Subgroups — #43252927

Order 28544.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₂₃⋊C₃₂	C₂	1	2	yes	maximal
2	C₄×C₁₆×C₂₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂₂₃⋊C₃₂	C₄	1	4	yes	complemented
4	C₂×C₁₆×C₂₂₃	C₄	1	1	yes	characteristic
5	C₂×C₁₆×C₂₂₃	C₂²	1	1	yes	characteristic
6	C₄×C₈×C₂₂₃	C₄	1	1	yes	characteristic
7	C₂×C₈×C₂₂₃	C₈	1	1	yes	characteristic
8	C₂×C₈×C₂₂₃	C₂×C₄	1	1	yes	characteristic
9	C₄²×C₂₂₃	C₈	1	1	yes	characteristic
10	C₁₆×C₂₂₃	C₈	1	2	yes	—
11	C₁₆×C₂₂₃	C₂×C₄	1	2	yes	—
12	C₂×C₄×C₂₂₃	C₁₆	1	2	yes	—
13	C₂×C₄×C₂₂₃	C₂×C₈	1	1	yes	characteristic
14	C₈×C₂₂₃	C₁₆	1	2	yes	—
15	C₈×C₂₂₃	C₄²	1	1	yes	characteristic
16	C₈×C₂₂₃	C₂×C₈	1	1	yes	characteristic
17	C₂²×C₂₂₃	C₂×C₁₆	1	1	yes	characteristic
18	C₄×C₂₂₃	—	2	2	no	—
19	C₄×C₂₂₃	C₄×C₈	1	1	yes	characteristic
20	C₄×C₂₂₃	C₂×C₁₆	1	1	yes	characteristic
21	C₂×C₂₂₃	C₄×C₁₆	1	1	yes	characteristic
22	C₂×C₂₂₃	M₆₄	1	2	yes	—
23	C₂₂₃	#1397	1	1	yes	characteristic, complemented, Sylow
24	#1397	—	223	1	no	maximal, Sylow
25	C₂×C₃₂	—	223	2	no	—
26	C₄×C₁₆	D₂₂₃	1	1	yes	characteristic
27	C₂×C₁₆	Dic₂₂₃	1	1	yes	characteristic
28	C₂×C₁₆	C₂×D₂₂₃	1	1	yes	characteristic
29	C₄×C₈	Dic₂₂₃	1	1	yes	characteristic
30	C₃₂	—	223	4	no	—
31	C₂×C₈	C₂₂₃⋊C₈	1	1	yes	characteristic
32	C₂×C₈	C₂×Dic₂₂₃	1	1	yes	characteristic
33	C₄²	C₂₂₃⋊C₈	1	1	yes	characteristic
34	C₁₆	C₂₂₃⋊C₈	1	2	yes	—
35	C₁₆	C₄×D₂₂₃	1	2	yes	—
36	C₂×C₄	C₂₂₃⋊C₁₆	1	2	yes	—
37	C₂×C₄	C₂×C₂₂₃⋊C₈	1	1	yes	characteristic
38	C₈	C₂₂₃⋊C₁₆	1	2	yes	—
39	C₈	C₄×Dic₂₂₃	1	1	yes	characteristic
40	C₈	C₂×C₂₂₃⋊C₈	1	1	yes	characteristic
41	C₂²	C₂×C₂₂₃⋊C₁₆	1	1	yes	characteristic
42	C₄	—	2	2	no	—
43	C₄	C₄×C₂₂₃⋊C₈	1	1	yes	characteristic
44	C₄	C₂×C₂₂₃⋊C₁₆	1	1	yes	characteristic
45	C₂	C₄×C₂₂₃⋊C₁₆	1	1	yes	characteristic
46	C₂	#28383414	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₁₆	C₂×D₂₂₃
commutator	C₂×C₂₂₃	C₄×C₁₆
Fitting	C₄×C₁₆×C₂₂₃	C₂
socle, prime radical	C₂²×C₂₂₃	C₂×C₁₆
cosocle	C₂×C₁₆×C₂₂₃	C₂²
hypercenter	C₄×C₁₆	D₂₂₃
hypocenter	C₂₂₃	#1397
LCS	G ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂×C₁₆ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
3. C₂₂₃⋊C₃₂	C₄	semidirect
23. C₂₂₃	#1397	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂₂₃⋊C₃₂	2
C₄×C₁₆×C₂₂₃	2
#1397	223