Subgroups — #27299126

Order 13568.

514 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810577	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810402	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810402	C₂	1	1	yes	maximal, characteristic
4	C₂×#16810398	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₁₆×Dic₅₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#20272793	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1374	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#11719912	C₄	1	1	yes	characteristic
9	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
10	C₂×#13133174	C₄	1	1	yes	characteristic
11	C₂²×#11719908	C₂²	1	1	yes	characteristic
12	C₂²×#11719908	C₂²	1	1	yes	characteristic
13	#16810577	—	2	2	no	—
14	#16810577	—	2	2	no	—
15	C₂×#13133128	C₄	1	1	yes	characteristic
16	C₂×#13133127	C₄	1	1	yes	characteristic
17	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
18	#16810402	C₄	1	4	yes	complemented
19	#16810402	C₄	1	4	yes	complemented
20	#16810398	—	2	2	no	—
21	C₁₆×Dic₅₃	—	2	2	no	—
22	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
23	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
25	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
26	C₂×#11719912	—	2	2	no	—
27	C₂×#11719912	—	2	2	no	—
28	C₂×#11719912	C₈	1	4	yes	—
29	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
30	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
31	C₂×C₈×D₅₃	—	2	1	no	—
32	C₂×C₈×D₅₃	—	2	1	no	—
33	C₂×C₈×D₅₃	—	2	1	no	—
34	C₂×C₈×D₅₃	—	2	1	no	—
35	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
36	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
37	#13133174	—	2	2	no	—
38	#13133174	—	2	2	no	—
39	C₂×#11719919	C₈	1	2	yes	—
40	C₂×#11719921	C₈	1	2	yes	—
41	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
42	C₂×#11719908	—	2	1	no	—
43	C₂×#11719908	—	2	1	no	—
44	C₂×#11719908	—	2	1	no	—
45	C₂×#11719908	—	2	1	no	—
46	C₂×#11719908	C₂×C₄	1	2	yes	—
47	C₂×#11719908	C₂×C₄	1	2	yes	—
48	#13133128	—	2	2	no	—
49	#13133127	—	2	2	no	—
50	C₈×Dic₅₃	D₄	1	2	yes	—
51	C₈×Dic₅₃	D₄	1	2	yes	—
52	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
53	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
55	C₂×C₁₆×C₅₃	—	2	1	no	—
56	C₂×C₁₆×C₅₃	—	2	1	no	—
57	C₂×C₁₆×C₅₃	—	2	1	no	—
58	C₂×C₁₆×C₅₃	—	2	1	no	—
59	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
60	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
61	C₄×C₈×C₅₃	—	2	1	no	—
62	C₄×C₈×C₅₃	—	2	1	no	—
63	C₂³×D₅₃	C₂×C₈	1	1	yes	characteristic
64	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
65	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
66	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
67	C₂²×Dic₅₃	C₂×C₈	1	2	yes	—
68	C₂×C₄×D₅₃	—	2	1	no	—
69	C₂×C₄×D₅₃	—	2	1	no	—
70	C₂×C₄×D₅₃	—	2	1	no	—
71	C₂×C₄×D₅₃	—	2	1	no	—
72	C₂×C₄×D₅₃	C₄²	1	2	yes	—
73	C₂×C₄×D₅₃	C₂×C₈	1	2	yes	—
74	C₂×C₈×C₅₃	—	2	1	no	—
75	C₂×C₈×C₅₃	—	2	1	no	—
76	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
77	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
80	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
81	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
82	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
83	C₄²×C₅₃	—	2	1	no	—
84	C₄²×C₅₃	—	2	1	no	—
85	#11719921	—	2	4	no	—
86	#11719919	—	2	4	no	—
87	C₄×Dic₅₃	C₂²⋊C₄	1	2	yes	—
88	C₄×Dic₅₃	M₁₆	1	2	yes	—
89	C₂×#88322	—	2	1	no	—
90	C₂×#88322	—	2	1	no	—
91	C₂×#88322	C₄²	1	2	yes	—
92	C₂×#88322	C₂²⋊C₄	1	2	yes	—
93	C₂×#88322	C₂×C₈	1	2	yes	—
94	C₂×#88322	M₁₆	1	2	yes	—
95	#11719912	—	2	8	no	—
96	C₈×D₅₃	—	2	2	no	—
97	C₈×D₅₃	—	2	2	no	—
98	C₈×D₅₃	—	2	2	no	—
99	C₈×D₅₃	—	2	2	no	—
100	C₁₆×C₅₃	—	2	2	no	—
101	C₁₆×C₅₃	—	2	2	no	—
102	#11719908	—	2	2	no	—
103	#11719908	—	2	2	no	—
104	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
105	C₂²×D₅₃	—	2	1	no	—
106	C₂²×D₅₃	—	2	1	no	—
107	C₂²×D₅₃	—	2	2	no	—
108	C₂²×D₅₃	—	2	2	no	—
109	C₂²×D₅₃	C₄×C₈	1	2	yes	—
110	C₂×C₄×C₅₃	—	2	2	no	—
111	C₂×C₄×C₅₃	—	2	2	no	—
112	C₂×C₄×C₅₃	—	2	2	no	—
113	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
114	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
116	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
117	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
118	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
119	C₂×Dic₅₃	—	2	1	no	—
120	C₂×Dic₅₃	—	2	1	no	—
121	C₂×Dic₅₃	—	2	2	no	—
122	C₂×Dic₅₃	—	2	2	no	—
123	C₂×Dic₅₃	C₄×C₈	1	2	yes	—
124	C₂×Dic₅₃	C₂²⋊C₈	1	4	yes	—
125	C₄×D₅₃	—	2	2	no	—
126	C₄×D₅₃	—	2	2	no	—
127	C₄×D₅₃	—	2	2	no	—
128	C₄×D₅₃	—	2	2	no	—
129	C₈×C₅₃	—	2	2	no	—
130	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
131	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
132	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
133	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
134	#88322	—	2	2	no	—
135	#88322	C₂²⋊C₈	1	4	yes	—
136	#12600	—	53	1	no	maximal, Sylow
137	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
138	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
139	C₂²×C₅₃	#431	1	1	yes	characteristic
140	C₂²×C₅₃	#504	1	2	yes	—
141	C₂²×C₅₃	#504	1	2	yes	—
142	C₂×D₅₃	—	2	2	no	—
143	C₂×D₅₃	—	2	2	no	—
144	C₂×D₅₃	—	2	2	no	—
145	C₂×D₅₃	—	4	2	no	—
146	C₂×D₅₃	—	4	2	no	—
147	C₄×C₅₃	—	4	2	no	—
148	C₄×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
149	C₄×C₅₃	C₄×M₁₆	1	1	yes	characteristic
150	C₄×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
151	C₄×C₅₃	#431	1	1	yes	characteristic
152	Dic₅₃	—	2	4	no	—
153	Dic₅₃	—	2	2	no	—
154	C₂×#405	—	53	1	no	—
155	C₂×C₂²⋊C₁₆	—	53	1	no	—
156	C₂×C₂²⋊C₁₆	—	53	1	no	—
157	C₂×#346	—	53	1	no	—
158	C₂×C₄×C₁₆	—	53	1	no	—
159	#1374	—	53	1	no	—
160	#1374	D₅₃	1	1	yes	characteristic, complemented
161	C₂×C₅₃	C₄×C₂²⋊C₈	1	1	yes	characteristic
162	C₂×C₅₃	#2108	1	1	yes	characteristic
163	C₂×C₅₃	#2108	1	1	yes	characteristic
164	C₂×C₅₃	#2113	1	2	yes	—
165	C₂×C₅₃	#2164	1	2	yes	—
166	D₅₃	—	4	2	no	—
167	C₂²×M₁₆	—	53	1	no	—
168	C₂³×C₈	—	53	1	no	—
169	C₂×#117	—	53	1	no	—
170	C₂²×C₁₆	—	53	1	no	—
171	C₂²×C₁₆	—	53	1	no	—
172	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
173	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
174	#405	—	106	2	no	—
175	#405	—	106	2	no	—
176	C₂×#97	—	53	1	no	—
177	C₂×#97	—	53	1	no	—
178	C₂×C₄×C₈	—	53	1	no	—
179	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
180	C₂²⋊C₁₆	—	53	4	no	—
181	C₂²⋊C₁₆	—	53	4	no	—
182	#346	—	106	2	no	—
183	C₄×C₁₆	—	106	2	no	—
184	C₅₃	#12600	1	1	yes	characteristic, complemented, Sylow
185	C₂³×C₄	—	53	1	no	—
186	C₂×M₁₆	—	53	4	no	—
187	C₂×M₁₆	—	106	2	no	—
188	C₂×M₁₆	—	106	2	no	—
189	C₂²×C₈	—	53	1	no	—
190	C₂²×C₈	—	53	1	no	—
191	C₂²×C₈	—	53	2	no	—
192	C₂²×C₈	—	53	2	no	—
193	C₂²×C₈	—	106	1	no	—
194	C₂²×C₈	—	106	1	no	—
195	C₂²×C₈	—	106	1	no	—
196	C₂²×C₈	—	106	1	no	—
197	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
198	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
199	#117	—	106	2	no	—
200	#117	—	106	2	no	—
201	C₂×C₄⋊C₄	—	53	2	no	—
202	C₂×C₂²⋊C₄	—	53	2	no	—
203	C₂×C₄²	—	53	1	no	—
204	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
205	C₂×C₁₆	—	2	1	no	—
206	C₂×C₁₆	—	2	1	no	—
207	C₂×C₁₆	—	2	1	no	—
208	C₂×C₁₆	—	2	1	no	—
209	C₂×C₁₆	—	53	2	no	—
210	C₂×C₁₆	—	53	2	no	—
211	C₂×C₁₆	—	106	1	no	—
212	C₂×C₁₆	—	106	1	no	—
213	C₂×C₁₆	—	106	1	no	—
214	C₂×C₁₆	—	106	1	no	—
215	C₂×C₁₆	C₄×D₅₃	1	2	yes	—
216	C₂×C₁₆	C₄×D₅₃	1	2	yes	—
217	#97	—	106	2	no	—
218	#97	—	106	2	no	—
219	C₄×C₈	—	2	1	no	—
220	C₄×C₈	—	2	1	no	—
221	C₄×C₈	—	53	2	no	—
222	C₄×C₈	—	53	2	no	—
223	C₂⁴	—	53	1	no	—
224	C₂²×C₄	—	53	2	no	—
225	C₂²×C₄	—	53	1	no	—
226	C₂²×C₄	—	53	2	no	—
227	C₂²×C₄	—	53	2	no	—
228	C₂²×C₄	—	106	1	no	—
229	C₂²×C₄	—	106	1	no	—
230	C₂²×C₄	—	106	1	no	—
231	C₂²×C₄	—	106	1	no	—
232	C₂²×C₄	C₈×D₅₃	1	2	yes	—
233	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
234	M₁₆	—	106	8	no	—
235	C₂×C₈	—	2	1	no	—
236	C₂×C₈	—	2	1	no	—
237	C₂×C₈	—	53	2	no	—
238	C₂×C₈	—	53	2	no	—
239	C₂×C₈	—	53	2	no	—
240	C₂×C₈	—	53	2	no	—
241	C₂×C₈	—	106	1	no	—
242	C₂×C₈	—	106	1	no	—
243	C₂×C₈	—	106	2	no	—
244	C₂×C₈	—	106	2	no	—
245	C₂×C₈	—	106	2	no	—
246	C₂×C₈	—	106	2	no	—
247	C₂×C₈	C₈×D₅₃	1	2	yes	—
248	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
249	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
250	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
251	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
252	C₂×C₈	#11719947	1	1	yes	characteristic
253	C₂×C₈	#11719949	1	1	yes	characteristic
254	C₄⋊C₄	—	106	4	no	—
255	C₂²⋊C₄	—	106	4	no	—
256	C₄²	—	2	1	no	—
257	C₄²	—	2	1	no	—
258	C₄²	—	53	2	no	—
259	C₄²	—	53	2	no	—
260	C₁₆	—	2	2	no	—
261	C₁₆	—	2	2	no	—
262	C₁₆	—	106	2	no	—
263	C₁₆	—	106	2	no	—
264	C₂³	—	53	2	no	—
265	C₂³	—	106	2	no	—
266	C₂³	—	106	2	no	—
267	C₂³	—	106	1	no	—
268	C₂³	—	106	1	no	—
269	C₂³	C₂×C₈×D₅₃	1	1	yes	characteristic
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	53	2	no	—
274	C₂×C₄	—	53	4	no	—
275	C₂×C₄	—	106	2	no	—
276	C₂×C₄	—	106	2	no	—
277	C₂×C₄	—	106	1	no	—
278	C₂×C₄	—	106	1	no	—
279	C₂×C₄	—	106	2	no	—
280	C₂×C₄	—	106	2	no	—
281	C₂×C₄	—	106	2	no	—
282	C₂×C₄	—	106	2	no	—
283	C₂×C₄	C₄²×D₅₃	1	1	yes	characteristic
284	C₂×C₄	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
285	C₂×C₄	#13133175	1	1	yes	characteristic
286	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
287	C₂×C₄	M₁₆×D₅₃	1	1	yes	characteristic
288	C₂×C₄	M₁₆×D₅₃	1	1	yes	characteristic
289	C₈	—	2	2	no	—
290	C₈	—	53	4	no	—
291	C₈	—	106	2	no	—
292	C₈	#13133177	1	1	yes	characteristic
293	C₈	#13133177	1	1	yes	characteristic
294	C₈	#13133179	1	1	yes	characteristic
295	C₈	#13133179	1	1	yes	characteristic
296	C₂²	—	106	2	no	—
297	C₂²	—	106	2	no	—
298	C₂²	—	106	2	no	—
299	C₂²	—	212	2	no	—
300	C₂²	—	212	2	no	—
301	C₂²	C₄×C₈×D₅₃	1	1	yes	characteristic
302	C₂²	C₂²⋊C₈×D₅₃	1	1	yes	characteristic
303	C₂²	#16810675	1	1	yes	characteristic
304	C₂²	#16811039	1	2	yes	—
305	C₂²	#16811041	1	2	yes	—
306	C₄	—	4	2	no	—
307	C₄	—	106	2	no	—
308	C₄	—	106	4	no	—
309	C₄	#16810470	1	1	yes	characteristic
310	C₄	#16810575	1	1	yes	characteristic
311	C₄	C₂²⋊C₈×D₅₃	1	1	yes	characteristic
312	C₄	#16810675	1	1	yes	characteristic
313	C₂	—	212	2	no	—
314	C₂	#20274472	1	1	yes	characteristic
315	C₂	#20276310	1	1	yes	characteristic
316	C₂	#20276313	1	1	yes	characteristic
317	C₂	#20276327	1	2	yes	—
318	C₂	#20276521	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂×C₄×C₈
Fitting	C₅₃×#1374	C₂
socle	C₂³×C₅₃	C₂²×C₈
prime radical	C₂³×D₅₃	C₂×C₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#1374	D₅₃
hypocenter	C₅₃	#12600
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₈ ◁ #1374

Complements

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

N	H ≅ G/N	type
4. C₂×#16810398	C₂	semidirect
5. C₂×C₁₆×Dic₅₃	C₂	semidirect
6. #20272793	C₂	semidirect
7. C₅₃×#1374	C₂	semidirect
18. #16810402	C₄	semidirect
19. #16810402	C₄	semidirect
160. #1374	D₅₃	semidirect
184. C₅₃	#12600	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810577	2
C₂×#16810402	2
C₂×#16810398	2
C₂×C₁₆×Dic₅₃	2
#20272793	2
C₅₃×#1374	2
#12600	53