Subgroups — #27387645

Order 13568.

874 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133153	C₂	1	1	yes	maximal, characteristic
2	#20283304	C₂	1	2	yes	maximal
3	C₂×#16810661	C₂	1	1	yes	maximal, characteristic
4	#20275500	C₂	1	4	yes	maximal, complemented
5	#20275495	C₂	1	4	yes	maximal, complemented
6	#20274453	C₂	1	2	yes	maximal
7	C₂×C₅₃×#405	C₂	1	1	yes	maximal, characteristic
8	C₂²×#11719917	C₂²	1	2	yes	—
9	C₂×#13133176	C₂²	1	2	yes	—
10	C₂×#13133153	C₄	1	8	yes	complemented
11	C₂×#13133153	C₂²	1	2	yes	—
12	C₂×#13133153	C₂²	1	2	yes	—
13	#16810666	C₂²	1	8	yes	—
14	#16810663	—	2	4	no	—
15	#16810661	—	2	1	no	—
16	#16810661	—	2	1	no	—
17	#16810661	—	2	1	no	—
18	#16810661	—	2	1	no	—
19	#16810659	—	2	4	no	—
20	C₂×#13133151	C₂²	1	2	yes	—
21	C₂×#13133149	C₂²	1	2	yes	—
22	#16810391	C₂²	1	8	yes	—
23	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
24	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
25	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
26	C₅₃×#405	—	2	1	no	—
27	C₅₃×#405	—	2	1	no	—
28	C₅₃×#405	C₂²	1	2	yes	—
29	C₅₃×#405	C₂²	1	2	yes	—
30	C₂×#97×C₅₃	—	2	1	no	—
31	C₂×C₄×C₈×C₅₃	—	2	1	no	—
32	C₂×#11719917	—	2	2	no	—
33	C₂×#11719917	—	2	4	no	—
34	C₂×#11719917	C₂×C₄	1	8	yes	—
35	C₂×#11719917	C₂³	1	4	yes	—
36	C₂²×#88322	—	2	2	no	—
37	#13133176	—	2	2	no	—
38	#13133176	—	2	2	no	—
39	#13133176	—	2	2	no	—
40	#13133176	—	2	2	no	—
41	C₂×C₄⋊Dic₅₃	C₂³	1	4	yes	—
42	C₂×C₂²⋊Dic₅₃	—	2	2	no	—
43	C₂×C₄×Dic₅₃	—	2	2	no	—
44	#13133153	C₂×C₄	1	8	yes	complemented
45	#13133153	C₂×C₄	1	8	yes	complemented
46	#13133151	—	2	2	no	—
47	#13133151	—	2	2	no	—
48	#13133150	—	2	8	no	—
49	#13133149	—	2	2	no	—
50	#13133149	—	2	2	no	—
51	#13133148	—	2	8	no	—
52	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
53	C₂×M₁₆×C₅₃	—	2	1	no	—
54	C₂×M₁₆×C₅₃	—	2	2	no	—
55	C₂×M₁₆×C₅₃	Q₈	1	4	yes	—
56	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
57	C₂²×C₈×C₅₃	—	2	1	no	—
58	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
59	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
60	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
61	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
62	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
63	C₂²×C₈×C₅₃	Q₈	1	4	yes	—
64	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
65	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
66	#117×C₅₃	—	2	1	no	—
67	#117×C₅₃	—	2	1	no	—
68	#117×C₅₃	D₄	1	2	yes	—
69	#117×C₅₃	C₂³	1	2	yes	—
70	C₂×C₄⋊C₄×C₅₃	D₄	1	2	yes	—
71	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
72	C₂×C₄²×C₅₃	—	2	1	no	—
73	#97×C₅₃	—	2	2	no	—
74	#97×C₅₃	—	2	2	no	—
75	C₄×C₈×C₅₃	—	2	2	no	—
76	C₄×C₈×C₅₃	—	2	2	no	—
77	C₂⁴×C₅₃	#41	1	1	yes	characteristic
78	C₂²×C₄×C₅₃	—	2	2	no	—
79	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
80	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
85	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
86	C₂²×C₄×C₅₃	#41	1	2	yes	—
87	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
88	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
89	C₂²×Dic₅₃	—	2	4	no	—
90	M₁₆×C₅₃	—	2	4	no	—
91	M₁₆×C₅₃	—	2	2	no	—
92	M₁₆×C₅₃	C₂×Q₈	1	4	yes	—
93	C₂×C₈×C₅₃	—	2	4	no	—
94	C₂×C₈×C₅₃	—	2	2	no	—
95	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
96	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
97	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
98	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
99	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
100	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
101	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
103	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
104	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
106	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
107	C₄⋊C₄×C₅₃	—	2	2	no	—
108	C₄⋊C₄×C₅₃	C₂×D₄	1	4	yes	—
109	C₂²⋊C₄×C₅₃	—	2	2	no	—
110	C₂²⋊C₄×C₅₃	—	2	2	no	—
111	C₄²×C₅₃	—	2	2	no	—
112	C₄²×C₅₃	—	2	2	no	—
113	C₂²⋊Dic₅₃	—	4	2	no	—
114	C₂²⋊Dic₅₃	—	4	2	no	—
115	C₄⋊Dic₅₃	—	2	4	no	—
116	C₄⋊Dic₅₃	—	2	4	no	—
117	C₄×Dic₅₃	—	4	2	no	—
118	C₄×Dic₅₃	—	4	2	no	—
119	#11719917	—	2	8	no	—
120	#11719917	—	2	4	no	—
121	#11719917	C₂²×C₄	1	8	yes	—
122	C₂×#88322	—	2	8	no	—
123	C₂×#88322	—	2	4	no	—
124	C₂³×C₅₃	—	4	2	no	—
125	C₂³×C₅₃	C₄×Q₈	1	2	yes	—
126	C₂³×C₅₃	#121	1	1	yes	characteristic
127	C₂³×C₅₃	#121	1	1	yes	characteristic
128	C₂³×C₅₃	#125	1	2	yes	—
129	C₂³×C₅₃	C₂×#41	1	1	yes	characteristic
130	C₂×C₄×C₅₃	—	2	4	no	—
131	C₂×C₄×C₅₃	—	4	4	no	—
132	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
133	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
134	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
135	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
136	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
137	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
138	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
139	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
140	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
141	C₂×C₄×C₅₃	#125	1	2	yes	—
142	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
143	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
144	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
145	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
146	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
147	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
148	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
149	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
150	C₂×Dic₅₃	—	4	4	no	—
151	C₂×Dic₅₃	—	8	4	no	—
152	C₈×C₅₃	—	2	4	no	—
153	C₈×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
154	C₈×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
155	C₈×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
156	C₈×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
157	#88322	—	2	8	no	—
158	#21181	—	53	1	no	maximal, Sylow
159	C₂²×C₅₃	—	2	2	no	—
160	C₂²×C₅₃	—	2	2	no	—
161	C₂²×C₅₃	—	2	2	no	—
162	C₂²×C₅₃	—	4	2	no	—
163	C₂²×C₅₃	—	4	2	no	—
164	C₂²×C₅₃	#389	1	2	yes	—
165	C₂²×C₅₃	#389	1	2	yes	—
166	C₂²×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
167	C₂²×C₅₃	C₂×#121	1	1	yes	characteristic
168	C₂²×C₅₃	C₂×#125	1	1	yes	characteristic
169	C₄×C₅₃	—	4	4	no	—
170	C₄×C₅₃	#389	1	2	yes	—
171	C₄×C₅₃	#389	1	2	yes	—
172	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
173	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
174	C₄×C₅₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
175	C₄×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
176	Dic₅₃	—	8	4	no	—
177	C₂²×#108	—	53	1	no	—
178	#2911	—	53	2	no	—
179	C₂×#427	—	53	1	no	—
180	C₂×#405	D₅₃	1	1	yes	characteristic
181	#1952	—	53	4	no	—
182	#1951	—	53	4	no	—
183	#1742	—	53	2	no	—
184	C₂×C₅₃	—	2	1	no	—
185	C₂×C₅₃	—	2	1	no	—
186	C₂×C₅₃	—	2	2	no	—
187	C₂×C₅₃	—	2	2	no	—
188	C₂×C₅₃	C₂×#389	1	1	yes	characteristic
189	C₂×C₅₃	#3067	1	2	yes	—
190	C₂²×M₁₆	—	53	2	no	—
191	C₂²×M₁₆	C₂×D₅₃	1	1	yes	characteristic
192	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
193	C₂×#117	—	53	2	no	—
194	C₂×#117	C₂×D₅₃	1	1	yes	characteristic
195	C₂×#108	—	53	8	no	—
196	C₂×#108	—	53	2	no	—
197	C₂×#108	—	53	2	no	—
198	#428	—	53	8	no	—
199	#428	—	106	4	no	—
200	#427	—	106	1	no	—
201	#427	—	106	1	no	—
202	#427	—	106	1	no	—
203	#427	—	106	1	no	—
204	#427	—	106	4	no	—
205	C₂×#107	—	53	2	no	—
206	C₂×#106	—	53	2	no	—
207	#405	—	2	1	no	—
208	#405	—	2	1	no	—
209	#405	C₂×D₅₃	1	2	yes	—
210	#405	C₂×D₅₃	1	2	yes	—
211	C₂×#97	—	2	1	no	—
212	C₂×C₄×C₈	—	2	1	no	—
213	#343	—	53	8	no	—
214	C₅₃	#21181	1	1	yes	characteristic, complemented, Sylow
215	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
216	C₂×M₁₆	—	2	2	no	—
217	C₂×M₁₆	—	2	1	no	—
218	C₂×M₁₆	—	53	8	no	—
219	C₂×M₁₆	—	53	4	no	—
220	C₂×M₁₆	—	106	2	no	—
221	C₂×M₁₆	—	106	4	no	—
222	C₂×M₁₆	Dic₁₀₆	1	4	yes	—
223	C₂×M₁₆	C₂²×D₅₃	1	2	yes	—
224	C₂²×C₈	—	2	1	no	—
225	C₂²×C₈	—	106	2	no	—
226	C₂²×C₈	Dic₁₀₆	1	4	yes	—
227	C₂²×C₈	C₄×D₅₃	1	4	yes	—
228	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
229	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
230	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
231	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
232	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
233	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
234	#117	—	2	1	no	—
235	#117	—	2	1	no	—
236	#117	—	106	2	no	—
237	#117	—	106	2	no	—
238	#117	—	106	2	no	—
239	#117	—	106	2	no	—
240	#117	D₂₁₂	1	2	yes	—
241	#117	C₂²×D₅₃	1	2	yes	—
242	C₂×C₄⋊C₄	—	53	4	no	—
243	C₂×C₄⋊C₄	D₂₁₂	1	2	yes	—
244	C₂×C₂²⋊C₄	—	2	1	no	—
245	C₂×C₂²⋊C₄	—	106	2	no	—
246	C₂×C₄²	—	2	1	no	—
247	C₂×C₄²	—	106	2	no	—
248	#108	—	53	8	no	—
249	#108	—	53	8	no	—
250	#107	—	106	2	no	—
251	#107	—	106	2	no	—
252	#107	—	106	8	no	—
253	#106	—	106	2	no	—
254	#106	—	106	2	no	—
255	#106	—	106	8	no	—
256	#97	—	2	2	no	—
257	#97	—	2	2	no	—
258	C₄×C₈	—	2	2	no	—
259	C₄×C₈	—	2	2	no	—
260	C₂⁴	#11719945	1	1	yes	characteristic
261	C₂²×C₄	—	2	2	no	—
262	C₂²×C₄	—	106	4	no	—
263	C₂²×C₄	C₂×Dic₁₀₆	1	2	yes	—
264	C₂²×C₄	C₂×Dic₁₀₆	1	2	yes	—
265	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
266	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
267	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
268	C₂²×C₄	#11719945	1	2	yes	—
269	C₂²×C₄	#11719945	1	1	yes	characteristic
270	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
271	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
272	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
273	M₁₆	—	2	4	no	—
274	M₁₆	—	2	2	no	—
275	M₁₆	—	53	8	no	—
276	M₁₆	—	106	8	no	—
277	M₁₆	—	106	4	no	—
278	M₁₆	C₂×Dic₁₀₆	1	4	yes	—
279	C₂×C₈	—	2	4	no	—
280	C₂×C₈	—	2	2	no	—
281	C₂×C₈	—	106	8	no	—
282	C₂×C₈	—	106	4	no	—
283	C₂×C₈	#11719919	1	4	yes	—
284	C₂×C₈	#11719919	1	4	yes	—
285	C₂×C₈	#11719919	1	4	yes	—
286	C₂×C₈	#11719919	1	4	yes	—
287	C₂×C₈	C₂×Dic₁₀₆	1	2	yes	—
288	C₂×C₈	C₂×Dic₁₀₆	1	2	yes	—
289	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
290	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
291	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
292	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
293	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
294	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
295	C₄⋊C₄	—	2	2	no	—
296	C₄⋊C₄	—	106	4	no	—
297	C₄⋊C₄	—	106	4	no	—
298	C₄⋊C₄	C₂×D₂₁₂	1	4	yes	—
299	C₂²⋊C₄	—	2	2	no	—
300	C₂²⋊C₄	—	2	2	no	—
301	C₂²⋊C₄	—	212	2	no	—
302	C₂²⋊C₄	—	212	2	no	—
303	C₄²	—	2	2	no	—
304	C₄²	—	2	2	no	—
305	C₄²	—	212	2	no	—
306	C₄²	—	212	2	no	—
307	C₂³	—	4	2	no	—
308	C₂³	C₄×Dic₁₀₆	1	2	yes	—
309	C₂³	#13133191	1	1	yes	characteristic
310	C₂³	#13133191	1	1	yes	characteristic
311	C₂³	#13133209	1	2	yes	—
312	C₂³	C₂×#11719945	1	1	yes	characteristic
313	C₂×C₄	—	2	4	no	—
314	C₂×C₄	—	4	4	no	—
315	C₂×C₄	—	424	4	no	—
316	C₂×C₄	—	212	4	no	—
317	C₂×C₄	C₂×#11719919	1	2	yes	—
318	C₂×C₄	C₂×#11719919	1	2	yes	—
319	C₂×C₄	C₄×D₂₁₂	1	2	yes	—
320	C₂×C₄	C₄×D₂₁₂	1	2	yes	—
321	C₂×C₄	C₄×Dic₁₀₆	1	2	yes	—
322	C₂×C₄	#13133191	1	1	yes	characteristic
323	C₂×C₄	#13133191	1	1	yes	characteristic
324	C₂×C₄	#13133196	1	2	yes	—
325	C₂×C₄	#13133196	1	2	yes	—
326	C₂×C₄	#13133209	1	2	yes	—
327	C₂×C₄	#13133217	1	2	yes	—
328	C₂×C₄	#13133217	1	2	yes	—
329	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
330	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
331	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
332	C₂×C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
333	C₂×C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
334	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
335	C₈	—	2	4	no	—
336	C₈	—	106	8	no	—
337	C₈	C₂×#11719919	1	2	yes	—
338	C₈	C₂×#11719919	1	2	yes	—
339	C₈	C₂×#11719919	1	2	yes	—
340	C₈	C₂×#11719919	1	2	yes	—
341	C₂²	—	2	2	no	—
342	C₂²	—	2	2	no	—
343	C₂²	—	2	2	no	—
344	C₂²	—	4	2	no	—
345	C₂²	—	4	2	no	—
346	C₂²	#16810521	1	2	yes	—
347	C₂²	#16810521	1	2	yes	—
348	C₂²	C₂×C₄×Dic₁₀₆	1	1	yes	characteristic
349	C₂²	C₂×#13133191	1	1	yes	characteristic
350	C₂²	C₂×#13133209	1	1	yes	characteristic
351	C₄	—	4	4	no	—
352	C₄	—	424	4	no	—
353	C₄	#16810521	1	2	yes	—
354	C₄	#16810521	1	2	yes	—
355	C₄	C₂²×#11719919	1	1	yes	characteristic
356	C₄	C₂×C₄×D₂₁₂	1	1	yes	characteristic
357	C₄	C₂×#13133196	1	1	yes	characteristic
358	C₄	C₂×#13133217	1	1	yes	characteristic
359	C₂	—	2	2	no	—
360	C₂	—	2	2	no	—
361	C₂	—	2	1	no	—
362	C₂	—	2	1	no	—
363	C₂	C₂×#16810521	1	1	yes	characteristic
364	C₂	#20284435	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#13133191
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂×C₅₃×#405	C₂
socle	C₂²×C₅₃	C₂×#121
prime radical	C₂⁴×C₅₃	#41
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₂×#405	D₅₃
hypocenter	C₅₃	#21181
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
4. #20275500	C₂	semidirect
5. #20275495	C₂	semidirect
10. C₂×#13133153	C₄	semidirect
44. #13133153	C₂×C₄	semidirect
45. #13133153	C₂×C₄	semidirect
214. C₅₃	#21181	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13133153	2
#20283304	2
C₂×#16810661	2
#20275500	2
#20275495	2
#20274453	2
C₂×C₅₃×#405	2
#21181	53