Subgroups — #27402365

Order 13568.

862 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	M₁₆×D₂₁₂	C₂	1	2	yes	maximal
2	C₂×#16810702	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810670	C₂	1	2	yes	maximal, complemented
4	#20275490	C₂	1	2	yes	maximal, complemented
5	#20274921	C₂	1	4	yes	maximal, complemented
6	#20274887	C₂	1	1	yes	maximal, characteristic, complemented
7	#20274754	C₂	1	2	yes	maximal, complemented
8	C₄×#108×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#11719912	C₂²	1	2	yes	—
10	C₂×M₁₆×D₅₃	C₂²	1	4	yes	—
11	C₂×C₄×D₂₁₂	C₂²	1	1	yes	characteristic
12	#16810702	—	2	2	no	—
13	#16810702	C₄	1	4	yes	—
14	#16810702	C₄	1	4	yes	—
15	#16810696	—	2	4	no	—
16	C₈×D₂₁₂	—	2	2	no	—
17	#16810670	—	2	8	no	—
18	C₂×#13133153	C₂²	1	2	yes	—
19	C₂×#13133153	C₂²	1	2	yes	—
20	C₂×#13133152	C₂²	1	4	yes	—
21	#16810647	C₂²	1	2	yes	—
22	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
23	#16810586	C₂²	1	4	yes	—
24	C₂×#13133129	C₂²	1	2	yes	—
25	#16810387	C₂²	1	4	yes	—
26	C₂×#108×C₅₃	C₂²	1	1	yes	characteristic
27	C₂×#108×C₅₃	C₂²	1	1	yes	characteristic
28	C₄×M₁₆×C₅₃	C₂²	1	2	yes	—
29	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
30	C₅₃×#341	C₂²	1	2	yes	—
31	C₂²×D₂₁₂	Q₈	1	1	yes	characteristic
32	C₂²×C₄×D₅₃	C₂³	1	2	yes	—
33	C₂×#11719917	C₂³	1	4	yes	—
34	C₂×#11719912	—	2	4	no	—
35	C₂×#11719912	—	2	4	no	—
36	C₂×#11719912	—	2	4	no	—
37	C₂×#11719912	C₂×C₄	1	8	yes	—
38	M₁₆×D₅₃	—	2	8	no	—
39	M₁₆×D₅₃	—	4	4	no	—
40	C₂²×#88322	C₂³	1	2	yes	—
41	C₂×C₈×D₅₃	—	2	4	no	—
42	C₄×D₂₁₂	C₂×C₄	1	2	yes	—
43	C₄×D₂₁₂	C₂×C₄	1	2	yes	—
44	C₄×D₂₁₂	D₄	1	2	yes	—
45	C₄×D₂₁₂	D₄	1	2	yes	—
46	C₂×C₄⋊Dic₅₃	Q₈	1	1	yes	characteristic
47	C₂×#11719921	Q₈	1	2	yes	—
48	#13133153	—	2	4	no	—
49	#13133153	—	4	2	no	—
50	#13133152	—	2	8	no	—
51	C₄⋊#88322	—	2	2	no	—
52	C₄⋊#88322	—	2	2	no	—
53	C₄⋊#88322	C₂×C₄	1	2	yes	—
54	C₄⋊#88322	C₂×C₄	1	2	yes	—
55	#13133129	—	2	4	no	—
56	#13133129	—	2	4	no	—
57	#13133129	C₂×C₄	1	8	yes	—
58	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
59	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
60	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
61	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
62	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
63	#108×C₅₃	—	2	2	no	—
64	#108×C₅₃	D₄	1	4	yes	complemented
65	#97×C₅₃	—	2	2	no	—
66	C₄×C₈×C₅₃	—	2	2	no	—
67	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
68	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
69	C₂³×D₅₃	C₂×Q₈	1	2	yes	—
70	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
71	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
72	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
73	C₂²×Dic₅₃	C₂×Q₈	1	2	yes	—
74	C₂×D₂₁₂	—	4	2	no	—
75	C₂×D₂₁₂	C₄⋊C₄	1	2	yes	—
76	C₂×D₂₁₂	C₄⋊C₄	1	2	yes	—
77	C₂×C₄×D₅₃	—	2	2	no	—
78	C₂×C₄×D₅₃	—	2	2	no	—
79	C₂×C₄×D₅₃	—	2	2	no	—
80	C₂×C₄×D₅₃	—	2	2	no	—
81	C₂×C₄×D₅₃	C₂²×C₄	1	4	yes	—
82	C₂×C₄×D₅₃	C₂×D₄	1	4	yes	—
83	M₁₆×C₅₃	—	2	2	no	—
84	M₁₆×C₅₃	—	2	2	no	—
85	M₁₆×C₅₃	—	2	2	no	—
86	M₁₆×C₅₃	C₂×D₄	1	4	yes	—
87	C₂×C₈×C₅₃	—	2	2	no	—
88	C₂×C₈×C₅₃	—	2	1	no	—
89	C₂×C₈×C₅₃	—	2	1	no	—
90	C₂×C₈×C₅₃	—	2	2	no	—
91	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
92	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
93	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
94	C₂×C₈×C₅₃	#41	1	2	yes	—
95	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
96	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
97	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
98	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
99	#11719921	C₄⋊C₄	1	4	yes	—
100	#11719921	C₄⋊C₄	1	4	yes	—
101	C₄⋊Dic₅₃	C₄⋊C₄	1	2	yes	—
102	C₄⋊Dic₅₃	C₄⋊C₄	1	2	yes	—
103	#11719917	—	2	4	no	—
104	#11719917	—	2	4	no	—
105	C₂×#88322	—	2	2	no	—
106	C₂×#88322	—	2	2	no	—
107	C₂×#88322	—	2	4	no	—
108	C₂×#88322	C₂²×C₄	1	4	yes	—
109	#11719912	—	2	16	no	—
110	#11719912	—	4	8	no	—
111	C₈×D₅₃	—	4	8	no	—
112	C₂³×C₅₃	C₂²×Q₈	1	1	yes	characteristic
113	C₂²×D₅₃	—	2	2	no	—
114	C₂²×D₅₃	—	2	2	no	—
115	C₂²×D₅₃	—	4	2	no	—
116	C₂²×D₅₃	—	4	2	no	—
117	C₂²×D₅₃	C₂×C₄⋊C₄	1	4	yes	—
118	C₂×C₄×C₅₃	—	2	2	no	—
119	C₂×C₄×C₅₃	—	4	1	no	—
120	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
125	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
126	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
127	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
128	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
129	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
130	C₂×Dic₅₃	—	2	2	no	—
131	C₂×Dic₅₃	—	2	2	no	—
132	C₂×Dic₅₃	C₂×C₄⋊C₄	1	4	yes	—
133	D₂₁₂	—	2	4	no	—
134	D₂₁₂	—	4	2	no	—
135	C₄×D₅₃	—	2	4	no	—
136	C₄×D₅₃	—	2	4	no	—
137	C₄×D₅₃	—	2	4	no	—
138	C₄×D₅₃	—	2	4	no	—
139	C₈×C₅₃	—	2	2	no	—
140	C₈×C₅₃	—	2	4	no	—
141	C₈×C₅₃	—	4	2	no	—
142	C₈×C₅₃	C₄×D₄	1	4	yes	—
143	#88322	—	2	4	no	—
144	#88322	—	4	4	no	—
145	#22669	—	53	1	no	maximal, Sylow
146	C₂²×C₅₃	—	2	1	no	—
147	C₂²×C₅₃	—	2	1	no	—
148	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
149	C₂²×C₅₃	D₄×Q₈	1	1	yes	characteristic
150	C₂²×C₅₃	#554	1	1	yes	characteristic
151	C₂×D₅₃	—	2	4	no	—
152	C₂×D₅₃	—	2	4	no	—
153	C₂×D₅₃	—	2	4	no	—
154	C₂×D₅₃	—	8	2	no	—
155	C₂×D₅₃	—	8	2	no	—
156	C₄×C₅₃	—	2	1	no	—
157	C₄×C₅₃	—	2	1	no	—
158	C₄×C₅₃	—	2	1	no	—
159	C₄×C₅₃	#430	1	2	yes	complemented
160	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
161	C₄×C₅₃	#520	1	1	yes	characteristic
162	C₄×C₅₃	D₄²	1	1	yes	characteristic
163	C₄×C₅₃	#547	1	1	yes	characteristic
164	Dic₅₃	—	2	4	no	—
165	D₄×M₁₆	—	53	2	no	—
166	C₂×#436	—	53	1	no	—
167	C₂×#430	—	53	2	no	—
168	#1950	—	53	2	no	—
169	#1855	—	53	4	no	—
170	#1848	—	53	1	no	—
171	#1811	—	53	2	no	—
172	C₄×#108	D₅₃	1	1	yes	characteristic, complemented
173	C₂×C₅₃	—	2	1	no	—
174	C₂×C₅₃	—	2	1	no	—
175	C₂×C₅₃	D₄×C₄⋊C₄	1	1	yes	characteristic
176	C₂×C₅₃	C₂×#430	1	1	yes	characteristic
177	C₂×C₅₃	#2916	1	1	yes	characteristic
178	D₅₃	—	4	4	no	—
179	C₂²×M₁₆	—	53	2	no	—
180	C₂²×M₁₆	—	53	4	no	—
181	C₂×C₄×D₄	—	53	1	no	—
182	#436	—	53	4	no	—
183	#436	—	53	4	no	—
184	#436	—	106	2	no	—
185	#435	—	106	4	no	—
186	C₈×D₄	—	106	2	no	—
187	#430	—	106	8	no	—
188	C₂×#108	—	53	4	no	—
189	C₂×#108	—	53	2	no	—
190	C₂×#108	—	53	2	no	—
191	C₂×#108	C₂×D₅₃	1	1	yes	characteristic
192	C₂×#108	C₂×D₅₃	1	1	yes	characteristic
193	#423	—	53	2	no	—
194	C₂×C₄⋊C₈	—	53	1	no	—
195	#407	—	53	4	no	—
196	C₂×C₂²⋊C₈	—	53	2	no	—
197	C₄×M₁₆	C₂×D₅₃	1	2	yes	—
198	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
199	#341	—	53	4	no	—
200	#341	C₂×D₅₃	1	2	yes	—
201	C₅₃	#22669	1	1	yes	characteristic, complemented, Sylow
202	C₂²×D₄	—	53	1	no	—
203	C₂³×C₄	—	53	2	no	—
204	C₂×M₁₆	—	53	8	no	—
205	C₂×M₁₆	—	53	4	no	—
206	C₂×M₁₆	—	106	4	no	—
207	C₂×M₁₆	—	106	4	no	—
208	C₂×M₁₆	—	106	4	no	—
209	C₂×M₁₆	—	106	8	no	—
210	C₂×M₁₆	—	212	4	no	—
211	C₂×M₁₆	C₂²×D₅₃	1	2	yes	—
212	C₂×M₁₆	C₂²×D₅₃	1	2	yes	—
213	C₂²×C₈	—	53	2	no	—
214	C₂²×C₈	—	106	4	no	—
215	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
216	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
217	C₄×D₄	—	53	2	no	—
218	C₄×D₄	—	53	2	no	—
219	C₄×D₄	—	53	2	no	—
220	C₄×D₄	—	53	2	no	—
221	C₂×C₄⋊C₄	—	53	1	no	—
222	C₂×C₂²⋊C₄	—	53	2	no	—
223	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
224	#108	—	2	2	no	—
225	#108	—	106	8	no	—
226	#108	—	106	4	no	—
227	#108	—	212	2	no	—
228	#108	D₂₁₂	1	4	yes	complemented
229	C₄⋊C₈	—	53	2	no	—
230	C₄⋊C₈	—	53	2	no	—
231	C₄⋊C₈	—	106	2	no	—
232	C₄⋊C₈	—	106	2	no	—
233	C₂²⋊C₈	—	53	8	no	—
234	C₂²⋊C₈	—	106	4	no	—
235	C₂²⋊C₈	—	106	4	no	—
236	#97	—	2	2	no	—
237	C₄×C₈	—	2	2	no	—
238	C₄×C₈	C₄×D₅₃	1	2	yes	—
239	C₄×C₈	C₄×D₅₃	1	2	yes	—
240	C₂⁴	—	53	2	no	—
241	C₂×D₄	—	53	2	no	—
242	C₂×D₄	—	53	2	no	—
243	C₂×D₄	—	212	2	no	—
244	C₂²×C₄	—	53	2	no	—
245	C₂²×C₄	—	53	4	no	—
246	C₂²×C₄	—	53	4	no	—
247	C₂²×C₄	—	106	2	no	—
248	C₂²×C₄	—	106	2	no	—
249	C₂²×C₄	—	106	2	no	—
250	C₂²×C₄	—	106	2	no	—
251	C₂²×C₄	Q₈×D₅₃	1	1	yes	characteristic
252	C₂²×C₄	Q₈×D₅₃	1	1	yes	characteristic
253	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
254	M₁₆	—	2	2	no	—
255	M₁₆	—	2	2	no	—
256	M₁₆	—	2	2	no	—
257	M₁₆	—	106	16	no	—
258	M₁₆	—	106	4	no	—
259	M₁₆	—	106	4	no	—
260	M₁₆	—	212	8	no	—
261	M₁₆	C₂×D₂₁₂	1	4	yes	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	2	1	no	—
264	C₂×C₈	—	2	1	no	—
265	C₂×C₈	—	2	2	no	—
266	C₂×C₈	—	53	4	no	—
267	C₂×C₈	—	106	2	no	—
268	C₂×C₈	—	106	2	no	—
269	C₂×C₈	—	106	4	no	—
270	C₂×C₈	—	212	8	no	—
271	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
272	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
273	C₂×C₈	C₂×D₂₁₂	1	2	yes	—
274	C₂×C₈	#11719945	1	2	yes	—
275	C₄⋊C₄	—	53	2	no	—
276	C₄⋊C₄	—	53	2	no	—
277	C₂²⋊C₄	—	53	4	no	—
278	C₂²⋊C₄	—	53	4	no	—
279	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
280	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
281	C₄²	D₄×D₅₃	1	1	yes	characteristic
282	C₄²	D₄×D₅₃	1	1	yes	characteristic
283	C₂³	—	53	4	no	—
284	C₂³	—	106	2	no	—
285	C₂³	—	106	2	no	—
286	C₂³	—	212	2	no	—
287	C₂³	—	212	2	no	—
288	C₂³	C₂×Q₈×D₅₃	1	1	yes	characteristic
289	D₄	—	106	4	no	—
290	D₄	—	212	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	4	1	no	—
293	C₂×C₄	—	53	4	no	—
294	C₂×C₄	—	106	2	no	—
295	C₂×C₄	—	106	2	no	—
296	C₂×C₄	—	106	4	no	—
297	C₂×C₄	—	106	4	no	—
298	C₂×C₄	—	106	4	no	—
299	C₂×C₄	—	106	4	no	—
300	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
301	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
302	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
303	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
304	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
305	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
306	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
307	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
308	C₂×C₄	#13133263	1	1	yes	characteristic
309	C₂×C₄	#13133264	1	1	yes	characteristic
310	C₈	—	2	2	no	—
311	C₈	—	2	4	no	—
312	C₈	—	4	2	no	—
313	C₈	—	106	4	no	—
314	C₈	—	212	4	no	—
315	C₈	C₄×D₂₁₂	1	4	yes	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	106	4	no	—
319	C₂²	—	106	4	no	—
320	C₂²	—	106	4	no	—
321	C₂²	—	424	2	no	—
322	C₂²	—	424	2	no	—
323	C₂²	C₂×C₄⋊C₄×D₅₃	1	1	yes	characteristic
324	C₂²	Q₈×D₂₁₂	1	1	yes	characteristic
325	C₂²	#16811333	1	1	yes	characteristic
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	2	1	no	—
329	C₄	—	106	4	no	—
330	C₄	#16810670	1	2	yes	complemented
331	C₄	C₂×C₄×D₂₁₂	1	1	yes	characteristic
332	C₄	#16811103	1	1	yes	characteristic
333	C₄	D₄×D₂₁₂	1	1	yes	characteristic
334	C₄	#16811278	1	1	yes	characteristic
335	C₂	—	2	1	no	—
336	C₂	—	2	1	no	—
337	C₂	—	212	4	no	—
338	C₂	C₄⋊C₄×D₂₁₂	1	1	yes	characteristic
339	C₂	C₂×#16810670	1	1	yes	characteristic
340	C₂	#20283323	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₄×#108×C₅₃	C₂
socle	C₂²×C₅₃	C₂²×C₄⋊C₄
prime radical	C₂²×D₂₁₂	Q₈
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₄×#108	D₅₃
hypocenter	C₅₃	#22669
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#108

Complements

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

N	H ≅ G/N	type
3. C₂×#16810670	C₂	semidirect
4. #20275490	C₂	semidirect
5. #20274921	C₂	semidirect
6. #20274887	C₂	semidirect
7. #20274754	C₂	semidirect
8. C₄×#108×C₅₃	C₂	semidirect
64. #108×C₅₃	D₄	semidirect
159. C₄×C₅₃	#430	semidirect
172. C₄×#108	D₅₃	semidirect
201. C₅₃	#22669	semidirect
228. #108	D₂₁₂	semidirect
330. C₄	#16810670	semidirect

Maximal subgroups

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

subgroup	index
M₁₆×D₂₁₂	2
C₂×#16810702	2
C₂×#16810670	2
#20275490	2
#20274921	2
#20274887	2
#20274754	2
C₄×#108×C₅₃	2
#22669	53