Subgroups — #508×C₂₂₃ #28383220

Order 14272.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#135×C₂₂₃	C₂	1	2	yes	maximal, complemented
2	Q₃₂×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	SD₃₂×C₂₂₃	C₂	1	2	yes	maximal, complemented
4	D₁₆×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₁₆×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#41×C₂₂₃	—	2	2	no	—
7	Q₁₆×C₂₂₃	C₂²	1	2	yes	—
8	SD₁₆×C₂₂₃	—	2	2	no	—
9	D₈×C₂₂₃	C₂²	1	2	yes	—
10	C₂×C₈×C₂₂₃	C₂²	1	1	yes	characteristic
11	C₁₆×C₂₂₃	C₂²	1	1	yes	characteristic
12	C₁₆×C₂₂₃	C₂²	1	1	yes	characteristic
13	Q₈×C₂₂₃	—	2	2	no	—
14	D₄×C₂₂₃	—	2	2	no	—
15	D₄×C₂₂₃	—	4	2	no	—
16	C₂×C₄×C₂₂₃	—	4	2	no	—
17	C₂×C₄×C₂₂₃	D₄	1	1	yes	characteristic
18	C₈×C₂₂₃	D₄	1	1	yes	characteristic
19	C₈×C₂₂₃	C₂³	1	1	yes	characteristic
20	C₂²×C₂₂₃	—	4	2	no	—
21	C₂²×C₂₂₃	D₈	1	1	yes	characteristic
22	C₄×C₂₂₃	—	4	2	no	—
23	C₄×C₂₂₃	D₈	1	1	yes	characteristic
24	C₄×C₂₂₃	C₂×D₄	1	1	yes	characteristic
25	C₂×C₂₂₃	—	2	1	no	—
26	C₂×C₂₂₃	—	8	2	no	—
27	C₂×C₂₂₃	C₂×D₈	1	1	yes	characteristic
28	C₂₂₃	#508	1	1	yes	characteristic, direct product factor, Sylow
29	#508	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
30	#135	C₂×C₂₂₃	1	2	yes	complemented
31	Q₃₂	C₂×C₂₂₃	1	1	yes	characteristic, complemented
32	SD₃₂	C₂×C₂₂₃	1	2	yes	complemented
33	D₁₆	C₂×C₂₂₃	1	1	yes	characteristic, complemented
34	C₂×C₁₆	C₂×C₂₂₃	1	1	yes	characteristic, complemented
35	#41	—	2	2	no	—
36	Q₁₆	C₂²×C₂₂₃	1	2	yes	—
37	SD₁₆	—	2	2	no	—
38	D₈	C₂²×C₂₂₃	1	2	yes	—
39	C₂×C₈	C₂²×C₂₂₃	1	1	yes	characteristic
40	C₁₆	C₂²×C₂₂₃	1	1	yes	characteristic
41	C₁₆	C₂²×C₂₂₃	1	1	yes	characteristic
42	Q₈	—	2	2	no	—
43	D₄	—	2	2	no	—
44	D₄	—	4	2	no	—
45	C₂×C₄	—	4	2	no	—
46	C₂×C₄	D₄×C₂₂₃	1	1	yes	characteristic
47	C₈	D₄×C₂₂₃	1	1	yes	characteristic
48	C₈	C₂³×C₂₂₃	1	1	yes	characteristic
49	C₂²	—	4	2	no	—
50	C₂²	D₈×C₂₂₃	1	1	yes	characteristic
51	C₄	—	4	2	no	—
52	C₄	D₈×C₂₂₃	1	1	yes	characteristic
53	C₄	C₂×D₄×C₂₂₃	1	1	yes	characteristic
54	C₂	—	2	1	no	—
55	C₂	—	8	2	no	—
56	C₂	C₂×D₈×C₂₂₃	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₂₂₃	D₈
commutator, Frattini, cosocle	C₈	C₂³×C₂₂₃
socle	C₂×C₂₂₃	C₂×D₈
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₂₂₃ ◁ C₂×C₄×C₂₂₃ ◁ C₂×C₈×C₂₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. #135×C₂₂₃	C₂	semidirect
2. Q₃₂×C₂₂₃	C₂	semidirect
3. SD₃₂×C₂₂₃	C₂	semidirect
4. D₁₆×C₂₂₃	C₂	semidirect
5. C₂×C₁₆×C₂₂₃	C₂	semidirect
28. C₂₂₃	#508	direct (N × H)
29. #508	C₂₂₃	direct (N × H)
30. #135	C₂×C₂₂₃	semidirect
31. Q₃₂	C₂×C₂₂₃	semidirect
32. SD₃₂	C₂×C₂₂₃	semidirect
33. D₁₆	C₂×C₂₂₃	semidirect
34. C₂×C₁₆	C₂×C₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
#135×C₂₂₃	2
Q₃₂×C₂₂₃	2
SD₃₂×C₂₂₃	2
D₁₆×C₂₂₃	2
C₂×C₁₆×C₂₂₃	2
#508	223

Subgroups — #508×C223 #28383220

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #508×C₂₂₃ #28383220