Subgroups — #43252865

Order 28544.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#28383333	C₂	1	2	yes	maximal
2	C₄×C₁₆×C₂₂₃	C₂	1	1	yes	maximal, characteristic
3	C₄⋊(C₂₂₃⋊C₈)	—	2	2	no	—
4	C₂×C₁₆×C₂₂₃	C₄	1	1	yes	characteristic
5	C₂×C₁₆×C₂₂₃	C₄	1	1	yes	characteristic
6	C₄×C₈×C₂₂₃	C₂²	1	1	yes	characteristic
7	C₂×C₂₂₃⋊C₈	—	4	2	no	—
8	C₂×C₈×C₂₂₃	C₂×C₄	1	1	yes	characteristic
9	C₂×C₈×C₂₂₃	Q₈	1	1	yes	characteristic
10	C₄²×C₂₂₃	D₄	1	1	yes	characteristic
11	C₁₆×C₂₂₃	—	2	1	no	—
12	C₁₆×C₂₂₃	C₈	1	2	yes	complemented
13	C₂₂₃⋊C₈	—	8	2	no	—
14	C₂×C₄×C₂₂₃	C₄⋊C₄	1	1	yes	characteristic
15	C₂×C₄×C₂₂₃	D₈	1	1	yes	characteristic
16	C₂×C₄×C₂₂₃	Q₁₆	1	1	yes	characteristic
17	C₈×C₂₂₃	—	2	1	no	—
18	C₈×C₂₂₃	C₂×C₈	1	1	yes	characteristic
19	C₈×C₂₂₃	M₁₆	1	1	yes	characteristic
20	C₂²×C₂₂₃	#107	1	1	yes	characteristic
21	C₄×C₂₂₃	—	2	1	no	—
22	C₄×C₂₂₃	C₄⋊C₈	1	1	yes	characteristic
23	C₄×C₂₂₃	#108	1	1	yes	characteristic
24	C₄×C₂₂₃	D₁₆	1	1	yes	characteristic
25	C₄×C₂₂₃	Q₃₂	1	1	yes	characteristic
26	C₂×C₂₂₃	#335	1	1	yes	characteristic
27	C₂×C₂₂₃	#366	1	1	yes	characteristic
28	C₂×C₂₂₃	#368	1	1	yes	characteristic
29	C₂₂₃	#1371	1	1	yes	characteristic, complemented, Sylow
30	#1371	—	223	1	no	maximal, Sylow
31	C₄×C₁₆	D₂₂₃	1	1	yes	characteristic
32	#335	—	223	2	no	—
33	C₂×C₁₆	Dic₂₂₃	1	1	yes	characteristic
34	C₂×C₁₆	Dic₂₂₃	1	1	yes	characteristic
35	C₄⋊C₈	—	446	2	no	—
36	C₄×C₈	C₂×D₂₂₃	1	1	yes	characteristic
37	C₂×C₈	—	892	2	no	—
38	C₂×C₈	C₂×Dic₂₂₃	1	1	yes	characteristic
39	C₂×C₈	Dic₄₄₆	1	1	yes	characteristic
40	C₄²	D₈₉₂	1	1	yes	characteristic
41	C₁₆	—	2	1	no	—
42	C₁₆	C₂₂₃⋊C₈	1	2	yes	complemented
43	C₂×C₄	C₄⋊Dic₂₂₃	1	1	yes	characteristic
44	C₂×C₄	D₁₇₈₄	1	1	yes	characteristic
45	C₂×C₄	Dic₈₉₂	1	1	yes	characteristic
46	C₈	—	2	1	no	—
47	C₈	—	1784	2	no	—
48	C₈	C₂×C₂₂₃⋊C₈	1	1	yes	characteristic
49	C₈	#16813154	1	1	yes	characteristic
50	C₂²	#20297238	1	1	yes	characteristic
51	C₄	—	2	1	no	—
52	C₄	C₄⋊(C₂₂₃⋊C₈)	1	1	yes	characteristic
53	C₄	#20297240	1	1	yes	characteristic
54	C₄	D₃₅₆₈	1	1	yes	characteristic
55	C₄	Dic₁₇₈₄	1	1	yes	characteristic
56	C₂	#28383333	1	1	yes	characteristic
57	C₂	#28383406	1	1	yes	characteristic
58	C₂	#28383410	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₁₇₈₄
commutator	C₈×C₂₂₃	C₂×C₈
Frattini	C₄×C₈	C₂×D₂₂₃
Fitting	C₄×C₁₆×C₂₂₃	C₂
socle, prime radical	C₂²×C₂₂₃	#107
cosocle	C₄×C₈×C₂₂₃	C₂²
hypercenter	C₄×C₁₆	D₂₂₃
hypocenter	C₂₂₃	#1371
LCS	G ▷ C₈×C₂₂₃ ▷ C₄×C₂₂₃ ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
12. C₁₆×C₂₂₃	C₈	semidirect
29. C₂₂₃	#1371	semidirect
42. C₁₆	C₂₂₃⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#28383333	2
C₄×C₁₆×C₂₂₃	2
#1371	223