Subgroups — #28311412

Order 13568.

298 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20289753	C₂	1	6	yes	maximal
2	#20274402	C₂	1	1	yes	maximal, characteristic
3	C₂²×#11719910	C₂²	1	4	yes	—
4	C₂×#13133128	C₄	1	3	yes	—
5	C₂×C₄×#88322	C₂²	1	3	yes	—
6	C₅₃×#376	C₄	1	1	yes	characteristic
7	C₂×#11719910	—	2	12	no	—
8	C₂²×#88322	C₂×C₄	1	3	yes	—
9	C₂²×#88322	C₂³	1	1	yes	characteristic
10	#13133128	—	2	6	no	—
11	C₄×#88322	—	2	6	no	—
12	C₂×C₄²×C₅₃	C₂×C₄	1	3	yes	—
13	#95×C₅₃	C₈	1	4	yes	—
14	C₂²×C₄×C₅₃	C₂×C₈	1	6	yes	—
15	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
16	C₄²×C₅₃	—	2	3	no	—
17	C₄²×C₅₃	—	2	3	no	—
18	C₂×#88322	—	2	3	no	—
19	C₂×#88322	—	2	3	no	—
20	C₂×#88322	M₁₆	1	6	yes	—
21	C₂×#88322	#41	1	6	yes	—
22	#11719910	—	4	4	no	—
23	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
24	C₂×C₄×C₅₃	—	2	6	no	—
25	C₂×C₄×C₅₃	—	2	12	no	—
26	C₂×C₄×C₅₃	#117	1	3	yes	—
27	C₂×C₄×C₅₃	C₂×M₁₆	1	3	yes	—
28	#88322	—	2	6	no	—
29	#88322	#126	1	4	yes	—
30	#12895	—	53	1	no	maximal, Sylow
31	C₂²×C₅₃	#431	1	3	yes	—
32	C₂²×C₅₃	#504	1	4	yes	—
33	C₄×C₅₃	—	4	6	no	—
34	C₄×C₅₃	#383	1	1	yes	characteristic
35	C₄×C₅₃	#432	1	3	yes	—
36	#1726	—	53	1	no	—
37	#1374	—	53	6	no	—
38	C₂×C₅₃	#1839	1	1	yes	characteristic
39	C₂×C₅₃	#2164	1	6	yes	—
40	C₂²×C₁₆	—	53	4	no	—
41	C₂×#97	—	53	3	no	—
42	C₂×C₄×C₈	—	53	3	no	—
43	#376	#6157	1	1	yes	characteristic
44	C₅₃	#12895	1	1	yes	characteristic, complemented, Sylow
45	C₂²×C₈	—	53	1	no	—
46	C₂²×C₈	—	53	3	no	—
47	C₂×C₄²	C₂×#6157	1	3	yes	—
48	C₂×C₁₆	—	106	12	no	—
49	#97	—	106	6	no	—
50	C₄×C₈	—	106	6	no	—
51	#95	#88324	1	4	yes	—
52	C₂²×C₄	C₂×#88324	1	6	yes	—
53	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
54	C₂×C₈	—	53	6	no	—
55	C₂×C₈	—	53	6	no	—
56	C₂×C₈	—	106	3	no	—
57	C₂×C₈	—	106	3	no	—
58	C₄²	—	2	3	no	—
59	C₄²	—	2	3	no	—
60	C₁₆	—	212	4	no	—
61	C₂³	C₂²×#88324	1	1	yes	characteristic
62	C₂×C₄	—	2	6	no	—
63	C₂×C₄	—	2	12	no	—
64	C₂×C₄	#13133296	1	3	yes	—
65	C₂×C₄	D₅₃⋊M₁₆	1	3	yes	—
66	C₈	—	53	4	no	—
67	C₈	—	106	6	no	—
68	C₂²	#16811639	1	3	yes	—
69	C₂²	#16811681	1	4	yes	—
70	C₄	—	4	6	no	—
71	C₄	#16811568	1	1	yes	characteristic
72	C₄	#16811646	1	3	yes	—
73	C₂	#20290642	1	1	yes	characteristic
74	C₂	#20291080	1	6	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#6157
commutator, socle, prime radical	C₂³×C₅₃	C₂²×C₈
Fitting	C₅₃×#376	C₄
cosocle	C₂²×#88322	C₂³
hypercenter	#376	#6157
hypocenter	C₅₃	#12895
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ #376

Complements

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

N	H ≅ G/N	type
44. C₅₃	#12895	semidirect

Maximal subgroups

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

subgroup	index
#20289753	2
#20274402	2
#12895	53