Subgroups — M₆₄×Dic₅₃ #27304267

Order 13568.

250 conjugacy classes of subgroups.

The subgroup lattice has 150 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	1	yes	maximal, characteristic
2	C₂×C₁₆×Dic₅₃	C₂	1	1	yes	maximal, characteristic
3	#20272848	C₂	1	2	yes	maximal, complemented
4	C₃₂×Dic₅₃	C₂	1	2	yes	maximal, complemented
5	C₂×C₅₃×M₆₄	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11719908	C₄	1	1	yes	characteristic
7	C₂×C₈×Dic₅₃	C₄	1	1	yes	characteristic
8	#16810457	C₄	1	4	yes	complemented
9	C₂×#13133123	C₂²	1	2	yes	—
10	C₁₆×Dic₅₃	C₄	1	1	yes	characteristic
11	C₁₆×Dic₅₃	C₄	1	1	yes	characteristic
12	C₁₆×Dic₅₃	C₂²	1	1	yes	characteristic
13	C₁₆×Dic₅₃	C₂²	1	1	yes	characteristic
14	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
15	C₅₃×M₆₄	C₄	1	4	yes	complemented
16	C₂×C₃₂×C₅₃	C₂²	1	2	yes	—
17	C₂²×#88322	C₈	1	1	yes	characteristic
18	C₂×C₄×Dic₅₃	C₈	1	1	yes	characteristic
19	C₂×#11719908	C₈	1	4	yes	—
20	C₂×#11719908	C₂×C₄	1	1	yes	characteristic
21	C₂×#11719908	C₂×C₄	1	1	yes	characteristic
22	C₈×Dic₅₃	C₈	1	1	yes	characteristic
23	C₈×Dic₅₃	C₈	1	1	yes	characteristic
24	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
25	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
26	#13133123	C₂×C₄	1	4	yes	complemented
27	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
30	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
31	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
32	C₃₂×C₅₃	C₂×C₄	1	4	yes	complemented
33	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
34	C₂²×Dic₅₃	C₁₆	1	2	yes	—
35	C₂×C₈×C₅₃	C₄²	1	2	yes	—
36	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
37	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
38	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
39	C₄×Dic₅₃	C₁₆	1	2	yes	—
40	C₄×Dic₅₃	C₂×C₈	1	1	yes	characteristic
41	C₄×Dic₅₃	C₂×C₈	1	1	yes	characteristic
42	C₂×#88322	C₁₆	1	4	yes	—
43	C₂×#88322	C₂×C₈	1	1	yes	characteristic
44	C₂×#88322	C₂×C₈	1	1	yes	characteristic
45	C₁₆×C₅₃	C₄²	1	2	yes	—
46	C₁₆×C₅₃	C₂²×C₄	1	1	yes	characteristic
47	C₁₆×C₅₃	C₂²×C₄	1	1	yes	characteristic
48	#11719908	C₂×C₈	1	2	yes	—
49	#11719908	C₂×C₈	1	2	yes	—
50	C₂³×C₅₃	C₂×C₁₆	1	1	yes	characteristic
51	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
52	C₂×C₄×C₅₃	C₂×C₁₆	1	1	yes	characteristic
53	C₂×C₄×C₅₃	C₂×C₁₆	1	2	yes	—
54	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
55	C₂×Dic₅₃	—	2	4	no	—
56	C₂×Dic₅₃	C₂×C₁₆	1	2	yes	—
57	C₂×Dic₅₃	C₂×C₁₆	1	2	yes	—
58	C₈×C₅₃	C₄×C₈	1	2	yes	—
59	C₈×C₅₃	C₂×C₄²	1	1	yes	characteristic
60	C₈×C₅₃	C₂²×C₈	1	1	yes	characteristic
61	#88322	C₂×C₁₆	1	2	yes	—
62	#88322	C₂×C₁₆	1	2	yes	—
63	C₄×M₆₄	—	53	1	no	maximal, Sylow
64	C₂²×C₅₃	—	2	2	no	—
65	C₂²×C₅₃	C₄×C₁₆	1	2	yes	—
66	C₂²×C₅₃	C₂²×C₁₆	1	1	yes	characteristic
67	C₄×C₅₃	C₄×C₁₆	1	2	yes	—
68	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
69	C₄×C₅₃	C₂²×C₁₆	1	1	yes	characteristic
70	Dic₅₃	—	2	2	no	—
71	Dic₅₃	M₆₄	1	4	yes	direct product factor
72	C₂×M₆₄	—	53	1	no	—
73	C₂×M₆₄	D₅₃	1	1	yes	characteristic
74	C₂×C₄×C₁₆	—	53	1	no	—
75	#1397	—	53	2	no	—
76	C₄×C₃₂	—	53	2	no	—
77	C₂×C₅₃	—	2	2	no	—
78	C₂×C₅₃	C₂×C₄×C₁₆	1	1	yes	characteristic
79	C₂×C₅₃	C₂×M₆₄	1	2	yes	—
80	C₂²×C₁₆	—	53	1	no	—
81	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
82	C₂×C₄×C₈	—	53	1	no	—
83	M₆₄	—	53	4	no	—
84	M₆₄	Dic₅₃	1	4	yes	direct product factor
85	C₂×C₃₂	—	53	2	no	—
86	C₂×C₃₂	C₂×D₅₃	1	2	yes	—
87	C₄×C₁₆	—	53	1	no	—
88	C₄×C₁₆	—	53	1	no	—
89	C₄×C₁₆	—	53	1	no	—
90	C₄×C₁₆	—	53	1	no	—
91	C₅₃	C₄×M₆₄	1	1	yes	characteristic, complemented, Sylow
92	C₂²×C₈	—	53	1	no	—
93	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
94	C₂×C₄²	—	53	1	no	—
95	C₂×C₁₆	—	53	1	no	—
96	C₂×C₁₆	—	53	1	no	—
97	C₂×C₁₆	—	53	4	no	—
98	C₂×C₁₆	C₄×D₅₃	1	2	yes	—
99	C₂×C₁₆	C₄×D₅₃	1	1	yes	characteristic
100	C₂×C₁₆	C₂×Dic₅₃	1	2	yes	—
101	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
102	C₄×C₈	—	53	1	no	—
103	C₄×C₈	—	53	1	no	—
104	C₄×C₈	—	53	1	no	—
105	C₄×C₈	—	53	1	no	—
106	C₃₂	—	53	4	no	—
107	C₃₂	C₂×Dic₅₃	1	4	yes	complemented
108	C₂²×C₄	—	53	2	no	—
109	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
110	C₂×C₈	—	53	4	no	—
111	C₂×C₈	—	53	1	no	—
112	C₂×C₈	—	53	1	no	—
113	C₂×C₈	C₈×D₅₃	1	1	yes	characteristic
114	C₂×C₈	C₈×D₅₃	1	2	yes	—
115	C₂×C₈	C₄×Dic₅₃	1	2	yes	—
116	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
117	C₄²	—	53	2	no	—
118	C₄²	—	53	1	no	—
119	C₄²	—	53	1	no	—
120	C₁₆	—	53	2	no	—
121	C₁₆	—	53	2	no	—
122	C₁₆	C₄×Dic₅₃	1	2	yes	—
123	C₁₆	C₂×C₄×D₅₃	1	1	yes	characteristic
124	C₁₆	C₂²×Dic₅₃	1	1	yes	characteristic
125	C₂³	C₁₆×D₅₃	1	1	yes	characteristic
126	C₂×C₄	—	53	2	no	—
127	C₂×C₄	—	53	2	no	—
128	C₂×C₄	—	106	4	no	—
129	C₂×C₄	C₈×Dic₅₃	1	2	yes	—
130	C₂×C₄	C₁₆×D₅₃	1	2	yes	—
131	C₂×C₄	C₁₆×D₅₃	1	1	yes	characteristic
132	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
133	C₈	—	53	2	no	—
134	C₈	—	53	2	no	—
135	C₈	C₈×Dic₅₃	1	2	yes	—
136	C₈	C₂×C₄×Dic₅₃	1	1	yes	characteristic
137	C₈	C₂×C₈×D₅₃	1	1	yes	characteristic
138	C₂²	—	2	2	no	—
139	C₂²	C₁₆×Dic₅₃	1	2	yes	—
140	C₂²	C₂×C₁₆×D₅₃	1	1	yes	characteristic
141	C₄	—	53	4	no	—
142	C₄	—	106	2	no	—
143	C₄	C₁₆×Dic₅₃	1	2	yes	—
144	C₄	C₂×C₈×Dic₅₃	1	1	yes	characteristic
145	C₄	C₂×C₁₆×D₅₃	1	1	yes	characteristic
146	C₂	—	2	2	no	—
147	C₂	C₂×C₁₆×Dic₅₃	1	1	yes	characteristic
148	C₂	M₆₄×D₅₃	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
3. #20272848	C₂	semidirect
4. C₃₂×Dic₅₃	C₂	semidirect
8. #16810457	C₄	semidirect
15. C₅₃×M₆₄	C₄	semidirect
26. #13133123	C₂×C₄	semidirect
32. C₃₂×C₅₃	C₂×C₄	semidirect
71. Dic₅₃	M₆₄	direct (N × H)
84. M₆₄	Dic₅₃	direct (N × H)
91. C₅₃	C₄×M₆₄	semidirect
107. C₃₂	C₂×Dic₅₃	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
C₂×C₁₆×Dic₅₃	2
#20272848	2
C₃₂×Dic₅₃	2
C₂×C₅₃×M₆₄	2
C₄×M₆₄	53

Subgroups — M64×Dic53 #27304267

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — M₆₄×Dic₅₃ #27304267