Subgroups — C₂×H(3,ℤ/4) #1730

Order 128.

202 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#117	C₂	1	2	yes	maximal
3	H(3,ℤ/4)	C₂	1	4	yes	maximal, direct product factor
4	C₂³×C₄	C₂²	1	1	yes	characteristic
5	C₂×M₁₆	—	2	2	no	—
6	C₂×M₁₆	—	2	2	no	—
7	C₂×M₁₆	C₄	1	2	yes	complemented
8	C₂×M₁₆	C₂²	1	2	yes	—
9	C₂²×C₈	C₄	1	2	yes	complemented
10	#117	—	2	2	no	—
11	#117	—	2	2	no	—
12	#117	C₄	1	4	yes	complemented
13	#117	C₂²	1	4	yes	—
14	C₂×C₄⋊C₄	—	2	2	no	—
15	C₂×C₂²⋊C₄	—	2	2	no	—
16	C₂×C₄²	C₄	1	4	yes	complemented
17	C₂⁴	D₄	1	1	yes	characteristic
18	C₂²×C₄	—	2	4	no	—
19	C₂²×C₄	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄	C₂×C₄	1	2	yes	—
21	C₂²×C₄	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄	C₂×C₄	1	2	yes	—
23	C₂²×C₄	D₄	1	1	yes	characteristic
24	C₂²×C₄	D₄	1	2	yes	—
25	C₂²×C₄	D₄	1	2	yes	—
26	C₂²×C₄	Q₈	1	1	yes	characteristic
27	C₂²×C₄	Q₈	1	1	yes	characteristic
28	C₂²×C₄	C₂³	1	1	yes	characteristic
29	M₁₆	—	2	4	no	—
30	M₁₆	—	2	4	no	—
31	C₂×C₈	—	2	2	no	—
32	C₂×C₈	—	2	2	no	—
33	C₂×C₈	C₂×C₄	1	4	yes	complemented
34	C₄⋊C₄	—	2	4	no	—
35	C₄⋊C₄	—	2	4	no	—
36	C₂²⋊C₄	—	2	4	no	—
37	C₂²⋊C₄	—	2	4	no	—
38	C₄²	—	2	4	no	—
39	C₄²	C₂×C₄	1	8	yes	complemented
40	C₂³	—	4	2	no	—
41	C₂³	C₂²⋊C₄	1	2	yes	—
42	C₂³	C₂²⋊C₄	1	2	yes	—
43	C₂³	C₄⋊C₄	1	1	yes	characteristic
44	C₂³	C₄⋊C₄	1	1	yes	characteristic
45	C₂³	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄	—	2	8	no	—
47	C₂×C₄	—	4	8	no	—
48	C₂×C₄	C₄²	1	2	yes	—
49	C₂×C₄	C₄²	1	1	yes	characteristic
50	C₂×C₄	C₄²	1	1	yes	characteristic
51	C₂×C₄	C₂²⋊C₄	1	2	yes	—
52	C₂×C₄	C₂²⋊C₄	1	2	yes	—
53	C₂×C₄	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×C₄	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₄	C₂²⋊C₄	1	2	yes	—
56	C₂×C₄	C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₄	C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₄	C₄⋊C₄	1	2	yes	—
59	C₂×C₄	C₄⋊C₄	1	2	yes	—
60	C₂×C₄	C₄⋊C₄	1	2	yes	—
61	C₂×C₄	C₄⋊C₄	1	2	yes	—
62	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₄	C₂²×C₄	1	2	yes	—
64	C₂×C₄	C₂×D₄	1	2	yes	—
65	C₂×C₄	C₂×Q₈	1	1	yes	characteristic
66	C₈	—	2	4	no	—
67	C₂²	—	2	4	no	—
68	C₂²	—	2	2	no	—
69	C₂²	—	4	2	no	—
70	C₂²	—	4	1	no	—
71	C₂²	—	4	1	no	—
72	C₂²	#95	1	1	yes	characteristic
73	C₂²	#95	1	2	yes	—
74	C₂²	#95	1	1	yes	characteristic
75	C₂²	C₂×C₂²⋊C₄	1	2	yes	—
76	C₂²	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₄	—	4	8	no	—
78	C₄	#95	1	2	yes	—
79	C₄	#95	1	1	yes	characteristic
80	C₄	#95	1	1	yes	characteristic
81	C₄	C₂×C₄²	1	1	yes	characteristic
82	C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
83	C₄	C₂×C₄⋊C₄	1	2	yes	—
84	C₂	—	2	2	no	—
85	C₂	—	2	2	no	—
86	C₂	—	2	1	no	—
87	C₂	—	2	1	no	—
88	C₂	H(3,ℤ/4)	1	2	yes	direct product factor
89	C₂	C₂×#95	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₄²
commutator	C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄	C₂³
socle	C₂²	#95
prime radical	C₂⁴	D₄
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
3. H(3,ℤ/4)	C₂	direct (N × H)
7. C₂×M₁₆	C₄	semidirect
9. C₂²×C₈	C₄	semidirect
12. #117	C₄	semidirect
16. C₂×C₄²	C₄	semidirect
33. C₂×C₈	C₂×C₄	semidirect
39. C₄²	C₂×C₄	semidirect
88. C₂	H(3,ℤ/4)	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×M₁₆	2
C₂×#117	2
H(3,ℤ/4)	2

Subgroups — C2×H(3,ℤ/4) #1730

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×H(3,ℤ/4) #1730