Subgroups — C₂₂₃⋊#564 #28384335

Order 14272.

172 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20297296	C₂	1	2	yes	maximal
2	#20297295	C₂	1	4	yes	maximal
3	#20297294	C₂	1	4	yes	maximal
4	#20297294	C₂	1	4	yes	maximal
5	#125×C₂₂₃	C₂	1	1	yes	maximal, characteristic
6	C₄⋊Dic₂₂₃	C₂²	1	2	yes	—
7	C₄⋊Dic₂₂₃	C₂²	1	4	yes	—
8	C₄⋊Dic₂₂₃	C₂²	1	2	yes	—
9	Dic₂₂₃⋊C₄	C₂²	1	4	yes	—
10	Dic₂₂₃⋊C₄	C₂²	1	4	yes	—
11	Dic₂₂₃⋊C₄	C₂²	1	4	yes	—
12	Dic₂₂₃⋊C₄	C₂²	1	4	yes	—
13	C₄×Dic₂₂₃	C₂²	1	4	yes	—
14	C₄⋊C₄×C₂₂₃	C₂²	1	2	yes	—
15	C₄⋊C₄×C₂₂₃	C₂²	1	4	yes	—
16	C₄²×C₂₂₃	C₂²	1	1	yes	characteristic
17	C₂×Dic₂₂₃	C₂³	1	4	yes	—
18	C₂×Dic₂₂₃	C₂³	1	4	yes	—
19	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
20	C₂×C₄×C₂₂₃	C₂³	1	2	yes	—
21	C₂×C₄×C₂₂₃	C₂³	1	4	yes	—
22	C₂²×C₂₂₃	C₂⁴	1	1	yes	characteristic
23	C₄×C₂₂₃	—	2	2	no	—
24	C₄×C₂₂₃	—	2	4	no	—
25	C₄×C₂₂₃	—	2	1	no	—
26	Dic₂₂₃	—	2	4	no	—
27	Dic₂₂₃	—	2	4	no	—
28	C₂×C₂₂₃	2₋¹⁺⁴	1	2	yes	—
29	C₂×C₂₂₃	2₋¹⁺⁴	1	1	yes	characteristic
30	C₂₂₃	#564	1	1	yes	characteristic, complemented, Sylow
31	#564	—	223	1	no	maximal, Sylow
32	#125	—	223	4	no	—
33	#125	—	223	4	no	—
34	#125	—	223	4	no	—
35	#125	—	223	2	no	—
36	#125	D₂₂₃	1	1	yes	characteristic
37	C₄⋊C₄	—	223	4	no	—
38	C₄⋊C₄	—	223	4	no	—
39	C₄⋊C₄	—	223	4	no	—
40	C₄⋊C₄	—	223	4	no	—
41	C₄⋊C₄	—	223	4	no	—
42	C₄⋊C₄	—	223	2	no	—
43	C₄⋊C₄	—	223	2	no	—
44	C₄⋊C₄	C₂×D₂₂₃	1	2	yes	—
45	C₄⋊C₄	C₂×D₂₂₃	1	4	yes	—
46	C₄²	—	223	4	no	—
47	C₄²	C₂×D₂₂₃	1	1	yes	characteristic
48	C₂×C₄	—	223	4	no	—
49	C₂×C₄	—	223	4	no	—
50	C₂×C₄	C₂²×D₂₂₃	1	4	yes	—
51	C₂×C₄	C₂²×D₂₂₃	1	2	yes	—
52	C₂×C₄	C₂²×D₂₂₃	1	1	yes	characteristic
53	C₂²	C₂³×D₂₂₃	1	1	yes	characteristic
54	C₄	—	2	4	no	—
55	C₄	—	2	2	no	—
56	C₄	—	2	1	no	—
57	C₄	—	446	4	no	—
58	C₄	—	446	4	no	—
59	C₂	#20297351	1	2	yes	—
60	C₂	#20297352	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₂₃
commutator, socle, prime radical, cosocle	C₂²×C₂₂₃	C₂⁴
Fitting	#125×C₂₂₃	C₂
hypercenter	#125	D₂₂₃
hypocenter	C₂₂₃	#564
LCS	G ▷ C₂²×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂² ◁ #125

Complements

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

N	H ≅ G/N	type
30. C₂₂₃	#564	semidirect

Maximal subgroups

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

subgroup	index
#20297296	2
#20297295	2
#20297294	2
#125×C₂₂₃	2
#564	223

Subgroups — C223⋊#564 #28384335

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₂₃⋊#564 #28384335