Subgroups — C₅×#8786 #11931506

Order 1280.

378 conjugacy classes of subgroups.

The subgroup lattice has 190 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#1842	C₂	1	1	yes	maximal, characteristic
2	C₅×#1831	C₂	1	1	yes	maximal, characteristic
3	C₄×C₅×#97	C₂	1	1	yes	maximal, characteristic
4	C₅×#1277	C₂	1	4	yes	maximal
5	C₂×C₅×C₄⋊C₈	—	2	2	no	—
6	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
7	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
8	C₂×C₅×#97	C₄	1	2	yes	—
9	C₂×C₅×#97	C₄	1	1	yes	characteristic
10	C₂×C₅×#97	C₄	1	1	yes	characteristic
11	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
12	C₄³×C₅	C₂²	1	1	yes	characteristic
13	C₅×#336	—	2	2	no	—
14	#8786	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
15	C₂²×C₅×C₈	—	2	2	no	—
16	C₂²×C₅×C₈	—	2	2	no	—
17	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
18	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
19	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄²×C₅	D₄	1	2	yes	—
22	C₂×C₄²×C₅	Q₈	1	2	yes	—
23	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
24	C₅×C₄⋊C₈	—	2	4	no	—
25	C₅×C₄⋊C₈	—	2	4	no	—
26	C₅×C₄⋊C₈	—	4	4	no	—
27	C₅×#97	—	2	4	no	—
28	C₅×#97	—	2	2	no	—
29	C₅×#97	C₈	1	4	yes	complemented
30	C₄×C₅×C₈	—	2	2	no	—
31	C₄×C₅×C₈	—	2	2	no	—
32	#1842	C₂×C₅	1	1	yes	characteristic
33	#1831	C₂×C₅	1	1	yes	characteristic
34	C₄×#97	C₂×C₅	1	1	yes	characteristic
35	#1277	C₂×C₅	1	4	yes	—
36	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
37	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
38	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₅×C₈	—	2	2	no	—
42	C₂×C₅×C₈	—	2	2	no	—
43	C₂×C₅×C₈	—	2	2	no	—
44	C₂×C₅×C₈	—	2	2	no	—
45	C₂×C₅×C₈	—	4	2	no	—
46	C₂×C₅×C₈	—	4	2	no	—
47	C₂×C₅×C₈	—	4	2	no	—
48	C₂×C₅×C₈	—	4	4	no	—
49	C₂×C₅×C₈	—	4	2	no	—
50	C₂×C₅×C₈	C₂×C₈	1	4	yes	—
51	C₂×C₅×C₈	M₁₆	1	4	yes	—
52	C₄²×C₅	—	2	2	no	—
53	C₄²×C₅	—	2	2	no	—
54	C₄²×C₅	—	2	2	no	—
55	C₄²×C₅	—	4	2	no	—
56	C₄²×C₅	C₂×C₈	1	2	yes	—
57	C₄²×C₅	M₁₆	1	2	yes	—
58	C₄²×C₅	#41	1	2	yes	—
59	C₄²×C₅	#41	1	2	yes	—
60	C₂×C₄⋊C₈	—	2	2	no	—
61	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
62	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
63	C₂×#97	C₄×C₅	1	2	yes	—
64	C₂×#97	C₄×C₅	1	1	yes	characteristic
65	C₂×#97	C₄×C₅	1	1	yes	characteristic
66	C₂×C₄×C₈	C₂²×C₅	1	2	yes	—
67	C₄³	C₂²×C₅	1	1	yes	characteristic
68	#336	—	2	2	no	—
69	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₅	—	2	2	no	—
71	C₂×C₄×C₅	—	2	2	no	—
72	C₂×C₄×C₅	—	2	2	no	—
73	C₂×C₄×C₅	—	2	2	no	—
74	C₂×C₄×C₅	—	2	2	no	—
75	C₂×C₄×C₅	—	2	2	no	—
76	C₂×C₄×C₅	C₄⋊C₈	1	4	yes	—
77	C₂×C₄×C₅	#117	1	2	yes	—
78	C₂×C₄×C₅	#124	1	1	yes	characteristic
79	C₂×C₄×C₅	#124	1	1	yes	characteristic
80	C₂×C₄×C₅	#125	1	2	yes	—
81	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
82	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
83	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
84	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
85	C₂×C₄×C₅	#137	1	2	yes	—
86	C₅×C₈	—	2	4	no	—
87	C₅×C₈	—	2	4	no	—
88	C₅×C₈	—	8	2	no	—
89	C₅×C₈	—	8	2	no	—
90	C₂²×C₈	—	2	2	no	—
91	C₂²×C₈	—	2	2	no	—
92	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
93	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
94	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
95	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
96	C₂×C₄²	C₅×D₄	1	2	yes	—
97	C₂×C₄²	C₅×Q₈	1	2	yes	—
98	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
99	C₄⋊C₈	—	2	4	no	—
100	C₄⋊C₈	—	2	4	no	—
101	C₄⋊C₈	—	4	4	no	—
102	#97	—	2	4	no	—
103	#97	—	2	2	no	—
104	#97	C₅×C₈	1	4	yes	complemented
105	C₄×C₈	—	2	2	no	—
106	C₄×C₈	—	2	2	no	—
107	C₂²×C₅	#382	1	1	yes	characteristic
108	C₂²×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
109	C₂²×C₅	#423	1	1	yes	characteristic
110	C₂²×C₅	#428	1	2	yes	—
111	C₂²×C₅	#430	1	2	yes	—
112	C₄×C₅	—	2	4	no	—
113	C₄×C₅	—	2	2	no	—
114	C₄×C₅	—	4	2	no	—
115	C₄×C₅	#431	1	2	yes	—
116	C₄×C₅	#432	1	2	yes	—
117	C₄×C₅	#489	1	2	yes	—
118	C₄×C₅	#490	1	1	yes	characteristic
119	C₄×C₅	#491	1	1	yes	characteristic
120	C₂²×C₄	C₅×C₄⋊C₄	1	2	yes	—
121	C₂²×C₄	C₅×C₄⋊C₄	1	2	yes	—
122	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
123	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
124	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
125	C₂×C₈	—	2	2	no	—
126	C₂×C₈	—	2	2	no	—
127	C₂×C₈	—	2	2	no	—
128	C₂×C₈	—	2	2	no	—
129	C₂×C₈	—	4	2	no	—
130	C₂×C₈	—	4	2	no	—
131	C₂×C₈	—	4	4	no	—
132	C₂×C₈	—	4	2	no	—
133	C₂×C₈	—	4	2	no	—
134	C₂×C₈	C₂×C₅×C₈	1	4	yes	—
135	C₂×C₈	C₅×M₁₆	1	4	yes	—
136	C₄²	—	2	2	no	—
137	C₄²	—	2	2	no	—
138	C₄²	—	2	2	no	—
139	C₄²	—	4	2	no	—
140	C₄²	C₂×C₅×C₈	1	2	yes	—
141	C₄²	C₅×M₁₆	1	2	yes	—
142	C₄²	C₅×#41	1	2	yes	—
143	C₄²	C₅×#41	1	2	yes	—
144	C₂×C₅	#1565	1	2	yes	—
145	C₂×C₅	#1569	1	2	yes	—
146	C₂×C₅	#1831	1	1	yes	characteristic
147	C₂×C₅	#1836	1	1	yes	characteristic
148	C₂×C₅	#1838	1	1	yes	characteristic
149	C₂³	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	C₅×C₄⋊C₈	1	4	yes	—
157	C₂×C₄	C₅×#117	1	2	yes	—
158	C₂×C₄	C₅×#124	1	1	yes	characteristic
159	C₂×C₄	C₅×#124	1	1	yes	characteristic
160	C₂×C₄	C₅×#125	1	2	yes	—
161	C₂×C₄	C₂²×C₅×C₈	1	1	yes	characteristic
162	C₂×C₄	C₂×C₅×M₁₆	1	2	yes	—
163	C₂×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
164	C₂×C₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
165	C₂×C₄	C₅×#137	1	2	yes	—
166	C₈	—	2	4	no	—
167	C₈	—	2	4	no	—
168	C₈	—	8	2	no	—
169	C₈	—	8	2	no	—
170	C₅	#8786	1	1	yes	characteristic, direct product factor, Sylow
171	C₂²	C₅×#382	1	1	yes	characteristic
172	C₂²	C₂×C₅×C₄⋊C₈	1	1	yes	characteristic
173	C₂²	C₅×#423	1	1	yes	characteristic
174	C₂²	C₅×#428	1	2	yes	—
175	C₂²	C₅×#430	1	2	yes	—
176	C₄	—	2	4	no	—
177	C₄	—	2	2	no	—
178	C₄	—	4	2	no	—
179	C₄	C₅×#431	1	2	yes	—
180	C₄	C₅×#432	1	2	yes	—
181	C₄	C₅×#489	1	2	yes	—
182	C₄	C₅×#490	1	1	yes	characteristic
183	C₄	C₅×#491	1	1	yes	characteristic
184	C₂	C₅×#1565	1	2	yes	—
185	C₂	C₅×#1569	1	2	yes	—
186	C₂	C₅×#1831	1	1	yes	characteristic
187	C₂	C₅×#1836	1	1	yes	characteristic
188	C₂	C₅×#1838	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×C₄⋊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
14. #8786	C₅	direct (N × H)
29. C₅×#97	C₈	semidirect
104. #97	C₅×C₈	semidirect
170. C₅	#8786	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×#1842	2
C₅×#1831	2
C₄×C₅×#97	2
C₅×#1277	2
#8786	5

Subgroups — C5×#8786 #11931506

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#8786 #11931506