Subgroups — #45687455

Order 33344.

126 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#22552348	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₅₂₁	C₂	1	1	yes	maximal, characteristic
3	#30659637	C₂	1	4	yes	maximal, complemented
4	C₂×M₁₆×C₅₂₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₅₂₁	C₄	1	2	yes	—
6	#22552348	—	2	2	no	—
7	C₂×#16861960	C₂²	1	2	yes	—
8	C₄×Dic₅₂₁	C₄	1	1	yes	characteristic
9	C₄×Dic₅₂₁	C₄	1	1	yes	characteristic
10	C₄×Dic₅₂₁	C₂²	1	2	yes	—
11	C₂²×C₄×C₅₂₁	C₂²	1	1	yes	characteristic
12	M₁₆×C₅₂₁	—	2	2	no	—
13	C₂×C₈×C₅₂₁	C₂²	1	2	yes	—
14	C₂×Dic₅₂₁	—	2	4	no	—
15	C₂×Dic₅₂₁	C₂×C₄	1	2	yes	—
16	C₂×Dic₅₂₁	C₂×C₄	1	2	yes	—
17	#16861960	—	2	2	no	—
18	C₂³×C₅₂₁	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×C₅₂₁	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×C₅₂₁	D₄	1	2	yes	—
21	C₂×C₄×C₅₂₁	Q₈	1	2	yes	—
22	C₂×C₄×C₅₂₁	C₂³	1	1	yes	characteristic
23	C₈×C₅₂₁	—	2	2	no	—
24	C₂²×C₅₂₁	—	2	2	no	—
25	C₂²×C₅₂₁	C₄⋊C₄	1	2	yes	—
26	C₂²×C₅₂₁	C₂²×C₄	1	1	yes	characteristic
27	C₄×C₅₂₁	C₄⋊C₄	1	2	yes	—
28	C₄×C₅₂₁	C₂×D₄	1	1	yes	characteristic
29	C₄×C₅₂₁	C₂×Q₈	1	1	yes	characteristic
30	Dic₅₂₁	—	2	2	no	—
31	Dic₅₂₁	M₁₆	1	4	yes	complemented
32	C₂×C₅₂₁	—	2	2	no	—
33	C₂×C₅₂₁	C₂×C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₅₂₁	C₂×M₁₆	1	2	yes	—
35	C₅₂₁	#423	1	1	yes	characteristic, complemented, Sylow
36	#423	—	521	1	no	maximal, Sylow
37	C₂×M₁₆	—	521	1	no	—
38	C₂×M₁₆	D₅₂₁	1	1	yes	characteristic
39	C₂×C₄²	—	521	1	no	—
40	C₄⋊C₈	—	521	4	no	—
41	C₂²×C₄	—	521	2	no	—
42	C₂²×C₄	C₂×D₅₂₁	1	1	yes	characteristic
43	M₁₆	—	2	2	no	—
44	M₁₆	—	1042	2	no	—
45	C₂×C₈	—	521	2	no	—
46	C₂×C₈	C₂×D₅₂₁	1	2	yes	—
47	C₄²	—	521	1	no	—
48	C₄²	—	521	1	no	—
49	C₄²	—	521	2	no	—
50	C₂³	C₄×D₅₂₁	1	1	yes	characteristic
51	C₂×C₄	—	521	2	no	—
52	C₂×C₄	—	521	2	no	—
53	C₂×C₄	—	1042	4	no	—
54	C₂×C₄	C₄×D₅₂₁	1	1	yes	characteristic
55	C₂×C₄	C₂²⋊D₅₂₁	1	2	yes	—
56	C₂×C₄	Dic₁₀₄₂	1	2	yes	—
57	C₂×C₄	C₂²×D₅₂₁	1	1	yes	characteristic
58	C₈	—	2	2	no	—
59	C₈	—	1042	2	no	—
60	C₂²	—	2	2	no	—
61	C₂²	#22552343	1	2	yes	—
62	C₂²	C₂×C₄×D₅₂₁	1	1	yes	characteristic
63	C₄	—	521	4	no	—
64	C₄	—	1042	2	no	—
65	C₄	#22552343	1	2	yes	—
66	C₄	C₂×C₂²⋊D₅₂₁	1	1	yes	characteristic
67	C₄	C₂×Dic₁₀₄₂	1	1	yes	characteristic
68	C₂	—	2	2	no	—
69	C₂	C₂×#22552343	1	1	yes	characteristic
70	C₂	M₁₆×D₅₂₁	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₅₂₁
commutator, socle	C₂²×C₅₂₁	C₂²×C₄
Fitting	C₂×M₁₆×C₅₂₁	C₂
prime radical	C₂³×C₅₂₁	C₂×C₄
cosocle	C₂×C₄×C₅₂₁	C₂³
hypercenter	C₂×M₁₆	D₅₂₁
hypocenter	C₅₂₁	#423
LCS	G ▷ C₂²×C₅₂₁ ▷ C₅₂₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
3. #30659637	C₂	semidirect
31. Dic₅₂₁	M₁₆	semidirect
35. C₅₂₁	#423	semidirect

Maximal subgroups

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

subgroup	index
C₂×#22552348	2
C₂×C₄×Dic₅₂₁	2
#30659637	2
C₂×M₁₆×C₅₂₁	2
#423	521