Subgroups — #27484047

Order 13568.

612 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	#20285229	C₂	1	1	yes	maximal, characteristic, complemented
2	#20275887	C₂	1	4	yes	maximal, complemented
3	#20275614	C₂	1	2	yes	maximal
4	#20274840	C₂	1	2	yes	maximal, complemented
5	#20274769	C₂	1	2	yes	maximal, complemented
6	#20274761	C₂	1	2	yes	maximal, complemented
7	#20274594	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅₃×#443	C₂	1	1	yes	maximal, characteristic
9	#16811180	C₂²	1	2	yes	—
10	#16811028	—	2	1	no	—
11	#16811025	—	2	1	no	—
12	#16811025	—	2	1	no	—
13	#16811021	—	2	1	no	—
14	C₂×#13133153	C₂²	1	1	yes	characteristic
15	#16810661	C₂²	1	2	yes	—
16	#16810661	C₂²	1	2	yes	—
17	#16810661	C₂²	1	1	yes	characteristic
18	#16810640	C₂²	1	4	yes	—
19	#16810597	C₂²	1	4	yes	—
20	#16810578	C₂²	1	2	yes	—
21	#16810395	C₂²	1	4	yes	—
22	#16810383	C₂²	1	2	yes	—
23	#16810380	C₂²	1	4	yes	—
24	C₂×#135×C₅₃	C₂²	1	1	yes	characteristic
25	C₂×#131×C₅₃	C₂²	1	2	yes	—
26	C₅₃×#443	—	2	1	no	—
27	C₅₃×#443	—	2	1	no	—
28	C₅₃×#443	—	2	1	no	—
29	C₅₃×#443	—	2	1	no	—
30	C₂×#108×C₅₃	C₂²	1	1	yes	characteristic
31	C₂×#104×C₅₃	C₂²	1	2	yes	—
32	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
33	C₂×#11719917	C₂³	1	2	yes	—
34	#13133217	—	2	2	no	—
35	#13133209	—	2	2	no	—
36	#13133196	—	4	2	no	—
37	C₄×Dic₁₀₆	—	4	2	no	—
38	#13133176	C₂³	1	2	yes	—
39	#13133176	C₂³	1	4	yes	—
40	#13133153	—	2	1	no	—
41	#13133153	—	2	1	no	—
42	#13133151	—	2	2	no	—
43	#13133151	—	2	2	no	—
44	#13133151	—	2	1	no	—
45	#13133149	—	2	2	no	—
46	#13133149	—	2	2	no	—
47	#13133149	—	2	1	no	—
48	#13133144	—	4	4	no	—
49	#13133132	—	4	4	no	—
50	#13133127	—	2	2	no	—
51	C₈×Dic₅₃	—	2	2	no	—
52	C₂×#41×C₅₃	C₂³	1	2	yes	—
53	#135×C₅₃	—	2	1	no	—
54	#135×C₅₃	—	2	1	no	—
55	#135×C₅₃	—	2	1	no	—
56	#135×C₅₃	—	2	1	no	—
57	C₂×Q₁₆×C₅₃	Q₈	1	1	yes	characteristic, complemented
58	C₂×SD₁₆×C₅₃	Q₈	1	2	yes	complemented
59	C₂×D₈×C₅₃	Q₈	1	1	yes	characteristic, complemented
60	#131×C₅₃	—	2	2	no	—
61	#131×C₅₃	—	2	2	no	—
62	#131×C₅₃	—	2	2	no	—
63	#131×C₅₃	—	2	2	no	—
64	C₂×M₁₆×C₅₃	—	2	2	no	—
65	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
66	C₂²×C₈×C₅₃	—	2	2	no	—
67	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
68	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
69	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
70	#108×C₅₃	—	2	1	no	—
71	#108×C₅₃	—	2	1	no	—
72	#104×C₅₃	—	2	2	no	—
73	#104×C₅₃	—	2	2	no	—
74	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
75	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
76	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
77	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
78	#41×C₅₃	—	2	2	no	—
79	#41×C₅₃	—	2	2	no	—
80	#41×C₅₃	—	2	2	no	—
81	#41×C₅₃	—	2	2	no	—
82	C₂×Q₈×C₅₃	C₂×Q₈	1	2	yes	—
83	C₂×D₄×C₅₃	—	2	2	no	—
84	C₂×D₄×C₅₃	C₂×Q₈	1	2	yes	—
85	C₂²×C₄×C₅₃	—	2	1	no	—
86	C₂²×C₄×C₅₃	—	2	2	no	—
87	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
88	C₂×Dic₁₀₆	—	4	2	no	—
89	Q₁₆×C₅₃	—	2	1	no	—
90	Q₁₆×C₅₃	—	2	1	no	—
91	SD₁₆×C₅₃	—	2	2	no	—
92	SD₁₆×C₅₃	—	2	2	no	—
93	D₈×C₅₃	—	2	1	no	—
94	D₈×C₅₃	—	2	1	no	—
95	M₁₆×C₅₃	—	2	2	no	—
96	M₁₆×C₅₃	—	2	2	no	—
97	M₁₆×C₅₃	—	4	2	no	—
98	M₁₆×C₅₃	—	4	2	no	—
99	C₂×C₈×C₅₃	—	2	2	no	—
100	C₂×C₈×C₅₃	—	4	2	no	—
101	C₂×C₈×C₅₃	—	4	2	no	—
102	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
103	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
104	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
107	C₂×C₈×C₅₃	C₂×Q₈	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	#41	1	1	yes	characteristic
111	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
112	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
113	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
114	C₄²×C₅₃	—	2	1	no	—
115	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
117	C₂²⋊Dic₅₃	—	2	2	no	—
118	C₂²⋊Dic₅₃	—	2	4	no	—
119	C₄⋊Dic₅₃	—	2	4	no	—
120	C₄⋊Dic₅₃	—	2	2	no	—
121	C₄⋊Dic₅₃	—	4	2	no	—
122	#11719919	—	4	4	no	—
123	C₄×Dic₅₃	—	2	2	no	—
124	C₄×Dic₅₃	—	2	4	no	—
125	#11719917	—	2	2	no	—
126	#11719917	—	2	2	no	—
127	C₂×#88322	—	2	2	no	—
128	C₂³×C₅₃	—	2	2	no	—
129	C₂³×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
130	Q₈×C₅₃	—	2	2	no	—
131	Q₈×C₅₃	—	2	2	no	—
132	D₄×C₅₃	—	2	2	no	—
133	D₄×C₅₃	—	2	2	no	—
134	D₄×C₅₃	—	4	2	no	—
135	D₄×C₅₃	—	4	2	no	—
136	C₂×C₄×C₅₃	—	2	1	no	—
137	C₂×C₄×C₅₃	—	2	1	no	—
138	C₂×C₄×C₅₃	—	2	2	no	—
139	C₂×C₄×C₅₃	—	4	2	no	—
140	C₂×C₄×C₅₃	—	4	2	no	—
141	C₂×C₄×C₅₃	—	4	2	no	—
142	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
143	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
144	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
145	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
146	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
147	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
148	C₂×Dic₅₃	—	4	4	no	—
149	C₂×Dic₅₃	—	4	2	no	—
150	Dic₁₀₆	—	8	4	no	—
151	C₈×C₅₃	—	4	2	no	—
152	C₈×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
153	C₈×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
154	C₈×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
155	C₈×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
156	C₈×C₅₃	#124	1	1	yes	characteristic
157	C₈×C₅₃	#124	1	1	yes	characteristic
158	C₈×C₅₃	#124	1	1	yes	characteristic
159	C₈×C₅₃	#124	1	1	yes	characteristic
160	#88322	—	4	2	no	—
161	#29154	—	53	1	no	maximal, Sylow
162	C₂²×C₅₃	—	4	1	no	—
163	C₂²×C₅₃	—	4	2	no	—
164	C₂²×C₅₃	—	8	2	no	—
165	C₂²×C₅₃	#535	1	1	yes	characteristic
166	C₂²×C₅₃	#554	1	1	yes	characteristic
167	C₂²×C₅₃	#555	1	1	yes	characteristic
168	C₄×C₅₃	—	2	2	no	—
169	C₄×C₅₃	—	4	1	no	—
170	C₄×C₅₃	—	4	2	no	—
171	C₄×C₅₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
172	C₄×C₅₃	C₂×#124	1	1	yes	characteristic
173	C₄×C₅₃	#554	1	1	yes	characteristic
174	C₄×C₅₃	#555	1	1	yes	characteristic
175	Dic₅₃	—	8	4	no	—
176	Dic₅₃	—	8	2	no	—
177	#3160	—	53	1	no	—
178	C₂×#443	D₅₃	1	1	yes	characteristic
179	#2031	—	53	4	no	—
180	#1976	—	53	2	no	—
181	#1833	—	53	2	no	—
182	#1814	—	53	2	no	—
183	#1812	—	53	2	no	—
184	#1772	—	53	1	no	—
185	C₂×C₅₃	—	2	1	no	—
186	C₂×C₅₃	—	2	1	no	—
187	C₂×C₅₃	—	2	1	no	—
188	C₂×C₅₃	—	8	2	no	—
189	C₂×C₅₃	#2563	1	1	yes	characteristic
190	C₂×#135	C₂×D₅₃	1	1	yes	characteristic
191	C₂×#131	C₂×D₅₃	1	2	yes	—
192	#533	—	53	2	no	—
193	#500	—	106	1	no	—
194	#499	—	106	1	no	—
195	#499	—	106	1	no	—
196	#498	—	106	1	no	—
197	#443	—	2	1	no	—
198	#443	—	2	1	no	—
199	#443	—	2	1	no	—
200	#443	—	2	1	no	—
201	C₂×#108	—	53	1	no	—
202	C₂×#108	C₂×D₅₃	1	1	yes	characteristic
203	#427	—	53	1	no	—
204	#427	—	53	2	no	—
205	#427	—	53	2	no	—
206	#421	—	53	4	no	—
207	C₂×#104	C₂×D₅₃	1	2	yes	—
208	#410	—	53	4	no	—
209	#405	—	53	2	no	—
210	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
211	#344	—	53	4	no	—
212	#339	—	53	2	no	—
213	H(3,ℤ/4)	—	53	4	no	—
214	C₅₃	#29154	1	1	yes	characteristic, complemented, Sylow
215	C₂×#41	C₂²×D₅₃	1	2	yes	—
216	#135	—	2	1	no	—
217	#135	—	2	1	no	—
218	#135	—	2	1	no	—
219	#135	—	2	1	no	—
220	C₂×Q₁₆	Dic₁₀₆	1	1	yes	characteristic, complemented
221	C₂×SD₁₆	Dic₁₀₆	1	2	yes	complemented
222	C₂×D₈	Dic₁₀₆	1	1	yes	characteristic, complemented
223	#131	—	2	2	no	—
224	#131	—	2	2	no	—
225	#131	—	2	2	no	—
226	#131	—	2	2	no	—
227	C₂×M₁₆	—	2	2	no	—
228	C₂×M₁₆	—	53	2	no	—
229	C₂×M₁₆	C₂²×D₅₃	1	2	yes	—
230	C₂²×C₈	—	2	2	no	—
231	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
232	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
233	C₄⋊Q₈	—	106	2	no	—
234	#125	—	106	2	no	—
235	C₂²⋊Q₈	—	212	2	no	—
236	C₄×Q₈	—	212	2	no	—
237	#117	—	53	4	no	—
238	#117	—	53	2	no	—
239	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
240	#108	—	2	1	no	—
241	#108	—	2	1	no	—
242	#108	—	106	1	no	—
243	#108	—	106	1	no	—
244	#107	—	106	1	no	—
245	#107	—	106	2	no	—
246	#107	—	106	2	no	—
247	#106	—	106	1	no	—
248	#106	—	106	2	no	—
249	#106	—	106	2	no	—
250	#104	—	2	2	no	—
251	#104	—	2	2	no	—
252	#104	—	212	4	no	—
253	#99	—	212	4	no	—
254	#97	—	106	2	no	—
255	C₄×C₈	—	106	2	no	—
256	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
257	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
258	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
259	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
260	#41	—	2	2	no	—
261	#41	—	2	2	no	—
262	#41	—	2	2	no	—
263	#41	—	2	2	no	—
264	C₂×Q₈	—	212	2	no	—
265	C₂×Q₈	C₂×Dic₁₀₆	1	2	yes	—
266	C₂×D₄	—	2	2	no	—
267	C₂×D₄	C₂×Dic₁₀₆	1	2	yes	—
268	C₂²×C₄	—	2	1	no	—
269	C₂²×C₄	—	2	2	no	—
270	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
271	Q₁₆	—	2	1	no	—
272	Q₁₆	—	2	1	no	—
273	SD₁₆	—	2	2	no	—
274	SD₁₆	—	2	2	no	—
275	D₈	—	2	1	no	—
276	D₈	—	2	1	no	—
277	M₁₆	—	2	2	no	—
278	M₁₆	—	2	2	no	—
279	M₁₆	—	4	2	no	—
280	M₁₆	—	4	2	no	—
281	M₁₆	—	106	2	no	—
282	M₁₆	—	106	2	no	—
283	C₂×C₈	—	2	2	no	—
284	C₂×C₈	—	4	2	no	—
285	C₂×C₈	—	4	2	no	—
286	C₂×C₈	—	106	2	no	—
287	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
288	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
289	C₂×C₈	#11719945	1	1	yes	characteristic
290	C₂×C₈	#11719945	1	1	yes	characteristic
291	C₂×C₈	#11719947	1	1	yes	characteristic
292	C₂×C₈	#11719947	1	1	yes	characteristic
293	C₂×C₈	#11719947	1	1	yes	characteristic
294	C₂×C₈	#11719947	1	1	yes	characteristic
295	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
296	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
297	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
298	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
299	C₄⋊C₄	—	106	4	no	—
300	C₄⋊C₄	—	106	2	no	—
301	C₄⋊C₄	—	212	4	no	—
302	C₄⋊C₄	—	212	2	no	—
303	C₂²⋊C₄	—	106	2	no	—
304	C₂²⋊C₄	—	106	4	no	—
305	C₄²	—	2	1	no	—
306	C₄²	—	106	2	no	—
307	C₄²	—	106	4	no	—
308	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
309	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
310	C₂³	—	2	2	no	—
311	C₂³	#13133263	1	1	yes	characteristic
312	Q₈	—	2	2	no	—
313	Q₈	—	2	2	no	—
314	Q₈	—	424	4	no	—
315	D₄	—	2	2	no	—
316	D₄	—	2	2	no	—
317	D₄	—	4	2	no	—
318	D₄	—	4	2	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	4	2	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	212	2	no	—
326	C₂×C₄	—	212	4	no	—
327	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
328	C₂×C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
329	C₂×C₄	C₂×#11719947	1	1	yes	characteristic
330	C₂×C₄	C₂×#11719947	1	1	yes	characteristic
331	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
332	C₂×C₄	#13133263	1	1	yes	characteristic
333	C₈	—	4	2	no	—
334	C₈	—	212	2	no	—
335	C₈	#13133196	1	1	yes	characteristic
336	C₈	#13133196	1	1	yes	characteristic
337	C₈	#13133196	1	1	yes	characteristic
338	C₈	#13133196	1	1	yes	characteristic
339	C₈	#13133204	1	1	yes	characteristic
340	C₈	#13133204	1	1	yes	characteristic
341	C₈	#13133204	1	1	yes	characteristic
342	C₈	#13133204	1	1	yes	characteristic
343	C₂²	—	4	1	no	—
344	C₂²	—	4	2	no	—
345	C₂²	—	8	2	no	—
346	C₂²	#16811187	1	1	yes	characteristic
347	C₂²	#16811330	1	1	yes	characteristic
348	C₂²	#16811338	1	1	yes	characteristic
349	C₄	—	2	2	no	—
350	C₄	—	4	1	no	—
351	C₄	—	4	2	no	—
352	C₄	—	424	2	no	—
353	C₄	—	424	4	no	—
354	C₄	C₂×#13133196	1	1	yes	characteristic
355	C₄	C₂×#13133204	1	1	yes	characteristic
356	C₄	#16811330	1	1	yes	characteristic
357	C₄	#16811338	1	1	yes	characteristic
358	C₂	—	2	1	no	—
359	C₂	—	2	1	no	—
360	C₂	—	2	1	no	—
361	C₂	—	8	2	no	—
362	C₂	#20279888	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#16811338
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂×C₅₃×#443	C₂
socle	C₂×C₅₃	#2563
prime radical	C₂×#135×C₅₃	C₂²
hypercenter	C₂×#443	D₅₃
hypocenter	C₅₃	#29154
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#443

Complements

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

N	H ≅ G/N	type
1. #20285229	C₂	semidirect
2. #20275887	C₂	semidirect
4. #20274840	C₂	semidirect
5. #20274769	C₂	semidirect
6. #20274761	C₂	semidirect
7. #20274594	C₂	semidirect
57. C₂×Q₁₆×C₅₃	Q₈	semidirect
58. C₂×SD₁₆×C₅₃	Q₈	semidirect
59. C₂×D₈×C₅₃	Q₈	semidirect
214. C₅₃	#29154	semidirect
220. C₂×Q₁₆	Dic₁₀₆	semidirect
221. C₂×SD₁₆	Dic₁₀₆	semidirect
222. C₂×D₈	Dic₁₀₆	semidirect

Maximal subgroups

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

subgroup	index
#20285229	2
#20275887	2
#20275614	2
#20274840	2
#20274769	2
#20274761	2
#20274594	2
C₂×C₅₃×#443	2
#29154	53