Subgroups — #28299319

Order 13568.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811462	C₂	1	2	yes	maximal, complemented
2	C₂×C₃₂×D₅₃	C₂	1	1	yes	maximal, characteristic
3	#16811462	—	2	2	no	—
4	C₂×C₁₆×D₅₃	C₄	1	1	yes	characteristic
5	C₂×#13133123	C₂²	1	1	yes	characteristic
6	C₃₂×D₅₃	—	2	2	no	—
7	C₂×C₃₂×C₅₃	C₄	1	1	yes	characteristic
8	C₂×C₈×D₅₃	C₈	1	1	yes	characteristic
9	C₂×#11719908	C₈	1	1	yes	characteristic
10	C₁₆×D₅₃	—	2	1	no	—
11	C₁₆×D₅₃	—	2	1	no	—
12	#13133123	D₄	1	2	yes	—
13	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
14	C₃₂×C₅₃	—	2	1	no	—
15	C₂×C₄×D₅₃	C₁₆	1	1	yes	characteristic
16	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
17	C₂×#88322	C₁₆	1	1	yes	characteristic
18	C₈×D₅₃	—	2	1	no	—
19	C₈×D₅₃	—	2	1	no	—
20	C₁₆×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
21	C₁₆×C₅₃	M₁₆	1	1	yes	characteristic
22	#11719908	—	2	1	no	—
23	C₂²×D₅₃	C₃₂	1	1	yes	characteristic
24	C₂×C₄×C₅₃	C₂×C₁₆	1	1	yes	characteristic
25	C₂×Dic₅₃	C₃₂	1	1	yes	characteristic
26	C₄×D₅₃	—	2	1	no	—
27	C₄×D₅₃	—	2	1	no	—
28	C₈×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
29	C₈×C₅₃	M₃₂	1	1	yes	characteristic
30	#88322	—	2	1	no	—
31	C₂²⋊C₆₄	—	53	1	no	maximal, Sylow
32	C₂²×C₅₃	C₂×C₃₂	1	1	yes	characteristic
33	C₂×D₅₃	—	2	1	no	—
34	C₂×D₅₃	—	2	1	no	—
35	C₂×D₅₃	C₆₄	1	2	yes	complemented
36	C₄×C₅₃	C₂²⋊C₁₆	1	1	yes	characteristic
37	C₄×C₅₃	M₆₄	1	1	yes	characteristic
38	Dic₅₃	—	2	1	no	—
39	C₂²×C₃₂	—	53	1	no	—
40	C₂×C₆₄	—	53	2	no	—
41	C₂×C₅₃	C₂²⋊C₃₂	1	1	yes	characteristic
42	C₂×C₅₃	C₂×C₆₄	1	1	yes	characteristic
43	C₂×C₅₃	M₁₂₈	1	1	yes	characteristic
44	D₅₃	—	2	2	no	—
45	C₂²×C₁₆	—	53	1	no	—
46	C₂×C₃₂	—	53	1	no	—
47	C₂×C₃₂	—	106	2	no	—
48	C₂×C₃₂	#6157	1	1	yes	characteristic
49	C₆₄	—	106	2	no	—
50	C₅₃	C₂²⋊C₆₄	1	1	yes	characteristic, complemented, Sylow
51	C₂²×C₈	—	53	1	no	—
52	C₂×C₁₆	—	53	1	no	—
53	C₂×C₁₆	—	106	1	no	—
54	C₂×C₁₆	—	106	1	no	—
55	C₂×C₁₆	C₂×#6157	1	1	yes	characteristic
56	C₃₂	—	2	1	no	—
57	C₃₂	—	53	2	no	—
58	C₂²×C₄	—	53	1	no	—
59	C₂×C₈	—	53	1	no	—
60	C₂×C₈	—	106	1	no	—
61	C₂×C₈	—	106	1	no	—
62	C₂×C₈	D₅₃⋊C₈	1	1	yes	characteristic
63	C₁₆	—	106	1	no	—
64	C₁₆	#11719936	1	1	yes	characteristic
65	C₁₆	C₂²⋊#6157	1	1	yes	characteristic
66	C₂³	—	53	1	no	—
67	C₂×C₄	—	53	1	no	—
68	C₂×C₄	—	106	1	no	—
69	C₂×C₄	—	106	1	no	—
70	C₂×C₄	D₅₃⋊C₁₆	1	1	yes	characteristic
71	C₈	—	106	1	no	—
72	C₈	#13133274	1	1	yes	characteristic
73	C₈	#13133292	1	1	yes	characteristic
74	C₂²	—	53	2	no	—
75	C₂²	—	106	1	no	—
76	C₂²	—	106	1	no	—
77	C₂²	D₅₃⋊C₃₂	1	1	yes	characteristic
78	C₄	—	106	1	no	—
79	C₄	#16811518	1	1	yes	characteristic
80	C₄	#16811556	1	1	yes	characteristic
81	C₂	—	106	2	no	—
82	C₂	#20289829	1	1	yes	characteristic
83	C₂	D₅₃⋊C₆₄	1	1	yes	characteristic
84	C₂	#20289885	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₁₆	C₂×#6157
commutator	C₂×C₅₃	C₂×C₆₄
Fitting	C₂×C₃₂×C₅₃	C₄
socle	C₂²×C₅₃	C₂×C₃₂
prime radical	C₂²×D₅₃	C₃₂
cosocle	C₂×#13133123	C₂²
hypercenter	C₂×C₃₂	#6157
hypocenter	C₅₃	C₂²⋊C₆₄
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂×C₃₂

Complements

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

N	H ≅ G/N	type
1. C₂×#16811462	C₂	semidirect
35. C₂×D₅₃	C₆₄	semidirect
50. C₅₃	C₂²⋊C₆₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16811462	2
C₂×C₃₂×D₅₃	2
C₂²⋊C₆₄	53