Subgroups — #28383343

Order 14272.

114 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊Dic₂₂₃	C₂	1	2	yes	maximal
2	C₂²×C₈×C₂₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂²×Dic₂₂₃	—	2	2	no	—
4	C₄⋊Dic₂₂₃	—	2	2	no	—
5	C₄⋊Dic₂₂₃	C₄	1	4	yes	complemented
6	C₂²×C₄×C₂₂₃	C₂²	1	1	yes	characteristic
7	C₂×C₈×C₂₂₃	—	2	2	no	—
8	C₂×C₈×C₂₂₃	C₄	1	2	yes	complemented
9	C₂×Dic₂₂₃	—	2	4	no	—
10	C₂×Dic₂₂₃	—	4	2	no	—
11	C₂×Dic₂₂₃	—	4	2	no	—
12	C₂³×C₂₂₃	D₄	1	1	yes	characteristic
13	C₂×C₄×C₂₂₃	C₂×C₄	1	2	yes	—
14	C₂×C₄×C₂₂₃	C₂×C₄	1	1	yes	characteristic
15	C₂×C₄×C₂₂₃	D₄	1	2	yes	—
16	C₂×C₄×C₂₂₃	Q₈	1	1	yes	characteristic
17	C₈×C₂₂₃	—	2	2	no	—
18	C₂²×C₂₂₃	C₂²⋊C₄	1	2	yes	—
19	C₂²×C₂₂₃	C₄⋊C₄	1	1	yes	characteristic
20	C₂²×C₂₂₃	D₈	1	1	yes	characteristic
21	C₂²×C₂₂₃	SD₁₆	1	2	yes	—
22	C₂²×C₂₂₃	Q₁₆	1	1	yes	characteristic
23	C₄×C₂₂₃	C₄²	1	1	yes	characteristic
24	C₄×C₂₂₃	C₂²⋊C₄	1	1	yes	characteristic
25	C₄×C₂₂₃	C₄⋊C₄	1	2	yes	—
26	Dic₂₂₃	—	4	4	no	—
27	C₂×C₂₂₃	#95	1	1	yes	characteristic
28	C₂×C₂₂₃	#102	1	2	yes	—
29	C₂×C₂₂₃	#103	1	2	yes	—
30	C₂×C₂₂₃	#106	1	1	yes	characteristic
31	C₂×C₂₂₃	#107	1	1	yes	characteristic
32	C₂₂₃	#340	1	1	yes	characteristic, complemented, Sylow
33	#340	—	223	1	no	maximal, Sylow
34	C₂²×C₈	D₂₂₃	1	1	yes	characteristic
35	C₂×C₄⋊C₄	—	223	2	no	—
36	C₂²×C₄	—	446	2	no	—
37	C₂²×C₄	C₂×D₂₂₃	1	1	yes	characteristic
38	C₂×C₈	—	2	2	no	—
39	C₂×C₈	Dic₂₂₃	1	2	yes	complemented
40	C₄⋊C₄	—	446	2	no	—
41	C₄⋊C₄	—	223	4	no	—
42	C₂³	D₈₉₂	1	1	yes	characteristic
43	C₂×C₄	—	892	2	no	—
44	C₂×C₄	—	892	2	no	—
45	C₂×C₄	—	446	4	no	—
46	C₂×C₄	C₄×D₂₂₃	1	2	yes	—
47	C₂×C₄	C₂×Dic₂₂₃	1	1	yes	characteristic
48	C₂×C₄	C₂²⋊D₂₂₃	1	2	yes	—
49	C₂×C₄	Dic₄₄₆	1	1	yes	characteristic
50	C₈	—	2	2	no	—
51	C₂²	(C₂×D₂₂₃)⋊C₄	1	2	yes	—
52	C₂²	C₄⋊Dic₂₂₃	1	1	yes	characteristic
53	C₂²	D₁₇₈₄	1	1	yes	characteristic
54	C₂²	#16813159	1	2	yes	—
55	C₂²	Dic₈₉₂	1	1	yes	characteristic
56	C₄	—	892	4	no	—
57	C₄	C₄×Dic₂₂₃	1	1	yes	characteristic
58	C₄	C₂²⋊Dic₂₂₃	1	1	yes	characteristic
59	C₄	Dic₂₂₃⋊C₄	1	2	yes	—
60	C₂	C₂₂₃⋊#95	1	1	yes	characteristic
61	C₂	#20297226	1	2	yes	—
62	C₂	#20297229	1	2	yes	—
63	C₂	#20297236	1	1	yes	characteristic
64	C₂	#20297238	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₈₉₂
commutator	C₄×C₂₂₃	C₄²
Frattini	C₂²×C₄	C₂×D₂₂₃
Fitting	C₂²×C₈×C₂₂₃	C₂
socle, prime radical	C₂³×C₂₂₃	D₄
cosocle	C₂²×C₄×C₂₂₃	C₂²
hypercenter	C₂²×C₈	D₂₂₃
hypocenter	C₂₂₃	#340
LCS	G ▷ C₄×C₂₂₃ ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
5. C₄⋊Dic₂₂₃	C₄	semidirect
8. C₂×C₈×C₂₂₃	C₄	semidirect
32. C₂₂₃	#340	semidirect
39. C₂×C₈	Dic₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊Dic₂₂₃	2
C₂²×C₈×C₂₂₃	2
#340	223