Subgroups — C₂×C₈×Dic₂₂₃ #28383525

Order 14272.

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

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

Complements

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

N	H ≅ G/N	type
3. C₈×Dic₂₂₃	C₂	direct (N × H)
6. C₂×C₂₂₃⋊C₈	C₄	semidirect
11. C₂×C₈×C₂₂₃	C₄	semidirect
13. C₂×Dic₂₂₃	C₈	direct (N × H)
15. C₂₂₃⋊C₈	C₂×C₄	semidirect
21. C₈×C₂₂₃	C₂×C₄	semidirect
28. Dic₂₂₃	C₂×C₈	direct (N × H)
29. C₂×C₂₂₃	C₄×C₈	semidirect
32. C₂₂₃	C₂×C₄×C₈	semidirect
42. C₂×C₈	Dic₂₂₃	direct (N × H)
54. C₈	C₂×Dic₂₂₃	direct (N × H)
62. C₂	C₈×Dic₂₂₃	direct (N × H)

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
C₂×C₄×Dic₂₂₃	2
C₈×Dic₂₂₃	2
C₂²×C₈×C₂₂₃	2
C₂×C₄×C₈	223

Subgroups — C2×C8×Dic223 #28383525

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₈×Dic₂₂₃ #28383525