Subgroups — C₂×Q₈×Dic₂₂₃ #28384046

Order 14272.

298 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×Dic₂₂₃	C₂	1	8	yes	maximal, direct product factor
2	C₂×C₄⋊Dic₂₂₃	C₂	1	3	yes	maximal
3	C₂×C₄×Dic₂₂₃	C₂	1	3	yes	maximal
4	C₂²×Q₈×C₂₂₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₂₂₃	C₂²	1	3	yes	—
6	C₂²×Dic₂₂₃	C₂²	1	1	yes	characteristic
7	C₄⋊Dic₂₂₃	C₂²	1	12	yes	—
8	C₄×Dic₂₂₃	C₂²	1	12	yes	—
9	C₂×Q₈×C₂₂₃	C₄	1	8	yes	complemented
10	C₂×Q₈×C₂₂₃	C₂²	1	4	yes	—
11	C₂²×C₄×C₂₂₃	C₂²	1	3	yes	—
12	C₂×Dic₂₂₃	—	2	6	no	—
13	C₂×Dic₂₂₃	Q₈	1	4	yes	direct product factor
14	C₂×Dic₂₂₃	C₂³	1	6	yes	—
15	C₂×Dic₂₂₃	C₂³	1	2	yes	—
16	C₂³×C₂₂₃	C₂³	1	1	yes	characteristic
17	Q₈×C₂₂₃	C₂×C₄	1	16	yes	complemented
18	C₂×C₄×C₂₂₃	C₂×C₄	1	12	yes	—
19	C₂×C₄×C₂₂₃	C₂³	1	6	yes	—
20	C₂²×C₂₂₃	C₂²×C₄	1	2	yes	—
21	C₂²×C₂₂₃	C₂×Q₈	1	2	yes	—
22	C₂²×C₂₂₃	#41	1	2	yes	—
23	C₂²×C₂₂₃	C₂⁴	1	1	yes	characteristic
24	C₄×C₂₂₃	C₂²×C₄	1	12	yes	—
25	Dic₂₂₃	—	2	6	no	—
26	Dic₂₂₃	C₂×Q₈	1	4	yes	direct product factor
27	C₂×C₂₂₃	C₄×Q₈	1	4	yes	complemented
28	C₂×C₂₂₃	C₂³×C₄	1	1	yes	characteristic
29	C₂×C₂₂₃	C₂²×Q₈	1	1	yes	characteristic
30	C₂×C₂₂₃	C₂×#41	1	1	yes	characteristic
31	C₂₂₃	C₂×C₄×Q₈	1	1	yes	characteristic, complemented, Sylow
32	C₂×C₄×Q₈	—	223	1	no	maximal, Sylow
33	C₂²×Q₈	D₂₂₃	1	1	yes	characteristic
34	C₄×Q₈	—	223	8	no	—
35	C₂×C₄⋊C₄	—	223	3	no	—
36	C₂×C₄²	—	223	3	no	—
37	C₂×Q₈	Dic₂₂₃	1	8	yes	direct product factor
38	C₂×Q₈	C₂×D₂₂₃	1	4	yes	—
39	C₂²×C₄	—	223	3	no	—
40	C₂²×C₄	—	223	1	no	—
41	C₂²×C₄	C₂×D₂₂₃	1	3	yes	—
42	C₄⋊C₄	—	223	12	no	—
43	C₄²	—	223	12	no	—
44	C₂³	C₂²×D₂₂₃	1	1	yes	characteristic
45	Q₈	C₂×Dic₂₂₃	1	16	yes	direct product factor
46	C₂×C₄	—	446	6	no	—
47	C₂×C₄	—	223	6	no	—
48	C₂×C₄	—	223	4	no	—
49	C₂×C₄	—	223	2	no	—
50	C₂×C₄	C₂×Dic₂₂₃	1	12	yes	—
51	C₂×C₄	C₂²×D₂₂₃	1	6	yes	—
52	C₂²	C₂²×Dic₂₂₃	1	2	yes	—
53	C₂²	Q₈×D₂₂₃	1	2	yes	—
54	C₂²	#16813171	1	2	yes	—
55	C₂²	C₂³×D₂₂₃	1	1	yes	characteristic
56	C₄	—	446	6	no	—
57	C₄	—	223	4	no	—
58	C₄	C₂²×Dic₂₂₃	1	12	yes	—
59	C₂	Q₈×Dic₂₂₃	1	4	yes	direct product factor
60	C₂	C₂³×Dic₂₂₃	1	1	yes	characteristic
61	C₂	C₂×Q₈×D₂₂₃	1	1	yes	characteristic
62	C₂	C₂×#16813171	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×D₂₂₃
commutator	C₂×C₂₂₃	C₂³×C₄
Frattini	C₂²	C₂³×D₂₂₃
Fitting	C₂²×Q₈×C₂₂₃	C₂
socle, prime radical	C₂³×C₂₂₃	C₂³
cosocle	C₂²×C₂₂₃	C₂⁴
hypercenter	C₂²×Q₈	D₂₂₃
hypocenter	C₂₂₃	C₂×C₄×Q₈
LCS	G ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂³ ◁ C₂²×Q₈

Complements

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

N	H ≅ G/N	type
1. Q₈×Dic₂₂₃	C₂	direct (N × H)
9. C₂×Q₈×C₂₂₃	C₄	semidirect
13. C₂×Dic₂₂₃	Q₈	direct (N × H)
17. Q₈×C₂₂₃	C₂×C₄	semidirect
26. Dic₂₂₃	C₂×Q₈	direct (N × H)
27. C₂×C₂₂₃	C₄×Q₈	semidirect
31. C₂₂₃	C₂×C₄×Q₈	semidirect
37. C₂×Q₈	Dic₂₂₃	direct (N × H)
45. Q₈	C₂×Dic₂₂₃	direct (N × H)
59. C₂	Q₈×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
Q₈×Dic₂₂₃	2
C₂×C₄⋊Dic₂₂₃	2
C₂×C₄×Dic₂₂₃	2
C₂²×Q₈×C₂₂₃	2
C₂×C₄×Q₈	223

Subgroups — C2×Q8×Dic223 #28384046

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×Q₈×Dic₂₂₃ #28384046