Subgroups — #27537199

Order 13568.

600 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	#20276522	C₂	1	4	yes	maximal, complemented
2	#20276518	C₂	1	2	yes	maximal, complemented
3	#20276514	C₂	1	2	yes	maximal, complemented
4	#20276474	C₂	1	2	yes	maximal, complemented
5	#20276326	C₂	1	1	yes	maximal, characteristic
6	C₂×C₂²⋊#11719908	C₂	1	1	yes	maximal, characteristic
7	#20276312	C₂	1	2	yes	maximal, complemented
8	C₂×C₅₃×#405	C₂	1	1	yes	maximal, characteristic
9	C₂×#13133157	C₂²	1	2	yes	—
10	C₂²×#11719908	C₂²	1	2	yes	—
11	C₄⋊#11719908	C₂²	1	4	yes	complemented
12	C₄⋊#11719908	C₂²	1	4	yes	complemented
13	C₂²⋊#11719908	C₂²	1	1	yes	characteristic
14	C₂²⋊#11719908	C₂²	1	2	yes	—
15	C₂²⋊#11719908	C₂²	1	1	yes	characteristic
16	C₂²⋊#11719908	C₂²	1	2	yes	—
17	C₂²⋊#11719908	C₂²	1	2	yes	—
18	#16810399	C₂²	1	4	yes	complemented
19	C₄×#11719908	C₂²	1	4	yes	complemented
20	C₂²×M₁₆×C₅₃	C₄	1	1	yes	characteristic
21	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
22	C₂×#117×C₅₃	C₄	1	1	yes	characteristic
23	C₅₃×#405	C₄	1	1	yes	characteristic
24	C₅₃×#405	C₄	1	2	yes	—
25	C₅₃×#405	C₄	1	1	yes	characteristic
26	C₅₃×#405	C₂²	1	2	yes	—
27	C₅₃×#405	C₂²	1	1	yes	characteristic
28	C₅₃×#405	C₂²	1	1	yes	characteristic
29	C₂×#97×C₅₃	C₄	1	2	yes	—
30	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
31	#13133157	—	2	4	no	—
32	C₂×#11719908	—	2	4	no	—
33	C₂×#11719908	C₂³	1	2	yes	—
34	C₂×#11719908	C₂³	1	2	yes	—
35	C₂×#11719908	C₂³	1	4	yes	—
36	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
37	C₂×M₁₆×C₅₃	—	2	4	no	—
38	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
39	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
40	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₈×C₅₃	—	2	4	no	—
42	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
46	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
47	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
48	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
49	#117×C₅₃	C₈	1	2	yes	—
50	#117×C₅₃	C₈	1	1	yes	characteristic
51	#117×C₅₃	C₈	1	1	yes	characteristic
52	#117×C₅₃	C₂×C₄	1	2	yes	—
53	#117×C₅₃	C₂×C₄	1	1	yes	characteristic
54	#117×C₅₃	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄⋊C₄×C₅₃	C₈	1	1	yes	characteristic
56	C₂×C₄⋊C₄×C₅₃	C₈	1	1	yes	characteristic
57	C₂×C₂²⋊C₄×C₅₃	C₈	1	2	yes	—
58	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
59	#97×C₅₃	C₂×C₄	1	2	yes	—
60	#97×C₅₃	C₂×C₄	1	2	yes	—
61	#97×C₅₃	C₂×C₄	1	2	yes	—
62	#97×C₅₃	C₂×C₄	1	2	yes	—
63	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
64	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
65	C₄×C₈×C₅₃	C₂³	1	2	yes	—
66	C₄×C₈×C₅₃	C₂³	1	2	yes	—
67	C₂⁴×C₅₃	C₂×C₈	1	1	yes	characteristic
68	C₂²×C₄×C₅₃	—	2	2	no	—
69	C₂²×C₄×C₅₃	—	2	2	no	—
70	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
71	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
72	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
73	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
74	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
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	M₁₆×C₅₃	—	2	2	no	—
79	M₁₆×C₅₃	—	2	4	no	—
80	M₁₆×C₅₃	M₁₆	1	4	yes	—
81	C₂×C₈×C₅₃	—	2	2	no	—
82	C₂×C₈×C₅₃	—	2	2	no	—
83	C₂×C₈×C₅₃	—	2	2	no	—
84	C₂×C₈×C₅₃	—	2	4	no	—
85	C₂×C₈×C₅₃	—	2	2	no	—
86	C₂×C₈×C₅₃	—	2	2	no	—
87	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
89	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
92	C₂×C₈×C₅₃	#41	1	4	yes	—
93	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
94	C₄⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
95	C₄⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
96	C₄⋊C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
97	C₄⋊C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
98	C₄⋊C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
99	C₄⋊C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
100	C₂²⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
101	C₂²⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
102	C₂²⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
103	C₂²⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
104	C₄²×C₅₃	C₂×C₈	1	2	yes	—
105	C₄²×C₅₃	C₂×C₈	1	2	yes	—
106	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
107	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
108	#11719908	—	2	2	no	—
109	#11719908	—	2	2	no	—
110	#11719908	—	2	4	no	—
111	C₂³×C₅₃	—	2	2	no	—
112	C₂³×C₅₃	—	2	2	no	—
113	C₂³×C₅₃	—	4	1	no	—
114	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
115	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
116	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
117	C₂×C₄×C₅₃	—	2	2	no	—
118	C₂×C₄×C₅₃	—	2	4	no	—
119	C₂×C₄×C₅₃	—	2	1	no	—
120	C₂×C₄×C₅₃	—	2	1	no	—
121	C₂×C₄×C₅₃	—	2	1	no	—
122	C₂×C₄×C₅₃	—	2	1	no	—
123	C₂×C₄×C₅₃	—	2	2	no	—
124	C₂×C₄×C₅₃	—	2	2	no	—
125	C₂×C₄×C₅₃	—	4	2	no	—
126	C₂×C₄×C₅₃	#117	1	2	yes	—
127	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
128	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
129	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
130	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
131	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
132	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
133	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
134	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
135	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
136	C₈×C₅₃	—	2	1	no	—
137	C₈×C₅₃	—	2	1	no	—
138	C₈×C₅₃	—	2	2	no	—
139	C₈×C₅₃	#117	1	2	yes	—
140	C₈×C₅₃	C₂×M₁₆	1	4	yes	—
141	C₈×C₅₃	C₂×#41	1	2	yes	—
142	#33422	—	53	1	no	maximal, Sylow
143	C₂²×C₅₃	—	2	1	no	—
144	C₂²×C₅₃	—	2	1	no	—
145	C₂²×C₅₃	—	2	1	no	—
146	C₂²×C₅₃	—	2	1	no	—
147	C₂²×C₅₃	—	2	1	no	—
148	C₂²×C₅₃	—	2	1	no	—
149	C₂²×C₅₃	—	2	1	no	—
150	C₂²×C₅₃	—	4	4	no	—
151	C₂²×C₅₃	#431	1	2	yes	—
152	C₂²×C₅₃	#504	1	2	yes	—
153	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
154	C₄×C₅₃	—	2	2	no	—
155	C₄×C₅₃	—	2	2	no	—
156	C₄×C₅₃	—	2	2	no	—
157	C₄×C₅₃	—	2	2	no	—
158	C₄×C₅₃	—	2	1	no	—
159	C₄×C₅₃	—	2	1	no	—
160	C₄×C₅₃	#431	1	2	yes	—
161	C₄×C₅₃	C₂×#117	1	1	yes	characteristic
162	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
163	C₂×#405	D₅₃	1	1	yes	characteristic
164	#2164	—	53	4	no	—
165	#2163	—	53	2	no	—
166	#2162	—	53	2	no	—
167	#2151	—	53	2	no	—
168	#2112	—	53	1	no	—
169	C₂×C₂²⋊C₁₆	—	53	1	no	—
170	#2108	—	53	2	no	—
171	C₂×C₅₃	—	2	2	no	—
172	C₂×C₅₃	—	2	2	no	—
173	C₂×C₅₃	—	4	1	no	—
174	C₂×C₅₃	C₂×#431	1	1	yes	characteristic
175	C₂×C₅₃	C₂×#504	1	1	yes	characteristic
176	C₂×C₅₃	#3407	1	1	yes	characteristic
177	C₂²×M₁₆	Dic₅₃	1	1	yes	characteristic
178	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
179	C₂×#117	Dic₅₃	1	1	yes	characteristic
180	C₂×M₃₂	—	53	2	no	—
181	C₂²×C₁₆	—	53	2	no	—
182	#405	Dic₅₃	1	1	yes	characteristic
183	#405	Dic₅₃	1	2	yes	—
184	#405	Dic₅₃	1	1	yes	characteristic
185	#405	C₂×D₅₃	1	1	yes	characteristic
186	#405	C₂×D₅₃	1	1	yes	characteristic
187	#405	C₂×D₅₃	1	2	yes	—
188	C₂×#97	Dic₅₃	1	2	yes	—
189	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
190	C₄⋊C₁₆	—	53	4	no	—
191	C₄⋊C₁₆	—	53	4	no	—
192	C₂²⋊C₁₆	—	53	1	no	—
193	C₂²⋊C₁₆	—	53	1	no	—
194	C₂²⋊C₁₆	—	53	2	no	—
195	C₂²⋊C₁₆	—	53	2	no	—
196	C₂²⋊C₁₆	—	53	2	no	—
197	#346	—	53	4	no	—
198	C₄×C₁₆	—	53	4	no	—
199	C₅₃	#33422	1	1	yes	characteristic, complemented, Sylow
200	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
201	C₂×M₁₆	—	2	4	no	—
202	C₂×M₁₆	C₂×Dic₅₃	1	1	yes	characteristic
203	C₂×M₁₆	C₂×Dic₅₃	1	2	yes	—
204	C₂×M₁₆	C₂×Dic₅₃	1	1	yes	characteristic
205	C₂²×C₈	—	2	4	no	—
206	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
207	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
208	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
209	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
210	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
211	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
212	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
213	#117	#88322	1	1	yes	characteristic
214	#117	#88322	1	1	yes	characteristic
215	#117	#88322	1	2	yes	—
216	#117	C₂×Dic₅₃	1	2	yes	—
217	#117	C₂×Dic₅₃	1	1	yes	characteristic
218	#117	C₂×Dic₅₃	1	1	yes	characteristic
219	C₂×C₄⋊C₄	#88322	1	1	yes	characteristic
220	C₂×C₄⋊C₄	#88322	1	1	yes	characteristic
221	C₂×C₂²⋊C₄	#88322	1	2	yes	—
222	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
223	M₃₂	—	106	4	no	—
224	C₂×C₁₆	—	53	2	no	—
225	C₂×C₁₆	—	53	4	no	—
226	C₂×C₁₆	—	53	2	no	—
227	C₂×C₁₆	—	106	4	no	—
228	#97	C₂×Dic₅₃	1	2	yes	—
229	#97	C₂×Dic₅₃	1	2	yes	—
230	#97	C₂×Dic₅₃	1	2	yes	—
231	#97	C₂×Dic₅₃	1	2	yes	—
232	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
233	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
234	C₄×C₈	C₂²×D₅₃	1	2	yes	—
235	C₄×C₈	C₂²×D₅₃	1	2	yes	—
236	C₂⁴	C₂×#88322	1	1	yes	characteristic
237	C₂²×C₄	—	2	2	no	—
238	C₂²×C₄	—	2	2	no	—
239	C₂²×C₄	C₂×#88322	1	2	yes	—
240	C₂²×C₄	C₂×#88322	1	1	yes	characteristic
241	C₂²×C₄	C₂×#88322	1	1	yes	characteristic
242	C₂²×C₄	C₂×#88322	1	1	yes	characteristic
243	C₂²×C₄	C₂×#88322	1	2	yes	—
244	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
245	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
246	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
247	M₁₆	—	2	2	no	—
248	M₁₆	—	2	4	no	—
249	M₁₆	#11719917	1	4	yes	—
250	C₂×C₈	—	2	2	no	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	2	4	no	—
254	C₂×C₈	—	2	2	no	—
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	#11719945	1	4	yes	—
257	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
258	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
259	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
260	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
261	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
262	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
263	C₄⋊C₄	C₂×#88322	1	2	yes	—
264	C₄⋊C₄	C₂×#88322	1	2	yes	—
265	C₄⋊C₄	C₂×#88322	1	1	yes	characteristic
266	C₄⋊C₄	C₂×#88322	1	1	yes	characteristic
267	C₄⋊C₄	C₂×#88322	1	1	yes	characteristic
268	C₄⋊C₄	C₂×#88322	1	1	yes	characteristic
269	C₂²⋊C₄	C₂×#88322	1	2	yes	—
270	C₂²⋊C₄	C₂×#88322	1	2	yes	—
271	C₂²⋊C₄	C₂×#88322	1	2	yes	—
272	C₂²⋊C₄	C₂×#88322	1	2	yes	—
273	C₄²	C₂×#88322	1	2	yes	—
274	C₄²	C₂×#88322	1	2	yes	—
275	C₄²	C₂²×Dic₅₃	1	2	yes	—
276	C₄²	C₂²×Dic₅₃	1	2	yes	—
277	C₁₆	—	106	2	no	—
278	C₁₆	—	106	2	no	—
279	C₁₆	—	106	4	no	—
280	C₂³	—	2	2	no	—
281	C₂³	—	2	2	no	—
282	C₂³	—	4	1	no	—
283	C₂³	C₂²×#88322	1	1	yes	characteristic
284	C₂³	C₂²×#88322	1	1	yes	characteristic
285	C₂³	C₂²×#88322	1	1	yes	characteristic
286	C₂×C₄	—	2	4	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	#13133176	1	2	yes	—
296	C₂×C₄	C₂²×#88322	1	2	yes	—
297	C₂×C₄	C₂²×#88322	1	2	yes	—
298	C₂×C₄	C₂²×#88322	1	2	yes	—
299	C₂×C₄	C₂²×#88322	1	2	yes	—
300	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
301	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
302	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
303	C₂×C₄	C₂×#11719917	1	2	yes	—
304	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
305	C₈	—	2	1	no	—
306	C₈	—	2	1	no	—
307	C₈	—	2	2	no	—
308	C₈	#13133176	1	2	yes	—
309	C₈	C₂×#11719917	1	4	yes	—
310	C₈	C₂×#11719945	1	2	yes	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	2	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	4	4	no	—
319	C₂²	#16810676	1	2	yes	—
320	C₂²	#16811040	1	2	yes	—
321	C₂²	C₂³×#88322	1	1	yes	characteristic
322	C₄	—	2	2	no	—
323	C₄	—	2	2	no	—
324	C₄	—	2	2	no	—
325	C₄	—	2	2	no	—
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	#16810676	1	2	yes	—
329	C₄	C₂×#13133176	1	1	yes	characteristic
330	C₄	C₂²×#11719917	1	1	yes	characteristic
331	C₂	—	2	2	no	—
332	C₂	—	2	2	no	—
333	C₂	—	4	1	no	—
334	C₂	C₂×#16810676	1	1	yes	characteristic
335	C₂	C₂×#16811040	1	1	yes	characteristic
336	C₂	#20287818	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₈
Fitting	C₂×C₅₃×#405	C₂
prime radical	C₂⁴×C₅₃	C₂×C₈
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₂×#405	D₅₃
hypocenter	C₅₃	#33422
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
1. #20276522	C₂	semidirect
2. #20276518	C₂	semidirect
3. #20276514	C₂	semidirect
4. #20276474	C₂	semidirect
7. #20276312	C₂	semidirect
11. C₄⋊#11719908	C₂²	semidirect
12. C₄⋊#11719908	C₂²	semidirect
18. #16810399	C₂²	semidirect
19. C₄×#11719908	C₂²	semidirect
199. C₅₃	#33422	semidirect

Maximal subgroups

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

subgroup	index
#20276522	2
#20276518	2
#20276514	2
#20276474	2
#20276326	2
C₂×C₂²⋊#11719908	2
#20276312	2
C₂×C₅₃×#405	2
#33422	53