Subgroups — C₅×#8811 #11931531

Order 1280.

362 conjugacy classes of subgroups.

The subgroup lattice has 150 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#1831	C₂	1	2	yes	maximal
2	C₅×#1744	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#335	C₂	1	2	yes	maximal
4	C₂×C₅×#334	C₂	1	2	yes	maximal
5	C₂×C₅×C₄⋊C₈	C₂²	1	4	yes	—
6	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
7	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₄³×C₅	C₂²	1	1	yes	characteristic
9	C₅×#336	—	2	4	no	—
10	C₅×#336	C₄	1	4	yes	—
11	C₅×#335	—	2	4	no	—
12	C₅×#334	—	2	4	no	—
13	#8811	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
14	C₂²×C₅×C₈	—	2	4	no	—
15	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
16	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
17	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
18	C₂×C₄²×C₅	D₄	1	2	yes	—
19	C₂×C₄²×C₅	Q₈	1	2	yes	—
20	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
21	C₅×C₄⋊C₈	—	2	4	no	—
22	C₅×C₄⋊C₈	—	2	4	no	—
23	C₄×C₅×C₈	D₄	1	2	yes	—
24	C₄×C₅×C₈	D₄	1	2	yes	—
25	C₄×C₅×C₈	Q₈	1	2	yes	—
26	C₄×C₅×C₈	Q₈	1	2	yes	—
27	#1831	C₂×C₅	1	2	yes	—
28	#1744	C₂×C₅	1	1	yes	characteristic
29	C₂×#335	C₂×C₅	1	2	yes	—
30	C₂×#334	C₂×C₅	1	2	yes	—
31	C₂²×C₄×C₅	C₄⋊C₄	1	4	yes	—
32	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
34	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
35	C₂×C₅×C₈	—	2	4	no	—
36	C₂×C₅×C₈	—	2	4	no	—
37	C₂×C₅×C₈	—	4	4	no	—
38	C₂×C₅×C₈	—	4	8	no	—
39	C₂×C₅×C₈	C₄⋊C₄	1	4	yes	—
40	C₂×C₅×C₈	C₄⋊C₄	1	4	yes	—
41	C₄²×C₅	—	2	4	no	—
42	C₄²×C₅	—	2	4	no	—
43	C₄²×C₅	—	4	2	no	—
44	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
45	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
46	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
47	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₄⋊C₈	C₂²×C₅	1	4	yes	—
49	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
50	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
51	C₄³	C₂²×C₅	1	1	yes	characteristic
52	#336	—	2	4	no	—
53	#336	C₄×C₅	1	4	yes	—
54	#335	—	2	4	no	—
55	#334	—	2	4	no	—
56	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₄×C₅	—	2	2	no	—
58	C₂×C₄×C₅	—	2	4	no	—
59	C₂×C₄×C₅	—	2	4	no	—
60	C₂×C₄×C₅	—	2	4	no	—
61	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₄×C₅	#127	1	1	yes	characteristic
64	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
65	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
66	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
67	C₂×C₄×C₅	#131	1	4	yes	—
68	C₂×C₄×C₅	#135	1	4	yes	—
69	C₅×C₈	—	2	4	no	—
70	C₅×C₈	—	2	4	no	—
71	C₅×C₈	—	8	4	no	—
72	C₂²×C₈	—	2	4	no	—
73	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
74	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
75	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
76	C₂×C₄²	C₅×D₄	1	2	yes	—
77	C₂×C₄²	C₅×Q₈	1	2	yes	—
78	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
79	C₄⋊C₈	—	2	4	no	—
80	C₄⋊C₈	—	2	4	no	—
81	C₄×C₈	C₅×D₄	1	2	yes	—
82	C₄×C₈	C₅×D₄	1	2	yes	—
83	C₄×C₈	C₅×Q₈	1	2	yes	—
84	C₄×C₈	C₅×Q₈	1	2	yes	—
85	C₂²×C₅	#384	1	1	yes	characteristic
86	C₂²×C₅	#424	1	2	yes	—
87	C₂²×C₅	#427	1	2	yes	—
88	C₂²×C₅	#430	1	2	yes	—
89	C₄×C₅	—	2	1	no	—
90	C₄×C₅	—	2	1	no	—
91	C₄×C₅	—	4	4	no	—
92	C₄×C₅	#424	1	2	yes	—
93	C₄×C₅	#424	1	2	yes	—
94	C₄×C₅	#495	1	2	yes	—
95	C₄×C₅	#499	1	2	yes	—
96	C₂²×C₄	C₅×C₄⋊C₄	1	4	yes	—
97	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
98	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
99	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
100	C₂×C₈	—	2	4	no	—
101	C₂×C₈	—	2	4	no	—
102	C₂×C₈	—	4	4	no	—
103	C₂×C₈	—	4	8	no	—
104	C₂×C₈	C₅×C₄⋊C₄	1	4	yes	—
105	C₂×C₈	C₅×C₄⋊C₄	1	4	yes	—
106	C₄²	—	2	4	no	—
107	C₄²	—	2	4	no	—
108	C₄²	—	4	2	no	—
109	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
110	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
111	C₄²	C₂×C₅×Q₈	1	1	yes	characteristic
112	C₄²	C₂×C₅×Q₈	1	1	yes	characteristic
113	C₂×C₅	#1572	1	4	yes	—
114	C₂×C₅	#1843	1	1	yes	characteristic
115	C₂×C₅	#1846	1	1	yes	characteristic
116	C₂×C₅	#1849	1	1	yes	characteristic
117	C₂³	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	2	4	no	—
120	C₂×C₄	—	2	4	no	—
121	C₂×C₄	—	2	4	no	—
122	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
123	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
124	C₂×C₄	C₅×#127	1	1	yes	characteristic
125	C₂×C₄	C₅×C₄⋊Q₈	1	1	yes	characteristic
126	C₂×C₄	C₅×C₄⋊Q₈	1	1	yes	characteristic
127	C₂×C₄	C₅×C₄⋊Q₈	1	1	yes	characteristic
128	C₂×C₄	C₅×#131	1	4	yes	—
129	C₂×C₄	C₅×#135	1	4	yes	—
130	C₈	—	2	4	no	—
131	C₈	—	2	4	no	—
132	C₈	—	8	4	no	—
133	C₅	#8811	1	1	yes	characteristic, direct product factor, Sylow
134	C₂²	C₅×#384	1	1	yes	characteristic
135	C₂²	C₅×#424	1	2	yes	—
136	C₂²	C₅×#427	1	2	yes	—
137	C₂²	C₅×#430	1	2	yes	—
138	C₄	—	2	1	no	—
139	C₄	—	2	1	no	—
140	C₄	—	4	4	no	—
141	C₄	C₅×#424	1	2	yes	—
142	C₄	C₅×#424	1	2	yes	—
143	C₄	C₅×#495	1	2	yes	—
144	C₄	C₅×#499	1	2	yes	—
145	C₂	C₅×#1572	1	4	yes	—
146	C₂	C₅×#1843	1	1	yes	characteristic
147	C₂	C₅×#1846	1	1	yes	characteristic
148	C₂	C₅×#1849	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₅	C₂×D₄
commutator	C₂²×C₄	C₂²×C₄×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
socle, prime radical	C₂³×C₅	C₂×C₄⋊C₄
LCS	G ▷ C₂²×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
13. #8811	C₅	direct (N × H)
133. C₅	#8811	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×#1831	2
C₅×#1744	2
C₂×C₅×#335	2
C₂×C₅×#334	2
#8811	5

Subgroups — C5×#8811 #11931531

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#8811 #11931531