Subgroups — #2380

Order 128.

324 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#127	C₂	1	1	yes	maximal, characteristic
2	#386	C₂	1	12	yes	maximal, complemented
3	#383	C₂	1	2	yes	maximal, complemented
4	C₂²×D₄	C₂²	1	6	yes	—
5	C₂³×C₄	C₂²	1	4	yes	—
6	#127	C₄	1	8	yes	complemented
7	C₂×C₂²⋊C₄	C₂²	1	12	yes	—
8	C₂×C₄²	C₂²	1	1	yes	characteristic
9	#95	C₂²	1	12	yes	—
10	C₂⁴	C₂³	1	4	yes	—
11	C₂×D₄	—	4	6	no	—
12	C₂×D₄	—	4	6	no	—
13	C₂×D₄	C₂×C₄	1	24	yes	complemented
14	C₂²×C₄	—	4	12	no	—
15	C₂²×C₄	C₂³	1	8	yes	—
16	C₂²×C₄	C₂³	1	3	yes	—
17	C₂²⋊C₄	—	4	12	no	—
18	C₄²	C₂×C₄	1	4	yes	complemented
19	C₂³	—	2	12	no	—
20	C₂³	—	2	12	no	—
21	C₂³	C₂²×C₄	1	8	yes	—
22	C₂³	C₂⁴	1	1	yes	characteristic
23	D₄	—	4	24	no	—
24	C₂×C₄	—	2	6	no	—
25	C₂×C₄	—	2	24	no	—
26	C₂×C₄	—	4	16	no	—
27	C₂×C₄	C₂²×C₄	1	6	yes	—
28	C₂²	—	2	24	no	—
29	C₂²	—	4	4	no	—
30	C₂²	—	4	12	no	—
31	C₂²	C₂³×C₄	1	1	yes	characteristic
32	C₂²	H(5,𝔽₂)	1	6	yes	—
33	C₄	—	2	6	no	—
34	C₄	—	4	8	no	—
35	C₂	—	4	8	no	—
36	C₂	#518	1	3	yes	—
37	C₂	#560	1	4	yes	—

	subgroup	G/H
center, Frattini, socle, cosocle	C₂³	C₂⁴
commutator	C₂²	C₂³×C₄
prime radical	C₂×#127	C₂
LCS	G ▷ C₂² ▷ 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.