Subgroups — #27289899

Order 13568.

762 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810647	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#11719917	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810381	C₂	1	4	yes	maximal
4	C₂×C₅₃×#423	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11719917	C₂²	1	2	yes	—
6	C₂²×#11719917	C₂²	1	2	yes	—
7	#16810647	—	2	1	no	—
8	#16810647	—	2	1	no	—
9	#16810647	—	2	2	no	—
10	C₂×C₄⋊#88322	C₄	1	4	yes	—
11	C₄×#11719917	—	2	1	no	—
12	C₄×#11719917	—	2	1	no	—
13	C₄×#11719917	—	2	1	no	—
14	C₄×#11719917	—	2	1	no	—
15	C₂×#13133128	C₄	1	2	yes	—
16	C₂×C₄×#88322	C₄	1	2	yes	—
17	#16810381	—	2	8	no	—
18	C₂²×M₁₆×C₅₃	C₂²	1	2	yes	—
19	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
20	C₅₃×#423	—	2	1	no	—
21	C₅₃×#423	—	2	1	no	—
22	C₅₃×#423	—	2	2	no	—
23	C₂×C₄⋊C₈×C₅₃	C₄	1	4	yes	—
24	C₂×#11719917	—	2	2	no	—
25	C₂×#11719917	—	2	2	no	—
26	C₂×#11719917	—	2	4	no	—
27	C₂×#11719917	—	2	2	no	—
28	C₂×#11719917	—	2	2	no	—
29	C₂×#11719917	—	2	2	no	—
30	C₂×#11719917	—	2	2	no	—
31	C₂×#11719917	—	2	4	no	—
32	C₂×#11719917	D₄	1	4	yes	—
33	C₂×#11719917	Q₈	1	4	yes	—
34	C₂²×#88322	C₂×C₄	1	4	yes	—
35	C₂²×#88322	C₂×C₄	1	4	yes	—
36	C₄⋊#88322	—	2	4	no	—
37	C₄⋊#88322	—	2	4	no	—
38	#13133128	—	2	2	no	—
39	#13133128	—	2	2	no	—
40	C₄×#88322	—	2	2	no	—
41	C₄×#88322	—	2	2	no	—
42	C₂³×C₄×C₅₃	Q₈	1	2	yes	—
43	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
44	C₂×M₁₆×C₅₃	—	2	4	no	—
45	C₂×M₁₆×C₅₃	—	2	2	no	—
46	C₂×M₁₆×C₅₃	—	2	2	no	—
47	C₂×M₁₆×C₅₃	—	2	4	no	—
48	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
49	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
52	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
53	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
54	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
55	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
56	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
57	C₂×C₄²×C₅₃	D₄	1	2	yes	—
58	C₄⋊C₈×C₅₃	—	2	4	no	—
59	C₄⋊C₈×C₅₃	—	2	4	no	—
60	C₂⁴×C₅₃	C₂×Q₈	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	—	4	4	no	—
62	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
63	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
64	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
65	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
66	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
67	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
68	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
75	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
76	C₂²×C₄×C₅₃	#41	1	2	yes	—
77	C₂²×C₄×C₅₃	#41	1	2	yes	—
78	M₁₆×C₅₃	—	4	4	no	—
79	M₁₆×C₅₃	—	4	2	no	—
80	M₁₆×C₅₃	—	4	2	no	—
81	C₂×C₈×C₅₃	—	2	4	no	—
82	C₂×C₈×C₅₃	—	2	8	no	—
83	C₄²×C₅₃	C₄²	1	2	yes	—
84	C₄²×C₅₃	C₄²	1	2	yes	—
85	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
86	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
87	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
88	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
89	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
90	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
91	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
92	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
93	#11719917	—	2	4	no	—
94	#11719917	—	2	4	no	—
95	#11719917	—	2	4	no	—
96	#11719917	—	2	4	no	—
97	#11719917	—	4	4	no	—
98	#11719917	—	4	2	no	—
99	#11719917	—	4	2	no	—
100	C₂×#88322	—	2	4	no	—
101	C₂×#88322	—	2	4	no	—
102	C₂×#88322	—	2	4	no	—
103	C₂×#88322	—	2	8	no	—
104	C₂×#88322	C₄⋊C₄	1	8	yes	—
105	C₂³×C₅₃	—	4	2	no	—
106	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
108	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₂³×C₅₃	#125	1	2	yes	—
110	C₂³×C₅₃	C₄⋊Q₈	1	2	yes	—
111	C₂×C₄×C₅₃	—	2	4	no	—
112	C₂×C₄×C₅₃	—	2	4	no	—
113	C₂×C₄×C₅₃	—	2	4	no	—
114	C₂×C₄×C₅₃	—	4	2	no	—
115	C₂×C₄×C₅₃	—	4	2	no	—
116	C₂×C₄×C₅₃	—	4	2	no	—
117	C₂×C₄×C₅₃	#95	1	2	yes	—
118	C₂×C₄×C₅₃	#95	1	2	yes	—
119	C₂×C₄×C₅₃	#95	1	2	yes	—
120	C₂×C₄×C₅₃	#95	1	2	yes	—
121	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
122	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
126	C₂×C₄×C₅₃	#117	1	2	yes	—
127	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
128	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
129	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
130	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
131	C₂×C₄×C₅₃	#121	1	2	yes	—
132	C₂×C₄×C₅₃	#121	1	2	yes	—
133	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
134	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
135	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
136	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
137	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
138	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
139	C₈×C₅₃	—	4	4	no	—
140	#88322	—	2	8	no	—
141	#88322	—	4	4	no	—
142	C₄⋊#338	—	53	1	no	maximal, Sylow
143	C₂²×C₅₃	—	2	2	no	—
144	C₂²×C₅₃	—	2	2	no	—
145	C₂²×C₅₃	—	2	2	no	—
146	C₂²×C₅₃	—	4	2	no	—
147	C₂²×C₅₃	—	4	2	no	—
148	C₂²×C₅₃	C₂×#95	1	1	yes	characteristic
149	C₂²×C₅₃	#382	1	1	yes	characteristic
150	C₂²×C₅₃	#384	1	1	yes	characteristic
151	C₂²×C₅₃	#389	1	2	yes	—
152	C₂²×C₅₃	#389	1	2	yes	—
153	C₄×C₅₃	—	2	2	no	—
154	C₄×C₅₃	—	2	2	no	—
155	C₄×C₅₃	—	2	2	no	—
156	C₄×C₅₃	#338	1	4	yes	complemented
157	C₄×C₅₃	C₄×C₄⋊C₄	1	2	yes	—
158	C₄×C₅₃	#380	1	2	yes	—
159	C₄×C₅₃	#390	1	1	yes	characteristic
160	C₄×C₅₃	#390	1	1	yes	characteristic
161	C₄×C₅₃	#391	1	1	yes	characteristic
162	C₄×C₅₃	#391	1	1	yes	characteristic
163	C₂×#423	—	53	1	no	—
164	C₂×#423	D₅₃	1	1	yes	characteristic
165	C₂×C₄×M₁₆	—	53	1	no	—
166	C₂×#338	—	53	4	no	—
167	C₂×C₅₃	—	2	1	no	—
168	C₂×C₅₃	—	2	1	no	—
169	C₂×C₅₃	—	2	2	no	—
170	C₂×C₅₃	—	2	2	no	—
171	C₂×C₅₃	C₄⋊#95	1	1	yes	characteristic
172	C₂×C₅₃	C₂×#338	1	2	yes	—
173	C₂²×M₁₆	—	53	2	no	—
174	C₂²×M₁₆	—	53	2	no	—
175	C₂²×M₁₆	C₂×D₅₃	1	2	yes	—
176	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
177	#423	—	2	1	no	—
178	#423	—	2	1	no	—
179	#423	—	2	2	no	—
180	#423	—	106	1	no	—
181	#423	—	106	1	no	—
182	#423	—	106	2	no	—
183	C₂×C₄⋊C₈	—	53	4	no	—
184	C₂×C₄⋊C₈	Dic₅₃	1	4	yes	—
185	C₄×M₁₆	—	106	1	no	—
186	C₄×M₁₆	—	106	1	no	—
187	C₄×M₁₆	—	106	1	no	—
188	C₄×M₁₆	—	106	1	no	—
189	C₂×#97	—	53	2	no	—
190	C₂×C₄×C₈	—	53	2	no	—
191	#338	—	106	8	no	—
192	C₅₃	C₄⋊#338	1	1	yes	characteristic, complemented, Sylow
193	C₂³×C₄	Dic₁₀₆	1	2	yes	—
194	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
195	C₂×M₁₆	—	2	4	no	—
196	C₂×M₁₆	—	2	4	no	—
197	C₂×M₁₆	—	2	2	no	—
198	C₂×M₁₆	—	2	2	no	—
199	C₂×M₁₆	—	53	4	no	—
200	C₂×M₁₆	—	53	4	no	—
201	C₂×M₁₆	—	106	2	no	—
202	C₂×M₁₆	—	106	2	no	—
203	C₂×M₁₆	—	106	2	no	—
204	C₂×M₁₆	—	106	2	no	—
205	C₂×M₁₆	—	106	2	no	—
206	C₂×M₁₆	—	106	4	no	—
207	C₂×M₁₆	—	106	4	no	—
208	C₂×M₁₆	—	106	2	no	—
209	C₂²×C₈	—	53	4	no	—
210	C₂²×C₈	—	53	4	no	—
211	C₂²×C₈	C₂×Dic₅₃	1	4	yes	—
212	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
213	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
214	C₂×C₄²	C₄×D₅₃	1	2	yes	—
215	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
216	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
217	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
218	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
219	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
220	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
221	C₄⋊C₈	—	2	4	no	—
222	C₄⋊C₈	—	2	4	no	—
223	C₄⋊C₈	—	106	4	no	—
224	C₄⋊C₈	—	106	4	no	—
225	#97	—	106	2	no	—
226	#97	—	106	2	no	—
227	C₄×C₈	—	106	2	no	—
228	C₄×C₈	—	106	2	no	—
229	C₂⁴	C₂×Dic₁₀₆	1	1	yes	characteristic
230	C₂²×C₄	—	4	4	no	—
231	C₂²×C₄	#11719919	1	2	yes	—
232	C₂²×C₄	#11719919	1	2	yes	—
233	C₂²×C₄	#11719919	1	2	yes	—
234	C₂²×C₄	#11719919	1	2	yes	—
235	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
236	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
237	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
238	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
239	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
240	C₂²×C₄	D₄×D₅₃	1	2	yes	—
241	C₂²×C₄	#11719947	1	2	yes	—
242	C₂²×C₄	Q₈×D₅₃	1	2	yes	—
243	C₂²×C₄	#11719949	1	2	yes	—
244	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
245	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
246	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
247	M₁₆	—	4	4	no	—
248	M₁₆	—	4	2	no	—
249	M₁₆	—	4	2	no	—
250	M₁₆	—	106	4	no	—
251	M₁₆	—	106	4	no	—
252	M₁₆	—	106	4	no	—
253	M₁₆	—	106	4	no	—
254	M₁₆	—	212	4	no	—
255	M₁₆	—	212	2	no	—
256	M₁₆	—	212	2	no	—
257	C₂×C₈	—	2	4	no	—
258	C₂×C₈	—	2	8	no	—
259	C₂×C₈	—	53	8	no	—
260	C₂×C₈	—	106	4	no	—
261	C₂×C₈	—	106	4	no	—
262	C₂×C₈	—	106	4	no	—
263	C₂×C₈	—	106	8	no	—
264	C₄²	C₄×Dic₅₃	1	2	yes	—
265	C₄²	C₄×Dic₅₃	1	2	yes	—
266	C₄²	#11719921	1	1	yes	characteristic
267	C₄²	#11719921	1	1	yes	characteristic
268	C₄²	#11719921	1	1	yes	characteristic
269	C₄²	#11719921	1	1	yes	characteristic
270	C₄²	#11719921	1	2	yes	—
271	C₄²	#11719921	1	2	yes	—
272	C₄²	C₂²⋊Dic₅₃	1	2	yes	—
273	C₄²	C₂²⋊Dic₅₃	1	2	yes	—
274	C₂³	—	4	2	no	—
275	C₂³	C₂×#11719919	1	1	yes	characteristic
276	C₂³	C₂×#11719919	1	1	yes	characteristic
277	C₂³	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
278	C₂³	#13133208	1	2	yes	—
279	C₂³	#13133216	1	2	yes	—
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	2	4	no	—
282	C₂×C₄	—	2	4	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	#13133124	1	2	yes	—
287	C₂×C₄	#13133124	1	2	yes	—
288	C₂×C₄	#13133124	1	2	yes	—
289	C₂×C₄	#13133124	1	2	yes	—
290	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
291	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
292	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
293	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
294	C₂×C₄	C₄⋊C₄×D₅₃	1	2	yes	—
295	C₂×C₄	#13133175	1	2	yes	—
296	C₂×C₄	#13133179	1	2	yes	—
297	C₂×C₄	D₄×Dic₅₃	1	2	yes	—
298	C₂×C₄	#13133182	1	2	yes	—
299	C₂×C₄	Q₈×Dic₅₃	1	2	yes	—
300	C₂×C₄	#13133188	1	2	yes	—
301	C₂×C₄	#13133192	1	2	yes	—
302	C₂×C₄	#13133193	1	2	yes	—
303	C₂×C₄	#13133197	1	2	yes	—
304	C₂×C₄	#13133204	1	1	yes	characteristic
305	C₂×C₄	#13133206	1	1	yes	characteristic
306	C₂×C₄	#13133213	1	1	yes	characteristic
307	C₂×C₄	#13133215	1	1	yes	characteristic
308	C₈	—	4	4	no	—
309	C₈	—	106	8	no	—
310	C₈	—	212	4	no	—
311	C₂²	—	2	2	no	—
312	C₂²	—	2	2	no	—
313	C₂²	—	2	2	no	—
314	C₂²	—	4	2	no	—
315	C₂²	—	4	2	no	—
316	C₂²	C₂×#13133124	1	1	yes	characteristic
317	C₂²	#16810488	1	1	yes	characteristic
318	C₂²	#16810492	1	1	yes	characteristic
319	C₂²	#16810519	1	2	yes	—
320	C₂²	#16810522	1	2	yes	—
321	C₄	—	2	2	no	—
322	C₄	—	2	2	no	—
323	C₄	—	2	2	no	—
324	C₄	#16810381	1	4	yes	complemented
325	C₄	C₄⋊C₄×Dic₅₃	1	2	yes	—
326	C₄	#16810480	1	2	yes	—
327	C₄	#16810524	1	1	yes	characteristic
328	C₄	#16810526	1	1	yes	characteristic
329	C₄	#16810528	1	1	yes	characteristic
330	C₄	#16810530	1	1	yes	characteristic
331	C₂	—	2	2	no	—
332	C₂	—	2	2	no	—
333	C₂	—	2	1	no	—
334	C₂	—	2	1	no	—
335	C₂	#20272944	1	1	yes	characteristic
336	C₂	C₂×#16810381	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₄×Dic₅₃
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₂×C₅₃×#423	C₂
socle	C₂²×C₅₃	C₂×#95
prime radical	C₂⁴×C₅₃	C₂×Q₈
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	C₂×#423	D₅₃
hypocenter	C₅₃	C₄⋊#338
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#423

Complements

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

N	H ≅ G/N	type
156. C₄×C₅₃	#338	semidirect
192. C₅₃	C₄⋊#338	semidirect
324. C₄	#16810381	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810647	2
C₂×C₄×#11719917	2
C₂×#16810381	2
C₂×C₅₃×#423	2
C₄⋊#338	53