Subgroups — #16811254

Order 3392.

206 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13133216	C₂	1	4	yes	maximal, complemented
2	#13133208	C₂	1	4	yes	maximal, complemented
3	#13133196	C₂	1	4	yes	maximal
4	#13133176	C₂	1	2	yes	maximal
5	C₂×C₄⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄×C₅₃	C₂²	1	2	yes	—
7	C₂²×C₄×C₅₃	C₂²	1	1	yes	characteristic
8	C₂×Dic₁₀₆	C₂²	1	4	yes	—
9	C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
10	C₂²⋊Dic₅₃	C₂²	1	4	yes	—
11	C₄⋊Dic₅₃	C₂²	1	2	yes	—
12	C₄⋊Dic₅₃	C₂²	1	2	yes	—
13	C₄⋊Dic₅₃	C₂²	1	4	yes	—
14	#11719919	C₂²	1	8	yes	—
15	C₄×Dic₅₃	C₂²	1	4	yes	—
16	C₂³×C₅₃	C₂³	1	1	yes	characteristic
17	C₂×C₄×C₅₃	—	4	2	no	—
18	C₂×C₄×C₅₃	Q₈	1	4	yes	complemented
19	C₂×C₄×C₅₃	C₂³	1	4	yes	—
20	C₂×C₄×C₅₃	C₂³	1	2	yes	—
21	C₂×Dic₅₃	C₂³	1	4	yes	—
22	C₂×Dic₅₃	C₂³	1	4	yes	—
23	Dic₁₀₆	—	4	4	no	—
24	C₂²×C₅₃	—	2	1	no	—
25	C₂²×C₅₃	—	2	1	no	—
26	C₂²×C₅₃	C₂×Q₈	1	2	yes	—
27	C₂²×C₅₃	C₂⁴	1	1	yes	characteristic
28	C₄×C₅₃	—	2	4	no	—
29	C₄×C₅₃	C₂×Q₈	1	4	yes	—
30	Dic₅₃	—	2	4	no	—
31	Dic₅₃	—	2	4	no	—
32	C₂×C₅₃	—	2	2	no	—
33	C₂×C₅₃	C₂²×Q₈	1	1	yes	characteristic
34	C₂×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
35	C₂×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
36	#544	—	53	1	no	maximal, Sylow
37	C₅₃	#544	1	1	yes	characteristic, complemented, Sylow
38	C₄⋊Q₈	—	53	4	no	—
39	#125	—	53	4	no	—
40	C₂²⋊Q₈	—	53	4	no	—
41	#117	—	53	2	no	—
42	C₂×C₄⋊C₄	D₅₃	1	1	yes	characteristic
43	C₂×Q₈	—	53	4	no	—
44	C₂²×C₄	C₂×D₅₃	1	2	yes	—
45	C₂²×C₄	C₂×D₅₃	1	1	yes	characteristic
46	C₄⋊C₄	—	53	2	no	—
47	C₄⋊C₄	—	53	2	no	—
48	C₄⋊C₄	—	53	4	no	—
49	C₄⋊C₄	—	53	8	no	—
50	C₄⋊C₄	C₂×D₅₃	1	4	yes	—
51	C₂²⋊C₄	—	53	4	no	—
52	C₄²	—	53	4	no	—
53	C₂³	C₂²×D₅₃	1	1	yes	characteristic
54	Q₈	—	212	4	no	—
55	C₂×C₄	—	4	2	no	—
56	C₂×C₄	—	53	4	no	—
57	C₂×C₄	—	53	4	no	—
58	C₂×C₄	Dic₁₀₆	1	4	yes	complemented
59	C₂×C₄	C₂²×D₅₃	1	4	yes	—
60	C₂×C₄	C₂²×D₅₃	1	2	yes	—
61	C₂²	—	2	1	no	—
62	C₂²	—	2	1	no	—
63	C₂²	C₂×Dic₁₀₆	1	2	yes	—
64	C₂²	C₂³×D₅₃	1	1	yes	characteristic
65	C₄	—	2	4	no	—
66	C₄	—	106	4	no	—
67	C₄	—	106	4	no	—
68	C₄	C₂×Dic₁₀₆	1	4	yes	—
69	C₂	—	2	2	no	—
70	C₂	C₂²×Dic₁₀₆	1	1	yes	characteristic
71	C₂	#13133262	1	1	yes	characteristic
72	C₂	#13133265	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₅₃
commutator, socle, cosocle	C₂²×C₅₃	C₂⁴
Fitting	C₂×C₄⋊C₄×C₅₃	C₂
prime radical	C₂³×C₅₃	C₂³
hypercenter	C₂×C₄⋊C₄	D₅₃
hypocenter	C₅₃	#544
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
1. #13133216	C₂	semidirect
2. #13133208	C₂	semidirect
18. C₂×C₄×C₅₃	Q₈	semidirect
37. C₅₃	#544	semidirect
58. C₂×C₄	Dic₁₀₆	semidirect

Maximal subgroups

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

subgroup	index
#13133216	2
#13133208	2
#13133196	2
#13133176	2
C₂×C₄⋊C₄×C₅₃	2
#544	53