Subgroups — #27276894

Order 13568.

664 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20274867	C₂	1	1	yes	maximal, characteristic, complemented
2	#20274832	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#16810367	C₂	1	2	yes	maximal
4	#20272598	C₂	1	2	yes	maximal, complemented
5	C₄×#127×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#88322	—	2	1	no	—
7	C₂×C₄⋊#88322	—	2	1	no	—
8	C₂×C₄⋊#88322	C₂²	1	2	yes	—
9	C₂×C₄⋊#88322	C₂²	1	2	yes	—
10	#16810378	—	2	1	no	—
11	#16810378	—	2	1	no	—
12	#16810367	—	2	4	no	—
13	C₂×#127×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₄×C₅₃	—	2	1	no	—
15	C₂×C₄×D₄×C₅₃	—	2	1	no	—
16	C₂×C₄×D₄×C₅₃	C₄	1	2	yes	—
17	C₅₃×#390	—	2	1	no	—
18	C₅₃×#390	—	2	1	no	—
19	C₅₃×#390	C₄	1	2	yes	—
20	C₅₃×#384	C₂²	1	1	yes	characteristic
21	C₄³×C₅₃	C₂²	1	1	yes	characteristic
22	C₂²×#88322	—	2	2	no	—
23	C₂²×#88322	—	2	2	no	—
24	C₄⋊#88322	—	2	2	no	—
25	C₄⋊#88322	—	2	2	no	—
26	C₄⋊#88322	—	2	2	no	—
27	C₄⋊#88322	—	2	2	no	—
28	C₄⋊#88322	—	4	2	no	—
29	C₄⋊#88322	—	4	2	no	—
30	C₂²×D₄×C₅₃	—	2	1	no	—
31	C₂²×D₄×C₅₃	—	2	1	no	—
32	C₂²×D₄×C₅₃	C₂×C₄	1	2	yes	—
33	C₂³×C₄×C₅₃	—	2	2	no	—
34	#127×C₅₃	—	2	1	no	—
35	#127×C₅₃	—	2	1	no	—
36	#127×C₅₃	D₄	1	2	yes	—
37	#127×C₅₃	Q₈	1	2	yes	—
38	C₄×D₄×C₅₃	—	4	4	no	—
39	C₄×D₄×C₅₃	—	4	4	no	—
40	C₄×D₄×C₅₃	—	4	2	no	—
41	C₄×D₄×C₅₃	—	4	2	no	—
42	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
43	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
44	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
45	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
46	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
47	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
48	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
51	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
52	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
53	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
54	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
55	C₂⁴×C₅₃	—	2	2	no	—
56	C₂×D₄×C₅₃	—	2	2	no	—
57	C₂×D₄×C₅₃	—	2	2	no	—
58	C₂×D₄×C₅₃	—	2	2	no	—
59	C₂×D₄×C₅₃	—	2	2	no	—
60	C₂×D₄×C₅₃	—	2	2	no	—
61	C₂×D₄×C₅₃	—	4	2	no	—
62	C₂×D₄×C₅₃	—	4	2	no	—
63	C₂×D₄×C₅₃	—	4	1	no	—
64	C₂×D₄×C₅₃	—	4	1	no	—
65	C₂×D₄×C₅₃	—	4	2	no	—
66	C₂×D₄×C₅₃	—	4	2	no	—
67	C₂×D₄×C₅₃	—	4	1	no	—
68	C₂×D₄×C₅₃	—	4	1	no	—
69	C₂×D₄×C₅₃	C₄⋊C₄	1	4	yes	—
70	C₂²×C₄×C₅₃	—	2	2	no	—
71	C₂²×C₄×C₅₃	—	4	4	no	—
72	C₂²×C₄×C₅₃	—	4	4	no	—
73	C₂²×C₄×C₅₃	—	4	4	no	—
74	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
75	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
76	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
77	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
78	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
81	C₄⋊C₄×C₅₃	—	4	2	no	—
82	C₄⋊C₄×C₅₃	—	4	1	no	—
83	C₄⋊C₄×C₅₃	—	4	1	no	—
84	C₄⋊C₄×C₅₃	—	4	2	no	—
85	C₂²⋊C₄×C₅₃	—	4	4	no	—
86	C₂²⋊C₄×C₅₃	—	4	4	no	—
87	C₂²⋊C₄×C₅₃	—	4	4	no	—
88	C₄²×C₅₃	—	2	2	no	—
89	C₄²×C₅₃	—	2	2	no	—
90	C₄²×C₅₃	—	2	2	no	—
91	C₄²×C₅₃	—	2	1	no	—
92	C₄²×C₅₃	—	2	1	no	—
93	C₄²×C₅₃	—	4	1	no	—
94	C₄²×C₅₃	D₈	1	2	yes	—
95	C₄²×C₅₃	SD₁₆	1	2	yes	—
96	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
97	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
98	C₄²×C₅₃	#41	1	1	yes	characteristic
99	C₄²×C₅₃	#41	1	1	yes	characteristic
100	C₂×#88322	—	4	2	no	—
101	C₂×#88322	—	4	2	no	—
102	C₂×#88322	—	4	2	no	—
103	C₂×#88322	—	4	2	no	—
104	C₂×#88322	—	4	2	no	—
105	C₂×#88322	—	4	2	no	—
106	C₂³×C₅₃	—	2	4	no	—
107	C₂³×C₅₃	—	4	2	no	—
108	C₂³×C₅₃	—	4	2	no	—
109	C₂³×C₅₃	—	4	2	no	—
110	C₂³×C₅₃	—	4	2	no	—
111	C₂³×C₅₃	—	4	2	no	—
112	C₂³×C₅₃	—	4	2	no	—
113	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	D₄×C₅₃	—	4	4	no	—
115	D₄×C₅₃	—	4	4	no	—
116	D₄×C₅₃	—	4	2	no	—
117	D₄×C₅₃	—	4	2	no	—
118	D₄×C₅₃	—	4	4	no	—
119	D₄×C₅₃	—	4	2	no	—
120	D₄×C₅₃	—	4	2	no	—
121	D₄×C₅₃	—	4	4	no	—
122	C₂×C₄×C₅₃	—	2	1	no	—
123	C₂×C₄×C₅₃	—	2	1	no	—
124	C₂×C₄×C₅₃	—	2	1	no	—
125	C₂×C₄×C₅₃	—	2	1	no	—
126	C₂×C₄×C₅₃	—	2	1	no	—
127	C₂×C₄×C₅₃	—	2	1	no	—
128	C₂×C₄×C₅₃	—	2	1	no	—
129	C₂×C₄×C₅₃	—	2	1	no	—
130	C₂×C₄×C₅₃	—	2	1	no	—
131	C₂×C₄×C₅₃	—	2	1	no	—
132	C₂×C₄×C₅₃	—	2	1	no	—
133	C₂×C₄×C₅₃	—	4	2	no	—
134	C₂×C₄×C₅₃	—	4	2	no	—
135	C₂×C₄×C₅₃	—	4	2	no	—
136	C₂×C₄×C₅₃	—	8	8	no	—
137	C₂×C₄×C₅₃	#100	1	2	yes	—
138	C₂×C₄×C₅₃	#102	1	2	yes	—
139	C₂×C₄×C₅₃	#102	1	2	yes	—
140	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
141	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
142	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
143	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
144	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
145	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
146	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
147	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
148	C₂×C₄×C₅₃	#135	1	2	yes	—
149	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
150	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
151	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
152	#88322	—	8	2	no	—
153	#88322	—	8	2	no	—
154	#8588	—	53	1	no	maximal, Sylow
155	C₂²×C₅₃	—	4	4	no	—
156	C₂²×C₅₃	—	4	4	no	—
157	C₂²×C₅₃	—	4	4	no	—
158	C₂²×C₅₃	—	8	2	no	—
159	C₂²×C₅₃	—	8	2	no	—
160	C₂²×C₅₃	—	8	2	no	—
161	C₂²×C₅₃	—	8	2	no	—
162	C₂²×C₅₃	#380	1	1	yes	characteristic
163	C₂²×C₅₃	C₂×#100	1	1	yes	characteristic
164	C₂²×C₅₃	#413	1	1	yes	characteristic
165	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
166	C₂²×C₅₃	#416	1	1	yes	characteristic
167	C₂²×C₅₃	#417	1	1	yes	characteristic
168	C₂²×C₅₃	#418	1	1	yes	characteristic
169	C₄×C₅₃	—	2	2	no	—
170	C₄×C₅₃	—	2	1	no	—
171	C₄×C₅₃	—	2	1	no	—
172	C₄×C₅₃	—	2	2	no	—
173	C₄×C₅₃	—	2	2	no	—
174	C₄×C₅₃	—	4	1	no	—
175	C₄×C₅₃	—	8	2	no	—
176	C₄×C₅₃	#459	1	1	yes	characteristic
177	C₄×C₅₃	#461	1	1	yes	characteristic
178	C₄×C₅₃	#463	1	2	yes	—
179	C₄×C₅₃	#474	1	1	yes	characteristic
180	C₄×C₅₃	#476	1	1	yes	characteristic
181	C₄×C₅₃	#478	1	2	yes	—
182	C₄×#127	D₅₃	1	1	yes	characteristic
183	#1842	—	53	1	no	—
184	#1831	—	53	1	no	—
185	C₂×#331	—	53	2	no	—
186	#1296	—	53	2	no	—
187	C₂×C₅₃	—	8	4	no	—
188	C₂×C₅₃	#1540	1	1	yes	characteristic
189	C₂×C₅₃	#1545	1	1	yes	characteristic
190	C₂×C₅₃	#1551	1	1	yes	characteristic
191	C₂×C₅₃	#1557	1	1	yes	characteristic
192	C₂×C₅₃	#1800	1	1	yes	characteristic
193	C₂×C₅₃	#1802	1	1	yes	characteristic
194	C₂×C₅₃	#1804	1	1	yes	characteristic
195	C₂×#127	C₂×D₅₃	1	1	yes	characteristic
196	C₂×C₄×D₄	—	2	1	no	—
197	C₂×C₄×D₄	—	2	1	no	—
198	C₂×C₄×D₄	Dic₅₃	1	2	yes	—
199	C₂×C₄⋊C₈	—	53	2	no	—
200	C₂×C₄⋊C₈	—	53	2	no	—
201	C₂×C₄⋊C₈	—	106	1	no	—
202	C₂×C₄⋊C₈	—	106	1	no	—
203	#390	—	2	1	no	—
204	#390	—	2	1	no	—
205	#390	Dic₅₃	1	2	yes	—
206	#384	C₂×D₅₃	1	1	yes	characteristic
207	C₄³	C₂×D₅₃	1	1	yes	characteristic
208	#336	—	106	1	no	—
209	#336	—	106	1	no	—
210	#331	—	106	4	no	—
211	C₅₃	#8588	1	1	yes	characteristic, complemented, Sylow
212	C₂²×D₄	—	2	1	no	—
213	C₂²×D₄	—	2	1	no	—
214	C₂²×D₄	C₂×Dic₅₃	1	2	yes	—
215	C₂³×C₄	—	2	2	no	—
216	C₂²×C₈	—	106	2	no	—
217	C₂²×C₈	—	106	2	no	—
218	#127	—	2	1	no	—
219	#127	—	2	1	no	—
220	#127	Dic₁₀₆	1	2	yes	—
221	#127	D₂₁₂	1	2	yes	—
222	C₄×D₄	—	4	4	no	—
223	C₄×D₄	—	4	4	no	—
224	C₄×D₄	—	4	2	no	—
225	C₄×D₄	—	4	2	no	—
226	C₂×C₄⋊C₄	—	2	1	no	—
227	C₂×C₄⋊C₄	—	2	1	no	—
228	C₂×C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
229	C₂×C₂²⋊C₄	—	2	2	no	—
230	C₂×C₂²⋊C₄	—	2	2	no	—
231	C₂×C₂²⋊C₄	—	2	2	no	—
232	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
233	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
234	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
235	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
236	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
237	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
238	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
239	C₄⋊C₈	—	106	2	no	—
240	C₄⋊C₈	—	106	2	no	—
241	C₄⋊C₈	—	106	2	no	—
242	C₄⋊C₈	—	106	2	no	—
243	C₄⋊C₈	—	212	2	no	—
244	C₄⋊C₈	—	212	2	no	—
245	C₂⁴	—	2	2	no	—
246	C₂×D₄	—	2	2	no	—
247	C₂×D₄	—	2	2	no	—
248	C₂×D₄	—	2	2	no	—
249	C₂×D₄	—	2	2	no	—
250	C₂×D₄	—	2	2	no	—
251	C₂×D₄	—	4	2	no	—
252	C₂×D₄	—	4	2	no	—
253	C₂×D₄	—	4	1	no	—
254	C₂×D₄	—	4	1	no	—
255	C₂×D₄	—	4	2	no	—
256	C₂×D₄	—	4	1	no	—
257	C₂×D₄	—	4	2	no	—
258	C₂×D₄	—	4	1	no	—
259	C₂×D₄	C₄⋊Dic₅₃	1	4	yes	—
260	C₂²×C₄	—	2	2	no	—
261	C₂²×C₄	—	4	4	no	—
262	C₂²×C₄	—	4	4	no	—
263	C₂²×C₄	—	4	4	no	—
264	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
265	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
266	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
267	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
268	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
269	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
270	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
271	C₂×C₈	—	212	2	no	—
272	C₂×C₈	—	212	2	no	—
273	C₂×C₈	—	212	2	no	—
274	C₂×C₈	—	212	2	no	—
275	C₂×C₈	—	212	2	no	—
276	C₂×C₈	—	212	2	no	—
277	C₄⋊C₄	—	4	2	no	—
278	C₄⋊C₄	—	4	1	no	—
279	C₄⋊C₄	—	4	1	no	—
280	C₄⋊C₄	—	4	2	no	—
281	C₂²⋊C₄	—	4	4	no	—
282	C₂²⋊C₄	—	4	4	no	—
283	C₂²⋊C₄	—	4	4	no	—
284	C₄²	—	2	2	no	—
285	C₄²	—	2	2	no	—
286	C₄²	—	2	1	no	—
287	C₄²	—	2	2	no	—
288	C₄²	—	2	1	no	—
289	C₄²	—	4	1	no	—
290	C₄²	#11719922	1	2	yes	—
291	C₄²	#11719923	1	2	yes	—
292	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
293	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
294	C₄²	#11719945	1	1	yes	characteristic
295	C₄²	#11719945	1	1	yes	characteristic
296	C₂³	—	2	4	no	—
297	C₂³	—	4	2	no	—
298	C₂³	—	4	2	no	—
299	C₂³	—	4	2	no	—
300	C₂³	—	4	2	no	—
301	C₂³	—	4	2	no	—
302	C₂³	—	4	2	no	—
303	C₂³	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
304	D₄	—	4	4	no	—
305	D₄	—	4	4	no	—
306	D₄	—	4	2	no	—
307	D₄	—	4	2	no	—
308	D₄	—	4	4	no	—
309	D₄	—	4	2	no	—
310	D₄	—	4	2	no	—
311	D₄	—	4	4	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	2	1	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	4	2	no	—
326	C₂×C₄	—	8	8	no	—
327	C₂×C₄	#13133134	1	2	yes	—
328	C₂×C₄	#13133138	1	2	yes	—
329	C₂×C₄	#13133138	1	2	yes	—
330	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
331	C₂×C₄	#13133176	1	1	yes	characteristic
332	C₂×C₄	#13133191	1	1	yes	characteristic
333	C₂×C₄	#13133191	1	1	yes	characteristic
334	C₂×C₄	#13133196	1	1	yes	characteristic
335	C₂×C₄	#13133196	1	1	yes	characteristic
336	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
337	C₂×C₄	C₂×#11719923	1	1	yes	characteristic
338	C₂×C₄	#13133237	1	2	yes	—
339	C₂×C₄	#13133242	1	1	yes	characteristic
340	C₂×C₄	#13133243	1	2	yes	—
341	C₂×C₄	#13133247	1	1	yes	characteristic
342	C₈	—	424	2	no	—
343	C₈	—	424	2	no	—
344	C₂²	—	4	4	no	—
345	C₂²	—	4	4	no	—
346	C₂²	—	4	4	no	—
347	C₂²	—	8	2	no	—
348	C₂²	—	8	2	no	—
349	C₂²	—	8	2	no	—
350	C₂²	—	8	2	no	—
351	C₂²	#16810483	1	1	yes	characteristic
352	C₂²	C₂×#13133134	1	1	yes	characteristic
353	C₂²	#16810607	1	1	yes	characteristic
354	C₂²	C₂×#13133138	1	1	yes	characteristic
355	C₂²	#16810619	1	1	yes	characteristic
356	C₂²	#16810624	1	1	yes	characteristic
357	C₂²	#16810628	1	1	yes	characteristic
358	C₄	—	2	2	no	—
359	C₄	—	2	1	no	—
360	C₄	—	2	1	no	—
361	C₄	—	2	2	no	—
362	C₄	—	2	2	no	—
363	C₄	—	4	1	no	—
364	C₄	—	8	2	no	—
365	C₄	#16810818	1	1	yes	characteristic
366	C₄	#16810828	1	1	yes	characteristic
367	C₄	#16810838	1	2	yes	—
368	C₄	#16810903	1	1	yes	characteristic
369	C₄	#16810913	1	1	yes	characteristic
370	C₄	#16810923	1	2	yes	—
371	C₂	—	8	4	no	—
372	C₂	#20273364	1	1	yes	characteristic
373	C₂	#20273383	1	1	yes	characteristic
374	C₂	#20273410	1	1	yes	characteristic
375	C₂	#20273437	1	1	yes	characteristic
376	C₂	#20274707	1	1	yes	characteristic
377	C₂	#20274715	1	1	yes	characteristic
378	C₂	#20274725	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₅₃
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₄×#127×C₅₃	C₂
socle	C₂³×C₅₃	C₂×C₂²⋊C₄
prime radical	C₂×#127×C₅₃	C₂²
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₄×#127	D₅₃
hypocenter	C₅₃	#8588
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×#127

Complements

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

N	H ≅ G/N	type
1. #20274867	C₂	semidirect
2. #20274832	C₂	semidirect
4. #20272598	C₂	semidirect
211. C₅₃	#8588	semidirect

Maximal subgroups

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

subgroup	index
#20274867	2
#20274832	2
C₂×#16810367	2
#20272598	2
C₄×#127×C₅₃	2
#8588	53