Subgroups — #43252714

Order 28544.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#28383356	C₂	1	1	yes	maximal, characteristic
2	#28383356	C₂	1	1	yes	maximal, characteristic
3	#322×C₂₂₃	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂₂₃⋊C₁₆	C₄	1	2	yes	—
5	C₂×C₂₂₃⋊C₁₆	C₄	1	2	yes	—
6	C₄×C₈×C₂₂₃	C₄	1	2	yes	—
7	C₄×C₈×C₂₂₃	C₂²	1	1	yes	characteristic
8	C₂₂₃⋊C₁₆	—	2	2	no	—
9	C₂₂₃⋊C₁₆	C₈	1	4	yes	complemented
10	C₂×C₈×C₂₂₃	—	2	2	no	—
11	C₂×C₈×C₂₂₃	C₂×C₄	1	1	yes	characteristic
12	C₂×C₈×C₂₂₃	C₂×C₄	1	1	yes	characteristic
13	C₄²×C₂₂₃	C₂×C₄	1	1	yes	characteristic
14	C₂×C₄×C₂₂₃	C₄²	1	1	yes	characteristic
15	C₂×C₄×C₂₂₃	M₁₆	1	2	yes	—
16	C₈×C₂₂₃	—	4	2	no	—
17	C₈×C₂₂₃	C₂×C₈	1	2	yes	—
18	C₈×C₂₂₃	M₁₆	1	2	yes	—
19	C₂²×C₂₂₃	#97	1	1	yes	characteristic
20	C₄×C₂₂₃	—	2	2	no	—
21	C₄×C₂₂₃	C₄×C₈	1	1	yes	characteristic
22	C₄×C₂₂₃	#97	1	1	yes	characteristic
23	C₂×C₂₂₃	#322	1	1	yes	characteristic
24	C₂×C₂₂₃	#347	1	2	yes	—
25	C₂₂₃	#1313	1	1	yes	characteristic, complemented, Sylow
26	#1313	—	223	1	no	maximal, Sylow
27	#346	—	223	1	no	—
28	#346	—	223	1	no	—
29	#322	D₂₂₃	1	1	yes	characteristic
30	C₂×C₁₆	—	223	2	no	—
31	C₂×C₁₆	—	223	2	no	—
32	C₄×C₈	Dic₂₂₃	1	2	yes	—
33	C₄×C₈	C₂×D₂₂₃	1	1	yes	characteristic
34	C₂×C₈	—	2	2	no	—
35	C₂×C₈	C₄×D₂₂₃	1	1	yes	characteristic
36	C₂×C₈	C₄×D₂₂₃	1	1	yes	characteristic
37	C₄²	C₂×Dic₂₂₃	1	1	yes	characteristic
38	C₁₆	—	446	2	no	—
39	C₁₆	—	223	4	no	—
40	C₂×C₄	C₄×Dic₂₂₃	1	1	yes	characteristic
41	C₂×C₄	#16813154	1	2	yes	—
42	C₈	—	4	2	no	—
43	C₈	C₈×D₂₂₃	1	2	yes	—
44	C₈	#16813153	1	2	yes	—
45	C₂²	#20297215	1	1	yes	characteristic
46	C₄	—	2	2	no	—
47	C₄	C₈×Dic₂₂₃	1	1	yes	characteristic
48	C₄	#20297214	1	1	yes	characteristic
49	C₂	#28383302	1	1	yes	characteristic
50	C₂	#28383358	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₄×Dic₂₂₃
commutator	C₄×C₂₂₃	C₄×C₈
Frattini	C₄×C₈	C₂×D₂₂₃
Fitting	#322×C₂₂₃	C₂
socle, prime radical	C₂²×C₂₂₃	#97
cosocle	C₄×C₈×C₂₂₃	C₂²
hypercenter	#322	D₂₂₃
hypocenter	C₂₂₃	#1313
LCS	G ▷ C₄×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂×C₄ ◁ #322

Complements

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

N	H ≅ G/N	type
9. C₂₂₃⋊C₁₆	C₈	semidirect
25. C₂₂₃	#1313	semidirect

Maximal subgroups

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

subgroup	index
#28383356	2
#322×C₂₂₃	2
#1313	223