Subgroups — #16810913

Order 3392.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13133217	C₂	1	1	yes	maximal, characteristic, complemented
2	#13133148	C₂	1	2	yes	maximal, complemented
3	C₄⋊#88322	C₂	1	1	yes	maximal, characteristic, complemented
4	#13133138	C₂	1	2	yes	maximal
5	C₄×D₄×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×D₄×C₅₃	C₂²	1	1	yes	characteristic
7	C₂²×C₄×C₅₃	—	2	1	no	—
8	C₂×Dic₁₀₆	—	2	1	no	—
9	C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
10	C₂²⋊C₄×C₅₃	—	2	1	no	—
11	C₄²×C₅₃	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₅₃	—	2	1	no	—
13	C₄⋊Dic₅₃	C₂²	1	2	yes	—
14	C₂×#88322	C₂²	1	2	yes	—
15	C₂³×C₅₃	—	2	1	no	—
16	D₄×C₅₃	—	2	1	no	—
17	D₄×C₅₃	Q₈	1	2	yes	complemented
18	C₂×C₄×C₅₃	—	2	1	no	—
19	C₂×C₄×C₅₃	—	4	2	no	—
20	C₂×C₄×C₅₃	D₄	1	1	yes	characteristic
21	C₂×C₄×C₅₃	D₄	1	1	yes	characteristic
22	C₂×C₄×C₅₃	C₂³	1	1	yes	characteristic
23	C₂×Dic₅₃	—	2	2	no	—
24	Dic₁₀₆	—	4	2	no	—
25	#88322	—	2	2	no	—
26	C₂²×C₅₃	—	2	2	no	—
27	C₂²×C₅₃	—	4	1	no	—
28	C₂²×C₅₃	—	4	1	no	—
29	C₂²×C₅₃	C₂×D₄	1	1	yes	characteristic
30	C₄×C₅₃	—	2	1	no	—
31	C₄×C₅₃	—	4	1	no	—
32	C₄×C₅₃	SD₁₆	1	2	yes	—
33	C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
34	C₄×C₅₃	#41	1	1	yes	characteristic
35	Dic₅₃	—	4	2	no	—
36	C₂×C₅₃	—	4	2	no	—
37	C₂×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
38	C₂×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
39	C₂×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
40	#476	—	53	1	no	maximal, Sylow
41	C₅₃	#476	1	1	yes	characteristic, complemented, Sylow
42	C₄⋊Q₈	—	53	1	no	—
43	C₄×D₄	D₅₃	1	1	yes	characteristic
44	#106	—	53	2	no	—
45	C₄⋊C₈	—	53	1	no	—
46	#102	—	53	2	no	—
47	C₂×Q₈	—	106	1	no	—
48	C₂×D₄	C₂×D₅₃	1	1	yes	characteristic
49	C₂²×C₄	—	2	1	no	—
50	C₂×C₈	—	53	2	no	—
51	C₄⋊C₄	—	53	2	no	—
52	C₄⋊C₄	—	106	1	no	—
53	C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
54	C₂²⋊C₄	—	2	1	no	—
55	C₄²	C₂×D₅₃	1	1	yes	characteristic
56	C₂³	—	2	1	no	—
57	Q₈	—	212	2	no	—
58	D₄	—	2	1	no	—
59	D₄	Dic₁₀₆	1	2	yes	complemented
60	C₂×C₄	—	2	1	no	—
61	C₂×C₄	—	4	2	no	—
62	C₂×C₄	—	106	2	no	—
63	C₂×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
64	C₂×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
65	C₂×C₄	C₂²×D₅₃	1	1	yes	characteristic
66	C₈	—	106	2	no	—
67	C₂²	—	2	2	no	—
68	C₂²	—	4	1	no	—
69	C₂²	—	4	1	no	—
70	C₂²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
71	C₄	—	2	1	no	—
72	C₄	—	4	1	no	—
73	C₄	—	212	2	no	—
74	C₄	#11719923	1	2	yes	—
75	C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
76	C₄	#11719945	1	1	yes	characteristic
77	C₂	—	4	2	no	—
78	C₂	#13133196	1	1	yes	characteristic
79	C₂	C₂×#11719923	1	1	yes	characteristic
80	C₂	#13133242	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₅₃
commutator, cosocle	C₂×C₄×C₅₃	C₂³
Frattini	C₂×C₄	C₂²×D₅₃
Fitting	C₄×D₄×C₅₃	C₂
socle	C₂²×C₅₃	C₂×D₄
prime radical	C₂×D₄×C₅₃	C₂²
hypercenter	C₄×D₄	D₅₃
hypocenter	C₅₃	#476
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×D₄

Complements

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

N	H ≅ G/N	type
1. #13133217	C₂	semidirect
2. #13133148	C₂	semidirect
3. C₄⋊#88322	C₂	semidirect
17. D₄×C₅₃	Q₈	semidirect
41. C₅₃	#476	semidirect
59. D₄	Dic₁₀₆	semidirect

Maximal subgroups

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

subgroup	index
#13133217	2
#13133148	2
C₄⋊#88322	2
#13133138	2
C₄×D₄×C₅₃	2
#476	53