Subgroups — #27353980

Order 13568.

706 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20284724	C₂	1	1	yes	maximal, characteristic
2	#20283649	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₈×D₂₁₂	C₂	1	1	yes	maximal, characteristic
4	#20273553	C₂	1	4	yes	maximal, complemented
5	C₈×#11719913	C₂	1	4	yes	maximal, complemented
6	#20273486	C₂	1	1	yes	maximal, characteristic, complemented
7	#20273141	C₂	1	1	yes	maximal, characteristic
8	#20273119	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#1450	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×C₈×D₅₃	—	2	1	no	—
11	#16811180	C₂²	1	1	yes	characteristic
12	C₂×C₄×D₂₁₂	C₂²	1	1	yes	characteristic
13	#16810954	C₄	1	2	yes	—
14	#16810914	C₄	1	2	yes	—
15	#16810829	C₄	1	2	yes	—
16	#16810776	C₄	1	2	yes	—
17	#16810753	—	2	1	no	—
18	#16810753	C₂²	1	2	yes	complemented
19	C₈×Dic₁₀₆	—	2	1	no	—
20	C₈×Dic₁₀₆	C₂²	1	2	yes	complemented
21	C₄×#11719913	C₂²	1	4	yes	—
22	C₈×D₂₁₂	—	2	1	no	—
23	C₈×D₂₁₂	—	2	1	no	—
24	C₈×D₂₁₂	C₂²	1	2	yes	—
25	C₈×D₂₁₂	C₂²	1	2	yes	—
26	#16810649	—	2	1	no	—
27	#16810647	C₂²	1	1	yes	characteristic
28	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
29	C₂×#13133129	—	2	1	no	—
30	#16810578	—	2	1	no	—
31	#16810374	C₂²	1	4	yes	complemented
32	#16810355	C₂²	1	4	yes	complemented
33	#16810352	C₂²	1	4	yes	—
34	C₅₃×#431	C₂²	1	1	yes	characteristic
35	C₅₃×#431	C₂²	1	1	yes	characteristic
36	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
37	C₅₃×#322	C₂²	1	2	yes	complemented
38	C₈²×C₅₃	C₂²	1	2	yes	complemented
39	C₂²×D₂₁₂	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₄×D₅₃	—	2	1	no	—
41	C₂×#11719913	C₂×C₄	1	4	yes	—
42	C₂×#11719917	—	2	1	no	—
43	C₂²×#88322	—	2	1	no	—
44	C₂×C₈×D₅₃	—	2	2	no	—
45	C₂×C₈×D₅₃	—	2	2	no	—
46	C₂×C₈×D₅₃	—	4	4	no	—
47	#13133217	C₂×C₄	1	1	yes	characteristic
48	#13133217	C₂×C₄	1	1	yes	characteristic
49	#13133209	—	2	1	no	—
50	#13133196	—	2	1	no	—
51	#13133196	C₂×C₄	1	2	yes	—
52	C₄×Dic₁₀₆	—	2	1	no	—
53	C₄×Dic₁₀₆	C₂³	1	2	yes	—
54	C₄×D₂₁₂	—	2	1	no	—
55	C₄×D₂₁₂	—	2	1	no	—
56	C₄×D₂₁₂	C₂×C₄	1	2	yes	—
57	C₄×D₂₁₂	C₂³	1	2	yes	—
58	#13133176	—	2	1	no	—
59	C₂×C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
60	C₂×#11719921	—	2	1	no	—
61	#13133149	C₂×C₄	1	4	yes	—
62	C₄⋊#88322	—	2	1	no	—
63	C₄⋊#88322	—	2	1	no	—
64	C₄⋊#88322	C₂³	1	2	yes	—
65	C₄⋊#88322	C₂³	1	2	yes	—
66	#13133146	—	2	2	no	—
67	#13133146	—	2	2	no	—
68	#13133142	C₂×C₄	1	4	yes	—
69	#13133139	C₂×C₄	1	4	yes	—
70	#13133129	—	2	2	no	—
71	#13133129	—	2	2	no	—
72	#13133127	—	2	2	no	—
73	C₈×Dic₅₃	—	2	2	no	—
74	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
75	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
76	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
77	C₄⋊C₈×C₅₃	—	2	1	no	—
78	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
79	C₂²⋊C₈×C₅₃	—	2	1	no	—
80	C₂²⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
81	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
82	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
83	C₄×C₈×C₅₃	C₂³	1	2	yes	—
84	C₄×C₈×C₅₃	C₂³	1	2	yes	—
85	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
86	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
87	C₂³×D₅₃	—	2	1	no	—
88	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
89	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
91	C₂²×Dic₅₃	—	2	1	no	—
92	C₂×D₂₁₂	—	2	1	no	—
93	C₂×D₂₁₂	—	2	4	no	—
94	C₂×D₂₁₂	C₂²×C₄	1	2	yes	—
95	C₂×C₄×D₅₃	—	2	2	no	—
96	C₂×C₄×D₅₃	—	2	2	no	—
97	C₂×C₄×D₅₃	—	4	2	no	—
98	C₂×C₄×D₅₃	—	4	2	no	—
99	C₂×Dic₁₀₆	—	2	1	no	—
100	C₂×Dic₁₀₆	C₂²×C₄	1	2	yes	—
101	C₂×C₈×C₅₃	—	2	2	no	—
102	C₂×C₈×C₅₃	—	2	2	no	—
103	C₂×C₈×C₅₃	SD₁₆	1	4	yes	complemented
104	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
105	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
110	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
111	C₄²×C₅₃	#41	1	1	yes	characteristic
112	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
113	C₂²⋊Dic₅₃	—	2	2	no	—
114	#11719921	—	2	2	no	—
115	#11719921	—	2	2	no	—
116	C₄⋊Dic₅₃	—	2	1	no	—
117	C₄⋊Dic₅₃	—	2	1	no	—
118	C₄⋊Dic₅₃	—	2	1	no	—
119	C₄⋊Dic₅₃	—	2	1	no	—
120	C₄⋊Dic₅₃	C₂²×C₄	1	2	yes	—
121	C₄⋊Dic₅₃	C₂²×C₄	1	2	yes	—
122	#11719919	—	2	2	no	—
123	#11719919	—	2	2	no	—
124	C₄×Dic₅₃	—	2	2	no	—
125	#11719917	—	4	2	no	—
126	C₂×#88322	—	2	2	no	—
127	C₂×#88322	—	2	2	no	—
128	C₂×#88322	—	4	2	no	—
129	#11719913	—	2	8	no	—
130	C₈×D₅₃	—	4	4	no	—
131	C₈×D₅₃	—	4	4	no	—
132	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
133	C₂²×D₅₃	—	2	4	no	—
134	C₂²×D₅₃	—	2	2	no	—
135	C₂²×D₅₃	—	4	2	no	—
136	C₂²×D₅₃	—	4	2	no	—
137	C₂×C₄×C₅₃	—	2	2	no	—
138	C₂×C₄×C₅₃	—	4	1	no	—
139	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
140	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
141	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
142	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
143	C₂×C₄×C₅₃	#135	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	C₂×#41	1	1	yes	characteristic
147	C₂×Dic₅₃	—	2	2	no	—
148	C₂×Dic₅₃	—	2	2	no	—
149	C₂×Dic₅₃	—	2	2	no	—
150	C₂×Dic₅₃	—	4	2	no	—
151	D₂₁₂	—	2	4	no	—
152	D₂₁₂	—	2	2	no	—
153	D₂₁₂	#131	1	4	yes	—
154	C₄×D₅₃	—	4	2	no	—
155	C₄×D₅₃	—	4	2	no	—
156	C₄×D₅₃	—	4	4	no	—
157	Dic₁₀₆	—	2	4	no	—
158	Dic₁₀₆	—	4	2	no	—
159	C₈×C₅₃	—	2	4	no	—
160	C₈×C₅₃	—	2	1	no	—
161	C₈×C₅₃	—	2	1	no	—
162	C₈×C₅₃	—	4	2	no	—
163	C₈×C₅₃	C₂×SD₁₆	1	2	yes	—
164	C₈×C₅₃	C₂×SD₁₆	1	2	yes	—
165	#88322	—	4	2	no	—
166	#88322	—	4	2	no	—
167	#17902	—	53	1	no	maximal, Sylow
168	C₂²×C₅₃	—	2	1	no	—
169	C₂²×C₅₃	—	2	1	no	—
170	C₂²×C₅₃	C₄×SD₁₆	1	2	yes	—
171	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
172	C₂×D₅₃	—	2	4	no	—
173	C₂×D₅₃	—	4	8	no	—
174	C₂×D₅₃	—	4	2	no	—
175	C₂×D₅₃	—	4	2	no	—
176	C₄×C₅₃	—	2	1	no	—
177	C₄×C₅₃	—	2	1	no	—
178	C₄×C₅₃	—	2	1	no	—
179	C₄×C₅₃	C₄×SD₁₆	1	2	yes	—
180	C₄×C₅₃	C₂×#131	1	2	yes	—
181	C₄×C₅₃	C₂²×SD₁₆	1	1	yes	characteristic
182	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
183	Dic₅₃	—	4	2	no	—
184	Dic₅₃	—	4	2	no	—
185	Dic₅₃	—	4	2	no	—
186	#3101	—	53	1	no	—
187	#2960	—	53	1	no	—
188	C₂×C₈×D₄	—	53	1	no	—
189	#1582	—	53	4	no	—
190	C₈×SD₁₆	—	53	4	no	—
191	#1570	—	53	1	no	—
192	#1486	—	53	1	no	—
193	#1481	—	53	1	no	—
194	#1450	D₅₃	1	1	yes	characteristic, complemented
195	C₂×C₅₃	—	2	2	no	—
196	C₂×C₅₃	#2929	1	1	yes	characteristic
197	C₂×C₅₃	C₂×C₄×SD₁₆	1	1	yes	characteristic
198	C₂×C₅₃	#2955	1	1	yes	characteristic
199	D₅₃	—	4	4	no	—
200	C₂³×C₈	—	106	1	no	—
201	#533	—	53	1	no	—
202	C₂×C₄×D₄	—	53	1	no	—
203	#483	—	53	2	no	—
204	#476	—	53	2	no	—
205	#461	—	53	2	no	—
206	#450	—	53	2	no	—
207	#446	—	53	2	no	—
208	#446	—	106	1	no	—
209	C₈×Q₈	—	53	2	no	—
210	C₈×Q₈	—	106	1	no	—
211	C₄×SD₁₆	—	53	4	no	—
212	C₈×D₄	—	53	2	no	—
213	C₈×D₄	—	53	2	no	—
214	C₈×D₄	—	106	1	no	—
215	C₈×D₄	—	106	1	no	—
216	#431	C₂×D₅₃	1	1	yes	characteristic
217	#431	C₂×D₅₃	1	1	yes	characteristic
218	#424	—	106	1	no	—
219	#423	—	53	1	no	—
220	C₂×C₄⋊C₈	—	53	1	no	—
221	C₂×C₂²⋊C₈	—	106	1	no	—
222	#405	—	106	1	no	—
223	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
224	#334	—	53	4	no	—
225	Q₈⋊C₈	—	53	4	no	—
226	D₄⋊C₈	—	53	4	no	—
227	#322	C₂×D₅₃	1	2	yes	complemented
228	C₈²	C₂×D₅₃	1	2	yes	complemented
229	C₅₃	#17902	1	1	yes	characteristic, complemented, Sylow
230	C₂²×D₄	—	53	1	no	—
231	C₂³×C₄	—	106	1	no	—
232	C₂×SD₁₆	—	53	4	no	—
233	C₂×M₁₆	—	106	1	no	—
234	C₂²×C₈	—	106	1	no	—
235	C₂²×C₈	—	106	2	no	—
236	C₂²×C₈	—	106	2	no	—
237	C₂²×C₈	—	212	4	no	—
238	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
239	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
240	C₄⋊Q₈	—	53	1	no	—
241	C₄⋊Q₈	—	53	1	no	—
242	#125	—	106	1	no	—
243	C₂²⋊Q₈	—	53	2	no	—
244	C₂²⋊Q₈	—	106	1	no	—
245	C₄×Q₈	—	53	2	no	—
246	C₄×Q₈	—	106	1	no	—
247	C₄×D₄	—	53	2	no	—
248	C₄×D₄	—	53	2	no	—
249	C₄×D₄	—	106	1	no	—
250	C₄×D₄	—	106	1	no	—
251	#117	—	106	1	no	—
252	C₂×C₄⋊C₄	—	53	1	no	—
253	C₂×C₂²⋊C₄	—	106	1	no	—
254	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
255	#106	—	53	4	no	—
256	C₄⋊C₈	—	2	1	no	—
257	C₄⋊C₈	—	53	2	no	—
258	C₄⋊C₈	—	53	2	no	—
259	C₄⋊C₈	—	106	2	no	—
260	C₄⋊C₈	—	106	2	no	—
261	C₄⋊C₈	—	106	1	no	—
262	C₄⋊C₈	—	106	1	no	—
263	C₄⋊C₈	C₄×D₅₃	1	2	yes	—
264	#103	—	53	4	no	—
265	#102	—	53	4	no	—
266	C₂²⋊C₈	—	2	1	no	—
267	C₂²⋊C₈	—	106	2	no	—
268	C₂²⋊C₈	—	106	2	no	—
269	C₂²⋊C₈	C₄×D₅₃	1	2	yes	—
270	#97	—	106	2	no	—
271	C₄×C₈	—	106	2	no	—
272	C₄×C₈	D₂₁₂	1	1	yes	characteristic
273	C₄×C₈	D₂₁₂	1	1	yes	characteristic
274	C₄×C₈	C₂²×D₅₃	1	2	yes	—
275	C₄×C₈	C₂²×D₅₃	1	2	yes	—
276	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
277	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
278	C₂⁴	—	106	1	no	—
279	C₂×Q₈	—	53	2	no	—
280	C₂×Q₈	—	106	1	no	—
281	C₂×D₄	—	53	2	no	—
282	C₂×D₄	—	106	4	no	—
283	C₂×D₄	—	106	1	no	—
284	C₂²×C₄	—	106	1	no	—
285	C₂²×C₄	—	106	2	no	—
286	C₂²×C₄	—	106	2	no	—
287	C₂²×C₄	—	212	2	no	—
288	C₂²×C₄	—	212	2	no	—
289	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
290	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
291	C₂²×C₄	#11719945	1	1	yes	characteristic
292	SD₁₆	—	106	8	no	—
293	M₁₆	—	212	2	no	—
294	C₂×C₈	—	2	2	no	—
295	C₂×C₈	—	2	2	no	—
296	C₂×C₈	—	106	2	no	—
297	C₂×C₈	—	106	2	no	—
298	C₂×C₈	—	212	2	no	—
299	C₂×C₈	—	212	4	no	—
300	C₂×C₈	—	212	4	no	—
301	C₂×C₈	#11719913	1	4	yes	complemented
302	C₂×C₈	C₂×C₄×D₅₃	1	4	yes	—
303	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
304	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
305	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
306	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
307	C₄⋊C₄	—	53	2	no	—
308	C₄⋊C₄	—	53	2	no	—
309	C₄⋊C₄	—	106	1	no	—
310	C₄⋊C₄	—	106	1	no	—
311	C₄⋊C₄	—	106	1	no	—
312	C₄⋊C₄	—	106	2	no	—
313	C₄⋊C₄	—	106	1	no	—
314	C₄⋊C₄	—	106	2	no	—
315	C₂²⋊C₄	—	106	2	no	—
316	C₂²⋊C₄	—	106	2	no	—
317	C₂²⋊C₄	—	106	2	no	—
318	C₄²	—	106	2	no	—
319	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
320	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
321	C₄²	#11719945	1	1	yes	characteristic
322	C₄²	C₂³×D₅₃	1	1	yes	characteristic
323	C₂³	—	106	4	no	—
324	C₂³	—	106	2	no	—
325	C₂³	—	212	2	no	—
326	C₂³	—	212	2	no	—
327	C₂³	C₄×D₂₁₂	1	1	yes	characteristic
328	Q₈	—	106	4	no	—
329	Q₈	—	212	2	no	—
330	D₄	—	53	4	no	—
331	D₄	—	106	4	no	—
332	D₄	—	106	2	no	—
333	C₂×C₄	—	2	2	no	—
334	C₂×C₄	—	4	1	no	—
335	C₂×C₄	—	106	2	no	—
336	C₂×C₄	—	106	2	no	—
337	C₂×C₄	—	106	2	no	—
338	C₂×C₄	—	212	2	no	—
339	C₂×C₄	—	212	2	no	—
340	C₂×C₄	—	212	2	no	—
341	C₂×C₄	—	212	4	no	—
342	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
343	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
344	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
345	C₂×C₄	C₂×#11719913	1	2	yes	—
346	C₂×C₄	#13133239	1	2	yes	—
347	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
348	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
349	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
350	C₈	—	2	4	no	—
351	C₈	—	2	1	no	—
352	C₈	—	2	1	no	—
353	C₈	—	4	2	no	—
354	C₈	—	212	2	no	—
355	C₈	—	212	2	no	—
356	C₈	C₂×#11719913	1	2	yes	—
357	C₈	C₂×#11719913	1	2	yes	—
358	C₂²	—	2	1	no	—
359	C₂²	—	2	1	no	—
360	C₂²	—	106	4	no	—
361	C₂²	—	212	8	no	—
362	C₂²	—	212	2	no	—
363	C₂²	—	212	2	no	—
364	C₂²	C₄×#11719913	1	2	yes	—
365	C₂²	C₂×C₄×D₂₁₂	1	1	yes	characteristic
366	C₄	—	2	1	no	—
367	C₄	—	2	1	no	—
368	C₄	—	2	1	no	—
369	C₄	—	212	2	no	—
370	C₄	—	212	2	no	—
371	C₄	—	212	2	no	—
372	C₄	C₄×#11719913	1	2	yes	—
373	C₄	#131×D₅₃	1	2	yes	—
374	C₄	C₂²×#11719913	1	1	yes	characteristic
375	C₄	C₂×#13133239	1	1	yes	characteristic
376	C₂	—	2	2	no	—
377	C₂	—	212	4	no	—
378	C₂	#20283411	1	1	yes	characteristic
379	C₂	C₂×C₄×#11719913	1	1	yes	characteristic
380	C₂	#20283619	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#1450	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂²×D₂₁₂	C₂×C₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#1450	D₅₃
hypocenter	C₅₃	#17902
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #1450

Complements

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

N	H ≅ G/N	type
2. #20283649	C₂	semidirect
4. #20273553	C₂	semidirect
5. C₈×#11719913	C₂	semidirect
6. #20273486	C₂	semidirect
8. #20273119	C₂	semidirect
9. C₅₃×#1450	C₂	semidirect
18. #16810753	C₂²	semidirect
20. C₈×Dic₁₀₆	C₂²	semidirect
31. #16810374	C₂²	semidirect
32. #16810355	C₂²	semidirect
37. C₅₃×#322	C₂²	semidirect
38. C₈²×C₅₃	C₂²	semidirect
103. C₂×C₈×C₅₃	SD₁₆	semidirect
194. #1450	D₅₃	semidirect
227. #322	C₂×D₅₃	semidirect
228. C₈²	C₂×D₅₃	semidirect
229. C₅₃	#17902	semidirect
301. C₂×C₈	#11719913	semidirect

Maximal subgroups

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

subgroup	index
#20284724	2
#20283649	2
C₂×C₈×D₂₁₂	2
#20273553	2
C₈×#11719913	2
#20273486	2
#20273141	2
#20273119	2
C₅₃×#1450	2
#17902	53