Subgroups — C₄×C₅₃×#401 #27221378

Order 13568.

1222 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#401	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1443	C₂	1	7	yes	maximal
3	C₄×#95×C₅₃	C₂	1	7	yes	maximal
4	C₂²×C₄²×C₅₃	C₂²	1	7	yes	—
5	C₅₃×#401	C₄	1	8	yes	direct product factor
6	C₂×#95×C₅₃	C₂²	1	21	yes	—
7	C₂×#95×C₅₃	C₂²	1	7	yes	—
8	C₂³×C₄×C₅₃	C₂³	1	7	yes	—
9	C₂³×C₄×C₅₃	C₂³	1	7	yes	—
10	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
11	C₂×C₄²×C₅₃	—	2	42	no	—
12	#95×C₅₃	—	2	42	no	—
13	#95×C₅₃	C₂×C₄	1	28	yes	—
14	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
15	C₂²×C₄×C₅₃	—	2	21	no	—
16	C₂²×C₄×C₅₃	—	2	21	no	—
17	C₂²×C₄×C₅₃	—	2	21	no	—
18	C₂²×C₄×C₅₃	—	2	21	no	—
19	C₂²×C₄×C₅₃	C₂²×C₄	1	14	yes	—
20	C₂²×C₄×C₅₃	#41	1	14	yes	—
21	C₂²×C₄×C₅₃	#41	1	14	yes	—
22	C₄²×C₅₃	—	4	28	no	—
23	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
24	C₂³×C₅₃	C₂×#41	1	7	yes	—
25	C₂³×C₅₃	C₂×#41	1	7	yes	—
26	C₂×C₄×C₅₃	—	2	42	no	—
27	C₂×C₄×C₅₃	—	2	42	no	—
28	C₂×C₄×C₅₃	—	4	7	no	—
29	C₂×C₄×C₅₃	—	4	7	no	—
30	C₂×C₄×C₅₃	—	4	21	no	—
31	C₂×C₄×C₅₃	—	4	21	no	—
32	C₂×C₄×C₅₃	#126	1	28	yes	—
33	C₄×#401	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
34	C₂²×C₅₃	C₄×#41	1	7	yes	—
35	C₂²×C₅₃	C₂×#126	1	7	yes	—
36	C₂²×C₅₃	#529	1	21	yes	—
37	C₄×C₅₃	—	4	14	no	—
38	C₄×C₅₃	—	4	14	no	—
39	C₄×C₅₃	#401	1	8	yes	direct product factor
40	C₂×#401	C₂×C₅₃	1	1	yes	characteristic
41	#1443	C₂×C₅₃	1	7	yes	—
42	C₄×#95	C₂×C₅₃	1	7	yes	—
43	C₂×C₅₃	C₄×#126	1	7	yes	—
44	C₂×C₅₃	C₂×#401	1	1	yes	characteristic
45	C₂×C₅₃	#2401	1	7	yes	—
46	C₂²×C₄²	C₂²×C₅₃	1	7	yes	—
47	#401	C₄×C₅₃	1	8	yes	direct product factor
48	C₂×#95	C₂²×C₅₃	1	21	yes	—
49	C₂×#95	C₂²×C₅₃	1	7	yes	—
50	C₅₃	C₄×#401	1	1	yes	characteristic, direct product factor, Sylow
51	C₂³×C₄	C₂³×C₅₃	1	7	yes	—
52	C₂³×C₄	C₂³×C₅₃	1	7	yes	—
53	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
54	C₂×C₄²	—	2	42	no	—
55	#95	—	2	42	no	—
56	#95	C₂×C₄×C₅₃	1	28	yes	—
57	C₂⁴	C₂⁴×C₅₃	1	1	yes	characteristic
58	C₂²×C₄	—	2	21	no	—
59	C₂²×C₄	—	2	21	no	—
60	C₂²×C₄	—	2	21	no	—
61	C₂²×C₄	—	2	21	no	—
62	C₂²×C₄	C₂²×C₄×C₅₃	1	14	yes	—
63	C₂²×C₄	#41×C₅₃	1	14	yes	—
64	C₂²×C₄	#41×C₅₃	1	14	yes	—
65	C₄²	—	4	28	no	—
66	C₂³	C₂³×C₄×C₅₃	1	1	yes	characteristic
67	C₂³	C₂×#41×C₅₃	1	7	yes	—
68	C₂³	C₂×#41×C₅₃	1	7	yes	—
69	C₂×C₄	—	2	42	no	—
70	C₂×C₄	—	2	42	no	—
71	C₂×C₄	—	4	7	no	—
72	C₂×C₄	—	4	7	no	—
73	C₂×C₄	—	4	21	no	—
74	C₂×C₄	—	4	21	no	—
75	C₂×C₄	#126×C₅₃	1	28	yes	—
76	C₂²	C₄×#41×C₅₃	1	7	yes	—
77	C₂²	C₂×#126×C₅₃	1	7	yes	—
78	C₂²	C₅₃×#529	1	21	yes	—
79	C₄	—	4	14	no	—
80	C₄	—	4	14	no	—
81	C₄	C₅₃×#401	1	8	yes	direct product factor
82	C₂	C₄×#126×C₅₃	1	7	yes	—
83	C₂	C₂×C₅₃×#401	1	1	yes	characteristic
84	C₂	C₅₃×#2401	1	7	yes	—

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

Complements

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

N	H ≅ G/N	type
5. C₅₃×#401	C₄	direct (N × H)
33. C₄×#401	C₅₃	direct (N × H)
39. C₄×C₅₃	#401	direct (N × H)
47. #401	C₄×C₅₃	direct (N × H)
50. C₅₃	C₄×#401	direct (N × H)
81. C₄	C₅₃×#401	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#401	2
C₅₃×#1443	2
C₄×#95×C₅₃	2
C₄×#401	53

Subgroups — C4×C53×#401 #27221378

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₅₃×#401 #27221378