Subgroups — #27538834

Order 13568.

524 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20285227	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16810739	C₂	1	1	yes	maximal, characteristic
3	#20276890	C₂	1	4	yes	maximal, complemented
4	#20276885	C₂	1	4	yes	maximal, complemented
5	#20276489	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#16810443	C₂	1	1	yes	maximal, characteristic, complemented
7	#20276452	C₂	1	1	yes	maximal, characteristic
8	#20276450	C₂	1	1	yes	maximal, characteristic
9	C₅₃×#2155	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#13133239	C₂²	1	1	yes	characteristic
11	C₂×#13133225	—	2	1	no	—
12	#16811176	—	2	1	no	—
13	C₂×#13133157	C₂²	1	1	yes	characteristic
14	#16811027	—	2	1	no	—
15	#16811024	—	2	1	no	—
16	#16811023	C₂²	1	2	yes	complemented
17	#16811020	C₂²	1	2	yes	complemented
18	#16810739	—	2	1	no	—
19	#16810739	—	2	1	no	—
20	#16810739	C₂²	1	2	yes	—
21	#16810739	C₂²	1	2	yes	—
22	#16810661	C₂²	1	1	yes	characteristic
23	#16810659	—	2	1	no	—
24	C₂×#13133143	—	2	1	no	—
25	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
26	#16810447	C₂²	1	4	yes	complemented
27	#16810443	C₂²	1	2	yes	complemented
28	#16810443	C₂²	1	2	yes	complemented
29	#16810434	C₂²	1	4	yes	—
30	#16810432	C₂²	1	4	yes	—
31	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
32	C₂×#108×C₅₃	C₂²	1	1	yes	characteristic
33	C₅₃×#427	C₂²	1	1	yes	characteristic
34	C₅₃×#365	C₂²	1	4	yes	complemented
35	C₂×#11719945	—	2	1	no	—
36	#13133239	—	2	1	no	—
37	#13133239	—	2	1	no	—
38	#13133239	Q₈	1	2	yes	complemented
39	#13133239	C₂³	1	2	yes	—
40	C₂×Dic₂₁₂	Q₈	1	1	yes	characteristic, complemented
41	C₂×#11719913	—	2	1	no	—
42	C₂×D₄₂₄	Q₈	1	1	yes	characteristic, complemented
43	#13133225	—	2	2	no	—
44	#13133225	—	2	2	no	—
45	#13133225	—	4	1	no	—
46	#13133225	—	4	1	no	—
47	C₂×#11719912	—	4	1	no	—
48	C₂²×#88322	D₄	1	1	yes	characteristic
49	C₂×C₈×D₅₃	—	4	1	no	—
50	#13133216	—	2	2	no	—
51	#13133208	—	2	2	no	—
52	#13133194	—	4	2	no	—
53	C₄×Dic₁₀₆	—	4	2	no	—
54	#13133176	—	2	1	no	—
55	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
56	#13133157	—	2	1	no	—
57	#13133157	C₂³	1	2	yes	—
58	C₂×#11719908	—	2	1	no	—
59	#13133151	—	2	1	no	—
60	#13133150	—	2	2	no	—
61	#13133149	C₂³	1	2	yes	—
62	#13133148	—	2	2	no	—
63	#13133143	—	2	2	no	—
64	#13133143	—	2	2	no	—
65	C₈×Dic₅₃	D₄	1	1	yes	characteristic
66	C₈×Dic₅₃	D₄	1	1	yes	characteristic
67	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
68	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
69	C₂×M₁₆×C₅₃	—	2	1	no	—
70	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
71	#117×C₅₃	—	2	1	no	—
72	M₃₂×C₅₃	—	2	1	no	—
73	M₃₂×C₅₃	C₂³	1	2	yes	—
74	C₂×C₁₆×C₅₃	—	2	1	no	—
75	#108×C₅₃	—	2	1	no	—
76	#108×C₅₃	C₂³	1	2	yes	—
77	#107×C₅₃	—	2	1	no	—
78	#106×C₅₃	C₂³	1	2	yes	—
79	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
80	C₂×C₂²⋊D₅₃	—	4	1	no	—
81	C₂²×Dic₅₃	—	2	1	no	—
82	#11719945	—	2	2	no	—
83	#11719945	—	2	2	no	—
84	#11719945	—	4	1	no	—
85	#11719945	—	4	1	no	—
86	C₂×D₂₁₂	—	2	1	no	—
87	C₂×C₄×D₅₃	—	4	1	no	—
88	C₂×Dic₁₀₆	—	2	1	no	—
89	C₂×Dic₁₀₆	—	4	2	no	—
90	M₁₆×C₅₃	—	2	2	no	—
91	M₁₆×C₅₃	—	4	1	no	—
92	C₂×C₈×C₅₃	—	4	1	no	—
93	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
95	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
96	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
97	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
98	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
99	C₄⋊C₄×C₅₃	—	2	2	no	—
100	C₂²⋊C₄×C₅₃	—	4	1	no	—
101	C₄²×C₅₃	—	4	1	no	—
102	C₂²⋊Dic₅₃	—	4	1	no	—
103	C₄⋊Dic₅₃	—	2	2	no	—
104	C₄⋊Dic₅₃	—	4	2	no	—
105	#11719919	—	4	2	no	—
106	#11719919	—	4	2	no	—
107	C₄×Dic₅₃	—	2	1	no	—
108	C₄×Dic₅₃	—	4	1	no	—
109	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
110	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
111	C₂×#88322	SD₁₆	1	1	yes	characteristic
112	C₂×#88322	SD₁₆	1	1	yes	characteristic
113	C₂×#88322	SD₁₆	1	1	yes	characteristic
114	C₂×#88322	SD₁₆	1	1	yes	characteristic
115	C₂×#88322	C₂×D₄	1	1	yes	characteristic
116	C₂×#88322	C₂×D₄	1	1	yes	characteristic
117	Dic₂₁₂	—	2	1	no	—
118	Dic₂₁₂	C₂×Q₈	1	2	yes	—
119	D₄₂₄	—	2	1	no	—
120	D₄₂₄	C₂×Q₈	1	2	yes	—
121	#11719913	—	2	2	no	—
122	#11719913	—	2	2	no	—
123	#11719912	—	4	2	no	—
124	#11719912	—	4	2	no	—
125	C₈×D₅₃	—	4	2	no	—
126	C₈×D₅₃	—	4	2	no	—
127	C₁₆×C₅₃	—	2	2	no	—
128	#11719908	—	2	2	no	—
129	C₂³×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
130	C₂²×D₅₃	—	4	1	no	—
131	C₂×C₄×C₅₃	—	4	2	no	—
132	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
133	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
134	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
135	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
136	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
137	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
138	C₂²⋊D₅₃	—	4	2	no	—
139	C₂²⋊D₅₃	—	4	2	no	—
140	C₂×Dic₅₃	—	2	1	no	—
141	C₂×Dic₅₃	—	2	1	no	—
142	C₂×Dic₅₃	—	4	2	no	—
143	C₂×Dic₅₃	—	4	1	no	—
144	C₂×Dic₅₃	—	4	2	no	—
145	D₂₁₂	—	2	2	no	—
146	D₂₁₂	—	4	1	no	—
147	C₄×D₅₃	—	4	2	no	—
148	C₄×D₅₃	—	4	2	no	—
149	Dic₁₀₆	—	2	2	no	—
150	Dic₁₀₆	—	4	1	no	—
151	Dic₁₀₆	—	8	4	no	—
152	C₈×C₅₃	—	4	2	no	—
153	C₈×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
154	C₈×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
155	C₈×C₅₃	C₂²×Q₈	1	1	yes	characteristic
156	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
157	#88322	C₂×SD₁₆	1	1	yes	characteristic
158	#88322	C₂×SD₁₆	1	1	yes	characteristic
159	#88322	C₂×SD₁₆	1	1	yes	characteristic
160	#88322	C₂×SD₁₆	1	1	yes	characteristic
161	#33637	—	53	1	no	maximal, Sylow
162	C₂²×C₅₃	—	4	1	no	—
163	C₂²×C₅₃	#476	1	1	yes	characteristic
164	C₂²×C₅₃	#476	1	1	yes	characteristic
165	C₂²×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
166	C₂×D₅₃	—	4	2	no	—
167	C₂×D₅₃	—	8	2	no	—
168	C₄×C₅₃	—	8	2	no	—
169	C₄×C₅₃	#476	1	1	yes	characteristic
170	C₄×C₅₃	#476	1	1	yes	characteristic
171	C₄×C₅₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
172	C₄×C₅₃	C₂²×SD₁₆	1	1	yes	characteristic
173	Dic₅₃	—	2	2	no	—
174	Dic₅₃	—	4	1	no	—
175	Dic₅₃	—	4	2	no	—
176	Dic₅₃	—	8	2	no	—
177	#3160	—	53	1	no	—
178	C₂×#443	—	53	1	no	—
179	#2231	—	53	4	no	—
180	#2230	—	53	4	no	—
181	#2155	—	53	1	no	—
182	#2155	D₅₃	1	1	yes	characteristic, complemented
183	C₂×#364	—	53	1	no	—
184	#2145	—	53	1	no	—
185	#2145	—	53	1	no	—
186	C₂×C₅₃	—	2	1	no	—
187	C₂×C₅₃	—	2	1	no	—
188	C₂×C₅₃	—	2	1	no	—
189	C₂×C₅₃	C₂×#476	1	1	yes	characteristic
190	D₅₃	—	8	2	no	—
191	C₂×#135	—	53	1	no	—
192	C₂×#131	—	106	1	no	—
193	#533	—	106	1	no	—
194	C₂×M₃₂	—	53	1	no	—
195	C₂×M₃₂	C₂×D₅₃	1	1	yes	characteristic
196	#500	—	106	1	no	—
197	#499	—	53	2	no	—
198	#499	—	106	1	no	—
199	#498	—	53	2	no	—
200	#443	—	53	2	no	—
201	#443	—	53	2	no	—
202	#443	—	106	1	no	—
203	#443	—	106	1	no	—
204	C₂×#108	C₂×D₅₃	1	1	yes	characteristic
205	#427	—	53	1	no	—
206	#427	—	106	1	no	—
207	#427	C₂×D₅₃	1	1	yes	characteristic
208	C₂×#104	—	106	1	no	—
209	C₂×C₄×C₈	—	53	1	no	—
210	#365	—	53	4	no	—
211	#365	C₂×D₅₃	1	4	yes	complemented
212	#364	—	53	2	no	—
213	#364	—	53	2	no	—
214	#360	—	53	4	no	—
215	#360	—	53	4	no	—
216	C₅₃	#33637	1	1	yes	characteristic, complemented, Sylow
217	C₂×#41	—	106	1	no	—
218	#135	—	53	2	no	—
219	#135	—	53	2	no	—
220	#135	—	106	1	no	—
221	#135	—	106	1	no	—
222	C₂×Q₁₆	—	53	1	no	—
223	C₂×SD₁₆	—	106	1	no	—
224	C₂×D₈	—	53	1	no	—
225	#131	—	106	2	no	—
226	#131	—	106	2	no	—
227	#131	—	212	1	no	—
228	#131	—	212	1	no	—
229	C₂×M₁₆	—	2	1	no	—
230	C₂×M₁₆	—	212	1	no	—
231	C₂²×C₈	—	53	1	no	—
232	C₂²×C₈	—	212	1	no	—
233	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
234	C₄⋊Q₈	—	106	2	no	—
235	#125	—	106	2	no	—
236	C₂²⋊Q₈	—	212	2	no	—
237	C₄×Q₈	—	212	2	no	—
238	#117	—	2	1	no	—
239	#117	—	106	1	no	—
240	C₂×C₄²	—	53	1	no	—
241	M₃₂	—	2	1	no	—
242	M₃₂	—	53	2	no	—
243	M₃₂	—	106	1	no	—
244	M₃₂	C₂²×D₅₃	1	2	yes	—
245	C₂×C₁₆	—	2	1	no	—
246	C₂×C₁₆	—	106	1	no	—
247	#108	—	2	1	no	—
248	#108	C₂²×D₅₃	1	2	yes	—
249	#107	—	2	1	no	—
250	#107	—	106	2	no	—
251	#107	—	106	1	no	—
252	#106	—	53	2	no	—
253	#106	—	106	2	no	—
254	#106	C₂²×D₅₃	1	2	yes	—
255	#104	—	106	2	no	—
256	#104	—	106	2	no	—
257	C₄×C₈	—	53	1	no	—
258	C₄×C₈	—	53	1	no	—
259	C₄×C₈	—	53	1	no	—
260	C₄×C₈	—	53	1	no	—
261	#41	—	106	2	no	—
262	#41	—	106	2	no	—
263	#41	—	212	1	no	—
264	#41	—	212	1	no	—
265	C₂×Q₈	—	106	1	no	—
266	C₂×Q₈	—	212	2	no	—
267	C₂×D₄	—	106	1	no	—
268	C₂×D₄	—	212	1	no	—
269	C₂²×C₄	—	106	1	no	—
270	C₂²×C₄	—	212	1	no	—
271	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
272	Q₁₆	—	53	2	no	—
273	Q₁₆	—	106	1	no	—
274	SD₁₆	—	106	2	no	—
275	SD₁₆	—	106	2	no	—
276	D₈	—	53	2	no	—
277	D₈	—	106	1	no	—
278	M₁₆	—	2	2	no	—
279	M₁₆	—	4	1	no	—
280	M₁₆	—	212	2	no	—
281	M₁₆	—	212	2	no	—
282	C₂×C₈	—	4	1	no	—
283	C₂×C₈	—	53	1	no	—
284	C₂×C₈	—	53	1	no	—
285	C₂×C₈	—	53	1	no	—
286	C₂×C₈	—	53	1	no	—
287	C₂×C₈	—	53	1	no	—
288	C₂×C₈	—	53	1	no	—
289	C₂×C₈	—	212	2	no	—
290	C₂×C₈	—	212	2	no	—
291	C₂×C₈	#11719945	1	1	yes	characteristic
292	C₂×C₈	#11719945	1	1	yes	characteristic
293	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
294	C₂×C₈	Q₈×D₅₃	1	1	yes	characteristic
295	C₂×C₈	Q₈×D₅₃	1	1	yes	characteristic
296	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
297	C₄⋊C₄	—	2	2	no	—
298	C₄⋊C₄	—	106	2	no	—
299	C₄⋊C₄	—	212	2	no	—
300	C₄⋊C₄	—	212	2	no	—
301	C₄⋊C₄	—	212	2	no	—
302	C₂²⋊C₄	—	4	1	no	—
303	C₂²⋊C₄	—	212	1	no	—
304	C₄²	—	4	1	no	—
305	C₄²	—	53	1	no	—
306	C₄²	—	53	1	no	—
307	C₄²	—	106	1	no	—
308	C₄²	—	212	1	no	—
309	C₁₆	—	2	2	no	—
310	C₁₆	—	106	2	no	—
311	C₂³	—	212	1	no	—
312	C₂³	#13133244	1	1	yes	characteristic
313	Q₈	—	106	2	no	—
314	Q₈	—	424	4	no	—
315	Q₈	—	212	1	no	—
316	D₄	—	106	2	no	—
317	D₄	—	212	2	no	—
318	D₄	—	212	2	no	—
319	D₄	—	212	1	no	—
320	C₂×C₄	—	4	2	no	—
321	C₂×C₄	—	106	1	no	—
322	C₂×C₄	—	106	1	no	—
323	C₂×C₄	—	212	2	no	—
324	C₂×C₄	—	212	2	no	—
325	C₂×C₄	—	212	1	no	—
326	C₂×C₄	—	212	2	no	—
327	C₂×C₄	—	212	2	no	—
328	C₂×C₄	#13133195	1	1	yes	characteristic
329	C₂×C₄	#13133195	1	1	yes	characteristic
330	C₂×C₄	SD₁₆×D₅₃	1	1	yes	characteristic
331	C₂×C₄	SD₁₆×D₅₃	1	1	yes	characteristic
332	C₂×C₄	#13133244	1	1	yes	characteristic
333	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
334	C₈	—	4	2	no	—
335	C₈	—	53	1	no	—
336	C₈	—	53	1	no	—
337	C₈	—	53	1	no	—
338	C₈	—	53	1	no	—
339	C₈	#13133195	1	1	yes	characteristic
340	C₈	#13133195	1	1	yes	characteristic
341	C₈	C₂×Q₈×D₅₃	1	1	yes	characteristic
342	C₈	C₂×#11719945	1	1	yes	characteristic
343	C₂²	—	4	1	no	—
344	C₂²	—	424	2	no	—
345	C₂²	—	212	2	no	—
346	C₂²	#16810912	1	1	yes	characteristic
347	C₂²	#16810912	1	1	yes	characteristic
348	C₂²	C₂×#13133244	1	1	yes	characteristic
349	C₄	—	8	2	no	—
350	C₄	—	106	2	no	—
351	C₄	—	424	2	no	—
352	C₄	—	212	1	no	—
353	C₄	—	212	2	no	—
354	C₄	#16810912	1	1	yes	characteristic
355	C₄	#16810912	1	1	yes	characteristic
356	C₄	C₂×#13133195	1	1	yes	characteristic
357	C₄	C₂×SD₁₆×D₅₃	1	1	yes	characteristic
358	C₂	—	2	1	no	—
359	C₂	—	2	1	no	—
360	C₂	—	2	1	no	—
361	C₂	—	424	2	no	—
362	C₂	C₂×#16810912	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#16810912
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2155	C₂
socle	C₂×C₅₃	C₂×#476
prime radical	C₂×#13133239	C₂²
hypercenter	#2155	D₅₃
hypocenter	C₅₃	#33637
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2155

Complements

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

N	H ≅ G/N	type
1. #20285227	C₂	semidirect
3. #20276890	C₂	semidirect
4. #20276885	C₂	semidirect
5. #20276489	C₂	semidirect
6. C₂×#16810443	C₂	semidirect
9. C₅₃×#2155	C₂	semidirect
16. #16811023	C₂²	semidirect
17. #16811020	C₂²	semidirect
26. #16810447	C₂²	semidirect
27. #16810443	C₂²	semidirect
28. #16810443	C₂²	semidirect
34. C₅₃×#365	C₂²	semidirect
38. #13133239	Q₈	semidirect
40. C₂×Dic₂₁₂	Q₈	semidirect
42. C₂×D₄₂₄	Q₈	semidirect
182. #2155	D₅₃	semidirect
211. #365	C₂×D₅₃	semidirect
216. C₅₃	#33637	semidirect

Maximal subgroups

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

subgroup	index
#20285227	2
C₂×#16810739	2
#20276890	2
#20276885	2
#20276489	2
C₂×#16810443	2
#20276452	2
#20276450	2
C₅₃×#2155	2
#33637	53