Subgroups — #2846

Order 128.

1711 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁶	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#120	C₂	1	7	yes	maximal, complemented
3	#379	C₂	1	7	yes	maximal, complemented
4	C₂⁵	—	2	28	no	—
5	C₂⁵	C₂²	1	7	yes	—
6	C₂²×D₄	C₂²	1	7	yes	complemented
7	#120	—	2	28	no	—
8	C₂×C₂²⋊C₄	C₂²	1	21	yes	complemented
9	C₂⁴	—	2	224	no	—
10	C₂⁴	—	2	84	no	—
11	C₂⁴	D₄	1	28	yes	complemented
12	C₂⁴	C₂³	1	7	yes	—
13	C₂⁴	C₂³	1	1	yes	characteristic, complemented
14	C₂×D₄	—	2	42	no	—
15	C₂²×C₄	C₂³	1	7	yes	complemented
16	C₂²⋊C₄	—	2	42	no	—
17	C₂³	—	2	256	no	—
18	C₂³	—	2	224	no	—
19	C₂³	—	2	168	no	—
20	C₂³	—	2	28	no	—
21	C₂³	—	2	7	no	—
22	C₂³	C₂×D₄	1	42	yes	—
23	C₂³	C₂⁴	1	1	yes	characteristic
24	D₄	—	4	28	no	—
25	C₂×C₄	—	2	21	no	—
26	C₂²	—	2	224	no	—
27	C₂²	—	2	84	no	—
28	C₂²	—	4	7	no	—
29	C₂²	#120	1	28	yes	complemented
30	C₂²	C₂²×D₄	1	7	yes	—
31	C₄	—	4	7	no	—
32	C₂	—	2	28	no	—
33	C₂	—	8	1	no	—
34	C₂	C₂×#120	1	7	yes	—

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