Subgroups — #27537565

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×#16811059	C₂	1	1	yes	maximal, characteristic
2	#20283633	C₂	1	1	yes	maximal, characteristic, complemented
3	#20276611	C₂	1	8	yes	maximal, complemented
4	#20276487	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#16810436	C₂	1	1	yes	maximal, characteristic, complemented
6	#20276450	C₂	1	1	yes	maximal, characteristic, complemented
7	#20276320	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#2948	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#13133239	C₂²	1	1	yes	characteristic
10	#16811389	—	2	1	no	—
11	#16811059	C₄	1	4	yes	complemented
12	#16811059	C₂²	1	4	yes	—
13	C₂×#13133161	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	#16810746	C₂²	1	4	yes	complemented
18	#16810741	—	2	2	no	—
19	C₂×#13133153	C₂²	1	1	yes	characteristic
20	#16810641	—	2	1	no	—
21	#16810445	C₂²	1	4	yes	complemented
22	#16810436	C₂²	1	4	yes	complemented
23	#16810432	C₂²	1	4	yes	complemented
24	#16810401	C₂²	1	4	yes	complemented
25	C₂²×D₈×C₅₃	C₂²	1	1	yes	characteristic
26	C₅₃×#518	—	2	1	no	—
27	C₅₃×#442	C₂²	1	4	yes	complemented
28	C₄×D₈×C₅₃	—	2	2	no	—
29	C₅₃×#427	C₂²	1	1	yes	characteristic
30	C₅₃×#418	—	2	1	no	—
31	C₅₃×#405	C₂²	1	1	yes	characteristic
32	C₂×#11719945	—	2	1	no	—
33	#13133239	—	2	1	no	—
34	#13133239	—	2	1	no	—
35	#13133239	C₂×C₄	1	2	yes	complemented
36	#13133239	C₂³	1	2	yes	—
37	C₂×Dic₂₁₂	C₂×C₄	1	1	yes	characteristic, complemented
38	C₂×#11719913	—	2	1	no	—
39	C₂×D₄₂₄	C₂×C₄	1	1	yes	characteristic, complemented
40	#13133225	—	4	2	no	—
41	#13133223	—	2	4	no	—
42	C₂×#11719917	—	2	1	no	—
43	M₁₆×D₅₃	—	4	2	no	—
44	#13133161	C₂×C₄	1	8	yes	complemented
45	#13133158	C₂×C₄	1	8	yes	complemented
46	#13133157	—	2	1	no	—
47	#13133157	C₂×C₄	1	2	yes	complemented
48	#13133157	C₂³	1	2	yes	—
49	#13133157	C₂³	1	2	yes	—
50	C₂×#11719908	—	2	1	no	—
51	C₂×#11719908	C₂×C₄	1	2	yes	complemented
52	#13133153	—	2	1	no	—
53	#13133153	C₂³	1	2	yes	—
54	#13133145	—	2	4	no	—
55	C₂²×D₄×C₅₃	—	2	1	no	—
56	C₂×D₈×C₅₃	—	2	1	no	—
57	C₂×D₈×C₅₃	—	2	1	no	—
58	C₂×D₈×C₅₃	—	2	1	no	—
59	C₂×D₈×C₅₃	C₂×C₄	1	2	yes	—
60	C₂×D₈×C₅₃	C₂×C₄	1	2	yes	—
61	C₂×D₈×C₅₃	C₂³	1	2	yes	—
62	C₂×M₁₆×C₅₃	D₄	1	1	yes	characteristic
63	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
64	C₄×D₄×C₅₃	—	2	4	no	—
65	C₄×D₄×C₅₃	—	4	2	no	—
66	#117×C₅₃	—	2	1	no	—
67	#117×C₅₃	D₄	1	1	yes	characteristic
68	C₂×C₂²⋊C₄×C₅₃	—	4	2	no	—
69	#107×C₅₃	—	2	1	no	—
70	#106×C₅₃	C₂³	1	2	yes	—
71	#102×C₅₃	—	2	4	no	—
72	#97×C₅₃	C₂³	1	2	yes	—
73	C₄×C₈×C₅₃	D₄	1	2	yes	—
74	C₂⁴×C₅₃	—	4	1	no	—
75	C₂×D₄×C₅₃	—	2	2	no	—
76	C₂×D₄×C₅₃	—	2	2	no	—
77	C₂×D₄×C₅₃	—	2	2	no	—
78	C₂×D₄×C₅₃	—	4	1	no	—
79	C₂×D₄×C₅₃	—	4	1	no	—
80	C₂×D₄×C₅₃	—	4	1	no	—
81	C₂²×C₄×C₅₃	—	4	4	no	—
82	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
83	C₂×C₂²⋊D₅₃	—	4	1	no	—
84	#11719945	—	2	2	no	—
85	#11719945	—	2	2	no	—
86	#11719945	—	4	1	no	—
87	#11719945	—	4	1	no	—
88	C₂×D₂₁₂	—	2	1	no	—
89	C₂×C₄×D₅₃	—	4	1	no	—
90	C₂×Dic₁₀₆	—	2	1	no	—
91	D₈×C₅₃	—	2	2	no	—
92	D₈×C₅₃	—	2	2	no	—
93	D₈×C₅₃	—	2	2	no	—
94	D₈×C₅₃	C₂²×C₄	1	4	yes	—
95	M₁₆×C₅₃	—	2	1	no	—
96	M₁₆×C₅₃	—	2	1	no	—
97	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
100	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
102	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
103	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
104	C₄⋊C₄×C₅₃	—	2	1	no	—
105	C₄⋊C₄×C₅₃	—	2	2	no	—
106	C₂²⋊C₄×C₅₃	—	2	1	no	—
107	C₂²⋊C₄×C₅₃	—	4	4	no	—
108	C₂²⋊C₄×C₅₃	—	4	1	no	—
109	C₄²×C₅₃	—	4	1	no	—
110	C₄²×C₅₃	C₂×D₄	1	2	yes	—
111	#11719917	—	2	2	no	—
112	#11719917	—	4	1	no	—
113	C₂×#88322	—	4	1	no	—
114	Dic₂₁₂	—	2	1	no	—
115	Dic₂₁₂	C₂²×C₄	1	2	yes	—
116	D₄₂₄	—	2	1	no	—
117	D₄₂₄	C₂²×C₄	1	2	yes	—
118	#11719913	—	2	2	no	—
119	#11719913	—	2	2	no	—
120	#11719912	—	4	4	no	—
121	C₈×D₅₃	—	4	4	no	—
122	#11719908	—	2	2	no	—
123	#11719908	C₂²×C₄	1	4	yes	—
124	C₂³×C₅₃	—	4	2	no	—
125	C₂³×C₅₃	—	4	2	no	—
126	C₂³×C₅₃	—	4	2	no	—
127	C₂³×C₅₃	—	16	2	no	—
128	C₂³×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
129	C₂²×D₅₃	—	4	1	no	—
130	D₄×C₅₃	—	2	4	no	—
131	D₄×C₅₃	—	4	2	no	—
132	D₄×C₅₃	—	4	2	no	—
133	D₄×C₅₃	—	4	2	no	—
134	C₂×C₄×C₅₃	—	2	2	no	—
135	C₂×C₄×C₅₃	—	4	2	no	—
136	C₂×C₄×C₅₃	—	16	4	no	—
137	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
138	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
139	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
140	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
141	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
142	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
143	C₂²⋊D₅₃	—	4	2	no	—
144	C₂²⋊D₅₃	—	4	2	no	—
145	C₂×Dic₅₃	—	4	1	no	—
146	D₂₁₂	—	2	2	no	—
147	D₂₁₂	—	4	1	no	—
148	C₄×D₅₃	—	4	2	no	—
149	C₄×D₅₃	—	4	2	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	—	4	4	no	—
161	C₂²×C₅₃	—	8	4	no	—
162	C₂²×C₅₃	—	16	2	no	—
163	C₂²×C₅₃	—	16	2	no	—
164	C₂²×C₅₃	#442	1	1	yes	characteristic
165	C₂²×C₅₃	#442	1	1	yes	characteristic
166	C₂²×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
167	C₂×D₅₃	—	4	2	no	—
168	C₂×D₅₃	—	16	1	no	—
169	C₄×C₅₃	—	4	2	no	—
170	C₄×C₅₃	—	8	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	—	4	2	no	—
176	C₂×#509	—	53	1	no	—
177	#2957	—	53	1	no	—
178	#2948	D₅₃	1	1	yes	characteristic, complemented
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	4	no	—
188	C₂×C₅₃	C₂×#442	1	1	yes	characteristic
189	D₅₃	—	8	2	no	—
190	C₂×#135	—	53	1	no	—
191	C₂²×D₈	C₂×D₅₃	1	1	yes	characteristic
192	#568	—	106	1	no	—
193	#518	—	2	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	—	53	4	no	—
201	#443	—	106	2	no	—
202	#442	C₂×D₅₃	1	4	yes	complemented
203	C₄×D₈	—	2	2	no	—
204	C₂×#108	—	53	1	no	—
205	#427	C₂×D₅₃	1	1	yes	characteristic
206	#421	—	106	1	no	—
207	#418	—	2	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	—	106	1	no	—
215	C₂²×D₄	—	2	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	—	2	1	no	—
223	C₂×D₈	—	2	1	no	—
224	C₂×D₈	—	2	1	no	—
225	C₂×D₈	—	53	1	no	—
226	C₂×D₈	C₄×D₅₃	1	2	yes	—
227	C₂×D₈	C₄×D₅₃	1	2	yes	—
228	C₂×D₈	C₂²×D₅₃	1	2	yes	—
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	C₂²⋊D₅₃	1	1	yes	characteristic
234	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
235	C₄×D₄	—	2	4	no	—
236	C₄×D₄	—	4	2	no	—
237	#117	—	2	1	no	—
238	#117	C₂²⋊D₅₃	1	1	yes	characteristic
239	C₂×C₂²⋊C₄	—	4	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	—	53	2	no	—
249	#108	—	106	1	no	—
250	#107	—	2	1	no	—
251	#106	C₂²×D₅₃	1	2	yes	—
252	#104	—	106	4	no	—
253	#102	—	2	4	no	—
254	#97	C₂²×D₅₃	1	2	yes	—
255	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
256	C₂⁴	—	4	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	—	2	2	no	—
263	C₂×D₄	—	2	2	no	—
264	C₂×D₄	—	2	2	no	—
265	C₂×D₄	—	4	1	no	—
266	C₂×D₄	—	4	1	no	—
267	C₂×D₄	—	4	1	no	—
268	C₂×D₄	—	106	1	no	—
269	C₂×D₄	—	212	1	no	—
270	C₂²×C₄	—	4	4	no	—
271	C₂²×C₄	—	212	1	no	—
272	C₂²×C₄	C₂×C₂²⋊D₅₃	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	—	2	2	no	—
278	D₈	—	2	2	no	—
279	D₈	—	2	2	no	—
280	D₈	—	53	2	no	—
281	D₈	—	106	1	no	—
282	D₈	C₂×C₄×D₅₃	1	4	yes	—
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	#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	—	2	2	no	—
299	C₂²⋊C₄	—	2	1	no	—
300	C₂²⋊C₄	—	4	4	no	—
301	C₂²⋊C₄	—	4	1	no	—
302	C₄²	—	4	1	no	—
303	C₄²	C₂×C₂²⋊D₅₃	1	2	yes	—
304	C₁₆	—	53	4	no	—
305	C₁₆	—	106	2	no	—
306	C₂³	—	4	2	no	—
307	C₂³	—	4	2	no	—
308	C₂³	—	4	2	no	—
309	C₂³	—	16	2	no	—
310	C₂³	—	212	1	no	—
311	C₂³	#13133243	1	1	yes	characteristic
312	Q₈	—	106	2	no	—
313	Q₈	—	212	1	no	—
314	D₄	—	2	4	no	—
315	D₄	—	4	2	no	—
316	D₄	—	4	2	no	—
317	D₄	—	4	2	no	—
318	D₄	—	106	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	—	4	2	no	—
324	C₂×C₄	—	16	4	no	—
325	C₂×C₄	—	212	1	no	—
326	C₂×C₄	—	212	2	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	#13133242	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	—	212	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	4	no	—
342	C₂²	—	8	4	no	—
343	C₂²	—	16	2	no	—
344	C₂²	—	16	2	no	—
345	C₂²	—	848	1	no	—
346	C₂²	—	212	2	no	—
347	C₂²	#16810733	1	1	yes	characteristic
348	C₂²	#16810733	1	1	yes	characteristic
349	C₂²	C₂×#13133243	1	1	yes	characteristic
350	C₄	—	4	2	no	—
351	C₄	—	8	2	no	—
352	C₄	—	212	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×#13133242	1	1	yes	characteristic
357	C₂	—	2	1	no	—
358	C₂	—	2	1	no	—
359	C₂	—	2	1	no	—
360	C₂	—	8	4	no	—
361	C₂	—	424	2	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×#2948	C₂
socle	C₂×C₅₃	C₂×#442
prime radical	C₂×#16811059	C₂
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	#2948	D₅₃
hypocenter	C₅₃	#33471
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #405 ◁ #2948

Complements

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

N	H ≅ G/N	type
2. #20283633	C₂	semidirect
3. #20276611	C₂	semidirect
4. #20276487	C₂	semidirect
5. C₂×#16810436	C₂	semidirect
6. #20276450	C₂	semidirect
7. #20276320	C₂	semidirect
8. C₅₃×#2948	C₂	semidirect
11. #16811059	C₄	semidirect
13. C₂×#13133161	C₄	semidirect
14. C₂×#13133158	C₄	semidirect
17. #16810746	C₂²	semidirect
21. #16810445	C₂²	semidirect
22. #16810436	C₂²	semidirect
23. #16810432	C₂²	semidirect
24. #16810401	C₂²	semidirect
27. C₅₃×#442	C₂²	semidirect
35. #13133239	C₂×C₄	semidirect
37. C₂×Dic₂₁₂	C₂×C₄	semidirect
39. C₂×D₄₂₄	C₂×C₄	semidirect
44. #13133161	C₂×C₄	semidirect
45. #13133158	C₂×C₄	semidirect
47. #13133157	C₂×C₄	semidirect
51. C₂×#11719908	C₂×C₄	semidirect
178. #2948	D₅₃	semidirect
202. #442	C₂×D₅₃	semidirect
213. C₅₃	#33471	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16811059	2
#20283633	2
#20276611	2
#20276487	2
C₂×#16810436	2
#20276450	2
#20276320	2
C₅₃×#2948	2
#33471	53