Subgroups — #28299362

Order 13568.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20289847	C₂	1	2	yes	maximal
2	#20274322	C₂	1	1	yes	maximal, characteristic
3	#16811528	—	2	2	no	—
4	#16811000	C₂²	1	1	yes	characteristic
5	#16810971	—	2	2	no	—
6	C₈×#88322	C₄	1	1	yes	characteristic, complemented
7	C₅₃×#494	C₄	1	1	yes	characteristic, complemented
8	#13133276	—	4	2	no	—
9	C₂×D₄₂₄	—	4	1	no	—
10	#13133214	D₄	1	2	yes	—
11	#13133137	—	4	4	no	—
12	C₄×#88322	—	2	1	no	—
13	C₂×Q₁₆×C₅₃	—	4	1	no	—
14	C₄⋊Q₈×C₅₃	—	2	1	no	—
15	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
16	C₂×Q₈×C₅₃	—	4	1	no	—
17	C₂×D₂₁₂	—	2	2	no	—
18	C₂×D₂₁₂	—	4	2	no	—
19	C₂²⋊#6157	—	8	2	no	—
20	Q₁₆×C₅₃	—	8	2	no	—
21	C₂×C₈×C₅₃	—	2	1	no	—
22	C₄⋊C₄×C₅₃	—	4	2	no	—
23	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
24	C₂×#88322	—	2	2	no	—
25	D₄₂₄	—	8	2	no	—
26	C₂²×D₅₃	—	4	2	no	—
27	C₂×#6157	—	16	2	no	—
28	Q₈×C₅₃	—	8	2	no	—
29	C₂×C₄×C₅₃	—	2	1	no	—
30	C₂×C₄×C₅₃	—	8	1	no	—
31	C₂×C₄×C₅₃	#99	1	1	yes	characteristic
32	D₂₁₂	—	8	2	no	—
33	D₂₁₂	—	8	4	no	—
34	C₈×C₅₃	—	2	2	no	—
35	#88322	—	4	2	no	—
36	#7530	—	53	1	no	maximal, Sylow
37	C₂²×C₅₃	#353	1	1	yes	characteristic
38	C₂×D₅₃	—	8	2	no	—
39	C₂×D₅₃	—	16	2	no	—
40	#6157	—	32	2	no	—
41	C₄×C₅₃	—	2	2	no	—
42	C₄×C₅₃	—	2	1	no	—
43	C₄×C₅₃	—	16	1	no	—
44	#1709	—	53	1	no	—
45	#1408	—	53	2	no	—
46	C₂×C₅₃	—	2	1	no	—
47	C₂×C₅₃	#1408	1	1	yes	characteristic
48	D₅₃	—	16	2	no	—
49	#494	#6157	1	1	yes	characteristic, complemented
50	#493	—	53	1	no	—
51	#486	—	106	2	no	—
52	#353	—	106	2	no	—
53	C₈²	—	53	1	no	—
54	C₅₃	#7530	1	1	yes	characteristic, complemented, Sylow
55	C₂×Q₁₆	—	4	1	no	—
56	C₂×D₈	—	212	1	no	—
57	C₄⋊Q₈	—	2	1	no	—
58	#127	—	53	2	no	—
59	#102	—	212	4	no	—
60	#99	—	212	2	no	—
61	C₄×C₈	—	106	1	no	—
62	C₄×C₈	C₂×#6157	1	1	yes	characteristic
63	C₂×Q₈	—	4	1	no	—
64	C₂×D₄	—	106	2	no	—
65	C₂×D₄	—	212	2	no	—
66	Q₁₆	—	8	2	no	—
67	D₈	—	424	2	no	—
68	C₂×C₈	—	2	1	no	—
69	C₂×C₈	—	106	2	no	—
70	C₄⋊C₄	—	4	2	no	—
71	C₂²⋊C₄	—	424	2	no	—
72	C₄²	C₂²⋊#6157	1	1	yes	characteristic
73	C₂³	—	212	2	no	—
74	Q₈	—	8	2	no	—
75	D₄	—	424	4	no	—
76	D₄	—	424	2	no	—
77	C₂×C₄	—	2	1	no	—
78	C₂×C₄	—	8	1	no	—
79	C₂×C₄	—	848	2	no	—
80	C₂×C₄	#13133275	1	1	yes	characteristic
81	C₈	—	2	2	no	—
82	C₈	—	212	2	no	—
83	C₂²	—	848	2	no	—
84	C₂²	—	424	2	no	—
85	C₂²	#16811527	1	1	yes	characteristic
86	C₄	—	2	2	no	—
87	C₄	—	2	1	no	—
88	C₄	—	16	1	no	—
89	C₄	—	1696	2	no	—
90	C₂	—	2	1	no	—
91	C₂	—	848	2	no	—
92	C₂	#20289846	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20289846
commutator	C₄×C₈×C₅₃	C₂×C₄
Frattini	C₄×C₈	C₂×#6157
Fitting	C₅₃×#494	C₄
socle	C₂×C₅₃	#1408
prime radical, cosocle	#16811000	C₂²
hypercenter	#494	#6157
hypocenter	C₅₃	#7530
LCS	G ▷ C₄×C₈×C₅₃ ▷ C₄²×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ #494

Complements

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

N	H ≅ G/N	type
6. C₈×#88322	C₄	semidirect
7. C₅₃×#494	C₄	semidirect
49. #494	#6157	semidirect
54. C₅₃	#7530	semidirect

Maximal subgroups

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

subgroup	index
#20289847	2
#20274322	2
#7530	53