Subgroups — B(4,𝔽₂) #457

Order 64.

99 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	H(5,𝔽₂)	C₂	1	1	yes	maximal, characteristic, complemented
2	#120	C₂	1	3	yes	maximal, complemented
3	#99	C₂	1	3	yes	maximal, complemented
4	C₂⁴	C₂²	1	1	yes	characteristic, complemented
5	#41	—	2	3	no	—
6	C₂×D₄	—	2	3	no	—
7	C₂×D₄	—	2	3	no	—
8	C₂×D₄	C₂²	1	3	yes	—
9	C₂²⋊C₄	—	2	3	no	—
10	C₂²⋊C₄	C₂²	1	3	yes	complemented
11	C₂³	—	2	1	no	—
12	C₂³	—	2	3	no	—
13	C₂³	—	4	2	no	—
14	C₂³	D₄	1	3	yes	complemented
15	C₂³	C₂³	1	1	yes	characteristic
16	Q₈	—	2	1	no	—
17	D₄	—	2	3	no	—
18	D₄	—	2	6	no	—
19	D₄	—	4	6	no	—
20	C₂×C₄	—	2	3	no	—
21	C₂×C₄	—	2	3	no	—
22	C₂×C₄	D₄	1	3	yes	complemented
23	C₂²	—	2	6	no	—
24	C₂²	—	2	3	no	—
25	C₂²	—	4	1	no	—
26	C₂²	—	4	3	no	—
27	C₂²	—	4	3	no	—
28	C₂²	—	4	1	no	—
29	C₂²	—	8	1	no	—
30	C₂²	C₂×D₄	1	3	yes	—
31	C₄	—	2	3	no	—
32	C₄	—	4	3	no	—
33	C₂	—	2	3	no	—
34	C₂	—	4	2	no	—
35	C₂	—	4	3	no	—
36	C₂	#120	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#120
commutator, Frattini, cosocle	C₂³	C₂³
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₂ ▷ 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.