Subgroups — #27403095

Order 13568.

642 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	M₁₆×Dic₁₀₆	C₂	1	1	yes	maximal, characteristic
2	#20283666	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810747	C₂	1	1	yes	maximal, characteristic
4	#20275491	C₂	1	1	yes	maximal, characteristic
5	#20275489	C₂	1	2	yes	maximal
6	#20275487	C₂	1	2	yes	maximal
7	#20274852	C₂	1	1	yes	maximal, characteristic
8	#20274851	C₂	1	2	yes	maximal
9	#108×Dic₅₃	C₂	1	2	yes	maximal
10	C₄×#13133152	C₂	1	1	yes	maximal, characteristic
11	C₅₃×#1950	C₂	1	1	yes	maximal, characteristic
12	C₂×C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic
13	#16810753	—	2	1	no	—
14	#16810752	—	2	2	no	—
15	#16810751	—	2	2	no	—
16	#16810750	—	2	1	no	—
17	C₈×Dic₁₀₆	—	2	1	no	—
18	#16810747	—	2	1	no	—
19	#16810747	C₄	1	4	yes	—
20	#16810747	C₄	1	2	yes	—
21	#16810747	C₄	1	2	yes	—
22	C₂×#13133153	C₂²	1	2	yes	—
23	C₂×#13133152	C₂²	1	1	yes	characteristic
24	C₂×#13133152	C₂²	1	1	yes	characteristic
25	#16810647	C₂²	1	1	yes	characteristic
26	#16810646	C₂²	1	2	yes	—
27	#16810646	C₂²	1	2	yes	—
28	C₂×#13133146	C₂²	1	2	yes	—
29	C₄×#11719917	C₂²	1	1	yes	characteristic
30	M₁₆×Dic₅₃	C₂²	1	2	yes	—
31	M₁₆×Dic₅₃	C₂²	1	2	yes	—
32	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
33	C₂×C₈×Dic₅₃	C₂²	1	2	yes	—
34	#16810388	C₂²	1	2	yes	—
35	#16810387	C₂²	1	2	yes	—
36	#16810387	C₂²	1	1	yes	characteristic
37	#16810387	C₂²	1	2	yes	—
38	#16810387	C₂²	1	1	yes	characteristic
39	C₂×#108×C₅₃	C₂²	1	2	yes	—
40	C₅₃×#423	C₂²	1	1	yes	characteristic
41	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
42	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
43	C₅₃×#341	C₂²	1	2	yes	—
44	C₂²×Dic₁₀₆	Q₈	1	1	yes	characteristic
45	C₂×#11719917	C₂³	1	2	yes	—
46	C₂×#11719917	C₂³	1	1	yes	characteristic
47	C₂×#11719917	C₂³	1	1	yes	characteristic
48	C₂²×#88322	C₂³	1	1	yes	characteristic
49	C₂²×#88322	C₂³	1	1	yes	characteristic
50	C₄×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
51	C₄×Dic₁₀₆	C₂×C₄	1	2	yes	—
52	C₄×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
53	C₄×Dic₁₀₆	D₄	1	2	yes	—
54	C₄×Dic₁₀₆	D₄	1	1	yes	characteristic
55	C₄×Dic₁₀₆	D₄	1	1	yes	characteristic
56	C₂×C₄⋊Dic₅₃	Q₈	1	1	yes	characteristic
57	C₂×#11719919	Q₈	1	2	yes	—
58	C₂×C₄×Dic₅₃	C₂³	1	2	yes	—
59	#13133153	—	2	4	no	—
60	#13133152	—	2	2	no	—
61	#13133152	—	2	2	no	—
62	C₄⋊#88322	—	2	1	no	—
63	C₄⋊#88322	—	2	1	no	—
64	#13133146	—	2	2	no	—
65	#13133146	—	2	2	no	—
66	#13133146	—	2	2	no	—
67	#13133146	—	2	2	no	—
68	#13133146	C₂×C₄	1	4	yes	—
69	#13133146	C₂×C₄	1	4	yes	—
70	#13133128	—	2	1	no	—
71	#13133127	—	2	2	no	—
72	#13133127	—	2	2	no	—
73	C₄×#88322	—	2	1	no	—
74	C₄×#88322	C₂×C₄	1	2	yes	—
75	C₄×#88322	C₂×C₄	1	2	yes	—
76	C₈×Dic₅₃	—	2	2	no	—
77	C₈×Dic₅₃	—	2	2	no	—
78	C₈×Dic₅₃	C₂×C₄	1	4	yes	—
79	C₈×Dic₅₃	C₂×C₄	1	4	yes	—
80	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
81	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
82	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
83	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
84	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
85	#108×C₅₃	—	2	4	no	—
86	C₄⋊C₈×C₅₃	—	2	1	no	—
87	C₄⋊C₈×C₅₃	—	2	1	no	—
88	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
89	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
90	#97×C₅₃	—	2	1	no	—
91	C₄×C₈×C₅₃	—	2	1	no	—
92	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
93	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
94	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
95	C₂²×Dic₅₃	C₂×Q₈	1	2	yes	—
96	C₂²×Dic₅₃	C₂×Q₈	1	2	yes	—
97	C₂×Dic₁₀₆	—	2	4	no	—
98	C₂×Dic₁₀₆	C₄⋊C₄	1	1	yes	characteristic
99	C₂×Dic₁₀₆	C₄⋊C₄	1	2	yes	—
100	C₂×Dic₁₀₆	C₄⋊C₄	1	1	yes	characteristic
101	M₁₆×C₅₃	—	2	1	no	—
102	M₁₆×C₅₃	—	2	2	no	—
103	M₁₆×C₅₃	—	2	1	no	—
104	M₁₆×C₅₃	—	2	2	no	—
105	M₁₆×C₅₃	—	2	1	no	—
106	M₁₆×C₅₃	#41	1	2	yes	—
107	C₂×C₈×C₅₃	—	2	1	no	—
108	C₂×C₈×C₅₃	—	2	2	no	—
109	C₂×C₈×C₅₃	—	2	2	no	—
110	C₂×C₈×C₅₃	—	2	1	no	—
111	C₂×C₈×C₅₃	—	2	2	no	—
112	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
113	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
114	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
115	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
117	C₄⋊Dic₅₃	C₄⋊C₄	1	2	yes	—
118	C₄⋊Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
119	C₄⋊Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
120	#11719919	C₄⋊C₄	1	2	yes	—
121	#11719919	C₄⋊C₄	1	2	yes	—
122	#11719919	C₄⋊C₄	1	2	yes	—
123	#11719919	C₄⋊C₄	1	2	yes	—
124	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
125	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
126	C₄×Dic₅₃	C₂×D₄	1	2	yes	—
127	C₄×Dic₅₃	C₂×D₄	1	2	yes	—
128	#11719917	—	2	1	no	—
129	#11719917	—	2	1	no	—
130	#11719917	—	2	2	no	—
131	#11719917	—	2	1	no	—
132	#11719917	—	2	2	no	—
133	#11719917	#41	1	2	yes	—
134	C₂×#88322	—	2	1	no	—
135	C₂×#88322	—	2	1	no	—
136	C₂×#88322	—	2	1	no	—
137	C₂×#88322	—	2	1	no	—
138	C₂×#88322	—	2	1	no	—
139	C₂×#88322	—	2	2	no	—
140	C₂×#88322	—	2	1	no	—
141	C₂×#88322	C₂²×C₄	1	2	yes	—
142	C₂×#88322	C₂²×C₄	1	2	yes	—
143	C₂³×C₅₃	C₂²×Q₈	1	1	yes	characteristic
144	C₂×C₄×C₅₃	—	2	2	no	—
145	C₂×C₄×C₅₃	—	4	1	no	—
146	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
147	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
148	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
149	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
150	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
151	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
152	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
153	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
154	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
155	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
156	C₂×Dic₅₃	—	2	4	no	—
157	C₂×Dic₅₃	—	4	2	no	—
158	C₂×Dic₅₃	C₂×C₄⋊C₄	1	2	yes	—
159	C₂×Dic₅₃	C₂×C₄⋊C₄	1	2	yes	—
160	C₂×Dic₅₃	C₂×C₄⋊C₄	1	2	yes	—
161	C₂×Dic₅₃	C₂×C₄⋊C₄	1	2	yes	—
162	Dic₁₀₆	—	2	2	no	—
163	Dic₁₀₆	—	2	4	no	—
164	Dic₁₀₆	#108	1	4	yes	complemented
165	C₈×C₅₃	—	2	4	no	—
166	C₈×C₅₃	—	2	2	no	—
167	C₈×C₅₃	—	4	2	no	—
168	C₈×C₅₃	—	4	1	no	—
169	#88322	—	2	2	no	—
170	#88322	—	2	2	no	—
171	#88322	—	2	2	no	—
172	#88322	—	4	1	no	—
173	#88322	—	4	2	no	—
174	#22746	—	53	1	no	maximal, Sylow
175	C₂²×C₅₃	—	2	1	no	—
176	C₂²×C₅₃	—	2	1	no	—
177	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
178	C₂²×C₅₃	#557	1	1	yes	characteristic
179	C₂²×C₅₃	#557	1	1	yes	characteristic
180	C₄×C₅₃	—	2	1	no	—
181	C₄×C₅₃	—	2	1	no	—
182	C₄×C₅₃	—	2	1	no	—
183	C₄×C₅₃	C₂×#108	1	2	yes	—
184	C₄×C₅₃	C₄×#41	1	1	yes	characteristic
185	C₄×C₅₃	#519	1	1	yes	characteristic
186	C₄×C₅₃	#548	1	1	yes	characteristic
187	C₄×C₅₃	#548	1	1	yes	characteristic
188	Dic₅₃	—	2	2	no	—
189	Dic₅₃	—	2	2	no	—
190	Dic₅₃	—	2	2	no	—
191	Dic₅₃	C₂×#108	1	4	yes	—
192	Q₈×M₁₆	—	53	1	no	—
193	Q₈×M₁₆	—	53	1	no	—
194	C₂×C₈×Q₈	—	53	1	no	—
195	#1950	—	53	2	no	—
196	#1950	—	53	1	no	—
197	#1950	—	53	2	no	—
198	#1950	D₅₃	1	1	yes	characteristic
199	#1837	—	53	1	no	—
200	#1837	—	53	2	no	—
201	C₄×#108	—	53	2	no	—
202	C₄×#108	—	53	1	no	—
203	C₂×C₅₃	—	2	1	no	—
204	C₂×C₅₃	—	2	1	no	—
205	C₂×C₅₃	#2351	1	1	yes	characteristic
206	C₂×C₅₃	C₂²×#108	1	1	yes	characteristic
207	C₂×C₅₃	#2916	1	1	yes	characteristic
208	C₂×C₄×Q₈	—	53	1	no	—
209	#446	—	106	1	no	—
210	#446	—	106	2	no	—
211	#446	—	106	2	no	—
212	#446	—	106	1	no	—
213	C₈×Q₈	—	53	2	no	—
214	C₈×Q₈	—	53	4	no	—
215	C₈×Q₈	—	53	2	no	—
216	C₈×Q₈	—	106	1	no	—
217	C₈×Q₈	—	106	1	no	—
218	C₂×#108	—	53	1	no	—
219	C₂×#108	—	53	1	no	—
220	C₂×#108	—	53	2	no	—
221	C₂×#108	C₂×D₅₃	1	2	yes	—
222	#423	—	53	2	no	—
223	#423	—	53	2	no	—
224	#423	—	53	1	no	—
225	#423	C₂×D₅₃	1	1	yes	characteristic
226	C₂×C₄⋊C₈	—	53	2	no	—
227	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
228	C₄×M₁₆	—	53	2	no	—
229	C₄×M₁₆	—	53	2	no	—
230	C₄×M₁₆	—	53	1	no	—
231	C₄×M₁₆	C₂×D₅₃	1	1	yes	characteristic
232	C₂×C₄×C₈	—	53	2	no	—
233	C₂×C₄×C₈	—	53	1	no	—
234	#341	—	53	2	no	—
235	#341	—	53	1	no	—
236	#341	—	53	2	no	—
237	#341	—	53	1	no	—
238	#341	—	53	2	no	—
239	#341	C₂×D₅₃	1	2	yes	—
240	C₅₃	#22746	1	1	yes	characteristic, complemented, Sylow
241	C₂²×Q₈	—	53	1	no	—
242	C₂×M₁₆	—	53	1	no	—
243	C₂×M₁₆	—	53	2	no	—
244	C₂×M₁₆	—	53	1	no	—
245	C₂×M₁₆	C₂²×D₅₃	1	2	yes	—
246	C₂×M₁₆	C₂²×D₅₃	1	1	yes	characteristic
247	C₂×M₁₆	C₂²×D₅₃	1	1	yes	characteristic
248	C₂²×C₈	—	53	1	no	—
249	C₂²×C₈	—	53	1	no	—
250	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
251	C₄×Q₈	—	53	1	no	—
252	C₄×Q₈	—	53	2	no	—
253	C₄×Q₈	—	53	1	no	—
254	C₄×Q₈	—	53	1	no	—
255	C₄×Q₈	—	53	2	no	—
256	C₄×Q₈	—	53	1	no	—
257	C₂×C₄⋊C₄	—	53	2	no	—
258	C₂×C₄⋊C₄	—	53	1	no	—
259	C₂×C₄²	—	53	2	no	—
260	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
261	#108	—	2	4	no	—
262	#108	—	106	4	no	—
263	#108	—	106	2	no	—
264	#108	—	106	2	no	—
265	C₄⋊C₈	—	2	1	no	—
266	C₄⋊C₈	—	2	1	no	—
267	C₄⋊C₈	—	53	4	no	—
268	C₄⋊C₈	—	53	4	no	—
269	C₄⋊C₈	—	106	2	no	—
270	C₄⋊C₈	—	106	2	no	—
271	C₄⋊C₈	—	106	2	no	—
272	C₄⋊C₈	—	106	2	no	—
273	C₄⋊C₈	—	106	1	no	—
274	C₄⋊C₈	—	106	1	no	—
275	C₄⋊C₈	C₄×D₅₃	1	2	yes	—
276	C₄⋊C₈	C₄×D₅₃	1	2	yes	—
277	#97	—	2	1	no	—
278	#97	—	106	2	no	—
279	#97	—	106	2	no	—
280	#97	—	106	1	no	—
281	C₄×C₈	—	2	1	no	—
282	C₄×C₈	—	53	4	no	—
283	C₄×C₈	—	53	4	no	—
284	C₄×C₈	—	53	2	no	—
285	C₄×C₈	—	53	2	no	—
286	C₄×C₈	—	106	2	no	—
287	C₄×C₈	—	106	2	no	—
288	C₄×C₈	—	106	1	no	—
289	C₂×Q₈	—	53	1	no	—
290	C₂×Q₈	—	53	2	no	—
291	C₂×Q₈	—	53	1	no	—
292	C₂×Q₈	—	106	4	no	—
293	C₂²×C₄	—	53	2	no	—
294	C₂²×C₄	—	53	2	no	—
295	C₂²×C₄	Q₈×D₅₃	1	1	yes	characteristic
296	C₂²×C₄	Q₈×D₅₃	1	1	yes	characteristic
297	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
298	M₁₆	—	2	1	no	—
299	M₁₆	—	2	1	no	—
300	M₁₆	—	2	2	no	—
301	M₁₆	—	2	2	no	—
302	M₁₆	—	2	1	no	—
303	M₁₆	—	53	2	no	—
304	M₁₆	—	106	1	no	—
305	M₁₆	—	106	2	no	—
306	M₁₆	—	106	1	no	—
307	M₁₆	—	106	1	no	—
308	M₁₆	—	106	2	no	—
309	M₁₆	#11719947	1	2	yes	—
310	C₂×C₈	—	2	1	no	—
311	C₂×C₈	—	2	2	no	—
312	C₂×C₈	—	2	2	no	—
313	C₂×C₈	—	2	1	no	—
314	C₂×C₈	—	2	2	no	—
315	C₂×C₈	—	53	2	no	—
316	C₂×C₈	—	53	2	no	—
317	C₂×C₈	—	106	1	no	—
318	C₂×C₈	—	106	1	no	—
319	C₂×C₈	—	106	1	no	—
320	C₂×C₈	—	106	1	no	—
321	C₂×C₈	—	106	1	no	—
322	C₂×C₈	—	106	1	no	—
323	C₂×C₈	—	106	2	no	—
324	C₂×C₈	C₂×C₄×D₅₃	1	4	yes	—
325	C₄⋊C₄	—	53	2	no	—
326	C₄⋊C₄	—	53	2	no	—
327	C₄⋊C₄	—	53	2	no	—
328	C₄⋊C₄	—	53	2	no	—
329	C₄⋊C₄	—	53	2	no	—
330	C₄⋊C₄	—	53	1	no	—
331	C₄⋊C₄	—	53	1	no	—
332	C₄²	—	53	2	no	—
333	C₄²	—	53	2	no	—
334	C₄²	—	53	2	no	—
335	C₄²	—	53	2	no	—
336	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
337	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
338	C₄²	D₄×D₅₃	1	1	yes	characteristic
339	C₄²	D₄×D₅₃	1	1	yes	characteristic
340	C₂³	C₂×Q₈×D₅₃	1	1	yes	characteristic
341	Q₈	—	53	4	no	—
342	Q₈	—	106	4	no	—
343	Q₈	—	106	2	no	—
344	C₂×C₄	—	2	2	no	—
345	C₂×C₄	—	4	1	no	—
346	C₂×C₄	—	53	2	no	—
347	C₂×C₄	—	53	2	no	—
348	C₂×C₄	—	53	2	no	—
349	C₂×C₄	—	53	2	no	—
350	C₂×C₄	—	106	4	no	—
351	C₂×C₄	—	212	2	no	—
352	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
353	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
354	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
355	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
356	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
357	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
358	C₂×C₄	#41×D₅₃	1	1	yes	characteristic
359	C₂×C₄	C₂×#11719947	1	1	yes	characteristic
360	C₂×C₄	#13133264	1	1	yes	characteristic
361	C₂×C₄	#13133265	1	1	yes	characteristic
362	C₈	—	2	2	no	—
363	C₈	—	2	4	no	—
364	C₈	—	4	2	no	—
365	C₈	—	4	1	no	—
366	C₈	—	106	2	no	—
367	C₈	—	106	2	no	—
368	C₈	—	106	2	no	—
369	C₈	—	212	1	no	—
370	C₈	—	212	2	no	—
371	C₂²	—	2	1	no	—
372	C₂²	—	2	1	no	—
373	C₂²	C₂×C₄⋊C₄×D₅₃	1	1	yes	characteristic
374	C₂²	#16811348	1	1	yes	characteristic
375	C₂²	#16811350	1	1	yes	characteristic
376	C₄	—	2	1	no	—
377	C₄	—	2	1	no	—
378	C₄	—	2	1	no	—
379	C₄	—	53	4	no	—
380	C₄	—	106	2	no	—
381	C₄	—	106	2	no	—
382	C₄	—	106	2	no	—
383	C₄	#108×D₅₃	1	2	yes	—
384	C₄	C₄×#11719947	1	1	yes	characteristic
385	C₄	#16811099	1	1	yes	characteristic
386	C₄	#16811279	1	1	yes	characteristic
387	C₄	#16811283	1	1	yes	characteristic
388	C₂	—	2	1	no	—
389	C₂	—	2	1	no	—
390	C₂	#20277634	1	1	yes	characteristic
391	C₂	C₂×#108×D₅₃	1	1	yes	characteristic
392	C₂	#20283325	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1950	C₂
socle	C₂²×C₅₃	C₂²×C₄⋊C₄
prime radical	C₂³×C₅₃	C₂²×Q₈
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#1950	D₅₃
hypocenter	C₅₃	#22746
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1950

Complements

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

N	H ≅ G/N	type
164. Dic₁₀₆	#108	semidirect
240. C₅₃	#22746	semidirect

Maximal subgroups

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

subgroup	index
M₁₆×Dic₁₀₆	2
#20283666	2
C₂×#16810747	2
#20275491	2
#20275489	2
#20275487	2
#20274852	2
#20274851	2
#108×Dic₅₃	2
C₄×#13133152	2
C₅₃×#1950	2
#22746	53