Subgroups — #27274304

Order 13568.

698 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283348	C₂	1	1	yes	maximal, characteristic
2	#20283348	C₂	1	1	yes	maximal, characteristic
3	#20272608	C₂	1	4	yes	maximal, complemented
4	C₂×C₅₃×#376	C₂	1	1	yes	maximal, characteristic
5	#16810680	—	2	1	no	—
6	#16810680	C₄	1	2	yes	complemented
7	#16810680	C₄	1	2	yes	complemented
8	#16810680	C₂²	1	2	yes	—
9	#16810676	—	2	1	no	—
10	#16810676	—	2	1	no	—
11	#16810676	—	2	1	no	—
12	#16810676	C₄	1	2	yes	complemented
13	#16810647	—	2	1	no	—
14	#16810647	C₄	1	2	yes	complemented
15	#16810587	—	2	1	no	—
16	#16810587	—	2	1	no	—
17	C₄×#11719917	—	2	1	no	—
18	C₄×#11719917	C₂²	1	2	yes	—
19	C₂²×C₄²×C₅₃	—	2	1	no	—
20	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
21	C₅₃×#376	—	2	1	no	—
22	C₅₃×#376	—	2	1	no	—
23	C₅₃×#376	C₄	1	2	yes	—
24	C₅₃×#376	C₂²	1	2	yes	—
25	C₂×#95×C₅₃	—	2	1	no	—
26	C₂×#95×C₅₃	C₄	1	2	yes	—
27	C₂×#11719917	—	2	2	no	—
28	C₂×#11719917	—	2	2	no	—
29	C₄⋊#88322	—	2	2	no	—
30	C₄⋊#88322	—	2	2	no	—
31	C₄⋊#88322	—	2	2	no	—
32	C₄⋊#88322	C₂×C₄	1	4	yes	complemented
33	C₂²⋊#88322	—	2	2	no	—
34	C₂²⋊#88322	—	2	2	no	—
35	C₂²⋊#88322	—	2	2	no	—
36	C₂²⋊#88322	—	2	2	no	—
37	#13133128	—	2	2	no	—
38	#13133128	C₂×C₄	1	4	yes	complemented
39	C₄×#88322	—	2	2	no	—
40	C₄×#88322	—	2	2	no	—
41	C₂³×C₄×C₅₃	—	2	2	no	—
42	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
43	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
44	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₅₃	—	2	8	no	—
46	C₂×C₄²×C₅₃	—	2	2	no	—
47	C₂×C₄²×C₅₃	—	2	2	no	—
48	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
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	1	yes	characteristic
52	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
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	Q₈	1	1	yes	characteristic
58	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
59	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
60	#95×C₅₃	—	2	2	no	—
61	#95×C₅₃	—	2	2	no	—
62	#95×C₅₃	—	2	2	no	—
63	#95×C₅₃	C₂×C₄	1	4	yes	—
64	C₂⁴×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₅₃	—	2	4	no	—
66	C₂²×C₄×C₅₃	—	2	8	no	—
67	C₂²×C₄×C₅₃	—	2	4	no	—
68	C₂²×C₄×C₅₃	—	4	2	no	—
69	C₂²×C₄×C₅₃	—	4	2	no	—
70	C₂²×C₄×C₅₃	—	4	4	no	—
71	C₂²×C₄×C₅₃	—	4	4	no	—
72	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
73	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
74	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
75	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
76	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
77	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
78	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
79	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
80	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
81	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
82	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
83	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
84	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
85	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
86	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
87	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
88	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
90	C₄²×C₅₃	—	2	8	no	—
91	C₄²×C₅₃	—	2	8	no	—
92	C₄²×C₅₃	C₄²	1	1	yes	characteristic
93	C₄²×C₅₃	C₄²	1	1	yes	characteristic
94	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
95	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
96	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
97	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
98	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
99	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
100	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
101	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
102	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
103	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
104	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
105	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
106	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
107	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
108	#11719917	—	4	4	no	—
109	#11719917	—	4	4	no	—
110	C₂×#88322	—	2	4	no	—
111	C₂×#88322	—	2	4	no	—
112	C₂³×C₅₃	—	2	2	no	—
113	C₂³×C₅₃	—	2	2	no	—
114	C₂³×C₅₃	—	2	2	no	—
115	C₂³×C₅₃	#95	1	1	yes	characteristic
116	C₂³×C₅₃	#95	1	1	yes	characteristic
117	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₅₃	—	2	2	no	—
119	C₂×C₄×C₅₃	—	2	2	no	—
120	C₂×C₄×C₅₃	—	2	2	no	—
121	C₂×C₄×C₅₃	—	2	2	no	—
122	C₂×C₄×C₅₃	—	2	2	no	—
123	C₂×C₄×C₅₃	—	2	2	no	—
124	C₂×C₄×C₅₃	—	2	2	no	—
125	C₂×C₄×C₅₃	—	2	2	no	—
126	C₂×C₄×C₅₃	—	2	2	no	—
127	C₂×C₄×C₅₃	—	2	8	no	—
128	C₂×C₄×C₅₃	—	4	1	no	—
129	C₂×C₄×C₅₃	—	4	1	no	—
130	C₂×C₄×C₅₃	—	4	1	no	—
131	C₂×C₄×C₅₃	—	4	1	no	—
132	C₂×C₄×C₅₃	—	4	1	no	—
133	C₂×C₄×C₅₃	—	4	1	no	—
134	C₂×C₄×C₅₃	—	4	1	no	—
135	C₂×C₄×C₅₃	—	4	1	no	—
136	C₂×C₄×C₅₃	—	4	16	no	—
137	C₂×C₄×C₅₃	—	4	4	no	—
138	C₂×C₄×C₅₃	—	4	4	no	—
139	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
140	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
141	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
142	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
143	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
144	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
145	C₂×C₄×C₅₃	#104	1	4	yes	—
146	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
147	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
148	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
149	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
150	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
151	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
152	#88322	—	4	4	no	—
153	#88322	—	4	4	no	—
154	#8072	—	53	1	no	maximal, Sylow
155	C₂²×C₅₃	—	2	8	no	—
156	C₂²×C₅₃	—	2	1	no	—
157	C₂²×C₅₃	—	2	1	no	—
158	C₂²×C₅₃	—	2	1	no	—
159	C₂²×C₅₃	—	2	1	no	—
160	C₂²×C₅₃	—	2	1	no	—
161	C₂²×C₅₃	—	2	1	no	—
162	C₂²×C₅₃	#339	1	2	yes	—
163	C₂²×C₅₃	#344	1	2	yes	—
164	C₂²×C₅₃	C₂×#95	1	1	yes	characteristic
165	C₂²×C₅₃	#421	1	2	yes	—
166	C₄×C₅₃	—	2	1	no	—
167	C₄×C₅₃	—	2	1	no	—
168	C₄×C₅₃	—	2	1	no	—
169	C₄×C₅₃	—	2	1	no	—
170	C₄×C₅₃	—	2	1	no	—
171	C₄×C₅₃	—	2	1	no	—
172	C₄×C₅₃	—	2	1	no	—
173	C₄×C₅₃	—	2	1	no	—
174	C₄×C₅₃	—	2	1	no	—
175	C₄×C₅₃	—	2	1	no	—
176	C₄×C₅₃	—	4	8	no	—
177	C₄×C₅₃	#339	1	2	yes	—
178	C₄×C₅₃	C₂×#104	1	2	yes	—
179	#2920	—	53	1	no	—
180	#2920	—	53	1	no	—
181	C₂×#376	D₅₃	1	1	yes	characteristic
182	#1300	—	53	4	no	—
183	C₂×C₅₃	—	2	2	no	—
184	C₂×C₅₃	—	2	2	no	—
185	C₂×C₅₃	—	2	2	no	—
186	C₂×C₅₃	C₂×#339	1	1	yes	characteristic
187	C₂×C₅₃	#1733	1	1	yes	characteristic
188	C₂×C₅₃	C₂×#344	1	1	yes	characteristic
189	C₂²×C₄²	—	2	1	no	—
190	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
191	#432	—	53	2	no	—
192	#432	—	53	2	no	—
193	#432	—	53	2	no	—
194	#432	—	106	1	no	—
195	#431	—	53	2	no	—
196	#431	—	106	1	no	—
197	#431	—	106	1	no	—
198	#431	—	106	1	no	—
199	#423	—	53	2	no	—
200	#423	—	106	1	no	—
201	#407	—	106	1	no	—
202	#407	—	106	1	no	—
203	C₄×M₁₆	—	53	2	no	—
204	C₄×M₁₆	—	106	1	no	—
205	#376	—	2	1	no	—
206	#376	—	2	1	no	—
207	#376	Dic₅₃	1	2	yes	—
208	#376	C₂×D₅₃	1	2	yes	—
209	C₂×#95	—	2	1	no	—
210	C₂×#95	Dic₅₃	1	2	yes	—
211	C₅₃	#8072	1	1	yes	characteristic, complemented, Sylow
212	C₂³×C₄	—	2	2	no	—
213	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
214	C₂³×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
215	C₂³×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
216	C₂×M₁₆	—	106	2	no	—
217	C₂×M₁₆	—	106	2	no	—
218	C₂×C₄²	—	2	8	no	—
219	C₂×C₄²	—	2	2	no	—
220	C₂×C₄²	—	2	2	no	—
221	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
222	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
223	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
224	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
225	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
226	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
227	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
228	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
229	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
230	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
231	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
232	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
233	C₄⋊C₈	—	53	4	no	—
234	C₄⋊C₈	—	106	2	no	—
235	C₄⋊C₈	—	106	2	no	—
236	C₄⋊C₈	—	106	2	no	—
237	C₂²⋊C₈	—	106	2	no	—
238	C₂²⋊C₈	—	106	2	no	—
239	C₂²⋊C₈	—	106	2	no	—
240	C₂²⋊C₈	—	106	2	no	—
241	#97	—	53	4	no	—
242	#97	—	106	2	no	—
243	C₄×C₈	—	106	2	no	—
244	C₄×C₈	—	106	2	no	—
245	#95	—	2	2	no	—
246	#95	—	2	2	no	—
247	#95	—	2	2	no	—
248	#95	C₂×Dic₅₃	1	4	yes	—
249	C₂⁴	C₂²⋊Dic₅₃	1	1	yes	characteristic
250	C₂²×C₄	—	2	4	no	—
251	C₂²×C₄	—	2	8	no	—
252	C₂²×C₄	—	2	4	no	—
253	C₂²×C₄	—	4	2	no	—
254	C₂²×C₄	—	4	2	no	—
255	C₂²×C₄	—	4	4	no	—
256	C₂²×C₄	—	4	4	no	—
257	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
258	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
259	C₂²×C₄	#11719919	1	1	yes	characteristic
260	C₂²×C₄	#11719919	1	1	yes	characteristic
261	C₂²×C₄	#11719919	1	1	yes	characteristic
262	C₂²×C₄	#11719919	1	1	yes	characteristic
263	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
264	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
265	C₂²×C₄	#11719921	1	1	yes	characteristic
266	C₂²×C₄	#11719921	1	1	yes	characteristic
267	C₂²×C₄	#11719921	1	1	yes	characteristic
268	C₂²×C₄	#11719921	1	1	yes	characteristic
269	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
270	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
271	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
272	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
273	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
274	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
275	M₁₆	—	212	4	no	—
276	M₁₆	—	212	4	no	—
277	C₂×C₈	—	106	4	no	—
278	C₂×C₈	—	106	4	no	—
279	C₄²	—	2	8	no	—
280	C₄²	—	2	8	no	—
281	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
282	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
283	C₄²	#11719919	1	1	yes	characteristic
284	C₄²	#11719919	1	1	yes	characteristic
285	C₄²	#11719919	1	1	yes	characteristic
286	C₄²	#11719919	1	1	yes	characteristic
287	C₄²	C₄⋊Dic₅₃	1	1	yes	characteristic
288	C₄²	C₄⋊Dic₅₃	1	1	yes	characteristic
289	C₄²	#11719921	1	1	yes	characteristic
290	C₄²	#11719921	1	1	yes	characteristic
291	C₄²	#11719921	1	1	yes	characteristic
292	C₄²	#11719921	1	1	yes	characteristic
293	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
294	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
295	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
296	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
297	C₂³	—	2	2	no	—
298	C₂³	—	2	2	no	—
299	C₂³	—	2	2	no	—
300	C₂³	#13133124	1	1	yes	characteristic
301	C₂³	#13133124	1	1	yes	characteristic
302	C₂³	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	8	no	—
313	C₂×C₄	—	4	1	no	—
314	C₂×C₄	—	4	1	no	—
315	C₂×C₄	—	4	1	no	—
316	C₂×C₄	—	4	1	no	—
317	C₂×C₄	—	4	1	no	—
318	C₂×C₄	—	4	1	no	—
319	C₂×C₄	—	4	1	no	—
320	C₂×C₄	—	4	1	no	—
321	C₂×C₄	—	4	16	no	—
322	C₂×C₄	—	4	4	no	—
323	C₂×C₄	—	4	4	no	—
324	C₂×C₄	#13133124	1	1	yes	characteristic
325	C₂×C₄	#13133124	1	1	yes	characteristic
326	C₂×C₄	#13133124	1	1	yes	characteristic
327	C₂×C₄	#13133124	1	1	yes	characteristic
328	C₂×C₄	#13133124	1	1	yes	characteristic
329	C₂×C₄	#13133124	1	1	yes	characteristic
330	C₂×C₄	#13133145	1	4	yes	—
331	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
332	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
333	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
334	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
335	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
336	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
337	C₈	—	212	4	no	—
338	C₈	—	212	4	no	—
339	C₂²	—	2	8	no	—
340	C₂²	—	2	1	no	—
341	C₂²	—	2	1	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	2	1	no	—
344	C₂²	—	2	1	no	—
345	C₂²	—	2	1	no	—
346	C₂²	#16810383	1	2	yes	—
347	C₂²	#16810393	1	2	yes	—
348	C₂²	C₂×#13133124	1	1	yes	characteristic
349	C₂²	#16810640	1	2	yes	—
350	C₄	—	2	1	no	—
351	C₄	—	2	1	no	—
352	C₄	—	2	1	no	—
353	C₄	—	2	1	no	—
354	C₄	—	2	1	no	—
355	C₄	—	2	1	no	—
356	C₄	—	2	1	no	—
357	C₄	—	2	1	no	—
358	C₄	—	2	1	no	—
359	C₄	—	2	1	no	—
360	C₄	—	4	8	no	—
361	C₄	#16810383	1	2	yes	—
362	C₄	C₂×#13133145	1	2	yes	—
363	C₂	—	2	2	no	—
364	C₂	—	2	2	no	—
365	C₂	—	2	2	no	—
366	C₂	C₂×#16810383	1	1	yes	characteristic
367	C₂	#20274424	1	1	yes	characteristic
368	C₂	C₂×#16810393	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#11719921
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₂×C₅₃×#376	C₂
socle	C₂²×C₅₃	C₂×#95
prime radical	C₂⁴×C₅₃	C₂²⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂×#376	D₅₃
hypocenter	C₅₃	#8072
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂×#376

Complements

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

N	H ≅ G/N	type
3. #20272608	C₂	semidirect
6. #16810680	C₄	semidirect
7. #16810680	C₄	semidirect
12. #16810676	C₄	semidirect
14. #16810647	C₄	semidirect
32. C₄⋊#88322	C₂×C₄	semidirect
38. #13133128	C₂×C₄	semidirect
211. C₅₃	#8072	semidirect

Maximal subgroups

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

subgroup	index
#20283348	2
#20272608	2
C₂×C₅₃×#376	2
#8072	53