Subgroups — #27539447

Order 13568.

578 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	#20285226	C₂	1	1	yes	maximal, characteristic
2	#494×D₅₃	C₂	1	1	yes	maximal, characteristic
3	#20276959	C₂	1	4	yes	maximal, complemented
4	#20276944	C₂	1	1	yes	maximal, characteristic, complemented
5	#20276943	C₂	1	1	yes	maximal, characteristic, complemented
6	#20276942	C₂	1	1	yes	maximal, characteristic, complemented
7	#20276433	C₂	1	4	yes	maximal
8	C₄×#13133156	C₂	1	1	yes	maximal, characteristic
9	C₅₃×#2241	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×Q₁₆×D₅₃	—	2	1	no	—
11	C₂×Q₁₆×D₅₃	C₂²	1	2	yes	—
12	#16811178	—	2	1	no	—
13	C₄⋊Q₈×D₅₃	—	2	1	no	—
14	C₂×#13133156	C₂²	1	2	yes	—
15	#16811028	C₂²	1	1	yes	characteristic
16	#16811024	—	2	1	no	—
17	#16811024	C₂²	1	2	yes	—
18	#16811019	—	2	1	no	—
19	#16811003	C₂²	1	1	yes	characteristic
20	#16811002	—	2	1	no	—
21	#16811001	—	2	1	no	—
22	#16811001	C₂²	1	2	yes	—
23	#16810660	—	2	1	no	—
24	#16810660	C₂²	1	2	yes	—
25	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
26	#16810428	C₂²	1	4	yes	—
27	#16810427	C₂²	1	4	yes	—
28	#16810426	C₂²	1	4	yes	—
29	#16810398	C₂²	1	2	yes	—
30	C₄×#11719908	C₂²	1	1	yes	characteristic
31	C₅₃×#500	C₂²	1	1	yes	characteristic
32	C₅₃×#494	C₂²	1	1	yes	characteristic
33	C₅₃×#358	C₂²	1	4	yes	—
34	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
35	C₂×Q₈×D₅₃	—	2	2	no	—
36	C₂×Dic₂₁₂	—	2	1	no	—
37	C₂×Dic₂₁₂	C₂³	1	2	yes	—
38	C₂×#11719925	—	2	2	no	—
39	C₂×#11719925	—	2	2	no	—
40	Q₁₆×D₅₃	—	2	8	no	—
41	Q₁₆×D₅₃	—	4	4	no	—
42	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
43	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
44	#13133217	—	2	1	no	—
45	#13133217	—	2	1	no	—
46	#13133216	—	4	1	no	—
47	#13133216	—	4	1	no	—
48	#13133215	—	2	2	no	—
49	#13133208	—	2	2	no	—
50	#13133197	—	4	1	no	—
51	#13133193	—	4	1	no	—
52	Q₈×Dic₅₃	—	4	1	no	—
53	#13133182	—	4	1	no	—
54	#13133175	—	2	2	no	—
55	C₄⋊C₄×D₅₃	—	4	1	no	—
56	C₄⋊C₄×D₅₃	—	4	1	no	—
57	C₄²×D₅₃	D₄	1	1	yes	characteristic
58	#13133156	—	2	4	no	—
59	C₂×#11719908	C₂³	1	2	yes	—
60	#13133151	—	2	1	no	—
61	#13133151	C₂³	1	2	yes	—
62	#13133148	—	2	2	no	—
63	#13133148	—	2	2	no	—
64	C₄×#88322	D₄	1	1	yes	characteristic
65	C₈×Dic₅₃	D₄	1	1	yes	characteristic
66	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
67	C₂×Q₁₆×C₅₃	—	2	1	no	—
68	C₂×Q₁₆×C₅₃	C₂³	1	2	yes	—
69	C₄⋊Q₈×C₅₃	—	2	1	no	—
70	C₄⋊Q₈×C₅₃	—	2	1	no	—
71	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
72	#107×C₅₃	—	2	1	no	—
73	#107×C₅₃	C₂³	1	2	yes	—
74	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
75	C₂×Q₈×C₅₃	—	2	2	no	—
76	C₂×Q₈×C₅₃	—	4	1	no	—
77	Q₈×D₅₃	—	4	4	no	—
78	Q₈×D₅₃	—	4	4	no	—
79	C₂×C₄×D₅₃	—	4	2	no	—
80	C₂×C₄×D₅₃	D₈	1	1	yes	characteristic
81	C₂×C₄×D₅₃	D₈	1	1	yes	characteristic
82	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
83	C₂×Dic₁₀₆	—	2	2	no	—
84	C₂×Dic₁₀₆	—	4	1	no	—
85	C₂×Dic₁₀₆	—	4	2	no	—
86	Q₁₆×C₅₃	—	2	4	no	—
87	Q₁₆×C₅₃	—	4	2	no	—
88	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
90	C₄⋊C₄×C₅₃	—	2	2	no	—
91	C₄⋊C₄×C₅₃	—	4	1	no	—
92	C₄⋊C₄×C₅₃	—	4	1	no	—
93	C₄⋊C₄×C₅₃	—	4	1	no	—
94	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
95	#11719925	—	4	4	no	—
96	#11719925	—	4	4	no	—
97	#11719921	—	4	2	no	—
98	C₄⋊Dic₅₃	—	2	2	no	—
99	C₄⋊Dic₅₃	—	4	1	no	—
100	C₄⋊Dic₅₃	—	4	1	no	—
101	C₄⋊Dic₅₃	—	4	1	no	—
102	C₄⋊Dic₅₃	—	4	2	no	—
103	#11719919	—	4	2	no	—
104	#11719919	—	4	2	no	—
105	#11719919	—	4	2	no	—
106	C₄×Dic₅₃	—	4	2	no	—
107	C₄×Dic₅₃	D₈	1	1	yes	characteristic
108	C₄×Dic₅₃	D₈	1	1	yes	characteristic
109	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
110	C₂×#88322	C₂×D₄	1	1	yes	characteristic
111	C₂×#88322	C₂×D₄	1	1	yes	characteristic
112	Dic₂₁₂	—	2	4	no	—
113	Dic₂₁₂	—	4	2	no	—
114	C₈×D₅₃	SD₁₆	1	4	yes	—
115	C₈×D₅₃	#41	1	4	yes	—
116	C₁₆×C₅₃	—	2	2	no	—
117	#11719908	—	2	2	no	—
118	C₂²×D₅₃	C₂×D₈	1	1	yes	characteristic
119	Q₈×C₅₃	—	4	2	no	—
120	Q₈×C₅₃	—	4	2	no	—
121	Q₈×C₅₃	—	8	2	no	—
122	C₂×C₄×C₅₃	—	4	2	no	—
123	C₂×C₄×C₅₃	—	4	2	no	—
124	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
125	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
126	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
127	C₂×Dic₅₃	—	4	2	no	—
128	C₂×Dic₅₃	—	4	2	no	—
129	C₂×Dic₅₃	C₂×D₈	1	1	yes	characteristic
130	C₂×Dic₅₃	C₂×D₈	1	1	yes	characteristic
131	C₂×Dic₅₃	C₂×D₈	1	1	yes	characteristic
132	C₄×D₅₃	—	2	2	no	—
133	C₄×D₅₃	—	2	2	no	—
134	C₄×D₅₃	—	8	4	no	—
135	C₄×D₅₃	#124	1	2	yes	—
136	C₄×D₅₃	C₂×SD₁₆	1	2	yes	—
137	Dic₁₀₆	—	4	2	no	—
138	Dic₁₀₆	—	4	2	no	—
139	Dic₁₀₆	—	8	2	no	—
140	Dic₁₀₆	—	8	4	no	—
141	C₈×C₅₃	C₂×SD₁₆	1	2	yes	—
142	C₈×C₅₃	C₂×#41	1	2	yes	—
143	#88322	#124	1	2	yes	—
144	#88322	C₂×SD₁₆	1	2	yes	—
145	#33705	—	53	1	no	maximal, Sylow
146	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
147	C₂×D₅₃	—	2	1	no	—
148	C₂×D₅₃	—	2	1	no	—
149	C₂×D₅₃	#486	1	2	yes	—
150	C₄×C₅₃	—	8	2	no	—
151	C₄×C₅₃	—	8	2	no	—
152	C₄×C₅₃	#510	1	2	yes	—
153	C₄×C₅₃	#510	1	2	yes	—
154	C₄×C₅₃	C₂×#124	1	1	yes	characteristic
155	C₄×C₅₃	C₂²×SD₁₆	1	1	yes	characteristic
156	Dic₅₃	—	2	1	no	—
157	Dic₅₃	—	2	1	no	—
158	Dic₅₃	—	8	2	no	—
159	Dic₅₃	—	8	2	no	—
160	Dic₅₃	#486	1	2	yes	—
161	#3160	—	53	1	no	—
162	C₂×#494	—	53	1	no	—
163	#2245	—	53	4	no	—
164	#2241	—	53	1	no	—
165	#2241	—	53	1	no	—
166	#2241	—	53	1	no	—
167	#2241	D₅₃	1	1	yes	characteristic, complemented
168	#2141	—	53	4	no	—
169	C₄×M₃₂	—	53	1	no	—
170	C₂×C₅₃	C₂×#486	1	1	yes	characteristic
171	C₂×C₅₃	C₂×#510	1	1	yes	characteristic
172	C₂×C₅₃	C₂×#510	1	1	yes	characteristic
173	D₅₃	—	2	2	no	—
174	C₂²×Q₁₆	—	53	2	no	—
175	C₂²×Q₁₆	—	106	1	no	—
176	#533	—	106	1	no	—
177	C₂×C₄⋊Q₈	—	106	1	no	—
178	C₂×M₃₂	—	53	2	no	—
179	#500	—	53	1	no	—
180	#500	C₂×D₅₃	1	1	yes	characteristic
181	#499	—	53	2	no	—
182	#499	—	106	1	no	—
183	#498	—	106	1	no	—
184	#494	—	53	2	no	—
185	#494	—	53	1	no	—
186	#494	—	106	1	no	—
187	#494	—	106	1	no	—
188	#494	C₂×D₅₃	1	1	yes	characteristic
189	#427	—	53	2	no	—
190	#427	—	106	1	no	—
191	C₂×C₄×C₈	—	53	1	no	—
192	#358	—	53	4	no	—
193	#358	—	53	4	no	—
194	#358	—	53	4	no	—
195	#358	C₂×D₅₃	1	4	yes	—
196	#346	—	53	2	no	—
197	C₄×C₁₆	—	53	1	no	—
198	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic
199	C₅₃	#33705	1	1	yes	characteristic, complemented, Sylow
200	C₂²×Q₈	—	106	2	no	—
201	C₂×Q₁₆	—	2	1	no	—
202	C₂×Q₁₆	—	53	2	no	—
203	C₂×Q₁₆	—	106	8	no	—
204	C₂×Q₁₆	—	106	2	no	—
205	C₂×Q₁₆	—	106	2	no	—
206	C₂×Q₁₆	—	106	1	no	—
207	C₂×Q₁₆	—	212	4	no	—
208	C₂×Q₁₆	C₂²×D₅₃	1	2	yes	—
209	C₂²×C₈	—	53	1	no	—
210	C₂²×C₈	—	53	1	no	—
211	C₄⋊Q₈	—	2	1	no	—
212	C₄⋊Q₈	—	2	1	no	—
213	C₄⋊Q₈	—	106	2	no	—
214	C₄⋊Q₈	—	106	1	no	—
215	C₄⋊Q₈	—	106	1	no	—
216	C₄⋊Q₈	—	212	1	no	—
217	C₄⋊Q₈	—	212	1	no	—
218	#125	—	106	2	no	—
219	C₂²⋊Q₈	—	212	1	no	—
220	C₂²⋊Q₈	—	212	1	no	—
221	C₄×Q₈	—	212	1	no	—
222	C₄×Q₈	—	212	1	no	—
223	#117	—	106	2	no	—
224	C₂×C₄⋊C₄	—	212	1	no	—
225	C₂×C₄⋊C₄	—	212	1	no	—
226	C₂×C₄²	—	53	1	no	—
227	M₃₂	—	106	4	no	—
228	C₂×C₁₆	—	53	2	no	—
229	C₂×C₁₆	C₂²×D₅₃	1	2	yes	—
230	#107	—	2	1	no	—
231	#107	—	53	2	no	—
232	#107	—	106	1	no	—
233	#107	C₂²×D₅₃	1	2	yes	—
234	#106	—	106	2	no	—
235	#106	—	106	2	no	—
236	C₄×C₈	—	53	1	no	—
237	C₄×C₈	—	53	1	no	—
238	C₄×C₈	—	53	1	no	—
239	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
240	C₂×Q₈	—	2	2	no	—
241	C₂×Q₈	—	4	1	no	—
242	C₂×Q₈	—	106	2	no	—
243	C₂×Q₈	—	212	2	no	—
244	C₂×Q₈	—	212	4	no	—
245	C₂×Q₈	—	212	4	no	—
246	C₂×Q₈	—	212	1	no	—
247	C₂²×C₄	—	53	1	no	—
248	C₂²×C₄	—	53	1	no	—
249	C₂²×C₄	—	53	1	no	—
250	C₂²×C₄	—	212	2	no	—
251	Q₁₆	—	2	4	no	—
252	Q₁₆	—	4	2	no	—
253	Q₁₆	—	106	4	no	—
254	Q₁₆	—	212	4	no	—
255	Q₁₆	—	212	4	no	—
256	Q₁₆	—	212	2	no	—
257	C₂×C₈	—	53	1	no	—
258	C₂×C₈	—	53	1	no	—
259	C₂×C₈	—	53	4	no	—
260	C₂×C₈	—	53	4	no	—
261	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
262	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	4	1	no	—
265	C₄⋊C₄	—	4	1	no	—
266	C₄⋊C₄	—	4	1	no	—
267	C₄⋊C₄	—	106	2	no	—
268	C₄⋊C₄	—	212	2	no	—
269	C₄⋊C₄	—	212	2	no	—
270	C₄⋊C₄	—	212	2	no	—
271	C₄⋊C₄	—	212	2	no	—
272	C₄⋊C₄	—	212	1	no	—
273	C₄⋊C₄	—	212	1	no	—
274	C₄⋊C₄	—	212	1	no	—
275	C₂²⋊C₄	—	212	2	no	—
276	C₄²	—	53	1	no	—
277	C₄²	—	53	1	no	—
278	C₄²	—	53	1	no	—
279	C₄²	—	212	2	no	—
280	C₄²	D₄×D₅₃	1	1	yes	characteristic
281	C₁₆	—	2	2	no	—
282	C₁₆	—	106	2	no	—
283	C₂³	—	53	1	no	—
284	Q₈	—	4	2	no	—
285	Q₈	—	4	2	no	—
286	Q₈	—	8	2	no	—
287	Q₈	—	424	4	no	—
288	Q₈	—	424	2	no	—
289	Q₈	—	212	2	no	—
290	Q₈	—	212	2	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	53	1	no	—
294	C₂×C₄	—	53	2	no	—
295	C₂×C₄	—	53	2	no	—
296	C₂×C₄	—	53	1	no	—
297	C₂×C₄	—	53	1	no	—
298	C₂×C₄	—	106	2	no	—
299	C₂×C₄	—	106	2	no	—
300	C₂×C₄	—	424	4	no	—
301	C₂×C₄	—	212	2	no	—
302	C₂×C₄	—	212	2	no	—
303	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
304	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
305	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
306	C₈	—	53	2	no	—
307	C₈	—	53	2	no	—
308	C₈	SD₁₆×D₅₃	1	2	yes	—
309	C₈	#41×D₅₃	1	2	yes	—
310	C₂²	—	53	2	no	—
311	C₂²	—	106	1	no	—
312	C₂²	—	106	1	no	—
313	C₂²	C₂×D₈×D₅₃	1	1	yes	characteristic
314	C₄	—	8	2	no	—
315	C₄	—	8	2	no	—
316	C₄	—	53	2	no	—
317	C₄	—	106	1	no	—
318	C₄	—	106	1	no	—
319	C₄	—	424	2	no	—
320	C₄	—	424	2	no	—
321	C₄	#16811062	1	2	yes	—
322	C₄	#16811065	1	2	yes	—
323	C₄	#124×D₅₃	1	1	yes	characteristic
324	C₄	C₂×SD₁₆×D₅₃	1	1	yes	characteristic
325	C₂	—	106	2	no	—
326	C₂	#486×D₅₃	1	1	yes	characteristic
327	C₂	C₂×#16811062	1	1	yes	characteristic
328	C₂	C₂×#16811065	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₅₃
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2241	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	C₂²×D₅₃	C₂×D₈
hypercenter	#2241	D₅₃
hypocenter	C₅₃	#33705
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2241

Complements

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

N	H ≅ G/N	type
3. #20276959	C₂	semidirect
4. #20276944	C₂	semidirect
5. #20276943	C₂	semidirect
6. #20276942	C₂	semidirect
9. C₅₃×#2241	C₂	semidirect
167. #2241	D₅₃	semidirect
199. C₅₃	#33705	semidirect

Maximal subgroups

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

subgroup	index
#20285226	2
#494×D₅₃	2
#20276959	2
#20276944	2
#20276943	2
#20276942	2
#20276433	2
C₄×#13133156	2
C₅₃×#2241	2
#33705	53