Subgroups — #28383328

Order 14272.

64 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20297304	C₂	1	1	yes	maximal, characteristic
2	C₄⋊(C₂₂₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₈×C₂₂₃	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₄₄₆	C₄	1	2	yes	—
5	C₂×C₂₂₃⋊C₈	—	2	1	no	—
6	C₄⋊Dic₂₂₃	—	2	1	no	—
7	C₄⋊Dic₂₂₃	—	2	1	no	—
8	C₂×C₈×C₂₂₃	—	2	1	no	—
9	C₄²×C₂₂₃	C₂²	1	1	yes	characteristic
10	Dic₄₄₆	—	4	2	no	—
11	C₂×Dic₂₂₃	—	2	2	no	—
12	C₂₂₃⋊C₈	—	4	1	no	—
13	C₂×C₄×C₂₂₃	C₂×C₄	1	1	yes	characteristic
14	C₂×C₄×C₂₂₃	D₄	1	1	yes	characteristic
15	C₂×C₄×C₂₂₃	D₄	1	1	yes	characteristic
16	C₈×C₂₂₃	—	4	1	no	—
17	C₂²×C₂₂₃	C₂²⋊C₄	1	1	yes	characteristic
18	C₄×C₂₂₃	—	2	1	no	—
19	C₄×C₂₂₃	SD₁₆	1	1	yes	characteristic
20	C₄×C₂₂₃	SD₁₆	1	1	yes	characteristic
21	C₄×C₂₂₃	Q₁₆	1	1	yes	characteristic
22	C₄×C₂₂₃	Q₁₆	1	1	yes	characteristic
23	Dic₂₂₃	—	4	2	no	—
24	C₂×C₂₂₃	#100	1	1	yes	characteristic
25	C₂×C₂₂₃	#103	1	1	yes	characteristic
26	C₂×C₂₂₃	#103	1	1	yes	characteristic
27	C₂₂₃	#333	1	1	yes	characteristic, complemented, Sylow
28	#333	—	223	1	no	maximal, Sylow
29	C₄⋊Q₈	—	223	1	no	—
30	C₄⋊C₈	—	223	1	no	—
31	C₄⋊C₈	D₂₂₃	1	1	yes	characteristic
32	C₂×Q₈	—	223	2	no	—
33	C₂×C₈	—	2	1	no	—
34	C₂×C₈	—	446	1	no	—
35	C₄⋊C₄	—	446	1	no	—
36	C₄⋊C₄	—	446	1	no	—
37	C₄²	C₂×D₂₂₃	1	1	yes	characteristic
38	Q₈	—	892	2	no	—
39	C₂×C₄	—	446	2	no	—
40	C₂×C₄	C₄×D₂₂₃	1	1	yes	characteristic
41	C₂×C₄	C₂²⋊D₂₂₃	1	1	yes	characteristic
42	C₂×C₄	D₈₉₂	1	1	yes	characteristic
43	C₈	—	4	1	no	—
44	C₈	—	892	1	no	—
45	C₂²	(C₂×D₂₂₃)⋊C₄	1	1	yes	characteristic
46	C₄	—	2	1	no	—
47	C₄	—	892	2	no	—
48	C₄	#16813158	1	1	yes	characteristic
49	C₄	#16813159	1	1	yes	characteristic
50	C₄	#16813160	1	1	yes	characteristic
51	C₄	Dic₈₉₂	1	1	yes	characteristic
52	C₂	#20297220	1	1	yes	characteristic
53	C₂	#20297227	1	1	yes	characteristic
54	C₂	#20297229	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	(C₂×D₂₂₃)⋊C₄
commutator	C₂×C₄×C₂₂₃	C₂×C₄
Frattini	C₄²	C₂×D₂₂₃
Fitting	C₄⋊C₈×C₂₂₃	C₂
socle, prime radical	C₂²×C₂₂₃	C₂²⋊C₄
cosocle	C₄²×C₂₂₃	C₂²
hypercenter	C₄⋊C₈	D₂₂₃
hypocenter	C₂₂₃	#333
LCS	G ▷ C₂×C₄×C₂₂₃ ▷ C₂²×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
27. C₂₂₃	#333	semidirect

Maximal subgroups

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

subgroup	index
#20297304	2
C₄⋊(C₂₂₃⋊C₈)	2
C₄⋊C₈×C₂₂₃	2
#333	223