Subgroups — #43252864

Order 28544.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₂₂₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	#28383333	C₂	1	1	yes	maximal, characteristic
3	#335×C₂₂₃	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂₂₃⋊C₁₆	C₄	1	1	yes	characteristic
5	C₂×C₂₂₃⋊C₁₆	C₄	1	1	yes	characteristic
6	C₄⋊(C₂₂₃⋊C₈)	—	2	1	no	—
7	C₄⋊C₈×C₂₂₃	—	2	1	no	—
8	C₄×C₈×C₂₂₃	C₂²	1	1	yes	characteristic
9	C₂×C₂₂₃⋊C₈	—	4	1	no	—
10	C₂₂₃⋊C₁₆	—	2	1	no	—
11	C₂₂₃⋊C₁₆	C₈	1	2	yes	complemented
12	C₂×C₈×C₂₂₃	—	4	1	no	—
13	C₂×C₈×C₂₂₃	C₂×C₄	1	1	yes	characteristic
14	C₂×C₈×C₂₂₃	Q₈	1	1	yes	characteristic
15	C₄²×C₂₂₃	D₄	1	1	yes	characteristic
16	C₂₂₃⋊C₈	—	8	1	no	—
17	C₂×C₄×C₂₂₃	C₄⋊C₄	1	1	yes	characteristic
18	C₂×C₄×C₂₂₃	D₈	1	1	yes	characteristic
19	C₂×C₄×C₂₂₃	Q₁₆	1	1	yes	characteristic
20	C₈×C₂₂₃	—	2	1	no	—
21	C₈×C₂₂₃	—	8	1	no	—
22	C₈×C₂₂₃	C₂×C₈	1	1	yes	characteristic
23	C₈×C₂₂₃	M₁₆	1	1	yes	characteristic
24	C₂²×C₂₂₃	#107	1	1	yes	characteristic
25	C₄×C₂₂₃	—	2	1	no	—
26	C₄×C₂₂₃	C₄⋊C₈	1	1	yes	characteristic
27	C₄×C₂₂₃	#108	1	1	yes	characteristic
28	C₄×C₂₂₃	D₁₆	1	1	yes	characteristic
29	C₄×C₂₂₃	Q₃₂	1	1	yes	characteristic
30	C₂×C₂₂₃	#335	1	1	yes	characteristic
31	C₂×C₂₂₃	#366	1	1	yes	characteristic
32	C₂×C₂₂₃	#368	1	1	yes	characteristic
33	C₂₂₃	#1371	1	1	yes	characteristic, complemented, Sylow
34	#1371	—	223	1	no	maximal, Sylow
35	C₄×C₁₆	—	223	1	no	—
36	#335	—	223	1	no	—
37	#335	D₂₂₃	1	1	yes	characteristic
38	C₂×C₁₆	—	223	1	no	—
39	C₂×C₁₆	—	223	1	no	—
40	C₄⋊C₈	—	2	1	no	—
41	C₄⋊C₈	—	446	1	no	—
42	C₄×C₈	C₂×D₂₂₃	1	1	yes	characteristic
43	C₂×C₈	—	4	1	no	—
44	C₂×C₈	—	892	1	no	—
45	C₂×C₈	C₄×D₂₂₃	1	1	yes	characteristic
46	C₂×C₈	Dic₄₄₆	1	1	yes	characteristic
47	C₄²	C₂²⋊D₂₂₃	1	1	yes	characteristic
48	C₁₆	—	446	1	no	—
49	C₁₆	—	223	2	no	—
50	C₂×C₄	Dic₂₂₃⋊C₄	1	1	yes	characteristic
51	C₂×C₄	#16813155	1	1	yes	characteristic
52	C₂×C₄	#16813160	1	1	yes	characteristic
53	C₈	—	2	1	no	—
54	C₈	—	8	1	no	—
55	C₈	—	1784	1	no	—
56	C₈	C₈×D₂₂₃	1	1	yes	characteristic
57	C₈	#16813153	1	1	yes	characteristic
58	C₂²	#20297237	1	1	yes	characteristic
59	C₄	—	2	1	no	—
60	C₄	Dic₂₂₃⋊C₈	1	1	yes	characteristic
61	C₄	#20297239	1	1	yes	characteristic
62	C₄	#20297245	1	1	yes	characteristic
63	C₄	#20297250	1	1	yes	characteristic
64	C₂	#28383332	1	1	yes	characteristic
65	C₂	#28383405	1	1	yes	characteristic
66	C₂	#28383409	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#16813155
commutator	C₈×C₂₂₃	C₂×C₈
Frattini	C₄×C₈	C₂×D₂₂₃
Fitting	#335×C₂₂₃	C₂
socle, prime radical	C₂²×C₂₂₃	#107
cosocle	C₄×C₈×C₂₂₃	C₂²
hypercenter	#335	D₂₂₃
hypocenter	C₂₂₃	#1371
LCS	G ▷ C₈×C₂₂₃ ▷ C₄×C₂₂₃ ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ #335

Complements

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

N	H ≅ G/N	type
11. C₂₂₃⋊C₁₆	C₈	semidirect
33. C₂₂₃	#1371	semidirect

Maximal subgroups

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

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