Subgroups — #28383345

Order 14272.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16813154	C₂	1	2	yes	maximal
2	C₂²×C₈×C₂₂₃	C₂	1	1	yes	maximal, characteristic
3	#16813154	—	2	2	no	—
4	#16813154	C₄	1	4	yes	—
5	C₂×C₂₂₃⋊C₈	—	2	2	no	—
6	C₂²×C₄×C₂₂₃	C₂²	1	1	yes	characteristic
7	C₂×C₈×C₂₂₃	—	2	2	no	—
8	C₂×C₈×C₂₂₃	C₄	1	2	yes	—
9	C₂₂₃⋊C₈	—	2	4	no	—
10	C₂³×C₂₂₃	Q₈	1	1	yes	characteristic
11	C₂×C₄×C₂₂₃	C₂×C₄	1	2	yes	—
12	C₂×C₄×C₂₂₃	C₂×C₄	1	1	yes	characteristic
13	C₂×C₄×C₂₂₃	D₄	1	2	yes	—
14	C₂×C₄×C₂₂₃	D₄	1	1	yes	characteristic
15	C₈×C₂₂₃	—	2	2	no	—
16	C₂²×C₂₂₃	—	2	2	no	—
17	C₂²×C₂₂₃	C₄⋊C₄	1	2	yes	—
18	C₂²×C₂₂₃	C₄⋊C₄	1	1	yes	characteristic
19	C₄×C₂₂₃	C₄²	1	1	yes	characteristic
20	C₄×C₂₂₃	C₂²⋊C₄	1	2	yes	—
21	C₄×C₂₂₃	C₂²⋊C₄	1	1	yes	characteristic
22	C₂×C₂₂₃	—	2	2	no	—
23	C₂×C₂₂₃	#95	1	1	yes	characteristic
24	C₂×C₂₂₃	#108	1	2	yes	—
25	C₂₂₃	#341	1	1	yes	characteristic, complemented, Sylow
26	#341	—	223	1	no	maximal, Sylow
27	C₂×M₁₆	—	223	2	no	—
28	C₂²×C₈	D₂₂₃	1	1	yes	characteristic
29	C₂²×C₄	C₂×D₂₂₃	1	1	yes	characteristic
30	M₁₆	—	446	2	no	—
31	M₁₆	—	223	4	no	—
32	C₂×C₈	—	2	2	no	—
33	C₂×C₈	—	446	2	no	—
34	C₂×C₈	Dic₂₂₃	1	2	yes	—
35	C₂³	Dic₄₄₆	1	1	yes	characteristic
36	C₂×C₄	C₄×D₂₂₃	1	2	yes	—
37	C₂×C₄	C₂×Dic₂₂₃	1	1	yes	characteristic
38	C₂×C₄	C₂²⋊D₂₂₃	1	2	yes	—
39	C₂×C₄	D₈₉₂	1	1	yes	characteristic
40	C₈	—	2	2	no	—
41	C₈	—	446	4	no	—
42	C₂²	—	2	2	no	—
43	C₂²	Dic₂₂₃⋊C₄	1	2	yes	—
44	C₂²	C₄⋊Dic₂₂₃	1	1	yes	characteristic
45	C₄	C₄×Dic₂₂₃	1	1	yes	characteristic
46	C₄	(C₂×D₂₂₃)⋊C₄	1	2	yes	—
47	C₄	C₂²⋊Dic₂₂₃	1	1	yes	characteristic
48	C₂	—	2	2	no	—
49	C₂	C₂₂₃⋊#95	1	1	yes	characteristic
50	C₂	#20297240	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₈₉₂
commutator	C₄×C₂₂₃	C₄²
Frattini	C₂²×C₄	C₂×D₂₂₃
Fitting	C₂²×C₈×C₂₂₃	C₂
socle	C₂²×C₂₂₃	C₄⋊C₄
prime radical	C₂³×C₂₂₃	Q₈
cosocle	C₂²×C₄×C₂₂₃	C₂²
hypercenter	C₂²×C₈	D₂₂₃
hypocenter	C₂₂₃	#341
LCS	G ▷ C₄×C₂₂₃ ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
25. C₂₂₃	#341	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16813154	2
C₂²×C₈×C₂₂₃	2
#341	223