Subgroups — #27304575

Order 13568.

218 conjugacy classes of subgroups.

The subgroup lattice has 124 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810457	C₂	1	2	yes	maximal
2	C₄⋊#13133123	C₂	1	4	yes	maximal, complemented
3	C₂×C₄×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
4	#16810457	—	2	4	no	—
5	C₂×#13133123	C₂²	1	4	yes	—
6	C₂²×C₁₆×C₅₃	C₄	1	1	yes	characteristic
7	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
9	C₄×C₁₆×C₅₃	C₄	1	2	yes	—
10	C₄×C₁₆×C₅₃	C₂²	1	2	yes	—
11	#13133123	—	2	4	no	—
12	C₂²×C₈×C₅₃	C₈	1	1	yes	characteristic
13	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
14	C₂×C₄²×C₅₃	C₈	1	1	yes	characteristic
15	C₂×C₁₆×C₅₃	—	2	2	no	—
16	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
17	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
18	C₂×C₁₆×C₅₃	D₄	1	2	yes	—
19	C₂×C₁₆×C₅₃	Q₈	1	2	yes	—
20	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
21	C₄×C₈×C₅₃	C₈	1	2	yes	—
22	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₅₃	C₁₆	1	2	yes	—
24	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
25	C₂×C₈×C₅₃	—	2	2	no	—
26	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
27	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
28	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
29	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
30	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
31	C₄²×C₅₃	C₁₆	1	1	yes	characteristic
32	C₄²×C₅₃	C₁₆	1	1	yes	characteristic
33	C₄²×C₅₃	C₂×C₈	1	2	yes	—
34	C₁₆×C₅₃	—	2	2	no	—
35	C₁₆×C₅₃	C₄⋊C₄	1	2	yes	—
36	C₁₆×C₅₃	C₂×D₄	1	1	yes	characteristic
37	C₁₆×C₅₃	C₂×Q₈	1	1	yes	characteristic
38	C₂³×C₅₃	C₂×C₁₆	1	1	yes	characteristic
39	C₂×C₄×C₅₃	—	2	4	no	—
40	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
41	C₂×C₄×C₅₃	C₂×C₁₆	1	2	yes	—
42	C₂×C₄×C₅₃	C₂×C₁₆	1	1	yes	characteristic
43	C₂×C₄×C₅₃	C₂×C₁₆	1	2	yes	—
44	C₂×C₄×C₅₃	M₃₂	1	2	yes	—
45	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
46	C₈×C₅₃	—	2	2	no	—
47	C₈×C₅₃	C₄⋊C₈	1	2	yes	—
48	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₈×C₅₃	C₂×M₁₆	1	1	yes	characteristic
50	#13627	—	53	1	no	maximal, Sylow
51	C₂²×C₅₃	—	2	2	no	—
52	C₂²×C₅₃	C₄⋊C₁₆	1	2	yes	—
53	C₂²×C₅₃	C₂²×C₁₆	1	1	yes	characteristic
54	C₄×C₅₃	—	2	2	no	—
55	C₄×C₅₃	C₄⋊C₁₆	1	2	yes	—
56	C₄×C₅₃	M₆₄	1	4	yes	complemented
57	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
58	C₄×C₅₃	C₂×M₃₂	1	1	yes	characteristic
59	C₂×M₆₄	—	53	2	no	—
60	C₂×C₄×C₁₆	D₅₃	1	1	yes	characteristic
61	C₄⋊C₃₂	—	53	4	no	—
62	C₂×C₅₃	—	2	2	no	—
63	C₂×C₅₃	C₂×C₄⋊C₁₆	1	1	yes	characteristic
64	C₂×C₅₃	C₂×M₆₄	1	2	yes	—
65	C₂²×C₁₆	Dic₅₃	1	1	yes	characteristic
66	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
67	C₂×C₄×C₈	Dic₅₃	1	1	yes	characteristic
68	M₆₄	—	106	4	no	—
69	C₂×C₃₂	—	53	4	no	—
70	C₄×C₁₆	Dic₅₃	1	2	yes	—
71	C₄×C₁₆	C₂×D₅₃	1	2	yes	—
72	C₅₃	#13627	1	1	yes	characteristic, complemented, Sylow
73	C₂²×C₈	#88322	1	1	yes	characteristic
74	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
75	C₂×C₄²	#88322	1	1	yes	characteristic
76	C₂×C₁₆	—	2	2	no	—
77	C₂×C₁₆	Dic₁₀₆	1	2	yes	—
78	C₂×C₁₆	D₂₁₂	1	2	yes	—
79	C₂×C₁₆	C₂×Dic₅₃	1	2	yes	—
80	C₂×C₁₆	C₂×Dic₅₃	1	1	yes	characteristic
81	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
82	C₄×C₈	#88322	1	2	yes	—
83	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
84	C₃₂	—	106	4	no	—
85	C₂²×C₄	#11719908	1	2	yes	—
86	C₂²×C₄	C₂×#88322	1	1	yes	characteristic
87	C₂×C₈	—	2	2	no	—
88	C₂×C₈	C₂×#88322	1	2	yes	—
89	C₂×C₈	C₂×#88322	1	1	yes	characteristic
90	C₂×C₈	#11719917	1	2	yes	—
91	C₂×C₈	C₄⋊Dic₅₃	1	2	yes	—
92	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
93	C₄²	#11719908	1	1	yes	characteristic
94	C₄²	#11719908	1	1	yes	characteristic
95	C₄²	C₂×#88322	1	2	yes	—
96	C₁₆	—	2	2	no	—
97	C₁₆	C₄⋊Dic₅₃	1	2	yes	—
98	C₁₆	C₂×Dic₁₀₆	1	1	yes	characteristic
99	C₁₆	C₂×D₂₁₂	1	1	yes	characteristic
100	C₂³	C₂×#11719908	1	1	yes	characteristic
101	C₂×C₄	—	2	4	no	—
102	C₂×C₄	C₄⋊#88322	1	2	yes	—
103	C₂×C₄	C₂×#11719908	1	2	yes	—
104	C₂×C₄	C₂×#11719908	1	1	yes	characteristic
105	C₂×C₄	C₂×#11719908	1	2	yes	—
106	C₂×C₄	#13133157	1	2	yes	—
107	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
108	C₈	—	2	2	no	—
109	C₈	C₄⋊#88322	1	2	yes	—
110	C₈	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
111	C₈	C₂×#11719917	1	1	yes	characteristic
112	C₂²	—	2	2	no	—
113	C₂²	C₄⋊#11719908	1	2	yes	—
114	C₂²	C₂²×#11719908	1	1	yes	characteristic
115	C₄	—	2	2	no	—
116	C₄	C₄⋊#11719908	1	2	yes	—
117	C₄	#16810457	1	4	yes	complemented
118	C₄	C₂×C₄⋊#88322	1	1	yes	characteristic
119	C₄	C₂×#13133157	1	1	yes	characteristic
120	C₂	—	2	2	no	—
121	C₂	C₂×C₄⋊#11719908	1	1	yes	characteristic
122	C₂	C₂×#16810457	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₁₆	C₂²×D₅₃
commutator, socle	C₂²×C₅₃	C₂²×C₁₆
Fitting	C₂×C₄×C₁₆×C₅₃	C₂
prime radical	C₂³×C₅₃	C₂×C₁₆
cosocle	C₂×C₁₆×C₅₃	C₂³
hypercenter	C₂×C₄×C₁₆	D₅₃
hypocenter	C₅₃	#13627
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
2. C₄⋊#13133123	C₂	semidirect
56. C₄×C₅₃	M₆₄	semidirect
72. C₅₃	#13627	semidirect
117. C₄	#16810457	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810457	2
C₄⋊#13133123	2
C₂×C₄×C₁₆×C₅₃	2
#13627	53