Subgroups — #27538781

Order 13568.

556 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20285254	C₂	1	1	yes	maximal, characteristic, complemented
2	#20276872	C₂	1	2	yes	maximal, complemented
3	#20276872	C₂	1	2	yes	maximal, complemented
4	#20276862	C₂	1	2	yes	maximal, complemented
5	#20276852	C₂	1	2	yes	maximal, complemented
6	#20276504	C₂	1	1	yes	maximal, characteristic, complemented
7	#20276497	C₂	1	1	yes	maximal, characteristic, complemented
8	#20276474	C₂	1	1	yes	maximal, characteristic, complemented
9	#20276431	C₂	1	1	yes	maximal, characteristic
10	C₂×#16810424	C₂	1	1	yes	maximal, characteristic
11	C₅₃×#2948	C₂	1	1	yes	maximal, characteristic
12	#16811251	—	2	1	no	—
13	C₂×#13133157	C₂²	1	1	yes	characteristic
14	C₂²×#11719908	C₂²	1	1	yes	characteristic
15	#16811032	—	2	2	no	—
16	#16811028	C₂²	1	2	yes	complemented
17	#16811025	C₂²	1	2	yes	complemented
18	#16810665	—	2	1	no	—
19	#16810661	C₂²	1	1	yes	characteristic
20	C₂×#13133151	C₂²	1	1	yes	characteristic
21	#16810450	C₂²	1	1	yes	characteristic, complemented
22	#16810450	C₂²	1	1	yes	characteristic, complemented
23	#16810450	C₂²	1	2	yes	complemented
24	#16810448	C₂²	1	1	yes	characteristic, complemented
25	#16810448	C₂²	1	1	yes	characteristic, complemented
26	#16810448	C₂²	1	2	yes	complemented
27	C₄⋊#11719908	C₂²	1	2	yes	complemented
28	C₄⋊#11719908	C₂²	1	2	yes	complemented
29	#16810424	C₂²	1	2	yes	—
30	#16810424	C₂²	1	2	yes	—
31	#16810424	C₂²	1	2	yes	—
32	#16810424	C₂²	1	2	yes	—
33	C₂²×D₈×C₅₃	C₂²	1	1	yes	characteristic
34	C₅₃×#518	—	2	1	no	—
35	C₅₃×#442	—	2	2	no	—
36	C₄×D₈×C₅₃	C₂²	1	2	yes	—
37	C₄×D₈×C₅₃	C₂²	1	2	yes	—
38	C₅₃×#427	C₂²	1	1	yes	characteristic
39	C₅₃×#418	—	2	1	no	—
40	C₅₃×#405	C₂²	1	1	yes	characteristic
41	#13133217	—	2	2	no	—
42	#13133216	—	4	1	no	—
43	#13133209	—	2	2	no	—
44	#13133208	—	4	1	no	—
45	#13133194	—	4	2	no	—
46	#13133176	—	2	1	no	—
47	C₂×C₄⋊Dic₅₃	—	2	1	no	—
48	#13133157	—	2	2	no	—
49	C₂×#11719908	—	2	2	no	—
50	C₂×#11719908	C₂³	1	1	yes	characteristic
51	C₂×#11719908	C₂³	1	1	yes	characteristic
52	C₂×#11719908	C₂³	1	2	yes	—
53	#13133151	—	2	2	no	—
54	#13133151	—	2	1	no	—
55	#13133151	C₂³	1	1	yes	characteristic
56	#13133151	C₂³	1	2	yes	—
57	#13133151	C₂³	1	1	yes	characteristic
58	#13133149	—	2	2	no	—
59	#13133149	—	2	1	no	—
60	C₂²×D₄×C₅₃	—	2	1	no	—
61	C₂×D₈×C₅₃	—	2	1	no	—
62	C₂×D₈×C₅₃	—	2	2	no	—
63	C₂×D₈×C₅₃	Q₈	1	4	yes	complemented
64	C₂×D₈×C₅₃	C₂³	1	2	yes	—
65	C₂×M₁₆×C₅₃	D₄	1	1	yes	characteristic
66	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
67	C₄×D₄×C₅₃	—	2	2	no	—
68	C₄×D₄×C₅₃	—	2	2	no	—
69	C₄×D₄×C₅₃	—	4	1	no	—
70	C₄×D₄×C₅₃	—	4	1	no	—
71	#117×C₅₃	—	2	1	no	—
72	#117×C₅₃	D₄	1	1	yes	characteristic
73	C₂×C₂²⋊C₄×C₅₃	—	4	2	no	—
74	#107×C₅₃	C₂³	1	1	yes	characteristic
75	#107×C₅₃	C₂³	1	1	yes	characteristic
76	#106×C₅₃	—	2	1	no	—
77	#102×C₅₃	—	2	2	no	—
78	#102×C₅₃	—	2	2	no	—
79	#97×C₅₃	D₄	1	2	yes	—
80	C₄×C₈×C₅₃	C₂³	1	2	yes	—
81	C₂⁴×C₅₃	—	4	1	no	—
82	C₂×D₄×C₅₃	—	2	4	no	—
83	C₂×D₄×C₅₃	—	2	2	no	—
84	C₂×D₄×C₅₃	—	4	1	no	—
85	C₂×D₄×C₅₃	—	4	2	no	—
86	C₂²×C₄×C₅₃	—	4	2	no	—
87	C₂²×C₄×C₅₃	—	4	2	no	—
88	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
89	C₂²×Dic₅₃	—	4	1	no	—
90	C₂×Dic₁₀₆	—	4	2	no	—
91	D₈×C₅₃	—	2	4	no	—
92	D₈×C₅₃	—	2	2	no	—
93	D₈×C₅₃	C₂×Q₈	1	4	yes	—
94	M₁₆×C₅₃	—	2	2	no	—
95	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
96	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
98	C₂×C₈×C₅₃	#41	1	2	yes	—
99	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
100	C₄⋊C₄×C₅₃	—	2	1	no	—
101	C₄⋊C₄×C₅₃	—	2	1	no	—
102	C₄⋊C₄×C₅₃	D₈	1	1	yes	characteristic
103	C₄⋊C₄×C₅₃	D₈	1	1	yes	characteristic
104	C₂²⋊C₄×C₅₃	—	4	1	no	—
105	C₂²⋊C₄×C₅₃	—	4	2	no	—
106	C₂²⋊C₄×C₅₃	—	4	2	no	—
107	C₂²⋊C₄×C₅₃	D₈	1	2	yes	—
108	C₄²×C₅₃	—	4	1	no	—
109	C₄²×C₅₃	C₂×D₄	1	2	yes	—
110	C₂²⋊Dic₅₃	—	4	1	no	—
111	C₄⋊Dic₅₃	—	2	1	no	—
112	C₄⋊Dic₅₃	—	2	2	no	—
113	C₄⋊Dic₅₃	—	2	1	no	—
114	C₄⋊Dic₅₃	—	4	1	no	—
115	C₄⋊Dic₅₃	—	4	2	no	—
116	#11719919	—	4	2	no	—
117	#11719919	—	4	2	no	—
118	C₄×Dic₅₃	—	4	1	no	—
119	#11719908	—	2	1	no	—
120	#11719908	—	2	1	no	—
121	#11719908	—	2	2	no	—
122	C₂³×C₅₃	—	4	2	no	—
123	C₂³×C₅₃	—	4	4	no	—
124	C₂³×C₅₃	—	16	1	no	—
125	C₂³×C₅₃	—	16	1	no	—
126	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
127	D₄×C₅₃	—	2	4	no	—
128	D₄×C₅₃	—	4	2	no	—
129	D₄×C₅₃	—	4	4	no	—
130	C₂×C₄×C₅₃	—	4	1	no	—
131	C₂×C₄×C₅₃	—	4	1	no	—
132	C₂×C₄×C₅₃	—	16	2	no	—
133	C₂×C₄×C₅₃	—	16	2	no	—
134	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
135	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
136	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
137	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
138	C₂×C₄×C₅₃	#137	1	2	yes	—
139	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
140	C₂×Dic₅₃	—	4	1	no	—
141	C₂×Dic₅₃	—	4	2	no	—
142	C₂×Dic₅₃	—	4	1	no	—
143	C₂×Dic₅₃	—	16	1	no	—
144	Dic₁₀₆	—	16	2	no	—
145	C₈×C₅₃	—	2	2	no	—
146	C₈×C₅₃	C₂²⋊Q₈	1	2	yes	—
147	C₈×C₅₃	C₂²×Q₈	1	1	yes	characteristic
148	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
149	#33631	—	53	1	no	maximal, Sylow
150	C₂²×C₅₃	—	2	1	no	—
151	C₂²×C₅₃	—	2	1	no	—
152	C₂²×C₅₃	—	4	4	no	—
153	C₂²×C₅₃	—	8	2	no	—
154	C₂²×C₅₃	—	8	2	no	—
155	C₂²×C₅₃	—	16	4	no	—
156	C₂²×C₅₃	#474	1	2	yes	—
157	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
158	C₄×C₅₃	—	2	2	no	—
159	C₄×C₅₃	—	2	2	no	—
160	C₄×C₅₃	—	8	1	no	—
161	C₄×C₅₃	—	8	1	no	—
162	C₄×C₅₃	#474	1	2	yes	—
163	C₄×C₅₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
164	C₄×C₅₃	C₂×#137	1	1	yes	characteristic
165	Dic₅₃	—	8	1	no	—
166	Dic₅₃	—	8	2	no	—
167	Dic₅₃	—	8	1	no	—
168	#3164	—	53	1	no	—
169	#2948	D₅₃	1	1	yes	characteristic
170	#2228	—	53	2	no	—
171	#2228	—	53	2	no	—
172	#2226	—	53	2	no	—
173	#2224	—	53	2	no	—
174	#2159	—	53	1	no	—
175	#2158	—	53	1	no	—
176	#2151	—	53	1	no	—
177	#2140	—	53	1	no	—
178	C₂×#357	—	53	1	no	—
179	C₂×C₅₃	—	2	2	no	—
180	C₂×C₅₃	—	8	4	no	—
181	C₂×C₅₃	C₂×#474	1	1	yes	characteristic
182	C₂×C₅₃	#3415	1	1	yes	characteristic
183	C₂×C₅₃	#3416	1	1	yes	characteristic
184	C₂²×D₈	C₂×D₅₃	1	1	yes	characteristic
185	#544	—	106	1	no	—
186	#518	—	2	1	no	—
187	C₂×M₃₂	—	53	1	no	—
188	C₂²×C₁₆	—	53	1	no	—
189	#501	—	106	2	no	—
190	#500	—	53	2	no	—
191	#499	—	53	2	no	—
192	#442	—	2	2	no	—
193	C₄×D₈	C₂×D₅₃	1	2	yes	—
194	C₄×D₈	C₂×D₅₃	1	2	yes	—
195	#428	—	106	1	no	—
196	#427	—	53	1	no	—
197	#427	C₂×D₅₃	1	1	yes	characteristic
198	C₂×#107	—	53	1	no	—
199	#418	—	2	1	no	—
200	#405	C₂×D₅₃	1	1	yes	characteristic
201	#367	—	53	1	no	—
202	#367	—	53	1	no	—
203	#367	—	53	2	no	—
204	#366	—	53	1	no	—
205	#366	—	53	1	no	—
206	#366	—	53	2	no	—
207	C₄⋊C₁₆	—	53	2	no	—
208	C₄⋊C₁₆	—	53	2	no	—
209	#357	—	53	2	no	—
210	#357	—	53	2	no	—
211	#357	—	53	2	no	—
212	#357	—	53	2	no	—
213	C₅₃	#33631	1	1	yes	characteristic, complemented, Sylow
214	C₂²×D₄	—	2	1	no	—
215	C₂×D₈	—	2	2	no	—
216	C₂×D₈	—	2	1	no	—
217	C₂×D₈	Dic₁₀₆	1	4	yes	complemented
218	C₂×D₈	C₂²×D₅₃	1	2	yes	—
219	C₂×M₁₆	C₂²⋊D₅₃	1	1	yes	characteristic
220	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
221	C₄⋊Q₈	—	106	2	no	—
222	C₄⋊Q₈	—	212	1	no	—
223	#125	—	106	2	no	—
224	#125	—	212	1	no	—
225	C₂²⋊Q₈	—	212	2	no	—
226	C₄×D₄	—	2	2	no	—
227	C₄×D₄	—	2	2	no	—
228	C₄×D₄	—	4	1	no	—
229	C₄×D₄	—	4	1	no	—
230	#117	—	2	1	no	—
231	#117	—	106	1	no	—
232	#117	C₂²⋊D₅₃	1	1	yes	characteristic
233	C₂×C₄⋊C₄	—	106	1	no	—
234	C₂×C₂²⋊C₄	—	4	2	no	—
235	M₃₂	—	106	2	no	—
236	C₂×C₁₆	—	53	1	no	—
237	C₂×C₁₆	—	53	2	no	—
238	C₂×C₁₆	—	53	1	no	—
239	C₂×C₁₆	—	106	2	no	—
240	#107	—	53	2	no	—
241	#107	—	53	1	no	—
242	#107	—	53	1	no	—
243	#107	—	106	1	no	—
244	#107	—	106	2	no	—
245	#107	C₂²×D₅₃	1	1	yes	characteristic
246	#107	C₂²×D₅₃	1	1	yes	characteristic
247	#106	—	2	1	no	—
248	#106	—	106	1	no	—
249	#106	—	106	2	no	—
250	#102	—	2	2	no	—
251	#102	—	2	2	no	—
252	#97	C₂²⋊D₅₃	1	2	yes	—
253	C₄×C₈	C₂²×D₅₃	1	2	yes	—
254	C₂⁴	—	4	1	no	—
255	C₂×Q₈	—	212	2	no	—
256	C₂×D₄	—	2	4	no	—
257	C₂×D₄	—	2	2	no	—
258	C₂×D₄	—	4	1	no	—
259	C₂×D₄	—	4	2	no	—
260	C₂²×C₄	—	4	2	no	—
261	C₂²×C₄	—	4	2	no	—
262	C₂²×C₄	—	212	1	no	—
263	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
264	D₈	—	2	4	no	—
265	D₈	—	2	2	no	—
266	D₈	C₂×Dic₁₀₆	1	4	yes	—
267	M₁₆	—	2	2	no	—
268	C₂×C₈	C₂×Dic₁₀₆	1	2	yes	—
269	C₂×C₈	#11719945	1	2	yes	—
270	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
271	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
272	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
273	C₄⋊C₄	—	2	1	no	—
274	C₄⋊C₄	—	2	1	no	—
275	C₄⋊C₄	—	106	1	no	—
276	C₄⋊C₄	—	106	2	no	—
277	C₄⋊C₄	—	106	1	no	—
278	C₄⋊C₄	—	212	1	no	—
279	C₄⋊C₄	—	212	2	no	—
280	C₄⋊C₄	—	212	2	no	—
281	C₄⋊C₄	—	212	2	no	—
282	C₄⋊C₄	#11719922	1	1	yes	characteristic
283	C₄⋊C₄	#11719922	1	1	yes	characteristic
284	C₂²⋊C₄	—	4	1	no	—
285	C₂²⋊C₄	—	4	2	no	—
286	C₂²⋊C₄	—	4	2	no	—
287	C₂²⋊C₄	—	212	1	no	—
288	C₂²⋊C₄	#11719922	1	2	yes	—
289	C₄²	—	4	1	no	—
290	C₄²	—	212	1	no	—
291	C₄²	C₂×C₂²⋊D₅₃	1	2	yes	—
292	C₁₆	—	106	1	no	—
293	C₁₆	—	106	1	no	—
294	C₁₆	—	106	2	no	—
295	C₂³	—	4	2	no	—
296	C₂³	—	4	4	no	—
297	C₂³	—	16	1	no	—
298	C₂³	—	16	1	no	—
299	C₂³	C₂×#11719922	1	1	yes	characteristic
300	Q₈	—	848	2	no	—
301	D₄	—	2	4	no	—
302	D₄	—	4	4	no	—
303	D₄	—	4	2	no	—
304	C₂×C₄	—	4	1	no	—
305	C₂×C₄	—	4	1	no	—
306	C₂×C₄	—	16	2	no	—
307	C₂×C₄	—	16	2	no	—
308	C₂×C₄	—	848	1	no	—
309	C₂×C₄	—	212	1	no	—
310	C₂×C₄	—	212	1	no	—
311	C₂×C₄	—	212	2	no	—
312	C₂×C₄	#13133196	1	2	yes	—
313	C₂×C₄	C₂×#11719922	1	2	yes	—
314	C₂×C₄	C₂×#11719922	1	2	yes	—
315	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
316	C₂×C₄	#13133247	1	2	yes	—
317	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
318	C₈	—	2	2	no	—
319	C₈	#13133196	1	2	yes	—
320	C₈	C₂²×Dic₁₀₆	1	1	yes	characteristic
321	C₈	C₂×#11719945	1	1	yes	characteristic
322	C₂²	—	2	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	4	4	no	—
325	C₂²	—	8	2	no	—
326	C₂²	—	8	2	no	—
327	C₂²	—	16	4	no	—
328	C₂²	#16810903	1	2	yes	—
329	C₂²	C₂²×#11719922	1	1	yes	characteristic
330	C₄	—	2	2	no	—
331	C₄	—	2	2	no	—
332	C₄	—	8	1	no	—
333	C₄	—	8	1	no	—
334	C₄	—	424	2	no	—
335	C₄	—	424	1	no	—
336	C₄	—	424	1	no	—
337	C₄	#16810903	1	2	yes	—
338	C₄	C₂×#13133196	1	1	yes	characteristic
339	C₄	C₂×#13133247	1	1	yes	characteristic
340	C₂	—	2	2	no	—
341	C₂	—	8	4	no	—
342	C₂	C₂×#16810903	1	1	yes	characteristic
343	C₂	#20287854	1	1	yes	characteristic
344	C₂	#20287862	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#11719922
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2948	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	C₂²×D₈×C₅₃	C₂²
hypercenter	#2948	D₅₃
hypocenter	C₅₃	#33631
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #2948

Complements

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

N	H ≅ G/N	type
1. #20285254	C₂	semidirect
2. #20276872	C₂	semidirect
3. #20276872	C₂	semidirect
4. #20276862	C₂	semidirect
5. #20276852	C₂	semidirect
6. #20276504	C₂	semidirect
7. #20276497	C₂	semidirect
8. #20276474	C₂	semidirect
16. #16811028	C₂²	semidirect
17. #16811025	C₂²	semidirect
21. #16810450	C₂²	semidirect
22. #16810450	C₂²	semidirect
23. #16810450	C₂²	semidirect
24. #16810448	C₂²	semidirect
25. #16810448	C₂²	semidirect
26. #16810448	C₂²	semidirect
27. C₄⋊#11719908	C₂²	semidirect
28. C₄⋊#11719908	C₂²	semidirect
63. C₂×D₈×C₅₃	Q₈	semidirect
213. C₅₃	#33631	semidirect
217. C₂×D₈	Dic₁₀₆	semidirect

Maximal subgroups

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

subgroup	index
#20285254	2
#20276872	2
#20276862	2
#20276852	2
#20276504	2
#20276497	2
#20276474	2
#20276431	2
C₂×#16810424	2
C₅₃×#2948	2
#33631	53