Subgroups — #27355888

Order 13568.

600 conjugacy classes of subgroups.

The subgroup lattice has 362 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	#20283713	C₂	1	2	yes	maximal, complemented
3	#20273604	C₂	1	2	yes	maximal, complemented
4	#20273564	C₂	1	2	yes	maximal, complemented
5	#20273543	C₂	1	2	yes	maximal, complemented
6	#20273514	C₂	1	2	yes	maximal, complemented
7	#20273121	C₂	1	2	yes	maximal, complemented
8	C₈×#11719917	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#1564	C₂	1	1	yes	maximal, characteristic, complemented
10	#16811157	C₂²	1	2	yes	—
11	#16811025	C₄	1	1	yes	characteristic
12	#16811007	C₄	1	1	yes	characteristic
13	#16810978	C₄	1	2	yes	—
14	#16810864	C₄	1	1	yes	characteristic
15	#16810859	C₄	1	1	yes	characteristic
16	#16810852	C₄	1	2	yes	—
17	#16810750	C₂²	1	2	yes	—
18	#16810747	C₂²	1	2	yes	—
19	C₄×Dic₂₁₂	C₂²	1	1	yes	characteristic
20	C₄×#11719913	C₂²	1	2	yes	—
21	C₄×D₄₂₄	C₂²	1	1	yes	characteristic
22	#16810700	C₂²	1	2	yes	—
23	#16810698	—	2	2	no	—
24	#16810697	—	2	2	no	—
25	#16810688	C₂²	1	2	yes	—
26	#16810682	—	2	2	no	—
27	#16810676	C₂²	1	1	yes	characteristic
28	#16810577	—	2	2	no	—
29	C₄×#11719917	C₂²	1	1	yes	characteristic
30	#16810353	C₂²	1	2	yes	—
31	#16810353	C₂²	1	2	yes	—
32	#16810350	C₂²	1	2	yes	—
33	#16810350	C₂²	1	2	yes	—
34	#16810345	C₂²	1	2	yes	—
35	C₈×#88322	C₂²	1	2	yes	—
36	C₅₃×#431	C₂²	1	2	yes	—
37	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
38	C₅₃×#335	C₂²	1	2	yes	—
39	C₅₃×#334	C₂²	1	1	yes	characteristic
40	C₅₃×#334	C₂²	1	1	yes	characteristic
41	C₂×Dic₂₁₂	C₂×C₄	1	1	yes	characteristic
42	C₂×#11719913	C₂×C₄	1	2	yes	—
43	C₂×D₄₂₄	C₂×C₄	1	1	yes	characteristic
44	C₂×#11719917	D₄	1	2	yes	—
45	C₂×#11719912	—	2	2	no	—
46	C₂×C₈×D₅₃	—	2	2	no	—
47	#13133211	—	2	2	no	—
48	#13133209	C₂×C₄	1	2	yes	—
49	#13133205	C₂×C₄	1	2	yes	—
50	#13133200	—	2	2	no	—
51	#13133196	C₂×C₄	1	2	yes	—
52	#13133191	C₂×C₄	1	2	yes	—
53	C₄×Dic₁₀₆	C₂³	1	2	yes	—
54	C₄×D₂₁₂	C₂³	1	2	yes	—
55	C₄×C₂²⋊D₅₃	—	2	2	no	—
56	#13133174	—	2	2	no	—
57	#13133151	C₂×C₄	1	1	yes	characteristic
58	#13133151	C₂×C₄	1	1	yes	characteristic
59	#13133149	C₂×C₄	1	2	yes	—
60	C₄⋊#88322	—	2	1	no	—
61	#13133146	—	2	2	no	—
62	#13133146	—	2	2	no	—
63	#13133142	C₂×C₄	1	2	yes	—
64	#13133142	C₂×C₄	1	2	yes	—
65	#13133139	C₂×C₄	1	2	yes	—
66	#13133139	C₂×C₄	1	2	yes	—
67	C₂²⋊#88322	—	2	1	no	—
68	#13133129	—	2	2	no	—
69	#13133129	—	2	2	no	—
70	#13133128	D₄	1	1	yes	characteristic
71	#13133128	D₄	1	1	yes	characteristic
72	#13133127	—	2	2	no	—
73	C₄×#88322	C₂³	1	2	yes	—
74	C₄×#88322	C₂³	1	2	yes	—
75	C₈×Dic₅₃	—	2	2	no	—
76	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
77	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
78	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
79	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
80	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
81	C₂²⋊C₈×C₅₃	—	2	2	no	—
82	C₄×C₈×C₅₃	—	2	2	no	—
83	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
84	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
85	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
86	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
87	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
88	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
90	C₂×C₂²⋊D₅₃	—	2	2	no	—
91	C₂×D₂₁₂	C₂²×C₄	1	2	yes	—
92	C₂×C₄×D₅₃	—	2	2	no	—
93	C₂×Dic₁₀₆	C₂²×C₄	1	2	yes	—
94	C₂×C₈×C₅₃	—	2	2	no	—
95	C₂×C₈×C₅₃	—	2	2	no	—
96	C₂×C₈×C₅₃	—	2	2	no	—
97	C₂×C₈×C₅₃	—	4	1	no	—
98	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
102	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
103	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
104	C₄²×C₅₃	#41	1	1	yes	characteristic
105	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
106	C₂²⋊Dic₅₃	—	2	2	no	—
107	#11719921	—	2	2	no	—
108	#11719921	—	2	2	no	—
109	#11719921	—	2	2	no	—
110	#11719921	—	2	2	no	—
111	C₄⋊Dic₅₃	C₂²×C₄	1	2	yes	—
112	C₄⋊Dic₅₃	C₂²×C₄	1	2	yes	—
113	#11719919	—	2	2	no	—
114	#11719919	—	2	2	no	—
115	#11719919	—	2	2	no	—
116	#11719919	—	2	2	no	—
117	C₄×Dic₅₃	—	2	2	no	—
118	#11719917	—	2	4	no	—
119	C₂×#88322	—	2	2	no	—
120	C₂×#88322	C₂×D₄	1	2	yes	—
121	C₂×#88322	C₂×D₄	1	2	yes	—
122	Dic₂₁₂	—	4	1	no	—
123	D₄₂₄	—	4	1	no	—
124	#11719913	—	2	4	no	—
125	#11719912	—	4	4	no	—
126	C₈×D₅₃	—	4	4	no	—
127	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
128	C₂²×D₅₃	—	2	2	no	—
129	C₂×C₄×C₅₃	—	2	1	no	—
130	C₂×C₄×C₅₃	—	2	1	no	—
131	C₂×C₄×C₅₃	—	2	2	no	—
132	C₂×C₄×C₅₃	—	4	1	no	—
133	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
134	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
135	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
136	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
137	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
138	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
139	C₂²⋊D₅₃	—	8	2	no	—
140	C₂×Dic₅₃	—	2	2	no	—
141	C₂×Dic₅₃	—	2	2	no	—
142	C₂×Dic₅₃	—	2	2	no	—
143	D₂₁₂	—	2	2	no	—
144	D₂₁₂	—	2	2	no	—
145	C₄×D₅₃	—	4	2	no	—
146	C₄×D₅₃	—	4	2	no	—
147	Dic₁₀₆	—	2	2	no	—
148	Dic₁₀₆	—	2	2	no	—
149	C₈×C₅₃	—	2	1	no	—
150	C₈×C₅₃	—	2	1	no	—
151	C₈×C₅₃	—	2	1	no	—
152	C₈×C₅₃	—	4	2	no	—
153	C₈×C₅₃	—	4	2	no	—
154	C₈×C₅₃	#131	1	2	yes	—
155	#88322	—	2	2	no	—
156	#88322	—	4	2	no	—
157	#88322	#135	1	4	yes	—
158	#18070	—	53	1	no	maximal, Sylow
159	C₂²×C₅₃	—	2	1	no	—
160	C₂²×C₅₃	—	2	1	no	—
161	C₂²×C₅₃	—	2	1	no	—
162	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
163	C₂×D₅₃	—	4	2	no	—
164	C₂×D₅₃	—	4	2	no	—
165	C₂×D₅₃	—	4	2	no	—
166	C₄×C₅₃	—	2	1	no	—
167	C₄×C₅₃	—	2	1	no	—
168	C₄×C₅₃	—	2	1	no	—
169	C₄×C₅₃	—	2	1	no	—
170	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
171	C₄×C₅₃	#568	1	1	yes	characteristic
172	C₄×C₅₃	C₂×#135	1	2	yes	—
173	Dic₅₃	—	4	2	no	—
174	Dic₅₃	—	4	2	no	—
175	Dic₅₃	—	4	2	no	—
176	#3168	—	53	1	no	—
177	#2969	—	53	2	no	—
178	#1590	—	53	2	no	—
179	#1584	—	53	2	no	—
180	#1581	—	53	2	no	—
181	C₈×SD₁₆	—	53	2	no	—
182	#1564	D₅₃	1	1	yes	characteristic, complemented
183	#1482	—	53	2	no	—
184	C₈×M₁₆	—	53	1	no	—
185	C₂×C₅₃	—	4	1	no	—
186	C₂×C₅₃	#2935	1	1	yes	characteristic
187	C₂×C₅₃	C₄×#135	1	1	yes	characteristic
188	C₂×C₅₃	#2957	1	1	yes	characteristic
189	D₅₃	—	8	2	no	—
190	#529	—	53	2	no	—
191	#499	—	53	1	no	—
192	#495	—	53	1	no	—
193	#488	—	53	2	no	—
194	#467	—	53	1	no	—
195	#466	—	53	1	no	—
196	#465	—	53	2	no	—
197	#446	—	53	2	no	—
198	C₈×Q₈	—	53	2	no	—
199	C₄×Q₁₆	—	53	1	no	—
200	C₄×SD₁₆	—	53	2	no	—
201	C₄×D₈	—	53	1	no	—
202	#436	—	53	2	no	—
203	#436	—	106	2	no	—
204	#435	—	106	2	no	—
205	C₈×D₄	—	53	2	no	—
206	#433	—	106	2	no	—
207	#431	—	53	1	no	—
208	#431	C₂×D₅₃	1	2	yes	—
209	#405	—	106	2	no	—
210	C₄×M₁₆	—	53	1	no	—
211	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
212	#335	C₂×D₅₃	1	2	yes	—
213	#334	C₂×D₅₃	1	1	yes	characteristic
214	#334	C₂×D₅₃	1	1	yes	characteristic
215	Q₈⋊C₈	—	53	2	no	—
216	Q₈⋊C₈	—	53	2	no	—
217	D₄⋊C₈	—	53	2	no	—
218	D₄⋊C₈	—	53	2	no	—
219	#322	—	53	2	no	—
220	C₈²	—	53	2	no	—
221	C₅₃	#18070	1	1	yes	characteristic, complemented, Sylow
222	C₂×Q₁₆	—	53	1	no	—
223	C₂×SD₁₆	—	53	2	no	—
224	C₂×D₈	—	53	1	no	—
225	C₂×M₁₆	—	53	2	no	—
226	C₂×M₁₆	—	106	2	no	—
227	C₂²×C₈	—	106	2	no	—
228	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
229	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
230	#126	—	106	2	no	—
231	#125	—	53	2	no	—
232	#124	—	53	2	no	—
233	#123	—	106	2	no	—
234	C₂²⋊Q₈	—	53	2	no	—
235	#121	—	53	2	no	—
236	C₄×Q₈	—	53	2	no	—
237	C₄×D₄	—	53	2	no	—
238	C₄×D₄	—	106	2	no	—
239	#117	—	106	2	no	—
240	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
241	#107	—	53	1	no	—
242	#107	—	53	1	no	—
243	#106	—	53	2	no	—
244	C₄⋊C₈	—	106	2	no	—
245	C₄⋊C₈	—	106	2	no	—
246	C₄⋊C₈	—	106	1	no	—
247	C₄⋊C₈	C₂²×D₅₃	1	2	yes	—
248	C₄⋊C₈	C₂²×D₅₃	1	2	yes	—
249	#103	—	53	2	no	—
250	#103	—	53	2	no	—
251	#102	—	53	2	no	—
252	#102	—	53	2	no	—
253	C₂²⋊C₈	—	2	2	no	—
254	C₂²⋊C₈	—	106	2	no	—
255	C₂²⋊C₈	—	106	2	no	—
256	C₂²⋊C₈	—	106	1	no	—
257	#97	—	53	1	no	—
258	#97	—	53	1	no	—
259	#97	—	106	2	no	—
260	C₄×C₈	—	2	2	no	—
261	C₄×C₈	—	53	2	no	—
262	C₄×C₈	—	53	2	no	—
263	C₄×C₈	—	106	2	no	—
264	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
265	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
266	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
267	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
268	C₂×Q₈	—	53	2	no	—
269	C₂×D₄	—	53	2	no	—
270	C₂×D₄	—	106	2	no	—
271	C₂²×C₄	—	106	2	no	—
272	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
273	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
274	C₂²×C₄	#11719949	1	1	yes	characteristic
275	Q₁₆	—	212	1	no	—
276	SD₁₆	—	106	4	no	—
277	D₈	—	212	1	no	—
278	M₁₆	—	106	4	no	—
279	M₁₆	—	212	4	no	—
280	C₂×C₈	—	2	2	no	—
281	C₂×C₈	—	2	2	no	—
282	C₂×C₈	—	2	2	no	—
283	C₂×C₈	—	4	1	no	—
284	C₂×C₈	—	53	2	no	—
285	C₂×C₈	—	53	2	no	—
286	C₂×C₈	—	106	2	no	—
287	C₂×C₈	—	212	4	no	—
288	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
289	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
290	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
291	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
292	C₄⋊C₄	—	53	2	no	—
293	C₄⋊C₄	—	53	2	no	—
294	C₄⋊C₄	—	106	2	no	—
295	C₄⋊C₄	—	106	2	no	—
296	C₄⋊C₄	—	106	2	no	—
297	C₄⋊C₄	—	106	2	no	—
298	C₂²⋊C₄	—	106	2	no	—
299	C₂²⋊C₄	—	106	2	no	—
300	C₂²⋊C₄	—	106	2	no	—
301	C₂²⋊C₄	—	106	2	no	—
302	C₂²⋊C₄	—	106	2	no	—
303	C₄²	—	106	2	no	—
304	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
305	C₄²	D₄×D₅₃	1	1	yes	characteristic
306	C₄²	#11719949	1	1	yes	characteristic
307	C₄²	C₂³×D₅₃	1	1	yes	characteristic
308	C₂³	—	106	2	no	—
309	C₂³	#13133179	1	1	yes	characteristic
310	Q₈	—	106	2	no	—
311	Q₈	—	106	2	no	—
312	D₄	—	106	2	no	—
313	D₄	—	106	2	no	—
314	D₄	—	424	2	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	4	1	no	—
319	C₂×C₄	—	106	2	no	—
320	C₂×C₄	—	106	2	no	—
321	C₂×C₄	—	106	2	no	—
322	C₂×C₄	—	212	2	no	—
323	C₂×C₄	—	212	2	no	—
324	C₂×C₄	#13133179	1	1	yes	characteristic
325	C₂×C₄	#13133179	1	1	yes	characteristic
326	C₂×C₄	#13133179	1	1	yes	characteristic
327	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
328	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
329	C₂×C₄	C₂×#11719949	1	1	yes	characteristic
330	C₈	—	2	1	no	—
331	C₈	—	2	1	no	—
332	C₈	—	2	1	no	—
333	C₈	—	4	2	no	—
334	C₈	—	4	2	no	—
335	C₈	—	53	4	no	—
336	C₈	—	106	2	no	—
337	C₈	—	212	2	no	—
338	C₈	#13133223	1	2	yes	—
339	C₂²	—	2	1	no	—
340	C₂²	—	2	1	no	—
341	C₂²	—	2	1	no	—
342	C₂²	—	212	2	no	—
343	C₂²	—	212	2	no	—
344	C₂²	—	212	2	no	—
345	C₂²	C₂×#13133179	1	1	yes	characteristic
346	C₄	—	2	1	no	—
347	C₄	—	2	1	no	—
348	C₄	—	2	1	no	—
349	C₄	—	2	1	no	—
350	C₄	—	212	2	no	—
351	C₄	—	212	2	no	—
352	C₄	—	212	2	no	—
353	C₄	C₂×#13133223	1	1	yes	characteristic
354	C₄	#16811389	1	1	yes	characteristic
355	C₄	#135×D₅₃	1	2	yes	—
356	C₂	—	4	1	no	—
357	C₂	—	424	2	no	—
358	C₂	#20283458	1	1	yes	characteristic
359	C₂	#20283486	1	1	yes	characteristic
360	C₂	#20283631	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#1564	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	#16810859	C₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#1564	D₅₃
hypocenter	C₅₃	#18070
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1564

Complements

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

N	H ≅ G/N	type
2. #20283713	C₂	semidirect
3. #20273604	C₂	semidirect
4. #20273564	C₂	semidirect
5. #20273543	C₂	semidirect
6. #20273514	C₂	semidirect
7. #20273121	C₂	semidirect
8. C₈×#11719917	C₂	semidirect
9. C₅₃×#1564	C₂	semidirect
182. #1564	D₅₃	semidirect
221. C₅₃	#18070	semidirect

Maximal subgroups

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

subgroup	index
#20285285	2
#20283713	2
#20273604	2
#20273564	2
#20273543	2
#20273514	2
#20273121	2
C₈×#11719917	2
C₅₃×#1564	2
#18070	53