Subgroups — C₂²×H(5,𝔽₂) #3591

Order 128.

3280 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×H(5,𝔽₂)	C₂	1	48	yes	maximal, direct product factor
2	C₂²×#41	C₂	1	6	yes	maximal, complemented
3	C₂³×D₄	C₂	1	9	yes	maximal, complemented
4	C₂⁵	C₂²	1	6	yes	complemented
5	H(5,𝔽₂)	C₂²	1	256	yes	direct product factor
6	C₂×#41	C₂²	1	144	yes	complemented
7	C₂²×Q₈	C₂²	1	2	yes	complemented
8	C₂²×D₄	C₂²	1	18	yes	complemented
9	C₂²×D₄	C₂²	1	216	yes	complemented
10	C₂³×C₄	C₂²	1	9	yes	complemented
11	C₂⁴	—	4	24	no	—
12	C₂⁴	C₂³	1	9	yes	—
13	C₂⁴	C₂³	1	72	yes	complemented
14	#41	C₂³	1	384	yes	complemented
15	C₂×Q₈	C₂³	1	24	yes	complemented
16	C₂×D₄	C₂³	1	216	yes	complemented
17	C₂×D₄	C₂³	1	576	yes	complemented
18	C₂²×C₄	C₂³	1	6	yes	—
19	C₂²×C₄	C₂³	1	108	yes	complemented
20	C₂³	—	2	36	no	—
21	C₂³	—	4	144	no	—
22	C₂³	C₂⁴	1	54	yes	—
23	C₂³	C₂⁴	1	1	yes	characteristic
24	C₂³	C₂⁴	1	96	yes	complemented
25	Q₈	C₂⁴	1	32	yes	complemented
26	D₄	C₂⁴	1	288	yes	complemented
27	C₂×C₄	C₂⁴	1	36	yes	—
28	C₂×C₄	C₂⁴	1	144	yes	complemented
29	C₂²	—	2	108	no	—
30	C₂²	—	4	96	no	—
31	C₂²	H(5,𝔽₂)	1	4	yes	direct product factor
32	C₂²	C₂⁵	1	36	yes	—
33	C₂²	C₂⁵	1	3	yes	—
34	C₄	C₂⁵	1	24	yes	—
35	C₂	—	2	36	no	—
36	C₂	C₂×H(5,𝔽₂)	1	6	yes	direct product factor
37	C₂	C₂⁶	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³	C₂⁴
commutator, Frattini, cosocle	C₂	C₂⁶
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ 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.