Subgroups — #27537628

Order 13568.

526 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	#20276624	C₂	1	1	yes	maximal, characteristic, complemented
2	Q₈×#11719908	C₂	1	1	yes	maximal, characteristic, complemented
3	#20276547	C₂	1	1	yes	maximal, characteristic, complemented
4	#20276541	C₂	1	2	yes	maximal, complemented
5	D₄×#11719908	C₂	1	2	yes	maximal, complemented
6	D₄×#11719908	C₂	1	1	yes	maximal, characteristic, complemented
7	#20276522	C₂	1	2	yes	maximal, complemented
8	#20276518	C₂	1	1	yes	maximal, characteristic, complemented
9	#20276312	C₂	1	2	yes	maximal, complemented
10	C₂×C₄×#11719908	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#2966	C₂	1	1	yes	maximal, characteristic
12	C₂×#13133157	C₂²	1	2	yes	—
13	C₂²×#11719908	C₂²	1	1	yes	characteristic
14	C₂²×#11719908	C₂²	1	2	yes	—
15	C₂²×#11719908	C₂²	1	1	yes	characteristic
16	C₄⋊#11719908	C₂²	1	1	yes	characteristic
17	C₄⋊#11719908	C₂²	1	1	yes	characteristic
18	C₄⋊#11719908	C₂²	1	2	yes	—
19	C₄⋊#11719908	C₂²	1	2	yes	—
20	C₂²⋊#11719908	C₂²	1	2	yes	—
21	C₂²⋊#11719908	C₂²	1	2	yes	—
22	C₂²⋊#11719908	C₂²	1	2	yes	—
23	#16810399	C₂²	1	2	yes	—
24	#16810399	C₂²	1	2	yes	—
25	C₄×#11719908	C₂²	1	1	yes	characteristic
26	C₄×#11719908	C₂²	1	1	yes	characteristic
27	C₄×#11719908	C₂²	1	2	yes	—
28	C₄×#11719908	C₂²	1	1	yes	characteristic
29	C₄×#11719908	C₂²	1	1	yes	characteristic
30	C₅₃×#529	C₄	1	1	yes	characteristic
31	C₅₃×#446	C₄	1	1	yes	characteristic
32	C₈×Q₈×C₅₃	C₂²	1	1	yes	characteristic
33	C₅₃×#436	C₄	1	1	yes	characteristic
34	C₅₃×#435	C₄	1	2	yes	—
35	C₈×D₄×C₅₃	C₂²	1	2	yes	—
36	C₈×D₄×C₅₃	C₂²	1	1	yes	characteristic
37	C₅₃×#433	C₄	1	2	yes	—
38	C₅₃×#432	C₄	1	1	yes	characteristic
39	C₅₃×#405	C₂²	1	2	yes	—
40	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
41	#13133157	—	2	4	no	—
42	C₂×#11719908	—	2	2	no	—
43	C₂×#11719908	—	2	4	no	—
44	C₂×#11719908	—	2	2	no	—
45	C₂×#11719908	C₂³	1	1	yes	characteristic
46	C₂×#11719908	C₂³	1	1	yes	characteristic
47	C₂×#11719908	C₂³	1	2	yes	—
48	C₂×#11719908	C₂³	1	2	yes	—
49	C₂×#11719908	C₂³	1	1	yes	characteristic
50	C₂×#11719908	C₂³	1	1	yes	characteristic
51	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
52	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
53	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
54	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
55	#126×C₅₃	C₈	1	2	yes	—
56	#125×C₅₃	C₈	1	1	yes	characteristic
57	#124×C₅₃	C₈	1	1	yes	characteristic
58	#123×C₅₃	C₈	1	2	yes	—
59	C₂²⋊Q₈×C₅₃	C₈	1	1	yes	characteristic
60	#121×C₅₃	C₈	1	1	yes	characteristic
61	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
62	C₄×D₄×C₅₃	C₂×C₄	1	2	yes	—
63	C₄×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
64	#117×C₅₃	C₂×C₄	1	2	yes	—
65	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
66	C₄⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
67	C₄⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
68	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
69	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
70	C₂²⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
71	C₂²⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
72	C₂²⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
73	#97×C₅₃	C₂×C₄	1	2	yes	—
74	#97×C₅₃	C₂×C₄	1	2	yes	—
75	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
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	1	yes	characteristic
79	C₄×C₈×C₅₃	C₂³	1	2	yes	—
80	C₂×Q₈×C₅₃	C₂×C₈	1	1	yes	characteristic
81	C₂×D₄×C₅₃	C₂×C₈	1	2	yes	—
82	C₂×D₄×C₅₃	C₂×C₈	1	1	yes	characteristic
83	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
84	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
85	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
86	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
87	M₁₆×C₅₃	—	2	4	no	—
88	C₂×C₈×C₅₃	—	2	2	no	—
89	C₂×C₈×C₅₃	—	2	1	no	—
90	C₂×C₈×C₅₃	—	2	1	no	—
91	C₂×C₈×C₅₃	—	2	2	no	—
92	C₂×C₈×C₅₃	—	2	2	no	—
93	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
96	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
97	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
99	C₄⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
100	C₄⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
101	C₄⋊C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
102	C₄⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
103	C₄⋊C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
104	C₄⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
105	C₂²⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
106	C₂²⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
107	C₂²⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
108	C₂²⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
109	C₂²⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
110	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
111	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
112	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
113	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
114	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
115	#11719908	—	2	1	no	—
116	#11719908	—	2	1	no	—
117	#11719908	—	2	2	no	—
118	#11719908	—	2	2	no	—
119	#11719908	#41	1	2	yes	—
120	#11719908	#41	1	2	yes	—
121	C₂³×C₅₃	C₂²×C₈	1	2	yes	—
122	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
123	Q₈×C₅₃	—	2	2	no	—
124	D₄×C₅₃	—	2	4	no	—
125	D₄×C₅₃	—	2	2	no	—
126	C₂×C₄×C₅₃	—	2	2	no	—
127	C₂×C₄×C₅₃	—	2	1	no	—
128	C₂×C₄×C₅₃	—	2	2	no	—
129	C₂×C₄×C₅₃	—	2	1	no	—
130	C₂×C₄×C₅₃	—	2	2	no	—
131	C₂×C₄×C₅₃	—	2	2	no	—
132	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
133	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
134	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
135	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
136	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
137	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
138	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
139	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
140	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
141	C₈×C₅₃	—	2	1	no	—
142	C₈×C₅₃	—	2	2	no	—
143	C₈×C₅₃	—	2	1	no	—
144	C₈×C₅₃	—	2	2	no	—
145	C₈×C₅₃	#131	1	2	yes	—
146	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
147	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
148	#33480	—	53	1	no	maximal, Sylow
149	C₂²×C₅₃	—	2	1	no	—
150	C₂²×C₅₃	—	2	1	no	—
151	C₂²×C₅₃	—	2	2	no	—
152	C₂²×C₅₃	—	2	2	no	—
153	C₂²×C₅₃	—	2	2	no	—
154	C₂²×C₅₃	#504	1	2	yes	—
155	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
156	C₄×C₅₃	—	2	1	no	—
157	C₄×C₅₃	—	2	1	no	—
158	C₄×C₅₃	—	2	1	no	—
159	C₄×C₅₃	—	2	2	no	—
160	C₄×C₅₃	—	2	2	no	—
161	C₄×C₅₃	—	2	1	no	—
162	C₄×C₅₃	—	2	2	no	—
163	C₄×C₅₃	#504	1	2	yes	—
164	C₄×C₅₃	C₄×#41	1	1	yes	characteristic
165	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
166	#2966	D₅₃	1	1	yes	characteristic
167	#2183	—	53	1	no	—
168	Q₈×C₁₆	—	53	1	no	—
169	#2169	—	53	1	no	—
170	#2168	—	53	2	no	—
171	D₄×C₁₆	—	53	1	no	—
172	D₄×C₁₆	—	53	2	no	—
173	#2164	—	53	2	no	—
174	#2163	—	53	1	no	—
175	#2108	—	53	2	no	—
176	C₂×C₄×C₁₆	—	53	1	no	—
177	C₂×C₅₃	—	2	2	no	—
178	C₂×C₅₃	—	4	2	no	—
179	C₂×C₅₃	C₈×#41	1	1	yes	characteristic
180	C₂×C₅₃	C₂×#504	1	1	yes	characteristic
181	C₂×C₅₃	C₂×#504	1	1	yes	characteristic
182	#529	Dic₅₃	1	1	yes	characteristic
183	C₂×M₃₂	—	53	2	no	—
184	C₂²×C₁₆	—	53	1	no	—
185	C₂²×C₁₆	—	53	1	no	—
186	C₂²×C₁₆	—	53	2	no	—
187	#446	Dic₅₃	1	1	yes	characteristic
188	C₈×Q₈	C₂×D₅₃	1	1	yes	characteristic
189	#436	Dic₅₃	1	1	yes	characteristic
190	#435	Dic₅₃	1	2	yes	—
191	C₈×D₄	C₂×D₅₃	1	2	yes	—
192	C₈×D₄	C₂×D₅₃	1	1	yes	characteristic
193	#433	Dic₅₃	1	2	yes	—
194	#432	Dic₅₃	1	1	yes	characteristic
195	#405	C₂×D₅₃	1	2	yes	—
196	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
197	C₄⋊C₁₆	—	53	1	no	—
198	C₄⋊C₁₆	—	53	1	no	—
199	C₄⋊C₁₆	—	53	2	no	—
200	C₄⋊C₁₆	—	53	2	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	—	53	2	no	—
206	C₄×C₁₆	—	53	1	no	—
207	C₄×C₁₆	—	53	1	no	—
208	C₄×C₁₆	—	53	1	no	—
209	C₄×C₁₆	—	53	1	no	—
210	C₄×C₁₆	—	53	2	no	—
211	C₅₃	#33480	1	1	yes	characteristic, complemented, Sylow
212	C₂×M₁₆	C₂×Dic₅₃	1	2	yes	—
213	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
214	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
215	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
216	#126	#88322	1	2	yes	—
217	#125	#88322	1	1	yes	characteristic
218	#124	#88322	1	1	yes	characteristic
219	#123	#88322	1	2	yes	—
220	C₂²⋊Q₈	#88322	1	1	yes	characteristic
221	#121	#88322	1	1	yes	characteristic
222	C₄×Q₈	C₂×Dic₅₃	1	1	yes	characteristic
223	C₄×D₄	C₂×Dic₅₃	1	2	yes	—
224	C₄×D₄	C₂×Dic₅₃	1	1	yes	characteristic
225	#117	C₂×Dic₅₃	1	2	yes	—
226	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
227	M₃₂	—	106	4	no	—
228	C₂×C₁₆	—	53	1	no	—
229	C₂×C₁₆	—	53	1	no	—
230	C₂×C₁₆	—	53	1	no	—
231	C₂×C₁₆	—	53	1	no	—
232	C₂×C₁₆	—	53	2	no	—
233	C₂×C₁₆	—	53	2	no	—
234	C₂×C₁₆	—	106	2	no	—
235	C₂×C₁₆	—	106	4	no	—
236	C₂×C₁₆	—	106	2	no	—
237	C₄⋊C₈	C₂×Dic₅₃	1	1	yes	characteristic
238	C₄⋊C₈	C₂×Dic₅₃	1	1	yes	characteristic
239	C₄⋊C₈	C₂×Dic₅₃	1	2	yes	—
240	C₄⋊C₈	C₂×Dic₅₃	1	2	yes	—
241	C₂²⋊C₈	C₂×Dic₅₃	1	2	yes	—
242	C₂²⋊C₈	C₂×Dic₅₃	1	2	yes	—
243	C₂²⋊C₈	C₂×Dic₅₃	1	2	yes	—
244	#97	C₂×Dic₅₃	1	2	yes	—
245	#97	C₂×Dic₅₃	1	2	yes	—
246	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
247	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
248	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
249	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
250	C₄×C₈	C₂²×D₅₃	1	2	yes	—
251	C₂×Q₈	C₂×#88322	1	1	yes	characteristic
252	C₂×D₄	C₂×#88322	1	2	yes	—
253	C₂×D₄	C₂×#88322	1	1	yes	characteristic
254	C₂²×C₄	C₂×#88322	1	1	yes	characteristic
255	C₂²×C₄	C₂×#88322	1	1	yes	characteristic
256	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
257	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
258	M₁₆	—	2	4	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	2	1	no	—
261	C₂×C₈	—	2	1	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
265	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
266	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
267	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
268	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
269	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
270	C₄⋊C₄	C₂×#88322	1	2	yes	—
271	C₄⋊C₄	C₂×#88322	1	2	yes	—
272	C₄⋊C₄	C₂×#88322	1	1	yes	characteristic
273	C₄⋊C₄	C₂×#88322	1	2	yes	—
274	C₄⋊C₄	C₂×#88322	1	1	yes	characteristic
275	C₄⋊C₄	C₂×#88322	1	2	yes	—
276	C₂²⋊C₄	C₂×#88322	1	2	yes	—
277	C₂²⋊C₄	C₂×#88322	1	2	yes	—
278	C₂²⋊C₄	C₂×#88322	1	2	yes	—
279	C₂²⋊C₄	C₂×#88322	1	2	yes	—
280	C₂²⋊C₄	C₂×#88322	1	2	yes	—
281	C₄²	C₂×#88322	1	1	yes	characteristic
282	C₄²	C₂×#88322	1	1	yes	characteristic
283	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
284	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
285	C₄²	C₂²×Dic₅₃	1	2	yes	—
286	C₁₆	—	53	2	no	—
287	C₁₆	—	53	2	no	—
288	C₁₆	—	106	1	no	—
289	C₁₆	—	106	1	no	—
290	C₁₆	—	106	2	no	—
291	C₁₆	—	106	2	no	—
292	C₂³	C₂²×#88322	1	2	yes	—
293	C₂³	C₂²×#88322	1	1	yes	characteristic
294	Q₈	—	2	2	no	—
295	D₄	—	2	4	no	—
296	D₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
304	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
305	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
306	C₂×C₄	C₂²×#88322	1	2	yes	—
307	C₂×C₄	C₂²×#88322	1	2	yes	—
308	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
309	C₂×C₄	C₂²×#88322	1	2	yes	—
310	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
311	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
312	C₈	—	2	1	no	—
313	C₈	—	2	2	no	—
314	C₈	—	2	1	no	—
315	C₈	—	2	2	no	—
316	C₈	#13133224	1	2	yes	—
317	C₈	#41×D₅₃	1	1	yes	characteristic
318	C₈	#41×D₅₃	1	1	yes	characteristic
319	C₂²	—	2	1	no	—
320	C₂²	—	2	1	no	—
321	C₂²	—	2	2	no	—
322	C₂²	—	2	2	no	—
323	C₂²	—	2	2	no	—
324	C₂²	#16811040	1	2	yes	—
325	C₂²	C₂³×#88322	1	1	yes	characteristic
326	C₄	—	2	1	no	—
327	C₄	—	2	2	no	—
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	2	1	no	—
331	C₄	—	2	2	no	—
332	C₄	—	2	2	no	—
333	C₄	#16811040	1	2	yes	—
334	C₄	#41×Dic₅₃	1	1	yes	characteristic
335	C₄	C₂×#13133224	1	1	yes	characteristic
336	C₂	—	2	2	no	—
337	C₂	—	4	2	no	—
338	C₂	#41×#88322	1	1	yes	characteristic
339	C₂	C₂×#16811040	1	1	yes	characteristic
340	C₂	C₂×#16811040	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₈
Fitting	C₅₃×#2966	C₂
prime radical	#121×C₅₃	C₈
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	#2966	D₅₃
hypocenter	C₅₃	#33480
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ #2966

Complements

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

N	H ≅ G/N	type
1. #20276624	C₂	semidirect
2. Q₈×#11719908	C₂	semidirect
3. #20276547	C₂	semidirect
4. #20276541	C₂	semidirect
5. D₄×#11719908	C₂	semidirect
6. D₄×#11719908	C₂	semidirect
7. #20276522	C₂	semidirect
8. #20276518	C₂	semidirect
9. #20276312	C₂	semidirect
10. C₂×C₄×#11719908	C₂	semidirect
211. C₅₃	#33480	semidirect

Maximal subgroups

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

subgroup	index
#20276624	2
Q₈×#11719908	2
#20276547	2
#20276541	2
D₄×#11719908	2
#20276522	2
#20276518	2
#20276312	2
C₂×C₄×#11719908	2
C₅₃×#2966	2
#33480	53