Subgroups — #14583381

Order 2752.

294 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10621902	C₂	1	1	yes	maximal, characteristic
2	Q₈×Dic₄₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×D₁₇₂	C₂	1	3	yes	maximal, complemented
4	#13063287	C₂	1	3	yes	maximal, complemented
5	#13063282	C₂	1	3	yes	maximal, complemented
6	C₄⋊C₄×D₄₃	C₂	1	3	yes	maximal, complemented
7	C₄×Q₈×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Q₈×C₄₃	C₂²	1	1	yes	characteristic
9	#10621902	C₄	1	8	yes	complemented
10	C₂×D₁₇₂	C₂²	1	3	yes	—
11	C₂×C₄×D₄₃	C₂²	1	6	yes	—
12	C₂×C₄×D₄₃	C₂²	1	3	yes	—
13	C₄⋊C₄×C₄₃	C₂²	1	3	yes	—
14	C₄²×C₄₃	C₂²	1	3	yes	—
15	(C₂×D₄₃)⋊C₄	C₂²	1	3	yes	—
16	(C₂×D₄₃)⋊C₄	C₂²	1	3	yes	—
17	C₄⋊Dic₄₃	C₂²	1	3	yes	—
18	Dic₄₃⋊C₄	C₂²	1	3	yes	—
19	Dic₄₃⋊C₄	C₂²	1	1	yes	characteristic
20	C₄×Dic₄₃	C₂²	1	3	yes	—
21	C₂²×D₄₃	C₂³	1	3	yes	—
22	Q₈×C₄₃	C₂×C₄	1	4	yes	complemented
23	C₂×C₄×C₄₃	C₂³	1	1	yes	characteristic
24	C₂×C₄×C₄₃	C₂³	1	3	yes	—
25	C₂×C₄×C₄₃	C₂³	1	3	yes	—
26	C₂×Dic₄₃	C₂³	1	1	yes	characteristic
27	C₂×Dic₄₃	C₂³	1	3	yes	—
28	C₂×Dic₄₃	C₂³	1	1	yes	characteristic
29	D₁₇₂	C₂×C₄	1	12	yes	complemented
30	C₄×D₄₃	—	4	6	no	—
31	C₄×D₄₃	C₂×C₄	1	12	yes	complemented
32	C₂²×C₄₃	C₂⁴	1	1	yes	characteristic
33	C₂×D₄₃	—	2	3	no	—
34	C₂×D₄₃	—	2	3	no	—
35	C₂×D₄₃	C₂²×C₄	1	6	yes	—
36	C₄×C₄₃	—	2	3	no	—
37	C₄×C₄₃	—	2	1	no	—
38	C₄×C₄₃	C₂²×C₄	1	6	yes	—
39	Dic₄₃	—	2	3	no	—
40	Dic₄₃	—	2	1	no	—
41	Dic₄₃	C₂²×C₄	1	2	yes	—
42	C₂×C₄₃	C₂³×C₄	1	1	yes	characteristic
43	C₂×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
44	C₂×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
45	D₄₃	—	2	6	no	—
46	#520	—	43	1	no	maximal, Sylow
47	C₄₃	#520	1	1	yes	characteristic, complemented, Sylow
48	C₂×#41	—	43	1	no	—
49	C₄×Q₈	—	43	1	no	—
50	C₄×Q₈	D₄₃	1	1	yes	characteristic, complemented
51	C₄×D₄	—	43	3	no	—
52	C₄×D₄	—	43	3	no	—
53	#117	—	43	3	no	—
54	C₂×C₄⋊C₄	—	43	3	no	—
55	#41	—	43	8	no	—
56	C₂×Q₈	C₂×D₄₃	1	1	yes	characteristic
57	C₂×D₄	—	43	3	no	—
58	C₂²×C₄	—	43	3	no	—
59	C₂²×C₄	—	43	6	no	—
60	C₄⋊C₄	—	43	3	no	—
61	C₄⋊C₄	—	43	3	no	—
62	C₄⋊C₄	—	43	1	no	—
63	C₄⋊C₄	C₂×D₄₃	1	3	yes	—
64	C₂²⋊C₄	—	43	3	no	—
65	C₂²⋊C₄	—	43	3	no	—
66	C₄²	—	43	3	no	—
67	C₄²	C₂×D₄₃	1	3	yes	—
68	C₂³	—	43	3	no	—
69	Q₈	C₄×D₄₃	1	4	yes	complemented
70	D₄	—	43	12	no	—
71	C₂×C₄	—	43	1	no	—
72	C₂×C₄	—	43	12	no	—
73	C₂×C₄	—	43	1	no	—
74	C₂×C₄	—	43	3	no	—
75	C₂×C₄	—	172	6	no	—
76	C₂×C₄	C₂²×D₄₃	1	1	yes	characteristic
77	C₂×C₄	C₂²×D₄₃	1	3	yes	—
78	C₂×C₄	C₂²×D₄₃	1	3	yes	—
79	C₂²	—	43	6	no	—
80	C₂²	—	86	3	no	—
81	C₂²	—	86	3	no	—
82	C₂²	C₂³×D₄₃	1	1	yes	characteristic
83	C₄	—	2	3	no	—
84	C₄	—	2	1	no	—
85	C₄	—	43	2	no	—
86	C₄	—	86	3	no	—
87	C₄	—	86	1	no	—
88	C₄	C₂×C₄×D₄₃	1	6	yes	—
89	C₂	—	86	6	no	—
90	C₂	C₂²×C₄×D₄₃	1	1	yes	characteristic
91	C₂	#13063371	1	1	yes	characteristic
92	C₂	#13063373	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₄₃
commutator	C₂×C₄₃	C₂³×C₄
Fitting	C₄×Q₈×C₄₃	C₂
socle, cosocle	C₂²×C₄₃	C₂⁴
prime radical	C₂×#10621902	C₂
hypercenter	C₄×Q₈	D₄₃
hypocenter	C₄₃	#520
LCS	G ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
2. Q₈×Dic₄₃	C₂	semidirect
3. C₄×D₁₇₂	C₂	semidirect
4. #13063287	C₂	semidirect
5. #13063282	C₂	semidirect
6. C₄⋊C₄×D₄₃	C₂	semidirect
7. C₄×Q₈×C₄₃	C₂	semidirect
9. #10621902	C₄	semidirect
22. Q₈×C₄₃	C₂×C₄	semidirect
29. D₁₇₂	C₂×C₄	semidirect
31. C₄×D₄₃	C₂×C₄	semidirect
47. C₄₃	#520	semidirect
50. C₄×Q₈	D₄₃	semidirect
69. Q₈	C₄×D₄₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10621902	2
Q₈×Dic₄₃	2
C₄×D₁₇₂	2
#13063287	2
#13063282	2
C₄⋊C₄×D₄₃	2
C₄×Q₈×C₄₃	2
#520	43