Subgroups — #24909736

Order 10624.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19128106	C₂	1	1	yes	maximal, characteristic
2	#19127526	C₂	1	4	yes	maximal
3	C₄×C₈₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	#500×C₈₃	C₂	1	1	yes	maximal, characteristic
5	#14581792	—	2	1	no	—
6	C₂×C₈₃⋊C₁₆	C₂²	1	2	yes	—
7	#14581726	—	2	1	no	—
8	#14581726	C₂²	1	2	yes	—
9	C₄⋊Q₈×C₈₃	—	2	1	no	—
10	#107×C₈₃	—	2	1	no	—
11	#107×C₈₃	C₂²	1	2	yes	—
12	C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
13	C₂×Dic₁₆₆	—	4	1	no	—
14	C₄⋊Dic₈₃	—	2	2	no	—
15	C₄⋊Dic₈₃	—	4	1	no	—
16	C₈₃⋊C₁₆	Q₈	1	4	yes	complemented
17	C₂×Q₈×C₈₃	—	4	1	no	—
18	C₂×C₈×C₈₃	D₄	1	1	yes	characteristic
19	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
20	C₄⋊C₄×C₈₃	—	2	2	no	—
21	C₄⋊C₄×C₈₃	—	4	1	no	—
22	C₄²×C₈₃	D₄	1	1	yes	characteristic
23	Q₈×C₈₃	—	8	2	no	—
24	C₂×C₄×C₈₃	—	4	2	no	—
25	C₂×C₄×C₈₃	D₈	1	1	yes	characteristic
26	C₂×C₄×C₈₃	D₈	1	1	yes	characteristic
27	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
28	C₂×Dic₈₃	—	4	2	no	—
29	Dic₁₆₆	—	8	2	no	—
30	C₈×C₈₃	Q₁₆	1	2	yes	—
31	C₈×C₈₃	C₂×Q₈	1	2	yes	—
32	C₂²×C₈₃	C₂×D₈	1	1	yes	characteristic
33	C₄×C₈₃	—	8	2	no	—
34	C₄×C₈₃	D₁₆	1	2	yes	—
35	C₄×C₈₃	Q₃₂	1	2	yes	—
36	C₄×C₈₃	C₄⋊Q₈	1	1	yes	characteristic
37	C₄×C₈₃	C₂×Q₁₆	1	1	yes	characteristic
38	Dic₈₃	—	8	2	no	—
39	C₂×C₈₃	#500	1	1	yes	characteristic
40	C₂×C₈₃	C₂×D₁₆	1	1	yes	characteristic
41	C₂×C₈₃	C₂×Q₃₂	1	1	yes	characteristic
42	#2252	—	83	1	no	maximal, Sylow
43	C₈₃	#2252	1	1	yes	characteristic, complemented, Sylow
44	#500	—	83	1	no	—
45	#500	D₈₃	1	1	yes	characteristic
46	#366	—	83	4	no	—
47	C₄×C₁₆	—	83	1	no	—
48	C₄⋊Q₈	—	2	1	no	—
49	C₄⋊Q₈	—	166	1	no	—
50	C₂×C₁₆	—	83	2	no	—
51	#107	—	2	1	no	—
52	#107	—	83	2	no	—
53	#107	—	166	1	no	—
54	#107	C₂×D₈₃	1	2	yes	—
55	C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
56	C₂×Q₈	—	4	1	no	—
57	C₂×Q₈	—	332	1	no	—
58	C₂×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
59	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
60	C₄⋊C₄	—	2	2	no	—
61	C₄⋊C₄	—	4	1	no	—
62	C₄⋊C₄	—	332	1	no	—
63	C₄⋊C₄	—	166	2	no	—
64	C₄²	C₂²⋊D₈₃	1	1	yes	characteristic
65	C₁₆	—	83	4	no	—
66	Q₈	—	8	2	no	—
67	Q₈	—	664	2	no	—
68	C₂×C₄	—	4	2	no	—
69	C₂×C₄	—	332	2	no	—
70	C₂×C₄	#13050575	1	1	yes	characteristic
71	C₂×C₄	#13050575	1	1	yes	characteristic
72	C₂×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
73	C₈	#13050580	1	2	yes	—
74	C₈	Q₈×D₈₃	1	2	yes	—
75	C₂²	C₂×#13050575	1	1	yes	characteristic
76	C₄	—	8	2	no	—
77	C₄	—	664	2	no	—
78	C₄	#14581733	1	2	yes	—
79	C₄	#14581738	1	2	yes	—
80	C₄	#14581790	1	1	yes	characteristic
81	C₄	C₂×#13050580	1	1	yes	characteristic
82	C₂	#19128104	1	1	yes	characteristic
83	C₂	C₂×#14581733	1	1	yes	characteristic
84	C₂	C₂×#14581738	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#13050575
commutator, cosocle	C₂×C₈×C₈₃	C₂³
Frattini	C₂×C₈	C₂²×D₈₃
Fitting	#500×C₈₃	C₂
socle, prime radical	C₂²×C₈₃	C₂×D₈
hypercenter	#500	D₈₃
hypocenter	C₈₃	#2252
LCS	G ▷ C₂×C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #500

Complements

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

N	H ≅ G/N	type
16. C₈₃⋊C₁₆	Q₈	semidirect
43. C₈₃	#2252	semidirect

Maximal subgroups

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

subgroup	index
#19128106	2
#19127526	2
C₄×C₈₃⋊C₁₆	2
#500×C₈₃	2
#2252	83