Subgroups — C₄×C₈×D₂₂₃ #28383524

Order 14272.

162 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

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

Complements

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

N	H ≅ G/N	type
3. C₄×C₂₂₃⋊C₈	C₂	semidirect
4. C₈×Dic₂₂₃	C₂	semidirect
5. C₄×C₈×C₂₂₃	C₂	semidirect
9. C₈×D₂₂₃	C₄	direct (N × H)
17. C₄×D₂₂₃	C₈	direct (N × H)
19. C₂₂₃⋊C₈	C₂×C₄	semidirect
22. C₈×C₂₂₃	C₂×C₄	semidirect
26. C₄×C₂₂₃	C₂×C₈	semidirect
29. Dic₂₂₃	C₂×C₈	semidirect
32. D₂₂₃	C₄×C₈	direct (N × H)
33. C₂₂₃	C₂×C₄×C₈	semidirect
39. C₄×C₈	D₂₂₃	direct (N × H)
56. C₈	C₄×D₂₂₃	direct (N × H)
62. C₄	C₈×D₂₂₃	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₈×D₂₂₃	2
C₄²×D₂₂₃	2
C₄×C₂₂₃⋊C₈	2
C₈×Dic₂₂₃	2
C₄×C₈×C₂₂₃	2
C₂×C₄×C₈	223

Subgroups — C4×C8×D223 #28383524

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₈×D₂₂₃ #28383524