Subgroups — C₂²×M₁₂₈ #513306

Order 512.

302 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂×M₁₂₈	C₂	1	12	yes	maximal, direct product factor
3	C₂²×C₆₄	C₂	1	2	yes	maximal, complemented
4	C₂³×C₁₆	C₄	1	1	yes	characteristic
5	C₂²×C₃₂	C₄	1	1	yes	characteristic
6	C₂²×C₃₂	C₄	1	6	yes	—
7	C₂²×C₃₂	C₂²	1	6	yes	—
8	C₂²×C₃₂	C₂²	1	1	yes	characteristic
9	M₁₂₈	C₂²	1	16	yes	direct product factor
10	C₂×C₆₄	C₂²	1	12	yes	complemented
11	C₂³×C₈	C₈	1	1	yes	characteristic
12	C₂²×C₁₆	C₈	1	1	yes	characteristic
13	C₂²×C₁₆	C₈	1	6	yes	—
14	C₂²×C₁₆	C₂×C₄	1	6	yes	—
15	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
16	C₂×C₃₂	C₂×C₄	1	6	yes	—
17	C₂×C₃₂	C₂×C₄	1	12	yes	—
18	C₂×C₃₂	C₂×C₄	1	3	yes	—
19	C₂×C₃₂	C₂³	1	3	yes	—
20	C₂×C₃₂	C₂³	1	4	yes	—
21	C₆₄	C₂³	1	8	yes	complemented
22	C₂³×C₄	C₁₆	1	1	yes	characteristic
23	C₂²×C₈	C₁₆	1	6	yes	—
24	C₂²×C₈	C₁₆	1	1	yes	characteristic
25	C₂²×C₈	C₂×C₈	1	6	yes	—
26	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
27	C₂×C₁₆	C₂×C₈	1	6	yes	—
28	C₂×C₁₆	C₂×C₈	1	12	yes	—
29	C₂×C₁₆	C₂×C₈	1	3	yes	—
30	C₂×C₁₆	C₂²×C₄	1	4	yes	—
31	C₂×C₁₆	C₂²×C₄	1	3	yes	—
32	C₃₂	C₂²×C₄	1	4	yes	—
33	C₃₂	C₂²×C₄	1	3	yes	—
34	C₃₂	C₂⁴	1	1	yes	characteristic
35	C₂⁴	C₃₂	1	1	yes	characteristic
36	C₂²×C₄	C₃₂	1	6	yes	—
37	C₂²×C₄	C₃₂	1	1	yes	characteristic
38	C₂²×C₄	C₂×C₁₆	1	6	yes	—
39	C₂²×C₄	C₂×C₁₆	1	1	yes	characteristic
40	C₂×C₈	C₂×C₁₆	1	12	yes	—
41	C₂×C₈	C₂×C₁₆	1	6	yes	—
42	C₂×C₈	C₂×C₁₆	1	3	yes	—
43	C₂×C₈	C₂²×C₈	1	4	yes	—
44	C₂×C₈	C₂²×C₈	1	3	yes	—
45	C₁₆	C₂²×C₈	1	4	yes	—
46	C₁₆	C₂²×C₈	1	3	yes	—
47	C₁₆	C₂³×C₄	1	1	yes	characteristic
48	C₂³	—	2	4	no	—
49	C₂³	C₂×C₃₂	1	6	yes	—
50	C₂³	C₂×C₃₂	1	1	yes	characteristic
51	C₂×C₄	C₂×C₃₂	1	12	yes	—
52	C₂×C₄	C₂×C₃₂	1	3	yes	—
53	C₂×C₄	C₂×C₃₂	1	6	yes	—
54	C₂×C₄	C₂²×C₁₆	1	4	yes	—
55	C₂×C₄	C₂²×C₁₆	1	3	yes	—
56	C₈	C₂²×C₁₆	1	3	yes	—
57	C₈	C₂²×C₁₆	1	4	yes	—
58	C₈	C₂³×C₈	1	1	yes	characteristic
59	C₂²	—	2	12	no	—
60	C₂²	M₁₂₈	1	4	yes	direct product factor
61	C₂²	C₂²×C₃₂	1	4	yes	—
62	C₂²	C₂²×C₃₂	1	3	yes	—
63	C₄	C₂²×C₃₂	1	3	yes	—
64	C₄	C₂²×C₃₂	1	4	yes	—
65	C₄	C₂³×C₁₆	1	1	yes	characteristic
66	C₂	—	2	4	no	—
67	C₂	C₂×M₁₂₈	1	6	yes	direct product factor
68	C₂	C₂³×C₃₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃₂	C₂²
commutator	C₂	C₂³×C₃₂
Frattini, cosocle	C₃₂	C₂⁴
socle	C₂³	C₂×C₃₂
prime radical	C₂⁴	C₃₂
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₂ ◁ G
LECS	G ▷ C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₂×M₁₂₈	C₂	direct (N × H)
3. C₂²×C₆₄	C₂	semidirect
9. M₁₂₈	C₂²	direct (N × H)
10. C₂×C₆₄	C₂²	semidirect
21. C₆₄	C₂³	semidirect
60. C₂²	M₁₂₈	direct (N × H)
67. C₂	C₂×M₁₂₈	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂³×C₃₂	2
C₂×M₁₂₈	2
C₂²×C₆₄	2

Subgroups — C22×M128 #513306

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×M₁₂₈ #513306