Subgroups — #28313710

Order 13568.

1278 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20289891	C₂	1	14	yes	maximal
2	C₄³×D₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂×(C₂×#6157)⋊C₄	C₂²	1	28	yes	—
4	C₂×C₄²×D₅₃	C₂²	1	7	yes	—
5	C₄²×Dic₅₃	C₄	1	7	yes	complemented
6	C₄³×C₅₃	C₄	1	1	yes	characteristic, complemented
7	C₂³×#6157	C₂³	1	8	yes	—
8	(C₂×#6157)⋊C₄	—	2	56	no	—
9	C₂²×C₄×D₅₃	C₂³	1	7	yes	—
10	C₂×C₄×Dic₅₃	C₂×C₄	1	7	yes	—
11	C₂×C₄×Dic₅₃	C₂×C₄	1	14	yes	—
12	C₄²×D₅₃	—	2	28	no	—
13	C₂×C₄²×C₅₃	C₂×C₄	1	7	yes	—
14	C₂³×D₅₃	C₂⁴	1	1	yes	characteristic
15	C₂²×#6157	—	2	56	no	—
16	C₂²×C₄×C₅₃	C₂²×C₄	1	7	yes	—
17	C₂²×Dic₅₃	C₂²×C₄	1	7	yes	—
18	C₂×C₄×D₅₃	—	2	28	no	—
19	C₂×C₄×D₅₃	#41	1	28	yes	—
20	C₄²×C₅₃	—	2	14	no	—
21	C₄×Dic₅₃	—	2	14	no	—
22	C₄×Dic₅₃	—	2	28	no	—
23	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
24	C₂²×D₅₃	C₂×#41	1	14	yes	—
25	C₂×#6157	—	4	56	no	—
26	C₂×C₄×C₅₃	—	2	14	no	—
27	C₂×C₄×C₅₃	#117	1	14	yes	—
28	C₂×Dic₅₃	—	2	14	no	—
29	C₂×Dic₅₃	#117	1	14	yes	—
30	C₄×D₅₃	—	2	56	no	—
31	#14691	—	53	1	no	maximal, Sylow
32	C₂²×C₅₃	C₂×#117	1	7	yes	—
33	C₂×D₅₃	#529	1	28	yes	—
34	#6157	—	8	8	no	—
35	C₄×C₅₃	—	2	14	no	—
36	Dic₅₃	—	2	14	no	—
37	C₂×C₄³	—	53	1	no	—
38	#1434	—	53	14	no	—
39	C₂×C₅₃	#2283	1	7	yes	—
40	D₅₃	#2866	1	8	yes	—
41	C₂²×C₄²	—	53	7	no	—
42	C₂×#95	—	53	28	no	—
43	C₄³	—	53	7	no	—
44	C₄³	#6157	1	1	yes	characteristic, complemented
45	C₅₃	#14691	1	1	yes	characteristic, complemented, Sylow
46	C₂³×C₄	—	53	7	no	—
47	C₂³×C₄	—	53	8	no	—
48	C₂×C₄²	—	53	14	no	—
49	C₂×C₄²	—	53	7	no	—
50	C₂×C₄²	—	106	28	no	—
51	C₂×C₄²	C₂×#6157	1	7	yes	—
52	#95	—	106	56	no	—
53	C₂⁴	—	53	1	no	—
54	C₂²×C₄	—	53	7	no	—
55	C₂²×C₄	—	53	28	no	—
56	C₂²×C₄	—	106	28	no	—
57	C₂²×C₄	—	106	56	no	—
58	C₂²×C₄	C₂²×#6157	1	7	yes	—
59	C₄²	—	2	14	no	—
60	C₄²	—	106	28	no	—
61	C₄²	—	106	14	no	—
62	C₂³	—	53	14	no	—
63	C₂³	C₂³×#6157	1	1	yes	characteristic
64	C₂×C₄	—	2	14	no	—
65	C₂×C₄	—	53	14	no	—
66	C₂×C₄	—	106	56	no	—
67	C₂×C₄	—	106	14	no	—
68	C₂×C₄	—	212	56	no	—
69	C₂×C₄	#13133296	1	14	yes	—
70	C₂²	—	53	28	no	—
71	C₂²	C₂×#13133296	1	7	yes	—
72	C₄	—	2	14	no	—
73	C₄	—	106	14	no	—
74	C₄	—	424	8	no	—
75	C₂	—	53	8	no	—
76	C₂	#20291162	1	7	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×#6157
commutator, socle	C₂³×C₅₃	C₂³×C₄
Fitting	C₄³×C₅₃	C₄
prime radical, cosocle	C₂³×D₅₃	C₂⁴
hypercenter	C₄³	#6157
hypocenter	C₅₃	#14691
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₄³

Complements

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

N	H ≅ G/N	type
5. C₄²×Dic₅₃	C₄	semidirect
6. C₄³×C₅₃	C₄	semidirect
44. C₄³	#6157	semidirect
45. C₅₃	#14691	semidirect

Maximal subgroups

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

subgroup	index
#20289891	2
C₄³×D₅₃	2
#14691	53