Subgroups — #12019783

Order 1280.

218 conjugacy classes of subgroups.

The subgroup lattice has 124 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×#64719	C₂	1	2	yes	maximal
3	C₄⋊#3926	C₂	1	4	yes	maximal, complemented
4	C₂²×C₅×C₁₆	C₄	1	1	yes	characteristic
5	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
6	C₂×C₄×C₅×C₈	C₄	1	1	yes	characteristic
7	C₄×C₅×C₁₆	C₄	1	2	yes	—
8	C₄×C₅×C₁₆	C₂²	1	2	yes	—
9	#64719	—	2	4	no	—
10	C₂×#3926	C₂²	1	4	yes	—
11	#13627	—	5	1	no	maximal, Sylow
12	C₂²×C₅×C₈	C₈	1	1	yes	characteristic
13	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
14	C₂×C₄²×C₅	C₈	1	1	yes	characteristic
15	C₂×C₅×C₁₆	—	2	2	no	—
16	C₂×C₅×C₁₆	C₂×C₄	1	2	yes	—
17	C₂×C₅×C₁₆	C₂×C₄	1	1	yes	characteristic
18	C₂×C₅×C₁₆	D₄	1	2	yes	—
19	C₂×C₅×C₁₆	Q₈	1	2	yes	—
20	C₂×C₅×C₁₆	C₂³	1	1	yes	characteristic
21	C₄×C₅×C₈	C₈	1	2	yes	—
22	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
23	#3926	—	2	4	no	—
24	C₂×M₆₄	—	5	2	no	—
25	C₂×C₄×C₁₆	D₅	1	1	yes	characteristic
26	C₄⋊C₃₂	—	5	4	no	—
27	C₂²×C₄×C₅	C₁₆	1	2	yes	—
28	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
29	C₂×C₅×C₈	—	2	2	no	—
30	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
31	C₂×C₅×C₈	C₂×C₈	1	2	yes	—
32	C₂×C₅×C₈	C₂×C₈	1	1	yes	characteristic
33	C₂×C₅×C₈	M₁₆	1	2	yes	—
34	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
35	C₄²×C₅	C₁₆	1	1	yes	characteristic
36	C₄²×C₅	C₁₆	1	1	yes	characteristic
37	C₄²×C₅	C₂×C₈	1	2	yes	—
38	C₅×C₁₆	—	2	2	no	—
39	C₅×C₁₆	C₄⋊C₄	1	2	yes	—
40	C₅×C₁₆	C₂×D₄	1	1	yes	characteristic
41	C₅×C₁₆	C₂×Q₈	1	1	yes	characteristic
42	C₂²×C₁₆	Dic₅	1	1	yes	characteristic
43	C₂²×C₁₆	C₂×D₅	1	1	yes	characteristic
44	C₂×C₄×C₈	Dic₅	1	1	yes	characteristic
45	M₆₄	—	10	4	no	—
46	C₂×C₃₂	—	5	4	no	—
47	C₄×C₁₆	Dic₅	1	2	yes	—
48	C₄×C₁₆	C₂×D₅	1	2	yes	—
49	C₂³×C₅	C₂×C₁₆	1	1	yes	characteristic
50	C₂×C₄×C₅	—	2	4	no	—
51	C₂×C₄×C₅	C₄⋊C₈	1	2	yes	—
52	C₂×C₄×C₅	C₂×C₁₆	1	2	yes	—
53	C₂×C₄×C₅	C₂×C₁₆	1	1	yes	characteristic
54	C₂×C₄×C₅	C₂×C₁₆	1	2	yes	—
55	C₂×C₄×C₅	M₃₂	1	2	yes	—
56	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
57	C₅×C₈	—	2	2	no	—
58	C₅×C₈	C₄⋊C₈	1	2	yes	—
59	C₅×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₅×C₈	C₂×M₁₆	1	1	yes	characteristic
61	C₂²×C₈	#168	1	1	yes	characteristic
62	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
63	C₂×C₄²	#168	1	1	yes	characteristic
64	C₂×C₁₆	—	2	2	no	—
65	C₂×C₁₆	Dic₁₀	1	2	yes	—
66	C₂×C₁₆	D₂₀	1	2	yes	—
67	C₂×C₁₆	C₂×Dic₅	1	2	yes	—
68	C₂×C₁₆	C₂×Dic₅	1	1	yes	characteristic
69	C₂×C₁₆	C₂²×D₅	1	1	yes	characteristic
70	C₄×C₈	#168	1	2	yes	—
71	C₄×C₈	C₂×Dic₅	1	2	yes	—
72	C₃₂	—	10	4	no	—
73	C₂²×C₅	—	2	2	no	—
74	C₂²×C₅	C₄⋊C₁₆	1	2	yes	—
75	C₂²×C₅	C₂²×C₁₆	1	1	yes	characteristic
76	C₄×C₅	—	2	2	no	—
77	C₄×C₅	C₄⋊C₁₆	1	2	yes	—
78	C₄×C₅	M₆₄	1	4	yes	complemented
79	C₄×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
80	C₄×C₅	C₂×M₃₂	1	1	yes	characteristic
81	C₂²×C₄	#672	1	2	yes	—
82	C₂²×C₄	C₂×#168	1	1	yes	characteristic
83	C₂×C₈	—	2	2	no	—
84	C₂×C₈	C₂×#168	1	2	yes	—
85	C₂×C₈	C₂×#168	1	1	yes	characteristic
86	C₂×C₈	#681	1	2	yes	—
87	C₂×C₈	C₄⋊Dic₅	1	2	yes	—
88	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
89	C₄²	#672	1	1	yes	characteristic
90	C₄²	#672	1	1	yes	characteristic
91	C₄²	C₂×#168	1	2	yes	—
92	C₁₆	—	2	2	no	—
93	C₁₆	C₄⋊Dic₅	1	2	yes	—
94	C₁₆	C₂×Dic₁₀	1	1	yes	characteristic
95	C₁₆	C₂×D₂₀	1	1	yes	characteristic
96	C₂×C₅	—	2	2	no	—
97	C₂×C₅	C₂×C₄⋊C₁₆	1	1	yes	characteristic
98	C₂×C₅	C₂×M₆₄	1	2	yes	—
99	C₂³	C₂×#672	1	1	yes	characteristic
100	C₂×C₄	—	2	4	no	—
101	C₂×C₄	C₄⋊#168	1	2	yes	—
102	C₂×C₄	C₂×#672	1	2	yes	—
103	C₂×C₄	C₂×#672	1	1	yes	characteristic
104	C₂×C₄	C₂×#672	1	2	yes	—
105	C₂×C₄	#3944	1	2	yes	—
106	C₂×C₄	C₂²×#168	1	1	yes	characteristic
107	C₈	—	2	2	no	—
108	C₈	C₄⋊#168	1	2	yes	—
109	C₈	C₂×#681	1	1	yes	characteristic
110	C₈	C₂×C₄⋊Dic₅	1	1	yes	characteristic
111	C₅	#13627	1	1	yes	characteristic, complemented, Sylow
112	C₂²	—	2	2	no	—
113	C₂²	C₄⋊#672	1	2	yes	—
114	C₂²	C₂²×#672	1	1	yes	characteristic
115	C₄	—	2	2	no	—
116	C₄	C₄⋊#672	1	2	yes	—
117	C₄	#64719	1	4	yes	complemented
118	C₄	C₂×C₄⋊#168	1	1	yes	characteristic
119	C₄	C₂×#3944	1	1	yes	characteristic
120	C₂	—	2	2	no	—
121	C₂	C₂×C₄⋊#672	1	1	yes	characteristic
122	C₂	C₂×#64719	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₁₆	C₂²×D₅
commutator, socle	C₂²×C₅	C₂²×C₁₆
Fitting	C₂×C₄×C₅×C₁₆	C₂
prime radical	C₂³×C₅	C₂×C₁₆
cosocle	C₂×C₅×C₁₆	C₂³
hypercenter	C₂×C₄×C₁₆	D₅
hypocenter	C₅	#13627
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
3. C₄⋊#3926	C₂	semidirect
78. C₄×C₅	M₆₄	semidirect
111. C₅	#13627	semidirect
117. C₄	#64719	semidirect

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×#64719	2
C₄⋊#3926	2
#13627	5