Subgroups — #27276885

Order 13568.

516 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20274867	C₂	1	1	yes	maximal, characteristic
2	#20274867	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810369	C₂	1	2	yes	maximal
4	#20272598	C₂	1	2	yes	maximal
5	C₄×C₄⋊Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#88322	—	2	1	no	—
7	C₂×C₄⋊#88322	—	2	1	no	—
8	C₂×C₄⋊#88322	—	2	1	no	—
9	C₂×C₄⋊#88322	—	2	1	no	—
10	C₂×C₄⋊#88322	C₂²	1	2	yes	—
11	C₂×C₄⋊#88322	C₂²	1	2	yes	—
12	#16810369	—	2	4	no	—
13	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×Q₈×C₅₃	—	2	1	no	—
15	C₅₃×#391	—	2	1	no	—
16	C₅₃×#389	—	2	1	no	—
17	C₅₃×#389	C₄	1	2	yes	—
18	C₅₃×#384	C₂²	1	1	yes	characteristic
19	C₄×C₄⋊C₄×C₅₃	—	2	1	no	—
20	C₄×C₄⋊C₄×C₅₃	C₄	1	2	yes	—
21	C₄³×C₅₃	C₂²	1	1	yes	characteristic
22	C₂²×#88322	—	2	2	no	—
23	C₂²×#88322	—	2	2	no	—
24	C₄⋊#88322	—	2	2	no	—
25	C₄⋊#88322	—	2	2	no	—
26	C₄⋊#88322	—	2	2	no	—
27	C₄⋊#88322	—	2	2	no	—
28	C₄⋊#88322	—	4	2	no	—
29	C₄⋊#88322	—	4	2	no	—
30	C₄⋊#88322	—	4	2	no	—
31	C₄⋊#88322	—	4	2	no	—
32	C₂²×Q₈×C₅₃	—	2	1	no	—
33	C₄⋊Q₈×C₅₃	—	2	1	no	—
34	C₄⋊Q₈×C₅₃	—	2	1	no	—
35	C₄⋊Q₈×C₅₃	D₄	1	2	yes	—
36	C₄⋊Q₈×C₅₃	Q₈	1	2	yes	—
37	C₄×Q₈×C₅₃	—	4	2	no	—
38	C₄×Q₈×C₅₃	—	4	2	no	—
39	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
40	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
41	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
42	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
43	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
44	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₅₃	—	2	2	no	—
46	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
49	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
50	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
51	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
52	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
53	#95×C₅₃	—	2	2	no	—
54	#95×C₅₃	—	2	2	no	—
55	C₂×Q₈×C₅₃	—	2	2	no	—
56	C₂×Q₈×C₅₃	—	2	2	no	—
57	C₂×Q₈×C₅₃	—	4	1	no	—
58	C₂×Q₈×C₅₃	—	4	1	no	—
59	C₂×Q₈×C₅₃	—	4	1	no	—
60	C₂×Q₈×C₅₃	—	4	1	no	—
61	C₂²×C₄×C₅₃	—	2	2	no	—
62	C₂²×C₄×C₅₃	—	2	2	no	—
63	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
70	C₄⋊C₄×C₅₃	—	2	2	no	—
71	C₄⋊C₄×C₅₃	—	2	2	no	—
72	C₄⋊C₄×C₅₃	—	2	2	no	—
73	C₄⋊C₄×C₅₃	—	4	2	no	—
74	C₄⋊C₄×C₅₃	—	4	4	no	—
75	C₄⋊C₄×C₅₃	—	4	1	no	—
76	C₄⋊C₄×C₅₃	—	4	1	no	—
77	C₄⋊C₄×C₅₃	—	4	1	no	—
78	C₄⋊C₄×C₅₃	—	4	1	no	—
79	C₄⋊C₄×C₅₃	C₄⋊C₄	1	4	yes	—
80	C₄²×C₅₃	—	2	2	no	—
81	C₄²×C₅₃	—	2	1	no	—
82	C₄²×C₅₃	—	2	1	no	—
83	C₄²×C₅₃	—	2	1	no	—
84	C₄²×C₅₃	—	2	1	no	—
85	C₄²×C₅₃	—	4	1	no	—
86	C₄²×C₅₃	—	4	4	no	—
87	C₄²×C₅₃	D₈	1	2	yes	—
88	C₄²×C₅₃	SD₁₆	1	2	yes	—
89	C₄²×C₅₃	SD₁₆	1	2	yes	—
90	C₄²×C₅₃	Q₁₆	1	2	yes	—
91	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
92	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
93	C₄²×C₅₃	#41	1	1	yes	characteristic
94	C₄²×C₅₃	#41	1	1	yes	characteristic
95	C₂×#88322	—	4	2	no	—
96	C₂×#88322	—	4	2	no	—
97	C₂×#88322	—	4	2	no	—
98	C₂×#88322	—	4	2	no	—
99	C₂×#88322	—	4	2	no	—
100	C₂×#88322	—	4	2	no	—
101	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	Q₈×C₅₃	—	4	2	no	—
103	Q₈×C₅₃	—	4	2	no	—
104	Q₈×C₅₃	—	4	2	no	—
105	Q₈×C₅₃	—	4	2	no	—
106	C₂×C₄×C₅₃	—	2	1	no	—
107	C₂×C₄×C₅₃	—	2	1	no	—
108	C₂×C₄×C₅₃	—	2	1	no	—
109	C₂×C₄×C₅₃	—	2	1	no	—
110	C₂×C₄×C₅₃	—	2	1	no	—
111	C₂×C₄×C₅₃	—	2	1	no	—
112	C₂×C₄×C₅₃	—	2	4	no	—
113	C₂×C₄×C₅₃	—	2	1	no	—
114	C₂×C₄×C₅₃	—	2	1	no	—
115	C₂×C₄×C₅₃	—	2	1	no	—
116	C₂×C₄×C₅₃	—	4	2	no	—
117	C₂×C₄×C₅₃	—	4	2	no	—
118	C₂×C₄×C₅₃	—	4	2	no	—
119	C₂×C₄×C₅₃	—	4	2	no	—
120	C₂×C₄×C₅₃	—	4	2	no	—
121	C₂×C₄×C₅₃	#101	1	2	yes	—
122	C₂×C₄×C₅₃	#102	1	2	yes	—
123	C₂×C₄×C₅₃	#102	1	2	yes	—
124	C₂×C₄×C₅₃	#103	1	2	yes	—
125	C₂×C₄×C₅₃	#103	1	2	yes	—
126	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
127	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
128	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
129	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
130	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
131	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
132	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
133	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
134	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
135	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
136	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
137	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
138	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
139	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
140	#88322	—	8	2	no	—
141	#88322	—	8	2	no	—
142	#8586	—	53	1	no	maximal, Sylow
143	C₂²×C₅₃	#380	1	1	yes	characteristic
144	C₂²×C₅₃	C₂×#101	1	1	yes	characteristic
145	C₂²×C₅₃	#413	1	1	yes	characteristic
146	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
147	C₂²×C₅₃	C₂×#103	1	1	yes	characteristic
148	C₂²×C₅₃	#417	1	1	yes	characteristic
149	C₂²×C₅₃	#417	1	1	yes	characteristic
150	C₄×C₅₃	—	2	1	no	—
151	C₄×C₅₃	—	2	1	no	—
152	C₄×C₅₃	—	2	2	no	—
153	C₄×C₅₃	—	2	2	no	—
154	C₄×C₅₃	—	4	4	no	—
155	C₄×C₅₃	—	4	1	no	—
156	C₄×C₅₃	—	8	2	no	—
157	C₄×C₅₃	#332	1	4	yes	complemented
158	C₄×C₅₃	#459	1	1	yes	characteristic
159	C₄×C₅₃	#460	1	1	yes	characteristic
160	C₄×C₅₃	#461	1	1	yes	characteristic
161	C₄×C₅₃	#462	1	1	yes	characteristic
162	C₄×C₅₃	#474	1	1	yes	characteristic
163	C₄×C₅₃	#475	1	1	yes	characteristic
164	C₄×C₅₃	#476	1	1	yes	characteristic
165	C₄×C₅₃	#477	1	1	yes	characteristic
166	C₄×C₄⋊Q₈	D₅₃	1	1	yes	characteristic
167	#1842	—	53	1	no	—
168	#1842	—	53	1	no	—
169	C₂×#332	—	53	2	no	—
170	#1296	—	53	2	no	—
171	C₂×C₅₃	C₂×#332	1	1	yes	characteristic
172	C₂×C₅₃	#1547	1	1	yes	characteristic
173	C₂×C₅₃	#1550	1	1	yes	characteristic
174	C₂×C₅₃	#1561	1	1	yes	characteristic
175	C₂×C₅₃	#1801	1	1	yes	characteristic
176	C₂×C₅₃	#1802	1	1	yes	characteristic
177	C₂×C₅₃	#1803	1	1	yes	characteristic
178	C₂×C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic
179	C₂×C₄×Q₈	—	2	1	no	—
180	C₂×C₄⋊C₈	—	53	2	no	—
181	C₂×C₄⋊C₈	—	53	2	no	—
182	C₂×C₄⋊C₈	—	106	1	no	—
183	C₂×C₄⋊C₈	—	106	1	no	—
184	C₂×C₄⋊C₈	—	106	1	no	—
185	C₂×C₄⋊C₈	—	106	1	no	—
186	#391	—	2	1	no	—
187	#389	—	2	1	no	—
188	#389	Dic₅₃	1	2	yes	—
189	#384	C₂×D₅₃	1	1	yes	characteristic
190	C₄×C₄⋊C₄	—	2	1	no	—
191	C₄×C₄⋊C₄	Dic₅₃	1	2	yes	—
192	C₄³	C₂×D₅₃	1	1	yes	characteristic
193	#332	—	106	4	no	—
194	C₅₃	#8586	1	1	yes	characteristic, complemented, Sylow
195	C₂²×Q₈	—	2	1	no	—
196	C₂²×C₈	—	106	2	no	—
197	C₂²×C₈	—	106	2	no	—
198	C₄⋊Q₈	—	2	1	no	—
199	C₄⋊Q₈	—	2	1	no	—
200	C₄⋊Q₈	Dic₁₀₆	1	2	yes	—
201	C₄⋊Q₈	D₂₁₂	1	2	yes	—
202	C₄×Q₈	—	4	2	no	—
203	C₄×Q₈	—	4	2	no	—
204	C₂×C₄⋊C₄	—	2	1	no	—
205	C₂×C₄⋊C₄	—	2	1	no	—
206	C₂×C₄⋊C₄	—	2	1	no	—
207	C₂×C₄⋊C₄	—	2	2	no	—
208	C₂×C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
209	C₂×C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
210	C₂×C₄²	—	2	2	no	—
211	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
212	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
213	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
214	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
215	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
216	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
217	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
218	C₄⋊C₈	—	106	2	no	—
219	C₄⋊C₈	—	106	2	no	—
220	C₄⋊C₈	—	106	2	no	—
221	C₄⋊C₈	—	106	2	no	—
222	C₄⋊C₈	—	212	2	no	—
223	C₄⋊C₈	—	212	2	no	—
224	C₄⋊C₈	—	212	2	no	—
225	C₄⋊C₈	—	212	2	no	—
226	#95	—	2	2	no	—
227	#95	—	2	2	no	—
228	C₂×Q₈	—	2	2	no	—
229	C₂×Q₈	—	2	2	no	—
230	C₂×Q₈	—	4	1	no	—
231	C₂×Q₈	—	4	1	no	—
232	C₂×Q₈	—	4	1	no	—
233	C₂×Q₈	—	4	1	no	—
234	C₂²×C₄	—	2	2	no	—
235	C₂²×C₄	—	2	2	no	—
236	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
237	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
238	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
239	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
240	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
241	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
242	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
243	C₂×C₈	—	212	2	no	—
244	C₂×C₈	—	212	2	no	—
245	C₂×C₈	—	212	2	no	—
246	C₂×C₈	—	212	2	no	—
247	C₂×C₈	—	212	2	no	—
248	C₂×C₈	—	212	2	no	—
249	C₄⋊C₄	—	2	2	no	—
250	C₄⋊C₄	—	2	2	no	—
251	C₄⋊C₄	—	2	2	no	—
252	C₄⋊C₄	—	4	2	no	—
253	C₄⋊C₄	—	4	4	no	—
254	C₄⋊C₄	—	4	1	no	—
255	C₄⋊C₄	—	4	1	no	—
256	C₄⋊C₄	—	4	1	no	—
257	C₄⋊C₄	—	4	1	no	—
258	C₄⋊C₄	C₄⋊Dic₅₃	1	4	yes	—
259	C₄²	—	2	2	no	—
260	C₄²	—	2	1	no	—
261	C₄²	—	2	1	no	—
262	C₄²	—	2	1	no	—
263	C₄²	—	2	1	no	—
264	C₄²	—	4	1	no	—
265	C₄²	—	4	4	no	—
266	C₄²	#11719922	1	2	yes	—
267	C₄²	#11719923	1	2	yes	—
268	C₄²	#11719924	1	2	yes	—
269	C₄²	#11719925	1	2	yes	—
270	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
271	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
272	C₄²	#11719945	1	1	yes	characteristic
273	C₄²	#11719945	1	1	yes	characteristic
274	C₂³	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
275	Q₈	—	4	2	no	—
276	Q₈	—	4	2	no	—
277	Q₈	—	4	2	no	—
278	Q₈	—	4	2	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	4	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	#13133136	1	2	yes	—
295	C₂×C₄	#13133138	1	2	yes	—
296	C₂×C₄	#13133138	1	2	yes	—
297	C₂×C₄	#13133141	1	2	yes	—
298	C₂×C₄	#13133141	1	2	yes	—
299	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
300	C₂×C₄	#13133176	1	1	yes	characteristic
301	C₂×C₄	#13133191	1	1	yes	characteristic
302	C₂×C₄	#13133191	1	1	yes	characteristic
303	C₂×C₄	#13133196	1	1	yes	characteristic
304	C₂×C₄	#13133196	1	1	yes	characteristic
305	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
306	C₂×C₄	C₂×#11719924	1	1	yes	characteristic
307	C₂×C₄	C₂×#11719923	1	1	yes	characteristic
308	C₂×C₄	C₂×#11719925	1	1	yes	characteristic
309	C₂×C₄	#13133242	1	1	yes	characteristic
310	C₂×C₄	#13133242	1	1	yes	characteristic
311	C₂×C₄	#13133247	1	1	yes	characteristic
312	C₂×C₄	#13133247	1	1	yes	characteristic
313	C₈	—	424	2	no	—
314	C₈	—	424	2	no	—
315	C₂²	#16810483	1	1	yes	characteristic
316	C₂²	C₂×#13133136	1	1	yes	characteristic
317	C₂²	#16810607	1	1	yes	characteristic
318	C₂²	C₂×#13133138	1	1	yes	characteristic
319	C₂²	C₂×#13133141	1	1	yes	characteristic
320	C₂²	#16810624	1	1	yes	characteristic
321	C₂²	#16810624	1	1	yes	characteristic
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	2	2	no	—
325	C₄	—	2	2	no	—
326	C₄	—	4	4	no	—
327	C₄	—	4	1	no	—
328	C₄	—	8	2	no	—
329	C₄	#16810369	1	4	yes	complemented
330	C₄	#16810818	1	1	yes	characteristic
331	C₄	#16810823	1	1	yes	characteristic
332	C₄	#16810828	1	1	yes	characteristic
333	C₄	#16810833	1	1	yes	characteristic
334	C₄	#16810903	1	1	yes	characteristic
335	C₄	#16810908	1	1	yes	characteristic
336	C₄	#16810913	1	1	yes	characteristic
337	C₄	#16810918	1	1	yes	characteristic
338	C₂	C₂×#16810369	1	1	yes	characteristic
339	C₂	#20273392	1	1	yes	characteristic
340	C₂	#20273405	1	1	yes	characteristic
341	C₂	#20273453	1	1	yes	characteristic
342	C₂	#20274711	1	1	yes	characteristic
343	C₂	#20274715	1	1	yes	characteristic
344	C₂	#20274720	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₅₃
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₄×C₄⋊Q₈×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₄×C₄⋊Q₈	D₅₃
hypocenter	C₅₃	#8586
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
157. C₄×C₅₃	#332	semidirect
194. C₅₃	#8586	semidirect
329. C₄	#16810369	semidirect

Maximal subgroups

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

subgroup	index
#20274867	2
C₂×#16810369	2
#20272598	2
C₄×C₄⋊Q₈×C₅₃	2
#8586	53