Subgroups — #14583534

Order 2752.

428 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₄×D₄₃	C₂	1	4	yes	maximal, complemented
2	#13063296	C₂	1	4	yes	maximal, complemented
3	#13063293	C₂	1	4	yes	maximal, complemented
4	C₄×D₁₇₂	C₂	1	2	yes	maximal, complemented
5	#127×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×D₄₃	C₂²	1	4	yes	complemented
7	C₂×D₄×C₄₃	C₂²	1	4	yes	complemented
8	C₂×D₄×C₄₃	C₂²	1	2	yes	complemented
9	C₂×C₂²⋊D₄₃	C₂²	1	8	yes	complemented
10	D₄×D₄₃	—	2	16	no	—
11	C₂×D₁₇₂	C₂²	1	2	yes	—
12	C₂×C₄×D₄₃	C₂²	1	4	yes	complemented
13	C₄²×C₄₃	C₂²	1	1	yes	characteristic, complemented
14	C₂²⋊Dic₄₃	C₂²	1	4	yes	complemented
15	(C₂×D₄₃)⋊C₄	C₂²	1	4	yes	complemented
16	C₄⋊Dic₄₃	C₂²	1	2	yes	complemented
17	C₂³×C₄₃	C₂³	1	4	yes	—
18	C₂²×D₄₃	—	2	16	no	—
19	C₂²×D₄₃	—	4	4	no	—
20	C₂²×D₄₃	C₂³	1	4	yes	—
21	D₄×C₄₃	—	2	8	no	—
22	D₄×C₄₃	—	4	2	no	—
23	C₂×C₄×C₄₃	C₂³	1	1	yes	characteristic
24	C₂×C₄×C₄₃	C₂³	1	2	yes	—
25	C₂²⋊D₄₃	—	2	16	no	—
26	C₂×Dic₄₃	C₂³	1	4	yes	—
27	D₁₇₂	D₄	1	8	yes	complemented
28	C₄×D₄₃	—	2	8	no	—
29	C₂²×C₄₃	—	2	8	no	—
30	C₂²×C₄₃	—	2	4	no	—
31	C₂²×C₄₃	C₂⁴	1	1	yes	characteristic
32	C₂×D₄₃	—	2	4	no	—
33	C₂×D₄₃	—	2	4	no	—
34	C₂×D₄₃	—	4	16	no	—
35	C₂×D₄₃	C₂×D₄	1	8	yes	complemented
36	C₄×C₄₃	—	2	1	no	—
37	C₄×C₄₃	C₂×D₄	1	4	yes	complemented
38	Dic₄₃	—	2	4	no	—
39	C₂×C₄₃	—	4	4	no	—
40	C₂×C₄₃	C₂²×D₄	1	2	yes	—
41	C₂×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
42	D₄₃	—	2	8	no	—
43	D₄²	—	43	1	no	maximal, Sylow
44	C₄₃	D₄²	1	1	yes	characteristic, complemented, Sylow
45	C₂²×D₄	—	43	4	no	—
46	#127	D₄₃	1	1	yes	characteristic, complemented
47	#121	—	43	4	no	—
48	#120	—	43	4	no	—
49	C₄×D₄	—	43	2	no	—
50	C₂⁴	—	43	4	no	—
51	C₂×D₄	—	43	8	no	—
52	C₂×D₄	—	43	2	no	—
53	C₂×D₄	—	86	16	no	—
54	C₂×D₄	C₂×D₄₃	1	4	yes	complemented
55	C₂×D₄	C₂×D₄₃	1	2	yes	complemented
56	C₂²×C₄	—	43	4	no	—
57	C₄⋊C₄	—	43	2	no	—
58	C₂²⋊C₄	—	43	4	no	—
59	C₂²⋊C₄	—	43	4	no	—
60	C₄²	C₂×D₄₃	1	1	yes	characteristic, complemented
61	C₂³	—	43	4	no	—
62	C₂³	—	86	16	no	—
63	C₂³	—	172	4	no	—
64	C₂³	C₂²×D₄₃	1	4	yes	—
65	D₄	—	2	8	no	—
66	D₄	—	4	2	no	—
67	D₄	—	43	8	no	—
68	D₄	—	86	16	no	—
69	C₂×C₄	—	43	4	no	—
70	C₂×C₄	—	86	8	no	—
71	C₂×C₄	C₂²×D₄₃	1	1	yes	characteristic
72	C₂×C₄	C₂²×D₄₃	1	2	yes	—
73	C₂²	—	2	8	no	—
74	C₂²	—	2	4	no	—
75	C₂²	—	43	8	no	—
76	C₂²	—	86	4	no	—
77	C₂²	—	86	4	no	—
78	C₂²	—	172	16	no	—
79	C₂²	C₂³×D₄₃	1	1	yes	characteristic
80	C₄	—	2	1	no	—
81	C₄	—	86	4	no	—
82	C₄	D₄×D₄₃	1	4	yes	complemented
83	C₂	—	4	4	no	—
84	C₂	—	86	8	no	—
85	C₂	C₂×D₄×D₄₃	1	2	yes	—
86	C₂	#13063370	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₄₃
commutator, socle, cosocle	C₂²×C₄₃	C₂⁴
Fitting	#127×C₄₃	C₂
hypercenter	#127	D₄₃
hypocenter	C₄₃	D₄²
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ #127

Complements

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

N	H ≅ G/N	type
1. C₂×D₄×D₄₃	C₂	semidirect
2. #13063296	C₂	semidirect
3. #13063293	C₂	semidirect
4. C₄×D₁₇₂	C₂	semidirect
5. #127×C₄₃	C₂	semidirect
6. C₂³×D₄₃	C₂²	semidirect
7. C₂×D₄×C₄₃	C₂²	semidirect
8. C₂×D₄×C₄₃	C₂²	semidirect
9. C₂×C₂²⋊D₄₃	C₂²	semidirect
12. C₂×C₄×D₄₃	C₂²	semidirect
13. C₄²×C₄₃	C₂²	semidirect
14. C₂²⋊Dic₄₃	C₂²	semidirect
15. (C₂×D₄₃)⋊C₄	C₂²	semidirect
16. C₄⋊Dic₄₃	C₂²	semidirect
27. D₁₇₂	D₄	semidirect
35. C₂×D₄₃	C₂×D₄	semidirect
37. C₄×C₄₃	C₂×D₄	semidirect
44. C₄₃	D₄²	semidirect
46. #127	D₄₃	semidirect
54. C₂×D₄	C₂×D₄₃	semidirect
55. C₂×D₄	C₂×D₄₃	semidirect
60. C₄²	C₂×D₄₃	semidirect
82. C₄	D₄×D₄₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₄×D₄₃	2
#13063296	2
#13063293	2
C₄×D₁₇₂	2
#127×C₄₃	2
D₄²	43