Subgroups — #28383608

Order 14272.

114 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂₂₃⋊C₈	C₂	1	1	yes	maximal, characteristic
2	#20297263	C₂	1	1	yes	maximal, characteristic
3	Dic₂₂₃⋊C₈	C₂	1	4	yes	maximal, complemented
4	C₂×M₁₆×C₂₂₃	C₂	1	1	yes	maximal, characteristic
5	C₂×C₂₂₃⋊C₈	—	2	2	no	—
6	C₂×C₂₂₃⋊C₈	C₂²	1	2	yes	—
7	C₄⋊Dic₂₂₃	C₄	1	2	yes	—
8	C₂²⋊Dic₂₂₃	C₄	1	2	yes	—
9	C₄×Dic₂₂₃	C₂²	1	2	yes	—
10	C₂²×C₄×C₂₂₃	C₂²	1	1	yes	characteristic
11	M₁₆×C₂₂₃	—	2	2	no	—
12	C₂×C₈×C₂₂₃	C₂²	1	2	yes	—
13	C₂×Dic₂₂₃	C₂×C₄	1	4	yes	—
14	C₂₂₃⋊C₈	—	2	2	no	—
15	C₂³×C₂₂₃	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄×C₂₂₃	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄×C₂₂₃	D₄	1	2	yes	—
18	C₂×C₄×C₂₂₃	Q₈	1	2	yes	—
19	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
20	C₈×C₂₂₃	—	2	2	no	—
21	C₂²×C₂₂₃	—	2	2	no	—
22	C₂²×C₂₂₃	C₄⋊C₄	1	2	yes	—
23	C₂²×C₂₂₃	C₂²×C₄	1	1	yes	characteristic
24	C₄×C₂₂₃	C₄⋊C₄	1	2	yes	—
25	C₄×C₂₂₃	C₂×D₄	1	1	yes	characteristic
26	C₄×C₂₂₃	C₂×Q₈	1	1	yes	characteristic
27	Dic₂₂₃	—	2	4	no	—
28	C₂×C₂₂₃	—	2	2	no	—
29	C₂×C₂₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
30	C₂×C₂₂₃	#131	1	2	yes	—
31	C₂₂₃	#424	1	1	yes	characteristic, complemented, Sylow
32	#424	—	223	1	no	maximal, Sylow
33	C₂×M₁₆	D₂₂₃	1	1	yes	characteristic
34	C₂²×C₈	—	223	1	no	—
35	#117	—	223	1	no	—
36	C₄⋊C₈	—	223	4	no	—
37	C₂²×C₄	C₂×D₂₂₃	1	1	yes	characteristic
38	M₁₆	—	2	2	no	—
39	C₂×C₈	—	446	2	no	—
40	C₂×C₈	—	223	2	no	—
41	C₂×C₈	C₂×D₂₂₃	1	2	yes	—
42	C₄⋊C₄	—	223	2	no	—
43	C₂²⋊C₄	—	223	2	no	—
44	C₄²	—	223	2	no	—
45	C₂³	C₄×D₂₂₃	1	1	yes	characteristic
46	C₂×C₄	—	223	4	no	—
47	C₂×C₄	C₄×D₂₂₃	1	1	yes	characteristic
48	C₂×C₄	C₂²⋊D₂₂₃	1	2	yes	—
49	C₂×C₄	Dic₄₄₆	1	2	yes	—
50	C₂×C₄	C₂²×D₂₂₃	1	1	yes	characteristic
51	C₈	—	2	2	no	—
52	C₈	—	446	2	no	—
53	C₂²	—	2	2	no	—
54	C₂²	Dic₂₂₃⋊C₄	1	2	yes	—
55	C₂²	C₂×C₄×D₂₂₃	1	1	yes	characteristic
56	C₄	—	446	4	no	—
57	C₄	Dic₂₂₃⋊C₄	1	2	yes	—
58	C₄	C₂×C₂²⋊D₂₂₃	1	1	yes	characteristic
59	C₄	C₂×Dic₄₄₆	1	1	yes	characteristic
60	C₂	—	2	2	no	—
61	C₂	C₂×Dic₂₂₃⋊C₄	1	1	yes	characteristic
62	C₂	#20297312	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₂₂₃
commutator, socle	C₂²×C₂₂₃	C₂²×C₄
Fitting	C₂×M₁₆×C₂₂₃	C₂
prime radical	C₂³×C₂₂₃	C₂×C₄
cosocle	C₂×C₄×C₂₂₃	C₂³
hypercenter	C₂×M₁₆	D₂₂₃
hypocenter	C₂₂₃	#424
LCS	G ▷ C₂²×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
3. Dic₂₂₃⋊C₈	C₂	semidirect
31. C₂₂₃	#424	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₂₂₃⋊C₈	2
#20297263	2
Dic₂₂₃⋊C₈	2
C₂×M₁₆×C₂₂₃	2
#424	223