Subgroups — #27353330

Order 13568.

702 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	C₈×#11719945	C₂	1	1	yes	maximal, characteristic
2	#20283284	C₂	1	2	yes	maximal
3	#20273491	C₂	1	2	yes	maximal, complemented
4	#20273488	C₂	1	4	yes	maximal, complemented
5	#20273472	C₂	1	2	yes	maximal, complemented
6	C₂×#16810376	C₂	1	1	yes	maximal, characteristic, complemented
7	#335×D₅₃	C₂	1	2	yes	maximal, complemented
8	C₅₃×#1567	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#13133225	—	2	2	no	—
10	C₄×#11719945	C₂²	1	1	yes	characteristic
11	C₈×Dic₁₀₆	C₄	1	1	yes	characteristic
12	C₈×Dic₁₀₆	C₄	1	1	yes	characteristic
13	C₈×D₂₁₂	C₄	1	1	yes	characteristic
14	C₈×D₂₁₂	C₄	1	1	yes	characteristic
15	C₈×C₂²⋊D₅₃	C₄	1	2	yes	—
16	C₈×C₂²⋊D₅₃	C₄	1	2	yes	—
17	#16810676	C₂²	1	1	yes	characteristic
18	#16810674	C₂²	1	2	yes	—
19	#16810651	—	2	2	no	—
20	#16810648	—	2	4	no	—
21	#16810645	C₂²	1	4	yes	—
22	C₂×C₄⋊#88322	C₂²	1	2	yes	—
23	C₄⋊C₈×D₅₃	C₂²	1	4	yes	—
24	C₄×C₈×D₅₃	C₂²	1	2	yes	—
25	#16810376	C₂²	1	1	yes	characteristic
26	#16810376	C₂²	1	1	yes	characteristic
27	#16810376	C₂²	1	2	yes	—
28	#16810375	C₂²	1	4	yes	—
29	#16810375	C₂²	1	4	yes	—
30	C₅₃×#423	C₂²	1	2	yes	—
31	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
32	C₅₃×#335	C₂²	1	2	yes	—
33	C₅₃×#335	C₂²	1	2	yes	—
34	C₂×#11719945	D₄	1	1	yes	characteristic
35	#13133225	—	4	8	no	—
36	C₂×#11719912	—	2	4	no	—
37	C₂²×#88322	—	2	2	no	—
38	C₂×C₈×D₅₃	—	2	4	no	—
39	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
40	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
41	C₄×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
42	C₄×Dic₁₀₆	Q₈	1	1	yes	characteristic
43	C₄×D₂₁₂	C₂×C₄	1	1	yes	characteristic
44	C₄×D₂₁₂	Q₈	1	1	yes	characteristic
45	C₄×C₂²⋊D₅₃	C₂×C₄	1	2	yes	—
46	C₄×C₂²⋊D₅₃	Q₈	1	2	yes	—
47	#13133176	D₄	1	1	yes	characteristic
48	#13133174	D₄	1	2	yes	—
49	C₄²×D₅₃	C₂³	1	2	yes	—
50	C₄⋊#88322	—	2	2	no	—
51	C₄⋊#88322	C₂×C₄	1	1	yes	characteristic
52	C₄⋊#88322	C₂×C₄	1	1	yes	characteristic
53	C₄⋊#88322	C₂³	1	2	yes	—
54	C₄⋊#88322	C₂³	1	2	yes	—
55	#13133146	—	2	4	no	—
56	#13133146	—	2	4	no	—
57	#13133146	C₂×C₄	1	2	yes	—
58	#13133146	C₂×C₄	1	2	yes	—
59	C₂²⋊#88322	C₂×C₄	1	2	yes	—
60	#13133129	C₂×C₄	1	2	yes	—
61	#13133129	C₂×C₄	1	2	yes	—
62	C₄×#88322	C₂³	1	2	yes	—
63	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
64	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
65	C₂×M₁₆×C₅₃	—	2	2	no	—
66	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
67	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
68	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
69	C₄⋊C₈×C₅₃	—	2	2	no	—
70	C₄⋊C₈×C₅₃	C₂³	1	4	yes	—
71	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
72	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
73	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
75	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
78	C₂×C₂²⋊D₅₃	C₄⋊C₄	1	2	yes	—
79	#11719945	D₈	1	4	yes	—
80	#11719945	Q₁₆	1	4	yes	—
81	C₂×D₂₁₂	C₄⋊C₄	1	1	yes	characteristic
82	C₂×C₄×D₅₃	C₂²×C₄	1	2	yes	—
83	C₂×C₄×D₅₃	C₂×D₄	1	2	yes	—
84	C₂×C₄×D₅₃	C₂×Q₈	1	2	yes	—
85	C₂×Dic₁₀₆	C₄⋊C₄	1	1	yes	characteristic
86	M₁₆×C₅₃	—	4	4	no	—
87	C₂×C₈×C₅₃	—	2	2	no	—
88	C₂×C₈×C₅₃	—	2	4	no	—
89	C₂×C₈×C₅₃	—	4	1	no	—
90	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
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	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
95	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
96	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
97	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
98	C₂²⋊Dic₅₃	C₄⋊C₄	1	2	yes	—
99	#11719921	C₄⋊C₄	1	2	yes	—
100	#11719921	C₄⋊C₄	1	2	yes	—
101	C₄⋊Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
102	C₄⋊Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
103	#11719919	C₄⋊C₄	1	2	yes	—
104	#11719919	C₄⋊C₄	1	2	yes	—
105	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
106	C₄×Dic₅₃	C₂×D₄	1	2	yes	—
107	C₄×Dic₅₃	C₂×Q₈	1	2	yes	—
108	C₂×#88322	—	2	2	no	—
109	C₂×#88322	—	2	2	no	—
110	C₂×#88322	—	4	4	no	—
111	C₂×#88322	C₂²×C₄	1	2	yes	—
112	C₂×#88322	C₂²×C₄	1	2	yes	—
113	#11719912	—	4	8	no	—
114	C₈×D₅₃	—	2	4	no	—
115	C₈×D₅₃	—	4	2	no	—
116	C₈×D₅₃	—	4	8	no	—
117	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
118	C₂²×D₅₃	C₂×C₄⋊C₄	1	2	yes	—
119	C₂×C₄×C₅₃	—	2	1	no	—
120	C₂×C₄×C₅₃	—	2	1	no	—
121	C₂×C₄×C₅₃	—	4	1	no	—
122	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
126	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
127	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
128	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
129	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
130	C₂²⋊D₅₃	#107	1	8	yes	—
131	C₂×Dic₅₃	C₂×C₄⋊C₄	1	2	yes	—
132	C₂×Dic₅₃	C₂×C₄⋊C₄	1	2	yes	—
133	C₂×Dic₅₃	C₂×C₄⋊C₄	1	2	yes	—
134	D₂₁₂	#107	1	4	yes	—
135	C₄×D₅₃	—	2	2	no	—
136	C₄×D₅₃	—	2	2	no	—
137	C₄×D₅₃	—	4	2	no	—
138	C₄×D₅₃	C₂×D₈	1	4	yes	—
139	C₄×D₅₃	C₂×Q₁₆	1	4	yes	—
140	Dic₁₀₆	#107	1	4	yes	—
141	C₈×C₅₃	—	2	1	no	—
142	C₈×C₅₃	—	2	1	no	—
143	C₈×C₅₃	—	2	1	no	—
144	C₈×C₅₃	—	4	4	no	—
145	C₈×C₅₃	#131	1	2	yes	complemented
146	#88322	—	2	2	no	—
147	#88322	—	2	2	no	—
148	#88322	—	4	2	no	—
149	#88322	—	4	2	no	—
150	#17828	—	53	1	no	maximal, Sylow
151	C₂²×C₅₃	—	2	1	no	—
152	C₂²×C₅₃	—	2	1	no	—
153	C₂²×C₅₃	C₂×#107	1	2	yes	—
154	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
155	C₂×D₅₃	—	2	2	no	—
156	C₂×D₅₃	—	2	2	no	—
157	C₂×D₅₃	C₂×#107	1	4	yes	—
158	C₄×C₅₃	—	2	1	no	—
159	C₄×C₅₃	—	2	1	no	—
160	C₄×C₅₃	—	2	1	no	—
161	C₄×C₅₃	C₂×#107	1	2	yes	—
162	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
163	C₄×C₅₃	#568	1	1	yes	characteristic
164	C₄×C₅₃	C₂²×D₈	1	1	yes	characteristic
165	C₄×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
166	Dic₅₃	—	2	2	no	—
167	Dic₅₃	—	2	2	no	—
168	Dic₅₃	C₂×#107	1	4	yes	—
169	C₈×#41	—	53	1	no	—
170	#2906	—	53	2	no	—
171	#1571	—	53	2	no	—
172	#1571	—	53	4	no	—
173	#1567	—	53	2	no	—
174	#1567	D₅₃	1	1	yes	characteristic, complemented
175	C₂×#335	—	53	1	no	—
176	C₂×#335	—	53	2	no	—
177	C₂×C₅₃	—	2	2	no	—
178	C₂×C₅₃	#2906	1	1	yes	characteristic
179	C₂×C₅₃	C₂²×#107	1	1	yes	characteristic
180	C₂×C₅₃	#2916	1	1	yes	characteristic
181	D₅₃	—	2	4	no	—
182	C₂×#131	—	106	2	no	—
183	C₄×#41	—	53	1	no	—
184	C₈×Q₈	—	53	1	no	—
185	C₈×Q₈	—	53	1	no	—
186	C₈×D₄	—	53	2	no	—
187	C₈×D₄	—	53	2	no	—
188	C₈×D₄	—	53	1	no	—
189	C₈×D₄	—	53	1	no	—
190	#431	—	53	1	no	—
191	#431	—	53	2	no	—
192	#424	—	106	4	no	—
193	#424	—	106	2	no	—
194	#423	—	53	4	no	—
195	#423	C₂×D₅₃	1	2	yes	—
196	C₂×C₄⋊C₈	—	53	2	no	—
197	C₂×C₄⋊C₈	—	53	4	no	—
198	C₂×C₄×C₈	—	53	2	no	—
199	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
200	#335	—	53	4	no	—
201	#335	—	53	4	no	—
202	#335	—	53	1	no	—
203	#335	—	53	1	no	—
204	#335	—	53	2	no	—
205	#335	C₂×D₅₃	1	2	yes	—
206	#335	C₂×D₅₃	1	2	yes	—
207	C₅₃	#17828	1	1	yes	characteristic, complemented, Sylow
208	C₂×#41	—	53	1	no	—
209	#131	—	212	8	no	—
210	C₂×M₁₆	—	2	2	no	—
211	C₂×M₁₆	—	106	4	no	—
212	C₂²×C₈	—	53	2	no	—
213	C₂²×C₈	—	53	2	no	—
214	C₂²×C₈	—	106	4	no	—
215	C₂²×C₈	—	106	2	no	—
216	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
217	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
218	C₄×Q₈	—	53	1	no	—
219	C₄×Q₈	—	53	1	no	—
220	C₄×D₄	—	53	2	no	—
221	C₄×D₄	—	53	2	no	—
222	C₄×D₄	—	53	1	no	—
223	C₄×D₄	—	53	1	no	—
224	#117	—	53	1	no	—
225	#117	—	53	2	no	—
226	C₂×C₄²	—	53	2	no	—
227	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
228	C₄⋊C₈	—	2	2	no	—
229	C₄⋊C₈	—	53	2	no	—
230	C₄⋊C₈	—	53	2	no	—
231	C₄⋊C₈	—	53	1	no	—
232	C₄⋊C₈	—	53	1	no	—
233	C₄⋊C₈	—	53	2	no	—
234	C₄⋊C₈	—	53	2	no	—
235	C₄⋊C₈	—	106	4	no	—
236	C₄⋊C₈	—	106	4	no	—
237	C₄⋊C₈	—	106	2	no	—
238	C₄⋊C₈	C₂²×D₅₃	1	4	yes	—
239	C₂²⋊C₈	—	53	2	no	—
240	C₂²⋊C₈	—	53	2	no	—
241	C₂²⋊C₈	—	53	2	no	—
242	C₄×C₈	—	53	2	no	—
243	C₄×C₈	—	53	2	no	—
244	C₄×C₈	—	53	2	no	—
245	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
246	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
247	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
248	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
249	#41	—	53	4	no	—
250	#41	—	53	4	no	—
251	C₂×Q₈	—	53	1	no	—
252	C₂×D₄	—	53	2	no	—
253	C₂×D₄	—	53	1	no	—
254	C₂²×C₄	—	53	2	no	—
255	C₂²×C₄	—	53	2	no	—
256	C₂²×C₄	—	53	2	no	—
257	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
258	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
259	C₂²×C₄	Q₈×D₅₃	1	1	yes	characteristic
260	M₁₆	—	4	4	no	—
261	M₁₆	—	212	8	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	2	4	no	—
264	C₂×C₈	—	4	1	no	—
265	C₂×C₈	—	53	2	no	—
266	C₂×C₈	—	53	2	no	—
267	C₂×C₈	—	106	4	no	—
268	C₂×C₈	—	106	2	no	—
269	C₂×C₈	—	106	2	no	—
270	C₂×C₈	—	212	2	no	—
271	C₂×C₈	—	212	8	no	—
272	C₂×C₈	—	212	4	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	C₂×C₄×D₅₃	1	1	yes	characteristic
277	C₄⋊C₄	—	53	2	no	—
278	C₄⋊C₄	—	53	2	no	—
279	C₄⋊C₄	—	53	1	no	—
280	C₄⋊C₄	—	53	1	no	—
281	C₂²⋊C₄	—	53	2	no	—
282	C₂²⋊C₄	—	53	2	no	—
283	C₂²⋊C₄	—	53	2	no	—
284	C₄²	—	53	2	no	—
285	C₄²	—	53	2	no	—
286	C₄²	—	53	2	no	—
287	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
288	C₄²	D₄×D₅₃	1	1	yes	characteristic
289	C₄²	Q₈×D₅₃	1	1	yes	characteristic
290	C₄²	C₂³×D₅₃	1	1	yes	characteristic
291	C₂³	—	53	2	no	—
292	C₂³	C₄⋊C₄×D₅₃	1	1	yes	characteristic
293	Q₈	—	53	4	no	—
294	D₄	—	53	8	no	—
295	D₄	—	53	4	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	4	1	no	—
299	C₂×C₄	—	53	2	no	—
300	C₂×C₄	—	53	4	no	—
301	C₂×C₄	—	53	4	no	—
302	C₂×C₄	—	53	2	no	—
303	C₂×C₄	—	53	2	no	—
304	C₂×C₄	—	106	2	no	—
305	C₂×C₄	—	106	2	no	—
306	C₂×C₄	—	212	2	no	—
307	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
308	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
309	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
310	C₂×C₄	D₈×D₅₃	1	2	yes	—
311	C₂×C₄	Q₁₆×D₅₃	1	2	yes	—
312	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
313	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
314	C₂×C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
315	C₈	—	2	1	no	—
316	C₈	—	2	1	no	—
317	C₈	—	2	1	no	—
318	C₈	—	4	4	no	—
319	C₈	—	106	2	no	—
320	C₈	—	106	2	no	—
321	C₈	—	212	2	no	—
322	C₈	—	212	2	no	—
323	C₈	#13133225	1	2	yes	complemented
324	C₂²	—	2	1	no	—
325	C₂²	—	2	1	no	—
326	C₂²	—	53	4	no	—
327	C₂²	—	106	2	no	—
328	C₂²	—	106	2	no	—
329	C₂²	#107×D₅₃	1	2	yes	—
330	C₂²	C₂×C₄⋊C₄×D₅₃	1	1	yes	characteristic
331	C₄	—	2	1	no	—
332	C₄	—	2	1	no	—
333	C₄	—	2	1	no	—
334	C₄	—	53	4	no	—
335	C₄	—	106	2	no	—
336	C₄	—	106	2	no	—
337	C₄	#107×D₅₃	1	2	yes	—
338	C₄	C₂×#13133225	1	1	yes	characteristic
339	C₄	#16811389	1	1	yes	characteristic
340	C₄	C₂×D₈×D₅₃	1	1	yes	characteristic
341	C₄	C₂×Q₁₆×D₅₃	1	1	yes	characteristic
342	C₂	—	2	2	no	—
343	C₂	—	106	4	no	—
344	C₂	#20283284	1	1	yes	characteristic
345	C₂	C₂×#107×D₅₃	1	1	yes	characteristic
346	C₂	#20283323	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×#1567	C₂
socle	C₂²×C₅₃	C₂²×C₄⋊C₄
prime radical	C₂×#11719945	D₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#1567	D₅₃
hypocenter	C₅₃	#17828
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1567

Complements

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

N	H ≅ G/N	type
3. #20273491	C₂	semidirect
4. #20273488	C₂	semidirect
5. #20273472	C₂	semidirect
6. C₂×#16810376	C₂	semidirect
7. #335×D₅₃	C₂	semidirect
8. C₅₃×#1567	C₂	semidirect
145. C₈×C₅₃	#131	semidirect
174. #1567	D₅₃	semidirect
207. C₅₃	#17828	semidirect
323. C₈	#13133225	semidirect

Maximal subgroups

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

subgroup	index
C₈×#11719945	2
#20283284	2
#20273491	2
#20273488	2
#20273472	2
C₂×#16810376	2
#335×D₅₃	2
C₅₃×#1567	2
#17828	53