Subgroups — #28384017

Order 14272.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20297327	C₂	1	1	yes	maximal, characteristic, complemented
2	D₈×D₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#20297247	C₂	1	1	yes	maximal, characteristic, complemented
4	D₃₅₆₈	C₂	1	1	yes	maximal, characteristic, complemented
5	#20297245	C₂	1	1	yes	maximal, characteristic, complemented
6	#20297243	C₂	1	1	yes	maximal, characteristic, complemented
7	SD₃₂×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
8	#16813171	—	2	1	no	—
9	D₄×D₂₂₃	—	2	1	no	—
10	#16813157	—	2	1	no	—
11	D₁₇₈₄	C₂²	1	1	yes	characteristic
12	D₁₇₈₄	C₂²	1	1	yes	characteristic, complemented
13	#16813155	—	2	1	no	—
14	C₈×D₂₂₃	C₂²	1	1	yes	characteristic
15	C₂₂₃⋊C₁₆	C₂²	1	1	yes	characteristic, complemented
16	Q₁₆×C₂₂₃	C₂²	1	1	yes	characteristic, complemented
17	D₈×C₂₂₃	C₂²	1	1	yes	characteristic
18	C₁₆×C₂₂₃	C₂²	1	1	yes	characteristic, complemented
19	C₂²×D₂₂₃	—	4	1	no	—
20	D₈₉₂	—	2	1	no	—
21	D₈₉₂	—	2	1	no	—
22	D₈₉₂	—	4	1	no	—
23	C₂²⋊D₂₂₃	—	4	1	no	—
24	C₄×D₂₂₃	—	4	1	no	—
25	C₄×D₂₂₃	D₄	1	1	yes	characteristic
26	C₂₂₃⋊C₈	D₄	1	1	yes	characteristic
27	Q₈×C₂₂₃	—	2	1	no	—
28	D₄×C₂₂₃	—	2	1	no	—
29	C₈×C₂₂₃	C₂³	1	1	yes	characteristic
30	C₂²×C₂₂₃	—	4	1	no	—
31	C₂×D₂₂₃	—	4	1	no	—
32	C₂×D₂₂₃	—	4	1	no	—
33	C₂×D₂₂₃	—	8	2	no	—
34	C₂×D₂₂₃	D₈	1	1	yes	characteristic
35	C₄×C₂₂₃	—	4	1	no	—
36	C₄×C₂₂₃	C₂×D₄	1	1	yes	characteristic
37	Dic₂₂₃	D₈	1	1	yes	characteristic
38	C₂×C₂₂₃	—	8	1	no	—
39	C₂×C₂₂₃	C₂×D₈	1	1	yes	characteristic
40	D₂₂₃	—	2	1	no	—
41	D₂₂₃	—	8	1	no	—
42	D₂₂₃	—	8	1	no	—
43	C₂₂₃	#509	1	1	yes	characteristic, complemented, Sylow
44	#509	—	223	1	no	maximal, Sylow
45	#135	—	223	1	no	—
46	C₂×D₈	—	223	1	no	—
47	SD₃₂	—	223	1	no	—
48	SD₃₂	D₂₂₃	1	1	yes	characteristic, complemented
49	D₁₆	—	223	1	no	—
50	D₁₆	—	223	1	no	—
51	M₃₂	—	223	1	no	—
52	#41	—	446	1	no	—
53	C₂×D₄	—	446	1	no	—
54	Q₁₆	C₂×D₂₂₃	1	1	yes	characteristic, complemented
55	SD₁₆	—	446	1	no	—
56	D₈	—	446	1	no	—
57	D₈	—	223	1	no	—
58	D₈	—	223	1	no	—
59	D₈	C₂×D₂₂₃	1	1	yes	characteristic
60	C₂×C₈	—	223	1	no	—
61	C₁₆	—	223	1	no	—
62	C₁₆	C₂×D₂₂₃	1	1	yes	characteristic, complemented
63	C₂³	—	892	1	no	—
64	Q₈	—	2	1	no	—
65	D₄	—	2	1	no	—
66	D₄	—	892	1	no	—
67	D₄	—	892	1	no	—
68	D₄	—	446	1	no	—
69	D₄	—	446	1	no	—
70	C₂×C₄	—	892	1	no	—
71	C₂×C₄	—	223	1	no	—
72	C₈	—	223	1	no	—
73	C₈	C₂²×D₂₂₃	1	1	yes	characteristic
74	C₂²	—	4	1	no	—
75	C₂²	—	892	1	no	—
76	C₂²	—	892	1	no	—
77	C₂²	—	223	1	no	—
78	C₂²	—	1784	2	no	—
79	C₄	—	4	1	no	—
80	C₄	—	223	1	no	—
81	C₄	D₄×D₂₂₃	1	1	yes	characteristic
82	C₂	—	8	1	no	—
83	C₂	—	446	1	no	—
84	C₂	—	1784	1	no	—
85	C₂	—	1784	1	no	—
86	C₂	D₈×D₂₂₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₈×D₂₂₃
commutator, cosocle	C₈×C₂₂₃	C₂³
Frattini	C₈	C₂²×D₂₂₃
Fitting	SD₃₂×C₂₂₃	C₂
socle	C₂×C₂₂₃	C₂×D₈
hypercenter	SD₃₂	D₂₂₃
hypocenter	C₂₂₃	#509
LCS	G ▷ C₈×C₂₂₃ ▷ C₄×C₂₂₃ ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ SD₃₂

Complements

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

N	H ≅ G/N	type
1. #20297327	C₂	semidirect
2. D₈×D₂₂₃	C₂	semidirect
3. #20297247	C₂	semidirect
4. D₃₅₆₈	C₂	semidirect
5. #20297245	C₂	semidirect
6. #20297243	C₂	semidirect
7. SD₃₂×C₂₂₃	C₂	semidirect
12. D₁₇₈₄	C₂²	semidirect
15. C₂₂₃⋊C₁₆	C₂²	semidirect
16. Q₁₆×C₂₂₃	C₂²	semidirect
18. C₁₆×C₂₂₃	C₂²	semidirect
43. C₂₂₃	#509	semidirect
48. SD₃₂	D₂₂₃	semidirect
54. Q₁₆	C₂×D₂₂₃	semidirect
62. C₁₆	C₂×D₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
#20297327	2
D₈×D₂₂₃	2
#20297247	2
D₃₅₆₈	2
#20297245	2
#20297243	2
SD₃₂×C₂₂₃	2
#509	223