Subgroups — #27472855

Order 13568.

658 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20284516	C₂	1	2	yes	maximal
2	C₂×C₄×Dic₂₁₂	C₂	1	1	yes	maximal, characteristic
3	#20276220	C₂	1	2	yes	maximal
4	#20275983	C₂	1	2	yes	maximal
5	#20275370	C₂	1	4	yes	maximal
6	#20274853	C₂	1	1	yes	maximal, characteristic
7	#20274708	C₂	1	2	yes	maximal
8	C₅₃×#2960	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₂₁₂	C₂²	1	1	yes	characteristic
10	C₂×C₄×Dic₁₀₆	C₂²	1	2	yes	—
11	#16810927	—	2	4	no	—
12	#16810915	—	2	2	no	—
13	#16810905	—	2	2	no	—
14	C₄×Dic₂₁₂	—	2	1	no	—
15	C₄×Dic₂₁₂	—	2	1	no	—
16	C₄×Dic₂₁₂	—	2	1	no	—
17	C₄×Dic₂₁₂	—	2	1	no	—
18	#16810666	C₂²	1	4	yes	—
19	C₂×#13133151	C₂²	1	1	yes	characteristic
20	#16810647	C₂²	1	2	yes	—
21	#16810631	C₂²	1	4	yes	—
22	C₂×#13133142	C₂²	1	2	yes	—
23	C₂×#13133135	C₂²	1	4	yes	—
24	#16810391	C₂²	1	4	yes	—
25	#16810388	C₂²	1	2	yes	—
26	#16810388	C₂²	1	2	yes	—
27	C₅₃×#533	C₂²	1	1	yes	characteristic
28	C₅₃×#446	—	2	1	no	—
29	C₅₃×#446	—	2	1	no	—
30	C₈×Q₈×C₅₃	—	2	1	no	—
31	C₈×Q₈×C₅₃	—	2	1	no	—
32	C₅₃×#424	C₂²	1	2	yes	—
33	C₅₃×#423	C₂²	1	1	yes	characteristic
34	C₅₃×#405	C₂²	1	2	yes	—
35	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
36	C₂²×Dic₁₀₆	C₂³	1	2	yes	—
37	C₂×Dic₂₁₂	—	2	1	no	—
38	C₂×Dic₂₁₂	—	2	1	no	—
39	C₂×Dic₂₁₂	—	2	4	no	—
40	C₂×#11719917	C₂³	1	4	yes	—
41	C₄×Dic₁₀₆	—	2	2	no	—
42	C₄×Dic₁₀₆	—	2	2	no	—
43	C₄×Dic₁₀₆	—	2	2	no	—
44	C₄×Dic₁₀₆	—	2	2	no	—
45	C₂×C₄⋊Dic₅₃	C₂³	1	2	yes	—
46	C₂×#11719919	—	2	2	no	—
47	C₂×C₄×Dic₅₃	—	2	2	no	—
48	#13133151	—	2	1	no	—
49	#13133151	—	2	1	no	—
50	#13133150	—	2	4	no	—
51	#13133148	—	2	4	no	—
52	C₄⋊#88322	—	2	2	no	—
53	C₄⋊#88322	—	2	2	no	—
54	#13133142	—	2	2	no	—
55	#13133142	—	2	2	no	—
56	#13133140	—	2	4	no	—
57	#13133140	—	2	4	no	—
58	#13133135	—	2	8	no	—
59	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
60	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
61	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
62	C₄⋊Q₈×C₅₃	D₄	1	1	yes	characteristic
63	C₄⋊Q₈×C₅₃	D₄	1	1	yes	characteristic
64	#125×C₅₃	D₄	1	2	yes	—
65	C₂²⋊Q₈×C₅₃	—	2	1	no	—
66	C₂²⋊Q₈×C₅₃	—	2	1	no	—
67	C₄×Q₈×C₅₃	—	2	1	no	—
68	C₄×Q₈×C₅₃	—	2	1	no	—
69	#117×C₅₃	C₂³	1	2	yes	—
70	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
71	C₄⋊C₈×C₅₃	—	2	2	no	—
72	C₄⋊C₈×C₅₃	—	2	2	no	—
73	C₄⋊C₈×C₅₃	—	2	1	no	—
74	C₄⋊C₈×C₅₃	—	2	1	no	—
75	#97×C₅₃	—	2	2	no	—
76	C₄×C₈×C₅₃	—	2	2	no	—
77	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
78	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
79	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
80	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
81	C₂×Q₈×C₅₃	—	2	1	no	—
82	C₂×Q₈×C₅₃	—	2	1	no	—
83	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
84	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
85	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
86	C₂²×Dic₅₃	—	2	2	no	—
87	C₂²×Dic₅₃	—	2	2	no	—
88	C₂×Dic₁₀₆	—	2	2	no	—
89	C₂×Dic₁₀₆	—	2	2	no	—
90	C₂×Dic₁₀₆	—	2	2	no	—
91	C₂×Dic₁₀₆	—	2	2	no	—
92	C₂×Dic₁₀₆	—	2	2	no	—
93	C₂×Dic₁₀₆	#41	1	4	yes	—
94	M₁₆×C₅₃	—	2	4	no	—
95	C₂×C₈×C₅₃	—	2	2	no	—
96	C₂×C₈×C₅₃	—	2	2	no	—
97	C₂×C₈×C₅₃	—	2	2	no	—
98	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
102	C₄⋊C₄×C₅₃	—	2	2	no	—
103	C₄⋊C₄×C₅₃	—	2	2	no	—
104	C₄⋊C₄×C₅₃	—	2	1	no	—
105	C₄⋊C₄×C₅₃	—	2	1	no	—
106	C₄⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
107	C₄⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
108	C₂²⋊C₄×C₅₃	—	2	2	no	—
109	C₄²×C₅₃	—	2	2	no	—
110	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
111	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
112	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
113	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
114	C₄⋊Dic₅₃	—	2	2	no	—
115	C₄⋊Dic₅₃	—	2	2	no	—
116	#11719919	—	4	2	no	—
117	#11719919	—	4	2	no	—
118	C₄×Dic₅₃	—	4	2	no	—
119	C₄×Dic₅₃	—	4	2	no	—
120	#11719917	—	2	4	no	—
121	#11719917	—	2	4	no	—
122	C₂×#88322	—	2	4	no	—
123	Dic₂₁₂	—	2	4	no	—
124	Dic₂₁₂	—	2	4	no	—
125	C₂³×C₅₃	C₂×#41	1	1	yes	characteristic
126	Q₈×C₅₃	—	4	2	no	—
127	Q₈×C₅₃	—	4	2	no	—
128	C₂×C₄×C₅₃	—	2	2	no	—
129	C₂×C₄×C₅₃	—	2	2	no	—
130	C₂×C₄×C₅₃	—	2	2	no	—
131	C₂×C₄×C₅₃	—	4	1	no	—
132	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
133	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
134	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
135	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
136	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
137	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
138	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
139	C₂×C₄×C₅₃	2₋¹⁺⁴	1	2	yes	—
140	C₂×Dic₅₃	—	2	4	no	—
141	C₂×Dic₅₃	—	4	2	no	—
142	C₂×Dic₅₃	—	4	2	no	—
143	C₂×Dic₅₃	—	4	2	no	—
144	C₂×Dic₅₃	—	8	2	no	—
145	Dic₁₀₆	—	2	4	no	—
146	Dic₁₀₆	—	2	4	no	—
147	Dic₁₀₆	—	4	2	no	—
148	Dic₁₀₆	—	4	2	no	—
149	C₈×C₅₃	—	2	1	no	—
150	C₈×C₅₃	—	2	1	no	—
151	C₈×C₅₃	—	2	1	no	—
152	C₈×C₅₃	—	2	1	no	—
153	C₈×C₅₃	—	4	2	no	—
154	#88322	—	4	4	no	—
155	#28219	—	53	1	no	maximal, Sylow
156	C₂²×C₅₃	—	2	1	no	—
157	C₂²×C₅₃	—	2	1	no	—
158	C₂²×C₅₃	C₂×#121	1	1	yes	characteristic
159	C₂²×C₅₃	#552	1	2	yes	—
160	C₄×C₅₃	—	2	1	no	—
161	C₄×C₅₃	—	2	1	no	—
162	C₄×C₅₃	—	2	1	no	—
163	C₄×C₅₃	—	4	2	no	—
164	C₄×C₅₃	—	4	2	no	—
165	C₄×C₅₃	#473	1	2	yes	—
166	C₄×C₅₃	#525	1	1	yes	characteristic
167	C₄×C₅₃	#536	1	1	yes	characteristic
168	C₄×C₅₃	#548	1	2	yes	—
169	Dic₅₃	—	4	4	no	—
170	Dic₅₃	—	8	2	no	—
171	#3078	—	53	2	no	—
172	#2960	D₅₃	1	1	yes	characteristic
173	C₂×C₄×Q₁₆	—	53	1	no	—
174	#2089	—	53	2	no	—
175	#2047	—	53	2	no	—
176	#1930	—	53	4	no	—
177	#1837	—	53	1	no	—
178	#1801	—	53	2	no	—
179	C₂×C₅₃	—	2	2	no	—
180	C₂×C₅₃	#2536	1	1	yes	characteristic
181	C₂×C₅₃	C₂×#473	1	1	yes	characteristic
182	C₂×C₅₃	#3067	1	1	yes	characteristic
183	C₂²×Q₁₆	—	53	1	no	—
184	#533	C₂×D₅₃	1	1	yes	characteristic
185	C₂×C₄×Q₈	—	53	2	no	—
186	#479	—	106	4	no	—
187	#477	—	106	2	no	—
188	#475	—	106	2	no	—
189	#446	—	2	1	no	—
190	#446	—	2	1	no	—
191	C₈×Q₈	—	2	1	no	—
192	C₈×Q₈	—	2	1	no	—
193	C₄×Q₁₆	—	106	1	no	—
194	C₄×Q₁₆	—	106	1	no	—
195	C₄×Q₁₆	—	106	1	no	—
196	C₄×Q₁₆	—	106	1	no	—
197	#428	—	53	4	no	—
198	C₂×#107	—	53	1	no	—
199	#424	C₂×D₅₃	1	2	yes	—
200	#423	—	53	2	no	—
201	#423	C₂×D₅₃	1	1	yes	characteristic
202	#419	—	53	4	no	—
203	C₂×#103	—	53	2	no	—
204	C₂×#101	—	53	4	no	—
205	#405	C₂×D₅₃	1	2	yes	—
206	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
207	#343	—	53	4	no	—
208	#341	—	53	2	no	—
209	#341	—	53	2	no	—
210	C₅₃	#28219	1	1	yes	characteristic, complemented, Sylow
211	C₂²×Q₈	—	53	2	no	—
212	C₂×Q₁₆	—	106	4	no	—
213	C₂×Q₁₆	—	106	1	no	—
214	C₂×Q₁₆	—	106	1	no	—
215	C₂×M₁₆	—	53	4	no	—
216	C₂×M₁₆	C₂²×D₅₃	1	2	yes	—
217	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
218	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
219	C₄⋊Q₈	D₂₁₂	1	1	yes	characteristic
220	C₄⋊Q₈	D₂₁₂	1	1	yes	characteristic
221	#125	D₂₁₂	1	2	yes	—
222	C₂²⋊Q₈	—	2	1	no	—
223	C₂²⋊Q₈	—	2	1	no	—
224	C₄×Q₈	—	2	1	no	—
225	C₄×Q₈	—	2	1	no	—
226	C₄×Q₈	—	106	2	no	—
227	C₄×Q₈	—	106	2	no	—
228	C₄×Q₈	—	106	2	no	—
229	C₄×Q₈	—	106	2	no	—
230	#117	C₂²×D₅₃	1	2	yes	—
231	C₂×C₄⋊C₄	—	53	2	no	—
232	C₂×C₄⋊C₄	—	106	2	no	—
233	C₂×C₄²	—	106	2	no	—
234	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
235	#107	—	106	1	no	—
236	#107	—	106	1	no	—
237	#107	—	106	4	no	—
238	#106	—	106	4	no	—
239	C₄⋊C₈	—	2	2	no	—
240	C₄⋊C₈	—	2	2	no	—
241	C₄⋊C₈	—	2	1	no	—
242	C₄⋊C₈	—	2	1	no	—
243	C₄⋊C₈	—	106	2	no	—
244	C₄⋊C₈	—	106	2	no	—
245	#103	—	106	2	no	—
246	#103	—	106	2	no	—
247	#103	—	106	4	no	—
248	#103	—	106	4	no	—
249	#101	—	106	8	no	—
250	#97	—	2	2	no	—
251	C₄×C₈	—	2	2	no	—
252	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
253	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
254	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
255	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
256	C₂×Q₈	—	2	1	no	—
257	C₂×Q₈	—	2	1	no	—
258	C₂×Q₈	—	53	4	no	—
259	C₂×Q₈	—	106	2	no	—
260	C₂×Q₈	—	106	2	no	—
261	C₂×Q₈	—	106	2	no	—
262	C₂×Q₈	—	106	2	no	—
263	C₂×Q₈	—	106	2	no	—
264	C₂²×C₄	—	106	2	no	—
265	C₂²×C₄	—	106	2	no	—
266	C₂²×C₄	#11719945	1	1	yes	characteristic
267	C₂²×C₄	#11719945	1	1	yes	characteristic
268	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
269	Q₁₆	—	106	4	no	—
270	Q₁₆	—	106	4	no	—
271	M₁₆	—	2	4	no	—
272	M₁₆	—	106	4	no	—
273	M₁₆	—	106	4	no	—
274	C₂×C₈	—	2	2	no	—
275	C₂×C₈	—	2	2	no	—
276	C₂×C₈	—	2	2	no	—
277	C₂×C₈	—	106	4	no	—
278	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
279	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
280	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
281	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
282	C₄⋊C₄	—	2	2	no	—
283	C₄⋊C₄	—	2	2	no	—
284	C₄⋊C₄	—	2	1	no	—
285	C₄⋊C₄	—	2	1	no	—
286	C₄⋊C₄	—	106	2	no	—
287	C₄⋊C₄	—	106	2	no	—
288	C₄⋊C₄	—	212	2	no	—
289	C₄⋊C₄	—	212	2	no	—
290	C₄⋊C₄	C₂×D₂₁₂	1	2	yes	—
291	C₄⋊C₄	C₂×D₂₁₂	1	2	yes	—
292	C₂²⋊C₄	—	2	2	no	—
293	C₄²	—	2	2	no	—
294	C₄²	—	212	2	no	—
295	C₄²	—	212	2	no	—
296	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
297	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
298	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
299	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
300	C₂³	C₂×#11719945	1	1	yes	characteristic
301	Q₈	—	4	2	no	—
302	Q₈	—	4	2	no	—
303	Q₈	—	106	4	no	—
304	Q₈	—	106	4	no	—
305	Q₈	—	212	2	no	—
306	Q₈	—	212	2	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	4	1	no	—
311	C₂×C₄	—	106	4	no	—
312	C₂×C₄	—	424	2	no	—
313	C₂×C₄	—	212	2	no	—
314	C₂×C₄	—	212	2	no	—
315	C₂×C₄	—	212	2	no	—
316	C₂×C₄	#13133191	1	1	yes	characteristic
317	C₂×C₄	#13133191	1	1	yes	characteristic
318	C₂×C₄	#13133191	1	1	yes	characteristic
319	C₂×C₄	#13133191	1	1	yes	characteristic
320	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
321	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
322	C₂×C₄	#41×D₅₃	1	2	yes	—
323	C₂×C₄	#13133265	1	2	yes	—
324	C₈	—	2	1	no	—
325	C₈	—	2	1	no	—
326	C₈	—	2	1	no	—
327	C₈	—	2	1	no	—
328	C₈	—	4	2	no	—
329	C₈	—	212	4	no	—
330	C₂²	—	2	1	no	—
331	C₂²	—	2	1	no	—
332	C₂²	C₂×#13133191	1	1	yes	characteristic
333	C₂²	#16811314	1	2	yes	—
334	C₄	—	2	1	no	—
335	C₄	—	2	1	no	—
336	C₄	—	2	1	no	—
337	C₄	—	4	2	no	—
338	C₄	—	4	2	no	—
339	C₄	—	424	2	no	—
340	C₄	—	212	4	no	—
341	C₄	#16810898	1	2	yes	—
342	C₄	#16811130	1	1	yes	characteristic
343	C₄	#16811197	1	1	yes	characteristic
344	C₄	#16811285	1	2	yes	—
345	C₂	—	2	2	no	—
346	C₂	#20279620	1	1	yes	characteristic
347	C₂	C₂×#16810898	1	1	yes	characteristic
348	C₂	#20284435	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#13133191
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#2960	C₂
socle	C₂²×C₅₃	C₂×#121
prime radical	C₂³×C₅₃	C₂×#41
hypercenter	#2960	D₅₃
hypocenter	C₅₃	#28219
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960

Complements

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

N	H ≅ G/N	type
210. C₅₃	#28219	semidirect

Maximal subgroups

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

subgroup	index
#20284516	2
C₂×C₄×Dic₂₁₂	2
#20276220	2
#20275983	2
#20275370	2
#20274853	2
#20274708	2
C₅₃×#2960	2
#28219	53