Subgroups — #45687191

Order 33344.

110 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	#117×C₅₂₁	C₂	1	1	yes	maximal, characteristic
4	C₂²×Dic₅₂₁	—	2	1	no	—
5	#22552348	—	2	1	no	—
6	#22552348	C₄	1	2	yes	complemented
7	C₂×#16861960	—	2	1	no	—
8	C₄×Dic₅₂₁	—	2	1	no	—
9	C₄×Dic₅₂₁	C₄	1	2	yes	complemented
10	C₂²×C₄×C₅₂₁	C₂²	1	1	yes	characteristic
11	C₄⋊C₄×C₅₂₁	C₄	1	2	yes	complemented
12	C₂²⋊C₄×C₅₂₁	—	2	1	no	—
13	C₄²×C₅₂₁	—	2	1	no	—
14	C₂×Dic₅₂₁	—	2	4	no	—
15	C₂×Dic₅₂₁	—	2	2	no	—
16	#16861960	—	2	2	no	—
17	C₂³×C₅₂₁	D₄	1	1	yes	characteristic
18	C₂×C₄×C₅₂₁	—	2	2	no	—
19	C₂×C₄×C₅₂₁	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×C₅₂₁	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×C₅₂₁	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×C₅₂₁	D₄	1	1	yes	characteristic
23	C₂×C₄×C₅₂₁	D₄	1	1	yes	characteristic
24	C₂×C₄×C₅₂₁	Q₈	1	1	yes	characteristic
25	C₂²×C₅₂₁	—	2	2	no	—
26	C₂²×C₅₂₁	C₂²⋊C₄	1	1	yes	characteristic
27	C₂²×C₅₂₁	C₂²⋊C₄	1	1	yes	characteristic
28	C₂²×C₅₂₁	C₄⋊C₄	1	1	yes	characteristic
29	C₄×C₅₂₁	—	4	2	no	—
30	C₄×C₅₂₁	C₄²	1	1	yes	characteristic
31	C₄×C₅₂₁	C₂²⋊C₄	1	1	yes	characteristic
32	C₄×C₅₂₁	C₄⋊C₄	1	1	yes	characteristic
33	C₄×C₅₂₁	C₄⋊C₄	1	1	yes	characteristic
34	Dic₅₂₁	—	2	4	no	—
35	C₂×C₅₂₁	—	2	1	no	—
36	C₂×C₅₂₁	—	2	1	no	—
37	C₂×C₅₂₁	#95	1	1	yes	characteristic
38	C₂×C₅₂₁	#104	1	2	yes	—
39	C₅₂₁	#339	1	1	yes	characteristic, complemented, Sylow
40	#339	—	521	1	no	maximal, Sylow
41	C₂×M₁₆	—	521	1	no	—
42	#117	D₅₂₁	1	1	yes	characteristic
43	C₂×C₄²	—	521	1	no	—
44	C₂²×C₄	—	1042	1	no	—
45	C₂²×C₄	C₂×D₅₂₁	1	1	yes	characteristic
46	M₁₆	—	521	2	no	—
47	M₁₆	—	1042	1	no	—
48	C₂×C₈	—	1042	1	no	—
49	C₄⋊C₄	Dic₅₂₁	1	2	yes	complemented
50	C₂²⋊C₄	—	2	1	no	—
51	C₄²	—	2	1	no	—
52	C₄²	—	521	2	no	—
53	C₄²	—	1042	1	no	—
54	C₂³	C₂²⋊D₅₂₁	1	1	yes	characteristic
55	C₂×C₄	—	2	2	no	—
56	C₂×C₄	—	1042	4	no	—
57	C₂×C₄	—	1042	2	no	—
58	C₂×C₄	C₄×D₅₂₁	1	1	yes	characteristic
59	C₂×C₄	C₄×D₅₂₁	1	1	yes	characteristic
60	C₂×C₄	C₂×Dic₅₂₁	1	1	yes	characteristic
61	C₂×C₄	C₂²⋊D₅₂₁	1	1	yes	characteristic
62	C₂×C₄	D₂₀₈₄	1	1	yes	characteristic
63	C₂×C₄	Dic₁₀₄₂	1	1	yes	characteristic
64	C₈	—	1042	2	no	—
65	C₂²	—	2	2	no	—
66	C₂²	#22552341	1	1	yes	characteristic
67	C₂²	C₂²⋊Dic₅₂₁	1	1	yes	characteristic
68	C₂²	#22552343	1	1	yes	characteristic
69	C₄	—	4	2	no	—
70	C₄	—	1042	4	no	—
71	C₄	C₄×Dic₅₂₁	1	1	yes	characteristic
72	C₄	#22552341	1	1	yes	characteristic
73	C₄	#22552343	1	1	yes	characteristic
74	C₄	C₄⋊Dic₅₂₁	1	1	yes	characteristic
75	C₂	—	2	1	no	—
76	C₂	—	2	1	no	—
77	C₂	#30659615	1	1	yes	characteristic
78	C₂	#30659635	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²⋊D₅₂₁
commutator	C₄×C₅₂₁	C₄²
Frattini	C₂²×C₄	C₂×D₅₂₁
Fitting	#117×C₅₂₁	C₂
socle	C₂²×C₅₂₁	C₂²⋊C₄
prime radical	C₂³×C₅₂₁	D₄
cosocle	C₂²×C₄×C₅₂₁	C₂²
hypercenter	#117	D₅₂₁
hypocenter	C₅₂₁	#339
LCS	G ▷ C₄×C₅₂₁ ▷ C₂×C₅₂₁ ▷ C₅₂₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #117

Complements

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

N	H ≅ G/N	type
6. #22552348	C₄	semidirect
9. C₄×Dic₅₂₁	C₄	semidirect
11. C₄⋊C₄×C₅₂₁	C₄	semidirect
39. C₅₂₁	#339	semidirect
49. C₄⋊C₄	Dic₅₂₁	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
#117×C₅₂₁	2
#339	521