Subgroups — #27271042

Order 13568.

386 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂²⋊#11719908	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810377	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅₃×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	C₂³×#88322	C₄	1	1	yes	characteristic
5	C₂²×C₄×Dic₅₃	C₄	1	1	yes	characteristic
6	C₂²×#11719908	—	2	1	no	—
7	C₂²⋊#11719908	—	2	1	no	—
8	C₂²⋊#11719908	C₄	1	2	yes	complemented
9	#16810377	—	2	1	no	—
10	#16810377	—	2	1	no	—
11	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
12	C₂²×C₁₆×C₅₃	—	2	1	no	—
13	C₅₃×C₂²⋊C₁₆	—	2	1	no	—
14	C₅₃×C₂²⋊C₁₆	C₄	1	2	yes	complemented
15	C₂³×Dic₅₃	C₈	1	2	yes	—
16	C₂²×#88322	—	2	1	no	—
17	C₂²×#88322	—	2	1	no	—
18	C₂²×#88322	—	4	1	no	—
19	C₂²×#88322	—	4	1	no	—
20	C₂²×#88322	C₈	1	2	yes	—
21	C₂×C₄×Dic₅₃	—	2	1	no	—
22	C₂×C₄×Dic₅₃	—	2	1	no	—
23	C₂×C₄×Dic₅₃	—	4	1	no	—
24	C₂×C₄×Dic₅₃	—	4	1	no	—
25	C₂×#11719908	—	2	2	no	—
26	C₂×#11719908	—	4	1	no	—
27	C₂×#11719908	—	4	1	no	—
28	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₅₃	—	4	1	no	—
30	C₂²×C₈×C₅₃	—	4	1	no	—
31	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
35	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₅₃	Q₈	1	1	yes	characteristic
37	C₂×C₁₆×C₅₃	—	2	2	no	—
38	C₂×C₁₆×C₅₃	—	4	1	no	—
39	C₂×C₁₆×C₅₃	—	4	1	no	—
40	C₂⁴×C₅₃	C₂×C₈	1	1	yes	characteristic
41	C₂²×C₄×C₅₃	—	2	1	no	—
42	C₂²×C₄×C₅₃	—	2	1	no	—
43	C₂²×C₄×C₅₃	—	2	1	no	—
44	C₂²×C₄×C₅₃	—	2	1	no	—
45	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
46	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
47	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	M₁₆	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	M₁₆	1	1	yes	characteristic
51	C₂²×Dic₅₃	—	2	2	no	—
52	C₂²×Dic₅₃	—	2	2	no	—
53	C₂²×Dic₅₃	—	4	2	no	—
54	C₂²×Dic₅₃	—	4	2	no	—
55	C₂²×Dic₅₃	C₁₆	1	4	yes	complemented
56	C₂×C₈×C₅₃	—	2	1	no	—
57	C₂×C₈×C₅₃	—	2	1	no	—
58	C₂×C₈×C₅₃	—	2	1	no	—
59	C₂×C₈×C₅₃	—	2	1	no	—
60	C₂×C₈×C₅₃	—	4	1	no	—
61	C₂×C₈×C₅₃	—	4	1	no	—
62	C₂×C₈×C₅₃	—	4	1	no	—
63	C₂×C₈×C₅₃	—	4	1	no	—
64	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
68	C₄×Dic₅₃	—	4	2	no	—
69	C₄×Dic₅₃	—	4	1	no	—
70	C₄×Dic₅₃	—	4	1	no	—
71	C₂×#88322	—	2	2	no	—
72	C₂×#88322	—	4	2	no	—
73	C₂×#88322	—	4	1	no	—
74	C₂×#88322	—	4	1	no	—
75	C₂×#88322	—	4	2	no	—
76	C₁₆×C₅₃	—	4	2	no	—
77	#11719908	—	4	2	no	—
78	C₂³×C₅₃	—	2	1	no	—
79	C₂³×C₅₃	—	2	1	no	—
80	C₂³×C₅₃	—	2	2	no	—
81	C₂³×C₅₃	C₄×C₈	1	1	yes	characteristic
82	C₂³×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
83	C₂³×C₅₃	C₄⋊C₈	1	1	yes	characteristic
84	C₂³×C₅₃	C₂×C₁₆	1	2	yes	—
85	C₂³×C₅₃	M₃₂	1	2	yes	—
86	C₂×C₄×C₅₃	—	2	1	no	—
87	C₂×C₄×C₅₃	—	2	1	no	—
88	C₂×C₄×C₅₃	—	2	1	no	—
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	—	2	1	no	—
94	C₂×C₄×C₅₃	—	2	1	no	—
95	C₂×C₄×C₅₃	—	2	1	no	—
96	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
97	C₂×C₄×C₅₃	#97	1	1	yes	characteristic
98	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
99	C₂×C₄×C₅₃	#99	1	1	yes	characteristic
100	C₂×C₄×C₅₃	#99	1	1	yes	characteristic
101	C₂×C₄×C₅₃	#100	1	1	yes	characteristic
102	C₂×C₄×C₅₃	#101	1	1	yes	characteristic
103	C₂×C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
104	C₂×Dic₅₃	—	2	4	no	—
105	C₂×Dic₅₃	—	4	4	no	—
106	C₂×Dic₅₃	—	4	2	no	—
107	C₂×Dic₅₃	—	4	2	no	—
108	C₂×Dic₅₃	—	4	4	no	—
109	C₈×C₅₃	—	2	1	no	—
110	C₈×C₅₃	—	2	1	no	—
111	C₈×C₅₃	—	2	1	no	—
112	C₈×C₅₃	—	2	1	no	—
113	#88322	—	4	2	no	—
114	#7183	—	53	1	no	maximal, Sylow
115	C₂²×C₅₃	—	2	1	no	—
116	C₂²×C₅₃	—	2	1	no	—
117	C₂²×C₅₃	—	2	2	no	—
118	C₂²×C₅₃	—	2	2	no	—
119	C₂²×C₅₃	—	2	2	no	—
120	C₂²×C₅₃	—	2	2	no	—
121	C₂²×C₅₃	—	2	1	no	—
122	C₂²×C₅₃	—	2	1	no	—
123	C₂²×C₅₃	#323	1	1	yes	characteristic
124	C₂²×C₅₃	#324	1	1	yes	characteristic
125	C₂²×C₅₃	#336	1	1	yes	characteristic
126	C₂²×C₅₃	C₄×C₁₆	1	1	yes	characteristic
127	C₂²×C₅₃	#346	1	1	yes	characteristic
128	C₂²×C₅₃	C₂²⋊C₁₆	1	2	yes	—
129	C₂²×C₅₃	#349	1	2	yes	—
130	C₂²×C₅₃	C₄⋊C₁₆	1	2	yes	—
131	C₄×C₅₃	—	2	1	no	—
132	C₄×C₅₃	—	2	1	no	—
133	C₄×C₅₃	#323	1	1	yes	characteristic
134	C₄×C₅₃	#324	1	1	yes	characteristic
135	C₄×C₅₃	#342	1	1	yes	characteristic
136	C₄×C₅₃	#343	1	1	yes	characteristic
137	Dic₅₃	—	4	4	no	—
138	C₂×C₂²⋊C₁₆	—	53	1	no	—
139	C₂×C₂²⋊C₁₆	D₅₃	1	1	yes	characteristic
140	C₂×#336	—	53	1	no	—
141	C₂×C₅₃	—	2	1	no	—
142	C₂×C₅₃	—	2	1	no	—
143	C₂×C₅₃	—	2	2	no	—
144	C₂×C₅₃	#1280	1	1	yes	characteristic
145	C₂×C₅₃	#1314	1	2	yes	—
146	C₂×C₅₃	#1322	1	2	yes	—
147	C₂×C₅₃	#1374	1	1	yes	characteristic
148	C₂×C₅₃	#1377	1	1	yes	characteristic
149	C₂³×C₈	—	53	1	no	—
150	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
151	C₂²×C₄²	—	53	1	no	—
152	C₂²×C₁₆	—	2	1	no	—
153	C₂²×C₁₆	—	106	1	no	—
154	C₂²⋊C₁₆	—	2	1	no	—
155	C₂²⋊C₁₆	—	53	2	no	—
156	C₂²⋊C₁₆	—	106	1	no	—
157	C₂²⋊C₁₆	Dic₅₃	1	2	yes	complemented
158	#336	—	106	1	no	—
159	#336	—	106	1	no	—
160	C₅₃	#7183	1	1	yes	characteristic, complemented, Sylow
161	C₂³×C₄	—	53	2	no	—
162	C₂³×C₄	C₄×D₅₃	1	1	yes	characteristic
163	C₂²×C₈	—	4	1	no	—
164	C₂²×C₈	—	4	1	no	—
165	C₂²×C₈	—	53	2	no	—
166	C₂²×C₈	—	106	1	no	—
167	C₂²×C₈	—	106	1	no	—
168	C₂²×C₈	—	212	1	no	—
169	C₂²×C₈	—	212	1	no	—
170	C₂²×C₈	Dic₁₀₆	1	1	yes	characteristic
171	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
172	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
173	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
174	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
175	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
176	C₂×C₄²	—	106	1	no	—
177	C₂×C₄²	—	106	1	no	—
178	C₂×C₄²	—	212	1	no	—
179	C₂×C₄²	—	212	1	no	—
180	C₂×C₁₆	—	2	2	no	—
181	C₂×C₁₆	—	4	1	no	—
182	C₂×C₁₆	—	4	1	no	—
183	C₂×C₁₆	—	106	2	no	—
184	C₂×C₁₆	—	212	1	no	—
185	C₂×C₁₆	—	212	1	no	—
186	C₂⁴	C₈×D₅₃	1	1	yes	characteristic
187	C₂²×C₄	—	2	1	no	—
188	C₂²×C₄	—	2	1	no	—
189	C₂²×C₄	—	2	1	no	—
190	C₂²×C₄	—	2	1	no	—
191	C₂²×C₄	—	53	4	no	—
192	C₂²×C₄	—	106	2	no	—
193	C₂²×C₄	—	106	2	no	—
194	C₂²×C₄	—	212	2	no	—
195	C₂²×C₄	—	212	2	no	—
196	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
197	C₂²×C₄	#11719912	1	1	yes	characteristic
198	C₂²×C₄	#11719912	1	1	yes	characteristic
199	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
200	C₂²×C₄	#11719919	1	1	yes	characteristic
201	C₂²×C₄	#11719921	1	1	yes	characteristic
202	C₂×C₈	—	2	1	no	—
203	C₂×C₈	—	2	1	no	—
204	C₂×C₈	—	2	1	no	—
205	C₂×C₈	—	2	1	no	—
206	C₂×C₈	—	4	1	no	—
207	C₂×C₈	—	4	1	no	—
208	C₂×C₈	—	4	1	no	—
209	C₂×C₈	—	4	1	no	—
210	C₂×C₈	—	106	2	no	—
211	C₂×C₈	—	212	2	no	—
212	C₂×C₈	—	212	1	no	—
213	C₂×C₈	—	212	1	no	—
214	C₂×C₈	—	212	2	no	—
215	C₂×C₈	#11719919	1	1	yes	characteristic
216	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
217	C₂×C₈	#11719921	1	1	yes	characteristic
218	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
219	C₄²	—	212	2	no	—
220	C₄²	—	212	1	no	—
221	C₄²	—	212	1	no	—
222	C₁₆	—	4	2	no	—
223	C₁₆	—	212	2	no	—
224	C₂³	—	2	1	no	—
225	C₂³	—	2	1	no	—
226	C₂³	—	2	2	no	—
227	C₂³	C₈×Dic₅₃	1	1	yes	characteristic
228	C₂³	#13133129	1	1	yes	characteristic
229	C₂³	#13133146	1	1	yes	characteristic
230	C₂³	C₁₆×D₅₃	1	2	yes	—
231	C₂³	#13133156	1	2	yes	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	106	4	no	—
243	C₂×C₄	—	212	4	no	—
244	C₂×C₄	—	212	2	no	—
245	C₂×C₄	—	212	2	no	—
246	C₂×C₄	—	212	4	no	—
247	C₂×C₄	#13133124	1	1	yes	characteristic
248	C₂×C₄	#13133127	1	1	yes	characteristic
249	C₂×C₄	#13133129	1	1	yes	characteristic
250	C₂×C₄	#13133131	1	1	yes	characteristic
251	C₂×C₄	#13133131	1	1	yes	characteristic
252	C₂×C₄	#13133133	1	1	yes	characteristic
253	C₂×C₄	#13133135	1	1	yes	characteristic
254	C₂×C₄	#13133146	1	1	yes	characteristic
255	C₈	—	2	1	no	—
256	C₈	—	2	1	no	—
257	C₈	—	2	1	no	—
258	C₈	—	2	1	no	—
259	C₈	—	212	2	no	—
260	C₂²	—	2	2	no	—
261	C₂²	—	2	2	no	—
262	C₂²	—	2	1	no	—
263	C₂²	—	2	1	no	—
264	C₂²	—	2	1	no	—
265	C₂²	—	2	1	no	—
266	C₂²	—	2	2	no	—
267	C₂²	—	2	2	no	—
268	C₂²	#16810346	1	1	yes	characteristic
269	C₂²	#16810348	1	1	yes	characteristic
270	C₂²	#16810377	1	1	yes	characteristic
271	C₂²	C₁₆×Dic₅₃	1	1	yes	characteristic
272	C₂²	#16810398	1	1	yes	characteristic
273	C₂²	#16810402	1	2	yes	—
274	C₂²	#16810404	1	2	yes	—
275	C₂²	#16810441	1	2	yes	—
276	C₄	—	2	1	no	—
277	C₄	—	2	1	no	—
278	C₄	—	212	4	no	—
279	C₄	#16810346	1	1	yes	characteristic
280	C₄	#16810348	1	1	yes	characteristic
281	C₄	#16810389	1	1	yes	characteristic
282	C₄	#16810391	1	1	yes	characteristic
283	C₂	—	2	1	no	—
284	C₂	—	2	1	no	—
285	C₂	—	2	2	no	—
286	C₂	#20272557	1	1	yes	characteristic
287	C₂	#20272638	1	2	yes	—
288	C₂	#20272654	1	2	yes	—
289	C₂	#20272792	1	1	yes	characteristic
290	C₂	#20272798	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	#11719921
commutator	C₂²×C₅₃	C₄×C₁₆
Frattini	C₂³×C₈	C₂×D₅₃
Fitting	C₂×C₅₃×C₂²⋊C₁₆	C₂
socle	C₂³×C₅₃	C₂²⋊C₈
prime radical	C₂⁴×C₅₃	C₂×C₈
cosocle	C₂³×C₈×C₅₃	C₂²
hypercenter	C₂×C₂²⋊C₁₆	D₅₃
hypocenter	C₅₃	#7183
LCS	G ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₈ ◁ C₂×C₂²⋊C₁₆

Complements

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

N	H ≅ G/N	type
8. C₂²⋊#11719908	C₄	semidirect
14. C₅₃×C₂²⋊C₁₆	C₄	semidirect
55. C₂²×Dic₅₃	C₁₆	semidirect
157. C₂²⋊C₁₆	Dic₅₃	semidirect
160. C₅₃	#7183	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂²⋊#11719908	2
C₂×#16810377	2
C₂×C₅₃×C₂²⋊C₁₆	2
#7183	53