Subgroups — #27298947

Order 13568.

586 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133157	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810578	C₂	1	1	yes	maximal, characteristic
3	#20272796	C₂	1	4	yes	maximal, complemented
4	C₂²×M₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11719917	C₄	1	1	yes	characteristic
6	C₂×#13133176	C₄	1	1	yes	characteristic
7	C₂×#13133157	—	2	2	no	—
8	C₂×#13133157	—	2	1	no	—
9	C₂×#13133157	C₄	1	4	yes	complemented
10	C₂×#13133157	C₂²	1	2	yes	—
11	C₂²×#11719908	—	2	1	no	—
12	#16810578	—	2	2	no	—
13	#16810578	—	2	2	no	—
14	C₂×#13133127	C₄	1	2	yes	—
15	C₂×C₈×Dic₅₃	C₂²	1	2	yes	—
16	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×M₃₂×C₅₃	—	2	2	no	—
18	C₂×M₃₂×C₅₃	—	2	1	no	—
19	C₂×M₃₂×C₅₃	C₄	1	4	yes	complemented
20	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
21	C₂²×C₁₆×C₅₃	—	2	1	no	—
22	C₂×#11719917	—	2	2	no	—
23	C₂×#11719917	—	2	2	no	—
24	C₂×#11719917	C₈	1	2	yes	—
25	C₂×#11719917	C₈	1	2	yes	—
26	C₂²×#88322	C₂×C₄	1	2	yes	—
27	#13133176	—	2	2	no	—
28	#13133176	C₈	1	4	yes	—
29	C₂×C₄⋊Dic₅₃	—	2	1	no	—
30	C₂×C₂²⋊Dic₅₃	—	2	1	no	—
31	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
32	#13133157	—	2	4	no	—
33	#13133157	—	2	2	no	—
34	#13133157	C₂×C₄	1	4	yes	—
35	C₂×#11719908	—	2	4	no	—
36	C₂×#11719908	—	2	2	no	—
37	#13133127	—	2	4	no	—
38	C₈×Dic₅₃	—	2	2	no	—
39	C₈×Dic₅₃	—	2	2	no	—
40	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
42	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
43	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₈×C₅₃	D₄	1	2	yes	—
45	C₂²×C₈×C₅₃	D₄	1	2	yes	—
46	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
47	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
48	C₂²×C₈×C₅₃	Q₈	1	1	yes	characteristic
49	C₂²×C₈×C₅₃	Q₈	1	1	yes	characteristic
50	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
51	M₃₂×C₅₃	—	2	4	no	—
52	M₃₂×C₅₃	—	2	2	no	—
53	M₃₂×C₅₃	C₂×C₄	1	4	yes	—
54	C₂×C₁₆×C₅₃	—	2	4	no	—
55	C₂×C₁₆×C₅₃	—	2	2	no	—
56	C₂⁴×C₅₃	M₁₆	1	1	yes	characteristic
57	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
58	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
59	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
63	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
64	C₂²×C₄×C₅₃	M₁₆	1	1	yes	characteristic
65	C₂²×C₄×C₅₃	M₁₆	1	2	yes	—
66	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
67	C₂²×Dic₅₃	—	2	2	no	—
68	C₂×C₈×C₅₃	C₄²	1	2	yes	—
69	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
70	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
71	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
72	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
73	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
74	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
76	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
77	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
78	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
79	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
80	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
83	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
84	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
85	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
86	C₂²⋊Dic₅₃	—	2	4	no	—
87	C₄⋊Dic₅₃	—	2	4	no	—
88	C₄×Dic₅₃	—	2	2	no	—
89	C₄×Dic₅₃	C₂×C₈	1	4	yes	—
90	#11719917	—	2	4	no	—
91	#11719917	—	2	4	no	—
92	C₂×#88322	—	2	2	no	—
93	C₂×#88322	—	2	2	no	—
94	C₂×#88322	C₂×C₈	1	4	yes	—
95	C₁₆×C₅₃	—	2	4	no	—
96	#11719908	—	2	4	no	—
97	C₂³×C₅₃	—	4	2	no	—
98	C₂³×C₅₃	#97	1	2	yes	—
99	C₂³×C₅₃	C₂²⋊C₈	1	2	yes	—
100	C₂³×C₅₃	C₄⋊C₈	1	1	yes	characteristic
101	C₂³×C₅₃	C₄⋊C₈	1	1	yes	characteristic
102	C₂³×C₅₃	C₂×M₁₆	1	1	yes	characteristic
103	C₂×C₄×C₅₃	#95	1	2	yes	—
104	C₂×C₄×C₅₃	#95	1	2	yes	—
105	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
106	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
107	C₂×C₄×C₅₃	#97	1	2	yes	—
108	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
109	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
110	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
111	C₂×C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
112	C₂×C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
113	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
114	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
115	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
116	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
119	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
120	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
121	C₂×Dic₅₃	—	2	4	no	—
122	C₂×Dic₅₃	—	4	4	no	—
123	C₈×C₅₃	#95	1	2	yes	—
124	C₈×C₅₃	#95	1	2	yes	—
125	C₈×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
126	C₈×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
127	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
128	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
129	#88322	—	2	4	no	—
130	#12549	—	53	1	no	maximal, Sylow
131	C₂²×C₅₃	—	2	2	no	—
132	C₂²×C₅₃	—	2	2	no	—
133	C₂²×C₅₃	—	2	2	no	—
134	C₂²×C₅₃	—	4	2	no	—
135	C₂²×C₅₃	—	4	2	no	—
136	C₂²×C₅₃	#336	1	2	yes	—
137	C₂²×C₅₃	#336	1	2	yes	—
138	C₂²×C₅₃	C₂×#97	1	1	yes	characteristic
139	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
140	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
141	C₄×C₅₃	#336	1	2	yes	—
142	C₄×C₅₃	#336	1	2	yes	—
143	C₄×C₅₃	C₂×#95	1	1	yes	characteristic
144	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
145	C₄×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
146	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
147	Dic₅₃	—	4	4	no	—
148	C₂²×M₃₂	—	53	1	no	—
149	C₂²×M₃₂	D₅₃	1	1	yes	characteristic
150	C₂×#405	—	53	1	no	—
151	#1376	—	53	4	no	—
152	C₂×C₅₃	—	2	1	no	—
153	C₂×C₅₃	—	2	2	no	—
154	C₂×C₅₃	—	2	2	no	—
155	C₂×C₅₃	—	2	1	no	—
156	C₂×C₅₃	C₂×#336	1	1	yes	characteristic
157	C₂×C₅₃	#2153	1	2	yes	—
158	C₂²×M₁₆	—	53	1	no	—
159	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
160	C₂×#117	—	53	1	no	—
161	C₂×M₃₂	—	2	1	no	—
162	C₂×M₃₂	—	2	2	no	—
163	C₂×M₃₂	—	53	4	no	—
164	C₂×M₃₂	—	53	2	no	—
165	C₂×M₃₂	—	106	2	no	—
166	C₂×M₃₂	—	106	1	no	—
167	C₂×M₃₂	Dic₅₃	1	4	yes	complemented
168	C₂×M₃₂	C₂×D₅₃	1	2	yes	—
169	C₂²×C₁₆	—	2	1	no	—
170	C₂²×C₁₆	—	106	1	no	—
171	#405	—	106	2	no	—
172	#405	—	106	2	no	—
173	C₂×#97	—	53	2	no	—
174	C₂×C₄×C₈	—	53	2	no	—
175	C₅₃	#12549	1	1	yes	characteristic, complemented, Sylow
176	C₂³×C₄	C₄×D₅₃	1	1	yes	characteristic
177	C₂×M₁₆	—	53	2	no	—
178	C₂×M₁₆	—	53	2	no	—
179	C₂×M₁₆	—	106	2	no	—
180	C₂×M₁₆	—	106	2	no	—
181	C₂²×C₈	—	53	2	no	—
182	C₂²×C₈	Dic₁₀₆	1	1	yes	characteristic
183	C₂²×C₈	Dic₁₀₆	1	1	yes	characteristic
184	C₂²×C₈	C₄×D₅₃	1	2	yes	—
185	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
186	C₂²×C₈	D₂₁₂	1	2	yes	—
187	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
188	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
189	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
190	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
191	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
192	#117	—	53	4	no	—
193	#117	—	106	2	no	—
194	C₂×C₄⋊C₄	—	106	1	no	—
195	C₂×C₂²⋊C₄	—	106	1	no	—
196	C₂×C₄²	—	53	2	no	—
197	M₃₂	—	2	2	no	—
198	M₃₂	—	2	4	no	—
199	M₃₂	—	53	4	no	—
200	M₃₂	—	106	2	no	—
201	M₃₂	—	106	4	no	—
202	M₃₂	C₂×Dic₅₃	1	4	yes	—
203	C₂×C₁₆	—	2	2	no	—
204	C₂×C₁₆	—	2	4	no	—
205	C₂×C₁₆	—	106	2	no	—
206	C₂×C₁₆	—	106	4	no	—
207	#97	—	106	4	no	—
208	C₄×C₈	—	106	2	no	—
209	C₄×C₈	—	106	2	no	—
210	C₂⁴	#11719912	1	1	yes	characteristic
211	C₂²×C₄	—	106	2	no	—
212	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
213	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
214	C₂²×C₄	C₈×D₅₃	1	2	yes	—
215	C₂²×C₄	#11719912	1	1	yes	characteristic
216	C₂²×C₄	#11719912	1	2	yes	—
217	C₂²×C₄	C₄×Dic₅₃	1	2	yes	—
218	C₂²×C₄	#11719919	1	1	yes	characteristic
219	C₂²×C₄	#11719919	1	1	yes	characteristic
220	C₂²×C₄	#11719921	1	2	yes	—
221	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
222	M₁₆	—	106	4	no	—
223	M₁₆	—	106	4	no	—
224	C₂×C₈	—	53	4	no	—
225	C₂×C₈	—	106	2	no	—
226	C₂×C₈	—	106	2	no	—
227	C₂×C₈	C₄×Dic₅₃	1	2	yes	—
228	C₂×C₈	#11719919	1	1	yes	characteristic
229	C₂×C₈	#11719919	1	1	yes	characteristic
230	C₂×C₈	#11719919	1	2	yes	—
231	C₂×C₈	#11719919	1	2	yes	—
232	C₂×C₈	C₄⋊Dic₅₃	1	2	yes	—
233	C₂×C₈	C₄⋊Dic₅₃	1	2	yes	—
234	C₂×C₈	#11719921	1	2	yes	—
235	C₂×C₈	#11719921	1	2	yes	—
236	C₂×C₈	#11719921	1	2	yes	—
237	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
238	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
239	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
240	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
241	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
242	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
243	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
244	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
245	C₄⋊C₄	—	106	4	no	—
246	C₂²⋊C₄	—	106	4	no	—
247	C₄²	—	53	4	no	—
248	C₄²	—	106	2	no	—
249	C₁₆	—	2	4	no	—
250	C₁₆	—	106	4	no	—
251	C₂³	—	4	2	no	—
252	C₂³	#13133127	1	2	yes	—
253	C₂³	#13133129	1	2	yes	—
254	C₂³	#13133146	1	1	yes	characteristic
255	C₂³	#13133146	1	1	yes	characteristic
256	C₂³	C₂×#11719912	1	1	yes	characteristic
257	C₂×C₄	—	106	4	no	—
258	C₂×C₄	—	212	4	no	—
259	C₂×C₄	#13133124	1	2	yes	—
260	C₂×C₄	#13133124	1	2	yes	—
261	C₂×C₄	C₈×Dic₅₃	1	2	yes	—
262	C₂×C₄	C₈×Dic₅₃	1	2	yes	—
263	C₂×C₄	#13133127	1	2	yes	—
264	C₂×C₄	#13133129	1	2	yes	—
265	C₂×C₄	#13133129	1	2	yes	—
266	C₂×C₄	#13133129	1	2	yes	—
267	C₂×C₄	#13133146	1	1	yes	characteristic
268	C₂×C₄	#13133146	1	1	yes	characteristic
269	C₂×C₄	#13133146	1	2	yes	—
270	C₂×C₄	#13133146	1	2	yes	—
271	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
272	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
273	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
274	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
275	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
276	C₂×C₄	C₂×#11719912	1	1	yes	characteristic
277	C₈	—	106	4	no	—
278	C₈	#13133124	1	2	yes	—
279	C₈	#13133124	1	2	yes	—
280	C₈	C₂×#11719921	1	1	yes	characteristic
281	C₈	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
282	C₈	C₂×#11719919	1	1	yes	characteristic
283	C₈	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
284	C₂²	—	2	2	no	—
285	C₂²	—	2	2	no	—
286	C₂²	—	2	2	no	—
287	C₂²	—	4	2	no	—
288	C₂²	—	4	2	no	—
289	C₂²	#16810377	1	2	yes	—
290	C₂²	#16810377	1	2	yes	—
291	C₂²	C₂×#13133127	1	1	yes	characteristic
292	C₂²	C₂×#13133129	1	1	yes	characteristic
293	C₂²	C₂×#13133146	1	1	yes	characteristic
294	C₄	—	212	4	no	—
295	C₄	#16810377	1	2	yes	—
296	C₄	#16810377	1	2	yes	—
297	C₄	C₂×#13133124	1	1	yes	characteristic
298	C₄	C₂×C₈×Dic₅₃	1	1	yes	characteristic
299	C₄	C₂×#13133129	1	1	yes	characteristic
300	C₄	C₂×#13133146	1	1	yes	characteristic
301	C₂	—	2	1	no	—
302	C₂	—	2	1	no	—
303	C₂	—	2	2	no	—
304	C₂	—	2	2	no	—
305	C₂	C₂×#16810377	1	1	yes	characteristic
306	C₂	#20276480	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₅₃
commutator	C₄×C₅₃	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂²×M₃₂×C₅₃	C₂
socle	C₂²×C₅₃	C₂×C₄⋊C₈
prime radical	C₂⁴×C₅₃	M₁₆
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂²×M₃₂	D₅₃
hypocenter	C₅₃	#12549
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ C₂²×M₃₂

Complements

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

N	H ≅ G/N	type
3. #20272796	C₂	semidirect
9. C₂×#13133157	C₄	semidirect
19. C₂×M₃₂×C₅₃	C₄	semidirect
167. C₂×M₃₂	Dic₅₃	semidirect
175. C₅₃	#12549	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13133157	2
C₂×#16810578	2
#20272796	2
C₂²×M₃₂×C₅₃	2
#12549	53