Subgroups — #14582768

Order 2752.

186 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊Dic₄₃	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₄₃⋊C₄	C₂	1	4	yes	maximal
3	C₂×C₄×Dic₄₃	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊C₄×C₄₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄×C₄₃	C₂²	1	2	yes	—
6	C₂²×C₄×C₄₃	C₂²	1	1	yes	characteristic
7	C₂²×Dic₄₃	C₂²	1	2	yes	—
8	C₂²×Dic₄₃	C₂²	1	2	yes	—
9	C₄⋊C₄×C₄₃	—	2	2	no	—
10	C₄⋊Dic₄₃	—	2	2	no	—
11	Dic₄₃⋊C₄	—	2	8	no	—
12	C₄×Dic₄₃	C₄	1	4	yes	complemented
13	C₂³×C₄₃	C₂³	1	1	yes	characteristic
14	C₂×C₄×C₄₃	—	2	4	no	—
15	C₂×C₄×C₄₃	—	2	2	no	—
16	C₂×C₄×C₄₃	C₂×C₄	1	2	yes	—
17	C₂×C₄×C₄₃	D₄	1	2	yes	—
18	C₂×C₄×C₄₃	Q₈	1	2	yes	—
19	C₂×Dic₄₃	—	2	4	no	—
20	C₂×Dic₄₃	—	2	2	no	—
21	C₂×Dic₄₃	C₂×C₄	1	4	yes	—
22	C₂×Dic₄₃	D₄	1	4	yes	—
23	C₂×Dic₄₃	Q₈	1	4	yes	—
24	C₂²×C₄₃	C₂²×C₄	1	1	yes	characteristic
25	C₂²×C₄₃	C₂×D₄	1	2	yes	—
26	C₂²×C₄₃	C₂×D₄	1	1	yes	characteristic
27	C₂²×C₄₃	C₂×Q₈	1	1	yes	characteristic
28	C₂²×C₄₃	C₂×Q₈	1	2	yes	—
29	C₄×C₄₃	—	4	2	no	—
30	C₄×C₄₃	C₄⋊C₄	1	4	yes	complemented
31	Dic₄₃	—	4	2	no	—
32	Dic₄₃	C₄⋊C₄	1	8	yes	complemented
33	C₂×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
34	C₂×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₄₃	#127	1	1	yes	characteristic
36	C₂×C₄₃	C₄⋊Q₈	1	1	yes	characteristic
37	C₂×C₄₃	C₄⋊Q₈	1	2	yes	—
38	#384	—	43	1	no	maximal, Sylow
39	C₄₃	#384	1	1	yes	characteristic, complemented, Sylow
40	C₂×C₄⋊C₄	—	43	4	no	—
41	C₂×C₄⋊C₄	—	43	1	no	—
42	C₂×C₄⋊C₄	D₄₃	1	1	yes	characteristic
43	C₂×C₄²	—	43	1	no	—
44	C₂²×C₄	—	43	2	no	—
45	C₂²×C₄	—	43	2	no	—
46	C₂²×C₄	C₂×D₄₃	1	2	yes	—
47	C₂²×C₄	C₂×D₄₃	1	1	yes	characteristic
48	C₄⋊C₄	—	2	2	no	—
49	C₄⋊C₄	—	86	8	no	—
50	C₄⋊C₄	—	86	2	no	—
51	C₄²	—	43	4	no	—
52	C₂³	C₂²×D₄₃	1	1	yes	characteristic
53	C₂×C₄	—	2	4	no	—
54	C₂×C₄	—	2	2	no	—
55	C₂×C₄	—	43	4	no	—
56	C₂×C₄	—	43	4	no	—
57	C₂×C₄	—	43	4	no	—
58	C₂×C₄	—	86	4	no	—
59	C₂×C₄	—	86	2	no	—
60	C₂×C₄	Dic₈₆	1	2	yes	—
61	C₂×C₄	C₄×D₄₃	1	2	yes	—
62	C₂×C₄	C₂²⋊D₄₃	1	2	yes	—
63	C₂²	C₂×Dic₈₆	1	1	yes	characteristic
64	C₂²	C₂×C₄×D₄₃	1	1	yes	characteristic
65	C₂²	D₄×D₄₃	1	2	yes	—
66	C₂²	Q₈×D₄₃	1	2	yes	—
67	C₂²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
68	C₄	—	4	2	no	—
69	C₄	—	43	8	no	—
70	C₄	—	172	2	no	—
71	C₄	Dic₄₃⋊C₄	1	4	yes	complemented
72	C₂	C₄⋊C₄×D₄₃	1	2	yes	—
73	C₂	C₂×Dic₄₃⋊C₄	1	1	yes	characteristic
74	C₂	#13063321	1	1	yes	characteristic
75	C₂	#13063323	1	1	yes	characteristic
76	C₂	#13063324	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₄₃
commutator	C₂²×C₄₃	C₂²×C₄
Fitting	C₂×C₄⋊C₄×C₄₃	C₂
socle, prime radical, cosocle	C₂³×C₄₃	C₂³
hypercenter	C₂×C₄⋊C₄	D₄₃
hypocenter	C₄₃	#384
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
12. C₄×Dic₄₃	C₄	semidirect
30. C₄×C₄₃	C₄⋊C₄	semidirect
32. Dic₄₃	C₄⋊C₄	semidirect
39. C₄₃	#384	semidirect
71. C₄	Dic₄₃⋊C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊Dic₄₃	2
C₂×Dic₄₃⋊C₄	2
C₂×C₄×Dic₄₃	2
C₂×C₄⋊C₄×C₄₃	2
#384	43