Subgroups — #28383337

Order 14272.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20297263	C₂	1	2	yes	maximal
2	C₂×M₁₆×C₂₂₃	C₂	1	1	yes	maximal, characteristic
3	C₄⋊Dic₂₂₃	—	2	2	no	—
4	C₂²⋊Dic₂₂₃	—	2	2	no	—
5	C₄×Dic₂₂₃	C₄	1	4	yes	complemented
6	C₂²×C₄×C₂₂₃	C₂²	1	1	yes	characteristic
7	M₁₆×C₂₂₃	—	2	2	no	—
8	C₂×C₈×C₂₂₃	C₄	1	2	yes	complemented
9	C₂×Dic₂₂₃	—	2	4	no	—
10	C₂³×C₂₂₃	D₄	1	1	yes	characteristic
11	C₂×C₄×C₂₂₃	C₂×C₄	1	2	yes	—
12	C₂×C₄×C₂₂₃	C₂×C₄	1	1	yes	characteristic
13	C₂×C₄×C₂₂₃	D₄	1	2	yes	—
14	C₂×C₄×C₂₂₃	Q₈	1	1	yes	characteristic
15	C₈×C₂₂₃	—	2	2	no	—
16	C₂²×C₂₂₃	—	4	1	no	—
17	C₂²×C₂₂₃	C₂²⋊C₄	1	2	yes	—
18	C₂²×C₂₂₃	C₄⋊C₄	1	1	yes	characteristic
19	C₄×C₂₂₃	C₄²	1	1	yes	characteristic
20	C₄×C₂₂₃	C₂²⋊C₄	1	1	yes	characteristic
21	C₄×C₂₂₃	C₄⋊C₄	1	2	yes	—
22	Dic₂₂₃	—	4	4	no	—
23	C₂×C₂₂₃	—	2	2	no	—
24	C₂×C₂₂₃	—	2	1	no	—
25	C₂×C₂₂₃	#95	1	1	yes	characteristic
26	C₂₂₃	H(3,ℤ/4)	1	1	yes	characteristic, complemented, Sylow
27	H(3,ℤ/4)	—	223	1	no	maximal, Sylow
28	C₂×M₁₆	D₂₂₃	1	1	yes	characteristic
29	#117	—	223	2	no	—
30	C₂²×C₄	C₂×D₂₂₃	1	1	yes	characteristic
31	M₁₆	—	2	2	no	—
32	C₂×C₈	Dic₂₂₃	1	2	yes	complemented
33	C₄⋊C₄	—	446	2	no	—
34	C₂²⋊C₄	—	446	2	no	—
35	C₄²	—	223	4	no	—
36	C₂³	D₈₉₂	1	1	yes	characteristic
37	C₂×C₄	—	446	4	no	—
38	C₂×C₄	C₄×D₂₂₃	1	2	yes	—
39	C₂×C₄	C₂×Dic₂₂₃	1	1	yes	characteristic
40	C₂×C₄	C₂²⋊D₂₂₃	1	2	yes	—
41	C₂×C₄	Dic₄₄₆	1	1	yes	characteristic
42	C₈	—	2	2	no	—
43	C₂²	—	4	1	no	—
44	C₂²	(C₂×D₂₂₃)⋊C₄	1	2	yes	—
45	C₂²	C₄⋊Dic₂₂₃	1	1	yes	characteristic
46	C₄	—	892	4	no	—
47	C₄	C₄×Dic₂₂₃	1	1	yes	characteristic
48	C₄	C₂²⋊Dic₂₂₃	1	1	yes	characteristic
49	C₄	Dic₂₂₃⋊C₄	1	2	yes	—
50	C₂	—	2	2	no	—
51	C₂	—	2	1	no	—
52	C₂	C₂₂₃⋊#95	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₄×Dic₂₂₃
commutator	C₄×C₂₂₃	C₄²
Frattini	C₂²×C₄	C₂×D₂₂₃
Fitting	C₂×M₁₆×C₂₂₃	C₂
socle	C₂×C₂₂₃	#95
prime radical	C₂³×C₂₂₃	D₄
cosocle	C₂²×C₄×C₂₂₃	C₂²
hypercenter	C₂×M₁₆	D₂₂₃
hypocenter	C₂₂₃	H(3,ℤ/4)
LCS	G ▷ C₄×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
5. C₄×Dic₂₂₃	C₄	semidirect
8. C₂×C₈×C₂₂₃	C₄	semidirect
26. C₂₂₃	H(3,ℤ/4)	semidirect
32. C₂×C₈	Dic₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
#20297263	2
C₂×M₁₆×C₂₂₃	2
H(3,ℤ/4)	223