Subgroups — #14679

Order 256.

15431 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁷	C₂	1	1	yes	maximal, characteristic
2	C₂×#379	C₂	1	14	yes	maximal, complemented
3	C₂⁶	—	2	56	no	—
4	C₂⁶	C₄	1	8	yes	complemented
5	C₂⁶	C₂²	1	7	yes	—
6	C₂²×C₂²⋊C₄	C₂²	1	28	yes	complemented
7	#379	—	2	56	no	—
8	C₂⁵	—	2	896	no	—
9	C₂⁵	—	2	224	no	—
10	C₂⁵	—	2	168	no	—
11	C₂⁵	C₂×C₄	1	28	yes	—
12	C₂⁵	D₄	1	56	yes	—
13	C₂⁵	C₂³	1	7	yes	—
14	C₂³×C₄	C₂³	1	8	yes	complemented
15	C₂×C₂²⋊C₄	—	2	168	no	—
16	C₂⁴	—	2	2048	no	—
17	C₂⁴	—	2	896	no	—
18	C₂⁴	—	2	1792	no	—
19	C₂⁴	—	2	336	no	—
20	C₂⁴	—	2	672	no	—
21	C₂⁴	—	2	56	no	—
22	C₂⁴	C₂²⋊C₄	1	112	yes	complemented
23	C₂⁴	C₂²×C₄	1	14	yes	—
24	C₂⁴	C₂×D₄	1	84	yes	—
25	C₂⁴	C₂⁴	1	1	yes	characteristic
26	C₂²×C₄	—	2	56	no	—
27	C₂²⋊C₄	—	4	112	no	—
28	C₂³	—	2	2048	no	—
29	C₂³	—	2	1792	no	—
30	C₂³	—	2	896	no	—
31	C₂³	—	2	56	no	—
32	C₂³	—	2	336	no	—
33	C₂³	—	2	672	no	—
34	C₂³	C₂×C₂²⋊C₄	1	84	yes	—
35	C₂³	#120	1	112	yes	—
36	C₂³	C₂³×C₄	1	1	yes	characteristic
37	C₂³	C₂²×D₄	1	14	yes	—
38	C₂×C₄	—	4	56	no	—
39	C₂²	—	2	896	no	—
40	C₂²	—	2	224	no	—
41	C₂²	—	2	168	no	—
42	C₂²	#379	1	56	yes	complemented
43	C₂²	C₂²×C₂²⋊C₄	1	7	yes	—
44	C₂²	C₂×#120	1	28	yes	—
45	C₄	—	8	8	no	—
46	C₂	—	2	56	no	—
47	C₂	C₂×#379	1	7	yes	—
48	C₂	#2846	1	8	yes	—

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁴	C₂⁴
commutator	C₂³	C₂³×C₄
prime radical	C₂⁷	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

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

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

subgroup	index
C₂⁷	2
C₂×#379	2