Subgroups — #28383436

Order 14272.

506 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂²⋊Dic₂₂₃	C₂	1	6	yes	maximal, complemented
2	C₂⁵×C₂₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂²×Dic₂₂₃	C₂²	1	4	yes	complemented
4	C₂²⋊Dic₂₂₃	—	2	12	no	—
5	C₂⁴×C₂₂₃	—	2	12	no	—
6	C₂⁴×C₂₂₃	C₄	1	4	yes	complemented
7	C₂⁴×C₂₂₃	C₂²	1	3	yes	—
8	C₂×Dic₂₂₃	—	2	12	no	—
9	C₂³×C₂₂₃	—	2	32	no	—
10	C₂³×C₂₂₃	—	2	24	no	—
11	C₂³×C₂₂₃	—	2	12	no	—
12	C₂³×C₂₂₃	C₂×C₄	1	6	yes	—
13	C₂³×C₂₂₃	D₄	1	12	yes	—
14	C₂³×C₂₂₃	C₂³	1	1	yes	characteristic
15	C₂²×C₂₂₃	—	2	32	no	—
16	C₂²×C₂₂₃	—	2	12	no	—
17	C₂²×C₂₂₃	—	2	24	no	—
18	C₂²×C₂₂₃	C₂²⋊C₄	1	12	yes	complemented
19	C₂²×C₂₂₃	C₂²×C₄	1	1	yes	characteristic
20	C₂²×C₂₂₃	C₂×D₄	1	6	yes	—
21	Dic₂₂₃	—	4	4	no	—
22	C₂×C₂₂₃	—	2	12	no	—
23	C₂×C₂₂₃	C₂×C₂²⋊C₄	1	3	yes	—
24	C₂×C₂₂₃	#120	1	4	yes	—
25	C₂₂₃	#379	1	1	yes	characteristic, complemented, Sylow
26	#379	—	223	1	no	maximal, Sylow
27	C₂⁵	D₂₂₃	1	1	yes	characteristic
28	C₂×C₂²⋊C₄	—	223	6	no	—
29	C₂⁴	—	2	12	no	—
30	C₂⁴	Dic₂₂₃	1	4	yes	complemented
31	C₂⁴	C₂×D₂₂₃	1	3	yes	—
32	C₂²×C₄	—	223	4	no	—
33	C₂²⋊C₄	—	446	12	no	—
34	C₂³	—	2	32	no	—
35	C₂³	—	2	24	no	—
36	C₂³	—	2	12	no	—
37	C₂³	C₂×Dic₂₂₃	1	6	yes	—
38	C₂³	C₂²⋊D₂₂₃	1	12	yes	—
39	C₂³	C₂²×D₂₂₃	1	1	yes	characteristic
40	C₂×C₄	—	446	12	no	—
41	C₂²	—	2	32	no	—
42	C₂²	—	2	12	no	—
43	C₂²	—	2	24	no	—
44	C₂²	C₂²⋊Dic₂₂₃	1	12	yes	complemented
45	C₂²	C₂²×Dic₂₂₃	1	1	yes	characteristic
46	C₂²	C₂×C₂²⋊D₂₂₃	1	6	yes	—
47	C₄	—	892	4	no	—
48	C₂	—	2	12	no	—
49	C₂	C₂×C₂²⋊Dic₂₂₃	1	3	yes	—
50	C₂	#20297273	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₂₂₃
commutator	C₂²×C₂₂₃	C₂²×C₄
Fitting, prime radical	C₂⁵×C₂₂₃	C₂
socle, cosocle	C₂³×C₂₂₃	C₂³
hypercenter	C₂⁵	D₂₂₃
hypocenter	C₂₂₃	#379
LCS	G ▷ C₂²×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂³ ◁ C₂⁵

Complements

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

N	H ≅ G/N	type
1. C₂×C₂²⋊Dic₂₂₃	C₂	semidirect
3. C₂²×Dic₂₂₃	C₂²	semidirect
6. C₂⁴×C₂₂₃	C₄	semidirect
18. C₂²×C₂₂₃	C₂²⋊C₄	semidirect
25. C₂₂₃	#379	semidirect
30. C₂⁴	Dic₂₂₃	semidirect
44. C₂²	C₂²⋊Dic₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂²⋊Dic₂₂₃	2
C₂⁵×C₂₂₃	2
#379	223