Subgroups — C₂²×#127 #3440

Order 128.

1500 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₄	C₂	1	6	yes	maximal, complemented
2	C₂×#127	C₂	1	24	yes	maximal, direct product factor
3	C₂²×C₄²	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂⁵	C₂²	1	4	yes	—
5	C₂²×D₄	—	2	48	no	—
6	C₂²×D₄	C₂²	1	72	yes	complemented
7	C₂³×C₄	C₂²	1	3	yes	—
8	#127	C₂²	1	64	yes	direct product factor
9	C₂×C₄²	C₂²	1	12	yes	complemented
10	C₂⁴	—	2	48	no	—
11	C₂⁴	C₂³	1	24	yes	—
12	C₂⁴	C₂³	1	1	yes	characteristic
13	C₂×D₄	—	2	288	no	—
14	C₂×D₄	C₂³	1	96	yes	complemented
15	C₂²×C₄	D₄	1	24	yes	complemented
16	C₂²×C₄	C₂³	1	18	yes	—
17	C₄²	C₂³	1	16	yes	complemented
18	C₂³	—	2	144	no	—
19	C₂³	—	4	64	no	—
20	C₂³	C₂×D₄	1	12	yes	—
21	C₂³	C₂⁴	1	16	yes	—
22	C₂³	C₂⁴	1	3	yes	—
23	D₄	—	2	192	no	—
24	C₂×C₄	C₂×D₄	1	72	yes	complemented
25	C₂×C₄	C₂⁴	1	12	yes	—
26	C₂²	—	2	48	no	—
27	C₂²	—	4	96	no	—
28	C₂²	#127	1	16	yes	direct product factor
29	C₂²	C₂²×D₄	1	18	yes	—
30	C₂²	C₂⁵	1	1	yes	characteristic
31	C₄	C₂²×D₄	1	24	yes	complemented
32	C₂	—	4	16	no	—
33	C₂	C₂×#127	1	12	yes	direct product factor
34	C₂	C₂³×D₄	1	3	yes	—

	subgroup	G/H
center, socle	C₂⁴	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁵
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂² ▷ 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.