Subgroups — #27541827

Order 13568.

518 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20287607	C₂	1	1	yes	maximal, characteristic, complemented
2	#20276956	C₂	1	2	yes	maximal, complemented
3	#20276938	C₂	1	1	yes	maximal, characteristic, complemented
4	#20276872	C₂	1	2	yes	maximal, complemented
5	#20276852	C₂	1	1	yes	maximal, characteristic, complemented
6	#20276752	C₂	1	1	yes	maximal, characteristic, complemented
7	#20276752	C₂	1	2	yes	maximal, complemented
8	#20276624	C₂	1	1	yes	maximal, characteristic, complemented
9	#20276579	C₂	1	2	yes	maximal, complemented
10	C₄×#13133161	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#3391	C₂	1	1	yes	maximal, characteristic
12	#16811351	—	2	1	no	—
13	C₂×#13133161	C₂²	1	1	yes	characteristic
14	C₂×#13133161	C₂²	1	2	yes	—
15	#16811028	C₂²	1	1	yes	characteristic
16	#16811025	C₂²	1	2	yes	—
17	#16810931	—	2	2	no	—
18	#16810909	—	2	1	no	—
19	C₄×Dic₂₁₂	C₂²	1	1	yes	characteristic
20	C₄×Dic₂₁₂	C₂²	1	2	yes	—
21	#16810450	C₂²	1	1	yes	characteristic
22	#16810448	C₂²	1	2	yes	—
23	C₄⋊#11719908	C₂²	1	1	yes	characteristic
24	C₄⋊#11719908	C₂²	1	2	yes	—
25	#16810426	C₂²	1	1	yes	characteristic
26	#16810424	C₂²	1	2	yes	—
27	#16810424	C₂²	1	2	yes	—
28	#16810424	C₂²	1	2	yes	—
29	#16810424	C₂²	1	1	yes	characteristic
30	#16810424	C₂²	1	2	yes	—
31	#16810399	C₂²	1	2	yes	—
32	C₄×#11719908	C₂²	1	1	yes	characteristic
33	C₅₃×#550	—	2	1	no	—
34	C₅₃×#493	C₂²	1	1	yes	characteristic
35	C₅₃×#493	C₂²	1	2	yes	—
36	C₅₃×#459	—	2	1	no	—
37	C₅₃×#459	—	2	2	no	—
38	C₈×Q₈×C₅₃	C₂²	1	1	yes	characteristic
39	C₄×D₈×C₅₃	C₂²	1	1	yes	characteristic
40	C₄×D₈×C₅₃	C₂²	1	2	yes	—
41	C₂×Dic₂₁₂	C₂³	1	1	yes	characteristic
42	#13133217	—	2	1	no	—
43	#13133215	—	4	1	no	—
44	#13133209	—	2	2	no	—
45	#13133207	—	4	2	no	—
46	C₄×Dic₁₀₆	—	2	1	no	—
47	C₄×Dic₁₀₆	—	2	2	no	—
48	#13133182	—	4	1	no	—
49	#13133161	—	2	2	no	—
50	#13133161	—	2	4	no	—
51	C₂×#11719908	C₂³	1	1	yes	characteristic
52	C₂×#11719908	C₂³	1	2	yes	—
53	C₂×#11719908	C₂³	1	1	yes	characteristic
54	#13133151	—	2	1	no	—
55	#13133151	C₂³	1	1	yes	characteristic
56	#13133151	C₂³	1	2	yes	—
57	#13133149	—	2	2	no	—
58	#13133142	—	2	2	no	—
59	#13133142	—	2	1	no	—
60	C₂×#41×C₅₃	—	4	1	no	—
61	C₂×D₈×C₅₃	—	2	2	no	—
62	C₂×D₈×C₅₃	—	2	1	no	—
63	C₂×D₈×C₅₃	C₂³	1	2	yes	—
64	C₂×D₈×C₅₃	C₂³	1	1	yes	characteristic
65	#127×C₅₃	—	2	2	no	—
66	#127×C₅₃	—	2	1	no	—
67	#121×C₅₃	—	4	1	no	—
68	#121×C₅₃	—	4	2	no	—
69	C₄×Q₈×C₅₃	D₄	1	1	yes	characteristic
70	C₄×D₄×C₅₃	—	2	2	no	—
71	C₄×D₄×C₅₃	—	2	1	no	—
72	#107×C₅₃	C₂³	1	1	yes	characteristic
73	C₄⋊C₈×C₅₃	D₄	1	1	yes	characteristic
74	C₄⋊C₈×C₅₃	D₄	1	2	yes	—
75	#102×C₅₃	—	2	2	no	—
76	#102×C₅₃	—	2	1	no	—
77	C₄×C₈×C₅₃	C₂³	1	2	yes	—
78	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
79	#41×C₅₃	—	8	4	no	—
80	C₂×Q₈×C₅₃	D₈	1	1	yes	characteristic
81	C₂×D₄×C₅₃	—	2	2	no	—
82	C₂×D₄×C₅₃	—	2	1	no	—
83	C₂×D₄×C₅₃	—	4	2	no	—
84	C₂×D₄×C₅₃	—	4	1	no	—
85	C₂×D₄×C₅₃	—	4	2	no	—
86	C₂×D₄×C₅₃	—	4	1	no	—
87	C₂²×C₄×C₅₃	—	4	2	no	—
88	C₂²×C₄×C₅₃	—	4	1	no	—
89	C₂×Dic₁₀₆	—	2	1	no	—
90	C₂×Dic₁₀₆	—	4	1	no	—
91	D₈×C₅₃	—	2	2	no	—
92	D₈×C₅₃	—	2	1	no	—
93	D₈×C₅₃	—	2	2	no	—
94	D₈×C₅₃	—	2	1	no	—
95	D₈×C₅₃	—	8	2	no	—
96	D₈×C₅₃	—	8	1	no	—
97	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
98	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
100	C₄⋊C₄×C₅₃	—	2	1	no	—
101	C₄⋊C₄×C₅₃	D₈	1	1	yes	characteristic
102	C₄⋊C₄×C₅₃	D₈	1	2	yes	—
103	C₂²⋊C₄×C₅₃	—	4	2	no	—
104	C₂²⋊C₄×C₅₃	—	4	1	no	—
105	C₄²×C₅₃	C₂×D₄	1	2	yes	—
106	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
107	C₄⋊Dic₅₃	—	2	1	no	—
108	C₄⋊Dic₅₃	—	2	2	no	—
109	C₄⋊Dic₅₃	—	4	1	no	—
110	#11719919	—	4	2	no	—
111	#11719919	—	4	1	no	—
112	#11719919	—	4	2	no	—
113	#11719919	—	4	2	no	—
114	C₄×Dic₅₃	—	4	2	no	—
115	C₄×Dic₅₃	—	4	1	no	—
116	Dic₂₁₂	—	2	1	no	—
117	Dic₂₁₂	#41	1	2	yes	complemented
118	#11719908	—	2	1	no	—
119	#11719908	—	2	2	no	—
120	#11719908	—	2	1	no	—
121	C₂³×C₅₃	—	4	2	no	—
122	C₂³×C₅₃	—	4	1	no	—
123	Q₈×C₅₃	—	2	2	no	—
124	D₄×C₅₃	—	4	2	no	—
125	D₄×C₅₃	—	4	1	no	—
126	D₄×C₅₃	—	4	2	no	—
127	D₄×C₅₃	—	4	1	no	—
128	D₄×C₅₃	—	8	4	no	—
129	D₄×C₅₃	—	8	2	no	—
130	D₄×C₅₃	—	16	2	no	—
131	D₄×C₅₃	—	16	1	no	—
132	C₂×C₄×C₅₃	—	4	1	no	—
133	C₂×C₄×C₅₃	—	8	4	no	—
134	C₂×C₄×C₅₃	—	8	2	no	—
135	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
136	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
137	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
138	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
139	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
140	C₂×Dic₅₃	—	4	1	no	—
141	C₂×Dic₅₃	—	4	2	no	—
142	C₂×Dic₅₃	—	4	1	no	—
143	Dic₁₀₆	—	2	2	no	—
144	Dic₁₀₆	—	4	1	no	—
145	Dic₁₀₆	—	8	2	no	—
146	C₈×C₅₃	—	2	2	no	—
147	C₈×C₅₃	—	2	1	no	—
148	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
149	C₈×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
150	#33935	—	53	1	no	maximal, Sylow
151	C₂²×C₅₃	—	8	2	no	—
152	C₂²×C₅₃	—	8	1	no	—
153	C₂²×C₅₃	—	8	2	no	—
154	C₂²×C₅₃	—	8	1	no	—
155	C₂²×C₅₃	—	8	2	no	—
156	C₂²×C₅₃	—	8	1	no	—
157	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
158	C₄×C₅₃	—	2	2	no	—
159	C₄×C₅₃	—	2	2	no	—
160	C₄×C₅₃	—	2	1	no	—
161	C₄×C₅₃	—	8	1	no	—
162	C₄×C₅₃	#509	1	2	yes	—
163	C₄×C₅₃	#548	1	1	yes	characteristic
164	C₄×C₅₃	#577	1	1	yes	characteristic
165	Dic₅₃	—	4	2	no	—
166	Dic₅₃	—	8	2	no	—
167	Dic₅₃	—	8	1	no	—
168	#3391	D₅₃	1	1	yes	characteristic
169	#3383	—	53	1	no	—
170	#2244	—	53	2	no	—
171	#2240	—	53	1	no	—
172	#2228	—	53	2	no	—
173	#2224	—	53	1	no	—
174	#2207	—	53	2	no	—
175	#2207	—	53	1	no	—
176	#2183	—	53	1	no	—
177	#2175	—	53	2	no	—
178	C₄×SD₃₂	—	53	1	no	—
179	C₂×C₅₃	—	16	2	no	—
180	C₂×C₅₃	—	16	1	no	—
181	C₂×C₅₃	#3358	1	1	yes	characteristic
182	C₂×C₅₃	C₂×#509	1	1	yes	characteristic
183	C₂×C₅₃	#3416	1	1	yes	characteristic
184	#557	—	106	1	no	—
185	#550	—	2	1	no	—
186	C₂×SD₃₂	—	53	1	no	—
187	C₂×SD₃₂	—	53	2	no	—
188	#500	—	53	1	no	—
189	#499	—	53	2	no	—
190	#493	C₂×D₅₃	1	1	yes	characteristic
191	#493	C₂×D₅₃	1	2	yes	—
192	#479	—	106	2	no	—
193	#475	—	106	1	no	—
194	#459	—	2	1	no	—
195	#459	—	2	2	no	—
196	C₈×Q₈	C₂×D₅₃	1	1	yes	characteristic
197	C₄×Q₁₆	—	53	1	no	—
198	C₄×Q₁₆	—	53	2	no	—
199	C₄×D₈	C₂×D₅₃	1	1	yes	characteristic
200	C₄×D₈	C₂×D₅₃	1	2	yes	—
201	#367	—	53	1	no	—
202	#366	—	53	2	no	—
203	C₄⋊C₁₆	—	53	1	no	—
204	C₄⋊C₁₆	—	53	2	no	—
205	#358	—	53	1	no	—
206	#357	—	53	1	no	—
207	#357	—	53	2	no	—
208	#357	—	53	2	no	—
209	#357	—	53	2	no	—
210	#357	—	53	2	no	—
211	#346	—	53	2	no	—
212	C₄×C₁₆	—	53	1	no	—
213	C₅₃	#33935	1	1	yes	characteristic, complemented, Sylow
214	C₂×#41	—	4	1	no	—
215	C₂×Q₁₆	—	53	1	no	—
216	C₂×D₈	—	2	2	no	—
217	C₂×D₈	—	2	1	no	—
218	C₂×D₈	C₂²×D₅₃	1	1	yes	characteristic
219	C₂×D₈	C₂²×D₅₃	1	2	yes	—
220	C₄⋊Q₈	—	106	1	no	—
221	C₄⋊Q₈	—	212	1	no	—
222	#127	—	2	2	no	—
223	#127	—	2	1	no	—
224	#125	—	106	2	no	—
225	#125	—	212	2	no	—
226	#121	—	4	1	no	—
227	#121	—	4	2	no	—
228	C₄×Q₈	—	106	1	no	—
229	C₄×Q₈	—	106	2	no	—
230	C₄×Q₈	—	212	1	no	—
231	C₄×Q₈	C₂²⋊D₅₃	1	1	yes	characteristic
232	C₄×D₄	—	2	2	no	—
233	C₄×D₄	—	2	1	no	—
234	SD₃₂	—	106	2	no	—
235	SD₃₂	—	106	4	no	—
236	C₂×C₁₆	—	53	1	no	—
237	C₂×C₁₆	—	53	1	no	—
238	C₂×C₁₆	—	53	2	no	—
239	#107	—	53	2	no	—
240	#107	—	53	1	no	—
241	#107	—	106	1	no	—
242	#107	C₂²×D₅₃	1	1	yes	characteristic
243	#106	—	106	2	no	—
244	C₄⋊C₈	C₂²⋊D₅₃	1	1	yes	characteristic
245	C₄⋊C₈	C₂²⋊D₅₃	1	2	yes	—
246	#103	—	106	2	no	—
247	#103	—	106	1	no	—
248	#102	—	2	2	no	—
249	#102	—	2	1	no	—
250	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
251	C₄×C₈	C₂²×D₅₃	1	2	yes	—
252	#41	—	8	4	no	—
253	C₂×Q₈	—	106	1	no	—
254	C₂×Q₈	—	212	1	no	—
255	C₂×Q₈	#11719922	1	1	yes	characteristic
256	C₂×D₄	—	2	2	no	—
257	C₂×D₄	—	2	1	no	—
258	C₂×D₄	—	4	2	no	—
259	C₂×D₄	—	4	1	no	—
260	C₂×D₄	—	4	2	no	—
261	C₂×D₄	—	4	1	no	—
262	C₂²×C₄	—	4	2	no	—
263	C₂²×C₄	—	4	1	no	—
264	Q₁₆	—	53	2	no	—
265	Q₁₆	—	106	1	no	—
266	D₈	—	2	1	no	—
267	D₈	—	2	2	no	—
268	D₈	—	2	1	no	—
269	D₈	—	2	2	no	—
270	D₈	—	8	2	no	—
271	D₈	—	8	1	no	—
272	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
273	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
274	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
275	C₄⋊C₄	—	2	1	no	—
276	C₄⋊C₄	—	106	1	no	—
277	C₄⋊C₄	—	106	2	no	—
278	C₄⋊C₄	—	212	1	no	—
279	C₄⋊C₄	—	212	2	no	—
280	C₄⋊C₄	—	212	2	no	—
281	C₄⋊C₄	—	212	2	no	—
282	C₄⋊C₄	—	212	1	no	—
283	C₄⋊C₄	#11719922	1	1	yes	characteristic
284	C₄⋊C₄	#11719922	1	2	yes	—
285	C₂²⋊C₄	—	4	2	no	—
286	C₂²⋊C₄	—	4	1	no	—
287	C₄²	—	212	2	no	—
288	C₄²	—	212	1	no	—
289	C₄²	C₂×C₂²⋊D₅₃	1	2	yes	—
290	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
291	C₁₆	—	106	1	no	—
292	C₁₆	—	106	2	no	—
293	C₁₆	—	106	1	no	—
294	C₂³	—	4	2	no	—
295	C₂³	—	4	1	no	—
296	Q₈	—	2	2	no	—
297	Q₈	—	106	2	no	—
298	Q₈	—	424	2	no	—
299	Q₈	—	212	1	no	—
300	D₄	—	4	1	no	—
301	D₄	—	4	2	no	—
302	D₄	—	4	1	no	—
303	D₄	—	4	2	no	—
304	D₄	—	8	4	no	—
305	D₄	—	8	2	no	—
306	D₄	—	16	2	no	—
307	D₄	—	16	1	no	—
308	C₂×C₄	—	4	1	no	—
309	C₂×C₄	—	8	4	no	—
310	C₂×C₄	—	8	2	no	—
311	C₂×C₄	—	212	1	no	—
312	C₂×C₄	—	212	2	no	—
313	C₂×C₄	—	212	1	no	—
314	C₂×C₄	C₂×#11719922	1	2	yes	—
315	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
316	C₂×C₄	C₂×#11719922	1	2	yes	—
317	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
318	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
319	C₈	—	2	2	no	—
320	C₈	—	2	1	no	—
321	C₈	#41×D₅₃	1	1	yes	characteristic
322	C₈	#13133264	1	1	yes	characteristic
323	C₂²	—	8	2	no	—
324	C₂²	—	8	1	no	—
325	C₂²	—	8	2	no	—
326	C₂²	—	8	1	no	—
327	C₂²	—	8	2	no	—
328	C₂²	—	8	1	no	—
329	C₂²	C₂²×#11719922	1	1	yes	characteristic
330	C₄	—	2	2	no	—
331	C₄	—	2	2	no	—
332	C₄	—	2	1	no	—
333	C₄	—	8	1	no	—
334	C₄	—	424	2	no	—
335	C₄	—	424	1	no	—
336	C₄	—	212	2	no	—
337	C₄	#16811059	1	2	yes	—
338	C₄	#16811280	1	1	yes	characteristic
339	C₄	#16811434	1	1	yes	characteristic
340	C₂	—	16	2	no	—
341	C₂	—	16	1	no	—
342	C₂	#20287346	1	1	yes	characteristic
343	C₂	C₂×#16811059	1	1	yes	characteristic
344	C₂	#20287862	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#11719922
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting, prime radical	C₅₃×#3391	C₂
socle	C₂²×C₅₃	C₂²×D₈
hypercenter	#3391	D₅₃
hypocenter	C₅₃	#33935
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ #3391

Complements

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

N	H ≅ G/N	type
1. #20287607	C₂	semidirect
2. #20276956	C₂	semidirect
3. #20276938	C₂	semidirect
4. #20276872	C₂	semidirect
5. #20276852	C₂	semidirect
6. #20276752	C₂	semidirect
7. #20276752	C₂	semidirect
8. #20276624	C₂	semidirect
9. #20276579	C₂	semidirect
10. C₄×#13133161	C₂	semidirect
117. Dic₂₁₂	#41	semidirect
213. C₅₃	#33935	semidirect

Maximal subgroups

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

subgroup	index
#20287607	2
#20276956	2
#20276938	2
#20276872	2
#20276852	2
#20276752	2
#20276624	2
#20276579	2
C₄×#13133161	2
C₅₃×#3391	2
#33935	53