Subgroups — #28298178

Order 13568.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20289737	C₂	1	2	yes	maximal
2	#20272631	C₂	1	1	yes	maximal, characteristic
3	C₄⋊#11719910	—	2	2	no	—
4	C₄×#11719908	C₄	1	1	yes	characteristic
5	C₈×#88322	C₂²	1	1	yes	characteristic
6	C₄×C₁₆×C₅₃	C₄	1	1	yes	characteristic
7	C₂×#11719910	—	4	2	no	—
8	C₂×#11719908	—	2	1	no	—
9	C₄×#88322	D₄	1	1	yes	characteristic
10	C₄×#88322	Q₈	1	1	yes	characteristic
11	C₂×C₁₆×C₅₃	C₈	1	1	yes	characteristic
12	C₂×C₁₆×C₅₃	C₈	1	1	yes	characteristic
13	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
14	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
15	C₂×C₈×C₅₃	M₁₆	1	1	yes	characteristic
16	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
17	C₂×#88322	—	2	1	no	—
18	C₂×#88322	SD₁₆	1	2	yes	—
19	#11719910	—	8	2	no	—
20	C₁₆×C₅₃	—	2	1	no	—
21	C₁₆×C₅₃	C₁₆	1	2	yes	complemented
22	#11719908	—	2	2	no	—
23	C₂×C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
24	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
25	C₂×C₄×C₅₃	#108	1	1	yes	characteristic
26	C₈×C₅₃	—	2	1	no	—
27	C₈×C₅₃	C₂×C₁₆	1	1	yes	characteristic
28	C₈×C₅₃	M₃₂	1	1	yes	characteristic
29	#88322	—	2	2	no	—
30	#88322	—	4	1	no	—
31	#7099	—	53	1	no	maximal, Sylow
32	C₂²×C₅₃	#334	1	1	yes	characteristic
33	C₄×C₅₃	—	2	1	no	—
34	C₄×C₅₃	C₄⋊C₁₆	1	1	yes	characteristic
35	C₄×C₅₃	#364	1	1	yes	characteristic
36	C₄×C₅₃	#365	1	2	yes	—
37	#1367	—	53	2	no	—
38	#1311	—	53	1	no	—
39	C₂×C₅₃	#1367	1	1	yes	characteristic
40	C₂×C₅₃	#1368	1	1	yes	characteristic
41	C₂×C₅₃	#1369	1	1	yes	characteristic
42	C₄⋊C₁₆	—	106	2	no	—
43	C₄×C₁₆	—	53	1	no	—
44	C₄×C₁₆	#6157	1	1	yes	characteristic
45	C₈²	—	53	1	no	—
46	C₅₃	#7099	1	1	yes	characteristic, complemented, Sylow
47	C₂×C₁₆	—	106	1	no	—
48	C₂×C₁₆	—	212	2	no	—
49	C₂×C₁₆	#88324	1	1	yes	characteristic
50	C₂×C₁₆	#88324	1	1	yes	characteristic
51	C₄×C₈	—	53	1	no	—
52	C₄×C₈	—	53	1	no	—
53	C₄×C₈	C₂×#6157	1	1	yes	characteristic
54	C₂×C₈	—	53	2	no	—
55	C₂×C₈	—	106	1	no	—
56	C₂×C₈	#11719936	1	1	yes	characteristic
57	C₂×C₈	C₂×#88324	1	1	yes	characteristic
58	C₄²	C₄⋊#6157	1	1	yes	characteristic
59	C₁₆	—	2	1	no	—
60	C₁₆	—	106	2	no	—
61	C₁₆	—	424	2	no	—
62	C₁₆	#11719910	1	2	yes	complemented
63	C₂×C₄	C₄⋊#88324	1	1	yes	characteristic
64	C₂×C₄	#13133285	1	1	yes	characteristic
65	C₂×C₄	#13133287	1	1	yes	characteristic
66	C₈	—	2	1	no	—
67	C₈	—	106	2	no	—
68	C₈	—	212	1	no	—
69	C₈	C₂×#11719910	1	1	yes	characteristic
70	C₈	#13133291	1	1	yes	characteristic
71	C₂²	#16811484	1	1	yes	characteristic
72	C₄	—	2	1	no	—
73	C₄	C₄⋊#11719910	1	1	yes	characteristic
74	C₄	#16811545	1	1	yes	characteristic
75	C₄	#16811547	1	2	yes	—
76	C₂	#20289737	1	1	yes	characteristic
77	C₂	#20289739	1	1	yes	characteristic
78	C₂	#20289742	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13133285
commutator	C₈×C₅₃	C₂×C₁₆
Frattini	C₄×C₈	C₂×#6157
Fitting	C₄×C₁₆×C₅₃	C₄
socle, prime radical	C₂²×C₅₃	#334
cosocle	C₈×#88322	C₂²
hypercenter	C₄×C₁₆	#6157
hypocenter	C₅₃	#7099
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
21. C₁₆×C₅₃	C₁₆	semidirect
46. C₅₃	#7099	semidirect
62. C₁₆	#11719910	semidirect

Maximal subgroups

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

subgroup	index
#20289737	2
#20272631	2
#7099	53