Subgroups — #28383357

Order 14272.

46 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20297244	C₂	1	1	yes	maximal, characteristic
2	#20297214	C₂	1	1	yes	maximal, characteristic
3	M₃₂×C₂₂₃	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂₂₃⋊C₈	C₄	1	1	yes	characteristic
5	C₄×Dic₂₂₃	C₄	1	1	yes	characteristic
6	C₂₂₃⋊C₁₆	C₄	1	2	yes	complemented
7	C₂×C₈×C₂₂₃	C₂²	1	1	yes	characteristic
8	C₁₆×C₂₂₃	C₄	1	2	yes	complemented
9	C₂×Dic₂₂₃	—	2	1	no	—
10	C₂₂₃⋊C₈	—	2	1	no	—
11	C₂×C₄×C₂₂₃	C₂×C₄	1	1	yes	characteristic
12	C₈×C₂₂₃	C₂×C₄	1	1	yes	characteristic
13	C₈×C₂₂₃	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₂₂₃	M₁₆	1	1	yes	characteristic
15	C₄×C₂₂₃	C₄²	1	1	yes	characteristic
16	C₄×C₂₂₃	M₁₆	1	1	yes	characteristic
17	Dic₂₂₃	—	4	1	no	—
18	C₂×C₂₂₃	—	2	1	no	—
19	C₂×C₂₂₃	#97	1	1	yes	characteristic
20	C₂₂₃	#347	1	1	yes	characteristic, complemented, Sylow
21	#347	—	223	1	no	maximal, Sylow
22	M₃₂	—	223	1	no	—
23	M₃₂	D₂₂₃	1	1	yes	characteristic
24	#97	—	223	1	no	—
25	C₂×C₈	—	223	1	no	—
26	C₂×C₈	C₂×D₂₂₃	1	1	yes	characteristic
27	C₄²	—	223	1	no	—
28	C₁₆	—	223	2	no	—
29	C₁₆	Dic₂₂₃	1	2	yes	complemented
30	C₂×C₄	—	446	1	no	—
31	C₂×C₄	C₄×D₂₂₃	1	1	yes	characteristic
32	C₈	—	446	1	no	—
33	C₈	C₄×D₂₂₃	1	1	yes	characteristic
34	C₈	C₂×Dic₂₂₃	1	1	yes	characteristic
35	C₂²	#16813153	1	1	yes	characteristic
36	C₄	—	892	1	no	—
37	C₄	C₄×Dic₂₂₃	1	1	yes	characteristic
38	C₄	#16813153	1	1	yes	characteristic
39	C₂	—	2	1	no	—
40	C₂	#20297214	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₄×Dic₂₂₃
commutator	C₄×C₂₂₃	C₄²
Frattini	C₂×C₈	C₂×D₂₂₃
Fitting	M₃₂×C₂₂₃	C₂
socle	C₂×C₂₂₃	#97
prime radical	C₂²×C₂₂₃	M₁₆
cosocle	C₂×C₈×C₂₂₃	C₂²
hypercenter	M₃₂	D₂₂₃
hypocenter	C₂₂₃	#347
LCS	G ▷ C₄×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₄ ◁ M₃₂

Complements

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

N	H ≅ G/N	type
6. C₂₂₃⋊C₁₆	C₄	semidirect
8. C₁₆×C₂₂₃	C₄	semidirect
20. C₂₂₃	#347	semidirect
29. C₁₆	Dic₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
#20297244	2
#20297214	2
M₃₂×C₂₂₃	2
#347	223