Subgroups — #27354783

Order 13568.

592 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20285285	C₂	1	1	yes	maximal, characteristic
2	#20283866	C₂	1	2	yes	maximal, complemented
3	#20273619	C₂	1	2	yes	maximal, complemented
4	#20273613	C₂	1	1	yes	maximal, characteristic, complemented
5	#20273601	C₂	1	1	yes	maximal, characteristic, complemented
6	#20273529	C₂	1	2	yes	maximal, complemented
7	C₈×Dic₂₁₂	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈×D₄₂₄	C₂	1	1	yes	maximal, characteristic, complemented
9	#20273474	C₂	1	1	yes	maximal, characteristic, complemented
10	#20273181	C₂	1	2	yes	maximal, complemented
11	C₅₃×#1450	C₂	1	1	yes	maximal, characteristic, complemented
12	#16811157	C₂²	1	2	yes	—
13	#16811025	C₄	1	1	yes	characteristic
14	#16811007	C₄	1	1	yes	characteristic
15	#16810978	C₄	1	2	yes	—
16	#16810864	C₄	1	1	yes	characteristic
17	#16810859	C₄	1	1	yes	characteristic
18	#16810852	C₄	1	2	yes	—
19	#16810753	C₂²	1	2	yes	—
20	C₈×Dic₁₀₆	C₂²	1	2	yes	—
21	C₄×Dic₂₁₂	C₂²	1	1	yes	characteristic
22	C₄×#11719913	C₂²	1	2	yes	—
23	C₄×D₄₂₄	C₂²	1	1	yes	characteristic
24	#16810702	C₂²	1	2	yes	—
25	#16810698	—	2	2	no	—
26	#16810697	—	2	2	no	—
27	C₈×D₂₁₂	C₂²	1	2	yes	—
28	#16810683	—	2	2	no	—
29	#16810648	—	2	2	no	—
30	#16810647	C₂²	1	2	yes	—
31	#16810376	C₂²	1	1	yes	characteristic
32	#16810376	C₂²	1	1	yes	characteristic
33	#16810376	C₂²	1	2	yes	—
34	#16810355	C₂²	1	2	yes	—
35	#16810355	C₂²	1	2	yes	—
36	#16810352	C₂²	1	2	yes	—
37	#16810352	C₂²	1	2	yes	—
38	C₅₃×#431	C₂²	1	1	yes	characteristic
39	C₅₃×#431	C₂²	1	1	yes	characteristic
40	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
41	C₅₃×#322	C₂²	1	1	yes	characteristic
42	C₅₃×#322	C₂²	1	1	yes	characteristic
43	C₈²×C₅₃	C₂²	1	1	yes	characteristic
44	C₈²×C₅₃	C₂²	1	1	yes	characteristic
45	C₂×Dic₂₁₂	C₂×C₄	1	1	yes	characteristic
46	C₂×#11719913	C₂×C₄	1	2	yes	—
47	C₂×D₄₂₄	C₂×C₄	1	1	yes	characteristic
48	C₂×#11719917	—	2	2	no	—
49	C₂×#11719912	—	2	2	no	—
50	C₂×C₈×D₅₃	—	2	2	no	—
51	#13133211	—	2	2	no	—
52	#13133209	C₂×C₄	1	2	yes	—
53	#13133205	C₂×C₄	1	2	yes	—
54	#13133200	—	2	2	no	—
55	#13133196	C₂×C₄	1	2	yes	—
56	#13133191	C₂×C₄	1	2	yes	—
57	C₄×Dic₁₀₆	C₂³	1	2	yes	—
58	C₄×D₂₁₂	C₂³	1	2	yes	—
59	C₄×C₂²⋊D₅₃	—	2	2	no	—
60	#13133174	—	2	2	no	—
61	#13133151	C₂×C₄	1	1	yes	characteristic
62	#13133151	C₂×C₄	1	1	yes	characteristic
63	#13133149	C₂×C₄	1	2	yes	—
64	C₄⋊#88322	—	2	2	no	—
65	C₄⋊#88322	C₂³	1	2	yes	—
66	C₄⋊#88322	C₂³	1	2	yes	—
67	#13133146	—	2	2	no	—
68	#13133146	—	2	2	no	—
69	#13133142	C₂×C₄	1	2	yes	—
70	#13133142	C₂×C₄	1	2	yes	—
71	#13133139	C₂×C₄	1	2	yes	—
72	#13133139	C₂×C₄	1	2	yes	—
73	#13133129	—	2	2	no	—
74	#13133129	—	2	2	no	—
75	#13133127	—	2	2	no	—
76	C₈×Dic₅₃	—	2	2	no	—
77	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
78	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
79	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
80	C₄⋊C₈×C₅₃	—	2	1	no	—
81	C₄⋊C₈×C₅₃	D₄	1	1	yes	characteristic
82	C₄⋊C₈×C₅₃	D₄	1	1	yes	characteristic
83	C₂²⋊C₈×C₅₃	—	2	1	no	—
84	C₂²⋊C₈×C₅₃	D₄	1	1	yes	characteristic
85	C₂²⋊C₈×C₅₃	D₄	1	1	yes	characteristic
86	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
87	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
88	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
89	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
90	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
91	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
92	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
93	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
94	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
95	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
97	C₂×C₂²⋊D₅₃	—	2	2	no	—
98	C₂×D₂₁₂	C₂²×C₄	1	2	yes	—
99	C₂×C₄×D₅₃	—	2	2	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	—	2	1	no	—
104	C₂×C₈×C₅₃	—	2	1	no	—
105	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
110	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
111	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
112	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
113	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
114	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₅₃	#41	1	1	yes	characteristic
116	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
117	C₂²⋊Dic₅₃	—	2	2	no	—
118	#11719921	—	2	2	no	—
119	#11719921	—	2	2	no	—
120	#11719921	—	2	2	no	—
121	#11719921	—	2	2	no	—
122	C₄⋊Dic₅₃	C₂²×C₄	1	2	yes	—
123	C₄⋊Dic₅₃	C₂²×C₄	1	2	yes	—
124	#11719919	—	2	2	no	—
125	#11719919	—	2	2	no	—
126	#11719919	—	2	2	no	—
127	#11719919	—	2	2	no	—
128	C₄×Dic₅₃	—	2	2	no	—
129	#11719917	—	8	2	no	—
130	C₂×#88322	—	2	2	no	—
131	C₂×#88322	—	2	2	no	—
132	Dic₂₁₂	—	2	2	no	—
133	D₄₂₄	—	2	2	no	—
134	#11719913	—	2	4	no	—
135	#11719912	—	8	2	no	—
136	C₈×D₅₃	—	8	2	no	—
137	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
138	C₂²×D₅₃	—	2	2	no	—
139	C₂×C₄×C₅₃	—	2	1	no	—
140	C₂×C₄×C₅₃	—	2	1	no	—
141	C₂×C₄×C₅₃	—	2	1	no	—
142	C₂×C₄×C₅₃	—	2	1	no	—
143	C₂×C₄×C₅₃	—	4	1	no	—
144	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
145	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
146	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
147	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
148	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
149	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
150	C₂²⋊D₅₃	—	8	2	no	—
151	C₂×Dic₅₃	—	2	2	no	—
152	C₂×Dic₅₃	—	2	2	no	—
153	C₂×Dic₅₃	—	2	2	no	—
154	D₂₁₂	—	2	2	no	—
155	D₂₁₂	—	2	2	no	—
156	C₄×D₅₃	—	4	2	no	—
157	C₄×D₅₃	—	4	2	no	—
158	Dic₁₀₆	—	2	2	no	—
159	Dic₁₀₆	—	2	2	no	—
160	C₈×C₅₃	—	2	1	no	—
161	C₈×C₅₃	—	2	1	no	—
162	C₈×C₅₃	—	2	1	no	—
163	C₈×C₅₃	—	2	1	no	—
164	C₈×C₅₃	—	2	1	no	—
165	C₈×C₅₃	—	2	1	no	—
166	C₈×C₅₃	—	4	1	no	—
167	C₈×C₅₃	—	4	1	no	—
168	C₈×C₅₃	#131	1	2	yes	—
169	C₈×C₅₃	#131	1	2	yes	—
170	#88322	—	4	2	no	—
171	#88322	—	4	2	no	—
172	#17973	—	53	1	no	maximal, Sylow
173	C₂²×C₅₃	—	2	1	no	—
174	C₂²×C₅₃	—	2	1	no	—
175	C₂²×C₅₃	—	2	1	no	—
176	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
177	C₂×D₅₃	—	4	2	no	—
178	C₂×D₅₃	—	4	2	no	—
179	C₂×D₅₃	—	4	2	no	—
180	C₄×C₅₃	—	2	1	no	—
181	C₄×C₅₃	—	2	1	no	—
182	C₄×C₅₃	—	2	1	no	—
183	C₄×C₅₃	—	2	1	no	—
184	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
185	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
186	C₄×C₅₃	#576	1	1	yes	characteristic
187	C₄×C₅₃	#578	1	1	yes	characteristic
188	Dic₅₃	—	4	2	no	—
189	Dic₅₃	—	4	2	no	—
190	Dic₅₃	—	4	2	no	—
191	#3168	—	53	1	no	—
192	#2993	—	53	2	no	—
193	#1592	—	53	2	no	—
194	#1591	—	53	1	no	—
195	#1589	—	53	1	no	—
196	SD₁₆⋊C₈	—	53	2	no	—
197	C₈×Q₁₆	—	53	1	no	—
198	C₈×D₈	—	53	1	no	—
199	#1567	—	53	1	no	—
200	#1495	—	53	2	no	—
201	#1450	D₅₃	1	1	yes	characteristic, complemented
202	C₂×C₅₃	—	4	1	no	—
203	C₂×C₅₃	#2931	1	1	yes	characteristic
204	C₂×C₅₃	#2951	1	1	yes	characteristic
205	C₂×C₅₃	#2957	1	1	yes	characteristic
206	D₅₃	—	8	2	no	—
207	#529	—	53	2	no	—
208	#499	—	53	1	no	—
209	#495	—	53	1	no	—
210	#488	—	53	2	no	—
211	#467	—	53	1	no	—
212	#466	—	53	1	no	—
213	#465	—	53	2	no	—
214	#446	—	53	2	no	—
215	C₈×Q₈	—	53	2	no	—
216	C₄×Q₁₆	—	53	1	no	—
217	C₄×SD₁₆	—	53	2	no	—
218	C₄×D₈	—	53	1	no	—
219	#436	—	53	2	no	—
220	#436	—	106	2	no	—
221	#435	—	106	2	no	—
222	C₈×D₄	—	53	2	no	—
223	#433	—	106	2	no	—
224	#431	C₂×D₅₃	1	1	yes	characteristic
225	#431	C₂×D₅₃	1	1	yes	characteristic
226	#424	—	106	2	no	—
227	#423	—	53	2	no	—
228	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
229	#335	—	53	1	no	—
230	#335	—	53	1	no	—
231	#335	—	53	2	no	—
232	Q₈⋊C₈	—	53	2	no	—
233	Q₈⋊C₈	—	53	2	no	—
234	D₄⋊C₈	—	53	2	no	—
235	D₄⋊C₈	—	53	2	no	—
236	#322	C₂×D₅₃	1	1	yes	characteristic
237	#322	C₂×D₅₃	1	1	yes	characteristic
238	C₈²	C₂×D₅₃	1	1	yes	characteristic
239	C₈²	C₂×D₅₃	1	1	yes	characteristic
240	C₅₃	#17973	1	1	yes	characteristic, complemented, Sylow
241	C₂×Q₁₆	—	53	1	no	—
242	C₂×SD₁₆	—	53	2	no	—
243	C₂×D₈	—	53	1	no	—
244	C₂×M₁₆	—	106	2	no	—
245	C₂×M₁₆	—	106	2	no	—
246	C₂²×C₈	—	106	2	no	—
247	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
248	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
249	#126	—	106	2	no	—
250	#125	—	53	2	no	—
251	#124	—	53	2	no	—
252	#123	—	106	2	no	—
253	C₂²⋊Q₈	—	53	2	no	—
254	#121	—	53	2	no	—
255	C₄×Q₈	—	53	2	no	—
256	C₄×D₄	—	53	2	no	—
257	C₄×D₄	—	106	2	no	—
258	#117	—	106	2	no	—
259	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
260	#107	—	53	1	no	—
261	#107	—	53	1	no	—
262	#106	—	53	2	no	—
263	C₄⋊C₈	—	2	1	no	—
264	C₄⋊C₈	—	53	2	no	—
265	C₄⋊C₈	—	53	2	no	—
266	C₄⋊C₈	—	106	2	no	—
267	C₄⋊C₈	—	106	2	no	—
268	C₄⋊C₈	—	106	2	no	—
269	C₄⋊C₈	D₂₁₂	1	1	yes	characteristic
270	C₄⋊C₈	D₂₁₂	1	1	yes	characteristic
271	#103	—	53	2	no	—
272	#103	—	53	2	no	—
273	#102	—	53	2	no	—
274	#102	—	53	2	no	—
275	C₂²⋊C₈	—	2	1	no	—
276	C₂²⋊C₈	—	106	2	no	—
277	C₂²⋊C₈	—	106	2	no	—
278	C₂²⋊C₈	D₂₁₂	1	1	yes	characteristic
279	C₂²⋊C₈	D₂₁₂	1	1	yes	characteristic
280	#97	—	106	2	no	—
281	C₄×C₈	—	106	2	no	—
282	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
283	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
284	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
285	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
286	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
287	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
288	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
289	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
290	C₂×Q₈	—	53	2	no	—
291	C₂×D₄	—	53	2	no	—
292	C₂×D₄	—	106	2	no	—
293	C₂²×C₄	—	106	2	no	—
294	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
295	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
296	C₂²×C₄	#11719945	1	1	yes	characteristic
297	Q₁₆	—	106	2	no	—
298	SD₁₆	—	106	4	no	—
299	D₈	—	106	2	no	—
300	M₁₆	—	424	2	no	—
301	M₁₆	—	424	2	no	—
302	C₂×C₈	—	2	2	no	—
303	C₂×C₈	—	2	2	no	—
304	C₂×C₈	—	2	1	no	—
305	C₂×C₈	—	2	1	no	—
306	C₂×C₈	—	106	2	no	—
307	C₂×C₈	—	106	2	no	—
308	C₂×C₈	—	424	2	no	—
309	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
310	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
311	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
312	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
313	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
314	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
315	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
316	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
317	C₄⋊C₄	—	53	2	no	—
318	C₄⋊C₄	—	53	2	no	—
319	C₄⋊C₄	—	106	2	no	—
320	C₄⋊C₄	—	106	2	no	—
321	C₄⋊C₄	—	106	2	no	—
322	C₄⋊C₄	—	106	2	no	—
323	C₂²⋊C₄	—	106	2	no	—
324	C₂²⋊C₄	—	106	2	no	—
325	C₂²⋊C₄	—	106	2	no	—
326	C₂²⋊C₄	—	106	2	no	—
327	C₂²⋊C₄	—	106	2	no	—
328	C₄²	—	106	2	no	—
329	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
330	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
331	C₄²	#11719945	1	1	yes	characteristic
332	C₄²	C₂³×D₅₃	1	1	yes	characteristic
333	C₂³	—	106	2	no	—
334	C₂³	C₄×D₂₁₂	1	1	yes	characteristic
335	Q₈	—	106	2	no	—
336	Q₈	—	106	2	no	—
337	D₄	—	106	2	no	—
338	D₄	—	106	2	no	—
339	D₄	—	424	2	no	—
340	C₂×C₄	—	2	1	no	—
341	C₂×C₄	—	2	1	no	—
342	C₂×C₄	—	2	1	no	—
343	C₂×C₄	—	2	1	no	—
344	C₂×C₄	—	4	1	no	—
345	C₂×C₄	—	106	2	no	—
346	C₂×C₄	—	106	2	no	—
347	C₂×C₄	—	106	2	no	—
348	C₂×C₄	—	212	2	no	—
349	C₂×C₄	—	212	2	no	—
350	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
351	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
352	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
353	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
354	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
355	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
356	C₈	—	2	1	no	—
357	C₈	—	2	1	no	—
358	C₈	—	2	1	no	—
359	C₈	—	2	1	no	—
360	C₈	—	2	1	no	—
361	C₈	—	2	1	no	—
362	C₈	—	4	1	no	—
363	C₈	—	4	1	no	—
364	C₈	—	212	2	no	—
365	C₈	—	212	2	no	—
366	C₈	#13133223	1	2	yes	—
367	C₈	#13133223	1	2	yes	—
368	C₂²	—	2	1	no	—
369	C₂²	—	2	1	no	—
370	C₂²	—	2	1	no	—
371	C₂²	—	212	2	no	—
372	C₂²	—	212	2	no	—
373	C₂²	—	212	2	no	—
374	C₂²	C₂×C₄×D₂₁₂	1	1	yes	characteristic
375	C₄	—	2	1	no	—
376	C₄	—	2	1	no	—
377	C₄	—	2	1	no	—
378	C₄	—	2	1	no	—
379	C₄	—	212	2	no	—
380	C₄	—	212	2	no	—
381	C₄	—	212	2	no	—
382	C₄	C₂×#13133223	1	1	yes	characteristic
383	C₄	C₂×#13133223	1	1	yes	characteristic
384	C₄	#16811428	1	1	yes	characteristic
385	C₄	#16811440	1	1	yes	characteristic
386	C₂	—	4	1	no	—
387	C₂	—	424	2	no	—
388	C₂	#20283428	1	1	yes	characteristic
389	C₂	#20283592	1	1	yes	characteristic
390	C₂	#20283633	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#1450	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	#16810859	C₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#1450	D₅₃
hypocenter	C₅₃	#17973
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ #1450

Complements

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

N	H ≅ G/N	type
2. #20283866	C₂	semidirect
3. #20273619	C₂	semidirect
4. #20273613	C₂	semidirect
5. #20273601	C₂	semidirect
6. #20273529	C₂	semidirect
7. C₈×Dic₂₁₂	C₂	semidirect
8. C₈×D₄₂₄	C₂	semidirect
9. #20273474	C₂	semidirect
10. #20273181	C₂	semidirect
11. C₅₃×#1450	C₂	semidirect
201. #1450	D₅₃	semidirect
240. C₅₃	#17973	semidirect

Maximal subgroups

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

subgroup	index
#20285285	2
#20283866	2
#20273619	2
#20273613	2
#20273601	2
#20273529	2
C₈×Dic₂₁₂	2
C₈×D₄₂₄	2
#20273474	2
#20273181	2
C₅₃×#1450	2
#17973	53