Subgroups — #401×#6157 #28312885

Order 13568.

1222 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20289902	C₂	1	7	yes	maximal
2	#95×#6157	C₂	1	7	yes	maximal
3	#401×D₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₄×#6157	C₂²	1	7	yes	—
5	C₂×(C₂×#6157)⋊C₄	C₂²	1	21	yes	—
6	C₅₃⋊#401	C₄	1	7	yes	complemented
7	#95×D₅₃	C₂²	1	7	yes	—
8	C₅₃×#401	C₄	1	1	yes	characteristic, complemented
9	C₂³×#6157	C₂³	1	7	yes	—
10	C₂³×#6157	C₂³	1	1	yes	characteristic
11	C₂×C₄×#6157	—	2	42	no	—
12	(C₂×#6157)⋊C₄	—	2	42	no	—
13	C₂²×C₄×D₅₃	C₂³	1	7	yes	—
14	#13133124	C₂×C₄	1	21	yes	—
15	#95×C₅₃	C₂×C₄	1	7	yes	—
16	C₂³×D₅₃	C₂⁴	1	1	yes	characteristic
17	C₂²×#6157	—	2	21	no	—
18	C₂²×#6157	—	2	21	no	—
19	C₂²×#6157	#41	1	14	yes	—
20	C₂²×#6157	#41	1	14	yes	—
21	C₂²×C₄×C₅₃	C₂²×C₄	1	7	yes	—
22	C₂²×Dic₅₃	C₂²×C₄	1	7	yes	—
23	C₂×C₄×D₅₃	—	2	21	no	—
24	C₂×C₄×D₅₃	—	2	21	no	—
25	C₄×#6157	—	4	28	no	—
26	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
27	C₂²×D₅₃	C₂×#41	1	7	yes	—
28	C₂²×D₅₃	C₂×#41	1	7	yes	—
29	C₂×#6157	—	2	42	no	—
30	C₂×#6157	—	4	7	no	—
31	C₂×#6157	—	4	21	no	—
32	C₂×#6157	#126	1	28	yes	—
33	C₂×C₄×C₅₃	—	2	21	no	—
34	C₂×Dic₅₃	—	2	21	no	—
35	C₄×D₅₃	—	4	7	no	—
36	C₄×D₅₃	—	4	21	no	—
37	C₄×#401	—	53	1	no	maximal, Sylow
38	C₂²×C₅₃	C₄×#41	1	7	yes	—
39	C₂×D₅₃	C₂×#126	1	7	yes	—
40	C₂×D₅₃	#529	1	21	yes	—
41	#6157	—	4	14	no	—
42	#6157	#401	1	8	yes	direct product factor
43	C₄×C₅₃	—	4	7	no	—
44	Dic₅₃	—	4	7	no	—
45	C₂×#401	—	53	1	no	—
46	#1443	—	53	7	no	—
47	C₄×#95	—	53	7	no	—
48	C₂×C₅₃	C₄×#126	1	7	yes	—
49	D₅₃	C₂×#401	1	1	yes	characteristic
50	D₅₃	#2401	1	7	yes	—
51	C₂²×C₄²	—	53	7	no	—
52	#401	—	53	7	no	—
53	#401	#6157	1	1	yes	characteristic, direct product factor
54	C₂×#95	—	53	7	no	—
55	C₂×#95	—	53	21	no	—
56	C₅₃	C₄×#401	1	1	yes	characteristic, complemented, Sylow
57	C₂³×C₄	—	53	1	no	—
58	C₂³×C₄	—	53	7	no	—
59	C₂³×C₄	—	53	7	no	—
60	C₂×C₄²	—	106	42	no	—
61	#95	—	53	21	no	—
62	#95	—	106	42	no	—
63	#95	C₂×#6157	1	7	yes	—
64	C₂⁴	—	53	1	no	—
65	C₂²×C₄	—	53	14	no	—
66	C₂²×C₄	—	53	14	no	—
67	C₂²×C₄	—	53	7	no	—
68	C₂²×C₄	—	106	21	no	—
69	C₂²×C₄	—	106	21	no	—
70	C₂²×C₄	—	106	21	no	—
71	C₂²×C₄	—	106	21	no	—
72	C₂²×C₄	C₂²×#6157	1	7	yes	—
73	C₄²	—	212	28	no	—
74	C₂³	—	53	7	no	—
75	C₂³	—	53	7	no	—
76	C₂³	C₂³×#6157	1	1	yes	characteristic
77	C₂×C₄	—	2	21	no	—
78	C₂×C₄	—	53	28	no	—
79	C₂×C₄	—	106	42	no	—
80	C₂×C₄	—	106	21	no	—
81	C₂×C₄	—	212	7	no	—
82	C₂×C₄	—	212	21	no	—
83	C₂×C₄	—	212	7	no	—
84	C₂×C₄	—	212	21	no	—
85	C₂²	—	53	7	no	—
86	C₂²	—	53	21	no	—
87	C₂²	#41×#6157	1	7	yes	—
88	C₄	—	4	7	no	—
89	C₄	—	53	8	no	—
90	C₄	—	212	14	no	—
91	C₄	—	212	7	no	—
92	C₂	—	53	1	no	—
93	C₂	—	53	7	no	—
94	C₂	#126×#6157	1	7	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×#6157
commutator, socle	C₂³×C₅₃	C₂³×C₄
Fitting	C₅₃×#401	C₄
prime radical, cosocle	C₂³×D₅₃	C₂⁴
hypercenter	#401	#6157
hypocenter	C₅₃	C₄×#401
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ #401

Complements

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

N	H ≅ G/N	type
6. C₅₃⋊#401	C₄	semidirect
8. C₅₃×#401	C₄	semidirect
42. #6157	#401	direct (N × H)
53. #401	#6157	direct (N × H)
56. C₅₃	C₄×#401	semidirect

Maximal subgroups

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

subgroup	index
#20289902	2
#95×#6157	2
#401×D₅₃	2
C₄×#401	53