Subgroups — #27299286

Order 13568.

460 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20276400	C₂	1	1	yes	maximal, characteristic
2	#20276380	C₂	1	1	yes	maximal, characteristic
3	#20274515	C₂	1	2	yes	maximal, complemented
4	#20272844	C₂	1	2	yes	maximal, complemented
5	C₅₃×#2931	C₂	1	1	yes	maximal, characteristic
6	#16810607	C₂²	1	2	yes	—
7	#16810597	C₂²	1	2	yes	—
8	#16810578	—	2	2	no	—
9	#16810422	C₄	1	2	yes	complemented
10	#16810420	C₄	1	2	yes	complemented
11	#16810418	C₄	1	2	yes	—
12	#16810416	C₄	1	2	yes	—
13	#16810394	—	2	2	no	—
14	C₂×#131×C₅₃	C₄	1	1	yes	characteristic, complemented
15	C₂×#131×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×#532	C₂²	1	1	yes	characteristic
17	C₅₃×#436	—	2	2	no	—
18	C₈×D₄×C₅₃	—	2	2	no	—
19	C₅₃×#423	C₄	1	1	yes	characteristic, complemented
20	C₅₃×#408	C₄	1	1	yes	characteristic, complemented
21	C₅₃×#408	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic, complemented
23	C₂×#11719917	—	2	2	no	—
24	#13133176	—	2	2	no	—
25	#13133136	C₂×C₄	1	4	yes	—
26	#13133134	—	2	2	no	—
27	#13133132	—	2	2	no	—
28	#13133132	C₂×C₄	1	4	yes	—
29	#13133127	—	4	2	no	—
30	C₈×Dic₅₃	—	4	2	no	—
31	C₂×#41×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂×#41×C₅₃	C₂³	1	1	yes	characteristic
33	#131×C₅₃	—	4	1	no	—
34	#131×C₅₃	—	4	1	no	—
35	#131×C₅₃	—	4	1	no	—
36	#131×C₅₃	—	4	1	no	—
37	C₂×M₁₆×C₅₃	—	2	1	no	—
38	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
39	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
40	C₂×M₁₆×C₅₃	D₄	1	2	yes	—
41	C₂²×C₈×C₅₃	—	2	1	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	2	yes	—
45	C₄⋊Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
46	#127×C₅₃	C₂×C₄	1	1	yes	characteristic
47	#124×C₅₃	C₂×C₄	1	1	yes	characteristic
48	#124×C₅₃	C₂×C₄	1	1	yes	characteristic
49	#121×C₅₃	—	2	2	no	—
50	C₄×D₄×C₅₃	—	2	2	no	—
51	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
52	C₄⋊C₈×C₅₃	—	2	2	no	—
53	C₂²⋊C₈×C₅₃	—	2	2	no	—
54	C₂²⋊C₈×C₅₃	—	2	2	no	—
55	C₄×C₈×C₅₃	—	2	1	no	—
56	C₄×C₈×C₅₃	—	2	1	no	—
57	#41×C₅₃	—	4	1	no	—
58	#41×C₅₃	—	4	1	no	—
59	#41×C₅₃	—	4	1	no	—
60	#41×C₅₃	—	4	1	no	—
61	C₂×Q₈×C₅₃	C₄²	1	1	yes	characteristic
62	C₂×Q₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
63	C₂×D₄×C₅₃	—	2	1	no	—
64	C₂×D₄×C₅₃	—	2	2	no	—
65	C₂×D₄×C₅₃	C₄²	1	1	yes	characteristic
66	C₂×D₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
67	C₂×D₄×C₅₃	#41	1	2	yes	—
68	C₂²×C₄×C₅₃	—	2	1	no	—
69	C₂²×C₄×C₅₃	—	2	1	no	—
70	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
72	M₁₆×C₅₃	—	4	1	no	—
73	M₁₆×C₅₃	—	4	2	no	—
74	M₁₆×C₅₃	—	4	2	no	—
75	M₁₆×C₅₃	—	4	1	no	—
76	C₂×C₈×C₅₃	—	2	1	no	—
77	C₂×C₈×C₅₃	—	2	1	no	—
78	C₂×C₈×C₅₃	—	2	2	no	—
79	C₂×C₈×C₅₃	—	2	2	no	—
80	C₂×C₈×C₅₃	—	2	1	no	—
81	C₂×C₈×C₅₃	—	2	1	no	—
82	C₂×C₈×C₅₃	—	4	1	no	—
83	C₂×C₈×C₅₃	—	4	1	no	—
84	C₂×C₈×C₅₃	—	4	1	no	—
85	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
87	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
88	C₄⋊C₄×C₅₃	—	2	2	no	—
89	C₂²⋊C₄×C₅₃	—	2	2	no	—
90	C₂²⋊C₄×C₅₃	—	2	2	no	—
91	C₄²×C₅₃	—	2	1	no	—
92	C₄²×C₅₃	C₄²	1	1	yes	characteristic
93	C₄²×C₅₃	C₄²	1	1	yes	characteristic
94	C₂²⋊Dic₅₃	—	2	4	no	—
95	C₄⋊Dic₅₃	—	4	2	no	—
96	C₄×Dic₅₃	—	4	2	no	—
97	#11719917	—	2	4	no	—
98	#11719917	—	4	2	no	—
99	C₂×#88322	—	4	2	no	—
100	C₂³×C₅₃	—	2	1	no	—
101	C₂³×C₅₃	#117	1	1	yes	characteristic
102	C₂³×C₅₃	C₄×D₄	1	2	yes	—
103	Q₈×C₅₃	—	4	1	no	—
104	Q₈×C₅₃	—	4	1	no	—
105	D₄×C₅₃	—	4	1	no	—
106	D₄×C₅₃	—	4	2	no	—
107	D₄×C₅₃	—	4	1	no	—
108	D₄×C₅₃	—	4	4	no	—
109	D₄×C₅₃	—	4	2	no	—
110	C₂×C₄×C₅₃	—	2	1	no	—
111	C₂×C₄×C₅₃	—	2	1	no	—
112	C₂×C₄×C₅₃	—	2	2	no	—
113	C₂×C₄×C₅₃	—	2	1	no	—
114	C₂×C₄×C₅₃	—	2	2	no	—
115	C₂×C₄×C₅₃	—	2	1	no	—
116	C₂×C₄×C₅₃	—	2	1	no	—
117	C₂×C₄×C₅₃	—	4	2	no	—
118	C₂×C₄×C₅₃	—	4	1	no	—
119	C₂×C₄×C₅₃	—	4	1	no	—
120	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
121	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
123	C₂×Dic₅₃	—	4	4	no	—
124	C₈×C₅₃	—	2	2	no	—
125	C₈×C₅₃	—	2	1	no	—
126	C₈×C₅₃	—	4	1	no	—
127	C₈×C₅₃	—	4	1	no	—
128	C₈×C₅₃	#99	1	1	yes	characteristic
129	C₈×C₅₃	#99	1	1	yes	characteristic
130	#88322	—	4	4	no	—
131	#12633	—	53	1	no	maximal, Sylow
132	C₂²×C₅₃	—	2	2	no	—
133	C₂²×C₅₃	—	2	1	no	—
134	C₂²×C₅₃	—	4	1	no	—
135	C₂²×C₅₃	—	4	2	no	—
136	C₂²×C₅₃	—	4	2	no	—
137	C₂²×C₅₃	—	4	1	no	—
138	C₂²×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
139	C₄×C₅₃	—	2	2	no	—
140	C₄×C₅₃	—	2	2	no	—
141	C₄×C₅₃	—	2	1	no	—
142	C₄×C₅₃	—	4	1	no	—
143	C₄×C₅₃	—	4	1	no	—
144	C₄×C₅₃	C₂×#99	1	1	yes	characteristic
145	C₄×C₅₃	#410	1	1	yes	characteristic
146	Dic₅₃	—	8	4	no	—
147	#2931	D₅₃	1	1	yes	characteristic
148	#2132	—	53	1	no	—
149	#2126	—	53	1	no	—
150	#1757	—	53	2	no	—
151	#1395	—	53	2	no	—
152	C₂×C₅₃	—	2	1	no	—
153	C₂×C₅₃	—	4	2	no	—
154	C₂×C₅₃	—	4	1	no	—
155	C₂×C₅₃	—	8	1	no	—
156	C₂×C₅₃	C₄×#99	1	1	yes	characteristic
157	C₂×#131	Dic₅₃	1	1	yes	characteristic, complemented
158	C₂×#131	C₂×D₅₃	1	1	yes	characteristic
159	#532	C₂×D₅₃	1	1	yes	characteristic
160	#436	—	2	2	no	—
161	C₈×D₄	—	2	2	no	—
162	#423	Dic₅₃	1	1	yes	characteristic, complemented
163	#413	—	53	2	no	—
164	#410	—	53	2	no	—
165	#408	Dic₅₃	1	1	yes	characteristic, complemented
166	#408	C₂×D₅₃	1	1	yes	characteristic
167	#405	—	106	2	no	—
168	C₂×C₄×C₈	Dic₅₃	1	1	yes	characteristic, complemented
169	#356	—	53	2	no	—
170	#355	—	53	2	no	—
171	#354	—	53	2	no	—
172	#353	—	53	2	no	—
173	#344	—	106	2	no	—
174	C₅₃	#12633	1	1	yes	characteristic, complemented, Sylow
175	C₂×#41	C₂×Dic₅₃	1	1	yes	characteristic
176	C₂×#41	C₂²×D₅₃	1	1	yes	characteristic
177	#131	—	4	1	no	—
178	#131	—	4	1	no	—
179	#131	—	4	1	no	—
180	#131	—	4	1	no	—
181	C₂×M₁₆	—	2	1	no	—
182	C₂×M₁₆	—	106	2	no	—
183	C₂×M₁₆	C₂×Dic₅₃	1	1	yes	characteristic
184	C₂×M₁₆	C₂×Dic₅₃	1	1	yes	characteristic
185	C₂×M₁₆	C₂²⋊D₅₃	1	2	yes	—
186	C₂²×C₈	—	2	1	no	—
187	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
188	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
189	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
190	C₄⋊Q₈	C₄×D₅₃	1	1	yes	characteristic
191	#127	C₄×D₅₃	1	1	yes	characteristic
192	#124	C₄×D₅₃	1	1	yes	characteristic
193	#124	C₄×D₅₃	1	1	yes	characteristic
194	#121	—	2	2	no	—
195	C₄×D₄	—	2	2	no	—
196	#117	—	106	2	no	—
197	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
198	C₄⋊C₈	—	2	2	no	—
199	#101	—	53	4	no	—
200	#100	—	106	2	no	—
201	#99	—	53	4	no	—
202	#99	—	106	2	no	—
203	C₂²⋊C₈	—	2	2	no	—
204	C₂²⋊C₈	—	2	2	no	—
205	#97	—	212	2	no	—
206	C₄×C₈	—	2	1	no	—
207	C₄×C₈	—	2	1	no	—
208	C₄×C₈	—	212	2	no	—
209	#41	—	4	1	no	—
210	#41	—	4	1	no	—
211	#41	—	4	1	no	—
212	#41	—	4	1	no	—
213	C₂×Q₈	C₄×Dic₅₃	1	1	yes	characteristic
214	C₂×Q₈	C₂×C₄×D₅₃	1	1	yes	characteristic
215	C₂×D₄	—	2	1	no	—
216	C₂×D₄	—	2	2	no	—
217	C₂×D₄	C₄×Dic₅₃	1	1	yes	characteristic
218	C₂×D₄	C₂×C₄×D₅₃	1	1	yes	characteristic
219	C₂×D₄	#11719945	1	2	yes	—
220	C₂²×C₄	—	2	1	no	—
221	C₂²×C₄	—	2	1	no	—
222	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
223	C₂²×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
224	M₁₆	—	4	1	no	—
225	M₁₆	—	4	2	no	—
226	M₁₆	—	4	2	no	—
227	M₁₆	—	4	1	no	—
228	M₁₆	—	106	4	no	—
229	M₁₆	—	212	2	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	1	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	2	1	no	—
234	C₂×C₈	—	2	1	no	—
235	C₂×C₈	—	2	1	no	—
236	C₂×C₈	—	4	1	no	—
237	C₂×C₈	—	4	1	no	—
238	C₂×C₈	—	4	1	no	—
239	C₂×C₈	—	212	2	no	—
240	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
241	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
242	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
243	C₄⋊C₄	—	2	2	no	—
244	C₄⋊C₄	—	212	2	no	—
245	C₂²⋊C₄	—	2	2	no	—
246	C₂²⋊C₄	—	2	2	no	—
247	C₂²⋊C₄	—	106	4	no	—
248	C₄²	—	2	1	no	—
249	C₄²	—	212	2	no	—
250	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
251	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
252	C₂³	—	2	1	no	—
253	C₂³	#13133176	1	1	yes	characteristic
254	C₂³	C₄×C₂²⋊D₅₃	1	2	yes	—
255	Q₈	—	4	1	no	—
256	Q₈	—	4	1	no	—
257	D₄	—	4	1	no	—
258	D₄	—	4	2	no	—
259	D₄	—	4	1	no	—
260	D₄	—	4	4	no	—
261	D₄	—	4	2	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	1	no	—
271	C₂×C₄	—	4	1	no	—
272	C₂×C₄	—	212	4	no	—
273	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
274	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
275	C₂×C₄	C₄×C₂²⋊D₅₃	1	2	yes	—
276	C₈	—	2	2	no	—
277	C₈	—	2	1	no	—
278	C₈	—	4	1	no	—
279	C₈	—	4	1	no	—
280	C₈	—	212	4	no	—
281	C₈	#13133132	1	1	yes	characteristic
282	C₈	#13133132	1	1	yes	characteristic
283	C₂²	—	2	2	no	—
284	C₂²	—	2	1	no	—
285	C₂²	—	4	1	no	—
286	C₂²	—	4	2	no	—
287	C₂²	—	4	2	no	—
288	C₂²	—	4	1	no	—
289	C₂²	C₄×C₂²⋊Dic₅₃	1	1	yes	characteristic
290	C₄	—	2	2	no	—
291	C₄	—	2	2	no	—
292	C₄	—	2	1	no	—
293	C₄	—	4	1	no	—
294	C₄	—	4	1	no	—
295	C₄	—	424	4	no	—
296	C₄	C₂×#13133132	1	1	yes	characteristic
297	C₄	#16810597	1	1	yes	characteristic
298	C₂	—	2	1	no	—
299	C₂	—	4	2	no	—
300	C₂	—	4	1	no	—
301	C₂	—	8	1	no	—
302	C₂	C₄×#13133132	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#13133132
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂×#41	C₂²×D₅₃
Fitting	C₅₃×#2931	C₂
socle	C₂×C₅₃	C₄×#99
prime radical	C₅₃×#532	C₂²
cosocle	C₂×#41×C₅₃	C₂³
hypercenter	#2931	D₅₃
hypocenter	C₅₃	#12633
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₂×#131 ◁ #2931

Complements

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

N	H ≅ G/N	type
3. #20274515	C₂	semidirect
4. #20272844	C₂	semidirect
9. #16810422	C₄	semidirect
10. #16810420	C₄	semidirect
14. C₂×#131×C₅₃	C₄	semidirect
19. C₅₃×#423	C₄	semidirect
20. C₅₃×#408	C₄	semidirect
22. C₂×C₄×C₈×C₅₃	C₄	semidirect
157. C₂×#131	Dic₅₃	semidirect
162. #423	Dic₅₃	semidirect
165. #408	Dic₅₃	semidirect
168. C₂×C₄×C₈	Dic₅₃	semidirect
174. C₅₃	#12633	semidirect

Maximal subgroups

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

subgroup	index
#20276400	2
#20276380	2
#20274515	2
#20272844	2
C₅₃×#2931	2
#12633	53