Subgroups — #28319502

Order 13568.

810 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20291392	C₂	1	2	yes	maximal
2	#20290654	C₂	1	2	yes	maximal, complemented
3	#20290639	C₂	1	2	yes	maximal, complemented
4	#20290594	C₂	1	4	yes	maximal
5	C₄×#13133287	C₂	1	4	yes	maximal, complemented
6	C₂×C₄×C₈×D₅₃	C₂	1	1	yes	maximal, characteristic
7	#16811645	—	2	2	no	—
8	#16811644	—	2	2	no	—
9	#16811642	—	2	2	no	—
10	#16811639	—	2	2	no	—
11	C₂×#13133287	C₂²	1	8	yes	—
12	#16811627	C₂²	1	4	yes	—
13	#16811594	C₂²	1	4	yes	—
14	C₄×#11719936	C₂²	1	4	yes	—
15	#16811499	C₂²	1	8	yes	—
16	C₂²×C₈×D₅₃	C₂²	1	2	yes	—
17	C₂×C₄²×D₅₃	C₂²	1	1	yes	characteristic
18	C₂×C₄×#88322	C₄	1	1	yes	characteristic
19	C₂×C₈×Dic₅₃	C₄	1	2	yes	—
20	C₄×C₈×D₅₃	C₄	1	1	yes	characteristic
21	C₄×C₈×D₅₃	C₄	1	2	yes	—
22	C₄×C₈×D₅₃	C₄	1	1	yes	characteristic
23	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
24	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
25	C₄×C₈×D₅₃	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
27	C₂×#11719936	C₂³	1	8	yes	—
28	#13133287	—	2	16	no	—
29	C₄⋊#88324	—	2	4	no	—
30	C₄⋊#88324	—	2	4	no	—
31	#13133274	—	2	4	no	—
32	C₂²⋊#88324	—	2	4	no	—
33	#13133272	—	2	4	no	—
34	C₄×#88324	—	2	4	no	—
35	C₂²×C₄×D₅₃	Q₈	1	1	yes	characteristic
36	C₂²×C₄×D₅₃	Q₈	1	1	yes	characteristic
37	C₂²×C₄×D₅₃	C₂³	1	1	yes	characteristic
38	C₂²×#88322	C₂×C₄	1	2	yes	—
39	C₂×C₈×D₅₃	—	2	8	no	—
40	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
41	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
42	C₂×C₈×D₅₃	C₂³	1	2	yes	—
43	C₂×C₈×D₅₃	C₂³	1	2	yes	—
44	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
46	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
47	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
48	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
49	C₄²×D₅₃	D₄	1	1	yes	characteristic
50	C₄²×D₅₃	D₄	1	1	yes	characteristic
51	C₄²×D₅₃	Q₈	1	1	yes	characteristic
52	C₄²×D₅₃	Q₈	1	1	yes	characteristic
53	C₄²×D₅₃	C₂³	1	1	yes	characteristic
54	C₄²×D₅₃	C₂³	1	1	yes	characteristic
55	C₄×#88322	C₂×C₄	1	1	yes	characteristic
56	C₄×#88322	C₂×C₄	1	2	yes	—
57	C₄×#88322	C₂×C₄	1	1	yes	characteristic
58	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
59	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
60	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
61	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
62	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
63	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
64	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
65	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
66	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
67	C₂³×D₅₃	C₂×Q₈	1	1	yes	characteristic
68	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
69	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
70	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
71	C₂²×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
72	C₂²×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
73	C₂²×Dic₅₃	C₂×D₄	1	1	yes	characteristic
74	C₂×C₄×D₅₃	—	2	4	no	—
75	C₂×C₄×D₅₃	—	2	4	no	—
76	C₂×C₄×D₅₃	—	2	4	no	—
77	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
78	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
79	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
80	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
81	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
83	C₂×C₄×D₅₃	C₂×Q₈	1	1	yes	characteristic
84	C₂×C₄×D₅₃	C₂×Q₈	1	1	yes	characteristic
85	C₂×C₄×D₅₃	C₂×Q₈	1	1	yes	characteristic
86	C₂×C₄×D₅₃	C₂×Q₈	1	1	yes	characteristic
87	C₂×C₄×D₅₃	C₂×Q₈	1	1	yes	characteristic
88	C₂×C₄×D₅₃	C₂⁴	1	1	yes	characteristic
89	C₂×#88324	—	2	8	no	—
90	#11719936	—	2	16	no	—
91	C₂×C₈×C₅₃	—	2	4	no	—
92	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
93	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
94	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
95	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
96	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
97	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
98	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
99	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
100	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
101	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
102	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
103	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
104	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
105	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
106	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
107	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
108	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
109	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
110	C₂×#88322	—	2	4	no	—
111	C₂×#88322	C₂²×C₄	1	2	yes	—
112	C₂×#88322	C₂²×C₄	1	2	yes	—
113	C₈×D₅₃	—	2	4	no	—
114	C₈×D₅₃	—	2	4	no	—
115	C₈×D₅₃	—	2	4	no	—
116	C₈×D₅₃	#41	1	8	yes	—
117	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
118	C₂²×D₅₃	—	2	2	no	—
119	C₂²×D₅₃	—	2	2	no	—
120	C₂²×D₅₃	—	2	2	no	—
121	C₂²×D₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
122	C₂²×D₅₃	C₂²×Q₈	1	1	yes	characteristic
123	C₂×C₄×C₅₃	—	2	2	no	—
124	C₂×C₄×C₅₃	—	2	2	no	—
125	C₂×C₄×C₅₃	#108	1	4	yes	complemented
126	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
127	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
128	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
129	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
130	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
131	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
132	C₂×Dic₅₃	—	2	2	no	—
133	C₂×Dic₅₃	—	2	2	no	—
134	C₂×Dic₅₃	—	2	2	no	—
135	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
136	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
137	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
138	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
139	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
140	C₂×Dic₅₃	C₂²×D₄	1	1	yes	characteristic
141	C₄×D₅₃	—	2	2	no	—
142	C₄×D₅₃	—	2	2	no	—
143	C₄×D₅₃	—	2	2	no	—
144	C₄×D₅₃	—	2	2	no	—
145	C₄×D₅₃	—	2	2	no	—
146	C₄×D₅₃	—	2	2	no	—
147	C₄×D₅₃	—	2	2	no	—
148	C₄×D₅₃	—	2	2	no	—
149	C₄×D₅₃	—	2	2	no	—
150	C₄×D₅₃	—	2	2	no	—
151	C₄×D₅₃	#108	1	4	yes	complemented
152	C₄×D₅₃	C₂×#41	1	4	yes	—
153	#88324	—	4	8	no	—
154	C₈×C₅₃	—	2	2	no	—
155	C₈×C₅₃	#117	1	4	yes	—
156	#88322	—	2	2	no	—
157	#88322	#117	1	4	yes	—
158	#20826	—	53	1	no	maximal, Sylow
159	C₂²×C₅₃	—	2	1	no	—
160	C₂²×C₅₃	—	2	1	no	—
161	C₂²×C₅₃	C₂×#108	1	2	yes	—
162	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
163	C₂×D₅₃	—	2	8	no	—
164	C₂×D₅₃	—	2	1	no	—
165	C₂×D₅₃	—	2	1	no	—
166	C₂×D₅₃	—	2	1	no	—
167	C₂×D₅₃	—	2	1	no	—
168	C₂×D₅₃	C₂×#108	1	2	yes	—
169	C₂×D₅₃	#533	1	2	yes	—
170	C₄×C₅₃	—	2	1	no	—
171	C₄×C₅₃	—	2	1	no	—
172	C₄×C₅₃	C₂×#108	1	2	yes	—
173	C₄×C₅₃	C₂×#108	1	2	yes	—
174	C₄×C₅₃	C₂×#108	1	2	yes	—
175	C₄×C₅₃	C₂×#117	1	2	yes	—
176	Dic₅₃	—	2	1	no	—
177	Dic₅₃	—	2	1	no	—
178	Dic₅₃	—	2	1	no	—
179	Dic₅₃	—	2	1	no	—
180	Dic₅₃	C₂×#108	1	2	yes	—
181	Dic₅₃	#532	1	2	yes	—
182	#2920	—	53	2	no	—
183	C₂²×C₄×C₈	—	53	1	no	—
184	#1848	—	53	2	no	—
185	#1837	—	53	2	no	—
186	#1810	—	53	4	no	—
187	C₄×#108	—	53	4	no	—
188	C₂×C₅₃	—	2	2	no	—
189	C₂×C₅₃	#2285	1	1	yes	characteristic
190	C₂×C₅₃	C₂²×#108	1	1	yes	characteristic
191	C₂×C₅₃	C₂²×#108	1	1	yes	characteristic
192	D₅₃	—	2	2	no	—
193	D₅₃	—	2	2	no	—
194	C₂³×C₈	—	53	2	no	—
195	C₂²×C₄²	—	53	1	no	—
196	#432	—	106	2	no	—
197	#432	—	106	2	no	—
198	#431	—	106	2	no	—
199	#431	—	106	2	no	—
200	C₂×#108	—	53	8	no	—
201	#423	—	53	4	no	—
202	#407	—	53	4	no	—
203	C₄×M₁₆	—	53	4	no	—
204	C₂×C₄×C₈	—	53	2	no	—
205	C₂×C₄×C₈	—	53	1	no	—
206	C₂×C₄×C₈	—	53	1	no	—
207	C₂×C₄×C₈	—	53	1	no	—
208	C₂×C₄×C₈	—	53	1	no	—
209	C₂×C₄×C₈	—	53	2	no	—
210	C₂×C₄×C₈	—	53	1	no	—
211	C₂×C₄×C₈	—	53	2	no	—
212	C₂×C₄×C₈	#6157	1	1	yes	characteristic
213	#341	—	53	8	no	—
214	C₅₃	#20826	1	1	yes	characteristic, complemented, Sylow
215	C₂³×C₄	—	53	1	no	—
216	C₂³×C₄	—	53	1	no	—
217	C₂³×C₄	—	53	1	no	—
218	C₂×M₁₆	—	53	8	no	—
219	C₂²×C₈	—	53	2	no	—
220	C₂²×C₈	—	53	2	no	—
221	C₂²×C₈	—	53	2	no	—
222	C₂²×C₈	—	53	2	no	—
223	C₂²×C₈	—	53	2	no	—
224	C₂²×C₈	—	106	8	no	—
225	C₂²×C₈	C₂×#6157	1	2	yes	—
226	C₂×C₄²	—	53	1	no	—
227	C₂×C₄²	—	53	1	no	—
228	C₂×C₄²	—	53	1	no	—
229	C₂×C₄²	—	53	1	no	—
230	C₂×C₄²	—	53	1	no	—
231	C₂×C₄²	—	53	1	no	—
232	C₂×C₄²	—	53	1	no	—
233	C₂×C₄²	—	53	1	no	—
234	C₂×C₄²	—	53	1	no	—
235	C₂×C₄²	—	53	1	no	—
236	C₂×C₄²	—	53	1	no	—
237	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
238	#108	—	106	16	no	—
239	C₄⋊C₈	—	106	4	no	—
240	C₄⋊C₈	—	106	4	no	—
241	C₂²⋊C₈	—	106	4	no	—
242	C₂²⋊C₈	—	106	4	no	—
243	#97	—	106	4	no	—
244	C₄×C₈	—	53	2	no	—
245	C₄×C₈	—	53	2	no	—
246	C₄×C₈	—	53	2	no	—
247	C₄×C₈	—	53	2	no	—
248	C₄×C₈	—	53	1	no	—
249	C₄×C₈	—	53	1	no	—
250	C₄×C₈	—	53	2	no	—
251	C₄×C₈	—	106	4	no	—
252	C₄×C₈	C₂×#6157	1	1	yes	characteristic
253	C₄×C₈	C₂×#6157	1	1	yes	characteristic
254	C₄×C₈	C₂×#6157	1	2	yes	—
255	C₂⁴	—	53	1	no	—
256	C₂²×C₄	—	53	1	no	—
257	C₂²×C₄	—	53	1	no	—
258	C₂²×C₄	—	53	1	no	—
259	C₂²×C₄	—	53	1	no	—
260	C₂²×C₄	—	53	1	no	—
261	C₂²×C₄	—	53	1	no	—
262	C₂²×C₄	—	53	1	no	—
263	C₂²×C₄	—	53	1	no	—
264	C₂²×C₄	—	53	1	no	—
265	C₂²×C₄	—	53	1	no	—
266	C₂²×C₄	—	53	1	no	—
267	C₂²×C₄	—	53	1	no	—
268	C₂²×C₄	—	53	1	no	—
269	C₂²×C₄	—	53	1	no	—
270	C₂²×C₄	—	53	1	no	—
271	C₂²×C₄	—	106	4	no	—
272	C₂²×C₄	—	106	4	no	—
273	C₂²×C₄	—	106	4	no	—
274	C₂²×C₄	C₄⋊#6157	1	1	yes	characteristic
275	C₂²×C₄	C₄⋊#6157	1	1	yes	characteristic
276	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
277	M₁₆	—	106	16	no	—
278	C₂×C₈	—	2	4	no	—
279	C₂×C₈	—	53	2	no	—
280	C₂×C₈	—	53	8	no	—
281	C₂×C₈	—	53	2	no	—
282	C₂×C₈	—	106	4	no	—
283	C₂×C₈	—	106	4	no	—
284	C₂×C₈	—	106	4	no	—
285	C₂×C₈	—	106	4	no	—
286	C₂×C₈	—	106	8	no	—
287	C₂×C₈	C₂²×#6157	1	2	yes	—
288	C₂×C₈	C₂²×#6157	1	2	yes	—
289	C₄²	—	53	1	no	—
290	C₄²	—	53	1	no	—
291	C₄²	—	53	1	no	—
292	C₄²	—	53	1	no	—
293	C₄²	—	53	1	no	—
294	C₄²	—	53	1	no	—
295	C₄²	—	53	1	no	—
296	C₄²	—	53	1	no	—
297	C₄²	—	53	1	no	—
298	C₄²	—	53	1	no	—
299	C₄²	—	53	1	no	—
300	C₄²	—	53	1	no	—
301	C₄²	C₄⋊#6157	1	1	yes	characteristic
302	C₄²	C₄⋊#6157	1	1	yes	characteristic
303	C₄²	C₂²×#6157	1	1	yes	characteristic
304	C₄²	C₂²×#6157	1	1	yes	characteristic
305	C₂³	—	53	1	no	—
306	C₂³	—	53	1	no	—
307	C₂³	—	106	2	no	—
308	C₂³	—	106	2	no	—
309	C₂³	—	106	2	no	—
310	C₂³	C₂×C₄⋊#6157	1	1	yes	characteristic
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	53	1	no	—
314	C₂×C₄	—	53	1	no	—
315	C₂×C₄	—	53	4	no	—
316	C₂×C₄	—	53	4	no	—
317	C₂×C₄	—	53	1	no	—
318	C₂×C₄	—	53	1	no	—
319	C₂×C₄	—	53	1	no	—
320	C₂×C₄	—	53	1	no	—
321	C₂×C₄	—	106	2	no	—
322	C₂×C₄	—	106	2	no	—
323	C₂×C₄	—	106	2	no	—
324	C₂×C₄	—	106	2	no	—
325	C₂×C₄	—	106	2	no	—
326	C₂×C₄	—	106	2	no	—
327	C₂×C₄	—	106	2	no	—
328	C₂×C₄	—	106	2	no	—
329	C₂×C₄	—	106	2	no	—
330	C₂×C₄	—	106	2	no	—
331	C₂×C₄	—	106	2	no	—
332	C₂×C₄	—	106	2	no	—
333	C₂×C₄	—	106	2	no	—
334	C₂×C₄	#13133287	1	4	yes	complemented
335	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
336	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
337	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
338	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
339	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
340	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
341	C₈	—	2	2	no	—
342	C₈	—	53	4	no	—
343	C₈	—	106	2	no	—
344	C₈	—	212	8	no	—
345	C₈	#13133296	1	4	yes	—
346	C₂²	—	2	1	no	—
347	C₂²	—	2	1	no	—
348	C₂²	—	53	2	no	—
349	C₂²	—	53	2	no	—
350	C₂²	—	106	8	no	—
351	C₂²	—	106	1	no	—
352	C₂²	—	106	1	no	—
353	C₂²	—	106	1	no	—
354	C₂²	—	106	1	no	—
355	C₂²	C₂×#13133287	1	2	yes	—
356	C₂²	C₂²×C₄⋊#6157	1	1	yes	characteristic
357	C₄	—	2	1	no	—
358	C₄	—	2	1	no	—
359	C₄	—	53	2	no	—
360	C₄	—	53	2	no	—
361	C₄	—	106	1	no	—
362	C₄	—	106	1	no	—
363	C₄	—	106	1	no	—
364	C₄	—	106	1	no	—
365	C₄	C₂×#13133287	1	2	yes	—
366	C₄	C₂×#13133287	1	2	yes	—
367	C₄	D₅₃⋊#108	1	2	yes	—
368	C₄	C₂×#13133296	1	2	yes	—
369	C₂	—	2	2	no	—
370	C₂	—	106	2	no	—
371	C₂	—	106	2	no	—
372	C₂	#20291164	1	1	yes	characteristic
373	C₂	C₂²×#13133287	1	1	yes	characteristic
374	C₂	C₂×D₅₃⋊#108	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₄⋊#6157
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×#6157
Fitting	C₂×C₄×C₈×C₅₃	C₄
socle	C₂²×C₅₃	C₂²×C₄⋊C₄
prime radical	C₂³×D₅₃	C₂×Q₈
cosocle	C₂×C₄×D₅₃	C₂⁴
hypercenter	C₂×C₄×C₈	#6157
hypocenter	C₅₃	#20826
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #20290654	C₂	semidirect
3. #20290639	C₂	semidirect
5. C₄×#13133287	C₂	semidirect
125. C₂×C₄×C₅₃	#108	semidirect
151. C₄×D₅₃	#108	semidirect
214. C₅₃	#20826	semidirect
334. C₂×C₄	#13133287	semidirect

Maximal subgroups

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

subgroup	index
#20291392	2
#20290654	2
#20290639	2
#20290594	2
C₄×#13133287	2
C₂×C₄×C₈×D₅₃	2
#20826	53