Subgroups — #27540901

Order 13568.

636 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20287865	C₂	1	1	yes	maximal, characteristic, complemented
2	#20286517	C₂	1	2	yes	maximal, complemented
3	#20276966	C₂	1	1	yes	maximal, characteristic, complemented
4	#20276840	C₂	1	1	yes	maximal, characteristic, complemented
5	#20276837	C₂	1	1	yes	maximal, characteristic, complemented
6	#20276794	C₂	1	2	yes	maximal, complemented
7	#20276693	C₂	1	2	yes	maximal, complemented
8	#20276690	C₂	1	2	yes	maximal, complemented
9	#20276596	C₂	1	1	yes	maximal, characteristic, complemented
10	#20276550	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#3417	C₂	1	1	yes	maximal, characteristic, complemented
12	#16811438	—	2	2	no	—
13	#16811437	C₂²	1	2	yes	—
14	#16811063	C₂²	1	2	yes	—
15	#16811062	—	2	2	no	—
16	#16811052	—	2	1	no	—
17	C₂×Dic₄₂₄	C₂²	1	1	yes	characteristic
18	C₂×#13133163	C₂²	1	2	yes	—
19	Q₃₂×D₅₃	—	2	1	no	—
20	#16811039	C₂²	1	1	yes	characteristic
21	#16811001	C₂²	1	2	yes	—
22	#16810898	C₂²	1	2	yes	—
23	#16810896	—	2	2	no	—
24	#16810895	C₂²	1	2	yes	—
25	#16810806	—	2	2	no	—
26	#16810804	—	2	2	no	—
27	#16810740	C₂²	1	2	yes	—
28	#16810738	C₂²	1	1	yes	characteristic
29	#16810453	C₂²	1	1	yes	characteristic
30	#16810443	C₂²	1	1	yes	characteristic
31	#16810440	C₂²	1	2	yes	—
32	#16810439	C₂²	1	2	yes	—
33	#16810430	C₂²	1	2	yes	—
34	#16810408	C₂²	1	1	yes	characteristic
35	#16810407	C₂²	1	1	yes	characteristic
36	C₁₆×Dic₅₃	C₂²	1	1	yes	characteristic
37	C₅₃×#578	C₂²	1	2	yes	—
38	C₅₃×#510	—	2	2	no	—
39	C₅₃×#510	C₂²	1	2	yes	—
40	C₅₃×#508	—	2	1	no	—
41	C₅₃×#508	C₂²	1	1	yes	characteristic
42	C₂×Q₃₂×C₅₃	—	2	1	no	—
43	C₂×Q₃₂×C₅₃	C₂²	1	1	yes	characteristic
44	C₅₃×#504	C₂²	1	1	yes	characteristic
45	#13133264	—	2	2	no	—
46	#13133248	—	2	2	no	—
47	#13133248	—	4	2	no	—
48	#13133247	—	2	2	no	—
49	#13133247	—	2	2	no	—
50	#13133247	—	4	2	no	—
51	#13133246	—	4	2	no	—
52	#13133237	—	4	2	no	—
53	#13133237	D₄	1	1	yes	characteristic
54	#13133236	—	2	2	no	—
55	C₂×Dic₂₁₂	C₂³	1	2	yes	—
56	C₂×#11719925	—	2	2	no	—
57	C₂×#11719925	—	4	2	no	—
58	Q₁₆×D₅₃	—	2	2	no	—
59	#13133224	—	2	2	no	—
60	#13133223	C₂³	1	1	yes	characteristic
61	#13133215	—	2	2	no	—
62	Dic₄₂₄	—	2	1	no	—
63	Dic₄₂₄	—	2	1	no	—
64	#13133163	—	2	2	no	—
65	#13133163	—	4	2	no	—
66	#13133161	—	2	2	no	—
67	#13133157	C₂³	1	1	yes	characteristic
68	#13133156	—	2	1	no	—
69	C₂×#11719908	C₂³	1	1	yes	characteristic
70	C₁₆×D₅₃	—	2	1	no	—
71	#13133153	C₂³	1	2	yes	—
72	#13133152	D₄	1	1	yes	characteristic
73	#13133144	—	2	2	no	—
74	#13133144	D₄	1	1	yes	characteristic
75	#13133143	D₄	1	1	yes	characteristic
76	#13133136	—	2	2	no	—
77	#13133135	—	2	2	no	—
78	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
79	2₋¹⁺⁴×C₅₃	—	2	2	no	—
80	#137×C₅₃	—	2	2	no	—
81	#137×C₅₃	—	4	2	no	—
82	#135×C₅₃	—	2	2	no	—
83	#135×C₅₃	C₂³	1	2	yes	—
84	C₂×Q₁₆×C₅₃	—	2	2	no	—
85	C₂×Q₁₆×C₅₃	C₂³	1	2	yes	—
86	#131×C₅₃	C₂³	1	1	yes	characteristic
87	Q₃₂×C₅₃	—	2	2	no	—
88	Q₃₂×C₅₃	—	2	1	no	—
89	Q₃₂×C₅₃	—	2	1	no	—
90	Q₃₂×C₅₃	D₄	1	1	yes	characteristic
91	SD₃₂×C₅₃	—	2	2	no	—
92	SD₃₂×C₅₃	D₄	1	2	yes	—
93	D₁₆×C₅₃	D₄	1	1	yes	characteristic
94	M₃₂×C₅₃	—	2	1	no	—
95	M₃₂×C₅₃	C₂³	1	1	yes	characteristic
96	C₂×C₁₆×C₅₃	—	2	1	no	—
97	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
98	#41×C₅₃	—	2	2	no	—
99	#41×C₅₃	—	4	2	no	—
100	#41×C₅₃	—	4	2	no	—
101	#41×C₅₃	—	8	2	no	—
102	#41×C₅₃	C₂×D₄	1	1	yes	characteristic
103	C₂×Q₈×C₅₃	—	2	2	no	—
104	C₂×Q₈×C₅₃	—	4	2	no	—
105	C₂×Q₈×C₅₃	—	4	2	no	—
106	Q₈×D₅₃	—	4	2	no	—
107	#11719947	—	4	2	no	—
108	#11719947	—	8	2	no	—
109	#11719945	—	4	2	no	—
110	#11719945	C₂×D₄	1	1	yes	characteristic
111	C₂×Dic₁₀₆	—	2	2	no	—
112	C₂×Dic₁₀₆	—	4	2	no	—
113	Q₁₆×C₅₃	—	2	2	no	—
114	Q₁₆×C₅₃	—	2	2	no	—
115	Q₁₆×C₅₃	—	4	2	no	—
116	Q₁₆×C₅₃	C₂×D₄	1	2	yes	—
117	SD₁₆×C₅₃	—	2	2	no	—
118	SD₁₆×C₅₃	—	4	2	no	—
119	D₈×C₅₃	C₂×D₄	1	2	yes	—
120	M₁₆×C₅₃	—	2	1	no	—
121	M₁₆×C₅₃	C₂×D₄	1	1	yes	characteristic
122	C₂×C₈×C₅₃	—	2	1	no	—
123	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
124	#11719925	—	4	2	no	—
125	#11719925	—	4	2	no	—
126	#11719925	—	4	2	no	—
127	#11719925	—	4	2	no	—
128	#11719925	—	4	2	no	—
129	#11719925	D₈	1	1	yes	characteristic
130	#11719924	—	4	2	no	—
131	#11719924	D₈	1	1	yes	characteristic
132	#11719923	—	4	2	no	—
133	#11719923	—	4	2	no	—
134	#11719923	D₈	1	1	yes	characteristic
135	#11719922	D₈	1	1	yes	characteristic
136	#11719919	—	8	2	no	—
137	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
138	#11719917	—	2	2	no	—
139	#11719917	—	4	2	no	—
140	#11719917	C₂×D₄	1	1	yes	characteristic
141	C₂×#88322	—	4	2	no	—
142	C₂×#88322	C₂×D₄	1	1	yes	characteristic
143	Dic₂₁₂	—	2	2	no	—
144	Dic₂₁₂	—	2	2	no	—
145	#11719912	—	2	1	no	—
146	C₈×D₅₃	—	2	1	no	—
147	C₁₆×C₅₃	—	2	1	no	—
148	C₁₆×C₅₃	C₂×D₄	1	1	yes	characteristic
149	C₁₆×C₅₃	C₂×D₄	1	1	yes	characteristic
150	#11719908	—	2	1	no	—
151	#11719908	—	2	1	no	—
152	Q₈×C₅₃	—	2	2	no	—
153	Q₈×C₅₃	—	4	2	no	—
154	Q₈×C₅₃	—	4	2	no	—
155	Q₈×C₅₃	—	8	2	no	—
156	Q₈×C₅₃	C₂×D₈	1	1	yes	characteristic
157	D₄×C₅₃	—	2	1	no	—
158	D₄×C₅₃	—	2	2	no	—
159	D₄×C₅₃	—	4	2	no	—
160	D₄×C₅₃	—	8	2	no	—
161	D₄×C₅₃	C₂×D₈	1	1	yes	characteristic
162	C₂×C₄×C₅₃	—	2	1	no	—
163	C₂×C₄×C₅₃	—	4	2	no	—
164	C₂×C₄×C₅₃	—	4	2	no	—
165	C₂×C₄×C₅₃	—	8	2	no	—
166	C₂×C₄×C₅₃	—	8	2	no	—
167	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
168	C₂²⋊D₅₃	—	2	1	no	—
169	C₂²⋊D₅₃	—	8	2	no	—
170	C₂×Dic₅₃	—	2	1	no	—
171	C₂×Dic₅₃	—	4	2	no	—
172	C₂×Dic₅₃	—	8	2	no	—
173	D₂₁₂	C₂×D₈	1	1	yes	characteristic
174	C₄×D₅₃	—	2	1	no	—
175	C₄×D₅₃	—	8	2	no	—
176	Dic₁₀₆	—	4	2	no	—
177	Dic₁₀₆	—	4	2	no	—
178	Dic₁₀₆	—	8	2	no	—
179	Dic₁₀₆	C₂×D₈	1	1	yes	characteristic
180	C₈×C₅₃	—	2	1	no	—
181	C₈×C₅₃	C₂²×D₄	1	1	yes	characteristic
182	C₈×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
183	#88322	—	2	1	no	—
184	#88322	—	4	2	no	—
185	#88322	C₂×D₈	1	1	yes	characteristic
186	#88322	C₂×D₈	1	1	yes	characteristic
187	#33860	—	53	1	no	maximal, Sylow
188	C₂²×C₅₃	—	2	1	no	—
189	C₂²×C₅₃	—	4	2	no	—
190	C₂²×C₅₃	#576	1	1	yes	characteristic
191	C₂×D₅₃	—	2	1	no	—
192	C₄×C₅₃	—	2	1	no	—
193	C₄×C₅₃	—	4	2	no	—
194	C₄×C₅₃	—	8	2	no	—
195	C₄×C₅₃	D₄²	1	1	yes	characteristic
196	C₄×C₅₃	C₂²×D₈	1	1	yes	characteristic
197	Dic₅₃	—	2	2	no	—
198	Dic₅₃	—	2	1	no	—
199	Dic₅₃	—	8	2	no	—
200	#3417	—	53	1	no	—
201	#3417	D₅₃	1	1	yes	characteristic, complemented
202	#3293	—	53	2	no	—
203	#2247	—	53	1	no	—
204	#2222	—	53	1	no	—
205	#2222	—	53	1	no	—
206	#2214	—	53	2	no	—
207	#2195	—	53	2	no	—
208	#2195	—	53	2	no	—
209	#2178	—	53	1	no	—
210	#2170	—	53	1	no	—
211	C₂×C₅₃	—	2	1	no	—
212	C₂×C₅₃	—	4	1	no	—
213	C₂×C₅₃	—	8	2	no	—
214	C₂×C₅₃	D₄×D₈	1	1	yes	characteristic
215	D₅₃	—	4	1	no	—
216	#578	—	53	2	no	—
217	#578	—	106	2	no	—
218	#578	C₂×D₅₃	1	2	yes	—
219	#510	—	2	2	no	—
220	#510	—	53	2	no	—
221	#510	—	106	2	no	—
222	#510	C₂×D₅₃	1	2	yes	—
223	#508	—	2	1	no	—
224	#508	—	106	1	no	—
225	#508	C₂×D₅₃	1	1	yes	characteristic
226	C₂×Q₃₂	—	2	1	no	—
227	C₂×Q₃₂	—	53	2	no	—
228	C₂×Q₃₂	—	53	1	no	—
229	C₂×Q₃₂	—	106	1	no	—
230	C₂×Q₃₂	C₂×D₅₃	1	1	yes	characteristic
231	#504	—	53	1	no	—
232	#504	C₂×D₅₃	1	1	yes	characteristic
233	#494	—	53	2	no	—
234	#473	—	53	2	no	—
235	#473	—	53	2	no	—
236	#473	—	106	2	no	—
237	#456	—	106	2	no	—
238	#456	—	106	2	no	—
239	#443	—	53	1	no	—
240	#443	—	53	2	no	—
241	#368	—	53	1	no	—
242	#364	—	53	1	no	—
243	#362	—	53	2	no	—
244	#362	—	53	2	no	—
245	#359	—	53	2	no	—
246	#350	—	53	1	no	—
247	#350	—	53	1	no	—
248	C₄×C₁₆	—	53	1	no	—
249	C₅₃	#33860	1	1	yes	characteristic, complemented, Sylow
250	2₋¹⁺⁴	—	2	2	no	—
251	2₋¹⁺⁴	—	106	2	no	—
252	#137	—	2	2	no	—
253	#137	—	4	2	no	—
254	#137	—	106	2	no	—
255	#137	—	106	2	no	—
256	#137	—	106	2	no	—
257	#137	—	212	2	no	—
258	#137	—	212	2	no	—
259	#137	—	212	2	no	—
260	#135	—	2	2	no	—
261	#135	—	53	1	no	—
262	#135	—	106	2	no	—
263	#135	—	212	2	no	—
264	#135	C₂²×D₅₃	1	2	yes	—
265	C₂×Q₁₆	—	2	2	no	—
266	C₂×Q₁₆	—	53	2	no	—
267	C₂×Q₁₆	—	106	2	no	—
268	C₂×Q₁₆	—	106	2	no	—
269	C₂×Q₁₆	—	212	2	no	—
270	C₂×Q₁₆	C₂²×D₅₃	1	2	yes	—
271	#131	—	53	1	no	—
272	#131	—	106	2	no	—
273	#131	C₂²×D₅₃	1	1	yes	characteristic
274	C₄⋊Q₈	—	106	2	no	—
275	Q₃₂	—	2	2	no	—
276	Q₃₂	—	2	1	no	—
277	Q₃₂	—	2	1	no	—
278	Q₃₂	—	106	2	no	—
279	Q₃₂	—	106	1	no	—
280	Q₃₂	—	106	1	no	—
281	Q₃₂	—	212	2	no	—
282	Q₃₂	C₂²⋊D₅₃	1	1	yes	characteristic
283	SD₃₂	—	2	2	no	—
284	SD₃₂	—	106	2	no	—
285	SD₃₂	C₂²⋊D₅₃	1	2	yes	—
286	D₁₆	C₂²⋊D₅₃	1	1	yes	characteristic
287	M₃₂	—	2	1	no	—
288	M₃₂	—	53	1	no	—
289	M₃₂	—	106	1	no	—
290	M₃₂	C₂²×D₅₃	1	1	yes	characteristic
291	C₂×C₁₆	—	2	1	no	—
292	C₂×C₁₆	—	53	1	no	—
293	C₂×C₁₆	—	106	1	no	—
294	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
295	#108	—	53	1	no	—
296	#108	—	53	2	no	—
297	#104	—	53	1	no	—
298	#104	—	53	1	no	—
299	#104	—	106	2	no	—
300	#101	—	106	2	no	—
301	#101	—	106	2	no	—
302	C₄×C₈	—	53	1	no	—
303	#41	—	2	2	no	—
304	#41	—	4	2	no	—
305	#41	—	4	2	no	—
306	#41	—	8	2	no	—
307	#41	—	53	1	no	—
308	#41	—	424	2	no	—
309	#41	—	212	2	no	—
310	#41	—	212	2	no	—
311	#41	D₄×D₅₃	1	1	yes	characteristic
312	C₂×Q₈	—	2	2	no	—
313	C₂×Q₈	—	4	2	no	—
314	C₂×Q₈	—	4	2	no	—
315	C₂×Q₈	—	106	2	no	—
316	C₂×Q₈	—	212	2	no	—
317	C₂×Q₈	—	212	2	no	—
318	Q₁₆	—	2	2	no	—
319	Q₁₆	—	2	2	no	—
320	Q₁₆	—	4	2	no	—
321	Q₁₆	—	53	1	no	—
322	Q₁₆	—	106	2	no	—
323	Q₁₆	—	106	2	no	—
324	Q₁₆	—	212	2	no	—
325	Q₁₆	—	212	2	no	—
326	Q₁₆	—	212	2	no	—
327	Q₁₆	—	212	2	no	—
328	Q₁₆	—	212	2	no	—
329	Q₁₆	C₂×C₂²⋊D₅₃	1	2	yes	—
330	SD₁₆	—	2	2	no	—
331	SD₁₆	—	4	2	no	—
332	SD₁₆	—	53	1	no	—
333	SD₁₆	—	53	1	no	—
334	SD₁₆	—	212	2	no	—
335	SD₁₆	—	212	2	no	—
336	SD₁₆	—	212	2	no	—
337	D₈	—	53	1	no	—
338	D₈	C₂×C₂²⋊D₅₃	1	2	yes	—
339	M₁₆	—	2	1	no	—
340	M₁₆	—	53	1	no	—
341	M₁₆	—	106	1	no	—
342	M₁₆	—	106	2	no	—
343	M₁₆	—	212	2	no	—
344	M₁₆	D₄×D₅₃	1	1	yes	characteristic
345	C₂×C₈	—	2	1	no	—
346	C₂×C₈	—	53	1	no	—
347	C₂×C₈	—	106	1	no	—
348	C₂×C₈	—	212	2	no	—
349	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
350	C₄⋊C₄	—	424	2	no	—
351	C₄²	—	53	1	no	—
352	C₁₆	—	2	1	no	—
353	C₁₆	—	106	1	no	—
354	C₁₆	—	106	1	no	—
355	C₁₆	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
356	C₁₆	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
357	Q₈	—	2	2	no	—
358	Q₈	—	4	2	no	—
359	Q₈	—	4	2	no	—
360	Q₈	—	8	2	no	—
361	Q₈	—	53	1	no	—
362	Q₈	—	424	2	no	—
363	Q₈	—	212	2	no	—
364	Q₈	—	212	2	no	—
365	Q₈	D₈×D₅₃	1	1	yes	characteristic
366	D₄	—	2	1	no	—
367	D₄	—	2	2	no	—
368	D₄	—	4	2	no	—
369	D₄	—	8	2	no	—
370	D₄	—	53	1	no	—
371	D₄	—	106	1	no	—
372	D₄	—	424	2	no	—
373	D₄	D₈×D₅₃	1	1	yes	characteristic
374	C₂×C₄	—	2	1	no	—
375	C₂×C₄	—	4	2	no	—
376	C₂×C₄	—	4	2	no	—
377	C₂×C₄	—	8	2	no	—
378	C₂×C₄	—	8	2	no	—
379	C₂×C₄	—	106	1	no	—
380	C₂×C₄	—	106	1	no	—
381	C₂×C₄	—	424	2	no	—
382	C₂×C₄	—	424	2	no	—
383	C₂×C₄	—	212	2	no	—
384	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
385	C₈	—	2	1	no	—
386	C₈	—	53	1	no	—
387	C₈	—	53	1	no	—
388	C₈	—	106	1	no	—
389	C₈	—	212	2	no	—
390	C₈	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
391	C₈	#13133262	1	1	yes	characteristic
392	C₂²	—	2	1	no	—
393	C₂²	—	4	2	no	—
394	C₂²	—	106	1	no	—
395	C₂²	#16811424	1	1	yes	characteristic
396	C₄	—	2	1	no	—
397	C₄	—	4	2	no	—
398	C₄	—	8	2	no	—
399	C₄	—	106	2	no	—
400	C₄	—	106	1	no	—
401	C₄	—	424	2	no	—
402	C₄	D₄×C₂²⋊D₅₃	1	1	yes	characteristic
403	C₄	C₂×D₈×D₅₃	1	1	yes	characteristic
404	C₂	—	2	1	no	—
405	C₂	—	4	1	no	—
406	C₂	—	8	2	no	—
407	C₂	—	212	1	no	—
408	C₂	D₈×C₂²⋊D₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₈×C₂²⋊D₅₃
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#3417	C₂
socle	C₂×C₅₃	D₄×D₈
hypercenter	#3417	D₅₃
hypocenter	C₅₃	#33860
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #131 ◁ #3417

Complements

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

N	H ≅ G/N	type
1. #20287865	C₂	semidirect
2. #20286517	C₂	semidirect
3. #20276966	C₂	semidirect
4. #20276840	C₂	semidirect
5. #20276837	C₂	semidirect
6. #20276794	C₂	semidirect
7. #20276693	C₂	semidirect
8. #20276690	C₂	semidirect
9. #20276596	C₂	semidirect
10. #20276550	C₂	semidirect
11. C₅₃×#3417	C₂	semidirect
201. #3417	D₅₃	semidirect
249. C₅₃	#33860	semidirect

Maximal subgroups

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

subgroup	index
#20287865	2
#20286517	2
#20276966	2
#20276840	2
#20276837	2
#20276794	2
#20276693	2
#20276690	2
#20276596	2
#20276550	2
C₅₃×#3417	2
#33860	53