Subgroups — C₅×#9872 #11932592

Order 1280.

458 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₅×#125	C₂	1	1	yes	maximal, characteristic
2	C₅×#1842	C₂	1	1	yes	maximal, characteristic
3	C₅×#1744	C₂	1	1	yes	maximal, characteristic
4	C₅×#1279	C₂	1	4	yes	maximal
5	C₂×C₅×#125	C₄	1	1	yes	characteristic
6	C₂×C₅×C₄⋊C₈	—	2	2	no	—
7	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
8	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
9	C₅×#389	C₄	1	2	yes	—
10	C₅×#387	—	2	2	no	—
11	C₅×#382	C₄	1	1	yes	characteristic
12	C₄×C₅×C₄⋊C₄	—	2	2	no	—
13	C₄×C₅×C₄⋊C₄	C₂²	1	2	yes	—
14	C₄³×C₅	C₂²	1	1	yes	characteristic
15	C₅×#336	—	2	2	no	—
16	#9872	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
17	C₂²×C₅×C₈	—	2	2	no	—
18	C₂²×C₅×C₈	—	2	2	no	—
19	C₅×#125	—	2	2	no	—
20	C₅×#125	C₈	1	4	yes	complemented
21	C₂×C₅×C₄⋊C₄	—	2	2	no	—
22	C₂×C₅×C₄⋊C₄	—	2	2	no	—
23	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
24	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
25	C₂×C₄²×C₅	—	2	2	no	—
26	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₅	D₄	1	2	yes	—
29	C₂×C₄²×C₅	D₄	1	2	yes	—
30	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
31	C₅×C₄⋊C₈	—	2	4	no	—
32	C₅×C₄⋊C₈	—	4	4	no	—
33	C₄×C₅×C₈	—	2	2	no	—
34	C₄×C₅×C₈	Q₈	1	4	yes	—
35	C₅×#95	—	2	2	no	—
36	C₅×#95	—	2	2	no	—
37	C₅×#95	—	2	2	no	—
38	C₄×#125	C₂×C₅	1	1	yes	characteristic
39	#1842	C₂×C₅	1	1	yes	characteristic
40	#1744	C₂×C₅	1	1	yes	characteristic
41	#1279	C₂×C₅	1	4	yes	—
42	C₂²×C₄×C₅	—	2	2	no	—
43	C₂²×C₄×C₅	—	2	2	no	—
44	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
45	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
46	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
49	C₂×C₅×C₈	—	2	4	no	—
50	C₂×C₅×C₈	—	4	4	no	—
51	C₂×C₅×C₈	—	4	2	no	—
52	C₂×C₅×C₈	—	4	4	no	—
53	C₅×C₄⋊C₄	—	2	8	no	—
54	C₅×C₄⋊C₄	—	2	2	no	—
55	C₅×C₄⋊C₄	—	2	4	no	—
56	C₅×C₄⋊C₄	—	4	4	no	—
57	C₅×C₄⋊C₄	C₂×C₈	1	4	yes	—
58	C₄²×C₅	—	2	2	no	—
59	C₄²×C₅	—	2	2	no	—
60	C₄²×C₅	—	2	2	no	—
61	C₄²×C₅	—	4	2	no	—
62	C₄²×C₅	—	4	2	no	—
63	C₄²×C₅	—	4	2	no	—
64	C₄²×C₅	C₂×C₈	1	2	yes	—
65	C₄²×C₅	M₁₆	1	2	yes	—
66	C₄²×C₅	C₂×Q₈	1	2	yes	—
67	C₄²×C₅	#41	1	2	yes	—
68	C₂×#125	C₄×C₅	1	1	yes	characteristic
69	C₂×C₄⋊C₈	—	2	2	no	—
70	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
71	C₂×C₄×C₈	C₂²×C₅	1	2	yes	—
72	#389	C₄×C₅	1	2	yes	—
73	#387	—	2	2	no	—
74	#382	C₄×C₅	1	1	yes	characteristic
75	C₄×C₄⋊C₄	—	2	2	no	—
76	C₄×C₄⋊C₄	C₂²×C₅	1	2	yes	—
77	C₄³	C₂²×C₅	1	1	yes	characteristic
78	#336	—	2	2	no	—
79	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₅	—	2	2	no	—
81	C₂×C₄×C₅	—	2	2	no	—
82	C₂×C₄×C₅	—	2	2	no	—
83	C₂×C₄×C₅	—	2	4	no	—
84	C₂×C₄×C₅	—	2	2	no	—
85	C₂×C₄×C₅	—	2	2	no	—
86	C₂×C₄×C₅	—	2	2	no	—
87	C₂×C₄×C₅	—	4	2	no	—
88	C₂×C₄×C₅	—	4	2	no	—
89	C₂×C₄×C₅	—	4	2	no	—
90	C₂×C₄×C₅	—	4	2	no	—
91	C₂×C₄×C₅	—	4	2	no	—
92	C₂×C₄×C₅	C₂²⋊C₈	1	4	yes	—
93	C₂×C₄×C₅	C₄×Q₈	1	2	yes	—
94	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
95	C₂×C₄×C₅	#124	1	1	yes	characteristic
96	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
97	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
98	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
99	C₂×C₄×C₅	#131	1	2	yes	—
100	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
101	C₂×C₄×C₅	#137	1	2	yes	—
102	C₅×C₈	—	4	4	no	—
103	C₅×C₈	—	8	2	no	—
104	C₂²×C₈	—	2	2	no	—
105	C₂²×C₈	—	2	2	no	—
106	#125	—	2	2	no	—
107	#125	C₅×C₈	1	4	yes	complemented
108	C₂×C₄⋊C₄	—	2	2	no	—
109	C₂×C₄⋊C₄	—	2	2	no	—
110	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
111	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
112	C₂×C₄²	—	2	2	no	—
113	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
114	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
115	C₂×C₄²	C₅×D₄	1	2	yes	—
116	C₂×C₄²	C₅×D₄	1	2	yes	—
117	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
118	C₄⋊C₈	—	2	4	no	—
119	C₄⋊C₈	—	4	4	no	—
120	C₄×C₈	—	2	2	no	—
121	C₄×C₈	C₅×Q₈	1	4	yes	—
122	#95	—	2	2	no	—
123	#95	—	2	2	no	—
124	#95	—	2	2	no	—
125	C₂²×C₅	#391	1	1	yes	characteristic
126	C₂²×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
127	C₂²×C₅	#408	1	1	yes	characteristic
128	C₂²×C₅	#417	1	2	yes	—
129	C₂²×C₅	#421	1	2	yes	—
130	C₄×C₅	—	2	4	no	—
131	C₄×C₅	—	2	2	no	—
132	C₄×C₅	—	4	4	no	—
133	C₄×C₅	—	4	2	no	—
134	C₄×C₅	—	8	2	no	—
135	C₄×C₅	C₈×Q₈	1	2	yes	—
136	C₄×C₅	#446	1	2	yes	—
137	C₄×C₅	#490	1	1	yes	characteristic
138	C₄×C₅	#491	1	1	yes	characteristic
139	C₄×C₅	#501	1	2	yes	—
140	C₂²×C₄	—	2	2	no	—
141	C₂²×C₄	—	2	2	no	—
142	C₂²×C₄	C₅×C₂²⋊C₄	1	2	yes	—
143	C₂²×C₄	C₅×C₂²⋊C₄	1	2	yes	—
144	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
145	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
146	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
147	C₂×C₈	—	2	4	no	—
148	C₂×C₈	—	4	4	no	—
149	C₂×C₈	—	4	2	no	—
150	C₂×C₈	—	4	4	no	—
151	C₄⋊C₄	—	2	2	no	—
152	C₄⋊C₄	—	2	8	no	—
153	C₄⋊C₄	—	2	4	no	—
154	C₄⋊C₄	—	4	4	no	—
155	C₄⋊C₄	C₂×C₅×C₈	1	4	yes	—
156	C₄²	—	2	2	no	—
157	C₄²	—	2	2	no	—
158	C₄²	—	2	2	no	—
159	C₄²	—	4	2	no	—
160	C₄²	—	4	2	no	—
161	C₄²	—	4	2	no	—
162	C₄²	C₂×C₅×C₈	1	2	yes	—
163	C₄²	C₅×M₁₆	1	2	yes	—
164	C₄²	C₂×C₅×Q₈	1	2	yes	—
165	C₄²	C₅×#41	1	2	yes	—
166	C₂×C₅	#1477	1	2	yes	—
167	C₂×C₅	#1495	1	2	yes	—
168	C₂×C₅	#1984	1	1	yes	characteristic
169	C₂×C₅	#1992	1	1	yes	characteristic
170	C₂×C₅	#1995	1	1	yes	characteristic
171	C₂³	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	4	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	4	2	no	—
180	C₂×C₄	—	4	2	no	—
181	C₂×C₄	—	4	2	no	—
182	C₂×C₄	—	4	2	no	—
183	C₂×C₄	—	4	2	no	—
184	C₂×C₄	C₅×C₂²⋊C₈	1	4	yes	—
185	C₂×C₄	C₄×C₅×Q₈	1	2	yes	—
186	C₂×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
187	C₂×C₄	C₅×#124	1	1	yes	characteristic
188	C₂×C₄	C₅×C₄⋊Q₈	1	1	yes	characteristic
189	C₂×C₄	C₂²×C₅×C₈	1	1	yes	characteristic
190	C₂×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
191	C₂×C₄	C₅×#131	1	2	yes	—
192	C₂×C₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
193	C₂×C₄	C₅×#137	1	2	yes	—
194	C₈	—	4	4	no	—
195	C₈	—	8	2	no	—
196	C₅	#9872	1	1	yes	characteristic, direct product factor, Sylow
197	C₂²	C₅×#391	1	1	yes	characteristic
198	C₂²	C₂×C₅×C₂²⋊C₈	1	1	yes	characteristic
199	C₂²	C₅×#408	1	1	yes	characteristic
200	C₂²	C₅×#417	1	2	yes	—
201	C₂²	C₅×#421	1	2	yes	—
202	C₄	—	2	4	no	—
203	C₄	—	2	2	no	—
204	C₄	—	4	4	no	—
205	C₄	—	4	2	no	—
206	C₄	—	8	2	no	—
207	C₄	C₅×C₈×Q₈	1	2	yes	—
208	C₄	C₅×#446	1	2	yes	—
209	C₄	C₅×#490	1	1	yes	characteristic
210	C₄	C₅×#491	1	1	yes	characteristic
211	C₄	C₅×#501	1	2	yes	—
212	C₂	C₅×#1477	1	2	yes	—
213	C₂	C₅×#1495	1	2	yes	—
214	C₂	C₅×#1984	1	1	yes	characteristic
215	C₂	C₅×#1992	1	1	yes	characteristic
216	C₂	C₅×#1995	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₅	C₂×D₄
commutator	C₂×C₄	C₂²×C₅×C₈
Frattini, cosocle	C₂×C₄²	C₂³×C₅
socle, prime radical	C₂³×C₅	C₂×C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅ ◁ C₄³×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
16. #9872	C₅	direct (N × H)
20. C₅×#125	C₈	semidirect
107. #125	C₅×C₈	semidirect
196. C₅	#9872	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₄×C₅×#125	2
C₅×#1842	2
C₅×#1744	2
C₅×#1279	2
#9872	5

Subgroups — C5×#9872 #11932592

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#9872 #11932592