Subgroups — #28319373

Order 13568.

684 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20290646	C₂	1	4	yes	maximal, complemented
2	#20290645	C₂	1	4	yes	maximal, complemented
3	C₂×#16811505	C₂	1	2	yes	maximal
4	#20290396	C₂	1	2	yes	maximal
5	D₅₃⋊#338	C₂	1	1	yes	maximal, characteristic
6	#20290387	C₂	1	1	yes	maximal, characteristic
7	#20283131	C₂	1	1	yes	maximal, characteristic
8	C₂×D₅₃⋊M₁₆	C₂²	1	2	yes	—
9	C₂×#13133296	C₂²	1	2	yes	—
10	#16811505	C₂²	1	4	yes	—
11	#16811505	C₂²	1	4	yes	—
12	#16811494	C₂²	1	8	yes	—
13	#16811492	C₂²	1	4	yes	—
14	#16811491	C₂²	1	4	yes	—
15	C₂²×#11719912	C₂²	1	1	yes	characteristic
16	C₂×M₁₆×D₅₃	C₂²	1	1	yes	characteristic
17	#117×D₅₃	C₂²	1	1	yes	characteristic
18	#16810581	C₄	1	1	yes	characteristic, complemented
19	#16810580	C₄	1	2	yes	complemented
20	#16810579	C₄	1	1	yes	characteristic, complemented
21	#16810578	C₄	1	1	yes	characteristic, complemented
22	#16810577	C₄	1	2	yes	complemented
23	#16810575	C₂²	1	2	yes	—
24	C₄×#11719912	C₂²	1	2	yes	—
25	C₅₃×#405	C₄	1	1	yes	characteristic, complemented
26	C₂×#11719936	—	2	2	no	—
27	D₅₃⋊M₁₆	—	2	2	no	—
28	D₅₃⋊M₁₆	—	2	2	no	—
29	D₅₃⋊M₁₆	—	2	2	no	—
30	D₅₃⋊M₁₆	C₂³	1	4	yes	—
31	C₂×#11719940	—	2	2	no	—
32	C₂×D₅₃⋊C₈	—	2	2	no	—
33	#13133296	—	2	2	no	—
34	#13133296	—	2	2	no	—
35	#13133296	—	2	2	no	—
36	#13133296	C₂³	1	4	yes	—
37	C₂×C₄⋊#6157	—	2	2	no	—
38	C₂×C₂²⋊#6157	—	2	2	no	—
39	C₂×C₄×#6157	—	2	2	no	—
40	C₂²×C₄×D₅₃	C₂³	1	1	yes	characteristic
41	C₂×#11719917	C₂×C₄	1	1	yes	characteristic
42	C₂×#11719912	—	2	2	no	—
43	C₂×#11719912	—	2	2	no	—
44	C₂×#11719912	C₂×C₄	1	1	yes	characteristic
45	C₂×#11719912	C₂×C₄	1	1	yes	characteristic
46	C₂×#11719912	C₂³	1	1	yes	characteristic
47	C₂×#11719912	C₂³	1	2	yes	—
48	C₂×#11719912	C₂³	1	1	yes	characteristic
49	M₁₆×D₅₃	—	2	2	no	—
50	M₁₆×D₅₃	—	2	2	no	—
51	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
52	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
53	#13133176	C₂×C₄	1	1	yes	characteristic
54	#13133175	—	2	1	no	—
55	#13133174	C₂×C₄	1	2	yes	—
56	#13133173	—	2	1	no	—
57	C₄⋊C₄×D₅₃	—	2	1	no	—
58	C₂²⋊C₄×D₅₃	—	2	1	no	—
59	C₄²×D₅₃	C₂³	1	2	yes	—
60	#13133128	C₂×C₄	1	2	yes	complemented
61	#13133127	C₂×C₄	1	2	yes	complemented
62	#13133127	C₂×C₄	1	2	yes	complemented
63	C₄×#88322	C₂×C₄	1	2	yes	complemented
64	C₈×Dic₅₃	C₂×C₄	1	2	yes	complemented
65	C₈×Dic₅₃	C₂×C₄	1	2	yes	complemented
66	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
67	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
68	#117×C₅₃	C₂×C₄	1	1	yes	characteristic
69	#97×C₅₃	C₂×C₄	1	2	yes	complemented
70	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	complemented
71	C₂³×D₅₃	#41	1	1	yes	characteristic
72	C₂²×#6157	—	4	2	no	—
73	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
74	C₂²×Dic₅₃	#41	1	1	yes	characteristic
75	C₂×C₄×D₅₃	—	2	2	no	—
76	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₄×D₅₃	#41	1	2	yes	—
78	C₂×C₄×D₅₃	#41	1	2	yes	—
79	C₂×C₄×D₅₃	#41	1	2	yes	—
80	C₂×C₄×D₅₃	#41	1	2	yes	—
81	C₂×C₄×D₅₃	#41	1	1	yes	characteristic
82	C₂×C₄×D₅₃	#41	1	1	yes	characteristic
83	C₂×C₄×D₅₃	C₂⁴	1	1	yes	characteristic
84	C₂²⋊#6157	—	2	4	no	—
85	C₂²⋊#6157	—	4	2	no	—
86	#11719940	—	2	4	no	—
87	#11719940	—	4	2	no	—
88	C₂×#88324	—	4	2	no	—
89	C₄⋊#6157	—	2	4	no	—
90	C₄⋊#6157	—	4	2	no	—
91	C₄×#6157	—	2	4	no	—
92	C₄×#6157	—	4	2	no	—
93	#11719936	—	2	4	no	—
94	#11719936	—	4	2	no	—
95	D₅₃⋊C₈	—	2	4	no	—
96	D₅₃⋊C₈	—	4	2	no	—
97	M₁₆×C₅₃	—	2	2	no	—
98	C₂×C₈×C₅₃	—	2	2	no	—
99	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
101	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
102	C₄⋊C₄×C₅₃	—	2	1	no	—
103	C₂²⋊C₄×C₅₃	—	2	1	no	—
104	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
105	C₂²⋊Dic₅₃	—	2	1	no	—
106	#11719921	—	2	2	no	—
107	C₄⋊Dic₅₃	—	2	1	no	—
108	#11719919	—	2	2	no	—
109	C₄×Dic₅₃	—	2	2	no	—
110	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
111	#11719917	—	2	2	no	—
112	C₂×#88322	—	2	2	no	—
113	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
114	C₂×#88322	C₂²×C₄	1	2	yes	—
115	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
116	#11719912	—	2	1	no	—
117	#11719912	—	2	1	no	—
118	#11719912	—	2	1	no	—
119	#11719912	—	2	1	no	—
120	#11719912	—	2	4	no	—
121	#11719912	—	2	2	no	—
122	#11719912	—	2	2	no	—
123	C₈×D₅₃	—	2	2	no	—
124	C₈×D₅₃	—	2	2	no	—
125	C₂³×C₅₃	#117	1	1	yes	characteristic
126	C₂²×D₅₃	—	8	1	no	—
127	C₂²×D₅₃	#117	1	1	yes	characteristic
128	C₂²×D₅₃	#126	1	2	yes	—
129	C₂²×D₅₃	#126	1	2	yes	—
130	C₂²×D₅₃	C₂×#41	1	1	yes	characteristic
131	C₂×#6157	—	4	4	no	—
132	C₂×#6157	—	8	4	no	—
133	C₂×C₄×C₅₃	—	2	2	no	—
134	C₂×C₄×C₅₃	#117	1	2	yes	—
135	C₂×C₄×C₅₃	#117	1	2	yes	—
136	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
137	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
138	C₂×Dic₅₃	—	2	2	no	—
139	C₂×Dic₅₃	#117	1	1	yes	characteristic
140	C₂×Dic₅₃	#126	1	2	yes	—
141	C₂×Dic₅₃	#126	1	2	yes	—
142	C₂×Dic₅₃	C₂×#41	1	1	yes	characteristic
143	C₄×D₅₃	—	8	2	no	—
144	C₄×D₅₃	#117	1	1	yes	characteristic
145	C₄×D₅₃	#117	1	1	yes	characteristic
146	C₄×D₅₃	#117	1	1	yes	characteristic
147	C₄×D₅₃	#117	1	1	yes	characteristic
148	C₄×D₅₃	#126	1	2	yes	—
149	C₄×D₅₃	#126	1	2	yes	—
150	C₄×D₅₃	#126	1	2	yes	—
151	C₄×D₅₃	#126	1	2	yes	—
152	C₄×D₅₃	C₂×#41	1	2	yes	—
153	C₄×D₅₃	C₂×#41	1	2	yes	—
154	#88324	—	4	4	no	—
155	C₈×C₅₃	—	2	2	no	—
156	C₈×C₅₃	—	2	1	no	—
157	C₈×C₅₃	—	2	1	no	—
158	#88322	—	2	2	no	—
159	#88322	—	2	1	no	—
160	#88322	—	2	1	no	—
161	#20776	—	53	1	no	maximal, Sylow
162	C₂²×C₅₃	—	4	1	no	—
163	C₂²×C₅₃	#383	1	2	yes	—
164	C₂²×C₅₃	C₂×#117	1	1	yes	characteristic
165	C₂×D₅₃	—	4	1	no	—
166	C₂×D₅₃	—	4	2	no	—
167	C₂×D₅₃	—	4	1	no	—
168	C₂×D₅₃	—	4	2	no	—
169	C₂×D₅₃	#383	1	1	yes	characteristic
170	C₂×D₅₃	#383	1	1	yes	characteristic
171	C₂×D₅₃	C₂×#126	1	2	yes	—
172	#6157	—	8	4	no	—
173	C₄×C₅₃	—	4	2	no	—
174	C₄×C₅₃	#383	1	2	yes	—
175	C₄×C₅₃	C₂×#117	1	1	yes	characteristic
176	C₄×C₅₃	C₂×#117	1	1	yes	characteristic
177	Dic₅₃	—	4	2	no	—
178	Dic₅₃	#383	1	1	yes	characteristic
179	Dic₅₃	#383	1	1	yes	characteristic
180	Dic₅₃	C₂×#126	1	2	yes	—
181	#2873	—	53	1	no	—
182	#1841	—	53	4	no	—
183	#1841	—	53	4	no	—
184	C₂×#344	—	53	2	no	—
185	#1733	—	53	2	no	—
186	C₂×#338	—	53	1	no	—
187	C₂×H(3,ℤ/4)	—	53	1	no	—
188	C₂×C₅₃	—	2	1	no	—
189	C₂×C₅₃	—	2	2	no	—
190	C₂×C₅₃	C₂×#383	1	1	yes	characteristic
191	D₅₃	—	2	2	no	—
192	D₅₃	—	2	1	no	—
193	D₅₃	—	2	1	no	—
194	C₂²×M₁₆	—	53	1	no	—
195	C₂²×M₁₆	—	53	1	no	—
196	C₂²×M₁₆	—	53	2	no	—
197	C₂×#117	—	53	1	no	—
198	C₂×#117	—	53	2	no	—
199	#405	—	53	1	no	—
200	#405	—	53	1	no	—
201	#405	—	53	1	no	—
202	#405	—	53	2	no	—
203	#405	—	53	2	no	—
204	#405	#6157	1	1	yes	characteristic, complemented
205	C₄×M₁₆	—	53	2	no	—
206	C₄×M₁₆	—	53	2	no	—
207	#344	—	53	4	no	—
208	#344	—	53	4	no	—
209	#339	—	53	8	no	—
210	#338	—	53	4	no	—
211	H(3,ℤ/4)	—	53	4	no	—
212	C₅₃	#20776	1	1	yes	characteristic, complemented, Sylow
213	C₂³×C₄	—	53	1	no	—
214	C₂×M₁₆	—	53	1	no	—
215	C₂×M₁₆	—	53	2	no	—
216	C₂×M₁₆	—	53	1	no	—
217	C₂×M₁₆	—	53	1	no	—
218	C₂×M₁₆	—	53	1	no	—
219	C₂×M₁₆	—	53	1	no	—
220	C₂×M₁₆	—	53	4	no	—
221	C₂×M₁₆	—	106	2	no	—
222	C₂×M₁₆	—	106	2	no	—
223	C₂×M₁₆	—	106	2	no	—
224	C₂×M₁₆	—	106	2	no	—
225	C₂×M₁₆	—	106	2	no	—
226	C₂×M₁₆	—	106	2	no	—
227	C₂×M₁₆	—	106	2	no	—
228	C₂×M₁₆	—	106	2	no	—
229	C₂×M₁₆	—	106	2	no	—
230	C₂×M₁₆	C₂×#6157	1	1	yes	characteristic
231	C₂²×C₈	—	53	1	no	—
232	C₂²×C₈	—	53	2	no	—
233	C₂²×C₈	—	106	2	no	—
234	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
235	#117	—	53	1	no	—
236	#117	—	53	2	no	—
237	#117	—	53	4	no	—
238	#117	—	106	1	no	—
239	#117	—	106	1	no	—
240	#117	—	106	2	no	—
241	#117	—	106	2	no	—
242	#117	—	106	2	no	—
243	#117	C₂×#6157	1	1	yes	characteristic
244	C₂×C₄⋊C₄	—	106	1	no	—
245	C₂×C₄⋊C₄	—	106	2	no	—
246	C₂×C₂²⋊C₄	—	106	1	no	—
247	C₂×C₂²⋊C₄	—	106	2	no	—
248	C₂×C₄²	—	53	2	no	—
249	C₂×C₄²	—	106	2	no	—
250	#97	—	53	2	no	—
251	#97	—	53	2	no	—
252	#97	—	53	2	no	—
253	#97	C₂×#6157	1	2	yes	complemented
254	C₄×C₈	—	53	2	no	—
255	C₄×C₈	—	53	2	no	—
256	C₄×C₈	—	53	2	no	—
257	C₄×C₈	C₂×#6157	1	2	yes	complemented
258	C₂⁴	—	53	1	no	—
259	C₂²×C₄	—	53	1	no	—
260	C₂²×C₄	—	53	2	no	—
261	C₂²×C₄	—	53	1	no	—
262	C₂²×C₄	—	53	2	no	—
263	C₂²×C₄	—	53	2	no	—
264	C₂²×C₄	—	53	2	no	—
265	C₂²×C₄	—	53	1	no	—
266	C₂²×C₄	—	53	1	no	—
267	C₂²×C₄	—	53	1	no	—
268	C₂²×C₄	—	106	2	no	—
269	C₂²×C₄	—	212	2	no	—
270	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
271	M₁₆	—	2	2	no	—
272	M₁₆	—	106	4	no	—
273	M₁₆	—	106	1	no	—
274	M₁₆	—	106	1	no	—
275	M₁₆	—	106	1	no	—
276	M₁₆	—	106	1	no	—
277	M₁₆	—	106	2	no	—
278	M₁₆	—	106	2	no	—
279	M₁₆	—	106	2	no	—
280	M₁₆	—	106	4	no	—
281	M₁₆	—	106	4	no	—
282	M₁₆	—	212	2	no	—
283	M₁₆	—	212	2	no	—
284	C₂×C₈	—	2	2	no	—
285	C₂×C₈	—	53	2	no	—
286	C₂×C₈	—	53	1	no	—
287	C₂×C₈	—	53	1	no	—
288	C₂×C₈	—	106	2	no	—
289	C₂×C₈	—	106	2	no	—
290	C₂×C₈	—	106	2	no	—
291	C₂×C₈	—	106	4	no	—
292	C₂×C₈	—	212	2	no	—
293	C₂×C₈	—	212	2	no	—
294	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
295	C₂×C₈	C₂²×#6157	1	2	yes	—
296	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
297	C₄⋊C₄	—	2	1	no	—
298	C₄⋊C₄	—	106	2	no	—
299	C₄⋊C₄	—	106	4	no	—
300	C₄⋊C₄	—	106	1	no	—
301	C₄⋊C₄	—	212	2	no	—
302	C₂²⋊C₄	—	2	1	no	—
303	C₂²⋊C₄	—	106	2	no	—
304	C₂²⋊C₄	—	106	1	no	—
305	C₂²⋊C₄	—	106	4	no	—
306	C₂²⋊C₄	—	212	2	no	—
307	C₄²	—	53	2	no	—
308	C₄²	—	106	2	no	—
309	C₄²	—	106	4	no	—
310	C₄²	—	212	2	no	—
311	C₄²	C₂²×#6157	1	2	yes	—
312	C₂³	—	53	2	no	—
313	C₂³	—	53	2	no	—
314	C₂³	—	53	1	no	—
315	C₂³	—	53	1	no	—
316	C₂³	—	424	1	no	—
317	C₂³	#13133296	1	1	yes	characteristic
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	53	2	no	—
320	C₂×C₄	—	53	2	no	—
321	C₂×C₄	—	53	1	no	—
322	C₂×C₄	—	53	1	no	—
323	C₂×C₄	—	53	2	no	—
324	C₂×C₄	—	53	2	no	—
325	C₂×C₄	—	53	2	no	—
326	C₂×C₄	—	53	2	no	—
327	C₂×C₄	—	53	1	no	—
328	C₂×C₄	—	53	1	no	—
329	C₂×C₄	—	53	2	no	—
330	C₂×C₄	—	53	1	no	—
331	C₂×C₄	—	53	1	no	—
332	C₂×C₄	—	53	2	no	—
333	C₂×C₄	—	106	2	no	—
334	C₂×C₄	—	424	2	no	—
335	C₂×C₄	—	424	4	no	—
336	C₂×C₄	—	212	4	no	—
337	C₂×C₄	#13133296	1	1	yes	characteristic
338	C₂×C₄	#13133296	1	2	yes	—
339	C₂×C₄	#13133296	1	2	yes	—
340	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
341	C₈	—	2	2	no	—
342	C₈	—	2	1	no	—
343	C₈	—	2	1	no	—
344	C₈	—	106	2	no	—
345	C₈	—	106	1	no	—
346	C₈	—	106	1	no	—
347	C₈	—	212	4	no	—
348	C₂²	—	4	1	no	—
349	C₂²	—	53	2	no	—
350	C₂²	—	53	1	no	—
351	C₂²	—	53	1	no	—
352	C₂²	—	212	1	no	—
353	C₂²	—	212	2	no	—
354	C₂²	—	212	1	no	—
355	C₂²	—	212	2	no	—
356	C₂²	#16811568	1	2	yes	—
357	C₂²	C₂×#13133296	1	1	yes	characteristic
358	C₄	—	4	2	no	—
359	C₄	—	53	2	no	—
360	C₄	—	53	1	no	—
361	C₄	—	53	1	no	—
362	C₄	—	424	4	no	—
363	C₄	—	212	2	no	—
364	C₄	#16811568	1	2	yes	—
365	C₄	C₂×#13133296	1	1	yes	characteristic
366	C₄	C₂×#13133296	1	1	yes	characteristic
367	C₂	—	2	1	no	—
368	C₂	—	2	2	no	—
369	C₂	—	106	2	no	—
370	C₂	—	106	1	no	—
371	C₂	—	106	1	no	—
372	C₂	C₂×#16811568	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×#13133296
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×#6157
Fitting	C₅₃×#405	C₄
socle	C₂×C₅₃	C₂×#383
prime radical	C₂³×D₅₃	#41
cosocle	C₂×C₄×D₅₃	C₂⁴
hypercenter	#405	#6157
hypocenter	C₅₃	#20776
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #405

Complements

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

N	H ≅ G/N	type
1. #20290646	C₂	semidirect
2. #20290645	C₂	semidirect
18. #16810581	C₄	semidirect
19. #16810580	C₄	semidirect
20. #16810579	C₄	semidirect
21. #16810578	C₄	semidirect
22. #16810577	C₄	semidirect
25. C₅₃×#405	C₄	semidirect
60. #13133128	C₂×C₄	semidirect
61. #13133127	C₂×C₄	semidirect
62. #13133127	C₂×C₄	semidirect
63. C₄×#88322	C₂×C₄	semidirect
64. C₈×Dic₅₃	C₂×C₄	semidirect
65. C₈×Dic₅₃	C₂×C₄	semidirect
69. #97×C₅₃	C₂×C₄	semidirect
70. C₄×C₈×C₅₃	C₂×C₄	semidirect
204. #405	#6157	semidirect
212. C₅₃	#20776	semidirect
253. #97	C₂×#6157	semidirect
257. C₄×C₈	C₂×#6157	semidirect

Maximal subgroups

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

subgroup	index
#20290646	2
#20290645	2
C₂×#16811505	2
#20290396	2
D₅₃⋊#338	2
#20290387	2
#20283131	2
#20776	53