Subgroups — #27542159

Order 13568.

510 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20287725	C₂	1	1	yes	maximal, characteristic
2	#20287719	C₂	1	1	yes	maximal, characteristic, complemented
3	#20276983	C₂	1	2	yes	maximal, complemented
4	#20276972	C₂	1	1	yes	maximal, characteristic
5	#20276779	C₂	1	2	yes	maximal, complemented
6	#20276759	C₂	1	2	yes	maximal, complemented
7	#20276756	C₂	1	1	yes	maximal, characteristic
8	#20276620	C₂	1	1	yes	maximal, characteristic, complemented
9	#20276582	C₂	1	2	yes	maximal, complemented
10	C₄×#13133163	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#2209	C₂	1	1	yes	maximal, characteristic, complemented
12	#16811358	—	2	1	no	—
13	#16811348	—	2	1	no	—
14	C₂×#13133163	C₂²	1	1	yes	characteristic
15	C₂×#13133162	C₂²	1	2	yes	—
16	C₂×#13133160	C₂²	1	2	yes	—
17	C₂×#13133160	C₂²	1	2	yes	—
18	#16811008	C₂²	1	2	yes	—
19	#16811001	C₂²	1	2	yes	—
20	#16811000	C₂²	1	1	yes	characteristic
21	#16810841	—	2	2	no	—
22	#16810831	—	2	2	no	—
23	#16810830	—	2	1	no	—
24	#16810823	—	2	1	no	—
25	C₈×Dic₁₀₆	C₂²	1	1	yes	characteristic
26	C₄×Dic₂₁₂	C₂²	1	1	yes	characteristic
27	Q₁₆×Dic₅₃	C₂²	1	2	yes	—
28	#16810705	C₂²	1	2	yes	—
29	#16810448	C₂²	1	1	yes	characteristic
30	#16810441	C₂²	1	2	yes	—
31	#16810427	C₂²	1	1	yes	characteristic
32	#16810426	C₂²	1	1	yes	characteristic
33	#16810425	C₂²	1	2	yes	—
34	#16810398	C₂²	1	2	yes	—
35	C₄×#11719908	C₂²	1	1	yes	characteristic
36	C₂×Q₃₂×C₅₃	C₂²	1	2	yes	—
37	C₅₃×#494	C₂²	1	1	yes	characteristic
38	C₄×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
39	C₅₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
40	C₅₃×#358	C₂²	1	2	yes	—
41	C₂×Dic₂₁₂	C₂³	1	1	yes	characteristic
42	C₂×#11719925	—	2	2	no	—
43	C₂×#11719924	—	2	2	no	—
44	C₂×D₄₂₄	—	2	1	no	—
45	C₂×D₄₂₄	C₂³	1	2	yes	—
46	#13133215	—	2	2	no	—
47	#13133214	—	2	1	no	—
48	#13133208	—	4	1	no	—
49	#13133207	—	4	2	no	—
50	#13133206	—	2	2	no	—
51	#13133205	—	4	1	no	—
52	#13133202	—	4	2	no	—
53	Q₈×Dic₅₃	—	2	2	no	—
54	C₄×Dic₁₀₆	—	2	1	no	—
55	C₄×Dic₁₀₆	D₄	1	1	yes	characteristic
56	#13133182	—	4	1	no	—
57	C₄×D₂₁₂	—	4	1	no	—
58	#13133179	—	2	2	no	—
59	#13133163	—	2	2	no	—
60	#13133162	—	4	2	no	—
61	#13133160	—	2	4	no	—
62	#13133160	—	4	2	no	—
63	C₂×#11719908	C₂³	1	1	yes	characteristic
64	C₂×#11719908	C₂³	1	1	yes	characteristic
65	#13133151	C₂³	1	1	yes	characteristic
66	C₄⋊#88322	D₄	1	1	yes	characteristic
67	#13133146	D₄	1	2	yes	—
68	#13133142	—	2	1	no	—
69	#13133141	—	2	2	no	—
70	#13133137	—	2	2	no	—
71	C₈×Dic₅₃	C₂³	1	2	yes	—
72	C₂×Q₁₆×C₅₃	—	2	1	no	—
73	C₂×Q₁₆×C₅₃	C₂³	1	1	yes	characteristic
74	C₂×Q₁₆×C₅₃	C₂³	1	2	yes	—
75	C₄⋊Q₈×C₅₃	—	2	1	no	—
76	C₄×Q₈×C₅₃	—	2	1	no	—
77	Q₃₂×C₅₃	—	4	2	no	—
78	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
79	#107×C₅₃	C₂³	1	1	yes	characteristic
80	#103×C₅₃	—	2	1	no	—
81	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
82	C₂×Q₈×C₅₃	—	2	1	no	—
83	C₂×Q₈×C₅₃	—	2	2	no	—
84	C₂×D₂₁₂	—	2	2	no	—
85	C₂×D₂₁₂	—	4	1	no	—
86	C₂×D₂₁₂	—	4	1	no	—
87	C₂×C₄×D₅₃	—	4	2	no	—
88	C₂×Dic₁₀₆	—	2	1	no	—
89	C₂×Dic₁₀₆	D₈	1	1	yes	characteristic
90	Q₁₆×C₅₃	—	2	2	no	—
91	Q₁₆×C₅₃	—	2	1	no	—
92	Q₁₆×C₅₃	—	2	1	no	—
93	Q₁₆×C₅₃	—	2	2	no	—
94	Q₁₆×C₅₃	—	8	1	no	—
95	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
97	C₄⋊C₄×C₅₃	—	2	1	no	—
98	C₄⋊C₄×C₅₃	—	4	1	no	—
99	C₄⋊C₄×C₅₃	—	4	1	no	—
100	C₄⋊C₄×C₅₃	—	4	1	no	—
101	C₄²×C₅₃	—	4	1	no	—
102	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
103	#11719925	—	8	2	no	—
104	#11719924	—	8	2	no	—
105	#11719921	—	4	2	no	—
106	#11719921	—	4	2	no	—
107	#11719921	—	4	2	no	—
108	C₄⋊Dic₅₃	—	2	1	no	—
109	C₄⋊Dic₅₃	—	4	2	no	—
110	C₄⋊Dic₅₃	D₈	1	1	yes	characteristic
111	#11719919	—	4	1	no	—
112	#11719919	—	4	2	no	—
113	#11719919	—	4	2	no	—
114	#11719919	D₈	1	2	yes	—
115	C₄×Dic₅₃	—	4	1	no	—
116	C₄×Dic₅₃	—	4	2	no	—
117	C₄×Dic₅₃	C₂×D₄	1	2	yes	—
118	C₂×#88322	C₂×D₄	1	2	yes	—
119	Dic₂₁₂	—	2	1	no	—
120	Dic₂₁₂	—	2	1	no	—
121	D₄₂₄	—	2	2	no	—
122	D₄₂₄	—	2	2	no	—
123	D₄₂₄	—	8	1	no	—
124	C₁₆×C₅₃	—	2	2	no	—
125	#11719908	—	2	1	no	—
126	#11719908	#41	1	2	yes	—
127	C₂²×D₅₃	—	4	2	no	—
128	Q₈×C₅₃	—	4	1	no	—
129	Q₈×C₅₃	—	4	2	no	—
130	Q₈×C₅₃	—	4	1	no	—
131	Q₈×C₅₃	—	4	2	no	—
132	C₂×C₄×C₅₃	—	4	1	no	—
133	C₂×C₄×C₅₃	—	4	2	no	—
134	C₂×C₄×C₅₃	—	4	1	no	—
135	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
136	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
137	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
138	C₂×Dic₅₃	—	4	1	no	—
139	C₂×Dic₅₃	—	4	1	no	—
140	C₂×Dic₅₃	C₂×D₈	1	2	yes	—
141	C₂×Dic₅₃	C₂×D₈	1	2	yes	—
142	D₂₁₂	—	4	2	no	—
143	D₂₁₂	—	4	2	no	—
144	D₂₁₂	—	8	2	no	—
145	D₂₁₂	—	16	1	no	—
146	C₄×D₅₃	—	16	2	no	—
147	Dic₁₀₆	—	2	2	no	—
148	Dic₁₀₆	—	4	1	no	—
149	Dic₁₀₆	—	4	1	no	—
150	C₈×C₅₃	—	2	1	no	—
151	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
152	C₈×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
153	#88322	—	2	2	no	—
154	#33967	—	53	1	no	maximal, Sylow
155	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
156	C₂×D₅₃	—	8	2	no	—
157	C₂×D₅₃	—	8	2	no	—
158	C₂×D₅₃	—	8	2	no	—
159	C₄×C₅₃	—	2	1	no	—
160	C₄×C₅₃	—	8	2	no	—
161	C₄×C₅₃	—	8	1	no	—
162	C₄×C₅₃	—	8	1	no	—
163	C₄×C₅₃	#510	1	2	yes	—
164	C₄×C₅₃	#550	1	1	yes	characteristic
165	C₄×C₅₃	#576	1	1	yes	characteristic
166	Dic₅₃	—	2	2	no	—
167	Dic₅₃	—	2	2	no	—
168	Dic₅₃	—	8	1	no	—
169	Dic₅₃	—	8	1	no	—
170	#3396	—	53	1	no	—
171	#3395	—	53	1	no	—
172	#2251	—	53	2	no	—
173	#2248	—	53	1	no	—
174	#2211	—	53	2	no	—
175	#2209	D₅₃	1	1	yes	characteristic, complemented
176	#2208	—	53	2	no	—
177	#2207	—	53	1	no	—
178	#2183	—	53	1	no	—
179	#2175	—	53	2	no	—
180	C₄×Q₃₂	—	53	1	no	—
181	C₂×C₅₃	#3391	1	1	yes	characteristic
182	C₂×C₅₃	C₂×#510	1	1	yes	characteristic
183	C₂×C₅₃	#3416	1	1	yes	characteristic
184	D₅₃	—	16	2	no	—
185	#559	—	106	1	no	—
186	#557	—	106	1	no	—
187	C₂×Q₃₂	—	53	1	no	—
188	C₂×Q₃₂	C₂×D₅₃	1	2	yes	—
189	C₂×SD₃₂	—	53	2	no	—
190	C₂×SD₃₂	—	53	2	no	—
191	C₂×SD₃₂	—	53	2	no	—
192	#495	—	53	2	no	—
193	#494	—	53	2	no	—
194	#494	C₂×D₅₃	1	1	yes	characteristic
195	#493	—	53	1	no	—
196	#463	—	106	2	no	—
197	#462	—	106	1	no	—
198	#462	—	106	2	no	—
199	#460	—	106	1	no	—
200	C₈×Q₈	—	53	1	no	—
201	C₄×Q₁₆	—	53	2	no	—
202	C₄×Q₁₆	—	53	1	no	—
203	C₄×Q₁₆	C₂×D₅₃	1	1	yes	characteristic
204	C₄×D₈	—	53	2	no	—
205	#366	—	53	1	no	—
206	C₄⋊C₁₆	—	53	2	no	—
207	C₄⋊C₁₆	C₂×D₅₃	1	1	yes	characteristic
208	#358	—	53	1	no	—
209	#358	—	53	1	no	—
210	#358	C₂×D₅₃	1	2	yes	—
211	#357	—	53	2	no	—
212	#346	—	53	2	no	—
213	C₄×C₁₆	—	53	1	no	—
214	C₅₃	#33967	1	1	yes	characteristic, complemented, Sylow
215	C₂×Q₁₆	—	2	1	no	—
216	C₂×Q₁₆	—	53	1	no	—
217	C₂×Q₁₆	—	106	2	no	—
218	C₂×Q₁₆	C₂²×D₅₃	1	1	yes	characteristic
219	C₂×Q₁₆	C₂²×D₅₃	1	2	yes	—
220	C₂×SD₁₆	—	106	2	no	—
221	C₂×D₈	—	53	2	no	—
222	C₂×D₈	—	106	1	no	—
223	C₄⋊Q₈	—	2	1	no	—
224	C₄⋊Q₈	—	106	2	no	—
225	#127	—	106	1	no	—
226	#125	—	212	2	no	—
227	#125	—	212	1	no	—
228	#124	—	106	2	no	—
229	#124	—	212	1	no	—
230	#123	—	212	2	no	—
231	C₄×Q₈	—	2	1	no	—
232	C₄×Q₈	—	53	1	no	—
233	C₄×Q₈	—	106	2	no	—
234	C₄×Q₈	—	106	1	no	—
235	C₄×Q₈	—	212	1	no	—
236	C₄×D₄	—	106	2	no	—
237	C₄×D₄	—	212	1	no	—
238	Q₃₂	—	4	2	no	—
239	Q₃₂	—	106	2	no	—
240	SD₃₂	—	106	4	no	—
241	SD₃₂	—	212	2	no	—
242	SD₃₂	—	212	2	no	—
243	C₂×C₁₆	—	53	1	no	—
244	C₂×C₁₆	—	53	1	no	—
245	C₂×C₁₆	C₂²×D₅₃	1	2	yes	—
246	#107	—	53	1	no	—
247	#107	C₂²×D₅₃	1	1	yes	characteristic
248	C₄⋊C₈	—	53	2	no	—
249	C₄⋊C₈	—	53	1	no	—
250	#103	—	2	1	no	—
251	#103	—	106	2	no	—
252	#103	—	106	1	no	—
253	#102	—	106	2	no	—
254	C₄×C₈	—	53	2	no	—
255	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
256	C₂×Q₈	—	2	1	no	—
257	C₂×Q₈	—	2	2	no	—
258	C₂×Q₈	—	53	1	no	—
259	C₂×Q₈	—	106	1	no	—
260	C₂×D₄	—	106	2	no	—
261	C₂×D₄	—	212	1	no	—
262	C₂×D₄	—	212	1	no	—
263	C₂²×C₄	—	212	2	no	—
264	Q₁₆	—	2	2	no	—
265	Q₁₆	—	2	1	no	—
266	Q₁₆	—	2	2	no	—
267	Q₁₆	—	2	1	no	—
268	Q₁₆	—	8	1	no	—
269	Q₁₆	—	106	1	no	—
270	Q₁₆	—	106	1	no	—
271	Q₁₆	—	424	2	no	—
272	SD₁₆	—	424	2	no	—
273	D₈	—	106	2	no	—
274	D₈	—	106	2	no	—
275	D₈	—	424	1	no	—
276	C₂×C₈	—	53	2	no	—
277	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
278	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
279	C₄⋊C₄	—	2	1	no	—
280	C₄⋊C₄	—	4	1	no	—
281	C₄⋊C₄	—	4	1	no	—
282	C₄⋊C₄	—	4	1	no	—
283	C₄⋊C₄	—	53	1	no	—
284	C₄⋊C₄	—	53	2	no	—
285	C₄⋊C₄	—	106	1	no	—
286	C₄⋊C₄	—	212	2	no	—
287	C₄⋊C₄	—	212	2	no	—
288	C₄⋊C₄	—	212	2	no	—
289	C₄⋊C₄	—	212	1	no	—
290	C₂²⋊C₄	—	212	2	no	—
291	C₂²⋊C₄	—	212	2	no	—
292	C₂²⋊C₄	—	212	2	no	—
293	C₄²	—	4	1	no	—
294	C₄²	—	53	2	no	—
295	C₄²	—	212	2	no	—
296	C₄²	—	212	1	no	—
297	C₄²	D₄×D₅₃	1	1	yes	characteristic
298	C₁₆	—	2	2	no	—
299	C₁₆	—	53	2	no	—
300	C₁₆	—	106	1	no	—
301	C₂³	—	212	2	no	—
302	Q₈	—	4	2	no	—
303	Q₈	—	4	1	no	—
304	Q₈	—	4	1	no	—
305	Q₈	—	4	2	no	—
306	Q₈	—	106	2	no	—
307	Q₈	—	212	1	no	—
308	Q₈	—	212	1	no	—
309	D₄	—	848	1	no	—
310	D₄	—	424	2	no	—
311	D₄	—	212	2	no	—
312	D₄	—	212	2	no	—
313	C₂×C₄	—	4	1	no	—
314	C₂×C₄	—	4	1	no	—
315	C₂×C₄	—	4	2	no	—
316	C₂×C₄	—	53	2	no	—
317	C₂×C₄	—	53	2	no	—
318	C₂×C₄	—	848	2	no	—
319	C₂×C₄	—	212	1	no	—
320	C₂×C₄	—	212	1	no	—
321	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
322	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
323	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
324	C₈	—	2	1	no	—
325	C₈	—	106	2	no	—
326	C₈	#41×D₅₃	1	1	yes	characteristic
327	C₈	#13133262	1	1	yes	characteristic
328	C₂²	—	424	2	no	—
329	C₂²	—	424	2	no	—
330	C₂²	—	424	2	no	—
331	C₂²	C₂×D₈×D₅₃	1	1	yes	characteristic
332	C₄	—	2	1	no	—
333	C₄	—	8	1	no	—
334	C₄	—	8	1	no	—
335	C₄	—	8	2	no	—
336	C₄	—	106	2	no	—
337	C₄	—	106	2	no	—
338	C₄	—	424	1	no	—
339	C₄	—	424	1	no	—
340	C₄	#16811065	1	2	yes	—
341	C₄	#16811291	1	1	yes	characteristic
342	C₄	#16811427	1	1	yes	characteristic
343	C₂	—	848	2	no	—
344	C₂	#20287686	1	1	yes	characteristic
345	C₂	C₂×#16811065	1	1	yes	characteristic
346	C₂	#20287860	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₅₃
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2209	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	#16811000	C₂²
hypercenter	#2209	D₅₃
hypocenter	C₅₃	#33967
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2209

Complements

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

N	H ≅ G/N	type
2. #20287719	C₂	semidirect
3. #20276983	C₂	semidirect
5. #20276779	C₂	semidirect
6. #20276759	C₂	semidirect
8. #20276620	C₂	semidirect
9. #20276582	C₂	semidirect
10. C₄×#13133163	C₂	semidirect
11. C₅₃×#2209	C₂	semidirect
175. #2209	D₅₃	semidirect
214. C₅₃	#33967	semidirect

Maximal subgroups

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

subgroup	index
#20287725	2
#20287719	2
#20276983	2
#20276972	2
#20276779	2
#20276759	2
#20276756	2
#20276620	2
#20276582	2
C₄×#13133163	2
C₅₃×#2209	2
#33967	53