Subgroups — #28311287

Order 13568.

542 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	C₂×#16811518	C₂	1	2	yes	maximal
2	#20289754	C₂	1	2	yes	maximal, complemented
3	#20289753	C₂	1	2	yes	maximal, complemented
4	C₂×C₄×C₈×D₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11719910	C₂²	1	4	yes	—
6	#16811518	—	2	4	no	—
7	C₂²×C₈×D₅₃	C₄	1	1	yes	characteristic
8	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
9	C₂×C₄²×D₅₃	C₄	1	1	yes	characteristic
10	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
11	C₂×C₈×Dic₅₃	C₄	1	1	yes	characteristic
12	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
13	C₄×C₈×D₅₃	—	2	4	no	—
14	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
15	C₂×#11719910	—	2	4	no	—
16	C₂×#11719910	—	2	4	no	—
17	C₂×#11719910	—	2	4	no	—
18	C₂²×C₄×D₅₃	C₈	1	1	yes	characteristic
19	C₂²×C₄×D₅₃	C₈	1	1	yes	characteristic
20	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
22	C₂²×#88322	C₂³	1	1	yes	characteristic
23	C₂×C₈×D₅₃	—	2	2	no	—
24	C₂×C₈×D₅₃	—	2	2	no	—
25	C₂×C₈×D₅₃	—	2	2	no	—
26	C₂×C₈×D₅₃	—	2	2	no	—
27	C₂×C₈×D₅₃	—	2	2	no	—
28	C₂×C₈×D₅₃	D₄	1	4	yes	—
29	C₂×C₄×Dic₅₃	C₈	1	1	yes	characteristic
30	C₂×C₄×Dic₅₃	C₈	1	1	yes	characteristic
31	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
32	C₄²×D₅₃	—	2	2	no	—
33	C₄²×D₅₃	—	2	2	no	—
34	C₄×#88322	—	2	2	no	—
35	C₈×Dic₅₃	—	2	2	no	—
36	C₈×Dic₅₃	—	2	2	no	—
37	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
40	C₄×C₈×C₅₃	—	2	2	no	—
41	C₂³×D₅₃	C₂×C₈	1	1	yes	characteristic
42	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
43	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
44	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
45	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
46	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
47	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
48	C₂×C₄×D₅₃	—	2	2	no	—
49	C₂×C₄×D₅₃	—	2	2	no	—
50	C₂×C₄×D₅₃	—	2	2	no	—
51	C₂×C₄×D₅₃	—	2	2	no	—
52	C₂×C₄×D₅₃	—	2	1	no	—
53	C₂×C₄×D₅₃	—	2	1	no	—
54	C₂×C₄×D₅₃	—	2	1	no	—
55	C₂×C₄×D₅₃	—	2	1	no	—
56	C₂×C₄×D₅₃	—	2	2	no	—
57	C₂×C₄×D₅₃	C₁₆	1	4	yes	complemented
58	C₂×C₄×D₅₃	C₂²⋊C₄	1	2	yes	—
59	C₂×C₄×D₅₃	M₁₆	1	2	yes	—
60	C₂×C₈×C₅₃	—	2	1	no	—
61	C₂×C₈×C₅₃	—	2	1	no	—
62	C₂×C₈×C₅₃	—	2	1	no	—
63	C₂×C₈×C₅₃	—	2	1	no	—
64	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
65	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
66	C₄²×C₅₃	—	2	1	no	—
67	C₄²×C₅₃	—	2	1	no	—
68	C₄×Dic₅₃	—	2	1	no	—
69	C₄×Dic₅₃	—	2	1	no	—
70	C₄×Dic₅₃	—	2	2	no	—
71	C₄×Dic₅₃	—	2	2	no	—
72	C₂×#88322	—	2	1	no	—
73	C₂×#88322	—	2	1	no	—
74	C₂×#88322	M₁₆	1	2	yes	—
75	C₂×#88322	C₂×D₄	1	2	yes	—
76	C₂×#88322	#41	1	2	yes	—
77	C₂×#88322	#41	1	2	yes	—
78	C₈×D₅₃	—	2	8	no	—
79	C₈×D₅₃	—	2	8	no	—
80	#11719910	—	4	4	no	—
81	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
82	C₂²×D₅₃	—	2	1	no	—
83	C₂²×D₅₃	—	2	1	no	—
84	C₂²×D₅₃	—	2	1	no	—
85	C₂²×D₅₃	—	2	1	no	—
86	C₂²×D₅₃	C₂²⋊C₈	1	2	yes	—
87	C₂²×D₅₃	C₂×C₁₆	1	2	yes	—
88	C₂²×D₅₃	M₃₂	1	2	yes	—
89	C₂×C₄×C₅₃	—	2	1	no	—
90	C₂×C₄×C₅₃	—	2	1	no	—
91	C₂×C₄×C₅₃	—	2	1	no	—
92	C₂×C₄×C₅₃	—	2	1	no	—
93	C₂×C₄×C₅₃	C₂×C₁₆	1	2	yes	—
94	C₂×C₄×C₅₃	M₃₂	1	2	yes	—
95	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
97	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
98	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
99	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
100	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
101	C₂×Dic₅₃	—	2	1	no	—
102	C₂×Dic₅₃	—	2	1	no	—
103	C₂×Dic₅₃	—	2	1	no	—
104	C₂×Dic₅₃	—	2	1	no	—
105	C₂×Dic₅₃	—	2	1	no	—
106	C₂×Dic₅₃	—	2	1	no	—
107	C₂×Dic₅₃	C₂²⋊C₈	1	2	yes	—
108	C₂×Dic₅₃	C₂×C₁₆	1	2	yes	—
109	C₂×Dic₅₃	M₃₂	1	2	yes	—
110	C₄×D₅₃	—	2	8	no	—
111	C₄×D₅₃	—	2	8	no	—
112	C₄×D₅₃	—	2	2	no	—
113	C₄×D₅₃	—	2	2	no	—
114	C₄×D₅₃	—	2	2	no	—
115	C₄×D₅₃	—	2	2	no	—
116	C₈×C₅₃	—	2	2	no	—
117	C₈×C₅₃	—	2	2	no	—
118	#88322	—	2	2	no	—
119	#88322	#123	1	4	yes	—
120	#12844	—	53	1	no	maximal, Sylow
121	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
122	C₂²×C₅₃	#431	1	1	yes	characteristic
123	C₂²×C₅₃	#431	1	1	yes	characteristic
124	C₂²×C₅₃	C₂²×C₁₆	1	1	yes	characteristic
125	C₂²×C₅₃	C₂×M₃₂	1	1	yes	characteristic
126	C₂²×C₅₃	C₂×M₃₂	1	1	yes	characteristic
127	C₂²×C₅₃	C₂×M₃₂	1	1	yes	characteristic
128	C₂×D₅₃	—	2	2	no	—
129	C₂×D₅₃	—	2	2	no	—
130	C₂×D₅₃	—	2	2	no	—
131	C₂×D₅₃	—	2	2	no	—
132	C₂×D₅₃	—	2	2	no	—
133	C₂×D₅₃	—	2	2	no	—
134	C₂×D₅₃	C₂²⋊C₁₆	1	4	yes	—
135	C₄×C₅₃	—	2	2	no	—
136	C₄×C₅₃	—	2	2	no	—
137	C₄×C₅₃	#381	1	1	yes	characteristic
138	C₄×C₅₃	#407	1	1	yes	characteristic
139	C₄×C₅₃	#432	1	1	yes	characteristic
140	C₄×C₅₃	#432	1	1	yes	characteristic
141	Dic₅₃	—	2	2	no	—
142	Dic₅₃	—	2	2	no	—
143	Dic₅₃	—	2	2	no	—
144	C₂²×C₄×C₈	—	53	1	no	—
145	C₂×C₂²⋊C₁₆	—	53	2	no	—
146	#1374	—	53	2	no	—
147	#1374	—	53	2	no	—
148	C₂×C₅₃	#1817	1	1	yes	characteristic
149	C₂×C₅₃	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
150	C₂×C₅₃	#2112	1	1	yes	characteristic
151	C₂×C₅₃	#2162	1	1	yes	characteristic
152	C₂×C₅₃	#2162	1	1	yes	characteristic
153	C₂×C₅₃	#2163	1	1	yes	characteristic
154	C₂×C₅₃	#2163	1	1	yes	characteristic
155	D₅₃	—	2	4	no	—
156	C₂³×C₈	—	53	1	no	—
157	C₂³×C₈	—	53	1	no	—
158	C₂²×C₄²	—	53	1	no	—
159	C₂²×C₁₆	—	53	4	no	—
160	C₂×C₄×C₈	—	53	1	no	—
161	C₂×C₄×C₈	—	53	1	no	—
162	C₂×C₄×C₈	—	53	1	no	—
163	C₂×C₄×C₈	—	106	4	no	—
164	C₂×C₄×C₈	#6157	1	1	yes	characteristic
165	C₂²⋊C₁₆	—	106	4	no	—
166	C₅₃	#12844	1	1	yes	characteristic, complemented, Sylow
167	C₂³×C₄	—	53	1	no	—
168	C₂³×C₄	—	53	1	no	—
169	C₂³×C₄	—	53	1	no	—
170	C₂²×C₈	—	53	1	no	—
171	C₂²×C₈	—	53	1	no	—
172	C₂²×C₈	—	53	4	no	—
173	C₂²×C₈	—	106	2	no	—
174	C₂²×C₈	—	106	2	no	—
175	C₂²×C₈	—	106	2	no	—
176	C₂²×C₈	—	106	2	no	—
177	C₂²×C₈	—	106	2	no	—
178	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
179	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
180	C₂×C₄²	—	53	1	no	—
181	C₂×C₄²	—	53	1	no	—
182	C₂×C₄²	—	53	1	no	—
183	C₂×C₄²	—	106	2	no	—
184	C₂×C₄²	—	106	2	no	—
185	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
186	C₂×C₁₆	—	106	4	no	—
187	C₂×C₁₆	—	106	4	no	—
188	C₂×C₁₆	—	106	4	no	—
189	C₄×C₈	—	2	2	no	—
190	C₄×C₈	—	106	2	no	—
191	C₄×C₈	—	106	2	no	—
192	C₄×C₈	—	106	2	no	—
193	C₂⁴	—	53	1	no	—
194	C₂²×C₄	—	53	1	no	—
195	C₂²×C₄	—	53	1	no	—
196	C₂²×C₄	—	53	1	no	—
197	C₂²×C₄	—	53	4	no	—
198	C₂²×C₄	—	53	2	no	—
199	C₂²×C₄	—	53	2	no	—
200	C₂²×C₄	—	106	2	no	—
201	C₂²×C₄	—	106	2	no	—
202	C₂²×C₄	—	106	2	no	—
203	C₂²×C₄	—	106	2	no	—
204	C₂²×C₄	—	106	2	no	—
205	C₂²×C₄	—	106	1	no	—
206	C₂²×C₄	—	106	1	no	—
207	C₂²×C₄	—	106	1	no	—
208	C₂²×C₄	—	106	1	no	—
209	C₂²×C₄	D₅₃⋊C₈	1	1	yes	characteristic
210	C₂²×C₄	D₅₃⋊C₈	1	1	yes	characteristic
211	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
212	C₂×C₈	—	2	1	no	—
213	C₂×C₈	—	2	1	no	—
214	C₂×C₈	—	2	1	no	—
215	C₂×C₈	—	2	1	no	—
216	C₂×C₈	—	53	2	no	—
217	C₂×C₈	—	53	2	no	—
218	C₂×C₈	—	53	2	no	—
219	C₂×C₈	—	53	2	no	—
220	C₂×C₈	—	106	8	no	—
221	C₂×C₈	—	106	8	no	—
222	C₂×C₈	—	106	1	no	—
223	C₂×C₈	—	106	1	no	—
224	C₂×C₈	#11719936	1	2	yes	—
225	C₂×C₈	C₂²⋊#6157	1	2	yes	—
226	C₄²	—	2	1	no	—
227	C₄²	—	2	1	no	—
228	C₄²	—	106	2	no	—
229	C₄²	—	106	2	no	—
230	C₄²	—	106	1	no	—
231	C₄²	—	106	1	no	—
232	C₁₆	—	212	4	no	—
233	C₂³	—	53	2	no	—
234	C₂³	—	53	2	no	—
235	C₂³	—	53	2	no	—
236	C₂³	—	106	1	no	—
237	C₂³	—	106	1	no	—
238	C₂³	—	106	1	no	—
239	C₂³	—	106	1	no	—
240	C₂³	C₂×D₅₃⋊C₈	1	1	yes	characteristic
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	53	2	no	—
246	C₂×C₄	—	53	2	no	—
247	C₂×C₄	—	53	2	no	—
248	C₂×C₄	—	106	8	no	—
249	C₂×C₄	—	106	8	no	—
250	C₂×C₄	—	106	1	no	—
251	C₂×C₄	—	106	1	no	—
252	C₂×C₄	—	106	1	no	—
253	C₂×C₄	—	106	1	no	—
254	C₂×C₄	—	106	1	no	—
255	C₂×C₄	—	106	1	no	—
256	C₂×C₄	—	106	2	no	—
257	C₂×C₄	—	106	2	no	—
258	C₂×C₄	—	106	2	no	—
259	C₂×C₄	—	106	2	no	—
260	C₂×C₄	D₅₃⋊C₁₆	1	2	yes	—
261	C₂×C₄	#13133292	1	2	yes	—
262	C₂×C₄	C₂×C₂²⋊#6157	1	1	yes	characteristic
263	C₂×C₄	#13133296	1	1	yes	characteristic
264	C₂×C₄	#13133296	1	1	yes	characteristic
265	C₂×C₄	D₅₃⋊M₁₆	1	1	yes	characteristic
266	C₂×C₄	D₅₃⋊M₁₆	1	1	yes	characteristic
267	C₂×C₄	C₂×#11719936	1	1	yes	characteristic
268	C₈	—	2	2	no	—
269	C₈	—	2	2	no	—
270	C₈	—	53	4	no	—
271	C₈	—	106	2	no	—
272	C₂²	—	53	4	no	—
273	C₂²	—	106	2	no	—
274	C₂²	—	106	2	no	—
275	C₂²	—	106	2	no	—
276	C₂²	—	106	2	no	—
277	C₂²	—	106	2	no	—
278	C₂²	—	106	2	no	—
279	C₂²	D₅₃⋊(C₂²⋊C₈)	1	1	yes	characteristic
280	C₂²	#16811641	1	1	yes	characteristic
281	C₂²	#16811642	1	1	yes	characteristic
282	C₂²	C₂×D₅₃⋊C₁₆	1	1	yes	characteristic
283	C₂²	D₅₃⋊M₃₂	1	1	yes	characteristic
284	C₂²	D₅₃⋊M₃₂	1	1	yes	characteristic
285	C₂²	C₂×#13133292	1	1	yes	characteristic
286	C₄	—	2	2	no	—
287	C₄	—	2	2	no	—
288	C₄	—	106	2	no	—
289	C₄	—	106	2	no	—
290	C₄	—	106	2	no	—
291	C₄	#16811566	1	1	yes	characteristic
292	C₄	#16811594	1	1	yes	characteristic
293	C₄	#16811645	1	1	yes	characteristic
294	C₄	#16811646	1	1	yes	characteristic
295	C₂	—	106	4	no	—
296	C₂	#20290607	1	1	yes	characteristic
297	C₂	D₅₃⋊(C₂²⋊C₁₆)	1	1	yes	characteristic
298	C₂	#20290943	1	1	yes	characteristic
299	C₂	#20291074	1	1	yes	characteristic
300	C₂	#20291075	1	1	yes	characteristic
301	C₂	#20291078	1	1	yes	characteristic
302	C₂	#20291079	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#6157
commutator	C₂²×C₅₃	C₂²×C₁₆
Fitting	C₂×C₄×C₈×C₅₃	C₄
socle	C₂³×C₅₃	C₂²×C₈
prime radical	C₂³×D₅₃	C₂×C₈
cosocle	C₂²×#88322	C₂³
hypercenter	C₂×C₄×C₈	#6157
hypocenter	C₅₃	#12844
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #20289754	C₂	semidirect
3. #20289753	C₂	semidirect
57. C₂×C₄×D₅₃	C₁₆	semidirect
166. C₅₃	#12844	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16811518	2
#20289754	2
#20289753	2
C₂×C₄×C₈×D₅₃	2
#12844	53