Subgroups — #28384055

Order 14272.

338 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₄×Dic₂₂₃	C₂	1	8	yes	maximal, complemented
2	#20297263	C₂	1	2	yes	maximal, complemented
3	C₂×C₂²⋊Dic₂₂₃	C₂	1	4	yes	maximal, complemented
4	C₂²×D₄×C₂₂₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₂₂₃	C₂²	1	8	yes	complemented
6	C₄⋊Dic₂₂₃	C₂²	1	4	yes	complemented
7	C₂²⋊Dic₂₂₃	C₂²	1	4	yes	—
8	C₂²⋊Dic₂₂₃	C₂²	1	8	yes	complemented
9	C₄×Dic₂₂₃	C₂²	1	4	yes	complemented
10	C₂⁴×C₂₂₃	C₂²	1	2	yes	—
11	C₂×D₄×C₂₂₃	C₄	1	8	yes	complemented
12	C₂×D₄×C₂₂₃	C₂²	1	4	yes	—
13	C₂²×C₄×C₂₂₃	C₂²	1	1	yes	characteristic
14	C₂×Dic₂₂₃	—	4	8	no	—
15	C₂×Dic₂₂₃	C₂³	1	8	yes	—
16	C₂³×C₂₂₃	—	4	4	no	—
17	C₂³×C₂₂₃	C₂×C₄	1	8	yes	complemented
18	C₂³×C₂₂₃	C₂³	1	1	yes	characteristic
19	C₂³×C₂₂₃	C₂³	1	4	yes	—
20	D₄×C₂₂₃	C₂×C₄	1	16	yes	complemented
21	C₂×C₄×C₂₂₃	C₂×C₄	1	4	yes	complemented
22	C₂×C₄×C₂₂₃	C₂³	1	2	yes	—
23	C₂²×C₂₂₃	—	2	2	no	—
24	C₂²×C₂₂₃	—	2	8	no	—
25	C₂²×C₂₂₃	—	4	8	no	—
26	C₂²×C₂₂₃	C₂²×C₄	1	2	yes	—
27	C₂²×C₂₂₃	C₂²×C₄	1	8	yes	—
28	C₂²×C₂₂₃	C₂⁴	1	1	yes	characteristic
29	C₄×C₂₂₃	C₂²×C₄	1	4	yes	—
30	Dic₂₂₃	—	2	8	no	—
31	C₂×C₂₂₃	—	2	2	no	—
32	C₂×C₂₂₃	—	2	8	no	—
33	C₂×C₂₂₃	C₂³×C₄	1	1	yes	characteristic
34	C₂×C₂₂₃	H(5,𝔽₂)	1	2	yes	—
35	C₂₂₃	#518	1	1	yes	characteristic, complemented, Sylow
36	#518	—	223	1	no	maximal, Sylow
37	C₂²×D₄	D₂₂₃	1	1	yes	characteristic
38	C₄×D₄	—	223	8	no	—
39	#117	—	223	2	no	—
40	C₂×C₂²⋊C₄	—	223	4	no	—
41	C₂⁴	C₂×D₂₂₃	1	2	yes	—
42	C₂×D₄	Dic₂₂₃	1	8	yes	complemented
43	C₂×D₄	C₂×D₂₂₃	1	4	yes	—
44	C₂²×C₄	—	223	8	no	—
45	C₂²×C₄	C₂×D₂₂₃	1	1	yes	characteristic
46	C₄⋊C₄	—	223	4	no	—
47	C₂²⋊C₄	—	223	4	no	—
48	C₂²⋊C₄	—	223	8	no	—
49	C₄²	—	223	4	no	—
50	C₂³	—	4	4	no	—
51	C₂³	C₂×Dic₂₂₃	1	8	yes	complemented
52	C₂³	C₂²×D₂₂₃	1	1	yes	characteristic
53	C₂³	C₂²×D₂₂₃	1	4	yes	—
54	D₄	C₂×Dic₂₂₃	1	16	yes	complemented
55	C₂×C₄	—	892	8	no	—
56	C₂×C₄	—	223	8	no	—
57	C₂×C₄	C₂×Dic₂₂₃	1	4	yes	complemented
58	C₂×C₄	C₂²×D₂₂₃	1	2	yes	—
59	C₂²	—	2	8	no	—
60	C₂²	—	2	2	no	—
61	C₂²	—	4	8	no	—
62	C₂²	C₂²×Dic₂₂₃	1	2	yes	—
63	C₂²	C₂²×Dic₂₂₃	1	8	yes	—
64	C₂²	C₂³×D₂₂₃	1	1	yes	characteristic
65	C₄	—	446	8	no	—
66	C₄	C₂²×Dic₂₂₃	1	4	yes	—
67	C₂	—	2	2	no	—
68	C₂	—	2	8	no	—
69	C₂	C₂³×Dic₂₂₃	1	1	yes	characteristic
70	C₂	#20297349	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₂₃
commutator	C₂×C₂₂₃	C₂³×C₄
Fitting, prime radical	C₂²×D₄×C₂₂₃	C₂
socle, cosocle	C₂²×C₂₂₃	C₂⁴
hypercenter	C₂²×D₄	D₂₂₃
hypocenter	C₂₂₃	#518
LCS	G ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂² ◁ C₂²×D₄

Complements

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

N	H ≅ G/N	type
1. D₄×Dic₂₂₃	C₂	semidirect
2. #20297263	C₂	semidirect
3. C₂×C₂²⋊Dic₂₂₃	C₂	semidirect
5. C₂²×Dic₂₂₃	C₂²	semidirect
6. C₄⋊Dic₂₂₃	C₂²	semidirect
8. C₂²⋊Dic₂₂₃	C₂²	semidirect
9. C₄×Dic₂₂₃	C₂²	semidirect
11. C₂×D₄×C₂₂₃	C₄	semidirect
17. C₂³×C₂₂₃	C₂×C₄	semidirect
20. D₄×C₂₂₃	C₂×C₄	semidirect
21. C₂×C₄×C₂₂₃	C₂×C₄	semidirect
35. C₂₂₃	#518	semidirect
42. C₂×D₄	Dic₂₂₃	semidirect
51. C₂³	C₂×Dic₂₂₃	semidirect
54. D₄	C₂×Dic₂₂₃	semidirect
57. C₂×C₄	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
#20297263	2
C₂×C₂²⋊Dic₂₂₃	2
C₂²×D₄×C₂₂₃	2
#518	223