Subgroups — C₁₆×M₁₆×C₅₃ #27218919

Order 13568.

306 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#2162	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₈×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1312	C₂	1	2	yes	maximal, complemented
5	C₈×C₁₆×C₅₃	C₂	1	2	yes	maximal, complemented
6	C₂²×C₁₆×C₅₃	C₄	1	2	yes	—
7	C₅₃×#431	C₄	1	1	yes	characteristic
8	C₄×M₁₆×C₅₃	C₄	1	1	yes	characteristic
9	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
10	C₅₃×C₄⋊C₁₆	C₄	1	2	yes	—
11	C₅₃×C₂²⋊C₁₆	C₄	1	2	yes	—
12	C₄×C₁₆×C₅₃	C₄	1	2	yes	—
13	C₄×C₁₆×C₅₃	C₂²	1	2	yes	—
14	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
15	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×#322	C₄	1	2	yes	—
17	C₈²×C₅₃	C₂²	1	2	yes	—
18	C₂×M₁₆×C₅₃	C₈	1	2	yes	—
19	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₁₆×C₅₃	—	2	4	no	—
23	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
24	C₂×C₁₆×C₅₃	C₂×C₄	1	4	yes	—
25	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
26	C₄⋊C₈×C₅₃	C₈	1	2	yes	—
27	C₂²⋊C₈×C₅₃	C₈	1	2	yes	—
28	#97×C₅₃	C₈	1	2	yes	—
29	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
33	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
34	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
35	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
36	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
37	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
38	M₁₆×C₅₃	C₁₆	1	8	yes	direct product factor
39	C₂×C₈×C₅₃	—	2	2	no	—
40	C₂×C₈×C₅₃	—	2	2	no	—
41	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
42	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
43	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	—
44	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	—
45	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
47	C₄²×C₅₃	C₄²	1	1	yes	characteristic
48	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
49	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
50	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
51	C₁₆×C₅₃	—	2	2	no	—
52	C₁₆×C₅₃	—	2	4	no	—
53	C₁₆×C₅₃	M₁₆	1	4	yes	direct product factor
54	C₂³×C₅₃	C₄×C₈	1	1	yes	characteristic
55	C₂×C₄×C₅₃	—	2	1	no	—
56	C₂×C₄×C₅₃	—	2	1	no	—
57	C₂×C₄×C₅₃	—	2	2	no	—
58	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
59	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
60	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
61	C₂×C₄×C₅₃	C₂×C₁₆	1	4	yes	—
62	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
63	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
64	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
65	C₈×C₅₃	—	2	1	no	—
66	C₈×C₅₃	—	2	1	no	—
67	C₈×C₅₃	—	2	4	no	—
68	C₈×C₅₃	C₂×C₁₆	1	8	yes	complemented
69	C₈×C₅₃	C₂×M₁₆	1	2	yes	—
70	C₈×C₅₃	#131	1	2	yes	—
71	C₁₆×M₁₆	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
72	C₂²×C₅₃	—	2	1	no	—
73	C₂²×C₅₃	—	2	1	no	—
74	C₂²×C₅₃	C₄×C₁₆	1	2	yes	—
75	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
76	C₄×C₅₃	—	2	1	no	—
77	C₄×C₅₃	—	2	1	no	—
78	C₄×C₅₃	C₄×C₁₆	1	2	yes	—
79	C₄×C₅₃	C₄×M₁₆	1	1	yes	characteristic
80	C₄×C₅₃	#405	1	1	yes	characteristic
81	C₄×C₅₃	C₂²×C₁₆	1	2	yes	—
82	C₄×C₅₃	#504	1	2	yes	—
83	#2162	C₂×C₅₃	1	1	yes	characteristic
84	C₂×C₄×C₁₆	C₂×C₅₃	1	1	yes	characteristic
85	C₈×M₁₆	C₂×C₅₃	1	1	yes	characteristic
86	#1312	C₂×C₅₃	1	2	yes	complemented
87	C₈×C₁₆	C₂×C₅₃	1	2	yes	complemented
88	C₂×C₅₃	—	2	2	no	—
89	C₂×C₅₃	C₈×M₁₆	1	1	yes	characteristic
90	C₂×C₅₃	C₂×C₄×C₁₆	1	1	yes	characteristic
91	C₂×C₅₃	#2108	1	1	yes	characteristic
92	C₂²×C₁₆	C₄×C₅₃	1	2	yes	—
93	#431	C₄×C₅₃	1	1	yes	characteristic
94	C₄×M₁₆	C₄×C₅₃	1	1	yes	characteristic
95	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
96	C₄⋊C₁₆	C₄×C₅₃	1	2	yes	—
97	C₂²⋊C₁₆	C₄×C₅₃	1	2	yes	—
98	C₄×C₁₆	C₄×C₅₃	1	2	yes	—
99	C₄×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
100	C₄×C₁₆	C₂²×C₅₃	1	2	yes	—
101	C₄×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
102	#322	C₄×C₅₃	1	2	yes	—
103	C₈²	C₂²×C₅₃	1	2	yes	—
104	C₅₃	C₁₆×M₁₆	1	1	yes	characteristic, direct product factor, Sylow
105	C₂×M₁₆	C₈×C₅₃	1	2	yes	—
106	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
107	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
108	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
109	C₂×C₁₆	—	2	4	no	—
110	C₂×C₁₆	C₂×C₄×C₅₃	1	2	yes	—
111	C₂×C₁₆	C₂×C₄×C₅₃	1	4	yes	—
112	C₂×C₁₆	C₂×C₄×C₅₃	1	2	yes	—
113	C₄⋊C₈	C₈×C₅₃	1	2	yes	—
114	C₂²⋊C₈	C₈×C₅₃	1	2	yes	—
115	#97	C₈×C₅₃	1	2	yes	—
116	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
117	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
118	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
119	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
120	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
121	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
122	C₂²×C₄	C₄²×C₅₃	1	1	yes	characteristic
123	C₂²×C₄	C₂×C₈×C₅₃	1	1	yes	characteristic
124	C₂²×C₄	C₂×C₈×C₅₃	1	1	yes	characteristic
125	M₁₆	C₁₆×C₅₃	1	8	yes	direct product factor
126	C₂×C₈	—	2	2	no	—
127	C₂×C₈	—	2	2	no	—
128	C₂×C₈	C₄²×C₅₃	1	1	yes	characteristic
129	C₂×C₈	C₄²×C₅₃	1	1	yes	characteristic
130	C₂×C₈	C₂×C₈×C₅₃	1	4	yes	—
131	C₂×C₈	C₂×C₈×C₅₃	1	4	yes	—
132	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
133	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
134	C₄²	C₄²×C₅₃	1	1	yes	characteristic
135	C₄²	C₂×C₈×C₅₃	1	1	yes	characteristic
136	C₄²	C₂×C₈×C₅₃	1	1	yes	characteristic
137	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
138	C₁₆	—	2	2	no	—
139	C₁₆	—	2	4	no	—
140	C₁₆	M₁₆×C₅₃	1	4	yes	direct product factor
141	C₂³	C₄×C₈×C₅₃	1	1	yes	characteristic
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	C₄×C₈×C₅₃	1	1	yes	characteristic
146	C₂×C₄	C₄×C₈×C₅₃	1	1	yes	characteristic
147	C₂×C₄	C₄×C₈×C₅₃	1	1	yes	characteristic
148	C₂×C₄	C₂×C₁₆×C₅₃	1	4	yes	—
149	C₂×C₄	C₂×C₄²×C₅₃	1	1	yes	characteristic
150	C₂×C₄	C₂²×C₈×C₅₃	1	1	yes	characteristic
151	C₂×C₄	C₂²×C₈×C₅₃	1	1	yes	characteristic
152	C₈	—	2	4	no	—
153	C₈	—	2	1	no	—
154	C₈	—	2	1	no	—
155	C₈	C₂×C₁₆×C₅₃	1	8	yes	complemented
156	C₈	C₂×M₁₆×C₅₃	1	2	yes	—
157	C₈	#131×C₅₃	1	2	yes	—
158	C₂²	—	2	1	no	—
159	C₂²	—	2	1	no	—
160	C₂²	C₄×C₁₆×C₅₃	1	2	yes	—
161	C₂²	C₂×C₄×C₈×C₅₃	1	1	yes	characteristic
162	C₄	—	2	1	no	—
163	C₄	—	2	1	no	—
164	C₄	C₄×C₁₆×C₅₃	1	2	yes	—
165	C₄	C₄×M₁₆×C₅₃	1	1	yes	characteristic
166	C₄	C₅₃×#405	1	1	yes	characteristic
167	C₄	C₂²×C₁₆×C₅₃	1	2	yes	—
168	C₄	C₅₃×#504	1	2	yes	—
169	C₂	—	2	2	no	—
170	C₂	C₈×M₁₆×C₅₃	1	1	yes	characteristic
171	C₂	C₂×C₄×C₁₆×C₅₃	1	1	yes	characteristic
172	C₂	C₅₃×#2108	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₁₆×C₅₃	C₂²
commutator	C₂	C₂×C₄×C₁₆×C₅₃
Frattini, cosocle	C₄×C₈	C₂³×C₅₃
socle	C₂²×C₅₃	C₂×C₄×C₈
prime radical	C₂³×C₅₃	C₄×C₈
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₆×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₅₃×#1312	C₂	semidirect
5. C₈×C₁₆×C₅₃	C₂	semidirect
38. M₁₆×C₅₃	C₁₆	direct (N × H)
53. C₁₆×C₅₃	M₁₆	direct (N × H)
68. C₈×C₅₃	C₂×C₁₆	semidirect
71. C₁₆×M₁₆	C₅₃	direct (N × H)
86. #1312	C₂×C₅₃	semidirect
87. C₈×C₁₆	C₂×C₅₃	semidirect
104. C₅₃	C₁₆×M₁₆	direct (N × H)
125. M₁₆	C₁₆×C₅₃	direct (N × H)
140. C₁₆	M₁₆×C₅₃	direct (N × H)
155. C₈	C₂×C₁₆×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#2162	2
C₂×C₄×C₁₆×C₅₃	2
C₈×M₁₆×C₅₃	2
C₅₃×#1312	2
C₈×C₁₆×C₅₃	2
C₁₆×M₁₆	53

Subgroups — C16×M16×C53 #27218919

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₆×M₁₆×C₅₃ #27218919