Subgroups — #27537568

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	#20283632	C₂	1	1	yes	maximal, characteristic, complemented
2	#20283567	C₂	1	1	yes	maximal, characteristic, complemented
3	#20276614	C₂	1	8	yes	maximal, complemented
4	#20276488	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#16810436	C₂	1	1	yes	maximal, characteristic, complemented
6	#20276451	C₂	1	1	yes	maximal, characteristic, complemented
7	#20276319	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅₃×#509	C₂	1	1	yes	maximal, characteristic
9	#16811390	—	2	1	no	—
10	#16811098	—	2	1	no	—
11	C₂×#13133157	C₂²	1	1	yes	characteristic
12	#16810745	C₂²	1	4	yes	complemented
13	#16810740	—	2	2	no	—
14	#16810735	C₂²	1	4	yes	complemented
15	D₈×Dic₅₃	—	2	2	no	—
16	C₂×#13133153	C₂²	1	1	yes	characteristic
17	#16810661	C₂²	1	1	yes	characteristic
18	#16810640	—	2	1	no	—
19	#16810628	—	2	1	no	—
20	#16810578	C₂²	1	1	yes	characteristic
21	#16810446	C₂²	1	4	yes	complemented
22	#16810436	C₂²	1	4	yes	complemented
23	#16810433	C₂²	1	4	yes	complemented
24	#16810400	C₂²	1	4	yes	complemented
25	C₂×#135×C₅₃	C₂²	1	1	yes	characteristic
26	C₂²×D₈×C₅₃	C₂²	1	1	yes	characteristic
27	C₅₃×#509	C₄	1	4	yes	complemented
28	C₅₃×#509	C₂²	1	4	yes	—
29	C₂×SD₃₂×C₅₃	C₄	1	2	yes	complemented
30	C₂×D₁₆×C₅₃	C₄	1	2	yes	complemented
31	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
32	#13133224	—	2	4	no	—
33	#13133224	—	4	2	no	—
34	C₂×#11719917	—	2	1	no	—
35	C₂×#11719917	—	4	2	no	—
36	C₂×#11719917	D₄	1	1	yes	characteristic
37	D₄×Dic₅₃	—	2	4	no	—
38	D₄×Dic₅₃	—	4	2	no	—
39	#13133176	—	2	1	no	—
40	#13133176	D₄	1	1	yes	characteristic
41	C₂×C₂²⋊Dic₅₃	—	4	2	no	—
42	#13133157	—	2	1	no	—
43	#13133157	C₂³	1	2	yes	—
44	C₂×#11719908	—	2	1	no	—
45	#13133153	—	2	1	no	—
46	#13133153	C₂³	1	2	yes	—
47	#13133151	—	2	1	no	—
48	#13133149	C₂³	1	2	yes	—
49	#13133144	—	2	4	no	—
50	#13133138	—	2	4	no	—
51	#13133127	C₂³	1	2	yes	—
52	C₈×Dic₅₃	D₄	1	2	yes	—
53	C₂×#41×C₅₃	—	2	1	no	—
54	C₂²×D₄×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	—	2	1	no	—
62	C₂×D₈×C₅₃	—	2	1	no	—
63	C₂×D₈×C₅₃	—	2	1	no	—
64	C₂×D₈×C₅₃	C₂×C₄	1	2	yes	—
65	C₂×D₈×C₅₃	C₂×C₄	1	2	yes	—
66	C₂×D₈×C₅₃	C₂×C₄	1	1	yes	characteristic, complemented
67	C₂×D₈×C₅₃	C₂³	1	2	yes	—
68	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
69	SD₃₂×C₅₃	C₂×C₄	1	8	yes	complemented
70	D₁₆×C₅₃	C₂×C₄	1	8	yes	complemented
71	M₃₂×C₅₃	C₂×C₄	1	2	yes	complemented
72	M₃₂×C₅₃	C₂³	1	2	yes	—
73	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	complemented
74	C₂⁴×C₅₃	—	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	2	no	—
81	C₂×D₄×C₅₃	—	2	2	no	—
82	C₂×D₄×C₅₃	—	2	1	no	—
83	C₂×D₄×C₅₃	—	2	2	no	—
84	C₂×D₄×C₅₃	—	4	1	no	—
85	C₂×D₄×C₅₃	—	4	1	no	—
86	C₂×D₄×C₅₃	—	4	1	no	—
87	C₂×D₄×C₅₃	—	4	1	no	—
88	C₂²×C₄×C₅₃	—	4	1	no	—
89	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
90	C₂²×Dic₅₃	—	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	2	no	—
96	D₈×C₅₃	—	2	2	no	—
97	D₈×C₅₃	—	2	1	no	—
98	D₈×C₅₃	—	2	2	no	—
99	D₈×C₅₃	C₂²×C₄	1	2	yes	—
100	D₈×C₅₃	C₂²×C₄	1	4	yes	—
101	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
104	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
105	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
106	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
107	C₂²⋊Dic₅₃	—	2	1	no	—
108	C₂²⋊Dic₅₃	—	4	4	no	—
109	C₂²⋊Dic₅₃	—	4	1	no	—
110	C₄⋊Dic₅₃	—	2	1	no	—
111	C₄⋊Dic₅₃	—	2	2	no	—
112	C₄×Dic₅₃	—	4	1	no	—
113	C₄×Dic₅₃	C₂×D₄	1	2	yes	—
114	#11719917	—	2	1	no	—
115	#11719917	—	2	1	no	—
116	#11719917	—	2	2	no	—
117	#11719917	—	4	4	no	—
118	#11719917	—	4	1	no	—
119	C₂×#88322	—	4	4	no	—
120	C₂×#88322	—	4	1	no	—
121	C₂×#88322	C₂×D₄	1	2	yes	—
122	C₁₆×C₅₃	C₂²×C₄	1	4	yes	—
123	#11719908	—	2	2	no	—
124	C₂³×C₅₃	—	4	2	no	—
125	C₂³×C₅₃	—	4	1	no	—
126	C₂³×C₅₃	—	4	2	no	—
127	C₂³×C₅₃	—	4	2	no	—
128	C₂³×C₅₃	—	16	2	no	—
129	C₂³×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
130	Q₈×C₅₃	—	2	2	no	—
131	Q₈×C₅₃	—	4	1	no	—
132	D₄×C₅₃	—	2	4	no	—
133	D₄×C₅₃	—	2	2	no	—
134	D₄×C₅₃	—	4	2	no	—
135	D₄×C₅₃	—	4	2	no	—
136	D₄×C₅₃	—	4	2	no	—
137	D₄×C₅₃	—	4	2	no	—
138	D₄×C₅₃	—	4	2	no	—
139	D₄×C₅₃	—	4	1	no	—
140	C₂×C₄×C₅₃	—	4	1	no	—
141	C₂×C₄×C₅₃	—	4	2	no	—
142	C₂×C₄×C₅₃	—	4	2	no	—
143	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
144	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
145	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
146	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
147	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
148	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
149	C₂×Dic₅₃	—	2	2	no	—
150	C₂×Dic₅₃	—	4	2	no	—
151	C₂×Dic₅₃	—	16	4	no	—
152	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
153	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
154	C₈×C₅₃	C₂³×C₄	1	1	yes	characteristic
155	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
156	#88322	—	2	2	no	—
157	#88322	—	4	2	no	—
158	#33471	—	53	1	no	maximal, Sylow
159	C₂²×C₅₃	—	4	1	no	—
160	C₂²×C₅₃	—	4	2	no	—
161	C₂²×C₅₃	—	4	4	no	—
162	C₂²×C₅₃	—	8	4	no	—
163	C₂²×C₅₃	—	16	1	no	—
164	C₂²×C₅₃	—	16	2	no	—
165	C₂²×C₅₃	—	16	2	no	—
166	C₂²×C₅₃	#442	1	1	yes	characteristic
167	C₂²×C₅₃	#442	1	1	yes	characteristic
168	C₂²×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
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	D₅₃	1	1	yes	characteristic
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	—	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	—	8	2	no	—
189	C₂×C₅₃	C₂×#442	1	1	yes	characteristic
190	C₂×#135	C₂×D₅₃	1	1	yes	characteristic
191	C₂²×D₈	C₂×D₅₃	1	1	yes	characteristic
192	#568	—	106	1	no	—
193	#518	—	106	1	no	—
194	#509	Dic₅₃	1	4	yes	complemented
195	#509	C₂×D₅₃	1	4	yes	—
196	C₂×SD₃₂	Dic₅₃	1	2	yes	complemented
197	C₂×D₁₆	Dic₅₃	1	2	yes	complemented
198	C₂×M₃₂	—	53	1	no	—
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	—	53	1	no	—
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	—	2	1	no	—
216	#135	—	2	1	no	—
217	#135	—	2	1	no	—
218	#135	C₂×Dic₅₃	1	2	yes	complemented
219	#135	C₂²×D₅₃	1	2	yes	—
220	C₂×Q₁₆	C₂×Dic₅₃	1	1	yes	characteristic, complemented
221	C₂×SD₁₆	—	2	1	no	—
222	C₂×D₈	—	2	1	no	—
223	C₂×D₈	—	2	1	no	—
224	C₂×D₈	—	2	1	no	—
225	C₂×D₈	C₂×Dic₅₃	1	2	yes	—
226	C₂×D₈	C₂×Dic₅₃	1	2	yes	—
227	C₂×D₈	C₂×Dic₅₃	1	1	yes	characteristic, complemented
228	C₂×D₈	C₂²×D₅₃	1	2	yes	—
229	#131	—	106	4	no	—
230	#131	—	212	2	no	—
231	C₂×M₁₆	—	53	1	no	—
232	C₂×M₁₆	—	106	1	no	—
233	C₂×M₁₆	—	212	2	no	—
234	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
235	C₄×D₄	—	106	4	no	—
236	C₄×D₄	—	212	2	no	—
237	#117	—	53	1	no	—
238	#117	—	106	1	no	—
239	C₂×C₂²⋊C₄	—	212	2	no	—
240	SD₃₂	C₂×Dic₅₃	1	8	yes	complemented
241	D₁₆	C₂×Dic₅₃	1	8	yes	complemented
242	M₃₂	—	53	2	no	—
243	M₃₂	—	106	1	no	—
244	M₃₂	C₂×Dic₅₃	1	2	yes	complemented
245	M₃₂	C₂²×D₅₃	1	2	yes	—
246	C₂×C₁₆	—	106	1	no	—
247	C₂×C₁₆	C₂×Dic₅₃	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	—	53	2	no	—
255	C₄×C₈	—	53	2	no	—
256	C₂⁴	—	4	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	2	no	—
263	C₂×D₄	—	2	2	no	—
264	C₂×D₄	—	2	1	no	—
265	C₂×D₄	—	2	2	no	—
266	C₂×D₄	—	4	1	no	—
267	C₂×D₄	—	4	1	no	—
268	C₂×D₄	—	4	1	no	—
269	C₂×D₄	—	4	1	no	—
270	C₂²×C₄	—	4	1	no	—
271	C₂²×C₄	—	212	4	no	—
272	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
273	Q₁₆	—	2	1	no	—
274	Q₁₆	C₂²×Dic₅₃	1	2	yes	—
275	SD₁₆	—	2	2	no	—
276	SD₁₆	—	2	2	no	—
277	D₈	—	2	2	no	—
278	D₈	—	2	2	no	—
279	D₈	—	2	1	no	—
280	D₈	—	2	2	no	—
281	D₈	C₂²×Dic₅₃	1	2	yes	—
282	D₈	C₂²×Dic₅₃	1	4	yes	—
283	M₁₆	—	106	1	no	—
284	M₁₆	—	106	1	no	—
285	M₁₆	—	106	2	no	—
286	M₁₆	—	212	4	no	—
287	M₁₆	—	212	1	no	—
288	C₂×C₈	—	53	2	no	—
289	C₂×C₈	—	212	4	no	—
290	C₂×C₈	—	212	1	no	—
291	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
292	C₂×C₈	#11719947	1	1	yes	characteristic
293	C₂×C₈	#11719947	1	1	yes	characteristic
294	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
295	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
296	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
297	C₄⋊C₄	—	106	1	no	—
298	C₄⋊C₄	—	106	2	no	—
299	C₂²⋊C₄	—	106	1	no	—
300	C₂²⋊C₄	—	212	4	no	—
301	C₂²⋊C₄	—	212	1	no	—
302	C₄²	—	53	2	no	—
303	C₄²	—	212	1	no	—
304	C₁₆	—	106	2	no	—
305	C₁₆	C₂²×Dic₅₃	1	4	yes	—
306	C₂³	—	4	2	no	—
307	C₂³	—	4	1	no	—
308	C₂³	—	4	2	no	—
309	C₂³	—	4	2	no	—
310	C₂³	—	16	2	no	—
311	C₂³	#13133241	1	1	yes	characteristic
312	Q₈	—	2	2	no	—
313	Q₈	—	4	1	no	—
314	D₄	—	2	4	no	—
315	D₄	—	2	2	no	—
316	D₄	—	4	2	no	—
317	D₄	—	4	2	no	—
318	D₄	—	4	2	no	—
319	D₄	—	4	2	no	—
320	D₄	—	4	2	no	—
321	D₄	—	4	1	no	—
322	C₂×C₄	—	4	1	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	848	4	no	—
326	C₂×C₄	—	106	2	no	—
327	C₂×C₄	—	212	2	no	—
328	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
329	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
330	C₂×C₄	#13133241	1	1	yes	characteristic
331	C₂×C₄	#13133241	1	1	yes	characteristic
332	C₂×C₄	#13133241	1	1	yes	characteristic
333	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
334	C₈	—	106	2	no	—
335	C₈	—	212	2	no	—
336	C₈	D₄×Dic₅₃	1	1	yes	characteristic
337	C₈	D₄×Dic₅₃	1	1	yes	characteristic
338	C₈	C₂³×Dic₅₃	1	1	yes	characteristic
339	C₈	C₂×#11719947	1	1	yes	characteristic
340	C₂²	—	4	1	no	—
341	C₂²	—	4	2	no	—
342	C₂²	—	4	4	no	—
343	C₂²	—	8	4	no	—
344	C₂²	—	16	2	no	—
345	C₂²	—	16	2	no	—
346	C₂²	—	16	1	no	—
347	C₂²	#16810735	1	1	yes	characteristic
348	C₂²	#16810735	1	1	yes	characteristic
349	C₂²	C₂×#13133241	1	1	yes	characteristic
350	C₄	—	4	2	no	—
351	C₄	—	424	2	no	—
352	C₄	—	212	2	no	—
353	C₄	#16810735	1	1	yes	characteristic
354	C₄	#16810735	1	1	yes	characteristic
355	C₄	C₂×D₄×Dic₅₃	1	1	yes	characteristic
356	C₄	C₂×#13133241	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	—	8	2	no	—
362	C₂	C₂×#16810735	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#16810735
commutator	C₈×C₅₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅₃
Fitting, prime radical	C₂×C₅₃×#509	C₂
socle	C₂×C₅₃	C₂×#442
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₂×#509	D₅₃
hypocenter	C₅₃	#33471
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#509

Complements

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

N	H ≅ G/N	type
1. #20283632	C₂	semidirect
2. #20283567	C₂	semidirect
3. #20276614	C₂	semidirect
4. #20276488	C₂	semidirect
5. C₂×#16810436	C₂	semidirect
6. #20276451	C₂	semidirect
7. #20276319	C₂	semidirect
12. #16810745	C₂²	semidirect
14. #16810735	C₂²	semidirect
21. #16810446	C₂²	semidirect
22. #16810436	C₂²	semidirect
23. #16810433	C₂²	semidirect
24. #16810400	C₂²	semidirect
27. C₅₃×#509	C₄	semidirect
29. C₂×SD₃₂×C₅₃	C₄	semidirect
30. C₂×D₁₆×C₅₃	C₄	semidirect
57. #135×C₅₃	C₂×C₄	semidirect
59. C₂×Q₁₆×C₅₃	C₂×C₄	semidirect
66. C₂×D₈×C₅₃	C₂×C₄	semidirect
69. SD₃₂×C₅₃	C₂×C₄	semidirect
70. D₁₆×C₅₃	C₂×C₄	semidirect
71. M₃₂×C₅₃	C₂×C₄	semidirect
73. C₂×C₁₆×C₅₃	C₂×C₄	semidirect
194. #509	Dic₅₃	semidirect
196. C₂×SD₃₂	Dic₅₃	semidirect
197. C₂×D₁₆	Dic₅₃	semidirect
213. C₅₃	#33471	semidirect
218. #135	C₂×Dic₅₃	semidirect
220. C₂×Q₁₆	C₂×Dic₅₃	semidirect
227. C₂×D₈	C₂×Dic₅₃	semidirect
240. SD₃₂	C₂×Dic₅₃	semidirect
241. D₁₆	C₂×Dic₅₃	semidirect
244. M₃₂	C₂×Dic₅₃	semidirect
247. C₂×C₁₆	C₂×Dic₅₃	semidirect

Maximal subgroups

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

subgroup	index
#20283632	2
#20283567	2
#20276614	2
#20276488	2
C₂×#16810436	2
#20276451	2
#20276319	2
C₂×C₅₃×#509	2
#33471	53