Subgroups — #27304606

Order 13568.

178 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810453	C₂	1	2	yes	maximal
2	#20272919	C₂	1	4	yes	maximal, complemented
3	C₂×C₅₃×M₆₄	C₂	1	1	yes	maximal, characteristic
4	C₂×#13133153	—	2	2	no	—
5	#16810453	—	2	2	no	—
6	#16810453	C₂²	1	4	yes	—
7	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
8	C₅₃×M₆₄	C₄	1	4	yes	—
9	C₂×C₃₂×C₅₃	C₂²	1	2	yes	—
10	C₂×#11719917	—	4	2	no	—
11	#13133153	—	2	4	no	—
12	#13133153	—	4	2	no	—
13	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
14	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
15	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
16	C₂×C₁₆×C₅₃	Q₈	1	2	yes	—
17	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
18	C₃₂×C₅₃	C₂×C₄	1	4	yes	—
19	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
20	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
21	C₂×C₈×C₅₃	D₈	1	1	yes	characteristic
22	C₂×C₈×C₅₃	Q₁₆	1	2	yes	—
23	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
24	#11719917	—	4	4	no	—
25	#11719917	—	8	2	no	—
26	C₂×#88322	—	8	2	no	—
27	C₁₆×C₅₃	C₄⋊C₄	1	2	yes	—
28	C₁₆×C₅₃	C₂²×C₄	1	1	yes	characteristic
29	C₁₆×C₅₃	C₂×Q₈	1	1	yes	characteristic
30	C₂³×C₅₃	Q₃₂	1	1	yes	characteristic
31	C₂×C₄×C₅₃	#107	1	2	yes	—
32	C₂×C₄×C₅₃	D₁₆	1	2	yes	—
33	C₂×C₄×C₅₃	Q₃₂	1	1	yes	characteristic
34	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
35	C₈×C₅₃	#107	1	2	yes	—
36	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
37	C₈×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
38	#88322	—	8	4	no	—
39	#13636	—	53	1	no	maximal, Sylow
40	C₂²×C₅₃	—	4	1	no	—
41	C₂²×C₅₃	#366	1	2	yes	—
42	C₂²×C₅₃	C₂×Q₃₂	1	1	yes	characteristic
43	C₄×C₅₃	#366	1	2	yes	—
44	C₄×C₅₃	C₂×#107	1	1	yes	characteristic
45	C₄×C₅₃	C₂×D₁₆	1	1	yes	characteristic
46	C₂×M₆₄	D₅₃	1	1	yes	characteristic
47	C₂×#368	—	53	2	no	—
48	#1425	—	53	4	no	—
49	C₂×C₅₃	—	2	2	no	—
50	C₂×C₅₃	—	2	1	no	—
51	C₂×C₅₃	C₂×#366	1	1	yes	characteristic
52	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
53	C₂×#108	—	106	2	no	—
54	M₆₄	Dic₅₃	1	4	yes	—
55	C₂×C₃₂	C₂×D₅₃	1	2	yes	—
56	#368	—	53	4	no	—
57	#368	—	106	2	no	—
58	C₅₃	#13636	1	1	yes	characteristic, complemented, Sylow
59	C₂×M₁₆	—	212	2	no	—
60	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
61	C₂×C₁₆	Dic₁₀₆	1	2	yes	—
62	C₂×C₁₆	D₂₁₂	1	1	yes	characteristic
63	C₂×C₁₆	C₂×Dic₅₃	1	2	yes	—
64	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
65	#108	—	106	4	no	—
66	#108	—	212	2	no	—
67	C₃₂	C₂×Dic₅₃	1	4	yes	—
68	C₂²×C₄	D₄₂₄	1	1	yes	characteristic
69	M₁₆	—	424	2	no	—
70	M₁₆	—	212	4	no	—
71	C₂×C₈	—	424	2	no	—
72	C₂×C₈	D₄₂₄	1	1	yes	characteristic
73	C₂×C₈	Dic₂₁₂	1	2	yes	—
74	C₂×C₈	C₄⋊Dic₅₃	1	2	yes	—
75	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
76	C₁₆	C₄⋊Dic₅₃	1	2	yes	—
77	C₁₆	C₂×Dic₁₀₆	1	1	yes	characteristic
78	C₁₆	C₂²×Dic₅₃	1	1	yes	characteristic
79	C₂³	Dic₄₂₄	1	1	yes	characteristic
80	C₂×C₄	#13133151	1	2	yes	—
81	C₂×C₄	D₈₄₈	1	2	yes	—
82	C₂×C₄	Dic₄₂₄	1	1	yes	characteristic
83	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
84	C₈	—	424	4	no	—
85	C₈	#13133151	1	2	yes	—
86	C₈	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
87	C₈	C₂×Dic₂₁₂	1	1	yes	characteristic
88	C₂²	—	4	1	no	—
89	C₂²	#16810449	1	2	yes	—
90	C₂²	C₂×Dic₄₂₄	1	1	yes	characteristic
91	C₄	#16810449	1	2	yes	—
92	C₄	C₂×#13133151	1	1	yes	characteristic
93	C₄	C₂×D₈₄₈	1	1	yes	characteristic
94	C₂	—	2	1	no	—
95	C₂	—	2	2	no	—
96	C₂	C₂×#16810449	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₈₄₈
commutator	C₁₆×C₅₃	C₂²×C₄
Frattini	C₂×C₁₆	C₂²×D₅₃
Fitting	C₂×C₅₃×M₆₄	C₂
socle	C₂×C₅₃	C₂×#366
prime radical	C₂³×C₅₃	Q₃₂
cosocle	C₂×C₁₆×C₅₃	C₂³
hypercenter	C₂×M₆₄	D₅₃
hypocenter	C₅₃	#13636
LCS	G ▷ C₁₆×C₅₃ ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆ ◁ C₂×M₆₄

Complements

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

N	H ≅ G/N	type
2. #20272919	C₂	semidirect
58. C₅₃	#13636	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810453	2
#20272919	2
C₂×C₅₃×M₆₄	2
#13636	53