Subgroups — #27292718

Order 13568.

730 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133152	C₂	1	2	yes	maximal
2	C₂×#16810647	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810387	C₂	1	2	yes	maximal
4	C₂×#16810378	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1728	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11719917	C₂²	1	2	yes	—
7	C₂³×#88322	C₂²	1	2	yes	—
8	C₂×#13133152	—	2	4	no	—
9	C₂×#13133152	—	2	4	no	—
10	C₂×#13133152	—	2	4	no	—
11	#16810647	—	2	4	no	—
12	C₂×C₄⋊#88322	—	2	2	no	—
13	#16810387	—	2	4	no	—
14	#16810378	C₄	1	2	yes	—
15	#16810378	C₄	1	2	yes	—
16	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
17	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
18	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
19	C₅₃×#336	—	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	D₄	1	4	yes	—
26	C₂²×#88322	—	2	2	no	—
27	C₂²×#88322	—	2	2	no	—
28	C₂²×#88322	—	2	2	no	—
29	C₂²×#88322	—	2	2	no	—
30	C₂²×#88322	—	2	2	no	—
31	C₂²×#88322	C₂×C₄	1	4	yes	—
32	C₂²×#88322	D₄	1	4	yes	—
33	C₂²×#88322	Q₈	1	4	yes	—
34	#13133152	—	2	16	no	—
35	C₄⋊#88322	—	2	8	no	—
36	C₂³×C₄×C₅₃	Q₈	1	2	yes	—
37	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
38	C₂×M₁₆×C₅₃	—	2	2	no	—
39	C₂×M₁₆×C₅₃	—	2	2	no	—
40	C₂×M₁₆×C₅₃	—	2	2	no	—
41	C₂×M₁₆×C₅₃	—	2	1	no	—
42	C₂×M₁₆×C₅₃	—	2	1	no	—
43	C₂×M₁₆×C₅₃	—	2	2	no	—
44	C₂×M₁₆×C₅₃	D₄	1	2	yes	—
45	C₂×M₁₆×C₅₃	Q₈	1	2	yes	—
46	C₂²×C₈×C₅₃	—	2	1	no	—
47	C₂²×C₈×C₅₃	—	2	1	no	—
48	C₂×C₄²×C₅₃	—	2	2	no	—
49	C₂×C₄²×C₅₃	—	2	2	no	—
50	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄²×C₅₃	D₄	1	2	yes	—
53	C₂⁴×C₅₃	C₂×Q₈	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	—	2	4	no	—
55	C₂²×C₄×C₅₃	—	2	4	no	—
56	C₂²×C₄×C₅₃	—	4	2	no	—
57	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
58	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
59	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
61	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
65	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
66	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
67	C₂²×C₄×C₅₃	#41	1	2	yes	—
68	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
69	M₁₆×C₅₃	—	2	4	no	—
70	M₁₆×C₅₃	—	2	4	no	—
71	M₁₆×C₅₃	—	4	2	no	—
72	M₁₆×C₅₃	—	4	2	no	—
73	C₂×C₈×C₅₃	—	2	2	no	—
74	C₂×C₈×C₅₃	—	2	2	no	—
75	C₂×C₈×C₅₃	—	4	1	no	—
76	C₂×C₈×C₅₃	—	4	1	no	—
77	C₂×C₈×C₅₃	—	4	1	no	—
78	C₂×C₈×C₅₃	—	4	1	no	—
79	C₄²×C₅₃	—	2	2	no	—
80	C₄²×C₅₃	—	2	2	no	—
81	C₄²×C₅₃	—	2	2	no	—
82	C₄²×C₅₃	—	2	2	no	—
83	#11719917	—	2	4	no	—
84	#11719917	—	2	4	no	—
85	#11719917	—	4	2	no	—
86	#11719917	—	4	2	no	—
87	C₂×#88322	—	2	4	no	—
88	C₂×#88322	—	2	4	no	—
89	C₂×#88322	—	2	4	no	—
90	C₂×#88322	—	2	4	no	—
91	C₂×#88322	—	2	4	no	—
92	C₂×#88322	—	2	4	no	—
93	C₂×#88322	—	2	4	no	—
94	C₂×#88322	—	4	2	no	—
95	C₂×#88322	—	4	2	no	—
96	C₂×#88322	C₄⋊C₄	1	8	yes	—
97	C₂³×C₅₃	—	2	2	no	—
98	C₂³×C₅₃	—	2	2	no	—
99	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
100	C₂³×C₅₃	C₂²⋊Q₈	1	2	yes	—
101	C₂³×C₅₃	#125	1	1	yes	characteristic
102	C₂³×C₅₃	C₄⋊Q₈	1	2	yes	—
103	C₂³×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
104	C₂×C₄×C₅₃	—	2	2	no	—
105	C₂×C₄×C₅₃	—	2	2	no	—
106	C₂×C₄×C₅₃	—	2	2	no	—
107	C₂×C₄×C₅₃	—	2	2	no	—
108	C₂×C₄×C₅₃	—	2	2	no	—
109	C₂×C₄×C₅₃	—	4	8	no	—
110	C₂×C₄×C₅₃	#108	1	4	yes	complemented
111	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
112	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
113	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
114	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
115	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
116	C₂×C₄×C₅₃	#120	1	2	yes	—
117	C₂×C₄×C₅₃	#121	1	2	yes	—
118	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
119	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
120	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
121	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
122	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
123	C₂×C₄×C₅₃	#123	1	1	yes	characteristic
124	C₂×C₄×C₅₃	#123	1	2	yes	—
125	C₂×C₄×C₅₃	#125	1	2	yes	—
126	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
127	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
128	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
129	C₈×C₅₃	—	4	2	no	—
130	C₈×C₅₃	—	4	2	no	—
131	#88322	—	2	8	no	—
132	#88322	—	4	4	no	—
133	#11326	—	53	1	no	maximal, Sylow
134	C₂²×C₅₃	—	2	2	no	—
135	C₂²×C₅₃	—	2	2	no	—
136	C₂²×C₅₃	—	2	4	no	—
137	C₂²×C₅₃	—	2	2	no	—
138	C₂²×C₅₃	—	2	2	no	—
139	C₂²×C₅₃	#384	1	1	yes	characteristic
140	C₂²×C₅₃	#385	1	1	yes	characteristic
141	C₂²×C₅₃	#389	1	1	yes	characteristic
142	C₂²×C₅₃	#395	1	2	yes	—
143	C₂²×C₅₃	#398	1	2	yes	—
144	C₂²×C₅₃	C₂×#108	1	2	yes	—
145	C₂²×C₅₃	#430	1	2	yes	—
146	C₄×C₅₃	—	2	4	no	—
147	C₄×C₅₃	—	4	2	no	—
148	C₄×C₅₃	#385	1	2	yes	—
149	C₄×C₅₃	#389	1	2	yes	—
150	C₄×C₅₃	#392	1	1	yes	characteristic
151	C₄×C₅₃	#395	1	1	yes	characteristic
152	C₄×C₅₃	#398	1	1	yes	characteristic
153	C₄×C₅₃	#399	1	1	yes	characteristic
154	C₂²×#108	—	53	2	no	—
155	C₂×#423	—	53	1	no	—
156	C₂×#341	—	53	2	no	—
157	#1728	D₅₃	1	1	yes	characteristic
158	C₂×#336	—	53	1	no	—
159	C₂×C₅₃	—	2	2	no	—
160	C₂×C₅₃	—	2	2	no	—
161	C₂×C₅₃	#1444	1	1	yes	characteristic
162	C₂×C₅₃	#1848	1	1	yes	characteristic
163	C₂×C₅₃	#1849	1	1	yes	characteristic
164	C₂×C₅₃	#1855	1	2	yes	—
165	C₂×C₅₃	#1950	1	2	yes	—
166	C₂²×M₁₆	—	53	2	no	—
167	C₂²×M₁₆	C₂×D₅₃	1	1	yes	characteristic
168	C₂²×M₁₆	C₂×D₅₃	1	1	yes	characteristic
169	C₂³×C₈	—	53	2	no	—
170	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
171	C₂×#108	—	106	4	no	—
172	C₂×#108	—	106	4	no	—
173	C₂×#108	—	106	4	no	—
174	#423	—	106	4	no	—
175	C₂×C₄⋊C₈	—	106	2	no	—
176	#341	—	106	4	no	—
177	#336	—	2	2	no	—
178	#336	—	53	2	no	—
179	#336	—	53	2	no	—
180	C₅₃	#11326	1	1	yes	characteristic, complemented, Sylow
181	C₂³×C₄	Dic₁₀₆	1	2	yes	—
182	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
183	C₂×M₁₆	—	2	1	no	—
184	C₂×M₁₆	—	2	1	no	—
185	C₂×M₁₆	—	2	2	no	—
186	C₂×M₁₆	—	2	2	no	—
187	C₂×M₁₆	—	2	2	no	—
188	C₂×M₁₆	—	2	2	no	—
189	C₂×M₁₆	—	53	4	no	—
190	C₂×M₁₆	—	106	2	no	—
191	C₂×M₁₆	—	106	2	no	—
192	C₂×M₁₆	—	106	2	no	—
193	C₂×M₁₆	—	106	2	no	—
194	C₂×M₁₆	—	106	2	no	—
195	C₂×M₁₆	Dic₁₀₆	1	2	yes	—
196	C₂×M₁₆	D₂₁₂	1	2	yes	—
197	C₂²×C₈	—	2	1	no	—
198	C₂²×C₈	—	2	1	no	—
199	C₂²×C₈	—	53	4	no	—
200	C₂²×C₈	—	53	4	no	—
201	C₂²×C₈	—	53	4	no	—
202	C₂²×C₈	—	106	2	no	—
203	C₂²×C₈	—	106	2	no	—
204	C₂²×C₈	—	106	2	no	—
205	C₂²×C₈	—	106	2	no	—
206	C₂²×C₈	—	106	2	no	—
207	C₂×C₄²	—	2	2	no	—
208	C₂×C₄²	—	2	2	no	—
209	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
210	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
211	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
212	#108	—	106	16	no	—
213	C₄⋊C₈	—	106	8	no	—
214	C₂⁴	C₂×Dic₁₀₆	1	1	yes	characteristic
215	C₂²×C₄	—	2	4	no	—
216	C₂²×C₄	—	2	4	no	—
217	C₂²×C₄	—	4	2	no	—
218	C₂²×C₄	#11719919	1	2	yes	—
219	C₂²×C₄	#11719919	1	2	yes	—
220	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
221	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
222	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
223	C₂²×C₄	#11719945	1	1	yes	characteristic
224	C₂²×C₄	#11719945	1	1	yes	characteristic
225	C₂²×C₄	D₄×D₅₃	1	2	yes	—
226	C₂²×C₄	D₄×D₅₃	1	2	yes	—
227	C₂²×C₄	#11719947	1	2	yes	—
228	C₂²×C₄	Q₈×D₅₃	1	2	yes	—
229	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
230	M₁₆	—	2	4	no	—
231	M₁₆	—	2	4	no	—
232	M₁₆	—	4	2	no	—
233	M₁₆	—	4	2	no	—
234	M₁₆	—	106	4	no	—
235	M₁₆	—	106	4	no	—
236	M₁₆	—	212	2	no	—
237	M₁₆	—	212	2	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	4	1	no	—
241	C₂×C₈	—	4	1	no	—
242	C₂×C₈	—	4	1	no	—
243	C₂×C₈	—	4	1	no	—
244	C₂×C₈	—	53	8	no	—
245	C₂×C₈	—	106	4	no	—
246	C₂×C₈	—	106	4	no	—
247	C₂×C₈	—	106	4	no	—
248	C₂×C₈	—	106	4	no	—
249	C₂×C₈	—	106	4	no	—
250	C₂×C₈	—	106	4	no	—
251	C₂×C₈	—	106	4	no	—
252	C₂×C₈	—	212	2	no	—
253	C₂×C₈	—	212	2	no	—
254	C₄²	—	2	2	no	—
255	C₄²	—	2	2	no	—
256	C₄²	—	2	2	no	—
257	C₄²	—	2	2	no	—
258	C₂³	—	2	2	no	—
259	C₂³	—	2	2	no	—
260	C₂³	C₂×#11719919	1	1	yes	characteristic
261	C₂³	#13133194	1	2	yes	—
262	C₂³	#13133209	1	1	yes	characteristic
263	C₂³	#13133216	1	2	yes	—
264	C₂³	#13133217	1	1	yes	characteristic
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	4	8	no	—
271	C₂×C₄	#13133152	1	4	yes	complemented
272	C₂×C₄	C₄⋊C₄×D₅₃	1	2	yes	—
273	C₂×C₄	#13133177	1	2	yes	—
274	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
275	C₂×C₄	C₄×Dic₁₀₆	1	1	yes	characteristic
276	C₂×C₄	#13133185	1	1	yes	characteristic
277	C₂×C₄	#13133187	1	2	yes	—
278	C₂×C₄	#13133190	1	2	yes	—
279	C₂×C₄	#13133191	1	1	yes	characteristic
280	C₂×C₄	#13133194	1	2	yes	—
281	C₂×C₄	#13133195	1	2	yes	—
282	C₂×C₄	#13133196	1	1	yes	characteristic
283	C₂×C₄	#13133197	1	2	yes	—
284	C₂×C₄	#13133198	1	1	yes	characteristic
285	C₂×C₄	#13133199	1	2	yes	—
286	C₂×C₄	#13133207	1	2	yes	—
287	C₂×C₄	#13133213	1	1	yes	characteristic
288	C₂×C₄	#13133215	1	1	yes	characteristic
289	C₂×C₄	#13133216	1	2	yes	—
290	C₈	—	4	2	no	—
291	C₈	—	4	2	no	—
292	C₈	—	106	8	no	—
293	C₈	—	212	4	no	—
294	C₂²	—	2	2	no	—
295	C₂²	—	2	2	no	—
296	C₂²	—	2	2	no	—
297	C₂²	—	2	2	no	—
298	C₂²	—	2	4	no	—
299	C₂²	#16810492	1	1	yes	characteristic
300	C₂²	#16810495	1	1	yes	characteristic
301	C₂²	#16810521	1	1	yes	characteristic
302	C₂²	#16810544	1	2	yes	—
303	C₂²	#16810557	1	2	yes	—
304	C₂²	C₂×#13133152	1	2	yes	—
305	C₂²	#16810671	1	2	yes	—
306	C₄	—	2	4	no	—
307	C₄	—	4	2	no	—
308	C₄	#16810497	1	2	yes	—
309	C₄	#16810520	1	2	yes	—
310	C₄	#16810532	1	1	yes	characteristic
311	C₄	#16810543	1	1	yes	characteristic
312	C₄	#16810553	1	1	yes	characteristic
313	C₄	#16810559	1	1	yes	characteristic
314	C₂	—	2	2	no	—
315	C₂	—	2	2	no	—
316	C₂	#20272979	1	1	yes	characteristic
317	C₂	#20274886	1	1	yes	characteristic
318	C₂	#20274889	1	1	yes	characteristic
319	C₂	#20274919	1	2	yes	—
320	C₂	#20275490	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₅₃
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₅₃×#1728	C₂
socle	C₂³×C₅₃	C₂×C₄⋊C₄
prime radical	C₂⁴×C₅₃	C₂×Q₈
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	#1728	D₅₃
hypocenter	C₅₃	#11326
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ #1728

Complements

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

N	H ≅ G/N	type
110. C₂×C₄×C₅₃	#108	semidirect
180. C₅₃	#11326	semidirect
271. C₂×C₄	#13133152	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13133152	2
C₂×#16810647	2
C₂×#16810387	2
C₂×#16810378	2
C₅₃×#1728	2
#11326	53