Subgroups — #27299131

Order 13568.

522 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810575	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810404	C₂	1	2	yes	maximal
3	M₃₂×Dic₅₃	C₂	1	2	yes	maximal, complemented
4	#20272799	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1377	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#11719912	C₄	1	1	yes	characteristic
7	C₂²×#11719912	C₂²	1	1	yes	characteristic
8	C₂×C₄²×D₅₃	C₄	1	1	yes	characteristic
9	C₂×#13133157	C₂²	1	2	yes	—
10	#16810575	—	2	2	no	—
11	#16810575	—	2	2	no	—
12	C₂×#13133128	C₄	1	1	yes	characteristic
13	C₂×C₈×Dic₅₃	C₄	1	1	yes	characteristic
14	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
15	#16810404	C₄	1	8	yes	complemented
16	#16810398	—	2	2	no	—
17	C₁₆×Dic₅₃	—	2	2	no	—
18	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
19	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
20	C₂²×C₄×D₅₃	C₈	1	2	yes	—
21	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
22	C₂×#11719912	—	2	2	no	—
23	C₂×#11719912	—	2	1	no	—
24	C₂×#11719912	—	2	1	no	—
25	C₂×#11719912	—	2	2	no	—
26	C₂×#11719912	—	2	2	no	—
27	C₂×#11719912	C₈	1	4	yes	—
28	C₂×#11719912	C₂×C₄	1	4	yes	—
29	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
30	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×Dic₅₃	C₈	1	2	yes	—
32	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
33	C₄²×D₅₃	—	2	1	no	—
34	C₄²×D₅₃	—	2	1	no	—
35	C₄²×D₅₃	—	2	2	no	—
36	#13133157	—	2	2	no	—
37	#13133157	C₂×C₄	1	4	yes	complemented
38	C₂×#11719908	—	2	2	no	—
39	#13133128	—	2	2	no	—
40	C₈×Dic₅₃	—	2	1	no	—
41	C₈×Dic₅₃	—	2	1	no	—
42	C₈×Dic₅₃	D₄	1	2	yes	—
43	C₈×Dic₅₃	D₄	1	2	yes	—
44	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
47	M₃₂×C₅₃	—	2	2	no	—
48	M₃₂×C₅₃	C₂×C₄	1	4	yes	complemented
49	C₂×C₁₆×C₅₃	—	2	2	no	—
50	#97×C₅₃	—	2	2	no	—
51	C₂³×D₅₃	C₂×C₈	1	1	yes	characteristic
52	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
53	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
54	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
55	C₂²×Dic₅₃	C₂×C₈	1	2	yes	—
56	C₂×C₄×D₅₃	—	2	1	no	—
57	C₂×C₄×D₅₃	—	2	1	no	—
58	C₂×C₄×D₅₃	—	2	1	no	—
59	C₂×C₄×D₅₃	—	2	1	no	—
60	C₂×C₄×D₅₃	—	2	2	no	—
61	C₂×C₄×D₅₃	—	2	2	no	—
62	C₂×C₄×D₅₃	—	4	4	no	—
63	C₂×C₄×D₅₃	C₄²	1	2	yes	—
64	C₂×C₄×D₅₃	C₂×C₈	1	2	yes	—
65	C₂×C₈×C₅₃	—	2	1	no	—
66	C₂×C₈×C₅₃	—	2	1	no	—
67	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
68	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
69	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
70	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
71	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
72	C₄²×C₅₃	—	2	1	no	—
73	C₄²×C₅₃	—	2	1	no	—
74	C₄×Dic₅₃	—	2	2	no	—
75	C₄×Dic₅₃	—	2	2	no	—
76	C₄×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
77	C₄×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
78	C₄×Dic₅₃	M₁₆	1	2	yes	—
79	C₂×#88322	—	2	1	no	—
80	C₂×#88322	—	2	1	no	—
81	C₂×#88322	C₄²	1	2	yes	—
82	C₂×#88322	C₂²⋊C₄	1	2	yes	—
83	C₂×#88322	C₂×C₈	1	2	yes	—
84	C₂×#88322	M₁₆	1	2	yes	—
85	#11719912	—	2	4	no	—
86	#11719912	—	2	4	no	—
87	#11719912	—	2	8	no	—
88	C₁₆×C₅₃	—	2	4	no	—
89	#11719908	—	2	4	no	—
90	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
91	C₂²×D₅₃	—	2	1	no	—
92	C₂²×D₅₃	—	2	1	no	—
93	C₂²×D₅₃	—	4	2	no	—
94	C₂²×D₅₃	C₄×C₈	1	2	yes	—
95	C₂×C₄×C₅₃	—	2	2	no	—
96	C₂×C₄×C₅₃	—	4	2	no	—
97	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
98	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
100	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
101	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
102	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
103	C₂×Dic₅₃	—	2	1	no	—
104	C₂×Dic₅₃	—	2	1	no	—
105	C₂×Dic₅₃	—	2	4	no	—
106	C₂×Dic₅₃	C₄×C₈	1	2	yes	—
107	C₂×Dic₅₃	C₂²⋊C₈	1	2	yes	—
108	C₂×Dic₅₃	C₂²⋊C₈	1	2	yes	—
109	C₄×D₅₃	—	2	2	no	—
110	C₄×D₅₃	—	2	2	no	—
111	C₄×D₅₃	—	2	2	no	—
112	C₄×D₅₃	—	2	2	no	—
113	C₄×D₅₃	—	4	4	no	—
114	C₄×D₅₃	—	4	4	no	—
115	C₈×C₅₃	—	2	2	no	—
116	C₈×C₅₃	C₄×D₄	1	2	yes	—
117	C₈×C₅₃	C₄×D₄	1	2	yes	—
118	#88322	—	2	2	no	—
119	#88322	C₂²⋊C₈	1	4	yes	—
120	C₄×#349	—	53	1	no	maximal, Sylow
121	C₂²×C₅₃	—	2	2	no	—
122	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
123	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
124	C₂²×C₅₃	#431	1	1	yes	characteristic
125	C₂×D₅₃	—	2	2	no	—
126	C₂×D₅₃	—	4	2	no	—
127	C₂×D₅₃	—	4	2	no	—
128	C₂×D₅₃	—	4	2	no	—
129	C₄×C₅₃	—	4	2	no	—
130	C₄×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
131	C₄×C₅₃	C₄×M₁₆	1	1	yes	characteristic
132	C₄×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
133	C₄×C₅₃	#431	1	1	yes	characteristic
134	Dic₅₃	—	2	2	no	—
135	Dic₅₃	—	2	2	no	—
136	Dic₅₃	#349	1	4	yes	complemented
137	C₂×C₄×M₁₆	—	53	1	no	—
138	C₂×#349	—	53	2	no	—
139	C₄×M₃₂	—	53	2	no	—
140	#1377	—	53	1	no	—
141	#1377	D₅₃	1	1	yes	characteristic, complemented
142	C₂×C₅₃	—	2	1	no	—
143	C₂×C₅₃	—	2	1	no	—
144	C₂×C₅₃	C₄×C₂²⋊C₈	1	1	yes	characteristic
145	C₂×C₅₃	C₂×#349	1	2	yes	—
146	D₅₃	—	4	2	no	—
147	C₂²×M₁₆	—	53	1	no	—
148	C₂²×M₁₆	—	53	1	no	—
149	C₂²×C₄²	—	53	1	no	—
150	C₂×M₃₂	—	53	2	no	—
151	C₂×M₃₂	C₂×D₅₃	1	2	yes	—
152	C₄×M₁₆	—	106	2	no	—
153	C₄×M₁₆	—	106	2	no	—
154	C₂×#97	—	53	1	no	—
155	C₂×#97	C₂×D₅₃	1	1	yes	characteristic
156	C₂×C₄×C₈	—	53	1	no	—
157	C₂×C₄×C₈	—	53	1	no	—
158	#349	—	53	8	no	—
159	#346	—	106	2	no	—
160	C₄×C₁₆	—	106	2	no	—
161	C₅₃	C₄×#349	1	1	yes	characteristic, complemented, Sylow
162	C₂³×C₄	—	53	1	no	—
163	C₂³×C₄	—	53	2	no	—
164	C₂×M₁₆	—	53	4	no	—
165	C₂×M₁₆	—	53	4	no	—
166	C₂×M₁₆	—	106	2	no	—
167	C₂×M₁₆	—	106	2	no	—
168	C₂×M₁₆	—	106	2	no	—
169	C₂×M₁₆	—	106	1	no	—
170	C₂×M₁₆	—	106	1	no	—
171	C₂²×C₈	—	53	1	no	—
172	C₂²×C₈	—	53	1	no	—
173	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
174	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
175	C₂×C₄²	—	53	1	no	—
176	C₂×C₄²	—	53	2	no	—
177	C₂×C₄²	—	106	1	no	—
178	C₂×C₄²	—	106	1	no	—
179	C₂×C₄²	—	106	2	no	—
180	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
181	M₃₂	—	2	2	no	—
182	M₃₂	—	53	4	no	—
183	M₃₂	—	106	2	no	—
184	M₃₂	C₄×D₅₃	1	4	yes	complemented
185	C₂×C₁₆	—	2	2	no	—
186	C₂×C₁₆	—	106	2	no	—
187	#97	—	2	2	no	—
188	#97	—	106	2	no	—
189	C₄×C₈	—	53	2	no	—
190	C₄×C₈	—	53	2	no	—
191	C₄×C₈	—	106	1	no	—
192	C₄×C₈	—	106	1	no	—
193	C₂⁴	—	53	1	no	—
194	C₂²×C₄	—	53	2	no	—
195	C₂²×C₄	—	53	1	no	—
196	C₂²×C₄	—	53	2	no	—
197	C₂²×C₄	—	53	2	no	—
198	C₂²×C₄	—	106	2	no	—
199	C₂²×C₄	—	106	2	no	—
200	C₂²×C₄	—	106	1	no	—
201	C₂²×C₄	—	106	1	no	—
202	C₂²×C₄	—	106	1	no	—
203	C₂²×C₄	—	106	1	no	—
204	C₂²×C₄	—	212	4	no	—
205	C₂²×C₄	C₈×D₅₃	1	2	yes	—
206	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
207	M₁₆	—	106	8	no	—
208	M₁₆	—	106	4	no	—
209	M₁₆	—	106	4	no	—
210	C₂×C₈	—	2	1	no	—
211	C₂×C₈	—	2	1	no	—
212	C₂×C₈	—	53	2	no	—
213	C₂×C₈	—	53	2	no	—
214	C₂×C₈	—	53	2	no	—
215	C₂×C₈	—	53	2	no	—
216	C₂×C₈	—	106	1	no	—
217	C₂×C₈	—	106	1	no	—
218	C₂×C₈	C₈×D₅₃	1	2	yes	—
219	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
220	C₂×C₈	D₄×D₅₃	1	2	yes	—
221	C₂×C₈	#11719947	1	1	yes	characteristic
222	C₂×C₈	#11719949	1	1	yes	characteristic
223	C₄²	—	2	1	no	—
224	C₄²	—	2	1	no	—
225	C₄²	—	53	1	no	—
226	C₄²	—	53	1	no	—
227	C₄²	—	53	2	no	—
228	C₄²	—	106	2	no	—
229	C₄²	—	106	2	no	—
230	C₁₆	—	2	4	no	—
231	C₁₆	—	106	4	no	—
232	C₂³	—	53	2	no	—
233	C₂³	—	106	1	no	—
234	C₂³	—	106	1	no	—
235	C₂³	—	212	2	no	—
236	C₂³	C₂×C₈×D₅₃	1	1	yes	characteristic
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	—	53	2	no	—
240	C₂×C₄	—	53	2	no	—
241	C₂×C₄	—	53	2	no	—
242	C₂×C₄	—	106	4	no	—
243	C₂×C₄	—	106	1	no	—
244	C₂×C₄	—	106	1	no	—
245	C₂×C₄	—	106	2	no	—
246	C₂×C₄	—	106	2	no	—
247	C₂×C₄	—	106	2	no	—
248	C₂×C₄	—	106	2	no	—
249	C₂×C₄	—	212	4	no	—
250	C₂×C₄	—	212	4	no	—
251	C₂×C₄	C₄²×D₅₃	1	1	yes	characteristic
252	C₂×C₄	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
253	C₂×C₄	#13133175	1	1	yes	characteristic
254	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
255	C₂×C₄	M₁₆×D₅₃	1	1	yes	characteristic
256	C₂×C₄	M₁₆×D₅₃	1	1	yes	characteristic
257	C₈	—	2	2	no	—
258	C₈	—	53	4	no	—
259	C₈	—	106	2	no	—
260	C₈	#13133177	1	2	yes	—
261	C₈	#13133179	1	2	yes	—
262	C₂²	—	2	2	no	—
263	C₂²	—	106	2	no	—
264	C₂²	—	212	2	no	—
265	C₂²	—	212	2	no	—
266	C₂²	—	212	2	no	—
267	C₂²	C₄×C₈×D₅₃	1	1	yes	characteristic
268	C₂²	C₂²⋊C₈×D₅₃	1	1	yes	characteristic
269	C₂²	#16810675	1	1	yes	characteristic
270	C₄	—	4	2	no	—
271	C₄	—	53	4	no	—
272	C₄	—	106	2	no	—
273	C₄	—	106	2	no	—
274	C₄	#16810470	1	1	yes	characteristic
275	C₄	#16810575	1	1	yes	characteristic
276	C₄	C₂²⋊C₈×D₅₃	1	1	yes	characteristic
277	C₄	#16810675	1	1	yes	characteristic
278	C₂	—	2	1	no	—
279	C₂	—	2	1	no	—
280	C₂	—	212	2	no	—
281	C₂	#20274472	1	1	yes	characteristic
282	C₂	#349×D₅₃	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²⋊C₄×D₅₃
commutator	C₂²×C₅₃	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#1377	C₂
socle	C₂²×C₅₃	C₂×C₂²⋊C₈
prime radical	C₂³×D₅₃	C₂×C₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#1377	D₅₃
hypocenter	C₅₃	C₄×#349
LCS	G ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #1377

Complements

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

N	H ≅ G/N	type
3. M₃₂×Dic₅₃	C₂	semidirect
4. #20272799	C₂	semidirect
5. C₅₃×#1377	C₂	semidirect
15. #16810404	C₄	semidirect
37. #13133157	C₂×C₄	semidirect
48. M₃₂×C₅₃	C₂×C₄	semidirect
136. Dic₅₃	#349	semidirect
141. #1377	D₅₃	semidirect
161. C₅₃	C₄×#349	semidirect
184. M₃₂	C₄×D₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810575	2
C₂×#16810404	2
M₃₂×Dic₅₃	2
#20272799	2
C₅₃×#1377	2
C₄×#349	53