Subgroups — #41002622

Order 26944.

58 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27212306	C₂	1	2	yes	maximal, complemented
2	C₂×M₁₆×C₄₂₁	C₂	1	1	yes	maximal, characteristic
3	#20269746	—	2	2	no	—
4	C₂²×C₄×C₄₂₁	C₄	1	1	yes	characteristic
5	M₁₆×C₄₂₁	—	2	2	no	—
6	C₂×C₈×C₄₂₁	C₄	1	1	yes	characteristic
7	C₂×C₈×C₄₂₁	C₂²	1	1	yes	characteristic
8	C₂³×C₄₂₁	C₈	1	1	yes	characteristic
9	C₂×C₄×C₄₂₁	—	2	1	no	—
10	C₂×C₄×C₄₂₁	—	2	1	no	—
11	C₂×C₄×C₄₂₁	C₈	1	1	yes	characteristic
12	C₂×C₄×C₄₂₁	C₂×C₄	1	1	yes	characteristic
13	C₈×C₄₂₁	—	2	1	no	—
14	C₈×C₄₂₁	D₄	1	2	yes	—
15	C₂²×C₄₂₁	—	2	1	no	—
16	C₂²×C₄₂₁	—	4	1	no	—
17	C₂²×C₄₂₁	C₂×C₈	1	1	yes	characteristic
18	C₄×C₄₂₁	—	2	1	no	—
19	C₄×C₄₂₁	C₂²⋊C₄	1	1	yes	characteristic
20	C₄×C₄₂₁	M₁₆	1	1	yes	characteristic
21	C₂×C₄₂₁	—	2	1	no	—
22	C₂×C₄₂₁	—	4	1	no	—
23	C₂×C₄₂₁	C₂²⋊C₈	1	1	yes	characteristic
24	C₄₂₁	#349	1	1	yes	characteristic, complemented, Sylow
25	#349	—	421	1	no	maximal, Sylow
26	C₂×M₁₆	D₄₂₁	1	1	yes	characteristic
27	M₃₂	—	421	2	no	—
28	C₂²×C₄	Dic₄₂₁	1	1	yes	characteristic
29	M₁₆	—	2	2	no	—
30	C₂×C₈	Dic₄₂₁	1	1	yes	characteristic
31	C₂×C₈	C₂×D₄₂₁	1	1	yes	characteristic
32	C₁₆	—	842	2	no	—
33	C₂³	#16809890	1	1	yes	characteristic
34	C₂×C₄	—	2	1	no	—
35	C₂×C₄	—	2	1	no	—
36	C₂×C₄	#16809890	1	1	yes	characteristic
37	C₂×C₄	C₂×Dic₄₂₁	1	1	yes	characteristic
38	C₈	—	2	1	no	—
39	C₈	C₂²⋊D₄₂₁	1	2	yes	—
40	C₂²	—	2	1	no	—
41	C₂²	—	4	1	no	—
42	C₂²	C₂×#16809890	1	1	yes	characteristic
43	C₄	—	2	1	no	—
44	C₄	C₂²⋊Dic₄₂₁	1	1	yes	characteristic
45	C₄	#20269755	1	1	yes	characteristic
46	C₂	—	2	1	no	—
47	C₂	—	4	1	no	—
48	C₂	C₂²⋊#16809890	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²⋊Dic₄₂₁
commutator	C₂²×C₄₂₁	C₂×C₈
Frattini	C₂×C₈	C₂×D₄₂₁
Fitting	C₂×M₁₆×C₄₂₁	C₂
socle	C₂×C₄₂₁	C₂²⋊C₈
prime radical	C₂³×C₄₂₁	C₈
cosocle	C₂×C₈×C₄₂₁	C₂²
hypercenter	C₂×M₁₆	D₄₂₁
hypocenter	C₄₂₁	#349
LCS	G ▷ C₂²×C₄₂₁ ▷ C₂×C₄₂₁ ▷ C₄₂₁
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
1. #27212306	C₂	semidirect
24. C₄₂₁	#349	semidirect

Maximal subgroups

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

subgroup	index
#27212306	2
C₂×M₁₆×C₄₂₁	2
#349	421