Subgroups — C₂²×#126 #3438

Order 128.

660 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#126	C₂	1	24	yes	maximal, direct product factor
2	C₂²×C₄⋊C₄	C₂	1	3	yes	maximal, complemented
3	C₂²×C₂²⋊C₄	C₂	1	3	yes	maximal
4	C₂²×C₄²	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂⁵	C₂²	1	1	yes	characteristic
6	C₂³×C₄	C₂²	1	3	yes	—
7	C₂³×C₄	C₂²	1	3	yes	—
8	#126	C₂²	1	64	yes	direct product factor
9	C₂×C₄⋊C₄	C₂²	1	36	yes	complemented
10	C₂×C₂²⋊C₄	C₂²	1	36	yes	—
11	C₂×C₄²	C₂²	1	12	yes	complemented
12	C₂⁴	—	2	12	no	—
13	C₂⁴	C₂³	1	6	yes	—
14	C₂⁴	C₂³	1	1	yes	characteristic
15	C₂²×C₄	—	2	12	no	—
16	C₂²×C₄	—	2	12	no	—
17	C₂²×C₄	C₂³	1	18	yes	—
18	C₂²×C₄	C₂³	1	18	yes	—
19	C₄⋊C₄	C₂³	1	48	yes	complemented
20	C₂²⋊C₄	C₂³	1	48	yes	—
21	C₄²	C₂³	1	16	yes	complemented
22	C₂³	—	2	36	no	—
23	C₂³	—	4	16	no	—
24	C₂³	#41	1	12	yes	—
25	C₂³	C₂⁴	1	4	yes	—
26	C₂³	C₂⁴	1	3	yes	—
27	C₂×C₄	—	2	36	no	—
28	C₂×C₄	—	2	36	no	—
29	C₂×C₄	C₂⁴	1	12	yes	—
30	C₂×C₄	C₂⁴	1	12	yes	—
31	C₂²	—	2	12	no	—
32	C₂²	—	4	24	no	—
33	C₂²	#126	1	16	yes	direct product factor
34	C₂²	C₂×#41	1	18	yes	—
35	C₂²	C₂⁵	1	1	yes	characteristic
36	C₄	—	2	12	no	—
37	C₄	—	2	12	no	—
38	C₂	—	4	4	no	—
39	C₂	C₂×#126	1	12	yes	direct product factor
40	C₂	C₂²×#41	1	3	yes	—

	subgroup	G/H
center, socle	C₂⁴	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁵
prime radical	C₂⁵	C₂²
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.