Subgroups — #27537569

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×#16811061	C₂	1	1	yes	maximal, characteristic
2	#20283633	C₂	1	1	yes	maximal, characteristic, complemented
3	#20283568	C₂	1	1	yes	maximal, characteristic, complemented
4	#20276615	C₂	1	8	yes	maximal, complemented
5	#20276488	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#16810437	C₂	1	1	yes	maximal, characteristic, complemented
7	#20276452	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#2110	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#13133239	C₂²	1	1	yes	characteristic
10	C₂²×D₄₂₄	C₂²	1	1	yes	characteristic
11	#16811389	—	2	1	no	—
12	#16811097	—	2	1	no	—
13	#16811061	C₄	1	4	yes	complemented
14	#16811061	C₂²	1	4	yes	—
15	C₂×#13133162	C₄	1	2	yes	complemented
16	C₂×D₈₄₈	C₄	1	2	yes	complemented
17	#16810746	C₂²	1	4	yes	complemented
18	#16810741	—	2	2	no	—
19	#16810736	C₂²	1	4	yes	complemented
20	C₄×D₄₂₄	—	2	2	no	—
21	C₂×#13133153	C₂²	1	1	yes	characteristic
22	#16810661	C₂²	1	1	yes	characteristic
23	#16810641	—	2	1	no	—
24	#16810629	—	2	1	no	—
25	#16810446	C₂²	1	4	yes	complemented
26	#16810437	C₂²	1	4	yes	complemented
27	#16810434	C₂²	1	4	yes	complemented
28	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
29	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
30	C₅₃×#405	C₂²	1	1	yes	characteristic
31	C₅₃×#347	C₂²	1	4	yes	complemented
32	C₂×#11719945	—	2	1	no	—
33	C₂²×D₂₁₂	—	2	1	no	—
34	#13133239	—	2	1	no	—
35	#13133239	—	2	1	no	—
36	#13133239	C₂×C₄	1	2	yes	complemented
37	#13133239	C₂³	1	2	yes	—
38	C₂×Dic₂₁₂	C₂×C₄	1	1	yes	characteristic, complemented
39	C₂×#11719913	—	2	1	no	—
40	C₂×D₄₂₄	—	2	1	no	—
41	C₂×D₄₂₄	—	2	1	no	—
42	C₂×D₄₂₄	—	2	1	no	—
43	C₂×D₄₂₄	C₂×C₄	1	2	yes	—
44	C₂×D₄₂₄	C₂×C₄	1	2	yes	—
45	C₂×D₄₂₄	C₂×C₄	1	1	yes	characteristic, complemented
46	C₂×D₄₂₄	C₂³	1	2	yes	—
47	#13133225	—	4	2	no	—
48	#13133223	—	2	4	no	—
49	C₂×#11719917	—	2	1	no	—
50	M₁₆×D₅₃	—	4	2	no	—
51	C₄×D₂₁₂	—	2	4	no	—
52	#13133179	—	4	2	no	—
53	#13133176	—	2	1	no	—
54	C₂²⋊C₄×D₅₃	—	4	2	no	—
55	#13133162	C₂×C₄	1	8	yes	complemented
56	D₈₄₈	C₂×C₄	1	8	yes	complemented
57	#13133153	—	2	1	no	—
58	#13133153	C₂³	1	2	yes	—
59	#13133151	—	2	1	no	—
60	#13133149	C₂³	1	2	yes	—
61	#13133145	—	2	4	no	—
62	#13133139	—	2	4	no	—
63	C₂×M₁₆×C₅₃	D₄	1	1	yes	characteristic
64	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
65	#117×C₅₃	D₄	1	1	yes	characteristic
66	M₃₂×C₅₃	—	2	1	no	—
67	M₃₂×C₅₃	C₂×C₄	1	2	yes	complemented
68	M₃₂×C₅₃	C₂³	1	2	yes	—
69	M₃₂×C₅₃	C₂³	1	2	yes	—
70	C₂×C₁₆×C₅₃	—	2	1	no	—
71	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	complemented
72	#97×C₅₃	C₂³	1	2	yes	—
73	C₄×C₈×C₅₃	D₄	1	2	yes	—
74	C₂³×D₅₃	—	4	1	no	—
75	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
76	C₂×C₂²⋊D₅₃	—	4	1	no	—
77	#11719945	—	2	2	no	—
78	#11719945	—	2	2	no	—
79	#11719945	—	4	1	no	—
80	#11719945	—	4	1	no	—
81	C₂×D₂₁₂	—	2	2	no	—
82	C₂×D₂₁₂	—	2	2	no	—
83	C₂×D₂₁₂	—	2	1	no	—
84	C₂×D₂₁₂	—	2	2	no	—
85	C₂×D₂₁₂	—	4	1	no	—
86	C₂×D₂₁₂	—	4	1	no	—
87	C₂×D₂₁₂	—	4	1	no	—
88	C₂×C₄×D₅₃	—	4	1	no	—
89	C₂×C₄×D₅₃	—	4	4	no	—
90	C₂×Dic₁₀₆	—	2	1	no	—
91	M₁₆×C₅₃	—	2	1	no	—
92	M₁₆×C₅₃	—	2	1	no	—
93	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
96	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
98	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
99	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
100	C₄⋊C₄×C₅₃	—	2	1	no	—
101	C₂²⋊C₄×C₅₃	—	2	1	no	—
102	C₄²×C₅₃	C₂×D₄	1	2	yes	—
103	C₂²⋊Dic₅₃	—	4	1	no	—
104	#11719921	—	4	4	no	—
105	C₄⋊Dic₅₃	—	2	2	no	—
106	C₄×Dic₅₃	—	4	1	no	—
107	#11719917	—	2	2	no	—
108	#11719917	—	4	1	no	—
109	C₂×#88322	—	4	1	no	—
110	Dic₂₁₂	—	2	1	no	—
111	Dic₂₁₂	C₂²×C₄	1	2	yes	—
112	D₄₂₄	—	2	2	no	—
113	D₄₂₄	—	2	2	no	—
114	D₄₂₄	—	2	1	no	—
115	D₄₂₄	—	2	2	no	—
116	D₄₂₄	C₂²×C₄	1	2	yes	—
117	D₄₂₄	C₂²×C₄	1	4	yes	—
118	#11719913	—	2	2	no	—
119	#11719913	—	2	2	no	—
120	#11719912	—	4	4	no	—
121	C₈×D₅₃	—	4	4	no	—
122	C₁₆×C₅₃	—	2	2	no	—
123	C₁₆×C₅₃	C₂²×C₄	1	4	yes	—
124	C₂³×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
125	C₂²×D₅₃	—	4	2	no	—
126	C₂²×D₅₃	—	4	1	no	—
127	C₂²×D₅₃	—	4	2	no	—
128	C₂²×D₅₃	—	4	2	no	—
129	C₂²×D₅₃	—	16	2	no	—
130	C₂×C₄×C₅₃	—	2	2	no	—
131	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
132	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
133	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
134	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
135	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
136	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
137	C₂²⋊D₅₃	—	4	2	no	—
138	C₂²⋊D₅₃	—	4	2	no	—
139	C₂×Dic₅₃	—	4	1	no	—
140	C₂×Dic₅₃	—	4	2	no	—
141	D₂₁₂	—	2	4	no	—
142	D₂₁₂	—	2	2	no	—
143	D₂₁₂	—	4	2	no	—
144	D₂₁₂	—	4	2	no	—
145	D₂₁₂	—	4	2	no	—
146	D₂₁₂	—	4	1	no	—
147	C₄×D₅₃	—	4	2	no	—
148	C₄×D₅₃	—	4	2	no	—
149	C₄×D₅₃	—	16	4	no	—
150	Dic₁₀₆	—	2	2	no	—
151	Dic₁₀₆	—	4	1	no	—
152	C₈×C₅₃	—	2	2	no	—
153	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
154	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
155	C₈×C₅₃	C₂³×C₄	1	1	yes	characteristic
156	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
157	#88322	—	4	2	no	—
158	#33471	—	53	1	no	maximal, Sylow
159	C₂²×C₅₃	—	4	1	no	—
160	C₂²×C₅₃	#442	1	1	yes	characteristic
161	C₂²×C₅₃	#442	1	1	yes	characteristic
162	C₂²×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
163	C₂×D₅₃	—	4	2	no	—
164	C₂×D₅₃	—	4	4	no	—
165	C₂×D₅₃	—	8	4	no	—
166	C₂×D₅₃	—	16	1	no	—
167	C₂×D₅₃	—	16	2	no	—
168	C₂×D₅₃	—	16	2	no	—
169	C₄×C₅₃	—	4	2	no	—
170	C₄×C₅₃	#442	1	1	yes	characteristic
171	C₄×C₅₃	#442	1	1	yes	characteristic
172	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
173	C₄×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
174	Dic₅₃	—	4	2	no	—
175	Dic₅₃	—	8	2	no	—
176	C₂×#509	—	53	1	no	—
177	#2957	—	53	1	no	—
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	D₅₃	1	1	yes	characteristic, complemented
184	C₂×C₅₃	—	2	1	no	—
185	C₂×C₅₃	—	2	1	no	—
186	C₂×C₅₃	—	2	1	no	—
187	C₂×C₅₃	C₂×#442	1	1	yes	characteristic
188	D₅₃	—	8	4	no	—
189	D₅₃	—	8	2	no	—
190	C₂×#135	—	53	1	no	—
191	C₂²×D₈	—	53	1	no	—
192	#568	—	106	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	C₂×D₅₃	1	1	yes	characteristic
199	C₂×M₃₂	C₂×D₅₃	1	1	yes	characteristic
200	#444	—	53	4	no	—
201	#443	—	106	2	no	—
202	#442	—	53	4	no	—
203	C₄×D₈	—	106	2	no	—
204	C₂×#108	—	53	1	no	—
205	#427	—	53	1	no	—
206	#421	—	106	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	C₂×D₅₃	1	4	yes	complemented
213	C₅₃	#33471	1	1	yes	characteristic, complemented, Sylow
214	C₂×#41	—	106	1	no	—
215	C₂²×D₄	—	106	1	no	—
216	#135	—	53	2	no	—
217	#135	—	53	2	no	—
218	#135	—	106	1	no	—
219	#135	—	106	1	no	—
220	C₂×Q₁₆	—	53	1	no	—
221	C₂×SD₁₆	—	106	1	no	—
222	C₂×D₈	—	53	2	no	—
223	C₂×D₈	—	53	2	no	—
224	C₂×D₈	—	53	1	no	—
225	C₂×D₈	—	53	2	no	—
226	C₂×D₈	—	106	1	no	—
227	C₂×D₈	—	106	1	no	—
228	C₂×D₈	—	106	1	no	—
229	#131	—	106	4	no	—
230	#131	—	212	2	no	—
231	C₂×M₁₆	—	106	1	no	—
232	C₂×M₁₆	—	212	2	no	—
233	C₂×M₁₆	D₂₁₂	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	D₂₁₂	1	1	yes	characteristic
239	C₂×C₂²⋊C₄	—	212	2	no	—
240	SD₃₂	—	53	8	no	—
241	D₁₆	—	53	8	no	—
242	M₃₂	—	2	1	no	—
243	M₃₂	C₄×D₅₃	1	2	yes	complemented
244	M₃₂	C₂²×D₅₃	1	2	yes	—
245	M₃₂	C₂²×D₅₃	1	2	yes	—
246	C₂×C₁₆	—	2	1	no	—
247	C₂×C₁₆	C₄×D₅₃	1	2	yes	complemented
248	#108	—	53	2	no	—
249	#108	—	106	1	no	—
250	#107	—	106	1	no	—
251	#106	—	53	2	no	—
252	#104	—	106	4	no	—
253	#102	—	106	4	no	—
254	#97	C₂²×D₅₃	1	2	yes	—
255	C₄×C₈	D₂₁₂	1	2	yes	—
256	C₂⁴	—	212	1	no	—
257	#41	—	106	2	no	—
258	#41	—	106	2	no	—
259	#41	—	212	1	no	—
260	#41	—	212	1	no	—
261	C₂×Q₈	—	106	1	no	—
262	C₂×D₄	—	106	2	no	—
263	C₂×D₄	—	106	2	no	—
264	C₂×D₄	—	106	1	no	—
265	C₂×D₄	—	106	2	no	—
266	C₂×D₄	—	212	1	no	—
267	C₂×D₄	—	212	1	no	—
268	C₂×D₄	—	212	1	no	—
269	C₂×D₄	—	212	1	no	—
270	C₂²×C₄	—	212	4	no	—
271	C₂²×C₄	—	212	1	no	—
272	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
273	Q₁₆	—	53	2	no	—
274	Q₁₆	—	106	1	no	—
275	SD₁₆	—	106	2	no	—
276	SD₁₆	—	106	2	no	—
277	D₈	—	53	2	no	—
278	D₈	—	53	4	no	—
279	D₈	—	106	2	no	—
280	D₈	—	106	2	no	—
281	D₈	—	106	1	no	—
282	D₈	—	106	2	no	—
283	M₁₆	—	2	1	no	—
284	M₁₆	—	2	1	no	—
285	M₁₆	—	106	2	no	—
286	M₁₆	—	212	4	no	—
287	M₁₆	—	212	1	no	—
288	C₂×C₈	—	212	4	no	—
289	C₂×C₈	—	212	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	C₂×D₂₁₂	1	2	yes	—
293	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
294	C₂×C₈	#11719945	1	1	yes	characteristic
295	C₂×C₈	#11719945	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×D₂₁₂	1	2	yes	—
304	C₁₆	—	2	2	no	—
305	C₁₆	C₂×C₄×D₅₃	1	4	yes	—
306	C₂³	—	848	2	no	—
307	C₂³	—	212	1	no	—
308	C₂³	—	212	2	no	—
309	C₂³	—	212	2	no	—
310	C₂³	—	212	2	no	—
311	C₂³	#13133245	1	1	yes	characteristic
312	Q₈	—	106	2	no	—
313	Q₈	—	212	1	no	—
314	D₄	—	106	4	no	—
315	D₄	—	106	2	no	—
316	D₄	—	212	2	no	—
317	D₄	—	212	2	no	—
318	D₄	—	212	2	no	—
319	D₄	—	212	2	no	—
320	D₄	—	212	2	no	—
321	D₄	—	212	1	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	848	4	no	—
324	C₂×C₄	—	212	2	no	—
325	C₂×C₄	—	212	1	no	—
326	C₂×C₄	—	212	2	no	—
327	C₂×C₄	—	212	2	no	—
328	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
329	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
330	C₂×C₄	#13133245	1	1	yes	characteristic
331	C₂×C₄	#13133245	1	1	yes	characteristic
332	C₂×C₄	#13133245	1	1	yes	characteristic
333	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
334	C₈	—	2	2	no	—
335	C₈	—	212	2	no	—
336	C₈	C₄×D₂₁₂	1	1	yes	characteristic
337	C₈	C₄×D₂₁₂	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	—	848	2	no	—
342	C₂²	—	848	2	no	—
343	C₂²	—	848	1	no	—
344	C₂²	—	424	4	no	—
345	C₂²	—	212	2	no	—
346	C₂²	—	212	4	no	—
347	C₂²	#16810736	1	1	yes	characteristic
348	C₂²	#16810736	1	1	yes	characteristic
349	C₂²	C₂×#13133245	1	1	yes	characteristic
350	C₄	—	4	2	no	—
351	C₄	—	424	2	no	—
352	C₄	—	212	2	no	—
353	C₄	#16810736	1	1	yes	characteristic
354	C₄	#16810736	1	1	yes	characteristic
355	C₄	C₂×C₄×D₂₁₂	1	1	yes	characteristic
356	C₄	C₂×#13133245	1	1	yes	characteristic
357	C₂	—	2	1	no	—
358	C₂	—	2	1	no	—
359	C₂	—	2	1	no	—
360	C₂	—	424	4	no	—
361	C₂	—	424	2	no	—
362	C₂	C₂×#16810736	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#16810736
commutator	C₈×C₅₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2110	C₂
socle	C₂×C₅₃	C₂×#442
prime radical	C₂×#16811061	C₂
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	#2110	D₅₃
hypocenter	C₅₃	#33471
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #405 ◁ #2110

Complements

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

N	H ≅ G/N	type
2. #20283633	C₂	semidirect
3. #20283568	C₂	semidirect
4. #20276615	C₂	semidirect
5. #20276488	C₂	semidirect
6. C₂×#16810437	C₂	semidirect
7. #20276452	C₂	semidirect
8. C₅₃×#2110	C₂	semidirect
13. #16811061	C₄	semidirect
15. C₂×#13133162	C₄	semidirect
16. C₂×D₈₄₈	C₄	semidirect
17. #16810746	C₂²	semidirect
19. #16810736	C₂²	semidirect
25. #16810446	C₂²	semidirect
26. #16810437	C₂²	semidirect
27. #16810434	C₂²	semidirect
31. C₅₃×#347	C₂²	semidirect
36. #13133239	C₂×C₄	semidirect
38. C₂×Dic₂₁₂	C₂×C₄	semidirect
45. C₂×D₄₂₄	C₂×C₄	semidirect
55. #13133162	C₂×C₄	semidirect
56. D₈₄₈	C₂×C₄	semidirect
67. M₃₂×C₅₃	C₂×C₄	semidirect
71. C₂×C₁₆×C₅₃	C₂×C₄	semidirect
183. #2110	D₅₃	semidirect
212. #347	C₂×D₅₃	semidirect
213. C₅₃	#33471	semidirect
243. M₃₂	C₄×D₅₃	semidirect
247. C₂×C₁₆	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×#16811061	2
#20283633	2
#20283568	2
#20276615	2
#20276488	2
C₂×#16810437	2
#20276452	2
C₅₃×#2110	2
#33471	53