Subgroups — #28383692

Order 14272.

132 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₄×Dic₂₂₃	C₂	1	2	yes	maximal, complemented
2	#20297236	C₂	1	1	yes	maximal, characteristic, complemented
3	#20297225	C₂	1	2	yes	maximal, complemented
4	#20297214	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₈×C₂₂₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₂₃	—	2	2	no	—
7	C₂×C₂₂₃⋊C₈	C₂²	1	1	yes	characteristic
8	C₄⋊Dic₂₂₃	C₂²	1	2	yes	—
9	C₂²⋊Dic₂₂₃	—	2	2	no	—
10	C₄×Dic₂₂₃	C₂²	1	1	yes	characteristic
11	C₂×D₄×C₂₂₃	C₂²	1	2	yes	—
12	D₈×C₂₂₃	C₄	1	4	yes	complemented
13	C₂×C₈×C₂₂₃	C₂²	1	1	yes	characteristic
14	C₂×Dic₂₂₃	—	2	2	no	—
15	C₂×Dic₂₂₃	—	4	4	no	—
16	C₂×Dic₂₂₃	D₄	1	2	yes	—
17	C₂₂₃⋊C₈	—	2	1	no	—
18	C₂³×C₂₂₃	—	2	2	no	—
19	D₄×C₂₂₃	—	2	2	no	—
20	D₄×C₂₂₃	C₂×C₄	1	4	yes	complemented
21	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
22	C₈×C₂₂₃	C₂×C₄	1	2	yes	complemented
23	C₂²×C₂₂₃	—	2	4	no	—
24	C₂²×C₂₂₃	—	4	2	no	—
25	C₂²×C₂₂₃	—	4	2	no	—
26	C₂²×C₂₂₃	C₂×D₄	1	1	yes	characteristic
27	C₄×C₂₂₃	C₂²×C₄	1	1	yes	characteristic
28	C₄×C₂₂₃	#41	1	1	yes	characteristic
29	Dic₂₂₃	—	2	2	no	—
30	Dic₂₂₃	—	4	2	no	—
31	C₂×C₂₂₃	—	4	4	no	—
32	C₂×C₂₂₃	C₄×D₄	1	1	yes	characteristic
33	C₂×C₂₂₃	AGL(1,ℤ/8)	1	2	yes	—
34	C₂₂₃	#442	1	1	yes	characteristic, complemented, Sylow
35	#442	—	223	1	no	maximal, Sylow
36	C₂×D₈	D₂₂₃	1	1	yes	characteristic
37	C₄×D₄	—	223	2	no	—
38	#106	—	223	1	no	—
39	#102	—	223	2	no	—
40	#97	—	223	1	no	—
41	C₂×D₄	C₂×D₂₂₃	1	2	yes	—
42	C₂²×C₄	—	446	2	no	—
43	D₈	Dic₂₂₃	1	4	yes	complemented
44	C₂×C₈	—	223	1	no	—
45	C₂×C₈	C₂×D₂₂₃	1	1	yes	characteristic
46	C₄⋊C₄	—	223	2	no	—
47	C₂²⋊C₄	—	446	2	no	—
48	C₄²	—	223	1	no	—
49	C₂³	—	2	2	no	—
50	D₄	—	2	2	no	—
51	D₄	C₂×Dic₂₂₃	1	4	yes	complemented
52	C₂×C₄	—	892	4	no	—
53	C₂×C₄	—	446	2	no	—
54	C₂×C₄	—	223	2	no	—
55	C₂×C₄	C₂²×D₂₂₃	1	1	yes	characteristic
56	C₈	—	446	1	no	—
57	C₈	C₂×Dic₂₂₃	1	2	yes	complemented
58	C₂²	—	2	4	no	—
59	C₂²	—	4	2	no	—
60	C₂²	—	4	2	no	—
61	C₂²	D₄×D₂₂₃	1	1	yes	characteristic
62	C₄	—	892	2	no	—
63	C₄	—	446	2	no	—
64	C₄	C₂²×Dic₂₂₃	1	1	yes	characteristic
65	C₄	#16813169	1	1	yes	characteristic
66	C₂	—	4	4	no	—
67	C₂	D₄×Dic₂₂₃	1	1	yes	characteristic
68	C₂	#20297328	1	2	yes	—

	subgroup	G/H
center	C₂²	D₄×D₂₂₃
commutator	C₄×C₂₂₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₂₂₃
Fitting, prime radical	C₂×D₈×C₂₂₃	C₂
socle	C₂²×C₂₂₃	C₂×D₄
cosocle	C₂×C₄×C₂₂₃	C₂³
hypercenter	C₂×D₈	D₂₂₃
hypocenter	C₂₂₃	#442
LCS	G ▷ C₄×C₂₂₃ ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×D₈

Complements

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

N	H ≅ G/N	type
1. D₄×Dic₂₂₃	C₂	semidirect
2. #20297236	C₂	semidirect
3. #20297225	C₂	semidirect
4. #20297214	C₂	semidirect
12. D₈×C₂₂₃	C₄	semidirect
20. D₄×C₂₂₃	C₂×C₄	semidirect
22. C₈×C₂₂₃	C₂×C₄	semidirect
34. C₂₂₃	#442	semidirect
43. D₈	Dic₂₂₃	semidirect
51. D₄	C₂×Dic₂₂₃	semidirect
57. C₈	C₂×Dic₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
D₄×Dic₂₂₃	2
#20297236	2
#20297225	2
#20297214	2
C₂×D₈×C₂₂₃	2
#442	223