Subgroups — C₂²×#117 #3421

Order 128.

940 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₄	C₂	1	1	yes	maximal, characteristic
2	C₂×#117	C₂	1	24	yes	maximal, direct product factor
3	C₂²×C₄⋊C₄	C₂	1	2	yes	maximal, complemented
4	C₂²×C₂²⋊C₄	C₂	1	2	yes	maximal
5	C₂²×C₄²	C₂	1	2	yes	maximal, complemented
6	C₂⁵	C₂²	1	1	yes	characteristic
7	C₂³×C₄	C₄	1	16	yes	complemented
8	C₂³×C₄	C₂²	1	4	yes	—
9	C₂³×C₄	C₂²	1	1	yes	characteristic
10	C₂³×C₄	C₂²	1	12	yes	—
11	C₂³×C₄	C₂²	1	1	yes	characteristic
12	#117	C₂²	1	64	yes	direct product factor
13	C₂×C₄⋊C₄	C₂²	1	24	yes	complemented
14	C₂×C₂²⋊C₄	C₂²	1	24	yes	—
15	C₂×C₄²	C₂²	1	24	yes	complemented
16	C₂⁴	—	2	8	no	—
17	C₂⁴	C₂×C₄	1	8	yes	—
18	C₂⁴	C₂³	1	6	yes	—
19	C₂⁴	C₂³	1	1	yes	characteristic
20	C₂²×C₄	—	2	16	no	—
21	C₂²×C₄	C₂×C₄	1	8	yes	—
22	C₂²×C₄	C₂×C₄	1	8	yes	—
23	C₂²×C₄	C₂×C₄	1	96	yes	complemented
24	C₂²×C₄	C₂³	1	24	yes	—
25	C₂²×C₄	C₂³	1	6	yes	—
26	C₂²×C₄	C₂³	1	6	yes	—
27	C₂²×C₄	C₂³	1	16	yes	—
28	C₄⋊C₄	C₂³	1	32	yes	complemented
29	C₂²⋊C₄	C₂³	1	32	yes	—
30	C₄²	C₂³	1	32	yes	complemented
31	C₂³	—	2	32	no	—
32	C₂³	—	2	24	no	—
33	C₂³	C₂²×C₄	1	24	yes	—
34	C₂³	C₂²×C₄	1	4	yes	—
35	C₂³	#41	1	8	yes	—
36	C₂³	C₂⁴	1	4	yes	—
37	C₂³	C₂⁴	1	3	yes	—
38	C₂×C₄	—	2	48	no	—
39	C₂×C₄	C₂²×C₄	1	24	yes	—
40	C₂×C₄	C₂²×C₄	1	24	yes	—
41	C₂×C₄	C₂²×C₄	1	64	yes	complemented
42	C₂×C₄	C₂⁴	1	16	yes	—
43	C₂×C₄	C₂⁴	1	4	yes	—
44	C₂×C₄	C₂⁴	1	4	yes	—
45	C₂²	—	2	48	no	—
46	C₂²	—	2	8	no	—
47	C₂²	#117	1	16	yes	direct product factor
48	C₂²	C₂³×C₄	1	8	yes	—
49	C₂²	C₂³×C₄	1	6	yes	—
50	C₂²	C₂×#41	1	12	yes	—
51	C₂²	C₂⁵	1	1	yes	characteristic
52	C₄	—	2	16	no	—
53	C₄	C₂³×C₄	1	8	yes	—
54	C₄	C₂³×C₄	1	8	yes	—
55	C₂	—	2	8	no	—
56	C₂	C₂×#117	1	12	yes	direct product factor
57	C₂	C₂⁴×C₄	1	1	yes	characteristic
58	C₂	C₂²×#41	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂²
commutator	C₂	C₂⁴×C₄
Frattini, cosocle	C₂²	C₂⁵
socle	C₂⁴	C₂³
prime radical	C₂⁵	C₂²
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
2. C₂×#117	C₂	direct (N × H)
3. C₂²×C₄⋊C₄	C₂	semidirect
5. C₂²×C₄²	C₂	semidirect
7. C₂³×C₄	C₄	semidirect
12. #117	C₂²	direct (N × H)
13. C₂×C₄⋊C₄	C₂²	semidirect
15. C₂×C₄²	C₂²	semidirect
23. C₂²×C₄	C₂×C₄	semidirect
28. C₄⋊C₄	C₂³	semidirect
30. C₄²	C₂³	semidirect
41. C₂×C₄	C₂²×C₄	semidirect
47. C₂²	#117	direct (N × H)
56. C₂	C₂×#117	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×C₄	2
C₂×#117	2
C₂²×C₄⋊C₄	2
C₂²×C₂²⋊C₄	2
C₂²×C₄²	2

Subgroups — C22×#117 #3421

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#117 #3421