Subgroups — #27300029

Order 13568.

450 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20276471	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄⋊#11719908	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#16810407	C₂	1	2	yes	maximal
4	#20272804	C₂	1	2	yes	maximal, complemented
5	C₄×#131×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133157	C₂²	1	2	yes	—
7	C₂²×#11719908	C₂²	1	2	yes	—
8	C₄⋊#11719908	—	2	1	no	—
9	C₄⋊#11719908	—	2	1	no	—
10	C₄⋊#11719908	—	2	1	no	—
11	C₄⋊#11719908	—	2	1	no	—
12	#16810407	—	2	4	no	—
13	C₂×#131×C₅₃	C₄	1	1	yes	characteristic
14	C₂×#131×C₅₃	C₂²	1	1	yes	characteristic
15	C₄×#41×C₅₃	C₄	1	1	yes	characteristic
16	C₅₃×#405	—	2	1	no	—
17	C₅₃×#405	—	2	1	no	—
18	C₅₃×#405	C₄	1	1	yes	characteristic
19	C₅₃×#405	C₂²	1	1	yes	characteristic
20	C₄×M₁₆×C₅₃	—	2	1	no	—
21	C₄×M₁₆×C₅₃	C₄	1	1	yes	characteristic
22	C₂×C₄×C₈×C₅₃	—	2	1	no	—
23	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
24	#13133157	—	2	2	no	—
25	#13133157	—	2	2	no	—
26	C₂×#11719908	—	2	2	no	—
27	C₂×#11719908	—	2	2	no	—
28	C₂×#11719908	—	2	2	no	—
29	C₂×#11719908	—	2	2	no	—
30	C₂×#41×C₅₃	C₂×C₄	1	1	yes	characteristic
31	#131×C₅₃	—	2	2	no	—
32	#131×C₅₃	—	2	1	no	—
33	#131×C₅₃	—	2	1	no	—
34	#131×C₅₃	—	2	2	no	—
35	#131×C₅₃	D₄	1	2	yes	—
36	#131×C₅₃	Q₈	1	2	yes	—
37	C₂×M₁₆×C₅₃	—	2	1	no	—
38	C₂×M₁₆×C₅₃	—	2	1	no	—
39	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
40	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₈×C₅₃	—	2	1	no	—
42	C₂²×C₈×C₅₃	—	2	1	no	—
43	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
45	C₄×Q₈×C₅₃	C₈	1	1	yes	characteristic
46	C₄×Q₈×C₅₃	C₈	1	1	yes	characteristic
47	C₄×D₄×C₅₃	—	2	1	no	—
48	C₄×D₄×C₅₃	—	2	1	no	—
49	C₄×D₄×C₅₃	C₈	1	1	yes	characteristic
50	C₄×D₄×C₅₃	C₈	1	1	yes	characteristic
51	#117×C₅₃	—	2	1	no	—
52	#117×C₅₃	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄²×C₅₃	—	2	1	no	—
54	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
55	#97×C₅₃	—	2	1	no	—
56	#97×C₅₃	—	2	1	no	—
57	#97×C₅₃	—	2	1	no	—
58	C₄×C₈×C₅₃	—	2	1	no	—
59	C₄×C₈×C₅₃	—	2	1	no	—
60	C₄×C₈×C₅₃	—	2	1	no	—
61	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
62	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
63	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
64	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
65	#41×C₅₃	—	2	1	no	—
66	#41×C₅₃	—	2	1	no	—
67	#41×C₅₃	C₄⋊C₄	1	2	yes	—
68	#41×C₅₃	M₁₆	1	2	yes	—
69	C₂×Q₈×C₅₃	C₂×C₈	1	1	yes	characteristic
70	C₂×D₄×C₅₃	—	2	1	no	—
71	C₂×D₄×C₅₃	C₂×C₈	1	1	yes	characteristic
72	C₂²×C₄×C₅₃	—	2	1	no	—
73	C₂²×C₄×C₅₃	—	2	1	no	—
74	C₂²×C₄×C₅₃	—	2	1	no	—
75	C₂²×C₄×C₅₃	M₁₆	1	1	yes	characteristic
76	C₂²×C₄×C₅₃	M₁₆	1	1	yes	characteristic
77	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
78	M₁₆×C₅₃	—	2	4	no	—
79	M₁₆×C₅₃	—	2	1	no	—
80	M₁₆×C₅₃	—	2	1	no	—
81	M₁₆×C₅₃	—	2	2	no	—
82	M₁₆×C₅₃	—	2	2	no	—
83	M₁₆×C₅₃	C₄⋊C₄	1	2	yes	—
84	M₁₆×C₅₃	M₁₆	1	2	yes	—
85	C₂×C₈×C₅₃	—	2	4	no	—
86	C₂×C₈×C₅₃	—	2	1	no	—
87	C₂×C₈×C₅₃	—	2	1	no	—
88	C₂×C₈×C₅₃	—	2	1	no	—
89	C₂×C₈×C₅₃	—	2	1	no	—
90	C₂×C₈×C₅₃	—	2	2	no	—
91	C₂×C₈×C₅₃	—	2	2	no	—
92	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
98	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
99	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
100	C₄⋊C₄×C₅₃	—	2	1	no	—
101	C₄⋊C₄×C₅₃	—	2	1	no	—
102	C₄⋊C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
103	C₄⋊C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
104	C₂²⋊C₄×C₅₃	—	2	1	no	—
105	C₂²⋊C₄×C₅₃	—	2	1	no	—
106	C₂²⋊C₄×C₅₃	—	2	1	no	—
107	C₄²×C₅₃	—	2	1	no	—
108	C₄²×C₅₃	—	2	1	no	—
109	C₄²×C₅₃	—	2	1	no	—
110	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
111	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
112	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
113	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
114	#11719908	—	4	2	no	—
115	#11719908	—	4	2	no	—
116	C₂³×C₅₃	—	2	1	no	—
117	C₂³×C₅₃	C₂×M₁₆	1	1	yes	characteristic
118	Q₈×C₅₃	—	2	1	no	—
119	Q₈×C₅₃	C₄⋊C₈	1	2	yes	—
120	D₄×C₅₃	—	2	2	no	—
121	D₄×C₅₃	—	2	2	no	—
122	D₄×C₅₃	—	2	1	no	—
123	D₄×C₅₃	C₄⋊C₈	1	2	yes	—
124	C₂×C₄×C₅₃	—	2	1	no	—
125	C₂×C₄×C₅₃	—	2	2	no	—
126	C₂×C₄×C₅₃	—	2	2	no	—
127	C₂×C₄×C₅₃	—	2	2	no	—
128	C₂×C₄×C₅₃	—	2	1	no	—
129	C₂×C₄×C₅₃	—	2	1	no	—
130	C₂×C₄×C₅₃	—	4	1	no	—
131	C₂×C₄×C₅₃	—	4	2	no	—
132	C₂×C₄×C₅₃	—	4	2	no	—
133	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
134	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
135	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
136	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
137	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
138	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
139	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
140	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
141	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
142	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
143	C₈×C₅₃	—	2	2	no	—
144	C₈×C₅₃	—	2	1	no	—
145	C₈×C₅₃	—	2	2	no	—
146	C₈×C₅₃	—	2	1	no	—
147	C₈×C₅₃	#121	1	1	yes	characteristic
148	C₈×C₅₃	#121	1	1	yes	characteristic
149	C₈×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
150	C₈×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
151	#12787	—	53	1	no	maximal, Sylow
152	C₂²×C₅₃	—	2	1	no	—
153	C₂²×C₅₃	—	2	1	no	—
154	C₂²×C₅₃	—	2	2	no	—
155	C₂²×C₅₃	—	4	1	no	—
156	C₂²×C₅₃	—	4	1	no	—
157	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
158	C₂²×C₅₃	#423	1	1	yes	characteristic
159	C₂²×C₅₃	#432	1	1	yes	characteristic
160	C₄×C₅₃	—	2	2	no	—
161	C₄×C₅₃	—	2	1	no	—
162	C₄×C₅₃	—	2	1	no	—
163	C₄×C₅₃	—	2	1	no	—
164	C₄×C₅₃	—	4	1	no	—
165	C₄×C₅₃	—	4	1	no	—
166	C₄×C₅₃	#350	1	2	yes	complemented
167	C₄×C₅₃	#380	1	1	yes	characteristic
168	C₄×C₅₃	#407	1	1	yes	characteristic
169	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
170	C₄×C₅₃	#431	1	1	yes	characteristic
171	C₄×#131	D₅₃	1	1	yes	characteristic
172	#2150	—	53	1	no	—
173	C₂×C₄⋊C₁₆	—	53	1	no	—
174	C₂×#350	—	53	2	no	—
175	#1379	—	53	2	no	—
176	C₂×C₅₃	—	2	1	no	—
177	C₂×C₅₃	—	2	1	no	—
178	C₂×C₅₃	—	4	2	no	—
179	C₂×C₅₃	#1797	1	1	yes	characteristic
180	C₂×C₅₃	C₂×#350	1	1	yes	characteristic
181	C₂×C₅₃	#2117	1	1	yes	characteristic
182	C₂×#131	Dic₅₃	1	1	yes	characteristic
183	C₂×#131	C₂×D₅₃	1	1	yes	characteristic
184	C₄×#41	Dic₅₃	1	1	yes	characteristic
185	C₂×M₃₂	—	53	2	no	—
186	C₂²×C₁₆	—	53	2	no	—
187	#405	—	2	1	no	—
188	#405	—	2	1	no	—
189	#405	Dic₅₃	1	1	yes	characteristic
190	#405	C₂×D₅₃	1	1	yes	characteristic
191	C₄×M₁₆	—	2	1	no	—
192	C₄×M₁₆	Dic₅₃	1	1	yes	characteristic
193	C₂×C₄×C₈	—	2	1	no	—
194	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
195	C₄⋊C₁₆	—	106	1	no	—
196	C₄⋊C₁₆	—	106	1	no	—
197	C₄⋊C₁₆	—	106	1	no	—
198	C₄⋊C₁₆	—	106	1	no	—
199	#350	—	106	4	no	—
200	C₅₃	#12787	1	1	yes	characteristic, complemented, Sylow
201	C₂×#41	C₂×Dic₅₃	1	1	yes	characteristic
202	#131	—	2	2	no	—
203	#131	—	2	2	no	—
204	#131	—	2	1	no	—
205	#131	—	2	1	no	—
206	#131	Dic₁₀₆	1	2	yes	—
207	#131	D₂₁₂	1	2	yes	—
208	C₂×M₁₆	—	2	1	no	—
209	C₂×M₁₆	—	2	1	no	—
210	C₂×M₁₆	C₂×Dic₅₃	1	1	yes	characteristic
211	C₂×M₁₆	C₂×Dic₅₃	1	1	yes	characteristic
212	C₂²×C₈	—	2	1	no	—
213	C₂²×C₈	—	2	1	no	—
214	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
215	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
216	C₄×Q₈	#88322	1	1	yes	characteristic
217	C₄×Q₈	#88322	1	1	yes	characteristic
218	C₄×D₄	—	2	1	no	—
219	C₄×D₄	—	2	1	no	—
220	C₄×D₄	#88322	1	1	yes	characteristic
221	C₄×D₄	#88322	1	1	yes	characteristic
222	#117	—	2	1	no	—
223	#117	C₂×Dic₅₃	1	1	yes	characteristic
224	C₂×C₄²	—	2	1	no	—
225	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
226	M₃₂	—	106	2	no	—
227	M₃₂	—	106	2	no	—
228	C₂×C₁₆	—	106	2	no	—
229	C₂×C₁₆	—	106	2	no	—
230	C₂×C₁₆	—	106	2	no	—
231	C₂×C₁₆	—	106	2	no	—
232	#97	—	2	1	no	—
233	#97	—	2	1	no	—
234	#97	—	2	1	no	—
235	C₄×C₈	—	2	1	no	—
236	C₄×C₈	—	2	1	no	—
237	C₄×C₈	—	2	1	no	—
238	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
239	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
240	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
241	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
242	#41	—	2	1	no	—
243	#41	—	2	1	no	—
244	#41	#11719917	1	2	yes	—
245	#41	C₄⋊Dic₅₃	1	2	yes	—
246	C₂×Q₈	C₂×#88322	1	1	yes	characteristic
247	C₂×D₄	—	2	1	no	—
248	C₂×D₄	C₂×#88322	1	1	yes	characteristic
249	C₂²×C₄	—	2	1	no	—
250	C₂²×C₄	—	2	1	no	—
251	C₂²×C₄	—	2	1	no	—
252	C₂²×C₄	#11719917	1	1	yes	characteristic
253	C₂²×C₄	#11719917	1	1	yes	characteristic
254	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
255	M₁₆	—	2	4	no	—
256	M₁₆	—	2	1	no	—
257	M₁₆	—	2	1	no	—
258	M₁₆	—	2	2	no	—
259	M₁₆	—	2	2	no	—
260	M₁₆	#11719917	1	2	yes	—
261	M₁₆	C₄⋊Dic₅₃	1	2	yes	—
262	C₂×C₈	—	2	4	no	—
263	C₂×C₈	—	2	1	no	—
264	C₂×C₈	—	2	1	no	—
265	C₂×C₈	—	2	1	no	—
266	C₂×C₈	—	2	1	no	—
267	C₂×C₈	—	2	2	no	—
268	C₂×C₈	—	2	2	no	—
269	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
270	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
271	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
272	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
273	C₂×C₈	#11719945	1	1	yes	characteristic
274	C₂×C₈	#11719945	1	1	yes	characteristic
275	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
276	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
277	C₄⋊C₄	—	2	1	no	—
278	C₄⋊C₄	—	2	1	no	—
279	C₄⋊C₄	C₂×#88322	1	1	yes	characteristic
280	C₄⋊C₄	C₂×#88322	1	1	yes	characteristic
281	C₂²⋊C₄	—	2	1	no	—
282	C₂²⋊C₄	—	2	1	no	—
283	C₂²⋊C₄	—	2	1	no	—
284	C₄²	—	2	1	no	—
285	C₄²	—	2	1	no	—
286	C₄²	—	2	1	no	—
287	C₄²	C₂×#88322	1	1	yes	characteristic
288	C₄²	C₂×#88322	1	1	yes	characteristic
289	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
290	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
291	C₁₆	—	212	2	no	—
292	C₁₆	—	212	2	no	—
293	C₂³	—	2	1	no	—
294	C₂³	C₂×#11719917	1	1	yes	characteristic
295	Q₈	—	2	1	no	—
296	Q₈	C₄⋊#88322	1	2	yes	—
297	D₄	—	2	2	no	—
298	D₄	—	2	2	no	—
299	D₄	—	2	1	no	—
300	D₄	C₄⋊#88322	1	2	yes	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	4	1	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	C₂²⋊#88322	1	1	yes	characteristic
311	C₂×C₄	C₂²⋊#88322	1	1	yes	characteristic
312	C₂×C₄	C₂²⋊#88322	1	1	yes	characteristic
313	C₂×C₄	C₂²⋊#88322	1	1	yes	characteristic
314	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
315	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
316	C₂×C₄	#13133176	1	1	yes	characteristic
317	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
318	C₂×C₄	C₂×#11719917	1	1	yes	characteristic
319	C₂×C₄	C₂×#11719917	1	1	yes	characteristic
320	C₈	—	2	1	no	—
321	C₈	—	2	2	no	—
322	C₈	—	2	1	no	—
323	C₈	—	2	2	no	—
324	C₈	#13133191	1	1	yes	characteristic
325	C₈	#13133191	1	1	yes	characteristic
326	C₈	#13133196	1	1	yes	characteristic
327	C₈	#13133196	1	1	yes	characteristic
328	C₂²	—	2	1	no	—
329	C₂²	—	2	1	no	—
330	C₂²	—	2	2	no	—
331	C₂²	—	4	1	no	—
332	C₂²	—	4	1	no	—
333	C₂²	C₂×C₂²⋊#88322	1	1	yes	characteristic
334	C₂²	#16810647	1	1	yes	characteristic
335	C₂²	#16810680	1	1	yes	characteristic
336	C₄	—	2	2	no	—
337	C₄	—	2	1	no	—
338	C₄	—	2	1	no	—
339	C₄	—	2	1	no	—
340	C₄	—	4	1	no	—
341	C₄	—	4	1	no	—
342	C₄	#16810407	1	2	yes	complemented
343	C₄	#16810483	1	1	yes	characteristic
344	C₄	#16810587	1	1	yes	characteristic
345	C₄	C₂×C₄⋊#88322	1	1	yes	characteristic
346	C₄	#16810676	1	1	yes	characteristic
347	C₂	—	2	1	no	—
348	C₂	—	2	1	no	—
349	C₂	—	4	2	no	—
350	C₂	#20274695	1	1	yes	characteristic
351	C₂	C₂×#16810407	1	1	yes	characteristic
352	C₂	#20276343	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₄×#131×C₅₃	C₂
socle	C₂²×C₅₃	C₂×C₂²⋊C₈
prime radical	C₂×#41×C₅₃	C₂×C₄
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₄×#131	D₅₃
hypocenter	C₅₃	#12787
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₄×#131

Complements

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

N	H ≅ G/N	type
1. #20276471	C₂	semidirect
2. C₂×C₄⋊#11719908	C₂	semidirect
4. #20272804	C₂	semidirect
166. C₄×C₅₃	#350	semidirect
200. C₅₃	#12787	semidirect
342. C₄	#16810407	semidirect

Maximal subgroups

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

subgroup	index
#20276471	2
C₂×C₄⋊#11719908	2
C₂×#16810407	2
#20272804	2
C₄×#131×C₅₃	2
#12787	53