Subgroups — #27271900

Order 13568.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₂×D₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810342	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅₃×C₆₄	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₁₆×D₅₃	C₄	1	1	yes	characteristic
5	C₂×#13133123	C₄	1	1	yes	characteristic
6	C₃₂×D₅₃	—	2	1	no	—
7	C₃₂×D₅₃	—	2	1	no	—
8	#16810342	—	2	1	no	—
9	C₂×C₃₂×C₅₃	C₂²	1	1	yes	characteristic
10	C₅₃×C₆₄	—	2	1	no	—
11	C₂×C₈×D₅₃	C₈	1	1	yes	characteristic
12	C₂×#11719908	C₈	1	1	yes	characteristic
13	C₁₆×D₅₃	—	2	1	no	—
14	C₁₆×D₅₃	—	2	1	no	—
15	#13133123	—	2	1	no	—
16	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
17	C₃₂×C₅₃	D₄	1	1	yes	characteristic
18	C₃₂×C₅₃	D₄	1	1	yes	characteristic
19	C₂×C₄×D₅₃	C₁₆	1	1	yes	characteristic
20	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
21	C₂×#88322	C₁₆	1	1	yes	characteristic
22	C₈×D₅₃	—	2	1	no	—
23	C₈×D₅₃	—	2	1	no	—
24	C₁₆×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
25	C₁₆×C₅₃	M₁₆	1	1	yes	characteristic
26	#11719908	—	2	1	no	—
27	C₂²×D₅₃	C₃₂	1	1	yes	characteristic
28	C₂×C₄×C₅₃	C₂×C₁₆	1	1	yes	characteristic
29	C₂×Dic₅₃	C₃₂	1	1	yes	characteristic
30	C₄×D₅₃	—	2	1	no	—
31	C₄×D₅₃	—	2	1	no	—
32	C₈×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
33	C₈×C₅₃	M₃₂	1	1	yes	characteristic
34	#88322	—	2	1	no	—
35	C₂²⋊C₆₄	—	53	1	no	maximal, Sylow
36	C₂²×C₅₃	C₂×C₃₂	1	1	yes	characteristic
37	C₂×D₅₃	—	2	1	no	—
38	C₂×D₅₃	—	2	1	no	—
39	C₂×D₅₃	C₆₄	1	2	yes	complemented
40	C₄×C₅₃	C₂²⋊C₁₆	1	1	yes	characteristic
41	C₄×C₅₃	M₆₄	1	1	yes	characteristic
42	Dic₅₃	—	2	1	no	—
43	C₂²×C₃₂	—	53	1	no	—
44	C₂×C₆₄	—	53	1	no	—
45	C₂×C₆₄	D₅₃	1	1	yes	characteristic, complemented
46	C₂×C₅₃	C₂²⋊C₃₂	1	1	yes	characteristic
47	C₂×C₅₃	C₂×C₆₄	1	1	yes	characteristic
48	C₂×C₅₃	M₁₂₈	1	1	yes	characteristic
49	D₅₃	—	2	2	no	—
50	C₂²×C₁₆	—	53	1	no	—
51	C₂×C₃₂	—	53	1	no	—
52	C₂×C₃₂	—	106	1	no	—
53	C₂×C₃₂	—	106	1	no	—
54	C₂×C₃₂	C₂×D₅₃	1	1	yes	characteristic
55	C₆₄	—	2	1	no	—
56	C₆₄	—	106	1	no	—
57	C₅₃	C₂²⋊C₆₄	1	1	yes	characteristic, complemented, Sylow
58	C₂²×C₈	—	53	1	no	—
59	C₂×C₁₆	—	53	1	no	—
60	C₂×C₁₆	—	106	1	no	—
61	C₂×C₁₆	—	106	1	no	—
62	C₂×C₁₆	C₄×D₅₃	1	1	yes	characteristic
63	C₃₂	—	106	1	no	—
64	C₃₂	D₂₁₂	1	1	yes	characteristic
65	C₃₂	C₂²⋊D₅₃	1	1	yes	characteristic
66	C₂²×C₄	—	53	1	no	—
67	C₂×C₈	—	53	1	no	—
68	C₂×C₈	—	106	1	no	—
69	C₂×C₈	—	106	1	no	—
70	C₂×C₈	C₈×D₅₃	1	1	yes	characteristic
71	C₁₆	—	106	1	no	—
72	C₁₆	#11719912	1	1	yes	characteristic
73	C₁₆	#11719921	1	1	yes	characteristic
74	C₂³	—	53	1	no	—
75	C₂×C₄	—	53	1	no	—
76	C₂×C₄	—	106	1	no	—
77	C₂×C₄	—	106	1	no	—
78	C₂×C₄	C₁₆×D₅₃	1	1	yes	characteristic
79	C₈	—	106	1	no	—
80	C₈	#13133129	1	1	yes	characteristic
81	C₈	#13133156	1	1	yes	characteristic
82	C₂²	—	53	2	no	—
83	C₂²	—	106	1	no	—
84	C₂²	—	106	1	no	—
85	C₂²	C₃₂×D₅₃	1	1	yes	characteristic
86	C₄	—	106	1	no	—
87	C₄	#16810402	1	1	yes	characteristic
88	C₄	#16810456	1	1	yes	characteristic
89	C₂	—	106	2	no	—
90	C₂	#20272852	1	1	yes	characteristic
91	C₂	C₆₄×D₅₃	1	1	yes	characteristic
92	C₂	#20272925	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₂×#16810342	C₂	semidirect
3. C₂×C₅₃×C₆₄	C₂	semidirect
39. C₂×D₅₃	C₆₄	semidirect
45. C₂×C₆₄	D₅₃	semidirect
57. 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×C₃₂×D₅₃	2
C₂×#16810342	2
C₂×C₅₃×C₆₄	2
C₂²⋊C₆₄	53