Subgroups — #27289952

Order 13568.

794 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	C₂²×C₄⋊#88322	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#11719917	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810387	C₂	1	4	yes	maximal
4	C₂×C₅₃×#423	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11719917	C₂²	1	2	yes	—
6	C₂³×#88322	C₂²	1	2	yes	—
7	C₂×C₄⋊#88322	—	2	2	no	—
8	C₂×C₄⋊#88322	—	2	2	no	—
9	C₂×C₄⋊#88322	C₄	1	2	yes	—
10	C₂×C₄⋊#88322	C₄	1	2	yes	—
11	C₄×#11719917	—	2	2	no	—
12	C₄×#11719917	C₄	1	4	yes	—
13	C₂×#13133128	—	2	1	no	—
14	C₂×C₄×#88322	—	2	1	no	—
15	#16810387	—	2	8	no	—
16	C₂²×M₁₆×C₅₃	C₂²	1	2	yes	—
17	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
18	C₅₃×#423	—	2	2	no	—
19	C₅₃×#423	C₄	1	4	yes	—
20	C₂×C₄⋊C₈×C₅₃	—	2	1	no	—
21	C₂×C₄⋊C₈×C₅₃	—	2	1	no	—
22	C₂×#11719917	—	2	2	no	—
23	C₂×#11719917	—	2	2	no	—
24	C₂×#11719917	—	2	2	no	—
25	C₂×#11719917	C₂×C₄	1	4	yes	—
26	C₂×#11719917	D₄	1	4	yes	—
27	C₂×#11719917	Q₈	1	4	yes	—
28	C₂²×#88322	—	2	2	no	—
29	C₂²×#88322	—	2	4	no	—
30	C₂²×#88322	—	2	2	no	—
31	C₂²×#88322	—	2	2	no	—
32	C₂²×#88322	—	2	4	no	—
33	C₂²×#88322	C₂×C₄	1	4	yes	—
34	C₄⋊#88322	—	2	4	no	—
35	C₄⋊#88322	—	2	4	no	—
36	#13133128	—	2	4	no	—
37	C₄×#88322	—	2	4	no	—
38	C₂³×C₄×C₅₃	Q₈	1	1	yes	characteristic
39	C₂³×C₄×C₅₃	Q₈	1	1	yes	characteristic
40	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
41	C₂×M₁₆×C₅₃	—	2	2	no	—
42	C₂×M₁₆×C₅₃	—	2	4	no	—
43	C₂×M₁₆×C₅₃	—	2	4	no	—
44	C₂×M₁₆×C₅₃	C₂×C₄	1	4	yes	—
45	C₂²×C₈×C₅₃	—	2	2	no	—
46	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
47	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
50	C₂×C₄²×C₅₃	D₄	1	2	yes	—
51	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
52	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
53	C₂×C₄²×C₅₃	D₄	1	2	yes	—
54	C₄⋊C₈×C₅₃	—	2	4	no	—
55	C₄⋊C₈×C₅₃	—	2	4	no	—
56	C₂⁴×C₅₃	C₂×Q₈	1	1	yes	characteristic
57	C₂²×C₄×C₅₃	—	2	4	no	—
58	C₂²×C₄×C₅₃	—	4	2	no	—
59	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
63	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
64	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
65	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
66	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
75	C₂²×C₄×C₅₃	#41	1	2	yes	—
76	C₂²×C₄×C₅₃	#41	1	2	yes	—
77	M₁₆×C₅₃	—	2	8	no	—
78	M₁₆×C₅₃	—	4	2	no	—
79	M₁₆×C₅₃	—	4	2	no	—
80	C₂×C₈×C₅₃	—	2	4	no	—
81	C₂×C₈×C₅₃	—	4	4	no	—
82	C₄²×C₅₃	C₄²	1	2	yes	—
83	C₄²×C₅₃	C₄²	1	2	yes	—
84	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
85	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
86	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
87	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
88	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
89	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
90	#11719917	—	2	4	no	—
91	#11719917	—	2	4	no	—
92	#11719917	—	2	4	no	—
93	#11719917	C₄⋊C₄	1	8	yes	—
94	C₂×#88322	—	2	4	no	—
95	C₂×#88322	—	2	4	no	—
96	C₂×#88322	—	2	4	no	—
97	C₂×#88322	—	2	4	no	—
98	C₂×#88322	—	2	8	no	—
99	C₂×#88322	—	4	4	no	—
100	C₂×#88322	—	4	2	no	—
101	C₂×#88322	—	4	2	no	—
102	C₂³×C₅₃	—	2	2	no	—
103	C₂³×C₅₃	—	2	2	no	—
104	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
105	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
106	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂³×C₅₃	#125	1	2	yes	—
108	C₂³×C₅₃	C₄⋊Q₈	1	2	yes	—
109	C₂×C₄×C₅₃	—	2	2	no	—
110	C₂×C₄×C₅₃	—	2	2	no	—
111	C₂×C₄×C₅₃	—	2	2	no	—
112	C₂×C₄×C₅₃	—	2	4	no	—
113	C₂×C₄×C₅₃	—	2	4	no	—
114	C₂×C₄×C₅₃	—	4	2	no	—
115	C₂×C₄×C₅₃	—	4	2	no	—
116	C₂×C₄×C₅₃	#95	1	2	yes	—
117	C₂×C₄×C₅₃	#95	1	2	yes	—
118	C₂×C₄×C₅₃	#95	1	2	yes	—
119	C₂×C₄×C₅₃	#95	1	2	yes	—
120	C₂×C₄×C₅₃	#108	1	4	yes	—
121	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
122	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
126	C₂×C₄×C₅₃	#117	1	2	yes	—
127	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
128	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
129	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
130	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
131	C₂×C₄×C₅₃	#121	1	2	yes	—
132	C₂×C₄×C₅₃	#121	1	2	yes	—
133	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
134	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
135	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
136	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
137	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
138	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
139	C₈×C₅₃	—	4	4	no	—
140	#88322	—	2	8	no	—
141	#88322	—	4	4	no	—
142	#10853	—	53	1	no	maximal, Sylow
143	C₂²×C₅₃	—	2	4	no	—
144	C₂²×C₅₃	—	2	4	no	—
145	C₂²×C₅₃	—	2	1	no	—
146	C₂²×C₅₃	—	2	1	no	—
147	C₂²×C₅₃	—	2	1	no	—
148	C₂²×C₅₃	—	2	1	no	—
149	C₂²×C₅₃	C₂×#95	1	1	yes	characteristic
150	C₂²×C₅₃	#382	1	1	yes	characteristic
151	C₂²×C₅₃	#384	1	1	yes	characteristic
152	C₂²×C₅₃	#389	1	2	yes	—
153	C₂²×C₅₃	#389	1	2	yes	—
154	C₂²×C₅₃	C₂×#108	1	2	yes	—
155	C₂²×C₅₃	#430	1	2	yes	—
156	C₄×C₅₃	—	2	2	no	—
157	C₄×C₅₃	—	2	2	no	—
158	C₄×C₅₃	—	2	2	no	—
159	C₄×C₅₃	#341	1	4	yes	complemented
160	C₄×C₅₃	C₄×C₄⋊C₄	1	2	yes	—
161	C₄×C₅₃	#380	1	2	yes	—
162	C₄×C₅₃	#390	1	1	yes	characteristic
163	C₄×C₅₃	#390	1	1	yes	characteristic
164	C₄×C₅₃	#391	1	1	yes	characteristic
165	C₄×C₅₃	#391	1	1	yes	characteristic
166	C₂×#423	D₅₃	1	1	yes	characteristic
167	C₂²×C₄⋊C₈	—	53	1	no	—
168	C₂×C₄×M₁₆	—	53	1	no	—
169	C₂×#341	—	53	4	no	—
170	C₂×C₅₃	—	2	1	no	—
171	C₂×C₅₃	—	2	1	no	—
172	C₂×C₅₃	—	2	2	no	—
173	C₂×C₅₃	C₄⋊#95	1	1	yes	characteristic
174	C₂×C₅₃	C₂×#341	1	1	yes	characteristic
175	C₂×C₅₃	#1738	1	1	yes	characteristic
176	C₂×C₅₃	#1950	1	4	yes	—
177	C₂²×M₁₆	—	53	2	no	—
178	C₂²×M₁₆	C₂×D₅₃	1	2	yes	—
179	C₂³×C₈	—	53	2	no	—
180	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
181	#423	—	2	2	no	—
182	#423	Dic₅₃	1	4	yes	—
183	C₂×C₄⋊C₈	—	2	1	no	—
184	C₂×C₄⋊C₈	—	2	1	no	—
185	C₂×C₄⋊C₈	—	53	2	no	—
186	C₂×C₄⋊C₈	—	53	2	no	—
187	C₂×C₄⋊C₈	—	106	2	no	—
188	C₂×C₄⋊C₈	—	106	2	no	—
189	C₄×M₁₆	—	53	4	no	—
190	C₄×M₁₆	—	106	2	no	—
191	C₂×#97	—	106	1	no	—
192	C₂×C₄×C₈	—	106	1	no	—
193	#341	—	106	8	no	—
194	C₅₃	#10853	1	1	yes	characteristic, complemented, Sylow
195	C₂³×C₄	Dic₁₀₆	1	1	yes	characteristic
196	C₂³×C₄	Dic₁₀₆	1	1	yes	characteristic
197	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
198	C₂×M₁₆	—	2	2	no	—
199	C₂×M₁₆	—	2	4	no	—
200	C₂×M₁₆	—	2	4	no	—
201	C₂×M₁₆	—	53	4	no	—
202	C₂×M₁₆	—	53	4	no	—
203	C₂×M₁₆	—	53	4	no	—
204	C₂×M₁₆	—	106	2	no	—
205	C₂×M₁₆	—	106	2	no	—
206	C₂×M₁₆	—	106	2	no	—
207	C₂×M₁₆	C₂×Dic₅₃	1	4	yes	—
208	C₂²×C₈	—	2	2	no	—
209	C₂²×C₈	—	53	4	no	—
210	C₂²×C₈	—	106	2	no	—
211	C₂²×C₈	—	106	2	no	—
212	C₂²×C₈	—	106	4	no	—
213	C₂²×C₈	—	106	4	no	—
214	C₂²×C₈	—	106	2	no	—
215	C₂×C₄²	C₄×D₅₃	1	2	yes	—
216	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
217	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
218	C₂×C₄²	D₂₁₂	1	2	yes	—
219	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
220	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
221	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
222	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
223	C₄⋊C₈	—	2	4	no	—
224	C₄⋊C₈	—	2	4	no	—
225	C₄⋊C₈	—	106	4	no	—
226	C₄⋊C₈	—	106	4	no	—
227	#97	—	106	4	no	—
228	C₄×C₈	—	106	4	no	—
229	C₂⁴	C₂×Dic₁₀₆	1	1	yes	characteristic
230	C₂²×C₄	—	2	4	no	—
231	C₂²×C₄	—	4	2	no	—
232	C₂²×C₄	#11719919	1	1	yes	characteristic
233	C₂²×C₄	#11719919	1	1	yes	characteristic
234	C₂²×C₄	#11719919	1	1	yes	characteristic
235	C₂²×C₄	#11719919	1	2	yes	—
236	C₂²×C₄	#11719919	1	2	yes	—
237	C₂²×C₄	#11719919	1	1	yes	characteristic
238	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
239	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
240	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
241	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
242	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
243	C₂²×C₄	D₄×D₅₃	1	2	yes	—
244	C₂²×C₄	#11719947	1	2	yes	—
245	C₂²×C₄	Q₈×D₅₃	1	2	yes	—
246	C₂²×C₄	#11719949	1	2	yes	—
247	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
248	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
249	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
250	M₁₆	—	2	8	no	—
251	M₁₆	—	4	2	no	—
252	M₁₆	—	4	2	no	—
253	M₁₆	—	53	8	no	—
254	M₁₆	—	106	4	no	—
255	M₁₆	—	106	4	no	—
256	M₁₆	—	106	4	no	—
257	C₂×C₈	—	2	4	no	—
258	C₂×C₈	—	4	4	no	—
259	C₂×C₈	—	106	4	no	—
260	C₂×C₈	—	106	4	no	—
261	C₂×C₈	—	106	4	no	—
262	C₂×C₈	—	106	4	no	—
263	C₂×C₈	—	106	8	no	—
264	C₂×C₈	—	212	4	no	—
265	C₂×C₈	—	212	2	no	—
266	C₂×C₈	—	212	2	no	—
267	C₄²	C₄×Dic₅₃	1	2	yes	—
268	C₄²	C₄×Dic₅₃	1	2	yes	—
269	C₄²	#11719921	1	2	yes	—
270	C₄²	#11719921	1	2	yes	—
271	C₄²	#11719921	1	2	yes	—
272	C₄²	#11719921	1	2	yes	—
273	C₄²	C₂²⋊Dic₅₃	1	2	yes	—
274	C₄²	C₂²⋊Dic₅₃	1	2	yes	—
275	C₂³	—	2	2	no	—
276	C₂³	—	2	2	no	—
277	C₂³	C₂×#11719919	1	1	yes	characteristic
278	C₂³	C₂×#11719919	1	1	yes	characteristic
279	C₂³	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
280	C₂³	#13133208	1	2	yes	—
281	C₂³	#13133216	1	2	yes	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	4	no	—
286	C₂×C₄	—	2	4	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	#13133124	1	2	yes	—
290	C₂×C₄	#13133124	1	2	yes	—
291	C₂×C₄	#13133124	1	2	yes	—
292	C₂×C₄	#13133124	1	2	yes	—
293	C₂×C₄	#13133152	1	4	yes	—
294	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
295	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
296	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
297	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
298	C₂×C₄	C₄⋊C₄×D₅₃	1	2	yes	—
299	C₂×C₄	#13133175	1	2	yes	—
300	C₂×C₄	#13133179	1	2	yes	—
301	C₂×C₄	D₄×Dic₅₃	1	2	yes	—
302	C₂×C₄	#13133182	1	2	yes	—
303	C₂×C₄	Q₈×Dic₅₃	1	2	yes	—
304	C₂×C₄	#13133188	1	2	yes	—
305	C₂×C₄	#13133192	1	2	yes	—
306	C₂×C₄	#13133193	1	2	yes	—
307	C₂×C₄	#13133197	1	2	yes	—
308	C₂×C₄	#13133204	1	1	yes	characteristic
309	C₂×C₄	#13133206	1	1	yes	characteristic
310	C₂×C₄	#13133213	1	1	yes	characteristic
311	C₂×C₄	#13133215	1	1	yes	characteristic
312	C₈	—	4	4	no	—
313	C₈	—	106	8	no	—
314	C₈	—	212	4	no	—
315	C₂²	—	2	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	—	2	4	no	—
320	C₂²	—	2	4	no	—
321	C₂²	C₂×#13133124	1	1	yes	characteristic
322	C₂²	#16810488	1	1	yes	characteristic
323	C₂²	#16810492	1	1	yes	characteristic
324	C₂²	#16810519	1	2	yes	—
325	C₂²	#16810522	1	2	yes	—
326	C₂²	C₂×#13133152	1	2	yes	—
327	C₂²	#16810671	1	2	yes	—
328	C₄	—	2	2	no	—
329	C₄	—	2	2	no	—
330	C₄	—	2	2	no	—
331	C₄	#16810387	1	4	yes	complemented
332	C₄	C₄⋊C₄×Dic₅₃	1	2	yes	—
333	C₄	#16810480	1	2	yes	—
334	C₄	#16810524	1	1	yes	characteristic
335	C₄	#16810526	1	1	yes	characteristic
336	C₄	#16810528	1	1	yes	characteristic
337	C₄	#16810530	1	1	yes	characteristic
338	C₂	—	2	2	no	—
339	C₂	—	2	1	no	—
340	C₂	—	2	1	no	—
341	C₂	#20272944	1	1	yes	characteristic
342	C₂	C₂×#16810387	1	1	yes	characteristic
343	C₂	#20274441	1	1	yes	characteristic
344	C₂	#20275488	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₂×C₅₃×#423	C₂
socle	C₂³×C₅₃	C₂×C₄⋊C₄
prime radical	C₂⁴×C₅₃	C₂×Q₈
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	C₂×#423	D₅₃
hypocenter	C₅₃	#10853
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×#423

Complements

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

N	H ≅ G/N	type
159. C₄×C₅₃	#341	semidirect
194. C₅₃	#10853	semidirect
331. C₄	#16810387	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄⋊#88322	2
C₂×C₄×#11719917	2
C₂×#16810387	2
C₂×C₅₃×#423	2
#10853	53