Subgroups — #27354236

Order 13568.

638 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20284724	C₂	1	1	yes	maximal, characteristic
2	#20283809	C₂	1	1	yes	maximal, characteristic, complemented
3	#20283738	C₂	1	1	yes	maximal, characteristic
4	#20273574	C₂	1	2	yes	maximal, complemented
5	#20273553	C₂	1	2	yes	maximal, complemented
6	#20273529	C₂	1	2	yes	maximal, complemented
7	C₈×#11719913	C₂	1	2	yes	maximal, complemented
8	#20273495	C₂	1	1	yes	maximal, characteristic, complemented
9	#20273159	C₂	1	1	yes	maximal, characteristic
10	#20273137	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#1453	C₂	1	1	yes	maximal, characteristic, complemented
12	#16811180	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₂₁₂	C₂²	1	1	yes	characteristic
14	#16810954	C₄	1	2	yes	—
15	#16810914	C₄	1	2	yes	—
16	#16810829	C₄	1	2	yes	—
17	#16810776	C₄	1	2	yes	—
18	#16810753	C₂²	1	2	yes	complemented
19	#16810752	—	2	2	no	—
20	C₈×Dic₁₀₆	C₂²	1	2	yes	complemented
21	C₄×#11719913	C₂²	1	2	yes	—
22	C₄×#11719913	C₂²	1	2	yes	—
23	C₈×D₂₁₂	C₂²	1	2	yes	—
24	C₈×D₂₁₂	C₂²	1	2	yes	—
25	#16810688	—	2	2	no	—
26	#16810681	—	2	2	no	—
27	#16810680	C₂²	1	1	yes	characteristic
28	#16810676	C₂²	1	1	yes	characteristic
29	C₂²⋊C₈×D₅₃	—	2	2	no	—
30	#16810376	C₂²	1	2	yes	complemented
31	#16810374	C₂²	1	2	yes	complemented
32	#16810355	C₂²	1	2	yes	complemented
33	#16810355	C₂²	1	2	yes	complemented
34	#16810352	C₂²	1	2	yes	—
35	#16810352	C₂²	1	2	yes	—
36	C₅₃×#431	C₂²	1	1	yes	characteristic
37	C₅₃×#431	C₂²	1	1	yes	characteristic
38	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
39	C₅₃×#322	C₂²	1	1	yes	characteristic, complemented
40	C₅₃×#322	C₂²	1	1	yes	characteristic, complemented
41	C₅₃×#322	C₂²	1	1	yes	characteristic, complemented
42	C₈²×C₅₃	C₂²	1	1	yes	characteristic, complemented
43	C₂²×D₂₁₂	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₄×D₅₃	—	2	1	no	—
45	C₂×#11719913	C₂×C₄	1	2	yes	—
46	C₂×#11719913	C₂×C₄	1	2	yes	—
47	C₂×C₈×D₅₃	—	2	2	no	—
48	C₂×C₈×D₅₃	—	2	2	no	—
49	#13133217	C₂×C₄	1	1	yes	characteristic
50	#13133217	C₂×C₄	1	1	yes	characteristic
51	#13133209	—	2	1	no	—
52	#13133196	—	2	1	no	—
53	#13133196	C₂×C₄	1	2	yes	—
54	C₄×Dic₁₀₆	—	2	1	no	—
55	C₄×Dic₁₀₆	C₂³	1	2	yes	—
56	C₄×D₂₁₂	—	2	1	no	—
57	C₄×D₂₁₂	—	2	1	no	—
58	C₄×D₂₁₂	C₂×C₄	1	2	yes	—
59	C₄×D₂₁₂	C₂³	1	2	yes	—
60	#13133176	—	2	1	no	—
61	C₂×C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
62	C₂×#11719921	—	2	1	no	—
63	#13133149	C₂×C₄	1	2	yes	—
64	#13133149	C₂×C₄	1	2	yes	—
65	C₄⋊#88322	C₂³	1	2	yes	—
66	C₄⋊#88322	C₂³	1	2	yes	—
67	#13133146	—	2	2	no	—
68	#13133146	—	2	2	no	—
69	#13133142	C₂×C₄	1	2	yes	—
70	#13133142	C₂×C₄	1	2	yes	—
71	#13133139	C₂×C₄	1	2	yes	—
72	#13133139	C₂×C₄	1	2	yes	—
73	C₂²⋊#88322	—	2	1	no	—
74	C₂²⋊#88322	—	2	1	no	—
75	#13133129	—	2	2	no	—
76	#13133129	—	2	2	no	—
77	#13133128	—	2	1	no	—
78	#13133127	—	2	2	no	—
79	C₄×#88322	—	2	1	no	—
80	C₈×Dic₅₃	—	2	2	no	—
81	C₂×M₁₆×C₅₃	—	2	1	no	—
82	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
83	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
84	C₄⋊C₈×C₅₃	D₄	1	2	yes	—
85	C₂²⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
86	C₂²⋊C₈×C₅₃	D₄	1	2	yes	—
87	#97×C₅₃	—	2	1	no	—
88	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
89	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
90	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
91	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
92	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
93	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
94	C₂³×D₅₃	—	2	1	no	—
95	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
96	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
97	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
98	C₂²×Dic₅₃	—	2	1	no	—
99	C₂×D₂₁₂	—	2	1	no	—
100	C₂×D₂₁₂	—	2	4	no	—
101	C₂×D₂₁₂	C₂²×C₄	1	2	yes	—
102	C₂×C₄×D₅₃	—	2	2	no	—
103	C₂×C₄×D₅₃	—	2	2	no	—
104	C₂×C₄×D₅₃	—	8	2	no	—
105	C₂×Dic₁₀₆	—	2	1	no	—
106	C₂×Dic₁₀₆	C₂²×C₄	1	2	yes	—
107	M₁₆×C₅₃	—	4	2	no	—
108	C₂×C₈×C₅₃	—	2	1	no	—
109	C₂×C₈×C₅₃	—	2	1	no	—
110	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
111	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
112	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
113	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
114	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
115	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₅₃	#41	1	1	yes	characteristic
117	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
118	C₂²⋊Dic₅₃	—	2	2	no	—
119	#11719921	—	2	2	no	—
120	#11719921	—	2	2	no	—
121	C₄⋊Dic₅₃	—	2	1	no	—
122	C₄⋊Dic₅₃	—	2	1	no	—
123	C₄⋊Dic₅₃	—	2	1	no	—
124	C₄⋊Dic₅₃	—	2	1	no	—
125	C₄⋊Dic₅₃	C₂²×C₄	1	2	yes	—
126	C₄⋊Dic₅₃	C₂²×C₄	1	2	yes	—
127	#11719919	—	2	2	no	—
128	#11719919	—	2	2	no	—
129	C₄×Dic₅₃	—	2	2	no	—
130	C₂×#88322	—	2	2	no	—
131	C₂×#88322	—	2	2	no	—
132	#11719913	—	2	4	no	—
133	#11719913	—	2	4	no	—
134	C₈×D₅₃	—	8	2	no	—
135	C₈×D₅₃	—	8	2	no	—
136	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
137	C₂²×D₅₃	—	2	2	no	—
138	C₂²×D₅₃	—	2	4	no	—
139	C₂²×D₅₃	—	8	1	no	—
140	C₂²×D₅₃	—	8	1	no	—
141	C₂×C₄×C₅₃	—	2	2	no	—
142	C₂×C₄×C₅₃	—	2	2	no	—
143	C₂×C₄×C₅₃	—	4	1	no	—
144	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
145	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
146	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
147	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
148	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
149	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
150	C₂×Dic₅₃	—	2	2	no	—
151	C₂×Dic₅₃	—	2	2	no	—
152	C₂×Dic₅₃	—	2	2	no	—
153	C₂×Dic₅₃	—	8	1	no	—
154	D₂₁₂	—	2	4	no	—
155	D₂₁₂	—	2	2	no	—
156	D₂₁₂	#131	1	4	yes	—
157	C₄×D₅₃	—	4	2	no	—
158	C₄×D₅₃	—	4	2	no	—
159	C₄×D₅₃	—	8	2	no	—
160	Dic₁₀₆	—	2	2	no	—
161	Dic₁₀₆	—	2	2	no	—
162	Dic₁₀₆	—	8	1	no	—
163	C₈×C₅₃	—	2	2	no	—
164	C₈×C₅₃	—	2	2	no	—
165	C₈×C₅₃	—	2	2	no	—
166	C₈×C₅₃	—	2	2	no	—
167	C₈×C₅₃	—	2	2	no	—
168	C₈×C₅₃	—	4	1	no	—
169	#88322	—	4	2	no	—
170	#88322	—	4	2	no	—
171	#17924	—	53	1	no	maximal, Sylow
172	C₂²×C₅₃	—	2	1	no	—
173	C₂²×C₅₃	—	2	1	no	—
174	C₂²×C₅₃	—	2	1	no	—
175	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
176	C₂×D₅₃	—	2	4	no	—
177	C₂×D₅₃	—	4	2	no	—
178	C₂×D₅₃	—	4	2	no	—
179	C₂×D₅₃	—	8	4	no	—
180	C₄×C₅₃	—	2	1	no	—
181	C₄×C₅₃	—	2	1	no	—
182	C₄×C₅₃	—	2	1	no	—
183	C₄×C₅₃	—	2	1	no	—
184	C₄×C₅₃	C₂×#131	1	2	yes	—
185	C₄×C₅₃	#577	1	2	yes	—
186	Dic₅₃	—	4	2	no	—
187	Dic₅₃	—	4	2	no	—
188	Dic₅₃	—	4	2	no	—
189	#3101	—	53	1	no	—
190	#2985	—	53	1	no	—
191	#2972	—	53	1	no	—
192	#1585	—	53	2	no	—
193	#1582	—	53	2	no	—
194	SD₁₆⋊C₈	—	53	2	no	—
195	C₈×SD₁₆	—	53	2	no	—
196	#1572	—	53	1	no	—
197	#1490	—	53	1	no	—
198	#1485	—	53	1	no	—
199	#1453	D₅₃	1	1	yes	characteristic, complemented
200	C₂×C₅₃	—	4	1	no	—
201	C₂×C₅₃	#2929	1	1	yes	characteristic
202	C₂×C₅₃	#2949	1	1	yes	characteristic
203	C₂×C₅₃	#2956	1	1	yes	characteristic
204	D₅₃	—	4	4	no	—
205	#533	—	53	1	no	—
206	C₂×C₄×D₄	—	53	1	no	—
207	#483	—	53	2	no	—
208	#476	—	53	2	no	—
209	#461	—	53	2	no	—
210	#450	—	53	2	no	—
211	#446	—	53	2	no	—
212	#446	—	106	2	no	—
213	C₈×Q₈	—	53	2	no	—
214	C₄×SD₁₆	—	53	2	no	—
215	C₄×SD₁₆	—	53	2	no	—
216	C₈×D₄	—	53	2	no	—
217	C₈×D₄	—	53	2	no	—
218	C₈×D₄	—	106	2	no	—
219	#433	—	106	2	no	—
220	#432	—	53	1	no	—
221	#431	—	53	1	no	—
222	#431	C₂×D₅₃	1	1	yes	characteristic
223	#431	C₂×D₅₃	1	1	yes	characteristic
224	C₂×C₂²⋊C₈	—	106	2	no	—
225	C₄×M₁₆	C₂×D₅₃	1	1	yes	characteristic
226	#335	—	53	2	no	—
227	#334	—	53	2	no	—
228	Q₈⋊C₈	—	53	2	no	—
229	Q₈⋊C₈	—	53	2	no	—
230	D₄⋊C₈	—	53	2	no	—
231	D₄⋊C₈	—	53	2	no	—
232	#322	C₂×D₅₃	1	1	yes	characteristic, complemented
233	#322	C₂×D₅₃	1	1	yes	characteristic, complemented
234	#322	C₂×D₅₃	1	1	yes	characteristic, complemented
235	C₈²	C₂×D₅₃	1	1	yes	characteristic, complemented
236	C₅₃	#17924	1	1	yes	characteristic, complemented, Sylow
237	C₂²×D₄	—	53	1	no	—
238	C₂³×C₄	—	106	1	no	—
239	C₂×SD₁₆	—	53	2	no	—
240	C₂×SD₁₆	—	53	2	no	—
241	C₂×M₁₆	—	2	1	no	—
242	C₂²×C₈	—	106	2	no	—
243	C₂²×C₈	—	106	2	no	—
244	C₄⋊Q₈	—	53	1	no	—
245	C₄⋊Q₈	—	53	1	no	—
246	#125	—	106	1	no	—
247	C₂²⋊Q₈	—	53	2	no	—
248	C₂²⋊Q₈	—	106	1	no	—
249	C₄×Q₈	—	53	2	no	—
250	C₄×Q₈	—	106	1	no	—
251	C₄×D₄	—	53	2	no	—
252	C₄×D₄	—	53	2	no	—
253	C₄×D₄	—	106	1	no	—
254	C₄×D₄	—	106	1	no	—
255	#117	—	106	1	no	—
256	C₂×C₄⋊C₄	—	53	1	no	—
257	C₂×C₂²⋊C₄	—	106	1	no	—
258	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
259	#106	—	53	2	no	—
260	#106	—	53	2	no	—
261	C₄⋊C₈	—	53	2	no	—
262	C₄⋊C₈	—	53	2	no	—
263	C₄⋊C₈	—	106	2	no	—
264	C₄⋊C₈	—	106	2	no	—
265	C₄⋊C₈	C₄×D₅₃	1	2	yes	—
266	C₄⋊C₈	D₂₁₂	1	2	yes	—
267	#103	—	53	2	no	—
268	#103	—	53	2	no	—
269	#102	—	53	2	no	—
270	#102	—	53	2	no	—
271	C₂²⋊C₈	—	106	1	no	—
272	C₂²⋊C₈	—	106	1	no	—
273	C₂²⋊C₈	—	106	2	no	—
274	C₂²⋊C₈	—	106	2	no	—
275	C₂²⋊C₈	C₄×D₅₃	1	2	yes	—
276	C₂²⋊C₈	D₂₁₂	1	2	yes	—
277	#97	—	2	1	no	—
278	#97	—	106	2	no	—
279	#97	—	106	1	no	—
280	C₄×C₈	—	106	2	no	—
281	C₄×C₈	—	106	1	no	—
282	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
283	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
284	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
285	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
286	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
287	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
288	C₂⁴	—	106	1	no	—
289	C₂×Q₈	—	53	2	no	—
290	C₂×Q₈	—	106	1	no	—
291	C₂×D₄	—	53	2	no	—
292	C₂×D₄	—	106	4	no	—
293	C₂×D₄	—	106	1	no	—
294	C₂²×C₄	—	106	1	no	—
295	C₂²×C₄	—	106	2	no	—
296	C₂²×C₄	—	106	2	no	—
297	C₂²×C₄	—	424	2	no	—
298	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
299	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
300	C₂²×C₄	#11719945	1	1	yes	characteristic
301	SD₁₆	—	106	4	no	—
302	SD₁₆	—	106	4	no	—
303	M₁₆	—	4	2	no	—
304	C₂×C₈	—	2	1	no	—
305	C₂×C₈	—	2	1	no	—
306	C₂×C₈	—	106	2	no	—
307	C₂×C₈	—	106	2	no	—
308	C₂×C₈	—	424	2	no	—
309	C₂×C₈	—	424	2	no	—
310	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
311	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
312	C₂×C₈	C₂×D₂₁₂	1	2	yes	—
313	C₂×C₈	C₂×D₂₁₂	1	2	yes	—
314	C₄⋊C₄	—	53	2	no	—
315	C₄⋊C₄	—	53	2	no	—
316	C₄⋊C₄	—	106	1	no	—
317	C₄⋊C₄	—	106	1	no	—
318	C₄⋊C₄	—	106	1	no	—
319	C₄⋊C₄	—	106	2	no	—
320	C₄⋊C₄	—	106	1	no	—
321	C₄⋊C₄	—	106	2	no	—
322	C₂²⋊C₄	—	106	2	no	—
323	C₂²⋊C₄	—	106	2	no	—
324	C₂²⋊C₄	—	106	2	no	—
325	C₄²	—	106	2	no	—
326	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
327	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
328	C₄²	#11719945	1	1	yes	characteristic
329	C₄²	C₂³×D₅₃	1	1	yes	characteristic
330	C₂³	—	106	4	no	—
331	C₂³	—	106	2	no	—
332	C₂³	—	424	1	no	—
333	C₂³	—	424	1	no	—
334	C₂³	C₄×D₂₁₂	1	1	yes	characteristic
335	Q₈	—	106	2	no	—
336	Q₈	—	106	2	no	—
337	Q₈	—	424	1	no	—
338	D₄	—	53	4	no	—
339	D₄	—	106	4	no	—
340	D₄	—	106	2	no	—
341	C₂×C₄	—	2	2	no	—
342	C₂×C₄	—	2	2	no	—
343	C₂×C₄	—	4	1	no	—
344	C₂×C₄	—	106	2	no	—
345	C₂×C₄	—	106	2	no	—
346	C₂×C₄	—	106	2	no	—
347	C₂×C₄	—	424	2	no	—
348	C₂×C₄	—	424	1	no	—
349	C₂×C₄	—	212	2	no	—
350	C₂×C₄	—	212	2	no	—
351	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
352	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
353	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
354	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
355	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
356	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
357	C₈	—	2	2	no	—
358	C₈	—	2	2	no	—
359	C₈	—	2	2	no	—
360	C₈	—	2	2	no	—
361	C₈	—	2	2	no	—
362	C₈	—	4	1	no	—
363	C₈	—	212	2	no	—
364	C₈	—	212	2	no	—
365	C₂²	—	2	1	no	—
366	C₂²	—	2	1	no	—
367	C₂²	—	2	1	no	—
368	C₂²	—	106	4	no	—
369	C₂²	—	424	4	no	—
370	C₂²	—	212	2	no	—
371	C₂²	—	212	2	no	—
372	C₂²	C₂×C₄×D₂₁₂	1	1	yes	characteristic
373	C₄	—	2	1	no	—
374	C₄	—	2	1	no	—
375	C₄	—	2	1	no	—
376	C₄	—	2	1	no	—
377	C₄	—	212	2	no	—
378	C₄	—	212	2	no	—
379	C₄	—	212	2	no	—
380	C₄	#131×D₅₃	1	2	yes	—
381	C₄	#16811435	1	2	yes	—
382	C₂	—	4	1	no	—
383	C₂	—	212	4	no	—
384	C₂	#20283411	1	1	yes	characteristic
385	C₂	#20283576	1	1	yes	characteristic
386	C₂	#20283626	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#1453	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂²×D₂₁₂	C₂×C₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#1453	D₅₃
hypocenter	C₅₃	#17924
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ #1453

Complements

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

N	H ≅ G/N	type
2. #20283809	C₂	semidirect
4. #20273574	C₂	semidirect
5. #20273553	C₂	semidirect
6. #20273529	C₂	semidirect
7. C₈×#11719913	C₂	semidirect
8. #20273495	C₂	semidirect
10. #20273137	C₂	semidirect
11. C₅₃×#1453	C₂	semidirect
18. #16810753	C₂²	semidirect
20. C₈×Dic₁₀₆	C₂²	semidirect
30. #16810376	C₂²	semidirect
31. #16810374	C₂²	semidirect
32. #16810355	C₂²	semidirect
33. #16810355	C₂²	semidirect
39. C₅₃×#322	C₂²	semidirect
40. C₅₃×#322	C₂²	semidirect
41. C₅₃×#322	C₂²	semidirect
42. C₈²×C₅₃	C₂²	semidirect
199. #1453	D₅₃	semidirect
232. #322	C₂×D₅₃	semidirect
233. #322	C₂×D₅₃	semidirect
234. #322	C₂×D₅₃	semidirect
235. C₈²	C₂×D₅₃	semidirect
236. C₅₃	#17924	semidirect

Maximal subgroups

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

subgroup	index
#20284724	2
#20283809	2
#20283738	2
#20273574	2
#20273553	2
#20273529	2
C₈×#11719913	2
#20273495	2
#20273159	2
#20273137	2
C₅₃×#1453	2
#17924	53