Subgroups — C₄×#64681 #12014122

Order 1280.

510 conjugacy classes of subgroups.

The subgroup lattice has 128 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄²×C₅×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#64681	C₂	1	4	yes	maximal
3	C₂×C₄×#672	C₂	1	2	yes	maximal
4	C₂×C₄×C₅×C₈	C₄	1	2	yes	—
5	C₂×C₄×C₅×C₈	C₄	1	1	yes	characteristic
6	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
7	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₄³×C₅	C₄	1	1	yes	characteristic
9	C₂²×#672	C₂²	1	4	yes	—
10	#64681	C₄	1	16	yes	direct product factor
11	C₄×#672	C₄	1	8	yes	complemented
12	C₄×#346	—	5	1	no	maximal, Sylow
13	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
14	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
16	C₂×C₄²×C₅	C₈	1	4	yes	—
17	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
18	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
19	C₄×C₅×C₈	C₈	1	8	yes	—
20	C₄×C₅×C₈	C₈	1	4	yes	—
21	C₄×C₅×C₈	C₂×C₄	1	4	yes	—
22	C₄×C₅×C₈	C₂×C₄	1	8	yes	—
23	C₂×#672	C₂×C₄	1	8	yes	—
24	C₂×#672	C₂×C₄	1	16	yes	—
25	C₂×#346	—	5	4	no	—
26	C₂×C₄×C₁₆	—	5	2	no	—
27	C₄²×C₈	D₅	1	1	yes	characteristic
28	C₂²×C₄×C₅	C₂×C₈	1	4	yes	—
29	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
30	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
31	C₂×C₅×C₈	C₄²	1	2	yes	—
32	C₂×C₅×C₈	C₄²	1	4	yes	—
33	C₂×C₅×C₈	C₂×C₈	1	4	yes	—
34	C₂×C₅×C₈	C₂×C₈	1	4	yes	—
35	C₂×C₅×C₈	C₂×C₈	1	4	yes	—
36	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
37	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
38	C₄²×C₅	—	2	8	no	—
39	C₄²×C₅	C₄²	1	4	yes	—
40	C₄²×C₅	C₄²	1	2	yes	—
41	C₄²×C₅	C₂×C₈	1	4	yes	—
42	C₄²×C₅	C₂×C₈	1	2	yes	—
43	#672	C₄²	1	16	yes	complemented
44	C₂²×C₁₆	—	5	4	no	—
45	C₂×C₄×C₈	Dic₅	1	2	yes	—
46	C₂×C₄×C₈	Dic₅	1	1	yes	characteristic
47	C₂×C₄×C₈	C₂×D₅	1	2	yes	—
48	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
49	C₄³	Dic₅	1	1	yes	characteristic
50	#346	—	5	16	no	—
51	C₄×C₁₆	—	5	8	no	—
52	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
53	C₂×C₄×C₅	—	2	8	no	—
54	C₂×C₄×C₅	C₄×C₈	1	8	yes	—
55	C₂×C₄×C₅	C₄×C₈	1	4	yes	—
56	C₂×C₄×C₅	M₃₂	1	8	yes	—
57	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
58	C₂×C₄×C₅	C₂×C₄²	1	2	yes	—
59	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
60	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
61	C₅×C₈	C₄×C₈	1	4	yes	—
62	C₅×C₈	C₄×C₈	1	8	yes	—
63	C₅×C₈	C₂×C₄²	1	4	yes	—
64	C₂²×C₈	C₂×Dic₅	1	2	yes	—
65	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
66	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
67	C₂×C₄²	#168	1	4	yes	—
68	C₂×C₄²	C₂×Dic₅	1	2	yes	—
69	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
70	C₂×C₁₆	—	5	8	no	—
71	C₂×C₁₆	—	5	16	no	—
72	C₄×C₈	#168	1	8	yes	—
73	C₄×C₈	#168	1	4	yes	—
74	C₄×C₈	C₄×D₅	1	4	yes	—
75	C₄×C₈	C₄×D₅	1	8	yes	—
76	C₂²×C₅	C₂×C₄×C₈	1	1	yes	characteristic
77	C₂²×C₅	C₂×C₄×C₈	1	2	yes	—
78	C₂²×C₅	C₂×M₃₂	1	4	yes	—
79	C₄×C₅	—	2	8	no	—
80	C₄×C₅	#346	1	8	yes	complemented
81	C₄×C₅	C₄³	1	1	yes	characteristic
82	C₄×C₅	C₂×C₄×C₈	1	1	yes	characteristic
83	C₄×C₅	C₂×C₄×C₈	1	2	yes	—
84	C₂²×C₄	C₂×#168	1	4	yes	—
85	C₂²×C₄	C₂×#168	1	2	yes	—
86	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
87	C₂×C₈	C₂×#168	1	4	yes	—
88	C₂×C₈	C₂×#168	1	4	yes	—
89	C₂×C₈	C₂×#168	1	4	yes	—
90	C₂×C₈	C₄×Dic₅	1	2	yes	—
91	C₂×C₈	C₄×Dic₅	1	4	yes	—
92	C₂×C₈	C₂×C₄×D₅	1	2	yes	—
93	C₂×C₈	C₂×C₄×D₅	1	4	yes	—
94	C₄²	—	2	8	no	—
95	C₄²	C₂×#168	1	4	yes	—
96	C₄²	C₂×#168	1	2	yes	—
97	C₄²	C₄×Dic₅	1	4	yes	—
98	C₄²	C₄×Dic₅	1	2	yes	—
99	C₁₆	—	5	16	no	—
100	C₂×C₅	C₄²×C₈	1	1	yes	characteristic
101	C₂×C₅	C₂×#346	1	2	yes	—
102	C₂×C₅	C₄×M₃₂	1	4	yes	—
103	C₂³	C₂²×#168	1	1	yes	characteristic
104	C₂×C₄	—	2	8	no	—
105	C₂×C₄	C₄×#168	1	8	yes	—
106	C₂×C₄	C₄×#168	1	4	yes	—
107	C₂×C₄	#3944	1	8	yes	—
108	C₂×C₄	C₂²×#168	1	1	yes	characteristic
109	C₂×C₄	C₂²×#168	1	2	yes	—
110	C₂×C₄	C₂×C₄×Dic₅	1	1	yes	characteristic
111	C₂×C₄	C₂×C₄×Dic₅	1	2	yes	—
112	C₈	C₄×#168	1	8	yes	—
113	C₈	C₄×#168	1	4	yes	—
114	C₈	C₄²×D₅	1	4	yes	—
115	C₅	C₄×#346	1	1	yes	characteristic, complemented, Sylow
116	C₂²	C₂×C₄×#168	1	1	yes	characteristic
117	C₂²	C₂×C₄×#168	1	2	yes	—
118	C₂²	C₂×#3944	1	4	yes	—
119	C₄	—	2	8	no	—
120	C₄	#64681	1	8	yes	direct product factor
121	C₄	C₂×C₄×#168	1	2	yes	—
122	C₄	C₂×C₄×#168	1	1	yes	characteristic
123	C₄	C₄²×Dic₅	1	1	yes	characteristic
124	C₂	C₄²×#168	1	1	yes	characteristic
125	C₂	C₂×#64681	1	2	yes	—
126	C₂	C₄×#3944	1	4	yes	—

	subgroup	G/H
center	C₂×C₄×C₈	C₂×D₅
commutator	C₂×C₅	C₄²×C₈
Frattini	C₂²×C₈	C₂²×D₅
Fitting	C₄²×C₅×C₈	C₂
socle, prime radical	C₂³×C₅	C₂²×C₈
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	C₄²×C₈	D₅
hypocenter	C₅	C₄×#346
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄×C₈ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
10. #64681	C₄	direct (N × H)
11. C₄×#672	C₄	semidirect
43. #672	C₄²	semidirect
80. C₄×C₅	#346	semidirect
115. C₅	C₄×#346	semidirect
120. C₄	#64681	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄²×C₅×C₈	2
C₂×#64681	2
C₂×C₄×#672	2
C₄×#346	5

Subgroups — C4×#64681 #12014122

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#64681 #12014122