Subgroups — #27301097

Order 13568.

508 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133153	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810439	C₂	1	2	yes	maximal, complemented
3	#20276326	C₂	1	1	yes	maximal, characteristic
4	#20272823	C₂	1	2	yes	maximal, complemented
5	C₂×C₅₃×#467	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11719917	—	2	1	no	—
7	C₂×#13133157	C₂²	1	2	yes	—
8	C₂×#13133153	—	2	4	no	—
9	C₂×#13133153	—	2	1	no	—
10	C₂×#13133153	C₂²	1	2	yes	—
11	#16810439	—	2	4	no	—
12	C₂²⋊#11719908	—	2	1	no	—
13	C₂²⋊#11719908	—	2	1	no	—
14	C₂²×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
15	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×C₂²⋊Q₈×C₅₃	—	2	1	no	—
17	C₅₃×#467	—	2	1	no	—
18	C₅₃×#467	—	2	1	no	—
19	C₅₃×#467	C₄	1	2	yes	—
20	C₅₃×#467	C₄	1	2	yes	—
21	C₂×#107×C₅₃	C₂²	1	1	yes	characteristic
22	C₂×#103×C₅₃	—	2	1	no	—
23	C₂×#11719917	—	2	4	no	—
24	C₂×#11719917	—	2	2	no	—
25	C₂×#11719917	—	4	2	no	—
26	C₂×#11719917	—	4	1	no	—
27	C₂²×#88322	—	4	1	no	—
28	#13133157	—	2	2	no	—
29	#13133157	—	2	2	no	—
30	C₂×#11719908	—	2	2	no	—
31	#13133153	—	2	4	no	—
32	#13133153	—	2	2	no	—
33	#13133153	D₄	1	4	yes	—
34	C₂²×Q₈×C₅₃	—	2	1	no	—
35	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
36	C₂×Q₁₆×C₅₃	—	2	1	no	—
37	C₂×Q₁₆×C₅₃	—	2	2	no	—
38	C₂×Q₁₆×C₅₃	—	2	2	no	—
39	C₂×Q₁₆×C₅₃	C₂×C₄	1	2	yes	—
40	C₂²×C₈×C₅₃	—	2	2	no	—
41	C₂²×C₈×C₅₃	—	2	2	no	—
42	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
45	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
47	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
48	C₂²⋊Q₈×C₅₃	—	2	2	no	—
49	C₂²⋊Q₈×C₅₃	—	2	2	no	—
50	C₂²⋊Q₈×C₅₃	—	4	1	no	—
51	C₂²⋊Q₈×C₅₃	—	4	1	no	—
52	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
53	C₂×C₄⋊C₄×C₅₃	—	4	1	no	—
54	C₂×C₂²⋊C₄×C₅₃	—	4	1	no	—
55	#107×C₅₃	—	2	1	no	—
56	#107×C₅₃	C₂×C₄	1	2	yes	—
57	#103×C₅₃	—	2	2	no	—
58	#103×C₅₃	—	2	2	no	—
59	C₂⁴×C₅₃	SD₁₆	1	1	yes	characteristic
60	C₂×Q₈×C₅₃	—	2	2	no	—
61	C₂×Q₈×C₅₃	—	4	1	no	—
62	C₂×Q₈×C₅₃	—	4	1	no	—
63	C₂×Q₈×C₅₃	—	4	1	no	—
64	C₂×Q₈×C₅₃	—	4	1	no	—
65	C₂×Q₈×C₅₃	—	4	1	no	—
66	C₂²×C₄×C₅₃	—	2	1	no	—
67	C₂²×C₄×C₅₃	—	2	1	no	—
68	C₂²×C₄×C₅₃	—	2	1	no	—
69	C₂²×C₄×C₅₃	—	2	1	no	—
70	C₂²×C₄×C₅₃	—	4	1	no	—
71	C₂²×C₄×C₅₃	—	4	1	no	—
72	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
73	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
74	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
75	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
76	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
77	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
78	Q₁₆×C₅₃	—	2	4	no	—
79	Q₁₆×C₅₃	—	4	2	no	—
80	C₂×C₈×C₅₃	—	2	1	no	—
81	C₂×C₈×C₅₃	—	2	1	no	—
82	C₂×C₈×C₅₃	—	2	2	no	—
83	C₂×C₈×C₅₃	—	2	2	no	—
84	C₂×C₈×C₅₃	—	2	2	no	—
85	C₂×C₈×C₅₃	—	2	2	no	—
86	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
90	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₅₃	#41	1	2	yes	—
92	C₄⋊C₄×C₅₃	—	2	2	no	—
93	C₄⋊C₄×C₅₃	—	4	2	no	—
94	C₄⋊C₄×C₅₃	—	4	2	no	—
95	C₄⋊C₄×C₅₃	—	4	1	no	—
96	C₂²⋊C₄×C₅₃	—	4	2	no	—
97	C₂²⋊C₄×C₅₃	—	4	2	no	—
98	#11719917	—	2	4	no	—
99	#11719917	—	4	4	no	—
100	#11719917	—	4	2	no	—
101	C₂×#88322	—	4	4	no	—
102	C₂×#88322	—	4	2	no	—
103	#11719908	—	4	2	no	—
104	C₂³×C₅₃	—	2	2	no	—
105	C₂³×C₅₃	—	2	1	no	—
106	C₂³×C₅₃	—	2	1	no	—
107	C₂³×C₅₃	—	4	1	no	—
108	C₂³×C₅₃	#102	1	1	yes	characteristic
109	C₂³×C₅₃	#102	1	1	yes	characteristic
110	C₂³×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
111	Q₈×C₅₃	—	4	2	no	—
112	Q₈×C₅₃	—	4	2	no	—
113	Q₈×C₅₃	—	8	1	no	—
114	Q₈×C₅₃	—	8	1	no	—
115	C₂×C₄×C₅₃	—	2	1	no	—
116	C₂×C₄×C₅₃	—	2	1	no	—
117	C₂×C₄×C₅₃	—	2	1	no	—
118	C₂×C₄×C₅₃	—	2	1	no	—
119	C₂×C₄×C₅₃	—	2	1	no	—
120	C₂×C₄×C₅₃	—	2	1	no	—
121	C₂×C₄×C₅₃	—	2	1	no	—
122	C₂×C₄×C₅₃	—	2	1	no	—
123	C₂×C₄×C₅₃	—	2	1	no	—
124	C₂×C₄×C₅₃	—	2	1	no	—
125	C₂×C₄×C₅₃	—	4	2	no	—
126	C₂×C₄×C₅₃	—	4	2	no	—
127	C₂×C₄×C₅₃	—	8	2	no	—
128	C₂×C₄×C₅₃	—	8	1	no	—
129	C₂×C₄×C₅₃	—	8	1	no	—
130	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
131	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
132	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
133	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
134	C₂×C₄×C₅₃	#123	1	1	yes	characteristic
135	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
136	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
137	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
138	C₈×C₅₃	—	2	1	no	—
139	C₈×C₅₃	—	2	1	no	—
140	C₈×C₅₃	C₄×D₄	1	2	yes	—
141	C₈×C₅₃	#121	1	2	yes	—
142	#88322	—	4	4	no	—
143	#13003	—	53	1	no	maximal, Sylow
144	C₂²×C₅₃	—	2	1	no	—
145	C₂²×C₅₃	—	2	1	no	—
146	C₂²×C₅₃	—	2	1	no	—
147	C₂²×C₅₃	—	2	1	no	—
148	C₂²×C₅₃	—	2	1	no	—
149	C₂²×C₅₃	—	2	1	no	—
150	C₂²×C₅₃	—	2	1	no	—
151	C₂²×C₅₃	—	4	2	no	—
152	C₂²×C₅₃	—	4	2	no	—
153	C₂²×C₅₃	#362	1	2	yes	complemented
154	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
155	C₂²×C₅₃	#448	1	1	yes	characteristic
156	C₂²×C₅₃	#481	1	1	yes	characteristic
157	C₄×C₅₃	—	2	1	no	—
158	C₄×C₅₃	—	2	1	no	—
159	C₄×C₅₃	—	8	2	no	—
160	C₄×C₅₃	—	8	2	no	—
161	C₄×C₅₃	#386	1	1	yes	characteristic
162	C₄×C₅₃	#417	1	1	yes	characteristic
163	C₄×C₅₃	#447	1	1	yes	characteristic
164	C₄×C₅₃	#480	1	1	yes	characteristic
165	C₂×#467	D₅₃	1	1	yes	characteristic
166	C₂²×#108	—	53	1	no	—
167	C₂×#362	—	53	2	no	—
168	#2112	—	53	1	no	—
169	#1387	—	53	2	no	—
170	C₂×C₅₃	—	2	2	no	—
171	C₂×C₅₃	—	2	1	no	—
172	C₂×C₅₃	—	2	1	no	—
173	C₂×C₅₃	—	4	1	no	—
174	C₂×C₅₃	#1874	1	1	yes	characteristic
175	C₂×C₅₃	C₂×#362	1	1	yes	characteristic
176	C₂×C₅₃	#2148	1	1	yes	characteristic
177	C₂²×Q₁₆	C₂×D₅₃	1	1	yes	characteristic
178	C₂²×M₁₆	—	106	1	no	—
179	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
180	C₂×C₂²⋊Q₈	—	2	1	no	—
181	C₂×M₃₂	—	53	2	no	—
182	#467	—	2	1	no	—
183	#467	—	2	1	no	—
184	#467	Dic₅₃	1	2	yes	—
185	#467	Dic₅₃	1	2	yes	—
186	C₂×#108	—	53	2	no	—
187	C₂×#108	—	106	4	no	—
188	C₂×#108	—	106	1	no	—
189	C₂×#107	C₂×D₅₃	1	1	yes	characteristic
190	C₂×#103	—	2	1	no	—
191	#362	—	106	4	no	—
192	C₂²⋊C₁₆	—	106	1	no	—
193	C₂²⋊C₁₆	—	106	1	no	—
194	C₅₃	#13003	1	1	yes	characteristic, complemented, Sylow
195	C₂²×Q₈	—	2	1	no	—
196	C₂³×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
197	C₂×Q₁₆	—	2	2	no	—
198	C₂×Q₁₆	—	2	2	no	—
199	C₂×Q₁₆	—	2	1	no	—
200	C₂×Q₁₆	C₂×Dic₅₃	1	2	yes	—
201	C₂×M₁₆	—	106	4	no	—
202	C₂×M₁₆	—	106	2	no	—
203	C₂×M₁₆	—	212	2	no	—
204	C₂×M₁₆	—	212	1	no	—
205	C₂²×C₈	—	2	2	no	—
206	C₂²×C₈	—	2	2	no	—
207	C₂²×C₈	—	212	1	no	—
208	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
209	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
210	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
211	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
212	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
213	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
214	C₂²⋊Q₈	—	2	2	no	—
215	C₂²⋊Q₈	—	2	2	no	—
216	C₂²⋊Q₈	—	4	1	no	—
217	C₂²⋊Q₈	—	4	1	no	—
218	C₂×C₄⋊C₄	—	2	1	no	—
219	C₂×C₄⋊C₄	—	4	1	no	—
220	C₂×C₂²⋊C₄	—	4	1	no	—
221	M₃₂	—	106	2	no	—
222	M₃₂	—	106	2	no	—
223	C₂×C₁₆	—	106	2	no	—
224	#108	—	53	4	no	—
225	#108	—	106	4	no	—
226	#108	—	106	2	no	—
227	#107	—	2	1	no	—
228	#107	C₂×Dic₅₃	1	2	yes	—
229	#103	—	2	2	no	—
230	#103	—	2	2	no	—
231	C₂⁴	#11719923	1	1	yes	characteristic
232	C₂×Q₈	—	2	2	no	—
233	C₂×Q₈	—	4	1	no	—
234	C₂×Q₈	—	4	1	no	—
235	C₂×Q₈	—	4	1	no	—
236	C₂×Q₈	—	4	1	no	—
237	C₂×Q₈	—	4	1	no	—
238	C₂²×C₄	—	2	1	no	—
239	C₂²×C₄	—	2	1	no	—
240	C₂²×C₄	—	2	1	no	—
241	C₂²×C₄	—	2	1	no	—
242	C₂²×C₄	—	4	1	no	—
243	C₂²×C₄	—	4	1	no	—
244	C₂²×C₄	#11719922	1	1	yes	characteristic
245	C₂²×C₄	#11719922	1	1	yes	characteristic
246	C₂²×C₄	#11719923	1	1	yes	characteristic
247	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
248	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
249	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
250	Q₁₆	—	2	4	no	—
251	Q₁₆	—	4	2	no	—
252	M₁₆	—	106	4	no	—
253	M₁₆	—	212	4	no	—
254	M₁₆	—	212	2	no	—
255	C₂×C₈	—	2	1	no	—
256	C₂×C₈	—	2	1	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	212	4	no	—
262	C₂×C₈	—	212	2	no	—
263	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
264	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
265	C₂×C₈	D₄×D₅₃	1	2	yes	—
266	C₂×C₈	#11719947	1	2	yes	—
267	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
268	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	—	4	2	no	—
271	C₄⋊C₄	—	4	2	no	—
272	C₄⋊C₄	—	4	1	no	—
273	C₂²⋊C₄	—	4	2	no	—
274	C₂²⋊C₄	—	4	2	no	—
275	C₁₆	—	212	2	no	—
276	C₂³	—	2	1	no	—
277	C₂³	—	2	1	no	—
278	C₂³	—	2	2	no	—
279	C₂³	—	4	1	no	—
280	C₂³	#13133138	1	1	yes	characteristic
281	C₂³	#13133138	1	1	yes	characteristic
282	C₂³	C₂×#11719923	1	1	yes	characteristic
283	Q₈	—	4	2	no	—
284	Q₈	—	4	2	no	—
285	Q₈	—	8	1	no	—
286	Q₈	—	8	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	8	2	no	—
300	C₂×C₄	—	8	1	no	—
301	C₂×C₄	—	8	1	no	—
302	C₂×C₄	#13133138	1	1	yes	characteristic
303	C₂×C₄	#13133138	1	1	yes	characteristic
304	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
305	C₂×C₄	#13133185	1	1	yes	characteristic
306	C₂×C₄	#13133198	1	1	yes	characteristic
307	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
308	C₂×C₄	#13133242	1	1	yes	characteristic
309	C₂×C₄	#13133247	1	1	yes	characteristic
310	C₈	—	2	1	no	—
311	C₈	—	2	1	no	—
312	C₈	—	212	4	no	—
313	C₈	D₄×Dic₅₃	1	2	yes	—
314	C₈	#13133189	1	2	yes	—
315	C₂²	—	2	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	—	2	1	no	—
320	C₂²	—	2	1	no	—
321	C₂²	—	2	1	no	—
322	C₂²	—	4	2	no	—
323	C₂²	—	4	2	no	—
324	C₂²	#16810439	1	2	yes	complemented
325	C₂²	C₂×#13133138	1	1	yes	characteristic
326	C₂²	#16810761	1	1	yes	characteristic
327	C₂²	#16810939	1	1	yes	characteristic
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	8	2	no	—
331	C₄	—	8	2	no	—
332	C₄	#16810503	1	1	yes	characteristic
333	C₄	#16810624	1	1	yes	characteristic
334	C₄	#16810756	1	1	yes	characteristic
335	C₄	#16810934	1	1	yes	characteristic
336	C₂	—	2	1	no	—
337	C₂	—	2	1	no	—
338	C₂	—	2	2	no	—
339	C₂	—	4	1	no	—
340	C₂	#20275033	1	1	yes	characteristic
341	C₂	C₂×#16810439	1	1	yes	characteristic
342	C₂	#20276464	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#13133138
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#467	C₂
socle	C₂²×C₅₃	C₂×#102
prime radical	C₂⁴×C₅₃	SD₁₆
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×#467	D₅₃
hypocenter	C₅₃	#13003
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂×#467

Complements

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

N	H ≅ G/N	type
2. C₂×#16810439	C₂	semidirect
4. #20272823	C₂	semidirect
153. C₂²×C₅₃	#362	semidirect
194. C₅₃	#13003	semidirect
324. C₂²	#16810439	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13133153	2
C₂×#16810439	2
#20276326	2
#20272823	2
C₂×C₅₃×#467	2
#13003	53