Subgroups — #42122679

Order 27712.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#28357299	C₂	1	2	yes	maximal
2	C₂×#20294080	C₂	1	1	yes	maximal, characteristic
3	C₄⋊#13181884	—	2	2	no	—
4	C₂²⋊#13181884	—	2	2	no	—
5	C₄×#13181884	C₄	1	4	yes	complemented
6	C₂×C₄×D₄₃₃	C₂²	1	1	yes	characteristic
7	#20294080	—	2	2	no	—
8	C₂×#16812358	C₄	1	1	yes	characteristic, complemented
9	C₂×C₈×C₄₃₃	C₄	1	1	yes	characteristic, complemented
10	C₂×#13181884	—	2	4	no	—
11	C₂²×D₄₃₃	D₄	1	1	yes	characteristic
12	C₂×Dic₄₃₃	Q₈	1	1	yes	characteristic
13	C₄×D₄₃₃	C₂×C₄	1	2	yes	—
14	C₄×D₄₃₃	D₄	1	2	yes	—
15	#16812358	—	2	1	no	—
16	C₂×C₄×C₄₃₃	C₂×C₄	1	1	yes	characteristic
17	C₈×C₄₃₃	—	2	1	no	—
18	C₂²×C₄₃₃	C₄⋊C₄	1	1	yes	characteristic
19	C₂×D₄₃₃	—	4	1	no	—
20	C₂×D₄₃₃	C₂²⋊C₄	1	2	yes	—
21	#13181884	—	4	4	no	—
22	C₄×C₄₃₃	C₄²	1	1	yes	characteristic
23	C₄×C₄₃₃	C₂²⋊C₄	1	1	yes	characteristic
24	Dic₄₃₃	C₄⋊C₄	1	2	yes	—
25	C₂×C₄₃₃	—	2	1	no	—
26	C₂×C₄₃₃	#95	1	1	yes	characteristic
27	D₄₃₃	—	2	2	no	—
28	C₄₃₃	H(3,ℤ/4)	1	1	yes	characteristic, complemented, Sylow
29	H(3,ℤ/4)	—	433	1	no	maximal, Sylow
30	C₂×M₁₆	—	433	1	no	—
31	#117	—	433	2	no	—
32	C₂²×C₄	—	433	1	no	—
33	M₁₆	—	866	2	no	—
34	C₂×C₈	—	433	1	no	—
35	C₂×C₈	#13181884	1	1	yes	characteristic, complemented
36	C₄⋊C₄	—	866	2	no	—
37	C₂²⋊C₄	—	866	2	no	—
38	C₄²	—	433	4	no	—
39	C₂³	—	433	1	no	—
40	C₂×C₄	—	866	4	no	—
41	C₂×C₄	—	433	1	no	—
42	C₂×C₄	—	433	2	no	—
43	C₂×C₄	—	433	2	no	—
44	C₂×C₄	C₂×#13181884	1	1	yes	characteristic
45	C₈	—	2	1	no	—
46	C₈	—	866	1	no	—
47	C₂²	—	433	2	no	—
48	C₂²	—	1732	1	no	—
49	C₂²	C₄⋊#13181884	1	1	yes	characteristic
50	C₄	—	433	2	no	—
51	C₄	—	1732	4	no	—
52	C₄	C₄×#13181884	1	1	yes	characteristic
53	C₄	C₂²⋊#13181884	1	1	yes	characteristic
54	C₂	—	2	1	no	—
55	C₂	—	866	2	no	—
56	C₂	(C₂×#13181884)⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₄×#13181884
commutator	C₄×C₄₃₃	C₄²
Frattini	C₂×C₄	C₂×#13181884
Fitting	C₂×C₈×C₄₃₃	C₄
socle	C₂×C₄₃₃	#95
prime radical	C₂²×D₄₃₃	D₄
cosocle	C₂×C₄×D₄₃₃	C₂²
hypercenter	C₂×C₈	#13181884
hypocenter	C₄₃₃	H(3,ℤ/4)
LCS	G ▷ C₄×C₄₃₃ ▷ C₄₃₃
UCS	C₁ ◁ C₄ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
5. C₄×#13181884	C₄	semidirect
8. C₂×#16812358	C₄	semidirect
9. C₂×C₈×C₄₃₃	C₄	semidirect
28. C₄₃₃	H(3,ℤ/4)	semidirect
35. C₂×C₈	#13181884	semidirect

Maximal subgroups

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

subgroup	index
#28357299	2
C₂×#20294080	2
H(3,ℤ/4)	433