C₈	#103	semidirect
209. C₈	C₅×#103	semidirect
210. C₅	#9823	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×#1990	2
C₅×#1956	2
C₅×#1845	2
C₂×C₅×C₈²	2
#9823	5

Subgroups — C5×#9823 #11932543

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#9823 #11932543