Subgroups — H(3,ℤ/4)×C₄₂₁ #41002309

Order 26944.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₁₆×C₄₂₁	C₂	1	1	yes	maximal, characteristic
2	#117×C₄₂₁	C₂	1	2	yes	maximal
3	C₂²×C₄×C₄₂₁	C₂²	1	1	yes	characteristic
4	M₁₆×C₄₂₁	—	2	2	no	—
5	C₂×C₈×C₄₂₁	C₄	1	2	yes	complemented
6	C₄⋊C₄×C₄₂₁	—	2	2	no	—
7	C₂²⋊C₄×C₄₂₁	—	2	2	no	—
8	C₄²×C₄₂₁	C₄	1	4	yes	complemented
9	C₂³×C₄₂₁	D₄	1	1	yes	characteristic
10	C₂×C₄×C₄₂₁	—	2	4	no	—
11	C₂×C₄×C₄₂₁	C₂×C₄	1	1	yes	characteristic
12	C₂×C₄×C₄₂₁	C₂×C₄	1	2	yes	—
13	C₂×C₄×C₄₂₁	D₄	1	2	yes	—
14	C₂×C₄×C₄₂₁	Q₈	1	1	yes	characteristic
15	C₈×C₄₂₁	—	2	2	no	—
16	C₂²×C₄₂₁	—	4	1	no	—
17	C₂²×C₄₂₁	C₂²⋊C₄	1	2	yes	—
18	C₂²×C₄₂₁	C₄⋊C₄	1	1	yes	characteristic
19	C₄×C₄₂₁	—	4	4	no	—
20	C₄×C₄₂₁	C₄²	1	1	yes	characteristic
21	C₄×C₄₂₁	C₂²⋊C₄	1	1	yes	characteristic
22	C₄×C₄₂₁	C₄⋊C₄	1	2	yes	—
23	C₂×C₄₂₁	—	2	2	no	—
24	C₂×C₄₂₁	—	2	1	no	—
25	C₂×C₄₂₁	#95	1	1	yes	characteristic
26	C₄₂₁	H(3,ℤ/4)	1	1	yes	characteristic, direct product factor, Sylow
27	H(3,ℤ/4)	C₄₂₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
28	C₂×M₁₆	C₂×C₄₂₁	1	1	yes	characteristic
29	#117	C₂×C₄₂₁	1	2	yes	—
30	C₂²×C₄	C₂²×C₄₂₁	1	1	yes	characteristic
31	M₁₆	—	2	2	no	—
32	C₂×C₈	C₄×C₄₂₁	1	2	yes	complemented
33	C₄⋊C₄	—	2	2	no	—
34	C₂²⋊C₄	—	2	2	no	—
35	C₄²	C₄×C₄₂₁	1	4	yes	complemented
36	C₂³	D₄×C₄₂₁	1	1	yes	characteristic
37	C₂×C₄	—	2	4	no	—
38	C₂×C₄	C₂×C₄×C₄₂₁	1	1	yes	characteristic
39	C₂×C₄	C₂×C₄×C₄₂₁	1	2	yes	—
40	C₂×C₄	D₄×C₄₂₁	1	2	yes	—
41	C₂×C₄	Q₈×C₄₂₁	1	1	yes	characteristic
42	C₈	—	2	2	no	—
43	C₂²	—	4	1	no	—
44	C₂²	C₂²⋊C₄×C₄₂₁	1	2	yes	—
45	C₂²	C₄⋊C₄×C₄₂₁	1	1	yes	characteristic
46	C₄	—	4	4	no	—
47	C₄	C₄²×C₄₂₁	1	1	yes	characteristic
48	C₄	C₂²⋊C₄×C₄₂₁	1	1	yes	characteristic
49	C₄	C₄⋊C₄×C₄₂₁	1	2	yes	—
50	C₂	—	2	2	no	—
51	C₂	—	2	1	no	—
52	C₂	#95×C₄₂₁	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₄₂₁	C₄²
commutator	C₄	C₄²×C₄₂₁
Frattini, cosocle	C₂²×C₄	C₂²×C₄₂₁
socle	C₂×C₄₂₁	#95
prime radical	C₂³×C₄₂₁	D₄
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₄×C₄₂₁ ◁ G

Complements

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

N	H ≅ G/N	type
5. C₂×C₈×C₄₂₁	C₄	semidirect
8. C₄²×C₄₂₁	C₄	semidirect
26. C₄₂₁	H(3,ℤ/4)	direct (N × H)
27. H(3,ℤ/4)	C₄₂₁	direct (N × H)
32. C₂×C₈	C₄×C₄₂₁	semidirect
35. C₄²	C₄×C₄₂₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×M₁₆×C₄₂₁	2
#117×C₄₂₁	2
H(3,ℤ/4)	421

Subgroups — H(3,ℤ/4)×C421 #41002309

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — H(3,ℤ/4)×C₄₂₁ #41002309