Subgroups — #27540890

Order 13568.

646 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20287853	C₂	1	1	yes	maximal, characteristic, complemented
2	#20286501	C₂	1	2	yes	maximal, complemented
3	#20276977	C₂	1	1	yes	maximal, characteristic, complemented
4	#20276837	C₂	1	1	yes	maximal, characteristic, complemented
5	#20276835	C₂	1	1	yes	maximal, characteristic, complemented
6	#20276786	C₂	1	2	yes	maximal, complemented
7	#20276683	C₂	1	2	yes	maximal, complemented
8	#20276679	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	#16811431	—	2	2	no	—
13	#16811425	C₂²	1	2	yes	—
14	#16811060	—	2	2	no	—
15	#16811058	C₂²	1	2	yes	—
16	#16811056	—	2	1	no	—
17	C₂×#13133160	C₂²	1	2	yes	complemented
18	C₂×D₈₄₈	C₂²	1	1	yes	characteristic
19	D₁₆×D₅₃	—	2	1	no	—
20	#16811039	C₂²	1	1	yes	characteristic
21	#16811008	C₂²	1	2	yes	complemented
22	#16810895	C₂²	1	2	yes	—
23	#16810893	C₂²	1	2	yes	complemented
24	#16810888	—	2	2	no	—
25	#16810799	—	2	2	no	—
26	#16810796	—	2	2	no	—
27	#16810740	C₂²	1	2	yes	complemented
28	#16810738	C₂²	1	1	yes	characteristic
29	#16810453	C₂²	1	1	yes	characteristic, complemented
30	#16810443	C₂²	1	1	yes	characteristic, complemented
31	#16810439	C₂²	1	2	yes	complemented
32	#16810437	C₂²	1	2	yes	complemented
33	#16810430	C₂²	1	2	yes	complemented
34	#16810408	C₂²	1	1	yes	characteristic
35	#16810407	C₂²	1	1	yes	characteristic, complemented
36	C₁₆×Dic₅₃	C₂²	1	1	yes	characteristic, complemented
37	C₅₃×#578	C₂²	1	2	yes	complemented
38	C₅₃×#510	—	2	2	no	—
39	C₅₃×#510	C₂²	1	2	yes	complemented
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, complemented
44	C₅₃×#504	C₂²	1	1	yes	characteristic, complemented
45	#13133263	—	2	2	no	—
46	#13133248	—	2	2	no	—
47	#13133248	—	4	2	no	—
48	#13133244	—	4	2	no	—
49	#13133242	—	2	2	no	—
50	#13133242	—	2	2	no	—
51	#13133242	—	4	2	no	—
52	#13133240	—	2	2	no	—
53	#13133237	—	4	2	no	—
54	#13133237	D₄	1	1	yes	characteristic
55	C₂×#11719924	—	2	2	no	—
56	C₂×#11719924	—	4	2	no	—
57	C₂×D₄₂₄	C₂³	1	2	yes	—
58	D₈×D₅₃	—	2	2	no	—
59	#13133224	—	2	2	no	—
60	#13133223	C₂³	1	1	yes	characteristic
61	#13133206	—	2	2	no	—
62	#13133160	—	2	2	no	—
63	#13133160	—	4	2	no	—
64	D₈₄₈	—	2	1	no	—
65	D₈₄₈	—	2	1	no	—
66	#13133158	—	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	#13133133	—	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	#11719949	—	4	2	no	—
107	D₄×D₅₃	—	4	2	no	—
108	D₄×D₅₃	—	8	2	no	—
109	#11719945	—	4	2	no	—
110	#11719945	C₂×D₄	1	1	yes	characteristic
111	C₂×D₂₁₂	—	2	2	no	—
112	C₂×D₂₁₂	—	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	D₈	1	1	yes	characteristic
126	#11719924	—	4	2	no	—
127	#11719924	—	4	2	no	—
128	#11719924	—	4	2	no	—
129	#11719924	—	4	2	no	—
130	#11719924	—	4	2	no	—
131	#11719924	D₈	1	1	yes	characteristic
132	#11719923	D₈	1	1	yes	characteristic
133	#11719922	—	4	2	no	—
134	#11719922	—	4	2	no	—
135	#11719922	D₈	1	1	yes	characteristic
136	#11719921	—	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	D₄₂₄	—	2	2	no	—
144	D₄₂₄	—	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	C₂²×D₅₃	—	4	2	no	—
153	C₂²×D₅₃	—	8	2	no	—
154	Q₈×C₅₃	—	2	2	no	—
155	Q₈×C₅₃	—	4	2	no	—
156	Q₈×C₅₃	—	4	2	no	—
157	Q₈×C₅₃	—	8	2	no	—
158	Q₈×C₅₃	C₂×D₈	1	1	yes	characteristic
159	D₄×C₅₃	—	2	1	no	—
160	D₄×C₅₃	—	2	2	no	—
161	D₄×C₅₃	—	4	2	no	—
162	D₄×C₅₃	—	8	2	no	—
163	D₄×C₅₃	C₂×D₈	1	1	yes	characteristic
164	C₂×C₄×C₅₃	—	2	1	no	—
165	C₂×C₄×C₅₃	—	4	2	no	—
166	C₂×C₄×C₅₃	—	4	2	no	—
167	C₂×C₄×C₅₃	—	8	2	no	—
168	C₂×C₄×C₅₃	—	8	2	no	—
169	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
170	C₂²⋊D₅₃	—	2	1	no	—
171	C₂²⋊D₅₃	—	8	2	no	—
172	C₂×Dic₅₃	—	2	1	no	—
173	D₂₁₂	—	4	2	no	—
174	D₂₁₂	—	4	2	no	—
175	D₂₁₂	—	8	2	no	—
176	D₂₁₂	C₂×D₈	1	1	yes	characteristic
177	C₄×D₅₃	—	2	1	no	—
178	C₄×D₅₃	—	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	#33859	—	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	C₂×D₅₃	—	8	2	no	—
193	C₂×D₅₃	—	16	2	no	—
194	C₂×D₅₃	—	32	2	no	—
195	C₄×C₅₃	—	2	1	no	—
196	C₄×C₅₃	—	4	2	no	—
197	C₄×C₅₃	—	8	2	no	—
198	C₄×C₅₃	D₄²	1	1	yes	characteristic
199	C₄×C₅₃	C₂²×D₈	1	1	yes	characteristic
200	Dic₅₃	—	2	1	no	—
201	Dic₅₃	—	4	1	no	—
202	#3417	D₅₃	1	1	yes	characteristic, complemented
203	#3415	—	53	1	no	—
204	#3290	—	53	2	no	—
205	#2249	—	53	1	no	—
206	#2222	—	53	1	no	—
207	#2221	—	53	1	no	—
208	#2212	—	53	2	no	—
209	#2194	—	53	2	no	—
210	#2193	—	53	2	no	—
211	#2178	—	53	1	no	—
212	#2170	—	53	1	no	—
213	C₂×C₅₃	—	2	1	no	—
214	C₂×C₅₃	—	4	1	no	—
215	C₂×C₅₃	—	8	2	no	—
216	C₂×C₅₃	D₄×D₈	1	1	yes	characteristic
217	D₅₃	—	4	1	no	—
218	D₅₃	—	16	2	no	—
219	#578	C₂×D₅₃	1	2	yes	complemented
220	#577	—	106	2	no	—
221	#576	—	53	2	no	—
222	#510	—	2	2	no	—
223	#510	C₂×D₅₃	1	2	yes	complemented
224	#509	—	53	2	no	—
225	#509	—	106	2	no	—
226	#508	—	2	1	no	—
227	#508	—	106	1	no	—
228	#508	C₂×D₅₃	1	1	yes	characteristic
229	C₂×Q₃₂	—	2	1	no	—
230	C₂×Q₃₂	C₂×D₅₃	1	1	yes	characteristic, complemented
231	C₂×SD₃₂	—	53	2	no	—
232	C₂×D₁₆	—	53	1	no	—
233	C₂×D₁₆	—	106	1	no	—
234	#504	—	53	1	no	—
235	#504	C₂×D₅₃	1	1	yes	characteristic, complemented
236	#495	—	53	2	no	—
237	#473	—	53	2	no	—
238	#472	—	53	2	no	—
239	#471	—	106	2	no	—
240	#455	—	106	2	no	—
241	#454	—	106	2	no	—
242	#443	—	53	1	no	—
243	#443	—	53	2	no	—
244	#368	—	53	1	no	—
245	#364	—	53	1	no	—
246	#362	—	53	2	no	—
247	#361	—	53	2	no	—
248	#359	—	53	2	no	—
249	#350	—	53	1	no	—
250	#350	—	53	1	no	—
251	C₄×C₁₆	—	53	1	no	—
252	C₅₃	#33859	1	1	yes	characteristic, complemented, Sylow
253	2₋¹⁺⁴	—	2	2	no	—
254	H(5,𝔽₂)	—	106	2	no	—
255	#137	—	2	2	no	—
256	#137	—	4	2	no	—
257	#137	—	106	2	no	—
258	#137	—	212	2	no	—
259	AGL(1,ℤ/8)	—	106	2	no	—
260	AGL(1,ℤ/8)	—	106	2	no	—
261	AGL(1,ℤ/8)	—	212	2	no	—
262	AGL(1,ℤ/8)	—	212	2	no	—
263	#135	—	2	2	no	—
264	#135	—	53	1	no	—
265	#135	—	106	2	no	—
266	#135	—	212	2	no	—
267	#135	C₂²×D₅₃	1	2	yes	—
268	C₂×Q₁₆	—	2	2	no	—
269	C₂×Q₁₆	C₂²×D₅₃	1	2	yes	—
270	C₂×SD₁₆	—	106	2	no	—
271	C₂×SD₁₆	—	212	2	no	—
272	C₂×D₈	—	53	2	no	—
273	C₂×D₈	—	106	2	no	—
274	#131	—	53	1	no	—
275	#131	—	106	2	no	—
276	#131	C₂²×D₅₃	1	1	yes	characteristic
277	#124	—	106	2	no	—
278	Q₃₂	—	2	2	no	—
279	Q₃₂	—	2	1	no	—
280	Q₃₂	—	2	1	no	—
281	Q₃₂	C₂²⋊D₅₃	1	1	yes	characteristic
282	SD₃₂	—	2	2	no	—
283	SD₃₂	—	106	2	no	—
284	SD₃₂	—	212	2	no	—
285	SD₃₂	C₂²⋊D₅₃	1	2	yes	—
286	D₁₆	—	106	2	no	—
287	D₁₆	—	106	1	no	—
288	D₁₆	—	106	1	no	—
289	D₁₆	C₂²⋊D₅₃	1	1	yes	characteristic
290	M₃₂	—	2	1	no	—
291	M₃₂	—	53	1	no	—
292	M₃₂	—	106	1	no	—
293	M₃₂	C₂²×D₅₃	1	1	yes	characteristic
294	C₂×C₁₆	—	2	1	no	—
295	C₂×C₁₆	—	53	1	no	—
296	C₂×C₁₆	—	106	1	no	—
297	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
298	#108	—	53	1	no	—
299	#108	—	53	2	no	—
300	#104	—	53	1	no	—
301	#104	—	53	1	no	—
302	#104	—	106	2	no	—
303	#101	—	106	2	no	—
304	#100	—	106	2	no	—
305	C₄×C₈	—	53	1	no	—
306	#41	—	2	2	no	—
307	#41	—	4	2	no	—
308	#41	—	4	2	no	—
309	#41	—	8	2	no	—
310	#41	—	53	1	no	—
311	#41	—	212	2	no	—
312	#41	—	212	2	no	—
313	#41	D₄×D₅₃	1	1	yes	characteristic
314	C₂×Q₈	—	2	2	no	—
315	C₂×Q₈	—	4	2	no	—
316	C₂×Q₈	—	4	2	no	—
317	C₂×D₄	—	106	2	no	—
318	C₂×D₄	—	424	2	no	—
319	C₂×D₄	—	212	2	no	—
320	C₂×D₄	—	212	2	no	—
321	Q₁₆	—	2	2	no	—
322	Q₁₆	—	2	2	no	—
323	Q₁₆	—	4	2	no	—
324	Q₁₆	—	53	1	no	—
325	Q₁₆	—	212	2	no	—
326	Q₁₆	C₂×C₂²⋊D₅₃	1	2	yes	—
327	SD₁₆	—	2	2	no	—
328	SD₁₆	—	4	2	no	—
329	SD₁₆	—	53	1	no	—
330	SD₁₆	—	53	1	no	—
331	SD₁₆	—	212	2	no	—
332	SD₁₆	—	212	2	no	—
333	SD₁₆	—	212	2	no	—
334	SD₁₆	—	212	2	no	—
335	SD₁₆	—	212	2	no	—
336	D₈	—	53	1	no	—
337	D₈	—	106	2	no	—
338	D₈	—	106	2	no	—
339	D₈	—	212	2	no	—
340	D₈	—	212	2	no	—
341	D₈	C₂×C₂²⋊D₅₃	1	2	yes	—
342	M₁₆	—	2	1	no	—
343	M₁₆	—	53	1	no	—
344	M₁₆	—	106	1	no	—
345	M₁₆	—	106	2	no	—
346	M₁₆	—	212	2	no	—
347	M₁₆	D₄×D₅₃	1	1	yes	characteristic
348	C₂×C₈	—	2	1	no	—
349	C₂×C₈	—	53	1	no	—
350	C₂×C₈	—	106	1	no	—
351	C₂×C₈	—	212	2	no	—
352	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
353	C₂²⋊C₄	—	424	2	no	—
354	C₄²	—	53	1	no	—
355	C₁₆	—	2	1	no	—
356	C₁₆	—	106	1	no	—
357	C₁₆	—	106	1	no	—
358	C₁₆	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
359	C₁₆	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
360	C₂³	—	424	2	no	—
361	C₂³	—	212	2	no	—
362	Q₈	—	2	2	no	—
363	Q₈	—	4	2	no	—
364	Q₈	—	4	2	no	—
365	Q₈	—	8	2	no	—
366	Q₈	—	53	1	no	—
367	Q₈	D₈×D₅₃	1	1	yes	characteristic
368	D₄	—	2	1	no	—
369	D₄	—	2	2	no	—
370	D₄	—	4	2	no	—
371	D₄	—	8	2	no	—
372	D₄	—	53	1	no	—
373	D₄	—	106	1	no	—
374	D₄	—	424	2	no	—
375	D₄	—	424	2	no	—
376	D₄	—	212	2	no	—
377	D₄	—	212	2	no	—
378	D₄	D₈×D₅₃	1	1	yes	characteristic
379	C₂×C₄	—	2	1	no	—
380	C₂×C₄	—	4	2	no	—
381	C₂×C₄	—	4	2	no	—
382	C₂×C₄	—	8	2	no	—
383	C₂×C₄	—	8	2	no	—
384	C₂×C₄	—	106	1	no	—
385	C₂×C₄	—	106	1	no	—
386	C₂×C₄	—	424	2	no	—
387	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
388	C₈	—	2	1	no	—
389	C₈	—	53	1	no	—
390	C₈	—	53	1	no	—
391	C₈	—	106	1	no	—
392	C₈	—	212	2	no	—
393	C₈	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
394	C₈	#13133262	1	1	yes	characteristic
395	C₂²	—	2	1	no	—
396	C₂²	—	4	2	no	—
397	C₂²	—	848	2	no	—
398	C₂²	—	106	1	no	—
399	C₂²	—	1696	2	no	—
400	C₂²	—	424	2	no	—
401	C₂²	#16811424	1	1	yes	characteristic
402	C₄	—	2	1	no	—
403	C₄	—	4	2	no	—
404	C₄	—	8	2	no	—
405	C₄	—	106	1	no	—
406	C₄	—	212	1	no	—
407	C₄	D₄×C₂²⋊D₅₃	1	1	yes	characteristic
408	C₄	C₂×D₈×D₅₃	1	1	yes	characteristic
409	C₂	—	2	1	no	—
410	C₂	—	4	1	no	—
411	C₂	—	8	2	no	—
412	C₂	—	848	2	no	—
413	C₂	—	212	1	no	—
414	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	#33859
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. #20287853	C₂	semidirect
2. #20286501	C₂	semidirect
3. #20276977	C₂	semidirect
4. #20276837	C₂	semidirect
5. #20276835	C₂	semidirect
6. #20276786	C₂	semidirect
7. #20276683	C₂	semidirect
8. #20276679	C₂	semidirect
9. #20276596	C₂	semidirect
10. #20276550	C₂	semidirect
11. C₅₃×#3417	C₂	semidirect
17. C₂×#13133160	C₂²	semidirect
21. #16811008	C₂²	semidirect
23. #16810893	C₂²	semidirect
27. #16810740	C₂²	semidirect
29. #16810453	C₂²	semidirect
30. #16810443	C₂²	semidirect
31. #16810439	C₂²	semidirect
32. #16810437	C₂²	semidirect
33. #16810430	C₂²	semidirect
35. #16810407	C₂²	semidirect
36. C₁₆×Dic₅₃	C₂²	semidirect
37. C₅₃×#578	C₂²	semidirect
39. C₅₃×#510	C₂²	semidirect
43. C₂×Q₃₂×C₅₃	C₂²	semidirect
44. C₅₃×#504	C₂²	semidirect
202. #3417	D₅₃	semidirect
219. #578	C₂×D₅₃	semidirect
223. #510	C₂×D₅₃	semidirect
230. C₂×Q₃₂	C₂×D₅₃	semidirect
235. #504	C₂×D₅₃	semidirect
252. C₅₃	#33859	semidirect

Maximal subgroups

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

subgroup	index
#20287853	2
#20286501	2
#20276977	2
#20276837	2
#20276835	2
#20276786	2
#20276683	2
#20276679	2
#20276596	2
#20276550	2
C₅₃×#3417	2
#33859	53