Subgroups — #27384559

Order 13568.

954 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810671	C₂	1	2	yes	maximal
2	C₂²×#13133153	C₂	1	1	yes	maximal, characteristic
3	#20283278	C₂	1	2	yes	maximal
4	C₂×#16810578	C₂	1	1	yes	maximal, characteristic
5	#20274888	C₂	1	4	yes	maximal, complemented
6	#20274885	C₂	1	4	yes	maximal, complemented
7	C₂²×#108×C₅₃	C₂	1	1	yes	maximal, characteristic
8	C₂²×#11719917	C₂²	1	2	yes	—
9	C₂²×#11719917	C₂²	1	1	yes	characteristic
10	C₂×#13133176	C₂²	1	1	yes	characteristic
11	#16810671	—	2	8	no	—
12	C₂×#13133153	—	2	4	no	—
13	C₂×#13133153	C₂²	1	2	yes	—
14	C₂×#13133153	C₂²	1	2	yes	—
15	C₂×#13133152	C₂²	1	8	yes	—
16	#16810651	—	2	4	no	—
17	#16810649	—	2	4	no	—
18	#16810646	C₂²	1	8	yes	—
19	#16810578	C₄	1	4	yes	—
20	#16810578	C₄	1	4	yes	—
21	C₂×#13133127	C₂²	1	2	yes	—
22	C₂×C₈×Dic₅₃	C₂²	1	2	yes	—
23	C₂²×M₁₆×C₅₃	C₂²	1	2	yes	—
24	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
25	C₂×#108×C₅₃	—	2	4	no	—
26	C₂×#108×C₅₃	C₂²	1	2	yes	—
27	C₂×#108×C₅₃	C₂²	1	2	yes	—
28	C₂×#11719917	—	2	2	no	—
29	C₂×#11719917	—	2	4	no	—
30	C₂×#11719917	—	2	4	no	—
31	C₂×#11719917	C₂×C₄	1	2	yes	—
32	C₂×#11719917	C₂×C₄	1	2	yes	—
33	C₂×#11719917	D₄	1	4	yes	—
34	C₂×#11719917	Q₈	1	4	yes	—
35	C₂×#11719917	C₂³	1	4	yes	—
36	C₂²×#88322	—	2	2	no	—
37	C₂²×#88322	C₂³	1	2	yes	—
38	#13133176	C₂×C₄	1	2	yes	—
39	#13133176	C₂×C₄	1	2	yes	—
40	#13133176	D₄	1	2	yes	—
41	#13133176	D₄	1	2	yes	—
42	C₂×C₄⋊Dic₅₃	Q₈	1	2	yes	—
43	C₂×C₂²⋊Dic₅₃	Q₈	1	2	yes	—
44	C₂×C₄×Dic₅₃	C₂³	1	2	yes	—
45	#13133153	—	2	4	no	—
46	#13133153	—	2	4	no	—
47	#13133152	—	2	16	no	—
48	#13133146	—	2	16	no	—
49	#13133127	C₂×C₄	1	8	yes	—
50	C₈×Dic₅₃	C₂×C₄	1	8	yes	—
51	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
52	C₂×M₁₆×C₅₃	—	2	2	no	—
53	C₂×M₁₆×C₅₃	—	2	4	no	—
54	C₂×M₁₆×C₅₃	—	2	4	no	—
55	C₂×M₁₆×C₅₃	C₂³	1	4	yes	—
56	C₂²×C₈×C₅₃	—	2	2	no	—
57	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
58	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
59	C₂²×C₈×C₅₃	D₄	1	4	yes	—
60	C₂²×C₈×C₅₃	Q₈	1	4	yes	—
61	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
62	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
63	#108×C₅₃	—	2	4	no	—
64	#108×C₅₃	—	2	4	no	—
65	C₂⁴×C₅₃	C₂×Q₈	1	1	yes	characteristic
66	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
69	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
70	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
73	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
74	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
75	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
76	C₂²×Dic₅₃	C₂×Q₈	1	4	yes	—
77	M₁₆×C₅₃	—	2	4	no	—
78	M₁₆×C₅₃	—	2	4	no	—
79	M₁₆×C₅₃	—	4	4	no	—
80	C₂×C₈×C₅₃	—	2	4	no	—
81	C₂×C₈×C₅₃	—	4	4	no	—
82	C₂×C₈×C₅₃	C₄⋊C₄	1	8	yes	—
83	C₂×C₈×C₅₃	C₄⋊C₄	1	8	yes	—
84	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
85	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
86	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
87	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
88	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
89	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
90	C₂²⋊Dic₅₃	C₄⋊C₄	1	4	yes	—
91	C₂²⋊Dic₅₃	C₄⋊C₄	1	4	yes	—
92	C₄⋊Dic₅₃	C₄⋊C₄	1	4	yes	—
93	C₄⋊Dic₅₃	C₄⋊C₄	1	4	yes	—
94	C₄×Dic₅₃	C₂²×C₄	1	4	yes	—
95	C₄×Dic₅₃	C₂×D₄	1	4	yes	—
96	#11719917	—	2	4	no	—
97	#11719917	—	2	4	no	—
98	#11719917	—	4	4	no	—
99	#11719917	C₄⋊C₄	1	8	yes	—
100	#11719917	C₄⋊C₄	1	8	yes	—
101	C₂×#88322	—	2	4	no	—
102	C₂×#88322	—	4	4	no	—
103	C₂×#88322	C₂²×C₄	1	4	yes	—
104	C₂×#88322	C₂×D₄	1	4	yes	—
105	C₂×#88322	C₂×Q₈	1	4	yes	—
106	C₂³×C₅₃	—	4	2	no	—
107	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
108	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₂³×C₅₃	C₄⋊Q₈	1	2	yes	—
110	C₂³×C₅₃	C₄⋊Q₈	1	2	yes	—
111	C₂³×C₅₃	C₂²×Q₈	1	1	yes	characteristic
112	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
113	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
114	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
117	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
118	C₂×C₄×C₅₃	#127	1	2	yes	—
119	C₂×C₄×C₅₃	#127	1	2	yes	—
120	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
121	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
122	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
123	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
124	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
125	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
126	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
127	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
128	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
129	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
130	C₂×Dic₅₃	—	4	4	no	—
131	C₂×Dic₅₃	C₂×C₄⋊C₄	1	8	yes	—
132	C₈×C₅₃	—	4	4	no	—
133	C₈×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
134	C₈×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
135	#88322	—	4	4	no	—
136	#88322	C₂×C₄⋊C₄	1	8	yes	—
137	#20807	—	53	1	no	maximal, Sylow
138	C₂²×C₅₃	—	2	2	no	—
139	C₂²×C₅₃	—	2	2	no	—
140	C₂²×C₅₃	—	2	2	no	—
141	C₂²×C₅₃	—	4	2	no	—
142	C₂²×C₅₃	—	4	2	no	—
143	C₂²×C₅₃	#384	1	2	yes	—
144	C₂²×C₅₃	#384	1	2	yes	—
145	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
146	C₂²×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
147	C₂²×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
148	C₄×C₅₃	#384	1	2	yes	—
149	C₄×C₅₃	#384	1	2	yes	—
150	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
151	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
152	C₄×C₅₃	C₂×#127	1	1	yes	characteristic
153	C₄×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
154	Dic₅₃	—	2	8	no	—
155	C₂×#430	—	53	2	no	—
156	C₂²×#108	—	53	1	no	—
157	C₂²×#108	D₅₃	1	1	yes	characteristic
158	#2905	—	53	2	no	—
159	C₂×#405	—	53	1	no	—
160	#1849	—	53	4	no	—
161	#1848	—	53	4	no	—
162	C₂×C₅₃	—	2	1	no	—
163	C₂×C₅₃	—	2	1	no	—
164	C₂×C₅₃	—	2	2	no	—
165	C₂×C₅₃	—	2	2	no	—
166	C₂×C₅₃	C₂×#384	1	1	yes	characteristic
167	C₂×C₅₃	#2916	1	2	yes	—
168	C₂²×M₁₆	—	53	1	no	—
169	C₂²×M₁₆	—	53	2	no	—
170	C₂²×M₁₆	C₂×D₅₃	1	2	yes	—
171	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
172	C₂×#117	—	53	1	no	—
173	#430	—	106	8	no	—
174	C₂×#108	—	2	4	no	—
175	C₂×#108	—	53	8	no	—
176	C₂×#108	—	53	2	no	—
177	C₂×#108	—	53	2	no	—
178	C₂×#108	—	106	4	no	—
179	C₂×#108	C₂×D₅₃	1	2	yes	—
180	C₂×#108	C₂×D₅₃	1	2	yes	—
181	#424	—	106	4	no	—
182	#424	—	106	4	no	—
183	#423	—	53	8	no	—
184	#405	—	53	4	no	—
185	#405	—	53	4	no	—
186	C₂×#97	—	53	2	no	—
187	C₂×C₄×C₈	—	53	2	no	—
188	C₅₃	#20807	1	1	yes	characteristic, complemented, Sylow
189	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
190	C₂×M₁₆	—	2	2	no	—
191	C₂×M₁₆	—	2	4	no	—
192	C₂×M₁₆	—	2	4	no	—
193	C₂×M₁₆	—	53	2	no	—
194	C₂×M₁₆	—	53	2	no	—
195	C₂×M₁₆	—	53	4	no	—
196	C₂×M₁₆	—	53	4	no	—
197	C₂×M₁₆	—	53	4	no	—
198	C₂×M₁₆	—	106	2	no	—
199	C₂×M₁₆	—	106	4	no	—
200	C₂×M₁₆	—	106	4	no	—
201	C₂×M₁₆	C₂²×D₅₃	1	4	yes	—
202	C₂²×C₈	—	2	2	no	—
203	C₂²×C₈	—	53	2	no	—
204	C₂²×C₈	—	106	2	no	—
205	C₂²×C₈	Dic₁₀₆	1	4	yes	—
206	C₂²×C₈	C₄×D₅₃	1	2	yes	—
207	C₂²×C₈	C₄×D₅₃	1	2	yes	—
208	C₂²×C₈	C₂²⋊D₅₃	1	4	yes	—
209	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
210	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
211	#117	—	53	2	no	—
212	#117	—	53	2	no	—
213	#117	—	53	2	no	—
214	#117	—	53	2	no	—
215	C₂×C₄⋊C₄	—	53	2	no	—
216	C₂×C₂²⋊C₄	—	53	2	no	—
217	C₂×C₄²	—	53	2	no	—
218	#108	—	2	4	no	—
219	#108	—	2	4	no	—
220	#108	—	106	16	no	—
221	#108	—	106	4	no	—
222	#108	—	106	4	no	—
223	C₄⋊C₈	—	106	16	no	—
224	#97	—	53	8	no	—
225	C₄×C₈	—	53	8	no	—
226	C₂⁴	Q₈×D₅₃	1	1	yes	characteristic
227	C₂²×C₄	—	53	4	no	—
228	C₂²×C₄	C₂×Dic₁₀₆	1	2	yes	—
229	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
230	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
231	C₂²×C₄	D₄×D₅₃	1	2	yes	—
232	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
233	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
234	C₂²×C₄	Q₈×D₅₃	1	1	yes	characteristic
235	C₂²×C₄	Q₈×D₅₃	1	2	yes	—
236	C₂²×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
237	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
238	M₁₆	—	2	4	no	—
239	M₁₆	—	2	4	no	—
240	M₁₆	—	4	4	no	—
241	M₁₆	—	53	8	no	—
242	M₁₆	—	53	8	no	—
243	M₁₆	—	106	4	no	—
244	M₁₆	—	106	4	no	—
245	M₁₆	—	212	4	no	—
246	C₂×C₈	—	2	4	no	—
247	C₂×C₈	—	4	4	no	—
248	C₂×C₈	—	53	4	no	—
249	C₂×C₈	—	53	4	no	—
250	C₂×C₈	—	53	4	no	—
251	C₂×C₈	—	106	4	no	—
252	C₂×C₈	—	212	4	no	—
253	C₂×C₈	#11719919	1	8	yes	—
254	C₂×C₈	#11719919	1	8	yes	—
255	C₂×C₈	C₂×Dic₁₀₆	1	2	yes	—
256	C₂×C₈	C₂×Dic₁₀₆	1	2	yes	—
257	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
258	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
259	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
260	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
261	C₄⋊C₄	—	53	4	no	—
262	C₄⋊C₄	—	53	4	no	—
263	C₂²⋊C₄	—	53	4	no	—
264	C₂²⋊C₄	—	53	4	no	—
265	C₄²	—	53	4	no	—
266	C₄²	—	53	4	no	—
267	C₂³	—	4	2	no	—
268	C₂³	C₄⋊C₄×D₅₃	1	1	yes	characteristic
269	C₂³	C₄⋊C₄×D₅₃	1	1	yes	characteristic
270	C₂³	#13133215	1	2	yes	—
271	C₂³	#13133216	1	2	yes	—
272	C₂³	C₂×Q₈×D₅₃	1	1	yes	characteristic
273	C₂×C₄	—	53	8	no	—
274	C₂×C₄	—	212	4	no	—
275	C₂×C₄	C₄⋊C₄×D₅₃	1	2	yes	—
276	C₂×C₄	C₄⋊C₄×D₅₃	1	2	yes	—
277	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
278	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
279	C₂×C₄	C₂×#11719919	1	2	yes	—
280	C₂×C₄	C₂×#11719919	1	2	yes	—
281	C₂×C₄	#13133213	1	2	yes	—
282	C₂×C₄	#13133213	1	2	yes	—
283	C₂×C₄	#13133215	1	2	yes	—
284	C₂×C₄	#13133216	1	2	yes	—
285	C₂×C₄	#13133216	1	2	yes	—
286	C₂×C₄	#13133216	1	2	yes	—
287	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
288	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
289	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
290	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
291	C₂×C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
292	C₂×C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
293	C₈	—	4	4	no	—
294	C₈	—	53	8	no	—
295	C₈	—	212	4	no	—
296	C₈	C₂×#11719919	1	4	yes	—
297	C₈	C₂×#11719919	1	4	yes	—
298	C₂²	—	2	2	no	—
299	C₂²	—	2	2	no	—
300	C₂²	—	2	2	no	—
301	C₂²	—	4	2	no	—
302	C₂²	—	4	2	no	—
303	C₂²	#16810492	1	2	yes	—
304	C₂²	#16810492	1	2	yes	—
305	C₂²	C₂×C₄⋊C₄×D₅₃	1	1	yes	characteristic
306	C₂²	C₂×#13133215	1	1	yes	characteristic
307	C₂²	C₂×#13133216	1	1	yes	characteristic
308	C₄	—	106	8	no	—
309	C₄	#16810492	1	2	yes	—
310	C₄	#16810492	1	2	yes	—
311	C₄	C₂×C₄⋊C₄×D₅₃	1	1	yes	characteristic
312	C₄	C₂²×#11719919	1	1	yes	characteristic
313	C₄	C₂×#13133213	1	1	yes	characteristic
314	C₄	C₂×#13133216	1	1	yes	characteristic
315	C₂	—	2	1	no	—
316	C₂	—	2	1	no	—
317	C₂	—	2	2	no	—
318	C₂	—	2	2	no	—
319	C₂	C₂×#16810492	1	1	yes	characteristic
320	C₂	#20283323	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂²×#108×C₅₃	C₂
socle	C₂²×C₅₃	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₅₃	C₂×Q₈
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₂²×#108	D₅₃
hypocenter	C₅₃	#20807
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×#108

Complements

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

N	H ≅ G/N	type
5. #20274888	C₂	semidirect
6. #20274885	C₂	semidirect
188. C₅₃	#20807	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810671	2
C₂²×#13133153	2
#20283278	2
C₂×#16810578	2
#20274888	2
#20274885	2
C₂²×#108×C₅₃	2
#20807	53