Subgroups — #27353840

Order 13568.

682 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283676	C₂	1	2	yes	maximal, complemented
2	C₄×#13133239	C₂	1	1	yes	maximal, characteristic
3	#20273611	C₂	1	2	yes	maximal, complemented
4	#20273604	C₂	1	4	yes	maximal, complemented
5	#20273599	C₂	1	2	yes	maximal, complemented
6	#20273095	C₂	1	2	yes	maximal, complemented
7	C₂×C₈×#88322	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#1564	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#13133239	C₄	1	1	yes	characteristic
10	C₄×#11719945	C₂²	1	2	yes	—
11	#16810750	C₂²	1	4	yes	—
12	C₄×Dic₂₁₂	C₄	1	1	yes	characteristic
13	C₄×Dic₂₁₂	C₂²	1	1	yes	characteristic
14	C₄×#11719913	C₄	1	2	yes	—
15	C₄×#11719913	C₂²	1	2	yes	—
16	C₄×D₄₂₄	C₄	1	1	yes	characteristic
17	C₄×D₄₂₄	C₂²	1	1	yes	characteristic
18	#16810700	C₂²	1	4	yes	—
19	#16810691	—	2	4	no	—
20	#16810674	—	2	2	no	—
21	#16810661	C₄	1	1	yes	characteristic
22	#16810620	C₄	1	2	yes	—
23	C₄×#11719912	—	2	2	no	—
24	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
25	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
26	#16810353	C₂²	1	4	yes	—
27	#16810350	C₂²	1	4	yes	—
28	C₈×#88322	C₂²	1	2	yes	—
29	C₈×#88322	C₂²	1	2	yes	—
30	C₅₃×#431	C₂²	1	2	yes	—
31	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
32	C₅₃×#335	C₂²	1	2	yes	—
33	C₅₃×#334	C₂²	1	2	yes	—
34	C₂×#11719945	C₂×C₄	1	2	yes	—
35	#13133239	—	2	4	no	—
36	C₂×Dic₂₁₂	C₂×C₄	1	1	yes	characteristic
37	C₂×#11719913	C₂×C₄	1	2	yes	—
38	C₂×D₄₂₄	C₂×C₄	1	1	yes	characteristic
39	C₂×#11719912	—	2	4	no	—
40	C₂²×#88322	—	2	1	no	—
41	C₂²×#88322	D₄	1	2	yes	—
42	C₄×Dic₁₀₆	C₂×C₄	1	2	yes	—
43	C₄×Dic₁₀₆	C₂³	1	2	yes	—
44	C₄×D₂₁₂	C₂×C₄	1	2	yes	—
45	C₄×D₂₁₂	C₂³	1	2	yes	—
46	C₄×C₂²⋊D₅₃	—	2	2	no	—
47	C₄×C₂²⋊D₅₃	—	2	2	no	—
48	#13133176	C₂×C₄	1	2	yes	—
49	#13133174	—	2	2	no	—
50	C₄²×D₅₃	—	2	2	no	—
51	#13133151	C₂×C₄	1	1	yes	characteristic
52	#13133151	C₂×C₄	1	1	yes	characteristic
53	#13133149	C₂×C₄	1	2	yes	—
54	#13133146	—	2	4	no	—
55	#13133142	C₂×C₄	1	2	yes	—
56	#13133142	C₂×C₄	1	2	yes	—
57	#13133139	C₂×C₄	1	2	yes	—
58	#13133139	C₂×C₄	1	2	yes	—
59	#13133129	—	2	4	no	—
60	#13133127	—	2	4	no	—
61	C₄×#88322	—	2	1	no	—
62	C₄×#88322	D₄	1	1	yes	characteristic
63	C₄×#88322	D₄	1	1	yes	characteristic
64	C₄×#88322	C₂³	1	2	yes	—
65	C₄×#88322	C₂³	1	2	yes	—
66	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
67	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
68	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
69	C₄⋊C₈×C₅₃	C₂³	1	4	yes	—
70	C₂²⋊C₈×C₅₃	—	2	2	no	—
71	C₄×C₈×C₅₃	—	2	2	no	—
72	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
73	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
74	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
75	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
76	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
79	C₂×C₂²⋊D₅₃	—	2	2	no	—
80	#11719945	—	2	4	no	—
81	#11719945	—	2	4	no	—
82	C₂×D₂₁₂	C₂²×C₄	1	2	yes	—
83	C₂×C₄×D₅₃	—	2	2	no	—
84	C₂×C₄×D₅₃	—	2	2	no	—
85	C₂×C₄×D₅₃	—	2	2	no	—
86	C₂×Dic₁₀₆	C₂²×C₄	1	2	yes	—
87	C₂×C₈×C₅₃	—	2	2	no	—
88	C₂×C₈×C₅₃	—	2	4	no	—
89	C₂×C₈×C₅₃	M₁₆	1	4	yes	complemented
90	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
94	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
95	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
96	C₄²×C₅₃	#41	1	1	yes	characteristic
97	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
98	C₂²⋊Dic₅₃	—	2	2	no	—
99	#11719921	—	2	2	no	—
100	#11719921	—	2	2	no	—
101	C₄⋊Dic₅₃	C₂²×C₄	1	2	yes	—
102	C₄⋊Dic₅₃	C₂²×C₄	1	2	yes	—
103	#11719919	—	2	2	no	—
104	#11719919	—	2	2	no	—
105	C₄×Dic₅₃	—	2	2	no	—
106	C₄×Dic₅₃	—	2	2	no	—
107	C₄×Dic₅₃	—	2	2	no	—
108	C₂×#88322	—	2	4	no	—
109	C₂×#88322	—	2	2	no	—
110	C₂×#88322	—	2	4	no	—
111	C₂×#88322	C₂×D₄	1	2	yes	—
112	C₂×#88322	C₂×D₄	1	2	yes	—
113	Dic₂₁₂	—	2	2	no	—
114	D₄₂₄	—	2	2	no	—
115	#11719913	—	2	4	no	—
116	#11719912	—	4	8	no	—
117	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
118	C₂²×D₅₃	—	2	2	no	—
119	C₂×C₄×C₅₃	—	2	2	no	—
120	C₂×C₄×C₅₃	—	2	2	no	—
121	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
122	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
123	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
124	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
125	C₂×C₄×C₅₃	#131	1	2	yes	—
126	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
127	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
128	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
129	C₂²⋊D₅₃	—	4	4	no	—
130	C₂×Dic₅₃	—	2	2	no	—
131	C₂×Dic₅₃	—	2	2	no	—
132	C₂×Dic₅₃	—	2	2	no	—
133	D₂₁₂	—	2	2	no	—
134	D₂₁₂	—	2	2	no	—
135	C₄×D₅₃	—	4	4	no	—
136	C₄×D₅₃	—	4	4	no	—
137	C₄×D₅₃	—	4	4	no	—
138	Dic₁₀₆	—	2	2	no	—
139	Dic₁₀₆	—	2	2	no	—
140	C₈×C₅₃	—	2	1	no	—
141	C₈×C₅₃	—	2	1	no	—
142	C₈×C₅₃	—	4	4	no	—
143	C₈×C₅₃	C₂×M₁₆	1	2	yes	—
144	C₈×C₅₃	C₂×M₁₆	1	2	yes	—
145	#88322	—	2	4	no	—
146	#88322	—	2	2	no	—
147	#88322	#135	1	4	yes	complemented
148	#17886	—	53	1	no	maximal, Sylow
149	C₂²×C₅₃	—	2	1	no	—
150	C₂²×C₅₃	—	2	1	no	—
151	C₂²×C₅₃	#436	1	2	yes	—
152	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
153	C₂×D₅₃	—	4	2	no	—
154	C₂×D₅₃	—	4	2	no	—
155	C₂×D₅₃	—	4	2	no	—
156	C₄×C₅₃	—	2	1	no	—
157	C₄×C₅₃	—	2	1	no	—
158	C₄×C₅₃	#436	1	2	yes	—
159	C₄×C₅₃	#443	1	2	yes	—
160	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
161	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
162	C₄×C₅₃	C₂×#135	1	2	yes	—
163	Dic₅₃	—	4	2	no	—
164	Dic₅₃	—	4	2	no	—
165	Dic₅₃	—	4	2	no	—
166	#2965	—	53	2	no	—
167	C₄×#135	—	53	1	no	—
168	#1591	—	53	2	no	—
169	#1590	—	53	4	no	—
170	#1589	—	53	2	no	—
171	#1564	D₅₃	1	1	yes	characteristic, complemented
172	#1476	—	53	2	no	—
173	C₂×C₈²	—	53	1	no	—
174	C₂×C₅₃	—	2	2	no	—
175	C₂×C₅₃	C₂×#436	1	1	yes	characteristic
176	C₂×C₅₃	C₄×#135	1	1	yes	characteristic
177	C₂×C₅₃	C₂×#443	1	1	yes	characteristic
178	D₅₃	—	8	2	no	—
179	C₂×#135	—	53	1	no	—
180	C₄×#41	—	53	2	no	—
181	#446	—	53	4	no	—
182	C₄×Q₁₆	—	53	1	no	—
183	C₄×Q₁₆	—	53	1	no	—
184	C₄×SD₁₆	—	53	2	no	—
185	C₄×SD₁₆	—	53	2	no	—
186	C₄×D₈	—	53	1	no	—
187	C₄×D₈	—	53	1	no	—
188	#436	—	53	4	no	—
189	#435	—	106	4	no	—
190	#431	—	106	2	no	—
191	#431	C₂×D₅₃	1	2	yes	—
192	#427	—	53	1	no	—
193	#416	—	53	2	no	—
194	C₄×M₁₆	—	106	2	no	—
195	C₂×C₄×C₈	—	53	1	no	—
196	C₂×C₄×C₈	—	53	1	no	—
197	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
198	#335	C₂×D₅₃	1	2	yes	—
199	#334	C₂×D₅₃	1	2	yes	—
200	Q₈⋊C₈	—	53	4	no	—
201	D₄⋊C₈	—	53	4	no	—
202	C₈²	—	53	2	no	—
203	C₈²	—	53	2	no	—
204	C₅₃	#17886	1	1	yes	characteristic, complemented, Sylow
205	C₂×#41	—	53	2	no	—
206	#135	—	106	4	no	—
207	C₂×Q₁₆	—	53	1	no	—
208	C₂×SD₁₆	—	53	2	no	—
209	C₂×D₈	—	53	1	no	—
210	C₂×M₁₆	—	106	4	no	—
211	C₂²×C₈	—	53	2	no	—
212	C₂²×C₈	—	106	1	no	—
213	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
214	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
215	C₄×Q₈	—	53	2	no	—
216	C₄×Q₈	—	53	2	no	—
217	C₄×D₄	—	53	2	no	—
218	C₄×D₄	—	53	2	no	—
219	C₄×D₄	—	106	2	no	—
220	C₄×D₄	—	106	2	no	—
221	#117	—	53	2	no	—
222	#117	—	106	2	no	—
223	C₂×C₄²	—	106	2	no	—
224	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
225	#107	—	53	1	no	—
226	#107	—	53	1	no	—
227	#106	—	53	2	no	—
228	C₄⋊C₈	—	106	4	no	—
229	C₄⋊C₈	C₂²×D₅₃	1	4	yes	—
230	#103	—	53	2	no	—
231	#103	—	53	2	no	—
232	#102	—	53	2	no	—
233	#102	—	53	2	no	—
234	C₂²⋊C₈	—	2	2	no	—
235	C₂²⋊C₈	—	106	4	no	—
236	#97	—	106	4	no	—
237	C₄×C₈	—	2	2	no	—
238	C₄×C₈	—	53	1	no	—
239	C₄×C₈	—	53	1	no	—
240	C₄×C₈	—	53	2	no	—
241	C₄×C₈	—	53	2	no	—
242	C₄×C₈	—	106	1	no	—
243	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
244	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
245	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
246	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
247	#41	—	106	4	no	—
248	#41	—	106	4	no	—
249	C₂×Q₈	—	53	2	no	—
250	C₂×D₄	—	53	2	no	—
251	C₂×D₄	—	106	2	no	—
252	C₂²×C₄	—	106	2	no	—
253	C₂²×C₄	—	106	2	no	—
254	C₂²×C₄	—	106	2	no	—
255	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
256	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
257	C₂²×C₄	#11719949	1	1	yes	characteristic
258	Q₁₆	—	106	2	no	—
259	SD₁₆	—	106	4	no	—
260	D₈	—	106	2	no	—
261	M₁₆	—	212	8	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	2	4	no	—
264	C₂×C₈	—	53	2	no	—
265	C₂×C₈	—	53	2	no	—
266	C₂×C₈	—	106	4	no	—
267	C₂×C₈	—	106	4	no	—
268	C₂×C₈	—	106	2	no	—
269	C₂×C₈	#11719912	1	4	yes	complemented
270	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
271	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
272	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
273	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
274	C₄⋊C₄	—	53	2	no	—
275	C₄⋊C₄	—	53	2	no	—
276	C₄⋊C₄	—	106	2	no	—
277	C₄⋊C₄	—	106	2	no	—
278	C₂²⋊C₄	—	106	2	no	—
279	C₂²⋊C₄	—	106	2	no	—
280	C₂²⋊C₄	—	106	2	no	—
281	C₄²	—	106	2	no	—
282	C₄²	—	106	2	no	—
283	C₄²	—	106	2	no	—
284	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
285	C₄²	D₄×D₅₃	1	1	yes	characteristic
286	C₄²	#11719949	1	1	yes	characteristic
287	C₄²	C₂³×D₅₃	1	1	yes	characteristic
288	C₂³	—	106	2	no	—
289	C₂³	#13133179	1	1	yes	characteristic
290	Q₈	—	106	2	no	—
291	Q₈	—	106	2	no	—
292	D₄	—	106	2	no	—
293	D₄	—	106	2	no	—
294	D₄	—	212	4	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	106	2	no	—
298	C₂×C₄	—	106	2	no	—
299	C₂×C₄	—	106	2	no	—
300	C₂×C₄	—	212	4	no	—
301	C₂×C₄	—	212	4	no	—
302	C₂×C₄	—	212	4	no	—
303	C₂×C₄	#13133179	1	1	yes	characteristic
304	C₂×C₄	#13133179	1	1	yes	characteristic
305	C₂×C₄	#13133179	1	1	yes	characteristic
306	C₂×C₄	C₂×#11719912	1	2	yes	—
307	C₂×C₄	#13133225	1	2	yes	—
308	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
309	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
310	C₂×C₄	C₂×#11719949	1	1	yes	characteristic
311	C₈	—	2	1	no	—
312	C₈	—	2	1	no	—
313	C₈	—	4	4	no	—
314	C₈	—	53	4	no	—
315	C₈	—	106	4	no	—
316	C₈	—	106	2	no	—
317	C₈	C₂×#11719912	1	2	yes	—
318	C₈	C₂×#11719912	1	2	yes	—
319	C₂²	—	2	1	no	—
320	C₂²	—	2	1	no	—
321	C₂²	—	212	2	no	—
322	C₂²	—	212	2	no	—
323	C₂²	—	212	2	no	—
324	C₂²	#16810700	1	2	yes	—
325	C₂²	C₂×#13133179	1	1	yes	characteristic
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	212	2	no	—
329	C₄	—	212	2	no	—
330	C₄	—	212	2	no	—
331	C₄	#16810700	1	2	yes	—
332	C₄	#16810739	1	2	yes	—
333	C₄	C₂²×#11719912	1	1	yes	characteristic
334	C₄	C₂×#13133225	1	1	yes	characteristic
335	C₄	#135×D₅₃	1	2	yes	—
336	C₂	—	2	2	no	—
337	C₂	—	424	2	no	—
338	C₂	C₂×#16810700	1	1	yes	characteristic
339	C₂	#20283486	1	1	yes	characteristic
340	C₂	C₂×#16810739	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#1564	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂×#13133239	C₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#1564	D₅₃
hypocenter	C₅₃	#17886
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ #1564

Complements

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

N	H ≅ G/N	type
1. #20283676	C₂	semidirect
3. #20273611	C₂	semidirect
4. #20273604	C₂	semidirect
5. #20273599	C₂	semidirect
6. #20273095	C₂	semidirect
7. C₂×C₈×#88322	C₂	semidirect
8. C₅₃×#1564	C₂	semidirect
89. C₂×C₈×C₅₃	M₁₆	semidirect
147. #88322	#135	semidirect
171. #1564	D₅₃	semidirect
204. C₅₃	#17886	semidirect
269. C₂×C₈	#11719912	semidirect

Maximal subgroups

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

subgroup	index
#20283676	2
C₄×#13133239	2
#20273611	2
#20273604	2
#20273599	2
#20273095	2
C₂×C₈×#88322	2
C₅₃×#1564	2
#17886	53