Subgroups — #27537567

Order 13568.

668 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811058	C₂	1	1	yes	maximal, characteristic
2	#20283568	C₂	1	1	yes	maximal, characteristic, complemented
3	#20276613	C₂	1	8	yes	maximal, complemented
4	#20276489	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#16810435	C₂	1	1	yes	maximal, characteristic, complemented
6	#20276451	C₂	1	1	yes	maximal, characteristic, complemented
7	#20276320	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#2957	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₄₂₄	C₂²	1	1	yes	characteristic
10	#16811097	—	2	1	no	—
11	#16811058	C₄	1	4	yes	complemented
12	#16811058	C₂²	1	4	yes	—
13	C₂×#13133160	C₄	1	2	yes	complemented
14	C₂×#13133158	C₄	1	2	yes	complemented
15	C₂×#13133157	C₂²	1	1	yes	characteristic
16	C₂×#13133157	C₂²	1	1	yes	characteristic
17	#16810736	C₂²	1	4	yes	complemented
18	C₄×D₄₂₄	—	2	2	no	—
19	#16810661	C₂²	1	1	yes	characteristic
20	#16810629	—	2	1	no	—
21	#16810447	C₂²	1	4	yes	complemented
22	#16810435	C₂²	1	4	yes	complemented
23	#16810433	C₂²	1	4	yes	complemented
24	#16810401	C₂²	1	4	yes	complemented
25	C₂×#135×C₅₃	C₂²	1	1	yes	characteristic
26	C₅₃×#568	—	2	1	no	—
27	C₅₃×#444	C₂²	1	4	yes	complemented
28	C₅₃×#443	—	2	2	no	—
29	C₂×#108×C₅₃	C₂²	1	1	yes	characteristic
30	C₅₃×#421	—	2	1	no	—
31	C₅₃×#405	C₂²	1	1	yes	characteristic
32	C₂²×D₂₁₂	—	2	1	no	—
33	C₂×D₄₂₄	—	2	1	no	—
34	C₂×D₄₂₄	—	2	1	no	—
35	C₂×D₄₂₄	—	2	1	no	—
36	C₂×D₄₂₄	C₂×C₄	1	2	yes	—
37	C₂×D₄₂₄	C₂×C₄	1	2	yes	—
38	C₂×D₄₂₄	C₂³	1	2	yes	—
39	C₄×D₂₁₂	—	2	4	no	—
40	#13133179	—	4	2	no	—
41	#13133176	—	2	1	no	—
42	C₂²⋊C₄×D₅₃	—	4	2	no	—
43	#13133160	C₂×C₄	1	8	yes	complemented
44	#13133158	C₂×C₄	1	8	yes	complemented
45	#13133157	—	2	1	no	—
46	#13133157	C₂×C₄	1	2	yes	complemented
47	#13133157	C₂³	1	2	yes	—
48	#13133157	C₂³	1	2	yes	—
49	C₂×#11719908	—	2	1	no	—
50	C₂×#11719908	C₂×C₄	1	2	yes	complemented
51	#13133151	—	2	1	no	—
52	#13133149	C₂³	1	2	yes	—
53	#13133139	—	2	4	no	—
54	C₂×#41×C₅₃	—	2	1	no	—
55	#135×C₅₃	—	2	1	no	—
56	#135×C₅₃	—	2	1	no	—
57	#135×C₅₃	C₂×C₄	1	2	yes	complemented
58	#135×C₅₃	C₂³	1	2	yes	—
59	C₂×Q₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic, complemented
60	C₂×SD₁₆×C₅₃	—	2	1	no	—
61	C₂×D₈×C₅₃	C₂×C₄	1	1	yes	characteristic, complemented
62	#131×C₅₃	—	2	4	no	—
63	#131×C₅₃	—	4	2	no	—
64	C₂×M₁₆×C₅₃	—	2	1	no	—
65	C₂×M₁₆×C₅₃	—	4	2	no	—
66	C₂×M₁₆×C₅₃	D₄	1	1	yes	characteristic
67	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
68	#117×C₅₃	D₄	1	1	yes	characteristic
69	#108×C₅₃	—	2	1	no	—
70	#108×C₅₃	C₂³	1	2	yes	—
71	#104×C₅₃	—	2	4	no	—
72	#97×C₅₃	C₂³	1	2	yes	—
73	C₄×C₈×C₅₃	D₄	1	2	yes	—
74	C₂³×D₅₃	—	4	1	no	—
75	#41×C₅₃	—	2	2	no	—
76	#41×C₅₃	—	2	2	no	—
77	#41×C₅₃	—	4	1	no	—
78	#41×C₅₃	—	4	1	no	—
79	C₂×Q₈×C₅₃	—	2	1	no	—
80	C₂×D₄×C₅₃	—	2	1	no	—
81	C₂×D₄×C₅₃	—	4	1	no	—
82	C₂²×C₄×C₅₃	—	4	1	no	—
83	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
84	C₂×D₂₁₂	—	2	2	no	—
85	C₂×D₂₁₂	—	2	2	no	—
86	C₂×D₂₁₂	—	2	2	no	—
87	C₂×D₂₁₂	—	4	1	no	—
88	C₂×D₂₁₂	—	4	1	no	—
89	C₂×D₂₁₂	—	4	1	no	—
90	C₂×C₄×D₅₃	—	4	4	no	—
91	Q₁₆×C₅₃	—	2	1	no	—
92	Q₁₆×C₅₃	C₂²×C₄	1	2	yes	—
93	SD₁₆×C₅₃	—	2	2	no	—
94	SD₁₆×C₅₃	—	2	2	no	—
95	D₈×C₅₃	—	2	1	no	—
96	D₈×C₅₃	C₂²×C₄	1	2	yes	—
97	M₁₆×C₅₃	—	2	1	no	—
98	M₁₆×C₅₃	—	2	1	no	—
99	M₁₆×C₅₃	—	2	2	no	—
100	M₁₆×C₅₃	—	4	4	no	—
101	M₁₆×C₅₃	—	4	1	no	—
102	C₂×C₈×C₅₃	—	4	4	no	—
103	C₂×C₈×C₅₃	—	4	1	no	—
104	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
107	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
109	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
110	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
111	C₄⋊C₄×C₅₃	—	2	1	no	—
112	C₂²⋊C₄×C₅₃	—	2	1	no	—
113	C₄²×C₅₃	C₂×D₄	1	2	yes	—
114	C₂²⋊Dic₅₃	—	4	1	no	—
115	#11719921	—	4	4	no	—
116	C₄⋊Dic₅₃	—	2	2	no	—
117	C₄×Dic₅₃	—	4	1	no	—
118	D₄₂₄	—	2	2	no	—
119	D₄₂₄	—	2	2	no	—
120	D₄₂₄	—	2	2	no	—
121	D₄₂₄	C₂²×C₄	1	4	yes	—
122	#11719908	—	2	2	no	—
123	#11719908	C₂²×C₄	1	4	yes	—
124	C₂³×C₅₃	—	4	1	no	—
125	C₂³×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
126	C₂²×D₅₃	—	4	2	no	—
127	C₂²×D₅₃	—	4	2	no	—
128	C₂²×D₅₃	—	4	2	no	—
129	C₂²×D₅₃	—	16	2	no	—
130	Q₈×C₅₃	—	2	2	no	—
131	Q₈×C₅₃	—	4	1	no	—
132	D₄×C₅₃	—	2	2	no	—
133	D₄×C₅₃	—	4	2	no	—
134	D₄×C₅₃	—	4	2	no	—
135	D₄×C₅₃	—	4	1	no	—
136	C₂×C₄×C₅₃	—	2	2	no	—
137	C₂×C₄×C₅₃	—	4	1	no	—
138	C₂×C₄×C₅₃	—	4	2	no	—
139	C₂×C₄×C₅₃	—	4	2	no	—
140	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
141	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
142	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
143	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
144	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
145	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
146	C₂×Dic₅₃	—	4	2	no	—
147	D₂₁₂	—	2	4	no	—
148	D₂₁₂	—	4	2	no	—
149	D₂₁₂	—	4	2	no	—
150	D₂₁₂	—	4	2	no	—
151	C₄×D₅₃	—	16	4	no	—
152	C₈×C₅₃	—	2	2	no	—
153	C₈×C₅₃	—	4	2	no	—
154	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
155	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
156	C₈×C₅₃	C₂³×C₄	1	1	yes	characteristic
157	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
158	#33471	—	53	1	no	maximal, Sylow
159	C₂²×C₅₃	—	4	1	no	—
160	C₂²×C₅₃	—	4	2	no	—
161	C₂²×C₅₃	—	16	1	no	—
162	C₂²×C₅₃	#442	1	1	yes	characteristic
163	C₂²×C₅₃	#442	1	1	yes	characteristic
164	C₂²×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
165	C₂×D₅₃	—	4	4	no	—
166	C₂×D₅₃	—	8	4	no	—
167	C₂×D₅₃	—	16	2	no	—
168	C₂×D₅₃	—	16	2	no	—
169	C₄×C₅₃	—	4	2	no	—
170	C₄×C₅₃	—	4	2	no	—
171	C₄×C₅₃	#442	1	1	yes	characteristic
172	C₄×C₅₃	#442	1	1	yes	characteristic
173	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
174	C₄×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
175	Dic₅₃	—	8	2	no	—
176	C₂×#509	—	53	1	no	—
177	#2957	D₅₃	1	1	yes	characteristic, complemented
178	#2948	—	53	1	no	—
179	AGL(1,ℤ/16)	—	53	8	no	—
180	#2155	—	53	1	no	—
181	C₂×#361	—	53	1	no	—
182	#2145	—	53	1	no	—
183	#2110	—	53	1	no	—
184	C₂×C₅₃	—	2	1	no	—
185	C₂×C₅₃	—	2	1	no	—
186	C₂×C₅₃	—	2	1	no	—
187	C₂×C₅₃	—	8	2	no	—
188	C₂×C₅₃	C₂×#442	1	1	yes	characteristic
189	D₅₃	—	8	4	no	—
190	C₂×#135	C₂×D₅₃	1	1	yes	characteristic
191	C₂²×D₈	—	53	1	no	—
192	#568	—	2	1	no	—
193	#518	—	106	1	no	—
194	#509	—	53	4	no	—
195	#509	—	53	4	no	—
196	C₂×SD₃₂	—	53	2	no	—
197	C₂×D₁₆	—	53	2	no	—
198	C₂×M₃₂	—	53	1	no	—
199	C₂×M₃₂	—	53	1	no	—
200	#444	C₂×D₅₃	1	4	yes	complemented
201	#443	—	2	2	no	—
202	#442	—	53	4	no	—
203	C₄×D₈	—	106	2	no	—
204	C₂×#108	C₂×D₅₃	1	1	yes	characteristic
205	#427	—	53	1	no	—
206	#421	—	2	1	no	—
207	#418	—	106	1	no	—
208	#405	C₂×D₅₃	1	1	yes	characteristic
209	#365	—	53	4	no	—
210	#361	—	53	4	no	—
211	#360	—	53	4	no	—
212	#347	—	53	4	no	—
213	C₅₃	#33471	1	1	yes	characteristic, complemented, Sylow
214	C₂×#41	—	2	1	no	—
215	C₂²×D₄	—	106	1	no	—
216	#135	—	2	1	no	—
217	#135	—	2	1	no	—
218	#135	C₄×D₅₃	1	2	yes	complemented
219	#135	C₂²×D₅₃	1	2	yes	—
220	C₂×Q₁₆	C₄×D₅₃	1	1	yes	characteristic, complemented
221	C₂×SD₁₆	—	2	1	no	—
222	C₂×D₈	—	53	2	no	—
223	C₂×D₈	—	53	2	no	—
224	C₂×D₈	—	53	2	no	—
225	C₂×D₈	—	106	1	no	—
226	C₂×D₈	—	106	1	no	—
227	C₂×D₈	—	106	1	no	—
228	C₂×D₈	C₄×D₅₃	1	1	yes	characteristic, complemented
229	#131	—	2	4	no	—
230	#131	—	4	2	no	—
231	C₂×M₁₆	—	2	1	no	—
232	C₂×M₁₆	—	4	2	no	—
233	C₂×M₁₆	C₂²⋊D₅₃	1	1	yes	characteristic
234	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
235	C₄×D₄	—	106	4	no	—
236	C₄×D₄	—	212	2	no	—
237	#117	—	106	1	no	—
238	#117	C₂²⋊D₅₃	1	1	yes	characteristic
239	C₂×C₂²⋊C₄	—	212	2	no	—
240	SD₃₂	—	53	8	no	—
241	D₁₆	—	53	8	no	—
242	M₃₂	—	53	2	no	—
243	M₃₂	—	53	2	no	—
244	M₃₂	—	53	2	no	—
245	M₃₂	—	106	1	no	—
246	C₂×C₁₆	—	53	2	no	—
247	C₂×C₁₆	—	106	1	no	—
248	#108	—	2	1	no	—
249	#108	C₂²×D₅₃	1	2	yes	—
250	#107	—	106	1	no	—
251	#106	—	53	2	no	—
252	#104	—	2	4	no	—
253	#102	—	106	4	no	—
254	#97	C₂²×D₅₃	1	2	yes	—
255	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
256	C₂⁴	—	212	1	no	—
257	#41	—	2	2	no	—
258	#41	—	2	2	no	—
259	#41	—	4	1	no	—
260	#41	—	4	1	no	—
261	C₂×Q₈	—	2	1	no	—
262	C₂×D₄	—	2	1	no	—
263	C₂×D₄	—	4	1	no	—
264	C₂×D₄	—	106	2	no	—
265	C₂×D₄	—	106	2	no	—
266	C₂×D₄	—	106	2	no	—
267	C₂×D₄	—	212	1	no	—
268	C₂×D₄	—	212	1	no	—
269	C₂×D₄	—	212	1	no	—
270	C₂²×C₄	—	4	1	no	—
271	C₂²×C₄	—	212	4	no	—
272	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
273	Q₁₆	—	2	1	no	—
274	Q₁₆	C₂×C₄×D₅₃	1	2	yes	—
275	SD₁₆	—	2	2	no	—
276	SD₁₆	—	2	2	no	—
277	D₈	—	2	1	no	—
278	D₈	—	53	4	no	—
279	D₈	—	106	2	no	—
280	D₈	—	106	2	no	—
281	D₈	—	106	2	no	—
282	D₈	C₂×C₄×D₅₃	1	2	yes	—
283	M₁₆	—	2	1	no	—
284	M₁₆	—	2	1	no	—
285	M₁₆	—	2	2	no	—
286	M₁₆	—	4	4	no	—
287	M₁₆	—	4	1	no	—
288	C₂×C₈	—	4	4	no	—
289	C₂×C₈	—	4	1	no	—
290	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
291	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
292	C₂×C₈	#11719945	1	1	yes	characteristic
293	C₂×C₈	#11719945	1	1	yes	characteristic
294	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
295	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
296	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
297	C₄⋊C₄	—	2	1	no	—
298	C₄⋊C₄	—	106	2	no	—
299	C₂²⋊C₄	—	2	1	no	—
300	C₂²⋊C₄	—	212	4	no	—
301	C₂²⋊C₄	—	212	1	no	—
302	C₄²	—	212	1	no	—
303	C₄²	C₂×C₂²⋊D₅₃	1	2	yes	—
304	C₁₆	—	53	4	no	—
305	C₁₆	—	106	2	no	—
306	C₂³	—	4	1	no	—
307	C₂³	—	848	2	no	—
308	C₂³	—	212	2	no	—
309	C₂³	—	212	2	no	—
310	C₂³	—	212	2	no	—
311	C₂³	#13133242	1	1	yes	characteristic
312	Q₈	—	2	2	no	—
313	Q₈	—	4	1	no	—
314	D₄	—	2	2	no	—
315	D₄	—	4	2	no	—
316	D₄	—	4	2	no	—
317	D₄	—	4	1	no	—
318	D₄	—	106	4	no	—
319	D₄	—	212	2	no	—
320	D₄	—	212	2	no	—
321	D₄	—	212	2	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	4	1	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	4	2	no	—
326	C₂×C₄	—	848	4	no	—
327	C₂×C₄	—	212	2	no	—
328	C₂×C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
329	C₂×C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
330	C₂×C₄	#13133242	1	1	yes	characteristic
331	C₂×C₄	#13133243	1	1	yes	characteristic
332	C₂×C₄	#13133243	1	1	yes	characteristic
333	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
334	C₈	—	2	2	no	—
335	C₈	—	4	2	no	—
336	C₈	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
337	C₈	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
338	C₈	C₂²×C₄×D₅₃	1	1	yes	characteristic
339	C₈	C₂×#11719945	1	1	yes	characteristic
340	C₂²	—	4	1	no	—
341	C₂²	—	4	2	no	—
342	C₂²	—	16	1	no	—
343	C₂²	—	848	2	no	—
344	C₂²	—	848	2	no	—
345	C₂²	—	424	4	no	—
346	C₂²	—	212	4	no	—
347	C₂²	#16810733	1	1	yes	characteristic
348	C₂²	#16810733	1	1	yes	characteristic
349	C₂²	C₂×#13133242	1	1	yes	characteristic
350	C₄	—	4	2	no	—
351	C₄	—	4	2	no	—
352	C₄	—	424	2	no	—
353	C₄	#16810733	1	1	yes	characteristic
354	C₄	#16810733	1	1	yes	characteristic
355	C₄	C₂×C₄×C₂²⋊D₅₃	1	1	yes	characteristic
356	C₄	C₂×#13133243	1	1	yes	characteristic
357	C₂	—	2	1	no	—
358	C₂	—	2	1	no	—
359	C₂	—	2	1	no	—
360	C₂	—	8	2	no	—
361	C₂	—	424	4	no	—
362	C₂	C₂×#16810733	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#16810733
commutator	C₈×C₅₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2957	C₂
socle	C₂×C₅₃	C₂×#442
prime radical	C₂×#16811058	C₂
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	#2957	D₅₃
hypocenter	C₅₃	#33471
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #405 ◁ #2957

Complements

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

N	H ≅ G/N	type
2. #20283568	C₂	semidirect
3. #20276613	C₂	semidirect
4. #20276489	C₂	semidirect
5. C₂×#16810435	C₂	semidirect
6. #20276451	C₂	semidirect
7. #20276320	C₂	semidirect
8. C₅₃×#2957	C₂	semidirect
11. #16811058	C₄	semidirect
13. C₂×#13133160	C₄	semidirect
14. C₂×#13133158	C₄	semidirect
17. #16810736	C₂²	semidirect
21. #16810447	C₂²	semidirect
22. #16810435	C₂²	semidirect
23. #16810433	C₂²	semidirect
24. #16810401	C₂²	semidirect
27. C₅₃×#444	C₂²	semidirect
43. #13133160	C₂×C₄	semidirect
44. #13133158	C₂×C₄	semidirect
46. #13133157	C₂×C₄	semidirect
50. C₂×#11719908	C₂×C₄	semidirect
57. #135×C₅₃	C₂×C₄	semidirect
59. C₂×Q₁₆×C₅₃	C₂×C₄	semidirect
61. C₂×D₈×C₅₃	C₂×C₄	semidirect
177. #2957	D₅₃	semidirect
200. #444	C₂×D₅₃	semidirect
213. C₅₃	#33471	semidirect
218. #135	C₄×D₅₃	semidirect
220. C₂×Q₁₆	C₄×D₅₃	semidirect
228. C₂×D₈	C₄×D₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16811058	2
#20283568	2
#20276613	2
#20276489	2
C₂×#16810435	2
#20276451	2
#20276320	2
C₅₃×#2957	2
#33471	53