Subgroups — #27289261

Order 13568.

842 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810578	C₂	1	2	yes	maximal
2	C₂×#16810381	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810380	C₂	1	1	yes	maximal, characteristic
4	C₂×#16810379	C₂	1	2	yes	maximal
5	C₂×C₅₃×#405	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11719917	C₂²	1	2	yes	—
7	C₂×#13133176	C₂²	1	2	yes	—
8	#16810578	—	2	4	no	—
9	#16810578	—	2	4	no	—
10	C₂×#13133127	C₄	1	4	yes	complemented
11	C₂×C₈×Dic₅₃	C₄	1	4	yes	complemented
12	#16810381	C₄	1	4	yes	complemented
13	#16810380	C₄	1	4	yes	complemented
14	#16810379	C₄	1	8	yes	complemented
15	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
16	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
18	C₅₃×#405	—	2	4	no	—
19	C₂×#97×C₅₃	C₄	1	2	yes	complemented
20	C₂×C₄×C₈×C₅₃	C₄	1	2	yes	complemented
21	C₂×#11719917	—	2	2	no	—
22	C₂×#11719917	—	2	2	no	—
23	C₂×#11719917	—	2	2	no	—
24	C₂×#11719917	—	2	2	no	—
25	C₂×#11719917	C₂×C₄	1	4	yes	—
26	C₂×#11719917	C₂×C₄	1	4	yes	—
27	C₂²×#88322	C₂×C₄	1	4	yes	—
28	#13133176	—	2	2	no	—
29	#13133176	—	2	2	no	—
30	#13133176	C₂×C₄	1	4	yes	—
31	#13133176	C₂×C₄	1	4	yes	—
32	C₂×C₄⋊Dic₅₃	—	2	2	no	—
33	C₂×C₂²⋊Dic₅₃	—	2	2	no	—
34	C₂×C₄×Dic₅₃	C₂×C₄	1	4	yes	—
35	#13133127	—	2	8	no	—
36	C₈×Dic₅₃	—	2	8	no	—
37	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
38	C₂×M₁₆×C₅₃	—	2	2	no	—
39	C₂×M₁₆×C₅₃	—	2	2	no	—
40	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
41	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
42	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
43	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
44	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
45	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
46	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₈×C₅₃	D₄	1	2	yes	—
48	C₂²×C₈×C₅₃	D₄	1	4	yes	—
49	C₂²×C₈×C₅₃	Q₈	1	2	yes	—
50	#117×C₅₃	—	2	2	no	—
51	#117×C₅₃	C₂×C₄	1	4	yes	—
52	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
53	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
54	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
55	#97×C₅₃	—	2	4	no	—
56	C₄×C₈×C₅₃	—	2	4	no	—
57	C₂⁴×C₅₃	#41	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	—	2	2	no	—
59	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
60	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
67	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
68	C₂²×C₄×C₅₃	#41	1	2	yes	—
69	C₂²×Dic₅₃	—	2	4	no	—
70	M₁₆×C₅₃	—	2	4	no	—
71	M₁₆×C₅₃	—	2	4	no	—
72	C₂×C₈×C₅₃	—	2	2	no	—
73	C₂×C₈×C₅₃	—	2	2	no	—
74	C₂×C₈×C₅₃	C₄²	1	4	yes	—
75	C₂×C₈×C₅₃	C₄²	1	2	yes	—
76	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
77	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
78	C₂×C₈×C₅₃	C₂²⋊C₄	1	4	yes	—
79	C₂×C₈×C₅₃	C₂²⋊C₄	1	4	yes	—
80	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
81	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
82	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
83	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
84	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
85	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
86	C₄⋊C₄×C₅₃	—	2	4	no	—
87	C₂²⋊C₄×C₅₃	—	2	4	no	—
88	C₄²×C₅₃	—	2	2	no	—
89	C₄²×C₅₃	C₄²	1	4	yes	—
90	C₂²⋊Dic₅₃	—	2	4	no	—
91	C₂²⋊Dic₅₃	—	2	4	no	—
92	C₄⋊Dic₅₃	—	2	4	no	—
93	C₄⋊Dic₅₃	—	2	4	no	—
94	C₄×Dic₅₃	—	2	4	no	—
95	C₄×Dic₅₃	C₄²	1	8	yes	—
96	#11719917	—	2	4	no	—
97	#11719917	—	2	4	no	—
98	#11719917	—	2	4	no	—
99	#11719917	—	2	4	no	—
100	C₂×#88322	—	2	4	no	—
101	C₂×#88322	—	2	4	no	—
102	C₂×#88322	C₄²	1	8	yes	—
103	C₂³×C₅₃	—	4	2	no	—
104	C₂³×C₅₃	#117	1	2	yes	—
105	C₂³×C₅₃	#117	1	1	yes	characteristic
106	C₂³×C₅₃	C₄×D₄	1	2	yes	—
107	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
108	C₂³×C₅₃	C₄×Q₈	1	1	yes	characteristic
109	C₂×C₄×C₅₃	—	2	4	no	—
110	C₂×C₄×C₅₃	—	4	4	no	—
111	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
112	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
113	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
114	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
115	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
116	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
117	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
119	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₅₃	#117	1	2	yes	—
121	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
122	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
123	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
124	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
125	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
126	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
127	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
128	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
129	C₂×Dic₅₃	—	2	8	no	—
130	C₂×Dic₅₃	—	4	8	no	—
131	C₈×C₅₃	—	2	4	no	—
132	C₈×C₅₃	#95	1	4	yes	—
133	C₈×C₅₃	#95	1	2	yes	—
134	C₈×C₅₃	#95	1	2	yes	—
135	#88322	—	2	8	no	—
136	#10699	—	53	1	no	maximal, Sylow
137	C₂²×C₅₃	—	2	4	no	—
138	C₂²×C₅₃	—	2	2	no	—
139	C₂²×C₅₃	—	4	2	no	—
140	C₂²×C₅₃	—	4	1	no	—
141	C₂²×C₅₃	—	4	1	no	—
142	C₂²×C₅₃	#376	1	1	yes	characteristic
143	C₂²×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
144	C₂²×C₅₃	C₄×C₂²⋊C₄	1	2	yes	—
145	C₂²×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
146	C₂²×C₅₃	C₄×C₄⋊C₄	1	2	yes	—
147	C₄×C₅₃	—	4	4	no	—
148	C₄×C₅₃	C₄³	1	1	yes	characteristic
149	C₄×C₅₃	C₂×#95	1	1	yes	characteristic
150	C₄×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
151	C₄×C₅₃	C₄×C₂²⋊C₄	1	2	yes	—
152	C₄×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
153	C₄×C₅₃	C₄×C₄⋊C₄	1	2	yes	—
154	Dic₅₃	—	4	8	no	—
155	C₂×#405	—	53	2	no	—
156	C₂×#405	D₅₃	1	1	yes	characteristic
157	C₂×#338	—	53	1	no	—
158	C₂×H(3,ℤ/4)	—	53	1	no	—
159	C₂×H(3,ℤ/4)	—	53	2	no	—
160	C₂×C₅₃	—	2	2	no	—
161	C₂×C₅₃	—	2	1	no	—
162	C₂×C₅₃	—	2	1	no	—
163	C₂×C₅₃	—	2	2	no	—
164	C₂×C₅₃	C₄×#95	1	1	yes	characteristic
165	C₂×C₅₃	#1747	1	2	yes	—
166	C₂²×M₁₆	—	53	2	no	—
167	C₂²×M₁₆	C₂×D₅₃	1	1	yes	characteristic
168	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
169	C₂×#117	—	53	2	no	—
170	C₂×#117	C₂×D₅₃	1	1	yes	characteristic
171	#405	—	2	4	no	—
172	#405	—	106	4	no	—
173	#405	—	106	4	no	—
174	C₂×#97	—	53	4	no	—
175	C₂×#97	Dic₅₃	1	2	yes	complemented
176	C₂×C₄×C₈	—	53	4	no	—
177	C₂×C₄×C₈	Dic₅₃	1	2	yes	complemented
178	#338	—	53	4	no	—
179	H(3,ℤ/4)	—	53	4	no	—
180	H(3,ℤ/4)	—	53	8	no	—
181	C₅₃	#10699	1	1	yes	characteristic, complemented, Sylow
182	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
183	C₂×M₁₆	—	2	2	no	—
184	C₂×M₁₆	—	2	2	no	—
185	C₂×M₁₆	—	53	4	no	—
186	C₂×M₁₆	—	53	4	no	—
187	C₂×M₁₆	—	106	2	no	—
188	C₂×M₁₆	—	106	2	no	—
189	C₂×M₁₆	—	106	2	no	—
190	C₂×M₁₆	—	106	2	no	—
191	C₂×M₁₆	C₄×D₅₃	1	2	yes	—
192	C₂×M₁₆	C₄×D₅₃	1	2	yes	—
193	C₂²×C₈	—	53	4	no	—
194	C₂²×C₈	Dic₁₀₆	1	2	yes	—
195	C₂²×C₈	C₄×D₅₃	1	2	yes	—
196	C₂²×C₈	C₄×D₅₃	1	2	yes	—
197	C₂²×C₈	D₂₁₂	1	2	yes	—
198	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
199	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
200	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
201	C₂²×C₈	C₂²⋊D₅₃	1	4	yes	—
202	#117	—	2	2	no	—
203	#117	—	53	4	no	—
204	#117	—	53	4	no	—
205	#117	—	106	2	no	—
206	#117	—	106	2	no	—
207	#117	C₄×D₅₃	1	4	yes	—
208	C₂×C₄⋊C₄	—	2	1	no	—
209	C₂×C₄⋊C₄	—	106	2	no	—
210	C₂×C₂²⋊C₄	—	2	1	no	—
211	C₂×C₂²⋊C₄	—	106	2	no	—
212	C₂×C₄²	—	53	4	no	—
213	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
214	#97	—	2	4	no	—
215	#97	—	106	8	no	—
216	C₄×C₈	—	2	4	no	—
217	C₄×C₈	—	106	8	no	—
218	C₂⁴	#11719945	1	1	yes	characteristic
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	106	4	no	—
221	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
222	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
223	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
224	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
225	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
226	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
227	C₂²×C₄	#11719945	1	1	yes	characteristic
228	C₂²×C₄	#11719945	1	2	yes	—
229	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
230	C₂²×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
231	M₁₆	—	2	4	no	—
232	M₁₆	—	2	4	no	—
233	M₁₆	—	106	4	no	—
234	M₁₆	—	106	4	no	—
235	M₁₆	—	106	4	no	—
236	M₁₆	—	106	4	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	53	8	no	—
240	C₂×C₈	—	106	4	no	—
241	C₂×C₈	—	106	4	no	—
242	C₂×C₈	C₄×Dic₅₃	1	4	yes	—
243	C₂×C₈	C₄×Dic₅₃	1	1	yes	characteristic
244	C₂×C₈	C₄×Dic₅₃	1	2	yes	—
245	C₂×C₈	C₄×Dic₅₃	1	1	yes	characteristic
246	C₂×C₈	#11719919	1	4	yes	—
247	C₂×C₈	#11719919	1	4	yes	—
248	C₂×C₈	C₄⋊Dic₅₃	1	2	yes	—
249	C₂×C₈	C₄⋊Dic₅₃	1	2	yes	—
250	C₂×C₈	#11719921	1	4	yes	—
251	C₂×C₈	#11719921	1	4	yes	—
252	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
253	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
254	C₄⋊C₄	—	2	4	no	—
255	C₄⋊C₄	—	106	4	no	—
256	C₄⋊C₄	—	106	4	no	—
257	C₂²⋊C₄	—	2	4	no	—
258	C₂²⋊C₄	—	106	4	no	—
259	C₂²⋊C₄	—	106	4	no	—
260	C₄²	—	2	2	no	—
261	C₄²	—	53	8	no	—
262	C₄²	—	106	4	no	—
263	C₄²	C₄×Dic₅₃	1	4	yes	—
264	C₂³	—	4	2	no	—
265	C₂³	#13133174	1	2	yes	—
266	C₂³	#13133176	1	1	yes	characteristic
267	C₂³	C₄×C₂²⋊D₅₃	1	2	yes	—
268	C₂³	C₄×D₂₁₂	1	1	yes	characteristic
269	C₂³	C₄×Dic₁₀₆	1	1	yes	characteristic
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	4	4	no	—
272	C₂×C₄	—	106	8	no	—
273	C₂×C₄	—	212	8	no	—
274	C₂×C₄	C₄²×D₅₃	1	2	yes	—
275	C₂×C₄	C₄²×D₅₃	1	2	yes	—
276	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
277	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
278	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
279	C₂×C₄	C₂×#11719921	1	2	yes	—
280	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
281	C₂×C₄	C₂×#11719919	1	2	yes	—
282	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
283	C₂×C₄	#13133174	1	2	yes	—
284	C₂×C₄	#13133176	1	1	yes	characteristic
285	C₂×C₄	C₄×C₂²⋊D₅₃	1	2	yes	—
286	C₂×C₄	C₄×C₂²⋊D₅₃	1	2	yes	—
287	C₂×C₄	C₄×C₂²⋊D₅₃	1	2	yes	—
288	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
289	C₂×C₄	C₄×D₂₁₂	1	2	yes	—
290	C₂×C₄	C₄×Dic₁₀₆	1	2	yes	—
291	C₂×C₄	C₄×Dic₁₀₆	1	1	yes	characteristic
292	C₈	—	2	4	no	—
293	C₈	—	106	8	no	—
294	C₈	#13133124	1	4	yes	—
295	C₈	#13133124	1	2	yes	—
296	C₈	#13133124	1	2	yes	—
297	C₂²	—	2	2	no	—
298	C₂²	—	2	4	no	—
299	C₂²	—	4	2	no	—
300	C₂²	—	4	1	no	—
301	C₂²	—	4	1	no	—
302	C₂²	#16810469	1	1	yes	characteristic
303	C₂²	C₄×C₂²⋊Dic₅₃	1	1	yes	characteristic
304	C₂²	C₄×#11719921	1	2	yes	—
305	C₂²	C₄×C₄⋊Dic₅₃	1	1	yes	characteristic
306	C₂²	C₄×#11719919	1	2	yes	—
307	C₄	—	4	4	no	—
308	C₄	—	212	8	no	—
309	C₄	C₄²×Dic₅₃	1	1	yes	characteristic
310	C₄	C₂×#13133124	1	1	yes	characteristic
311	C₄	C₄×C₂²⋊Dic₅₃	1	1	yes	characteristic
312	C₄	C₄×#11719921	1	2	yes	—
313	C₄	C₄×C₄⋊Dic₅₃	1	1	yes	characteristic
314	C₄	C₄×#11719919	1	2	yes	—
315	C₂	—	2	1	no	—
316	C₂	—	2	2	no	—
317	C₂	—	2	1	no	—
318	C₂	—	2	2	no	—
319	C₂	C₄×#13133124	1	1	yes	characteristic
320	C₂	#20274470	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₄×Dic₅₃
commutator	C₄×C₅₃	C₄³
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₂×C₅₃×#405	C₂
socle	C₂²×C₅₃	#376
prime radical	C₂⁴×C₅₃	#41
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	C₂×#405	D₅₃
hypocenter	C₅₃	#10699
LCS	G ▷ C₄×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
10. C₂×#13133127	C₄	semidirect
11. C₂×C₈×Dic₅₃	C₄	semidirect
12. #16810381	C₄	semidirect
13. #16810380	C₄	semidirect
14. #16810379	C₄	semidirect
19. C₂×#97×C₅₃	C₄	semidirect
20. C₂×C₄×C₈×C₅₃	C₄	semidirect
175. C₂×#97	Dic₅₃	semidirect
177. C₂×C₄×C₈	Dic₅₃	semidirect
181. C₅₃	#10699	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810578	2
C₂×#16810381	2
C₂×#16810380	2
C₂×#16810379	2
C₂×C₅₃×#405	2
#10699	53