Subgroups — #424×C₂₂₃ #28383136

Order 14272.

114 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₁₆×C₂₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₈×C₂₂₃	C₂	1	1	yes	maximal, characteristic
3	#117×C₂₂₃	C₂	1	1	yes	maximal, characteristic
4	C₄⋊C₈×C₂₂₃	C₂	1	4	yes	maximal, complemented
5	C₂²×C₄×C₂₂₃	C₂²	1	1	yes	characteristic
6	M₁₆×C₂₂₃	—	2	2	no	—
7	C₂×C₈×C₂₂₃	—	2	2	no	—
8	C₂×C₈×C₂₂₃	C₂²	1	2	yes	—
9	C₂×C₈×C₂₂₃	C₂²	1	2	yes	—
10	C₄⋊C₄×C₂₂₃	C₄	1	2	yes	—
11	C₂²⋊C₄×C₂₂₃	C₄	1	2	yes	—
12	C₄²×C₂₂₃	C₂²	1	2	yes	—
13	C₂³×C₂₂₃	C₂×C₄	1	1	yes	characteristic
14	C₂×C₄×C₂₂₃	C₂×C₄	1	4	yes	—
15	C₂×C₄×C₂₂₃	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄×C₂₂₃	D₄	1	2	yes	—
17	C₂×C₄×C₂₂₃	Q₈	1	2	yes	—
18	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
19	C₈×C₂₂₃	—	2	2	no	—
20	C₈×C₂₂₃	—	2	2	no	—
21	C₂²×C₂₂₃	—	2	2	no	—
22	C₂²×C₂₂₃	C₄⋊C₄	1	2	yes	—
23	C₂²×C₂₂₃	C₂²×C₄	1	1	yes	characteristic
24	C₄×C₂₂₃	—	2	4	no	—
25	C₄×C₂₂₃	C₄⋊C₄	1	2	yes	—
26	C₄×C₂₂₃	C₂×D₄	1	1	yes	characteristic
27	C₄×C₂₂₃	C₂×Q₈	1	1	yes	characteristic
28	C₂×C₂₂₃	—	2	2	no	—
29	C₂×C₂₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
30	C₂×C₂₂₃	#131	1	2	yes	—
31	C₂₂₃	#424	1	1	yes	characteristic, direct product factor, Sylow
32	#424	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
33	C₂×M₁₆	C₂×C₂₂₃	1	1	yes	characteristic
34	C₂²×C₈	C₂×C₂₂₃	1	1	yes	characteristic
35	#117	C₂×C₂₂₃	1	1	yes	characteristic
36	C₄⋊C₈	C₂×C₂₂₃	1	4	yes	complemented
37	C₂²×C₄	C₂²×C₂₂₃	1	1	yes	characteristic
38	M₁₆	—	2	2	no	—
39	C₂×C₈	—	2	2	no	—
40	C₂×C₈	C₂²×C₂₂₃	1	2	yes	—
41	C₂×C₈	C₂²×C₂₂₃	1	2	yes	—
42	C₄⋊C₄	C₄×C₂₂₃	1	2	yes	—
43	C₂²⋊C₄	C₄×C₂₂₃	1	2	yes	—
44	C₄²	C₂²×C₂₂₃	1	2	yes	—
45	C₂³	C₂×C₄×C₂₂₃	1	1	yes	characteristic
46	C₂×C₄	C₂×C₄×C₂₂₃	1	4	yes	—
47	C₂×C₄	C₂×C₄×C₂₂₃	1	1	yes	characteristic
48	C₂×C₄	D₄×C₂₂₃	1	2	yes	—
49	C₂×C₄	Q₈×C₂₂₃	1	2	yes	—
50	C₂×C₄	C₂³×C₂₂₃	1	1	yes	characteristic
51	C₈	—	2	2	no	—
52	C₈	—	2	2	no	—
53	C₂²	—	2	2	no	—
54	C₂²	C₄⋊C₄×C₂₂₃	1	2	yes	—
55	C₂²	C₂²×C₄×C₂₂₃	1	1	yes	characteristic
56	C₄	—	2	4	no	—
57	C₄	C₄⋊C₄×C₂₂₃	1	2	yes	—
58	C₄	C₂×D₄×C₂₂₃	1	1	yes	characteristic
59	C₄	C₂×Q₈×C₂₂₃	1	1	yes	characteristic
60	C₂	—	2	2	no	—
61	C₂	C₂×C₄⋊C₄×C₂₂₃	1	1	yes	characteristic
62	C₂	#131×C₂₂₃	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
4. C₄⋊C₈×C₂₂₃	C₂	semidirect
31. C₂₂₃	#424	direct (N × H)
32. #424	C₂₂₃	direct (N × H)
36. C₄⋊C₈	C₂×C₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×M₁₆×C₂₂₃	2
C₂²×C₈×C₂₂₃	2
#117×C₂₂₃	2
C₄⋊C₈×C₂₂₃	2
#424	223

Subgroups — #424×C223 #28383136

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #424×C₂₂₃ #28383136