Subgroups — #43252971

Order 28544.

82 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₂₃⋊C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
2	#28383406	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×D₁₆×C₂₂₃	C₂	1	1	yes	maximal, characteristic
4	#20297238	—	2	1	no	—
5	C₂₂₃⋊C₃₂	—	2	1	no	—
6	C₂×D₈×C₂₂₃	—	2	1	no	—
7	D₁₆×C₂₂₃	—	2	1	no	—
8	D₁₆×C₂₂₃	C₄	1	2	yes	complemented
9	C₂×C₁₆×C₂₂₃	C₂²	1	1	yes	characteristic
10	C₄⋊Dic₂₂₃	—	4	1	no	—
11	C₂×D₄×C₂₂₃	—	4	1	no	—
12	D₈×C₂₂₃	—	2	2	no	—
13	D₈×C₂₂₃	—	4	1	no	—
14	C₂×C₈×C₂₂₃	D₄	1	1	yes	characteristic
15	C₁₆×C₂₂₃	C₂×C₄	1	1	yes	characteristic
16	C₁₆×C₂₂₃	D₄	1	1	yes	characteristic
17	C₂×Dic₂₂₃	—	8	1	no	—
18	C₂³×C₂₂₃	—	8	1	no	—
19	D₄×C₂₂₃	—	4	2	no	—
20	D₄×C₂₂₃	—	8	1	no	—
21	C₂×C₄×C₂₂₃	D₈	1	1	yes	characteristic
22	C₈×C₂₂₃	C₂²⋊C₄	1	1	yes	characteristic
23	C₈×C₂₂₃	SD₁₆	1	1	yes	characteristic
24	C₂²×C₂₂₃	—	8	2	no	—
25	C₂²×C₂₂₃	—	16	1	no	—
26	C₂²×C₂₂₃	—	16	1	no	—
27	C₂²×C₂₂₃	D₁₆	1	1	yes	characteristic
28	C₄×C₂₂₃	#102	1	1	yes	characteristic
29	C₄×C₂₂₃	SD₃₂	1	1	yes	characteristic
30	Dic₂₂₃	—	16	1	no	—
31	C₂×C₂₂₃	—	16	2	no	—
32	C₂×C₂₂₃	#357	1	1	yes	characteristic
33	C₂×C₂₂₃	D₃₂	1	1	yes	characteristic
34	C₂×C₂₂₃	SD₆₄	1	1	yes	characteristic
35	C₂₂₃	#1415	1	1	yes	characteristic, complemented, Sylow
36	#1415	—	223	1	no	maximal, Sylow
37	C₂×D₁₆	D₂₂₃	1	1	yes	characteristic
38	C₂×C₃₂	—	223	1	no	—
39	#366	—	223	1	no	—
40	C₂×D₈	—	2	1	no	—
41	D₁₆	—	2	1	no	—
42	D₁₆	Dic₂₂₃	1	2	yes	complemented
43	C₂×C₁₆	C₂×D₂₂₃	1	1	yes	characteristic
44	#107	—	446	1	no	—
45	C₃₂	—	446	1	no	—
46	C₂×D₄	—	4	1	no	—
47	D₈	—	2	2	no	—
48	D₈	—	4	1	no	—
49	C₂×C₈	C₂²⋊D₂₂₃	1	1	yes	characteristic
50	C₄⋊C₄	—	892	1	no	—
51	C₁₆	C₂×Dic₂₂₃	1	1	yes	characteristic
52	C₁₆	C₂²⋊D₂₂₃	1	1	yes	characteristic
53	C₂³	—	8	1	no	—
54	D₄	—	4	2	no	—
55	D₄	—	8	1	no	—
56	C₂×C₄	—	1784	1	no	—
57	C₂×C₄	#16813155	1	1	yes	characteristic
58	C₈	C₂²⋊Dic₂₂₃	1	1	yes	characteristic
59	C₈	#16813158	1	1	yes	characteristic
60	C₂²	—	8	2	no	—
61	C₂²	—	16	1	no	—
62	C₂²	—	16	1	no	—
63	C₂²	#20297245	1	1	yes	characteristic
64	C₄	—	3568	1	no	—
65	C₄	#20297225	1	1	yes	characteristic
66	C₄	#20297248	1	1	yes	characteristic
67	C₂	—	16	2	no	—
68	C₂	#28383381	1	1	yes	characteristic
69	C₂	#28383415	1	1	yes	characteristic
70	C₂	#28383418	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#20297245
commutator	C₁₆×C₂₂₃	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₂₂₃
Fitting, prime radical	C₂×D₁₆×C₂₂₃	C₂
socle	C₂²×C₂₂₃	D₁₆
cosocle	C₂×C₁₆×C₂₂₃	C₂²
hypercenter	C₂×D₁₆	D₂₂₃
hypocenter	C₂₂₃	#1415
LCS	G ▷ C₁₆×C₂₂₃ ▷ C₈×C₂₂₃ ▷ C₄×C₂₂₃ ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×D₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×C₂₂₃⋊C₃₂	C₂	semidirect
2. #28383406	C₂	semidirect
8. D₁₆×C₂₂₃	C₄	semidirect
35. C₂₂₃	#1415	semidirect
42. D₁₆	Dic₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂₂₃⋊C₃₂	2
#28383406	2
C₂×D₁₆×C₂₂₃	2
#1415	223