Subgroups — #492×C₂₂₃ #28383204

Order 14272.

142 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×SD₁₆×C₂₂₃	C₂	1	4	yes	maximal, complemented
2	C₄⋊Q₈×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#127×C₂₂₃	C₂	1	1	yes	maximal, characteristic
4	C₄×C₈×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×Q₈×C₂₂₃	C₂²	1	2	yes	—
6	C₂×D₄×C₂₂₃	—	2	2	no	—
7	C₂×D₄×C₂₂₃	C₂²	1	2	yes	—
8	SD₁₆×C₂₂₃	—	2	8	no	—
9	C₂×C₈×C₂₂₃	C₂²	1	2	yes	—
10	C₄⋊C₄×C₂₂₃	—	2	2	no	—
11	C₄²×C₂₂₃	C₂²	1	1	yes	characteristic
12	C₂³×C₂₂₃	—	2	2	no	—
13	Q₈×C₂₂₃	—	2	4	no	—
14	D₄×C₂₂₃	—	2	4	no	—
15	D₄×C₂₂₃	—	4	4	no	—
16	C₂×C₄×C₂₂₃	—	2	2	no	—
17	C₂×C₄×C₂₂₃	D₄	1	2	yes	—
18	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
19	C₈×C₂₂₃	D₄	1	4	yes	complemented
20	C₂²×C₂₂₃	—	4	2	no	—
21	C₂²×C₂₂₃	—	4	4	no	—
22	C₂²×C₂₂₃	C₂×D₄	1	1	yes	characteristic
23	C₄×C₂₂₃	—	4	2	no	—
24	C₄×C₂₂₃	SD₁₆	1	4	yes	complemented
25	C₄×C₂₂₃	C₂×D₄	1	2	yes	—
26	C₂×C₂₂₃	—	8	2	no	—
27	C₂×C₂₂₃	#127	1	1	yes	characteristic
28	C₂×C₂₂₃	C₂×SD₁₆	1	2	yes	—
29	C₂₂₃	#492	1	1	yes	characteristic, direct product factor, Sylow
30	#492	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
31	C₂×SD₁₆	C₂×C₂₂₃	1	4	yes	complemented
32	C₄⋊Q₈	C₂×C₂₂₃	1	1	yes	characteristic, complemented
33	#127	C₂×C₂₂₃	1	1	yes	characteristic
34	C₄×C₈	C₂×C₂₂₃	1	1	yes	characteristic, complemented
35	C₂×Q₈	C₂²×C₂₂₃	1	2	yes	—
36	C₂×D₄	—	2	2	no	—
37	C₂×D₄	C₂²×C₂₂₃	1	2	yes	—
38	SD₁₆	—	2	8	no	—
39	C₂×C₈	C₂²×C₂₂₃	1	2	yes	—
40	C₄⋊C₄	—	2	2	no	—
41	C₄²	C₂²×C₂₂₃	1	1	yes	characteristic
42	C₂³	—	2	2	no	—
43	Q₈	—	2	4	no	—
44	D₄	—	2	4	no	—
45	D₄	—	4	4	no	—
46	C₂×C₄	—	2	2	no	—
47	C₂×C₄	D₄×C₂₂₃	1	2	yes	—
48	C₂×C₄	C₂³×C₂₂₃	1	1	yes	characteristic
49	C₈	D₄×C₂₂₃	1	4	yes	complemented
50	C₂²	—	4	2	no	—
51	C₂²	—	4	4	no	—
52	C₂²	C₂×D₄×C₂₂₃	1	1	yes	characteristic
53	C₄	—	4	2	no	—
54	C₄	SD₁₆×C₂₂₃	1	4	yes	complemented
55	C₄	C₂×D₄×C₂₂₃	1	2	yes	—
56	C₂	—	8	2	no	—
57	C₂	#127×C₂₂₃	1	1	yes	characteristic
58	C₂	C₂×SD₁₆×C₂₂₃	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₂₂₃	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂³×C₂₂₃
prime radical	#127×C₂₂₃	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₂₃ ◁ C₄²×C₂₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×SD₁₆×C₂₂₃	C₂	semidirect
2. C₄⋊Q₈×C₂₂₃	C₂	semidirect
4. C₄×C₈×C₂₂₃	C₂	semidirect
19. C₈×C₂₂₃	D₄	semidirect
24. C₄×C₂₂₃	SD₁₆	semidirect
29. C₂₂₃	#492	direct (N × H)
30. #492	C₂₂₃	direct (N × H)
31. C₂×SD₁₆	C₂×C₂₂₃	semidirect
32. C₄⋊Q₈	C₂×C₂₂₃	semidirect
34. C₄×C₈	C₂×C₂₂₃	semidirect
49. C₈	D₄×C₂₂₃	semidirect
54. C₄	SD₁₆×C₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×SD₁₆×C₂₂₃	2
C₄⋊Q₈×C₂₂₃	2
#127×C₂₂₃	2
C₄×C₈×C₂₂₃	2
#492	223

Subgroups — #492×C223 #28383204

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #492×C₂₂₃ #28383204