Subgroups — #27401695

Order 13568.

692 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283373	C₂	1	2	yes	maximal
2	#20275469	C₂	1	2	yes	maximal, complemented
3	#20275316	C₂	1	4	yes	maximal, complemented
4	#20274814	C₂	1	2	yes	maximal
5	#20274753	C₂	1	1	yes	maximal, characteristic
6	#20274601	C₂	1	2	yes	maximal, complemented
7	#20274491	C₂	1	1	yes	maximal, characteristic
8	C₂×C₅₃×#427	C₂	1	1	yes	maximal, characteristic
9	#16810684	—	2	4	no	—
10	#16810680	—	2	2	no	—
11	#16810676	—	2	2	no	—
12	C₂×#13133153	C₂²	1	2	yes	—
13	#16810650	C₂²	1	4	yes	—
14	#16810587	C₂²	1	2	yes	—
15	C₂×C₂²⋊#88322	C₂²	1	1	yes	characteristic
16	#16810579	C₂²	1	4	yes	—
17	C₂×#13133127	C₂²	1	2	yes	—
18	C₄×C₂²⋊Dic₅₃	C₂²	1	1	yes	characteristic
19	#16810388	C₂²	1	2	yes	—
20	#16810382	C₂²	1	4	yes	—
21	#16810382	C₂²	1	4	yes	—
22	#16810377	C₂²	1	2	yes	—
23	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
24	C₂×#117×C₅₃	C₂²	1	2	yes	—
25	C₅₃×#427	C₄	1	1	yes	characteristic, complemented
26	C₅₃×#427	C₄	1	1	yes	characteristic, complemented
27	C₅₃×#427	C₄	1	2	yes	complemented
28	C₅₃×#427	C₂²	1	1	yes	characteristic
29	C₅₃×#427	C₂²	1	2	yes	—
30	C₅₃×#427	C₂²	1	1	yes	characteristic
31	C₂×#107×C₅₃	C₄	1	1	yes	characteristic, complemented
32	C₂×#107×C₅₃	C₄	1	1	yes	characteristic, complemented
33	C₂×#106×C₅₃	C₄	1	2	yes	complemented
34	C₂×#11719917	C₂³	1	4	yes	—
35	C₂²×#88322	C₂³	1	2	yes	—
36	C₂×C₂²⋊Dic₅₃	—	2	1	no	—
37	C₂×C₄×Dic₅₃	C₂³	1	2	yes	—
38	#13133153	—	2	4	no	—
39	C₄⋊#88322	—	2	4	no	—
40	C₄⋊#88322	—	2	4	no	—
41	C₂²⋊#88322	—	2	2	no	—
42	C₂²⋊#88322	—	2	2	no	—
43	C₂²⋊#88322	D₄	1	2	yes	—
44	C₂²⋊#88322	D₄	1	2	yes	—
45	#13133128	—	2	4	no	—
46	#13133127	—	2	4	no	—
47	C₄×#88322	—	2	4	no	—
48	#13133124	—	2	1	no	—
49	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
50	C₂²×C₈×C₅₃	—	2	4	no	—
51	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
52	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
53	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
54	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
55	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
56	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
57	#117×C₅₃	—	2	2	no	—
58	#117×C₅₃	—	2	2	no	—
59	#117×C₅₃	C₂×C₄	1	2	yes	—
60	#117×C₅₃	C₂×C₄	1	2	yes	—
61	#117×C₅₃	C₂³	1	2	yes	—
62	#117×C₅₃	C₂³	1	2	yes	—
63	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
64	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
65	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
66	C₂×C₄²×C₅₃	—	2	2	no	—
67	#107×C₅₃	C₂×C₄	1	2	yes	—
68	#107×C₅₃	C₂×C₄	1	2	yes	—
69	#107×C₅₃	C₂×C₄	1	1	yes	characteristic
70	#107×C₅₃	C₂×C₄	1	1	yes	characteristic
71	#107×C₅₃	C₂×C₄	1	1	yes	characteristic
72	#107×C₅₃	C₂×C₄	1	1	yes	characteristic
73	#106×C₅₃	C₂×C₄	1	2	yes	—
74	#106×C₅₃	C₂×C₄	1	2	yes	—
75	#106×C₅₃	C₂×C₄	1	2	yes	—
76	#106×C₅₃	C₂×C₄	1	2	yes	—
77	C₂⁴×C₅₃	#41	1	1	yes	characteristic
78	C₂²×C₄×C₅₃	—	2	2	no	—
79	C₂²×C₄×C₅₃	—	2	2	no	—
80	C₂²×C₄×C₅₃	—	2	2	no	—
81	C₂²×C₄×C₅₃	—	2	2	no	—
82	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
87	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
88	C₂²×Dic₅₃	—	2	2	no	—
89	C₂×C₈×C₅₃	—	2	2	no	—
90	C₂×C₈×C₅₃	—	2	2	no	—
91	C₂×C₈×C₅₃	—	2	2	no	—
92	C₂×C₈×C₅₃	—	2	2	no	—
93	C₂×C₈×C₅₃	—	2	2	no	—
94	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₈×C₅₃	#41	1	4	yes	—
99	C₄⋊C₄×C₅₃	—	2	2	no	—
100	C₄⋊C₄×C₅₃	—	2	2	no	—
101	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
102	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
103	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
104	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
105	C₂²⋊C₄×C₅₃	—	2	2	no	—
106	C₂²⋊C₄×C₅₃	—	2	2	no	—
107	C₂²⋊C₄×C₅₃	—	2	2	no	—
108	C₂²⋊C₄×C₅₃	—	2	2	no	—
109	C₄²×C₅₃	—	2	2	no	—
110	C₄²×C₅₃	—	2	2	no	—
111	C₄²×C₅₃	—	2	2	no	—
112	C₄²×C₅₃	—	2	2	no	—
113	C₂²⋊Dic₅₃	—	4	2	no	—
114	C₄×Dic₅₃	—	2	2	no	—
115	C₄×Dic₅₃	—	2	2	no	—
116	#11719917	—	2	4	no	—
117	#11719917	—	2	4	no	—
118	C₂×#88322	—	2	2	no	—
119	C₂×#88322	—	2	2	no	—
120	C₂×#88322	—	2	4	no	—
121	C₂×#88322	C₂×D₄	1	4	yes	—
122	C₂³×C₅₃	—	2	2	no	—
123	C₂³×C₅₃	—	2	2	no	—
124	C₂³×C₅₃	—	4	1	no	—
125	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
126	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
127	C₂³×C₅₃	C₂×#41	1	1	yes	characteristic
128	C₂×C₄×C₅₃	—	2	2	no	—
129	C₂×C₄×C₅₃	—	2	1	no	—
130	C₂×C₄×C₅₃	—	2	1	no	—
131	C₂×C₄×C₅₃	—	2	1	no	—
132	C₂×C₄×C₅₃	—	2	1	no	—
133	C₂×C₄×C₅₃	—	2	2	no	—
134	C₂×C₄×C₅₃	—	2	2	no	—
135	C₂×C₄×C₅₃	—	2	2	no	—
136	C₂×C₄×C₅₃	—	2	2	no	—
137	C₂×C₄×C₅₃	—	2	2	no	—
138	C₂×C₄×C₅₃	—	2	2	no	—
139	C₂×C₄×C₅₃	—	4	4	no	—
140	C₂×C₄×C₅₃	—	8	2	no	—
141	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
142	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
143	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
144	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
145	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
146	C₂×C₄×C₅₃	2₋¹⁺⁴	1	2	yes	—
147	C₂×Dic₅₃	—	4	2	no	—
148	C₂×Dic₅₃	—	4	2	no	—
149	C₂×Dic₅₃	—	4	2	no	—
150	C₈×C₅₃	—	2	1	no	—
151	C₈×C₅₃	—	2	1	no	—
152	C₈×C₅₃	#117	1	2	yes	—
153	C₈×C₅₃	#117	1	2	yes	—
154	#88322	—	2	4	no	—
155	#88322	—	4	4	no	—
156	#22607	—	53	1	no	maximal, Sylow
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	—	2	1	no	—
163	C₂²×C₅₃	—	2	1	no	—
164	C₂²×C₅₃	—	4	4	no	—
165	C₂²×C₅₃	#444	1	2	yes	—
166	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
167	C₂²×C₅₃	#552	1	2	yes	—
168	C₄×C₅₃	—	2	1	no	—
169	C₄×C₅₃	—	2	1	no	—
170	C₄×C₅₃	—	4	2	no	—
171	C₄×C₅₃	—	4	2	no	—
172	C₄×C₅₃	—	4	2	no	—
173	C₄×C₅₃	—	4	2	no	—
174	C₄×C₅₃	C₂×#117	1	1	yes	characteristic
175	C₄×C₅₃	#519	1	1	yes	characteristic
176	C₄×C₅₃	#547	1	2	yes	—
177	Dic₅₃	—	8	2	no	—
178	#2924	—	53	2	no	—
179	C₂×#427	D₅₃	1	1	yes	characteristic
180	#1946	—	53	2	no	—
181	#1922	—	53	4	no	—
182	#1826	—	53	2	no	—
183	#1810	—	53	1	no	—
184	#1773	—	53	2	no	—
185	#1752	—	53	1	no	—
186	C₂×C₅₃	—	2	2	no	—
187	C₂×C₅₃	—	2	2	no	—
188	C₂×C₅₃	—	4	1	no	—
189	C₂×C₅₃	#2344	1	1	yes	characteristic
190	C₂×C₅₃	C₂×#444	1	1	yes	characteristic
191	C₂×C₅₃	#2957	1	1	yes	characteristic
192	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
193	C₂×#117	C₂×D₅₃	1	2	yes	—
194	#433	—	106	4	no	—
195	#432	—	106	2	no	—
196	#431	—	106	2	no	—
197	C₂×#108	—	53	2	no	—
198	#427	Dic₅₃	1	1	yes	characteristic, complemented
199	#427	Dic₅₃	1	1	yes	characteristic, complemented
200	#427	Dic₅₃	1	2	yes	complemented
201	#427	C₂×D₅₃	1	2	yes	—
202	#427	C₂×D₅₃	1	1	yes	characteristic
203	#427	C₂×D₅₃	1	1	yes	characteristic
204	C₂×#107	Dic₅₃	1	1	yes	characteristic, complemented
205	C₂×#107	Dic₅₃	1	1	yes	characteristic, complemented
206	C₂×#106	Dic₅₃	1	2	yes	complemented
207	#424	—	53	4	no	—
208	#407	—	53	2	no	—
209	C₂×C₂²⋊C₈	—	53	1	no	—
210	#405	—	53	4	no	—
211	C₂×#97	—	53	2	no	—
212	C₄×C₂²⋊C₄	—	53	1	no	—
213	#341	—	53	2	no	—
214	#339	—	53	4	no	—
215	#339	—	53	4	no	—
216	#336	—	53	2	no	—
217	C₅₃	#22607	1	1	yes	characteristic, complemented, Sylow
218	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
219	C₂×M₁₆	—	53	4	no	—
220	C₂²×C₈	—	2	4	no	—
221	C₂²×C₈	—	53	2	no	—
222	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
223	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
224	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
225	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
226	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
227	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
228	#117	—	2	2	no	—
229	#117	—	2	2	no	—
230	#117	C₂×Dic₅₃	1	2	yes	—
231	#117	C₂×Dic₅₃	1	2	yes	—
232	#117	C₂²×D₅₃	1	2	yes	—
233	#117	C₂²×D₅₃	1	2	yes	—
234	C₂×C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
235	C₂×C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
236	C₂×C₂²⋊C₄	—	2	2	no	—
237	C₂×C₂²⋊C₄	—	106	1	no	—
238	C₂×C₄²	—	2	2	no	—
239	C₂×C₄²	—	53	2	no	—
240	#108	—	106	4	no	—
241	#107	C₂×Dic₅₃	1	2	yes	—
242	#107	C₂×Dic₅₃	1	2	yes	—
243	#107	C₂×Dic₅₃	1	1	yes	characteristic
244	#107	C₂×Dic₅₃	1	1	yes	characteristic
245	#107	C₂×Dic₅₃	1	1	yes	characteristic
246	#107	C₂×Dic₅₃	1	1	yes	characteristic
247	#106	C₂×Dic₅₃	1	2	yes	—
248	#106	C₂×Dic₅₃	1	2	yes	—
249	#106	C₂×Dic₅₃	1	2	yes	—
250	#106	C₂×Dic₅₃	1	2	yes	—
251	C₄⋊C₈	—	106	4	no	—
252	C₄⋊C₈	—	106	4	no	—
253	C₂²⋊C₈	—	53	2	no	—
254	C₂²⋊C₈	—	53	2	no	—
255	C₂²⋊C₈	—	106	2	no	—
256	C₂²⋊C₈	—	106	2	no	—
257	#97	—	106	4	no	—
258	#97	—	106	4	no	—
259	C₄×C₈	—	106	4	no	—
260	#95	—	106	1	no	—
261	C₂⁴	#11719947	1	1	yes	characteristic
262	C₂²×C₄	—	2	2	no	—
263	C₂²×C₄	—	2	2	no	—
264	C₂²×C₄	—	2	2	no	—
265	C₂²×C₄	—	2	2	no	—
266	C₂²×C₄	—	106	2	no	—
267	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
268	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
269	C₂²×C₄	#11719947	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	C₂³×D₅₃	1	1	yes	characteristic
273	M₁₆	—	106	4	no	—
274	M₁₆	—	106	4	no	—
275	C₂×C₈	—	2	2	no	—
276	C₂×C₈	—	2	2	no	—
277	C₂×C₈	—	2	2	no	—
278	C₂×C₈	—	2	2	no	—
279	C₂×C₈	—	2	2	no	—
280	C₂×C₈	—	53	4	no	—
281	C₂×C₈	—	106	2	no	—
282	C₂×C₈	—	106	2	no	—
283	C₂×C₈	—	106	4	no	—
284	C₂×C₈	#11719945	1	4	yes	—
285	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
286	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
287	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
288	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
289	C₄⋊C₄	—	2	2	no	—
290	C₄⋊C₄	—	2	2	no	—
291	C₄⋊C₄	C₂²×Dic₅₃	1	2	yes	—
292	C₄⋊C₄	C₂²×Dic₅₃	1	2	yes	—
293	C₄⋊C₄	C₂²×Dic₅₃	1	2	yes	—
294	C₄⋊C₄	C₂²×Dic₅₃	1	2	yes	—
295	C₂²⋊C₄	—	2	2	no	—
296	C₂²⋊C₄	—	2	2	no	—
297	C₂²⋊C₄	—	2	2	no	—
298	C₂²⋊C₄	—	2	2	no	—
299	C₂²⋊C₄	—	212	2	no	—
300	C₄²	—	2	2	no	—
301	C₄²	—	2	2	no	—
302	C₄²	—	2	2	no	—
303	C₄²	—	2	2	no	—
304	C₄²	—	106	2	no	—
305	C₄²	—	106	2	no	—
306	C₂³	—	2	2	no	—
307	C₂³	—	2	2	no	—
308	C₂³	—	4	1	no	—
309	C₂³	D₄×Dic₅₃	1	1	yes	characteristic
310	C₂³	D₄×Dic₅₃	1	1	yes	characteristic
311	C₂³	C₂×#11719947	1	1	yes	characteristic
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	4	4	no	—
324	C₂×C₄	—	8	2	no	—
325	C₂×C₄	—	212	2	no	—
326	C₂×C₄	—	212	2	no	—
327	C₂×C₄	—	212	2	no	—
328	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
329	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
330	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
331	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
332	C₂×C₄	C₂×#11719945	1	2	yes	—
333	C₂×C₄	#13133265	1	2	yes	—
334	C₈	—	2	1	no	—
335	C₈	—	2	1	no	—
336	C₈	—	106	4	no	—
337	C₈	—	212	4	no	—
338	C₈	#13133176	1	2	yes	—
339	C₈	#13133176	1	2	yes	—
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	—	2	1	no	—
347	C₂²	—	4	4	no	—
348	C₂²	#16810745	1	2	yes	—
349	C₂²	C₂×D₄×Dic₅₃	1	1	yes	characteristic
350	C₂²	#16811308	1	2	yes	—
351	C₄	—	2	1	no	—
352	C₄	—	2	1	no	—
353	C₄	—	4	2	no	—
354	C₄	—	4	2	no	—
355	C₄	—	4	2	no	—
356	C₄	—	4	2	no	—
357	C₄	—	424	2	no	—
358	C₄	C₂×#13133176	1	1	yes	characteristic
359	C₄	#16811101	1	1	yes	characteristic
360	C₄	#16811274	1	2	yes	—
361	C₂	—	2	2	no	—
362	C₂	—	2	2	no	—
363	C₂	—	4	1	no	—
364	C₂	#20277584	1	1	yes	characteristic
365	C₂	C₂×#16810745	1	1	yes	characteristic
366	C₂	#20283632	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂×C₅₃×#427	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂⁴×C₅₃	#41
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₂×#427	D₅₃
hypocenter	C₅₃	#22607
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#427

Complements

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

N	H ≅ G/N	type
2. #20275469	C₂	semidirect
3. #20275316	C₂	semidirect
6. #20274601	C₂	semidirect
25. C₅₃×#427	C₄	semidirect
26. C₅₃×#427	C₄	semidirect
27. C₅₃×#427	C₄	semidirect
31. C₂×#107×C₅₃	C₄	semidirect
32. C₂×#107×C₅₃	C₄	semidirect
33. C₂×#106×C₅₃	C₄	semidirect
198. #427	Dic₅₃	semidirect
199. #427	Dic₅₃	semidirect
200. #427	Dic₅₃	semidirect
204. C₂×#107	Dic₅₃	semidirect
205. C₂×#107	Dic₅₃	semidirect
206. C₂×#106	Dic₅₃	semidirect
217. C₅₃	#22607	semidirect

Maximal subgroups

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

subgroup	index
#20283373	2
#20275469	2
#20275316	2
#20274814	2
#20274753	2
#20274601	2
#20274491	2
C₂×C₅₃×#427	2
#22607	53