Subgroups — #28383622

Order 14272.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20297263	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊Dic₂₂₃	C₂	1	1	yes	maximal, characteristic
3	#20297238	C₂	1	2	yes	maximal, complemented
4	#20297236	C₂	1	2	yes	maximal, complemented
5	C₂×M₁₆×C₂₂₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₂₃	—	2	1	no	—
7	C₄⋊Dic₂₂₃	—	2	1	no	—
8	C₄⋊Dic₂₂₃	C₂²	1	1	yes	characteristic
9	C₄⋊Dic₂₂₃	C₂²	1	2	yes	—
10	C₄⋊Dic₂₂₃	C₂²	1	1	yes	characteristic
11	C₂²⋊Dic₂₂₃	—	2	1	no	—
12	C₄×Dic₂₂₃	—	2	1	no	—
13	C₂²×C₄×C₂₂₃	C₂²	1	1	yes	characteristic
14	M₁₆×C₂₂₃	C₄	1	4	yes	complemented
15	C₂×C₈×C₂₂₃	C₂²	1	2	yes	—
16	C₂×Dic₂₂₃	—	2	1	no	—
17	C₂×Dic₂₂₃	—	2	1	no	—
18	C₂×Dic₂₂₃	—	2	2	no	—
19	C₂×Dic₂₂₃	—	4	2	no	—
20	C₂³×C₂₂₃	D₄	1	1	yes	characteristic
21	C₂×C₄×C₂₂₃	C₂×C₄	1	2	yes	—
22	C₂×C₄×C₂₂₃	D₄	1	1	yes	characteristic
23	C₂×C₄×C₂₂₃	Q₈	1	2	yes	—
24	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
25	C₈×C₂₂₃	C₂×C₄	1	4	yes	complemented
26	C₂²×C₂₂₃	—	2	1	no	—
27	C₂²×C₂₂₃	—	2	1	no	—
28	C₂²×C₂₂₃	C₄⋊C₄	1	2	yes	—
29	C₂²×C₂₂₃	C₂×D₄	1	1	yes	characteristic
30	C₄×C₂₂₃	C₄⋊C₄	1	2	yes	—
31	C₄×C₂₂₃	C₂²×C₄	1	1	yes	characteristic
32	C₄×C₂₂₃	C₂×Q₈	1	1	yes	characteristic
33	Dic₂₂₃	—	4	1	no	—
34	Dic₂₂₃	—	4	2	no	—
35	Dic₂₂₃	—	4	1	no	—
36	C₂×C₂₂₃	—	2	2	no	—
37	C₂×C₂₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₂₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
39	C₂×C₂₂₃	#137	1	1	yes	characteristic
40	C₂₂₃	#428	1	1	yes	characteristic, complemented, Sylow
41	#428	—	223	1	no	maximal, Sylow
42	C₂×M₁₆	D₂₂₃	1	1	yes	characteristic
43	#117	—	223	1	no	—
44	C₂×C₄⋊C₄	—	223	1	no	—
45	#107	—	223	2	no	—
46	#106	—	223	2	no	—
47	C₂²×C₄	—	446	1	no	—
48	C₂²×C₄	C₂×D₂₂₃	1	1	yes	characteristic
49	M₁₆	Dic₂₂₃	1	4	yes	complemented
50	C₂×C₈	C₂×D₂₂₃	1	2	yes	—
51	C₄⋊C₄	—	446	1	no	—
52	C₄⋊C₄	—	223	1	no	—
53	C₄⋊C₄	—	223	2	no	—
54	C₄⋊C₄	—	223	1	no	—
55	C₂²⋊C₄	—	446	1	no	—
56	C₄²	—	446	1	no	—
57	C₂³	D₈₉₂	1	1	yes	characteristic
58	C₂×C₄	—	892	2	no	—
59	C₂×C₄	—	446	1	no	—
60	C₂×C₄	—	446	1	no	—
61	C₂×C₄	—	446	2	no	—
62	C₂×C₄	C₂×Dic₂₂₃	1	2	yes	—
63	C₂×C₄	D₈₉₂	1	1	yes	characteristic
64	C₂×C₄	Dic₄₄₆	1	2	yes	—
65	C₂×C₄	C₂²×D₂₂₃	1	1	yes	characteristic
66	C₈	C₂×Dic₂₂₃	1	4	yes	complemented
67	C₂²	—	2	1	no	—
68	C₂²	—	2	1	no	—
69	C₂²	C₄⋊Dic₂₂₃	1	2	yes	—
70	C₂²	C₂×D₈₉₂	1	1	yes	characteristic
71	C₄	—	892	2	no	—
72	C₄	—	892	1	no	—
73	C₄	—	892	1	no	—
74	C₄	C₄⋊Dic₂₂₃	1	2	yes	—
75	C₄	C₂²×Dic₂₂₃	1	1	yes	characteristic
76	C₄	C₂×Dic₄₄₆	1	1	yes	characteristic
77	C₂	—	2	2	no	—
78	C₂	C₂×C₄⋊Dic₂₂₃	1	1	yes	characteristic
79	C₂	#20297332	1	1	yes	characteristic
80	C₂	#20297337	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈₉₂
commutator	C₄×C₂₂₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₂₂₃
Fitting	C₂×M₁₆×C₂₂₃	C₂
socle	C₂²×C₂₂₃	C₂×D₄
prime radical	C₂³×C₂₂₃	D₄
cosocle	C₂×C₄×C₂₂₃	C₂³
hypercenter	C₂×M₁₆	D₂₂₃
hypocenter	C₂₂₃	#428
LCS	G ▷ C₄×C₂₂₃ ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
3. #20297238	C₂	semidirect
4. #20297236	C₂	semidirect
14. M₁₆×C₂₂₃	C₄	semidirect
25. C₈×C₂₂₃	C₂×C₄	semidirect
40. C₂₂₃	#428	semidirect
49. M₁₆	Dic₂₂₃	semidirect
66. C₈	C₂×Dic₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
#20297263	2
C₂×C₄⋊Dic₂₂₃	2
#20297238	2
#20297236	2
C₂×M₁₆×C₂₂₃	2
#428	223