Subgroups — #27402612

Order 13568.

812 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283631	C₂	1	2	yes	maximal
2	C₂×#16810741	C₂	1	1	yes	maximal, characteristic
3	#20283323	C₂	1	2	yes	maximal, complemented
4	#20275503	C₂	1	2	yes	maximal, complemented
5	#20274954	C₂	1	4	yes	maximal, complemented
6	#20274770	C₂	1	2	yes	maximal, complemented
7	#20274594	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#1848	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#13133239	C₂²	1	1	yes	characteristic
10	#16811389	C₂²	1	4	yes	—
11	C₂×#13133223	C₂²	1	2	yes	—
12	#16810744	—	2	4	no	—
13	#16810741	C₄	1	2	yes	complemented
14	#16810741	C₄	1	2	yes	complemented
15	#16810741	C₄	1	2	yes	complemented
16	#16810741	C₄	1	2	yes	complemented
17	#16810739	—	2	4	no	—
18	#16810671	C₂²	1	4	yes	—
19	#16810670	—	2	4	no	—
20	C₂×#13133153	C₂²	1	1	yes	characteristic
21	C₂×#13133153	C₂²	1	1	yes	characteristic
22	#108×D₅₃	—	2	4	no	—
23	#16810661	C₂²	1	1	yes	characteristic
24	#16810639	C₂²	1	4	yes	—
25	C₂×#13133145	C₂²	1	2	yes	—
26	#16810578	C₂²	1	2	yes	—
27	#16810394	C₂²	1	4	yes	—
28	#16810383	C₂²	1	2	yes	—
29	C₂×#108×C₅₃	C₂²	1	2	yes	—
30	C₂×#108×C₅₃	C₂²	1	2	yes	—
31	C₅₃×#423	C₂²	1	2	yes	—
32	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
33	C₂×#11719945	C₂³	1	2	yes	—
34	#13133239	C₂×C₄	1	2	yes	—
35	#13133239	C₂×C₄	1	2	yes	—
36	#13133239	D₄	1	2	yes	—
37	#13133239	D₄	1	2	yes	—
38	C₂×Dic₂₁₂	Q₈	1	1	yes	characteristic
39	C₂×#11719913	Q₈	1	2	yes	—
40	C₂×D₄₂₄	Q₈	1	1	yes	characteristic
41	#13133225	—	2	8	no	—
42	#13133223	—	2	2	no	—
43	#13133223	—	2	2	no	—
44	#13133223	—	4	4	no	—
45	#13133223	C₂×C₄	1	4	yes	—
46	#13133223	C₂×C₄	1	4	yes	—
47	C₂×#11719917	C₂³	1	2	yes	—
48	C₂×#11719917	C₂³	1	2	yes	—
49	C₂×#11719912	—	2	2	no	—
50	M₁₆×D₅₃	—	2	8	no	—
51	C₂×C₈×D₅₃	—	2	2	no	—
52	#13133176	C₂³	1	2	yes	—
53	#13133153	C₂×C₄	1	2	yes	—
54	#13133153	C₂×C₄	1	2	yes	—
55	#13133153	D₄	1	2	yes	—
56	#13133153	D₄	1	2	yes	—
57	#13133152	—	2	8	no	—
58	#13133151	—	2	1	no	—
59	#13133149	—	2	1	no	—
60	#13133145	C₂×C₄	1	4	yes	—
61	#13133145	C₂×C₄	1	4	yes	—
62	#13133143	—	2	8	no	—
63	#13133127	—	2	2	no	—
64	C₈×Dic₅₃	—	2	2	no	—
65	C₂×M₁₆×C₅₃	C₂³	1	4	yes	—
66	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
67	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
68	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
69	#108×C₅₃	—	2	2	no	—
70	#108×C₅₃	—	2	2	no	—
71	#108×C₅₃	—	2	2	no	—
72	#108×C₅₃	—	2	2	no	—
73	C₄⋊C₈×C₅₃	—	4	2	no	—
74	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
75	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
76	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
77	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₅₃	—	2	1	no	—
79	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
80	C₂×C₂²⋊D₅₃	—	2	2	no	—
81	#11719945	—	2	2	no	—
82	#11719945	—	2	2	no	—
83	#11719945	C₂²×C₄	1	4	yes	—
84	#11719945	C₂×D₄	1	4	yes	—
85	C₂×D₂₁₂	C₂×Q₈	1	2	yes	—
86	C₂×C₄×D₅₃	—	2	2	no	—
87	C₂×Dic₁₀₆	C₂×Q₈	1	2	yes	—
88	M₁₆×C₅₃	—	2	4	no	—
89	M₁₆×C₅₃	—	2	4	no	—
90	C₂×C₈×C₅₃	—	2	4	no	—
91	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
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	1	yes	characteristic
96	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
100	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
101	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
102	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
103	C₄²×C₅₃	—	2	1	no	—
104	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
105	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
106	C₂²⋊Dic₅₃	—	2	2	no	—
107	C₄⋊Dic₅₃	—	2	2	no	—
108	C₄×Dic₅₃	—	2	2	no	—
109	#11719917	—	2	2	no	—
110	#11719917	—	2	2	no	—
111	#11719917	C₂²×C₄	1	4	yes	—
112	#11719917	C₂×D₄	1	4	yes	—
113	C₂×#88322	—	2	2	no	—
114	C₂×#88322	—	2	2	no	—
115	Dic₂₁₂	C₄⋊C₄	1	2	yes	—
116	Dic₂₁₂	C₄⋊C₄	1	2	yes	—
117	D₄₂₄	C₄⋊C₄	1	2	yes	—
118	D₄₂₄	C₄⋊C₄	1	2	yes	—
119	#11719913	C₄⋊C₄	1	4	yes	—
120	#11719913	C₄⋊C₄	1	4	yes	—
121	#11719912	—	2	4	no	—
122	#11719912	—	2	4	no	—
123	#11719912	—	4	8	no	—
124	C₈×D₅₃	—	2	4	no	—
125	C₈×D₅₃	—	2	4	no	—
126	C₈×D₅₃	—	4	8	no	—
127	C₂³×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
128	C₂²×D₅₃	—	2	2	no	—
129	C₂×C₄×C₅₃	—	2	1	no	—
130	C₂×C₄×C₅₃	—	2	1	no	—
131	C₂×C₄×C₅₃	—	4	2	no	—
132	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
133	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
134	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
135	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
136	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
137	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
138	C₂²⋊D₅₃	—	2	4	no	—
139	C₂²⋊D₅₃	—	2	4	no	—
140	C₂×Dic₅₃	—	2	2	no	—
141	C₂×Dic₅₃	—	4	2	no	—
142	D₂₁₂	—	2	2	no	—
143	D₂₁₂	C₂×C₄⋊C₄	1	4	yes	—
144	C₄×D₅₃	—	2	4	no	—
145	C₄×D₅₃	—	2	4	no	—
146	Dic₁₀₆	—	2	2	no	—
147	Dic₁₀₆	C₂×C₄⋊C₄	1	4	yes	—
148	C₈×C₅₃	—	4	4	no	—
149	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
150	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
151	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
152	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
153	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
154	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
155	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
156	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
157	#88322	—	2	4	no	—
158	#88322	—	2	4	no	—
159	#22694	—	53	1	no	maximal, Sylow
160	C₂²×C₅₃	—	4	1	no	—
161	C₂²×C₅₃	#520	1	1	yes	characteristic
162	C₂²×C₅₃	#547	1	1	yes	characteristic
163	C₂²×C₅₃	D₄×Q₈	1	1	yes	characteristic
164	C₂×D₅₃	—	2	4	no	—
165	C₂×D₅₃	—	8	2	no	—
166	C₄×C₅₃	—	2	2	no	—
167	C₄×C₅₃	—	4	1	no	—
168	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
169	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
170	C₄×C₅₃	D₄²	1	1	yes	characteristic
171	C₄×C₅₃	#554	1	1	yes	characteristic
172	Dic₅₃	—	2	4	no	—
173	Dic₅₃	—	8	2	no	—
174	#2957	—	53	2	no	—
175	C₂×#443	—	53	1	no	—
176	#2916	—	53	2	no	—
177	#1953	—	53	2	no	—
178	#1861	—	53	4	no	—
179	#1848	D₅₃	1	1	yes	characteristic, complemented
180	#1814	—	53	2	no	—
181	#1772	—	53	1	no	—
182	C₂×C₅₃	—	2	1	no	—
183	C₂×C₅₃	—	2	1	no	—
184	C₂×C₅₃	—	2	1	no	—
185	C₂×C₅₃	D₄×C₄⋊C₄	1	1	yes	characteristic
186	D₅₃	—	4	4	no	—
187	C₂×#135	—	53	1	no	—
188	#568	—	53	4	no	—
189	C₂×#131	—	53	2	no	—
190	#444	—	106	4	no	—
191	#443	—	53	2	no	—
192	#443	—	53	2	no	—
193	#443	—	53	2	no	—
194	#443	—	53	2	no	—
195	#443	—	106	4	no	—
196	#430	—	53	4	no	—
197	#430	—	106	4	no	—
198	C₂×#108	—	53	1	no	—
199	C₂×#108	—	53	1	no	—
200	C₂×#108	—	106	4	no	—
201	C₂×#108	C₂×D₅₃	1	2	yes	—
202	C₂×#108	C₂×D₅₃	1	2	yes	—
203	#427	—	53	1	no	—
204	#423	C₂×D₅₃	1	2	yes	—
205	#421	—	53	4	no	—
206	C₂×#104	—	53	2	no	—
207	#405	—	53	2	no	—
208	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
209	#344	—	53	4	no	—
210	#339	—	53	2	no	—
211	C₅₃	#22694	1	1	yes	characteristic, complemented, Sylow
212	C₂×#41	—	53	2	no	—
213	#135	—	53	2	no	—
214	#135	—	53	2	no	—
215	#135	—	53	2	no	—
216	#135	—	53	2	no	—
217	C₂×Q₁₆	—	53	1	no	—
218	C₂×SD₁₆	—	53	2	no	—
219	C₂×D₈	—	53	1	no	—
220	#131	—	53	4	no	—
221	#131	—	53	4	no	—
222	#131	—	106	2	no	—
223	#131	—	106	2	no	—
224	#131	—	106	8	no	—
225	#131	—	212	4	no	—
226	C₂×M₁₆	—	53	2	no	—
227	C₂×M₁₆	—	53	2	no	—
228	C₂×M₁₆	—	106	2	no	—
229	C₂×M₁₆	—	106	8	no	—
230	C₂×M₁₆	C₂²×D₅₃	1	4	yes	—
231	C₂²×C₈	—	106	2	no	—
232	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
233	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
234	#117	—	53	2	no	—
235	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
236	#108	—	2	2	no	—
237	#108	—	2	2	no	—
238	#108	—	2	2	no	—
239	#108	—	2	2	no	—
240	#108	—	53	2	no	—
241	#108	—	53	2	no	—
242	#108	—	53	2	no	—
243	#108	—	53	2	no	—
244	#108	—	106	8	no	—
245	#107	—	106	1	no	—
246	#106	—	106	1	no	—
247	C₄⋊C₈	—	4	2	no	—
248	#104	—	53	4	no	—
249	#104	—	53	4	no	—
250	#104	—	106	8	no	—
251	#97	—	106	2	no	—
252	C₄×C₈	—	106	2	no	—
253	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
254	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
255	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
256	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
257	#41	—	53	4	no	—
258	#41	—	53	4	no	—
259	#41	—	106	2	no	—
260	#41	—	106	2	no	—
261	C₂×Q₈	—	53	2	no	—
262	C₂×D₄	—	53	2	no	—
263	C₂×D₄	—	106	2	no	—
264	C₂²×C₄	—	2	1	no	—
265	C₂²×C₄	—	106	2	no	—
266	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
267	Q₁₆	—	53	2	no	—
268	Q₁₆	—	53	2	no	—
269	SD₁₆	—	53	4	no	—
270	SD₁₆	—	53	4	no	—
271	D₈	—	53	2	no	—
272	D₈	—	53	2	no	—
273	M₁₆	—	2	4	no	—
274	M₁₆	—	2	4	no	—
275	M₁₆	—	53	4	no	—
276	M₁₆	—	53	4	no	—
277	M₁₆	—	106	2	no	—
278	M₁₆	—	106	2	no	—
279	M₁₆	—	106	4	no	—
280	M₁₆	—	106	4	no	—
281	M₁₆	—	212	8	no	—
282	C₂×C₈	—	2	4	no	—
283	C₂×C₈	—	106	2	no	—
284	C₂×C₈	—	106	2	no	—
285	C₂×C₈	—	106	4	no	—
286	C₂×C₈	—	106	4	no	—
287	C₂×C₈	—	212	8	no	—
288	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
289	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
290	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
291	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
292	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
293	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
294	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
295	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
296	C₂×C₈	Q₈×D₅₃	1	1	yes	characteristic
297	C₂×C₈	Q₈×D₅₃	1	1	yes	characteristic
298	C₂×C₈	#11719949	1	1	yes	characteristic
299	C₂×C₈	#11719949	1	1	yes	characteristic
300	C₄⋊C₄	—	106	2	no	—
301	C₂²⋊C₄	—	106	2	no	—
302	C₄²	—	2	1	no	—
303	C₄²	—	106	2	no	—
304	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
305	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
306	C₂³	—	106	2	no	—
307	C₂³	#13133265	1	1	yes	characteristic
308	Q₈	—	53	4	no	—
309	Q₈	—	106	2	no	—
310	D₄	—	53	4	no	—
311	D₄	—	106	4	no	—
312	D₄	—	106	4	no	—
313	D₄	—	106	2	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	4	2	no	—
317	C₂×C₄	—	106	2	no	—
318	C₂×C₄	—	106	4	no	—
319	C₂×C₄	—	106	4	no	—
320	C₂×C₄	—	212	2	no	—
321	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
322	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
323	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
324	C₂×C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
325	C₂×C₄	C₂×#11719949	1	1	yes	characteristic
326	C₂×C₄	#13133262	1	1	yes	characteristic
327	C₈	—	4	4	no	—
328	C₈	—	106	4	no	—
329	C₈	—	106	4	no	—
330	C₈	C₄⋊C₄×D₅₃	1	1	yes	characteristic
331	C₈	C₄⋊C₄×D₅₃	1	1	yes	characteristic
332	C₈	C₄⋊C₄×D₅₃	1	1	yes	characteristic
333	C₈	C₄⋊C₄×D₅₃	1	1	yes	characteristic
334	C₈	#13133179	1	1	yes	characteristic
335	C₈	#13133179	1	1	yes	characteristic
336	C₈	#13133179	1	1	yes	characteristic
337	C₈	#13133179	1	1	yes	characteristic
338	C₂²	—	4	1	no	—
339	C₂²	—	106	4	no	—
340	C₂²	—	424	2	no	—
341	C₂²	#16811104	1	1	yes	characteristic
342	C₂²	#16811276	1	1	yes	characteristic
343	C₂²	#16811288	1	1	yes	characteristic
344	C₄	—	2	2	no	—
345	C₄	—	4	1	no	—
346	C₄	—	106	4	no	—
347	C₄	—	424	2	no	—
348	C₄	C₂×C₄⋊C₄×D₅₃	1	1	yes	characteristic
349	C₄	C₂×#13133179	1	1	yes	characteristic
350	C₄	#16811258	1	1	yes	characteristic
351	C₄	#16811334	1	1	yes	characteristic
352	C₂	—	2	1	no	—
353	C₂	—	2	1	no	—
354	C₂	—	2	1	no	—
355	C₂	—	212	4	no	—
356	C₂	#20277615	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#16811258
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1848	C₂
socle	C₂×C₅₃	D₄×C₄⋊C₄
prime radical	C₂×#13133239	C₂²
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#1848	D₅₃
hypocenter	C₅₃	#22694
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #1848

Complements

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

N	H ≅ G/N	type
3. #20283323	C₂	semidirect
4. #20275503	C₂	semidirect
5. #20274954	C₂	semidirect
6. #20274770	C₂	semidirect
7. #20274594	C₂	semidirect
8. C₅₃×#1848	C₂	semidirect
13. #16810741	C₄	semidirect
14. #16810741	C₄	semidirect
15. #16810741	C₄	semidirect
16. #16810741	C₄	semidirect
179. #1848	D₅₃	semidirect
211. C₅₃	#22694	semidirect

Maximal subgroups

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

subgroup	index
#20283631	2
C₂×#16810741	2
#20283323	2
#20275503	2
#20274954	2
#20274770	2
#20274594	2
C₅₃×#1848	2
#22694	53