Subgroups — #28383364

Order 14272.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20297244	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂₂₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#131×C₂₂₃	C₂	1	1	yes	maximal, characteristic
4	C₂₂₃⋊C₁₆	—	2	1	no	—
5	C₂₂₃⋊C₁₆	—	2	1	no	—
6	#41×C₂₂₃	C₄	1	1	yes	characteristic
7	M₁₆×C₂₂₃	—	2	1	no	—
8	M₁₆×C₂₂₃	C₄	1	1	yes	characteristic
9	C₂×C₈×C₂₂₃	—	2	1	no	—
10	C₂×C₈×C₂₂₃	C₂²	1	1	yes	characteristic
11	Q₈×C₂₂₃	C₈	1	1	yes	characteristic
12	D₄×C₂₂₃	—	2	1	no	—
13	D₄×C₂₂₃	C₈	1	1	yes	characteristic
14	C₂×C₄×C₂₂₃	—	2	1	no	—
15	C₂×C₄×C₂₂₃	C₂×C₄	1	1	yes	characteristic
16	C₈×C₂₂₃	—	2	1	no	—
17	C₈×C₂₂₃	D₄	1	1	yes	characteristic
18	C₈×C₂₂₃	D₄	1	1	yes	characteristic
19	C₂²×C₂₂₃	—	2	1	no	—
20	C₂²×C₂₂₃	M₁₆	1	1	yes	characteristic
21	C₄×C₂₂₃	—	2	1	no	—
22	C₄×C₂₂₃	C₂²⋊C₄	1	1	yes	characteristic
23	C₄×C₂₂₃	C₂×C₈	1	1	yes	characteristic
24	C₂×C₂₂₃	—	2	1	no	—
25	C₂×C₂₂₃	—	4	1	no	—
26	C₂×C₂₂₃	C₂²⋊C₈	1	1	yes	characteristic
27	C₂₂₃	#350	1	1	yes	characteristic, complemented, Sylow
28	#350	—	223	1	no	maximal, Sylow
29	#131	D₂₂₃	1	1	yes	characteristic
30	M₃₂	—	223	1	no	—
31	C₂×C₁₆	—	223	1	no	—
32	#41	Dic₂₂₃	1	1	yes	characteristic
33	M₁₆	—	2	1	no	—
34	M₁₆	Dic₂₂₃	1	1	yes	characteristic
35	C₂×C₈	—	2	1	no	—
36	C₂×C₈	C₂×D₂₂₃	1	1	yes	characteristic
37	C₁₆	—	446	1	no	—
38	C₁₆	—	446	1	no	—
39	Q₈	C₂₂₃⋊C₈	1	1	yes	characteristic
40	D₄	—	2	1	no	—
41	D₄	C₂₂₃⋊C₈	1	1	yes	characteristic
42	C₂×C₄	—	2	1	no	—
43	C₂×C₄	C₂×Dic₂₂₃	1	1	yes	characteristic
44	C₈	—	2	1	no	—
45	C₈	C₂²⋊D₂₂₃	1	1	yes	characteristic
46	C₈	C₂²⋊D₂₂₃	1	1	yes	characteristic
47	C₂²	—	2	1	no	—
48	C₂²	#16813154	1	1	yes	characteristic
49	C₄	—	2	1	no	—
50	C₄	C₂²⋊Dic₂₂₃	1	1	yes	characteristic
51	C₄	C₂×C₂₂₃⋊C₈	1	1	yes	characteristic
52	C₂	—	2	1	no	—
53	C₂	—	4	1	no	—
54	C₂	C₂²⋊(C₂₂₃⋊C₈)	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂²⋊D₂₂₃
commutator	C₄×C₂₂₃	C₂×C₈
Frattini	C₂×C₈	C₂×D₂₂₃
Fitting	#131×C₂₂₃	C₂
socle	C₂×C₂₂₃	C₂²⋊C₈
prime radical	#41×C₂₂₃	C₄
cosocle	C₂×C₈×C₂₂₃	C₂²
hypercenter	#131	D₂₂₃
hypocenter	C₂₂₃	#350
LCS	G ▷ C₄×C₂₂₃ ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ #131

Complements

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

N	H ≅ G/N	type
1. #20297244	C₂	semidirect
2. C₂×C₂₂₃⋊C₁₆	C₂	semidirect
27. C₂₂₃	#350	semidirect

Maximal subgroups

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

subgroup	index
#20297244	2
C₂×C₂₂₃⋊C₁₆	2
#131×C₂₂₃	2
#350	223