Subgroups — C₅×#13281 #11936001

Order 1280.

366 conjugacy classes of subgroups.

The subgroup lattice has 172 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#499	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₅×C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₅×#1846	C₂	1	1	yes	maximal, characteristic
4	C₅×#1380	C₂	1	4	yes	maximal
5	C₂×C₅×#125	—	2	1	no	—
6	C₂²×C₅×C₁₆	C₂²	1	2	yes	—
7	C₅×#499	—	2	2	no	—
8	C₅×#499	C₄	1	4	yes	complemented
9	C₂×C₅×#107	C₂²	1	2	yes	—
10	C₂×C₅×#107	C₂²	1	2	yes	—
11	C₂×C₅×#106	—	2	1	no	—
12	C₂×C₅×#106	—	2	1	no	—
13	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
14	C₅×#382	—	2	1	no	—
15	C₄×C₅×C₁₆	—	2	2	no	—
16	#13281	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
17	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
18	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
19	C₅×#125	—	2	4	no	—
20	C₅×#125	—	4	2	no	—
21	C₂×C₅×C₄⋊C₄	—	2	2	no	—
22	C₂×C₅×C₄⋊C₄	—	2	2	no	—
23	C₂×C₅×C₄⋊C₄	—	4	2	no	—
24	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
25	C₂×C₅×C₁₆	—	2	2	no	—
26	C₂×C₅×C₁₆	—	2	2	no	—
27	C₂×C₅×C₁₆	Q₈	1	4	yes	—
28	C₅×#107	—	2	2	no	—
29	C₅×#107	—	2	4	no	—
30	C₅×#107	C₂×C₄	1	4	yes	—
31	C₅×#106	—	2	4	no	—
32	C₅×#106	—	4	2	no	—
33	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
34	C₄×C₅×C₈	D₄	1	2	yes	—
35	C₅×#95	—	4	2	no	—
36	C₂×#499	C₂×C₅	1	1	yes	characteristic
37	C₂×C₄×C₁₆	C₂×C₅	1	1	yes	characteristic
38	#1846	C₂×C₅	1	1	yes	characteristic
39	#1380	C₂×C₅	1	4	yes	—
40	C₂²×C₄×C₅	—	4	2	no	—
41	C₂²×C₄×C₅	—	4	2	no	—
42	C₂²×C₄×C₅	D₈	1	2	yes	—
43	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
44	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
45	C₂×C₅×C₈	SD₁₆	1	2	yes	—
46	C₂×C₅×C₈	Q₁₆	1	2	yes	—
47	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
49	C₂×C₅×C₈	C₂×Q₈	1	2	yes	—
50	C₂×C₅×C₈	#41	1	2	yes	—
51	C₅×C₄⋊C₄	—	2	4	no	—
52	C₅×C₄⋊C₄	—	4	8	no	—
53	C₅×C₄⋊C₄	—	4	2	no	—
54	C₅×C₄⋊C₄	—	4	2	no	—
55	C₅×C₄⋊C₄	—	4	2	no	—
56	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
57	C₄²×C₅	SD₁₆	1	2	yes	—
58	C₅×C₁₆	—	2	4	no	—
59	C₂×#125	—	2	1	no	—
60	C₂²×C₁₆	C₂²×C₅	1	2	yes	—
61	#499	—	2	2	no	—
62	#499	C₄×C₅	1	4	yes	complemented
63	C₂×#107	C₂²×C₅	1	2	yes	—
64	C₂×#107	C₂²×C₅	1	2	yes	—
65	C₂×#106	—	2	1	no	—
66	C₂×#106	—	2	1	no	—
67	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
68	#382	—	2	1	no	—
69	C₄×C₁₆	—	2	2	no	—
70	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
71	C₂×C₄×C₅	—	2	4	no	—
72	C₂×C₄×C₅	—	4	4	no	—
73	C₂×C₄×C₅	—	8	4	no	—
74	C₂×C₄×C₅	—	8	2	no	—
75	C₂×C₄×C₅	—	8	4	no	—
76	C₂×C₄×C₅	#102	1	4	yes	—
77	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₅	#124	1	1	yes	characteristic
79	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
80	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
81	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
82	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
83	C₅×C₈	#103	1	4	yes	—
84	C₅×C₈	C₄×Q₈	1	2	yes	—
85	C₅×C₈	C₂²⋊Q₈	1	2	yes	—
86	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
87	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
88	#125	—	2	4	no	—
89	#125	—	4	2	no	—
90	C₂×C₄⋊C₄	—	2	2	no	—
91	C₂×C₄⋊C₄	—	2	2	no	—
92	C₂×C₄⋊C₄	—	4	2	no	—
93	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
94	C₂×C₁₆	—	2	2	no	—
95	C₂×C₁₆	—	2	2	no	—
96	C₂×C₁₆	C₅×Q₈	1	4	yes	—
97	#107	—	2	2	no	—
98	#107	—	2	4	no	—
99	#107	C₂×C₄×C₅	1	4	yes	—
100	#106	—	2	4	no	—
101	#106	—	4	2	no	—
102	C₄×C₈	C₂×C₄×C₅	1	2	yes	—
103	C₄×C₈	C₅×D₄	1	2	yes	—
104	#95	—	4	2	no	—
105	C₂²×C₅	C₂×#102	1	1	yes	characteristic
106	C₂²×C₅	#486	1	1	yes	characteristic
107	C₂²×C₅	#500	1	1	yes	characteristic
108	C₂²×C₅	#508	1	4	yes	—
109	C₄×C₅	—	2	4	no	—
110	C₄×C₅	—	8	4	no	—
111	C₄×C₅	—	16	2	no	—
112	C₄×C₅	#391	1	1	yes	characteristic
113	C₄×C₅	C₂×#103	1	1	yes	characteristic
114	C₄×C₅	#487	1	1	yes	characteristic
115	C₄×C₅	#498	1	1	yes	characteristic
116	C₂²×C₄	—	4	2	no	—
117	C₂²×C₄	—	4	2	no	—
118	C₂²×C₄	C₅×D₈	1	2	yes	—
119	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
120	C₂×C₈	C₅×C₂²⋊C₄	1	2	yes	—
121	C₂×C₈	C₅×SD₁₆	1	2	yes	—
122	C₂×C₈	C₅×Q₁₆	1	2	yes	—
123	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
124	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
125	C₂×C₈	C₂×C₅×Q₈	1	2	yes	—
126	C₂×C₈	C₅×#41	1	2	yes	—
127	C₄⋊C₄	—	2	4	no	—
128	C₄⋊C₄	—	4	2	no	—
129	C₄⋊C₄	—	4	8	no	—
130	C₄⋊C₄	—	4	2	no	—
131	C₄⋊C₄	—	4	2	no	—
132	C₄²	C₅×C₂²⋊C₄	1	2	yes	—
133	C₄²	C₅×SD₁₆	1	2	yes	—
134	C₁₆	—	2	4	no	—
135	C₂×C₅	#1990	1	1	yes	characteristic
136	C₂×C₅	#2138	1	2	yes	—
137	C₂×C₅	#2242	1	2	yes	—
138	C₂×C₅	#2253	1	2	yes	—
139	C₂³	C₂×C₅×D₈	1	1	yes	characteristic
140	C₂×C₄	—	2	4	no	—
141	C₂×C₄	—	4	4	no	—
142	C₂×C₄	—	8	4	no	—
143	C₂×C₄	—	8	2	no	—
144	C₂×C₄	—	8	4	no	—
145	C₂×C₄	C₅×#102	1	4	yes	—
146	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
147	C₂×C₄	C₅×#124	1	1	yes	characteristic
148	C₂×C₄	C₅×C₄⋊Q₈	1	1	yes	characteristic
149	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
150	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
151	C₂×C₄	C₂×C₅×Q₁₆	1	1	yes	characteristic
152	C₈	C₅×#103	1	4	yes	—
153	C₈	C₄×C₅×Q₈	1	2	yes	—
154	C₈	C₅×C₂²⋊Q₈	1	2	yes	—
155	C₅	#13281	1	1	yes	characteristic, direct product factor, Sylow
156	C₂²	C₂×C₅×#102	1	1	yes	characteristic
157	C₂²	C₅×#486	1	1	yes	characteristic
158	C₂²	C₅×#500	1	1	yes	characteristic
159	C₂²	C₅×#508	1	4	yes	—
160	C₄	—	2	4	no	—
161	C₄	—	8	4	no	—
162	C₄	—	16	2	no	—
163	C₄	C₅×#391	1	1	yes	characteristic
164	C₄	C₂×C₅×#103	1	1	yes	characteristic
165	C₄	C₅×#487	1	1	yes	characteristic
166	C₄	C₅×#498	1	1	yes	characteristic
167	C₂	C₅×#1990	1	1	yes	characteristic
168	C₂	C₅×#2138	1	2	yes	—
169	C₂	C₅×#2242	1	2	yes	—
170	C₂	C₅×#2253	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₅	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₂²×C₈	C₂³×C₅
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₂×C₄²×C₅ ◁ C₂×C₄×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
8. C₅×#499	C₄	semidirect
16. #13281	C₅	direct (N × H)
62. #499	C₄×C₅	semidirect
155. C₅	#13281	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅×#499	2
C₂×C₄×C₅×C₁₆	2
C₅×#1846	2
C₅×#1380	2
#13281	5

Subgroups — C5×#13281 #11936001

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#13281 #11936001