Subgroups — C₅₃×#2657 #20271594

Order 6784.

844 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#127×C₅₃	C₂	1	3	yes	maximal, complemented
2	C₂×#124×C₅₃	C₂	1	3	yes	maximal, complemented
3	C₅₃×#392	C₂	1	8	yes	maximal, complemented
4	C₅₃×#376	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×Q₈×C₅₃	C₂²	1	1	yes	characteristic
6	C₂²×D₄×C₅₃	C₂²	1	3	yes	—
7	C₂²×D₄×C₅₃	C₂²	1	12	yes	—
8	#127×C₅₃	—	2	12	no	—
9	#124×C₅₃	—	2	12	no	—
10	C₂×C₂²⋊C₄×C₅₃	C₂²	1	12	yes	—
11	C₂×C₄²×C₅₃	C₂²	1	3	yes	—
12	#95×C₅₃	C₂²	1	4	yes	—
13	C₂⁴×C₅₃	C₂³	1	6	yes	—
14	C₂×Q₈×C₅₃	—	2	6	no	—
15	C₂×D₄×C₅₃	—	2	6	no	—
16	C₂×D₄×C₅₃	—	2	24	no	—
17	C₂×D₄×C₅₃	—	2	12	no	—
18	C₂×D₄×C₅₃	—	4	24	no	—
19	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
20	C₂²×C₄×C₅₃	C₂³	1	6	yes	—
21	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
22	C₂²⋊C₄×C₅₃	—	2	24	no	—
23	C₄²×C₅₃	D₄	1	12	yes	complemented
24	C₂³×C₅₃	—	2	12	no	—
25	C₂³×C₅₃	—	2	6	no	—
26	C₂³×C₅₃	—	2	24	no	—
27	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
28	Q₈×C₅₃	—	4	4	no	—
29	D₄×C₅₃	—	4	48	no	—
30	D₄×C₅₃	—	4	12	no	—
31	C₂×C₄×C₅₃	—	2	12	no	—
32	C₂×C₄×C₅₃	—	2	6	no	—
33	C₂×C₄×C₅₃	C₂×D₄	1	12	yes	—
34	C₂×C₄×C₅₃	C₂×D₄	1	6	yes	—
35	C₂²×C₅₃	—	4	6	no	—
36	C₂²×C₅₃	—	4	24	no	—
37	C₂²×C₅₃	—	4	12	no	—
38	C₂²×C₅₃	C₂²×D₄	1	3	yes	—
39	C₂²×C₅₃	H(5,𝔽₂)	1	4	yes	—
40	C₄×C₅₃	—	2	12	no	—
41	C₄×C₅₃	—	4	2	no	—
42	C₄×C₅₃	#127	1	4	yes	—
43	#2657	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
44	C₂×C₅₃	—	8	6	no	—
45	C₂×C₅₃	C₂×#127	1	1	yes	characteristic
46	C₂×C₅₃	#535	1	6	yes	—
47	C₂×#127	C₂×C₅₃	1	3	yes	complemented
48	C₂×#124	C₂×C₅₃	1	3	yes	complemented
49	#392	C₂×C₅₃	1	8	yes	complemented
50	#376	C₂×C₅₃	1	1	yes	characteristic, complemented
51	C₅₃	#2657	1	1	yes	characteristic, direct product factor, Sylow
52	C₂²×Q₈	C₂²×C₅₃	1	1	yes	characteristic
53	C₂²×D₄	C₂²×C₅₃	1	3	yes	—
54	C₂²×D₄	C₂²×C₅₃	1	12	yes	—
55	#127	—	2	12	no	—
56	#124	—	2	12	no	—
57	C₂×C₂²⋊C₄	C₂²×C₅₃	1	12	yes	—
58	C₂×C₄²	C₂²×C₅₃	1	3	yes	—
59	#95	C₂²×C₅₃	1	4	yes	—
60	C₂⁴	C₂³×C₅₃	1	6	yes	—
61	C₂×Q₈	—	2	6	no	—
62	C₂×D₄	—	2	6	no	—
63	C₂×D₄	—	2	24	no	—
64	C₂×D₄	—	2	12	no	—
65	C₂×D₄	—	4	24	no	—
66	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
67	C₂²×C₄	C₂³×C₅₃	1	6	yes	—
68	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
69	C₂²⋊C₄	—	2	24	no	—
70	C₄²	D₄×C₅₃	1	12	yes	complemented
71	C₂³	—	2	6	no	—
72	C₂³	—	2	12	no	—
73	C₂³	—	2	24	no	—
74	C₂³	C₂⁴×C₅₃	1	1	yes	characteristic
75	Q₈	—	4	4	no	—
76	D₄	—	4	48	no	—
77	D₄	—	4	12	no	—
78	C₂×C₄	—	2	12	no	—
79	C₂×C₄	—	2	6	no	—
80	C₂×C₄	C₂×D₄×C₅₃	1	12	yes	—
81	C₂×C₄	C₂×D₄×C₅₃	1	6	yes	—
82	C₂²	—	4	6	no	—
83	C₂²	—	4	24	no	—
84	C₂²	—	4	12	no	—
85	C₂²	C₂²×D₄×C₅₃	1	3	yes	—
86	C₂²	H(5,𝔽₂)×C₅₃	1	4	yes	—
87	C₄	—	2	12	no	—
88	C₄	—	4	2	no	—
89	C₄	#127×C₅₃	1	4	yes	—
90	C₂	—	8	6	no	—
91	C₂	C₂×#127×C₅₃	1	1	yes	characteristic
92	C₂	C₅₃×#535	1	6	yes	—

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

Complements

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

N	H ≅ G/N	type
1. C₂×#127×C₅₃	C₂	semidirect
2. C₂×#124×C₅₃	C₂	semidirect
3. C₅₃×#392	C₂	semidirect
4. C₅₃×#376	C₂	semidirect
23. C₄²×C₅₃	D₄	semidirect
43. #2657	C₅₃	direct (N × H)
47. C₂×#127	C₂×C₅₃	semidirect
48. C₂×#124	C₂×C₅₃	semidirect
49. #392	C₂×C₅₃	semidirect
50. #376	C₂×C₅₃	semidirect
51. C₅₃	#2657	direct (N × H)
70. C₄²	D₄×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#127×C₅₃	2
C₂×#124×C₅₃	2
C₅₃×#392	2
C₅₃×#376	2
#2657	53

Subgroups — C53×#2657 #20271594

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#2657 #20271594