Subgroups — #28328242

Order 13568.

598 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20291082	C₂	1	2	yes	maximal, complemented
2	#20291080	C₂	1	2	yes	maximal, complemented
3	#20291075	C₂	1	2	yes	maximal, complemented
4	#20291073	C₂	1	2	yes	maximal, complemented
5	#20291051	C₂	1	2	yes	maximal, complemented
6	#20290946	C₂	1	2	yes	maximal
7	#20290922	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×D₅₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
9	#20283137	C₂	1	1	yes	maximal, characteristic
10	C₂×#13133292	C₂²	1	2	yes	—
11	C₂×D₅₃⋊C₁₆	C₂²	1	2	yes	—
12	#16811544	C₂²	1	4	yes	—
13	C₄⋊#11719910	C₂²	1	4	yes	—
14	#16811518	C₂²	1	4	yes	—
15	C₂²⋊#11719910	C₂²	1	4	yes	—
16	#16811511	C₂²	1	2	yes	—
17	#16811509	C₂²	1	2	yes	—
18	#16811509	C₂²	1	2	yes	—
19	C₄×#11719910	C₂²	1	2	yes	—
20	C₂×#13133225	C₂²	1	1	yes	characteristic
21	C₂×#13133223	C₄	1	1	yes	characteristic
22	#16811091	C₄	1	1	yes	characteristic
23	#16810580	C₂²	1	1	yes	characteristic
24	#16810580	C₂²	1	1	yes	characteristic
25	#16810579	C₂²	1	1	yes	characteristic
26	#16810577	C₄	1	1	yes	characteristic
27	#16810577	C₄	1	1	yes	characteristic
28	#16810575	C₄	1	1	yes	characteristic
29	#16810575	C₄	1	1	yes	characteristic
30	M₁₆×Dic₅₃	C₄	1	1	yes	characteristic
31	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
32	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
33	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
34	C₅₃×#405	C₄	1	1	yes	characteristic
35	#13133292	—	2	4	no	—
36	D₅₃⋊C₁₆	—	2	4	no	—
37	C₂×#11719910	C₂³	1	4	yes	—
38	C₂×#11719910	C₂³	1	4	yes	—
39	C₂×#11719945	C₂×C₄	1	1	yes	characteristic
40	#13133225	—	2	4	no	—
41	#13133223	—	2	4	no	—
42	C₂×#11719917	C₂×C₄	1	1	yes	characteristic
43	C₂×#11719912	C₂×C₄	1	1	yes	characteristic
44	C₂×#11719912	C₂×C₄	1	1	yes	characteristic
45	C₂×#11719912	C₂×C₄	1	1	yes	characteristic
46	C₂×#11719912	C₂×C₄	1	1	yes	characteristic
47	C₂²×#88322	C₂³	1	1	yes	characteristic
48	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
49	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
50	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
51	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
52	#13133182	C₈	1	2	yes	—
53	#13133179	C₈	1	2	yes	—
54	#13133177	C₈	1	2	yes	—
55	#13133177	C₈	1	2	yes	—
56	#13133174	C₂×C₄	1	1	yes	characteristic
57	#13133174	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
59	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
60	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
61	#13133128	C₂×C₄	1	1	yes	characteristic
62	#13133128	C₂×C₄	1	1	yes	characteristic
63	#13133127	C₂×C₄	1	1	yes	characteristic
64	#13133127	C₂×C₄	1	1	yes	characteristic
65	C₄×#88322	C₂³	1	1	yes	characteristic
66	C₄×#88322	C₂³	1	1	yes	characteristic
67	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
68	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
69	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
70	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
71	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
72	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
73	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
74	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
75	#117×C₅₃	C₂×C₄	1	1	yes	characteristic
76	#97×C₅₃	C₂×C₄	1	1	yes	characteristic
77	#97×C₅₃	C₂×C₄	1	1	yes	characteristic
78	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
79	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
80	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₂²⋊D₅₃	C₂×C₈	1	1	yes	characteristic
82	C₂×C₂²⋊D₅₃	C₂×C₈	1	1	yes	characteristic
83	C₂²×Dic₅₃	C₂×C₈	1	2	yes	—
84	#11719945	—	2	2	no	—
85	#11719945	—	2	2	no	—
86	C₂×D₂₁₂	C₂×C₈	1	1	yes	characteristic
87	C₂×C₄×D₅₃	C₂×C₈	1	2	yes	—
88	C₂×C₄×D₅₃	C₂×C₈	1	2	yes	—
89	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
91	C₂×Dic₁₀₆	C₂×C₈	1	1	yes	characteristic
92	M₁₆×C₅₃	—	2	2	no	—
93	C₂×C₈×C₅₃	—	2	2	no	—
94	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
98	C₄⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
99	C₂²⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
100	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
101	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
102	#11719921	C₂×C₈	1	2	yes	—
103	#11719921	C₂×C₈	1	2	yes	—
104	#11719919	C₂×C₈	1	2	yes	—
105	#11719919	C₂×C₈	1	2	yes	—
106	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
107	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
108	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
109	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
110	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
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	1	yes	characteristic
115	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
116	C₂×#88322	C₂⁴	1	1	yes	characteristic
117	#11719912	—	2	2	no	—
118	#11719912	—	2	2	no	—
119	#11719912	—	2	2	no	—
120	#11719912	—	2	2	no	—
121	C₈×D₅₃	—	2	2	no	—
122	C₈×D₅₃	—	2	2	no	—
123	C₈×D₅₃	M₁₆	1	4	yes	—
124	C₈×D₅₃	#41	1	4	yes	—
125	#11719910	—	2	4	no	—
126	#11719910	—	2	4	no	—
127	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
128	C₂²×D₅₃	C₂²×C₈	1	1	yes	characteristic
129	C₂²×D₅₃	C₂²×C₈	1	1	yes	characteristic
130	C₂×C₄×C₅₃	—	2	1	no	—
131	C₂×C₄×C₅₃	—	2	1	no	—
132	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
133	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
134	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
135	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
136	C₂²⋊D₅₃	—	2	2	no	—
137	C₂²⋊D₅₃	—	2	2	no	—
138	C₂×Dic₅₃	—	2	4	no	—
139	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
140	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
141	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
142	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
143	D₂₁₂	—	2	2	no	—
144	C₄×D₅₃	—	2	1	no	—
145	C₄×D₅₃	—	2	1	no	—
146	C₄×D₅₃	—	2	4	no	—
147	C₄×D₅₃	—	2	4	no	—
148	C₄×D₅₃	#117	1	2	yes	—
149	C₄×D₅₃	C₂×M₁₆	1	2	yes	—
150	Dic₁₀₆	—	2	2	no	—
151	C₈×C₅₃	—	2	1	no	—
152	C₈×C₅₃	—	2	1	no	—
153	C₈×C₅₃	#117	1	2	yes	—
154	C₈×C₅₃	C₂×M₁₆	1	2	yes	—
155	#88322	—	2	1	no	—
156	#88322	—	2	1	no	—
157	#88322	C₂×M₁₆	1	2	yes	—
158	#88322	C₂×#41	1	2	yes	—
159	#33420	—	53	1	no	maximal, Sylow
160	C₂²×C₅₃	—	2	2	no	—
161	C₂²×C₅₃	—	2	1	no	—
162	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
163	C₂×D₅₃	—	2	2	no	—
164	C₂×D₅₃	—	2	2	no	—
165	C₂×D₅₃	—	2	1	no	—
166	C₂×D₅₃	#431	1	2	yes	—
167	C₄×C₅₃	—	2	2	no	—
168	C₄×C₅₃	—	2	2	no	—
169	C₄×C₅₃	—	2	1	no	—
170	C₄×C₅₃	C₂×#117	1	1	yes	characteristic
171	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
172	Dic₅₃	—	2	2	no	—
173	Dic₅₃	—	2	1	no	—
174	Dic₅₃	#431	1	2	yes	—
175	Dic₅₃	#504	1	4	yes	—
176	C₄×#131	—	53	1	no	—
177	#2164	—	53	2	no	—
178	#2164	—	53	2	no	—
179	#2162	—	53	2	no	—
180	#2162	—	53	2	no	—
181	#2151	—	53	2	no	—
182	#2113	—	53	2	no	—
183	C₄×M₃₂	—	53	1	no	—
184	C₂×C₄×C₁₆	—	53	1	no	—
185	C₂×C₅₃	—	4	1	no	—
186	C₂×C₅₃	C₂×#431	1	1	yes	characteristic
187	C₂×C₅₃	C₂×#504	1	2	yes	—
188	D₅₃	—	2	2	no	—
189	D₅₃	—	4	1	no	—
190	C₂×#131	—	53	1	no	—
191	C₂×#131	—	53	1	no	—
192	C₄×#41	—	53	1	no	—
193	C₂×M₃₂	—	53	2	no	—
194	C₂²×C₁₆	—	53	2	no	—
195	#405	—	53	1	no	—
196	#405	—	53	1	no	—
197	#405	—	53	1	no	—
198	#405	—	53	1	no	—
199	#405	—	53	1	no	—
200	#405	#6157	1	1	yes	characteristic
201	C₄×M₁₆	—	53	1	no	—
202	C₄×M₁₆	—	53	1	no	—
203	C₄×M₁₆	—	53	1	no	—
204	C₂×C₄×C₈	—	53	1	no	—
205	C₂×C₄×C₈	—	53	1	no	—
206	C₂×C₄×C₈	—	53	1	no	—
207	C₄⋊C₁₆	—	53	4	no	—
208	C₄⋊C₁₆	—	53	4	no	—
209	C₂²⋊C₁₆	—	53	4	no	—
210	C₂²⋊C₁₆	—	53	4	no	—
211	#346	—	53	2	no	—
212	C₄×C₁₆	—	53	2	no	—
213	C₄×C₁₆	—	53	2	no	—
214	C₄×C₁₆	—	53	2	no	—
215	C₅₃	#33420	1	1	yes	characteristic, complemented, Sylow
216	C₂×#41	—	53	1	no	—
217	#131	—	106	4	no	—
218	#131	—	106	4	no	—
219	C₂×M₁₆	—	53	1	no	—
220	C₂×M₁₆	—	53	1	no	—
221	C₂×M₁₆	—	53	1	no	—
222	C₂×M₁₆	—	53	1	no	—
223	C₂×M₁₆	—	53	1	no	—
224	C₂×M₁₆	C₂×#6157	1	1	yes	characteristic
225	C₂²×C₈	—	53	1	no	—
226	C₂²×C₈	—	53	1	no	—
227	C₂²×C₈	—	53	1	no	—
228	C₂²×C₈	—	53	1	no	—
229	C₂²×C₈	—	53	1	no	—
230	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
231	C₄×Q₈	—	53	2	no	—
232	C₄×D₄	—	53	2	no	—
233	C₄×D₄	—	53	2	no	—
234	C₄×D₄	—	53	2	no	—
235	#117	—	53	1	no	—
236	#117	—	53	1	no	—
237	#117	C₂×#6157	1	1	yes	characteristic
238	C₂×C₄²	—	53	1	no	—
239	C₂×C₄²	—	53	1	no	—
240	C₂×C₄²	—	53	1	no	—
241	M₃₂	—	106	4	no	—
242	C₂×C₁₆	—	53	4	no	—
243	C₂×C₁₆	—	53	4	no	—
244	C₂×C₁₆	—	106	4	no	—
245	#97	—	53	1	no	—
246	#97	—	53	1	no	—
247	#97	—	53	1	no	—
248	#97	—	53	1	no	—
249	#97	C₂×#6157	1	1	yes	characteristic
250	#97	C₂×#6157	1	1	yes	characteristic
251	C₄×C₈	—	53	1	no	—
252	C₄×C₈	—	53	1	no	—
253	C₄×C₈	—	53	1	no	—
254	C₄×C₈	—	53	1	no	—
255	C₄×C₈	—	53	1	no	—
256	C₄×C₈	—	53	1	no	—
257	C₄×C₈	—	53	1	no	—
258	C₄×C₈	—	53	1	no	—
259	C₄×C₈	C₂×#6157	1	1	yes	characteristic
260	C₄×C₈	C₂×#6157	1	1	yes	characteristic
261	#41	—	106	2	no	—
262	#41	—	106	2	no	—
263	C₂×Q₈	—	53	1	no	—
264	C₂×D₄	—	53	1	no	—
265	C₂×D₄	—	53	1	no	—
266	C₂×D₄	—	53	1	no	—
267	C₂²×C₄	—	53	1	no	—
268	C₂²×C₄	—	53	2	no	—
269	C₂²×C₄	—	53	2	no	—
270	C₂²×C₄	—	53	2	no	—
271	C₂²×C₄	—	53	1	no	—
272	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
273	M₁₆	—	2	2	no	—
274	M₁₆	—	106	2	no	—
275	M₁₆	—	106	2	no	—
276	M₁₆	—	106	2	no	—
277	M₁₆	—	106	2	no	—
278	M₁₆	—	106	2	no	—
279	C₂×C₈	—	2	2	no	—
280	C₂×C₈	—	53	1	no	—
281	C₂×C₈	—	53	1	no	—
282	C₂×C₈	—	53	4	no	—
283	C₂×C₈	—	53	4	no	—
284	C₂×C₈	—	53	1	no	—
285	C₂×C₈	—	53	1	no	—
286	C₂×C₈	—	106	2	no	—
287	C₂×C₈	—	106	2	no	—
288	C₂×C₈	—	106	2	no	—
289	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
290	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
291	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
292	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
293	C₄⋊C₄	—	53	2	no	—
294	C₄⋊C₄	—	53	2	no	—
295	C₄⋊C₄	D₅₃⋊C₈	1	2	yes	—
296	C₂²⋊C₄	—	53	2	no	—
297	C₂²⋊C₄	—	53	2	no	—
298	C₂²⋊C₄	D₅₃⋊C₈	1	2	yes	—
299	C₄²	—	53	2	no	—
300	C₄²	—	53	2	no	—
301	C₄²	—	53	2	no	—
302	C₄²	—	53	1	no	—
303	C₄²	—	53	1	no	—
304	C₄²	C₂²×#6157	1	1	yes	characteristic
305	C₄²	C₂²×#6157	1	1	yes	characteristic
306	C₁₆	—	106	4	no	—
307	C₁₆	—	106	4	no	—
308	C₂³	—	53	1	no	—
309	C₂³	—	53	1	no	—
310	C₂³	C₂×D₅₃⋊C₈	1	1	yes	characteristic
311	Q₈	—	106	2	no	—
312	D₄	—	106	2	no	—
313	D₄	—	106	2	no	—
314	D₄	—	106	2	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	53	2	no	—
318	C₂×C₄	—	53	1	no	—
319	C₂×C₄	—	53	2	no	—
320	C₂×C₄	—	53	2	no	—
321	C₂×C₄	—	53	2	no	—
322	C₂×C₄	—	53	1	no	—
323	C₂×C₄	—	106	4	no	—
324	C₂×C₄	—	106	4	no	—
325	C₂×C₄	—	106	4	no	—
326	C₂×C₄	—	106	1	no	—
327	C₂×C₄	—	106	1	no	—
328	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
329	C₂×C₄	C₂×D₅₃⋊C₈	1	2	yes	—
330	C₂×C₄	C₂×D₅₃⋊C₈	1	2	yes	—
331	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
332	C₈	—	2	1	no	—
333	C₈	—	2	1	no	—
334	C₈	—	53	2	no	—
335	C₈	—	53	2	no	—
336	C₈	—	106	1	no	—
337	C₈	—	106	1	no	—
338	C₈	#13133296	1	2	yes	—
339	C₈	D₅₃⋊M₁₆	1	2	yes	—
340	C₂²	—	2	1	no	—
341	C₂²	—	2	2	no	—
342	C₂²	—	53	2	no	—
343	C₂²	—	106	2	no	—
344	C₂²	—	106	1	no	—
345	C₂²	—	106	2	no	—
346	C₂²	C₂²×D₅₃⋊C₈	1	1	yes	characteristic
347	C₄	—	2	1	no	—
348	C₄	—	2	2	no	—
349	C₄	—	2	2	no	—
350	C₄	—	53	2	no	—
351	C₄	—	53	4	no	—
352	C₄	—	106	1	no	—
353	C₄	—	106	2	no	—
354	C₄	C₂×#13133296	1	1	yes	characteristic
355	C₄	C₂×D₅₃⋊M₁₆	1	1	yes	characteristic
356	C₂	—	4	1	no	—
357	C₂	—	106	2	no	—
358	C₂	—	212	1	no	—
359	C₂	D₅₃⋊#431	1	1	yes	characteristic
360	C₂	D₅₃⋊#504	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂²×#6157
commutator, socle	C₂²×C₅₃	C₂³×C₈
Frattini	C₂×C₄	C₂³×#6157
Fitting	C₅₃×#405	C₄
prime radical	C₂×#11719945	C₂×C₄
cosocle	C₂×#88322	C₂⁴
hypercenter	#405	#6157
hypocenter	C₅₃	#33420
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ #405

Complements

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

N	H ≅ G/N	type
1. #20291082	C₂	semidirect
2. #20291080	C₂	semidirect
3. #20291075	C₂	semidirect
4. #20291073	C₂	semidirect
5. #20291051	C₂	semidirect
7. #20290922	C₂	semidirect
8. C₄×D₅₃⋊C₁₆	C₂	semidirect
215. C₅₃	#33420	semidirect

Maximal subgroups

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

subgroup	index
#20291082	2
#20291080	2
#20291075	2
#20291073	2
#20291051	2
#20290946	2
#20290922	2
C₄×D₅₃⋊C₁₆	2
#20283137	2
#33420	53