Subgroups — #27537707

Order 13568.

584 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283411	C₂	1	1	yes	maximal, characteristic
2	#20276542	C₂	1	2	yes	maximal, complemented
3	#20276539	C₂	1	2	yes	maximal, complemented
4	C₁₆×D₂₁₂	C₂	1	2	yes	maximal, complemented
5	#20276534	C₂	1	2	yes	maximal, complemented
6	#20276522	C₂	1	1	yes	maximal, characteristic, complemented
7	#20276327	C₂	1	2	yes	maximal, complemented
8	C₂²⋊C₁₆×D₅₃	C₂	1	2	yes	maximal, complemented
9	C₅₃×#2164	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#13133223	C₂²	1	1	yes	characteristic
11	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
12	#16811135	C₄	1	1	yes	characteristic
13	C₂×#13133156	C₂²	1	2	yes	complemented
14	C₂×C₁₆×D₅₃	C₂²	1	2	yes	—
15	C₂×C₁₆×D₅₃	C₂²	1	2	yes	—
16	C₂×C₁₆×D₅₃	C₂²	1	2	yes	complemented
17	#16810696	C₄	1	2	yes	—
18	#16810696	C₄	1	2	yes	—
19	C₈×D₂₁₂	C₂²	1	2	yes	—
20	C₈×D₂₁₂	C₂²	1	2	yes	—
21	#16810650	C₄	1	1	yes	characteristic
22	#16810591	C₄	1	1	yes	characteristic
23	#16810586	C₄	1	1	yes	characteristic
24	C₄⋊#11719908	C₂²	1	2	yes	complemented
25	C₂²⋊#11719908	C₂²	1	2	yes	—
26	#16810402	C₂²	1	2	yes	—
27	#16810402	C₂²	1	2	yes	—
28	#16810402	C₂²	1	2	yes	complemented
29	#16810402	C₂²	1	2	yes	complemented
30	#16810399	C₂²	1	1	yes	characteristic, complemented
31	C₄×#11719908	C₂²	1	1	yes	characteristic, complemented
32	C₅₃×#405	C₂²	1	1	yes	characteristic
33	C₅₃×C₄⋊C₁₆	C₂²	1	2	yes	complemented
34	C₅₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
35	C₅₃×#346	C₂²	1	1	yes	characteristic, complemented
36	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic, complemented
37	C₂×#11719945	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
39	#13133223	—	4	2	no	—
40	C₂×#11719917	C₂×C₄	1	1	yes	characteristic
41	C₂×#11719912	C₂×C₄	1	2	yes	—
42	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
43	C₂×C₈×D₅₃	—	4	2	no	—
44	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
45	C₂×C₈×D₅₃	C₂³	1	2	yes	—
46	C₂×C₈×D₅₃	C₂³	1	2	yes	—
47	#13133201	C₈	1	2	yes	—
48	#13133197	C₈	1	2	yes	—
49	#13133192	C₈	1	2	yes	—
50	#13133187	C₈	1	2	yes	—
51	C₄×D₂₁₂	C₂×C₄	1	2	yes	—
52	C₄×D₂₁₂	C₂×C₄	1	2	yes	—
53	#13133156	—	4	2	no	—
54	C₂×#11719908	C₂³	1	2	yes	—
55	C₂×#11719908	C₂³	1	2	yes	—
56	C₁₆×D₅₃	—	4	2	no	—
57	C₁₆×D₅₃	—	4	2	no	—
58	C₁₆×D₅₃	—	4	2	no	—
59	C₄⋊#88322	C₂×C₄	1	2	yes	—
60	C₄⋊#88322	C₂×C₄	1	2	yes	—
61	#13133129	C₂×C₄	1	2	yes	—
62	#13133129	C₂×C₄	1	2	yes	—
63	#13133129	C₂×C₄	1	2	yes	—
64	#13133129	C₂×C₄	1	2	yes	—
65	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
66	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
67	#117×C₅₃	C₂×C₄	1	1	yes	characteristic
68	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
69	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
70	#97×C₅₃	C₂×C₄	1	1	yes	characteristic
71	#97×C₅₃	C₂×C₄	1	1	yes	characteristic
72	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
73	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
74	C₂³×D₅₃	C₂×C₈	1	1	yes	characteristic
75	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₂²⋊D₅₃	C₂×C₈	1	2	yes	—
77	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
78	#11719945	—	4	1	no	—
79	#11719945	—	4	1	no	—
80	C₂×D₂₁₂	C₂×C₈	1	2	yes	—
81	C₂×D₂₁₂	C₂×C₈	1	2	yes	—
82	C₂×D₂₁₂	C₂×C₈	1	1	yes	characteristic
83	C₂×C₄×D₅₃	—	4	1	no	—
84	C₂×C₄×D₅₃	—	4	1	no	—
85	C₂×C₄×D₅₃	C₂×C₈	1	2	yes	—
86	C₂×C₄×D₅₃	C₂²×C₄	1	2	yes	—
87	C₂×C₄×D₅₃	C₂²×C₄	1	2	yes	—
88	C₂×Dic₁₀₆	C₂×C₈	1	1	yes	characteristic
89	M₁₆×C₅₃	—	4	1	no	—
90	C₂×C₈×C₅₃	—	2	2	no	—
91	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
95	C₄⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
96	C₂²⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
97	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
98	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
99	#11719921	C₂×C₈	1	2	yes	—
100	#11719921	C₂×C₈	1	2	yes	—
101	#11719921	C₂×C₈	1	2	yes	—
102	#11719921	C₂×C₈	1	2	yes	—
103	C₄⋊Dic₅₃	C₂×C₈	1	2	yes	—
104	C₄⋊Dic₅₃	C₂×C₈	1	2	yes	—
105	#11719917	—	4	1	no	—
106	C₂×#88322	—	4	1	no	—
107	C₂×#88322	C₂²×C₄	1	2	yes	—
108	C₂×#88322	C₂²×C₄	1	2	yes	—
109	#11719912	—	4	2	no	—
110	C₈×D₅₃	—	2	2	no	—
111	C₈×D₅₃	—	2	2	no	—
112	C₈×D₅₃	—	2	2	no	—
113	C₈×D₅₃	—	2	2	no	—
114	C₈×D₅₃	—	4	2	no	—
115	C₁₆×C₅₃	—	2	2	no	—
116	C₁₆×C₅₃	—	2	2	no	—
117	#11719908	—	2	2	no	—
118	#11719908	—	2	2	no	—
119	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
120	C₂²×D₅₃	—	2	4	no	—
121	C₂²×D₅₃	—	4	1	no	—
122	C₂²×D₅₃	C₂²×C₈	1	2	yes	—
123	C₂²×D₅₃	C₂²×C₈	1	2	yes	—
124	C₂×C₄×C₅₃	—	2	1	no	—
125	C₂×C₄×C₅₃	—	2	1	no	—
126	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
127	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
128	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
129	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
130	C₂²⋊D₅₃	—	4	2	no	—
131	C₂×Dic₅₃	—	4	1	no	—
132	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
133	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
134	D₂₁₂	—	2	4	no	—
135	D₂₁₂	—	2	4	no	—
136	D₂₁₂	—	4	1	no	—
137	C₄×D₅₃	—	2	2	no	—
138	C₄×D₅₃	—	2	2	no	—
139	C₄×D₅₃	—	2	2	no	—
140	C₄×D₅₃	—	2	2	no	—
141	C₄×D₅₃	—	4	2	no	—
142	Dic₁₀₆	—	4	1	no	—
143	C₈×C₅₃	—	2	1	no	—
144	C₈×C₅₃	—	2	1	no	—
145	C₈×C₅₃	—	2	1	no	—
146	C₈×C₅₃	H(5,𝔽₂)	1	2	yes	—
147	#88322	—	2	2	no	—
148	#88322	—	2	2	no	—
149	#33491	—	53	1	no	maximal, Sylow
150	C₂²×C₅₃	—	2	2	no	—
151	C₂²×C₅₃	—	2	1	no	—
152	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
153	C₂×D₅₃	—	2	2	no	—
154	C₂×D₅₃	—	2	2	no	—
155	C₂×D₅₃	—	2	2	no	—
156	C₂×D₅₃	—	2	2	no	—
157	C₂×D₅₃	—	2	2	no	—
158	C₂×D₅₃	—	4	4	no	—
159	C₂×D₅₃	#504	1	4	yes	—
160	C₄×C₅₃	—	2	2	no	—
161	C₄×C₅₃	—	2	2	no	—
162	C₄×C₅₃	—	2	1	no	—
163	C₄×C₅₃	#518	1	1	yes	characteristic
164	C₄×C₅₃	#568	1	1	yes	characteristic
165	Dic₅₃	—	2	2	no	—
166	Dic₅₃	—	2	2	no	—
167	#2929	—	53	1	no	—
168	#2168	—	53	2	no	—
169	#2168	—	53	2	no	—
170	D₄×C₁₆	—	53	2	no	—
171	D₄×C₁₆	—	53	2	no	—
172	#2164	—	53	1	no	—
173	#2164	D₅₃	1	1	yes	characteristic, complemented
174	#2113	—	53	2	no	—
175	C₂×C₂²⋊C₁₆	—	53	2	no	—
176	C₂×C₅₃	—	4	1	no	—
177	C₂×C₅₃	#2972	1	1	yes	characteristic
178	C₂×C₅₃	C₂×#504	1	2	yes	—
179	D₅₃	—	2	4	no	—
180	D₅₃	—	4	2	no	—
181	C₂×#131	—	53	1	no	—
182	C₂³×C₈	—	53	1	no	—
183	#525	—	53	1	no	—
184	C₂×M₃₂	—	53	2	no	—
185	C₂²×C₁₆	—	53	2	no	—
186	C₂²×C₁₆	—	53	2	no	—
187	C₂²×C₁₆	—	53	2	no	—
188	#435	—	53	2	no	—
189	#435	—	53	2	no	—
190	C₈×D₄	—	53	2	no	—
191	C₈×D₄	—	53	2	no	—
192	#424	—	53	1	no	—
193	#408	—	53	1	no	—
194	#407	—	53	1	no	—
195	#405	C₂×D₅₃	1	1	yes	characteristic
196	C₄⋊C₁₆	—	53	2	no	—
197	C₄⋊C₁₆	C₂×D₅₃	1	2	yes	complemented
198	C₂²⋊C₁₆	—	53	2	no	—
199	C₂²⋊C₁₆	—	53	2	no	—
200	C₂²⋊C₁₆	—	53	2	no	—
201	C₂²⋊C₁₆	—	53	2	no	—
202	C₂²⋊C₁₆	—	53	2	no	—
203	C₂²⋊C₁₆	C₂×D₅₃	1	2	yes	—
204	#346	—	53	1	no	—
205	#346	C₂×D₅₃	1	1	yes	characteristic, complemented
206	C₄×C₁₆	—	53	1	no	—
207	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic, complemented
208	C₅₃	#33491	1	1	yes	characteristic, complemented, Sylow
209	C₂×#41	—	53	1	no	—
210	C₂³×C₄	—	53	1	no	—
211	#131	—	212	2	no	—
212	C₂×M₁₆	—	53	1	no	—
213	C₂×M₁₆	—	53	2	no	—
214	C₂×M₁₆	C₄×D₅₃	1	1	yes	characteristic
215	C₂²×C₈	—	53	1	no	—
216	C₂²×C₈	—	53	2	no	—
217	C₂²×C₈	—	53	2	no	—
218	C₂²×C₈	—	53	2	no	—
219	C₂²×C₈	—	212	2	no	—
220	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
221	#123	—	53	2	no	—
222	C₂²⋊Q₈	—	53	2	no	—
223	#121	—	53	2	no	—
224	#120	—	53	2	no	—
225	C₄×D₄	—	53	2	no	—
226	C₄×D₄	—	53	2	no	—
227	#117	C₄×D₅₃	1	1	yes	characteristic
228	M₃₂	—	212	2	no	—
229	C₂×C₁₆	—	53	2	no	—
230	C₂×C₁₆	—	53	2	no	—
231	C₂×C₁₆	—	212	2	no	—
232	C₂×C₁₆	—	212	2	no	—
233	C₂×C₁₆	—	212	2	no	—
234	C₂×C₁₆	C₂²×D₅₃	1	2	yes	—
235	C₂×C₁₆	C₂²×D₅₃	1	2	yes	—
236	C₄⋊C₈	—	53	2	no	—
237	C₄⋊C₈	—	53	2	no	—
238	C₂²⋊C₈	—	53	2	no	—
239	C₂²⋊C₈	—	53	2	no	—
240	C₂²⋊C₈	—	53	2	no	—
241	C₂²⋊C₈	—	53	2	no	—
242	#97	C₄×D₅₃	1	1	yes	characteristic
243	#97	C₄×D₅₃	1	1	yes	characteristic
244	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
245	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
246	C₂⁴	—	53	1	no	—
247	#41	—	212	1	no	—
248	#41	—	212	1	no	—
249	C₂×Q₈	—	53	1	no	—
250	C₂×D₄	—	53	2	no	—
251	C₂×D₄	—	53	2	no	—
252	C₂×D₄	—	53	2	no	—
253	C₂×D₄	—	53	1	no	—
254	C₂²×C₄	—	53	1	no	—
255	C₂²×C₄	—	53	2	no	—
256	C₂²×C₄	—	53	2	no	—
257	C₂²×C₄	—	53	2	no	—
258	C₂²×C₄	—	212	1	no	—
259	C₂²×C₄	—	212	1	no	—
260	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
261	M₁₆	—	4	1	no	—
262	M₁₆	—	212	2	no	—
263	M₁₆	—	212	1	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	53	2	no	—
266	C₂×C₈	—	53	2	no	—
267	C₂×C₈	—	106	2	no	—
268	C₂×C₈	—	106	2	no	—
269	C₂×C₈	—	106	2	no	—
270	C₂×C₈	—	106	2	no	—
271	C₂×C₈	—	212	2	no	—
272	C₂×C₈	—	212	1	no	—
273	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
274	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
275	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
276	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
277	C₄⋊C₄	—	53	2	no	—
278	C₄⋊C₄	—	53	2	no	—
279	C₄⋊C₄	C₈×D₅₃	1	2	yes	—
280	C₂²⋊C₄	—	53	2	no	—
281	C₂²⋊C₄	—	53	2	no	—
282	C₂²⋊C₄	—	53	2	no	—
283	C₂²⋊C₄	—	53	2	no	—
284	C₂²⋊C₄	C₈×D₅₃	1	2	yes	—
285	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
286	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
287	C₁₆	—	2	2	no	—
288	C₁₆	—	2	2	no	—
289	C₁₆	—	106	2	no	—
290	C₁₆	—	106	2	no	—
291	C₂³	—	53	2	no	—
292	C₂³	—	53	2	no	—
293	C₂³	—	106	4	no	—
294	C₂³	—	212	1	no	—
295	C₂³	C₂×C₈×D₅₃	1	1	yes	characteristic
296	Q₈	—	212	1	no	—
297	D₄	—	106	4	no	—
298	D₄	—	106	4	no	—
299	D₄	—	212	1	no	—
300	D₄	—	212	2	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	53	2	no	—
304	C₂×C₄	—	53	2	no	—
305	C₂×C₄	—	106	2	no	—
306	C₂×C₄	—	106	2	no	—
307	C₂×C₄	—	106	2	no	—
308	C₂×C₄	—	106	2	no	—
309	C₂×C₄	—	212	2	no	—
310	C₂×C₄	—	212	1	no	—
311	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
312	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
313	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
314	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
315	C₈	—	2	1	no	—
316	C₈	—	2	1	no	—
317	C₈	—	2	1	no	—
318	C₈	—	106	2	no	—
319	C₈	—	106	2	no	—
320	C₈	#13133263	1	2	yes	—
321	C₂²	—	2	2	no	—
322	C₂²	—	2	1	no	—
323	C₂²	—	53	4	no	—
324	C₂²	—	106	2	no	—
325	C₂²	—	106	2	no	—
326	C₂²	—	106	2	no	—
327	C₂²	—	106	2	no	—
328	C₂²	—	106	2	no	—
329	C₂²	—	212	4	no	—
330	C₂²	C₂²×C₈×D₅₃	1	1	yes	characteristic
331	C₄	—	2	2	no	—
332	C₄	—	2	2	no	—
333	C₄	—	2	1	no	—
334	C₄	—	106	2	no	—
335	C₄	—	106	2	no	—
336	C₄	#16811097	1	1	yes	characteristic
337	C₄	#16811389	1	1	yes	characteristic
338	C₂	—	4	1	no	—
339	C₂	—	106	4	no	—
340	C₂	—	212	2	no	—
341	C₂	#20283738	1	1	yes	characteristic
342	C₂	#504×D₅₃	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₈
Fitting	C₅₃×#2164	C₂
prime radical	#16811135	C₄
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	#2164	D₅₃
hypocenter	C₅₃	#33491
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ #2164

Complements

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

N	H ≅ G/N	type
2. #20276542	C₂	semidirect
3. #20276539	C₂	semidirect
4. C₁₆×D₂₁₂	C₂	semidirect
5. #20276534	C₂	semidirect
6. #20276522	C₂	semidirect
7. #20276327	C₂	semidirect
8. C₂²⋊C₁₆×D₅₃	C₂	semidirect
9. C₅₃×#2164	C₂	semidirect
13. C₂×#13133156	C₂²	semidirect
16. C₂×C₁₆×D₅₃	C₂²	semidirect
24. C₄⋊#11719908	C₂²	semidirect
28. #16810402	C₂²	semidirect
29. #16810402	C₂²	semidirect
30. #16810399	C₂²	semidirect
31. C₄×#11719908	C₂²	semidirect
33. C₅₃×C₄⋊C₁₆	C₂²	semidirect
35. C₅₃×#346	C₂²	semidirect
36. C₄×C₁₆×C₅₃	C₂²	semidirect
173. #2164	D₅₃	semidirect
197. C₄⋊C₁₆	C₂×D₅₃	semidirect
205. #346	C₂×D₅₃	semidirect
207. C₄×C₁₆	C₂×D₅₃	semidirect
208. C₅₃	#33491	semidirect

Maximal subgroups

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

subgroup	index
#20283411	2
#20276542	2
#20276539	2
C₁₆×D₂₁₂	2
#20276534	2
#20276522	2
#20276327	2
C₂²⋊C₁₆×D₅₃	2
C₅₃×#2164	2
#33491	53