Subgroups — C₅₃×#2205 #20271142

Order 6784.

224 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈²×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅₃×#456	C₂	1	3	yes	maximal
3	C₅₃×#356	C₂	1	3	yes	maximal, complemented
4	2₋¹⁺⁴×C₅₃	C₂²	1	1	yes	characteristic
5	#137×C₅₃	—	2	3	no	—
6	C₄⋊Q₈×C₅₃	—	2	3	no	—
7	C₄⋊Q₈×C₅₃	C₂²	1	3	yes	—
8	C₄×Q₈×C₅₃	—	2	3	no	—
9	#104×C₅₃	—	2	3	no	—
10	#101×C₅₃	C₂²	1	3	yes	—
11	#41×C₅₃	—	2	3	no	—
12	#41×C₅₃	—	4	1	no	—
13	C₂×Q₈×C₅₃	—	2	1	no	—
14	C₂×Q₈×C₅₃	—	2	1	no	—
15	C₂×Q₈×C₅₃	—	4	1	no	—
16	C₂×Q₈×C₅₃	D₄	1	3	yes	—
17	C₂×Q₈×C₅₃	C₂³	1	1	yes	characteristic
18	Q₁₆×C₅₃	—	4	3	no	—
19	SD₁₆×C₅₃	—	4	3	no	—
20	M₁₆×C₅₃	—	2	3	no	—
21	C₄⋊C₄×C₅₃	—	2	3	no	—
22	C₄⋊C₄×C₅₃	—	2	6	no	—
23	C₄²×C₅₃	—	2	3	no	—
24	C₄²×C₅₃	D₄	1	3	yes	—
25	Q₈×C₅₃	—	2	4	no	—
26	Q₈×C₅₃	—	2	3	no	—
27	Q₈×C₅₃	—	4	1	no	—
28	Q₈×C₅₃	—	4	3	no	—
29	Q₈×C₅₃	—	8	1	no	—
30	D₄×C₅₃	—	2	3	no	—
31	D₄×C₅₃	—	4	1	no	—
32	C₂×C₄×C₅₃	—	2	3	no	—
33	C₂×C₄×C₅₃	—	2	3	no	—
34	C₂×C₄×C₅₃	—	4	3	no	—
35	C₂×C₄×C₅₃	C₂×D₄	1	3	yes	—
36	C₈×C₅₃	—	4	3	no	—
37	C₂²×C₅₃	—	4	1	no	—
38	C₂²×C₅₃	#120	1	1	yes	characteristic
39	C₄×C₅₃	—	2	6	no	—
40	C₄×C₅₃	—	2	3	no	—
41	C₄×C₅₃	—	4	3	no	—
42	C₄×C₅₃	—	4	1	no	—
43	#2205	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
44	C₂×C₅₃	—	2	1	no	—
45	C₂×C₅₃	—	8	1	no	—
46	C₂×C₅₃	B(4,𝔽₂)	1	1	yes	characteristic
47	Q₈²	C₂×C₅₃	1	1	yes	characteristic, complemented
48	#456	C₂×C₅₃	1	3	yes	—
49	#356	C₂×C₅₃	1	3	yes	complemented
50	C₅₃	#2205	1	1	yes	characteristic, direct product factor, Sylow
51	2₋¹⁺⁴	C₂²×C₅₃	1	1	yes	characteristic
52	#137	—	2	3	no	—
53	C₄⋊Q₈	—	2	3	no	—
54	C₄⋊Q₈	C₂²×C₅₃	1	3	yes	—
55	C₄×Q₈	—	2	3	no	—
56	#104	—	2	3	no	—
57	#101	C₂²×C₅₃	1	3	yes	—
58	#41	—	2	3	no	—
59	#41	—	4	1	no	—
60	C₂×Q₈	—	2	1	no	—
61	C₂×Q₈	—	2	1	no	—
62	C₂×Q₈	—	4	1	no	—
63	C₂×Q₈	D₄×C₅₃	1	3	yes	—
64	C₂×Q₈	C₂³×C₅₃	1	1	yes	characteristic
65	Q₁₆	—	4	3	no	—
66	SD₁₆	—	4	3	no	—
67	M₁₆	—	2	3	no	—
68	C₄⋊C₄	—	2	3	no	—
69	C₄⋊C₄	—	2	6	no	—
70	C₄²	—	2	3	no	—
71	C₄²	D₄×C₅₃	1	3	yes	—
72	Q₈	—	2	4	no	—
73	Q₈	—	2	3	no	—
74	Q₈	—	4	1	no	—
75	Q₈	—	4	3	no	—
76	Q₈	—	8	1	no	—
77	D₄	—	2	3	no	—
78	D₄	—	4	1	no	—
79	C₂×C₄	—	2	3	no	—
80	C₂×C₄	—	2	3	no	—
81	C₂×C₄	—	4	3	no	—
82	C₂×C₄	C₂×D₄×C₅₃	1	3	yes	—
83	C₈	—	4	3	no	—
84	C₂²	—	4	1	no	—
85	C₂²	#120×C₅₃	1	1	yes	characteristic
86	C₄	—	2	6	no	—
87	C₄	—	2	3	no	—
88	C₄	—	4	3	no	—
89	C₄	—	4	1	no	—
90	C₂	—	2	1	no	—
91	C₂	—	8	1	no	—
92	C₂	C₅₃×B(4,𝔽₂)	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₅₃	B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×Q₈	C₂³×C₅₃
prime radical	2₋¹⁺⁴×C₅₃	C₂²
LCS	G ▷ C₂×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅₃ ◁ C₂²×C₅₃ ◁ C₂×Q₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. Q₈²×C₅₃	C₂	semidirect
3. C₅₃×#356	C₂	semidirect
43. #2205	C₅₃	direct (N × H)
47. Q₈²	C₂×C₅₃	semidirect
49. #356	C₂×C₅₃	semidirect
50. C₅₃	#2205	direct (N × H)

Maximal subgroups

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

subgroup	index
Q₈²×C₅₃	2
C₅₃×#456	2
C₅₃×#356	2
#2205	53

Subgroups — C53×#2205 #20271142

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#2205 #20271142