Subgroups — #27537686

Order 13568.

626 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₂₁₂	C₂	1	1	yes	maximal, characteristic
2	#20276548	C₂	1	2	yes	maximal, complemented
3	#20276539	C₂	1	4	yes	maximal, complemented
4	C₁₆×D₂₁₂	C₂	1	2	yes	maximal, complemented
5	#20276518	C₂	1	1	yes	maximal, characteristic, complemented
6	#20276324	C₂	1	2	yes	maximal, complemented
7	C₂²⋊C₁₆×D₅₃	C₂	1	2	yes	maximal, complemented
8	C₅₃×#2162	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₈×D₅₃	C₂²	1	2	yes	—
10	C₂×C₄×D₂₁₂	C₄	1	1	yes	characteristic
11	C₂×#13133156	C₂²	1	4	yes	complemented
12	C₂×C₁₆×D₅₃	C₂²	1	4	yes	complemented
13	C₈×D₂₁₂	C₄	1	2	yes	—
14	C₈×D₂₁₂	C₄	1	1	yes	characteristic
15	C₈×D₂₁₂	C₄	1	1	yes	characteristic
16	C₈×D₂₁₂	C₂²	1	1	yes	characteristic
17	C₈×D₂₁₂	C₂²	1	2	yes	—
18	C₈×D₂₁₂	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊#88322	C₄	1	1	yes	characteristic
20	C₂×#13133129	C₄	1	2	yes	—
21	C₄⋊#11719908	C₂²	1	2	yes	complemented
22	C₂²⋊#11719908	C₂²	1	2	yes	—
23	#16810402	C₂²	1	4	yes	complemented
24	#16810402	C₂²	1	4	yes	complemented
25	#16810399	C₂²	1	2	yes	complemented
26	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
27	C₅₃×C₄⋊C₁₆	C₂²	1	2	yes	complemented
28	C₅₃×C₂²⋊C₁₆	C₂²	1	1	yes	characteristic
29	C₅₃×C₂²⋊C₁₆	C₂²	1	1	yes	characteristic
30	C₄×C₁₆×C₅₃	C₂²	1	2	yes	complemented
31	C₂²×D₂₁₂	C₈	1	1	yes	characteristic
32	C₂²×C₄×D₅₃	C₂×C₄	1	2	yes	—
33	C₂²×#88322	C₂×C₄	1	2	yes	—
34	C₂×C₈×D₅₃	—	4	4	no	—
35	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
36	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
37	C₂×C₈×D₅₃	C₂³	1	2	yes	—
38	C₂×C₈×D₅₃	C₂³	1	2	yes	—
39	C₄×D₂₁₂	C₈	1	1	yes	characteristic
40	C₄×D₂₁₂	C₈	1	2	yes	—
41	C₄×D₂₁₂	C₈	1	1	yes	characteristic
42	C₄×D₂₁₂	C₂×C₄	1	2	yes	—
43	C₄×D₂₁₂	C₂×C₄	1	1	yes	characteristic
44	C₄×D₂₁₂	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄⋊Dic₅₃	C₈	1	1	yes	characteristic
46	C₂×#11719921	C₈	1	2	yes	—
47	#13133156	—	4	4	no	—
48	C₂×#11719908	C₂³	1	4	yes	—
49	C₁₆×D₅₃	—	4	4	no	—
50	C₄⋊#88322	C₂×C₄	1	1	yes	characteristic
51	C₄⋊#88322	C₂×C₄	1	2	yes	—
52	C₄⋊#88322	C₂×C₄	1	1	yes	characteristic
53	#13133129	C₂×C₄	1	2	yes	—
54	#13133129	C₂×C₄	1	2	yes	—
55	#13133129	C₂×C₄	1	2	yes	—
56	#13133129	C₂×C₄	1	2	yes	—
57	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
58	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
59	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
60	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
61	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
62	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
63	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
64	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
65	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
66	C₂³×D₅₃	C₂×C₈	1	2	yes	—
67	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
68	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
69	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
70	C₂²×Dic₅₃	C₂×C₈	1	2	yes	—
71	C₂×D₂₁₂	—	2	4	no	—
72	C₂×D₂₁₂	C₂×C₈	1	2	yes	—
73	C₂×D₂₁₂	C₂×C₈	1	1	yes	characteristic
74	C₂×D₂₁₂	C₂×C₈	1	1	yes	characteristic
75	C₂×C₄×D₅₃	—	4	2	no	—
76	C₂×C₄×D₅₃	—	4	2	no	—
77	C₂×C₄×D₅₃	C₂×C₈	1	2	yes	—
78	C₂×C₄×D₅₃	C₂×C₈	1	2	yes	—
79	C₂×C₄×D₅₃	C₂²×C₄	1	2	yes	—
80	C₂×C₄×D₅₃	C₂²×C₄	1	2	yes	—
81	C₂×C₈×C₅₃	—	2	2	no	—
82	C₂×C₈×C₅₃	—	4	1	no	—
83	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
87	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
88	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
89	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
90	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
91	#11719921	C₂×C₈	1	2	yes	—
92	#11719921	C₂×C₈	1	2	yes	—
93	#11719921	C₂×C₈	1	2	yes	—
94	#11719921	C₂×C₈	1	2	yes	—
95	C₄⋊Dic₅₃	C₂×C₈	1	2	yes	—
96	C₄⋊Dic₅₃	C₂×C₈	1	1	yes	characteristic
97	C₄⋊Dic₅₃	C₂×C₈	1	1	yes	characteristic
98	C₂×#88322	—	4	2	no	—
99	C₂×#88322	C₂²×C₄	1	2	yes	—
100	C₂×#88322	C₂²×C₄	1	2	yes	—
101	C₈×D₅₃	—	2	4	no	—
102	C₈×D₅₃	—	2	4	no	—
103	C₈×D₅₃	—	4	2	no	—
104	C₈×D₅₃	—	4	2	no	—
105	C₁₆×C₅₃	—	2	2	no	—
106	C₁₆×C₅₃	—	2	2	no	—
107	#11719908	—	2	4	no	—
108	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
109	C₂²×D₅₃	—	2	4	no	—
110	C₂²×D₅₃	—	2	4	no	—
111	C₂²×D₅₃	—	4	2	no	—
112	C₂²×D₅₃	C₂²×C₈	1	2	yes	—
113	C₂²×D₅₃	C₂²×C₈	1	2	yes	—
114	C₂×C₄×C₅₃	—	2	2	no	—
115	C₂×C₄×C₅₃	—	2	1	no	—
116	C₂×C₄×C₅₃	—	2	1	no	—
117	C₂×C₄×C₅₃	—	2	2	no	—
118	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
119	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
120	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
121	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
122	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
123	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
124	C₂×Dic₅₃	—	4	2	no	—
125	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
126	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
127	D₂₁₂	—	2	4	no	—
128	D₂₁₂	—	2	2	no	—
129	D₂₁₂	M₃₂	1	4	yes	complemented
130	C₄×D₅₃	—	2	2	no	—
131	C₄×D₅₃	—	2	2	no	—
132	C₄×D₅₃	—	2	2	no	—
133	C₄×D₅₃	—	2	2	no	—
134	C₄×D₅₃	—	4	2	no	—
135	C₄×D₅₃	—	4	2	no	—
136	C₈×C₅₃	—	2	1	no	—
137	C₈×C₅₃	—	2	1	no	—
138	C₈×C₅₃	—	2	1	no	—
139	C₈×C₅₃	H(5,𝔽₂)	1	2	yes	—
140	#88322	—	2	2	no	—
141	#88322	—	2	2	no	—
142	#33488	—	53	1	no	maximal, Sylow
143	C₂²×C₅₃	—	2	1	no	—
144	C₂²×C₅₃	—	2	1	no	—
145	C₂²×C₅₃	#504	1	2	yes	—
146	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
147	C₂×D₅₃	—	2	2	no	—
148	C₂×D₅₃	—	2	2	no	—
149	C₂×D₅₃	—	2	2	no	—
150	C₂×D₅₃	—	2	2	no	—
151	C₂×D₅₃	—	2	2	no	—
152	C₂×D₅₃	—	4	8	no	—
153	C₂×D₅₃	C₂×M₃₂	1	4	yes	—
154	C₄×C₅₃	—	2	1	no	—
155	C₄×C₅₃	—	2	1	no	—
156	C₄×C₅₃	—	2	1	no	—
157	C₄×C₅₃	—	2	1	no	—
158	C₄×C₅₃	C₂×M₃₂	1	2	yes	—
159	C₄×C₅₃	#518	1	1	yes	characteristic
160	C₄×C₅₃	#568	1	1	yes	characteristic
161	Dic₅₃	—	2	2	no	—
162	Dic₅₃	—	2	2	no	—
163	C₂×C₈×D₄	—	53	1	no	—
164	#2169	—	53	2	no	—
165	#2168	—	53	4	no	—
166	D₄×C₁₆	—	53	2	no	—
167	#2163	—	53	1	no	—
168	#2162	D₅₃	1	1	yes	characteristic, complemented
169	#2112	—	53	2	no	—
170	C₂×C₂²⋊C₁₆	—	53	2	no	—
171	C₂×C₅₃	—	2	2	no	—
172	C₂×C₅₃	#2972	1	1	yes	characteristic
173	C₂×C₅₃	C₂²×M₃₂	1	1	yes	characteristic
174	C₂×C₅₃	C₂×#504	1	1	yes	characteristic
175	D₅₃	—	2	4	no	—
176	D₅₃	—	4	2	no	—
177	C₂³×C₈	—	53	2	no	—
178	C₂×C₄×D₄	—	53	1	no	—
179	C₂×M₃₂	—	53	4	no	—
180	C₂²×C₁₆	—	53	4	no	—
181	C₈×D₄	—	53	2	no	—
182	C₈×D₄	—	53	1	no	—
183	C₈×D₄	—	53	1	no	—
184	C₈×D₄	—	53	1	no	—
185	C₈×D₄	—	53	2	no	—
186	C₈×D₄	—	53	1	no	—
187	C₂×C₄⋊C₈	—	53	1	no	—
188	C₂×C₂²⋊C₈	—	53	2	no	—
189	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
190	C₄⋊C₁₆	—	53	2	no	—
191	C₄⋊C₁₆	C₂×D₅₃	1	2	yes	complemented
192	C₂²⋊C₁₆	—	53	2	no	—
193	C₂²⋊C₁₆	—	53	4	no	—
194	C₂²⋊C₁₆	—	53	4	no	—
195	C₂²⋊C₁₆	C₂×D₅₃	1	1	yes	characteristic
196	C₂²⋊C₁₆	C₂×D₅₃	1	1	yes	characteristic
197	#346	—	53	2	no	—
198	C₄×C₁₆	C₂×D₅₃	1	2	yes	complemented
199	C₅₃	#33488	1	1	yes	characteristic, complemented, Sylow
200	C₂²×D₄	—	53	1	no	—
201	C₂³×C₄	—	53	2	no	—
202	C₂²×C₈	—	53	2	no	—
203	C₂²×C₈	—	53	2	no	—
204	C₂²×C₈	—	53	2	no	—
205	C₂²×C₈	—	53	2	no	—
206	C₂²×C₈	—	53	2	no	—
207	C₂²×C₈	—	212	4	no	—
208	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
209	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
210	C₄×D₄	—	53	1	no	—
211	C₄×D₄	—	53	2	no	—
212	C₄×D₄	—	53	1	no	—
213	C₄×D₄	—	53	1	no	—
214	C₄×D₄	—	53	2	no	—
215	C₄×D₄	—	53	1	no	—
216	C₂×C₄⋊C₄	—	53	1	no	—
217	C₂×C₂²⋊C₄	—	53	2	no	—
218	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
219	M₃₂	—	212	4	no	—
220	C₂×C₁₆	—	53	4	no	—
221	C₂×C₁₆	—	212	4	no	—
222	C₂×C₁₆	C₂²×D₅₃	1	2	yes	—
223	C₂×C₁₆	C₂²×D₅₃	1	2	yes	—
224	C₄⋊C₈	—	53	1	no	—
225	C₄⋊C₈	—	53	2	no	—
226	C₄⋊C₈	—	53	1	no	—
227	C₂²⋊C₈	—	53	2	no	—
228	C₂²⋊C₈	—	53	2	no	—
229	C₂²⋊C₈	—	53	2	no	—
230	C₂²⋊C₈	—	53	2	no	—
231	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
232	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
233	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
234	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
235	C₂⁴	—	53	2	no	—
236	C₂×D₄	—	53	1	no	—
237	C₂×D₄	—	53	2	no	—
238	C₂×D₄	—	53	1	no	—
239	C₂×D₄	—	106	4	no	—
240	C₂²×C₄	—	53	2	no	—
241	C₂²×C₄	—	53	2	no	—
242	C₂²×C₄	—	53	2	no	—
243	C₂²×C₄	—	53	2	no	—
244	C₂²×C₄	—	53	2	no	—
245	C₂²×C₄	—	212	2	no	—
246	C₂²×C₄	—	212	2	no	—
247	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
248	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
249	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
250	C₂×C₈	—	2	2	no	—
251	C₂×C₈	—	4	1	no	—
252	C₂×C₈	—	53	2	no	—
253	C₂×C₈	—	53	2	no	—
254	C₂×C₈	—	106	4	no	—
255	C₂×C₈	—	106	4	no	—
256	C₂×C₈	—	212	2	no	—
257	C₂×C₈	—	212	2	no	—
258	C₂×C₈	—	212	2	no	—
259	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
260	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
261	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
262	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
263	C₄⋊C₄	—	53	2	no	—
264	C₄⋊C₄	—	53	1	no	—
265	C₄⋊C₄	—	53	1	no	—
266	C₂²⋊C₄	—	53	2	no	—
267	C₂²⋊C₄	—	53	2	no	—
268	C₂²⋊C₄	—	53	2	no	—
269	C₂²⋊C₄	—	53	2	no	—
270	C₄²	C₈×D₅₃	1	1	yes	characteristic
271	C₄²	C₈×D₅₃	1	1	yes	characteristic
272	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
273	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
274	C₁₆	—	2	2	no	—
275	C₁₆	—	2	2	no	—
276	C₁₆	—	106	4	no	—
277	C₂³	—	53	2	no	—
278	C₂³	—	53	2	no	—
279	C₂³	—	106	4	no	—
280	C₂³	—	106	4	no	—
281	C₂³	—	212	2	no	—
282	C₂³	C₂×C₈×D₅₃	1	1	yes	characteristic
283	D₄	—	53	4	no	—
284	D₄	—	106	4	no	—
285	D₄	—	106	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	53	2	no	—
291	C₂×C₄	—	53	2	no	—
292	C₂×C₄	—	106	2	no	—
293	C₂×C₄	—	106	2	no	—
294	C₂×C₄	—	106	2	no	—
295	C₂×C₄	—	106	2	no	—
296	C₂×C₄	—	212	2	no	—
297	C₂×C₄	—	212	2	no	—
298	C₂×C₄	—	212	2	no	—
299	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
300	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
301	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
302	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
303	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
304	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
305	C₈	—	2	1	no	—
306	C₈	—	2	1	no	—
307	C₈	—	2	1	no	—
308	C₈	—	106	2	no	—
309	C₈	—	106	2	no	—
310	C₈	#13133263	1	2	yes	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	53	4	no	—
314	C₂²	—	106	2	no	—
315	C₂²	—	106	2	no	—
316	C₂²	—	106	2	no	—
317	C₂²	—	106	2	no	—
318	C₂²	—	106	2	no	—
319	C₂²	—	212	8	no	—
320	C₂²	#16811039	1	2	yes	—
321	C₂²	C₂²×C₈×D₅₃	1	1	yes	characteristic
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	2	1	no	—
325	C₄	—	2	1	no	—
326	C₄	—	106	2	no	—
327	C₄	—	106	2	no	—
328	C₄	M₃₂×D₅₃	1	2	yes	—
329	C₄	#16811097	1	1	yes	characteristic
330	C₄	#16811389	1	1	yes	characteristic
331	C₂	—	2	2	no	—
332	C₂	—	106	4	no	—
333	C₂	—	212	2	no	—
334	C₂	#20283738	1	1	yes	characteristic
335	C₂	C₂×M₃₂×D₅₃	1	1	yes	characteristic
336	C₂	C₂×#16811039	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₈
Fitting	C₅₃×#2162	C₂
prime radical	C₂²×D₂₁₂	C₈
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	#2162	D₅₃
hypocenter	C₅₃	#33488
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ #2162

Complements

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

N	H ≅ G/N	type
2. #20276548	C₂	semidirect
3. #20276539	C₂	semidirect
4. C₁₆×D₂₁₂	C₂	semidirect
5. #20276518	C₂	semidirect
6. #20276324	C₂	semidirect
7. C₂²⋊C₁₆×D₅₃	C₂	semidirect
8. C₅₃×#2162	C₂	semidirect
11. C₂×#13133156	C₂²	semidirect
12. C₂×C₁₆×D₅₃	C₂²	semidirect
21. C₄⋊#11719908	C₂²	semidirect
23. #16810402	C₂²	semidirect
24. #16810402	C₂²	semidirect
25. #16810399	C₂²	semidirect
27. C₅₃×C₄⋊C₁₆	C₂²	semidirect
30. C₄×C₁₆×C₅₃	C₂²	semidirect
129. D₂₁₂	M₃₂	semidirect
168. #2162	D₅₃	semidirect
191. C₄⋊C₁₆	C₂×D₅₃	semidirect
198. C₄×C₁₆	C₂×D₅₃	semidirect
199. C₅₃	#33488	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×D₂₁₂	2
#20276548	2
#20276539	2
C₁₆×D₂₁₂	2
#20276518	2
#20276324	2
C₂²⋊C₁₆×D₅₃	2
C₅₃×#2162	2
#33488	53