Subgroups — #16810990

Order 3392.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13133215	C₂	1	1	yes	maximal, characteristic
2	#13133208	C₂	1	1	yes	maximal, characteristic
3	#13133142	C₂	1	1	yes	maximal, characteristic
4	#13133141	C₂	1	1	yes	maximal, characteristic
5	#13133140	C₂	1	1	yes	maximal, characteristic
6	#13133127	C₂	1	1	yes	maximal, characteristic
7	#103×C₅₃	C₂	1	1	yes	maximal, characteristic
8	C₂×Q₈×C₅₃	C₂²	1	1	yes	characteristic
9	C₂×Dic₁₀₆	C₂²	1	1	yes	characteristic
10	C₂×C₈×C₅₃	C₂²	1	1	yes	characteristic
11	C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₅₃	—	2	1	no	—
13	C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
14	#11719919	—	2	1	no	—
15	#11719919	—	2	1	no	—
16	#11719919	—	2	1	no	—
17	C₄×Dic₅₃	C₂²	1	1	yes	characteristic
18	C₂×#88322	C₂²	1	1	yes	characteristic
19	Q₈×C₅₃	—	2	1	no	—
20	Q₈×C₅₃	—	2	1	no	—
21	C₂×C₄×C₅₃	—	2	1	no	—
22	C₂×C₄×C₅₃	—	2	1	no	—
23	C₂×C₄×C₅₃	C₂³	1	1	yes	characteristic
24	C₂×Dic₅₃	—	2	1	no	—
25	C₂×Dic₅₃	—	2	1	no	—
26	C₂×Dic₅₃	D₄	1	1	yes	characteristic
27	C₂×Dic₅₃	D₄	1	1	yes	characteristic
28	Dic₁₀₆	—	2	1	no	—
29	Dic₁₀₆	—	2	1	no	—
30	C₈×C₅₃	—	2	1	no	—
31	#88322	—	2	1	no	—
32	C₂²×C₅₃	C₂×D₄	1	1	yes	characteristic
33	C₄×C₅₃	—	4	1	no	—
34	C₄×C₅₃	—	4	1	no	—
35	C₄×C₅₃	#41	1	1	yes	characteristic
36	C₄×C₅₃	#41	1	1	yes	characteristic
37	Dic₅₃	—	2	1	no	—
38	Dic₅₃	—	2	1	no	—
39	Dic₅₃	—	4	1	no	—
40	Dic₅₃	—	4	1	no	—
41	C₂×C₅₃	#124	1	1	yes	characteristic
42	C₂×C₅₃	#137	1	1	yes	characteristic
43	C₂×C₅₃	#137	1	1	yes	characteristic
44	#491	—	53	1	no	maximal, Sylow
45	C₅₃	#491	1	1	yes	characteristic, complemented, Sylow
46	C₄⋊Q₈	—	53	1	no	—
47	#125	—	53	1	no	—
48	#103	—	53	1	no	—
49	#103	—	53	1	no	—
50	#103	—	53	1	no	—
51	#103	D₅₃	1	1	yes	characteristic
52	#97	—	53	1	no	—
53	C₂×Q₈	—	53	1	no	—
54	C₂×Q₈	C₂×D₅₃	1	1	yes	characteristic
55	C₂×C₈	—	53	1	no	—
56	C₂×C₈	C₂×D₅₃	1	1	yes	characteristic
57	C₄⋊C₄	—	53	1	no	—
58	C₄⋊C₄	—	106	1	no	—
59	C₄⋊C₄	—	106	1	no	—
60	C₄⋊C₄	—	106	1	no	—
61	C₄⋊C₄	—	106	1	no	—
62	C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
63	C₄²	—	53	1	no	—
64	Q₈	—	2	1	no	—
65	Q₈	—	2	1	no	—
66	Q₈	—	106	1	no	—
67	Q₈	—	106	1	no	—
68	C₂×C₄	—	2	1	no	—
69	C₂×C₄	—	2	1	no	—
70	C₂×C₄	—	53	1	no	—
71	C₂×C₄	—	53	1	no	—
72	C₂×C₄	—	106	1	no	—
73	C₂×C₄	—	106	1	no	—
74	C₂×C₄	C₂²×D₅₃	1	1	yes	characteristic
75	C₈	—	2	1	no	—
76	C₈	—	106	1	no	—
77	C₂²	D₄×D₅₃	1	1	yes	characteristic
78	C₄	—	4	1	no	—
79	C₄	—	4	1	no	—
80	C₄	—	106	1	no	—
81	C₄	—	106	1	no	—
82	C₄	—	212	1	no	—
83	C₄	—	212	1	no	—
84	C₄	#11719945	1	1	yes	characteristic
85	C₄	#11719947	1	1	yes	characteristic
86	C₂	#13133203	1	1	yes	characteristic
87	C₂	#13133246	1	1	yes	characteristic
88	C₂	#13133249	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₅₃
commutator, cosocle	C₂×C₄×C₅₃	C₂³
Frattini	C₂×C₄	C₂²×D₅₃
Fitting	#103×C₅₃	C₂
socle, prime radical	C₂²×C₅₃	C₂×D₄
hypercenter	#103	D₅₃
hypocenter	C₅₃	#491
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #103

Complements

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

N	H ≅ G/N	type
45. C₅₃	#491	semidirect

Maximal subgroups

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

subgroup	index
#13133215	2
#13133208	2
#13133142	2
#13133141	2
#13133140	2
#13133127	2
#103×C₅₃	2
#491	53