Subgroups — #27276887

Order 13568.

516 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#13133217	C₂	1	1	yes	maximal, characteristic
2	#20274867	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810371	C₂	1	2	yes	maximal
4	#20272599	C₂	1	2	yes	maximal
5	C₅₃×#1842	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133217	C₂²	1	1	yes	characteristic
7	C₂×C₄×Dic₁₀₆	C₄	1	2	yes	—
8	C₂×C₄⋊#88322	—	2	1	no	—
9	C₂×C₄⋊#88322	—	2	1	no	—
10	C₂×C₄⋊#88322	C₂²	1	2	yes	—
11	#16810529	C₄	1	2	yes	—
12	#16810521	—	2	1	no	—
13	#16810521	—	2	1	no	—
14	#16810493	C₂²	1	1	yes	characteristic
15	C₄×C₄⋊Dic₅₃	—	2	1	no	—
16	C₄×C₄⋊Dic₅₃	—	2	1	no	—
17	#16810371	—	2	4	no	—
18	C₂×C₄⋊C₈×C₅₃	—	2	1	no	—
19	C₂×C₄⋊C₈×C₅₃	—	2	1	no	—
20	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
21	C₄³×C₅₃	C₂²	1	1	yes	characteristic
22	C₂²×Dic₁₀₆	C₂×C₄	1	2	yes	—
23	C₂²×#88322	—	2	2	no	—
24	#13133217	—	2	1	no	—
25	#13133217	—	2	1	no	—
26	#13133217	D₄	1	2	yes	—
27	#13133217	Q₈	1	2	yes	—
28	C₄×Dic₁₀₆	—	4	4	no	—
29	C₂×C₄⋊Dic₅₃	—	2	1	no	—
30	C₂×C₄⋊Dic₅₃	—	2	1	no	—
31	C₂×C₄⋊Dic₅₃	—	2	1	no	—
32	C₂×C₄⋊Dic₅₃	—	2	1	no	—
33	C₂×C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
34	C₂×#11719919	—	2	2	no	—
35	C₂×C₄×Dic₅₃	—	2	2	no	—
36	C₄⋊#88322	—	2	2	no	—
37	C₄⋊#88322	—	2	2	no	—
38	C₄⋊#88322	—	4	2	no	—
39	C₄⋊#88322	—	4	2	no	—
40	#13133124	—	2	2	no	—
41	#13133124	—	2	2	no	—
42	C₂²×C₈×C₅₃	—	2	2	no	—
43	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
48	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
49	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
50	C₄⋊C₈×C₅₃	—	2	2	no	—
51	C₄⋊C₈×C₅₃	—	2	2	no	—
52	C₄⋊C₈×C₅₃	—	4	2	no	—
53	C₄⋊C₈×C₅₃	—	4	2	no	—
54	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
61	C₂²×Dic₅₃	—	2	2	no	—
62	C₂²×Dic₅₃	—	2	2	no	—
63	C₂×Dic₁₀₆	—	2	2	no	—
64	C₂×Dic₁₀₆	—	4	2	no	—
65	C₂×Dic₁₀₆	—	4	2	no	—
66	C₂×Dic₁₀₆	C₄⋊C₄	1	4	yes	—
67	C₂×C₈×C₅₃	—	4	2	no	—
68	C₂×C₈×C₅₃	—	4	2	no	—
69	C₂×C₈×C₅₃	—	4	2	no	—
70	C₄²×C₅₃	—	2	2	no	—
71	C₄²×C₅₃	—	2	1	no	—
72	C₄²×C₅₃	—	2	1	no	—
73	C₄²×C₅₃	—	2	1	no	—
74	C₄²×C₅₃	—	2	1	no	—
75	C₄²×C₅₃	—	4	1	no	—
76	C₄²×C₅₃	SD₁₆	1	2	yes	—
77	C₄²×C₅₃	SD₁₆	1	2	yes	—
78	C₄²×C₅₃	Q₁₆	1	2	yes	—
79	C₄²×C₅₃	Q₁₆	1	2	yes	—
80	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
81	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
82	C₄²×C₅₃	#41	1	1	yes	characteristic
83	C₄²×C₅₃	#41	1	1	yes	characteristic
84	C₄⋊Dic₅₃	—	2	2	no	—
85	C₄⋊Dic₅₃	—	2	2	no	—
86	C₄⋊Dic₅₃	—	2	2	no	—
87	C₄⋊Dic₅₃	—	2	2	no	—
88	C₄⋊Dic₅₃	—	4	1	no	—
89	C₄⋊Dic₅₃	—	4	1	no	—
90	C₄⋊Dic₅₃	—	4	1	no	—
91	C₄⋊Dic₅₃	—	4	1	no	—
92	C₄⋊Dic₅₃	—	4	2	no	—
93	#11719919	—	4	4	no	—
94	C₄×Dic₅₃	—	4	4	no	—
95	C₂×#88322	—	4	2	no	—
96	C₂×#88322	—	4	2	no	—
97	C₂×#88322	—	4	2	no	—
98	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₅₃	—	2	1	no	—
100	C₂×C₄×C₅₃	—	2	1	no	—
101	C₂×C₄×C₅₃	—	2	1	no	—
102	C₂×C₄×C₅₃	—	2	1	no	—
103	C₂×C₄×C₅₃	—	2	1	no	—
104	C₂×C₄×C₅₃	—	2	1	no	—
105	C₂×C₄×C₅₃	—	2	1	no	—
106	C₂×C₄×C₅₃	—	2	1	no	—
107	C₂×C₄×C₅₃	—	2	1	no	—
108	C₂×C₄×C₅₃	#100	1	2	yes	—
109	C₂×C₄×C₅₃	#103	1	2	yes	—
110	C₂×C₄×C₅₃	#103	1	2	yes	—
111	C₂×C₄×C₅₃	#103	1	2	yes	—
112	C₂×C₄×C₅₃	#103	1	2	yes	—
113	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
115	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
116	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
117	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
118	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
119	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
120	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
121	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
122	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
123	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
124	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
125	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
126	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
127	C₂×Dic₅₃	—	2	4	no	—
128	C₂×Dic₅₃	—	4	2	no	—
129	C₂×Dic₅₃	—	4	2	no	—
130	C₂×Dic₅₃	—	4	2	no	—
131	C₂×Dic₅₃	—	4	2	no	—
132	C₂×Dic₅₃	—	4	2	no	—
133	Dic₁₀₆	—	4	4	no	—
134	Dic₁₀₆	—	4	4	no	—
135	C₈×C₅₃	—	8	2	no	—
136	#88322	—	8	2	no	—
137	#8587	—	53	1	no	maximal, Sylow
138	C₂²×C₅₃	#380	1	1	yes	characteristic
139	C₂²×C₅₃	C₂×#100	1	1	yes	characteristic
140	C₂²×C₅₃	#413	1	1	yes	characteristic
141	C₂²×C₅₃	C₂×#103	1	1	yes	characteristic
142	C₂²×C₅₃	C₂×#103	1	1	yes	characteristic
143	C₂²×C₅₃	#417	1	1	yes	characteristic
144	C₂²×C₅₃	#417	1	1	yes	characteristic
145	C₄×C₅₃	—	2	1	no	—
146	C₄×C₅₃	—	2	1	no	—
147	C₄×C₅₃	—	2	2	no	—
148	C₄×C₅₃	—	2	2	no	—
149	C₄×C₅₃	—	4	1	no	—
150	C₄×C₅₃	#333	1	4	yes	complemented
151	C₄×C₅₃	#460	1	1	yes	characteristic
152	C₄×C₅₃	#460	1	1	yes	characteristic
153	C₄×C₅₃	#462	1	1	yes	characteristic
154	C₄×C₅₃	#462	1	1	yes	characteristic
155	C₄×C₅₃	#475	1	1	yes	characteristic
156	C₄×C₅₃	#475	1	1	yes	characteristic
157	C₄×C₅₃	#477	1	1	yes	characteristic
158	C₄×C₅₃	#477	1	1	yes	characteristic
159	Dic₅₃	—	4	4	no	—
160	Dic₅₃	—	8	2	no	—
161	C₄×C₄⋊Q₈	—	53	1	no	—
162	#1842	—	53	1	no	—
163	#1842	D₅₃	1	1	yes	characteristic
164	C₂×#333	—	53	2	no	—
165	#1296	—	53	2	no	—
166	C₂×C₅₃	C₂×#333	1	1	yes	characteristic
167	C₂×C₅₃	#1550	1	1	yes	characteristic
168	C₂×C₅₃	#1550	1	1	yes	characteristic
169	C₂×C₅₃	#1560	1	1	yes	characteristic
170	C₂×C₅₃	#1800	1	1	yes	characteristic
171	C₂×C₅₃	#1803	1	1	yes	characteristic
172	C₂×C₅₃	#1803	1	1	yes	characteristic
173	C₂×C₄⋊Q₈	—	53	1	no	—
174	C₂×C₄×Q₈	—	53	2	no	—
175	C₂×C₄⋊C₈	—	2	1	no	—
176	C₂×C₄⋊C₈	—	2	1	no	—
177	C₂×C₄⋊C₈	—	53	2	no	—
178	C₂×C₄⋊C₈	—	106	1	no	—
179	C₂×C₄⋊C₈	—	106	1	no	—
180	C₂×C₄⋊C₈	C₂×D₅₃	1	2	yes	—
181	#391	—	53	2	no	—
182	#389	—	106	1	no	—
183	#389	—	106	1	no	—
184	#384	—	53	1	no	—
185	C₄×C₄⋊C₄	—	106	1	no	—
186	C₄×C₄⋊C₄	—	106	1	no	—
187	C₄³	C₂×D₅₃	1	1	yes	characteristic
188	#333	—	106	4	no	—
189	C₅₃	#8587	1	1	yes	characteristic, complemented, Sylow
190	C₂²×Q₈	—	53	2	no	—
191	C₂²×C₈	—	2	2	no	—
192	C₂²×C₈	—	106	2	no	—
193	C₄⋊Q₈	—	53	2	no	—
194	C₄⋊Q₈	—	53	2	no	—
195	C₄⋊Q₈	—	106	1	no	—
196	C₄⋊Q₈	—	106	1	no	—
197	C₄×Q₈	—	212	4	no	—
198	C₂×C₄⋊C₄	—	53	2	no	—
199	C₂×C₄⋊C₄	—	106	1	no	—
200	C₂×C₄⋊C₄	—	106	1	no	—
201	C₂×C₄⋊C₄	—	106	1	no	—
202	C₂×C₄⋊C₄	—	106	1	no	—
203	C₂×C₄⋊C₄	—	106	2	no	—
204	C₂×C₄²	—	106	2	no	—
205	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
206	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
207	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
208	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
209	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
210	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
211	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
212	C₄⋊C₈	—	2	2	no	—
213	C₄⋊C₈	—	2	2	no	—
214	C₄⋊C₈	—	4	2	no	—
215	C₄⋊C₈	—	4	2	no	—
216	C₄⋊C₈	—	106	2	no	—
217	C₄⋊C₈	—	106	2	no	—
218	C₄⋊C₈	—	212	2	no	—
219	C₄⋊C₈	—	212	2	no	—
220	#95	—	106	2	no	—
221	#95	—	106	2	no	—
222	C₂×Q₈	—	53	4	no	—
223	C₂×Q₈	—	106	2	no	—
224	C₂×Q₈	—	212	2	no	—
225	C₂×Q₈	—	212	2	no	—
226	C₂²×C₄	—	106	2	no	—
227	C₂²×C₄	—	106	2	no	—
228	C₂²×C₄	#11719921	1	1	yes	characteristic
229	C₂²×C₄	#11719921	1	1	yes	characteristic
230	C₂²×C₄	#11719921	1	1	yes	characteristic
231	C₂²×C₄	#11719921	1	1	yes	characteristic
232	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
233	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
234	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
235	C₂×C₈	—	4	2	no	—
236	C₂×C₈	—	4	2	no	—
237	C₂×C₈	—	4	2	no	—
238	C₂×C₈	—	212	2	no	—
239	C₂×C₈	—	212	2	no	—
240	C₂×C₈	—	212	2	no	—
241	C₄⋊C₄	—	106	2	no	—
242	C₄⋊C₄	—	106	2	no	—
243	C₄⋊C₄	—	106	2	no	—
244	C₄⋊C₄	—	106	2	no	—
245	C₄⋊C₄	—	212	2	no	—
246	C₄⋊C₄	—	212	4	no	—
247	C₄⋊C₄	—	212	1	no	—
248	C₄⋊C₄	—	212	1	no	—
249	C₄⋊C₄	—	212	1	no	—
250	C₄⋊C₄	—	212	1	no	—
251	C₄²	—	2	2	no	—
252	C₄²	—	2	1	no	—
253	C₄²	—	2	1	no	—
254	C₄²	—	2	1	no	—
255	C₄²	—	2	1	no	—
256	C₄²	—	4	1	no	—
257	C₄²	—	212	4	no	—
258	C₄²	#11719913	1	2	yes	—
259	C₄²	Dic₂₁₂	1	2	yes	—
260	C₄²	#11719923	1	2	yes	—
261	C₄²	#11719925	1	2	yes	—
262	C₄²	D₄×D₅₃	1	1	yes	characteristic
263	C₄²	#11719947	1	1	yes	characteristic
264	C₄²	Q₈×D₅₃	1	1	yes	characteristic
265	C₄²	#11719949	1	1	yes	characteristic
266	C₂³	C₂×#11719921	1	1	yes	characteristic
267	Q₈	—	212	4	no	—
268	Q₈	—	212	4	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	106	4	no	—
279	C₂×C₄	—	212	2	no	—
280	C₂×C₄	—	212	2	no	—
281	C₂×C₄	—	212	2	no	—
282	C₂×C₄	—	212	2	no	—
283	C₂×C₄	—	212	2	no	—
284	C₂×C₄	#13133133	1	2	yes	—
285	C₂×C₄	#13133140	1	2	yes	—
286	C₂×C₄	#13133140	1	2	yes	—
287	C₂×C₄	#13133142	1	2	yes	—
288	C₂×C₄	#13133142	1	2	yes	—
289	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
290	C₂×C₄	#13133175	1	1	yes	characteristic
291	C₂×C₄	#13133188	1	1	yes	characteristic
292	C₂×C₄	#13133192	1	1	yes	characteristic
293	C₂×C₄	#13133193	1	1	yes	characteristic
294	C₂×C₄	#13133197	1	1	yes	characteristic
295	C₂×C₄	C₂×#11719923	1	1	yes	characteristic
296	C₂×C₄	C₂×#11719913	1	1	yes	characteristic
297	C₂×C₄	C₂×#11719925	1	1	yes	characteristic
298	C₂×C₄	C₂×Dic₂₁₂	1	1	yes	characteristic
299	C₂×C₄	#13133243	1	1	yes	characteristic
300	C₂×C₄	#13133245	1	1	yes	characteristic
301	C₂×C₄	#13133248	1	1	yes	characteristic
302	C₂×C₄	#13133250	1	1	yes	characteristic
303	C₈	—	8	2	no	—
304	C₈	—	424	2	no	—
305	C₂²	#16810480	1	1	yes	characteristic
306	C₂²	C₂×#13133133	1	1	yes	characteristic
307	C₂²	#16810605	1	1	yes	characteristic
308	C₂²	C₂×#13133140	1	1	yes	characteristic
309	C₂²	C₂×#13133142	1	1	yes	characteristic
310	C₂²	#16810622	1	1	yes	characteristic
311	C₂²	#16810625	1	1	yes	characteristic
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	—	2	2	no	—
315	C₄	—	2	2	no	—
316	C₄	—	4	1	no	—
317	C₄	—	424	2	no	—
318	C₄	—	212	4	no	—
319	C₄	#16810371	1	4	yes	complemented
320	C₄	#16810820	1	1	yes	characteristic
321	C₄	#16810824	1	1	yes	characteristic
322	C₄	#16810830	1	1	yes	characteristic
323	C₄	#16810834	1	1	yes	characteristic
324	C₄	#16810905	1	1	yes	characteristic
325	C₄	#16810909	1	1	yes	characteristic
326	C₄	#16810915	1	1	yes	characteristic
327	C₄	#16810919	1	1	yes	characteristic
328	C₂	C₂×#16810371	1	1	yes	characteristic
329	C₂	#20273403	1	1	yes	characteristic
330	C₂	#20273406	1	1	yes	characteristic
331	C₂	#20273448	1	1	yes	characteristic
332	C₂	#20274704	1	1	yes	characteristic
333	C₂	#20274717	1	1	yes	characteristic
334	C₂	#20274721	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#11719921
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1842	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1842	D₅₃
hypocenter	C₅₃	#8587
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1842

Complements

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

N	H ≅ G/N	type
150. C₄×C₅₃	#333	semidirect
189. C₅₃	#8587	semidirect
319. C₄	#16810371	semidirect

Maximal subgroups

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

subgroup	index
C₄×#13133217	2
#20274867	2
C₂×#16810371	2
#20272599	2
C₅₃×#1842	2
#8587	53