Subgroups — #27540925

Order 13568.

622 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20287107	C₂	1	1	yes	maximal, characteristic, complemented
2	#20286390	C₂	1	1	yes	maximal, characteristic
3	#20276899	C₂	1	2	yes	maximal, complemented
4	#20276859	C₂	1	1	yes	maximal, characteristic, complemented
5	#20276810	C₂	1	2	yes	maximal, complemented
6	#20276752	C₂	1	1	yes	maximal, characteristic, complemented
7	#20276645	C₂	1	2	yes	maximal, complemented
8	C₄×#13133162	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₆×D₂₁₂	C₂	1	1	yes	maximal, characteristic, complemented
10	#357×D₅₃	C₂	1	2	yes	maximal, complemented
11	C₅₃×#2240	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×D₈×D₅₃	C₂²	1	2	yes	—
13	#16811334	—	2	1	no	—
14	#16811258	—	2	1	no	—
15	C₂×#13133162	C₂²	1	1	yes	characteristic
16	C₂×#13133161	C₂²	1	2	yes	complemented
17	C₂×C₁₆×D₅₃	C₂²	1	2	yes	complemented
18	#16810960	—	2	2	no	—
19	#16810915	—	2	1	no	—
20	#16810865	C₂²	1	2	yes	complemented
21	#16810856	C₂²	1	2	yes	—
22	#16810815	—	2	1	no	—
23	#16810755	—	2	2	no	—
24	C₄×Dic₂₁₂	C₂²	1	1	yes	characteristic, complemented
25	C₄×D₄₂₄	C₂²	1	1	yes	characteristic
26	C₈×D₂₁₂	C₂²	1	1	yes	characteristic
27	#107×D₅₃	C₂²	1	2	yes	complemented
28	#16810451	C₂²	1	1	yes	characteristic, complemented
29	#16810450	C₂²	1	2	yes	complemented
30	C₄⋊#11719908	C₂²	1	1	yes	characteristic, complemented
31	#16810428	C₂²	1	1	yes	characteristic, complemented
32	#16810425	C₂²	1	1	yes	characteristic
33	#16810424	C₂²	1	2	yes	complemented
34	#16810423	C₂²	1	2	yes	—
35	#16810402	C₂²	1	2	yes	complemented
36	C₅₃×#500	C₂²	1	1	yes	characteristic, complemented
37	C₅₃×#493	C₂²	1	1	yes	characteristic
38	C₅₃×#357	C₂²	1	2	yes	complemented
39	C₅₃×#357	C₂²	1	2	yes	complemented
40	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic, complemented
41	C₂×D₄×D₅₃	—	2	2	no	—
42	C₂×D₄×D₅₃	—	4	1	no	—
43	C₂×Dic₂₁₂	C₂³	1	1	yes	characteristic
44	C₂×D₄₂₄	C₂³	1	1	yes	characteristic
45	C₂×#11719922	—	2	2	no	—
46	D₈×D₅₃	—	2	4	no	—
47	D₈×D₅₃	—	4	2	no	—
48	C₂×C₈×D₅₃	C₂³	1	2	yes	—
49	#13133208	—	4	1	no	—
50	#13133197	—	2	2	no	—
51	#13133193	—	4	2	no	—
52	#13133188	—	2	2	no	—
53	#13133188	—	4	1	no	—
54	#13133185	—	4	2	no	—
55	C₄×Dic₁₀₆	—	2	1	no	—
56	C₄×D₂₁₂	—	2	1	no	—
57	C₄×D₂₁₂	D₄	1	1	yes	characteristic
58	#13133179	—	4	1	no	—
59	C₄⋊C₄×D₅₃	—	2	2	no	—
60	#13133162	—	2	2	no	—
61	#13133161	—	4	2	no	—
62	C₂×#11719908	C₂³	1	2	yes	—
63	C₁₆×D₅₃	—	4	2	no	—
64	#13133151	C₂³	1	1	yes	characteristic
65	#13133151	C₂³	1	1	yes	characteristic
66	#13133150	—	2	2	no	—
67	C₄⋊#88322	D₄	1	1	yes	characteristic
68	#13133142	—	2	1	no	—
69	#13133140	—	2	2	no	—
70	#13133139	—	2	1	no	—
71	#13133138	—	2	2	no	—
72	#13133129	D₄	1	2	yes	—
73	C₂×D₈×C₅₃	—	2	1	no	—
74	C₂×D₈×C₅₃	C₂³	1	2	yes	—
75	C₄⋊Q₈×C₅₃	—	2	1	no	—
76	#127×C₅₃	—	2	1	no	—
77	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
78	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
79	#107×C₅₃	—	2	1	no	—
80	#107×C₅₃	C₂³	1	2	yes	—
81	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
82	C₂³×D₅₃	—	4	2	no	—
83	C₂×Q₈×C₅₃	—	4	1	no	—
84	C₂×D₄×C₅₃	—	2	2	no	—
85	C₂×D₄×C₅₃	—	4	1	no	—
86	C₂×D₄×C₅₃	—	4	1	no	—
87	C₂×C₂²⋊D₅₃	—	4	2	no	—
88	C₂×C₂²⋊D₅₃	—	4	2	no	—
89	D₄×D₅₃	—	4	4	no	—
90	D₄×D₅₃	—	8	2	no	—
91	D₄×D₅₃	—	8	4	no	—
92	C₂×D₂₁₂	—	2	1	no	—
93	C₂×D₂₁₂	D₈	1	1	yes	characteristic
94	C₂×C₄×D₅₃	—	4	1	no	—
95	C₂×C₄×D₅₃	—	4	2	no	—
96	C₂×C₄×D₅₃	C₂×D₄	1	2	yes	—
97	C₂×Dic₁₀₆	—	2	1	no	—
98	D₈×C₅₃	—	2	2	no	—
99	D₈×C₅₃	—	2	2	no	—
100	D₈×C₅₃	—	8	1	no	—
101	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
103	C₄⋊C₄×C₅₃	—	2	2	no	—
104	C₄⋊C₄×C₅₃	—	4	1	no	—
105	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
106	C₂²⋊Dic₅₃	—	4	2	no	—
107	#11719922	—	4	4	no	—
108	#11719921	—	4	2	no	—
109	#11719921	—	4	1	no	—
110	#11719921	D₈	1	2	yes	—
111	C₄⋊Dic₅₃	—	2	1	no	—
112	C₄⋊Dic₅₃	—	2	1	no	—
113	C₄⋊Dic₅₃	—	4	2	no	—
114	C₄⋊Dic₅₃	D₈	1	1	yes	characteristic
115	#11719919	—	4	1	no	—
116	#11719919	—	4	2	no	—
117	C₄×Dic₅₃	—	4	1	no	—
118	C₂×#88322	C₂×D₄	1	2	yes	—
119	Dic₂₁₂	—	2	1	no	—
120	Dic₂₁₂	—	2	1	no	—
121	D₄₂₄	—	2	1	no	—
122	D₄₂₄	#41	1	2	yes	—
123	C₈×D₅₃	—	2	2	no	—
124	C₈×D₅₃	—	2	2	no	—
125	C₁₆×C₅₃	—	2	1	no	—
126	C₁₆×C₅₃	—	2	1	no	—
127	#11719908	—	2	2	no	—
128	C₂³×C₅₃	—	4	2	no	—
129	C₂²×D₅₃	—	4	1	no	—
130	C₂²×D₅₃	—	8	4	no	—
131	C₂²×D₅₃	—	8	4	no	—
132	C₂²×D₅₃	—	16	2	no	—
133	C₂²×D₅₃	C₂×D₈	1	2	yes	—
134	Q₈×C₅₃	—	16	1	no	—
135	D₄×C₅₃	—	4	2	no	—
136	D₄×C₅₃	—	4	2	no	—
137	D₄×C₅₃	—	8	2	no	—
138	D₄×C₅₃	—	16	1	no	—
139	C₂×C₄×C₅₃	—	4	2	no	—
140	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
141	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
142	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
143	C₂²⋊D₅₃	—	8	4	no	—
144	C₂²⋊D₅₃	—	8	4	no	—
145	C₂×Dic₅₃	—	4	1	no	—
146	C₂×Dic₅₃	—	4	1	no	—
147	C₂×Dic₅₃	—	4	1	no	—
148	C₂×Dic₅₃	C₂×D₈	1	2	yes	—
149	D₂₁₂	—	2	2	no	—
150	D₂₁₂	—	4	1	no	—
151	D₂₁₂	SD₃₂	1	4	yes	complemented
152	C₄×D₅₃	—	2	2	no	—
153	C₄×D₅₃	—	2	2	no	—
154	C₄×D₅₃	—	8	2	no	—
155	C₄×D₅₃	—	16	2	no	—
156	Dic₁₀₆	—	4	1	no	—
157	Dic₁₀₆	—	4	1	no	—
158	C₈×C₅₃	—	2	1	no	—
159	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
160	C₈×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
161	#88322	—	2	2	no	—
162	#33862	—	53	1	no	maximal, Sylow
163	C₂²×C₅₃	—	8	2	no	—
164	C₂²×C₅₃	—	8	2	no	—
165	C₂²×C₅₃	—	8	2	no	—
166	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
167	C₂×D₅₃	—	2	2	no	—
168	C₂×D₅₃	—	2	2	no	—
169	C₂×D₅₃	—	4	2	no	—
170	C₂×D₅₃	—	8	1	no	—
171	C₂×D₅₃	—	8	1	no	—
172	C₂×D₅₃	—	16	8	no	—
173	C₂×D₅₃	C₂×SD₃₂	1	4	yes	—
174	C₄×C₅₃	—	2	1	no	—
175	C₄×C₅₃	—	8	2	no	—
176	C₄×C₅₃	C₂×SD₃₂	1	2	yes	—
177	C₄×C₅₃	#546	1	1	yes	characteristic
178	C₄×C₅₃	#577	1	1	yes	characteristic
179	Dic₅₃	—	2	2	no	—
180	Dic₅₃	—	8	1	no	—
181	Dic₅₃	—	8	1	no	—
182	Dic₅₃	—	8	1	no	—
183	#3338	—	53	1	no	—
184	D₄×D₈	—	53	1	no	—
185	#2240	D₅₃	1	1	yes	characteristic, complemented
186	#2233	—	53	2	no	—
187	#2226	—	53	1	no	—
188	#2217	—	53	2	no	—
189	#2207	—	53	1	no	—
190	#2187	—	53	2	no	—
191	C₄×SD₃₂	—	53	1	no	—
192	D₄×C₁₆	—	53	1	no	—
193	C₂×#357	—	53	2	no	—
194	C₂×C₅₃	—	16	2	no	—
195	C₂×C₅₃	#3294	1	1	yes	characteristic
196	C₂×C₅₃	C₂²×SD₃₂	1	1	yes	characteristic
197	C₂×C₅₃	#3415	1	1	yes	characteristic
198	D₅₃	—	2	4	no	—
199	D₅₃	—	8	2	no	—
200	C₂²×D₈	—	53	2	no	—
201	#554	—	106	1	no	—
202	D₄²	—	106	1	no	—
203	C₂×SD₃₂	—	53	2	no	—
204	C₂×SD₃₂	—	53	1	no	—
205	C₂²×C₁₆	—	53	2	no	—
206	#500	C₂×D₅₃	1	1	yes	characteristic, complemented
207	#493	C₂×D₅₃	1	1	yes	characteristic
208	#484	—	106	2	no	—
209	#477	—	106	1	no	—
210	#467	—	53	2	no	—
211	#466	—	53	2	no	—
212	#459	—	106	1	no	—
213	#447	—	106	2	no	—
214	C₄×Q₁₆	—	53	1	no	—
215	C₄×D₈	—	53	1	no	—
216	C₈×D₄	—	53	1	no	—
217	C₂×#107	—	53	2	no	—
218	#367	—	53	1	no	—
219	#367	—	53	2	no	—
220	C₄⋊C₁₆	—	53	1	no	—
221	#358	—	53	1	no	—
222	#357	—	53	2	no	—
223	#357	—	53	1	no	—
224	#357	—	53	2	no	—
225	#357	C₂×D₅₃	1	2	yes	complemented
226	#357	C₂×D₅₃	1	2	yes	complemented
227	C₂²⋊C₁₆	—	53	2	no	—
228	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic, complemented
229	C₅₃	#33862	1	1	yes	characteristic, complemented, Sylow
230	C₂²×D₄	—	106	2	no	—
231	C₂²×D₄	—	212	1	no	—
232	C₂×Q₁₆	—	53	1	no	—
233	C₂×D₈	—	2	1	no	—
234	C₂×D₈	—	53	1	no	—
235	C₂×D₈	—	106	2	no	—
236	C₂×D₈	—	106	4	no	—
237	C₂×D₈	—	212	2	no	—
238	C₂×D₈	C₂²×D₅₃	1	2	yes	—
239	C₂²×C₈	—	53	2	no	—
240	C₄⋊Q₈	—	2	1	no	—
241	#127	—	2	1	no	—
242	#125	—	212	1	no	—
243	C₂²⋊Q₈	—	106	2	no	—
244	C₂²⋊Q₈	—	212	2	no	—
245	#121	—	106	2	no	—
246	#121	—	212	1	no	—
247	#120	—	212	2	no	—
248	C₄×Q₈	—	106	1	no	—
249	C₄×D₄	—	53	1	no	—
250	C₄×D₄	—	106	1	no	—
251	C₄×D₄	—	212	1	no	—
252	C₂×C₄⋊C₄	—	106	2	no	—
253	SD₃₂	—	106	2	no	—
254	SD₃₂	—	212	2	no	—
255	C₂×C₁₆	—	53	2	no	—
256	C₂×C₁₆	—	212	2	no	—
257	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
258	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
259	#107	—	2	1	no	—
260	#107	—	53	1	no	—
261	#107	—	53	1	no	—
262	#107	—	106	2	no	—
263	#107	C₂²×D₅₃	1	2	yes	—
264	C₄⋊C₈	—	53	1	no	—
265	#103	—	106	2	no	—
266	#103	—	106	1	no	—
267	#102	—	106	2	no	—
268	#102	—	106	1	no	—
269	C₂²⋊C₈	—	53	2	no	—
270	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
271	C₂⁴	—	212	2	no	—
272	C₂×Q₈	—	4	1	no	—
273	C₂×Q₈	—	106	1	no	—
274	C₂×D₄	—	2	2	no	—
275	C₂×D₄	—	4	1	no	—
276	C₂×D₄	—	4	1	no	—
277	C₂×D₄	—	53	1	no	—
278	C₂×D₄	—	106	1	no	—
279	C₂×D₄	—	424	2	no	—
280	C₂×D₄	—	424	4	no	—
281	C₂×D₄	—	212	2	no	—
282	C₂×D₄	—	212	4	no	—
283	C₂×D₄	—	212	2	no	—
284	C₂²×C₄	—	53	2	no	—
285	C₂²×C₄	—	212	2	no	—
286	C₂²×C₄	—	212	1	no	—
287	Q₁₆	—	106	1	no	—
288	Q₁₆	—	106	1	no	—
289	D₈	—	2	2	no	—
290	D₈	—	2	2	no	—
291	D₈	—	8	1	no	—
292	D₈	—	53	2	no	—
293	D₈	—	106	1	no	—
294	D₈	—	212	4	no	—
295	C₂×C₈	—	53	2	no	—
296	C₂×C₈	—	106	2	no	—
297	C₂×C₈	—	106	2	no	—
298	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
299	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
300	C₄⋊C₄	—	2	2	no	—
301	C₄⋊C₄	—	4	1	no	—
302	C₄⋊C₄	—	53	1	no	—
303	C₄⋊C₄	—	106	1	no	—
304	C₄⋊C₄	—	106	1	no	—
305	C₄⋊C₄	—	212	2	no	—
306	C₄⋊C₄	—	212	2	no	—
307	C₄⋊C₄	—	212	1	no	—
308	C₂²⋊C₄	—	53	2	no	—
309	C₂²⋊C₄	—	212	2	no	—
310	C₂²⋊C₄	—	212	2	no	—
311	C₂²⋊C₄	—	212	1	no	—
312	C₄²	—	212	1	no	—
313	C₄²	D₄×D₅₃	1	1	yes	characteristic
314	C₁₆	—	2	1	no	—
315	C₁₆	—	2	1	no	—
316	C₁₆	—	106	2	no	—
317	C₂³	—	4	2	no	—
318	C₂³	—	53	2	no	—
319	C₂³	—	848	2	no	—
320	C₂³	—	424	4	no	—
321	C₂³	—	424	4	no	—
322	C₂³	—	212	1	no	—
323	Q₈	—	16	1	no	—
324	Q₈	—	212	1	no	—
325	Q₈	—	212	1	no	—
326	D₄	—	4	2	no	—
327	D₄	—	4	2	no	—
328	D₄	—	8	2	no	—
329	D₄	—	16	1	no	—
330	D₄	—	53	4	no	—
331	D₄	—	106	2	no	—
332	D₄	—	424	4	no	—
333	D₄	—	424	4	no	—
334	D₄	—	212	1	no	—
335	C₂×C₄	—	4	2	no	—
336	C₂×C₄	—	53	2	no	—
337	C₂×C₄	—	848	2	no	—
338	C₂×C₄	—	106	2	no	—
339	C₂×C₄	—	106	2	no	—
340	C₂×C₄	—	424	2	no	—
341	C₂×C₄	—	212	1	no	—
342	C₂×C₄	—	212	1	no	—
343	C₂×C₄	—	212	1	no	—
344	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
345	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
346	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
347	C₈	—	2	1	no	—
348	C₈	—	106	2	no	—
349	C₈	#41×D₅₃	1	1	yes	characteristic
350	C₈	#13133263	1	1	yes	characteristic
351	C₂²	—	8	2	no	—
352	C₂²	—	8	2	no	—
353	C₂²	—	8	2	no	—
354	C₂²	—	53	4	no	—
355	C₂²	—	848	8	no	—
356	C₂²	—	106	2	no	—
357	C₂²	—	106	2	no	—
358	C₂²	—	424	1	no	—
359	C₂²	—	424	1	no	—
360	C₂²	—	212	2	no	—
361	C₂²	C₂×D₈×D₅₃	1	1	yes	characteristic
362	C₄	—	2	1	no	—
363	C₄	—	8	2	no	—
364	C₄	—	106	2	no	—
365	C₄	—	424	1	no	—
366	C₄	—	424	1	no	—
367	C₄	—	424	1	no	—
368	C₄	SD₃₂×D₅₃	1	2	yes	—
369	C₄	#16811266	1	1	yes	characteristic
370	C₄	#16811429	1	1	yes	characteristic
371	C₂	—	16	2	no	—
372	C₂	—	106	4	no	—
373	C₂	—	424	2	no	—
374	C₂	#20286527	1	1	yes	characteristic
375	C₂	C₂×SD₃₂×D₅₃	1	1	yes	characteristic
376	C₂	#20287852	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₅₃
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2240	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	#20286390	C₂
hypercenter	#2240	D₅₃
hypocenter	C₅₃	#33862
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2240

Complements

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

N	H ≅ G/N	type
1. #20287107	C₂	semidirect
3. #20276899	C₂	semidirect
4. #20276859	C₂	semidirect
5. #20276810	C₂	semidirect
6. #20276752	C₂	semidirect
7. #20276645	C₂	semidirect
8. C₄×#13133162	C₂	semidirect
9. C₁₆×D₂₁₂	C₂	semidirect
10. #357×D₅₃	C₂	semidirect
11. C₅₃×#2240	C₂	semidirect
16. C₂×#13133161	C₂²	semidirect
17. C₂×C₁₆×D₅₃	C₂²	semidirect
20. #16810865	C₂²	semidirect
24. C₄×Dic₂₁₂	C₂²	semidirect
27. #107×D₅₃	C₂²	semidirect
28. #16810451	C₂²	semidirect
29. #16810450	C₂²	semidirect
30. C₄⋊#11719908	C₂²	semidirect
31. #16810428	C₂²	semidirect
33. #16810424	C₂²	semidirect
35. #16810402	C₂²	semidirect
36. C₅₃×#500	C₂²	semidirect
38. C₅₃×#357	C₂²	semidirect
39. C₅₃×#357	C₂²	semidirect
40. C₄×C₁₆×C₅₃	C₂²	semidirect
151. D₂₁₂	SD₃₂	semidirect
185. #2240	D₅₃	semidirect
206. #500	C₂×D₅₃	semidirect
225. #357	C₂×D₅₃	semidirect
226. #357	C₂×D₅₃	semidirect
228. C₄×C₁₆	C₂×D₅₃	semidirect
229. C₅₃	#33862	semidirect

Maximal subgroups

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

subgroup	index
#20287107	2
#20286390	2
#20276899	2
#20276859	2
#20276810	2
#20276752	2
#20276645	2
C₄×#13133162	2
C₁₆×D₂₁₂	2
#357×D₅₃	2
C₅₃×#2240	2
#33862	53