Subgroups — #27392289

Order 13568.

738 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20285061	C₂	1	1	yes	maximal, characteristic
2	#20283655	C₂	1	1	yes	maximal, characteristic, complemented
3	#20283434	C₂	1	1	yes	maximal, characteristic
4	#20275692	C₂	1	1	yes	maximal, characteristic, complemented
5	#20275543	C₂	1	1	yes	maximal, characteristic
6	#20275426	C₂	1	2	yes	maximal, complemented
7	#20275419	C₂	1	2	yes	maximal, complemented
8	#20274856	C₂	1	1	yes	maximal, characteristic, complemented
9	#20274559	C₂	1	2	yes	maximal, complemented
10	#20274552	C₂	1	2	yes	maximal, complemented
11	C₄×#97×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13133223	C₂²	1	2	yes	—
13	#16811180	C₂²	1	1	yes	characteristic
14	#16811173	C₂²	1	1	yes	characteristic
15	#16810992	C₄	1	2	yes	complemented
16	#16810988	C₄	1	2	yes	complemented
17	#16810983	C₄	1	4	yes	complemented
18	#16810751	—	2	4	no	—
19	#16810696	—	2	4	no	—
20	#16810649	C₂²	1	2	yes	—
21	#16810647	C₂²	1	1	yes	characteristic
22	#16810647	C₂²	1	1	yes	characteristic
23	C₂×#13133145	C₂²	1	2	yes	—
24	C₂×#13133145	C₂²	1	2	yes	—
25	#16810620	C₂²	1	2	yes	—
26	#16810620	C₂²	1	2	yes	—
27	#16810589	C₂²	1	2	yes	—
28	#16810578	C₂²	1	2	yes	—
29	#16810388	C₂²	1	2	yes	—
30	#16810388	C₂²	1	2	yes	—
31	#16810383	C₂²	1	2	yes	—
32	#16810383	C₂²	1	2	yes	—
33	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
34	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
35	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
36	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
37	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
38	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
39	C₄³×C₅₃	C₂²	1	1	yes	characteristic
40	C₂×#11719945	C₂³	1	2	yes	—
41	#13133223	—	2	4	no	—
42	#13133223	—	2	4	no	—
43	C₂×#11719917	C₂³	1	2	yes	—
44	C₂×#11719917	C₂³	1	2	yes	—
45	C₂×#11719912	—	2	2	no	—
46	C₂×C₈×D₅₃	—	2	2	no	—
47	#13133217	C₂×C₄	1	2	yes	—
48	#13133217	C₂×C₄	1	1	yes	characteristic
49	#13133214	C₂×C₄	1	1	yes	characteristic
50	#13133209	C₂×C₄	1	2	yes	—
51	#13133205	C₂×C₄	1	2	yes	—
52	#13133196	—	2	2	no	—
53	#13133191	—	2	2	no	—
54	C₄×Dic₁₀₆	—	2	2	no	—
55	C₄×D₂₁₂	—	2	2	no	—
56	#13133176	C₂³	1	2	yes	—
57	C₄⋊#88322	—	2	2	no	—
58	C₄⋊#88322	—	2	2	no	—
59	#13133146	—	2	4	no	—
60	#13133145	—	2	4	no	—
61	#13133145	—	2	4	no	—
62	#13133142	C₂×C₄	1	4	yes	—
63	#13133142	C₂×C₄	1	4	yes	—
64	#13133139	C₂×C₄	1	4	yes	—
65	#13133139	C₂×C₄	1	4	yes	—
66	#13133129	—	2	4	no	—
67	#13133127	—	2	2	no	—
68	C₈×Dic₅₃	—	2	2	no	—
69	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
70	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
71	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
72	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
73	C₂×C₄²×C₅₃	—	2	2	no	—
74	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
75	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
76	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
77	#97×C₅₃	—	2	2	no	—
78	#97×C₅₃	—	2	2	no	—
79	#97×C₅₃	C₂×C₄	1	2	yes	—
80	#97×C₅₃	C₂×C₄	1	2	yes	—
81	#97×C₅₃	D₄	1	4	yes	—
82	C₄×C₈×C₅₃	—	2	2	no	—
83	C₄×C₈×C₅₃	—	2	2	no	—
84	C₂²×C₄×C₅₃	—	2	1	no	—
85	C₂²×C₄×C₅₃	—	2	1	no	—
86	C₂²×C₄×C₅₃	#41	1	2	yes	—
87	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
88	C₂×C₂²⋊D₅₃	—	2	2	no	—
89	#11719945	—	2	2	no	—
90	#11719945	—	2	2	no	—
91	#11719945	—	2	2	no	—
92	#11719945	—	2	2	no	—
93	C₂×D₂₁₂	—	2	2	no	—
94	C₂×D₂₁₂	C₂²×C₄	1	2	yes	—
95	C₂×C₄×D₅₃	—	2	2	no	—
96	C₂×Dic₁₀₆	—	2	2	no	—
97	C₂×Dic₁₀₆	C₂²×C₄	1	2	yes	—
98	C₂×C₈×C₅₃	—	2	1	no	—
99	C₂×C₈×C₅₃	—	2	1	no	—
100	C₂×C₈×C₅₃	—	2	1	no	—
101	C₂×C₈×C₅₃	—	2	1	no	—
102	C₂×C₈×C₅₃	—	2	1	no	—
103	C₂×C₈×C₅₃	—	2	1	no	—
104	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
105	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
106	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
107	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
108	C₂×C₈×C₅₃	#41	1	2	yes	—
109	C₂×C₈×C₅₃	#41	1	2	yes	—
110	C₄²×C₅₃	—	2	1	no	—
111	C₄²×C₅₃	—	2	1	no	—
112	C₄²×C₅₃	—	4	4	no	—
113	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
114	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
115	C₄²×C₅₃	C₂×D₄	1	2	yes	—
116	C₄²×C₅₃	#41	1	2	yes	—
117	C₄²×C₅₃	#41	1	2	yes	—
118	C₂²⋊Dic₅₃	—	2	2	no	—
119	#11719921	—	2	4	no	—
120	C₄⋊Dic₅₃	—	2	2	no	—
121	C₄⋊Dic₅₃	—	2	2	no	—
122	C₄⋊Dic₅₃	C₂²×C₄	1	4	yes	—
123	#11719919	—	2	4	no	—
124	C₄×Dic₅₃	—	2	2	no	—
125	#11719917	—	2	2	no	—
126	#11719917	—	2	2	no	—
127	#11719917	—	2	2	no	—
128	#11719917	—	2	2	no	—
129	C₂×#88322	—	2	2	no	—
130	C₂×#88322	—	2	2	no	—
131	#11719912	—	4	4	no	—
132	C₈×D₅₃	—	4	4	no	—
133	C₂³×C₅₃	C₂×#41	1	1	yes	characteristic
134	C₂²×D₅₃	—	2	2	no	—
135	C₂×C₄×C₅₃	—	2	4	no	—
136	C₂×C₄×C₅₃	—	2	2	no	—
137	C₂×C₄×C₅₃	—	2	2	no	—
138	C₂×C₄×C₅₃	—	4	2	no	—
139	C₂×C₄×C₅₃	—	4	2	no	—
140	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
141	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
142	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
143	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
144	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
145	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
146	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
147	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
148	C₂²⋊D₅₃	—	4	2	no	—
149	C₂²⋊D₅₃	—	4	2	no	—
150	C₂×Dic₅₃	—	2	4	no	—
151	C₂×Dic₅₃	—	2	2	no	—
152	D₂₁₂	—	2	2	no	—
153	D₂₁₂	—	2	2	no	—
154	D₂₁₂	—	8	2	no	—
155	C₄×D₅₃	—	4	2	no	—
156	C₄×D₅₃	—	4	2	no	—
157	Dic₁₀₆	—	2	2	no	—
158	Dic₁₀₆	—	2	2	no	—
159	Dic₁₀₆	—	8	2	no	—
160	C₈×C₅₃	—	2	2	no	—
161	C₈×C₅₃	—	2	2	no	—
162	C₈×C₅₃	—	2	2	no	—
163	C₈×C₅₃	#124	1	4	yes	—
164	#88322	—	4	2	no	—
165	#88322	—	4	2	no	—
166	#21662	—	53	1	no	maximal, Sylow
167	C₂²×C₅₃	—	2	2	no	—
168	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
169	C₂²×C₅₃	#529	1	1	yes	characteristic
170	C₂²×C₅₃	#529	1	1	yes	characteristic
171	C₂×D₅₃	—	4	4	no	—
172	C₂×D₅₃	—	4	2	no	—
173	C₄×C₅₃	—	2	4	no	—
174	C₄×C₅₃	—	2	2	no	—
175	C₄×C₅₃	—	2	2	no	—
176	C₄×C₅₃	—	4	2	no	—
177	C₄×C₅₃	C₄×#41	1	1	yes	characteristic
178	C₄×C₅₃	C₄×#41	1	1	yes	characteristic
179	C₄×C₅₃	C₂×#124	1	1	yes	characteristic
180	C₄×C₅₃	#532	1	1	yes	characteristic
181	Dic₅₃	—	4	4	no	—
182	Dic₅₃	—	4	2	no	—
183	#3135	—	53	1	no	—
184	#2961	—	53	1	no	—
185	#2932	—	53	1	no	—
186	#1994	—	53	1	no	—
187	#1962	—	53	1	no	—
188	#1938	—	53	2	no	—
189	#1937	—	53	2	no	—
190	#1838	—	53	1	no	—
191	#1765	—	53	2	no	—
192	#1764	—	53	2	no	—
193	C₄×#97	D₅₃	1	1	yes	characteristic, complemented
194	C₂×C₅₃	—	2	1	no	—
195	C₂×C₅₃	—	2	1	no	—
196	C₂×C₅₃	C₄×#124	1	1	yes	characteristic
197	C₂×C₅₃	#2956	1	2	yes	—
198	D₅₃	—	8	2	no	—
199	C₂×#131	—	53	2	no	—
200	#533	—	53	1	no	—
201	#532	—	53	1	no	—
202	#491	—	53	2	no	—
203	#490	—	53	2	no	—
204	#489	—	53	4	no	—
205	#446	—	106	4	no	—
206	#435	—	106	4	no	—
207	#424	—	53	2	no	—
208	#423	—	53	1	no	—
209	#423	—	53	1	no	—
210	C₂×#104	—	53	2	no	—
211	C₂×#104	—	53	2	no	—
212	#416	—	53	2	no	—
213	#416	—	53	2	no	—
214	#408	—	53	2	no	—
215	#405	—	53	2	no	—
216	C₂×#97	C₂×D₅₃	1	1	yes	characteristic
217	C₂×#97	C₂×D₅₃	1	1	yes	characteristic
218	C₂×#97	C₂×D₅₃	1	1	yes	characteristic
219	C₂×#97	C₂×D₅₃	1	1	yes	characteristic
220	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
221	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
222	C₄³	C₂×D₅₃	1	1	yes	characteristic
223	#341	—	53	2	no	—
224	#341	—	53	2	no	—
225	#339	—	53	2	no	—
226	#339	—	53	2	no	—
227	C₅₃	#21662	1	1	yes	characteristic, complemented, Sylow
228	C₂×#41	—	53	2	no	—
229	#131	—	106	4	no	—
230	#131	—	106	4	no	—
231	C₂×M₁₆	—	53	2	no	—
232	C₂×M₁₆	—	53	2	no	—
233	C₂×M₁₆	—	106	2	no	—
234	C₂²×C₈	—	106	2	no	—
235	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
236	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
237	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
238	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
239	C₄⋊Q₈	—	53	2	no	—
240	C₄⋊Q₈	—	53	1	no	—
241	#127	—	53	1	no	—
242	#125	—	53	2	no	—
243	#124	—	53	2	no	—
244	C₂²⋊Q₈	—	106	2	no	—
245	#121	—	106	2	no	—
246	C₄×Q₈	—	106	2	no	—
247	C₄×D₄	—	106	2	no	—
248	#117	—	53	2	no	—
249	C₂×C₄²	—	2	2	no	—
250	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
251	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
252	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
253	C₄⋊C₈	—	106	4	no	—
254	C₄⋊C₈	—	106	2	no	—
255	C₄⋊C₈	—	106	2	no	—
256	#104	—	106	4	no	—
257	#104	—	106	4	no	—
258	#103	—	53	4	no	—
259	#103	—	53	4	no	—
260	#102	—	53	4	no	—
261	#102	—	53	4	no	—
262	C₂²⋊C₈	—	106	4	no	—
263	#97	—	2	2	no	—
264	#97	—	2	2	no	—
265	#97	—	106	2	no	—
266	#97	C₄×D₅₃	1	2	yes	—
267	#97	C₄×D₅₃	1	2	yes	—
268	#97	D₂₁₂	1	4	yes	—
269	C₄×C₈	—	2	2	no	—
270	C₄×C₈	—	2	2	no	—
271	C₄×C₈	—	106	2	no	—
272	#41	—	106	2	no	—
273	#41	—	106	2	no	—
274	#41	—	106	2	no	—
275	#41	—	106	2	no	—
276	C₂×Q₈	—	53	2	no	—
277	C₂×Q₈	—	106	2	no	—
278	C₂×D₄	—	53	2	no	—
279	C₂×D₄	—	106	2	no	—
280	C₂×D₄	—	106	2	no	—
281	C₂²×C₄	—	2	1	no	—
282	C₂²×C₄	—	2	1	no	—
283	C₂²×C₄	—	106	2	no	—
284	C₂²×C₄	#11719945	1	2	yes	—
285	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
286	M₁₆	—	106	2	no	—
287	M₁₆	—	106	2	no	—
288	M₁₆	—	106	2	no	—
289	M₁₆	—	106	2	no	—
290	M₁₆	—	212	4	no	—
291	C₂×C₈	—	2	1	no	—
292	C₂×C₈	—	2	1	no	—
293	C₂×C₈	—	2	1	no	—
294	C₂×C₈	—	2	1	no	—
295	C₂×C₈	—	2	1	no	—
296	C₂×C₈	—	2	1	no	—
297	C₂×C₈	—	106	2	no	—
298	C₂×C₈	—	106	2	no	—
299	C₂×C₈	—	212	4	no	—
300	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
301	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
302	C₂×C₈	C₂×D₂₁₂	1	2	yes	—
303	C₂×C₈	C₂×D₂₁₂	1	2	yes	—
304	C₂×C₈	#11719945	1	2	yes	—
305	C₂×C₈	#11719945	1	2	yes	—
306	C₄⋊C₄	—	53	4	no	—
307	C₄⋊C₄	—	106	2	no	—
308	C₄⋊C₄	—	106	4	no	—
309	C₄⋊C₄	—	106	2	no	—
310	C₂²⋊C₄	—	106	2	no	—
311	C₂²⋊C₄	—	106	4	no	—
312	C₄²	—	2	1	no	—
313	C₄²	—	2	1	no	—
314	C₄²	—	4	4	no	—
315	C₄²	—	106	2	no	—
316	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
317	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
318	C₄²	C₂×D₂₁₂	1	2	yes	—
319	C₄²	#11719945	1	2	yes	—
320	C₄²	#11719945	1	2	yes	—
321	C₂³	—	106	2	no	—
322	C₂³	C₂×#11719945	1	1	yes	characteristic
323	Q₈	—	106	2	no	—
324	Q₈	—	106	2	no	—
325	Q₈	—	424	2	no	—
326	D₄	—	106	2	no	—
327	D₄	—	106	2	no	—
328	D₄	—	424	2	no	—
329	D₄	—	212	2	no	—
330	D₄	—	212	2	no	—
331	C₂×C₄	—	2	4	no	—
332	C₂×C₄	—	2	2	no	—
333	C₂×C₄	—	2	2	no	—
334	C₂×C₄	—	4	2	no	—
335	C₂×C₄	—	4	2	no	—
336	C₂×C₄	—	106	4	no	—
337	C₂×C₄	—	106	2	no	—
338	C₂×C₄	—	212	2	no	—
339	C₂×C₄	—	212	2	no	—
340	C₂×C₄	C₄×D₂₁₂	1	2	yes	—
341	C₂×C₄	C₄×D₂₁₂	1	2	yes	—
342	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
343	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
344	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
345	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
346	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
347	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
348	C₈	—	2	2	no	—
349	C₈	—	2	2	no	—
350	C₈	—	2	2	no	—
351	C₈	—	212	2	no	—
352	C₈	—	212	2	no	—
353	C₈	#13133205	1	4	yes	—
354	C₂²	—	2	2	no	—
355	C₂²	—	212	2	no	—
356	C₂²	—	212	4	no	—
357	C₂²	C₂×C₄×D₂₁₂	1	1	yes	characteristic
358	C₂²	#16811157	1	1	yes	characteristic
359	C₂²	#16811157	1	1	yes	characteristic
360	C₄	—	2	4	no	—
361	C₄	—	2	2	no	—
362	C₄	—	2	2	no	—
363	C₄	—	4	2	no	—
364	C₄	—	212	2	no	—
365	C₄	—	212	4	no	—
366	C₄	C₄×#11719945	1	1	yes	characteristic
367	C₄	C₄×#11719945	1	1	yes	characteristic
368	C₄	C₂×#13133205	1	1	yes	characteristic
369	C₄	#16811173	1	1	yes	characteristic
370	C₂	—	2	1	no	—
371	C₂	—	2	1	no	—
372	C₂	—	424	2	no	—
373	C₂	C₄×#13133205	1	1	yes	characteristic
374	C₂	#20283626	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₄×#97×C₅₃	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	#16811173	C₂²
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₄×#97	D₅₃
hypocenter	C₅₃	#21662
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97 ◁ C₄×#97

Complements

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

N	H ≅ G/N	type
2. #20283655	C₂	semidirect
4. #20275692	C₂	semidirect
6. #20275426	C₂	semidirect
7. #20275419	C₂	semidirect
8. #20274856	C₂	semidirect
9. #20274559	C₂	semidirect
10. #20274552	C₂	semidirect
11. C₄×#97×C₅₃	C₂	semidirect
15. #16810992	C₄	semidirect
16. #16810988	C₄	semidirect
17. #16810983	C₄	semidirect
193. C₄×#97	D₅₃	semidirect
227. C₅₃	#21662	semidirect

Maximal subgroups

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

subgroup	index
#20285061	2
#20283655	2
#20283434	2
#20275692	2
#20275543	2
#20275426	2
#20275419	2
#20274856	2
#20274559	2
#20274552	2
C₄×#97×C₅₃	2
#21662	53