Subgroups — #27537656

Order 13568.

516 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	#20283690	C₂	1	1	yes	maximal, characteristic
2	#20276623	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₆×Dic₁₀₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#20276548	C₂	1	1	yes	maximal, characteristic, complemented
5	#20276545	C₂	1	2	yes	maximal, complemented
6	#20276538	C₂	1	2	yes	maximal, complemented
7	C₁₆×D₂₁₂	C₂	1	1	yes	maximal, characteristic, complemented
8	#20276523	C₂	1	2	yes	maximal, complemented
9	#20276514	C₂	1	1	yes	maximal, characteristic, complemented
10	#20276313	C₂	1	2	yes	maximal, complemented
11	C₄×M₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	#16811157	C₄	1	1	yes	characteristic
13	C₂×#13133156	C₂²	1	2	yes	—
14	C₂×C₁₆×D₅₃	C₂²	1	2	yes	—
15	#16810753	C₄	1	1	yes	characteristic
16	C₈×Dic₁₀₆	C₂²	1	1	yes	characteristic
17	#16810702	C₄	1	1	yes	characteristic
18	#16810694	C₄	1	2	yes	—
19	C₈×D₂₁₂	C₂²	1	1	yes	characteristic
20	C₈×C₂²⋊D₅₃	C₂²	1	2	yes	—
21	#16810682	C₄	1	2	yes	—
22	#16810680	C₄	1	1	yes	characteristic
23	#16810577	C₂²	1	2	yes	—
24	C₄⋊#11719908	C₂²	1	1	yes	characteristic
25	C₄⋊#11719908	C₂²	1	1	yes	characteristic
26	#16810441	C₂²	1	2	yes	—
27	#16810441	C₂²	1	2	yes	—
28	C₂²⋊#11719908	C₂²	1	2	yes	—
29	#16810402	C₂²	1	2	yes	—
30	#16810402	C₂²	1	2	yes	—
31	#16810398	C₂²	1	2	yes	—
32	C₄×#11719908	C₂²	1	2	yes	—
33	C₁₆×Dic₅₃	C₂²	1	2	yes	—
34	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
35	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
36	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
37	C₅₃×#346	C₂²	1	1	yes	characteristic
38	C₅₃×#346	C₂²	1	1	yes	characteristic
39	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
40	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
41	C₂×#11719912	C₂×C₄	1	2	yes	—
42	C₂×C₈×D₅₃	C₂³	1	2	yes	—
43	#13133211	C₈	1	2	yes	—
44	#13133209	C₈	1	1	yes	characteristic
45	#13133205	C₈	1	1	yes	characteristic
46	#13133200	C₈	1	2	yes	—
47	#13133196	C₈	1	1	yes	characteristic
48	#13133191	C₈	1	1	yes	characteristic
49	C₄×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
50	C₄×D₂₁₂	C₂×C₄	1	1	yes	characteristic
51	C₄×C₂²⋊D₅₃	C₂×C₄	1	2	yes	—
52	#13133174	C₂×C₄	1	2	yes	—
53	#13133156	—	2	4	no	—
54	C₂×#11719908	C₂³	1	2	yes	—
55	C₂×#11719908	C₂³	1	2	yes	—
56	C₁₆×D₅₃	—	2	4	no	—
57	C₄⋊#88322	C₂×C₄	1	1	yes	characteristic
58	C₄⋊#88322	C₂×C₄	1	1	yes	characteristic
59	#13133146	C₂×C₄	1	2	yes	—
60	#13133146	C₂×C₄	1	2	yes	—
61	C₂²⋊#88322	C₂×C₄	1	2	yes	—
62	#13133129	C₂×C₄	1	2	yes	—
63	#13133129	C₂×C₄	1	2	yes	—
64	#13133128	C₂×C₄	1	2	yes	—
65	#13133127	C₂×C₄	1	2	yes	—
66	C₈×Dic₅₃	C₂³	1	2	yes	—
67	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
68	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
69	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
70	M₃₂×C₅₃	—	2	2	no	—
71	M₃₂×C₅₃	—	2	2	no	—
72	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
73	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
74	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
75	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
76	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
77	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
78	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
79	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
80	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
81	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
82	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₂²⋊D₅₃	C₂×C₈	1	2	yes	—
84	C₂×D₂₁₂	C₂×C₈	1	1	yes	characteristic
85	C₂×C₄×D₅₃	C₂²×C₄	1	2	yes	—
86	C₂×Dic₁₀₆	C₂×C₈	1	1	yes	characteristic
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	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
94	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
95	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
96	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
97	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
98	C₂²⋊Dic₅₃	C₂×C₈	1	2	yes	—
99	#11719921	C₂×C₈	1	2	yes	—
100	#11719921	C₂×C₈	1	2	yes	—
101	#11719921	C₂×C₈	1	2	yes	—
102	#11719921	C₂×C₈	1	2	yes	—
103	C₄⋊Dic₅₃	C₂×C₈	1	1	yes	characteristic
104	C₄⋊Dic₅₃	C₂×C₈	1	1	yes	characteristic
105	#11719919	C₂×C₈	1	2	yes	—
106	#11719919	C₂×C₈	1	2	yes	—
107	#11719919	C₂×C₈	1	2	yes	—
108	#11719919	C₂×C₈	1	2	yes	—
109	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
110	C₂×#88322	C₂²×C₄	1	2	yes	—
111	C₂×#88322	C₂²×C₄	1	2	yes	—
112	#11719912	—	2	4	no	—
113	C₈×D₅₃	—	2	2	no	—
114	C₈×D₅₃	—	2	2	no	—
115	C₁₆×C₅₃	—	2	1	no	—
116	C₁₆×C₅₃	—	2	1	no	—
117	C₁₆×C₅₃	#41	1	2	yes	—
118	C₁₆×C₅₃	#41	1	2	yes	—
119	#11719908	—	2	2	no	—
120	#11719908	—	2	2	no	—
121	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
122	C₂²×D₅₃	C₂²×C₈	1	2	yes	—
123	C₂×C₄×C₅₃	—	2	1	no	—
124	C₂×C₄×C₅₃	—	2	1	no	—
125	C₂×C₄×C₅₃	—	2	2	no	—
126	C₂×C₄×C₅₃	—	4	1	no	—
127	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
128	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
129	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
130	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
131	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
132	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
133	C₂²⋊D₅₃	—	4	2	no	—
134	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
135	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
136	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
137	D₂₁₂	—	2	2	no	—
138	C₄×D₅₃	—	2	2	no	—
139	C₄×D₅₃	—	2	2	no	—
140	Dic₁₀₆	—	2	2	no	—
141	C₈×C₅₃	—	2	1	no	—
142	C₈×C₅₃	—	2	1	no	—
143	C₈×C₅₃	#131	1	2	yes	—
144	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
145	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
146	#88322	—	2	2	no	—
147	#88322	—	2	2	no	—
148	#33483	—	53	1	no	maximal, Sylow
149	C₂²×C₅₃	—	2	1	no	—
150	C₂²×C₅₃	—	2	1	no	—
151	C₂²×C₅₃	—	2	1	no	—
152	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
153	C₂×D₅₃	—	2	2	no	—
154	C₂×D₅₃	—	2	2	no	—
155	C₂×D₅₃	—	2	2	no	—
156	C₄×C₅₃	—	2	1	no	—
157	C₄×C₅₃	—	2	1	no	—
158	C₄×C₅₃	—	2	1	no	—
159	C₄×C₅₃	—	2	1	no	—
160	C₄×C₅₃	#504	1	2	yes	—
161	C₄×C₅₃	C₄×#41	1	1	yes	characteristic
162	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
163	Dic₅₃	—	2	2	no	—
164	Dic₅₃	—	2	2	no	—
165	Dic₅₃	—	2	2	no	—
166	#2966	—	53	1	no	—
167	#2183	—	53	1	no	—
168	Q₈×C₁₆	—	53	1	no	—
169	#2169	—	53	2	no	—
170	#2169	—	53	1	no	—
171	#2168	—	53	2	no	—
172	D₄×C₁₆	—	53	1	no	—
173	#2164	—	53	2	no	—
174	#2162	—	53	1	no	—
175	#2108	—	53	2	no	—
176	C₄×M₃₂	D₅₃	1	1	yes	characteristic, complemented
177	C₂×C₅₃	—	4	1	no	—
178	C₂×C₅₃	C₈×#41	1	1	yes	characteristic
179	C₂×C₅₃	C₂×#504	1	1	yes	characteristic
180	C₂×C₅₃	#3407	1	1	yes	characteristic
181	D₅₃	—	4	2	no	—
182	#529	—	53	1	no	—
183	C₂×M₃₂	—	53	2	no	—
184	C₂×M₃₂	C₂×D₅₃	1	1	yes	characteristic
185	C₂×M₃₂	C₂×D₅₃	1	1	yes	characteristic
186	C₂²×C₁₆	—	53	2	no	—
187	#446	—	53	1	no	—
188	C₈×Q₈	—	53	1	no	—
189	#436	—	53	1	no	—
190	#435	—	53	2	no	—
191	C₈×D₄	—	53	2	no	—
192	C₈×D₄	—	53	1	no	—
193	#433	—	53	2	no	—
194	#432	—	53	1	no	—
195	#405	—	53	2	no	—
196	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
197	C₄⋊C₁₆	—	53	2	no	—
198	C₄⋊C₁₆	—	53	2	no	—
199	C₄⋊C₁₆	—	53	1	no	—
200	C₄⋊C₁₆	—	53	1	no	—
201	C₂²⋊C₁₆	—	53	2	no	—
202	C₂²⋊C₁₆	—	53	2	no	—
203	C₂²⋊C₁₆	—	53	2	no	—
204	#346	—	53	2	no	—
205	#346	C₂×D₅₃	1	1	yes	characteristic
206	#346	C₂×D₅₃	1	1	yes	characteristic
207	C₄×C₁₆	—	53	2	no	—
208	C₄×C₁₆	—	53	2	no	—
209	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic
210	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic
211	C₅₃	#33483	1	1	yes	characteristic, complemented, Sylow
212	C₂×M₁₆	—	53	2	no	—
213	C₂²×C₈	—	53	2	no	—
214	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
215	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
216	#126	—	53	2	no	—
217	#125	—	53	1	no	—
218	#124	—	53	1	no	—
219	#123	—	53	2	no	—
220	C₂²⋊Q₈	—	53	1	no	—
221	#121	—	53	1	no	—
222	C₄×Q₈	—	53	1	no	—
223	C₄×D₄	—	53	2	no	—
224	C₄×D₄	—	53	1	no	—
225	#117	—	53	2	no	—
226	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
227	M₃₂	—	2	2	no	—
228	M₃₂	—	2	2	no	—
229	M₃₂	—	106	4	no	—
230	C₂×C₁₆	—	53	2	no	—
231	C₂×C₁₆	—	53	2	no	—
232	C₂×C₁₆	—	106	4	no	—
233	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
234	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
235	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
236	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
237	C₄⋊C₈	—	53	2	no	—
238	C₄⋊C₈	—	53	2	no	—
239	C₄⋊C₈	—	53	1	no	—
240	C₄⋊C₈	—	53	1	no	—
241	C₂²⋊C₈	—	53	2	no	—
242	C₂²⋊C₈	—	53	2	no	—
243	C₂²⋊C₈	—	53	2	no	—
244	#97	—	53	2	no	—
245	#97	—	53	2	no	—
246	C₄×C₈	—	53	2	no	—
247	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
248	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
249	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
250	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
251	C₂×Q₈	—	53	1	no	—
252	C₂×D₄	—	53	2	no	—
253	C₂×D₄	—	53	1	no	—
254	C₂²×C₄	—	53	2	no	—
255	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
256	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
257	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
258	M₁₆	—	106	4	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	2	1	no	—
261	C₂×C₈	—	2	1	no	—
262	C₂×C₈	—	53	2	no	—
263	C₂×C₈	—	53	2	no	—
264	C₂×C₈	—	106	2	no	—
265	C₂×C₈	—	106	2	no	—
266	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
267	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
268	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
269	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
270	C₄⋊C₄	—	53	2	no	—
271	C₄⋊C₄	—	53	2	no	—
272	C₄⋊C₄	—	53	1	no	—
273	C₄⋊C₄	—	53	2	no	—
274	C₄⋊C₄	—	53	1	no	—
275	C₄⋊C₄	—	53	2	no	—
276	C₂²⋊C₄	—	53	2	no	—
277	C₂²⋊C₄	—	53	2	no	—
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	C₈×D₅₃	1	1	yes	characteristic
283	C₄²	C₈×D₅₃	1	1	yes	characteristic
284	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
285	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
286	C₁₆	—	2	1	no	—
287	C₁₆	—	2	1	no	—
288	C₁₆	—	106	2	no	—
289	C₁₆	—	106	2	no	—
290	C₁₆	#11719945	1	2	yes	—
291	C₁₆	#11719945	1	2	yes	—
292	C₂³	—	53	2	no	—
293	C₂³	C₂×C₈×D₅₃	1	1	yes	characteristic
294	Q₈	—	106	2	no	—
295	D₄	—	106	2	no	—
296	D₄	—	212	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	4	1	no	—
301	C₂×C₄	—	53	2	no	—
302	C₂×C₄	—	53	2	no	—
303	C₂×C₄	—	53	2	no	—
304	C₂×C₄	—	106	2	no	—
305	C₂×C₄	—	106	2	no	—
306	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
307	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
308	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
309	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
310	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
311	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
312	C₈	—	2	1	no	—
313	C₈	—	2	1	no	—
314	C₈	—	106	2	no	—
315	C₈	—	106	2	no	—
316	C₈	#13133223	1	2	yes	—
317	C₈	C₂×#11719945	1	1	yes	characteristic
318	C₈	C₂×#11719945	1	1	yes	characteristic
319	C₂²	—	2	1	no	—
320	C₂²	—	2	1	no	—
321	C₂²	—	2	1	no	—
322	C₂²	—	106	2	no	—
323	C₂²	—	106	2	no	—
324	C₂²	—	106	2	no	—
325	C₂²	C₂²×C₈×D₅₃	1	1	yes	characteristic
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	106	2	no	—
331	C₄	—	106	2	no	—
332	C₄	—	106	2	no	—
333	C₄	#16811041	1	2	yes	—
334	C₄	C₄×#11719945	1	1	yes	characteristic
335	C₄	C₂×#13133223	1	1	yes	characteristic
336	C₂	—	4	1	no	—
337	C₂	—	212	2	no	—
338	C₂	C₈×#11719945	1	1	yes	characteristic
339	C₂	C₂×#16811041	1	1	yes	characteristic
340	C₂	#20287817	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₈
Fitting	C₄×M₃₂×C₅₃	C₂
prime radical	#13133191	C₈
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₄×M₃₂	D₅₃
hypocenter	C₅₃	#33483
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
2. #20276623	C₂	semidirect
3. C₁₆×Dic₁₀₆	C₂	semidirect
4. #20276548	C₂	semidirect
5. #20276545	C₂	semidirect
6. #20276538	C₂	semidirect
7. C₁₆×D₂₁₂	C₂	semidirect
8. #20276523	C₂	semidirect
9. #20276514	C₂	semidirect
10. #20276313	C₂	semidirect
11. C₄×M₃₂×C₅₃	C₂	semidirect
176. C₄×M₃₂	D₅₃	semidirect
211. C₅₃	#33483	semidirect

Maximal subgroups

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

subgroup	index
#20283690	2
#20276623	2
C₁₆×Dic₁₀₆	2
#20276548	2
#20276545	2
#20276538	2
C₁₆×D₂₁₂	2
#20276523	2
#20276514	2
#20276313	2
C₄×M₃₂×C₅₃	2
#33483	53