Subgroups — #27387514

Order 13568.

882 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133153	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810661	C₂	1	1	yes	maximal, characteristic
3	#20283278	C₂	1	2	yes	maximal
4	#20275468	C₂	1	4	yes	maximal, complemented
5	#20275463	C₂	1	4	yes	maximal, complemented
6	#20274425	C₂	1	2	yes	maximal
7	C₂×C₅₃×#405	C₂	1	1	yes	maximal, characteristic
8	C₂²×#11719917	C₂²	1	2	yes	—
9	C₂×#13133176	C₂²	1	2	yes	—
10	C₂×#13133153	—	2	4	no	—
11	C₂×#13133153	C₂²	1	2	yes	—
12	C₂×#13133153	C₂²	1	2	yes	—
13	#16810661	C₄	1	2	yes	complemented
14	#16810661	C₄	1	2	yes	complemented
15	#16810661	C₂²	1	2	yes	—
16	#16810661	C₂²	1	2	yes	—
17	C₂×#13133151	C₄	1	2	yes	complemented
18	C₂×#13133149	C₄	1	2	yes	complemented
19	#16810651	—	2	4	no	—
20	#16810649	C₂²	1	8	yes	—
21	#16810646	—	2	4	no	—
22	#16810383	C₂²	1	8	yes	—
23	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
24	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
25	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
26	C₅₃×#405	—	2	4	no	—
27	C₂×#97×C₅₃	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
29	C₂×#11719917	—	2	2	no	—
30	C₂×#11719917	—	2	4	no	—
31	C₂×#11719917	—	2	4	no	—
32	C₂×#11719917	C₂³	1	4	yes	—
33	C₂²×#88322	—	2	2	no	—
34	#13133176	—	2	2	no	—
35	#13133176	—	2	2	no	—
36	#13133176	C₂×C₄	1	4	yes	—
37	#13133176	C₂³	1	4	yes	—
38	C₂×C₄⋊Dic₅₃	C₂×C₄	1	4	yes	—
39	C₂×C₂²⋊Dic₅₃	—	2	2	no	—
40	C₂×C₄×Dic₅₃	—	2	2	no	—
41	#13133153	—	2	4	no	—
42	#13133153	—	2	4	no	—
43	#13133151	C₂×C₄	1	4	yes	—
44	#13133151	C₂×C₄	1	4	yes	—
45	#13133149	C₂×C₄	1	4	yes	—
46	#13133149	C₂×C₄	1	4	yes	—
47	#13133146	—	2	16	no	—
48	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
49	C₂×M₁₆×C₅₃	—	2	2	no	—
50	C₂×M₁₆×C₅₃	—	2	2	no	—
51	C₂×M₁₆×C₅₃	D₄	1	2	yes	—
52	C₂×M₁₆×C₅₃	D₄	1	2	yes	—
53	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
54	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
55	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
56	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
57	C₂²×C₈×C₅₃	D₄	1	4	yes	—
58	C₂²×C₈×C₅₃	Q₈	1	4	yes	—
59	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
60	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
61	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
62	#117×C₅₃	—	2	2	no	—
63	#117×C₅₃	Q₈	1	4	yes	—
64	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
65	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
66	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
67	#97×C₅₃	—	2	4	no	—
68	C₄×C₈×C₅₃	—	2	4	no	—
69	C₂⁴×C₅₃	#41	1	1	yes	characteristic
70	C₂²×C₄×C₅₃	—	2	2	no	—
71	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
77	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
78	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
79	C₂²×C₄×C₅₃	#41	1	2	yes	—
80	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
81	C₂²×Dic₅₃	—	2	4	no	—
82	M₁₆×C₅₃	—	2	4	no	—
83	M₁₆×C₅₃	—	2	4	no	—
84	C₂×C₈×C₅₃	—	2	2	no	—
85	C₂×C₈×C₅₃	—	2	2	no	—
86	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
87	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
88	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
89	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
90	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
95	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
96	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
97	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
98	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
99	C₄⋊C₄×C₅₃	—	2	4	no	—
100	C₂²⋊C₄×C₅₃	—	2	4	no	—
101	C₄²×C₅₃	—	2	2	no	—
102	C₄²×C₅₃	C₂×Q₈	1	4	yes	—
103	C₂²⋊Dic₅₃	—	2	4	no	—
104	C₂²⋊Dic₅₃	—	2	4	no	—
105	C₄⋊Dic₅₃	—	2	4	no	—
106	C₄⋊Dic₅₃	C₂²×C₄	1	8	yes	—
107	C₄×Dic₅₃	—	2	4	no	—
108	C₄×Dic₅₃	—	2	4	no	—
109	#11719917	—	2	4	no	—
110	#11719917	—	2	4	no	—
111	#11719917	—	4	4	no	—
112	C₂×#88322	—	2	4	no	—
113	C₂×#88322	—	4	4	no	—
114	C₂³×C₅₃	—	4	2	no	—
115	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
116	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
117	C₂³×C₅₃	C₂²⋊Q₈	1	2	yes	—
118	C₂³×C₅₃	#125	1	2	yes	—
119	C₂³×C₅₃	C₂×#41	1	1	yes	characteristic
120	C₂×C₄×C₅₃	—	2	4	no	—
121	C₂×C₄×C₅₃	—	4	4	no	—
122	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
123	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
124	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
125	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
126	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
127	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
128	C₂×C₄×C₅₃	#121	1	2	yes	—
129	C₂×C₄×C₅₃	#121	1	2	yes	—
130	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
131	C₂×C₄×C₅₃	#125	1	2	yes	—
132	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
133	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
134	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
135	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
136	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
137	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
138	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
139	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
140	C₂×Dic₅₃	—	2	8	no	—
141	C₂×Dic₅₃	—	8	4	no	—
142	C₈×C₅₃	—	2	4	no	—
143	C₈×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
144	C₈×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
145	C₈×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
146	C₈×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
147	#88322	—	4	4	no	—
148	#21163	—	53	1	no	maximal, Sylow
149	C₂²×C₅₃	—	2	2	no	—
150	C₂²×C₅₃	—	2	2	no	—
151	C₂²×C₅₃	—	2	2	no	—
152	C₂²×C₅₃	—	4	2	no	—
153	C₂²×C₅₃	—	4	2	no	—
154	C₂²×C₅₃	#389	1	2	yes	—
155	C₂²×C₅₃	#389	1	2	yes	—
156	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
157	C₂²×C₅₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
158	C₂²×C₅₃	C₂×#125	1	1	yes	characteristic
159	C₄×C₅₃	—	4	4	no	—
160	C₄×C₅₃	#389	1	2	yes	—
161	C₄×C₅₃	#389	1	2	yes	—
162	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
163	C₄×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
164	C₄×C₅₃	C₂×#121	1	1	yes	characteristic
165	C₄×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
166	Dic₅₃	—	4	8	no	—
167	C₂²×#108	—	53	1	no	—
168	C₂×#427	—	53	1	no	—
169	#2905	—	53	2	no	—
170	C₂×#405	D₅₃	1	1	yes	characteristic
171	#1946	—	53	4	no	—
172	#1945	—	53	4	no	—
173	#1733	—	53	2	no	—
174	C₂×C₅₃	—	2	1	no	—
175	C₂×C₅₃	—	2	1	no	—
176	C₂×C₅₃	—	2	2	no	—
177	C₂×C₅₃	—	2	2	no	—
178	C₂×C₅₃	C₂×#389	1	1	yes	characteristic
179	C₂×C₅₃	#2957	1	2	yes	—
180	C₂²×M₁₆	—	53	2	no	—
181	C₂²×M₁₆	C₂×D₅₃	1	1	yes	characteristic
182	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
183	C₂×#117	—	53	2	no	—
184	C₂×#117	C₂×D₅₃	1	1	yes	characteristic
185	C₂×#108	—	53	2	no	—
186	C₂×#108	—	53	2	no	—
187	C₂×#108	—	106	4	no	—
188	#427	—	53	2	no	—
189	#427	—	53	2	no	—
190	#427	—	53	2	no	—
191	#427	—	53	2	no	—
192	C₂×#107	—	53	2	no	—
193	C₂×#106	—	53	2	no	—
194	#424	—	53	8	no	—
195	#424	—	106	4	no	—
196	#423	—	106	4	no	—
197	#405	—	2	4	no	—
198	C₂×#97	C₂×D₅₃	1	2	yes	—
199	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
200	#339	—	53	8	no	—
201	C₅₃	#21163	1	1	yes	characteristic, complemented, Sylow
202	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
203	C₂×M₁₆	—	2	2	no	—
204	C₂×M₁₆	—	2	2	no	—
205	C₂×M₁₆	—	53	4	no	—
206	C₂×M₁₆	—	106	2	no	—
207	C₂×M₁₆	—	106	4	no	—
208	C₂×M₁₆	—	106	4	no	—
209	C₂×M₁₆	D₂₁₂	1	2	yes	—
210	C₂×M₁₆	D₂₁₂	1	2	yes	—
211	C₂²×C₈	—	106	2	no	—
212	C₂²×C₈	Dic₁₀₆	1	4	yes	—
213	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
214	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
215	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
216	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
217	C₂²×C₈	C₂²⋊D₅₃	1	4	yes	—
218	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
219	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
220	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
221	#117	—	2	2	no	—
222	#117	—	53	4	no	—
223	#117	—	53	4	no	—
224	#117	—	106	2	no	—
225	#117	—	106	2	no	—
226	#117	Dic₁₀₆	1	4	yes	—
227	C₂×C₄⋊C₄	—	2	1	no	—
228	C₂×C₄⋊C₄	—	53	4	no	—
229	C₂×C₂²⋊C₄	—	2	1	no	—
230	C₂×C₂²⋊C₄	—	106	2	no	—
231	C₂×C₄²	—	106	2	no	—
232	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
233	#108	—	106	4	no	—
234	#108	—	106	4	no	—
235	#107	—	53	4	no	—
236	#107	—	53	4	no	—
237	#106	—	53	4	no	—
238	#106	—	53	4	no	—
239	C₄⋊C₈	—	106	16	no	—
240	#97	—	2	4	no	—
241	C₄×C₈	—	2	4	no	—
242	C₂⁴	#11719945	1	1	yes	characteristic
243	C₂²×C₄	—	2	2	no	—
244	C₂²×C₄	—	106	4	no	—
245	C₂²×C₄	C₂×Dic₁₀₆	1	2	yes	—
246	C₂²×C₄	C₂×Dic₁₀₆	1	2	yes	—
247	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
248	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
249	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
250	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
251	C₂²×C₄	#11719945	1	1	yes	characteristic
252	C₂²×C₄	#11719945	1	2	yes	—
253	C₂²×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
254	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
255	M₁₆	—	2	4	no	—
256	M₁₆	—	2	4	no	—
257	M₁₆	—	106	4	no	—
258	M₁₆	—	106	4	no	—
259	M₁₆	—	212	4	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	106	4	no	—
263	C₂×C₈	—	212	4	no	—
264	C₂×C₈	#11719919	1	4	yes	—
265	C₂×C₈	#11719919	1	4	yes	—
266	C₂×C₈	#11719919	1	4	yes	—
267	C₂×C₈	#11719919	1	4	yes	—
268	C₂×C₈	C₂×Dic₁₀₆	1	2	yes	—
269	C₂×C₈	C₂×Dic₁₀₆	1	2	yes	—
270	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
271	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
272	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
273	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
274	C₂×C₈	C₂×D₂₁₂	1	4	yes	—
275	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
276	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
277	C₄⋊C₄	—	2	4	no	—
278	C₄⋊C₄	—	53	8	no	—
279	C₄⋊C₄	—	106	4	no	—
280	C₂²⋊C₄	—	2	4	no	—
281	C₂²⋊C₄	—	106	4	no	—
282	C₂²⋊C₄	—	106	4	no	—
283	C₄²	—	2	2	no	—
284	C₄²	—	106	4	no	—
285	C₄²	—	106	4	no	—
286	C₄²	C₂×Dic₁₀₆	1	4	yes	—
287	C₂³	—	4	2	no	—
288	C₂³	C₄×D₂₁₂	1	1	yes	characteristic
289	C₂³	C₄×D₂₁₂	1	1	yes	characteristic
290	C₂³	#13133196	1	2	yes	—
291	C₂³	#13133209	1	2	yes	—
292	C₂³	C₂×#11719945	1	1	yes	characteristic
293	C₂×C₄	—	2	4	no	—
294	C₂×C₄	—	4	4	no	—
295	C₂×C₄	—	106	8	no	—
296	C₂×C₄	—	424	4	no	—
297	C₂×C₄	C₂×#11719919	1	2	yes	—
298	C₂×C₄	C₂×#11719919	1	2	yes	—
299	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
300	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
301	C₂×C₄	C₄×Dic₁₀₆	1	2	yes	—
302	C₂×C₄	C₄×Dic₁₀₆	1	2	yes	—
303	C₂×C₄	#13133191	1	2	yes	—
304	C₂×C₄	#13133191	1	2	yes	—
305	C₂×C₄	#13133196	1	2	yes	—
306	C₂×C₄	#13133209	1	2	yes	—
307	C₂×C₄	#13133217	1	2	yes	—
308	C₂×C₄	#13133217	1	2	yes	—
309	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
310	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
311	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
312	C₂×C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
313	C₂×C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
314	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
315	C₈	—	2	4	no	—
316	C₈	—	212	4	no	—
317	C₈	C₂×#11719919	1	2	yes	—
318	C₈	C₂×#11719919	1	2	yes	—
319	C₈	C₂×#11719919	1	2	yes	—
320	C₈	C₂×#11719919	1	2	yes	—
321	C₂²	—	2	2	no	—
322	C₂²	—	2	2	no	—
323	C₂²	—	2	2	no	—
324	C₂²	—	4	2	no	—
325	C₂²	—	4	2	no	—
326	C₂²	#16810521	1	2	yes	—
327	C₂²	#16810521	1	2	yes	—
328	C₂²	C₂×C₄×D₂₁₂	1	1	yes	characteristic
329	C₂²	C₂×#13133196	1	1	yes	characteristic
330	C₂²	C₂×#13133209	1	1	yes	characteristic
331	C₄	—	4	4	no	—
332	C₄	—	212	8	no	—
333	C₄	#16810521	1	2	yes	—
334	C₄	#16810521	1	2	yes	—
335	C₄	C₂²×#11719919	1	1	yes	characteristic
336	C₄	C₂×C₄×Dic₁₀₆	1	1	yes	characteristic
337	C₄	C₂×#13133191	1	1	yes	characteristic
338	C₄	C₂×#13133217	1	1	yes	characteristic
339	C₂	—	2	1	no	—
340	C₂	—	2	2	no	—
341	C₂	—	2	2	no	—
342	C₂	—	2	1	no	—
343	C₂	C₂×#16810521	1	1	yes	characteristic
344	C₂	#20283633	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂×C₅₃×#405	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂⁴×C₅₃	#41
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₂×#405	D₅₃
hypocenter	C₅₃	#21163
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
4. #20275468	C₂	semidirect
5. #20275463	C₂	semidirect
13. #16810661	C₄	semidirect
14. #16810661	C₄	semidirect
17. C₂×#13133151	C₄	semidirect
18. C₂×#13133149	C₄	semidirect
201. C₅₃	#21163	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13133153	2
C₂×#16810661	2
#20283278	2
#20275468	2
#20275463	2
#20274425	2
C₂×C₅₃×#405	2
#21163	53