Subgroups — C₂²×#123 #3434

Order 128.

984 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₄	C₂	1	1	yes	maximal, characteristic
2	C₂⁴×C₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#123	C₂	1	24	yes	maximal, direct product factor
4	C₂²×C₄⋊C₄	C₂	1	2	yes	maximal, complemented
5	C₂²×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₂²⋊C₄	C₂	1	2	yes	maximal, complemented
7	C₂⁵	C₂²	1	1	yes	characteristic
8	C₂⁵	C₂²	1	1	yes	characteristic
9	C₂²×D₄	—	2	8	no	—
10	C₂²×D₄	C₂²	1	12	yes	—
11	C₂³×C₄	—	2	8	no	—
12	C₂³×C₄	C₂²	1	2	yes	—
13	C₂³×C₄	C₂²	1	2	yes	—
14	C₂³×C₄	C₂²	1	1	yes	characteristic
15	C₂³×C₄	C₂²	1	12	yes	complemented
16	#123	C₂²	1	64	yes	direct product factor
17	C₂×C₄⋊C₄	C₂²	1	24	yes	complemented
18	C₂×C₂²⋊C₄	C₂²	1	12	yes	complemented
19	C₂×C₂²⋊C₄	C₂²	1	24	yes	complemented
20	C₂⁴	—	2	8	no	—
21	C₂⁴	—	2	4	no	—
22	C₂⁴	—	2	8	no	—
23	C₂⁴	D₄	1	8	yes	—
24	C₂⁴	C₂³	1	6	yes	—
25	C₂⁴	C₂³	1	6	yes	—
26	C₂⁴	C₂³	1	1	yes	characteristic
27	C₂×D₄	—	2	48	no	—
28	C₂×D₄	C₂³	1	16	yes	—
29	C₂²×C₄	—	2	8	no	—
30	C₂²×C₄	—	2	48	no	—
31	C₂²×C₄	—	2	4	no	—
32	C₂²×C₄	—	2	8	no	—
33	C₂²×C₄	C₂³	1	12	yes	—
34	C₂²×C₄	C₂³	1	12	yes	—
35	C₂²×C₄	C₂³	1	6	yes	—
36	C₂²×C₄	C₂³	1	16	yes	complemented
37	C₄⋊C₄	C₂³	1	32	yes	complemented
38	C₂²⋊C₄	C₂³	1	16	yes	complemented
39	C₂²⋊C₄	C₂³	1	32	yes	complemented
40	C₂³	—	2	32	no	—
41	C₂³	—	2	24	no	—
42	C₂³	—	2	12	no	—
43	C₂³	—	2	24	no	—
44	C₂³	—	4	16	no	—
45	C₂³	C₂×D₄	1	24	yes	—
46	C₂³	C₂×D₄	1	4	yes	—
47	C₂³	#41	1	8	yes	—
48	C₂³	C₂⁴	1	4	yes	—
49	C₂³	C₂⁴	1	4	yes	—
50	C₂³	C₂⁴	1	3	yes	—
51	D₄	—	2	32	no	—
52	C₂×C₄	—	2	24	no	—
53	C₂×C₄	—	2	32	no	—
54	C₂×C₄	—	2	12	no	—
55	C₂×C₄	—	2	24	no	—
56	C₂×C₄	C₂⁴	1	8	yes	—
57	C₂×C₄	C₂⁴	1	8	yes	—
58	C₂×C₄	C₂⁴	1	4	yes	—
59	C₂²	—	2	48	no	—
60	C₂²	—	2	8	no	—
61	C₂²	—	2	4	no	—
62	C₂²	—	2	8	no	—
63	C₂²	—	4	24	no	—
64	C₂²	#123	1	16	yes	direct product factor
65	C₂²	C₂²×D₄	1	8	yes	—
66	C₂²	C₂²×D₄	1	6	yes	—
67	C₂²	C₂×#41	1	12	yes	—
68	C₂²	C₂⁵	1	1	yes	characteristic
69	C₄	—	2	8	no	—
70	C₄	—	2	4	no	—
71	C₄	—	2	8	no	—
72	C₂	—	2	8	no	—
73	C₂	—	4	4	no	—
74	C₂	C₂×#123	1	12	yes	direct product factor
75	C₂	C₂³×D₄	1	1	yes	characteristic
76	C₂	C₂²×#41	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁴	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁵
prime radical	C₂³×D₄	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
2. C₂⁴×C₄	C₂	semidirect
3. C₂×#123	C₂	direct (N × H)
4. C₂²×C₄⋊C₄	C₂	semidirect
5. C₂²×C₂²⋊C₄	C₂	semidirect
6. C₂²×C₂²⋊C₄	C₂	semidirect
15. C₂³×C₄	C₂²	semidirect
16. #123	C₂²	direct (N × H)
17. C₂×C₄⋊C₄	C₂²	semidirect
18. C₂×C₂²⋊C₄	C₂²	semidirect
19. C₂×C₂²⋊C₄	C₂²	semidirect
36. C₂²×C₄	C₂³	semidirect
37. C₄⋊C₄	C₂³	semidirect
38. C₂²⋊C₄	C₂³	semidirect
39. C₂²⋊C₄	C₂³	semidirect
64. C₂²	#123	direct (N × H)
74. C₂	C₂×#123	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×D₄	2
C₂⁴×C₄	2
C₂×#123	2
C₂²×C₄⋊C₄	2
C₂²×C₂²⋊C₄	2

Subgroups — C22×#123 #3434

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#123 #3434