Subgroups — #27356151

Order 13568.

598 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20285285	C₂	1	1	yes	maximal, characteristic
2	#20283877	C₂	1	2	yes	maximal, complemented
3	#20273630	C₂	1	1	yes	maximal, characteristic, complemented
4	#20273625	C₂	1	1	yes	maximal, characteristic, complemented
5	#20273560	C₂	1	2	yes	maximal, complemented
6	#20273553	C₂	1	2	yes	maximal, complemented
7	#20273540	C₂	1	1	yes	maximal, characteristic, complemented
8	#20273535	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#16810374	C₂	1	1	yes	maximal, characteristic, complemented
10	#20273129	C₂	1	2	yes	maximal, complemented
11	C₅₃×#1451	C₂	1	1	yes	maximal, characteristic, complemented
12	#16811157	C₂²	1	2	yes	—
13	#16811025	C₄	1	1	yes	characteristic
14	#16811007	C₄	1	1	yes	characteristic
15	#16810978	C₄	1	2	yes	—
16	#16810864	C₄	1	1	yes	characteristic
17	#16810859	C₄	1	1	yes	characteristic
18	#16810852	C₄	1	2	yes	—
19	#16810753	C₂²	1	2	yes	—
20	C₈×Dic₁₀₆	C₂²	1	2	yes	—
21	C₄×Dic₂₁₂	C₂²	1	1	yes	characteristic
22	C₄×#11719913	C₂²	1	2	yes	—
23	C₄×D₄₂₄	C₂²	1	1	yes	characteristic
24	#16810702	C₂²	1	2	yes	—
25	#16810691	—	2	2	no	—
26	C₈×D₂₁₂	C₂²	1	2	yes	—
27	#16810686	—	2	2	no	—
28	#16810683	—	2	2	no	—
29	#16810648	—	2	2	no	—
30	C₂×C₄⋊#88322	C₂²	1	2	yes	—
31	#16810374	C₂²	1	1	yes	characteristic
32	#16810374	C₂²	1	2	yes	—
33	#16810374	C₂²	1	1	yes	characteristic
34	#16810355	C₂²	1	2	yes	—
35	#16810355	C₂²	1	2	yes	—
36	#16810352	C₂²	1	2	yes	—
37	#16810352	C₂²	1	2	yes	—
38	C₅₃×#431	C₂²	1	1	yes	characteristic
39	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
40	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
41	C₅₃×#322	C₂²	1	1	yes	characteristic
42	C₅₃×#322	C₂²	1	1	yes	characteristic
43	C₅₃×#322	C₂²	1	1	yes	characteristic
44	C₅₃×#322	C₂²	1	1	yes	characteristic
45	C₂×Dic₂₁₂	C₂×C₄	1	1	yes	characteristic
46	C₂×#11719913	C₂×C₄	1	2	yes	—
47	C₂×D₄₂₄	C₂×C₄	1	1	yes	characteristic
48	C₂×#11719912	—	2	2	no	—
49	C₂²×#88322	—	2	2	no	—
50	C₂×C₈×D₅₃	—	2	2	no	—
51	#13133211	—	2	2	no	—
52	#13133209	C₂×C₄	1	2	yes	—
53	#13133205	C₂×C₄	1	2	yes	—
54	#13133200	—	2	2	no	—
55	#13133196	C₂×C₄	1	2	yes	—
56	#13133191	C₂×C₄	1	2	yes	—
57	C₄×Dic₁₀₆	C₂³	1	2	yes	—
58	C₄×D₂₁₂	C₂³	1	2	yes	—
59	C₄×C₂²⋊D₅₃	—	2	2	no	—
60	#13133174	—	2	2	no	—
61	#13133151	C₂×C₄	1	1	yes	characteristic
62	#13133151	C₂×C₄	1	1	yes	characteristic
63	#13133149	C₂×C₄	1	2	yes	—
64	C₄⋊#88322	—	2	2	no	—
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	#13133129	—	2	2	no	—
74	#13133129	—	2	2	no	—
75	#13133127	—	2	2	no	—
76	C₈×Dic₅₃	—	2	2	no	—
77	C₂×M₁₆×C₅₃	—	2	1	no	—
78	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
79	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
80	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
81	C₄⋊C₈×C₅₃	D₄	1	2	yes	—
82	C₂²⋊C₈×C₅₃	D₄	1	2	yes	—
83	#97×C₅₃	—	2	1	no	—
84	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
85	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
86	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
87	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
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	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
93	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
95	C₂×C₂²⋊D₅₃	—	2	2	no	—
96	C₂×D₂₁₂	C₂²×C₄	1	2	yes	—
97	C₂×C₄×D₅₃	—	2	2	no	—
98	C₂×Dic₁₀₆	C₂²×C₄	1	2	yes	—
99	M₁₆×C₅₃	—	4	2	no	—
100	C₂×C₈×C₅₃	—	2	2	no	—
101	C₂×C₈×C₅₃	—	2	1	no	—
102	C₂×C₈×C₅₃	—	2	1	no	—
103	C₂×C₈×C₅₃	—	4	1	no	—
104	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
109	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
110	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
111	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
112	C₄²×C₅₃	#41	1	1	yes	characteristic
113	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
114	C₂²⋊Dic₅₃	—	2	2	no	—
115	#11719921	—	2	2	no	—
116	#11719921	—	2	2	no	—
117	#11719921	—	2	2	no	—
118	#11719921	—	2	2	no	—
119	C₄⋊Dic₅₃	C₂²×C₄	1	2	yes	—
120	C₄⋊Dic₅₃	C₂²×C₄	1	2	yes	—
121	#11719919	—	2	2	no	—
122	#11719919	—	2	2	no	—
123	#11719919	—	2	2	no	—
124	#11719919	—	2	2	no	—
125	C₄×Dic₅₃	—	2	2	no	—
126	C₂×#88322	—	2	2	no	—
127	C₂×#88322	—	2	2	no	—
128	C₂×#88322	—	8	2	no	—
129	Dic₂₁₂	—	2	2	no	—
130	D₄₂₄	—	2	2	no	—
131	#11719913	—	4	2	no	—
132	#11719912	—	8	2	no	—
133	C₈×D₅₃	—	8	2	no	—
134	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
135	C₂²×D₅₃	—	2	2	no	—
136	C₂×C₄×C₅₃	—	2	2	no	—
137	C₂×C₄×C₅₃	—	2	1	no	—
138	C₂×C₄×C₅₃	—	2	1	no	—
139	C₂×C₄×C₅₃	—	2	2	no	—
140	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
141	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
142	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
143	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
144	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
145	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
146	C₂²⋊D₅₃	—	4	4	no	—
147	C₂×Dic₅₃	—	2	2	no	—
148	C₂×Dic₅₃	—	2	2	no	—
149	C₂×Dic₅₃	—	2	2	no	—
150	D₂₁₂	—	2	2	no	—
151	D₂₁₂	—	2	2	no	—
152	C₄×D₅₃	—	4	2	no	—
153	C₄×D₅₃	—	4	2	no	—
154	Dic₁₀₆	—	2	2	no	—
155	Dic₁₀₆	—	2	2	no	—
156	C₈×C₅₃	—	2	1	no	—
157	C₈×C₅₃	—	2	1	no	—
158	C₈×C₅₃	—	2	1	no	—
159	C₈×C₅₃	—	2	2	no	—
160	C₈×C₅₃	—	2	2	no	—
161	C₈×C₅₃	—	4	1	no	—
162	C₈×C₅₃	—	4	1	no	—
163	C₈×C₅₃	#131	1	2	yes	—
164	#88322	—	4	2	no	—
165	#88322	—	4	2	no	—
166	#18093	—	53	1	no	maximal, Sylow
167	C₂²×C₅₃	—	2	1	no	—
168	C₂²×C₅₃	—	2	1	no	—
169	C₂²×C₅₃	#444	1	2	yes	—
170	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
171	C₂×D₅₃	—	4	2	no	—
172	C₂×D₅₃	—	4	2	no	—
173	C₂×D₅₃	—	4	2	no	—
174	C₄×C₅₃	—	2	1	no	—
175	C₄×C₅₃	—	2	1	no	—
176	C₄×C₅₃	—	2	1	no	—
177	C₄×C₅₃	—	2	1	no	—
178	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
179	C₄×C₅₃	#568	1	1	yes	characteristic
180	C₄×C₅₃	#577	1	2	yes	—
181	Dic₅₃	—	4	2	no	—
182	Dic₅₃	—	4	2	no	—
183	Dic₅₃	—	4	2	no	—
184	#3168	—	53	1	no	—
185	#2994	—	53	2	no	—
186	#1594	—	53	1	no	—
187	#1593	—	53	1	no	—
188	#1583	—	53	2	no	—
189	#1582	—	53	2	no	—
190	#1580	—	53	1	no	—
191	#1579	—	53	1	no	—
192	C₂×#334	—	53	1	no	—
193	#1483	—	53	2	no	—
194	#1451	D₅₃	1	1	yes	characteristic, complemented
195	C₂×C₅₃	—	2	2	no	—
196	C₂×C₅₃	#2935	1	1	yes	characteristic
197	C₂×C₅₃	#2952	1	1	yes	characteristic
198	C₂×C₅₃	C₂×#444	1	1	yes	characteristic
199	D₅₃	—	8	2	no	—
200	#529	—	53	2	no	—
201	#499	—	53	1	no	—
202	#495	—	53	1	no	—
203	#488	—	53	2	no	—
204	#467	—	53	1	no	—
205	#466	—	53	1	no	—
206	#465	—	53	2	no	—
207	#446	—	53	2	no	—
208	C₈×Q₈	—	53	2	no	—
209	C₄×Q₁₆	—	53	1	no	—
210	C₄×SD₁₆	—	53	2	no	—
211	C₄×D₈	—	53	1	no	—
212	#436	—	53	2	no	—
213	#435	—	106	2	no	—
214	C₈×D₄	—	53	2	no	—
215	C₈×D₄	—	106	2	no	—
216	#433	—	106	2	no	—
217	#431	C₂×D₅₃	1	1	yes	characteristic
218	#424	—	106	2	no	—
219	C₂×C₄⋊C₈	—	53	2	no	—
220	C₄×M₁₆	C₂×D₅₃	1	1	yes	characteristic
221	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
222	#334	—	53	1	no	—
223	#334	—	53	2	no	—
224	#334	—	53	1	no	—
225	Q₈⋊C₈	—	53	2	no	—
226	Q₈⋊C₈	—	53	2	no	—
227	D₄⋊C₈	—	53	2	no	—
228	D₄⋊C₈	—	53	2	no	—
229	#322	C₂×D₅₃	1	1	yes	characteristic
230	#322	C₂×D₅₃	1	1	yes	characteristic
231	#322	C₂×D₅₃	1	1	yes	characteristic
232	#322	C₂×D₅₃	1	1	yes	characteristic
233	C₅₃	#18093	1	1	yes	characteristic, complemented, Sylow
234	C₂×Q₁₆	—	53	1	no	—
235	C₂×SD₁₆	—	53	2	no	—
236	C₂×D₈	—	53	1	no	—
237	C₂×M₁₆	—	2	1	no	—
238	C₂×M₁₆	—	106	2	no	—
239	C₂²×C₈	—	106	2	no	—
240	C₂²×C₈	—	106	2	no	—
241	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
242	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
243	#126	—	106	2	no	—
244	#125	—	53	2	no	—
245	#124	—	53	2	no	—
246	#123	—	106	2	no	—
247	C₂²⋊Q₈	—	53	2	no	—
248	#121	—	53	2	no	—
249	C₄×Q₈	—	53	2	no	—
250	C₄×D₄	—	53	2	no	—
251	C₄×D₄	—	106	2	no	—
252	#117	—	106	2	no	—
253	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
254	#107	—	53	1	no	—
255	#107	—	53	1	no	—
256	#106	—	53	2	no	—
257	C₄⋊C₈	—	53	2	no	—
258	C₄⋊C₈	—	53	2	no	—
259	C₄⋊C₈	—	106	2	no	—
260	C₄⋊C₈	—	106	2	no	—
261	C₄⋊C₈	—	106	2	no	—
262	C₄⋊C₈	D₂₁₂	1	2	yes	—
263	#103	—	53	2	no	—
264	#103	—	53	2	no	—
265	#102	—	53	2	no	—
266	#102	—	53	2	no	—
267	C₂²⋊C₈	—	106	2	no	—
268	C₂²⋊C₈	—	106	2	no	—
269	C₂²⋊C₈	D₂₁₂	1	2	yes	—
270	#97	—	2	1	no	—
271	#97	—	106	2	no	—
272	C₄×C₈	—	106	2	no	—
273	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
274	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
275	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
276	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
277	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
278	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
279	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
280	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
281	C₂×Q₈	—	53	2	no	—
282	C₂×D₄	—	53	2	no	—
283	C₂×D₄	—	106	2	no	—
284	C₂²×C₄	—	106	2	no	—
285	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
286	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
287	C₂²×C₄	#11719945	1	1	yes	characteristic
288	Q₁₆	—	106	2	no	—
289	SD₁₆	—	212	2	no	—
290	D₈	—	106	2	no	—
291	M₁₆	—	4	2	no	—
292	M₁₆	—	424	2	no	—
293	C₂×C₈	—	2	2	no	—
294	C₂×C₈	—	2	1	no	—
295	C₂×C₈	—	2	1	no	—
296	C₂×C₈	—	4	1	no	—
297	C₂×C₈	—	106	2	no	—
298	C₂×C₈	—	106	2	no	—
299	C₂×C₈	—	424	2	no	—
300	C₂×C₈	—	424	2	no	—
301	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
302	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
303	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
304	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
305	C₂×C₈	C₂×D₂₁₂	1	2	yes	—
306	C₂×C₈	C₂×D₂₁₂	1	2	yes	—
307	C₄⋊C₄	—	53	2	no	—
308	C₄⋊C₄	—	53	2	no	—
309	C₄⋊C₄	—	106	2	no	—
310	C₄⋊C₄	—	106	2	no	—
311	C₄⋊C₄	—	106	2	no	—
312	C₄⋊C₄	—	106	2	no	—
313	C₂²⋊C₄	—	106	2	no	—
314	C₂²⋊C₄	—	106	2	no	—
315	C₂²⋊C₄	—	106	2	no	—
316	C₂²⋊C₄	—	106	2	no	—
317	C₂²⋊C₄	—	106	2	no	—
318	C₄²	—	106	2	no	—
319	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
320	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
321	C₄²	#11719945	1	1	yes	characteristic
322	C₄²	C₂³×D₅₃	1	1	yes	characteristic
323	C₂³	—	106	2	no	—
324	C₂³	C₄×D₂₁₂	1	1	yes	characteristic
325	Q₈	—	106	2	no	—
326	Q₈	—	106	2	no	—
327	D₄	—	106	2	no	—
328	D₄	—	106	2	no	—
329	D₄	—	212	4	no	—
330	C₂×C₄	—	2	2	no	—
331	C₂×C₄	—	2	1	no	—
332	C₂×C₄	—	2	1	no	—
333	C₂×C₄	—	2	2	no	—
334	C₂×C₄	—	106	2	no	—
335	C₂×C₄	—	106	2	no	—
336	C₂×C₄	—	106	2	no	—
337	C₂×C₄	—	212	2	no	—
338	C₂×C₄	—	212	2	no	—
339	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
340	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
341	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
342	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
343	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
344	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
345	C₈	—	2	1	no	—
346	C₈	—	2	1	no	—
347	C₈	—	2	1	no	—
348	C₈	—	2	2	no	—
349	C₈	—	2	2	no	—
350	C₈	—	4	1	no	—
351	C₈	—	4	1	no	—
352	C₈	—	212	2	no	—
353	C₈	—	212	2	no	—
354	C₈	#13133225	1	2	yes	—
355	C₂²	—	2	1	no	—
356	C₂²	—	2	1	no	—
357	C₂²	—	212	2	no	—
358	C₂²	—	212	2	no	—
359	C₂²	—	212	2	no	—
360	C₂²	#16810746	1	2	yes	—
361	C₂²	C₂×C₄×D₂₁₂	1	1	yes	characteristic
362	C₄	—	2	1	no	—
363	C₄	—	2	1	no	—
364	C₄	—	2	1	no	—
365	C₄	—	2	1	no	—
366	C₄	—	212	2	no	—
367	C₄	—	212	2	no	—
368	C₄	—	212	2	no	—
369	C₄	C₂×#13133225	1	1	yes	characteristic
370	C₄	#16811389	1	1	yes	characteristic
371	C₄	#16811435	1	2	yes	—
372	C₂	—	2	2	no	—
373	C₂	—	424	2	no	—
374	C₂	#20283461	1	1	yes	characteristic
375	C₂	#20283600	1	1	yes	characteristic
376	C₂	C₂×#16810746	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#1451	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	#16810859	C₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#1451	D₅₃
hypocenter	C₅₃	#18093
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ #1451

Complements

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

N	H ≅ G/N	type
2. #20283877	C₂	semidirect
3. #20273630	C₂	semidirect
4. #20273625	C₂	semidirect
5. #20273560	C₂	semidirect
6. #20273553	C₂	semidirect
7. #20273540	C₂	semidirect
8. #20273535	C₂	semidirect
9. C₂×#16810374	C₂	semidirect
10. #20273129	C₂	semidirect
11. C₅₃×#1451	C₂	semidirect
194. #1451	D₅₃	semidirect
233. C₅₃	#18093	semidirect

Maximal subgroups

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

subgroup	index
#20285285	2
#20283877	2
#20273630	2
#20273625	2
#20273560	2
#20273553	2
#20273540	2
#20273535	2
C₂×#16810374	2
#20273129	2
C₅₃×#1451	2
#18093	53