Subgroups — #27276990

Order 13568.

550 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20274744	C₂	1	2	yes	maximal, complemented
2	#20274703	C₂	1	2	yes	maximal
3	#20272583	C₂	1	2	yes	maximal, complemented
4	C₂×C₈×D₄×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133151	—	2	2	no	—
6	C₂×#13133132	—	2	2	no	—
7	#16810483	C₂²	1	4	yes	—
8	#16810390	—	2	2	no	—
9	#16810386	—	2	2	no	—
10	C₂³×C₈×C₅₃	C₄	1	1	yes	characteristic, complemented
11	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
12	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
13	C₈×D₄×C₅₃	—	2	2	no	—
14	C₈×D₄×C₅₃	—	2	2	no	—
15	C₂×C₄⋊C₈×C₅₃	C₄	1	1	yes	characteristic, complemented
16	C₂×C₂²⋊C₈×C₅₃	C₄	1	1	yes	characteristic, complemented
17	C₂×C₂²⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic, complemented
19	C₂×C₄⋊Dic₅₃	—	2	4	no	—
20	C₂×C₂²⋊Dic₅₃	—	2	4	no	—
21	#13133151	—	4	4	no	—
22	#13133132	—	4	4	no	—
23	#13133124	—	2	4	no	—
24	C₂²×D₄×C₅₃	D₄	1	1	yes	characteristic
25	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
27	C₂²×C₈×C₅₃	—	2	1	no	—
28	C₂²×C₈×C₅₃	—	2	1	no	—
29	C₂²×C₈×C₅₃	—	4	2	no	—
30	C₂²×C₈×C₅₃	—	4	2	no	—
31	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₅₃	D₄	1	2	yes	—
36	C₂²×C₈×C₅₃	D₄	1	2	yes	—
37	C₄×D₄×C₅₃	—	2	2	no	—
38	C₄×D₄×C₅₃	—	2	2	no	—
39	C₂×C₄⋊C₄×C₅₃	Q₈	1	1	yes	characteristic
40	C₂×C₂²⋊C₄×C₅₃	D₄	1	1	yes	characteristic
41	C₂×C₂²⋊C₄×C₅₃	Q₈	1	1	yes	characteristic
42	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
43	C₄⋊C₈×C₅₃	—	2	2	no	—
44	C₂²⋊C₈×C₅₃	—	2	2	no	—
45	C₂²⋊C₈×C₅₃	—	2	2	no	—
46	C₄×C₈×C₅₃	—	2	1	no	—
47	C₄×C₈×C₅₃	—	2	1	no	—
48	C₂⁴×C₅₃	C₄⋊C₄	1	1	yes	characteristic
49	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
50	C₂×D₄×C₅₃	—	2	2	no	—
51	C₂×D₄×C₅₃	—	2	2	no	—
52	C₂×D₄×C₅₃	D₈	1	2	yes	—
53	C₂×D₄×C₅₃	Q₁₆	1	2	yes	—
54	C₂²×C₄×C₅₃	—	2	1	no	—
55	C₂²×C₄×C₅₃	—	2	1	no	—
56	C₂²×C₄×C₅₃	—	4	1	no	—
57	C₂²×C₄×C₅₃	—	4	1	no	—
58	C₂²×C₄×C₅₃	—	4	1	no	—
59	C₂²×C₄×C₅₃	—	4	1	no	—
60	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
66	C₂²×C₄×C₅₃	#41	1	2	yes	—
67	C₂²×Dic₅₃	—	4	4	no	—
68	C₂×C₈×C₅₃	—	2	2	no	—
69	C₂×C₈×C₅₃	—	2	4	no	—
70	C₂×C₈×C₅₃	—	2	1	no	—
71	C₂×C₈×C₅₃	—	2	1	no	—
72	C₂×C₈×C₅₃	—	4	1	no	—
73	C₂×C₈×C₅₃	—	4	2	no	—
74	C₂×C₈×C₅₃	—	4	2	no	—
75	C₂×C₈×C₅₃	—	4	2	no	—
76	C₂×C₈×C₅₃	—	4	1	no	—
77	C₂×C₈×C₅₃	—	4	1	no	—
78	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
80	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
81	C₄⋊C₄×C₅₃	—	2	2	no	—
82	C₂²⋊C₄×C₅₃	—	2	2	no	—
83	C₂²⋊C₄×C₅₃	—	2	2	no	—
84	C₄²×C₅₃	—	2	1	no	—
85	C₄²×C₅₃	—	2	1	no	—
86	C₂²⋊Dic₅₃	—	4	4	no	—
87	C₂²⋊Dic₅₃	—	4	4	no	—
88	C₄⋊Dic₅₃	—	4	4	no	—
89	C₄⋊Dic₅₃	—	4	4	no	—
90	C₂³×C₅₃	—	2	1	no	—
91	C₂³×C₅₃	—	2	1	no	—
92	C₂³×C₅₃	—	4	1	no	—
93	C₂³×C₅₃	—	4	1	no	—
94	C₂³×C₅₃	—	4	1	no	—
95	C₂³×C₅₃	—	4	1	no	—
96	C₂³×C₅₃	#107	1	2	yes	complemented
97	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
98	C₂³×C₅₃	#121	1	2	yes	—
99	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
100	C₂³×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
101	C₂³×C₅₃	#135	1	2	yes	—
102	D₄×C₅₃	—	2	4	no	—
103	D₄×C₅₃	—	2	4	no	—
104	C₂×C₄×C₅₃	—	2	1	no	—
105	C₂×C₄×C₅₃	—	2	1	no	—
106	C₂×C₄×C₅₃	—	2	1	no	—
107	C₂×C₄×C₅₃	—	2	2	no	—
108	C₂×C₄×C₅₃	—	2	1	no	—
109	C₂×C₄×C₅₃	—	2	2	no	—
110	C₂×C₄×C₅₃	—	2	1	no	—
111	C₂×C₄×C₅₃	—	2	1	no	—
112	C₂×C₄×C₅₃	—	4	1	no	—
113	C₂×C₄×C₅₃	—	4	2	no	—
114	C₂×C₄×C₅₃	—	4	2	no	—
115	C₂×C₄×C₅₃	—	4	1	no	—
116	C₂×C₄×C₅₃	—	4	1	no	—
117	C₂×C₄×C₅₃	—	4	1	no	—
118	C₂×C₄×C₅₃	—	4	1	no	—
119	C₂×C₄×C₅₃	#107	1	2	yes	complemented
120	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
122	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
123	C₂×Dic₅₃	—	8	4	no	—
124	C₂×Dic₅₃	—	8	4	no	—
125	C₂×Dic₅₃	—	8	4	no	—
126	C₈×C₅₃	—	2	1	no	—
127	C₈×C₅₃	—	2	2	no	—
128	C₈×C₅₃	—	4	1	no	—
129	C₈×C₅₃	—	4	1	no	—
130	C₈×C₅₃	#99	1	2	yes	complemented
131	#8609	—	53	1	no	maximal, Sylow
132	C₂²×C₅₃	—	2	2	no	—
133	C₂²×C₅₃	—	2	1	no	—
134	C₂²×C₅₃	—	2	2	no	—
135	C₂²×C₅₃	—	2	1	no	—
136	C₂²×C₅₃	—	2	2	no	—
137	C₂²×C₅₃	—	4	1	no	—
138	C₂²×C₅₃	—	4	1	no	—
139	C₂²×C₅₃	—	4	2	no	—
140	C₂²×C₅₃	—	4	2	no	—
141	C₂²×C₅₃	—	4	2	no	—
142	C₂²×C₅₃	—	4	1	no	—
143	C₂²×C₅₃	—	4	1	no	—
144	C₂²×C₅₃	#380	1	1	yes	characteristic
145	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
146	C₂²×C₅₃	#427	1	1	yes	characteristic
147	C₂²×C₅₃	#466	1	2	yes	—
148	C₂²×C₅₃	#467	1	2	yes	—
149	C₄×C₅₃	—	2	1	no	—
150	C₄×C₅₃	—	2	2	no	—
151	C₄×C₅₃	—	2	2	no	—
152	C₄×C₅₃	—	4	1	no	—
153	C₄×C₅₃	—	4	1	no	—
154	C₄×C₅₃	C₂×#99	1	1	yes	characteristic
155	C₄×C₅₃	#410	1	1	yes	characteristic
156	Dic₅₃	—	16	4	no	—
157	C₂×C₈×D₄	D₅₃	1	1	yes	characteristic
158	#1808	—	53	2	no	—
159	#1799	—	53	2	no	—
160	#1289	—	53	2	no	—
161	C₂×C₅₃	—	2	2	no	—
162	C₂×C₅₃	—	2	1	no	—
163	C₂×C₅₃	—	2	1	no	—
164	C₂×C₅₃	—	2	2	no	—
165	C₂×C₅₃	—	4	2	no	—
166	C₂×C₅₃	#1799	1	1	yes	characteristic
167	C₂×C₅₃	#1808	1	1	yes	characteristic
168	C₂×C₅₃	#1812	1	1	yes	characteristic
169	C₂³×C₈	Dic₅₃	1	1	yes	characteristic, complemented
170	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
171	C₂×C₄×D₄	C₂×D₅₃	1	1	yes	characteristic
172	C₈×D₄	—	2	2	no	—
173	C₈×D₄	—	2	2	no	—
174	C₂×#107	—	106	2	no	—
175	C₂×C₄⋊C₈	Dic₅₃	1	1	yes	characteristic, complemented
176	C₂×#99	—	106	2	no	—
177	C₂×C₂²⋊C₈	Dic₅₃	1	1	yes	characteristic, complemented
178	C₂×C₂²⋊C₈	C₂×D₅₃	1	1	yes	characteristic
179	C₂×C₄×C₈	Dic₅₃	1	1	yes	characteristic, complemented
180	#380	—	53	4	no	—
181	#342	—	106	2	no	—
182	#340	—	106	2	no	—
183	C₅₃	#8609	1	1	yes	characteristic, complemented, Sylow
184	C₂²×D₄	D₂₁₂	1	1	yes	characteristic
185	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
186	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
187	C₂²×C₈	—	2	1	no	—
188	C₂²×C₈	—	2	1	no	—
189	C₂²×C₈	—	4	2	no	—
190	C₂²×C₈	—	4	2	no	—
191	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
192	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
193	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
194	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
195	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
196	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
197	C₄×D₄	—	2	2	no	—
198	C₄×D₄	—	2	2	no	—
199	C₂×C₄⋊C₄	—	106	4	no	—
200	C₂×C₄⋊C₄	Dic₁₀₆	1	1	yes	characteristic
201	C₂×C₂²⋊C₄	—	106	4	no	—
202	C₂×C₂²⋊C₄	Dic₁₀₆	1	1	yes	characteristic
203	C₂×C₂²⋊C₄	D₂₁₂	1	1	yes	characteristic
204	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
205	#107	—	212	4	no	—
206	C₄⋊C₈	—	2	2	no	—
207	#99	—	212	4	no	—
208	C₂²⋊C₈	—	2	2	no	—
209	C₂²⋊C₈	—	2	2	no	—
210	C₄×C₈	—	2	1	no	—
211	C₄×C₈	—	2	1	no	—
212	#95	—	106	4	no	—
213	C₂⁴	C₄⋊Dic₅₃	1	1	yes	characteristic
214	C₂⁴	C₂×D₂₁₂	1	1	yes	characteristic
215	C₂×D₄	—	2	2	no	—
216	C₂×D₄	—	2	2	no	—
217	C₂×D₄	D₄₂₄	1	2	yes	—
218	C₂×D₄	Dic₂₁₂	1	2	yes	—
219	C₂²×C₄	—	2	1	no	—
220	C₂²×C₄	—	2	1	no	—
221	C₂²×C₄	—	4	1	no	—
222	C₂²×C₄	—	4	1	no	—
223	C₂²×C₄	—	4	1	no	—
224	C₂²×C₄	—	4	1	no	—
225	C₂²×C₄	—	212	4	no	—
226	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
227	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
228	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
229	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
230	C₂²×C₄	#11719945	1	2	yes	—
231	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
232	C₂²×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
233	C₂×C₈	—	2	4	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	2	1	no	—
236	C₂×C₈	—	2	1	no	—
237	C₂×C₈	—	4	1	no	—
238	C₂×C₈	—	4	2	no	—
239	C₂×C₈	—	4	2	no	—
240	C₂×C₈	—	4	2	no	—
241	C₂×C₈	—	4	1	no	—
242	C₂×C₈	—	4	1	no	—
243	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
244	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
245	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
246	C₄⋊C₄	—	2	2	no	—
247	C₄⋊C₄	—	212	4	no	—
248	C₄⋊C₄	—	212	4	no	—
249	C₂²⋊C₄	—	2	2	no	—
250	C₂²⋊C₄	—	2	2	no	—
251	C₂²⋊C₄	—	212	4	no	—
252	C₂²⋊C₄	—	212	4	no	—
253	C₄²	—	2	1	no	—
254	C₄²	—	2	1	no	—
255	C₂³	—	2	1	no	—
256	C₂³	—	2	1	no	—
257	C₂³	—	4	1	no	—
258	C₂³	—	4	1	no	—
259	C₂³	—	4	1	no	—
260	C₂³	—	4	1	no	—
261	C₂³	#13133151	1	2	yes	complemented
262	C₂³	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
263	C₂³	#13133191	1	2	yes	—
264	C₂³	C₂×D₄₂₄	1	1	yes	characteristic
265	C₂³	C₂×Dic₂₁₂	1	1	yes	characteristic
266	C₂³	#13133239	1	2	yes	—
267	D₄	—	2	4	no	—
268	D₄	—	2	4	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	4	1	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	4	1	no	—
281	C₂×C₄	—	4	1	no	—
282	C₂×C₄	—	4	1	no	—
283	C₂×C₄	—	4	1	no	—
284	C₂×C₄	—	424	4	no	—
285	C₂×C₄	—	424	4	no	—
286	C₂×C₄	—	424	4	no	—
287	C₂×C₄	#13133151	1	2	yes	complemented
288	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
289	C₂×C₄	#13133176	1	1	yes	characteristic
290	C₂×C₄	#13133196	1	2	yes	—
291	C₈	—	2	1	no	—
292	C₈	—	2	2	no	—
293	C₈	—	4	1	no	—
294	C₈	—	4	1	no	—
295	C₈	#13133132	1	2	yes	complemented
296	C₂²	—	2	1	no	—
297	C₂²	—	2	2	no	—
298	C₂²	—	2	1	no	—
299	C₂²	—	2	2	no	—
300	C₂²	—	2	2	no	—
301	C₂²	—	4	1	no	—
302	C₂²	—	4	1	no	—
303	C₂²	—	4	2	no	—
304	C₂²	—	4	2	no	—
305	C₂²	—	4	2	no	—
306	C₂²	—	4	1	no	—
307	C₂²	—	4	1	no	—
308	C₂²	#16810483	1	1	yes	characteristic
309	C₂²	C₂×#13133151	1	1	yes	characteristic
310	C₂²	#16810661	1	1	yes	characteristic
311	C₂²	#16810859	1	2	yes	—
312	C₂²	#16810864	1	2	yes	—
313	C₄	—	2	1	no	—
314	C₄	—	2	2	no	—
315	C₄	—	2	2	no	—
316	C₄	—	4	1	no	—
317	C₄	—	4	1	no	—
318	C₄	—	848	4	no	—
319	C₄	C₂×#13133132	1	1	yes	characteristic
320	C₄	#16810597	1	1	yes	characteristic
321	C₂	—	2	2	no	—
322	C₂	—	2	1	no	—
323	C₂	—	2	1	no	—
324	C₂	—	2	2	no	—
325	C₂	—	4	2	no	—
326	C₂	#20274703	1	1	yes	characteristic
327	C₂	#20274744	1	1	yes	characteristic
328	C₂	#20274761	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16810483
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₂×C₈×D₄×C₅₃	C₂
socle	C₂²×C₅₃	#380
prime radical	C₂²×D₄×C₅₃	D₄
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	C₂×C₈×D₄	D₅₃
hypocenter	C₅₃	#8609
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×C₈×D₄

Complements

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

N	H ≅ G/N	type
1. #20274744	C₂	semidirect
3. #20272583	C₂	semidirect
10. C₂³×C₈×C₅₃	C₄	semidirect
15. C₂×C₄⋊C₈×C₅₃	C₄	semidirect
16. C₂×C₂²⋊C₈×C₅₃	C₄	semidirect
18. C₂×C₄×C₈×C₅₃	C₄	semidirect
96. C₂³×C₅₃	#107	semidirect
119. C₂×C₄×C₅₃	#107	semidirect
130. C₈×C₅₃	#99	semidirect
169. C₂³×C₈	Dic₅₃	semidirect
175. C₂×C₄⋊C₈	Dic₅₃	semidirect
177. C₂×C₂²⋊C₈	Dic₅₃	semidirect
179. C₂×C₄×C₈	Dic₅₃	semidirect
183. C₅₃	#8609	semidirect
261. C₂³	#13133151	semidirect
287. C₂×C₄	#13133151	semidirect
295. C₈	#13133132	semidirect

Maximal subgroups

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

subgroup	index
#20274744	2
#20274703	2
#20272583	2
C₂×C₈×D₄×C₅₃	2
#8609	53