Subgroups — C₅×#12888 #11935608

Order 1280.

342 conjugacy classes of subgroups.

The subgroup lattice has 142 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₅×C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₅×#1845	C₂	1	2	yes	maximal
3	C₅×#1380	C₂	1	4	yes	maximal
4	C₂²×C₅×C₁₆	C₂²	1	2	yes	—
5	C₂×C₅×#107	—	2	2	no	—
6	C₂×C₅×#107	C₂²	1	4	yes	—
7	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₅×#384	—	2	2	no	—
9	C₄×C₅×C₁₆	C₄	1	4	yes	complemented
10	#12888	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
11	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
12	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
13	C₂×C₅×C₄⋊C₄	—	2	4	no	—
14	C₂×C₅×C₄⋊C₄	—	4	2	no	—
15	C₂×C₅×C₄⋊C₄	—	4	2	no	—
16	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
17	C₂×C₅×C₁₆	—	2	4	no	—
18	C₂×C₅×C₁₆	C₂×C₄	1	4	yes	—
19	C₅×#107	—	2	8	no	—
20	C₅×#107	—	4	4	no	—
21	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
22	C₄×C₅×C₈	Q₈	1	2	yes	—
23	C₂×C₄×C₁₆	C₂×C₅	1	1	yes	characteristic
24	#1845	C₂×C₅	1	2	yes	—
25	#1380	C₂×C₅	1	4	yes	—
26	C₂²×C₄×C₅	—	4	4	no	—
27	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
28	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
29	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
30	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
31	C₂×C₅×C₈	SD₁₆	1	4	yes	—
32	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
33	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
34	C₂×C₅×C₈	#41	1	4	yes	—
35	C₅×C₄⋊C₄	—	4	4	no	—
36	C₅×C₄⋊C₄	—	4	4	no	—
37	C₅×C₄⋊C₄	—	8	4	no	—
38	C₅×C₄⋊C₄	—	8	4	no	—
39	C₄²×C₅	C₄⋊C₄	1	2	yes	—
40	C₄²×C₅	Q₁₆	1	2	yes	—
41	C₅×C₁₆	—	2	4	no	—
42	C₂²×C₁₆	C₂²×C₅	1	2	yes	—
43	C₂×#107	—	2	2	no	—
44	C₂×#107	C₂²×C₅	1	4	yes	—
45	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
46	#384	—	2	2	no	—
47	C₄×C₁₆	C₄×C₅	1	4	yes	complemented
48	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
49	C₂×C₄×C₅	—	8	4	no	—
50	C₂×C₄×C₅	—	8	4	no	—
51	C₂×C₄×C₅	—	8	4	no	—
52	C₂×C₄×C₅	#107	1	2	yes	—
53	C₂×C₄×C₅	#107	1	2	yes	—
54	C₂×C₄×C₅	D₁₆	1	2	yes	—
55	C₂×C₄×C₅	SD₃₂	1	4	yes	—
56	C₂×C₄×C₅	Q₃₂	1	2	yes	—
57	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₅	#124	1	2	yes	—
59	C₂×C₄×C₅	C₂×SD₁₆	1	2	yes	—
60	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
61	C₅×C₈	#106	1	4	yes	—
62	C₅×C₈	#117	1	2	yes	—
63	C₅×C₈	#125	1	2	yes	—
64	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
65	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
66	C₂×C₄⋊C₄	—	2	4	no	—
67	C₂×C₄⋊C₄	—	4	2	no	—
68	C₂×C₄⋊C₄	—	4	2	no	—
69	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
70	C₂×C₁₆	—	2	4	no	—
71	C₂×C₁₆	C₂×C₄×C₅	1	4	yes	—
72	#107	—	2	8	no	—
73	#107	—	4	4	no	—
74	C₄×C₈	C₂×C₄×C₅	1	2	yes	—
75	C₄×C₈	C₅×Q₈	1	2	yes	—
76	C₂²×C₅	C₂×#107	1	1	yes	characteristic
77	C₂²×C₅	#486	1	2	yes	—
78	C₂²×C₅	C₂×D₁₆	1	1	yes	characteristic
79	C₂²×C₅	C₂×SD₃₂	1	2	yes	—
80	C₂²×C₅	C₂×Q₃₂	1	1	yes	characteristic
81	C₄×C₅	—	16	4	no	—
82	C₄×C₅	#366	1	4	yes	—
83	C₄×C₅	#367	1	4	yes	—
84	C₄×C₅	#382	1	1	yes	characteristic
85	C₄×C₅	C₂×#106	1	1	yes	characteristic
86	C₄×C₅	#487	1	2	yes	—
87	C₂²×C₄	—	4	4	no	—
88	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
89	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
90	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
91	C₂×C₈	C₅×C₄⋊C₄	1	2	yes	—
92	C₂×C₈	C₅×SD₁₆	1	4	yes	—
93	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
94	C₂×C₈	C₂×C₅×Q₈	1	1	yes	characteristic
95	C₂×C₈	C₅×#41	1	4	yes	—
96	C₄⋊C₄	—	4	4	no	—
97	C₄⋊C₄	—	4	4	no	—
98	C₄⋊C₄	—	8	4	no	—
99	C₄⋊C₄	—	8	4	no	—
100	C₄²	C₅×C₄⋊C₄	1	2	yes	—
101	C₄²	C₅×Q₁₆	1	2	yes	—
102	C₁₆	—	2	4	no	—
103	C₂×C₅	#1834	1	1	yes	characteristic
104	C₂×C₅	C₂×#366	1	1	yes	characteristic
105	C₂×C₅	C₂×#367	1	1	yes	characteristic
106	C₂×C₅	#2240	1	2	yes	—
107	C₂×C₅	#2241	1	2	yes	—
108	C₂³	C₂×C₅×D₈	1	1	yes	characteristic
109	C₂×C₄	—	8	4	no	—
110	C₂×C₄	—	8	4	no	—
111	C₂×C₄	—	8	4	no	—
112	C₂×C₄	C₅×#107	1	2	yes	—
113	C₂×C₄	C₅×#107	1	2	yes	—
114	C₂×C₄	C₅×D₁₆	1	2	yes	—
115	C₂×C₄	C₅×SD₃₂	1	4	yes	—
116	C₂×C₄	C₅×Q₃₂	1	2	yes	—
117	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
118	C₂×C₄	C₅×#124	1	2	yes	—
119	C₂×C₄	C₂×C₅×SD₁₆	1	2	yes	—
120	C₂×C₄	C₂×C₅×Q₁₆	1	1	yes	characteristic
121	C₈	C₅×#106	1	4	yes	—
122	C₈	C₅×#117	1	2	yes	—
123	C₈	C₅×#125	1	2	yes	—
124	C₅	#12888	1	1	yes	characteristic, direct product factor, Sylow
125	C₂²	C₂×C₅×#107	1	1	yes	characteristic
126	C₂²	C₅×#486	1	2	yes	—
127	C₂²	C₂×C₅×D₁₆	1	1	yes	characteristic
128	C₂²	C₂×C₅×SD₃₂	1	2	yes	—
129	C₂²	C₂×C₅×Q₃₂	1	1	yes	characteristic
130	C₄	—	16	4	no	—
131	C₄	C₅×#366	1	4	yes	—
132	C₄	C₅×#367	1	4	yes	—
133	C₄	C₅×#382	1	1	yes	characteristic
134	C₄	C₂×C₅×#106	1	1	yes	characteristic
135	C₄	C₅×#487	1	2	yes	—
136	C₂	C₅×#1834	1	1	yes	characteristic
137	C₂	C₂×C₅×#366	1	1	yes	characteristic
138	C₂	C₂×C₅×#367	1	1	yes	characteristic
139	C₂	C₅×#2240	1	2	yes	—
140	C₂	C₅×#2241	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
9. C₄×C₅×C₁₆	C₄	semidirect
10. #12888	C₅	direct (N × H)
47. C₄×C₁₆	C₄×C₅	semidirect
124. C₅	#12888	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×C₄×C₅×C₁₆	2
C₅×#1845	2
C₅×#1380	2
#12888	5

Subgroups — C5×#12888 #11935608

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#12888 #11935608