Subgroups — #16811373

Order 3392.

196 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13133215	C₂	1	2	yes	maximal, complemented
2	#13133210	C₂	1	4	yes	maximal, complemented
3	#13133209	C₂	1	2	yes	maximal, complemented
4	#13133198	C₂	1	2	yes	maximal
5	#13133194	C₂	1	4	yes	maximal, complemented
6	#124×C₅₃	C₂	1	1	yes	maximal, characteristic
7	C₂×Q₈×C₅₃	C₂²	1	1	yes	characteristic
8	C₂×D₄×C₅₃	C₂²	1	1	yes	characteristic
9	C₂²×Dic₅₃	C₂²	1	2	yes	—
10	C₂×Dic₁₀₆	C₂²	1	2	yes	—
11	C₂²⋊C₄×C₅₃	C₂²	1	4	yes	—
12	C₄²×C₅₃	C₂²	1	1	yes	characteristic
13	C₂²⋊Dic₅₃	C₂²	1	4	yes	—
14	C₂²⋊Dic₅₃	C₂²	1	2	yes	—
15	C₄⋊Dic₅₃	C₂²	1	2	yes	—
16	C₄⋊Dic₅₃	C₂²	1	2	yes	—
17	#11719919	C₂²	1	4	yes	—
18	#11719919	C₂²	1	4	yes	—
19	#11719919	C₂²	1	4	yes	—
20	C₄×Dic₅₃	C₂²	1	2	yes	—
21	C₂³×C₅₃	C₂³	1	2	yes	—
22	Q₈×C₅₃	—	4	1	no	—
23	D₄×C₅₃	—	4	1	no	—
24	C₂×C₄×C₅₃	C₂³	1	2	yes	—
25	C₂×C₄×C₅₃	C₂³	1	1	yes	characteristic
26	C₂×C₄×C₅₃	C₂³	1	2	yes	—
27	C₂×Dic₅₃	—	4	2	no	—
28	C₂×Dic₅₃	C₂³	1	4	yes	—
29	C₂×Dic₅₃	C₂³	1	4	yes	—
30	Dic₁₀₆	—	4	2	no	—
31	C₂²×C₅₃	—	2	2	no	—
32	C₂²×C₅₃	—	2	4	no	—
33	C₂²×C₅₃	C₂⁴	1	1	yes	characteristic
34	C₄×C₅₃	—	2	2	no	—
35	C₄×C₅₃	—	2	1	no	—
36	C₄×C₅₃	—	2	2	no	—
37	Dic₅₃	—	2	4	no	—
38	Dic₅₃	—	2	4	no	—
39	C₂×C₅₃	—	4	2	no	—
40	C₂×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
41	C₂×C₅₃	2₋¹⁺⁴	1	2	yes	—
42	#563	—	53	1	no	maximal, Sylow
43	C₅₃	#563	1	1	yes	characteristic, complemented, Sylow
44	C₄⋊Q₈	—	53	2	no	—
45	#126	—	53	4	no	—
46	#125	—	53	2	no	—
47	#124	D₅₃	1	1	yes	characteristic
48	#123	—	53	2	no	—
49	C₂²⋊Q₈	—	53	4	no	—
50	C₂×Q₈	—	53	2	no	—
51	C₂×Q₈	C₂×D₅₃	1	1	yes	characteristic
52	C₂×D₄	C₂×D₅₃	1	1	yes	characteristic
53	C₂²×C₄	—	53	2	no	—
54	C₄⋊C₄	—	53	2	no	—
55	C₄⋊C₄	—	53	4	no	—
56	C₄⋊C₄	—	53	4	no	—
57	C₄⋊C₄	—	53	4	no	—
58	C₄⋊C₄	—	53	2	no	—
59	C₂²⋊C₄	—	53	4	no	—
60	C₂²⋊C₄	—	53	2	no	—
61	C₂²⋊C₄	C₂×D₅₃	1	4	yes	—
62	C₄²	—	53	2	no	—
63	C₄²	C₂×D₅₃	1	1	yes	characteristic
64	C₂³	C₂²×D₅₃	1	2	yes	—
65	Q₈	—	4	1	no	—
66	Q₈	—	212	2	no	—
67	D₄	—	4	1	no	—
68	C₂×C₄	—	53	4	no	—
69	C₂×C₄	—	53	4	no	—
70	C₂×C₄	—	212	2	no	—
71	C₂×C₄	C₂²×D₅₃	1	2	yes	—
72	C₂×C₄	C₂²×D₅₃	1	2	yes	—
73	C₂×C₄	C₂²×D₅₃	1	1	yes	characteristic
74	C₂²	—	2	2	no	—
75	C₂²	—	2	4	no	—
76	C₂²	C₂³×D₅₃	1	1	yes	characteristic
77	C₄	—	2	1	no	—
78	C₄	—	2	2	no	—
79	C₄	—	2	2	no	—
80	C₄	—	106	4	no	—
81	C₄	—	106	4	no	—
82	C₂	—	4	2	no	—
83	C₂	#13133262	1	1	yes	characteristic
84	C₂	#13133264	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₅₃
commutator, socle, cosocle	C₂²×C₅₃	C₂⁴
Fitting	#124×C₅₃	C₂
prime radical	C₂×D₄×C₅₃	C₂²
hypercenter	#124	D₅₃
hypocenter	C₅₃	#563
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ #124

Complements

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

N	H ≅ G/N	type
1. #13133215	C₂	semidirect
2. #13133210	C₂	semidirect
3. #13133209	C₂	semidirect
5. #13133194	C₂	semidirect
43. C₅₃	#563	semidirect

Maximal subgroups

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

subgroup	index
#13133215	2
#13133210	2
#13133209	2
#13133198	2
#13133194	2
#124×C₅₃	2
#563	53