Subgroups — #27277225

Order 13568.

466 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810688	C₂	1	1	yes	maximal, characteristic
2	#20274852	C₂	1	1	yes	maximal, characteristic, complemented
3	#20273461	C₂	1	1	yes	maximal, characteristic, complemented
4	#20273141	C₂	1	1	yes	maximal, characteristic
5	#20273139	C₂	1	1	yes	maximal, characteristic
6	#20272612	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1301	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₈×D₅₃	—	2	1	no	—
9	C₂×C₄×D₂₁₂	C₂²	1	1	yes	characteristic
10	#16810688	—	2	1	no	—
11	#16810688	—	2	1	no	—
12	#16810688	C₄	1	2	yes	—
13	#16810688	C₄	1	2	yes	—
14	#16810676	C₂²	1	1	yes	characteristic
15	#16810647	C₂²	1	1	yes	characteristic
16	C₂×#13133129	—	2	1	no	—
17	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
18	#16810387	—	2	1	no	—
19	#16810387	—	2	1	no	—
20	#16810375	—	2	1	no	—
21	#16810373	—	2	1	no	—
22	#16810352	—	2	2	no	—
23	#16810350	—	2	2	no	—
24	C₅₃×#431	C₂²	1	1	yes	characteristic
25	C₅₃×#423	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
27	C₂²×D₂₁₂	Q₈	1	1	yes	characteristic
28	C₂²×C₄×D₅₃	—	2	1	no	—
29	C₂×#11719917	—	2	1	no	—
30	C₂²×#88322	D₄	1	1	yes	characteristic
31	C₂²×#88322	D₄	1	1	yes	characteristic
32	C₂×C₈×D₅₃	—	2	2	no	—
33	C₂×C₈×D₅₃	—	2	2	no	—
34	C₂×C₈×D₅₃	—	4	2	no	—
35	C₂×C₈×D₅₃	—	4	2	no	—
36	C₄×D₂₁₂	—	2	1	no	—
37	C₄×D₂₁₂	—	2	1	no	—
38	C₄×D₂₁₂	C₂×C₄	1	2	yes	—
39	C₄×D₂₁₂	D₄	1	2	yes	—
40	C₂×C₄⋊Dic₅₃	Q₈	1	1	yes	characteristic
41	C₂×#11719921	—	2	1	no	—
42	C₄⋊#88322	—	2	1	no	—
43	C₄⋊#88322	—	2	1	no	—
44	C₄⋊#88322	—	2	1	no	—
45	C₂²⋊#88322	—	2	1	no	—
46	#13133129	—	2	2	no	—
47	#13133129	—	2	2	no	—
48	C₄×#88322	—	2	1	no	—
49	C₄×#88322	C₂×C₄	1	1	yes	characteristic
50	C₄×#88322	C₂×C₄	1	1	yes	characteristic
51	C₄×#88322	D₄	1	1	yes	characteristic
52	C₄×#88322	D₄	1	1	yes	characteristic
53	C₂×M₁₆×C₅₃	—	2	1	no	—
54	C₂²×C₈×C₅₃	—	2	1	no	—
55	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
56	C₄⋊C₈×C₅₃	—	2	1	no	—
57	C₄⋊C₈×C₅₃	—	2	1	no	—
58	C₄⋊C₈×C₅₃	—	2	1	no	—
59	C₄⋊C₈×C₅₃	—	2	1	no	—
60	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
61	C₂²⋊C₈×C₅₃	—	2	1	no	—
62	C₄×C₈×C₅₃	—	2	1	no	—
63	C₂³×D₅₃	—	2	1	no	—
64	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
66	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
67	C₂²×Dic₅₃	—	2	1	no	—
68	C₂×D₂₁₂	—	2	1	no	—
69	C₂×D₂₁₂	—	2	4	no	—
70	C₂×D₂₁₂	C₄⋊C₄	1	2	yes	—
71	C₂×C₄×D₅₃	—	2	2	no	—
72	C₂×C₄×D₅₃	—	2	2	no	—
73	C₂×C₄×D₅₃	—	4	1	no	—
74	C₂×C₄×D₅₃	—	4	1	no	—
75	C₂×C₄×D₅₃	—	4	1	no	—
76	C₂×C₄×D₅₃	—	4	1	no	—
77	M₁₆×C₅₃	—	4	1	no	—
78	M₁₆×C₅₃	—	4	1	no	—
79	C₂×C₈×C₅₃	—	2	2	no	—
80	C₂×C₈×C₅₃	—	4	1	no	—
81	C₂×C₈×C₅₃	—	4	1	no	—
82	C₂×C₈×C₅₃	—	4	1	no	—
83	C₂×C₈×C₅₃	—	4	1	no	—
84	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
85	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
86	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
87	C₄²×C₅₃	#41	1	1	yes	characteristic
88	#11719921	—	2	2	no	—
89	#11719921	—	2	2	no	—
90	C₄⋊Dic₅₃	—	2	1	no	—
91	C₄⋊Dic₅₃	C₄⋊C₄	1	2	yes	—
92	#11719917	—	4	1	no	—
93	#11719917	—	4	1	no	—
94	C₂×#88322	—	2	1	no	—
95	C₂×#88322	—	2	1	no	—
96	C₂×#88322	—	2	1	no	—
97	C₂×#88322	—	2	1	no	—
98	C₂×#88322	—	4	1	no	—
99	C₂×#88322	—	4	1	no	—
100	C₂×#88322	C₂²⋊C₄	1	1	yes	characteristic
101	C₂×#88322	C₂²⋊C₄	1	1	yes	characteristic
102	C₂×#88322	C₂²⋊C₄	1	1	yes	characteristic
103	C₂×#88322	C₂²⋊C₄	1	1	yes	characteristic
104	C₈×D₅₃	—	4	4	no	—
105	C₈×D₅₃	—	4	4	no	—
106	C₂³×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
107	C₂²×D₅₃	—	2	4	no	—
108	C₂²×D₅₃	—	2	2	no	—
109	C₂²×D₅₃	—	4	2	no	—
110	C₂²×D₅₃	—	4	1	no	—
111	C₂²×D₅₃	—	4	1	no	—
112	C₂×C₄×C₅₃	—	2	2	no	—
113	C₂×C₄×C₅₃	—	4	1	no	—
114	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
117	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
118	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
119	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
120	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
121	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
122	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
123	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
124	C₂×Dic₅₃	—	2	2	no	—
125	C₂×Dic₅₃	—	4	1	no	—
126	C₂×Dic₅₃	—	4	1	no	—
127	D₂₁₂	—	2	4	no	—
128	D₂₁₂	—	2	2	no	—
129	D₂₁₂	#108	1	4	yes	—
130	C₄×D₅₃	—	4	2	no	—
131	C₄×D₅₃	—	4	2	no	—
132	C₄×D₅₃	—	4	2	no	—
133	C₄×D₅₃	—	4	2	no	—
134	C₈×C₅₃	—	4	2	no	—
135	C₈×C₅₃	—	8	1	no	—
136	C₈×C₅₃	—	8	1	no	—
137	#88322	—	2	1	no	—
138	#88322	—	2	1	no	—
139	#88322	—	2	1	no	—
140	#88322	—	2	1	no	—
141	#88322	—	8	1	no	—
142	#88322	—	8	1	no	—
143	#8660	—	53	1	no	maximal, Sylow
144	C₂²×C₅₃	—	2	1	no	—
145	C₂²×C₅₃	—	2	1	no	—
146	C₂²×C₅₃	#380	1	1	yes	characteristic
147	C₂²×C₅₃	#478	1	1	yes	characteristic
148	C₂²×C₅₃	#478	1	1	yes	characteristic
149	C₂×D₅₃	—	2	4	no	—
150	C₂×D₅₃	—	4	4	no	—
151	C₂×D₅₃	—	4	4	no	—
152	C₂×D₅₃	—	4	2	no	—
153	C₂×D₅₃	—	4	2	no	—
154	C₄×C₅₃	—	2	1	no	—
155	C₄×C₅₃	—	2	1	no	—
156	C₄×C₅₃	—	2	1	no	—
157	C₄×C₅₃	#416	1	1	yes	characteristic
158	C₄×C₅₃	#418	1	1	yes	characteristic
159	C₄×C₅₃	C₂×#108	1	2	yes	—
160	C₄×C₅₃	#463	1	1	yes	characteristic
161	C₄×C₅₃	#463	1	1	yes	characteristic
162	Dic₅₃	—	4	2	no	—
163	C₂×C₈×D₄	—	53	1	no	—
164	#1837	—	53	1	no	—
165	#1564	—	53	1	no	—
166	#1486	—	53	1	no	—
167	#1486	—	53	1	no	—
168	#1301	—	53	1	no	—
169	#1301	D₅₃	1	1	yes	characteristic, complemented
170	C₂×C₅₃	—	2	1	no	—
171	C₂×C₅₃	—	2	1	no	—
172	C₂×C₅₃	#1804	1	1	yes	characteristic
173	C₂×C₅₃	#1810	1	1	yes	characteristic
174	C₂×C₅₃	#1814	1	1	yes	characteristic
175	D₅₃	—	4	4	no	—
176	C₂³×C₈	—	106	1	no	—
177	C₂×C₄×D₄	—	53	1	no	—
178	C₈×D₄	—	53	2	no	—
179	C₈×D₄	—	53	2	no	—
180	C₈×D₄	—	106	1	no	—
181	C₈×D₄	—	106	1	no	—
182	#431	—	53	1	no	—
183	#431	C₂×D₅₃	1	1	yes	characteristic
184	#423	—	53	1	no	—
185	#423	C₂×D₅₃	1	1	yes	characteristic
186	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
187	C₂×C₂²⋊C₈	—	106	1	no	—
188	C₂×C₄×C₈	—	53	1	no	—
189	#341	—	106	1	no	—
190	#341	—	106	1	no	—
191	#335	—	106	1	no	—
192	#334	—	106	1	no	—
193	D₄⋊C₈	—	106	2	no	—
194	D₄⋊C₈	—	106	2	no	—
195	C₅₃	#8660	1	1	yes	characteristic, complemented, Sylow
196	C₂²×D₄	—	53	1	no	—
197	C₂³×C₄	—	106	1	no	—
198	C₂×M₁₆	—	2	1	no	—
199	C₂×M₁₆	—	106	1	no	—
200	C₂²×C₈	—	2	1	no	—
201	C₂²×C₈	—	53	1	no	—
202	C₂²×C₈	—	53	1	no	—
203	C₂²×C₈	—	106	2	no	—
204	C₂²×C₈	—	106	2	no	—
205	C₂²×C₈	—	212	2	no	—
206	C₂²×C₈	—	212	2	no	—
207	C₄×D₄	—	53	2	no	—
208	C₄×D₄	—	53	2	no	—
209	C₄×D₄	—	106	1	no	—
210	C₄×D₄	—	106	1	no	—
211	C₂×C₄⋊C₄	—	53	1	no	—
212	C₂×C₂²⋊C₄	—	106	1	no	—
213	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
214	C₄⋊C₈	—	2	1	no	—
215	C₄⋊C₈	—	2	1	no	—
216	C₄⋊C₈	—	2	1	no	—
217	C₄⋊C₈	—	2	1	no	—
218	C₄⋊C₈	—	106	1	no	—
219	C₄⋊C₈	—	106	1	no	—
220	C₄⋊C₈	—	106	1	no	—
221	C₄⋊C₈	C₄×D₅₃	1	2	yes	—
222	C₂²⋊C₈	—	2	1	no	—
223	C₂²⋊C₈	—	106	2	no	—
224	C₂²⋊C₈	—	106	2	no	—
225	C₂²⋊C₈	—	106	1	no	—
226	C₄×C₈	—	2	1	no	—
227	C₄×C₈	—	53	1	no	—
228	C₄×C₈	—	53	1	no	—
229	C₄×C₈	—	53	1	no	—
230	C₄×C₈	—	53	1	no	—
231	C₄×C₈	—	106	1	no	—
232	C₂⁴	—	106	1	no	—
233	C₂×D₄	—	53	2	no	—
234	C₂×D₄	—	106	4	no	—
235	C₂×D₄	—	106	1	no	—
236	C₂²×C₄	—	106	1	no	—
237	C₂²×C₄	—	106	2	no	—
238	C₂²×C₄	—	106	2	no	—
239	C₂²×C₄	—	212	1	no	—
240	C₂²×C₄	—	212	1	no	—
241	C₂²×C₄	—	212	1	no	—
242	C₂²×C₄	—	212	1	no	—
243	C₂²×C₄	#11719945	1	1	yes	characteristic
244	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
245	C₂²×C₄	Q₈×D₅₃	1	1	yes	characteristic
246	M₁₆	—	4	1	no	—
247	M₁₆	—	4	1	no	—
248	M₁₆	—	212	1	no	—
249	M₁₆	—	212	1	no	—
250	C₂×C₈	—	2	2	no	—
251	C₂×C₈	—	4	1	no	—
252	C₂×C₈	—	4	1	no	—
253	C₂×C₈	—	4	1	no	—
254	C₂×C₈	—	4	1	no	—
255	C₂×C₈	—	53	1	no	—
256	C₂×C₈	—	53	1	no	—
257	C₂×C₈	—	53	1	no	—
258	C₂×C₈	—	53	1	no	—
259	C₂×C₈	—	106	1	no	—
260	C₂×C₈	—	106	1	no	—
261	C₂×C₈	—	106	1	no	—
262	C₂×C₈	—	106	1	no	—
263	C₂×C₈	—	212	4	no	—
264	C₂×C₈	—	212	4	no	—
265	C₂×C₈	—	212	1	no	—
266	C₂×C₈	—	212	1	no	—
267	C₄⋊C₄	—	53	2	no	—
268	C₄⋊C₄	—	106	1	no	—
269	C₂²⋊C₄	—	106	2	no	—
270	C₂²⋊C₄	—	106	2	no	—
271	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
272	C₄²	#11719945	1	1	yes	characteristic
273	C₄²	D₄×D₅₃	1	1	yes	characteristic
274	C₄²	D₄×D₅₃	1	1	yes	characteristic
275	C₂³	—	106	4	no	—
276	C₂³	—	106	2	no	—
277	C₂³	—	212	2	no	—
278	C₂³	—	212	1	no	—
279	C₂³	—	212	1	no	—
280	C₂³	#13133195	1	1	yes	characteristic
281	D₄	—	53	4	no	—
282	D₄	—	106	4	no	—
283	D₄	—	106	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	4	1	no	—
286	C₂×C₄	—	106	2	no	—
287	C₂×C₄	—	212	1	no	—
288	C₂×C₄	—	212	1	no	—
289	C₂×C₄	—	212	2	no	—
290	C₂×C₄	—	212	2	no	—
291	C₂×C₄	—	212	2	no	—
292	C₂×C₄	—	212	2	no	—
293	C₂×C₄	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
294	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
295	C₂×C₄	#13133174	1	1	yes	characteristic
296	C₂×C₄	#13133190	1	1	yes	characteristic
297	C₂×C₄	#13133190	1	1	yes	characteristic
298	C₂×C₄	#13133195	1	1	yes	characteristic
299	C₂×C₄	#13133238	1	1	yes	characteristic
300	C₂×C₄	#13133240	1	1	yes	characteristic
301	C₂×C₄	#13133241	1	1	yes	characteristic
302	C₂×C₄	#13133244	1	1	yes	characteristic
303	C₈	—	4	2	no	—
304	C₈	—	8	1	no	—
305	C₈	—	8	1	no	—
306	C₈	—	106	1	no	—
307	C₈	—	106	1	no	—
308	C₈	—	106	1	no	—
309	C₈	—	106	1	no	—
310	C₈	—	424	1	no	—
311	C₈	—	424	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	106	4	no	—
315	C₂²	—	212	4	no	—
316	C₂²	—	212	4	no	—
317	C₂²	—	212	2	no	—
318	C₂²	—	212	2	no	—
319	C₂²	#16810482	1	1	yes	characteristic
320	C₂²	#16810922	1	1	yes	characteristic
321	C₂²	#16810926	1	1	yes	characteristic
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	2	1	no	—
325	C₄	—	212	2	no	—
326	C₄	#16810618	1	1	yes	characteristic
327	C₄	#16810626	1	1	yes	characteristic
328	C₄	#108×D₅₃	1	2	yes	—
329	C₄	#16810837	1	1	yes	characteristic
330	C₄	#16810841	1	1	yes	characteristic
331	C₂	—	2	1	no	—
332	C₂	—	2	1	no	—
333	C₂	—	212	4	no	—
334	C₂	#20274724	1	1	yes	characteristic
335	C₂	#20274752	1	1	yes	characteristic
336	C₂	#20274768	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13133190
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1301	C₂
socle	C₂²×C₅₃	#380
prime radical	C₂²×D₂₁₂	Q₈
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1301	D₅₃
hypocenter	C₅₃	#8660
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1301

Complements

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

N	H ≅ G/N	type
2. #20274852	C₂	semidirect
3. #20273461	C₂	semidirect
6. #20272612	C₂	semidirect
7. C₅₃×#1301	C₂	semidirect
169. #1301	D₅₃	semidirect
195. C₅₃	#8660	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810688	2
#20274852	2
#20273461	2
#20273141	2
#20273139	2
#20272612	2
C₅₃×#1301	2
#8660	53