Subgroups — #45690644

Order 33472.

82 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#30660392	C₂	1	1	yes	maximal, characteristic
2	Q₁₆×D₅₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	Dic₄₁₈₄	C₂	1	1	yes	maximal, characteristic, complemented
4	#30660319	C₂	1	1	yes	maximal, characteristic, complemented
5	#30660317	C₂	1	1	yes	maximal, characteristic, complemented
6	#30660312	C₂	1	1	yes	maximal, characteristic, complemented
7	SD₃₂×C₅₂₃	C₂	1	1	yes	maximal, characteristic, complemented
8	#22552640	—	2	1	no	—
9	Q₈×D₅₂₃	—	2	1	no	—
10	Dic₂₀₉₂	C₂²	1	1	yes	characteristic
11	Dic₂₀₉₂	C₂²	1	1	yes	characteristic
12	#22552631	—	2	1	no	—
13	#22552629	—	2	1	no	—
14	C₈×D₅₂₃	C₂²	1	1	yes	characteristic
15	C₅₂₃⋊C₁₆	C₂²	1	1	yes	characteristic
16	Q₁₆×C₅₂₃	C₂²	1	1	yes	characteristic
17	D₈×C₅₂₃	C₂²	1	1	yes	characteristic
18	C₁₆×C₅₂₃	C₂²	1	1	yes	characteristic
19	Dic₁₀₄₆	—	2	1	no	—
20	Dic₁₀₄₆	—	2	1	no	—
21	Dic₁₀₄₆	—	4	1	no	—
22	C₂²⋊D₅₂₃	—	4	1	no	—
23	C₂×Dic₅₂₃	—	4	1	no	—
24	C₄×D₅₂₃	—	4	1	no	—
25	C₄×D₅₂₃	D₄	1	1	yes	characteristic
26	C₅₂₃⋊C₈	D₄	1	1	yes	characteristic
27	Q₈×C₅₂₃	—	2	1	no	—
28	D₄×C₅₂₃	—	2	1	no	—
29	C₈×C₅₂₃	C₂³	1	1	yes	characteristic
30	C₂²×C₅₂₃	—	4	1	no	—
31	C₂×D₅₂₃	D₈	1	1	yes	characteristic
32	C₄×C₅₂₃	—	4	1	no	—
33	C₄×C₅₂₃	C₂×D₄	1	1	yes	characteristic
34	Dic₅₂₃	—	4	1	no	—
35	Dic₅₂₃	—	4	1	no	—
36	Dic₅₂₃	D₈	1	1	yes	characteristic
37	C₂×C₅₂₃	—	8	1	no	—
38	C₂×C₅₂₃	C₂×D₈	1	1	yes	characteristic
39	D₅₂₃	—	2	1	no	—
40	C₅₂₃	#510	1	1	yes	characteristic, complemented, Sylow
41	#510	—	523	1	no	maximal, Sylow
42	#135	—	523	1	no	—
43	C₂×Q₁₆	—	523	1	no	—
44	Q₃₂	—	523	1	no	—
45	Q₃₂	—	523	1	no	—
46	SD₃₂	—	523	1	no	—
47	SD₃₂	D₅₂₃	1	1	yes	characteristic, complemented
48	M₃₂	—	523	1	no	—
49	#41	—	1046	1	no	—
50	C₂×Q₈	—	1046	1	no	—
51	Q₁₆	—	523	1	no	—
52	Q₁₆	—	523	1	no	—
53	Q₁₆	—	1046	1	no	—
54	Q₁₆	C₂×D₅₂₃	1	1	yes	characteristic
55	SD₁₆	—	1046	1	no	—
56	D₈	C₂×D₅₂₃	1	1	yes	characteristic
57	C₂×C₈	—	523	1	no	—
58	C₁₆	—	523	1	no	—
59	C₁₆	C₂×D₅₂₃	1	1	yes	characteristic
60	Q₈	—	2	1	no	—
61	Q₈	—	1046	1	no	—
62	Q₈	—	1046	1	no	—
63	Q₈	—	2092	1	no	—
64	D₄	—	2	1	no	—
65	D₄	—	2092	1	no	—
66	C₂×C₄	—	523	1	no	—
67	C₂×C₄	—	2092	1	no	—
68	C₂×C₄	—	2092	1	no	—
69	C₈	—	523	1	no	—
70	C₈	C₂²×D₅₂₃	1	1	yes	characteristic
71	C₂²	—	4	1	no	—
72	C₂²	—	523	1	no	—
73	C₄	—	4	1	no	—
74	C₄	—	523	1	no	—
75	C₄	—	2092	1	no	—
76	C₄	—	2092	1	no	—
77	C₄	D₄×D₅₂₃	1	1	yes	characteristic
78	C₂	—	8	1	no	—
79	C₂	—	1046	1	no	—
80	C₂	D₈×D₅₂₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₈×D₅₂₃
commutator, cosocle	C₈×C₅₂₃	C₂³
Frattini	C₈	C₂²×D₅₂₃
Fitting	SD₃₂×C₅₂₃	C₂
socle	C₂×C₅₂₃	C₂×D₈
prime radical	#30660392	C₂
hypercenter	SD₃₂	D₅₂₃
hypocenter	C₅₂₃	#510
LCS	G ▷ C₈×C₅₂₃ ▷ C₄×C₅₂₃ ▷ C₂×C₅₂₃ ▷ C₅₂₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ SD₃₂

Complements

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

N	H ≅ G/N	type
2. Q₁₆×D₅₂₃	C₂	semidirect
3. Dic₄₁₈₄	C₂	semidirect
4. #30660319	C₂	semidirect
5. #30660317	C₂	semidirect
6. #30660312	C₂	semidirect
7. SD₃₂×C₅₂₃	C₂	semidirect
40. C₅₂₃	#510	semidirect
47. SD₃₂	D₅₂₃	semidirect

Maximal subgroups

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

subgroup	index
#30660392	2
Q₁₆×D₅₂₃	2
Dic₄₁₈₄	2
#30660319	2
#30660317	2
#30660312	2
SD₃₂×C₅₂₃	2
#510	523