Subgroups — #30660402

Order 16736.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22552640	C₂	1	1	yes	maximal, characteristic
2	Q₈×D₅₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	Dic₂₀₉₂	C₂	1	1	yes	maximal, characteristic, complemented
4	#22552631	C₂	1	1	yes	maximal, characteristic, complemented
5	#22552629	C₂	1	1	yes	maximal, characteristic, complemented
6	#22552624	C₂	1	1	yes	maximal, characteristic, complemented
7	SD₁₆×C₅₂₃	C₂	1	1	yes	maximal, characteristic, complemented
8	Dic₁₀₄₆	—	2	1	no	—
9	Dic₁₀₄₆	C₂²	1	1	yes	characteristic
10	Dic₁₀₄₆	C₂²	1	1	yes	characteristic
11	C₂²⋊D₅₂₃	—	2	1	no	—
12	C₂×Dic₅₂₃	—	2	1	no	—
13	C₄×D₅₂₃	—	2	1	no	—
14	C₄×D₅₂₃	C₂²	1	1	yes	characteristic
15	C₅₂₃⋊C₈	C₂²	1	1	yes	characteristic
16	Q₈×C₅₂₃	C₂²	1	1	yes	characteristic
17	D₄×C₅₂₃	C₂²	1	1	yes	characteristic
18	C₈×C₅₂₃	C₂²	1	1	yes	characteristic
19	C₂²×C₅₂₃	—	2	1	no	—
20	C₂×D₅₂₃	D₄	1	1	yes	characteristic
21	C₄×C₅₂₃	—	2	1	no	—
22	C₄×C₅₂₃	C₂³	1	1	yes	characteristic
23	Dic₅₂₃	—	2	1	no	—
24	Dic₅₂₃	—	2	1	no	—
25	Dic₅₂₃	D₄	1	1	yes	characteristic
26	C₂×C₅₂₃	—	4	1	no	—
27	C₂×C₅₂₃	C₂×D₄	1	1	yes	characteristic
28	D₅₂₃	—	2	1	no	—
29	C₅₂₃	#137	1	1	yes	characteristic, complemented, Sylow
30	#137	—	523	1	no	maximal, Sylow
31	#41	—	523	1	no	—
32	C₂×Q₈	—	523	1	no	—
33	Q₁₆	—	523	1	no	—
34	Q₁₆	—	523	1	no	—
35	SD₁₆	—	523	1	no	—
36	SD₁₆	D₅₂₃	1	1	yes	characteristic, complemented
37	M₁₆	—	523	1	no	—
38	Q₈	—	523	1	no	—
39	Q₈	—	523	1	no	—
40	Q₈	—	1046	1	no	—
41	Q₈	C₂×D₅₂₃	1	1	yes	characteristic
42	D₄	—	1046	1	no	—
43	D₄	C₂×D₅₂₃	1	1	yes	characteristic
44	C₂×C₄	—	523	1	no	—
45	C₂×C₄	—	1046	1	no	—
46	C₂×C₄	—	1046	1	no	—
47	C₈	—	523	1	no	—
48	C₈	C₂×D₅₂₃	1	1	yes	characteristic
49	C₂²	—	2	1	no	—
50	C₂²	—	523	1	no	—
51	C₄	—	2	1	no	—
52	C₄	—	523	1	no	—
53	C₄	—	1046	1	no	—
54	C₄	—	1046	1	no	—
55	C₄	C₂²×D₅₂₃	1	1	yes	characteristic
56	C₂	—	4	1	no	—
57	C₂	—	1046	1	no	—
58	C₂	D₄×D₅₂₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄×D₅₂₃
commutator, cosocle	C₄×C₅₂₃	C₂³
Frattini	C₄	C₂²×D₅₂₃
Fitting	SD₁₆×C₅₂₃	C₂
socle	C₂×C₅₂₃	C₂×D₄
prime radical	#22552640	C₂
hypercenter	SD₁₆	D₅₂₃
hypocenter	C₅₂₃	#137
LCS	G ▷ C₄×C₅₂₃ ▷ C₂×C₅₂₃ ▷ C₅₂₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ SD₁₆

Complements

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

N	H ≅ G/N	type
2. Q₈×D₅₂₃	C₂	semidirect
3. Dic₂₀₉₂	C₂	semidirect
4. #22552631	C₂	semidirect
5. #22552629	C₂	semidirect
6. #22552624	C₂	semidirect
7. SD₁₆×C₅₂₃	C₂	semidirect
29. C₅₂₃	#137	semidirect
36. SD₁₆	D₅₂₃	semidirect

Maximal subgroups

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

subgroup	index
#22552640	2
Q₈×D₅₂₃	2
Dic₂₀₉₂	2
#22552631	2
#22552629	2
#22552624	2
SD₁₆×C₅₂₃	2
#137	523