Subgroups — #27540967

Order 13568.

580 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20287063	C₂	1	1	yes	maximal, characteristic
2	#20276942	C₂	1	1	yes	maximal, characteristic, complemented
3	#20276910	C₂	1	2	yes	maximal, complemented
4	#20276857	C₂	1	1	yes	maximal, characteristic, complemented
5	#20276813	C₂	1	2	yes	maximal, complemented
6	#20276757	C₂	1	1	yes	maximal, characteristic, complemented
7	#20276632	C₂	1	2	yes	maximal, complemented
8	C₄×#13133160	C₂	1	1	yes	maximal, characteristic, complemented
9	D₄×#11719908	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#16810427	C₂	1	2	yes	maximal, complemented
11	D₄×Q₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	#16811277	—	2	1	no	—
13	C₂×#13133160	C₂²	1	1	yes	characteristic
14	C₂×#13133160	C₂²	1	2	yes	—
15	C₂²×#11719908	C₂²	1	2	yes	—
16	#16811028	C₂²	1	1	yes	characteristic
17	#16810937	—	2	2	no	—
18	#16810904	—	2	1	no	—
19	#16810859	C₂²	1	2	yes	—
20	C₄×D₄₂₄	C₂²	1	1	yes	characteristic
21	C₂×#13133151	C₂²	1	2	yes	—
22	#16810450	C₂²	1	2	yes	—
23	#16810450	C₂²	1	1	yes	characteristic
24	C₄⋊#11719908	C₂²	1	1	yes	characteristic
25	#16810427	C₂²	1	2	yes	—
26	#16810427	C₂²	1	2	yes	—
27	#16810427	C₂²	1	1	yes	characteristic
28	#16810427	C₂²	1	2	yes	—
29	#16810427	C₂²	1	2	yes	—
30	#16810423	C₂²	1	1	yes	characteristic
31	C₂²⋊#11719908	C₂²	1	2	yes	—
32	C₄×#11719908	C₂²	1	1	yes	characteristic
33	C₂²×Q₁₆×C₅₃	C₂²	1	2	yes	—
34	D₄×Q₈×C₅₃	—	2	1	no	—
35	C₅₃×#494	C₂²	1	1	yes	characteristic
36	C₅₃×#467	C₂²	1	2	yes	—
37	C₅₃×#462	—	2	1	no	—
38	C₅₃×#451	—	2	2	no	—
39	C₄×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
40	C₈×D₄×C₅₃	C₂²	1	1	yes	characteristic
41	C₂×#11719947	—	4	1	no	—
42	C₂×D₄₂₄	C₂³	1	1	yes	characteristic
43	#13133217	—	2	1	no	—
44	#13133201	—	4	2	no	—
45	#13133197	—	4	1	no	—
46	#13133191	—	2	2	no	—
47	C₄×D₂₁₂	—	2	1	no	—
48	C₂×C₄⋊Dic₅₃	—	2	2	no	—
49	#13133160	—	2	2	no	—
50	#13133160	—	4	2	no	—
51	C₂×#11719908	—	4	2	no	—
52	C₂×#11719908	C₂³	1	1	yes	characteristic
53	C₂×#11719908	C₂³	1	2	yes	—
54	C₂×#11719908	C₂³	1	1	yes	characteristic
55	#13133151	—	2	1	no	—
56	#13133151	—	2	2	no	—
57	#13133151	C₂³	1	2	yes	—
58	#13133151	C₂³	1	1	yes	characteristic
59	#13133139	—	2	2	no	—
60	#13133139	—	2	1	no	—
61	C₂²×Q₈×C₅₃	—	2	2	no	—
62	C₂×Q₁₆×C₅₃	—	2	2	no	—
63	C₂×Q₁₆×C₅₃	—	2	1	no	—
64	C₂×Q₁₆×C₅₃	—	2	4	no	—
65	C₂×Q₁₆×C₅₃	—	4	2	no	—
66	C₂×Q₁₆×C₅₃	C₂³	1	2	yes	—
67	C₂×Q₁₆×C₅₃	C₂³	1	1	yes	characteristic
68	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
69	C₄⋊Q₈×C₅₃	—	2	1	no	—
70	C₄⋊Q₈×C₅₃	—	4	1	no	—
71	C₂²⋊Q₈×C₅₃	—	2	2	no	—
72	C₂²⋊Q₈×C₅₃	—	4	2	no	—
73	C₄×Q₈×C₅₃	—	2	1	no	—
74	C₄×D₄×C₅₃	—	4	1	no	—
75	C₄×D₄×C₅₃	D₄	1	1	yes	characteristic
76	#107×C₅₃	C₂³	1	1	yes	characteristic
77	C₄⋊C₈×C₅₃	D₄	1	1	yes	characteristic
78	#103×C₅₃	—	2	1	no	—
79	#103×C₅₃	—	2	2	no	—
80	C₂²⋊C₈×C₅₃	D₄	1	2	yes	—
81	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
82	C₂×Q₈×C₅₃	—	2	1	no	—
83	C₂×Q₈×C₅₃	—	2	2	no	—
84	C₂×Q₈×C₅₃	—	4	4	no	—
85	C₂×Q₈×C₅₃	—	4	2	no	—
86	C₂×Q₈×C₅₃	—	8	2	no	—
87	C₂×D₄×C₅₃	D₈	1	1	yes	characteristic
88	C₂²×C₄×C₅₃	—	4	2	no	—
89	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
90	C₂×C₂²⋊D₅₃	—	4	2	no	—
91	C₂²×Dic₅₃	—	4	2	no	—
92	#11719947	—	8	4	no	—
93	C₂×D₂₁₂	—	2	1	no	—
94	C₂×C₄×D₅₃	—	4	1	no	—
95	C₂×Dic₁₀₆	—	4	1	no	—
96	Q₁₆×C₅₃	—	2	2	no	—
97	Q₁₆×C₅₃	—	2	1	no	—
98	Q₁₆×C₅₃	—	2	2	no	—
99	Q₁₆×C₅₃	—	4	4	no	—
100	Q₁₆×C₅₃	—	8	1	no	—
101	Q₁₆×C₅₃	#41	1	2	yes	complemented
102	C₂×C₈×C₅₃	—	2	2	no	—
103	C₂×C₈×C₅₃	—	2	2	no	—
104	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
106	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
107	C₄⋊C₄×C₅₃	—	2	1	no	—
108	C₄⋊C₄×C₅₃	—	4	1	no	—
109	C₄⋊C₄×C₅₃	—	4	1	no	—
110	C₄⋊C₄×C₅₃	—	4	2	no	—
111	C₄⋊C₄×C₅₃	—	4	1	no	—
112	C₄⋊C₄×C₅₃	D₈	1	1	yes	characteristic
113	C₂²⋊C₄×C₅₃	—	4	2	no	—
114	C₂²⋊C₄×C₅₃	D₈	1	2	yes	—
115	C₄²×C₅₃	—	4	1	no	—
116	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
117	#11719921	—	4	1	no	—
118	#11719921	—	4	2	no	—
119	C₄⋊Dic₅₃	—	2	1	no	—
120	C₄⋊Dic₅₃	—	2	2	no	—
121	C₄⋊Dic₅₃	—	4	2	no	—
122	C₄⋊Dic₅₃	—	4	1	no	—
123	D₄₂₄	—	2	1	no	—
124	D₄₂₄	—	2	1	no	—
125	#11719908	—	2	1	no	—
126	#11719908	—	2	1	no	—
127	#11719908	—	2	2	no	—
128	C₂³×C₅₃	C₂×D₈	1	2	yes	—
129	C₂²×D₅₃	—	4	1	no	—
130	Q₈×C₅₃	—	2	2	no	—
131	Q₈×C₅₃	—	4	1	no	—
132	Q₈×C₅₃	—	4	2	no	—
133	Q₈×C₅₃	—	4	2	no	—
134	Q₈×C₅₃	—	8	4	no	—
135	D₄×C₅₃	SD₃₂	1	4	yes	complemented
136	C₂×C₄×C₅₃	—	2	2	no	—
137	C₂×C₄×C₅₃	—	2	2	no	—
138	C₂×C₄×C₅₃	—	4	1	no	—
139	C₂×C₄×C₅₃	—	4	2	no	—
140	C₂×C₄×C₅₃	—	4	1	no	—
141	C₂×C₄×C₅₃	—	8	4	no	—
142	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
143	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
144	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
145	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
146	C₂²⋊D₅₃	—	8	4	no	—
147	C₂×Dic₅₃	—	4	2	no	—
148	C₂×Dic₅₃	—	4	1	no	—
149	C₂×Dic₅₃	—	8	4	no	—
150	D₂₁₂	—	4	1	no	—
151	D₂₁₂	—	4	1	no	—
152	C₄×D₅₃	—	8	2	no	—
153	Dic₁₀₆	—	8	2	no	—
154	C₈×C₅₃	—	2	2	no	—
155	C₈×C₅₃	—	2	1	no	—
156	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
157	C₈×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
158	#33865	—	53	1	no	maximal, Sylow
159	C₂²×C₅₃	—	2	2	no	—
160	C₂²×C₅₃	—	2	2	no	—
161	C₂²×C₅₃	C₂×SD₃₂	1	4	yes	—
162	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
163	C₂×D₅₃	—	8	1	no	—
164	C₂×D₅₃	—	8	1	no	—
165	C₂×D₅₃	—	8	1	no	—
166	C₄×C₅₃	—	2	1	no	—
167	C₄×C₅₃	—	2	2	no	—
168	C₄×C₅₃	—	4	2	no	—
169	C₄×C₅₃	—	8	1	no	—
170	C₄×C₅₃	—	8	2	no	—
171	C₄×C₅₃	C₂×SD₃₂	1	2	yes	—
172	C₄×C₅₃	#546	1	1	yes	characteristic
173	C₄×C₅₃	#577	1	1	yes	characteristic
174	Dic₅₃	—	8	2	no	—
175	Dic₅₃	—	8	1	no	—
176	#3334	—	53	1	no	—
177	D₄×Q₁₆	D₅₃	1	1	yes	characteristic, complemented
178	#2241	—	53	1	no	—
179	#2235	—	53	2	no	—
180	#2225	—	53	1	no	—
181	#2217	—	53	2	no	—
182	#2208	—	53	1	no	—
183	#2185	—	53	2	no	—
184	C₄×SD₃₂	—	53	1	no	—
185	D₄×C₁₆	—	53	1	no	—
186	C₂×#358	—	53	2	no	—
187	C₂×C₅₃	—	2	4	no	—
188	C₂×C₅₃	#3294	1	1	yes	characteristic
189	C₂×C₅₃	C₂²×SD₃₂	1	1	yes	characteristic
190	C₂×C₅₃	#3417	1	1	yes	characteristic
191	D₅₃	—	16	1	no	—
192	C₂²×Q₁₆	C₂×D₅₃	1	2	yes	—
193	D₄×Q₈	—	2	1	no	—
194	#547	—	106	1	no	—
195	C₂×SD₃₂	—	53	1	no	—
196	C₂×SD₃₂	—	53	2	no	—
197	C₂²×C₁₆	—	53	2	no	—
198	#500	—	53	1	no	—
199	#494	C₂×D₅₃	1	1	yes	characteristic
200	#480	—	106	2	no	—
201	#474	—	106	1	no	—
202	#467	C₂×D₅₃	1	2	yes	—
203	#466	—	53	2	no	—
204	#462	—	2	1	no	—
205	#451	—	2	2	no	—
206	C₄×Q₁₆	C₂×D₅₃	1	1	yes	characteristic
207	C₄×D₈	—	53	1	no	—
208	C₈×D₄	C₂×D₅₃	1	1	yes	characteristic
209	C₂×#107	—	53	2	no	—
210	#367	—	53	1	no	—
211	#367	—	53	2	no	—
212	C₄⋊C₁₆	—	53	1	no	—
213	#358	—	53	1	no	—
214	#358	—	53	2	no	—
215	#358	—	53	2	no	—
216	#358	—	53	2	no	—
217	#358	—	53	2	no	—
218	#357	—	53	1	no	—
219	C₂²⋊C₁₆	—	53	2	no	—
220	C₄×C₁₆	—	53	1	no	—
221	C₅₃	#33865	1	1	yes	characteristic, complemented, Sylow
222	C₂×#41	—	212	1	no	—
223	C₂²×Q₈	—	2	2	no	—
224	C₂×Q₁₆	—	2	4	no	—
225	C₂×Q₁₆	—	2	2	no	—
226	C₂×Q₁₆	—	2	1	no	—
227	C₂×Q₁₆	—	4	2	no	—
228	C₂×Q₁₆	C₂²×D₅₃	1	2	yes	—
229	C₂×Q₁₆	C₂²×D₅₃	1	1	yes	characteristic
230	C₂×D₈	—	53	1	no	—
231	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
232	C₄⋊Q₈	—	2	1	no	—
233	C₄⋊Q₈	—	4	1	no	—
234	C₄⋊Q₈	—	106	1	no	—
235	#123	—	212	2	no	—
236	C₂²⋊Q₈	—	2	2	no	—
237	C₂²⋊Q₈	—	4	2	no	—
238	C₂²⋊Q₈	—	212	1	no	—
239	#121	—	106	2	no	—
240	C₄×Q₈	—	2	1	no	—
241	C₄×D₄	—	4	1	no	—
242	C₄×D₄	—	106	1	no	—
243	C₄×D₄	C₂²⋊D₅₃	1	1	yes	characteristic
244	C₂×C₄⋊C₄	—	106	2	no	—
245	SD₃₂	—	106	2	no	—
246	SD₃₂	—	212	2	no	—
247	C₂×C₁₆	—	53	1	no	—
248	C₂×C₁₆	—	53	1	no	—
249	C₂×C₁₆	—	53	2	no	—
250	C₂×C₁₆	—	212	2	no	—
251	#107	—	53	1	no	—
252	#107	—	53	2	no	—
253	#107	—	106	2	no	—
254	#107	—	106	1	no	—
255	#107	C₂²×D₅₃	1	1	yes	characteristic
256	C₄⋊C₈	C₂²⋊D₅₃	1	1	yes	characteristic
257	#103	—	2	1	no	—
258	#103	—	2	2	no	—
259	#102	—	106	1	no	—
260	#102	—	106	2	no	—
261	C₂²⋊C₈	C₂²⋊D₅₃	1	2	yes	—
262	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
263	#41	—	424	4	no	—
264	C₂×Q₈	—	2	1	no	—
265	C₂×Q₈	—	2	2	no	—
266	C₂×Q₈	—	4	4	no	—
267	C₂×Q₈	—	4	2	no	—
268	C₂×Q₈	—	8	2	no	—
269	C₂×Q₈	—	212	1	no	—
270	C₂×D₄	—	106	1	no	—
271	C₂×D₄	—	212	2	no	—
272	C₂×D₄	#11719922	1	1	yes	characteristic
273	C₂²×C₄	—	4	2	no	—
274	C₂²×C₄	—	212	2	no	—
275	C₂²×C₄	—	212	1	no	—
276	C₂²×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
277	Q₁₆	—	2	2	no	—
278	Q₁₆	—	2	1	no	—
279	Q₁₆	—	2	2	no	—
280	Q₁₆	—	4	4	no	—
281	Q₁₆	—	8	1	no	—
282	Q₁₆	#11719947	1	2	yes	complemented
283	D₈	—	106	1	no	—
284	D₈	—	106	1	no	—
285	C₂×C₈	—	2	2	no	—
286	C₂×C₈	—	2	2	no	—
287	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
288	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
289	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
290	C₄⋊C₄	—	2	1	no	—
291	C₄⋊C₄	—	4	1	no	—
292	C₄⋊C₄	—	4	1	no	—
293	C₄⋊C₄	—	4	2	no	—
294	C₄⋊C₄	—	4	1	no	—
295	C₄⋊C₄	—	106	1	no	—
296	C₄⋊C₄	—	106	2	no	—
297	C₄⋊C₄	—	212	2	no	—
298	C₄⋊C₄	—	212	1	no	—
299	C₄⋊C₄	#11719922	1	1	yes	characteristic
300	C₂²⋊C₄	—	4	2	no	—
301	C₂²⋊C₄	—	212	1	no	—
302	C₂²⋊C₄	—	212	2	no	—
303	C₂²⋊C₄	#11719922	1	2	yes	—
304	C₄²	—	4	1	no	—
305	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
306	C₁₆	—	106	1	no	—
307	C₁₆	—	106	1	no	—
308	C₁₆	—	106	2	no	—
309	C₂³	—	212	1	no	—
310	C₂³	C₂×#11719922	1	2	yes	—
311	Q₈	—	2	2	no	—
312	Q₈	—	4	1	no	—
313	Q₈	—	4	2	no	—
314	Q₈	—	4	2	no	—
315	Q₈	—	8	4	no	—
316	Q₈	—	424	2	no	—
317	D₄	—	424	4	no	—
318	D₄	—	212	1	no	—
319	D₄	—	212	1	no	—
320	D₄	#13133160	1	4	yes	complemented
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	4	1	no	—
324	C₂×C₄	—	4	1	no	—
325	C₂×C₄	—	4	2	no	—
326	C₂×C₄	—	8	4	no	—
327	C₂×C₄	—	424	4	no	—
328	C₂×C₄	—	424	2	no	—
329	C₂×C₄	—	212	1	no	—
330	C₂×C₄	—	212	2	no	—
331	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
332	C₂×C₄	C₂×#11719922	1	2	yes	—
333	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
334	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
335	C₈	—	2	2	no	—
336	C₈	—	2	1	no	—
337	C₈	C₂×#11719947	1	1	yes	characteristic
338	C₈	#13133262	1	1	yes	characteristic
339	C₂²	—	2	2	no	—
340	C₂²	—	2	2	no	—
341	C₂²	—	424	1	no	—
342	C₂²	—	424	1	no	—
343	C₂²	—	424	1	no	—
344	C₂²	C₂×#13133160	1	4	yes	—
345	C₂²	C₂²×#11719922	1	1	yes	characteristic
346	C₄	—	2	2	no	—
347	C₄	—	2	1	no	—
348	C₄	—	4	2	no	—
349	C₄	—	8	2	no	—
350	C₄	—	8	1	no	—
351	C₄	—	424	1	no	—
352	C₄	—	424	2	no	—
353	C₄	C₂×#13133160	1	2	yes	—
354	C₄	#16811267	1	1	yes	characteristic
355	C₄	#16811434	1	1	yes	characteristic
356	C₂	—	2	4	no	—
357	C₂	—	848	1	no	—
358	C₂	#20286528	1	1	yes	characteristic
359	C₂	C₂²×#13133160	1	1	yes	characteristic
360	C₂	#20287866	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#11719922
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	D₄×Q₁₆×C₅₃	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	#20287063	C₂
hypercenter	D₄×Q₁₆	D₅₃
hypocenter	C₅₃	#33865
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ D₄×Q₁₆

Complements

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

N	H ≅ G/N	type
2. #20276942	C₂	semidirect
3. #20276910	C₂	semidirect
4. #20276857	C₂	semidirect
5. #20276813	C₂	semidirect
6. #20276757	C₂	semidirect
7. #20276632	C₂	semidirect
8. C₄×#13133160	C₂	semidirect
9. D₄×#11719908	C₂	semidirect
10. C₂×#16810427	C₂	semidirect
11. D₄×Q₁₆×C₅₃	C₂	semidirect
101. Q₁₆×C₅₃	#41	semidirect
135. D₄×C₅₃	SD₃₂	semidirect
177. D₄×Q₁₆	D₅₃	semidirect
221. C₅₃	#33865	semidirect
282. Q₁₆	#11719947	semidirect
320. D₄	#13133160	semidirect

Maximal subgroups

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

subgroup	index
#20287063	2
#20276942	2
#20276910	2
#20276857	2
#20276813	2
#20276757	2
#20276632	2
C₄×#13133160	2
D₄×#11719908	2
C₂×#16810427	2
D₄×Q₁₆×C₅₃	2
#33865	53