Subgroups — 2₋¹⁺⁴×D₂₂₃ #28384412

Order 14272.

794 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20297352	C₂	1	5	yes	maximal, complemented
2	#20297351	C₂	1	10	yes	maximal, complemented
3	#41×D₂₂₃	C₂	1	10	yes	maximal, complemented
4	C₂×Q₈×D₂₂₃	C₂	1	5	yes	maximal, complemented
5	2₋¹⁺⁴×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#16813171	C₂²	1	10	yes	complemented
7	#16813170	C₂²	1	30	yes	complemented
8	#16813169	C₂²	1	30	yes	complemented
9	C₂×Dic₄₄₆	C₂²	1	15	yes	complemented
10	Q₈×D₂₂₃	C₂²	1	10	yes	—
11	Q₈×D₂₂₃	C₂²	1	20	yes	complemented
12	D₄×D₂₂₃	C₂²	1	10	yes	—
13	C₂×C₄×D₂₂₃	C₂²	1	15	yes	—
14	#41×C₂₂₃	C₂²	1	10	yes	complemented
15	C₂×Q₈×C₂₂₃	C₂²	1	5	yes	complemented
16	C₂²×D₂₂₃	C₂³	1	5	yes	—
17	Dic₄₄₆	C₂³	1	30	yes	—
18	D₈₉₂	C₂³	1	10	yes	—
19	C₂²⋊D₂₂₃	C₂³	1	20	yes	—
20	C₂×Dic₂₂₃	C₂³	1	15	yes	—
21	C₄×D₂₂₃	C₂³	1	10	yes	—
22	C₄×D₂₂₃	C₂³	1	30	yes	—
23	Q₈×C₂₂₃	C₂³	1	10	yes	—
24	D₄×C₂₂₃	C₂³	1	10	yes	—
25	C₂×C₄×C₂₂₃	C₂³	1	15	yes	—
26	C₂²×C₂₂₃	C₂⁴	1	5	yes	—
27	C₂×D₂₂₃	—	2	10	no	—
28	C₂×D₂₂₃	C₂⁴	1	5	yes	—
29	C₂×D₂₂₃	C₂⁴	1	1	yes	characteristic
30	C₄×C₂₂₃	C₂⁴	1	10	yes	—
31	Dic₂₂₃	C₂⁴	1	10	yes	—
32	C₂×C₂₂₃	—	2	5	no	—
33	C₂×C₂₂₃	C₂⁵	1	1	yes	characteristic
34	D₂₂₃	—	2	5	no	—
35	D₂₂₃	2₋¹⁺⁴	1	2	yes	direct product factor
36	C₂₂₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic, complemented, Sylow
37	C₂×2₋¹⁺⁴	—	223	1	no	maximal, Sylow
38	2₋¹⁺⁴	—	223	10	no	—
39	2₋¹⁺⁴	—	223	5	no	—
40	2₋¹⁺⁴	D₂₂₃	1	1	yes	characteristic, direct product factor
41	C₂×#41	—	223	10	no	—
42	C₂²×Q₈	—	223	5	no	—
43	#41	—	223	10	no	—
44	#41	—	223	30	no	—
45	#41	—	223	30	no	—
46	#41	C₂×D₂₂₃	1	10	yes	complemented
47	C₂×Q₈	—	223	10	no	—
48	C₂×Q₈	—	223	20	no	—
49	C₂×Q₈	—	223	15	no	—
50	C₂×Q₈	C₂×D₂₂₃	1	5	yes	complemented
51	C₂×D₄	—	223	10	no	—
52	C₂²×C₄	—	223	15	no	—
53	C₂³	—	223	5	no	—
54	Q₈	—	223	30	no	—
55	Q₈	C₂²×D₂₂₃	1	10	yes	—
56	D₄	—	223	20	no	—
57	D₄	—	223	10	no	—
58	D₄	C₂²×D₂₂₃	1	10	yes	—
59	C₂×C₄	—	223	15	no	—
60	C₂×C₄	—	223	10	no	—
61	C₂×C₄	—	223	30	no	—
62	C₂×C₄	C₂²×D₂₂₃	1	15	yes	—
63	C₂²	—	446	10	no	—
64	C₂²	—	223	5	no	—
65	C₂²	—	223	1	no	—
66	C₂²	C₂³×D₂₂₃	1	5	yes	—
67	C₄	—	223	10	no	—
68	C₄	C₂³×D₂₂₃	1	10	yes	—
69	C₂	—	2	5	no	—
70	C₂	—	446	5	no	—
71	C₂	—	223	2	no	—
72	C₂	C₂⁴×D₂₂₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂⁴×D₂₂₃
commutator, socle, cosocle	C₂×C₂₂₃	C₂⁵
Fitting	2₋¹⁺⁴×C₂₂₃	C₂
hypercenter	2₋¹⁺⁴	D₂₂₃
hypocenter	C₂₂₃	C₂×2₋¹⁺⁴
LCS	G ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂ ◁ 2₋¹⁺⁴

Complements

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

N	H ≅ G/N	type
1. #20297352	C₂	semidirect
2. #20297351	C₂	semidirect
3. #41×D₂₂₃	C₂	semidirect
4. C₂×Q₈×D₂₂₃	C₂	semidirect
5. 2₋¹⁺⁴×C₂₂₃	C₂	semidirect
6. #16813171	C₂²	semidirect
7. #16813170	C₂²	semidirect
8. #16813169	C₂²	semidirect
9. C₂×Dic₄₄₆	C₂²	semidirect
11. Q₈×D₂₂₃	C₂²	semidirect
14. #41×C₂₂₃	C₂²	semidirect
15. C₂×Q₈×C₂₂₃	C₂²	semidirect
35. D₂₂₃	2₋¹⁺⁴	direct (N × H)
36. C₂₂₃	C₂×2₋¹⁺⁴	semidirect
40. 2₋¹⁺⁴	D₂₂₃	direct (N × H)
46. #41	C₂×D₂₂₃	semidirect
50. C₂×Q₈	C₂×D₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
#20297352	2
#20297351	2
#41×D₂₂₃	2
C₂×Q₈×D₂₂₃	2
2₋¹⁺⁴×C₂₂₃	2
C₂×2₋¹⁺⁴	223

Subgroups — 2−1+4×D223 #28384412

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — 2₋¹⁺⁴×D₂₂₃ #28384412