Subgroups — C₅₃×#12051 #27219563

Order 13568.

224 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#2183	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅₃×#2169	C₂	1	1	yes	maximal, characteristic
3	C₈×SD₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1367	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×Q₈⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×D₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
7	C₈×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×M₃₂×C₅₃	—	2	1	no	—
9	C₈×Q₈×C₅₃	C₂²	1	1	yes	characteristic
10	C₄×SD₁₆×C₅₃	C₄	1	1	yes	characteristic
11	C₈×D₄×C₅₃	C₂²	1	1	yes	characteristic
12	C₅₃×C₄⋊C₁₆	—	2	1	no	—
13	C₅₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
14	C₅₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
15	C₅₃×C₂²⋊C₁₆	—	2	1	no	—
16	C₅₃×#346	—	2	1	no	—
17	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
18	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
19	C₅₃×#334	C₄	1	1	yes	characteristic
20	C₅₃×Q₈⋊C₈	C₄	1	1	yes	characteristic
21	C₅₃×D₄⋊C₈	C₄	1	1	yes	characteristic
22	C₈²×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×SD₁₆×C₅₃	C₈	1	1	yes	characteristic
24	C₂²×C₈×C₅₃	—	2	1	no	—
25	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₄×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
27	M₃₂×C₅₃	—	4	2	no	—
28	C₂×C₁₆×C₅₃	—	2	1	no	—
29	C₂×C₁₆×C₅₃	—	2	1	no	—
30	C₂×C₁₆×C₅₃	—	2	1	no	—
31	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
32	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
33	#106×C₅₃	C₈	1	1	yes	characteristic
34	C₄⋊C₈×C₅₃	—	2	1	no	—
35	C₄⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
36	C₄⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
37	#103×C₅₃	C₈	1	1	yes	characteristic
38	#102×C₅₃	C₈	1	1	yes	characteristic
39	C₂²⋊C₈×C₅₃	—	2	1	no	—
40	C₄×C₈×C₅₃	—	2	1	no	—
41	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
42	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
44	C₂×Q₈×C₅₃	C₂×C₈	1	1	yes	characteristic
45	C₂×D₄×C₅₃	C₂×C₈	1	1	yes	characteristic
46	C₂²×C₄×C₅₃	—	2	1	no	—
47	SD₁₆×C₅₃	—	2	2	no	—
48	C₂×C₈×C₅₃	—	2	1	no	—
49	C₂×C₈×C₅₃	—	2	1	no	—
50	C₂×C₈×C₅₃	—	2	1	no	—
51	C₂×C₈×C₅₃	—	4	1	no	—
52	C₂×C₈×C₅₃	—	4	1	no	—
53	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
54	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
55	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
56	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
57	C₄⋊C₄×C₅₃	—	2	1	no	—
58	C₄⋊C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
59	C₄⋊C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
60	C₂²⋊C₄×C₅₃	—	2	1	no	—
61	C₄²×C₅₃	—	2	1	no	—
62	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
63	C₁₆×C₅₃	—	2	2	no	—
64	C₁₆×C₅₃	—	2	1	no	—
65	C₁₆×C₅₃	—	4	1	no	—
66	C₁₆×C₅₃	—	4	1	no	—
67	C₁₆×C₅₃	SD₁₆	1	2	yes	complemented
68	C₂³×C₅₃	—	2	1	no	—
69	Q₈×C₅₃	—	2	1	no	—
70	Q₈×C₅₃	—	2	1	no	—
71	D₄×C₅₃	—	2	1	no	—
72	D₄×C₅₃	—	2	1	no	—
73	C₂×C₄×C₅₃	—	2	1	no	—
74	C₂×C₄×C₅₃	—	2	1	no	—
75	C₂×C₄×C₅₃	—	2	1	no	—
76	C₂×C₄×C₅₃	—	4	1	no	—
77	C₂×C₄×C₅₃	—	4	1	no	—
78	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
79	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
80	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
81	C₈×C₅₃	—	2	2	no	—
82	C₈×C₅₃	—	2	1	no	—
83	C₈×C₅₃	—	2	1	no	—
84	C₈×C₅₃	—	4	1	no	—
85	C₈×C₅₃	—	4	1	no	—
86	C₈×C₅₃	M₃₂	1	2	yes	complemented
87	C₈×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
88	C₈×C₅₃	#135	1	1	yes	characteristic
89	#12051	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
90	C₂²×C₅₃	—	4	1	no	—
91	C₂²×C₅₃	—	4	1	no	—
92	C₂²×C₅₃	—	4	1	no	—
93	C₂²×C₅₃	C₈×D₄	1	1	yes	characteristic
94	C₄×C₅₃	—	2	1	no	—
95	C₄×C₅₃	—	4	1	no	—
96	C₄×C₅₃	—	4	1	no	—
97	C₄×C₅₃	—	4	1	no	—
98	C₄×C₅₃	C₄×SD₁₆	1	1	yes	characteristic
99	C₄×C₅₃	#443	1	1	yes	characteristic
100	C₄×C₅₃	C₂×M₃₂	1	1	yes	characteristic
101	C₄×C₅₃	#504	1	1	yes	characteristic
102	#2183	C₂×C₅₃	1	1	yes	characteristic, complemented
103	#2169	C₂×C₅₃	1	1	yes	characteristic
104	C₈×SD₁₆	C₂×C₅₃	1	1	yes	characteristic
105	#1367	C₂×C₅₃	1	1	yes	characteristic, complemented
106	Q₈⋊C₁₆	C₂×C₅₃	1	1	yes	characteristic, complemented
107	D₄⋊C₁₆	C₂×C₅₃	1	1	yes	characteristic
108	C₈×C₁₆	C₂×C₅₃	1	1	yes	characteristic, complemented
109	C₂×C₅₃	—	8	1	no	—
110	C₂×C₅₃	C₈×SD₁₆	1	1	yes	characteristic
111	C₂×C₅₃	#2169	1	1	yes	characteristic
112	C₂×C₅₃	#2170	1	1	yes	characteristic
113	C₂×M₃₂	—	2	1	no	—
114	C₈×Q₈	C₂²×C₅₃	1	1	yes	characteristic
115	C₄×SD₁₆	C₄×C₅₃	1	1	yes	characteristic
116	C₈×D₄	C₂²×C₅₃	1	1	yes	characteristic
117	C₄⋊C₁₆	—	2	1	no	—
118	C₄⋊C₁₆	C₂²×C₅₃	1	1	yes	characteristic
119	C₄⋊C₁₆	C₂²×C₅₃	1	1	yes	characteristic
120	C₂²⋊C₁₆	—	2	1	no	—
121	#346	—	2	1	no	—
122	C₄×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
123	C₄×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
124	#334	C₄×C₅₃	1	1	yes	characteristic
125	Q₈⋊C₈	C₄×C₅₃	1	1	yes	characteristic
126	D₄⋊C₈	C₄×C₅₃	1	1	yes	characteristic
127	C₈²	C₂²×C₅₃	1	1	yes	characteristic
128	C₅₃	#12051	1	1	yes	characteristic, direct product factor, Sylow
129	C₂×SD₁₆	C₈×C₅₃	1	1	yes	characteristic
130	C₂²×C₈	—	2	1	no	—
131	C₄×Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
132	C₄×D₄	C₂×C₄×C₅₃	1	1	yes	characteristic
133	M₃₂	—	4	2	no	—
134	C₂×C₁₆	—	2	1	no	—
135	C₂×C₁₆	—	2	1	no	—
136	C₂×C₁₆	—	2	1	no	—
137	C₂×C₁₆	D₄×C₅₃	1	1	yes	characteristic
138	C₂×C₁₆	D₄×C₅₃	1	1	yes	characteristic
139	#106	C₈×C₅₃	1	1	yes	characteristic
140	C₄⋊C₈	—	2	1	no	—
141	C₄⋊C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
142	C₄⋊C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
143	#103	C₈×C₅₃	1	1	yes	characteristic
144	#102	C₈×C₅₃	1	1	yes	characteristic
145	C₂²⋊C₈	—	2	1	no	—
146	C₄×C₈	—	2	1	no	—
147	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
148	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
149	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
150	C₂×Q₈	C₂×C₈×C₅₃	1	1	yes	characteristic
151	C₂×D₄	C₂×C₈×C₅₃	1	1	yes	characteristic
152	C₂²×C₄	—	2	1	no	—
153	SD₁₆	—	2	2	no	—
154	C₂×C₈	—	2	1	no	—
155	C₂×C₈	—	2	1	no	—
156	C₂×C₈	—	2	1	no	—
157	C₂×C₈	—	4	1	no	—
158	C₂×C₈	—	4	1	no	—
159	C₂×C₈	C₂×C₈×C₅₃	1	1	yes	characteristic
160	C₂×C₈	C₂×C₈×C₅₃	1	1	yes	characteristic
161	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
162	C₂×C₈	#41×C₅₃	1	1	yes	characteristic
163	C₄⋊C₄	—	2	1	no	—
164	C₄⋊C₄	C₂×C₈×C₅₃	1	1	yes	characteristic
165	C₄⋊C₄	C₂×C₈×C₅₃	1	1	yes	characteristic
166	C₂²⋊C₄	—	2	1	no	—
167	C₄²	—	2	1	no	—
168	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
169	C₁₆	—	2	2	no	—
170	C₁₆	—	2	1	no	—
171	C₁₆	—	4	1	no	—
172	C₁₆	—	4	1	no	—
173	C₁₆	SD₁₆×C₅₃	1	2	yes	complemented
174	C₂³	—	2	1	no	—
175	Q₈	—	2	1	no	—
176	Q₈	—	2	1	no	—
177	D₄	—	2	1	no	—
178	D₄	—	2	1	no	—
179	C₂×C₄	—	2	1	no	—
180	C₂×C₄	—	2	1	no	—
181	C₂×C₄	—	2	1	no	—
182	C₂×C₄	—	4	1	no	—
183	C₂×C₄	—	4	1	no	—
184	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
185	C₂×C₄	C₂²×C₈×C₅₃	1	1	yes	characteristic
186	C₂×C₄	#131×C₅₃	1	1	yes	characteristic
187	C₈	—	2	2	no	—
188	C₈	—	2	1	no	—
189	C₈	—	2	1	no	—
190	C₈	—	4	1	no	—
191	C₈	—	4	1	no	—
192	C₈	M₃₂×C₅₃	1	2	yes	complemented
193	C₈	C₂×SD₁₆×C₅₃	1	1	yes	characteristic
194	C₈	#135×C₅₃	1	1	yes	characteristic
195	C₂²	—	4	1	no	—
196	C₂²	—	4	1	no	—
197	C₂²	—	4	1	no	—
198	C₂²	C₈×D₄×C₅₃	1	1	yes	characteristic
199	C₄	—	2	1	no	—
200	C₄	—	4	1	no	—
201	C₄	—	4	1	no	—
202	C₄	—	4	1	no	—
203	C₄	C₄×SD₁₆×C₅₃	1	1	yes	characteristic
204	C₄	C₅₃×#443	1	1	yes	characteristic
205	C₄	C₂×M₃₂×C₅₃	1	1	yes	characteristic
206	C₄	C₅₃×#504	1	1	yes	characteristic
207	C₂	—	8	1	no	—
208	C₂	C₈×SD₁₆×C₅₃	1	1	yes	characteristic
209	C₂	C₅₃×#2169	1	1	yes	characteristic
210	C₂	C₅₃×#2170	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈×C₅₃	C₂×D₄
commutator	C₂×C₄	C₂²×C₈×C₅₃
Frattini, cosocle	C₄×C₈	C₂³×C₅₃
socle	C₂²×C₅₃	C₈×D₄
prime radical	C₂×D₄×C₅₃	C₂×C₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₅₃ ◁ C₄×C₁₆×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅₃×#2183	C₂	semidirect
4. C₅₃×#1367	C₂	semidirect
5. C₅₃×Q₈⋊C₁₆	C₂	semidirect
7. C₈×C₁₆×C₅₃	C₂	semidirect
67. C₁₆×C₅₃	SD₁₆	semidirect
86. C₈×C₅₃	M₃₂	semidirect
89. #12051	C₅₃	direct (N × H)
102. #2183	C₂×C₅₃	semidirect
105. #1367	C₂×C₅₃	semidirect
106. Q₈⋊C₁₆	C₂×C₅₃	semidirect
108. C₈×C₁₆	C₂×C₅₃	semidirect
128. C₅₃	#12051	direct (N × H)
173. C₁₆	SD₁₆×C₅₃	semidirect
192. C₈	M₃₂×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#2183	2
C₅₃×#2169	2
C₈×SD₁₆×C₅₃	2
C₅₃×#1367	2
C₅₃×Q₈⋊C₁₆	2
C₅₃×D₄⋊C₁₆	2
C₈×C₁₆×C₅₃	2
#12051	53

Subgroups — C53×#12051 #27219563

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#12051 #27219563