Subgroups — #28384058

Order 14272.

266 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×Dic₂₂₃	C₂	1	8	yes	maximal, complemented
2	#20297263	C₂	1	6	yes	maximal
3	C₂²×Q₈×C₂₂₃	C₂	1	1	yes	maximal, characteristic
4	C₄⋊Dic₂₂₃	C₂²	1	12	yes	—
5	C₂²⋊Dic₂₂₃	C₂²	1	4	yes	—
6	C₄×Dic₂₂₃	C₂²	1	12	yes	—
7	C₂×Q₈×C₂₂₃	C₄	1	8	yes	complemented
8	C₂×Q₈×C₂₂₃	C₂²	1	4	yes	—
9	C₂²×C₄×C₂₂₃	C₂²	1	3	yes	—
10	C₂×Dic₂₂₃	C₂³	1	8	yes	—
11	C₂³×C₂₂₃	C₂³	1	1	yes	characteristic
12	Q₈×C₂₂₃	C₂×C₄	1	16	yes	—
13	C₂×C₄×C₂₂₃	C₂×C₄	1	12	yes	—
14	C₂×C₄×C₂₂₃	C₂³	1	6	yes	—
15	C₂²×C₂₂₃	—	2	2	no	—
16	C₂²×C₂₂₃	C₂²×C₄	1	2	yes	—
17	C₂²×C₂₂₃	C₂⁴	1	1	yes	characteristic
18	C₄×C₂₂₃	C₂²×C₄	1	12	yes	—
19	Dic₂₂₃	—	2	8	no	—
20	C₂×C₂₂₃	—	2	2	no	—
21	C₂×C₂₂₃	C₂³×C₄	1	1	yes	characteristic
22	C₂×C₂₂₃	2₋¹⁺⁴	1	2	yes	—
23	C₂₂₃	#519	1	1	yes	characteristic, complemented, Sylow
24	#519	—	223	1	no	maximal, Sylow
25	C₂²×Q₈	D₂₂₃	1	1	yes	characteristic
26	C₄×Q₈	—	223	8	no	—
27	#117	—	223	6	no	—
28	C₂×Q₈	Dic₂₂₃	1	8	yes	complemented
29	C₂×Q₈	C₂×D₂₂₃	1	4	yes	—
30	C₂²×C₄	C₂×D₂₂₃	1	3	yes	—
31	C₄⋊C₄	—	223	12	no	—
32	C₂²⋊C₄	—	223	4	no	—
33	C₄²	—	223	12	no	—
34	C₂³	C₂²×D₂₂₃	1	1	yes	characteristic
35	Q₈	C₂×Dic₂₂₃	1	16	yes	—
36	C₂×C₄	—	223	8	no	—
37	C₂×C₄	C₂×Dic₂₂₃	1	12	yes	—
38	C₂×C₄	C₂²×D₂₂₃	1	6	yes	—
39	C₂²	—	2	2	no	—
40	C₂²	C₂²×Dic₂₂₃	1	2	yes	—
41	C₂²	C₂³×D₂₂₃	1	1	yes	characteristic
42	C₄	—	446	8	no	—
43	C₄	C₂²×Dic₂₂₃	1	12	yes	—
44	C₂	—	2	2	no	—
45	C₂	C₂³×Dic₂₂₃	1	1	yes	characteristic
46	C₂	#20297352	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₂₃
commutator	C₂×C₂₂₃	C₂³×C₄
Fitting	C₂²×Q₈×C₂₂₃	C₂
socle, cosocle	C₂²×C₂₂₃	C₂⁴
prime radical	C₂³×C₂₂₃	C₂³
hypercenter	C₂²×Q₈	D₂₂₃
hypocenter	C₂₂₃	#519
LCS	G ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂² ◁ C₂²×Q₈

Complements

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

N	H ≅ G/N	type
1. Q₈×Dic₂₂₃	C₂	semidirect
7. C₂×Q₈×C₂₂₃	C₄	semidirect
23. C₂₂₃	#519	semidirect
28. C₂×Q₈	Dic₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
Q₈×Dic₂₂₃	2
#20297263	2
C₂²×Q₈×C₂₂₃	2
#519	223