Subgroups — #30660293

Order 16736.

50 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₂₀₉₂	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₂₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄⋊C₄×C₅₂₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×D₅₂₃	—	2	1	no	—
5	D₂₀₉₂	—	2	1	no	—
6	D₂₀₉₂	C₄	1	2	yes	complemented
7	C₅₂₃⋊C₈	—	2	1	no	—
8	C₂×C₄×C₅₂₃	—	2	1	no	—
9	C₂×C₄×C₅₂₃	C₂²	1	1	yes	characteristic
10	C₂²×C₅₂₃	D₄	1	1	yes	characteristic
11	C₂×D₅₂₃	—	2	2	no	—
12	C₂×D₅₂₃	—	4	1	no	—
13	C₂×D₅₂₃	—	4	1	no	—
14	C₄×C₅₂₃	—	4	1	no	—
15	C₄×C₅₂₃	C₂×C₄	1	1	yes	characteristic
16	C₄×C₅₂₃	D₄	1	1	yes	characteristic
17	C₂×C₅₂₃	C₂²⋊C₄	1	1	yes	characteristic
18	C₂×C₅₂₃	D₈	1	1	yes	characteristic
19	C₂×C₅₂₃	SD₁₆	1	1	yes	characteristic
20	D₅₂₃	—	4	2	no	—
21	C₅₂₃	#102	1	1	yes	characteristic, complemented, Sylow
22	#102	—	523	1	no	maximal, Sylow
23	C₂×D₄	—	523	1	no	—
24	C₂×C₈	—	523	1	no	—
25	C₄⋊C₄	D₅₂₃	1	1	yes	characteristic, complemented
26	C₂³	—	1046	1	no	—
27	D₄	—	523	2	no	—
28	D₄	—	1046	1	no	—
29	C₂×C₄	—	2	1	no	—
30	C₂×C₄	C₂×D₅₂₃	1	1	yes	characteristic
31	C₈	—	1046	1	no	—
32	C₂²	—	1046	2	no	—
33	C₂²	—	2092	1	no	—
34	C₂²	—	2092	1	no	—
35	C₂²	C₂²⋊D₅₂₃	1	1	yes	characteristic
36	C₄	—	4	1	no	—
37	C₄	C₄×D₅₂₃	1	1	yes	characteristic
38	C₄	D₂₀₉₂	1	1	yes	characteristic
39	C₂	—	2092	2	no	—
40	C₂	(C₂×D₅₂₃)⋊C₄	1	1	yes	characteristic
41	C₂	#22552626	1	1	yes	characteristic
42	C₂	#22552628	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²⋊D₅₂₃
commutator	C₄×C₅₂₃	C₂×C₄
Frattini	C₂×C₄	C₂×D₅₂₃
Fitting	C₄⋊C₄×C₅₂₃	C₂
socle	C₂²×C₅₂₃	D₄
prime radical	C₂×D₂₀₉₂	C₂
cosocle	C₂×C₄×C₅₂₃	C₂²
hypercenter	C₄⋊C₄	D₅₂₃
hypocenter	C₅₂₃	#102
LCS	G ▷ C₄×C₅₂₃ ▷ C₂×C₅₂₃ ▷ C₅₂₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₄⋊C₄

Complements

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

N	H ≅ G/N	type
2. C₂×C₅₂₃⋊C₈	C₂	semidirect
3. C₄⋊C₄×C₅₂₃	C₂	semidirect
6. D₂₀₉₂	C₄	semidirect
21. C₅₂₃	#102	semidirect
25. C₄⋊C₄	D₅₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₂₀₉₂	2
C₂×C₅₂₃⋊C₈	2
C₄⋊C₄×C₅₂₃	2
#102	523