Subgroups — C₅×#8780 #11931500

Order 1280.

310 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#1726	C₂	1	1	yes	maximal, characteristic
2	C₅×#1564	C₂	1	2	yes	maximal
3	C₅×#1281	C₂	1	2	yes	maximal
4	C₅×#1274	C₂	1	2	yes	maximal
5	C₅×#431	C₂²	1	4	yes	—
6	C₂×C₅×#97	C₄	1	1	yes	characteristic
7	C₂×C₅×#97	C₄	1	1	yes	characteristic
8	C₂×C₅×#97	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₅×C₈	C₄	1	1	yes	characteristic
10	C₂×C₄×C₅×C₈	C₄	1	1	yes	characteristic
11	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₅×#376	C₂²	1	1	yes	characteristic
13	C₅×#335	—	2	2	no	—
14	C₅×#334	—	2	2	no	—
15	#8780	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
16	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
23	C₅×C₄⋊C₈	—	2	4	no	—
24	C₅×C₂²⋊C₈	—	2	4	no	—
25	C₅×#97	—	2	2	no	—
26	C₅×#97	—	2	2	no	—
27	C₅×#97	—	2	2	no	—
28	C₄×C₅×C₈	—	2	2	no	—
29	C₄×C₅×C₈	—	2	4	no	—
30	C₄×C₅×C₈	C₈	1	4	yes	complemented
31	C₄×C₅×C₈	D₄	1	1	yes	characteristic
32	C₄×C₅×C₈	D₄	1	1	yes	characteristic
33	C₄×C₅×C₈	Q₈	1	1	yes	characteristic
34	C₄×C₅×C₈	Q₈	1	1	yes	characteristic
35	C₅×#95	—	2	2	no	—
36	#1726	C₂×C₅	1	1	yes	characteristic
37	#1564	C₂×C₅	1	2	yes	—
38	#1281	C₂×C₅	1	2	yes	—
39	#1274	C₂×C₅	1	2	yes	—
40	C₂²×C₄×C₅	—	2	1	no	—
41	C₂²×C₄×C₅	—	2	1	no	—
42	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₅	#41	1	2	yes	—
44	C₂×C₅×C₈	—	2	1	no	—
45	C₂×C₅×C₈	—	2	1	no	—
46	C₂×C₅×C₈	—	2	1	no	—
47	C₂×C₅×C₈	—	2	1	no	—
48	C₂×C₅×C₈	—	2	1	no	—
49	C₂×C₅×C₈	—	2	1	no	—
50	C₂×C₅×C₈	—	4	4	no	—
51	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₅×C₈	C₂×C₈	1	2	yes	—
56	C₂×C₅×C₈	C₂×C₈	1	2	yes	—
57	C₂×C₅×C₈	M₁₆	1	2	yes	—
58	C₂×C₅×C₈	M₁₆	1	2	yes	—
59	C₄²×C₅	—	2	1	no	—
60	C₄²×C₅	—	2	1	no	—
61	C₄²×C₅	C₂×C₈	1	2	yes	—
62	C₄²×C₅	M₁₆	1	2	yes	—
63	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
64	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
65	C₄²×C₅	#41	1	2	yes	—
66	#431	C₂²×C₅	1	4	yes	—
67	C₂×#97	C₄×C₅	1	1	yes	characteristic
68	C₂×#97	C₄×C₅	1	1	yes	characteristic
69	C₂×#97	C₂²×C₅	1	1	yes	characteristic
70	C₂×C₄×C₈	C₄×C₅	1	1	yes	characteristic
71	C₂×C₄×C₈	C₄×C₅	1	1	yes	characteristic
72	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
73	#376	C₂²×C₅	1	1	yes	characteristic
74	#335	—	2	2	no	—
75	#334	—	2	2	no	—
76	C₂³×C₅	#117	1	1	yes	characteristic
77	C₂×C₄×C₅	—	2	2	no	—
78	C₂×C₄×C₅	—	2	2	no	—
79	C₂×C₄×C₅	—	4	2	no	—
80	C₂×C₄×C₅	—	4	2	no	—
81	C₂×C₄×C₅	—	4	2	no	—
82	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₅	#117	1	1	yes	characteristic
84	C₂×C₄×C₅	#124	1	2	yes	—
85	C₂×C₄×C₅	#125	1	2	yes	—
86	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
87	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
88	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
89	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
90	C₅×C₈	—	2	1	no	—
91	C₅×C₈	—	2	1	no	—
92	C₅×C₈	—	2	2	no	—
93	C₅×C₈	—	2	2	no	—
94	C₅×C₈	—	8	4	no	—
95	C₅×C₈	C₄⋊C₈	1	2	yes	—
96	C₅×C₈	C₄⋊C₈	1	2	yes	—
97	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
98	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
99	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
100	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
101	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
102	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
103	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
104	C₄⋊C₈	—	2	4	no	—
105	C₂²⋊C₈	—	2	4	no	—
106	#97	—	2	2	no	—
107	#97	—	2	2	no	—
108	#97	—	2	2	no	—
109	C₄×C₈	—	2	2	no	—
110	C₄×C₈	—	2	4	no	—
111	C₄×C₈	C₅×C₈	1	4	yes	complemented
112	C₄×C₈	C₅×D₄	1	1	yes	characteristic
113	C₄×C₈	C₅×D₄	1	1	yes	characteristic
114	C₄×C₈	C₅×Q₈	1	1	yes	characteristic
115	C₄×C₈	C₅×Q₈	1	1	yes	characteristic
116	#95	—	2	2	no	—
117	C₂²×C₅	—	2	2	no	—
118	C₂²×C₅	#382	1	1	yes	characteristic
119	C₂²×C₅	#431	1	1	yes	characteristic
120	C₂²×C₅	#432	1	1	yes	characteristic
121	C₄×C₅	—	2	2	no	—
122	C₄×C₅	—	2	2	no	—
123	C₄×C₅	—	4	2	no	—
124	C₄×C₅	—	4	2	no	—
125	C₄×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
126	C₄×C₅	#423	1	1	yes	characteristic
127	C₄×C₅	#431	1	1	yes	characteristic
128	C₄×C₅	#432	1	1	yes	characteristic
129	C₂²×C₄	—	2	1	no	—
130	C₂²×C₄	—	2	1	no	—
131	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
132	C₂²×C₄	C₅×#41	1	2	yes	—
133	C₂×C₈	—	2	1	no	—
134	C₂×C₈	—	2	1	no	—
135	C₂×C₈	—	2	1	no	—
136	C₂×C₈	—	2	1	no	—
137	C₂×C₈	—	2	1	no	—
138	C₂×C₈	—	2	1	no	—
139	C₂×C₈	—	4	4	no	—
140	C₂×C₈	C₅×C₄⋊C₄	1	1	yes	characteristic
141	C₂×C₈	C₅×C₄⋊C₄	1	1	yes	characteristic
142	C₂×C₈	C₅×C₄⋊C₄	1	1	yes	characteristic
143	C₂×C₈	C₅×C₄⋊C₄	1	1	yes	characteristic
144	C₂×C₈	C₂×C₅×C₈	1	2	yes	—
145	C₂×C₈	C₂×C₅×C₈	1	2	yes	—
146	C₂×C₈	C₅×M₁₆	1	2	yes	—
147	C₂×C₈	C₅×M₁₆	1	2	yes	—
148	C₄²	—	2	1	no	—
149	C₄²	—	2	1	no	—
150	C₄²	C₂×C₅×C₈	1	2	yes	—
151	C₄²	C₅×M₁₆	1	2	yes	—
152	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
153	C₄²	C₂×C₅×Q₈	1	1	yes	characteristic
154	C₄²	C₅×#41	1	2	yes	—
155	C₂×C₅	—	2	1	no	—
156	C₂×C₅	—	2	1	no	—
157	C₂×C₅	#1831	1	1	yes	characteristic
158	C₂×C₅	#1833	1	2	yes	—
159	C₂³	C₅×#117	1	1	yes	characteristic
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	4	2	no	—
163	C₂×C₄	—	4	2	no	—
164	C₂×C₄	—	4	2	no	—
165	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
166	C₂×C₄	C₅×#117	1	1	yes	characteristic
167	C₂×C₄	C₅×#124	1	2	yes	—
168	C₂×C₄	C₅×#125	1	2	yes	—
169	C₂×C₄	C₂²×C₅×C₈	1	1	yes	characteristic
170	C₂×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
171	C₂×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
172	C₂×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
173	C₈	—	2	1	no	—
174	C₈	—	2	1	no	—
175	C₈	—	2	2	no	—
176	C₈	—	2	2	no	—
177	C₈	—	8	4	no	—
178	C₈	C₅×C₄⋊C₈	1	2	yes	—
179	C₈	C₅×C₄⋊C₈	1	2	yes	—
180	C₅	#8780	1	1	yes	characteristic, direct product factor, Sylow
181	C₂²	—	2	2	no	—
182	C₂²	C₅×#382	1	1	yes	characteristic
183	C₂²	C₅×#431	1	1	yes	characteristic
184	C₂²	C₅×#432	1	1	yes	characteristic
185	C₄	—	2	2	no	—
186	C₄	—	2	2	no	—
187	C₄	—	4	2	no	—
188	C₄	—	4	2	no	—
189	C₄	C₂×C₅×C₄⋊C₈	1	1	yes	characteristic
190	C₄	C₅×#423	1	1	yes	characteristic
191	C₄	C₅×#431	1	1	yes	characteristic
192	C₄	C₅×#432	1	1	yes	characteristic
193	C₂	—	2	1	no	—
194	C₂	—	2	1	no	—
195	C₂	C₅×#1831	1	1	yes	characteristic
196	C₂	C₅×#1833	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₅	#117
commutator	C₂×C₄	C₂²×C₅×C₈
Frattini, cosocle	C₂×C₄²	C₂³×C₅
socle	C₂²×C₅	#382
prime radical	C₂³×C₅	#117
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₅×#376 ◁ G

Complements

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

N	H ≅ G/N	type
15. #8780	C₅	direct (N × H)
30. C₄×C₅×C₈	C₈	semidirect
111. C₄×C₈	C₅×C₈	semidirect
180. C₅	#8780	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅×#1726	2
C₅×#1564	2
C₅×#1281	2
C₅×#1274	2
#8780	5

Subgroups — C5×#8780 #11931500

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#8780 #11931500