Subgroups — #27292975

Order 13568.

614 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×#16810672	C₂	1	1	yes	maximal, characteristic
2	#20274441	C₂	1	2	yes	maximal
3	C₂×#16810381	C₂	1	1	yes	maximal, characteristic
4	#20274409	C₂	1	2	yes	maximal
5	C₅₃×#1728	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11719917	C₂²	1	2	yes	—
7	C₂²×#11719917	C₂²	1	2	yes	—
8	#16810672	—	2	2	no	—
9	#16810672	C₄	1	4	yes	complemented
10	C₂×#13133153	—	2	2	no	—
11	#16810387	—	2	4	no	—
12	#16810381	—	2	2	no	—
13	#16810378	—	2	4	no	—
14	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
15	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
16	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×#336	—	2	2	no	—
18	C₂×#11719917	—	2	2	no	—
19	C₂×#11719917	—	2	2	no	—
20	C₂×#11719917	—	2	2	no	—
21	C₂×#11719917	—	2	2	no	—
22	C₂×#11719917	—	2	2	no	—
23	C₂×#11719917	—	2	2	no	—
24	C₂×#11719917	—	2	2	no	—
25	C₂×#11719917	—	2	2	no	—
26	C₂×#11719917	—	2	2	no	—
27	C₂×#11719917	—	2	2	no	—
28	C₂×#11719917	—	2	2	no	—
29	C₂×#11719917	C₂×C₄	1	4	yes	—
30	C₂²×#88322	—	2	2	no	—
31	C₂²×#88322	D₄	1	4	yes	—
32	#13133153	—	2	8	no	—
33	C₂³×C₄×C₅₃	Q₈	1	2	yes	—
34	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
35	C₂×M₁₆×C₅₃	—	2	2	no	—
36	C₂×M₁₆×C₅₃	—	2	2	no	—
37	C₂×M₁₆×C₅₃	—	2	2	no	—
38	C₂×M₁₆×C₅₃	—	2	1	no	—
39	C₂×M₁₆×C₅₃	—	2	2	no	—
40	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
41	C₂×M₁₆×C₅₃	D₄	1	2	yes	—
42	C₂×M₁₆×C₅₃	Q₈	1	2	yes	—
43	C₂²×C₈×C₅₃	—	2	1	no	—
44	C₂²×C₈×C₅₃	D₄	1	2	yes	—
45	C₂×C₄²×C₅₃	—	2	1	no	—
46	C₂×C₄²×C₅₃	—	2	2	no	—
47	C₂×C₄²×C₅₃	—	2	2	no	—
48	C₂×C₄²×C₅₃	—	2	1	no	—
49	C₂⁴×C₅₃	C₂×Q₈	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	—	2	4	no	—
51	C₂²×C₄×C₅₃	—	2	2	no	—
52	C₂²×C₄×C₅₃	—	4	4	no	—
53	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
57	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	#41	1	2	yes	—
59	C₂²×C₄×C₅₃	#41	1	2	yes	—
60	C₂²×C₄×C₅₃	#41	1	2	yes	—
61	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
63	M₁₆×C₅₃	—	2	4	no	—
64	M₁₆×C₅₃	—	2	4	no	—
65	M₁₆×C₅₃	—	2	4	no	—
66	M₁₆×C₅₃	—	4	2	no	—
67	C₂×C₈×C₅₃	—	2	2	no	—
68	C₂×C₈×C₅₃	—	2	2	no	—
69	C₂×C₈×C₅₃	—	2	2	no	—
70	C₂×C₈×C₅₃	—	4	1	no	—
71	C₂×C₈×C₅₃	—	4	1	no	—
72	C₂×C₈×C₅₃	C₂²⋊C₄	1	4	yes	—
73	C₄²×C₅₃	—	4	1	no	—
74	C₄²×C₅₃	—	4	2	no	—
75	C₄²×C₅₃	—	4	1	no	—
76	#11719917	—	2	4	no	—
77	#11719917	—	2	4	no	—
78	#11719917	—	4	2	no	—
79	#11719917	—	4	2	no	—
80	#11719917	—	4	2	no	—
81	#11719917	—	4	2	no	—
82	#11719917	—	4	2	no	—
83	#11719917	—	4	2	no	—
84	C₂×#88322	—	2	4	no	—
85	C₂×#88322	—	2	4	no	—
86	C₂×#88322	—	2	4	no	—
87	C₂×#88322	—	2	4	no	—
88	C₂×#88322	—	4	2	no	—
89	C₂×#88322	—	4	2	no	—
90	C₂³×C₅₃	—	4	2	no	—
91	C₂³×C₅₃	C₄×Q₈	1	1	yes	characteristic
92	C₂³×C₅₃	C₂²⋊Q₈	1	2	yes	—
93	C₂³×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
94	C₂³×C₅₃	#125	1	2	yes	—
95	C₂³×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
96	C₂×C₄×C₅₃	—	2	8	no	—
97	C₂×C₄×C₅₃	—	2	4	no	—
98	C₂×C₄×C₅₃	—	4	4	no	—
99	C₂×C₄×C₅₃	—	4	2	no	—
100	C₂×C₄×C₅₃	—	4	2	no	—
101	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₅₃	#117	1	2	yes	—
103	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
104	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
105	C₂×C₄×C₅₃	#121	1	2	yes	—
106	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
107	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
108	C₂×C₄×C₅₃	#123	1	1	yes	characteristic
109	C₂×C₄×C₅₃	#123	1	1	yes	characteristic
110	C₂×C₄×C₅₃	#123	1	2	yes	—
111	C₂×C₄×C₅₃	#123	1	2	yes	—
112	C₂×C₄×C₅₃	#123	1	2	yes	—
113	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
114	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
115	C₂×C₄×C₅₃	#125	1	2	yes	—
116	C₂×C₄×C₅₃	#126	1	2	yes	—
117	C₂×C₄×C₅₃	#126	1	2	yes	—
118	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
119	C₈×C₅₃	—	2	4	no	—
120	C₈×C₅₃	—	4	2	no	—
121	#88322	—	4	4	no	—
122	#88322	—	4	4	no	—
123	#11375	—	53	1	no	maximal, Sylow
124	C₂²×C₅₃	—	2	4	no	—
125	C₂²×C₅₃	—	2	2	no	—
126	C₂²×C₅₃	—	4	1	no	—
127	C₂²×C₅₃	—	4	2	no	—
128	C₂²×C₅₃	—	4	1	no	—
129	C₂²×C₅₃	#387	1	2	yes	—
130	C₂²×C₅₃	#391	1	1	yes	characteristic
131	C₂²×C₅₃	#398	1	2	yes	—
132	C₂²×C₅₃	#399	1	1	yes	characteristic
133	C₂²×C₅₃	#400	1	1	yes	characteristic
134	C₄×C₅₃	—	4	4	no	—
135	C₄×C₅₃	#381	1	1	yes	characteristic
136	C₄×C₅₃	#387	1	2	yes	—
137	C₄×C₅₃	#390	1	1	yes	characteristic
138	C₄×C₅₃	#397	1	2	yes	—
139	C₄×C₅₃	#399	1	1	yes	characteristic
140	C₄×C₅₃	#400	1	1	yes	characteristic
141	C₂×#430	—	53	1	no	—
142	#1738	—	53	2	no	—
143	C₂×#338	—	53	1	no	—
144	#1728	—	53	2	no	—
145	#1728	D₅₃	1	1	yes	characteristic
146	C₂×C₅₃	—	2	2	no	—
147	C₂×C₅₃	—	2	1	no	—
148	C₂×C₅₃	—	2	2	no	—
149	C₂×C₅₃	—	2	1	no	—
150	C₂×C₅₃	#1445	1	1	yes	characteristic
151	C₂×C₅₃	#2091	1	2	yes	—
152	C₂²×M₁₆	—	53	2	no	—
153	C₂²×M₁₆	—	53	2	no	—
154	C₂²×M₁₆	C₂×D₅₃	1	1	yes	characteristic
155	C₂²×M₁₆	C₂×D₅₃	1	1	yes	characteristic
156	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
157	#430	—	53	4	no	—
158	#430	—	106	2	no	—
159	C₂×#108	—	106	2	no	—
160	#341	—	106	4	no	—
161	#338	—	106	2	no	—
162	#336	—	2	2	no	—
163	#336	—	106	4	no	—
164	C₅₃	#11375	1	1	yes	characteristic, complemented, Sylow
165	C₂³×C₄	Dic₁₀₆	1	2	yes	—
166	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
167	C₂×M₁₆	—	2	2	no	—
168	C₂×M₁₆	—	2	2	no	—
169	C₂×M₁₆	—	2	1	no	—
170	C₂×M₁₆	—	2	2	no	—
171	C₂×M₁₆	—	2	2	no	—
172	C₂×M₁₆	—	53	4	no	—
173	C₂×M₁₆	—	106	2	no	—
174	C₂×M₁₆	—	106	2	no	—
175	C₂×M₁₆	—	106	2	no	—
176	C₂×M₁₆	—	106	2	no	—
177	C₂×M₁₆	—	106	2	no	—
178	C₂×M₁₆	—	106	2	no	—
179	C₂×M₁₆	—	106	2	no	—
180	C₂×M₁₆	—	106	2	no	—
181	C₂×M₁₆	—	106	2	no	—
182	C₂×M₁₆	—	106	2	no	—
183	C₂×M₁₆	—	106	2	no	—
184	C₂×M₁₆	Dic₁₀₆	1	2	yes	—
185	C₂×M₁₆	C₄×D₅₃	1	2	yes	—
186	C₂×M₁₆	D₂₁₂	1	2	yes	—
187	C₂²×C₈	—	2	1	no	—
188	C₂²×C₈	—	53	4	no	—
189	C₂²×C₈	—	106	2	no	—
190	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
191	C₂×C₄²	—	2	1	no	—
192	C₂×C₄²	—	2	2	no	—
193	C₂×C₄²	—	2	2	no	—
194	C₂×C₄²	—	2	1	no	—
195	#108	—	106	8	no	—
196	C₂⁴	C₂×Dic₁₀₆	1	1	yes	characteristic
197	C₂²×C₄	—	2	4	no	—
198	C₂²×C₄	—	2	2	no	—
199	C₂²×C₄	—	4	4	no	—
200	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
201	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
202	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
203	C₂²×C₄	#11719945	1	1	yes	characteristic
204	C₂²×C₄	#11719945	1	1	yes	characteristic
205	C₂²×C₄	D₄×D₅₃	1	2	yes	—
206	C₂²×C₄	#11719947	1	2	yes	—
207	C₂²×C₄	#11719947	1	2	yes	—
208	C₂²×C₄	#11719949	1	2	yes	—
209	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
210	M₁₆	—	2	4	no	—
211	M₁₆	—	2	4	no	—
212	M₁₆	—	2	4	no	—
213	M₁₆	—	4	2	no	—
214	M₁₆	—	106	4	no	—
215	M₁₆	—	106	4	no	—
216	M₁₆	—	212	2	no	—
217	M₁₆	—	212	2	no	—
218	M₁₆	—	212	2	no	—
219	M₁₆	—	212	2	no	—
220	M₁₆	—	212	2	no	—
221	M₁₆	—	212	2	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	4	1	no	—
226	C₂×C₈	—	4	1	no	—
227	C₂×C₈	—	106	4	no	—
228	C₂×C₈	—	106	4	no	—
229	C₂×C₈	—	106	4	no	—
230	C₂×C₈	—	106	4	no	—
231	C₂×C₈	—	212	2	no	—
232	C₂×C₈	—	212	2	no	—
233	C₂×C₈	#11719921	1	4	yes	—
234	C₄²	—	4	1	no	—
235	C₄²	—	4	2	no	—
236	C₄²	—	4	1	no	—
237	C₂³	—	4	2	no	—
238	C₂³	C₄×Dic₁₀₆	1	1	yes	characteristic
239	C₂³	#13133194	1	2	yes	—
240	C₂³	#13133196	1	1	yes	characteristic
241	C₂³	#13133208	1	2	yes	—
242	C₂³	#13133217	1	1	yes	characteristic
243	C₂×C₄	—	2	8	no	—
244	C₂×C₄	—	2	4	no	—
245	C₂×C₄	—	4	4	no	—
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
249	C₂×C₄	#13133173	1	2	yes	—
250	C₂×C₄	#13133179	1	2	yes	—
251	C₂×C₄	C₄×Dic₁₀₆	1	1	yes	characteristic
252	C₂×C₄	#13133192	1	2	yes	—
253	C₂×C₄	#13133194	1	2	yes	—
254	C₂×C₄	#13133196	1	1	yes	characteristic
255	C₂×C₄	#13133198	1	1	yes	characteristic
256	C₂×C₄	#13133198	1	1	yes	characteristic
257	C₂×C₄	#13133199	1	2	yes	—
258	C₂×C₄	#13133199	1	2	yes	—
259	C₂×C₄	#13133201	1	2	yes	—
260	C₂×C₄	#13133205	1	1	yes	characteristic
261	C₂×C₄	#13133206	1	1	yes	characteristic
262	C₂×C₄	#13133208	1	2	yes	—
263	C₂×C₄	#13133212	1	2	yes	—
264	C₂×C₄	#13133212	1	2	yes	—
265	C₂×C₄	#13133213	1	1	yes	characteristic
266	C₈	—	2	4	no	—
267	C₈	—	4	2	no	—
268	C₈	—	212	4	no	—
269	C₈	—	212	4	no	—
270	C₂²	—	2	4	no	—
271	C₂²	—	2	2	no	—
272	C₂²	—	4	1	no	—
273	C₂²	—	4	2	no	—
274	C₂²	—	4	1	no	—
275	C₂²	#16810506	1	2	yes	—
276	C₂²	#16810529	1	1	yes	characteristic
277	C₂²	#16810555	1	2	yes	—
278	C₂²	#16810559	1	1	yes	characteristic
279	C₂²	#16810562	1	1	yes	characteristic
280	C₄	—	4	4	no	—
281	C₄	#16810484	1	1	yes	characteristic
282	C₄	#16810506	1	2	yes	—
283	C₄	#16810524	1	1	yes	characteristic
284	C₄	#16810550	1	2	yes	—
285	C₄	#16810559	1	1	yes	characteristic
286	C₄	#16810562	1	1	yes	characteristic
287	C₂	—	2	2	no	—
288	C₂	—	2	1	no	—
289	C₂	—	2	2	no	—
290	C₂	—	2	1	no	—
291	C₂	#20272984	1	1	yes	characteristic
292	C₂	#20276231	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#13133213
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₅₃×#1728	C₂
socle	C₂²×C₅₃	#399
prime radical	C₂⁴×C₅₃	C₂×Q₈
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	#1728	D₅₃
hypocenter	C₅₃	#11375
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #1728

Complements

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

N	H ≅ G/N	type
9. #16810672	C₄	semidirect
164. C₅₃	#11375	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810672	2
#20274441	2
C₂×#16810381	2
#20274409	2
C₅₃×#1728	2
#11375	53