Subgroups — #27270654

Order 13568.

448 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	#20274695	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810378	C₂	1	1	yes	maximal, characteristic
3	C₄×C₂²⋊C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂³×#88322	—	2	1	no	—
5	C₂×C₄⋊#88322	C₄	1	2	yes	—
6	C₂×C₂²⋊#88322	—	2	1	no	—
7	#16810378	—	2	1	no	—
8	#16810378	—	2	1	no	—
9	#16810378	C₄	1	2	yes	—
10	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
11	C₂×C₂²⋊C₈×C₅₃	C₄	1	2	yes	—
12	C₂×C₄×C₈×C₅₃	—	2	1	no	—
13	C₅₃×#336	—	2	1	no	—
14	C₂²×#88322	—	2	4	no	—
15	C₂²×#88322	—	2	2	no	—
16	C₂²×#88322	—	2	2	no	—
17	C₂²×#88322	—	4	2	no	—
18	C₂²×#88322	—	4	1	no	—
19	C₂²×#88322	—	4	1	no	—
20	C₄⋊#88322	—	4	2	no	—
21	C₂²⋊#88322	—	4	1	no	—
22	C₂²⋊#88322	—	4	1	no	—
23	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
25	C₂³×C₄×C₅₃	Q₈	1	1	yes	characteristic
26	C₂²×C₈×C₅₃	—	2	2	no	—
27	C₂×C₄²×C₅₃	—	4	2	no	—
28	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
31	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
32	C₂²⋊C₈×C₅₃	—	2	2	no	—
33	C₂²⋊C₈×C₅₃	C₈	1	4	yes	complemented
34	C₄×C₈×C₅₃	—	4	2	no	—
35	C₂⁴×C₅₃	C₄⋊C₄	1	1	yes	characteristic
36	C₂²×C₄×C₅₃	—	2	2	no	—
37	C₂²×C₄×C₅₃	—	2	2	no	—
38	C₂²×C₄×C₅₃	—	2	2	no	—
39	C₂²×C₄×C₅₃	—	2	2	no	—
40	C₂²×C₄×C₅₃	—	4	1	no	—
41	C₂²×C₄×C₅₃	—	4	1	no	—
42	C₂²×C₄×C₅₃	—	4	1	no	—
43	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
44	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
45	C₂²×C₄×C₅₃	C₂²⋊C₄	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	C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
50	C₂²×C₄×C₅₃	M₁₆	1	2	yes	—
51	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
52	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
53	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	Q₁₆	1	1	yes	characteristic
55	C₂×C₈×C₅₃	—	2	4	no	—
56	C₂×C₈×C₅₃	—	4	2	no	—
57	C₂×C₈×C₅₃	—	4	2	no	—
58	C₄²×C₅₃	—	2	2	no	—
59	C₄²×C₅₃	—	2	2	no	—
60	C₄²×C₅₃	—	4	1	no	—
61	C₄²×C₅₃	—	4	1	no	—
62	C₂×#88322	—	2	4	no	—
63	C₂×#88322	—	4	2	no	—
64	C₂×#88322	—	4	4	no	—
65	C₂×#88322	—	4	4	no	—
66	C₂×#88322	—	4	2	no	—
67	C₂×#88322	—	4	2	no	—
68	C₂×#88322	—	4	2	no	—
69	C₂×#88322	—	4	2	no	—
70	C₂³×C₅₃	—	2	1	no	—
71	C₂³×C₅₃	—	2	1	no	—
72	C₂³×C₅₃	—	2	1	no	—
73	C₂³×C₅₃	—	2	1	no	—
74	C₂³×C₅₃	#95	1	1	yes	characteristic
75	C₂³×C₅₃	C₄⋊C₈	1	2	yes	—
76	C₂³×C₅₃	#106	1	1	yes	characteristic
77	C₂³×C₅₃	#107	1	1	yes	characteristic
78	C₂³×C₅₃	#108	1	2	yes	—
79	C₂×C₄×C₅₃	—	2	1	no	—
80	C₂×C₄×C₅₃	—	2	1	no	—
81	C₂×C₄×C₅₃	—	2	1	no	—
82	C₂×C₄×C₅₃	—	2	1	no	—
83	C₂×C₄×C₅₃	—	2	1	no	—
84	C₂×C₄×C₅₃	—	2	1	no	—
85	C₂×C₄×C₅₃	—	2	1	no	—
86	C₂×C₄×C₅₃	—	2	1	no	—
87	C₂×C₄×C₅₃	—	2	2	no	—
88	C₂×C₄×C₅₃	—	2	2	no	—
89	C₂×C₄×C₅₃	—	2	2	no	—
90	C₂×C₄×C₅₃	—	2	2	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	—	4	2	no	—
95	C₂×C₄×C₅₃	—	4	2	no	—
96	C₂×C₄×C₅₃	—	4	4	no	—
97	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
98	C₂×C₄×C₅₃	#97	1	1	yes	characteristic
99	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
100	C₂×C₄×C₅₃	#99	1	2	yes	—
101	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
102	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
103	C₂×C₄×C₅₃	#103	1	1	yes	characteristic
104	C₂×C₄×C₅₃	#103	1	1	yes	characteristic
105	C₂×C₄×C₅₃	#104	1	2	yes	—
106	C₂×C₄×C₅₃	#108	1	2	yes	—
107	C₈×C₅₃	—	4	4	no	—
108	#88322	—	4	4	no	—
109	#88322	—	8	2	no	—
110	#7029	—	53	1	no	maximal, Sylow
111	C₂²×C₅₃	—	2	2	no	—
112	C₂²×C₅₃	—	2	1	no	—
113	C₂²×C₅₃	—	2	1	no	—
114	C₂²×C₅₃	—	2	2	no	—
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	#334	1	2	yes	—
120	C₂²×C₅₃	#335	1	2	yes	—
121	C₂²×C₅₃	#336	1	1	yes	characteristic
122	C₂²×C₅₃	#339	1	1	yes	characteristic
123	C₂²×C₅₃	#340	1	1	yes	characteristic
124	C₂²×C₅₃	#341	1	1	yes	characteristic
125	C₂²×C₅₃	#341	1	1	yes	characteristic
126	C₂²×C₅₃	#342	1	1	yes	characteristic
127	C₂²×C₅₃	#344	1	1	yes	characteristic
128	C₄×C₅₃	—	2	2	no	—
129	C₄×C₅₃	—	2	1	no	—
130	C₄×C₅₃	—	2	1	no	—
131	C₄×C₅₃	—	2	1	no	—
132	C₄×C₅₃	—	2	1	no	—
133	C₄×C₅₃	—	2	2	no	—
134	C₄×C₅₃	—	4	1	no	—
135	C₄×C₅₃	#327	1	2	yes	—
136	C₄×C₅₃	#328	1	2	yes	—
137	#1797	—	53	1	no	—
138	C₄×C₂²⋊C₈	D₅₃	1	1	yes	characteristic
139	C₂×#336	—	53	1	no	—
140	C₂×C₅₃	—	2	2	no	—
141	C₂×C₅₃	—	2	2	no	—
142	C₂×C₅₃	#1277	1	1	yes	characteristic
143	C₂×C₅₃	#1278	1	1	yes	characteristic
144	C₂×C₅₃	#1282	1	1	yes	characteristic
145	C₂×C₅₃	#1294	1	1	yes	characteristic
146	C₂×C₅₃	#1298	1	1	yes	characteristic
147	C₂×C₅₃	#1301	1	1	yes	characteristic
148	C₂×C₅₃	#1302	1	1	yes	characteristic
149	C₂³×C₈	—	106	1	no	—
150	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
151	C₂×C₄⋊C₈	—	53	2	no	—
152	C₂×C₂²⋊C₈	—	106	1	no	—
153	C₂×C₂²⋊C₈	Dic₅₃	1	2	yes	—
154	C₂×C₄×C₈	—	2	1	no	—
155	#336	—	2	1	no	—
156	#336	—	53	2	no	—
157	#336	—	106	1	no	—
158	#336	—	106	1	no	—
159	C₅₃	#7029	1	1	yes	characteristic, complemented, Sylow
160	C₂³×C₄	Dic₁₀₆	1	1	yes	characteristic
161	C₂³×C₄	D₂₁₂	1	1	yes	characteristic
162	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
163	C₂²×C₈	—	2	2	no	—
164	C₂²×C₈	—	106	4	no	—
165	C₂²×C₈	—	106	2	no	—
166	C₂²×C₈	—	106	2	no	—
167	C₂²×C₈	—	212	2	no	—
168	C₂²×C₈	—	212	1	no	—
169	C₂²×C₈	—	212	1	no	—
170	C₂×C₄²	—	4	2	no	—
171	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
172	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
173	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
174	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
175	C₄⋊C₈	—	212	2	no	—
176	C₂²⋊C₈	—	2	2	no	—
177	C₂²⋊C₈	—	212	1	no	—
178	C₂²⋊C₈	—	212	1	no	—
179	C₂²⋊C₈	#88322	1	4	yes	complemented
180	C₄×C₈	—	4	2	no	—
181	C₂⁴	C₄⋊Dic₅₃	1	1	yes	characteristic
182	C₂²×C₄	—	2	2	no	—
183	C₂²×C₄	—	2	2	no	—
184	C₂²×C₄	—	2	2	no	—
185	C₂²×C₄	—	2	2	no	—
186	C₂²×C₄	—	4	1	no	—
187	C₂²×C₄	—	4	1	no	—
188	C₂²×C₄	—	4	1	no	—
189	C₂²×C₄	#11719913	1	1	yes	characteristic
190	C₂²×C₄	#11719913	1	1	yes	characteristic
191	C₂²×C₄	D₄₂₄	1	1	yes	characteristic
192	C₂²×C₄	Dic₂₁₂	1	1	yes	characteristic
193	C₂²×C₄	C₂×#88322	1	2	yes	—
194	C₂²×C₄	#11719917	1	2	yes	—
195	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
196	C₂²×C₄	#11719919	1	1	yes	characteristic
197	C₂²×C₄	#11719919	1	1	yes	characteristic
198	C₂²×C₄	#11719921	1	1	yes	characteristic
199	C₂²×C₄	#11719921	1	1	yes	characteristic
200	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
201	C₂×C₈	—	2	4	no	—
202	C₂×C₈	—	4	2	no	—
203	C₂×C₈	—	4	2	no	—
204	C₂×C₈	—	106	4	no	—
205	C₂×C₈	—	212	2	no	—
206	C₂×C₈	—	212	2	no	—
207	C₂×C₈	—	212	2	no	—
208	C₂×C₈	—	212	2	no	—
209	C₂×C₈	—	212	4	no	—
210	C₂×C₈	—	212	4	no	—
211	C₂×C₈	—	212	2	no	—
212	C₄²	—	2	2	no	—
213	C₄²	—	2	2	no	—
214	C₄²	—	4	1	no	—
215	C₄²	—	4	1	no	—
216	C₂³	—	2	1	no	—
217	C₂³	—	2	1	no	—
218	C₂³	—	2	1	no	—
219	C₂³	—	2	1	no	—
220	C₂³	#13133124	1	1	yes	characteristic
221	C₂³	C₄⋊#88322	1	2	yes	—
222	C₂³	#13133149	1	1	yes	characteristic
223	C₂³	#13133151	1	1	yes	characteristic
224	C₂³	#13133153	1	2	yes	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	4	2	no	—
241	C₂×C₄	—	4	2	no	—
242	C₂×C₄	—	4	4	no	—
243	C₂×C₄	C₄×#88322	1	1	yes	characteristic
244	C₂×C₄	#13133128	1	1	yes	characteristic
245	C₂×C₄	C₂²⋊#88322	1	2	yes	—
246	C₂×C₄	#13133131	1	2	yes	—
247	C₂×C₄	#13133139	1	1	yes	characteristic
248	C₂×C₄	#13133139	1	1	yes	characteristic
249	C₂×C₄	#13133142	1	1	yes	characteristic
250	C₂×C₄	#13133142	1	1	yes	characteristic
251	C₂×C₄	#13133143	1	2	yes	—
252	C₂×C₄	#13133152	1	2	yes	—
253	C₈	—	4	4	no	—
254	C₈	—	424	2	no	—
255	C₈	—	212	4	no	—
256	C₂²	—	2	2	no	—
257	C₂²	—	2	2	no	—
258	C₂²	—	2	1	no	—
259	C₂²	—	2	1	no	—
260	C₂²	—	2	2	no	—
261	C₂²	—	2	1	no	—
262	C₂²	—	2	1	no	—
263	C₂²	—	2	2	no	—
264	C₂²	#16810374	1	2	yes	—
265	C₂²	#16810376	1	2	yes	—
266	C₂²	#16810378	1	1	yes	characteristic
267	C₂²	#16810384	1	1	yes	characteristic
268	C₂²	#16810386	1	1	yes	characteristic
269	C₂²	#16810387	1	1	yes	characteristic
270	C₂²	#16810388	1	1	yes	characteristic
271	C₂²	#16810389	1	1	yes	characteristic
272	C₂²	#16810393	1	1	yes	characteristic
273	C₄	—	2	2	no	—
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	2	1	no	—
277	C₄	—	2	1	no	—
278	C₄	—	2	2	no	—
279	C₄	—	4	1	no	—
280	C₄	#16810358	1	2	yes	—
281	C₄	#16810361	1	2	yes	—
282	C₂	—	2	2	no	—
283	C₂	—	2	2	no	—
284	C₂	#20272550	1	1	yes	characteristic
285	C₂	#20272553	1	1	yes	characteristic
286	C₂	#20272563	1	1	yes	characteristic
287	C₂	#20272595	1	1	yes	characteristic
288	C₂	#20272603	1	1	yes	characteristic
289	C₂	#20272611	1	1	yes	characteristic
290	C₂	#20272615	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	#11719921
commutator	C₂×C₄×C₅₃	C₄×C₈
Frattini	C₂²×C₄²	C₂×D₅₃
Fitting	C₄×C₂²⋊C₈×C₅₃	C₂
socle	C₂³×C₅₃	#95
prime radical	C₂⁴×C₅₃	C₄⋊C₄
cosocle	C₂²×C₄²×C₅₃	C₂²
hypercenter	C₄×C₂²⋊C₈	D₅₃
hypocenter	C₅₃	#7029
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₄×C₂²⋊C₈

Complements

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

N	H ≅ G/N	type
33. C₂²⋊C₈×C₅₃	C₈	semidirect
159. C₅₃	#7029	semidirect
179. C₂²⋊C₈	#88322	semidirect

Maximal subgroups

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

subgroup	index
#20274695	2
C₂×#16810378	2
C₄×C₂²⋊C₈×C₅₃	2
#7029	53