Subgroups — #14582955

Order 2752.

110 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄²×D₄₃	C₂	1	1	yes	maximal, characteristic
2	C₄⋊(C₄₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
3	(C₂×D₄₃)⋊C₈	C₂	1	2	yes	maximal
4	#13063235	C₂	1	2	yes	maximal, complemented
5	C₄⋊C₈×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×D₄₃	C₄	1	1	yes	characteristic
7	C₂×C₄×D₄₃	C₄	1	1	yes	characteristic
8	C₂×C₄×D₄₃	C₂²	1	1	yes	characteristic
9	C₂×C₈×C₄₃	C₂²	1	2	yes	—
10	C₄²×C₄₃	C₂²	1	1	yes	characteristic
11	C₄×Dic₄₃	C₄	1	1	yes	characteristic
12	C₄×Dic₄₃	C₄	1	1	yes	characteristic
13	C₄×Dic₄₃	C₂²	1	1	yes	characteristic
14	C₂×C₄₃⋊C₈	C₂²	1	2	yes	—
15	C₂²×D₄₃	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×C₄₃	C₂³	1	1	yes	characteristic
19	C₂×Dic₄₃	C₂×C₄	1	1	yes	characteristic
20	C₂×Dic₄₃	C₂×C₄	1	1	yes	characteristic
21	C₂×Dic₄₃	C₂×C₄	1	1	yes	characteristic
22	C₄×D₄₃	—	2	2	no	—
23	C₄×D₄₃	—	2	2	no	—
24	C₄×D₄₃	—	2	1	no	—
25	C₄×D₄₃	—	2	1	no	—
26	C₈×C₄₃	—	2	2	no	—
27	C₄₃⋊C₈	—	2	2	no	—
28	C₂²×C₄₃	C₂²×C₄	1	1	yes	characteristic
29	C₂×D₄₃	—	2	1	no	—
30	C₂×D₄₃	—	2	1	no	—
31	C₂×D₄₃	M₁₆	1	2	yes	—
32	C₄×C₄₃	—	2	1	no	—
33	C₄×C₄₃	M₁₆	1	2	yes	—
34	C₄×C₄₃	#41	1	1	yes	characteristic
35	C₄×C₄₃	#41	1	1	yes	characteristic
36	Dic₄₃	—	2	1	no	—
37	Dic₄₃	—	2	1	no	—
38	Dic₄₃	—	2	1	no	—
39	C₂×C₄₃	#117	1	1	yes	characteristic
40	C₂×C₄₃	C₂×M₁₆	1	1	yes	characteristic
41	C₂×C₄₃	C₂×M₁₆	1	1	yes	characteristic
42	D₄₃	—	2	2	no	—
43	#432	—	43	1	no	maximal, Sylow
44	C₄₃	#432	1	1	yes	characteristic, complemented, Sylow
45	C₂×C₄²	—	43	1	no	—
46	C₄⋊C₈	—	43	1	no	—
47	C₄⋊C₈	D₄₃	1	1	yes	characteristic, complemented
48	C₂²⋊C₈	—	43	2	no	—
49	#97	—	43	2	no	—
50	C₂²×C₄	—	43	1	no	—
51	C₂²×C₄	—	43	1	no	—
52	C₂²×C₄	—	43	1	no	—
53	C₂×C₈	—	43	2	no	—
54	C₂×C₈	C₂×D₄₃	1	2	yes	—
55	C₄²	—	43	1	no	—
56	C₄²	—	43	1	no	—
57	C₄²	—	43	1	no	—
58	C₄²	C₂×D₄₃	1	1	yes	characteristic
59	C₂³	—	43	1	no	—
60	C₂×C₄	—	43	1	no	—
61	C₂×C₄	—	43	1	no	—
62	C₂×C₄	—	43	1	no	—
63	C₂×C₄	—	86	2	no	—
64	C₂×C₄	—	86	2	no	—
65	C₂×C₄	—	86	1	no	—
66	C₂×C₄	—	86	1	no	—
67	C₂×C₄	C₄×D₄₃	1	1	yes	characteristic
68	C₂×C₄	C₄×D₄₃	1	1	yes	characteristic
69	C₂×C₄	C₂²×D₄₃	1	1	yes	characteristic
70	C₈	—	2	2	no	—
71	C₈	—	86	2	no	—
72	C₂²	—	43	2	no	—
73	C₂²	—	86	1	no	—
74	C₂²	—	86	1	no	—
75	C₂²	C₂×C₄×D₄₃	1	1	yes	characteristic
76	C₄	—	2	1	no	—
77	C₄	—	86	1	no	—
78	C₄	—	86	1	no	—
79	C₄	—	86	1	no	—
80	C₄	#10621872	1	2	yes	—
81	C₄	#10621900	1	1	yes	characteristic
82	C₄	#10621902	1	1	yes	characteristic
83	C₂	—	86	2	no	—
84	C₂	#13063283	1	1	yes	characteristic
85	C₂	M₁₆×D₄₃	1	1	yes	characteristic
86	C₂	C₂×#10621872	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₄₃
commutator, socle	C₂²×C₄₃	C₂²×C₄
Fitting	C₄⋊C₈×C₄₃	C₂
prime radical	C₂²×D₄₃	C₂×C₄
cosocle	C₂×C₄×C₄₃	C₂³
hypercenter	C₄⋊C₈	D₄₃
hypocenter	C₄₃	#432
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
2. C₄⋊(C₄₃⋊C₈)	C₂	semidirect
4. #13063235	C₂	semidirect
5. C₄⋊C₈×C₄₃	C₂	semidirect
44. C₄₃	#432	semidirect
47. C₄⋊C₈	D₄₃	semidirect

Maximal subgroups

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

subgroup	index
C₄²×D₄₃	2
C₄⋊(C₄₃⋊C₈)	2
(C₂×D₄₃)⋊C₈	2
#13063235	2
C₄⋊C₈×C₄₃	2
#432	43