Subgroups — #43252709

Order 28544.

96 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	C₈×C₂₂₃⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₄×C₁₆×C₂₂₃	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂₂₃⋊C₁₆	C₄	1	2	yes	—
5	C₄×C₂₂₃⋊C₈	C₄	1	2	yes	—
6	C₂×C₁₆×C₂₂₃	C₄	1	2	yes	—
7	C₄×C₈×C₂₂₃	C₂²	1	1	yes	characteristic
8	C₂×C₂₂₃⋊C₈	C₈	1	4	yes	—
9	C₂₂₃⋊C₁₆	C₈	1	4	yes	complemented
10	C₂×C₈×C₂₂₃	C₂×C₄	1	1	yes	characteristic
11	C₂×C₈×C₂₂₃	C₂×C₄	1	1	yes	characteristic
12	C₄²×C₂₂₃	C₂×C₄	1	1	yes	characteristic
13	C₁₆×C₂₂₃	C₈	1	4	yes	complemented
14	C₂₂₃⋊C₈	—	2	4	no	—
15	C₂×C₄×C₂₂₃	C₄²	1	1	yes	characteristic
16	C₂×C₄×C₂₂₃	C₂×C₈	1	2	yes	—
17	C₈×C₂₂₃	C₂×C₈	1	2	yes	—
18	C₈×C₂₂₃	C₂×C₈	1	2	yes	—
19	C₂²×C₂₂₃	C₄×C₈	1	1	yes	characteristic
20	C₄×C₂₂₃	C₄×C₈	1	1	yes	characteristic
21	C₄×C₂₂₃	C₄×C₈	1	1	yes	characteristic
22	C₄×C₂₂₃	M₃₂	1	4	yes	—
23	C₂×C₂₂₃	C₈²	1	1	yes	characteristic
24	C₂×C₂₂₃	#346	1	2	yes	—
25	C₂₂₃	#1311	1	1	yes	characteristic, complemented, Sylow
26	#1311	—	223	1	no	maximal, Sylow
27	C₄×C₁₆	—	223	1	no	—
28	C₄×C₁₆	D₂₂₃	1	1	yes	characteristic
29	C₈²	—	223	1	no	—
30	C₂×C₁₆	—	223	2	no	—
31	C₂×C₁₆	Dic₂₂₃	1	2	yes	—
32	C₄×C₈	—	223	2	no	—
33	C₄×C₈	C₂×D₂₂₃	1	1	yes	characteristic
34	C₂×C₈	—	223	4	no	—
35	C₂×C₈	C₄×D₂₂₃	1	1	yes	characteristic
36	C₂×C₈	C₂×Dic₂₂₃	1	1	yes	characteristic
37	C₄²	C₄×D₂₂₃	1	1	yes	characteristic
38	C₁₆	—	223	4	no	—
39	C₁₆	C₂₂₃⋊C₈	1	4	yes	complemented
40	C₂×C₄	C₄×Dic₂₂₃	1	1	yes	characteristic
41	C₂×C₄	C₈×D₂₂₃	1	2	yes	—
42	C₈	—	446	4	no	—
43	C₈	C₈×D₂₂₃	1	2	yes	—
44	C₈	C₂×C₂₂₃⋊C₈	1	2	yes	—
45	C₂²	C₈×Dic₂₂₃	1	1	yes	characteristic
46	C₄	C₈×Dic₂₂₃	1	1	yes	characteristic
47	C₄	C₄×C₂₂₃⋊C₈	1	1	yes	characteristic
48	C₄	#20297243	1	4	yes	—
49	C₂	C₈×C₂₂₃⋊C₈	1	1	yes	characteristic
50	C₂	#28383355	1	2	yes	—

	subgroup	G/H
center, Frattini	C₄×C₈	C₂×D₂₂₃
commutator	C₂×C₂₂₃	C₈²
Fitting	C₄×C₁₆×C₂₂₃	C₂
socle, prime radical	C₂²×C₂₂₃	C₄×C₈
cosocle	C₄×C₈×C₂₂₃	C₂²
hypercenter	C₄×C₁₆	D₂₂₃
hypocenter	C₂₂₃	#1311
LCS	G ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₄×C₈ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
9. C₂₂₃⋊C₁₆	C₈	semidirect
13. C₁₆×C₂₂₃	C₈	semidirect
25. C₂₂₃	#1311	semidirect
39. 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
C₄×C₂₂₃⋊C₁₆	2
C₈×C₂₂₃⋊C₈	2
C₄×C₁₆×C₂₂₃	2
#1311	223