Subgroups — C₅₃×#2381 #20271318

Order 6784.

424 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#391	C₂	1	4	yes	maximal
3	C₅₃×#387	C₂	1	8	yes	maximal
4	C₅₃×#383	C₂	1	2	yes	maximal
5	C₂²×Q₈×C₅₃	C₂²	1	2	yes	—
6	C₄⋊Q₈×C₅₃	C₄	1	8	yes	complemented
7	C₂×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
8	C₂×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
9	C₂×C₄²×C₅₃	C₂²	1	4	yes	—
10	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
11	#95×C₅₃	C₂²	1	8	yes	—
12	#95×C₅₃	C₂²	1	4	yes	—
13	#95×C₅₃	C₂²	1	8	yes	—
14	C₂×Q₈×C₅₃	—	4	2	no	—
15	C₂×Q₈×C₅₃	—	4	2	no	—
16	C₂×Q₈×C₅₃	C₂×C₄	1	8	yes	—
17	C₂²×C₄×C₅₃	C₂³	1	8	yes	—
18	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
19	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
20	C₂²×C₄×C₅₃	C₂³	1	4	yes	—
21	C₄⋊C₄×C₅₃	—	4	4	no	—
22	C₄⋊C₄×C₅₃	C₂×C₄	1	16	yes	—
23	C₄²×C₅₃	—	4	4	no	—
24	C₄²×C₅₃	C₂×C₄	1	4	yes	—
25	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
26	Q₈×C₅₃	—	4	8	no	—
27	C₂×C₄×C₅₃	—	2	2	no	—
28	C₂×C₄×C₅₃	—	2	4	no	—
29	C₂×C₄×C₅₃	—	2	4	no	—
30	C₂×C₄×C₅₃	—	2	4	no	—
31	C₂×C₄×C₅₃	—	2	8	no	—
32	C₂×C₄×C₅₃	—	2	16	no	—
33	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
34	C₂×C₄×C₅₃	C₂²×C₄	1	8	yes	—
35	C₂×C₄×C₅₃	C₂²×C₄	1	4	yes	—
36	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
37	C₂²×C₅₃	H(5,𝔽₂)	1	2	yes	—
38	C₂²×C₅₃	2₋¹⁺⁴	1	4	yes	—
39	C₄×C₅₃	—	2	8	no	—
40	C₄×C₅₃	—	2	2	no	—
41	C₄×C₅₃	—	2	4	no	—
42	C₄×C₅₃	—	4	8	no	—
43	#2381	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
44	C₂×C₅₃	#518	1	1	yes	characteristic
45	C₂×C₅₃	#519	1	2	yes	—
46	C₂×C₅₃	#563	1	4	yes	—
47	C₂×C₄⋊Q₈	C₂×C₅₃	1	1	yes	characteristic
48	#391	C₂×C₅₃	1	4	yes	—
49	#387	C₂×C₅₃	1	8	yes	—
50	#383	C₂×C₅₃	1	2	yes	—
51	C₅₃	#2381	1	1	yes	characteristic, direct product factor, Sylow
52	C₂²×Q₈	C₂²×C₅₃	1	2	yes	—
53	C₄⋊Q₈	C₄×C₅₃	1	8	yes	complemented
54	C₂×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
55	C₂×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
56	C₂×C₄²	C₂²×C₅₃	1	4	yes	—
57	C₂×C₄²	C₂²×C₅₃	1	1	yes	characteristic
58	#95	C₂²×C₅₃	1	8	yes	—
59	#95	C₂²×C₅₃	1	4	yes	—
60	#95	C₂²×C₅₃	1	8	yes	—
61	C₂×Q₈	—	4	2	no	—
62	C₂×Q₈	—	4	2	no	—
63	C₂×Q₈	C₂×C₄×C₅₃	1	8	yes	—
64	C₂²×C₄	C₂³×C₅₃	1	8	yes	—
65	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
66	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
67	C₂²×C₄	C₂³×C₅₃	1	4	yes	—
68	C₄⋊C₄	—	4	4	no	—
69	C₄⋊C₄	C₂×C₄×C₅₃	1	16	yes	—
70	C₄²	—	4	4	no	—
71	C₄²	C₂×C₄×C₅₃	1	4	yes	—
72	C₂³	C₂⁴×C₅₃	1	1	yes	characteristic
73	Q₈	—	4	8	no	—
74	C₂×C₄	—	2	4	no	—
75	C₂×C₄	—	2	4	no	—
76	C₂×C₄	—	2	2	no	—
77	C₂×C₄	—	2	4	no	—
78	C₂×C₄	—	2	8	no	—
79	C₂×C₄	—	2	16	no	—
80	C₂×C₄	C₂²×C₄×C₅₃	1	4	yes	—
81	C₂×C₄	C₂²×C₄×C₅₃	1	2	yes	—
82	C₂×C₄	C₂²×C₄×C₅₃	1	8	yes	—
83	C₂²	C₂³×C₄×C₅₃	1	1	yes	characteristic
84	C₂²	H(5,𝔽₂)×C₅₃	1	2	yes	—
85	C₂²	2₋¹⁺⁴×C₅₃	1	4	yes	—
86	C₄	—	2	8	no	—
87	C₄	—	2	2	no	—
88	C₄	—	2	4	no	—
89	C₄	—	4	8	no	—
90	C₂	C₅₃×#518	1	1	yes	characteristic
91	C₂	C₅₃×#519	1	2	yes	—
92	C₂	C₅₃×#563	1	4	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₅₃	C₂⁴
commutator	C₂²	C₂³×C₄×C₅₃
Frattini, cosocle	C₂³	C₂⁴×C₅₃
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
6. C₄⋊Q₈×C₅₃	C₄	semidirect
43. #2381	C₅₃	direct (N × H)
51. C₅₃	#2381	direct (N × H)
53. C₄⋊Q₈	C₄×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊Q₈×C₅₃	2
C₅₃×#391	2
C₅₃×#387	2
C₅₃×#383	2
#2381	53

Subgroups — C53×#2381 #20271318

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#2381 #20271318