Subgroups — #27300437

Order 13568.

366 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810441	C₂	1	2	yes	maximal
2	C₄×C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
3	#20272793	C₂	1	1	yes	maximal, characteristic
4	#20272792	C₂	1	2	yes	maximal
5	C₅₃×#1374	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11719908	C₂²	1	2	yes	—
7	C₂×C₄×#88322	C₄	1	1	yes	characteristic
8	C₂×C₈×Dic₅₃	C₄	1	1	yes	characteristic
9	C₂×C₈×Dic₅₃	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
11	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
12	C₄²×Dic₅₃	C₄	1	1	yes	characteristic
13	#16810441	—	2	4	no	—
14	C₂²×C₁₆×C₅₃	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
17	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×Dic₅₃	C₈	1	1	yes	characteristic
19	C₂×C₄×Dic₅₃	C₈	1	1	yes	characteristic
20	C₂×C₄×Dic₅₃	C₈	1	1	yes	characteristic
21	C₂×C₄×Dic₅₃	C₈	1	1	yes	characteristic
22	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂×#11719908	—	2	2	no	—
25	C₂×#11719908	—	2	2	no	—
26	C₂×#11719908	—	2	2	no	—
27	C₄×#88322	—	2	2	no	—
28	C₈×Dic₅₃	—	2	2	no	—
29	C₈×Dic₅₃	—	2	2	no	—
30	C₈×Dic₅₃	—	2	2	no	—
31	C₈×Dic₅₃	D₄	1	2	yes	—
32	C₈×Dic₅₃	Q₈	1	2	yes	—
33	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
35	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₁₆×C₅₃	—	2	2	no	—
37	C₂×C₁₆×C₅₃	—	2	2	no	—
38	C₂×C₁₆×C₅₃	—	2	2	no	—
39	C₄×C₈×C₅₃	—	2	2	no	—
40	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
41	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
42	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
43	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
44	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
45	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
46	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
47	C₂×C₈×C₅₃	—	2	1	no	—
48	C₂×C₈×C₅₃	—	2	1	no	—
49	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
50	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
51	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₈×C₅₃	#41	1	2	yes	—
53	C₂×C₈×C₅₃	#41	1	2	yes	—
54	C₄²×C₅₃	—	2	1	no	—
55	C₄²×C₅₃	—	2	1	no	—
56	C₄×Dic₅₃	—	2	2	no	—
57	C₄×Dic₅₃	—	2	1	no	—
58	C₄×Dic₅₃	—	2	1	no	—
59	C₄×Dic₅₃	—	2	2	no	—
60	C₄×Dic₅₃	—	2	2	no	—
61	C₄×Dic₅₃	C₁₆	1	4	yes	complemented
62	C₄×Dic₅₃	C₄⋊C₄	1	2	yes	—
63	C₄×Dic₅₃	M₁₆	1	2	yes	—
64	C₂×#88322	—	2	1	no	—
65	C₂×#88322	—	2	1	no	—
66	C₂×#88322	—	2	1	no	—
67	C₂×#88322	—	2	1	no	—
68	C₂×#88322	C₄⋊C₄	1	2	yes	—
69	C₂×#88322	M₁₆	1	2	yes	—
70	C₁₆×C₅₃	—	4	2	no	—
71	#11719908	—	4	2	no	—
72	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
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	—	2	1	no	—
77	C₂×C₄×C₅₃	C₂×C₁₆	1	2	yes	—
78	C₂×C₄×C₅₃	M₃₂	1	2	yes	—
79	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
81	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
82	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
83	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
84	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
85	C₂×Dic₅₃	—	2	1	no	—
86	C₂×Dic₅₃	—	2	1	no	—
87	C₂×Dic₅₃	—	2	1	no	—
88	C₂×Dic₅₃	—	2	1	no	—
89	C₂×Dic₅₃	—	2	1	no	—
90	C₂×Dic₅₃	—	2	1	no	—
91	C₂×Dic₅₃	C₄⋊C₈	1	2	yes	—
92	C₂×Dic₅₃	C₄⋊C₈	1	2	yes	—
93	C₂×Dic₅₃	C₂×C₁₆	1	2	yes	—
94	C₂×Dic₅₃	C₂×C₁₆	1	2	yes	—
95	C₂×Dic₅₃	M₃₂	1	2	yes	—
96	C₂×Dic₅₃	M₃₂	1	2	yes	—
97	C₈×C₅₃	—	2	2	no	—
98	C₈×C₅₃	#124	1	2	yes	—
99	C₈×C₅₃	#125	1	2	yes	—
100	#88322	—	2	2	no	—
101	#88322	—	2	2	no	—
102	#12879	—	53	1	no	maximal, Sylow
103	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
104	C₂²×C₅₃	#431	1	1	yes	characteristic
105	C₂²×C₅₃	#431	1	1	yes	characteristic
106	C₂²×C₅₃	C₂²×C₁₆	1	1	yes	characteristic
107	C₂²×C₅₃	C₂×M₃₂	1	1	yes	characteristic
108	C₂²×C₅₃	C₂×M₃₂	1	1	yes	characteristic
109	C₂²×C₅₃	C₂×M₃₂	1	1	yes	characteristic
110	C₄×C₅₃	—	2	2	no	—
111	C₄×C₅₃	—	2	2	no	—
112	C₄×C₅₃	#382	1	1	yes	characteristic
113	C₄×C₅₃	#423	1	1	yes	characteristic
114	C₄×C₅₃	#432	1	1	yes	characteristic
115	C₄×C₅₃	#432	1	1	yes	characteristic
116	Dic₅₃	—	2	2	no	—
117	Dic₅₃	—	2	2	no	—
118	Dic₅₃	—	2	2	no	—
119	Dic₅₃	C₄⋊C₁₆	1	4	yes	—
120	C₂×C₄⋊C₁₆	—	53	2	no	—
121	C₄²×C₈	—	53	1	no	—
122	#1374	—	53	2	no	—
123	#1374	—	53	1	no	—
124	#1374	D₅₃	1	1	yes	characteristic
125	C₂×C₅₃	#1831	1	1	yes	characteristic
126	C₂×C₅₃	C₂×C₄⋊C₁₆	1	1	yes	characteristic
127	C₂×C₅₃	#2150	1	1	yes	characteristic
128	C₂×C₅₃	#2162	1	1	yes	characteristic
129	C₂×C₅₃	#2162	1	1	yes	characteristic
130	C₂×C₅₃	#2163	1	1	yes	characteristic
131	C₂×C₅₃	#2163	1	1	yes	characteristic
132	C₂²×C₁₆	—	53	2	no	—
133	C₂²×C₁₆	C₂×D₅₃	1	2	yes	—
134	C₂×C₄×C₈	—	53	1	no	—
135	C₂×C₄×C₈	—	53	1	no	—
136	C₂×C₄×C₈	—	53	1	no	—
137	C₂×C₄×C₈	—	53	1	no	—
138	C₂×C₄×C₈	—	53	1	no	—
139	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
140	C₄³	—	53	1	no	—
141	C₄⋊C₁₆	—	106	4	no	—
142	C₅₃	#12879	1	1	yes	characteristic, complemented, Sylow
143	C₂²×C₈	—	53	1	no	—
144	C₂²×C₈	—	53	1	no	—
145	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
146	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
147	C₂×C₄²	—	53	1	no	—
148	C₂×C₄²	—	53	1	no	—
149	C₂×C₄²	—	53	1	no	—
150	C₂×C₄²	—	53	1	no	—
151	C₂×C₄²	—	53	1	no	—
152	C₂×C₄²	—	53	1	no	—
153	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
154	C₂×C₁₆	—	2	2	no	—
155	C₂×C₁₆	—	2	2	no	—
156	C₂×C₁₆	—	2	2	no	—
157	C₂×C₁₆	—	106	2	no	—
158	C₂×C₁₆	—	106	2	no	—
159	C₂×C₁₆	—	106	2	no	—
160	C₄×C₈	—	2	2	no	—
161	C₄×C₈	—	53	2	no	—
162	C₄×C₈	—	53	2	no	—
163	C₄×C₈	—	106	2	no	—
164	C₄×C₈	—	106	2	no	—
165	C₄×C₈	—	106	2	no	—
166	C₄×C₈	—	106	2	no	—
167	C₂²×C₄	—	53	1	no	—
168	C₂²×C₄	—	53	1	no	—
169	C₂²×C₄	—	53	1	no	—
170	C₂²×C₄	—	53	1	no	—
171	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
172	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
173	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
174	C₂×C₈	—	2	1	no	—
175	C₂×C₈	—	2	1	no	—
176	C₂×C₈	—	53	2	no	—
177	C₂×C₈	—	53	2	no	—
178	C₂×C₈	—	106	1	no	—
179	C₂×C₈	—	106	1	no	—
180	C₂×C₈	—	106	1	no	—
181	C₂×C₈	—	106	1	no	—
182	C₂×C₈	#11719912	1	2	yes	—
183	C₂×C₈	#11719945	1	2	yes	—
184	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
185	C₂×C₈	#11719947	1	2	yes	—
186	C₂×C₈	Q₈×D₅₃	1	1	yes	characteristic
187	C₄²	—	2	1	no	—
188	C₄²	—	2	1	no	—
189	C₄²	—	53	4	no	—
190	C₄²	—	53	2	no	—
191	C₄²	—	53	2	no	—
192	C₄²	—	106	2	no	—
193	C₄²	—	106	2	no	—
194	C₄²	—	106	2	no	—
195	C₄²	—	106	1	no	—
196	C₄²	—	106	1	no	—
197	C₁₆	—	4	2	no	—
198	C₁₆	—	212	2	no	—
199	C₂³	C₂×C₈×D₅₃	1	1	yes	characteristic
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	53	2	no	—
205	C₂×C₄	—	53	2	no	—
206	C₂×C₄	—	53	2	no	—
207	C₂×C₄	—	53	2	no	—
208	C₂×C₄	—	53	2	no	—
209	C₂×C₄	—	53	2	no	—
210	C₂×C₄	—	106	1	no	—
211	C₂×C₄	—	106	1	no	—
212	C₂×C₄	—	106	1	no	—
213	C₂×C₄	—	106	1	no	—
214	C₂×C₄	—	106	1	no	—
215	C₂×C₄	—	106	1	no	—
216	C₂×C₄	C₁₆×D₅₃	1	2	yes	—
217	C₂×C₄	#13133156	1	2	yes	—
218	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
219	C₂×C₄	#13133173	1	1	yes	characteristic
220	C₂×C₄	#13133174	1	1	yes	characteristic
221	C₂×C₄	M₁₆×D₅₃	1	1	yes	characteristic
222	C₂×C₄	M₁₆×D₅₃	1	1	yes	characteristic
223	C₂×C₄	C₂×#11719912	1	1	yes	characteristic
224	C₈	—	2	2	no	—
225	C₈	—	106	2	no	—
226	C₈	—	106	2	no	—
227	C₈	#13133203	1	2	yes	—
228	C₈	#13133207	1	2	yes	—
229	C₂²	C₄⋊C₈×D₅₃	1	1	yes	characteristic
230	C₂²	#16810673	1	1	yes	characteristic
231	C₂²	#16810674	1	1	yes	characteristic
232	C₂²	C₂×C₁₆×D₅₃	1	1	yes	characteristic
233	C₂²	M₃₂×D₅₃	1	1	yes	characteristic
234	C₂²	M₃₂×D₅₃	1	1	yes	characteristic
235	C₂²	C₂×#13133156	1	1	yes	characteristic
236	C₄	—	2	2	no	—
237	C₄	—	2	2	no	—
238	C₄	—	53	4	no	—
239	C₄	—	106	2	no	—
240	C₄	—	106	2	no	—
241	C₄	—	106	2	no	—
242	C₄	#16810487	1	1	yes	characteristic
243	C₄	#16810645	1	1	yes	characteristic
244	C₄	#16810677	1	1	yes	characteristic
245	C₄	#16810678	1	1	yes	characteristic
246	C₂	#20274830	1	1	yes	characteristic
247	C₂	C₄⋊C₁₆×D₅₃	1	1	yes	characteristic
248	C₂	#20276469	1	1	yes	characteristic
249	C₂	#20276511	1	1	yes	characteristic
250	C₂	#20276512	1	1	yes	characteristic
251	C₂	#20276515	1	1	yes	characteristic
252	C₂	#20276516	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂²×C₁₆
Fitting	C₅₃×#1374	C₂
socle, prime radical	C₂³×C₅₃	C₂²×C₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#1374	D₅₃
hypocenter	C₅₃	#12879
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₈ ◁ #1374

Complements

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

N	H ≅ G/N	type
61. C₄×Dic₅₃	C₁₆	semidirect
142. C₅₃	#12879	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810441	2
C₄×C₈×Dic₅₃	2
#20272793	2
#20272792	2
C₅₃×#1374	2
#12879	53