Subgroups — #16810898

Order 3392.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13133247	C₂	1	2	yes	maximal
2	C₂×Dic₂₁₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#13133153	C₂	1	1	yes	maximal, characteristic, complemented
4	#13133135	C₂	1	2	yes	maximal, complemented
5	#131×C₅₃	C₂	1	1	yes	maximal, characteristic
6	#41×C₅₃	C₂²	1	1	yes	characteristic
7	C₂×Dic₁₀₆	C₂²	1	2	yes	—
8	M₁₆×C₅₃	—	2	1	no	—
9	M₁₆×C₅₃	C₂²	1	1	yes	characteristic
10	C₂×C₈×C₅₃	—	2	1	no	—
11	C₂×C₈×C₅₃	C₂²	1	1	yes	characteristic
12	#11719925	—	2	2	no	—
13	#11719923	—	2	2	no	—
14	#11719917	C₂²	1	2	yes	—
15	Dic₂₁₂	—	2	1	no	—
16	Dic₂₁₂	—	2	1	no	—
17	Q₈×C₅₃	D₄	1	1	yes	characteristic
18	D₄×C₅₃	—	2	1	no	—
19	D₄×C₅₃	D₄	1	1	yes	characteristic
20	C₂×C₄×C₅₃	—	2	1	no	—
21	C₂×C₄×C₅₃	C₂³	1	1	yes	characteristic
22	C₂×Dic₅₃	—	2	2	no	—
23	Dic₁₀₆	—	2	2	no	—
24	Dic₁₀₆	—	2	2	no	—
25	C₈×C₅₃	—	2	1	no	—
26	C₈×C₅₃	D₄	1	1	yes	characteristic
27	C₈×C₅₃	D₄	1	1	yes	characteristic
28	#88322	—	2	2	no	—
29	C₂²×C₅₃	—	2	1	no	—
30	C₂²×C₅₃	#41	1	1	yes	characteristic
31	C₄×C₅₃	—	2	1	no	—
32	C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
33	C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
34	Dic₅₃	—	4	2	no	—
35	C₂×C₅₃	—	2	1	no	—
36	C₂×C₅₃	—	4	1	no	—
37	C₂×C₅₃	#121	1	1	yes	characteristic
38	#473	—	53	1	no	maximal, Sylow
39	C₅₃	#473	1	1	yes	characteristic, complemented, Sylow
40	#137	—	53	2	no	—
41	C₂×Q₁₆	—	53	1	no	—
42	#131	D₅₃	1	1	yes	characteristic
43	#108	—	53	1	no	—
44	#101	—	53	2	no	—
45	#41	C₂×D₅₃	1	1	yes	characteristic
46	C₂×Q₈	—	53	2	no	—
47	Q₁₆	—	106	2	no	—
48	Q₁₆	—	106	1	no	—
49	Q₁₆	—	106	1	no	—
50	SD₁₆	—	106	2	no	—
51	M₁₆	—	2	1	no	—
52	M₁₆	—	53	2	no	—
53	M₁₆	C₂×D₅₃	1	1	yes	characteristic
54	C₂×C₈	—	2	1	no	—
55	C₂×C₈	C₂×D₅₃	1	1	yes	characteristic
56	Q₈	—	106	2	no	—
57	Q₈	—	106	2	no	—
58	Q₈	D₂₁₂	1	1	yes	characteristic
59	D₄	—	2	1	no	—
60	D₄	D₂₁₂	1	1	yes	characteristic
61	C₂×C₄	—	2	1	no	—
62	C₂×C₄	—	106	2	no	—
63	C₂×C₄	C₂²×D₅₃	1	1	yes	characteristic
64	C₈	—	2	1	no	—
65	C₈	—	106	2	no	—
66	C₈	C₂²⋊D₅₃	1	1	yes	characteristic
67	C₈	C₂²⋊D₅₃	1	1	yes	characteristic
68	C₂²	—	2	1	no	—
69	C₂²	#11719945	1	1	yes	characteristic
70	C₄	—	2	1	no	—
71	C₄	—	212	2	no	—
72	C₄	C₂×D₂₁₂	1	1	yes	characteristic
73	C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
74	C₂	—	2	1	no	—
75	C₂	—	4	1	no	—
76	C₂	#13133191	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#13133191
commutator, cosocle	C₂×C₄×C₅₃	C₂³
Frattini	C₂×C₄	C₂²×D₅₃
Fitting	#131×C₅₃	C₂
socle	C₂×C₅₃	#121
prime radical	#41×C₅₃	C₂²
hypercenter	#131	D₅₃
hypocenter	C₅₃	#473
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #131

Complements

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

N	H ≅ G/N	type
2. C₂×Dic₂₁₂	C₂	semidirect
3. #13133153	C₂	semidirect
4. #13133135	C₂	semidirect
39. C₅₃	#473	semidirect

Maximal subgroups

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

subgroup	index
#13133247	2
C₂×Dic₂₁₂	2
#13133153	2
#13133135	2
#131×C₅₃	2
#473	53