Subgroups — #16810895

Order 3392.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13133248	C₂	1	2	yes	maximal
2	#13133223	C₂	1	1	yes	maximal, characteristic
3	#13133153	C₂	1	1	yes	maximal, characteristic, complemented
4	#13133136	C₂	1	2	yes	maximal, complemented
5	C₂×Q₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Q₈×C₅₃	C₂²	1	2	yes	—
7	#11719945	C₂²	1	1	yes	characteristic
8	Q₁₆×C₅₃	—	2	1	no	—
9	Q₁₆×C₅₃	—	2	1	no	—
10	C₂×C₈×C₅₃	C₂²	1	1	yes	characteristic
11	#11719925	—	2	2	no	—
12	#11719924	—	2	2	no	—
13	#11719917	C₂²	1	1	yes	characteristic
14	#11719917	C₂²	1	2	yes	—
15	#11719912	—	2	1	no	—
16	C₈×D₅₃	—	2	1	no	—
17	Q₈×C₅₃	—	2	2	no	—
18	Q₈×C₅₃	—	2	2	no	—
19	C₂×C₄×C₅₃	—	2	2	no	—
20	C₂×C₄×C₅₃	C₂³	1	1	yes	characteristic
21	C₂²⋊D₅₃	—	2	1	no	—
22	D₂₁₂	D₄	1	1	yes	characteristic
23	C₄×D₅₃	—	2	1	no	—
24	Dic₁₀₆	D₄	1	1	yes	characteristic
25	C₈×C₅₃	D₄	1	1	yes	characteristic
26	C₈×C₅₃	D₄	1	1	yes	characteristic
27	#88322	—	2	1	no	—
28	#88322	—	2	2	no	—
29	C₂²×C₅₃	#41	1	1	yes	characteristic
30	C₂×D₅₃	—	2	1	no	—
31	C₄×C₅₃	—	4	2	no	—
32	C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
33	C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
34	Dic₅₃	—	2	1	no	—
35	C₂×C₅₃	—	2	1	no	—
36	C₂×C₅₃	#121	1	1	yes	characteristic
37	D₅₃	—	4	1	no	—
38	#473	—	53	1	no	maximal, Sylow
39	C₅₃	#473	1	1	yes	characteristic, complemented, Sylow
40	#137	—	53	2	no	—
41	C₂×Q₁₆	D₅₃	1	1	yes	characteristic, complemented
42	#131	—	53	1	no	—
43	#108	—	53	1	no	—
44	#101	—	53	2	no	—
45	#41	—	53	1	no	—
46	C₂×Q₈	C₂×D₅₃	1	2	yes	—
47	Q₁₆	—	2	1	no	—
48	Q₁₆	—	2	1	no	—
49	Q₁₆	—	106	2	no	—
50	SD₁₆	—	106	2	no	—
51	M₁₆	—	53	1	no	—
52	M₁₆	—	53	2	no	—
53	M₁₆	—	106	1	no	—
54	C₂×C₈	—	106	1	no	—
55	C₂×C₈	C₂×D₅₃	1	1	yes	characteristic
56	Q₈	—	2	2	no	—
57	Q₈	—	2	2	no	—
58	Q₈	—	53	1	no	—
59	D₄	—	53	1	no	—
60	D₄	—	106	1	no	—
61	C₂×C₄	—	2	2	no	—
62	C₂×C₄	—	106	1	no	—
63	C₂×C₄	C₂²×D₅₃	1	1	yes	characteristic
64	C₈	—	106	2	no	—
65	C₈	—	106	1	no	—
66	C₈	C₂²⋊D₅₃	1	1	yes	characteristic
67	C₈	C₂²⋊D₅₃	1	1	yes	characteristic
68	C₂²	—	106	1	no	—
69	C₂²	#11719947	1	1	yes	characteristic
70	C₄	—	4	2	no	—
71	C₄	—	106	1	no	—
72	C₄	D₄×D₅₃	1	1	yes	characteristic
73	C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
74	C₂	—	2	1	no	—
75	C₂	—	212	1	no	—
76	C₂	#13133188	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#13133188
commutator, cosocle	C₂×C₄×C₅₃	C₂³
Frattini	C₂×C₄	C₂²×D₅₃
Fitting	C₂×Q₁₆×C₅₃	C₂
socle	C₂×C₅₃	#121
prime radical	#11719945	C₂²
hypercenter	C₂×Q₁₆	D₅₃
hypocenter	C₅₃	#473
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×Q₁₆

Complements

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

N	H ≅ G/N	type
3. #13133153	C₂	semidirect
4. #13133136	C₂	semidirect
5. C₂×Q₁₆×C₅₃	C₂	semidirect
39. C₅₃	#473	semidirect
41. C₂×Q₁₆	D₅₃	semidirect

Maximal subgroups

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

subgroup	index
#13133248	2
#13133223	2
#13133153	2
#13133136	2
C₂×Q₁₆×C₅₃	2
#473	53