Subgroups — #45690329

Order 33472.

102 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×Dic₁₀₄₆	C₂	1	1	yes	maximal, characteristic
2	Dic₅₂₃⋊C₈	C₂	1	2	yes	maximal
3	C₄×C₅₂₃⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₈×Dic₅₂₃	C₂	1	2	yes	maximal
5	C₄⋊C₈×C₅₂₃	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₁₀₄₆	C₄	1	1	yes	characteristic
7	C₂×C₅₂₃⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₅₂₃⋊C₈	C₂²	1	1	yes	characteristic
9	C₄⋊Dic₅₂₃	C₄	1	1	yes	characteristic
10	Dic₅₂₃⋊C₄	C₄	1	2	yes	—
11	C₄×Dic₅₂₃	C₂²	1	2	yes	—
12	C₂×C₈×C₅₂₃	C₂²	1	2	yes	—
13	C₄²×C₅₂₃	C₂²	1	1	yes	characteristic
14	Dic₁₀₄₆	C₈	1	4	yes	complemented
15	C₂×Dic₅₂₃	C₂×C₄	1	2	yes	—
16	C₂×Dic₅₂₃	C₂×C₄	1	2	yes	—
17	C₅₂₃⋊C₈	—	2	1	no	—
18	C₅₂₃⋊C₈	Q₈	1	2	yes	complemented
19	C₂×C₄×C₅₂₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×C₅₂₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×C₅₂₃	C₂³	1	1	yes	characteristic
22	C₈×C₅₂₃	—	2	2	no	—
23	C₂²×C₅₂₃	C₂²×C₄	1	1	yes	characteristic
24	C₄×C₅₂₃	—	2	1	no	—
25	C₄×C₅₂₃	C₂×C₈	1	2	yes	—
26	C₄×C₅₂₃	C₂×Q₈	1	1	yes	characteristic
27	C₄×C₅₂₃	#41	1	1	yes	characteristic
28	Dic₅₂₃	—	2	2	no	—
29	Dic₅₂₃	C₂×C₈	1	4	yes	—
30	C₂×C₅₂₃	C₄×Q₈	1	1	yes	characteristic
31	C₂×C₅₂₃	C₂²×C₈	1	1	yes	characteristic
32	C₂×C₅₂₃	#131	1	1	yes	characteristic
33	C₅₂₃	C₈×Q₈	1	1	yes	characteristic, complemented, Sylow
34	C₈×Q₈	—	523	1	no	maximal, Sylow
35	C₄×Q₈	—	523	1	no	—
36	C₄⋊C₈	—	523	2	no	—
37	C₄⋊C₈	D₅₂₃	1	1	yes	characteristic
38	C₄×C₈	—	523	2	no	—
39	C₄×C₈	—	523	1	no	—
40	C₂×Q₈	—	523	1	no	—
41	C₂×C₈	—	523	1	no	—
42	C₂×C₈	—	523	1	no	—
43	C₂×C₈	C₂×D₅₂₃	1	2	yes	—
44	C₄⋊C₄	—	523	2	no	—
45	C₄⋊C₄	—	523	1	no	—
46	C₄²	—	523	2	no	—
47	C₄²	C₂×D₅₂₃	1	1	yes	characteristic
48	Q₈	—	523	4	no	—
49	C₂×C₄	—	523	2	no	—
50	C₂×C₄	—	523	2	no	—
51	C₂×C₄	C₄×D₅₂₃	1	1	yes	characteristic
52	C₂×C₄	C₄×D₅₂₃	1	1	yes	characteristic
53	C₂×C₄	C₂²×D₅₂₃	1	1	yes	characteristic
54	C₈	—	2	2	no	—
55	C₈	—	523	2	no	—
56	C₈	—	1046	1	no	—
57	C₂²	C₂×C₄×D₅₂₃	1	1	yes	characteristic
58	C₄	—	2	1	no	—
59	C₄	—	523	4	no	—
60	C₄	—	1046	2	no	—
61	C₄	C₈×D₅₂₃	1	2	yes	—
62	C₄	Q₈×D₅₂₃	1	1	yes	characteristic
63	C₄	#22552640	1	1	yes	characteristic
64	C₂	#30660338	1	1	yes	characteristic
65	C₂	C₂×C₈×D₅₂₃	1	1	yes	characteristic
66	C₂	#30660381	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₅₂₃
commutator	C₂×C₅₂₃	C₂²×C₈
Fitting	C₄⋊C₈×C₅₂₃	C₂
socle, prime radical	C₂²×C₅₂₃	C₂²×C₄
cosocle	C₂×C₄×C₅₂₃	C₂³
hypercenter	C₄⋊C₈	D₅₂₃
hypocenter	C₅₂₃	C₈×Q₈
LCS	G ▷ C₂×C₅₂₃ ▷ C₅₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
14. Dic₁₀₄₆	C₈	semidirect
18. C₅₂₃⋊C₈	Q₈	semidirect
33. C₅₂₃	C₈×Q₈	semidirect

Maximal subgroups

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

subgroup	index
C₄×Dic₁₀₄₆	2
Dic₅₂₃⋊C₈	2
C₄×C₅₂₃⋊C₈	2
C₈×Dic₅₂₃	2
C₄⋊C₈×C₅₂₃	2
C₈×Q₈	523