Subgroups — #27537336

Order 13568.

642 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20287818	C₂	1	1	yes	maximal, characteristic, complemented
2	#20287817	C₂	1	1	yes	maximal, characteristic, complemented
3	#20283616	C₂	1	1	yes	maximal, characteristic
4	#20276554	C₂	1	4	yes	maximal, complemented
5	#20276549	C₂	1	4	yes	maximal, complemented
6	#20276480	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#16810406	C₂	1	1	yes	maximal, characteristic, complemented
8	#20276311	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₅₃×#350	C₂	1	1	yes	maximal, characteristic, complemented
10	#16811421	C₄	1	1	yes	characteristic
11	C₂×#13133223	C₂²	1	1	yes	characteristic
12	#16811041	C₂²	1	4	yes	complemented
13	#16811040	—	2	2	no	—
14	#16811040	C₂²	1	4	yes	complemented
15	#16811039	—	2	2	no	—
16	C₂×#13133157	—	2	2	no	—
17	C₂×#13133157	C₂²	1	1	yes	characteristic
18	C₂×#13133157	C₂²	1	1	yes	characteristic
19	M₃₂×D₅₃	—	2	2	no	—
20	#16810743	C₄	1	4	yes	—
21	#16810738	C₂²	1	4	yes	—
22	#16810671	C₄	1	1	yes	characteristic
23	#16810640	C₄	1	1	yes	characteristic
24	#16810639	C₄	1	1	yes	characteristic
25	#16810578	C₂²	1	1	yes	characteristic
26	#16810443	C₂²	1	4	yes	complemented
27	#16810406	C₂²	1	2	yes	complemented
28	#16810406	C₂²	1	2	yes	complemented
29	#16810398	C₂²	1	2	yes	complemented
30	C₁₆×Dic₅₃	C₂²	1	2	yes	complemented
31	C₂×#131×C₅₃	C₂²	1	1	yes	characteristic
32	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
33	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
34	C₅₃×#350	C₂²	1	2	yes	complemented
35	C₅₃×#350	C₂²	1	2	yes	complemented
36	C₂×#11719945	C₂×C₄	1	1	yes	characteristic
37	#13133248	C₈	1	2	yes	—
38	#13133247	C₈	1	2	yes	—
39	#13133243	C₈	1	2	yes	—
40	#13133242	C₈	1	2	yes	—
41	#13133237	C₂×C₄	1	4	yes	—
42	#13133223	—	2	1	no	—
43	#13133223	—	2	1	no	—
44	#13133223	C₂×C₄	1	2	yes	—
45	#13133223	C₂³	1	2	yes	—
46	C₂×#11719917	C₂×C₄	1	1	yes	characteristic
47	C₂×#11719917	C₂×C₄	1	1	yes	characteristic
48	C₂×#11719912	—	2	1	no	—
49	C₂×C₈×D₅₃	—	2	1	no	—
50	#13133176	C₂×C₄	1	1	yes	characteristic
51	#13133157	—	2	1	no	—
52	#13133157	—	2	4	no	—
53	#13133157	D₄	1	2	yes	—
54	#13133157	D₄	1	2	yes	—
55	#13133157	C₂³	1	2	yes	—
56	#13133156	—	2	4	no	—
57	C₂×#11719908	—	2	1	no	—
58	C₂×#11719908	—	2	4	no	—
59	C₂×#11719908	C₂³	1	2	yes	—
60	C₁₆×D₅₃	—	2	4	no	—
61	#13133152	C₂×C₄	1	4	yes	—
62	#13133144	C₂×C₄	1	4	yes	—
63	#13133143	C₂×C₄	1	4	yes	—
64	#13133127	C₂×C₄	1	2	yes	—
65	C₈×Dic₅₃	C₂³	1	2	yes	—
66	C₂×#41×C₅₃	C₂×C₄	1	1	yes	characteristic
67	#131×C₅₃	—	2	1	no	—
68	#131×C₅₃	—	2	1	no	—
69	#131×C₅₃	C₂×C₄	1	2	yes	—
70	#131×C₅₃	C₂³	1	2	yes	—
71	C₂×M₁₆×C₅₃	—	2	1	no	—
72	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
73	C₂²×C₈×C₅₃	—	2	1	no	—
74	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
75	M₃₂×C₅₃	—	2	1	no	—
76	M₃₂×C₅₃	C₂³	1	2	yes	—
77	C₂×C₁₆×C₅₃	—	2	1	no	—
78	C₂×C₁₆×C₅₃	—	2	1	no	—
79	C₂×C₁₆×C₅₃	—	2	1	no	—
80	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
81	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
82	#41×C₅₃	—	2	1	no	—
83	#41×C₅₃	—	2	1	no	—
84	#41×C₅₃	C₂×C₈	1	2	yes	—
85	#41×C₅₃	C₂²×C₄	1	2	yes	—
86	C₂×Q₈×C₅₃	C₂×C₈	1	1	yes	characteristic
87	C₂×D₄×C₅₃	—	2	1	no	—
88	C₂×D₄×C₅₃	C₂×C₈	1	1	yes	characteristic
89	C₂²×C₄×C₅₃	—	2	1	no	—
90	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₂²⋊D₅₃	—	2	1	no	—
92	#11719945	—	2	1	no	—
93	#11719945	—	2	1	no	—
94	#11719945	C₂×C₈	1	2	yes	—
95	#11719945	C₂²×C₄	1	2	yes	—
96	C₂×D₂₁₂	C₂×C₈	1	1	yes	characteristic
97	C₂×C₄×D₅₃	—	2	1	no	—
98	C₂×Dic₁₀₆	C₂×C₈	1	1	yes	characteristic
99	M₁₆×C₅₃	—	2	1	no	—
100	M₁₆×C₅₃	—	2	2	no	—
101	M₁₆×C₅₃	—	2	2	no	—
102	M₁₆×C₅₃	C₂²×C₄	1	2	yes	—
103	C₂×C₈×C₅₃	—	2	1	no	—
104	C₂×C₈×C₅₃	—	2	2	no	—
105	C₂×C₈×C₅₃	—	2	2	no	—
106	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
109	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
110	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
111	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
112	C₂²⋊Dic₅₃	—	2	1	no	—
113	#11719925	C₂×C₈	1	4	yes	—
114	#11719924	C₂×C₈	1	4	yes	—
115	#11719923	C₂×C₈	1	4	yes	—
116	#11719922	C₂×C₈	1	4	yes	—
117	C₄⋊Dic₅₃	—	2	1	no	—
118	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
119	#11719917	—	2	1	no	—
120	#11719917	C₂×C₈	1	2	yes	—
121	#11719917	C₂×C₈	1	2	yes	—
122	#11719917	C₂²×C₄	1	2	yes	—
123	C₂×#88322	—	2	1	no	—
124	C₂×#88322	C₂²×C₄	1	2	yes	—
125	#11719912	—	2	2	no	—
126	#11719912	—	2	2	no	—
127	C₈×D₅₃	—	2	2	no	—
128	C₈×D₅₃	—	2	2	no	—
129	C₁₆×C₅₃	—	2	2	no	—
130	C₁₆×C₅₃	—	2	1	no	—
131	C₁₆×C₅₃	—	2	1	no	—
132	#11719908	—	2	2	no	—
133	#11719908	C₂×D₄	1	4	yes	—
134	C₂³×C₅₃	—	2	1	no	—
135	C₂³×C₅₃	#131	1	1	yes	characteristic
136	C₂²×D₅₃	—	2	1	no	—
137	Q₈×C₅₃	—	2	1	no	—
138	Q₈×C₅₃	C₂²×C₈	1	2	yes	—
139	D₄×C₅₃	—	2	2	no	—
140	D₄×C₅₃	—	2	2	no	—
141	D₄×C₅₃	—	2	1	no	—
142	D₄×C₅₃	C₂²×C₈	1	2	yes	—
143	C₂×C₄×C₅₃	—	2	1	no	—
144	C₂×C₄×C₅₃	—	2	2	no	—
145	C₂×C₄×C₅₃	—	2	2	no	—
146	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
147	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
148	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
149	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
150	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
151	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
152	C₂²⋊D₅₃	—	2	2	no	—
153	C₂²⋊D₅₃	—	2	2	no	—
154	C₂×Dic₅₃	—	2	1	no	—
155	C₂×Dic₅₃	—	2	2	no	—
156	D₂₁₂	—	2	1	no	—
157	D₂₁₂	C₂²×C₈	1	2	yes	—
158	C₄×D₅₃	—	2	2	no	—
159	C₄×D₅₃	—	2	2	no	—
160	Dic₁₀₆	—	2	1	no	—
161	Dic₁₀₆	C₂²×C₈	1	2	yes	—
162	C₈×C₅₃	—	2	2	no	—
163	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
164	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
165	C₈×C₅₃	C₂²×D₄	1	1	yes	characteristic
166	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
167	#88322	—	2	2	no	—
168	#88322	C₂²×C₈	1	4	yes	—
169	#33442	—	53	1	no	maximal, Sylow
170	C₂²×C₅₃	—	2	2	no	—
171	C₂²×C₅₃	—	4	1	no	—
172	C₂²×C₅₃	—	8	1	no	—
173	C₂²×C₅₃	C₈×D₄	1	1	yes	characteristic
174	C₂²×C₅₃	C₈×D₄	1	1	yes	characteristic
175	C₂²×C₅₃	C₂×#131	1	1	yes	characteristic
176	C₂×D₅₃	—	2	2	no	—
177	C₂×D₅₃	—	8	1	no	—
178	C₄×C₅₃	—	2	2	no	—
179	C₄×C₅₃	C₈×D₄	1	1	yes	characteristic
180	C₄×C₅₃	C₈×D₄	1	1	yes	characteristic
181	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
182	C₄×C₅₃	C₂³×C₈	1	1	yes	characteristic
183	Dic₅₃	—	2	2	no	—
184	Dic₅₃	—	4	2	no	—
185	#3407	—	53	1	no	—
186	#3407	—	53	1	no	—
187	#2955	—	53	1	no	—
188	#2171	—	53	4	no	—
189	#2170	—	53	4	no	—
190	#2153	—	53	1	no	—
191	C₂×#350	—	53	1	no	—
192	C₂×#350	D₅₃	1	1	yes	characteristic, complemented
193	#2108	—	53	1	no	—
194	C₂×C₅₃	—	2	1	no	—
195	C₂×C₅₃	—	2	1	no	—
196	C₂×C₅₃	—	2	1	no	—
197	C₂×C₅₃	—	4	2	no	—
198	C₂×C₅₃	C₂×C₈×D₄	1	1	yes	characteristic
199	D₅₃	—	4	2	no	—
200	#575	—	53	1	no	—
201	C₂×#131	—	53	1	no	—
202	C₂×#131	C₂×D₅₃	1	1	yes	characteristic
203	#504	—	53	4	no	—
204	#504	—	53	4	no	—
205	#504	—	106	2	no	—
206	#504	—	106	2	no	—
207	C₂×M₃₂	—	53	1	no	—
208	C₂×M₃₂	—	53	1	no	—
209	C₂×M₃₂	—	106	2	no	—
210	C₂×M₃₂	—	106	2	no	—
211	C₂×M₃₂	C₂×D₅₃	1	1	yes	characteristic
212	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
213	#444	—	53	4	no	—
214	#443	—	53	4	no	—
215	#430	—	53	1	no	—
216	#421	—	53	1	no	—
217	#421	—	53	1	no	—
218	#405	—	53	1	no	—
219	#364	—	53	4	no	—
220	#350	—	53	2	no	—
221	#350	—	53	2	no	—
222	#350	C₂×D₅₃	1	2	yes	complemented
223	#350	C₂×D₅₃	1	2	yes	complemented
224	#346	—	53	2	no	—
225	C₄×C₁₆	—	53	2	no	—
226	C₅₃	#33442	1	1	yes	characteristic, complemented, Sylow
227	C₂×#41	—	53	1	no	—
228	C₂×#41	C₄×D₅₃	1	1	yes	characteristic
229	#137	—	53	2	no	—
230	#137	—	53	2	no	—
231	AGL(1,ℤ/8)	—	53	2	no	—
232	AGL(1,ℤ/8)	—	53	2	no	—
233	#135	—	53	4	no	—
234	#131	—	2	1	no	—
235	#131	—	2	1	no	—
236	#131	—	53	2	no	—
237	#131	—	53	2	no	—
238	#131	—	106	1	no	—
239	#131	—	106	1	no	—
240	#131	C₄×D₅₃	1	2	yes	—
241	#131	C₂²×D₅₃	1	2	yes	—
242	C₂×M₁₆	—	2	1	no	—
243	C₂×M₁₆	—	53	1	no	—
244	C₂×M₁₆	—	53	1	no	—
245	C₂×M₁₆	—	106	1	no	—
246	C₂×M₁₆	C₄×D₅₃	1	1	yes	characteristic
247	C₂²×C₈	—	2	1	no	—
248	C₂²×C₈	—	106	1	no	—
249	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
250	#117	—	53	1	no	—
251	M₃₂	—	2	1	no	—
252	M₃₂	—	53	2	no	—
253	M₃₂	—	53	2	no	—
254	M₃₂	—	53	2	no	—
255	M₃₂	—	106	4	no	—
256	M₃₂	—	106	4	no	—
257	M₃₂	—	106	1	no	—
258	M₃₂	C₂²×D₅₃	1	2	yes	—
259	C₂×C₁₆	—	2	1	no	—
260	C₂×C₁₆	—	2	1	no	—
261	C₂×C₁₆	—	2	1	no	—
262	C₂×C₁₆	—	53	2	no	—
263	C₂×C₁₆	—	106	4	no	—
264	C₂×C₁₆	—	106	4	no	—
265	C₂×C₁₆	—	106	1	no	—
266	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
267	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
268	#108	—	53	4	no	—
269	#104	—	53	4	no	—
270	#104	—	53	4	no	—
271	#97	—	53	2	no	—
272	C₄×C₈	—	53	2	no	—
273	#41	—	2	1	no	—
274	#41	—	2	1	no	—
275	#41	—	53	2	no	—
276	#41	—	53	2	no	—
277	#41	—	106	1	no	—
278	#41	—	106	1	no	—
279	#41	C₈×D₅₃	1	2	yes	—
280	#41	C₂×C₄×D₅₃	1	2	yes	—
281	C₂×Q₈	—	53	1	no	—
282	C₂×Q₈	C₈×D₅₃	1	1	yes	characteristic
283	C₂×D₄	—	2	1	no	—
284	C₂×D₄	—	53	1	no	—
285	C₂×D₄	—	106	1	no	—
286	C₂×D₄	C₈×D₅₃	1	1	yes	characteristic
287	C₂²×C₄	—	2	1	no	—
288	C₂²×C₄	—	106	1	no	—
289	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
290	Q₁₆	—	53	4	no	—
291	SD₁₆	—	53	4	no	—
292	SD₁₆	—	53	4	no	—
293	D₈	—	53	4	no	—
294	M₁₆	—	2	1	no	—
295	M₁₆	—	2	2	no	—
296	M₁₆	—	2	2	no	—
297	M₁₆	—	53	2	no	—
298	M₁₆	—	53	2	no	—
299	M₁₆	—	53	2	no	—
300	M₁₆	—	106	1	no	—
301	M₁₆	—	106	2	no	—
302	M₁₆	—	106	2	no	—
303	M₁₆	C₂×C₄×D₅₃	1	2	yes	—
304	C₂×C₈	—	2	1	no	—
305	C₂×C₈	—	2	2	no	—
306	C₂×C₈	—	2	2	no	—
307	C₂×C₈	—	53	2	no	—
308	C₂×C₈	—	106	1	no	—
309	C₂×C₈	—	106	2	no	—
310	C₂×C₈	—	106	2	no	—
311	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
312	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
313	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
314	C₂×C₈	#11719947	1	1	yes	characteristic
315	C₂×C₈	#11719947	1	1	yes	characteristic
316	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
317	C₄⋊C₄	—	106	1	no	—
318	C₂²⋊C₄	—	106	1	no	—
319	C₄²	—	53	2	no	—
320	C₁₆	—	2	2	no	—
321	C₁₆	—	2	1	no	—
322	C₁₆	—	2	1	no	—
323	C₁₆	—	53	4	no	—
324	C₁₆	—	106	2	no	—
325	C₂³	—	2	1	no	—
326	C₂³	—	106	1	no	—
327	C₂³	#13133225	1	1	yes	characteristic
328	Q₈	—	2	1	no	—
329	Q₈	—	53	2	no	—
330	Q₈	—	106	1	no	—
331	Q₈	C₂×C₈×D₅₃	1	2	yes	—
332	D₄	—	2	2	no	—
333	D₄	—	2	2	no	—
334	D₄	—	2	1	no	—
335	D₄	—	53	2	no	—
336	D₄	—	106	2	no	—
337	D₄	—	106	2	no	—
338	D₄	—	106	1	no	—
339	D₄	C₂×C₈×D₅₃	1	2	yes	—
340	C₂×C₄	—	2	1	no	—
341	C₂×C₄	—	2	2	no	—
342	C₂×C₄	—	2	2	no	—
343	C₂×C₄	—	106	1	no	—
344	C₂×C₄	—	106	2	no	—
345	C₂×C₄	—	106	2	no	—
346	C₂×C₄	—	106	2	no	—
347	C₂×C₄	#13133177	1	1	yes	characteristic
348	C₂×C₄	#13133177	1	1	yes	characteristic
349	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
350	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
351	C₂×C₄	#13133225	1	1	yes	characteristic
352	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
353	C₈	—	2	2	no	—
354	C₈	—	53	4	no	—
355	C₈	—	106	2	no	—
356	C₈	#13133177	1	1	yes	characteristic
357	C₈	#13133177	1	1	yes	characteristic
358	C₈	C₂×D₄×D₅₃	1	1	yes	characteristic
359	C₈	C₂×#11719947	1	1	yes	characteristic
360	C₂²	—	2	2	no	—
361	C₂²	—	4	1	no	—
362	C₂²	—	8	1	no	—
363	C₂²	—	106	2	no	—
364	C₂²	—	424	1	no	—
365	C₂²	#16810686	1	1	yes	characteristic
366	C₂²	#16810686	1	1	yes	characteristic
367	C₂²	C₂×#13133225	1	1	yes	characteristic
368	C₄	—	2	2	no	—
369	C₄	—	106	2	no	—
370	C₄	—	212	2	no	—
371	C₄	#16810686	1	1	yes	characteristic
372	C₄	#16810686	1	1	yes	characteristic
373	C₄	C₂×#13133177	1	1	yes	characteristic
374	C₄	C₂²×C₈×D₅₃	1	1	yes	characteristic
375	C₂	—	2	1	no	—
376	C₂	—	2	1	no	—
377	C₂	—	2	1	no	—
378	C₂	—	4	2	no	—
379	C₂	—	212	2	no	—
380	C₂	C₂×#16810686	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂×D₄×D₅₃
commutator	C₄×C₅₃	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂×C₅₃×#350	C₂
socle	C₂×C₅₃	C₂×C₈×D₄
prime radical	#16811421	C₄
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₂×#350	D₅₃
hypocenter	C₅₃	#33442
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×#350

Complements

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

N	H ≅ G/N	type
1. #20287818	C₂	semidirect
2. #20287817	C₂	semidirect
4. #20276554	C₂	semidirect
5. #20276549	C₂	semidirect
6. #20276480	C₂	semidirect
7. C₂×#16810406	C₂	semidirect
8. #20276311	C₂	semidirect
9. C₂×C₅₃×#350	C₂	semidirect
12. #16811041	C₂²	semidirect
14. #16811040	C₂²	semidirect
26. #16810443	C₂²	semidirect
27. #16810406	C₂²	semidirect
28. #16810406	C₂²	semidirect
29. #16810398	C₂²	semidirect
30. C₁₆×Dic₅₃	C₂²	semidirect
34. C₅₃×#350	C₂²	semidirect
35. C₅₃×#350	C₂²	semidirect
192. C₂×#350	D₅₃	semidirect
222. #350	C₂×D₅₃	semidirect
223. #350	C₂×D₅₃	semidirect
226. C₅₃	#33442	semidirect

Maximal subgroups

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

subgroup	index
#20287818	2
#20287817	2
#20283616	2
#20276554	2
#20276549	2
#20276480	2
C₂×#16810406	2
#20276311	2
C₂×C₅₃×#350	2
#33442	53