Subgroups — #41003258

Order 26944.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27212372	C₂	1	1	yes	maximal, characteristic
2	#27212305	C₂	1	2	yes	maximal, complemented
3	C₂×#20269746	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₆×D₄₂₁	C₂	1	2	yes	maximal, complemented
5	M₃₂×C₄₂₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#20269771	C₄	1	1	yes	characteristic
7	#20269755	C₄	1	1	yes	characteristic
8	#20269754	C₄	1	2	yes	—
9	C₈×D₄₂₁	C₂²	1	2	yes	—
10	#20269746	C₂²	1	1	yes	characteristic
11	#20269746	C₂²	1	1	yes	characteristic
12	C₂×C₈×C₄₂₁	C₂²	1	1	yes	characteristic
13	C₁₆×C₄₂₁	C₂²	1	2	yes	—
14	Dic₈₄₂	C₈	1	1	yes	characteristic
15	D₁₆₈₄	C₈	1	1	yes	characteristic
16	C₂²⋊D₄₂₁	C₈	1	2	yes	—
17	C₄×D₄₂₁	C₂×C₄	1	2	yes	—
18	#16809890	C₂×C₄	1	2	yes	—
19	C₂×C₄×C₄₂₁	C₂×C₄	1	1	yes	characteristic
20	C₈×C₄₂₁	C₂×C₄	1	1	yes	characteristic
21	C₈×C₄₂₁	C₂³	1	1	yes	characteristic
22	C₂²×C₄₂₁	C₂×C₈	1	1	yes	characteristic
23	C₂×D₄₂₁	C₂×C₈	1	2	yes	—
24	C₄×C₄₂₁	C₂×C₈	1	1	yes	characteristic
25	C₄×C₄₂₁	C₂²×C₄	1	1	yes	characteristic
26	Dic₄₂₁	C₂×C₈	1	2	yes	—
27	C₂×C₄₂₁	—	2	1	no	—
28	C₂×C₄₂₁	C₂²×C₈	1	1	yes	characteristic
29	D₄₂₁	—	2	2	no	—
30	C₄₂₁	#504	1	1	yes	characteristic, complemented, Sylow
31	#504	—	421	1	no	maximal, Sylow
32	#131	—	421	1	no	—
33	M₃₂	—	421	2	no	—
34	M₃₂	D₄₂₁	1	1	yes	characteristic, complemented
35	C₂×C₁₆	—	421	1	no	—
36	C₂×C₁₆	—	421	2	no	—
37	#41	—	421	1	no	—
38	M₁₆	—	421	1	no	—
39	M₁₆	—	421	2	no	—
40	C₂×C₈	—	421	2	no	—
41	C₂×C₈	C₂×D₄₂₁	1	1	yes	characteristic
42	C₁₆	—	421	1	no	—
43	C₁₆	—	421	1	no	—
44	C₁₆	C₂×D₄₂₁	1	2	yes	—
45	Q₈	—	421	1	no	—
46	D₄	—	421	2	no	—
47	D₄	—	421	1	no	—
48	C₂×C₄	—	421	2	no	—
49	C₂×C₄	C₄×D₄₂₁	1	1	yes	characteristic
50	C₈	—	421	2	no	—
51	C₈	C₄×D₄₂₁	1	1	yes	characteristic
52	C₈	C₂²×D₄₂₁	1	1	yes	characteristic
53	C₂²	—	421	2	no	—
54	C₂²	C₈×D₄₂₁	1	1	yes	characteristic
55	C₄	—	421	2	no	—
56	C₄	C₈×D₄₂₁	1	1	yes	characteristic
57	C₄	C₂×C₄×D₄₂₁	1	1	yes	characteristic
58	C₂	—	2	1	no	—
59	C₂	—	842	2	no	—
60	C₂	C₂×C₈×D₄₂₁	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₈	C₂²×D₄₂₁
commutator, socle	C₂×C₄₂₁	C₂²×C₈
Fitting	M₃₂×C₄₂₁	C₂
prime radical	#20269771	C₄
cosocle	C₈×C₄₂₁	C₂³
hypercenter	M₃₂	D₄₂₁
hypocenter	C₄₂₁	#504
LCS	G ▷ C₂×C₄₂₁ ▷ C₄₂₁
UCS	C₁ ◁ C₈ ◁ M₃₂

Complements

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

N	H ≅ G/N	type
2. #27212305	C₂	semidirect
3. C₂×#20269746	C₂	semidirect
4. C₁₆×D₄₂₁	C₂	semidirect
5. M₃₂×C₄₂₁	C₂	semidirect
30. C₄₂₁	#504	semidirect
34. M₃₂	D₄₂₁	semidirect

Maximal subgroups

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

subgroup	index
#27212372	2
#27212305	2
C₂×#20269746	2
C₁₆×D₄₂₁	2
M₃₂×C₄₂₁	2
#504	421