Subgroups — C₄×C₁₆×Dic₅ #12014118

Order 1280.

558 conjugacy classes of subgroups.

The subgroup lattice has 146 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	C₂×C₁₆×Dic₅	C₂	1	4	yes	maximal
3	C₄×C₈×Dic₅	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×#672	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₅×C₁₆	C₂²	1	2	yes	—
6	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
7	C₂×C₈×Dic₅	C₄	1	2	yes	—
8	C₂×C₈×Dic₅	C₂²	1	2	yes	—
9	C₂²×#672	C₂²	1	2	yes	—
10	C₄²×Dic₅	C₄	1	1	yes	characteristic
11	C₂×C₄×#168	C₄	1	1	yes	characteristic
12	C₄×C₅×C₁₆	C₄	1	4	yes	complemented
13	C₁₆×Dic₅	C₄	1	16	yes	direct product factor
14	C₄×#672	C₄	1	4	yes	complemented
15	C₄²×C₁₆	—	5	1	no	maximal, Sylow
16	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
18	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×Dic₅	C₈	1	4	yes	—
20	C₂×C₄×Dic₅	C₂×C₄	1	2	yes	—
21	C₂²×#168	C₂×C₄	1	2	yes	—
22	C₂×C₅×C₁₆	C₂×C₄	1	8	yes	—
23	C₂×C₅×C₁₆	C₂×C₄	1	4	yes	—
24	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
25	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
26	C₈×Dic₅	C₈	1	8	yes	—
27	C₈×Dic₅	C₂×C₄	1	8	yes	—
28	C₂×#672	C₂×C₄	1	4	yes	—
29	C₂×#672	C₂×C₄	1	8	yes	—
30	C₄×#168	C₈	1	4	yes	—
31	C₂×C₄×C₁₆	—	5	4	no	—
32	C₂×C₄×C₁₆	—	5	1	no	—
33	C₂×C₄×C₁₆	D₅	1	1	yes	characteristic
34	C₄²×C₈	—	5	1	no	—
35	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
36	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
37	C₂²×Dic₅	C₂×C₈	1	4	yes	—
38	C₂×C₅×C₈	C₄²	1	2	yes	—
39	C₂×C₅×C₈	C₂×C₈	1	4	yes	—
40	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
42	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
43	C₄²×C₅	C₄²	1	2	yes	—
44	C₄²×C₅	C₂×C₈	1	2	yes	—
45	C₄×Dic₅	C₁₆	1	16	yes	direct product factor
46	C₄×Dic₅	C₄²	1	4	yes	—
47	C₄×Dic₅	C₂×C₈	1	4	yes	—
48	C₂×#168	C₄²	1	4	yes	—
49	C₂×#168	C₂×C₈	1	4	yes	—
50	C₂×#168	C₂×C₈	1	4	yes	—
51	C₅×C₁₆	C₄²	1	8	yes	complemented
52	#672	C₄²	1	8	yes	complemented
53	C₂²×C₁₆	—	5	2	no	—
54	C₂²×C₁₆	C₂×D₅	1	2	yes	—
55	C₂×C₄×C₈	—	5	2	no	—
56	C₂×C₄×C₈	—	5	2	no	—
57	C₂×C₄×C₈	—	5	1	no	—
58	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
59	C₄³	—	5	1	no	—
60	C₄×C₁₆	—	5	16	no	—
61	C₄×C₁₆	—	5	4	no	—
62	C₄×C₁₆	Dic₅	1	4	yes	direct product factor
63	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
64	C₂×C₄×C₅	C₄×C₈	1	4	yes	—
65	C₂×C₄×C₅	C₂×C₁₆	1	8	yes	—
66	C₂×C₄×C₅	C₂×C₄²	1	2	yes	—
67	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
68	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
69	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
70	C₂×Dic₅	C₄×C₈	1	8	yes	—
71	C₂×Dic₅	C₂×C₁₆	1	16	yes	—
72	C₅×C₈	C₄×C₈	1	4	yes	—
73	C₅×C₈	C₂×C₄²	1	2	yes	—
74	C₅×C₈	C₂×C₄²	1	2	yes	—
75	#168	C₄×C₈	1	8	yes	—
76	C₂²×C₈	—	5	2	no	—
77	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
78	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
79	C₂×C₄²	—	5	4	no	—
80	C₂×C₄²	—	5	2	no	—
81	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
82	C₂×C₁₆	—	5	4	no	—
83	C₂×C₁₆	—	5	8	no	—
84	C₂×C₁₆	C₄×D₅	1	8	yes	—
85	C₂×C₁₆	C₂×Dic₅	1	4	yes	—
86	C₄×C₈	—	5	8	no	—
87	C₄×C₈	—	5	8	no	—
88	C₄×C₈	—	5	4	no	—
89	C₄×C₈	C₄×D₅	1	2	yes	—
90	C₄×C₈	C₂×Dic₅	1	2	yes	—
91	C₂²×C₅	C₂×C₄×C₈	1	1	yes	characteristic
92	C₂²×C₅	C₂×C₄×C₈	1	2	yes	—
93	C₂²×C₅	C₂²×C₁₆	1	4	yes	—
94	C₄×C₅	C₄×C₁₆	1	8	yes	complemented
95	C₄×C₅	C₄³	1	1	yes	characteristic
96	C₄×C₅	C₂×C₄×C₈	1	1	yes	characteristic
97	C₄×C₅	C₂×C₄×C₈	1	2	yes	—
98	Dic₅	C₄×C₁₆	1	16	yes	direct product factor
99	C₂²×C₄	—	5	4	no	—
100	C₂²×C₄	C₈×D₅	1	2	yes	—
101	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
102	C₂×C₈	—	5	4	no	—
103	C₂×C₈	—	5	4	no	—
104	C₂×C₈	—	5	4	no	—
105	C₂×C₈	C₈×D₅	1	4	yes	—
106	C₂×C₈	C₄×Dic₅	1	2	yes	—
107	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
108	C₂×C₈	C₂×C₄×D₅	1	4	yes	—
109	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
110	C₄²	—	5	16	no	—
111	C₄²	—	5	4	no	—
112	C₄²	—	5	4	no	—
113	C₄²	C₈×D₅	1	2	yes	—
114	C₄²	C₄×Dic₅	1	2	yes	—
115	C₁₆	—	5	8	no	—
116	C₁₆	C₄×Dic₅	1	8	yes	direct product factor
117	C₂×C₅	C₄²×C₈	1	1	yes	characteristic
118	C₂×C₅	C₂×C₄×C₁₆	1	4	yes	—
119	C₂×C₅	C₂×C₄×C₁₆	1	2	yes	—
120	C₂³	C₂×C₈×D₅	1	1	yes	characteristic
121	C₂×C₄	—	5	16	no	—
122	C₂×C₄	—	5	8	no	—
123	C₂×C₄	D₅×C₁₆	1	8	yes	—
124	C₂×C₄	C₈×Dic₅	1	4	yes	—
125	C₂×C₄	C₄²×D₅	1	2	yes	—
126	C₂×C₄	C₂×C₈×D₅	1	2	yes	—
127	C₂×C₄	C₂×C₈×D₅	1	1	yes	characteristic
128	C₂×C₄	C₂×C₄×Dic₅	1	1	yes	characteristic
129	C₈	—	5	8	no	—
130	C₈	C₈×Dic₅	1	4	yes	—
131	C₈	C₄²×D₅	1	2	yes	—
132	C₈	C₂×C₄×Dic₅	1	2	yes	—
133	C₅	C₄²×C₁₆	1	1	yes	characteristic, complemented, Sylow
134	C₂²	C₄×C₈×D₅	1	2	yes	—
135	C₂²	C₂×D₅×C₁₆	1	4	yes	—
136	C₂²	C₂×C₈×Dic₅	1	1	yes	characteristic
137	C₄	—	5	16	no	—
138	C₄	C₁₆×Dic₅	1	8	yes	direct product factor
139	C₄	C₄×C₈×D₅	1	2	yes	—
140	C₄	C₄²×Dic₅	1	1	yes	characteristic
141	C₄	C₂×C₈×Dic₅	1	1	yes	characteristic
142	C₂	C₄×D₅×C₁₆	1	4	yes	—
143	C₂	C₄×C₈×Dic₅	1	1	yes	characteristic
144	C₂	C₂×C₁₆×Dic₅	1	2	yes	—

	subgroup	G/H
center, hypercenter	C₂×C₄×C₁₆	D₅
commutator, hypocenter	C₅	C₄²×C₁₆
Frattini	C₂²×C₈	C₂²×D₅
Fitting	C₂×C₄×C₅×C₁₆	C₂
socle, prime radical	C₂³×C₅	C₂²×C₈
cosocle	C₂²×C₅×C₈	C₂³
LCS	G ▷ C₅
UCS	C₁ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
12. C₄×C₅×C₁₆	C₄	semidirect
13. C₁₆×Dic₅	C₄	direct (N × H)
14. C₄×#672	C₄	semidirect
45. C₄×Dic₅	C₁₆	direct (N × H)
51. C₅×C₁₆	C₄²	semidirect
52. #672	C₄²	semidirect
62. C₄×C₁₆	Dic₅	direct (N × H)
94. C₄×C₅	C₄×C₁₆	semidirect
98. Dic₅	C₄×C₁₆	direct (N × H)
116. C₁₆	C₄×Dic₅	direct (N × H)
133. C₅	C₄²×C₁₆	semidirect
138. C₄	C₁₆×Dic₅	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
C₂×C₁₆×Dic₅	2
C₄×C₈×Dic₅	2
C₂×C₄×#672	2
C₄²×C₁₆	5

Subgroups — C4×C16×Dic5 #12014118

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₁₆×Dic₅ #12014118