Subgroups — #27387664

Order 13568.

844 conjugacy classes of subgroups.

The subgroup lattice has 314 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	C₂×#16810671	C₂	1	2	yes	maximal
3	#20283133	C₂	1	1	yes	maximal, characteristic
4	#20275503	C₂	1	8	yes	maximal, complemented
5	C₂×#16810394	C₂	1	2	yes	maximal
6	C₂×C₅₃×#430	C₂	1	1	yes	maximal, characteristic
7	C₂²×#11719917	C₂²	1	2	yes	—
8	C₂²×#11719917	C₂²	1	1	yes	characteristic
9	C₂×#13133176	C₂²	1	1	yes	characteristic
10	#16810672	C₄	1	8	yes	complemented
11	#16810671	—	2	4	no	—
12	#16810671	C₂²	1	8	yes	—
13	C₂×#13133153	C₂²	1	4	yes	—
14	C₂×#13133152	—	2	4	no	—
15	#16810581	—	2	2	no	—
16	#16810578	C₂²	1	4	yes	—
17	M₁₆×Dic₅₃	—	2	2	no	—
18	#16810394	C₂²	1	8	yes	—
19	C₂²×M₁₆×C₅₃	C₂²	1	2	yes	—
20	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
21	C₅₃×#430	—	2	2	no	—
22	C₅₃×#430	—	2	2	no	—
23	C₂×#108×C₅₃	C₂²	1	4	yes	—
24	C₂×#11719917	—	2	1	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	C₂×#11719917	C₂×C₄	1	4	yes	—
31	C₂×#11719917	D₄	1	4	yes	—
32	C₂×#11719917	C₂³	1	4	yes	—
33	C₂×#11719917	C₂³	1	2	yes	—
34	C₂²×#88322	—	2	1	no	—
35	C₂²×#88322	—	2	2	no	—
36	#13133176	—	2	1	no	—
37	#13133176	—	2	1	no	—
38	#13133176	Q₈	1	2	yes	—
39	#13133176	C₂³	1	2	yes	—
40	C₂×C₄⋊Dic₅₃	—	2	1	no	—
41	C₂×C₂²⋊Dic₅₃	Q₈	1	2	yes	—
42	C₂×C₄×Dic₅₃	—	2	1	no	—
43	#13133153	C₂×C₄	1	16	yes	complemented
44	#13133152	—	2	16	no	—
45	#13133127	—	2	4	no	—
46	C₈×Dic₅₃	—	2	4	no	—
47	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
48	C₂×M₁₆×C₅₃	—	2	2	no	—
49	C₂×M₁₆×C₅₃	—	2	2	no	—
50	C₂×M₁₆×C₅₃	—	2	2	no	—
51	C₂×M₁₆×C₅₃	—	2	2	no	—
52	C₂×M₁₆×C₅₃	—	2	2	no	—
53	C₂×M₁₆×C₅₃	C₂×C₄	1	4	yes	—
54	C₂×M₁₆×C₅₃	D₄	1	2	yes	—
55	C₂×M₁₆×C₅₃	D₄	1	2	yes	—
56	C₂×M₁₆×C₅₃	Q₈	1	4	yes	—
57	C₂×M₁₆×C₅₃	C₂³	1	4	yes	—
58	C₂²×C₈×C₅₃	—	2	2	no	—
59	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
60	#108×C₅₃	—	2	8	no	—
61	C₂⁴×C₅₃	C₂×Q₈	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
67	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
68	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
69	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
70	C₂²×C₄×C₅₃	#41	1	2	yes	—
71	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
72	C₂²×Dic₅₃	—	2	2	no	—
73	M₁₆×C₅₃	—	2	4	no	—
74	M₁₆×C₅₃	—	2	4	no	—
75	M₁₆×C₅₃	—	4	2	no	—
76	M₁₆×C₅₃	—	4	2	no	—
77	M₁₆×C₅₃	C₄⋊C₄	1	8	yes	—
78	M₁₆×C₅₃	C₄⋊C₄	1	8	yes	—
79	C₂×C₈×C₅₃	—	2	4	no	—
80	C₂×C₈×C₅₃	—	4	2	no	—
81	C₂×C₈×C₅₃	—	4	2	no	—
82	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
83	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
84	C₂×C₈×C₅₃	C₂×Q₈	1	4	yes	—
85	C₂²⋊Dic₅₃	—	2	2	no	—
86	C₂²⋊Dic₅₃	C₂×Q₈	1	4	yes	—
87	C₄⋊Dic₅₃	—	2	2	no	—
88	C₄⋊Dic₅₃	—	2	2	no	—
89	C₄×Dic₅₃	—	2	2	no	—
90	C₄×Dic₅₃	—	2	2	no	—
91	#11719917	—	2	4	no	—
92	#11719917	—	2	4	no	—
93	#11719917	—	2	4	no	—
94	#11719917	—	2	4	no	—
95	#11719917	—	2	2	no	—
96	#11719917	C₂²×C₄	1	8	yes	—
97	#11719917	C₂×D₄	1	4	yes	—
98	C₂×#88322	—	2	4	no	—
99	C₂×#88322	—	2	4	no	—
100	C₂×#88322	—	2	4	no	—
101	C₂×#88322	—	2	2	no	—
102	C₂×#88322	—	2	4	no	—
103	C₂³×C₅₃	—	8	1	no	—
104	C₂³×C₅₃	C₄×Q₈	1	2	yes	—
105	C₂³×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
106	C₂³×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
107	C₂³×C₅₃	C₄⋊Q₈	1	2	yes	—
108	C₂³×C₅₃	C₂²×Q₈	1	1	yes	characteristic
109	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
110	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
111	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
112	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
113	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
114	C₂×C₄×C₅₃	#121	1	2	yes	—
115	C₂×C₄×C₅₃	#121	1	2	yes	—
116	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
117	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
118	C₂×C₄×C₅₃	#125	1	2	yes	—
119	C₂×C₄×C₅₃	#125	1	2	yes	—
120	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
121	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
122	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
123	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
124	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
125	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
126	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
127	C₂×Dic₅₃	—	2	4	no	—
128	C₂×Dic₅₃	—	8	2	no	—
129	C₈×C₅₃	—	4	4	no	—
130	C₈×C₅₃	C₂×C₄⋊C₄	1	8	yes	—
131	#88322	—	2	8	no	—
132	#88322	—	2	4	no	—
133	#21184	—	53	1	no	maximal, Sylow
134	C₂²×C₅₃	—	4	1	no	—
135	C₂²×C₅₃	—	4	2	no	—
136	C₂²×C₅₃	—	4	1	no	—
137	C₂²×C₅₃	—	4	2	no	—
138	C₂²×C₅₃	—	4	1	no	—
139	C₂²×C₅₃	#389	1	2	yes	—
140	C₂²×C₅₃	#389	1	2	yes	—
141	C₂²×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
142	C₂²×C₅₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
143	C₂²×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
144	C₄×C₅₃	#389	1	2	yes	—
145	C₄×C₅₃	#389	1	2	yes	—
146	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
147	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
148	C₄×C₅₃	C₂×#121	1	1	yes	characteristic
149	C₄×C₅₃	C₂×#125	1	1	yes	characteristic
150	Dic₅₃	—	4	4	no	—
151	C₂×#430	—	53	1	no	—
152	C₂×#430	—	53	2	no	—
153	C₂×#430	D₅₃	1	1	yes	characteristic
154	#2873	—	53	1	no	—
155	#1953	—	53	8	no	—
156	C₂×#344	—	53	2	no	—
157	C₂×C₅₃	—	2	1	no	—
158	C₂×C₅₃	—	2	1	no	—
159	C₂×C₅₃	—	2	1	no	—
160	C₂×C₅₃	—	2	2	no	—
161	C₂×C₅₃	—	2	2	no	—
162	C₂×C₅₃	C₂×#389	1	1	yes	characteristic
163	C₂²×M₁₆	—	53	1	no	—
164	C₂²×M₁₆	—	53	2	no	—
165	C₂²×M₁₆	C₂×D₅₃	1	2	yes	—
166	C₂²×M₁₆	C₂×D₅₃	1	1	yes	characteristic
167	C₂×#117	—	53	1	no	—
168	#430	—	2	2	no	—
169	#430	—	2	2	no	—
170	#430	—	53	8	no	—
171	#430	—	53	8	no	—
172	#430	—	106	4	no	—
173	C₂×#108	—	53	4	no	—
174	C₂×#108	—	106	4	no	—
175	C₂×#108	C₂×D₅₃	1	4	yes	—
176	#405	—	53	4	no	—
177	#405	—	106	2	no	—
178	C₄×M₁₆	—	106	2	no	—
179	#344	—	53	8	no	—
180	C₅₃	#21184	1	1	yes	characteristic, complemented, Sylow
181	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
182	C₂×M₁₆	—	2	2	no	—
183	C₂×M₁₆	—	2	2	no	—
184	C₂×M₁₆	—	2	2	no	—
185	C₂×M₁₆	—	2	2	no	—
186	C₂×M₁₆	—	2	2	no	—
187	C₂×M₁₆	—	53	4	no	—
188	C₂×M₁₆	—	53	4	no	—
189	C₂×M₁₆	—	53	4	no	—
190	C₂×M₁₆	—	53	4	no	—
191	C₂×M₁₆	—	53	2	no	—
192	C₂×M₁₆	—	106	2	no	—
193	C₂×M₁₆	—	106	2	no	—
194	C₂×M₁₆	—	106	2	no	—
195	C₂×M₁₆	—	106	2	no	—
196	C₂×M₁₆	—	106	1	no	—
197	C₂×M₁₆	Dic₁₀₆	1	4	yes	—
198	C₂×M₁₆	C₄×D₅₃	1	4	yes	—
199	C₂×M₁₆	C₂²⋊D₅₃	1	2	yes	—
200	C₂×M₁₆	C₂²⋊D₅₃	1	2	yes	—
201	C₂×M₁₆	C₂²×D₅₃	1	4	yes	—
202	C₂²×C₈	—	2	2	no	—
203	C₂²×C₈	—	106	2	no	—
204	C₂²×C₈	—	106	1	no	—
205	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
206	#117	—	53	2	no	—
207	#117	—	53	2	no	—
208	#117	—	106	1	no	—
209	#117	—	106	1	no	—
210	C₂×C₄⋊C₄	—	106	1	no	—
211	C₂×C₂²⋊C₄	—	53	2	no	—
212	C₂×C₄²	—	106	1	no	—
213	#108	—	2	8	no	—
214	#108	—	53	16	no	—
215	#108	—	106	16	no	—
216	#97	—	106	4	no	—
217	C₄×C₈	—	106	4	no	—
218	C₂⁴	Q₈×D₅₃	1	1	yes	characteristic
219	C₂²×C₄	—	106	2	no	—
220	C₂²×C₄	C₂×Dic₁₀₆	1	2	yes	—
221	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
222	C₂²×C₄	D₄×D₅₃	1	2	yes	—
223	C₂²×C₄	#11719947	1	2	yes	—
224	C₂²×C₄	Q₈×D₅₃	1	1	yes	characteristic
225	C₂²×C₄	#11719949	1	1	yes	characteristic
226	C₂²×C₄	#11719949	1	1	yes	characteristic
227	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
228	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
229	C₂²×C₄	C₂³×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	—	53	8	no	—
235	M₁₆	—	53	4	no	—
236	M₁₆	—	106	4	no	—
237	M₁₆	—	106	4	no	—
238	M₁₆	—	106	4	no	—
239	M₁₆	—	106	2	no	—
240	M₁₆	—	106	4	no	—
241	M₁₆	#11719919	1	8	yes	—
242	M₁₆	#11719919	1	8	yes	—
243	C₂×C₈	—	2	4	no	—
244	C₂×C₈	—	4	2	no	—
245	C₂×C₈	—	4	2	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	2	no	—
251	C₂×C₈	C₂×Dic₁₀₆	1	4	yes	—
252	C₂×C₈	C₂×C₄×D₅₃	1	4	yes	—
253	C₂×C₈	C₂×C₂²⋊D₅₃	1	4	yes	—
254	C₄⋊C₄	—	106	2	no	—
255	C₄⋊C₄	—	106	2	no	—
256	C₂²⋊C₄	—	53	4	no	—
257	C₂²⋊C₄	—	106	2	no	—
258	C₄²	—	106	2	no	—
259	C₄²	—	106	2	no	—
260	C₂³	—	8	1	no	—
261	C₂³	#13133182	1	2	yes	—
262	C₂³	#13133193	1	1	yes	characteristic
263	C₂³	#13133193	1	1	yes	characteristic
264	C₂³	#13133216	1	2	yes	—
265	C₂³	C₂×Q₈×D₅₃	1	1	yes	characteristic
266	C₂×C₄	—	106	4	no	—
267	C₂×C₄	—	424	2	no	—
268	C₂×C₄	C₂×#11719919	1	2	yes	—
269	C₂×C₄	C₂×#11719919	1	2	yes	—
270	C₂×C₄	#13133179	1	2	yes	—
271	C₂×C₄	#13133179	1	2	yes	—
272	C₂×C₄	#13133182	1	2	yes	—
273	C₂×C₄	#13133188	1	2	yes	—
274	C₂×C₄	#13133188	1	2	yes	—
275	C₂×C₄	#13133193	1	1	yes	characteristic
276	C₂×C₄	#13133193	1	1	yes	characteristic
277	C₂×C₄	#13133208	1	2	yes	—
278	C₂×C₄	#13133208	1	2	yes	—
279	C₂×C₄	#13133216	1	2	yes	—
280	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
281	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
282	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
283	C₂×C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
284	C₂×C₄	C₂×#11719947	1	1	yes	characteristic
285	C₂×C₄	C₂×#11719949	1	1	yes	characteristic
286	C₈	—	4	4	no	—
287	C₈	—	106	8	no	—
288	C₈	—	106	4	no	—
289	C₈	C₂×#11719919	1	8	yes	—
290	C₂²	—	4	1	no	—
291	C₂²	—	4	2	no	—
292	C₂²	—	4	1	no	—
293	C₂²	—	4	2	no	—
294	C₂²	—	4	1	no	—
295	C₂²	#16810519	1	2	yes	—
296	C₂²	#16810519	1	2	yes	—
297	C₂²	C₂×#13133182	1	1	yes	characteristic
298	C₂²	C₂×#13133193	1	1	yes	characteristic
299	C₂²	C₂×#13133216	1	1	yes	characteristic
300	C₄	—	212	4	no	—
301	C₄	#16810519	1	2	yes	—
302	C₄	#16810519	1	2	yes	—
303	C₄	C₂²×#11719919	1	1	yes	characteristic
304	C₄	C₂×#13133179	1	1	yes	characteristic
305	C₄	C₂×#13133188	1	1	yes	characteristic
306	C₄	C₂×#13133208	1	1	yes	characteristic
307	C₂	—	2	1	no	—
308	C₂	—	2	2	no	—
309	C₂	—	2	2	no	—
310	C₂	—	2	1	no	—
311	C₂	—	2	1	no	—
312	C₂	C₂×#16810519	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×#13133188
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂×C₅₃×#430	C₂
socle	C₂×C₅₃	C₂×#389
prime radical	C₂⁴×C₅₃	C₂×Q₈
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₂×#430	D₅₃
hypocenter	C₅₃	#21184
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#430

Complements

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

N	H ≅ G/N	type
4. #20275503	C₂	semidirect
10. #16810672	C₄	semidirect
43. #13133153	C₂×C₄	semidirect
180. C₅₃	#21184	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
C₂×#16810671	2
#20283133	2
#20275503	2
C₂×#16810394	2
C₂×C₅₃×#430	2
#21184	53