Subgroups — #27274182

Order 13568.

706 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊#88322	C₂	1	1	yes	maximal, characteristic
2	#20277122	C₂	1	1	yes	maximal, characteristic
3	#20272599	C₂	1	4	yes	maximal, complemented
4	C₂×C₅₃×#431	C₂	1	1	yes	maximal, characteristic
5	C₂³×#88322	—	2	1	no	—
6	C₂×#13133176	—	2	1	no	—
7	C₂×C₄⋊#88322	—	2	2	no	—
8	C₂×C₄⋊#88322	—	2	1	no	—
9	C₂×C₄⋊#88322	C₄	1	4	yes	complemented
10	C₂×C₄⋊#88322	C₂²	1	2	yes	—
11	#16810493	C₂²	1	2	yes	—
12	#16810489	—	2	1	no	—
13	#16810483	—	2	1	no	—
14	#16810483	C₄	1	2	yes	—
15	C₄×C₄⋊Dic₅₃	—	2	1	no	—
16	C₄×C₄⋊Dic₅₃	C₄	1	2	yes	—
17	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
18	C₅₃×#431	—	2	2	no	—
19	C₅₃×#431	C₄	1	4	yes	complemented
20	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
21	C₂×C₂²⋊C₈×C₅₃	—	2	1	no	—
22	C₂×C₄×C₈×C₅₃	—	2	1	no	—
23	C₂²×#88322	—	2	4	no	—
24	C₂²×#88322	—	2	2	no	—
25	C₂²×#88322	—	4	2	no	—
26	C₂²×#88322	—	4	2	no	—
27	#13133176	—	4	2	no	—
28	#13133176	—	4	2	no	—
29	C₂×C₄⋊Dic₅₃	—	2	2	no	—
30	C₂×C₄⋊Dic₅₃	—	2	2	no	—
31	C₂×C₄⋊Dic₅₃	C₂×C₄	1	4	yes	—
32	C₂×C₂²⋊Dic₅₃	—	2	2	no	—
33	C₂×C₄×Dic₅₃	—	2	2	no	—
34	C₄⋊#88322	—	2	4	no	—
35	C₄⋊#88322	—	2	2	no	—
36	C₄⋊#88322	C₂×C₄	1	4	yes	—
37	#13133124	—	2	2	no	—
38	#13133124	—	2	2	no	—
39	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
40	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
41	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
42	C₂²×C₈×C₅₃	—	2	2	no	—
43	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₅₃	D₄	1	2	yes	—
47	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
48	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
49	C₂×C₄²×C₅₃	Q₈	1	2	yes	—
50	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
51	C₄⋊C₈×C₅₃	—	2	2	no	—
52	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
53	C₂²⋊C₈×C₅₃	—	2	4	no	—
54	C₄×C₈×C₅₃	—	2	4	no	—
55	C₂⁴×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	—	2	4	no	—
57	C₂²×C₄×C₅₃	—	4	2	no	—
58	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
59	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
62	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₅₃	C₂²⋊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	D₈	1	2	yes	—
68	C₂²×C₄×C₅₃	SD₁₆	1	2	yes	—
69	C₂²×C₄×C₅₃	SD₁₆	1	2	yes	—
70	C₂²×C₄×C₅₃	Q₁₆	1	2	yes	—
71	C₂²×C₄×C₅₃	C₂²×C₄	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×Dic₅₃	—	2	4	no	—
75	C₂×C₈×C₅₃	—	2	4	no	—
76	C₂×C₈×C₅₃	—	4	4	no	—
77	C₄²×C₅₃	C₄²	1	2	yes	—
78	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
79	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
80	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
81	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
82	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
83	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
84	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
85	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
86	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
87	C₂²⋊Dic₅₃	—	4	4	no	—
88	C₄⋊Dic₅₃	—	4	2	no	—
89	C₄⋊Dic₅₃	—	4	2	no	—
90	C₄⋊Dic₅₃	—	4	4	no	—
91	C₄⋊Dic₅₃	—	4	4	no	—
92	C₄×Dic₅₃	—	4	4	no	—
93	C₂×#88322	—	2	4	no	—
94	C₂×#88322	—	4	8	no	—
95	C₂×#88322	—	4	2	no	—
96	C₂×#88322	—	4	2	no	—
97	C₂³×C₅₃	—	2	2	no	—
98	C₂³×C₅₃	—	2	2	no	—
99	C₂³×C₅₃	#95	1	2	yes	—
100	C₂³×C₅₃	#102	1	2	yes	—
101	C₂³×C₅₃	#103	1	2	yes	—
102	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₅₃	—	2	2	no	—
104	C₂×C₄×C₅₃	—	2	2	no	—
105	C₂×C₄×C₅₃	—	2	2	no	—
106	C₂×C₄×C₅₃	—	2	4	no	—
107	C₂×C₄×C₅₃	—	2	4	no	—
108	C₂×C₄×C₅₃	—	4	2	no	—
109	C₂×C₄×C₅₃	—	4	2	no	—
110	C₂×C₄×C₅₃	#95	1	2	yes	—
111	C₂×C₄×C₅₃	#95	1	2	yes	—
112	C₂×C₄×C₅₃	#95	1	2	yes	—
113	C₂×C₄×C₅₃	#102	1	2	yes	—
114	C₂×C₄×C₅₃	#102	1	4	yes	—
115	C₂×C₄×C₅₃	#103	1	2	yes	—
116	C₂×C₄×C₅₃	#103	1	4	yes	—
117	C₂×C₄×C₅₃	#106	1	4	yes	—
118	C₂×C₄×C₅₃	#107	1	4	yes	—
119	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
120	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₅₃	C₂×C₂²⋊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	C₂×C₄⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
126	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
127	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
128	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
129	C₂×C₄×C₅₃	#135	1	4	yes	—
130	C₂×Dic₅₃	—	4	4	no	—
131	C₂×Dic₅₃	—	4	4	no	—
132	C₂×Dic₅₃	—	4	4	no	—
133	C₈×C₅₃	—	4	4	no	—
134	#88322	—	4	4	no	—
135	#8039	—	53	1	no	maximal, Sylow
136	C₂²×C₅₃	—	2	4	no	—
137	C₂²×C₅₃	—	2	4	no	—
138	C₂²×C₅₃	—	2	2	no	—
139	C₂²×C₅₃	—	2	2	no	—
140	C₂²×C₅₃	#340	1	4	yes	—
141	C₂²×C₅₃	C₂×#95	1	1	yes	characteristic
142	C₂²×C₅₃	#413	1	2	yes	—
143	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
144	C₂²×C₅₃	C₂×#103	1	1	yes	characteristic
145	C₂²×C₅₃	#416	1	2	yes	—
146	C₄×C₅₃	—	2	2	no	—
147	C₄×C₅₃	—	2	2	no	—
148	C₄×C₅₃	—	2	1	no	—
149	C₄×C₅₃	—	2	1	no	—
150	C₄×C₅₃	#340	1	4	yes	—
151	C₄×C₅₃	C₂×#102	1	1	yes	characteristic
152	C₄×C₅₃	C₂×#103	1	1	yes	characteristic
153	C₄×C₅₃	#416	1	2	yes	—
154	C₄×C₅₃	C₂×#106	1	1	yes	characteristic
155	C₄×C₅₃	C₂×#107	1	1	yes	characteristic
156	C₄×C₅₃	#427	1	2	yes	—
157	Dic₅₃	—	8	4	no	—
158	C₂×#431	D₅₃	1	1	yes	characteristic
159	C₂²×C₄⋊C₈	—	53	1	no	—
160	#2285	—	53	1	no	—
161	#1296	—	53	4	no	—
162	C₂×C₅₃	—	2	4	no	—
163	C₂×C₅₃	#1540	1	2	yes	—
164	C₂×C₅₃	#1542	1	2	yes	—
165	C₂×C₅₃	C₂×#340	1	1	yes	characteristic
166	C₂×C₅₃	#1735	1	1	yes	characteristic
167	C₂×C₅₃	#1742	1	1	yes	characteristic
168	C₂³×C₈	—	106	1	no	—
169	C₂×#117	—	106	1	no	—
170	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
171	#431	—	2	2	no	—
172	#431	Dic₅₃	1	4	yes	complemented
173	C₂×C₄⋊C₈	—	53	4	no	—
174	C₂×C₄⋊C₈	—	53	2	no	—
175	C₂×C₄⋊C₈	—	106	1	no	—
176	C₂×C₄⋊C₈	—	106	2	no	—
177	C₂×C₄⋊C₈	C₂×D₅₃	1	2	yes	—
178	C₂×C₂²⋊C₈	—	2	1	no	—
179	C₂×C₄×C₈	—	2	1	no	—
180	#384	—	53	2	no	—
181	#382	—	106	1	no	—
182	#380	—	53	2	no	—
183	#380	—	106	1	no	—
184	C₄×C₄⋊C₄	—	53	2	no	—
185	C₄×C₄⋊C₄	—	106	1	no	—
186	C₅₃	#8039	1	1	yes	characteristic, complemented, Sylow
187	C₂³×C₄	C₄×D₅₃	1	1	yes	characteristic
188	C₂³×C₄	D₂₁₂	1	1	yes	characteristic
189	C₂³×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
190	C₂²×C₈	—	2	2	no	—
191	C₂²×C₈	—	106	2	no	—
192	C₂²×C₈	—	106	4	no	—
193	C₂²×C₈	—	212	2	no	—
194	C₂²×C₈	—	212	2	no	—
195	#117	—	212	2	no	—
196	#117	—	212	2	no	—
197	C₂×C₄⋊C₄	—	53	4	no	—
198	C₂×C₄⋊C₄	—	106	2	no	—
199	C₂×C₄⋊C₄	—	106	2	no	—
200	C₂×C₂²⋊C₄	—	106	2	no	—
201	C₂×C₄²	—	106	2	no	—
202	C₂×C₄²	Dic₁₀₆	1	2	yes	—
203	C₂×C₄²	C₄×D₅₃	1	2	yes	—
204	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
205	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
206	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
207	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
208	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
209	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
210	C₄⋊C₈	—	2	2	no	—
211	C₄⋊C₈	—	53	4	no	—
212	C₄⋊C₈	—	106	2	no	—
213	C₄⋊C₈	—	106	4	no	—
214	C₄⋊C₈	C₂×Dic₅₃	1	4	yes	—
215	C₂²⋊C₈	—	2	4	no	—
216	C₄×C₈	—	2	4	no	—
217	#95	—	106	2	no	—
218	#95	—	106	2	no	—
219	C₂⁴	#11719921	1	1	yes	characteristic
220	C₂²×C₄	—	2	4	no	—
221	C₂²×C₄	—	4	2	no	—
222	C₂²×C₄	—	106	4	no	—
223	C₂²×C₄	#11719913	1	2	yes	—
224	C₂²×C₄	#11719913	1	2	yes	—
225	C₂²×C₄	D₄₂₄	1	2	yes	—
226	C₂²×C₄	Dic₂₁₂	1	2	yes	—
227	C₂²×C₄	C₄×Dic₅₃	1	2	yes	—
228	C₂²×C₄	#11719919	1	2	yes	—
229	C₂²×C₄	#11719919	1	2	yes	—
230	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
231	C₂²×C₄	#11719921	1	1	yes	characteristic
232	C₂²×C₄	#11719921	1	1	yes	characteristic
233	C₂²×C₄	#11719921	1	2	yes	—
234	C₂²×C₄	#11719921	1	1	yes	characteristic
235	C₂²×C₄	C₂²⋊Dic₅₃	1	2	yes	—
236	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
237	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
238	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
239	C₂×C₈	—	2	4	no	—
240	C₂×C₈	—	4	4	no	—
241	C₂×C₈	—	106	4	no	—
242	C₂×C₈	—	212	8	no	—
243	C₂×C₈	—	212	2	no	—
244	C₂×C₈	—	212	2	no	—
245	C₄⋊C₄	—	212	4	no	—
246	C₄⋊C₄	—	212	2	no	—
247	C₄⋊C₄	—	212	2	no	—
248	C₄⋊C₄	—	212	4	no	—
249	C₂²⋊C₄	—	212	4	no	—
250	C₄²	—	212	4	no	—
251	C₄²	C₄×Dic₅₃	1	2	yes	—
252	C₄²	#11719919	1	2	yes	—
253	C₄²	#11719919	1	2	yes	—
254	C₄²	C₄⋊Dic₅₃	1	2	yes	—
255	C₄²	#11719921	1	2	yes	—
256	C₄²	C₂²⋊Dic₅₃	1	2	yes	—
257	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
258	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
259	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
260	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
261	C₂³	—	2	2	no	—
262	C₂³	—	2	2	no	—
263	C₂³	#13133124	1	2	yes	—
264	C₂³	#13133139	1	2	yes	—
265	C₂³	#13133142	1	2	yes	—
266	C₂³	C₂×#11719921	1	1	yes	characteristic
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	—	212	4	no	—
275	C₂×C₄	—	212	4	no	—
276	C₂×C₄	—	212	4	no	—
277	C₂×C₄	#13133124	1	2	yes	—
278	C₂×C₄	#13133124	1	2	yes	—
279	C₂×C₄	#13133124	1	2	yes	—
280	C₂×C₄	#13133139	1	4	yes	—
281	C₂×C₄	#13133139	1	2	yes	—
282	C₂×C₄	#13133142	1	4	yes	—
283	C₂×C₄	#13133142	1	2	yes	—
284	C₂×C₄	#13133149	1	4	yes	—
285	C₂×C₄	#13133151	1	4	yes	—
286	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
287	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
288	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
289	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
290	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
291	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
292	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
293	C₂×C₄	C₂×#11719913	1	1	yes	characteristic
294	C₂×C₄	C₂×#11719913	1	1	yes	characteristic
295	C₂×C₄	C₂×Dic₂₁₂	1	1	yes	characteristic
296	C₂×C₄	#13133237	1	4	yes	—
297	C₈	—	4	4	no	—
298	C₈	—	212	4	no	—
299	C₂²	—	2	2	no	—
300	C₂²	—	2	4	no	—
301	C₂²	—	2	4	no	—
302	C₂²	—	2	2	no	—
303	C₂²	#16810386	1	4	yes	—
304	C₂²	C₂×#13133124	1	1	yes	characteristic
305	C₂²	#16810605	1	2	yes	—
306	C₂²	C₂×#13133139	1	1	yes	characteristic
307	C₂²	C₂×#13133142	1	1	yes	characteristic
308	C₂²	#16810617	1	2	yes	—
309	C₄	—	2	2	no	—
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	2	2	no	—
313	C₄	—	424	4	no	—
314	C₄	#16810386	1	4	yes	—
315	C₄	C₂×#13133139	1	1	yes	characteristic
316	C₄	C₂×#13133142	1	1	yes	characteristic
317	C₄	#16810619	1	2	yes	—
318	C₄	C₂×#13133149	1	1	yes	characteristic
319	C₄	C₂×#13133151	1	1	yes	characteristic
320	C₄	#16810659	1	2	yes	—
321	C₂	—	2	4	no	—
322	C₂	#20273362	1	2	yes	—
323	C₂	#20273370	1	2	yes	—
324	C₂	C₂×#16810386	1	1	yes	characteristic
325	C₂	#20274431	1	1	yes	characteristic
326	C₂	#20274453	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	#11719921
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₂×C₅₃×#431	C₂
socle	C₂³×C₅₃	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₅₃	C₂²⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂×#431	D₅₃
hypocenter	C₅₃	#8039
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×#431

Complements

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

N	H ≅ G/N	type
3. #20272599	C₂	semidirect
9. C₂×C₄⋊#88322	C₄	semidirect
19. C₅₃×#431	C₄	semidirect
172. #431	Dic₅₃	semidirect
186. C₅₃	#8039	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄⋊#88322	2
#20277122	2
#20272599	2
C₂×C₅₃×#431	2
#8039	53