Subgroups — #14583086

Order 2752.

198 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13063293	C₂	1	3	yes	maximal, complemented
2	#13063240	C₂	1	3	yes	maximal, complemented
3	H(5,𝔽₂)×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂³×D₄₃	C₂²	1	1	yes	characteristic, complemented
5	#41×C₄₃	—	2	3	no	—
6	C₂×D₄×C₄₃	—	2	3	no	—
7	C₂×D₄×C₄₃	C₂²	1	3	yes	—
8	C₂×C₂²⋊D₄₃	—	2	3	no	—
9	C₂²⋊Dic₄₃	C₂²	1	3	yes	complemented
10	(C₂×D₄₃)⋊C₄	—	2	3	no	—
11	C₂³×C₄₃	—	2	1	no	—
12	C₂³×C₄₃	D₄	1	3	yes	complemented
13	C₂³×C₄₃	C₂³	1	1	yes	characteristic
14	C₂²×D₄₃	—	2	3	no	—
15	C₂²×D₄₃	—	4	2	no	—
16	Q₈×C₄₃	—	2	1	no	—
17	D₄×C₄₃	—	2	3	no	—
18	D₄×C₄₃	—	2	6	no	—
19	C₂×C₄×C₄₃	—	2	3	no	—
20	C₂×C₄×C₄₃	D₄	1	3	yes	complemented
21	C₂²⋊D₄₃	—	4	6	no	—
22	C₂×Dic₄₃	—	2	3	no	—
23	C₂²×C₄₃	—	2	3	no	—
24	C₂²×C₄₃	—	4	1	no	—
25	C₂²×C₄₃	—	4	3	no	—
26	C₂²×C₄₃	—	8	1	no	—
27	C₂²×C₄₃	C₂×D₄	1	3	yes	—
28	C₂×D₄₃	—	2	6	no	—
29	C₂×D₄₃	—	4	3	no	—
30	C₂×D₄₃	—	4	1	no	—
31	C₄×C₄₃	—	2	3	no	—
32	Dic₄₃	—	4	3	no	—
33	C₂×C₄₃	—	2	3	no	—
34	C₂×C₄₃	—	4	3	no	—
35	C₂×C₄₃	#120	1	1	yes	characteristic
36	D₄₃	—	4	2	no	—
37	B(4,𝔽₂)	—	43	1	no	maximal, Sylow
38	C₄₃	B(4,𝔽₂)	1	1	yes	characteristic, complemented, Sylow
39	H(5,𝔽₂)	D₄₃	1	1	yes	characteristic, complemented
40	#120	—	43	3	no	—
41	#99	—	43	3	no	—
42	C₂⁴	—	43	1	no	—
43	#41	—	2	3	no	—
44	C₂×D₄	—	2	3	no	—
45	C₂×D₄	—	86	3	no	—
46	C₂×D₄	C₂×D₄₃	1	3	yes	—
47	C₂²⋊C₄	—	43	3	no	—
48	C₂²⋊C₄	—	86	3	no	—
49	C₂³	—	2	1	no	—
50	C₂³	—	86	3	no	—
51	C₂³	—	172	2	no	—
52	C₂³	C₂²⋊D₄₃	1	3	yes	complemented
53	C₂³	C₂²×D₄₃	1	1	yes	characteristic
54	Q₈	—	2	1	no	—
55	D₄	—	2	3	no	—
56	D₄	—	2	6	no	—
57	D₄	—	172	6	no	—
58	C₂×C₄	—	2	3	no	—
59	C₂×C₄	—	86	3	no	—
60	C₂×C₄	C₂²⋊D₄₃	1	3	yes	complemented
61	C₂²	—	2	3	no	—
62	C₂²	—	4	1	no	—
63	C₂²	—	4	3	no	—
64	C₂²	—	8	1	no	—
65	C₂²	—	86	6	no	—
66	C₂²	—	172	3	no	—
67	C₂²	—	172	1	no	—
68	C₂²	C₂×C₂²⋊D₄₃	1	3	yes	—
69	C₄	—	2	3	no	—
70	C₄	—	172	3	no	—
71	C₂	—	2	3	no	—
72	C₂	—	4	3	no	—
73	C₂	—	172	2	no	—
74	C₂	#13063294	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#13063294
commutator, cosocle	C₂³×C₄₃	C₂³
Frattini	C₂³	C₂²×D₄₃
Fitting	H(5,𝔽₂)×C₄₃	C₂
socle	C₂×C₄₃	#120
hypercenter	H(5,𝔽₂)	D₄₃
hypocenter	C₄₃	B(4,𝔽₂)
LCS	G ▷ C₂³×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ H(5,𝔽₂)

Complements

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

N	H ≅ G/N	type
1. #13063293	C₂	semidirect
2. #13063240	C₂	semidirect
3. H(5,𝔽₂)×C₄₃	C₂	semidirect
4. C₂³×D₄₃	C₂²	semidirect
9. C₂²⋊Dic₄₃	C₂²	semidirect
12. C₂³×C₄₃	D₄	semidirect
20. C₂×C₄×C₄₃	D₄	semidirect
38. C₄₃	B(4,𝔽₂)	semidirect
39. H(5,𝔽₂)	D₄₃	semidirect
52. C₂³	C₂²⋊D₄₃	semidirect
60. C₂×C₄	C₂²⋊D₄₃	semidirect

Maximal subgroups

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

subgroup	index
#13063293	2
#13063240	2
H(5,𝔽₂)×C₄₃	2
B(4,𝔽₂)	43