Subgroups — #27289969

Order 13568.

918 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810647	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810493	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810391	C₂	1	4	yes	maximal
4	C₂×C₄×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11719917	C₂²	1	2	yes	—
6	C₂²×C₄⋊Dic₅₃	C₄	1	4	yes	—
7	C₂²×C₄⋊Dic₅₃	C₂²	1	2	yes	—
8	#16810647	—	2	2	no	—
9	#16810647	C₄	1	4	yes	complemented
10	C₂×C₄⋊#88322	—	2	2	no	—
11	#16810493	—	2	2	no	—
12	#16810493	—	2	1	no	—
13	#16810493	—	2	1	no	—
14	#16810391	—	2	8	no	—
15	C₂²×M₁₆×C₅₃	C₂²	1	2	yes	—
16	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
17	C₄×M₁₆×C₅₃	—	2	2	no	—
18	C₄×M₁₆×C₅₃	C₄	1	4	yes	complemented
19	C₂×#97×C₅₃	—	2	1	no	—
20	C₂×C₄×C₈×C₅₃	—	2	1	no	—
21	C₂³×Dic₅₃	C₂×C₄	1	4	yes	—
22	C₂×#11719917	—	2	2	no	—
23	C₂×#11719917	—	2	4	no	—
24	C₂×#11719917	—	2	4	no	—
25	C₂×#11719917	C₂×C₄	1	4	yes	—
26	C₂²×#88322	—	2	2	no	—
27	C₂×C₄⋊Dic₅₃	—	2	4	no	—
28	C₂×C₄⋊Dic₅₃	—	2	4	no	—
29	C₂×C₄⋊Dic₅₃	—	2	4	no	—
30	C₂×C₄⋊Dic₅₃	—	2	2	no	—
31	C₂×C₄⋊Dic₅₃	—	2	2	no	—
32	C₂×C₄⋊Dic₅₃	—	2	4	no	—
33	C₂×C₄⋊Dic₅₃	—	4	8	no	—
34	C₄⋊#88322	—	2	8	no	—
35	C₂³×C₄×C₅₃	C₂×C₄	1	2	yes	—
36	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
37	C₂×M₁₆×C₅₃	—	2	2	no	—
38	C₂×M₁₆×C₅₃	—	2	2	no	—
39	C₂×M₁₆×C₅₃	—	2	2	no	—
40	C₂×M₁₆×C₅₃	C₂×C₄	1	4	yes	—
41	C₂×M₁₆×C₅₃	D₄	1	4	yes	—
42	C₂×M₁₆×C₅₃	Q₈	1	4	yes	—
43	C₂²×C₈×C₅₃	—	2	2	no	—
44	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₅₃	D₄	1	2	yes	—
47	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
48	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
49	C₂×C₄²×C₅₃	D₄	1	2	yes	—
50	C₂×C₄²×C₅₃	Q₈	1	2	yes	—
51	#97×C₅₃	—	2	4	no	—
52	C₄×C₈×C₅₃	—	2	4	no	—
53	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	—	2	4	no	—
55	C₂²×C₄×C₅₃	—	2	4	no	—
56	C₂²×C₄×C₅₃	C₄²	1	4	yes	—
57	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
58	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
59	C₂²×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
60	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
67	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
68	C₂²×C₄×C₅₃	#41	1	2	yes	—
69	C₂²×C₄×C₅₃	#41	1	2	yes	—
70	C₂²×Dic₅₃	—	2	4	no	—
71	C₂²×Dic₅₃	—	2	8	no	—
72	C₂²×Dic₅₃	—	4	8	no	—
73	M₁₆×C₅₃	—	2	4	no	—
74	M₁₆×C₅₃	—	2	4	no	—
75	M₁₆×C₅₃	—	2	4	no	—
76	M₁₆×C₅₃	C₄⋊C₄	1	8	yes	complemented
77	C₂×C₈×C₅₃	—	2	4	no	—
78	C₂×C₈×C₅₃	—	2	4	no	—
79	C₂×C₈×C₅₃	—	2	4	no	—
80	C₄²×C₅₃	C₂²⋊C₄	1	4	yes	—
81	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
82	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
83	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
84	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
85	C₄²×C₅₃	C₄⋊C₄	1	4	yes	—
86	C₄⋊Dic₅₃	—	4	8	no	—
87	C₄⋊Dic₅₃	—	4	8	no	—
88	C₄⋊Dic₅₃	—	4	4	no	—
89	C₄⋊Dic₅₃	—	4	2	no	—
90	C₄⋊Dic₅₃	—	4	2	no	—
91	#11719917	—	2	8	no	—
92	#11719917	—	4	4	no	—
93	C₂×#88322	—	2	4	no	—
94	C₂×#88322	—	4	2	no	—
95	C₂×#88322	—	4	2	no	—
96	C₂³×C₅₃	—	2	2	no	—
97	C₂³×C₅₃	—	2	2	no	—
98	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
99	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₂³×C₅₃	C₄×D₄	1	2	yes	—
102	C₂³×C₅₃	C₄×Q₈	1	2	yes	—
103	C₂×C₄×C₅₃	—	2	2	no	—
104	C₂×C₄×C₅₃	—	2	2	no	—
105	C₂×C₄×C₅₃	—	2	8	no	—
106	C₂×C₄×C₅₃	—	2	8	no	—
107	C₂×C₄×C₅₃	#95	1	4	yes	—
108	C₂×C₄×C₅₃	#95	1	4	yes	—
109	C₂×C₄×C₅₃	#100	1	4	yes	—
110	C₂×C₄×C₅₃	#101	1	4	yes	—
111	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
115	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₅₃	#117	1	2	yes	—
117	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
118	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
119	C₂×C₄×C₅₃	#121	1	2	yes	—
120	C₂×C₄×C₅₃	#121	1	2	yes	—
121	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
122	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
123	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
124	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
125	C₂×C₄×C₅₃	#125	1	2	yes	—
126	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
127	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
128	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
129	C₂×Dic₅₃	—	4	4	no	—
130	C₂×Dic₅₃	—	4	8	no	—
131	C₂×Dic₅₃	—	8	8	no	—
132	C₈×C₅₃	—	2	8	no	—
133	#88322	—	4	4	no	—
134	#10857	—	53	1	no	maximal, Sylow
135	C₂²×C₅₃	—	2	2	no	—
136	C₂²×C₅₃	—	2	2	no	—
137	C₂²×C₅₃	—	2	2	no	—
138	C₂²×C₅₃	—	2	2	no	—
139	C₂²×C₅₃	—	2	4	no	—
140	C₂²×C₅₃	C₂×#95	1	1	yes	characteristic
141	C₂²×C₅₃	C₄×C₄⋊C₄	1	2	yes	—
142	C₂²×C₅₃	#389	1	2	yes	—
143	C₂²×C₅₃	#390	1	1	yes	characteristic
144	C₂²×C₅₃	#391	1	1	yes	characteristic
145	C₂²×C₅₃	C₂×#100	1	2	yes	—
146	C₂²×C₅₃	C₂×#101	1	2	yes	—
147	C₄×C₅₃	—	2	4	no	—
148	C₄×C₅₃	#343	1	8	yes	complemented
149	C₄×C₅₃	#380	1	2	yes	—
150	C₄×C₅₃	#382	1	1	yes	characteristic
151	C₄×C₅₃	#384	1	1	yes	characteristic
152	C₄×C₅₃	#389	1	2	yes	—
153	C₄×C₅₃	#390	1	1	yes	characteristic
154	C₄×C₅₃	#391	1	1	yes	characteristic
155	Dic₅₃	—	8	4	no	—
156	C₂×#423	—	53	1	no	—
157	C₂×C₄×M₁₆	D₅₃	1	1	yes	characteristic
158	C₂×#384	—	53	1	no	—
159	C₂×#343	—	53	4	no	—
160	C₂×C₅₃	—	2	2	no	—
161	C₂×C₅₃	—	2	2	no	—
162	C₂×C₅₃	C₄⋊#95	1	1	yes	characteristic
163	C₂×C₅₃	C₂×#343	1	2	yes	—
164	C₂×C₅₃	#1965	1	1	yes	characteristic
165	C₂×C₅₃	#1966	1	1	yes	characteristic
166	C₂×C₅₃	#1997	1	2	yes	—
167	C₂²×M₁₆	—	53	2	no	—
168	C₂²×M₁₆	C₂×D₅₃	1	2	yes	—
169	C₂²×C₄⋊C₄	—	53	2	no	—
170	C₂²×C₄⋊C₄	—	53	4	no	—
171	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
172	#423	—	53	4	no	—
173	#423	—	106	2	no	—
174	C₂×C₄⋊C₈	—	106	2	no	—
175	C₄×M₁₆	—	2	2	no	—
176	C₄×M₁₆	Dic₅₃	1	4	yes	complemented
177	C₂×#97	—	2	1	no	—
178	C₂×C₄×C₈	—	2	1	no	—
179	#384	—	106	2	no	—
180	#384	—	106	1	no	—
181	#384	—	106	1	no	—
182	#343	—	106	8	no	—
183	C₅₃	#10857	1	1	yes	characteristic, complemented, Sylow
184	C₂³×C₄	—	53	4	no	—
185	C₂³×C₄	C₄×D₅₃	1	2	yes	—
186	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
187	C₂×M₁₆	—	2	2	no	—
188	C₂×M₁₆	—	2	2	no	—
189	C₂×M₁₆	—	2	2	no	—
190	C₂×M₁₆	—	53	4	no	—
191	C₂×M₁₆	—	106	2	no	—
192	C₂×M₁₆	—	106	4	no	—
193	C₂×M₁₆	—	106	4	no	—
194	C₂×M₁₆	Dic₁₀₆	1	4	yes	—
195	C₂×M₁₆	D₂₁₂	1	4	yes	—
196	C₂×M₁₆	C₂×Dic₅₃	1	4	yes	—
197	C₂²×C₈	—	2	2	no	—
198	C₂²×C₈	—	106	2	no	—
199	C₂×C₄⋊C₄	—	106	4	no	—
200	C₂×C₄⋊C₄	—	106	2	no	—
201	C₂×C₄⋊C₄	—	106	4	no	—
202	C₂×C₄⋊C₄	—	106	2	no	—
203	C₂×C₄⋊C₄	—	106	4	no	—
204	C₂×C₄⋊C₄	—	106	4	no	—
205	C₂×C₄⋊C₄	—	212	8	no	—
206	C₂×C₄²	Dic₁₀₆	1	2	yes	—
207	C₂×C₄²	C₄×D₅₃	1	2	yes	—
208	C₂×C₄²	D₂₁₂	1	2	yes	—
209	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
210	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
211	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
212	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
213	C₄⋊C₈	—	106	8	no	—
214	#97	—	2	4	no	—
215	C₄×C₈	—	2	4	no	—
216	C₂⁴	C₂×C₄×D₅₃	1	1	yes	characteristic
217	C₂²×C₄	—	2	4	no	—
218	C₂²×C₄	—	2	4	no	—
219	C₂²×C₄	—	106	8	no	—
220	C₂²×C₄	—	106	4	no	—
221	C₂²×C₄	—	212	8	no	—
222	C₂²×C₄	C₄×Dic₅₃	1	4	yes	—
223	C₂²×C₄	#11719919	1	4	yes	—
224	C₂²×C₄	#11719921	1	2	yes	—
225	C₂²×C₄	#11719921	1	2	yes	—
226	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
227	C₂²×C₄	C₂×Dic₁₀₆	1	2	yes	—
228	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
229	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
230	C₂²×C₄	C₂×D₂₁₂	1	2	yes	—
231	C₂²×C₄	#11719945	1	2	yes	—
232	C₂²×C₄	#11719945	1	2	yes	—
233	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
234	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
235	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
236	M₁₆	—	2	4	no	—
237	M₁₆	—	2	4	no	—
238	M₁₆	—	2	4	no	—
239	M₁₆	—	106	8	no	—
240	M₁₆	—	212	4	no	—
241	M₁₆	C₄⋊Dic₅₃	1	8	yes	complemented
242	C₂×C₈	—	2	4	no	—
243	C₂×C₈	—	2	4	no	—
244	C₂×C₈	—	2	4	no	—
245	C₂×C₈	—	106	4	no	—
246	C₂×C₈	—	212	2	no	—
247	C₂×C₈	—	212	2	no	—
248	C₄⋊C₄	—	212	8	no	—
249	C₄⋊C₄	—	212	8	no	—
250	C₄⋊C₄	—	212	4	no	—
251	C₄⋊C₄	—	212	2	no	—
252	C₄⋊C₄	—	212	2	no	—
253	C₄²	#11719919	1	2	yes	—
254	C₄²	#11719919	1	2	yes	—
255	C₄²	C₄⋊Dic₅₃	1	4	yes	—
256	C₄²	#11719921	1	4	yes	—
257	C₄²	C₂²⋊Dic₅₃	1	2	yes	—
258	C₄²	C₂²⋊Dic₅₃	1	2	yes	—
259	C₂³	—	2	2	no	—
260	C₂³	—	2	2	no	—
261	C₂³	C₂×C₄×Dic₅₃	1	1	yes	characteristic
262	C₂³	C₂×#11719921	1	1	yes	characteristic
263	C₂³	C₂×#11719919	1	1	yes	characteristic
264	C₂³	C₄×D₂₁₂	1	2	yes	—
265	C₂³	C₄×Dic₁₀₆	1	2	yes	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	8	no	—
269	C₂×C₄	—	2	8	no	—
270	C₂×C₄	—	424	8	no	—
271	C₂×C₄	—	212	4	no	—
272	C₂×C₄	—	212	8	no	—
273	C₂×C₄	#13133124	1	4	yes	—
274	C₂×C₄	#13133124	1	4	yes	—
275	C₂×C₄	#13133133	1	4	yes	—
276	C₂×C₄	#13133135	1	4	yes	—
277	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
278	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
279	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
280	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
281	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
282	C₂×C₄	#13133176	1	2	yes	—
283	C₂×C₄	C₄×D₂₁₂	1	2	yes	—
284	C₂×C₄	C₄×Dic₁₀₆	1	2	yes	—
285	C₂×C₄	#13133191	1	2	yes	—
286	C₂×C₄	#13133191	1	2	yes	—
287	C₂×C₄	#13133196	1	2	yes	—
288	C₂×C₄	#13133196	1	2	yes	—
289	C₂×C₄	#13133205	1	1	yes	characteristic
290	C₂×C₄	#13133205	1	1	yes	characteristic
291	C₂×C₄	#13133209	1	2	yes	—
292	C₂×C₄	#13133214	1	1	yes	characteristic
293	C₂×C₄	#13133217	1	2	yes	—
294	C₂×C₄	#13133217	1	1	yes	characteristic
295	C₈	—	2	8	no	—
296	C₈	—	212	4	no	—
297	C₂²	—	2	2	no	—
298	C₂²	—	2	2	no	—
299	C₂²	—	2	2	no	—
300	C₂²	—	2	2	no	—
301	C₂²	—	2	4	no	—
302	C₂²	C₂×#13133124	1	1	yes	characteristic
303	C₂²	C₄×C₄⋊Dic₅₃	1	2	yes	—
304	C₂²	#16810521	1	2	yes	—
305	C₂²	#16810525	1	1	yes	characteristic
306	C₂²	#16810529	1	1	yes	characteristic
307	C₂²	C₂×#13133133	1	2	yes	—
308	C₂²	C₂×#13133135	1	2	yes	—
309	C₄	—	2	4	no	—
310	C₄	—	424	4	no	—
311	C₄	#16810391	1	8	yes	complemented
312	C₄	#16810483	1	2	yes	—
313	C₄	#16810489	1	1	yes	characteristic
314	C₄	#16810493	1	1	yes	characteristic
315	C₄	#16810521	1	2	yes	—
316	C₄	#16810525	1	1	yes	characteristic
317	C₄	#16810529	1	1	yes	characteristic
318	C₂	—	2	2	no	—
319	C₂	—	2	2	no	—
320	C₂	#20272945	1	1	yes	characteristic
321	C₂	C₂×#16810391	1	2	yes	—
322	C₂	#20275555	1	1	yes	characteristic
323	C₂	#20275559	1	1	yes	characteristic
324	C₂	#20275704	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×#11719921
commutator	C₂³×C₅₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₂×C₄×M₁₆×C₅₃	C₂
socle	C₂³×C₅₃	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₅₃	C₂²×C₄
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	C₂×C₄×M₁₆	D₅₃
hypocenter	C₅₃	#10857
LCS	G ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
9. #16810647	C₄	semidirect
18. C₄×M₁₆×C₅₃	C₄	semidirect
76. M₁₆×C₅₃	C₄⋊C₄	semidirect
148. C₄×C₅₃	#343	semidirect
176. C₄×M₁₆	Dic₅₃	semidirect
183. C₅₃	#10857	semidirect
241. M₁₆	C₄⋊Dic₅₃	semidirect
311. C₄	#16810391	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810647	2
C₂×#16810493	2
C₂×#16810391	2
C₂×C₄×M₁₆×C₅₃	2
#10857	53