Subgroups — #2258×C₂₂₃ #43251273

Order 28544.

106 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#504×C₂₂₃	C₂	1	1	yes	maximal, characteristic
2	M₆₄×C₂₂₃	C₂	1	3	yes	maximal, complemented
3	C₂×C₃₂×C₂₂₃	C₂	1	3	yes	maximal, complemented
4	#131×C₂₂₃	C₄	1	1	yes	characteristic
5	M₃₂×C₂₂₃	C₄	1	3	yes	—
6	C₂×C₁₆×C₂₂₃	C₂²	1	3	yes	—
7	C₃₂×C₂₂₃	C₂²	1	3	yes	—
8	C₃₂×C₂₂₃	C₂²	1	1	yes	characteristic
9	#41×C₂₂₃	C₈	1	1	yes	characteristic
10	M₁₆×C₂₂₃	C₈	1	3	yes	—
11	C₂×C₈×C₂₂₃	C₂×C₄	1	3	yes	—
12	C₁₆×C₂₂₃	C₂×C₄	1	3	yes	—
13	C₁₆×C₂₂₃	C₂³	1	1	yes	characteristic
14	Q₈×C₂₂₃	C₁₆	1	1	yes	characteristic
15	D₄×C₂₂₃	C₁₆	1	3	yes	—
16	C₂×C₄×C₂₂₃	C₂×C₈	1	3	yes	—
17	C₈×C₂₂₃	C₂×C₈	1	3	yes	—
18	C₈×C₂₂₃	C₂²×C₄	1	1	yes	characteristic
19	C₂²×C₂₂₃	C₂×C₁₆	1	3	yes	—
20	C₄×C₂₂₃	C₂×C₁₆	1	3	yes	—
21	C₄×C₂₂₃	C₂²×C₈	1	1	yes	characteristic
22	C₂×C₂₂₃	—	2	3	no	—
23	C₂×C₂₂₃	C₂²×C₁₆	1	1	yes	characteristic
24	C₂₂₃	#2258	1	1	yes	characteristic, direct product factor, Sylow
25	#2258	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
26	#504	C₂×C₂₂₃	1	1	yes	characteristic
27	M₆₄	C₂×C₂₂₃	1	3	yes	complemented
28	C₂×C₃₂	C₂×C₂₂₃	1	3	yes	complemented
29	#131	C₄×C₂₂₃	1	1	yes	characteristic
30	M₃₂	C₄×C₂₂₃	1	3	yes	—
31	C₂×C₁₆	C₂²×C₂₂₃	1	3	yes	—
32	C₃₂	C₂²×C₂₂₃	1	1	yes	characteristic
33	C₃₂	C₂²×C₂₂₃	1	3	yes	—
34	#41	C₈×C₂₂₃	1	1	yes	characteristic
35	M₁₆	C₈×C₂₂₃	1	3	yes	—
36	C₂×C₈	C₂×C₄×C₂₂₃	1	3	yes	—
37	C₁₆	C₂×C₄×C₂₂₃	1	3	yes	—
38	C₁₆	C₂³×C₂₂₃	1	1	yes	characteristic
39	Q₈	C₁₆×C₂₂₃	1	1	yes	characteristic
40	D₄	C₁₆×C₂₂₃	1	3	yes	—
41	C₂×C₄	C₂×C₈×C₂₂₃	1	3	yes	—
42	C₈	C₂×C₈×C₂₂₃	1	3	yes	—
43	C₈	C₂²×C₄×C₂₂₃	1	1	yes	characteristic
44	C₂²	C₂×C₁₆×C₂₂₃	1	3	yes	—
45	C₄	C₂×C₁₆×C₂₂₃	1	3	yes	—
46	C₄	C₂²×C₈×C₂₂₃	1	1	yes	characteristic
47	C₂	—	2	3	no	—
48	C₂	C₂²×C₁₆×C₂₂₃	1	1	yes	characteristic

	subgroup	G/H
center	C₃₂×C₂₂₃	C₂²
commutator	C₂	C₂²×C₁₆×C₂₂₃
Frattini, cosocle	C₁₆	C₂³×C₂₂₃
socle	C₂×C₂₂₃	C₂²×C₁₆
prime radical	#41×C₂₂₃	C₈
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃₂×C₂₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. M₆₄×C₂₂₃	C₂	semidirect
3. C₂×C₃₂×C₂₂₃	C₂	semidirect
24. C₂₂₃	#2258	direct (N × H)
25. #2258	C₂₂₃	direct (N × H)
27. M₆₄	C₂×C₂₂₃	semidirect
28. C₂×C₃₂	C₂×C₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
#504×C₂₂₃	2
M₆₄×C₂₂₃	2
C₂×C₃₂×C₂₂₃	2
#2258	223

Subgroups — #2258×C223 #43251273

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2258×C₂₂₃ #43251273