Subgroups — #27539541

Order 13568.

858 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811060	C₂	1	2	yes	maximal, complemented
2	C₂×#16811056	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×D₁₆×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#20285197	C₂	1	2	yes	maximal, complemented
5	#20276982	C₂	1	2	yes	maximal, complemented
6	#20276980	C₂	1	2	yes	maximal, complemented
7	#20276977	C₂	1	1	yes	maximal, characteristic, complemented
8	#20276965	C₂	1	1	yes	maximal, characteristic, complemented
9	#20276963	C₂	1	1	yes	maximal, characteristic, complemented
10	#20276310	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#2249	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13133244	—	2	2	no	—
13	C₂×#13133240	C₂²	1	2	yes	—
14	C₂×D₈×D₅₃	C₂²	1	2	yes	—
15	#16811191	—	2	2	no	—
16	#16811060	—	2	8	no	—
17	#16811056	—	2	4	no	—
18	C₂×#13133160	C₂²	1	2	yes	complemented
19	C₂×#13133160	C₂²	1	2	yes	complemented
20	C₂×D₈₄₈	C₂²	1	1	yes	characteristic
21	C₂×D₈₄₈	C₂²	1	1	yes	characteristic, complemented
22	C₂×D₈₄₈	C₂²	1	2	yes	complemented
23	C₂×#13133158	C₂²	1	2	yes	—
24	C₂×#13133158	C₂²	1	2	yes	complemented
25	D₁₆×D₅₃	—	2	4	no	—
26	C₂×#13133156	C₂²	1	1	yes	characteristic
27	C₂×C₁₆×D₅₃	C₂²	1	1	yes	characteristic
28	#16811013	—	2	2	no	—
29	#16811009	—	2	2	no	—
30	#16811008	C₂²	1	2	yes	complemented
31	#16811000	C₂²	1	2	yes	complemented
32	#16810998	C₂²	1	2	yes	complemented
33	#16810577	C₂²	1	1	yes	characteristic
34	#16810399	C₂²	1	1	yes	characteristic, complemented
35	#16810398	C₂²	1	1	yes	characteristic, complemented
36	C₁₆×Dic₅₃	C₂²	1	1	yes	characteristic, complemented
37	C₂×Q₃₂×C₅₃	C₂²	1	1	yes	characteristic, complemented
38	C₂×SD₃₂×C₅₃	C₂²	1	2	yes	complemented
39	C₂×D₁₆×C₅₃	C₂²	1	1	yes	characteristic
40	C₅₃×#495	C₂²	1	2	yes	complemented
41	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic, complemented
42	C₂×#11719949	—	2	2	no	—
43	C₂×D₄×D₅₃	—	2	2	no	—
44	#13133244	—	4	8	no	—
45	#13133240	—	2	4	no	—
46	#13133240	—	2	4	no	—
47	C₂×#11719924	—	2	2	no	—
48	C₂×#11719924	—	2	2	no	—
49	C₂×D₄₂₄	—	2	2	no	—
50	C₂×D₄₂₄	C₂³	1	2	yes	—
51	C₂×D₄₂₄	C₂³	1	2	yes	—
52	C₂×#11719922	—	2	2	no	—
53	C₂×#11719922	—	2	2	no	—
54	D₈×D₅₃	—	2	4	no	—
55	D₈×D₅₃	—	2	4	no	—
56	C₂×#11719912	D₄	1	1	yes	characteristic
57	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
58	#13133214	—	2	2	no	—
59	#13133213	—	2	2	no	—
60	#13133206	—	2	2	no	—
61	#13133190	—	4	4	no	—
62	#13133187	—	4	4	no	—
63	#13133174	D₄	1	1	yes	characteristic
64	#13133160	—	2	4	no	—
65	#13133160	—	2	4	no	—
66	D₈₄₈	—	2	2	no	—
67	D₈₄₈	—	2	2	no	—
68	D₈₄₈	—	2	4	no	—
69	#13133158	—	2	4	no	—
70	#13133158	—	2	4	no	—
71	#13133156	D₄	1	4	yes	complemented
72	C₂×#11719908	C₂³	1	1	yes	characteristic
73	C₂×#11719908	C₂³	1	1	yes	characteristic
74	C₁₆×D₅₃	D₄	1	4	yes	complemented
75	#13133128	D₄	1	1	yes	characteristic
76	#13133127	D₄	1	1	yes	characteristic
77	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
78	C₂×Q₁₆×C₅₃	C₂³	1	2	yes	—
79	C₂×SD₁₆×C₅₃	—	2	2	no	—
80	C₂×D₈×C₅₃	C₂³	1	2	yes	—
81	#124×C₅₃	—	2	2	no	—
82	Q₃₂×C₅₃	—	2	2	no	—
83	SD₃₂×C₅₃	—	2	4	no	—
84	D₁₆×C₅₃	—	2	2	no	—
85	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
86	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
87	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
88	C₂³×D₅₃	—	4	2	no	—
89	C₂×Q₈×C₅₃	—	2	2	no	—
90	C₂×D₄×C₅₃	—	2	2	no	—
91	C₂×C₂²⋊D₅₃	—	4	2	no	—
92	C₂×C₂²⋊D₅₃	—	4	4	no	—
93	#11719949	—	4	4	no	—
94	#11719949	—	4	4	no	—
95	D₄×D₅₃	—	4	4	no	—
96	D₄×D₅₃	—	4	4	no	—
97	C₂×D₂₁₂	—	2	2	no	—
98	C₂×D₂₁₂	—	2	2	no	—
99	C₂×D₂₁₂	—	4	2	no	—
100	C₂×D₂₁₂	—	4	4	no	—
101	C₂×C₄×D₅₃	—	4	2	no	—
102	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
103	Q₁₆×C₅₃	—	2	2	no	—
104	Q₁₆×C₅₃	—	2	2	no	—
105	SD₁₆×C₅₃	—	4	4	no	—
106	D₈×C₅₃	—	2	2	no	—
107	D₈×C₅₃	—	2	2	no	—
108	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
109	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
110	C₂²⋊C₄×C₅₃	—	4	4	no	—
111	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
112	#11719924	—	4	4	no	—
113	#11719924	—	4	4	no	—
114	#11719922	—	4	4	no	—
115	#11719922	—	4	4	no	—
116	#11719921	—	4	4	no	—
117	#11719921	D₈	1	2	yes	—
118	#11719919	D₈	1	2	yes	—
119	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
120	C₂×#88322	C₂×D₄	1	1	yes	characteristic
121	C₂×#88322	C₂×D₄	1	1	yes	characteristic
122	D₄₂₄	—	2	2	no	—
123	D₄₂₄	—	2	2	no	—
124	D₄₂₄	—	2	2	no	—
125	D₄₂₄	—	2	2	no	—
126	D₄₂₄	—	4	4	no	—
127	#11719912	D₈	1	4	yes	—
128	C₈×D₅₃	C₂×D₄	1	2	yes	—
129	C₈×D₅₃	C₂×D₄	1	2	yes	—
130	C₁₆×C₅₃	C₂×D₄	1	2	yes	—
131	C₁₆×C₅₃	C₂×D₄	1	2	yes	—
132	#11719908	C₂×D₄	1	2	yes	—
133	#11719908	C₂×D₄	1	2	yes	—
134	C₂³×C₅₃	—	4	2	no	—
135	C₂²×D₅₃	—	4	2	no	—
136	C₂²×D₅₃	—	4	2	no	—
137	C₂²×D₅₃	—	8	4	no	—
138	C₂²×D₅₃	—	16	4	no	—
139	C₂²×D₅₃	C₂×D₈	1	1	yes	characteristic
140	Q₈×C₅₃	—	4	2	no	—
141	Q₈×C₅₃	—	4	2	no	—
142	D₄×C₅₃	—	4	2	no	—
143	D₄×C₅₃	—	4	2	no	—
144	C₂×C₄×C₅₃	—	4	2	no	—
145	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
146	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
147	C₂²⋊D₅₃	—	8	4	no	—
148	C₂²⋊D₅₃	—	16	4	no	—
149	C₂×Dic₅₃	C₂×D₈	1	2	yes	—
150	C₂×Dic₅₃	C₂×D₈	1	1	yes	characteristic
151	D₂₁₂	—	4	2	no	—
152	D₂₁₂	—	4	2	no	—
153	D₂₁₂	—	4	2	no	—
154	D₂₁₂	—	4	2	no	—
155	D₂₁₂	—	8	4	no	—
156	D₂₁₂	—	16	4	no	—
157	C₄×D₅₃	—	8	4	no	—
158	C₄×D₅₃	#127	1	2	yes	—
159	C₄×D₅₃	C₂×D₈	1	2	yes	—
160	C₈×C₅₃	C₂×D₈	1	2	yes	—
161	C₈×C₅₃	C₂²×D₄	1	1	yes	characteristic
162	C₈×C₅₃	C₂²×D₄	1	1	yes	characteristic
163	#88322	#127	1	2	yes	—
164	#88322	C₂×D₈	1	2	yes	—
165	#33720	—	53	1	no	maximal, Sylow
166	C₂²×C₅₃	—	8	2	no	—
167	C₂²×C₅₃	—	8	4	no	—
168	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
169	C₂×D₅₃	—	2	2	no	—
170	C₂×D₅₃	—	8	2	no	—
171	C₂×D₅₃	—	8	2	no	—
172	C₂×D₅₃	—	8	4	no	—
173	C₂×D₅₃	—	8	4	no	—
174	C₂×D₅₃	—	16	8	no	—
175	C₂×D₅₃	#493	1	2	yes	—
176	C₄×C₅₃	—	2	2	no	—
177	C₄×C₅₃	—	8	2	no	—
178	C₄×C₅₃	C₂×#127	1	1	yes	characteristic
179	C₄×C₅₃	C₂²×D₈	1	1	yes	characteristic
180	Dic₅₃	—	2	2	no	—
181	Dic₅₃	#493	1	2	yes	—
182	C₂×#509	—	53	2	no	—
183	C₂×#508	—	53	1	no	—
184	C₂²×D₁₆	—	53	1	no	—
185	#3155	—	53	2	no	—
186	#2250	—	53	2	no	—
187	#2250	—	53	2	no	—
188	#2249	—	53	1	no	—
189	#2249	D₅₃	1	1	yes	characteristic, complemented
190	#2246	—	53	1	no	—
191	#2246	—	53	1	no	—
192	#2108	—	53	1	no	—
193	C₂×C₅₃	—	16	2	no	—
194	C₂×C₅₃	C₂×#493	1	1	yes	characteristic
195	C₂×C₅₃	#3415	1	2	yes	—
196	D₅₃	—	2	2	no	—
197	D₅₃	—	16	2	no	—
198	D₅₃	—	16	2	no	—
199	C₂×AGL(1,ℤ/8)	—	106	2	no	—
200	C₂×#135	—	53	2	no	—
201	C₂²×D₈	—	53	2	no	—
202	#535	—	106	2	no	—
203	#509	—	106	8	no	—
204	#508	—	106	4	no	—
205	C₂×Q₃₂	C₂×D₅₃	1	1	yes	characteristic, complemented
206	C₂×SD₃₂	—	53	2	no	—
207	C₂×SD₃₂	—	53	2	no	—
208	C₂×SD₃₂	C₂×D₅₃	1	2	yes	complemented
209	C₂×D₁₆	—	53	2	no	—
210	C₂×D₁₆	—	53	2	no	—
211	C₂×D₁₆	—	53	1	no	—
212	C₂×D₁₆	—	53	1	no	—
213	C₂×D₁₆	—	53	2	no	—
214	C₂×D₁₆	—	106	4	no	—
215	C₂×D₁₆	C₂×D₅₃	1	1	yes	characteristic
216	C₂×M₃₂	—	53	1	no	—
217	C₂²×C₁₆	—	53	1	no	—
218	#496	—	106	2	no	—
219	#496	—	106	2	no	—
220	#495	—	53	2	no	—
221	#495	C₂×D₅₃	1	2	yes	complemented
222	#493	—	53	2	no	—
223	#493	—	53	2	no	—
224	#405	—	53	1	no	—
225	#346	—	53	1	no	—
226	#346	—	53	1	no	—
227	C₄×C₁₆	—	53	1	no	—
228	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic, complemented
229	C₅₃	#33720	1	1	yes	characteristic, complemented, Sylow
230	C₂×#41	—	106	2	no	—
231	C₂²×D₄	—	106	2	no	—
232	AGL(1,ℤ/8)	—	212	8	no	—
233	#135	—	106	4	no	—
234	#135	—	106	4	no	—
235	C₂×Q₁₆	C₂²×D₅₃	1	2	yes	—
236	C₂×SD₁₆	—	2	2	no	—
237	C₂×SD₁₆	—	106	2	no	—
238	C₂×SD₁₆	—	106	2	no	—
239	C₂×D₈	—	53	2	no	—
240	C₂×D₈	—	53	2	no	—
241	C₂×D₈	—	106	4	no	—
242	C₂×D₈	—	106	4	no	—
243	C₂×D₈	—	106	2	no	—
244	C₂×D₈	—	106	2	no	—
245	C₂×D₈	—	106	2	no	—
246	C₂×D₈	C₂²×D₅₃	1	2	yes	—
247	C₂×M₁₆	—	53	1	no	—
248	C₂²×C₈	—	53	1	no	—
249	#127	—	106	2	no	—
250	#127	—	106	2	no	—
251	#124	—	2	2	no	—
252	#124	—	106	2	no	—
253	#121	—	212	4	no	—
254	#120	—	212	4	no	—
255	#117	—	53	1	no	—
256	Q₃₂	—	2	2	no	—
257	SD₃₂	—	2	4	no	—
258	SD₃₂	—	106	4	no	—
259	SD₃₂	—	106	4	no	—
260	D₁₆	—	2	2	no	—
261	D₁₆	—	106	4	no	—
262	D₁₆	—	106	4	no	—
263	D₁₆	—	106	2	no	—
264	D₁₆	—	106	2	no	—
265	D₁₆	—	106	4	no	—
266	M₃₂	—	53	4	no	—
267	C₂×C₁₆	—	53	1	no	—
268	C₂×C₁₆	—	53	1	no	—
269	C₂×C₁₆	—	53	4	no	—
270	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
271	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
272	#97	—	53	1	no	—
273	#97	—	53	1	no	—
274	C₄×C₈	—	53	1	no	—
275	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
276	C₂⁴	—	212	2	no	—
277	#41	—	212	4	no	—
278	#41	—	212	4	no	—
279	C₂×Q₈	—	2	2	no	—
280	C₂×D₄	—	2	2	no	—
281	C₂×D₄	—	106	2	no	—
282	C₂×D₄	—	106	2	no	—
283	C₂×D₄	—	212	2	no	—
284	C₂×D₄	—	212	4	no	—
285	C₂×D₄	—	212	4	no	—
286	C₂×D₄	—	212	4	no	—
287	C₂×D₄	—	212	2	no	—
288	C₂×D₄	—	212	4	no	—
289	C₂²×C₄	—	53	1	no	—
290	C₂²×C₄	—	212	2	no	—
291	Q₁₆	—	2	2	no	—
292	Q₁₆	—	2	2	no	—
293	SD₁₆	—	4	4	no	—
294	SD₁₆	—	212	4	no	—
295	SD₁₆	—	212	4	no	—
296	D₈	—	2	2	no	—
297	D₈	—	2	2	no	—
298	D₈	—	106	2	no	—
299	D₈	—	106	2	no	—
300	D₈	—	106	2	no	—
301	D₈	—	106	2	no	—
302	D₈	—	212	4	no	—
303	D₈	—	212	4	no	—
304	D₈	—	212	4	no	—
305	M₁₆	—	53	4	no	—
306	C₂×C₈	—	53	1	no	—
307	C₂×C₈	—	53	1	no	—
308	C₂×C₈	—	53	2	no	—
309	C₂×C₈	—	53	2	no	—
310	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
311	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
312	C₄⋊C₄	—	53	2	no	—
313	C₂²⋊C₄	—	4	4	no	—
314	C₂²⋊C₄	—	53	2	no	—
315	C₂²⋊C₄	—	212	4	no	—
316	C₄²	—	53	1	no	—
317	C₄²	D₄×D₅₃	1	1	yes	characteristic
318	C₁₆	—	53	2	no	—
319	C₁₆	—	53	2	no	—
320	C₁₆	D₄×D₅₃	1	2	yes	—
321	C₁₆	D₄×D₅₃	1	2	yes	—
322	C₂³	—	4	2	no	—
323	C₂³	—	53	1	no	—
324	C₂³	—	848	4	no	—
325	C₂³	—	424	4	no	—
326	C₂³	—	212	2	no	—
327	C₂³	—	212	2	no	—
328	Q₈	—	4	2	no	—
329	Q₈	—	4	2	no	—
330	D₄	—	4	2	no	—
331	D₄	—	4	2	no	—
332	D₄	—	848	4	no	—
333	D₄	—	848	4	no	—
334	D₄	—	424	4	no	—
335	D₄	—	424	4	no	—
336	D₄	—	212	2	no	—
337	D₄	—	212	2	no	—
338	D₄	—	212	2	no	—
339	D₄	—	212	2	no	—
340	C₂×C₄	—	4	2	no	—
341	C₂×C₄	—	53	1	no	—
342	C₂×C₄	—	53	2	no	—
343	C₂×C₄	—	53	2	no	—
344	C₂×C₄	—	53	2	no	—
345	C₂×C₄	—	424	4	no	—
346	C₂×C₄	D₈×D₅₃	1	2	yes	—
347	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
348	C₈	—	53	2	no	—
349	C₈	—	53	2	no	—
350	C₈	D₈×D₅₃	1	2	yes	—
351	C₈	C₂×D₄×D₅₃	1	1	yes	characteristic
352	C₈	C₂×D₄×D₅₃	1	1	yes	characteristic
353	C₂²	—	8	4	no	—
354	C₂²	—	8	2	no	—
355	C₂²	—	53	2	no	—
356	C₂²	—	848	8	no	—
357	C₂²	—	106	2	no	—
358	C₂²	—	424	4	no	—
359	C₂²	—	424	4	no	—
360	C₂²	—	424	2	no	—
361	C₂²	—	424	2	no	—
362	C₂²	C₂×D₈×D₅₃	1	1	yes	characteristic
363	C₄	—	2	2	no	—
364	C₄	—	8	2	no	—
365	C₄	—	53	2	no	—
366	C₄	—	106	2	no	—
367	C₄	#127×D₅₃	1	1	yes	characteristic
368	C₄	C₂×D₈×D₅₃	1	1	yes	characteristic
369	C₂	—	16	2	no	—
370	C₂	—	848	2	no	—
371	C₂	—	848	2	no	—
372	C₂	—	106	2	no	—
373	C₂	#493×D₅₃	1	1	yes	characteristic
374	C₂	#20287853	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₈×D₅₃
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2249	C₂
socle	C₂²×C₅₃	C₂²×D₈
hypercenter	#2249	D₅₃
hypocenter	C₅₃	#33720
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2249

Complements

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

N	H ≅ G/N	type
1. C₂×#16811060	C₂	semidirect
2. C₂×#16811056	C₂	semidirect
3. C₂×D₁₆×D₅₃	C₂	semidirect
4. #20285197	C₂	semidirect
5. #20276982	C₂	semidirect
6. #20276980	C₂	semidirect
7. #20276977	C₂	semidirect
8. #20276965	C₂	semidirect
9. #20276963	C₂	semidirect
10. #20276310	C₂	semidirect
11. C₅₃×#2249	C₂	semidirect
18. C₂×#13133160	C₂²	semidirect
19. C₂×#13133160	C₂²	semidirect
21. C₂×D₈₄₈	C₂²	semidirect
22. C₂×D₈₄₈	C₂²	semidirect
24. C₂×#13133158	C₂²	semidirect
30. #16811008	C₂²	semidirect
31. #16811000	C₂²	semidirect
32. #16810998	C₂²	semidirect
34. #16810399	C₂²	semidirect
35. #16810398	C₂²	semidirect
36. C₁₆×Dic₅₃	C₂²	semidirect
37. C₂×Q₃₂×C₅₃	C₂²	semidirect
38. C₂×SD₃₂×C₅₃	C₂²	semidirect
40. C₅₃×#495	C₂²	semidirect
41. C₄×C₁₆×C₅₃	C₂²	semidirect
71. #13133156	D₄	semidirect
74. C₁₆×D₅₃	D₄	semidirect
189. #2249	D₅₃	semidirect
205. C₂×Q₃₂	C₂×D₅₃	semidirect
208. C₂×SD₃₂	C₂×D₅₃	semidirect
221. #495	C₂×D₅₃	semidirect
228. C₄×C₁₆	C₂×D₅₃	semidirect
229. C₅₃	#33720	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16811060	2
C₂×#16811056	2
C₂×D₁₆×D₅₃	2
#20285197	2
#20276982	2
#20276980	2
#20276977	2
#20276965	2
#20276963	2
#20276310	2
C₅₃×#2249	2
#33720	53