Subgroups — #14583337

Order 2752.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13063347	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₃₄₄	C₂	1	1	yes	maximal, characteristic, complemented
3	#13063270	C₂	1	2	yes	maximal, complemented
4	D₆₈₈	C₂	1	2	yes	maximal, complemented
5	M₃₂×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#10621898	—	2	1	no	—
7	C₂×D₁₇₂	—	2	1	no	—
8	C₂×C₈×C₄₃	C₂²	1	1	yes	characteristic
9	Dic₁₇₂	C₂²	1	1	yes	characteristic, complemented
10	D₃₄₄	—	2	1	no	—
11	D₃₄₄	C₂²	1	2	yes	—
12	D₃₄₄	C₂²	1	1	yes	characteristic, complemented
13	#10621873	—	2	1	no	—
14	C₁₆×C₄₃	C₂²	1	2	yes	complemented
15	C₂²×D₄₃	—	4	1	no	—
16	C₂×C₄×C₄₃	D₄	1	1	yes	characteristic
17	C₂²⋊D₄₃	—	4	1	no	—
18	D₁₇₂	—	2	1	no	—
19	D₁₇₂	—	2	2	no	—
20	D₁₇₂	—	4	1	no	—
21	C₄×D₄₃	—	4	1	no	—
22	Dic₈₆	—	2	1	no	—
23	C₈×C₄₃	D₄	1	1	yes	characteristic
24	C₈×C₄₃	C₂³	1	1	yes	characteristic
25	C₂²×C₄₃	D₈	1	1	yes	characteristic
26	C₂×D₄₃	—	4	2	no	—
27	C₂×D₄₃	—	4	1	no	—
28	C₂×D₄₃	—	8	2	no	—
29	C₄×C₄₃	D₈	1	1	yes	characteristic
30	C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
31	Dic₄₃	—	4	1	no	—
32	C₂×C₄₃	—	2	1	no	—
33	C₂×C₄₃	C₂×D₈	1	1	yes	characteristic
34	D₄₃	—	8	1	no	—
35	D₄₃	—	8	2	no	—
36	#509	—	43	1	no	maximal, Sylow
37	C₄₃	#509	1	1	yes	characteristic, complemented, Sylow
38	#135	—	43	1	no	—
39	C₂×D₈	—	43	1	no	—
40	SD₃₂	—	43	2	no	—
41	D₁₆	—	43	2	no	—
42	M₃₂	D₄₃	1	1	yes	characteristic, complemented
43	#41	—	86	1	no	—
44	C₂×D₄	—	86	1	no	—
45	Q₁₆	—	43	1	no	—
46	SD₁₆	—	86	1	no	—
47	D₈	—	43	1	no	—
48	D₈	—	43	2	no	—
49	D₈	—	86	1	no	—
50	C₂×C₈	C₂×D₄₃	1	1	yes	characteristic
51	C₁₆	C₂×D₄₃	1	2	yes	complemented
52	C₂³	—	172	1	no	—
53	Q₈	—	86	1	no	—
54	D₄	—	86	2	no	—
55	D₄	—	86	1	no	—
56	D₄	—	172	1	no	—
57	D₄	—	172	1	no	—
58	C₂×C₄	—	172	1	no	—
59	C₂×C₄	D₁₇₂	1	1	yes	characteristic
60	C₈	D₁₇₂	1	1	yes	characteristic
61	C₈	C₂²×D₄₃	1	1	yes	characteristic
62	C₂²	—	344	2	no	—
63	C₂²	—	172	1	no	—
64	C₂²	—	172	2	no	—
65	C₂²	D₃₄₄	1	1	yes	characteristic
66	C₄	—	172	1	no	—
67	C₄	D₃₄₄	1	1	yes	characteristic
68	C₄	C₂×D₁₇₂	1	1	yes	characteristic
69	C₂	—	2	1	no	—
70	C₂	—	344	2	no	—
71	C₂	—	344	1	no	—
72	C₂	C₂×D₃₄₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₃₄₄
commutator, cosocle	C₈×C₄₃	C₂³
Frattini	C₈	C₂²×D₄₃
Fitting	M₃₂×C₄₃	C₂
socle	C₂×C₄₃	C₂×D₈
hypercenter	M₃₂	D₄₃
hypocenter	C₄₃	#509
LCS	G ▷ C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ M₃₂

Complements

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

N	H ≅ G/N	type
1. #13063347	C₂	semidirect
2. C₂×D₃₄₄	C₂	semidirect
3. #13063270	C₂	semidirect
4. D₆₈₈	C₂	semidirect
5. M₃₂×C₄₃	C₂	semidirect
9. Dic₁₇₂	C₂²	semidirect
12. D₃₄₄	C₂²	semidirect
14. C₁₆×C₄₃	C₂²	semidirect
37. C₄₃	#509	semidirect
42. M₃₂	D₄₃	semidirect
51. C₁₆	C₂×D₄₃	semidirect

Maximal subgroups

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

subgroup	index
#13063347	2
C₂×D₃₄₄	2
#13063270	2
D₆₈₈	2
M₃₂×C₄₃	2
#509	43