Subgroups — C₂×#493 #3144

Order 128.

380 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₈	C₂	1	4	yes	maximal, complemented
2	C₂×#127	C₂	1	2	yes	maximal, complemented
3	#493	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₄×C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₄	—	2	4	no	—
6	C₂²×D₄	C₂²	1	4	yes	—
7	C₂×D₈	—	2	16	no	—
8	C₂×D₈	C₂²	1	16	yes	complemented
9	C₂²×C₈	C₂²	1	2	yes	—
10	#127	—	2	4	no	—
11	#127	C₂²	1	8	yes	complemented
12	C₂×C₄²	C₂²	1	1	yes	characteristic
13	C₄×C₈	C₂²	1	4	yes	complemented
14	C₂⁴	—	2	4	no	—
15	C₂×D₄	—	2	16	no	—
16	C₂×D₄	—	2	4	no	—
17	C₂×D₄	—	2	16	no	—
18	C₂×D₄	—	4	16	no	—
19	C₂×D₄	C₂³	1	8	yes	—
20	C₂²×C₄	D₄	1	2	yes	—
21	C₂²×C₄	C₂³	1	1	yes	characteristic
22	D₈	—	2	32	no	—
23	C₂×C₈	D₄	1	8	yes	complemented
24	C₂×C₈	C₂³	1	4	yes	—
25	C₄²	D₄	1	2	yes	—
26	C₄²	C₂³	1	2	yes	—
27	C₂³	—	2	8	no	—
28	C₂³	—	4	8	no	—
29	C₂³	—	4	16	no	—
30	C₂³	C₂×D₄	1	1	yes	characteristic
31	D₄	—	2	16	no	—
32	D₄	—	4	8	no	—
33	D₄	—	4	32	no	—
34	C₂×C₄	D₈	1	8	yes	complemented
35	C₂×C₄	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄	C₂×D₄	1	4	yes	—
37	C₂×C₄	C₂×D₄	1	4	yes	—
38	C₂×C₄	C₂⁴	1	1	yes	characteristic
39	C₈	C₂×D₄	1	8	yes	complemented
40	C₂²	—	4	8	no	—
41	C₂²	—	4	16	no	—
42	C₂²	—	8	16	no	—
43	C₂²	#127	1	2	yes	—
44	C₂²	C₂×D₈	1	4	yes	—
45	C₂²	C₂²×D₄	1	1	yes	characteristic
46	C₄	#127	1	2	yes	—
47	C₄	C₂×D₈	1	8	yes	complemented
48	C₄	C₂²×D₄	1	2	yes	—
49	C₂	—	8	8	no	—
50	C₂	#493	1	4	yes	direct product factor
51	C₂	C₂×#127	1	1	yes	characteristic
52	C₂	C₂²×D₈	1	2	yes	—

	subgroup	G/H
center, socle	C₂³	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄ ▷ 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.