Subgroups — #27290009

Order 13568.

746 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810661	C₂	1	2	yes	maximal
2	#20274448	C₂	1	2	yes	maximal
3	C₂×#16810380	C₂	1	2	yes	maximal
4	C₂×C₅₃×#405	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133176	C₂²	1	4	yes	—
6	#16810661	—	2	4	no	—
7	#16810661	C₄	1	8	yes	complemented
8	C₂×#13133151	—	2	2	no	—
9	C₂×#13133149	—	2	2	no	—
10	#16810390	—	2	4	no	—
11	#16810380	—	2	4	no	—
12	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
13	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
15	C₅₃×#405	—	2	2	no	—
16	C₅₃×#405	—	2	2	no	—
17	C₂×#97×C₅₃	C₄	1	2	yes	complemented
18	C₂×C₄×C₈×C₅₃	C₄	1	2	yes	complemented
19	#13133176	—	2	4	no	—
20	#13133176	—	2	4	no	—
21	#13133176	—	2	4	no	—
22	#13133176	C₂×C₄	1	8	yes	—
23	C₂×C₄⋊Dic₅₃	—	2	4	no	—
24	C₂×C₂²⋊Dic₅₃	—	2	4	no	—
25	C₂×C₄×Dic₅₃	—	2	4	no	—
26	#13133151	—	2	8	no	—
27	#13133149	—	2	8	no	—
28	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
29	C₂×M₁₆×C₅₃	—	2	2	no	—
30	C₂×M₁₆×C₅₃	—	2	2	no	—
31	C₂×M₁₆×C₅₃	D₄	1	2	yes	—
32	C₂×M₁₆×C₅₃	Q₈	1	2	yes	—
33	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
34	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
35	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₅₃	D₄	1	2	yes	—
38	C₂²×C₈×C₅₃	D₄	1	2	yes	—
39	C₂²×C₈×C₅₃	D₄	1	2	yes	—
40	C₂²×C₈×C₅₃	Q₈	1	2	yes	—
41	#117×C₅₃	—	2	2	no	—
42	#117×C₅₃	—	2	2	no	—
43	C₂×C₄⋊C₄×C₅₃	Q₈	1	2	yes	—
44	C₂×C₂²⋊C₄×C₅₃	D₄	1	2	yes	—
45	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
46	#97×C₅₃	—	2	4	no	—
47	C₄×C₈×C₅₃	—	2	2	no	—
48	C₄×C₈×C₅₃	—	2	2	no	—
49	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
51	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
53	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
54	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
57	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	#41	1	2	yes	—
60	C₂²×C₄×C₅₃	#41	1	2	yes	—
61	C₂²×Dic₅₃	—	4	4	no	—
62	M₁₆×C₅₃	—	2	4	no	—
63	M₁₆×C₅₃	—	2	4	no	—
64	C₂×C₈×C₅₃	—	2	2	no	—
65	C₂×C₈×C₅₃	—	2	2	no	—
66	C₂×C₈×C₅₃	C₄²	1	2	yes	—
67	C₂×C₈×C₅₃	C₄²	1	2	yes	—
68	C₂×C₈×C₅₃	C₂²⋊C₄	1	4	yes	—
69	C₂×C₈×C₅₃	C₂²⋊C₄	1	4	yes	—
70	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
73	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
74	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
75	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
76	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
77	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
78	C₄⋊C₄×C₅₃	—	2	4	no	—
79	C₂²⋊C₄×C₅₃	—	2	4	no	—
80	C₄²×C₅₃	—	2	2	no	—
81	C₄²×C₅₃	—	2	2	no	—
82	C₂²⋊Dic₅₃	—	2	8	no	—
83	C₂²⋊Dic₅₃	—	4	4	no	—
84	C₄⋊Dic₅₃	—	2	8	no	—
85	C₄⋊Dic₅₃	—	4	4	no	—
86	C₄×Dic₅₃	—	2	8	no	—
87	C₄×Dic₅₃	—	4	4	no	—
88	C₂³×C₅₃	—	4	2	no	—
89	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
90	C₂³×C₅₃	C₄×D₄	1	2	yes	—
91	C₂³×C₅₃	#121	1	2	yes	—
92	C₂³×C₅₃	#127	1	1	yes	characteristic
93	C₂³×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
94	C₂×C₄×C₅₃	—	2	4	no	—
95	C₂×C₄×C₅₃	—	4	4	no	—
96	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
97	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
98	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
100	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
103	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
104	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
105	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
106	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
107	C₂×C₄×C₅₃	#121	1	2	yes	—
108	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
109	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
110	C₂×C₄×C₅₃	#124	1	2	yes	—
111	C₂×C₄×C₅₃	#125	1	2	yes	—
112	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
113	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
114	C₂×Dic₅₃	—	4	8	no	—
115	C₂×Dic₅₃	—	8	8	no	—
116	C₈×C₅₃	—	2	4	no	—
117	C₈×C₅₃	#95	1	2	yes	—
118	C₈×C₅₃	#95	1	4	yes	—
119	C₈×C₅₃	#95	1	2	yes	—
120	#10866	—	53	1	no	maximal, Sylow
121	C₂²×C₅₃	—	2	4	no	—
122	C₂²×C₅₃	—	2	2	no	—
123	C₂²×C₅₃	—	4	2	no	—
124	C₂²×C₅₃	—	4	1	no	—
125	C₂²×C₅₃	—	4	1	no	—
126	C₂²×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
127	C₂²×C₅₃	#380	1	1	yes	characteristic
128	C₂²×C₅₃	#384	1	1	yes	characteristic
129	C₂²×C₅₃	#389	1	2	yes	—
130	C₂²×C₅₃	#390	1	2	yes	—
131	C₄×C₅₃	—	4	4	no	—
132	C₄×C₅₃	C₂×#95	1	1	yes	characteristic
133	C₄×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
134	C₄×C₅₃	#380	1	1	yes	characteristic
135	C₄×C₅₃	#382	1	1	yes	characteristic
136	C₄×C₅₃	#389	1	2	yes	—
137	C₄×C₅₃	#391	1	2	yes	—
138	Dic₅₃	—	8	8	no	—
139	C₂×#427	—	53	2	no	—
140	C₂×#405	D₅₃	1	1	yes	characteristic
141	#1740	—	53	2	no	—
142	C₂×H(3,ℤ/4)	—	53	2	no	—
143	C₂×C₅₃	—	2	2	no	—
144	C₂×C₅₃	—	2	1	no	—
145	C₂×C₅₃	—	2	1	no	—
146	C₂×C₅₃	—	2	2	no	—
147	C₂×C₅₃	C₄⋊#95	1	1	yes	characteristic
148	C₂×C₅₃	#1812	1	2	yes	—
149	C₂²×M₁₆	C₂×D₅₃	1	1	yes	characteristic
150	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
151	C₂×#117	—	53	4	no	—
152	C₂×#117	C₂×D₅₃	1	1	yes	characteristic
153	#427	—	53	8	no	—
154	#427	—	106	4	no	—
155	C₂×#107	—	106	2	no	—
156	C₂×#106	—	106	2	no	—
157	#405	—	2	2	no	—
158	#405	—	2	2	no	—
159	C₂×#97	Dic₅₃	1	2	yes	complemented
160	C₂×C₄×C₈	Dic₅₃	1	2	yes	complemented
161	#342	—	106	4	no	—
162	H(3,ℤ/4)	—	106	4	no	—
163	C₅₃	#10866	1	1	yes	characteristic, complemented, Sylow
164	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
165	C₂×M₁₆	—	2	2	no	—
166	C₂×M₁₆	—	2	2	no	—
167	C₂×M₁₆	Dic₁₀₆	1	2	yes	—
168	C₂×M₁₆	D₂₁₂	1	2	yes	—
169	C₂²×C₈	Dic₁₀₆	1	2	yes	—
170	C₂²×C₈	C₄×D₅₃	1	4	yes	—
171	C₂²×C₈	D₂₁₂	1	2	yes	—
172	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
173	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
174	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
175	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
176	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
177	#117	—	2	2	no	—
178	#117	—	2	2	no	—
179	#117	—	53	8	no	—
180	#117	—	106	4	no	—
181	#117	—	106	4	no	—
182	#117	—	106	4	no	—
183	C₂×C₄⋊C₄	—	106	4	no	—
184	C₂×C₄⋊C₄	Dic₁₀₆	1	2	yes	—
185	C₂×C₂²⋊C₄	—	106	4	no	—
186	C₂×C₂²⋊C₄	D₂₁₂	1	2	yes	—
187	C₂×C₄²	—	106	4	no	—
188	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
189	#107	—	106	8	no	—
190	#106	—	106	8	no	—
191	#97	—	2	4	no	—
192	C₄×C₈	—	2	2	no	—
193	C₄×C₈	—	2	2	no	—
194	C₂⁴	C₂×D₂₁₂	1	1	yes	characteristic
195	C₂²×C₄	—	212	4	no	—
196	C₂²×C₄	C₄⋊Dic₅₃	1	4	yes	—
197	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
198	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
199	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
200	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
201	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
202	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
203	C₂²×C₄	#11719945	1	2	yes	—
204	C₂²×C₄	#11719945	1	2	yes	—
205	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
206	C₂²×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
207	M₁₆	—	2	4	no	—
208	M₁₆	—	2	4	no	—
209	C₂×C₈	—	2	2	no	—
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	C₄×Dic₅₃	1	2	yes	—
212	C₂×C₈	C₄×Dic₅₃	1	2	yes	—
213	C₂×C₈	#11719919	1	4	yes	—
214	C₂×C₈	#11719919	1	4	yes	—
215	C₂×C₈	C₄⋊Dic₅₃	1	4	yes	—
216	C₂×C₈	C₄⋊Dic₅₃	1	2	yes	—
217	C₂×C₈	C₄⋊Dic₅₃	1	2	yes	—
218	C₂×C₈	#11719921	1	4	yes	—
219	C₂×C₈	#11719921	1	4	yes	—
220	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
221	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
222	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
223	C₄⋊C₄	—	2	4	no	—
224	C₄⋊C₄	—	106	8	no	—
225	C₄⋊C₄	—	212	4	no	—
226	C₂²⋊C₄	—	2	4	no	—
227	C₂²⋊C₄	—	106	8	no	—
228	C₂²⋊C₄	—	212	4	no	—
229	C₄²	—	2	2	no	—
230	C₄²	—	2	2	no	—
231	C₄²	—	106	8	no	—
232	C₄²	—	212	4	no	—
233	C₂³	—	4	2	no	—
234	C₂³	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
235	C₂³	C₄×D₂₁₂	1	2	yes	—
236	C₂³	#13133191	1	2	yes	—
237	C₂³	#13133214	1	1	yes	characteristic
238	C₂³	#13133217	1	1	yes	characteristic
239	C₂×C₄	—	2	4	no	—
240	C₂×C₄	—	4	4	no	—
241	C₂×C₄	—	424	8	no	—
242	C₂×C₄	—	212	8	no	—
243	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
244	C₂×C₄	C₂×#11719921	1	2	yes	—
245	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
246	C₂×C₄	C₂×#11719919	1	2	yes	—
247	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
248	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
249	C₂×C₄	#13133176	1	1	yes	characteristic
250	C₂×C₄	#13133176	1	1	yes	characteristic
251	C₂×C₄	C₄×D₂₁₂	1	2	yes	—
252	C₂×C₄	C₄×Dic₁₀₆	1	2	yes	—
253	C₂×C₄	C₄×Dic₁₀₆	1	2	yes	—
254	C₂×C₄	#13133191	1	2	yes	—
255	C₂×C₄	#13133196	1	2	yes	—
256	C₂×C₄	#13133196	1	2	yes	—
257	C₂×C₄	#13133205	1	2	yes	—
258	C₂×C₄	#13133209	1	2	yes	—
259	C₂×C₄	#13133217	1	1	yes	characteristic
260	C₂×C₄	#13133217	1	1	yes	characteristic
261	C₈	—	2	4	no	—
262	C₈	#13133124	1	2	yes	—
263	C₈	#13133124	1	4	yes	—
264	C₈	#13133124	1	2	yes	—
265	C₂²	—	2	2	no	—
266	C₂²	—	2	4	no	—
267	C₂²	—	4	2	no	—
268	C₂²	—	4	1	no	—
269	C₂²	—	4	1	no	—
270	C₂²	C₄×C₄⋊Dic₅₃	1	1	yes	characteristic
271	C₂²	#16810483	1	1	yes	characteristic
272	C₂²	#16810493	1	1	yes	characteristic
273	C₂²	#16810521	1	2	yes	—
274	C₂²	#16810525	1	2	yes	—
275	C₄	—	4	4	no	—
276	C₄	—	424	8	no	—
277	C₄	C₂×#13133124	1	1	yes	characteristic
278	C₄	C₄×C₄⋊Dic₅₃	1	1	yes	characteristic
279	C₄	#16810483	1	1	yes	characteristic
280	C₄	#16810489	1	1	yes	characteristic
281	C₄	#16810521	1	2	yes	—
282	C₄	#16810529	1	2	yes	—
283	C₂	—	2	2	no	—
284	C₂	—	2	2	no	—
285	C₂	—	2	1	no	—
286	C₂	—	2	1	no	—
287	C₂	#20272945	1	1	yes	characteristic
288	C₂	#20274761	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#13133176
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₂×C₅₃×#405	C₂
socle	C₂²×C₅₃	#380
prime radical	C₂⁴×C₅₃	C₂×D₄
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	C₂×#405	D₅₃
hypocenter	C₅₃	#10866
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₁₆ ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
7. #16810661	C₄	semidirect
17. C₂×#97×C₅₃	C₄	semidirect
18. C₂×C₄×C₈×C₅₃	C₄	semidirect
159. C₂×#97	Dic₅₃	semidirect
160. C₂×C₄×C₈	Dic₅₃	semidirect
163. C₅₃	#10866	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810661	2
#20274448	2
C₂×#16810380	2
C₂×C₅₃×#405	2
#10866	53