Subgroups — #27387616

Order 13568.

906 conjugacy classes of subgroups.

The subgroup lattice has 356 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	C₂×#16810647	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×#11719917	C₂	1	1	yes	maximal, characteristic
4	#20275488	C₂	1	8	yes	maximal, complemented
5	#20274441	C₂	1	2	yes	maximal
6	C₂×C₅₃×#423	C₂	1	1	yes	maximal, characteristic
7	C₂²×#11719917	C₂²	1	1	yes	characteristic
8	C₂²×#11719917	C₂²	1	1	yes	characteristic
9	C₂²×#11719917	C₂²	1	2	yes	—
10	#16810671	—	2	8	no	—
11	C₂×#13133152	C₂²	1	8	yes	—
12	#16810647	C₄	1	2	yes	—
13	#16810647	C₄	1	4	yes	—
14	#16810647	C₄	1	2	yes	—
15	C₂×C₄⋊#88322	C₂²	1	4	yes	—
16	C₄×#11719917	—	2	2	no	—
17	C₄×#11719917	C₂²	1	4	yes	—
18	C₂×#13133128	—	2	1	no	—
19	C₂×C₄×#88322	—	2	1	no	—
20	#16810387	C₂²	1	8	yes	—
21	C₂²×M₁₆×C₅₃	C₂²	1	2	yes	—
22	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
23	C₅₃×#423	—	2	1	no	—
24	C₅₃×#423	—	2	1	no	—
25	C₅₃×#423	C₂²	1	4	yes	—
26	C₂×C₄⋊C₈×C₅₃	—	2	2	no	—
27	C₂×#11719917	—	2	1	no	—
28	C₂×#11719917	—	2	1	no	—
29	C₂×#11719917	—	2	4	no	—
30	C₂×#11719917	—	2	2	no	—
31	C₂×#11719917	—	2	4	no	—
32	C₂×#11719917	—	2	2	no	—
33	C₂×#11719917	C₂×C₄	1	4	yes	—
34	C₂×#11719917	C₂×C₄	1	4	yes	—
35	C₂×#11719917	D₄	1	4	yes	—
36	C₂×#11719917	Q₈	1	4	yes	—
37	C₂×#11719917	C₂³	1	2	yes	—
38	C₂×#11719917	C₂³	1	2	yes	—
39	C₂²×#88322	—	2	1	no	—
40	C₂²×#88322	—	2	1	no	—
41	C₂²×#88322	C₂³	1	4	yes	—
42	#13133152	—	2	16	no	—
43	C₄⋊#88322	C₂×C₄	1	8	yes	—
44	C₄⋊#88322	C₂×C₄	1	8	yes	—
45	#13133128	—	2	4	no	—
46	C₄×#88322	—	2	4	no	—
47	C₂³×C₄×C₅₃	Q₈	1	2	yes	—
48	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
49	C₂×M₁₆×C₅₃	—	2	2	no	—
50	C₂×M₁₆×C₅₃	—	2	4	no	—
51	C₂×M₁₆×C₅₃	—	2	4	no	—
52	C₂×M₁₆×C₅₃	C₂³	1	4	yes	—
53	C₂²×C₈×C₅₃	—	2	2	no	—
54	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
56	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
58	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
59	C₂×C₄²×C₅₃	D₄	1	2	yes	—
60	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
61	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
62	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
63	C₄⋊C₈×C₅₃	—	2	4	no	—
64	C₄⋊C₈×C₅₃	—	2	4	no	—
65	C₂⁴×C₅₃	C₂×Q₈	1	1	yes	characteristic
66	C₂²×C₄×C₅₃	—	4	4	no	—
67	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
68	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
69	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
70	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
71	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
76	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
77	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
78	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
79	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
80	C₂²×C₄×C₅₃	#41	1	2	yes	—
81	C₂²×C₄×C₅₃	#41	1	2	yes	—
82	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
83	M₁₆×C₅₃	—	2	4	no	—
84	M₁₆×C₅₃	—	2	4	no	—
85	M₁₆×C₅₃	—	4	4	no	—
86	C₂×C₈×C₅₃	—	2	4	no	—
87	C₂×C₈×C₅₃	—	4	4	no	—
88	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
89	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
90	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
91	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
92	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
93	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
94	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
95	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
96	C₄²×C₅₃	C₂×D₄	1	2	yes	—
97	C₄²×C₅₃	C₂×D₄	1	2	yes	—
98	#11719917	—	2	8	no	—
99	#11719917	—	2	8	no	—
100	#11719917	—	2	4	no	—
101	#11719917	—	2	4	no	—
102	#11719917	—	2	2	no	—
103	#11719917	—	2	2	no	—
104	#11719917	C₂×D₄	1	4	yes	—
105	#11719917	C₂×Q₈	1	4	yes	—
106	C₂×#88322	—	2	4	no	—
107	C₂×#88322	—	2	4	no	—
108	C₂×#88322	—	2	2	no	—
109	C₂×#88322	—	2	2	no	—
110	C₂×#88322	—	2	4	no	—
111	C₂×#88322	—	2	4	no	—
112	C₂×#88322	C₂²×C₄	1	8	yes	—
113	C₂³×C₅₃	—	4	2	no	—
114	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
115	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
116	C₂³×C₅₃	#125	1	2	yes	—
117	C₂³×C₅₃	C₄⋊Q₈	1	2	yes	—
118	C₂³×C₅₃	C₂²×Q₈	1	1	yes	characteristic
119	C₂×C₄×C₅₃	—	2	4	no	—
120	C₂×C₄×C₅₃	—	2	4	no	—
121	C₂×C₄×C₅₃	—	2	4	no	—
122	C₂×C₄×C₅₃	—	4	2	no	—
123	C₂×C₄×C₅₃	—	4	2	no	—
124	C₂×C₄×C₅₃	—	4	2	no	—
125	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
126	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
127	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
128	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
129	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
130	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
131	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
132	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
133	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
134	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
135	C₂×C₄×C₅₃	#121	1	2	yes	—
136	C₂×C₄×C₅₃	#121	1	2	yes	—
137	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
138	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
139	C₂×C₄×C₅₃	#125	1	2	yes	—
140	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
141	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
142	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
143	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
144	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
145	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
146	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
147	C₈×C₅₃	—	4	4	no	—
148	#88322	—	2	8	no	—
149	#88322	—	2	4	no	—
150	#88322	—	2	4	no	—
151	#21177	—	53	1	no	maximal, Sylow
152	C₂²×C₅₃	—	2	2	no	—
153	C₂²×C₅₃	—	2	2	no	—
154	C₂²×C₅₃	—	2	2	no	—
155	C₂²×C₅₃	—	4	2	no	—
156	C₂²×C₅₃	—	4	2	no	—
157	C₂²×C₅₃	#389	1	2	yes	—
158	C₂²×C₅₃	#389	1	2	yes	—
159	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
160	C₂²×C₅₃	C₂×#125	1	1	yes	characteristic
161	C₂²×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
162	C₄×C₅₃	—	2	2	no	—
163	C₄×C₅₃	—	2	2	no	—
164	C₄×C₅₃	—	2	2	no	—
165	C₄×C₅₃	#389	1	2	yes	—
166	C₄×C₅₃	#389	1	2	yes	—
167	C₄×C₅₃	#430	1	4	yes	complemented
168	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
169	C₄×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
170	C₄×C₅₃	C₂×#121	1	1	yes	characteristic
171	C₄×C₅₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
172	C₂×#430	—	53	2	no	—
173	C₂×#423	—	53	1	no	—
174	C₂×#423	D₅₃	1	1	yes	characteristic
175	C₂×C₄×M₁₆	—	53	1	no	—
176	#1950	—	53	8	no	—
177	#1738	—	53	2	no	—
178	C₂×C₅₃	—	2	1	no	—
179	C₂×C₅₃	—	2	1	no	—
180	C₂×C₅₃	—	2	2	no	—
181	C₂×C₅₃	—	2	2	no	—
182	C₂×C₅₃	C₂×#389	1	1	yes	characteristic
183	C₂×C₅₃	C₂×#430	1	2	yes	—
184	C₂²×M₁₆	—	53	2	no	—
185	C₂²×M₁₆	—	53	1	no	—
186	C₂²×M₁₆	—	53	1	no	—
187	C₂²×M₁₆	C₂×D₅₃	1	2	yes	—
188	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
189	#430	—	106	8	no	—
190	C₂×#108	—	53	8	no	—
191	#423	—	2	1	no	—
192	#423	—	2	1	no	—
193	#423	—	53	2	no	—
194	#423	—	53	4	no	—
195	#423	—	53	2	no	—
196	#423	C₂×D₅₃	1	4	yes	—
197	C₂×C₄⋊C₈	—	2	2	no	—
198	C₂×C₄⋊C₈	—	53	4	no	—
199	C₄×M₁₆	—	53	4	no	—
200	C₄×M₁₆	—	106	2	no	—
201	C₂×#97	—	106	1	no	—
202	C₂×C₄×C₈	—	106	1	no	—
203	#341	—	53	8	no	—
204	C₅₃	#21177	1	1	yes	characteristic, complemented, Sylow
205	C₂³×C₄	Dic₁₀₆	1	2	yes	—
206	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
207	C₂×M₁₆	—	2	2	no	—
208	C₂×M₁₆	—	2	4	no	—
209	C₂×M₁₆	—	2	4	no	—
210	C₂×M₁₆	—	53	4	no	—
211	C₂×M₁₆	—	53	4	no	—
212	C₂×M₁₆	—	53	4	no	—
213	C₂×M₁₆	—	53	4	no	—
214	C₂×M₁₆	—	53	2	no	—
215	C₂×M₁₆	—	53	2	no	—
216	C₂×M₁₆	—	106	4	no	—
217	C₂×M₁₆	—	106	2	no	—
218	C₂×M₁₆	—	106	2	no	—
219	C₂×M₁₆	—	106	4	no	—
220	C₂×M₁₆	—	106	1	no	—
221	C₂×M₁₆	—	106	1	no	—
222	C₂×M₁₆	C₂²×D₅₃	1	4	yes	—
223	C₂²×C₈	—	2	2	no	—
224	C₂²×C₈	—	53	4	no	—
225	C₂²×C₈	—	106	1	no	—
226	C₂²×C₈	—	106	1	no	—
227	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
228	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
229	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
230	C₂×C₄²	C₄×D₅₃	1	2	yes	—
231	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
232	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
233	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
234	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
235	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
236	#108	—	106	16	no	—
237	C₄⋊C₈	—	2	4	no	—
238	C₄⋊C₈	—	2	4	no	—
239	C₄⋊C₈	—	53	8	no	—
240	C₄⋊C₈	—	53	8	no	—
241	#97	—	106	4	no	—
242	C₄×C₈	—	106	4	no	—
243	C₂⁴	C₂×Dic₁₀₆	1	1	yes	characteristic
244	C₂²×C₄	—	4	4	no	—
245	C₂²×C₄	#11719919	1	2	yes	—
246	C₂²×C₄	#11719919	1	2	yes	—
247	C₂²×C₄	#11719919	1	2	yes	—
248	C₂²×C₄	#11719919	1	2	yes	—
249	C₂²×C₄	C₂×Dic₁₀₆	1	2	yes	—
250	C₂²×C₄	C₂×Dic₁₀₆	1	2	yes	—
251	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
252	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
253	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
254	C₂²×C₄	D₄×D₅₃	1	2	yes	—
255	C₂²×C₄	#11719947	1	2	yes	—
256	C₂²×C₄	Q₈×D₅₃	1	2	yes	—
257	C₂²×C₄	#11719949	1	2	yes	—
258	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
259	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
260	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
261	M₁₆	—	2	4	no	—
262	M₁₆	—	2	4	no	—
263	M₁₆	—	4	4	no	—
264	M₁₆	—	53	4	no	—
265	M₁₆	—	53	4	no	—
266	M₁₆	—	106	8	no	—
267	M₁₆	—	106	8	no	—
268	M₁₆	—	106	4	no	—
269	M₁₆	—	106	4	no	—
270	M₁₆	—	106	2	no	—
271	M₁₆	—	106	2	no	—
272	C₂×C₈	—	2	4	no	—
273	C₂×C₈	—	4	4	no	—
274	C₂×C₈	—	53	8	no	—
275	C₂×C₈	—	106	4	no	—
276	C₂×C₈	—	106	4	no	—
277	C₂×C₈	—	106	4	no	—
278	C₂×C₈	—	106	2	no	—
279	C₂×C₈	—	106	2	no	—
280	C₂×C₈	—	106	4	no	—
281	C₄²	#11719919	1	1	yes	characteristic
282	C₄²	#11719919	1	1	yes	characteristic
283	C₄²	#11719919	1	1	yes	characteristic
284	C₄²	#11719919	1	2	yes	—
285	C₄²	#11719919	1	2	yes	—
286	C₄²	#11719919	1	1	yes	characteristic
287	C₄²	C₂×C₄×D₅₃	1	2	yes	—
288	C₄²	C₂×C₄×D₅₃	1	2	yes	—
289	C₄²	C₂×C₂²⋊D₅₃	1	2	yes	—
290	C₄²	C₂×C₂²⋊D₅₃	1	2	yes	—
291	C₂³	—	4	2	no	—
292	C₂³	C₂×#11719919	1	1	yes	characteristic
293	C₂³	C₂×#11719919	1	1	yes	characteristic
294	C₂³	#13133208	1	2	yes	—
295	C₂³	#13133216	1	2	yes	—
296	C₂³	C₂²×Dic₁₀₆	1	1	yes	characteristic
297	C₂×C₄	—	2	4	no	—
298	C₂×C₄	—	2	4	no	—
299	C₂×C₄	—	2	4	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	C₂×#11719919	1	2	yes	—
304	C₂×C₄	C₂×#11719919	1	2	yes	—
305	C₂×C₄	C₂×#11719919	1	2	yes	—
306	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
307	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
308	C₂×C₄	C₂×#11719919	1	2	yes	—
309	C₂×C₄	#13133179	1	2	yes	—
310	C₂×C₄	#13133179	1	2	yes	—
311	C₂×C₄	#13133182	1	2	yes	—
312	C₂×C₄	#13133182	1	2	yes	—
313	C₂×C₄	#13133188	1	2	yes	—
314	C₂×C₄	#13133188	1	2	yes	—
315	C₂×C₄	#13133193	1	2	yes	—
316	C₂×C₄	#13133193	1	2	yes	—
317	C₂×C₄	#13133208	1	2	yes	—
318	C₂×C₄	#13133216	1	2	yes	—
319	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
320	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
321	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
322	C₂×C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
323	C₂×C₄	C₂×#11719947	1	1	yes	characteristic
324	C₂×C₄	C₂×#11719949	1	1	yes	characteristic
325	C₈	—	4	4	no	—
326	C₈	—	106	8	no	—
327	C₈	—	106	4	no	—
328	C₈	—	106	4	no	—
329	C₂²	—	2	2	no	—
330	C₂²	—	2	2	no	—
331	C₂²	—	2	2	no	—
332	C₂²	—	4	2	no	—
333	C₂²	—	4	2	no	—
334	C₂²	#16810519	1	2	yes	—
335	C₂²	#16810519	1	2	yes	—
336	C₂²	C₂²×#11719919	1	1	yes	characteristic
337	C₂²	C₂×#13133208	1	1	yes	characteristic
338	C₂²	C₂×#13133216	1	1	yes	characteristic
339	C₄	—	2	2	no	—
340	C₄	—	2	2	no	—
341	C₄	—	2	2	no	—
342	C₄	#16810519	1	2	yes	—
343	C₄	#16810519	1	2	yes	—
344	C₄	#16810671	1	4	yes	complemented
345	C₄	C₂×#13133179	1	1	yes	characteristic
346	C₄	C₂×#13133182	1	1	yes	characteristic
347	C₄	C₂×#13133188	1	1	yes	characteristic
348	C₄	C₂×#13133193	1	1	yes	characteristic
349	C₂	—	2	2	no	—
350	C₂	—	2	2	no	—
351	C₂	—	2	1	no	—
352	C₂	—	2	1	no	—
353	C₂	C₂×#16810519	1	1	yes	characteristic
354	C₂	C₂×#16810671	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂×C₅₃×#423	C₂
socle	C₂²×C₅₃	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₅₃	C₂×Q₈
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₂×#423	D₅₃
hypocenter	C₅₃	#21177
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂×#423

Complements

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

N	H ≅ G/N	type
4. #20275488	C₂	semidirect
167. C₄×C₅₃	#430	semidirect
204. C₅₃	#21177	semidirect
344. C₄	#16810671	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
C₂×#16810647	2
C₂×C₄×#11719917	2
#20275488	2
#20274441	2
C₂×C₅₃×#423	2
#21177	53