Subgroups — #35292211

Order 21248.

74 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24905674	C₂	1	2	yes	maximal
2	C₈₃×#1397	C₂	1	1	yes	maximal, characteristic
3	C₈₃⋊C₆₄	C₄	1	4	yes	complemented
4	C₂×C₃₂×C₈₃	C₄	1	1	yes	characteristic
5	C₂×C₃₂×C₈₃	C₂²	1	1	yes	characteristic
6	C₄×C₁₆×C₈₃	C₄	1	1	yes	characteristic
7	C₂×C₁₆×C₈₃	C₈	1	1	yes	characteristic
8	C₂×C₁₆×C₈₃	C₂×C₄	1	1	yes	characteristic
9	C₄×C₈×C₈₃	C₈	1	1	yes	characteristic
10	C₃₂×C₈₃	C₈	1	2	yes	—
11	C₃₂×C₈₃	C₂×C₄	1	2	yes	—
12	C₂×C₈×C₈₃	C₁₆	1	1	yes	characteristic
13	C₂×C₈×C₈₃	C₂×C₈	1	1	yes	characteristic
14	C₄²×C₈₃	C₁₆	1	1	yes	characteristic
15	C₁₆×C₈₃	C₁₆	1	2	yes	—
16	C₁₆×C₈₃	C₄²	1	1	yes	characteristic
17	C₁₆×C₈₃	C₂×C₈	1	1	yes	characteristic
18	C₂×C₄×C₈₃	—	2	1	no	—
19	C₂×C₄×C₈₃	C₂×C₁₆	1	1	yes	characteristic
20	C₈×C₈₃	—	2	1	no	—
21	C₈×C₈₃	C₄×C₈	1	1	yes	characteristic
22	C₈×C₈₃	C₂×C₁₆	1	1	yes	characteristic
23	C₂²×C₈₃	M₆₄	1	1	yes	characteristic
24	C₄×C₈₃	—	4	1	no	—
25	C₄×C₈₃	C₄×C₁₆	1	1	yes	characteristic
26	C₄×C₈₃	M₆₄	1	1	yes	characteristic
27	#7511	—	83	1	no	maximal, Sylow
28	C₂×C₈₃	—	2	1	no	—
29	C₂×C₈₃	#1397	1	1	yes	characteristic
30	M₁₂₈	—	83	2	no	—
31	#1397	D₈₃	1	1	yes	characteristic
32	C₈₃	#7511	1	1	yes	characteristic, complemented, Sylow
33	C₂×C₃₂	Dic₈₃	1	1	yes	characteristic
34	C₂×C₃₂	C₂×D₈₃	1	1	yes	characteristic
35	C₄×C₁₆	Dic₈₃	1	1	yes	characteristic
36	C₆₄	—	83	4	no	—
37	C₂×C₁₆	C₈₃⋊C₈	1	1	yes	characteristic
38	C₂×C₁₆	C₂×Dic₈₃	1	1	yes	characteristic
39	C₄×C₈	C₈₃⋊C₈	1	1	yes	characteristic
40	C₃₂	C₈₃⋊C₈	1	2	yes	—
41	C₃₂	C₄×D₈₃	1	2	yes	—
42	C₂×C₈	C₈₃⋊C₁₆	1	1	yes	characteristic
43	C₂×C₈	C₂×C₈₃⋊C₈	1	1	yes	characteristic
44	C₄²	C₈₃⋊C₁₆	1	1	yes	characteristic
45	C₁₆	C₈₃⋊C₁₆	1	2	yes	—
46	C₁₆	C₄×Dic₈₃	1	1	yes	characteristic
47	C₁₆	C₂×C₈₃⋊C₈	1	1	yes	characteristic
48	C₂×C₄	—	2	1	no	—
49	C₂×C₄	C₂×C₈₃⋊C₁₆	1	1	yes	characteristic
50	C₈	—	2	1	no	—
51	C₈	C₄×C₈₃⋊C₈	1	1	yes	characteristic
52	C₈	C₂×C₈₃⋊C₁₆	1	1	yes	characteristic
53	C₂²	#19127535	1	1	yes	characteristic
54	C₄	—	4	1	no	—
55	C₄	C₄×C₈₃⋊C₁₆	1	1	yes	characteristic
56	C₄	#19127535	1	1	yes	characteristic
57	C₂	—	2	1	no	—
58	C₂	#24905597	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆	C₄×Dic₈₃
commutator	C₄×C₈₃	C₄×C₁₆
Frattini	C₂×C₃₂	C₂×D₈₃
Fitting	C₈₃×#1397	C₂
socle	C₂×C₈₃	#1397
prime radical	C₂²×C₈₃	M₆₄
cosocle	C₂×C₃₂×C₈₃	C₂²
hypercenter	#1397	D₈₃
hypocenter	C₈₃	#7511
LCS	G ▷ C₄×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₁₆ ◁ #1397

Complements

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

N	H ≅ G/N	type
3. C₈₃⋊C₆₄	C₄	semidirect
32. C₈₃	#7511	semidirect

Maximal subgroups

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

subgroup	index
#24905674	2
C₈₃×#1397	2
#7511	83