Subgroups — #28299135

Order 13568.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊#13133267	C₂	1	2	yes	maximal
2	C₈×#11719908	C₂	1	1	yes	maximal, characteristic
3	C₂×#13133267	—	2	2	no	—
4	C₄×#11719908	C₄	1	1	yes	characteristic
5	C₄×#11719908	C₂²	1	1	yes	characteristic
6	C₈²×C₅₃	C₄	1	1	yes	characteristic
7	#13133267	—	4	2	no	—
8	C₂×#11719908	—	2	1	no	—
9	C₂×#11719908	D₄	1	1	yes	characteristic
10	C₂×#11719908	Q₈	1	1	yes	characteristic
11	C₄×C₈×C₅₃	C₈	1	1	yes	characteristic
12	C₄×C₈×C₅₃	C₈	1	1	yes	characteristic
13	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
14	C₂×C₈×C₅₃	—	2	1	no	—
15	C₂×C₈×C₅₃	C₁₆	1	1	yes	characteristic
16	C₂×C₈×C₅₃	C₁₆	1	1	yes	characteristic
17	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
18	C₂×C₈×C₅₃	M₁₆	1	1	yes	characteristic
19	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
20	#11719908	—	2	2	no	—
21	#11719908	—	2	1	no	—
22	#11719908	SD₁₆	1	2	yes	—
23	C₂×C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
24	C₂×C₄×C₅₃	C₂×C₁₆	1	1	yes	characteristic
25	C₂×C₄×C₅₃	M₃₂	1	1	yes	characteristic
26	C₈×C₅₃	—	2	2	no	—
27	C₈×C₅₃	—	2	1	no	—
28	C₈×C₅₃	—	2	1	no	—
29	C₈×C₅₃	C₃₂	1	2	yes	complemented
30	C₈×C₅₃	#106	1	1	yes	characteristic
31	C₈×C₅₃	#108	1	1	yes	characteristic
32	#7455	—	53	1	no	maximal, Sylow
33	C₂²×C₅₃	C₄⋊C₁₆	1	1	yes	characteristic
34	C₄×C₅₃	—	2	1	no	—
35	C₄×C₅₃	#334	1	1	yes	characteristic
36	C₄×C₅₃	#364	1	1	yes	characteristic
37	C₄×C₅₃	C₂×C₃₂	1	1	yes	characteristic
38	C₄×C₅₃	M₆₄	1	1	yes	characteristic
39	C₄⋊C₃₂	—	53	2	no	—
40	C₈×C₁₆	—	53	1	no	—
41	C₂×C₅₃	#1367	1	1	yes	characteristic
42	C₂×C₅₃	C₄⋊C₃₂	1	1	yes	characteristic
43	C₂×C₅₃	#1422	1	1	yes	characteristic
44	C₂×C₃₂	—	106	2	no	—
45	C₄×C₁₆	—	53	1	no	—
46	C₄×C₁₆	—	53	1	no	—
47	C₈²	#6157	1	1	yes	characteristic
48	C₅₃	#7455	1	1	yes	characteristic, complemented, Sylow
49	C₂×C₁₆	—	53	1	no	—
50	C₂×C₁₆	—	53	1	no	—
51	C₂×C₁₆	—	106	1	no	—
52	C₄×C₈	#88324	1	1	yes	characteristic
53	C₄×C₈	#88324	1	1	yes	characteristic
54	C₄×C₈	C₂×#6157	1	1	yes	characteristic
55	C₃₂	—	212	2	no	—
56	C₂×C₈	—	2	1	no	—
57	C₂×C₈	#11719910	1	1	yes	characteristic
58	C₂×C₈	#11719910	1	1	yes	characteristic
59	C₂×C₈	#11719936	1	1	yes	characteristic
60	C₂×C₈	C₄⋊#6157	1	1	yes	characteristic
61	C₄²	C₂×#88324	1	1	yes	characteristic
62	C₁₆	—	53	2	no	—
63	C₁₆	—	106	2	no	—
64	C₁₆	—	106	1	no	—
65	C₂×C₄	C₄⋊#88324	1	1	yes	characteristic
66	C₂×C₄	C₂×#11719910	1	1	yes	characteristic
67	C₂×C₄	#13133291	1	1	yes	characteristic
68	C₈	—	2	2	no	—
69	C₈	—	2	1	no	—
70	C₈	—	2	1	no	—
71	C₈	#13133267	1	2	yes	complemented
72	C₈	#13133285	1	1	yes	characteristic
73	C₈	#13133287	1	1	yes	characteristic
74	C₂²	C₄⋊#11719910	1	1	yes	characteristic
75	C₄	—	2	1	no	—
76	C₄	#16811484	1	1	yes	characteristic
77	C₄	#16811545	1	1	yes	characteristic
78	C₄	C₂×#13133267	1	1	yes	characteristic
79	C₄	#16811555	1	1	yes	characteristic
80	C₂	#20289737	1	1	yes	characteristic
81	C₂	C₄⋊#13133267	1	1	yes	characteristic
82	C₂	#20289874	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₄⋊#6157
commutator	C₄×C₅₃	C₂×C₃₂
Frattini	C₄×C₈	C₂×#6157
Fitting	C₈²×C₅₃	C₄
socle, prime radical	C₂²×C₅₃	C₄⋊C₁₆
cosocle	C₄×#11719908	C₂²
hypercenter	C₈²	#6157
hypocenter	C₅₃	#7455
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₈²

Complements

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

N	H ≅ G/N	type
29. C₈×C₅₃	C₃₂	semidirect
48. C₅₃	#7455	semidirect
71. C₈	#13133267	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊#13133267	2
C₈×#11719908	2
#7455	53