Subgroups — C₅×#8781 #11931501

Order 1280.

310 conjugacy classes of subgroups.

The subgroup lattice has 186 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	#8781	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	C₈	1	4	yes	complemented
28	C₄×C₅×C₈	—	2	2	no	—
29	C₄×C₅×C₈	—	2	2	no	—
30	C₄×C₅×C₈	—	2	4	no	—
31	C₄×C₅×C₈	D₄	1	2	yes	—
32	C₄×C₅×C₈	Q₈	1	2	yes	—
33	C₅×#95	—	2	2	no	—
34	#1726	C₂×C₅	1	1	yes	characteristic
35	#1564	C₂×C₅	1	2	yes	—
36	#1281	C₂×C₅	1	2	yes	—
37	#1274	C₂×C₅	1	2	yes	—
38	C₂²×C₄×C₅	—	2	1	no	—
39	C₂²×C₄×C₅	—	2	1	no	—
40	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₅	#41	1	2	yes	—
42	C₂×C₅×C₈	—	2	1	no	—
43	C₂×C₅×C₈	—	2	1	no	—
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	—	4	4	no	—
49	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
50	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
51	C₂×C₅×C₈	C₂×C₈	1	2	yes	—
52	C₂×C₅×C₈	C₂×C₈	1	2	yes	—
53	C₂×C₅×C₈	M₁₆	1	2	yes	—
54	C₂×C₅×C₈	M₁₆	1	2	yes	—
55	C₄²×C₅	—	2	1	no	—
56	C₄²×C₅	—	2	1	no	—
57	C₄²×C₅	C₂×C₈	1	2	yes	—
58	C₄²×C₅	M₁₆	1	2	yes	—
59	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
60	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
61	C₄²×C₅	#41	1	2	yes	—
62	#431	C₂²×C₅	1	4	yes	—
63	C₂×#97	C₄×C₅	1	1	yes	characteristic
64	C₂×#97	C₄×C₅	1	1	yes	characteristic
65	C₂×#97	C₂²×C₅	1	1	yes	characteristic
66	C₂×C₄×C₈	C₄×C₅	1	1	yes	characteristic
67	C₂×C₄×C₈	C₄×C₅	1	1	yes	characteristic
68	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
69	#376	C₂²×C₅	1	1	yes	characteristic
70	#335	—	2	2	no	—
71	#334	—	2	2	no	—
72	C₂³×C₅	#117	1	1	yes	characteristic
73	C₂×C₄×C₅	—	2	2	no	—
74	C₂×C₄×C₅	—	2	2	no	—
75	C₂×C₄×C₅	—	4	2	no	—
76	C₂×C₄×C₅	—	4	2	no	—
77	C₂×C₄×C₅	—	4	2	no	—
78	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₅	#117	1	1	yes	characteristic
80	C₂×C₄×C₅	#124	1	2	yes	—
81	C₂×C₄×C₅	#125	1	2	yes	—
82	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
83	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
84	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
85	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
86	C₅×C₈	—	2	2	no	—
87	C₅×C₈	—	2	2	no	—
88	C₅×C₈	—	2	2	no	—
89	C₅×C₈	—	8	4	no	—
90	C₅×C₈	C₄⋊C₈	1	4	yes	—
91	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
92	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
93	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
94	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
95	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
96	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
97	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
98	C₄⋊C₈	—	2	4	no	—
99	C₂²⋊C₈	—	2	4	no	—
100	#97	—	2	2	no	—
101	#97	—	2	2	no	—
102	#97	C₅×C₈	1	4	yes	complemented
103	C₄×C₈	—	2	2	no	—
104	C₄×C₈	—	2	2	no	—
105	C₄×C₈	—	2	4	no	—
106	C₄×C₈	C₅×D₄	1	2	yes	—
107	C₄×C₈	C₅×Q₈	1	2	yes	—
108	#95	—	2	2	no	—
109	C₂²×C₅	—	2	2	no	—
110	C₂²×C₅	#382	1	1	yes	characteristic
111	C₂²×C₅	#431	1	1	yes	characteristic
112	C₂²×C₅	#432	1	1	yes	characteristic
113	C₄×C₅	—	2	2	no	—
114	C₄×C₅	—	2	2	no	—
115	C₄×C₅	—	4	2	no	—
116	C₄×C₅	—	4	2	no	—
117	C₄×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
118	C₄×C₅	#423	1	1	yes	characteristic
119	C₄×C₅	#431	1	1	yes	characteristic
120	C₄×C₅	#432	1	1	yes	characteristic
121	C₂²×C₄	—	2	1	no	—
122	C₂²×C₄	—	2	1	no	—
123	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
124	C₂²×C₄	C₅×#41	1	2	yes	—
125	C₂×C₈	—	2	1	no	—
126	C₂×C₈	—	2	1	no	—
127	C₂×C₈	—	2	1	no	—
128	C₂×C₈	—	2	1	no	—
129	C₂×C₈	—	2	1	no	—
130	C₂×C₈	—	2	1	no	—
131	C₂×C₈	—	4	4	no	—
132	C₂×C₈	C₅×C₄⋊C₄	1	2	yes	—
133	C₂×C₈	C₅×C₄⋊C₄	1	2	yes	—
134	C₂×C₈	C₂×C₅×C₈	1	2	yes	—
135	C₂×C₈	C₂×C₅×C₈	1	2	yes	—
136	C₂×C₈	C₅×M₁₆	1	2	yes	—
137	C₂×C₈	C₅×M₁₆	1	2	yes	—
138	C₄²	—	2	1	no	—
139	C₄²	—	2	1	no	—
140	C₄²	C₂×C₅×C₈	1	2	yes	—
141	C₄²	C₅×M₁₆	1	2	yes	—
142	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
143	C₄²	C₂×C₅×Q₈	1	1	yes	characteristic
144	C₄²	C₅×#41	1	2	yes	—
145	C₂×C₅	—	2	1	no	—
146	C₂×C₅	—	2	1	no	—
147	C₂×C₅	#1831	1	1	yes	characteristic
148	C₂×C₅	#1833	1	2	yes	—
149	C₂³	C₅×#117	1	1	yes	characteristic
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	4	2	no	—
153	C₂×C₄	—	4	2	no	—
154	C₂×C₄	—	4	2	no	—
155	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
156	C₂×C₄	C₅×#117	1	1	yes	characteristic
157	C₂×C₄	C₅×#124	1	2	yes	—
158	C₂×C₄	C₅×#125	1	2	yes	—
159	C₂×C₄	C₂²×C₅×C₈	1	1	yes	characteristic
160	C₂×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
161	C₂×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
162	C₂×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
163	C₈	—	2	2	no	—
164	C₈	—	2	2	no	—
165	C₈	—	2	2	no	—
166	C₈	—	8	4	no	—
167	C₈	C₅×C₄⋊C₈	1	4	yes	—
168	C₅	#8781	1	1	yes	characteristic, direct product factor, Sylow
169	C₂²	—	2	2	no	—
170	C₂²	C₅×#382	1	1	yes	characteristic
171	C₂²	C₅×#431	1	1	yes	characteristic
172	C₂²	C₅×#432	1	1	yes	characteristic
173	C₄	—	2	2	no	—
174	C₄	—	2	2	no	—
175	C₄	—	4	2	no	—
176	C₄	—	4	2	no	—
177	C₄	C₂×C₅×C₄⋊C₈	1	1	yes	characteristic
178	C₄	C₅×#423	1	1	yes	characteristic
179	C₄	C₅×#431	1	1	yes	characteristic
180	C₄	C₅×#432	1	1	yes	characteristic
181	C₂	—	2	1	no	—
182	C₂	—	2	1	no	—
183	C₂	C₅×#1831	1	1	yes	characteristic
184	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. #8781	C₅	direct (N × H)
27. C₅×#97	C₈	semidirect
102. #97	C₅×C₈	semidirect
168. C₅	#8781	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
#8781	5

Subgroups — C5×#8781 #11931501

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#8781 #11931501