Subgroups — #28298181

Order 13568.

102 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20289744	C₂	1	2	yes	maximal
2	C₁₆×#88322	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	D₈	1	1	yes	characteristic
19	C₂×#88322	Q₁₆	1	1	yes	characteristic
20	#11719910	—	8	2	no	—
21	C₁₆×C₅₃	—	2	1	no	—
22	C₁₆×C₅₃	C₁₆	1	2	yes	complemented
23	#11719908	—	2	2	no	—
24	C₂×C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
25	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
26	C₂×C₄×C₅₃	#108	1	1	yes	characteristic
27	C₈×C₅₃	—	2	1	no	—
28	C₈×C₅₃	C₂×C₁₆	1	1	yes	characteristic
29	C₈×C₅₃	M₃₂	1	1	yes	characteristic
30	#88322	—	2	2	no	—
31	#88322	—	2	1	no	—
32	#88322	D₁₆	1	1	yes	characteristic
33	#88322	Q₃₂	1	1	yes	characteristic
34	#7100	—	53	1	no	maximal, Sylow
35	C₂²×C₅₃	#335	1	1	yes	characteristic
36	C₄×C₅₃	—	2	1	no	—
37	C₄×C₅₃	C₄⋊C₁₆	1	1	yes	characteristic
38	C₄×C₅₃	#364	1	1	yes	characteristic
39	C₄×C₅₃	#366	1	1	yes	characteristic
40	C₄×C₅₃	#368	1	1	yes	characteristic
41	#1370	—	53	2	no	—
42	C₈×C₁₆	—	53	1	no	—
43	C₂×C₅₃	#1370	1	1	yes	characteristic
44	C₂×C₅₃	#1371	1	1	yes	characteristic
45	C₂×C₅₃	#1373	1	1	yes	characteristic
46	C₄⋊C₁₆	—	106	2	no	—
47	C₄×C₁₆	—	53	1	no	—
48	C₄×C₁₆	#6157	1	1	yes	characteristic
49	C₈²	—	53	1	no	—
50	C₅₃	#7100	1	1	yes	characteristic, complemented, Sylow
51	C₂×C₁₆	—	106	1	no	—
52	C₂×C₁₆	—	212	2	no	—
53	C₂×C₁₆	#88324	1	1	yes	characteristic
54	C₂×C₁₆	#88324	1	1	yes	characteristic
55	C₄×C₈	—	53	1	no	—
56	C₄×C₈	—	53	1	no	—
57	C₄×C₈	C₂×#6157	1	1	yes	characteristic
58	C₂×C₈	—	53	1	no	—
59	C₂×C₈	—	53	1	no	—
60	C₂×C₈	—	106	1	no	—
61	C₂×C₈	#11719936	1	1	yes	characteristic
62	C₂×C₈	C₂×#88324	1	1	yes	characteristic
63	C₄²	C₄⋊#6157	1	1	yes	characteristic
64	C₁₆	—	2	1	no	—
65	C₁₆	—	106	2	no	—
66	C₁₆	—	424	2	no	—
67	C₁₆	#11719910	1	2	yes	complemented
68	C₂×C₄	C₄⋊#88324	1	1	yes	characteristic
69	C₂×C₄	#13133286	1	1	yes	characteristic
70	C₂×C₄	#13133288	1	1	yes	characteristic
71	C₈	—	2	1	no	—
72	C₈	—	53	1	no	—
73	C₈	—	53	1	no	—
74	C₈	—	106	2	no	—
75	C₈	—	106	1	no	—
76	C₈	C₂×#11719910	1	1	yes	characteristic
77	C₈	#13133291	1	1	yes	characteristic
78	C₂²	#16811486	1	1	yes	characteristic
79	C₄	—	2	1	no	—
80	C₄	C₄⋊#11719910	1	1	yes	characteristic
81	C₄	#16811545	1	1	yes	characteristic
82	C₄	#16811549	1	1	yes	characteristic
83	C₄	#16811551	1	1	yes	characteristic
84	C₂	#20289744	1	1	yes	characteristic
85	C₂	#20289746	1	1	yes	characteristic
86	C₂	#20289750	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13133286
commutator	C₈×C₅₃	C₂×C₁₆
Frattini	C₄×C₈	C₂×#6157
Fitting	C₄×C₁₆×C₅₃	C₄
socle, prime radical	C₂²×C₅₃	#335
cosocle	C₈×#88322	C₂²
hypercenter	C₄×C₁₆	#6157
hypocenter	C₅₃	#7100
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
22. C₁₆×C₅₃	C₁₆	semidirect
50. C₅₃	#7100	semidirect
67. C₁₆	#11719910	semidirect

Maximal subgroups

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

subgroup	index
#20289744	2
C₁₆×#88322	2
#7100	53