Subgroups — #27305130

Order 13568.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20277008	C₂	1	1	yes	maximal, characteristic
2	#20272926	C₂	1	1	yes	maximal, characteristic, complemented
3	#20272925	C₂	1	2	yes	maximal, complemented
4	C₆₄×D₅₃	C₂	1	2	yes	maximal, complemented
5	C₂×C₅₃×C₆₄	C₂	1	1	yes	maximal, characteristic, complemented
6	#16811039	C₄	1	1	yes	characteristic
7	#16810457	C₄	1	1	yes	characteristic
8	#16810456	C₄	1	2	yes	—
9	C₃₂×D₅₃	C₂²	1	2	yes	—
10	#16810342	C₂²	1	2	yes	—
11	C₂×C₃₂×C₅₃	C₂²	1	1	yes	characteristic
12	C₅₃×C₆₄	C₂²	1	1	yes	characteristic
13	C₅₃×C₆₄	C₂²	1	1	yes	characteristic
14	#13133223	C₈	1	1	yes	characteristic
15	#13133157	C₈	1	1	yes	characteristic
16	#13133156	C₈	1	2	yes	—
17	C₁₆×D₅₃	C₂×C₄	1	2	yes	—
18	#13133123	C₂×C₄	1	2	yes	—
19	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
20	C₃₂×C₅₃	C₂×C₄	1	1	yes	characteristic
21	C₃₂×C₅₃	C₂³	1	1	yes	characteristic
22	#11719945	C₁₆	1	1	yes	characteristic
23	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
24	#11719917	C₁₆	1	1	yes	characteristic
25	#11719912	C₁₆	1	2	yes	—
26	C₈×D₅₃	C₂×C₈	1	2	yes	—
27	C₁₆×C₅₃	C₂×C₈	1	1	yes	characteristic
28	C₁₆×C₅₃	C₂²×C₄	1	1	yes	characteristic
29	#11719908	C₂×C₈	1	2	yes	—
30	C₂×C₄×C₅₃	C₂×C₁₆	1	1	yes	characteristic
31	C₂²⋊D₅₃	C₃₂	1	2	yes	—
32	D₂₁₂	C₃₂	1	1	yes	characteristic
33	C₄×D₅₃	C₂×C₁₆	1	2	yes	—
34	Dic₁₀₆	C₃₂	1	1	yes	characteristic
35	C₈×C₅₃	C₂×C₁₆	1	1	yes	characteristic
36	C₈×C₅₃	C₂²×C₈	1	1	yes	characteristic
37	#88322	C₂×C₁₆	1	2	yes	—
38	#13737	—	53	1	no	maximal, Sylow
39	C₂²×C₅₃	C₂×C₃₂	1	1	yes	characteristic
40	C₂×D₅₃	C₂×C₃₂	1	2	yes	—
41	C₄×C₅₃	C₂×C₃₂	1	1	yes	characteristic
42	C₄×C₅₃	C₂²×C₁₆	1	1	yes	characteristic
43	Dic₅₃	C₂×C₃₂	1	2	yes	—
44	#2258	—	53	1	no	—
45	M₁₂₈	—	53	1	no	—
46	M₁₂₈	—	53	2	no	—
47	C₂×C₆₄	—	53	2	no	—
48	C₂×C₆₄	D₅₃	1	1	yes	characteristic, complemented
49	C₂×C₅₃	—	2	1	no	—
50	C₂×C₅₃	C₂²×C₃₂	1	1	yes	characteristic
51	D₅₃	—	2	2	no	—
52	#504	—	53	1	no	—
53	M₆₄	—	53	1	no	—
54	M₆₄	—	53	2	no	—
55	C₂×C₃₂	—	53	2	no	—
56	C₂×C₃₂	C₂×D₅₃	1	1	yes	characteristic
57	C₆₄	—	53	2	no	—
58	C₆₄	C₂×D₅₃	1	1	yes	characteristic
59	C₆₄	C₂×D₅₃	1	1	yes	characteristic
60	C₅₃	#13737	1	1	yes	characteristic, complemented, Sylow
61	#131	—	53	1	no	—
62	M₃₂	—	53	1	no	—
63	M₃₂	—	53	2	no	—
64	C₂×C₁₆	—	53	2	no	—
65	C₂×C₁₆	C₄×D₅₃	1	1	yes	characteristic
66	C₃₂	—	53	2	no	—
67	C₃₂	C₄×D₅₃	1	1	yes	characteristic
68	C₃₂	C₂²×D₅₃	1	1	yes	characteristic
69	#41	—	53	1	no	—
70	M₁₆	—	53	1	no	—
71	M₁₆	—	53	2	no	—
72	C₂×C₈	—	53	2	no	—
73	C₂×C₈	C₈×D₅₃	1	1	yes	characteristic
74	C₁₆	—	53	2	no	—
75	C₁₆	C₈×D₅₃	1	1	yes	characteristic
76	C₁₆	C₂×C₄×D₅₃	1	1	yes	characteristic
77	Q₈	—	53	1	no	—
78	D₄	—	53	2	no	—
79	D₄	—	53	1	no	—
80	C₂×C₄	—	53	2	no	—
81	C₂×C₄	C₁₆×D₅₃	1	1	yes	characteristic
82	C₈	—	53	2	no	—
83	C₈	C₁₆×D₅₃	1	1	yes	characteristic
84	C₈	C₂×C₈×D₅₃	1	1	yes	characteristic
85	C₂²	—	53	2	no	—
86	C₂²	C₃₂×D₅₃	1	1	yes	characteristic
87	C₄	—	53	2	no	—
88	C₄	C₃₂×D₅₃	1	1	yes	characteristic
89	C₄	C₂×C₁₆×D₅₃	1	1	yes	characteristic
90	C₂	—	2	1	no	—
91	C₂	—	106	2	no	—
92	C₂	C₂×C₃₂×D₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₆₄	C₂×D₅₃
commutator, socle	C₂×C₅₃	C₂²×C₃₂
Frattini	C₃₂	C₂²×D₅₃
Fitting	C₂×C₅₃×C₆₄	C₂
prime radical	#11719945	C₁₆
cosocle	C₃₂×C₅₃	C₂³
hypercenter	C₂×C₆₄	D₅₃
hypocenter	C₅₃	#13737
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₆₄ ◁ C₂×C₆₄

Complements

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

N	H ≅ G/N	type
2. #20272926	C₂	semidirect
3. #20272925	C₂	semidirect
4. C₆₄×D₅₃	C₂	semidirect
5. C₂×C₅₃×C₆₄	C₂	semidirect
48. C₂×C₆₄	D₅₃	semidirect
60. C₅₃	#13737	semidirect

Maximal subgroups

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

subgroup	index
#20277008	2
#20272926	2
#20272925	2
C₆₄×D₅₃	2
C₂×C₅₃×C₆₄	2
#13737	53