Subgroups — #27537776

Order 13568.

536 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	C₈×#11719949	C₂	1	1	yes	maximal, characteristic
2	#20276624	C₂	1	1	yes	maximal, characteristic, complemented
3	#20276542	C₂	1	2	yes	maximal, complemented
4	#20276539	C₂	1	2	yes	maximal, complemented
5	C₁₆×D₂₁₂	C₂	1	1	yes	maximal, characteristic, complemented
6	#20276534	C₂	1	1	yes	maximal, characteristic, complemented
7	#20276516	C₂	1	2	yes	maximal, complemented
8	#20276512	C₂	1	1	yes	maximal, characteristic, complemented
9	#20276469	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄⋊C₁₆×D₅₃	C₂	1	2	yes	maximal, complemented
11	C₅₃×#2183	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×#11719949	C₄	1	1	yes	characteristic
13	C₂×#13133156	C₂²	1	2	yes	—
14	C₂×C₁₆×D₅₃	C₂²	1	2	yes	—
15	C₂×C₁₆×D₅₃	C₂²	1	2	yes	—
16	Q₈×#88322	C₄	1	1	yes	characteristic
17	C₈×D₂₁₂	C₂²	1	1	yes	characteristic
18	C₈×D₂₁₂	C₂²	1	2	yes	—
19	#16810688	C₄	1	1	yes	characteristic
20	#16810688	C₄	1	2	yes	—
21	#16810675	C₄	1	2	yes	—
22	#16810675	C₄	1	1	yes	characteristic
23	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
24	C₄×C₈×D₅₃	C₂²	1	2	yes	—
25	C₄⋊#11719908	C₂²	1	1	yes	characteristic
26	C₄⋊#11719908	C₂²	1	2	yes	—
27	#16810441	C₂²	1	1	yes	characteristic
28	#16810441	C₂²	1	1	yes	characteristic
29	#16810441	C₂²	1	2	yes	—
30	#16810402	C₂²	1	1	yes	characteristic
31	#16810402	C₂²	1	1	yes	characteristic
32	#16810402	C₂²	1	2	yes	—
33	#16810402	C₂²	1	2	yes	—
34	#16810399	C₂²	1	2	yes	—
35	C₄×#11719908	C₂²	1	1	yes	characteristic
36	C₈×Q₈×C₅₃	C₂²	1	1	yes	characteristic
37	C₅₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
38	C₅₃×C₄⋊C₁₆	C₂²	1	2	yes	—
39	C₅₃×#346	C₂²	1	2	yes	—
40	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
41	C₂×#11719949	C₈	1	1	yes	characteristic
42	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
44	C₂×C₈×D₅₃	C₂³	1	2	yes	—
45	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
46	Q₈×Dic₅₃	C₈	1	1	yes	characteristic
47	C₄×D₂₁₂	C₂×C₄	1	2	yes	—
48	C₄×D₂₁₂	C₂×C₄	1	1	yes	characteristic
49	#13133179	C₈	1	2	yes	—
50	#13133179	C₈	1	1	yes	characteristic
51	#13133175	C₈	1	2	yes	—
52	#13133175	C₈	1	1	yes	characteristic
53	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
54	C₄²×D₅₃	C₂×C₄	1	2	yes	—
55	#13133156	—	4	2	no	—
56	C₂×#11719908	C₂³	1	1	yes	characteristic
57	C₂×#11719908	C₂³	1	2	yes	—
58	C₂×#11719908	C₂³	1	1	yes	characteristic
59	C₁₆×D₅₃	—	4	2	no	—
60	C₁₆×D₅₃	—	4	2	no	—
61	C₄⋊#88322	C₂×C₄	1	1	yes	characteristic
62	C₄⋊#88322	C₂×C₄	1	2	yes	—
63	#13133129	C₂×C₄	1	2	yes	—
64	#13133129	C₂×C₄	1	2	yes	—
65	#13133129	C₂×C₄	1	2	yes	—
66	C₄×#88322	C₂×C₄	1	1	yes	characteristic
67	C₄×#88322	C₂×C₄	1	2	yes	—
68	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
69	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
70	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
71	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
72	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
73	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
74	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
75	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
76	C₄⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
77	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
78	C₄×C₈×C₅₃	C₂³	1	2	yes	—
79	C₂×Q₈×C₅₃	C₂×C₈	1	1	yes	characteristic
80	#11719949	—	2	4	no	—
81	C₂×D₂₁₂	C₂×C₈	1	2	yes	—
82	C₂×D₂₁₂	C₂×C₈	1	1	yes	characteristic
83	C₂×C₄×D₅₃	C₂×C₈	1	2	yes	—
84	C₂×C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
85	C₂×C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
86	C₂×C₄×D₅₃	C₂×C₈	1	2	yes	—
87	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₄×D₅₃	C₂²×C₄	1	2	yes	—
89	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
91	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
92	C₄⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
93	C₄⋊C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
94	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
95	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
96	#11719921	C₂×C₈	1	2	yes	—
97	#11719921	C₂×C₈	1	2	yes	—
98	#11719921	C₂×C₈	1	2	yes	—
99	C₄⋊Dic₅₃	C₂×C₈	1	2	yes	—
100	C₄⋊Dic₅₃	C₂×C₈	1	1	yes	characteristic
101	C₄×Dic₅₃	C₂×C₈	1	1	yes	characteristic
102	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
103	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
104	C₄×Dic₅₃	C₂×C₈	1	1	yes	characteristic
105	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
106	C₂×#88322	C₂²×C₄	1	2	yes	—
107	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
108	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
109	C₈×D₅₃	—	2	2	no	—
110	C₈×D₅₃	—	2	1	no	—
111	C₈×D₅₃	—	2	2	no	—
112	C₈×D₅₃	—	2	1	no	—
113	C₈×D₅₃	—	4	2	no	—
114	C₈×D₅₃	—	4	1	no	—
115	C₁₆×C₅₃	—	2	1	no	—
116	C₁₆×C₅₃	—	2	2	no	—
117	C₁₆×C₅₃	—	2	1	no	—
118	#11719908	—	2	1	no	—
119	#11719908	—	2	2	no	—
120	#11719908	—	2	1	no	—
121	C₂²×D₅₃	C₂²×C₈	1	1	yes	characteristic
122	C₂²×D₅₃	C₂²×C₈	1	2	yes	—
123	Q₈×C₅₃	—	2	2	no	—
124	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
125	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
126	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
127	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
128	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
129	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
130	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
131	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
132	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
133	D₂₁₂	—	2	4	no	—
134	D₂₁₂	M₃₂	1	4	yes	complemented
135	C₄×D₅₃	—	2	4	no	—
136	C₄×D₅₃	—	2	2	no	—
137	C₄×D₅₃	—	2	1	no	—
138	C₄×D₅₃	—	2	2	no	—
139	C₄×D₅₃	—	2	1	no	—
140	C₄×D₅₃	—	2	2	no	—
141	C₄×D₅₃	—	2	4	no	—
142	C₄×D₅₃	—	2	2	no	—
143	C₈×C₅₃	—	2	1	no	—
144	C₈×C₅₃	—	2	2	no	—
145	C₈×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
146	C₈×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
147	#88322	—	2	2	no	—
148	#88322	—	2	1	no	—
149	#88322	—	2	1	no	—
150	#33504	—	53	1	no	maximal, Sylow
151	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
152	C₂×D₅₃	—	2	2	no	—
153	C₂×D₅₃	—	2	2	no	—
154	C₂×D₅₃	—	2	1	no	—
155	C₂×D₅₃	—	2	1	no	—
156	C₂×D₅₃	—	2	1	no	—
157	C₂×D₅₃	C₂×M₃₂	1	4	yes	—
158	C₄×C₅₃	—	2	2	no	—
159	C₄×C₅₃	—	2	2	no	—
160	C₄×C₅₃	—	2	1	no	—
161	C₄×C₅₃	C₂×M₃₂	1	2	yes	—
162	C₄×C₅₃	#520	1	1	yes	characteristic
163	C₄×C₅₃	#568	1	1	yes	characteristic
164	Dic₅₃	—	2	1	no	—
165	Dic₅₃	—	2	2	no	—
166	Dic₅₃	—	2	1	no	—
167	Dic₅₃	#504	1	2	yes	—
168	C₈×#41	—	53	1	no	—
169	#2183	—	53	1	no	—
170	#2183	D₅₃	1	1	yes	characteristic, complemented
171	#2168	—	53	2	no	—
172	#2168	—	53	2	no	—
173	D₄×C₁₆	—	53	1	no	—
174	D₄×C₁₆	—	53	1	no	—
175	#2163	—	53	2	no	—
176	#2162	—	53	1	no	—
177	#2150	—	53	1	no	—
178	C₂×C₄⋊C₁₆	—	53	2	no	—
179	C₂×C₅₃	#2988	1	1	yes	characteristic
180	C₂×C₅₃	C₂²×M₃₂	1	1	yes	characteristic
181	C₂×C₅₃	C₂×#504	1	1	yes	characteristic
182	D₅₃	—	2	4	no	—
183	D₅₃	—	4	1	no	—
184	C₄×#41	—	53	1	no	—
185	C₂×M₃₂	—	53	2	no	—
186	C₂²×C₁₆	—	53	2	no	—
187	C₂²×C₁₆	—	53	2	no	—
188	C₈×Q₈	—	53	1	no	—
189	C₈×Q₈	C₂×D₅₃	1	1	yes	characteristic
190	C₈×D₄	—	53	2	no	—
191	C₈×D₄	—	53	1	no	—
192	C₈×D₄	—	53	1	no	—
193	C₈×D₄	—	53	2	no	—
194	#431	—	53	1	no	—
195	#431	—	53	2	no	—
196	C₂×C₄×C₈	—	53	2	no	—
197	C₂×C₄×C₈	—	53	1	no	—
198	C₄⋊C₁₆	—	53	1	no	—
199	C₄⋊C₁₆	—	53	2	no	—
200	C₄⋊C₁₆	—	53	1	no	—
201	C₄⋊C₁₆	—	53	2	no	—
202	C₄⋊C₁₆	—	53	1	no	—
203	C₄⋊C₁₆	C₂×D₅₃	1	2	yes	—
204	C₄⋊C₁₆	C₂×D₅₃	1	1	yes	characteristic
205	C₂²⋊C₁₆	—	53	1	no	—
206	C₂²⋊C₁₆	—	53	1	no	—
207	C₂²⋊C₁₆	—	53	2	no	—
208	C₂²⋊C₁₆	—	53	2	no	—
209	#346	—	53	2	no	—
210	#346	C₂×D₅₃	1	2	yes	—
211	C₄×C₁₆	—	53	1	no	—
212	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic
213	C₅₃	#33504	1	1	yes	characteristic, complemented, Sylow
214	C₂×#41	—	53	1	no	—
215	C₂²×C₈	—	53	2	no	—
216	C₂²×C₈	—	53	1	no	—
217	C₂²×C₈	—	53	2	no	—
218	C₂²×C₈	—	53	1	no	—
219	C₄×Q₈	—	53	1	no	—
220	C₄×Q₈	C₄×D₅₃	1	1	yes	characteristic
221	C₄×D₄	—	53	1	no	—
222	C₄×D₄	—	53	2	no	—
223	C₄×D₄	—	53	1	no	—
224	C₄×D₄	—	53	2	no	—
225	#117	—	53	2	no	—
226	#117	—	53	1	no	—
227	C₂×C₄²	—	53	2	no	—
228	C₂×C₄²	—	53	1	no	—
229	M₃₂	—	212	2	no	—
230	C₂×C₁₆	—	53	1	no	—
231	C₂×C₁₆	—	53	1	no	—
232	C₂×C₁₆	—	53	2	no	—
233	C₂×C₁₆	—	212	2	no	—
234	C₂×C₁₆	—	212	2	no	—
235	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
236	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
237	C₂×C₁₆	C₂²×D₅₃	1	2	yes	—
238	C₄⋊C₈	—	53	1	no	—
239	C₄⋊C₈	—	53	2	no	—
240	C₄⋊C₈	C₄×D₅₃	1	2	yes	—
241	C₄⋊C₈	C₄×D₅₃	1	1	yes	characteristic
242	C₂²⋊C₈	—	53	2	no	—
243	C₂²⋊C₈	—	53	2	no	—
244	C₂²⋊C₈	—	53	2	no	—
245	C₄×C₈	—	53	2	no	—
246	C₄×C₈	—	53	1	no	—
247	C₄×C₈	—	53	1	no	—
248	C₄×C₈	—	53	2	no	—
249	C₄×C₈	—	53	1	no	—
250	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
251	C₄×C₈	C₂²×D₅₃	1	2	yes	—
252	#41	—	106	4	no	—
253	C₂×Q₈	C₈×D₅₃	1	1	yes	characteristic
254	C₂×D₄	—	53	1	no	—
255	C₂×D₄	—	53	2	no	—
256	C₂²×C₄	—	53	2	no	—
257	C₂²×C₄	—	53	1	no	—
258	C₂²×C₄	—	53	2	no	—
259	C₂²×C₄	—	53	1	no	—
260	C₂²×C₄	—	53	2	no	—
261	C₂²×C₄	—	53	1	no	—
262	C₂×C₈	—	53	1	no	—
263	C₂×C₈	—	53	2	no	—
264	C₂×C₈	—	53	1	no	—
265	C₂×C₈	—	106	2	no	—
266	C₂×C₈	—	106	1	no	—
267	C₂×C₈	—	106	2	no	—
268	C₂×C₈	—	106	1	no	—
269	C₂×C₈	—	212	2	no	—
270	C₂×C₈	—	212	1	no	—
271	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
272	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
273	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
274	C₄⋊C₄	—	53	2	no	—
275	C₄⋊C₄	—	53	1	no	—
276	C₄⋊C₄	C₈×D₅₃	1	2	yes	—
277	C₄⋊C₄	C₈×D₅₃	1	1	yes	characteristic
278	C₂²⋊C₄	—	53	2	no	—
279	C₂²⋊C₄	—	53	2	no	—
280	C₂²⋊C₄	—	53	2	no	—
281	C₄²	—	53	2	no	—
282	C₄²	—	53	1	no	—
283	C₄²	—	53	2	no	—
284	C₄²	—	53	1	no	—
285	C₄²	—	53	1	no	—
286	C₄²	C₂×C₄×D₅₃	1	2	yes	—
287	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
288	C₁₆	—	2	1	no	—
289	C₁₆	—	2	2	no	—
290	C₁₆	—	2	1	no	—
291	C₁₆	—	106	1	no	—
292	C₁₆	—	106	2	no	—
293	C₁₆	—	106	1	no	—
294	C₂³	—	53	2	no	—
295	C₂³	—	53	1	no	—
296	Q₈	—	2	2	no	—
297	D₄	—	53	4	no	—
298	D₄	—	106	4	no	—
299	C₂×C₄	—	53	1	no	—
300	C₂×C₄	—	53	1	no	—
301	C₂×C₄	—	53	2	no	—
302	C₂×C₄	—	53	1	no	—
303	C₂×C₄	—	106	4	no	—
304	C₂×C₄	—	106	2	no	—
305	C₂×C₄	—	106	4	no	—
306	C₂×C₄	—	106	2	no	—
307	C₂×C₄	—	106	2	no	—
308	C₂×C₄	—	106	1	no	—
309	C₂×C₄	—	106	2	no	—
310	C₂×C₄	—	106	1	no	—
311	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
312	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
313	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
314	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
315	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
316	C₈	—	2	1	no	—
317	C₈	—	2	2	no	—
318	C₈	—	106	2	no	—
319	C₈	—	106	1	no	—
320	C₈	—	106	1	no	—
321	C₈	#13133263	1	1	yes	characteristic
322	C₈	#13133265	1	1	yes	characteristic
323	C₂²	—	53	4	no	—
324	C₂²	—	106	2	no	—
325	C₂²	—	106	2	no	—
326	C₂²	—	106	1	no	—
327	C₂²	—	106	1	no	—
328	C₂²	—	106	1	no	—
329	C₂²	C₂²×C₈×D₅₃	1	1	yes	characteristic
330	C₄	—	2	2	no	—
331	C₄	—	2	2	no	—
332	C₄	—	2	1	no	—
333	C₄	—	53	2	no	—
334	C₄	—	106	1	no	—
335	C₄	—	106	2	no	—
336	C₄	—	106	1	no	—
337	C₄	M₃₂×D₅₃	1	2	yes	—
338	C₄	#16811105	1	1	yes	characteristic
339	C₄	#16811388	1	1	yes	characteristic
340	C₂	—	106	4	no	—
341	C₂	—	212	1	no	—
342	C₂	#20283825	1	1	yes	characteristic
343	C₂	C₂×M₃₂×D₅₃	1	1	yes	characteristic
344	C₂	#504×D₅₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₈
Fitting	C₅₃×#2183	C₂
prime radical	C₂×#11719949	C₈
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	#2183	D₅₃
hypocenter	C₅₃	#33504
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ #2183

Complements

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

N	H ≅ G/N	type
2. #20276624	C₂	semidirect
3. #20276542	C₂	semidirect
4. #20276539	C₂	semidirect
5. C₁₆×D₂₁₂	C₂	semidirect
6. #20276534	C₂	semidirect
7. #20276516	C₂	semidirect
8. #20276512	C₂	semidirect
9. #20276469	C₂	semidirect
10. C₄⋊C₁₆×D₅₃	C₂	semidirect
11. C₅₃×#2183	C₂	semidirect
134. D₂₁₂	M₃₂	semidirect
170. #2183	D₅₃	semidirect
213. C₅₃	#33504	semidirect

Maximal subgroups

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

subgroup	index
C₈×#11719949	2
#20276624	2
#20276542	2
#20276539	2
C₁₆×D₂₁₂	2
#20276534	2
#20276516	2
#20276512	2
#20276469	2
C₄⋊C₁₆×D₅₃	2
C₅₃×#2183	2
#33504	53