Subgroups — #49123576

Order 36224.

112 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#31826897	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₄₅₂₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#31826304	C₂	1	2	yes	maximal, complemented
4	D₉₀₅₆	C₂	1	2	yes	maximal, complemented
5	M₆₄×C₂₈₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#22583632	—	2	1	no	—
7	C₂×D₂₂₆₄	—	2	1	no	—
8	Dic₂₂₆₄	C₂²	1	1	yes	characteristic, complemented
9	#22583555	—	2	1	no	—
10	D₄₅₂₈	—	2	1	no	—
11	D₄₅₂₈	C₂²	1	2	yes	—
12	D₄₅₂₈	C₂²	1	1	yes	characteristic, complemented
13	C₂×C₁₆×C₂₈₃	C₂²	1	1	yes	characteristic
14	C₃₂×C₂₈₃	C₂²	1	2	yes	complemented
15	#17985392	—	4	1	no	—
16	C₂×D₁₁₃₂	—	4	1	no	—
17	Dic₁₁₃₂	—	2	1	no	—
18	#17985381	—	4	1	no	—
19	D₂₂₆₄	—	2	2	no	—
20	D₂₂₆₄	—	2	1	no	—
21	D₂₂₆₄	—	4	1	no	—
22	C₂×C₈×C₂₈₃	D₄	1	1	yes	characteristic
23	C₁₆×C₂₈₃	D₄	1	1	yes	characteristic
24	C₁₆×C₂₈₃	C₂³	1	1	yes	characteristic
25	C₂²×D₂₈₃	—	8	1	no	—
26	Dic₅₆₆	—	4	1	no	—
27	D₁₁₃₂	—	4	2	no	—
28	D₁₁₃₂	—	4	1	no	—
29	D₁₁₃₂	—	8	1	no	—
30	C₂²⋊D₂₈₃	—	8	1	no	—
31	C₄×D₂₈₃	—	8	1	no	—
32	C₂×C₄×C₂₈₃	D₈	1	1	yes	characteristic
33	C₈×C₂₈₃	D₈	1	1	yes	characteristic
34	C₈×C₂₈₃	C₂×D₄	1	1	yes	characteristic
35	C₂²×C₂₈₃	D₁₆	1	1	yes	characteristic
36	C₂×D₂₈₃	—	8	1	no	—
37	C₂×D₂₈₃	—	8	2	no	—
38	C₂×D₂₈₃	—	16	2	no	—
39	C₄×C₂₈₃	D₁₆	1	1	yes	characteristic
40	C₄×C₂₈₃	C₂×D₈	1	1	yes	characteristic
41	Dic₂₈₃	—	8	1	no	—
42	C₂×C₂₈₃	—	2	1	no	—
43	C₂×C₂₈₃	C₂×D₁₆	1	1	yes	characteristic
44	D₂₈₃	—	16	1	no	—
45	D₂₈₃	—	16	2	no	—
46	C₂₈₃	#2263	1	1	yes	characteristic, complemented, Sylow
47	#2263	—	283	1	no	maximal, Sylow
48	#508	—	283	1	no	—
49	C₂×D₁₆	—	283	1	no	—
50	SD₆₄	—	283	2	no	—
51	D₃₂	—	283	2	no	—
52	M₆₄	D₂₈₃	1	1	yes	characteristic, complemented
53	#135	—	566	1	no	—
54	C₂×D₈	—	566	1	no	—
55	Q₃₂	—	283	1	no	—
56	SD₃₂	—	566	1	no	—
57	D₁₆	—	283	2	no	—
58	D₁₆	—	283	1	no	—
59	D₁₆	—	566	1	no	—
60	C₂×C₁₆	C₂×D₂₈₃	1	1	yes	characteristic
61	C₃₂	C₂×D₂₈₃	1	2	yes	complemented
62	#41	—	1132	1	no	—
63	C₂×D₄	—	1132	1	no	—
64	Q₁₆	—	566	1	no	—
65	SD₁₆	—	1132	1	no	—
66	D₈	—	566	2	no	—
67	D₈	—	566	1	no	—
68	D₈	—	1132	1	no	—
69	C₂×C₈	D₁₁₃₂	1	1	yes	characteristic
70	C₁₆	D₁₁₃₂	1	1	yes	characteristic
71	C₁₆	C₂²×D₂₈₃	1	1	yes	characteristic
72	C₂³	—	2264	1	no	—
73	Q₈	—	1132	1	no	—
74	D₄	—	1132	2	no	—
75	D₄	—	1132	1	no	—
76	D₄	—	2264	1	no	—
77	D₄	—	2264	1	no	—
78	C₂×C₄	—	2264	1	no	—
79	C₂×C₄	D₂₂₆₄	1	1	yes	characteristic
80	C₈	D₂₂₆₄	1	1	yes	characteristic
81	C₈	C₂×D₁₁₃₂	1	1	yes	characteristic
82	C₂²	—	4528	2	no	—
83	C₂²	—	2264	1	no	—
84	C₂²	—	2264	2	no	—
85	C₂²	D₄₅₂₈	1	1	yes	characteristic
86	C₄	—	2264	1	no	—
87	C₄	D₄₅₂₈	1	1	yes	characteristic
88	C₄	C₂×D₂₂₆₄	1	1	yes	characteristic
89	C₂	—	2	1	no	—
90	C₂	—	4528	2	no	—
91	C₂	—	4528	1	no	—
92	C₂	C₂×D₄₅₂₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₄₅₂₈
commutator, cosocle	C₁₆×C₂₈₃	C₂³
Frattini	C₁₆	C₂²×D₂₈₃
Fitting	M₆₄×C₂₈₃	C₂
socle	C₂×C₂₈₃	C₂×D₁₆
hypercenter	M₆₄	D₂₈₃
hypocenter	C₂₈₃	#2263
LCS	G ▷ C₁₆×C₂₈₃ ▷ C₈×C₂₈₃ ▷ C₄×C₂₈₃ ▷ C₂×C₂₈₃ ▷ C₂₈₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ M₆₄

Complements

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

N	H ≅ G/N	type
1. #31826897	C₂	semidirect
2. C₂×D₄₅₂₈	C₂	semidirect
3. #31826304	C₂	semidirect
4. D₉₀₅₆	C₂	semidirect
5. M₆₄×C₂₈₃	C₂	semidirect
8. Dic₂₂₆₄	C₂²	semidirect
12. D₄₅₂₈	C₂²	semidirect
14. C₃₂×C₂₈₃	C₂²	semidirect
46. C₂₈₃	#2263	semidirect
52. M₆₄	D₂₈₃	semidirect
61. C₃₂	C₂×D₂₈₃	semidirect

Maximal subgroups

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

subgroup	index
#31826897	2
C₂×D₄₅₂₈	2
#31826304	2
D₉₀₅₆	2
M₆₄×C₂₈₃	2
#2263	283