Subgroups — #27540899

Order 13568.

646 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20287868	C₂	1	1	yes	maximal, characteristic, complemented
2	#20287852	C₂	1	1	yes	maximal, characteristic, complemented
3	#20286496	C₂	1	2	yes	maximal, complemented
4	#20276973	C₂	1	1	yes	maximal, characteristic, complemented
5	#20276841	C₂	1	1	yes	maximal, characteristic, complemented
6	#20276783	C₂	1	2	yes	maximal, complemented
7	#20276689	C₂	1	2	yes	maximal, complemented
8	#20276677	C₂	1	2	yes	maximal, complemented
9	#20276595	C₂	1	1	yes	maximal, characteristic, complemented
10	#20276550	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#2221	C₂	1	1	yes	maximal, characteristic, complemented
12	#16811440	C₂²	1	2	yes	complemented
13	#16811433	—	2	2	no	—
14	#16811424	C₂²	1	2	yes	—
15	#16811066	—	2	2	no	—
16	#16811066	C₂²	1	2	yes	complemented
17	#16811059	C₂²	1	2	yes	—
18	#16811057	—	2	2	no	—
19	#16811055	—	2	1	no	—
20	#16811055	C₂²	1	1	yes	characteristic
21	#16811052	—	2	1	no	—
22	C₂×Dic₄₂₄	—	2	1	no	—
23	C₂×Dic₄₂₄	C₂²	1	1	yes	characteristic, complemented
24	C₂×#13133161	C₂²	1	2	yes	complemented
25	D₁₆×D₅₃	—	2	1	no	—
26	#16811039	C₂²	1	1	yes	characteristic
27	#16811006	C₂²	1	2	yes	complemented
28	#16810899	C₂²	1	2	yes	—
29	#16810885	—	2	2	no	—
30	#16810803	—	2	2	no	—
31	#16810794	—	2	2	no	—
32	#16810739	C₂²	1	2	yes	complemented
33	#16810738	C₂²	1	1	yes	characteristic
34	#16810443	C₂²	1	1	yes	characteristic, complemented
35	#16810438	C₂²	1	2	yes	complemented
36	#16810429	C₂²	1	2	yes	complemented
37	#16810408	C₂²	1	1	yes	characteristic
38	#16810407	C₂²	1	1	yes	characteristic, complemented
39	C₁₆×Dic₅₃	C₂²	1	1	yes	characteristic, complemented
40	C₂×D₁₆×C₅₃	C₂²	1	1	yes	characteristic
41	C₅₃×#504	C₂²	1	1	yes	characteristic, complemented
42	C₅₃×#472	C₂²	1	2	yes	complemented
43	C₅₃×#368	C₂²	1	1	yes	characteristic, complemented
44	C₅₃×#361	C₂²	1	2	yes	complemented
45	#13133264	—	2	2	no	—
46	#13133262	—	2	2	no	—
47	#13133250	—	2	2	no	—
48	#13133250	—	2	2	no	—
49	#13133250	—	4	2	no	—
50	#13133246	—	4	2	no	—
51	#13133243	—	2	2	no	—
52	#13133241	—	4	2	no	—
53	#13133241	—	4	2	no	—
54	#13133239	—	2	2	no	—
55	#13133239	C₂³	1	2	yes	—
56	#13133237	D₄	1	1	yes	characteristic
57	#13133236	—	2	2	no	—
58	#13133236	—	4	2	no	—
59	C₂×Dic₂₁₂	—	2	2	no	—
60	C₂×Dic₂₁₂	C₂³	1	2	yes	—
61	C₂×#11719923	—	2	2	no	—
62	SD₁₆×D₅₃	—	4	2	no	—
63	D₈×D₅₃	—	2	2	no	—
64	#13133225	—	2	2	no	—
65	#13133223	C₂³	1	1	yes	characteristic
66	#13133204	—	2	2	no	—
67	Dic₄₂₄	—	2	2	no	—
68	Dic₄₂₄	—	2	1	no	—
69	Dic₄₂₄	—	2	1	no	—
70	Dic₄₂₄	D₄	1	1	yes	characteristic
71	#13133162	—	2	2	no	—
72	#13133162	D₄	1	2	yes	—
73	#13133161	—	2	2	no	—
74	#13133161	—	4	2	no	—
75	D₈₄₈	D₄	1	1	yes	characteristic
76	#13133158	—	2	2	no	—
77	#13133157	C₂³	1	1	yes	characteristic
78	#13133156	—	2	1	no	—
79	C₂×#11719908	C₂³	1	1	yes	characteristic
80	C₁₆×D₅₃	—	2	1	no	—
81	#13133152	D₄	1	1	yes	characteristic
82	#13133144	D₄	1	1	yes	characteristic
83	#13133143	—	2	2	no	—
84	#13133143	D₄	1	1	yes	characteristic
85	#13133135	—	2	2	no	—
86	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
87	AGL(1,ℤ/8)×C₅₃	—	2	2	no	—
88	C₂×D₈×C₅₃	C₂³	1	2	yes	—
89	#131×C₅₃	C₂³	1	1	yes	characteristic
90	D₁₆×C₅₃	—	2	1	no	—
91	D₁₆×C₅₃	—	2	1	no	—
92	M₃₂×C₅₃	—	2	1	no	—
93	M₃₂×C₅₃	C₂³	1	1	yes	characteristic
94	C₂×C₁₆×C₅₃	—	2	1	no	—
95	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
96	#108×C₅₃	C₂³	1	2	yes	—
97	#100×C₅₃	—	2	2	no	—
98	#41×C₅₃	C₂×D₄	1	1	yes	characteristic
99	C₂×D₄×C₅₃	—	2	2	no	—
100	C₂×C₂²⋊D₅₃	—	8	2	no	—
101	Q₈×D₅₃	—	4	2	no	—
102	#11719947	—	4	2	no	—
103	#11719947	—	4	2	no	—
104	#11719947	—	4	2	no	—
105	#11719947	—	8	2	no	—
106	D₄×D₅₃	—	4	2	no	—
107	D₄×D₅₃	—	4	2	no	—
108	#11719945	—	2	2	no	—
109	#11719945	—	4	2	no	—
110	#11719945	C₂×D₄	1	1	yes	characteristic
111	C₂×Dic₁₀₆	—	2	2	no	—
112	C₂×Dic₁₀₆	—	4	2	no	—
113	SD₁₆×C₅₃	—	4	2	no	—
114	D₈×C₅₃	—	2	2	no	—
115	D₈×C₅₃	—	2	2	no	—
116	D₈×C₅₃	—	4	2	no	—
117	M₁₆×C₅₃	—	2	1	no	—
118	M₁₆×C₅₃	—	2	2	no	—
119	M₁₆×C₅₃	C₂×D₄	1	1	yes	characteristic
120	C₂×C₈×C₅₃	—	2	1	no	—
121	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
122	C₂²⋊Dic₅₃	—	8	2	no	—
123	#11719925	D₈	1	1	yes	characteristic
124	#11719924	D₈	1	1	yes	characteristic
125	#11719923	—	4	2	no	—
126	#11719923	—	4	2	no	—
127	#11719923	—	4	2	no	—
128	#11719923	D₈	1	1	yes	characteristic
129	#11719922	—	4	2	no	—
130	#11719922	D₈	1	1	yes	characteristic
131	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
132	#11719917	C₂×D₄	1	1	yes	characteristic
133	C₂×#88322	C₂×D₄	1	1	yes	characteristic
134	Dic₂₁₂	—	2	2	no	—
135	Dic₂₁₂	—	2	2	no	—
136	Dic₂₁₂	—	4	2	no	—
137	Dic₂₁₂	—	4	2	no	—
138	Dic₂₁₂	C₂×D₄	1	2	yes	—
139	D₄₂₄	C₂×D₄	1	2	yes	—
140	#11719913	—	2	2	no	—
141	#11719913	—	4	2	no	—
142	#11719913	—	4	2	no	—
143	#11719912	—	2	1	no	—
144	#11719912	—	4	2	no	—
145	C₈×D₅₃	—	2	1	no	—
146	C₈×D₅₃	—	4	2	no	—
147	C₁₆×C₅₃	—	2	1	no	—
148	C₁₆×C₅₃	C₂×D₄	1	1	yes	characteristic
149	C₁₆×C₅₃	C₂×D₄	1	1	yes	characteristic
150	#11719908	—	2	1	no	—
151	#11719908	—	2	1	no	—
152	C₂³×C₅₃	—	4	2	no	—
153	C₂²×D₅₃	—	8	2	no	—
154	Q₈×C₅₃	C₂×D₈	1	1	yes	characteristic
155	D₄×C₅₃	—	2	1	no	—
156	D₄×C₅₃	—	4	2	no	—
157	D₄×C₅₃	—	4	2	no	—
158	D₄×C₅₃	C₂×D₈	1	1	yes	characteristic
159	C₂×C₄×C₅₃	—	2	1	no	—
160	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
161	C₂²⋊D₅₃	—	2	1	no	—
162	C₂²⋊D₅₃	—	4	2	no	—
163	C₂²⋊D₅₃	—	8	2	no	—
164	C₂²⋊D₅₃	—	8	2	no	—
165	C₂²⋊D₅₃	—	8	2	no	—
166	C₂×Dic₅₃	—	2	1	no	—
167	C₂×Dic₅₃	—	4	2	no	—
168	C₂×Dic₅₃	—	8	2	no	—
169	C₂×Dic₅₃	—	8	2	no	—
170	D₂₁₂	—	2	2	no	—
171	D₂₁₂	C₂×D₈	1	1	yes	characteristic
172	C₄×D₅₃	—	2	1	no	—
173	C₄×D₅₃	—	4	2	no	—
174	C₄×D₅₃	—	8	2	no	—
175	Dic₁₀₆	—	2	2	no	—
176	Dic₁₀₆	—	4	2	no	—
177	Dic₁₀₆	—	4	2	no	—
178	Dic₁₀₆	—	8	2	no	—
179	Dic₁₀₆	C₂×D₈	1	1	yes	characteristic
180	C₈×C₅₃	—	2	1	no	—
181	C₈×C₅₃	—	4	2	no	—
182	C₈×C₅₃	C₂²×D₄	1	1	yes	characteristic
183	C₈×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
184	#88322	—	2	1	no	—
185	#88322	C₂×D₈	1	1	yes	characteristic
186	#88322	C₂×D₈	1	1	yes	characteristic
187	#33859	—	53	1	no	maximal, Sylow
188	C₂²×C₅₃	—	2	1	no	—
189	C₂²×C₅₃	—	8	2	no	—
190	C₂²×C₅₃	—	16	2	no	—
191	C₂²×C₅₃	#576	1	1	yes	characteristic
192	C₂×D₅₃	—	2	1	no	—
193	C₂×D₅₃	—	4	2	no	—
194	C₂×D₅₃	—	32	2	no	—
195	C₄×C₅₃	—	2	1	no	—
196	C₄×C₅₃	D₄²	1	1	yes	characteristic
197	C₄×C₅₃	C₂²×D₈	1	1	yes	characteristic
198	Dic₅₃	—	2	1	no	—
199	Dic₅₃	—	4	1	no	—
200	Dic₅₃	—	4	2	no	—
201	Dic₅₃	—	8	2	no	—
202	#3417	—	53	1	no	—
203	#3415	—	53	1	no	—
204	#3290	—	53	2	no	—
205	#2249	—	53	1	no	—
206	#2222	—	53	1	no	—
207	#2221	D₅₃	1	1	yes	characteristic, complemented
208	#2212	—	53	2	no	—
209	#2194	—	53	2	no	—
210	#2193	—	53	2	no	—
211	#2178	—	53	1	no	—
212	#2170	—	53	1	no	—
213	C₂×C₅₃	—	2	1	no	—
214	C₂×C₅₃	—	4	1	no	—
215	C₂×C₅₃	—	16	2	no	—
216	C₂×C₅₃	D₄×D₈	1	1	yes	characteristic
217	D₅₃	—	4	1	no	—
218	D₅₃	—	8	2	no	—
219	#578	—	53	2	no	—
220	#577	—	106	2	no	—
221	#576	—	53	2	no	—
222	#510	—	53	2	no	—
223	#510	—	106	2	no	—
224	#509	—	53	2	no	—
225	#509	—	106	2	no	—
226	#508	—	53	1	no	—
227	#508	—	106	1	no	—
228	#508	—	106	1	no	—
229	C₂×Q₃₂	—	53	1	no	—
230	C₂×Q₃₂	—	106	1	no	—
231	C₂×SD₃₂	—	53	2	no	—
232	C₂×D₁₆	—	106	1	no	—
233	C₂×D₁₆	C₂×D₅₃	1	1	yes	characteristic
234	#504	—	53	1	no	—
235	#504	C₂×D₅₃	1	1	yes	characteristic, complemented
236	#495	—	53	2	no	—
237	#473	—	53	2	no	—
238	#472	C₂×D₅₃	1	2	yes	complemented
239	#471	—	106	2	no	—
240	#455	—	106	2	no	—
241	#454	—	106	2	no	—
242	#443	—	53	1	no	—
243	#443	—	53	2	no	—
244	#368	C₂×D₅₃	1	1	yes	characteristic, complemented
245	#364	—	53	1	no	—
246	#362	—	53	2	no	—
247	#361	C₂×D₅₃	1	2	yes	complemented
248	#359	—	53	2	no	—
249	#350	—	53	1	no	—
250	#350	—	53	1	no	—
251	C₄×C₁₆	—	53	1	no	—
252	C₅₃	#33859	1	1	yes	characteristic, complemented, Sylow
253	2₋¹⁺⁴	—	106	2	no	—
254	H(5,𝔽₂)	—	106	2	no	—
255	#137	—	106	2	no	—
256	#137	—	106	2	no	—
257	#137	—	212	2	no	—
258	#137	—	212	2	no	—
259	AGL(1,ℤ/8)	—	2	2	no	—
260	AGL(1,ℤ/8)	—	106	2	no	—
261	AGL(1,ℤ/8)	—	212	2	no	—
262	AGL(1,ℤ/8)	—	212	2	no	—
263	#135	—	53	1	no	—
264	#135	—	53	2	no	—
265	#135	—	106	2	no	—
266	#135	—	106	2	no	—
267	#135	—	212	2	no	—
268	C₂×Q₁₆	—	53	2	no	—
269	C₂×Q₁₆	—	106	2	no	—
270	C₂×SD₁₆	—	106	2	no	—
271	C₂×SD₁₆	—	212	2	no	—
272	C₂×D₈	—	106	2	no	—
273	C₂×D₈	C₂²×D₅₃	1	2	yes	—
274	#131	—	53	1	no	—
275	#131	—	106	2	no	—
276	#131	C₂²×D₅₃	1	1	yes	characteristic
277	#124	—	106	2	no	—
278	Q₃₂	—	53	1	no	—
279	Q₃₂	—	106	2	no	—
280	Q₃₂	—	106	1	no	—
281	Q₃₂	—	106	1	no	—
282	SD₃₂	—	53	2	no	—
283	SD₃₂	—	106	2	no	—
284	SD₃₂	—	106	2	no	—
285	SD₃₂	—	212	2	no	—
286	D₁₆	—	2	1	no	—
287	D₁₆	—	2	1	no	—
288	D₁₆	—	53	1	no	—
289	D₁₆	—	106	2	no	—
290	M₃₂	—	2	1	no	—
291	M₃₂	—	53	1	no	—
292	M₃₂	—	106	1	no	—
293	M₃₂	C₂²×D₅₃	1	1	yes	characteristic
294	C₂×C₁₆	—	2	1	no	—
295	C₂×C₁₆	—	53	1	no	—
296	C₂×C₁₆	—	106	1	no	—
297	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
298	#108	—	53	1	no	—
299	#108	C₂²×D₅₃	1	2	yes	—
300	#104	—	53	1	no	—
301	#104	—	53	1	no	—
302	#104	—	106	2	no	—
303	#101	—	106	2	no	—
304	#100	—	2	2	no	—
305	C₄×C₈	—	53	1	no	—
306	#41	—	53	1	no	—
307	#41	—	106	2	no	—
308	#41	—	424	2	no	—
309	#41	—	212	2	no	—
310	#41	—	212	2	no	—
311	#41	—	212	2	no	—
312	#41	—	212	2	no	—
313	#41	D₄×D₅₃	1	1	yes	characteristic
314	C₂×Q₈	—	106	2	no	—
315	C₂×Q₈	—	212	2	no	—
316	C₂×Q₈	—	212	2	no	—
317	C₂×D₄	—	2	2	no	—
318	C₂×D₄	—	424	2	no	—
319	C₂×D₄	—	212	2	no	—
320	C₂×D₄	—	212	2	no	—
321	Q₁₆	—	53	1	no	—
322	Q₁₆	—	53	2	no	—
323	Q₁₆	—	106	2	no	—
324	Q₁₆	—	106	2	no	—
325	Q₁₆	—	212	2	no	—
326	Q₁₆	—	212	2	no	—
327	SD₁₆	—	4	2	no	—
328	SD₁₆	—	53	1	no	—
329	SD₁₆	—	53	1	no	—
330	SD₁₆	—	106	2	no	—
331	SD₁₆	—	212	2	no	—
332	SD₁₆	—	212	2	no	—
333	SD₁₆	—	212	2	no	—
334	SD₁₆	—	212	2	no	—
335	SD₁₆	—	212	2	no	—
336	D₈	—	2	2	no	—
337	D₈	—	2	2	no	—
338	D₈	—	4	2	no	—
339	D₈	—	53	1	no	—
340	D₈	—	53	2	no	—
341	D₈	—	212	2	no	—
342	M₁₆	—	2	1	no	—
343	M₁₆	—	2	2	no	—
344	M₁₆	—	53	1	no	—
345	M₁₆	—	106	1	no	—
346	M₁₆	—	212	2	no	—
347	M₁₆	D₄×D₅₃	1	1	yes	characteristic
348	C₂×C₈	—	2	1	no	—
349	C₂×C₈	—	53	1	no	—
350	C₂×C₈	—	106	1	no	—
351	C₂×C₈	—	212	2	no	—
352	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
353	C₂²⋊C₄	—	424	2	no	—
354	C₄²	—	53	1	no	—
355	C₁₆	—	2	1	no	—
356	C₁₆	—	106	1	no	—
357	C₁₆	—	106	1	no	—
358	C₁₆	D₄×D₅₃	1	1	yes	characteristic
359	C₁₆	D₄×D₅₃	1	1	yes	characteristic
360	C₂³	—	4	2	no	—
361	C₂³	—	424	2	no	—
362	Q₈	—	53	1	no	—
363	Q₈	—	106	2	no	—
364	Q₈	—	424	2	no	—
365	Q₈	—	212	2	no	—
366	Q₈	—	212	2	no	—
367	Q₈	D₈×D₅₃	1	1	yes	characteristic
368	D₄	—	2	1	no	—
369	D₄	—	4	2	no	—
370	D₄	—	4	2	no	—
371	D₄	—	53	1	no	—
372	D₄	—	106	1	no	—
373	D₄	—	106	2	no	—
374	D₄	—	424	2	no	—
375	D₄	—	424	2	no	—
376	D₄	—	424	2	no	—
377	D₄	—	212	2	no	—
378	D₄	D₈×D₅₃	1	1	yes	characteristic
379	C₂×C₄	—	2	1	no	—
380	C₂×C₄	—	106	1	no	—
381	C₂×C₄	—	106	1	no	—
382	C₂×C₄	—	424	2	no	—
383	C₂×C₄	—	424	2	no	—
384	C₂×C₄	—	424	2	no	—
385	C₂×C₄	—	212	2	no	—
386	C₂×C₄	—	212	2	no	—
387	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
388	C₈	—	2	1	no	—
389	C₈	—	4	2	no	—
390	C₈	—	53	1	no	—
391	C₈	—	53	1	no	—
392	C₈	—	106	1	no	—
393	C₈	C₂×D₄×D₅₃	1	1	yes	characteristic
394	C₈	#13133263	1	1	yes	characteristic
395	C₂²	—	2	1	no	—
396	C₂²	—	8	2	no	—
397	C₂²	—	16	2	no	—
398	C₂²	—	106	1	no	—
399	C₂²	—	1696	2	no	—
400	C₂²	—	212	2	no	—
401	C₂²	#16811425	1	1	yes	characteristic
402	C₄	—	2	1	no	—
403	C₄	—	106	1	no	—
404	C₄	—	424	2	no	—
405	C₄	—	212	1	no	—
406	C₄	—	212	2	no	—
407	C₄	#16811257	1	1	yes	characteristic
408	C₄	C₂×D₈×D₅₃	1	1	yes	characteristic
409	C₂	—	2	1	no	—
410	C₂	—	4	1	no	—
411	C₂	—	16	2	no	—
412	C₂	—	424	2	no	—
413	C₂	—	212	1	no	—
414	C₂	#20286389	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20286389
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2221	C₂
socle	C₂×C₅₃	D₄×D₈
hypercenter	#2221	D₅₃
hypocenter	C₅₃	#33859
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #131 ◁ #2221

Complements

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

N	H ≅ G/N	type
1. #20287868	C₂	semidirect
2. #20287852	C₂	semidirect
3. #20286496	C₂	semidirect
4. #20276973	C₂	semidirect
5. #20276841	C₂	semidirect
6. #20276783	C₂	semidirect
7. #20276689	C₂	semidirect
8. #20276677	C₂	semidirect
9. #20276595	C₂	semidirect
10. #20276550	C₂	semidirect
11. C₅₃×#2221	C₂	semidirect
12. #16811440	C₂²	semidirect
16. #16811066	C₂²	semidirect
23. C₂×Dic₄₂₄	C₂²	semidirect
24. C₂×#13133161	C₂²	semidirect
27. #16811006	C₂²	semidirect
32. #16810739	C₂²	semidirect
34. #16810443	C₂²	semidirect
35. #16810438	C₂²	semidirect
36. #16810429	C₂²	semidirect
38. #16810407	C₂²	semidirect
39. C₁₆×Dic₅₃	C₂²	semidirect
41. C₅₃×#504	C₂²	semidirect
42. C₅₃×#472	C₂²	semidirect
43. C₅₃×#368	C₂²	semidirect
44. C₅₃×#361	C₂²	semidirect
207. #2221	D₅₃	semidirect
235. #504	C₂×D₅₃	semidirect
238. #472	C₂×D₅₃	semidirect
244. #368	C₂×D₅₃	semidirect
247. #361	C₂×D₅₃	semidirect
252. C₅₃	#33859	semidirect

Maximal subgroups

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

subgroup	index
#20287868	2
#20287852	2
#20286496	2
#20276973	2
#20276841	2
#20276783	2
#20276689	2
#20276677	2
#20276595	2
#20276550	2
C₅₃×#2221	2
#33859	53