Subgroups — C₅×#8814 #11931534

Order 1280.

454 conjugacy classes of subgroups.

The subgroup lattice has 162 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#1845	C₂	1	2	yes	maximal
2	C₅×#1844	C₂	1	2	yes	maximal
3	C₅×#1834	C₂	1	2	yes	maximal
4	C₂×C₅×C₈²	C₂	1	1	yes	maximal, characteristic
5	C₂×C₅×#107	—	2	4	no	—
6	C₂×C₅×#106	—	2	4	no	—
7	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
10	C₅×#384	C₂²	1	4	yes	—
11	C₅×#340	—	2	4	no	—
12	C₅×C₈²	C₄	1	4	yes	complemented
13	#8814	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
14	C₂²×C₅×C₈	D₄	1	2	yes	—
15	C₂²×C₅×C₈	D₄	1	2	yes	—
16	C₂²×C₅×C₈	D₄	1	2	yes	—
17	C₂×C₅×C₄⋊C₄	—	2	4	no	—
18	C₂×C₅×C₄⋊C₄	—	2	4	no	—
19	C₂×C₅×C₄⋊C₄	—	2	4	no	—
20	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
21	C₅×#107	—	4	8	no	—
22	C₅×#106	—	4	8	no	—
23	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
24	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
25	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
26	C₄×C₅×C₈	Q₈	1	2	yes	—
27	C₄×C₅×C₈	Q₈	1	2	yes	—
28	C₄×C₅×C₈	Q₈	1	2	yes	—
29	#1845	C₂×C₅	1	2	yes	—
30	#1844	C₂×C₅	1	2	yes	—
31	#1834	C₂×C₅	1	2	yes	—
32	C₂×C₈²	C₂×C₅	1	1	yes	characteristic
33	C₂²×C₄×C₅	—	4	4	no	—
34	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
35	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
36	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
37	C₂×C₅×C₈	—	2	4	no	—
38	C₂×C₅×C₈	C₄⋊C₄	1	4	yes	—
39	C₂×C₅×C₈	C₄⋊C₄	1	4	yes	—
40	C₂×C₅×C₈	C₄⋊C₄	1	4	yes	—
41	C₂×C₅×C₈	D₈	1	4	yes	—
42	C₂×C₅×C₈	SD₁₆	1	8	yes	—
43	C₂×C₅×C₈	Q₁₆	1	4	yes	—
44	C₅×C₄⋊C₄	—	4	8	no	—
45	C₅×C₄⋊C₄	—	4	8	no	—
46	C₅×C₄⋊C₄	—	4	8	no	—
47	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
48	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
49	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
50	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
51	C₂×#107	—	2	4	no	—
52	C₂×#106	—	2	4	no	—
53	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
54	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
55	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
56	#384	C₂²×C₅	1	4	yes	—
57	#340	—	2	4	no	—
58	C₈²	C₄×C₅	1	4	yes	complemented
59	C₂³×C₅	#127	1	1	yes	characteristic
60	C₂×C₄×C₅	—	8	4	no	—
61	C₂×C₄×C₅	—	8	4	no	—
62	C₂×C₄×C₅	—	8	4	no	—
63	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
67	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
68	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
69	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
70	C₂×C₄×C₅	C₂×SD₁₆	1	4	yes	—
71	C₂×C₄×C₅	C₂×Q₁₆	1	2	yes	—
72	C₂×C₄×C₅	#135	1	4	yes	—
73	C₅×C₈	—	2	4	no	—
74	C₅×C₈	#106	1	8	yes	complemented
75	C₅×C₈	#107	1	8	yes	complemented
76	C₂²×C₈	C₅×D₄	1	2	yes	—
77	C₂²×C₈	C₅×D₄	1	2	yes	—
78	C₂²×C₈	C₅×D₄	1	2	yes	—
79	C₂×C₄⋊C₄	—	2	4	no	—
80	C₂×C₄⋊C₄	—	2	4	no	—
81	C₂×C₄⋊C₄	—	2	4	no	—
82	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
83	#107	—	4	8	no	—
84	#106	—	4	8	no	—
85	C₄×C₈	C₂×C₄×C₅	1	2	yes	—
86	C₄×C₈	C₂×C₄×C₅	1	2	yes	—
87	C₄×C₈	C₂×C₄×C₅	1	2	yes	—
88	C₄×C₈	C₅×Q₈	1	2	yes	—
89	C₄×C₈	C₅×Q₈	1	2	yes	—
90	C₄×C₈	C₅×Q₈	1	2	yes	—
91	C₂²×C₅	#384	1	1	yes	characteristic
92	C₂²×C₅	#492	1	2	yes	—
93	C₂²×C₅	#493	1	1	yes	characteristic
94	C₂²×C₅	#494	1	1	yes	characteristic
95	C₂²×C₅	#495	1	2	yes	—
96	C₄×C₅	—	16	4	no	—
97	C₄×C₅	C₂×#106	1	2	yes	—
98	C₄×C₅	C₂×#107	1	2	yes	—
99	C₄×C₅	#427	1	2	yes	—
100	C₄×C₅	#498	1	2	yes	—
101	C₄×C₅	#499	1	2	yes	—
102	C₄×C₅	#500	1	2	yes	—
103	C₂²×C₄	—	4	4	no	—
104	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
105	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
106	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
107	C₂×C₈	—	2	4	no	—
108	C₂×C₈	C₅×C₄⋊C₄	1	4	yes	—
109	C₂×C₈	C₅×C₄⋊C₄	1	4	yes	—
110	C₂×C₈	C₅×C₄⋊C₄	1	4	yes	—
111	C₂×C₈	C₅×D₈	1	4	yes	—
112	C₂×C₈	C₅×SD₁₆	1	8	yes	—
113	C₂×C₈	C₅×Q₁₆	1	4	yes	—
114	C₄⋊C₄	—	4	8	no	—
115	C₄⋊C₄	—	4	8	no	—
116	C₄⋊C₄	—	4	8	no	—
117	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
118	C₄²	C₂×C₅×Q₈	1	1	yes	characteristic
119	C₄²	C₂×C₅×Q₈	1	1	yes	characteristic
120	C₄²	C₂×C₅×Q₈	1	1	yes	characteristic
121	C₂×C₅	#1706	1	2	yes	—
122	C₂×C₅	#1707	1	2	yes	—
123	C₂×C₅	#1844	1	1	yes	characteristic
124	C₂×C₅	#1845	1	1	yes	characteristic
125	C₂×C₅	#1846	1	1	yes	characteristic
126	C₂³	C₅×#127	1	1	yes	characteristic
127	C₂×C₄	—	8	4	no	—
128	C₂×C₄	—	8	4	no	—
129	C₂×C₄	—	8	4	no	—
130	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
131	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
132	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
133	C₂×C₄	C₅×C₄⋊Q₈	1	1	yes	characteristic
134	C₂×C₄	C₅×C₄⋊Q₈	1	1	yes	characteristic
135	C₂×C₄	C₅×C₄⋊Q₈	1	1	yes	characteristic
136	C₂×C₄	C₂×C₅×D₈	1	2	yes	—
137	C₂×C₄	C₂×C₅×SD₁₆	1	4	yes	—
138	C₂×C₄	C₂×C₅×Q₁₆	1	2	yes	—
139	C₂×C₄	C₅×#135	1	4	yes	—
140	C₈	—	2	4	no	—
141	C₈	C₅×#106	1	8	yes	complemented
142	C₈	C₅×#107	1	8	yes	complemented
143	C₅	#8814	1	1	yes	characteristic, direct product factor, Sylow
144	C₂²	C₅×#384	1	1	yes	characteristic
145	C₂²	C₅×#492	1	2	yes	—
146	C₂²	C₅×#493	1	1	yes	characteristic
147	C₂²	C₅×#494	1	1	yes	characteristic
148	C₂²	C₅×#495	1	2	yes	—
149	C₄	—	16	4	no	—
150	C₄	C₂×C₅×#106	1	2	yes	—
151	C₄	C₂×C₅×#107	1	2	yes	—
152	C₄	C₅×#427	1	2	yes	—
153	C₄	C₅×#498	1	2	yes	—
154	C₄	C₅×#499	1	2	yes	—
155	C₄	C₅×#500	1	2	yes	—
156	C₂	C₅×#1706	1	2	yes	—
157	C₂	C₅×#1707	1	2	yes	—
158	C₂	C₅×#1844	1	1	yes	characteristic
159	C₂	C₅×#1845	1	1	yes	characteristic
160	C₂	C₅×#1846	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₅	#127
commutator	C₄²	C₂²×C₄×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₂×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
12. C₅×C₈²	C₄	semidirect
13. #8814	C₅	direct (N × H)
58. C₈²	C₄×C₅	semidirect
74. C₅×C₈	#106	semidirect
75. C₅×C₈	#107	semidirect
141. C₈	C₅×#106	semidirect
142. C₈	C₅×#107	semidirect
143. C₅	#8814	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×#1845	2
C₅×#1844	2
C₅×#1834	2
C₂×C₅×C₈²	2
#8814	5

Subgroups — C5×#8814 #11931534

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#8814 #11931534