Subgroups — #27540871

Order 13568.

652 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20287860	C₂	1	2	yes	maximal, complemented
2	#20286519	C₂	1	1	yes	maximal, characteristic, complemented
3	#20286515	C₂	1	1	yes	maximal, characteristic, complemented
4	#20276972	C₂	1	1	yes	maximal, characteristic, complemented
5	#20276848	C₂	1	2	yes	maximal, complemented
6	#20276792	C₂	1	1	yes	maximal, characteristic, complemented
7	#20276683	C₂	1	2	yes	maximal, complemented
8	#20276682	C₂	1	2	yes	maximal, complemented
9	#20276594	C₂	1	1	yes	maximal, characteristic, complemented
10	#20276551	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#2214	C₂	1	1	yes	maximal, characteristic, complemented
12	#16811436	—	2	1	no	—
13	#16811436	C₂²	1	2	yes	complemented
14	#16811428	—	2	1	no	—
15	#16811425	C₂²	1	2	yes	—
16	#16811065	—	2	2	no	—
17	#16811061	C₂²	1	2	yes	complemented
18	#16811060	—	2	2	no	—
19	#16811056	—	2	2	no	—
20	#16811053	C₂²	1	1	yes	characteristic
21	C₂×#13133160	C₂²	1	2	yes	complemented
22	C₂×#13133160	C₂²	1	2	yes	complemented
23	SD₃₂×D₅₃	—	2	2	no	—
24	#16811041	C₂²	1	2	yes	complemented
25	#16811000	C₂²	1	1	yes	characteristic
26	#16810895	—	2	1	no	—
27	#16810895	C₂²	1	2	yes	complemented
28	#16810894	C₂²	1	2	yes	—
29	#16810893	—	2	1	no	—
30	#16810805	—	2	2	no	—
31	#16810792	—	2	2	no	—
32	#16810741	C₂²	1	1	yes	characteristic
33	#16810741	C₂²	1	1	yes	characteristic, complemented
34	#16810452	C₂²	1	1	yes	characteristic, complemented
35	#16810437	C₂²	1	2	yes	complemented
36	#16810430	C₂²	1	1	yes	characteristic
37	#16810429	C₂²	1	1	yes	characteristic, complemented
38	#16810408	C₂²	1	2	yes	complemented
39	C₄×#11719908	C₂²	1	1	yes	characteristic, complemented
40	C₅₃×#510	C₂²	1	2	yes	—
41	C₅₃×#494	C₂²	1	1	yes	characteristic, complemented
42	C₅₃×#443	C₂²	1	1	yes	characteristic
43	C₅₃×#364	C₂²	1	1	yes	characteristic, complemented
44	C₅₃×#362	C₂²	1	2	yes	complemented
45	#13133265	—	2	2	no	—
46	#13133263	—	2	2	no	—
47	#13133249	—	4	2	no	—
48	#13133249	—	4	2	no	—
49	#13133248	—	2	2	no	—
50	#13133248	—	2	2	no	—
51	#13133245	—	2	2	no	—
52	#13133245	—	4	2	no	—
53	#13133244	—	4	2	no	—
54	#13133242	—	2	2	no	—
55	#13133240	—	2	2	no	—
56	#13133240	—	2	2	no	—
57	#13133240	—	4	1	no	—
58	#13133239	—	4	1	no	—
59	#13133239	D₄	1	1	yes	characteristic
60	#13133239	C₂³	1	1	yes	characteristic
61	Q₁₆×D₅₃	—	2	2	no	—
62	Q₁₆×D₅₃	—	4	1	no	—
63	C₂×D₄₂₄	—	2	1	no	—
64	C₂×D₄₂₄	—	4	1	no	—
65	C₂×D₄₂₄	C₂³	1	2	yes	—
66	D₈×D₅₃	—	2	2	no	—
67	#13133223	—	2	1	no	—
68	#13133223	C₂³	1	2	yes	—
69	#13133214	—	2	1	no	—
70	#13133163	D₄	1	1	yes	characteristic, complemented
71	#13133162	—	2	2	no	—
72	#13133161	D₄	1	1	yes	characteristic, complemented
73	#13133160	—	2	2	no	—
74	#13133160	—	2	2	no	—
75	#13133160	—	4	2	no	—
76	#13133160	D₄	1	1	yes	characteristic, complemented
77	D₈₄₈	—	2	2	no	—
78	#13133158	D₄	1	1	yes	characteristic, complemented
79	#13133156	—	2	2	no	—
80	C₂×#11719908	C₂³	1	1	yes	characteristic
81	C₂×#11719908	C₂³	1	1	yes	characteristic
82	C₁₆×D₅₃	—	2	2	no	—
83	#13133153	D₄	1	1	yes	characteristic
84	#13133153	C₂³	1	1	yes	characteristic
85	#13133145	—	2	1	no	—
86	#13133145	D₄	1	2	yes	—
87	#13133136	—	2	2	no	—
88	#13133133	—	2	2	no	—
89	#135×C₅₃	C₂³	1	1	yes	characteristic
90	C₂×Q₁₆×C₅₃	—	2	1	no	—
91	C₂×Q₁₆×C₅₃	C₂³	1	2	yes	—
92	#131×C₅₃	—	2	1	no	—
93	C₄⋊Q₈×C₅₃	—	2	1	no	—
94	Q₃₂×C₅₃	—	2	2	no	—
95	SD₃₂×C₅₃	—	2	2	no	—
96	M₃₂×C₅₃	C₂³	1	2	yes	—
97	#108×C₅₃	C₂³	1	1	yes	characteristic
98	#104×C₅₃	—	2	1	no	—
99	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
100	#41×C₅₃	—	2	1	no	—
101	C₂×Q₈×C₅₃	—	2	2	no	—
102	#11719949	—	4	2	no	—
103	#11719949	—	4	2	no	—
104	#11719949	—	4	2	no	—
105	Q₈×D₅₃	—	4	2	no	—
106	Q₈×D₅₃	—	4	2	no	—
107	D₄×D₅₃	—	4	2	no	—
108	D₄×D₅₃	—	8	2	no	—
109	#11719945	—	2	1	no	—
110	#11719945	—	4	2	no	—
111	#11719945	—	8	2	no	—
112	#11719945	C₂×D₄	1	2	yes	—
113	C₂×D₂₁₂	—	2	2	no	—
114	C₂×D₂₁₂	—	4	2	no	—
115	C₂×D₂₁₂	—	8	1	no	—
116	Q₁₆×C₅₃	—	2	2	no	—
117	Q₁₆×C₅₃	—	2	2	no	—
118	Q₁₆×C₅₃	—	4	1	no	—
119	Q₁₆×C₅₃	—	4	1	no	—
120	Q₁₆×C₅₃	C₂×D₄	1	1	yes	characteristic
121	SD₁₆×C₅₃	—	2	1	no	—
122	D₈×C₅₃	C₂×D₄	1	1	yes	characteristic
123	M₁₆×C₅₃	—	2	1	no	—
124	M₁₆×C₅₃	—	4	1	no	—
125	C₂×C₈×C₅₃	—	4	1	no	—
126	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
127	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
128	C₄⋊C₄×C₅₃	—	8	1	no	—
129	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
130	#11719925	—	4	2	no	—
131	#11719925	—	4	2	no	—
132	#11719924	—	4	2	no	—
133	#11719924	—	4	2	no	—
134	#11719924	—	4	2	no	—
135	#11719922	—	4	2	no	—
136	#11719917	—	2	1	no	—
137	#11719917	C₂×D₄	1	2	yes	—
138	Dic₂₁₂	D₈	1	1	yes	characteristic
139	Dic₂₁₂	C₂×D₄	1	1	yes	characteristic
140	D₄₂₄	—	2	2	no	—
141	D₄₂₄	—	2	2	no	—
142	D₄₂₄	—	4	2	no	—
143	D₄₂₄	—	4	1	no	—
144	D₄₂₄	—	4	1	no	—
145	D₄₂₄	D₈	1	1	yes	characteristic
146	D₄₂₄	C₂×D₄	1	1	yes	characteristic
147	#11719913	—	2	1	no	—
148	#11719913	—	4	2	no	—
149	#11719913	D₈	1	2	yes	—
150	#11719912	—	2	2	no	—
151	#11719912	—	4	1	no	—
152	C₈×D₅₃	—	2	2	no	—
153	C₈×D₅₃	—	4	1	no	—
154	C₁₆×C₅₃	—	2	2	no	—
155	#11719908	—	2	1	no	—
156	#11719908	C₂×D₄	1	1	yes	characteristic
157	#11719908	C₂×D₄	1	1	yes	characteristic
158	C₂²×D₅₃	—	4	2	no	—
159	C₂²×D₅₃	—	8	2	no	—
160	Q₈×C₅₃	—	2	1	no	—
161	Q₈×C₅₃	—	4	2	no	—
162	Q₈×C₅₃	—	4	2	no	—
163	D₄×C₅₃	—	2	1	no	—
164	D₄×C₅₃	—	4	1	no	—
165	C₂×C₄×C₅₃	—	2	1	no	—
166	C₂×C₄×C₅₃	—	4	1	no	—
167	C₂×C₄×C₅₃	—	4	2	no	—
168	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
169	C₂²⋊D₅₃	—	2	2	no	—
170	C₂²⋊D₅₃	—	4	1	no	—
171	C₂²⋊D₅₃	—	8	2	no	—
172	D₂₁₂	—	2	1	no	—
173	D₂₁₂	—	4	2	no	—
174	D₂₁₂	—	4	2	no	—
175	D₂₁₂	—	8	2	no	—
176	D₂₁₂	—	8	2	no	—
177	D₂₁₂	—	16	2	no	—
178	D₂₁₂	C₂×D₈	1	2	yes	—
179	C₄×D₅₃	—	2	2	no	—
180	C₄×D₅₃	—	4	1	no	—
181	C₄×D₅₃	—	8	2	no	—
182	C₄×D₅₃	—	8	2	no	—
183	C₄×D₅₃	—	8	2	no	—
184	Dic₁₀₆	—	2	1	no	—
185	Dic₁₀₆	—	8	2	no	—
186	Dic₁₀₆	C₂×D₈	1	2	yes	—
187	C₈×C₅₃	—	4	1	no	—
188	C₈×C₅₃	C₂×D₈	1	1	yes	characteristic
189	C₈×C₅₃	C₂×D₈	1	1	yes	characteristic
190	C₈×C₅₃	C₂²×D₄	1	1	yes	characteristic
191	C₈×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
192	#88322	—	2	2	no	—
193	#88322	—	4	1	no	—
194	#33856	—	53	1	no	maximal, Sylow
195	C₂²×C₅₃	—	4	1	no	—
196	C₂²×C₅₃	#576	1	1	yes	characteristic
197	C₂×D₅₃	—	2	2	no	—
198	C₂×D₅₃	—	4	1	no	—
199	C₂×D₅₃	—	8	2	no	—
200	C₂×D₅₃	—	16	2	no	—
201	C₂×D₅₃	—	32	2	no	—
202	C₄×C₅₃	—	2	2	no	—
203	C₄×C₅₃	—	4	1	no	—
204	C₄×C₅₃	—	8	2	no	—
205	C₄×C₅₃	D₄²	1	1	yes	characteristic
206	C₄×C₅₃	C₂²×D₈	1	1	yes	characteristic
207	Dic₅₃	—	2	2	no	—
208	Dic₅₃	—	4	1	no	—
209	#3416	—	53	2	no	—
210	#3293	—	53	1	no	—
211	#3292	—	53	1	no	—
212	#2248	—	53	1	no	—
213	#2223	—	53	2	no	—
214	#2214	D₅₃	1	1	yes	characteristic, complemented
215	#2213	—	53	1	no	—
216	#2194	—	53	2	no	—
217	#2193	—	53	2	no	—
218	#2178	—	53	1	no	—
219	#2170	—	53	1	no	—
220	C₂×C₅₃	—	2	1	no	—
221	C₂×C₅₃	—	8	1	no	—
222	C₂×C₅₃	D₄×D₈	1	1	yes	characteristic
223	D₅₃	—	4	2	no	—
224	D₅₃	—	8	1	no	—
225	D₅₃	—	16	2	no	—
226	#578	—	53	2	no	—
227	#578	—	106	1	no	—
228	#576	—	53	2	no	—
229	#576	—	106	1	no	—
230	#510	—	106	2	no	—
231	#510	C₂×D₅₃	1	2	yes	—
232	#509	—	53	2	no	—
233	#509	—	106	2	no	—
234	#508	—	53	1	no	—
235	#508	—	106	2	no	—
236	C₂×SD₃₂	—	53	2	no	—
237	C₂×SD₃₂	—	53	2	no	—
238	C₂×SD₃₂	—	106	2	no	—
239	#504	—	53	2	no	—
240	#494	C₂×D₅₃	1	1	yes	characteristic, complemented
241	#493	—	53	1	no	—
242	#473	—	53	2	no	—
243	#473	—	106	1	no	—
244	#472	—	53	2	no	—
245	#472	—	106	1	no	—
246	#456	—	106	2	no	—
247	#453	—	106	2	no	—
248	#443	—	53	1	no	—
249	#443	—	53	1	no	—
250	#443	C₂×D₅₃	1	1	yes	characteristic
251	#368	—	53	1	no	—
252	#364	C₂×D₅₃	1	1	yes	characteristic, complemented
253	#362	C₂×D₅₃	1	2	yes	complemented
254	#361	—	53	2	no	—
255	#359	—	53	1	no	—
256	#359	—	53	1	no	—
257	#350	—	53	2	no	—
258	C₄×C₁₆	—	53	1	no	—
259	C₅₃	#33856	1	1	yes	characteristic, complemented, Sylow
260	2₋¹⁺⁴	—	106	2	no	—
261	H(5,𝔽₂)	—	106	2	no	—
262	#137	—	106	2	no	—
263	#137	—	106	2	no	—
264	#137	—	212	2	no	—
265	#137	—	212	2	no	—
266	AGL(1,ℤ/8)	—	106	2	no	—
267	AGL(1,ℤ/8)	—	106	2	no	—
268	AGL(1,ℤ/8)	—	212	2	no	—
269	AGL(1,ℤ/8)	—	212	2	no	—
270	#135	—	53	1	no	—
271	#135	—	53	1	no	—
272	#135	—	106	2	no	—
273	#135	—	106	2	no	—
274	#135	—	212	1	no	—
275	#135	—	212	1	no	—
276	#135	C₂²×D₅₃	1	1	yes	characteristic
277	C₂×Q₁₆	—	2	1	no	—
278	C₂×Q₁₆	—	106	2	no	—
279	C₂×Q₁₆	—	212	1	no	—
280	C₂×Q₁₆	C₂²×D₅₃	1	2	yes	—
281	C₂×D₈	—	53	2	no	—
282	C₂×D₈	—	106	2	no	—
283	C₂×D₈	—	106	1	no	—
284	C₂×D₈	—	212	1	no	—
285	#131	—	2	1	no	—
286	#131	—	53	2	no	—
287	#131	—	106	1	no	—
288	C₄⋊Q₈	—	2	1	no	—
289	#127	—	106	1	no	—
290	Q₃₂	—	2	2	no	—
291	Q₃₂	—	53	1	no	—
292	SD₃₂	—	2	2	no	—
293	SD₃₂	—	53	1	no	—
294	SD₃₂	—	53	1	no	—
295	SD₃₂	—	106	2	no	—
296	SD₃₂	—	106	2	no	—
297	SD₃₂	—	106	2	no	—
298	SD₃₂	—	212	2	no	—
299	D₁₆	—	53	1	no	—
300	D₁₆	—	106	2	no	—
301	M₃₂	—	106	2	no	—
302	M₃₂	C₂²×D₅₃	1	2	yes	—
303	C₂×C₁₆	—	53	1	no	—
304	C₂×C₁₆	—	53	1	no	—
305	C₂×C₁₆	—	106	2	no	—
306	#108	—	53	1	no	—
307	#108	—	53	1	no	—
308	#108	C₂²×D₅₃	1	1	yes	characteristic
309	#104	—	2	1	no	—
310	#104	—	53	2	no	—
311	#104	—	106	1	no	—
312	#101	—	106	2	no	—
313	#100	—	106	2	no	—
314	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
315	#41	—	2	1	no	—
316	#41	—	53	2	no	—
317	#41	—	106	1	no	—
318	#41	—	424	2	no	—
319	#41	—	212	2	no	—
320	#41	—	212	2	no	—
321	#41	—	212	2	no	—
322	#41	—	212	2	no	—
323	C₂×Q₈	—	2	2	no	—
324	C₂×Q₈	—	212	2	no	—
325	C₂×Q₈	—	212	2	no	—
326	C₂×D₄	—	106	2	no	—
327	C₂×D₄	—	424	1	no	—
328	C₂×D₄	—	424	2	no	—
329	C₂×D₄	—	212	2	no	—
330	C₂×D₄	—	212	2	no	—
331	Q₁₆	—	2	2	no	—
332	Q₁₆	—	2	2	no	—
333	Q₁₆	—	4	1	no	—
334	Q₁₆	—	4	1	no	—
335	Q₁₆	—	53	1	no	—
336	Q₁₆	—	53	1	no	—
337	Q₁₆	—	212	2	no	—
338	Q₁₆	—	212	2	no	—
339	Q₁₆	D₄×D₅₃	1	1	yes	characteristic
340	SD₁₆	—	2	1	no	—
341	SD₁₆	—	53	2	no	—
342	SD₁₆	—	106	1	no	—
343	SD₁₆	—	212	2	no	—
344	SD₁₆	—	212	2	no	—
345	SD₁₆	—	212	2	no	—
346	SD₁₆	—	212	2	no	—
347	D₈	—	53	1	no	—
348	D₈	—	53	1	no	—
349	D₈	—	106	2	no	—
350	D₈	—	106	2	no	—
351	D₈	—	212	2	no	—
352	D₈	—	212	2	no	—
353	D₈	—	212	1	no	—
354	D₈	—	212	1	no	—
355	D₈	D₄×D₅₃	1	1	yes	characteristic
356	M₁₆	—	2	1	no	—
357	M₁₆	—	4	1	no	—
358	M₁₆	—	53	2	no	—
359	M₁₆	—	106	2	no	—
360	M₁₆	—	106	1	no	—
361	M₁₆	—	212	1	no	—
362	C₂×C₈	—	4	1	no	—
363	C₂×C₈	—	106	2	no	—
364	C₂×C₈	—	212	1	no	—
365	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
366	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
367	C₄⋊C₄	—	8	1	no	—
368	C₄²	D₄×D₅₃	1	1	yes	characteristic
369	C₁₆	—	2	2	no	—
370	C₁₆	—	53	1	no	—
371	C₁₆	—	53	1	no	—
372	C₁₆	—	106	1	no	—
373	C₂³	—	424	2	no	—
374	C₂³	—	212	2	no	—
375	Q₈	—	2	1	no	—
376	Q₈	—	4	2	no	—
377	Q₈	—	4	2	no	—
378	Q₈	—	53	2	no	—
379	Q₈	—	106	1	no	—
380	Q₈	—	424	2	no	—
381	D₄	—	2	1	no	—
382	D₄	—	4	1	no	—
383	D₄	—	53	2	no	—
384	D₄	—	848	2	no	—
385	D₄	—	106	2	no	—
386	D₄	—	106	1	no	—
387	D₄	—	424	2	no	—
388	D₄	—	424	2	no	—
389	D₄	—	424	2	no	—
390	D₄	—	212	1	no	—
391	D₄	—	212	2	no	—
392	D₄	—	212	2	no	—
393	C₂×C₄	—	2	1	no	—
394	C₂×C₄	—	4	1	no	—
395	C₂×C₄	—	4	2	no	—
396	C₂×C₄	—	106	2	no	—
397	C₂×C₄	—	424	2	no	—
398	C₂×C₄	—	424	2	no	—
399	C₂×C₄	—	424	2	no	—
400	C₂×C₄	—	212	1	no	—
401	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
402	C₈	—	4	1	no	—
403	C₈	—	106	2	no	—
404	C₈	—	212	1	no	—
405	C₈	D₈×D₅₃	1	1	yes	characteristic
406	C₈	D₈×D₅₃	1	1	yes	characteristic
407	C₈	C₂×D₄×D₅₃	1	1	yes	characteristic
408	C₈	#13133263	1	1	yes	characteristic
409	C₂²	—	4	1	no	—
410	C₂²	—	848	2	no	—
411	C₂²	—	106	2	no	—
412	C₂²	—	1696	2	no	—
413	C₂²	—	424	2	no	—
414	C₂²	—	212	1	no	—
415	C₂²	#16811427	1	1	yes	characteristic
416	C₄	—	2	2	no	—
417	C₄	—	4	1	no	—
418	C₄	—	8	2	no	—
419	C₄	—	106	2	no	—
420	C₄	—	212	1	no	—
421	C₄	#16811257	1	1	yes	characteristic
422	C₄	C₂×D₈×D₅₃	1	1	yes	characteristic
423	C₂	—	2	1	no	—
424	C₂	—	8	1	no	—
425	C₂	—	848	2	no	—
426	C₂	—	424	1	no	—
427	C₂	—	212	2	no	—
428	C₂	#20286387	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20286387
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2214	C₂
socle	C₂×C₅₃	D₄×D₈
hypercenter	#2214	D₅₃
hypocenter	C₅₃	#33856
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #2214

Complements

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

N	H ≅ G/N	type
1. #20287860	C₂	semidirect
2. #20286519	C₂	semidirect
3. #20286515	C₂	semidirect
4. #20276972	C₂	semidirect
5. #20276848	C₂	semidirect
6. #20276792	C₂	semidirect
7. #20276683	C₂	semidirect
8. #20276682	C₂	semidirect
9. #20276594	C₂	semidirect
10. #20276551	C₂	semidirect
11. C₅₃×#2214	C₂	semidirect
13. #16811436	C₂²	semidirect
17. #16811061	C₂²	semidirect
21. C₂×#13133160	C₂²	semidirect
22. C₂×#13133160	C₂²	semidirect
24. #16811041	C₂²	semidirect
27. #16810895	C₂²	semidirect
33. #16810741	C₂²	semidirect
34. #16810452	C₂²	semidirect
35. #16810437	C₂²	semidirect
37. #16810429	C₂²	semidirect
38. #16810408	C₂²	semidirect
39. C₄×#11719908	C₂²	semidirect
41. C₅₃×#494	C₂²	semidirect
43. C₅₃×#364	C₂²	semidirect
44. C₅₃×#362	C₂²	semidirect
70. #13133163	D₄	semidirect
72. #13133161	D₄	semidirect
76. #13133160	D₄	semidirect
78. #13133158	D₄	semidirect
214. #2214	D₅₃	semidirect
240. #494	C₂×D₅₃	semidirect
252. #364	C₂×D₅₃	semidirect
253. #362	C₂×D₅₃	semidirect
259. C₅₃	#33856	semidirect

Maximal subgroups

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

subgroup	index
#20287860	2
#20286519	2
#20286515	2
#20276972	2
#20276848	2
#20276792	2
#20276683	2
#20276682	2
#20276594	2
#20276551	2
C₅₃×#2214	2
#33856	53