Subgroups — #27299197

Order 13568.

426 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811039	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810408	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₆×Dic₅₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#20274554	C₂	1	1	yes	maximal, characteristic
5	#20272804	C₂	1	1	yes	maximal, characteristic, complemented
6	#20272796	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#2108	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13133223	C₂²	1	1	yes	characteristic
9	#16811039	—	2	2	no	—
10	#16811039	C₄	1	4	yes	complemented
11	C₂×#13133157	C₂²	1	1	yes	characteristic
12	C₂×#13133156	—	2	1	no	—
13	C₂²×#11719908	C₂²	1	1	yes	characteristic
14	C₂×C₁₆×D₅₃	—	2	1	no	—
15	C₂×#13133143	C₄	1	1	yes	characteristic
16	#16810617	C₄	1	1	yes	characteristic
17	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
18	#16810408	C₄	1	4	yes	complemented
19	C₁₆×Dic₅₃	—	2	2	no	—
20	#16810387	C₄	1	1	yes	characteristic
21	#16810382	C₄	1	1	yes	characteristic
22	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
23	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
24	C₅₃×#405	C₂²	1	1	yes	characteristic
25	C₅₃×#346	—	2	1	no	—
26	C₄×C₁₆×C₅₃	—	2	1	no	—
27	C₂×#11719945	C₂×C₄	1	1	yes	characteristic
28	#13133223	—	2	1	no	—
29	#13133223	—	2	1	no	—
30	#13133223	C₂×C₄	1	2	yes	—
31	#13133223	C₂×C₄	1	2	yes	—
32	C₂×#11719917	C₂×C₄	1	1	yes	characteristic
33	C₂×#11719912	—	2	1	no	—
34	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
35	C₂×C₈×D₅₃	—	2	1	no	—
36	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
37	#13133157	—	2	1	no	—
38	#13133157	C₂×C₄	1	2	yes	—
39	#13133156	—	2	4	no	—
40	C₂×#11719908	—	2	1	no	—
41	C₂×#11719908	—	2	1	no	—
42	C₂×#11719908	—	2	1	no	—
43	C₂×#11719908	C₂×C₄	1	2	yes	—
44	C₁₆×D₅₃	—	2	4	no	—
45	#13133143	C₈	1	4	yes	—
46	#13133140	C₈	1	2	yes	—
47	#13133137	C₈	1	2	yes	—
48	C₈×Dic₅₃	—	2	1	no	—
49	C₈×Dic₅₃	—	2	1	no	—
50	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
51	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
52	#117×C₅₃	C₂×C₄	1	1	yes	characteristic
53	M₃₂×C₅₃	—	2	1	no	—
54	M₃₂×C₅₃	C₂×C₄	1	2	yes	—
55	C₂×C₁₆×C₅₃	—	2	1	no	—
56	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
57	C₂×C₁₆×C₅₃	D₄	1	2	yes	—
58	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
59	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
60	#97×C₅₃	—	2	1	no	—
61	C₄×C₈×C₅₃	—	2	1	no	—
62	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₂²⋊D₅₃	—	2	1	no	—
64	C₂²×Dic₅₃	—	2	1	no	—
65	#11719945	—	2	1	no	—
66	#11719945	—	2	1	no	—
67	#11719945	C₄²	1	2	yes	—
68	#11719945	C₂×C₈	1	2	yes	—
69	C₂×D₂₁₂	C₂×C₈	1	1	yes	characteristic
70	C₂×C₄×D₅₃	—	2	1	no	—
71	C₂×Dic₁₀₆	C₂×C₈	1	1	yes	characteristic
72	M₁₆×C₅₃	—	2	1	no	—
73	M₁₆×C₅₃	C₂×C₈	1	2	yes	—
74	C₂×C₈×C₅₃	—	2	1	no	—
75	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
78	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
79	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
80	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
81	C₄⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
82	C₂²⋊C₄×C₅₃	—	2	1	no	—
83	C₄²×C₅₃	—	2	1	no	—
84	C₄×Dic₅₃	—	2	1	no	—
85	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
86	#11719917	—	2	1	no	—
87	#11719917	C₄²	1	2	yes	—
88	C₂×#88322	—	2	1	no	—
89	C₂×#88322	—	2	1	no	—
90	C₂×#88322	—	2	1	no	—
91	C₂×#88322	C₂×C₈	1	2	yes	—
92	#11719912	—	2	2	no	—
93	#11719912	—	2	2	no	—
94	C₈×D₅₃	—	2	2	no	—
95	C₈×D₅₃	—	2	2	no	—
96	C₁₆×C₅₃	—	2	2	no	—
97	C₁₆×C₅₃	C₂²⋊C₄	1	2	yes	—
98	C₁₆×C₅₃	C₂²⋊C₄	1	2	yes	—
99	#11719908	—	2	2	no	—
100	#11719908	—	2	2	no	—
101	C₂³×C₅₃	#131	1	1	yes	characteristic
102	C₂²×D₅₃	—	2	1	no	—
103	C₂×C₄×C₅₃	—	2	2	no	—
104	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
105	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
106	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
107	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
108	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
109	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
110	C₂²⋊D₅₃	—	2	2	no	—
111	C₂²⋊D₅₃	—	2	2	no	—
112	C₂×Dic₅₃	—	2	4	no	—
113	C₂×Dic₅₃	—	2	1	no	—
114	C₂×Dic₅₃	—	2	2	no	—
115	D₂₁₂	—	2	1	no	—
116	D₂₁₂	C₄×C₈	1	2	yes	—
117	C₄×D₅₃	—	2	2	no	—
118	C₄×D₅₃	—	2	2	no	—
119	Dic₁₀₆	—	2	1	no	—
120	Dic₁₀₆	C₄×C₈	1	2	yes	—
121	C₈×C₅₃	—	2	2	no	—
122	C₈×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	C₈×C₅₃	#117	1	1	yes	characteristic
124	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
125	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
126	#88322	—	2	2	no	—
127	#88322	—	2	2	no	—
128	#12614	—	53	1	no	maximal, Sylow
129	C₂²×C₅₃	—	2	2	no	—
130	C₂²×C₅₃	#405	1	1	yes	characteristic
131	C₂²×C₅₃	C₈×D₄	1	1	yes	characteristic
132	C₂²×C₅₃	C₈×D₄	1	1	yes	characteristic
133	C₂×D₅₃	—	2	2	no	—
134	C₂×D₅₃	—	4	2	no	—
135	C₄×C₅₃	—	4	2	no	—
136	C₄×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
137	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
138	C₄×C₅₃	C₈×D₄	1	1	yes	characteristic
139	C₄×C₅₃	C₈×D₄	1	1	yes	characteristic
140	Dic₅₃	—	2	4	no	—
141	Dic₅₃	—	2	2	no	—
142	C₂×#504	—	53	1	no	—
143	C₂×#350	—	53	1	no	—
144	#2108	D₅₃	1	1	yes	characteristic, complemented
145	C₂×C₄×C₁₆	—	53	1	no	—
146	#1764	—	53	1	no	—
147	#1379	—	53	1	no	—
148	#1376	—	53	1	no	—
149	C₂×C₅₃	—	2	1	no	—
150	C₂×C₅₃	—	2	1	no	—
151	C₂×C₅₃	C₈×C₂²⋊C₄	1	1	yes	characteristic
152	C₂×C₅₃	#2170	1	2	yes	—
153	D₅₃	—	4	2	no	—
154	C₂×#131	—	53	1	no	—
155	#504	—	53	4	no	—
156	#504	—	106	2	no	—
157	C₂×M₃₂	—	53	1	no	—
158	C₂×M₃₂	—	106	1	no	—
159	C₂×M₃₂	C₂×D₅₃	1	1	yes	characteristic
160	C₂²×C₁₆	—	53	1	no	—
161	C₂²×C₁₆	—	106	1	no	—
162	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
163	C₂×#104	—	53	1	no	—
164	#416	—	53	1	no	—
165	#405	C₂×D₅₃	1	1	yes	characteristic
166	C₂×C₄×C₈	—	53	1	no	—
167	#350	—	53	4	no	—
168	#346	—	2	1	no	—
169	C₄×C₁₆	—	2	1	no	—
170	C₄×C₁₆	—	106	2	no	—
171	#341	—	53	1	no	—
172	#339	—	53	1	no	—
173	C₅₃	#12614	1	1	yes	characteristic, complemented, Sylow
174	C₂×#41	—	53	1	no	—
175	#131	—	53	2	no	—
176	#131	—	53	2	no	—
177	#131	—	106	1	no	—
178	#131	—	106	1	no	—
179	C₂×M₁₆	—	53	1	no	—
180	C₂×M₁₆	—	106	1	no	—
181	C₂×M₁₆	C₄×D₅₃	1	1	yes	characteristic
182	C₂²×C₈	—	53	1	no	—
183	C₂²×C₈	—	106	1	no	—
184	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
185	#117	C₄×D₅₃	1	1	yes	characteristic
186	C₂×C₄²	—	53	1	no	—
187	M₃₂	—	2	1	no	—
188	M₃₂	—	53	2	no	—
189	M₃₂	—	106	1	no	—
190	M₃₂	—	106	4	no	—
191	M₃₂	C₄×D₅₃	1	2	yes	—
192	C₂×C₁₆	—	2	1	no	—
193	C₂×C₁₆	—	53	2	no	—
194	C₂×C₁₆	—	106	1	no	—
195	C₂×C₁₆	—	106	1	no	—
196	C₂×C₁₆	—	106	1	no	—
197	C₂×C₁₆	—	106	4	no	—
198	C₂×C₁₆	C₄×D₅₃	1	2	yes	—
199	C₂×C₁₆	D₂₁₂	1	2	yes	—
200	C₂×C₁₆	C₂²⋊D₅₃	1	1	yes	characteristic
201	C₂×C₁₆	C₂²⋊D₅₃	1	1	yes	characteristic
202	#104	—	53	4	no	—
203	#103	—	53	2	no	—
204	#102	—	53	2	no	—
205	#97	—	2	1	no	—
206	C₄×C₈	—	2	1	no	—
207	C₄×C₈	—	106	1	no	—
208	C₄×C₈	—	106	1	no	—
209	#41	—	53	2	no	—
210	#41	—	53	2	no	—
211	#41	—	106	1	no	—
212	#41	—	106	1	no	—
213	C₂×Q₈	—	53	1	no	—
214	C₂×D₄	—	53	1	no	—
215	C₂×D₄	—	106	1	no	—
216	C₂²×C₄	—	106	1	no	—
217	C₂²×C₄	—	106	1	no	—
218	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
219	M₁₆	—	2	1	no	—
220	M₁₆	—	53	2	no	—
221	M₁₆	—	106	1	no	—
222	M₁₆	—	106	2	no	—
223	M₁₆	—	106	2	no	—
224	M₁₆	C₈×D₅₃	1	2	yes	—
225	C₂×C₈	—	2	1	no	—
226	C₂×C₈	—	53	2	no	—
227	C₂×C₈	—	106	1	no	—
228	C₂×C₈	—	106	1	no	—
229	C₂×C₈	—	106	1	no	—
230	C₂×C₈	—	106	2	no	—
231	C₂×C₈	—	106	2	no	—
232	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
233	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
234	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
235	C₂×C₈	#11719945	1	1	yes	characteristic
236	C₂×C₈	#11719945	1	1	yes	characteristic
237	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
238	C₄⋊C₄	C₈×D₅₃	1	2	yes	—
239	C₂²⋊C₄	—	2	1	no	—
240	C₄²	—	2	1	no	—
241	C₄²	—	53	2	no	—
242	C₄²	—	106	1	no	—
243	C₁₆	—	2	2	no	—
244	C₁₆	—	106	2	no	—
245	C₁₆	—	106	2	no	—
246	C₁₆	#11719921	1	2	yes	—
247	C₁₆	#11719921	1	2	yes	—
248	C₂³	—	106	1	no	—
249	C₂³	#13133223	1	1	yes	characteristic
250	Q₈	—	53	2	no	—
251	Q₈	—	106	1	no	—
252	D₄	—	53	2	no	—
253	D₄	—	106	2	no	—
254	D₄	—	106	2	no	—
255	D₄	—	106	1	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	106	4	no	—
258	C₂×C₄	—	106	1	no	—
259	C₂×C₄	—	106	2	no	—
260	C₂×C₄	—	106	2	no	—
261	C₂×C₄	—	106	2	no	—
262	C₂×C₄	C₄²×D₅₃	1	1	yes	characteristic
263	C₂×C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
264	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
265	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
266	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
267	C₂×C₄	#13133223	1	1	yes	characteristic
268	C₈	—	2	2	no	—
269	C₈	—	106	2	no	—
270	C₈	—	106	2	no	—
271	C₈	C₂×#11719921	1	1	yes	characteristic
272	C₈	#13133174	1	1	yes	characteristic
273	C₈	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
274	C₈	C₄×D₂₁₂	1	1	yes	characteristic
275	C₂²	—	2	2	no	—
276	C₂²	—	106	2	no	—
277	C₂²	—	212	2	no	—
278	C₂²	#16810577	1	1	yes	characteristic
279	C₂²	C₈×C₂²⋊D₅₃	1	1	yes	characteristic
280	C₂²	C₈×D₂₁₂	1	1	yes	characteristic
281	C₄	—	4	2	no	—
282	C₄	—	106	4	no	—
283	C₄	—	106	2	no	—
284	C₄	C₄×#11719921	1	1	yes	characteristic
285	C₄	C₄×C₈×D₅₃	1	1	yes	characteristic
286	C₄	C₈×C₂²⋊D₅₃	1	1	yes	characteristic
287	C₄	C₈×D₂₁₂	1	1	yes	characteristic
288	C₂	—	2	1	no	—
289	C₂	—	2	1	no	—
290	C₂	—	212	2	no	—
291	C₂	C₈×#11719921	1	1	yes	characteristic
292	C₂	#20276550	1	2	yes	—

	subgroup	G/H
center	C₂×C₁₆	C₂²⋊D₅₃
commutator	C₄×C₅₃	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#2108	C₂
socle	C₂²×C₅₃	C₈×D₄
prime radical	C₂×#11719945	C₂×C₄
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#2108	D₅₃
hypocenter	C₅₃	#12614
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂²×C₁₆ ◁ #2108

Complements

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

N	H ≅ G/N	type
3. C₂×C₁₆×Dic₅₃	C₂	semidirect
5. #20272804	C₂	semidirect
6. #20272796	C₂	semidirect
7. C₅₃×#2108	C₂	semidirect
10. #16811039	C₄	semidirect
18. #16810408	C₄	semidirect
144. #2108	D₅₃	semidirect
173. C₅₃	#12614	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16811039	2
C₂×#16810408	2
C₂×C₁₆×Dic₅₃	2
#20274554	2
#20272804	2
#20272796	2
C₅₃×#2108	2
#12614	53