Subgroups — #27537604

Order 13568.

530 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×Q₈×D₅₃	C₂	1	1	yes	maximal, characteristic
2	#20276624	C₂	1	1	yes	maximal, characteristic, complemented
3	#20276622	C₂	1	2	yes	maximal, complemented
4	#20276621	C₂	1	2	yes	maximal, complemented
5	C₁₆×Dic₁₀₆	C₂	1	1	yes	maximal, characteristic, complemented
6	#20276617	C₂	1	1	yes	maximal, characteristic, complemented
7	#20276469	C₂	1	2	yes	maximal
8	#20276469	C₂	1	1	yes	maximal, characteristic
9	#20276308	C₂	1	2	yes	maximal
10	C₄×#13133156	C₂	1	1	yes	maximal, characteristic
11	C₅₃×#2183	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×Q₈×D₅₃	C₄	1	1	yes	characteristic
13	C₂×#13133156	C₂²	1	1	yes	characteristic
14	C₂×#13133156	C₂²	1	2	yes	—
15	C₂×#13133156	C₂²	1	1	yes	characteristic
16	Q₈×#88322	C₄	1	1	yes	characteristic
17	C₈×Dic₁₀₆	C₂²	1	1	yes	characteristic
18	C₈×Dic₁₀₆	C₂²	1	2	yes	—
19	#16810747	C₄	1	2	yes	—
20	#16810747	C₄	1	1	yes	characteristic
21	C₄⋊C₈×D₅₃	C₄	1	2	yes	—
22	C₄⋊C₈×D₅₃	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	2	yes	—
28	#16810441	C₂²	1	2	yes	—
29	#16810441	C₂²	1	2	yes	—
30	#16810399	C₂²	1	2	yes	—
31	#16810398	C₂²	1	1	yes	characteristic
32	#16810398	C₂²	1	1	yes	characteristic
33	C₄×#11719908	C₂²	1	1	yes	characteristic
34	C₁₆×Dic₅₃	C₂²	1	2	yes	—
35	C₁₆×Dic₅₃	C₂²	1	2	yes	—
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×Q₈×D₅₃	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	1	yes	characteristic
45	Q₈×Dic₅₃	C₈	1	1	yes	characteristic
46	C₄×Dic₁₀₆	C₂×C₄	1	2	yes	—
47	C₄×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
48	#13133182	C₈	1	2	yes	—
49	#13133182	C₈	1	1	yes	characteristic
50	C₄⋊C₄×D₅₃	C₈	1	1	yes	characteristic
51	C₄⋊C₄×D₅₃	C₈	1	2	yes	—
52	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
53	C₄²×D₅₃	C₂×C₄	1	2	yes	—
54	#13133156	—	2	2	no	—
55	#13133156	—	2	4	no	—
56	#13133156	Q₈	1	4	yes	complemented
57	C₂×#11719908	C₂³	1	2	yes	—
58	C₂×#11719908	C₂³	1	1	yes	characteristic
59	C₂×#11719908	C₂³	1	1	yes	characteristic
60	C₄⋊#88322	C₂×C₄	1	1	yes	characteristic
61	C₄⋊#88322	C₂×C₄	1	2	yes	—
62	#13133146	C₂×C₄	1	2	yes	—
63	#13133146	C₂×C₄	1	2	yes	—
64	#13133146	C₂×C₄	1	2	yes	—
65	C₄×#88322	C₂×C₄	1	1	yes	characteristic
66	C₄×#88322	C₂×C₄	1	2	yes	—
67	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
68	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
69	C₈×Dic₅₃	C₂³	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	Q₈×D₅₃	—	2	4	no	—
81	C₂×C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
82	C₂×C₄×D₅₃	C₂×C₈	1	2	yes	—
83	C₂×C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
84	C₂×C₄×D₅₃	C₂×C₈	1	2	yes	—
85	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
86	C₂×Dic₁₀₆	C₂×C₈	1	2	yes	—
87	C₂×Dic₁₀₆	C₂×C₈	1	1	yes	characteristic
88	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
90	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
91	C₄⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
92	C₄⋊C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
93	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
94	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
95	C₄⋊Dic₅₃	C₂×C₈	1	2	yes	—
96	C₄⋊Dic₅₃	C₂×C₈	1	1	yes	characteristic
97	#11719919	C₂×C₈	1	2	yes	—
98	#11719919	C₂×C₈	1	2	yes	—
99	#11719919	C₂×C₈	1	2	yes	—
100	C₄×Dic₅₃	C₂×C₈	1	1	yes	characteristic
101	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
102	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
103	C₄×Dic₅₃	C₂×C₈	1	1	yes	characteristic
104	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
105	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
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	4	no	—
111	C₈×D₅₃	C₂×Q₈	1	2	yes	—
112	C₈×D₅₃	#41	1	2	yes	—
113	C₁₆×C₅₃	—	2	1	no	—
114	C₁₆×C₅₃	—	2	2	no	—
115	C₁₆×C₅₃	C₂×Q₈	1	2	yes	complemented
116	#11719908	—	2	2	no	—
117	#11719908	—	2	1	no	—
118	#11719908	C₂×Q₈	1	2	yes	complemented
119	C₂²×D₅₃	C₂²×C₈	1	1	yes	characteristic
120	Q₈×C₅₃	—	2	2	no	—
121	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
122	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
123	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
124	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
125	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
126	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
127	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
128	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
129	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
130	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
131	C₄×D₅₃	—	2	2	no	—
132	C₄×D₅₃	—	2	4	no	—
133	C₄×D₅₃	—	4	1	no	—
134	C₄×D₅₃	—	4	2	no	—
135	C₄×D₅₃	C₄×Q₈	1	2	yes	—
136	C₄×D₅₃	#131	1	2	yes	—
137	Dic₁₀₆	—	2	4	no	—
138	Dic₁₀₆	M₃₂	1	4	yes	complemented
139	C₈×C₅₃	—	2	1	no	—
140	C₈×C₅₃	—	2	2	no	—
141	C₈×C₅₃	C₂²×Q₈	1	1	yes	characteristic
142	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
143	#88322	—	2	2	no	—
144	#88322	—	2	1	no	—
145	#88322	C₄×Q₈	1	2	yes	—
146	Q₈×M₃₂	—	53	1	no	maximal, Sylow
147	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
148	C₂×D₅₃	—	2	1	no	—
149	C₂×D₅₃	—	2	1	no	—
150	C₂×D₅₃	C₈×Q₈	1	2	yes	—
151	C₄×C₅₃	—	2	2	no	—
152	C₄×C₅₃	—	2	1	no	—
153	C₄×C₅₃	—	2	2	no	—
154	C₄×C₅₃	C₂×M₃₂	1	2	yes	—
155	C₄×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
156	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
157	Dic₅₃	—	2	1	no	—
158	Dic₅₃	—	2	2	no	—
159	Dic₅₃	—	2	1	no	—
160	Dic₅₃	C₈×Q₈	1	2	yes	—
161	Dic₅₃	C₂×M₃₂	1	4	yes	—
162	C₂×C₈×Q₈	—	53	1	no	—
163	#2183	—	53	1	no	—
164	#2183	—	53	2	no	—
165	#2183	—	53	2	no	—
166	#2183	D₅₃	1	1	yes	characteristic, complemented
167	Q₈×C₁₆	—	53	1	no	—
168	Q₈×C₁₆	—	53	1	no	—
169	#2150	—	53	2	no	—
170	#2150	—	53	1	no	—
171	C₄×M₃₂	—	53	2	no	—
172	C₄×M₃₂	—	53	1	no	—
173	C₂×C₅₃	C₂×C₈×Q₈	1	1	yes	characteristic
174	C₂×C₅₃	C₂²×M₃₂	1	1	yes	characteristic
175	C₂×C₅₃	#3407	1	1	yes	characteristic
176	D₅₃	—	2	2	no	—
177	C₂×C₄×Q₈	—	53	1	no	—
178	C₂×M₃₂	—	53	1	no	—
179	C₂×M₃₂	—	53	2	no	—
180	C₂×M₃₂	—	53	1	no	—
181	C₈×Q₈	—	53	1	no	—
182	C₈×Q₈	—	53	2	no	—
183	C₈×Q₈	—	53	1	no	—
184	C₈×Q₈	—	53	1	no	—
185	C₈×Q₈	—	53	2	no	—
186	C₈×Q₈	C₂×D₅₃	1	1	yes	characteristic
187	C₂×C₄⋊C₈	—	53	2	no	—
188	C₂×C₄⋊C₈	—	53	1	no	—
189	C₂×C₄×C₈	—	53	2	no	—
190	C₂×C₄×C₈	—	53	1	no	—
191	C₄⋊C₁₆	—	53	2	no	—
192	C₄⋊C₁₆	—	53	2	no	—
193	C₄⋊C₁₆	—	53	2	no	—
194	C₄⋊C₁₆	—	53	1	no	—
195	C₄⋊C₁₆	—	53	2	no	—
196	C₄⋊C₁₆	C₂×D₅₃	1	1	yes	characteristic
197	C₄⋊C₁₆	C₂×D₅₃	1	2	yes	—
198	#346	—	53	1	no	—
199	#346	—	53	1	no	—
200	#346	—	53	2	no	—
201	#346	C₂×D₅₃	1	2	yes	—
202	C₄×C₁₆	—	53	2	no	—
203	C₄×C₁₆	—	53	2	no	—
204	C₄×C₁₆	—	53	1	no	—
205	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic
206	C₅₃	Q₈×M₃₂	1	1	yes	characteristic, complemented, Sylow
207	C₂²×Q₈	—	53	1	no	—
208	C₂²×C₈	—	53	1	no	—
209	C₂²×C₈	—	53	2	no	—
210	C₂²×C₈	—	53	1	no	—
211	C₄×Q₈	—	53	1	no	—
212	C₄×Q₈	—	53	1	no	—
213	C₄×Q₈	—	53	2	no	—
214	C₄×Q₈	—	53	1	no	—
215	C₄×Q₈	—	53	2	no	—
216	C₄×Q₈	C₄×D₅₃	1	1	yes	characteristic
217	C₂×C₄⋊C₄	—	53	2	no	—
218	C₂×C₄⋊C₄	—	53	1	no	—
219	C₂×C₄²	—	53	2	no	—
220	C₂×C₄²	—	53	1	no	—
221	M₃₂	—	53	4	no	—
222	M₃₂	—	106	4	no	—
223	M₃₂	—	106	2	no	—
224	C₂×C₁₆	—	53	1	no	—
225	C₂×C₁₆	—	53	2	no	—
226	C₂×C₁₆	—	53	1	no	—
227	C₂×C₁₆	C₂²×D₅₃	1	2	yes	—
228	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
229	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
230	C₄⋊C₈	—	53	2	no	—
231	C₄⋊C₈	—	53	2	no	—
232	C₄⋊C₈	—	53	2	no	—
233	C₄⋊C₈	—	53	1	no	—
234	C₄⋊C₈	—	53	2	no	—
235	C₄⋊C₈	C₄×D₅₃	1	2	yes	—
236	C₄⋊C₈	C₄×D₅₃	1	1	yes	characteristic
237	C₄×C₈	—	53	2	no	—
238	C₄×C₈	—	53	1	no	—
239	C₄×C₈	—	53	2	no	—
240	C₄×C₈	—	53	1	no	—
241	C₄×C₈	—	53	2	no	—
242	C₄×C₈	—	53	1	no	—
243	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
244	C₄×C₈	C₂²×D₅₃	1	2	yes	—
245	C₂×Q₈	—	53	1	no	—
246	C₂×Q₈	—	53	2	no	—
247	C₂×Q₈	—	106	4	no	—
248	C₂×Q₈	C₈×D₅₃	1	1	yes	characteristic
249	C₂²×C₄	—	53	1	no	—
250	C₂²×C₄	—	53	2	no	—
251	C₂²×C₄	—	53	1	no	—
252	C₂²×C₄	—	53	2	no	—
253	C₂²×C₄	—	53	1	no	—
254	C₂×C₈	—	53	1	no	—
255	C₂×C₈	—	53	2	no	—
256	C₂×C₈	—	53	2	no	—
257	C₂×C₈	—	53	2	no	—
258	C₂×C₈	—	53	1	no	—
259	C₂×C₈	—	106	2	no	—
260	C₂×C₈	—	106	4	no	—
261	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
262	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
263	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
264	C₄⋊C₄	—	53	2	no	—
265	C₄⋊C₄	—	53	2	no	—
266	C₄⋊C₄	—	53	2	no	—
267	C₄⋊C₄	—	53	2	no	—
268	C₄⋊C₄	—	53	1	no	—
269	C₄⋊C₄	C₈×D₅₃	1	2	yes	—
270	C₄⋊C₄	C₈×D₅₃	1	1	yes	characteristic
271	C₄²	—	53	2	no	—
272	C₄²	—	53	2	no	—
273	C₄²	—	53	1	no	—
274	C₄²	—	53	1	no	—
275	C₄²	—	53	2	no	—
276	C₄²	—	53	1	no	—
277	C₄²	C₂×C₄×D₅₃	1	2	yes	—
278	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
279	C₁₆	—	2	2	no	—
280	C₁₆	—	2	1	no	—
281	C₁₆	—	53	2	no	—
282	C₁₆	—	106	2	no	—
283	C₁₆	—	106	1	no	—
284	C₁₆	Q₈×D₅₃	1	2	yes	complemented
285	C₂³	—	53	1	no	—
286	Q₈	—	2	2	no	—
287	Q₈	—	53	4	no	—
288	Q₈	—	106	4	no	—
289	C₂×C₄	—	53	1	no	—
290	C₂×C₄	—	53	2	no	—
291	C₂×C₄	—	53	2	no	—
292	C₂×C₄	—	53	2	no	—
293	C₂×C₄	—	53	2	no	—
294	C₂×C₄	—	53	1	no	—
295	C₂×C₄	—	53	1	no	—
296	C₂×C₄	—	106	4	no	—
297	C₂×C₄	—	106	2	no	—
298	C₂×C₄	—	212	2	no	—
299	C₂×C₄	—	212	1	no	—
300	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
301	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
302	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
303	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
304	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
305	C₈	—	2	1	no	—
306	C₈	—	2	2	no	—
307	C₈	—	53	2	no	—
308	C₈	—	106	2	no	—
309	C₈	—	106	1	no	—
310	C₈	C₂×Q₈×D₅₃	1	1	yes	characteristic
311	C₈	#41×D₅₃	1	1	yes	characteristic
312	C₂²	—	53	2	no	—
313	C₂²	—	106	1	no	—
314	C₂²	—	106	1	no	—
315	C₂²	C₂²×C₈×D₅₃	1	1	yes	characteristic
316	C₄	—	2	2	no	—
317	C₄	—	2	2	no	—
318	C₄	—	2	1	no	—
319	C₄	—	53	4	no	—
320	C₄	—	53	2	no	—
321	C₄	—	106	1	no	—
322	C₄	—	106	2	no	—
323	C₄	—	106	1	no	—
324	C₄	M₃₂×D₅₃	1	2	yes	—
325	C₄	C₄×Q₈×D₅₃	1	1	yes	characteristic
326	C₄	#131×D₅₃	1	1	yes	characteristic
327	C₂	—	106	2	no	—
328	C₂	C₈×Q₈×D₅₃	1	1	yes	characteristic
329	C₂	C₂×M₃₂×D₅₃	1	1	yes	characteristic
330	C₂	#20287816	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	C₂²×D₅₃	C₂²×C₈
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	#2183	D₅₃
hypocenter	C₅₃	Q₈×M₃₂
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. #20276622	C₂	semidirect
4. #20276621	C₂	semidirect
5. C₁₆×Dic₁₀₆	C₂	semidirect
6. #20276617	C₂	semidirect
11. C₅₃×#2183	C₂	semidirect
56. #13133156	Q₈	semidirect
115. C₁₆×C₅₃	C₂×Q₈	semidirect
118. #11719908	C₂×Q₈	semidirect
138. Dic₁₀₆	M₃₂	semidirect
166. #2183	D₅₃	semidirect
206. C₅₃	Q₈×M₃₂	semidirect
284. C₁₆	Q₈×D₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₈×Q₈×D₅₃	2
#20276624	2
#20276622	2
#20276621	2
C₁₆×Dic₁₀₆	2
#20276617	2
#20276469	2
#20276308	2
C₄×#13133156	2
C₅₃×#2183	2
Q₈×M₃₂	53