Subgroups — #28383326

Order 14272.

64 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊(C₂₂₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	C₄⋊(C₂₂₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₄⋊Q₈×C₂₂₃	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂₂₃⋊C₈	—	2	1	no	—
5	C₂×C₂₂₃⋊C₈	—	2	1	no	—
6	C₂×Q₈×C₂₂₃	—	2	1	no	—
7	C₄⋊C₄×C₂₂₃	—	2	1	no	—
8	C₄⋊C₄×C₂₂₃	C₄	1	2	yes	—
9	C₄²×C₂₂₃	C₂²	1	1	yes	characteristic
10	C₂₂₃⋊C₈	—	4	1	no	—
11	C₂₂₃⋊C₈	—	4	1	no	—
12	Q₈×C₂₂₃	—	4	1	no	—
13	Q₈×C₂₂₃	—	4	1	no	—
14	C₂×C₄×C₂₂₃	—	2	2	no	—
15	C₂×C₄×C₂₂₃	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄×C₂₂₃	D₄	1	1	yes	characteristic
17	C₂×C₄×C₂₂₃	D₄	1	1	yes	characteristic
18	C₂²×C₂₂₃	C₂²⋊C₄	1	1	yes	characteristic
19	C₄×C₂₂₃	—	2	1	no	—
20	C₄×C₂₂₃	—	4	2	no	—
21	C₄×C₂₂₃	D₈	1	1	yes	characteristic
22	C₄×C₂₂₃	SD₁₆	1	1	yes	characteristic
23	C₄×C₂₂₃	SD₁₆	1	1	yes	characteristic
24	C₄×C₂₂₃	Q₁₆	1	1	yes	characteristic
25	C₂×C₂₂₃	#101	1	1	yes	characteristic
26	C₂×C₂₂₃	#102	1	1	yes	characteristic
27	C₂×C₂₂₃	#103	1	1	yes	characteristic
28	C₂₂₃	#332	1	1	yes	characteristic, complemented, Sylow
29	#332	—	223	1	no	maximal, Sylow
30	C₄⋊Q₈	D₂₂₃	1	1	yes	characteristic
31	C₄⋊C₈	—	223	1	no	—
32	C₄⋊C₈	—	223	1	no	—
33	C₂×Q₈	—	2	1	no	—
34	C₂×C₈	—	446	1	no	—
35	C₂×C₈	—	446	1	no	—
36	C₄⋊C₄	—	2	1	no	—
37	C₄⋊C₄	Dic₂₂₃	1	2	yes	—
38	C₄²	C₂×D₂₂₃	1	1	yes	characteristic
39	Q₈	—	4	1	no	—
40	Q₈	—	4	1	no	—
41	C₂×C₄	—	2	2	no	—
42	C₂×C₄	C₂×Dic₂₂₃	1	1	yes	characteristic
43	C₂×C₄	C₂²⋊D₂₂₃	1	1	yes	characteristic
44	C₂×C₄	C₂²⋊D₂₂₃	1	1	yes	characteristic
45	C₈	—	892	1	no	—
46	C₈	—	892	1	no	—
47	C₂²	C₂²⋊Dic₂₂₃	1	1	yes	characteristic
48	C₄	—	2	1	no	—
49	C₄	—	4	2	no	—
50	C₄	#16813155	1	1	yes	characteristic
51	C₄	#16813157	1	1	yes	characteristic
52	C₄	#16813158	1	1	yes	characteristic
53	C₄	#16813160	1	1	yes	characteristic
54	C₂	#20297223	1	1	yes	characteristic
55	C₂	#20297225	1	1	yes	characteristic
56	C₂	#20297228	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²⋊Dic₂₂₃
commutator	C₂×C₄×C₂₂₃	C₂×C₄
Frattini	C₄²	C₂×D₂₂₃
Fitting	C₄⋊Q₈×C₂₂₃	C₂
socle, prime radical	C₂²×C₂₂₃	C₂²⋊C₄
cosocle	C₄²×C₂₂₃	C₂²
hypercenter	C₄⋊Q₈	D₂₂₃
hypocenter	C₂₂₃	#332
LCS	G ▷ C₂×C₄×C₂₂₃ ▷ C₂²×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
28. C₂₂₃	#332	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊(C₂₂₃⋊C₈)	2
C₄⋊Q₈×C₂₂₃	2
#332	223