Subgroups — #27540904

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	#20287868	C₂	1	1	yes	maximal, characteristic, complemented
2	#20287864	C₂	1	1	yes	maximal, characteristic, complemented
3	#20286518	C₂	1	2	yes	maximal, complemented
4	#20276966	C₂	1	1	yes	maximal, characteristic, complemented
5	#20276841	C₂	1	1	yes	maximal, characteristic, complemented
6	#20276795	C₂	1	2	yes	maximal, complemented
7	#20276696	C₂	1	2	yes	maximal, complemented
8	#20276691	C₂	1	2	yes	maximal, complemented
9	#20276595	C₂	1	1	yes	maximal, characteristic, complemented
10	#20276550	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#2222	C₂	1	1	yes	maximal, characteristic, complemented
12	#16811440	C₂²	1	2	yes	—
13	#16811439	—	2	2	no	—
14	#16811436	C₂²	1	2	yes	—
15	#16811066	—	2	2	no	—
16	#16811066	C₂²	1	2	yes	—
17	#16811065	—	2	2	no	—
18	#16811064	C₂²	1	2	yes	—
19	#16811056	—	2	1	no	—
20	#16811055	—	2	1	no	—
21	#16811055	C₂²	1	1	yes	characteristic
22	C₂×Dic₄₂₄	—	2	1	no	—
23	C₂×Dic₄₂₄	C₂²	1	1	yes	characteristic
24	C₂×#13133163	C₂²	1	2	yes	—
25	Q₃₂×D₅₃	—	2	1	no	—
26	#16811039	C₂²	1	1	yes	characteristic
27	#16811001	C₂²	1	2	yes	—
28	#16810899	C₂²	1	2	yes	—
29	#16810897	—	2	2	no	—
30	#16810808	—	2	2	no	—
31	#16810804	—	2	2	no	—
32	#16810739	C₂²	1	2	yes	—
33	#16810738	C₂²	1	1	yes	characteristic
34	#16810443	C₂²	1	1	yes	characteristic
35	#16810438	C₂²	1	2	yes	—
36	#16810429	C₂²	1	2	yes	—
37	#16810408	C₂²	1	1	yes	characteristic
38	#16810407	C₂²	1	1	yes	characteristic
39	C₁₆×Dic₅₃	C₂²	1	1	yes	characteristic
40	C₂×Q₃₂×C₅₃	C₂²	1	1	yes	characteristic
41	C₅₃×#504	C₂²	1	1	yes	characteristic
42	C₅₃×#473	C₂²	1	2	yes	—
43	C₅₃×#368	C₂²	1	1	yes	characteristic
44	C₅₃×#362	C₂²	1	2	yes	—
45	#13133265	—	2	2	no	—
46	#13133264	—	2	2	no	—
47	#13133250	—	2	2	no	—
48	#13133250	—	2	2	no	—
49	#13133250	—	4	2	no	—
50	#13133249	—	4	2	no	—
51	#13133249	—	4	2	no	—
52	#13133248	—	2	2	no	—
53	#13133246	—	4	2	no	—
54	#13133240	—	2	2	no	—
55	#13133239	—	2	2	no	—
56	#13133239	C₂³	1	2	yes	—
57	#13133238	—	4	2	no	—
58	#13133237	D₄	1	1	yes	characteristic
59	C₂×Dic₂₁₂	—	2	2	no	—
60	C₂×Dic₂₁₂	C₂³	1	2	yes	—
61	C₂×#11719925	—	2	2	no	—
62	Q₁₆×D₅₃	—	2	2	no	—
63	Q₁₆×D₅₃	—	4	2	no	—
64	#13133225	—	2	2	no	—
65	#13133223	C₂³	1	1	yes	characteristic
66	#13133215	—	2	2	no	—
67	Dic₄₂₄	—	2	2	no	—
68	Dic₄₂₄	—	2	1	no	—
69	Dic₄₂₄	—	2	1	no	—
70	Dic₄₂₄	D₄	1	1	yes	characteristic
71	#13133163	—	2	2	no	—
72	#13133163	—	4	2	no	—
73	#13133162	—	2	2	no	—
74	#13133162	D₄	1	2	yes	—
75	#13133160	—	2	2	no	—
76	D₈₄₈	D₄	1	1	yes	characteristic
77	#13133157	C₂³	1	1	yes	characteristic
78	#13133156	—	2	1	no	—
79	C₂×#11719908	C₂³	1	1	yes	characteristic
80	C₁₆×D₅₃	—	2	1	no	—
81	#13133152	D₄	1	1	yes	characteristic
82	#13133144	D₄	1	1	yes	characteristic
83	#13133143	—	2	2	no	—
84	#13133143	D₄	1	1	yes	characteristic
85	#13133135	—	2	2	no	—
86	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
87	#137×C₅₃	—	2	2	no	—
88	C₂×Q₁₆×C₅₃	C₂³	1	2	yes	—
89	#131×C₅₃	C₂³	1	1	yes	characteristic
90	Q₃₂×C₅₃	—	2	1	no	—
91	Q₃₂×C₅₃	—	2	1	no	—
92	M₃₂×C₅₃	—	2	1	no	—
93	M₃₂×C₅₃	C₂³	1	1	yes	characteristic
94	C₂×C₁₆×C₅₃	—	2	1	no	—
95	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
96	#108×C₅₃	C₂³	1	2	yes	—
97	#101×C₅₃	—	2	2	no	—
98	#41×C₅₃	C₂×D₄	1	1	yes	characteristic
99	C₂×Q₈×C₅₃	—	2	2	no	—
100	#11719949	—	4	2	no	—
101	#11719949	—	4	2	no	—
102	Q₈×D₅₃	—	4	2	no	—
103	Q₈×D₅₃	—	4	2	no	—
104	Q₈×D₅₃	—	4	2	no	—
105	#11719947	—	4	2	no	—
106	#11719947	—	8	2	no	—
107	#11719945	—	2	2	no	—
108	#11719945	—	4	2	no	—
109	#11719945	—	8	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	SD₁₆×C₅₃	—	4	2	no	—
117	M₁₆×C₅₃	—	2	1	no	—
118	M₁₆×C₅₃	—	2	2	no	—
119	M₁₆×C₅₃	C₂×D₄	1	1	yes	characteristic
120	C₂×C₈×C₅₃	—	2	1	no	—
121	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
122	#11719925	—	4	2	no	—
123	#11719925	—	4	2	no	—
124	#11719925	—	4	2	no	—
125	#11719925	D₈	1	1	yes	characteristic
126	#11719924	—	4	2	no	—
127	#11719924	D₈	1	1	yes	characteristic
128	#11719923	D₈	1	1	yes	characteristic
129	#11719922	D₈	1	1	yes	characteristic
130	#11719919	—	8	2	no	—
131	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
132	#11719917	C₂×D₄	1	1	yes	characteristic
133	C₂×#88322	C₂×D₄	1	1	yes	characteristic
134	Dic₂₁₂	—	2	2	no	—
135	Dic₂₁₂	—	2	2	no	—
136	Dic₂₁₂	—	4	2	no	—
137	Dic₂₁₂	—	4	2	no	—
138	Dic₂₁₂	C₂×D₄	1	2	yes	—
139	D₄₂₄	C₂×D₄	1	2	yes	—
140	#11719913	—	2	2	no	—
141	#11719913	—	4	2	no	—
142	#11719913	—	4	2	no	—
143	#11719912	—	2	1	no	—
144	#11719912	—	4	2	no	—
145	C₈×D₅₃	—	2	1	no	—
146	C₈×D₅₃	—	4	2	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	—	4	2	no	—
153	Q₈×C₅₃	—	4	2	no	—
154	Q₈×C₅₃	C₂×D₈	1	1	yes	characteristic
155	D₄×C₅₃	—	2	1	no	—
156	D₄×C₅₃	C₂×D₈	1	1	yes	characteristic
157	C₂×C₄×C₅₃	—	2	1	no	—
158	C₂×C₄×C₅₃	—	4	2	no	—
159	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
160	C₂²⋊D₅₃	—	2	1	no	—
161	C₂²⋊D₅₃	—	4	2	no	—
162	C₂²⋊D₅₃	—	8	2	no	—
163	C₂×Dic₅₃	—	2	1	no	—
164	C₂×Dic₅₃	—	4	2	no	—
165	C₂×Dic₅₃	—	8	2	no	—
166	D₂₁₂	—	2	2	no	—
167	D₂₁₂	—	8	2	no	—
168	D₂₁₂	C₂×D₈	1	1	yes	characteristic
169	C₄×D₅₃	—	2	1	no	—
170	C₄×D₅₃	—	4	2	no	—
171	C₄×D₅₃	—	8	2	no	—
172	C₄×D₅₃	—	8	2	no	—
173	C₄×D₅₃	—	8	2	no	—
174	Dic₁₀₆	—	2	2	no	—
175	Dic₁₀₆	—	4	2	no	—
176	Dic₁₀₆	—	4	2	no	—
177	Dic₁₀₆	—	8	2	no	—
178	Dic₁₀₆	—	8	2	no	—
179	Dic₁₀₆	C₂×D₈	1	1	yes	characteristic
180	C₈×C₅₃	—	2	1	no	—
181	C₈×C₅₃	—	4	2	no	—
182	C₈×C₅₃	C₂²×D₄	1	1	yes	characteristic
183	C₈×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
184	#88322	—	2	1	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	#576	1	1	yes	characteristic
190	C₂×D₅₃	—	2	1	no	—
191	C₂×D₅₃	—	4	2	no	—
192	C₄×C₅₃	—	2	1	no	—
193	C₄×C₅₃	—	8	2	no	—
194	C₄×C₅₃	D₄²	1	1	yes	characteristic
195	C₄×C₅₃	C₂²×D₈	1	1	yes	characteristic
196	Dic₅₃	—	2	2	no	—
197	Dic₅₃	—	2	1	no	—
198	Dic₅₃	—	4	2	no	—
199	Dic₅₃	—	8	2	no	—
200	#3417	—	53	1	no	—
201	#3417	—	53	1	no	—
202	#3293	—	53	2	no	—
203	#2247	—	53	1	no	—
204	#2222	—	53	1	no	—
205	#2222	D₅₃	1	1	yes	characteristic, complemented
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	D₄×D₈	1	1	yes	characteristic
214	D₅₃	—	4	1	no	—
215	D₅₃	—	8	2	no	—
216	#578	—	53	2	no	—
217	#578	—	53	2	no	—
218	#578	—	106	2	no	—
219	#510	—	53	2	no	—
220	#510	—	53	2	no	—
221	#510	—	106	2	no	—
222	#510	—	106	2	no	—
223	#508	—	53	1	no	—
224	#508	—	106	1	no	—
225	#508	—	106	1	no	—
226	C₂×Q₃₂	—	53	2	no	—
227	C₂×Q₃₂	—	53	1	no	—
228	C₂×Q₃₂	—	106	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	—	106	2	no	—
236	#473	C₂×D₅₃	1	2	yes	—
237	#456	—	106	2	no	—
238	#456	—	106	2	no	—
239	#443	—	53	1	no	—
240	#443	—	53	2	no	—
241	#368	C₂×D₅₃	1	1	yes	characteristic
242	#364	—	53	1	no	—
243	#362	—	53	2	no	—
244	#362	C₂×D₅₃	1	2	yes	—
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	—	106	2	no	—
251	2₋¹⁺⁴	—	106	2	no	—
252	#137	—	2	2	no	—
253	#137	—	106	2	no	—
254	#137	—	106	2	no	—
255	#137	—	106	2	no	—
256	#137	—	212	2	no	—
257	#137	—	212	2	no	—
258	#137	—	212	2	no	—
259	#137	—	212	2	no	—
260	#135	—	53	1	no	—
261	#135	—	53	2	no	—
262	#135	—	106	2	no	—
263	#135	—	106	2	no	—
264	#135	—	212	2	no	—
265	C₂×Q₁₆	—	53	2	no	—
266	C₂×Q₁₆	—	106	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	1	no	—
276	Q₃₂	—	2	1	no	—
277	Q₃₂	—	53	1	no	—
278	Q₃₂	—	106	2	no	—
279	Q₃₂	—	106	2	no	—
280	Q₃₂	—	106	1	no	—
281	Q₃₂	—	106	1	no	—
282	Q₃₂	—	212	2	no	—
283	SD₃₂	—	53	2	no	—
284	SD₃₂	—	106	2	no	—
285	SD₃₂	—	106	2	no	—
286	D₁₆	—	53	1	no	—
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	C₂²×D₅₃	1	2	yes	—
297	#104	—	53	1	no	—
298	#104	—	53	1	no	—
299	#104	—	106	2	no	—
300	#101	—	2	2	no	—
301	#101	—	106	2	no	—
302	C₄×C₈	—	53	1	no	—
303	#41	—	53	1	no	—
304	#41	—	106	2	no	—
305	#41	—	424	2	no	—
306	#41	—	424	2	no	—
307	#41	—	212	2	no	—
308	#41	—	212	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	—	106	2	no	—
314	C₂×Q₈	—	212	2	no	—
315	C₂×Q₈	—	212	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	—	53	2	no	—
323	Q₁₆	—	106	2	no	—
324	Q₁₆	—	106	2	no	—
325	Q₁₆	—	212	2	no	—
326	Q₁₆	—	212	2	no	—
327	Q₁₆	—	212	2	no	—
328	Q₁₆	—	212	2	no	—
329	Q₁₆	—	212	2	no	—
330	SD₁₆	—	4	2	no	—
331	SD₁₆	—	53	1	no	—
332	SD₁₆	—	53	1	no	—
333	SD₁₆	—	106	2	no	—
334	SD₁₆	—	212	2	no	—
335	SD₁₆	—	212	2	no	—
336	SD₁₆	—	212	2	no	—
337	D₈	—	53	1	no	—
338	D₈	—	53	2	no	—
339	M₁₆	—	2	1	no	—
340	M₁₆	—	2	2	no	—
341	M₁₆	—	53	1	no	—
342	M₁₆	—	106	1	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	D₄×D₅₃	1	1	yes	characteristic
356	C₁₆	D₄×D₅₃	1	1	yes	characteristic
357	Q₈	—	4	2	no	—
358	Q₈	—	4	2	no	—
359	Q₈	—	53	1	no	—
360	Q₈	—	106	2	no	—
361	Q₈	—	424	2	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	—	53	1	no	—
368	D₄	—	106	1	no	—
369	D₄	—	106	2	no	—
370	D₄	—	424	2	no	—
371	D₄	—	424	2	no	—
372	D₄	—	212	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	—	106	1	no	—
377	C₂×C₄	—	106	1	no	—
378	C₂×C₄	—	424	2	no	—
379	C₂×C₄	—	424	2	no	—
380	C₂×C₄	—	424	2	no	—
381	C₂×C₄	—	424	2	no	—
382	C₂×C₄	—	212	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	—	4	2	no	—
387	C₈	—	53	1	no	—
388	C₈	—	53	1	no	—
389	C₈	—	106	1	no	—
390	C₈	C₂×D₄×D₅₃	1	1	yes	characteristic
391	C₈	#13133263	1	1	yes	characteristic
392	C₂²	—	2	1	no	—
393	C₂²	—	106	1	no	—
394	C₂²	—	212	2	no	—
395	C₂²	#16811425	1	1	yes	characteristic
396	C₄	—	2	1	no	—
397	C₄	—	8	2	no	—
398	C₄	—	106	2	no	—
399	C₄	—	106	1	no	—
400	C₄	—	424	2	no	—
401	C₄	—	212	2	no	—
402	C₄	#16811257	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	—	424	2	no	—
407	C₂	—	212	1	no	—
408	C₂	#20286389	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20286389
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2222	C₂
socle	C₂×C₅₃	D₄×D₈
hypercenter	#2222	D₅₃
hypocenter	C₅₃	#33860
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #131 ◁ #2222

Complements

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

N	H ≅ G/N	type
1. #20287868	C₂	semidirect
2. #20287864	C₂	semidirect
3. #20286518	C₂	semidirect
4. #20276966	C₂	semidirect
5. #20276841	C₂	semidirect
6. #20276795	C₂	semidirect
7. #20276696	C₂	semidirect
8. #20276691	C₂	semidirect
9. #20276595	C₂	semidirect
10. #20276550	C₂	semidirect
11. C₅₃×#2222	C₂	semidirect
205. #2222	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
#20287868	2
#20287864	2
#20286518	2
#20276966	2
#20276841	2
#20276795	2
#20276696	2
#20276691	2
#20276595	2
#20276550	2
C₅₃×#2222	2
#33860	53