Subgroups — #27541699

Order 13568.

514 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20287139	C₂	1	1	yes	maximal, characteristic, complemented
2	#20287064	C₂	1	1	yes	maximal, characteristic, complemented
3	#20276997	C₂	1	1	yes	maximal, characteristic, complemented
4	#20276905	C₂	1	2	yes	maximal, complemented
5	#20276899	C₂	1	2	yes	maximal, complemented
6	#20276862	C₂	1	1	yes	maximal, characteristic, complemented
7	#20276852	C₂	1	1	yes	maximal, characteristic, complemented
8	#20276815	C₂	1	2	yes	maximal, complemented
9	#20276548	C₂	1	1	yes	maximal, characteristic, complemented
10	#20276500	C₂	1	2	yes	maximal, complemented
11	C₄×D₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	#16811338	—	2	1	no	—
13	#16811278	—	2	1	no	—
14	C₂×#13133156	C₂²	1	2	yes	—
15	#16811028	C₂²	1	1	yes	characteristic
16	#16811028	C₂²	1	1	yes	characteristic
17	#16810945	—	2	2	no	—
18	#16810935	—	2	2	no	—
19	#16810913	—	2	1	no	—
20	#16810903	—	2	1	no	—
21	#16810856	C₂²	1	2	yes	—
22	#16810856	C₂²	1	2	yes	—
23	C₈×D₂₁₂	C₂²	1	1	yes	characteristic
24	#16810660	C₂²	1	2	yes	—
25	#107×D₅₃	C₂²	1	2	yes	—
26	#16810451	C₂²	1	2	yes	—
27	#16810451	C₂²	1	2	yes	—
28	#16810450	C₂²	1	2	yes	—
29	#16810448	C₂²	1	2	yes	—
30	C₄⋊#11719908	C₂²	1	1	yes	characteristic
31	#16810424	C₂²	1	2	yes	—
32	#16810424	C₂²	1	2	yes	—
33	#16810402	C₂²	1	2	yes	—
34	C₂×D₁₆×C₅₃	C₂²	1	1	yes	characteristic
35	C₄×D₈×C₅₃	C₂²	1	1	yes	characteristic
36	C₄×D₈×C₅₃	C₂²	1	1	yes	characteristic
37	C₅₃×#366	C₂²	1	1	yes	characteristic
38	C₅₃×#357	C₂²	1	1	yes	characteristic
39	C₅₃×#357	C₂²	1	1	yes	characteristic
40	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
41	C₂×#11719945	—	4	1	no	—
42	C₂×C₈×D₅₃	C₂³	1	2	yes	—
43	#13133217	—	2	1	no	—
44	#13133217	—	2	1	no	—
45	#13133203	—	4	2	no	—
46	#13133199	—	4	2	no	—
47	#13133196	—	4	1	no	—
48	#13133191	—	4	1	no	—
49	#13133188	—	2	2	no	—
50	#13133188	—	2	2	no	—
51	C₄×D₂₁₂	D₄	1	1	yes	characteristic
52	#13133176	—	4	1	no	—
53	#13133175	—	2	2	no	—
54	C₄⋊C₄×D₅₃	—	2	2	no	—
55	#13133156	—	2	4	no	—
56	C₂×#11719908	C₂³	1	2	yes	—
57	#13133151	—	2	1	no	—
58	#13133151	—	2	1	no	—
59	#13133151	C₂³	1	2	yes	—
60	#13133151	C₂³	1	2	yes	—
61	#13133150	—	2	2	no	—
62	#13133148	—	2	2	no	—
63	C₄⋊#88322	D₄	1	1	yes	characteristic
64	#13133138	—	2	2	no	—
65	#13133138	—	2	2	no	—
66	#13133129	D₄	1	2	yes	—
67	C₂×D₈×C₅₃	C₂³	1	1	yes	characteristic
68	C₂×D₈×C₅₃	C₂³	1	1	yes	characteristic
69	C₄×D₄×C₅₃	—	2	1	no	—
70	C₄×D₄×C₅₃	—	2	1	no	—
71	D₁₆×C₅₃	—	2	2	no	—
72	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
73	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
74	#107×C₅₃	C₂³	1	1	yes	characteristic
75	#107×C₅₃	C₂³	1	1	yes	characteristic
76	#102×C₅₃	—	2	1	no	—
77	#102×C₅₃	—	2	1	no	—
78	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
79	C₂×D₄×C₅₃	—	2	1	no	—
80	C₂×D₄×C₅₃	—	2	1	no	—
81	C₂²×C₄×C₅₃	—	4	1	no	—
82	C₂²×C₄×C₅₃	—	4	1	no	—
83	C₂×C₂²⋊D₅₃	—	4	2	no	—
84	C₂×C₂²⋊D₅₃	—	4	2	no	—
85	#11719945	—	8	4	no	—
86	C₂×D₂₁₂	D₈	1	1	yes	characteristic
87	C₂×C₄×D₅₃	—	4	2	no	—
88	C₂×C₄×D₅₃	C₂×D₄	1	2	yes	—
89	C₂×Dic₁₀₆	—	4	1	no	—
90	C₂×Dic₁₀₆	—	4	1	no	—
91	D₈×C₅₃	—	2	1	no	—
92	D₈×C₅₃	—	2	1	no	—
93	D₈×C₅₃	—	2	1	no	—
94	D₈×C₅₃	—	2	1	no	—
95	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
97	C₄⋊C₄×C₅₃	—	2	1	no	—
98	C₄⋊C₄×C₅₃	—	2	1	no	—
99	C₂²⋊C₄×C₅₃	—	4	1	no	—
100	C₂²⋊C₄×C₅₃	—	4	1	no	—
101	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
102	C₂²⋊Dic₅₃	—	4	2	no	—
103	C₂²⋊Dic₅₃	—	4	2	no	—
104	#11719921	—	4	2	no	—
105	#11719921	D₈	1	2	yes	—
106	C₄⋊Dic₅₃	—	2	2	no	—
107	C₄⋊Dic₅₃	—	2	2	no	—
108	C₄⋊Dic₅₃	—	4	1	no	—
109	C₄⋊Dic₅₃	—	4	1	no	—
110	C₄⋊Dic₅₃	D₈	1	1	yes	characteristic
111	#11719919	—	4	2	no	—
112	C₄×Dic₅₃	—	4	2	no	—
113	C₂×#88322	C₂×D₄	1	2	yes	—
114	C₈×D₅₃	—	2	2	no	—
115	C₈×D₅₃	—	2	2	no	—
116	C₁₆×C₅₃	—	2	1	no	—
117	C₁₆×C₅₃	#41	1	2	yes	complemented
118	#11719908	—	2	2	no	—
119	C₂³×C₅₃	—	4	1	no	—
120	C₂³×C₅₃	—	4	1	no	—
121	C₂²×D₅₃	C₂×D₈	1	2	yes	—
122	D₄×C₅₃	—	4	1	no	—
123	D₄×C₅₃	—	4	1	no	—
124	D₄×C₅₃	—	4	1	no	—
125	D₄×C₅₃	—	4	1	no	—
126	C₂×C₄×C₅₃	—	4	1	no	—
127	C₂×C₄×C₅₃	—	4	1	no	—
128	C₂×C₄×C₅₃	—	8	2	no	—
129	C₂×C₄×C₅₃	—	8	2	no	—
130	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
131	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
132	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
133	C₂²⋊D₅₃	—	8	4	no	—
134	C₂²⋊D₅₃	—	16	2	no	—
135	C₂×Dic₅₃	—	4	2	no	—
136	C₂×Dic₅₃	—	4	2	no	—
137	C₂×Dic₅₃	C₂×D₈	1	2	yes	—
138	D₂₁₂	—	2	2	no	—
139	C₄×D₅₃	—	2	2	no	—
140	C₄×D₅₃	—	2	2	no	—
141	C₄×D₅₃	—	8	4	no	—
142	Dic₁₀₆	—	8	2	no	—
143	Dic₁₀₆	—	16	1	no	—
144	C₈×C₅₃	—	2	1	no	—
145	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
146	C₈×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
147	#88322	—	2	2	no	—
148	#33921	—	53	1	no	maximal, Sylow
149	C₂²×C₅₃	—	8	1	no	—
150	C₂²×C₅₃	—	8	1	no	—
151	C₂²×C₅₃	—	8	1	no	—
152	C₂²×C₅₃	—	8	1	no	—
153	C₂²×C₅₃	—	8	1	no	—
154	C₂²×C₅₃	—	8	1	no	—
155	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
156	C₂×D₅₃	—	2	2	no	—
157	C₂×D₅₃	—	2	2	no	—
158	C₂×D₅₃	—	2	2	no	—
159	C₄×C₅₃	—	2	1	no	—
160	C₄×C₅₃	—	8	1	no	—
161	C₄×C₅₃	—	8	1	no	—
162	C₄×C₅₃	#509	1	2	yes	—
163	C₄×C₅₃	#547	1	1	yes	characteristic
164	C₄×C₅₃	#578	1	1	yes	characteristic
165	Dic₅₃	—	2	2	no	—
166	Dic₅₃	—	8	2	no	—
167	Dic₅₃	—	8	2	no	—
168	#3341	—	53	1	no	—
169	#3334	—	53	1	no	—
170	#2254	—	53	1	no	—
171	#2234	—	53	2	no	—
172	#2233	—	53	2	no	—
173	#2226	—	53	1	no	—
174	#2224	—	53	1	no	—
175	#2218	—	53	2	no	—
176	C₄×D₁₆	D₅₃	1	1	yes	characteristic, complemented
177	#2169	—	53	1	no	—
178	#2159	—	53	2	no	—
179	C₂×C₅₃	—	16	1	no	—
180	C₂×C₅₃	—	16	1	no	—
181	C₂×C₅₃	#3334	1	1	yes	characteristic
182	C₂×C₅₃	C₂×#509	1	1	yes	characteristic
183	C₂×C₅₃	#3416	1	1	yes	characteristic
184	D₅₃	—	4	2	no	—
185	#555	—	106	1	no	—
186	#547	—	106	1	no	—
187	C₂×D₁₆	C₂×D₅₃	1	1	yes	characteristic
188	C₂×M₃₂	—	53	2	no	—
189	#500	—	53	1	no	—
190	#500	—	53	1	no	—
191	#482	—	106	2	no	—
192	#480	—	106	2	no	—
193	#476	—	106	1	no	—
194	#474	—	106	1	no	—
195	#466	—	53	2	no	—
196	#466	—	53	2	no	—
197	C₄×D₈	C₂×D₅₃	1	1	yes	characteristic
198	C₄×D₈	C₂×D₅₃	1	1	yes	characteristic
199	C₈×D₄	—	53	1	no	—
200	#427	—	53	2	no	—
201	C₂×#107	—	53	2	no	—
202	#367	—	53	2	no	—
203	#367	—	53	2	no	—
204	#367	—	53	2	no	—
205	#366	—	53	2	no	—
206	#366	C₂×D₅₃	1	1	yes	characteristic
207	C₄⋊C₁₆	—	53	1	no	—
208	#357	—	53	2	no	—
209	#357	—	53	2	no	—
210	#357	C₂×D₅₃	1	1	yes	characteristic
211	#357	C₂×D₅₃	1	1	yes	characteristic
212	C₂²⋊C₁₆	—	53	2	no	—
213	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic
214	C₅₃	#33921	1	1	yes	characteristic, complemented, Sylow
215	C₂×#41	—	212	1	no	—
216	C₂×D₈	C₂²×D₅₃	1	1	yes	characteristic
217	C₂×D₈	C₂²×D₅₃	1	1	yes	characteristic
218	C₂²×C₈	—	53	2	no	—
219	C₄⋊Q₈	—	106	1	no	—
220	C₄⋊Q₈	—	106	1	no	—
221	#124	—	212	2	no	—
222	#123	—	212	2	no	—
223	C₂²⋊Q₈	—	212	1	no	—
224	#121	—	106	2	no	—
225	#121	—	106	2	no	—
226	#121	—	212	1	no	—
227	C₄×D₄	—	2	1	no	—
228	C₄×D₄	—	2	1	no	—
229	C₄×D₄	—	53	1	no	—
230	#117	—	106	2	no	—
231	#117	—	212	1	no	—
232	C₂×C₄⋊C₄	—	106	2	no	—
233	D₁₆	—	2	2	no	—
234	M₃₂	—	106	4	no	—
235	C₂×C₁₆	—	53	2	no	—
236	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
237	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
238	#107	—	53	2	no	—
239	#107	—	53	2	no	—
240	#107	—	106	2	no	—
241	#107	—	106	1	no	—
242	#107	—	106	1	no	—
243	#107	C₂²×D₅₃	1	1	yes	characteristic
244	#107	C₂²×D₅₃	1	1	yes	characteristic
245	#106	—	106	2	no	—
246	C₄⋊C₈	—	53	1	no	—
247	#102	—	2	1	no	—
248	#102	—	2	1	no	—
249	#102	—	106	2	no	—
250	#102	—	106	2	no	—
251	C₂²⋊C₈	—	53	2	no	—
252	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
253	#41	—	424	4	no	—
254	C₂×Q₈	—	212	1	no	—
255	C₂×Q₈	—	212	1	no	—
256	C₂×D₄	—	2	1	no	—
257	C₂×D₄	—	2	1	no	—
258	C₂×D₄	—	53	1	no	—
259	C₂×D₄	—	212	2	no	—
260	C₂×D₄	—	212	2	no	—
261	C₂²×C₄	—	4	1	no	—
262	C₂²×C₄	—	4	1	no	—
263	C₂²×C₄	—	53	2	no	—
264	C₂²×C₄	—	212	2	no	—
265	D₈	—	2	1	no	—
266	D₈	—	2	1	no	—
267	D₈	—	2	1	no	—
268	D₈	—	2	1	no	—
269	C₂×C₈	—	53	2	no	—
270	C₂×C₈	—	106	2	no	—
271	C₂×C₈	—	106	2	no	—
272	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
273	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
274	C₄⋊C₄	—	2	1	no	—
275	C₄⋊C₄	—	2	1	no	—
276	C₄⋊C₄	—	53	1	no	—
277	C₄⋊C₄	—	106	2	no	—
278	C₄⋊C₄	—	106	2	no	—
279	C₄⋊C₄	—	212	2	no	—
280	C₄⋊C₄	—	212	1	no	—
281	C₄⋊C₄	—	212	1	no	—
282	C₂²⋊C₄	—	4	1	no	—
283	C₂²⋊C₄	—	4	1	no	—
284	C₂²⋊C₄	—	53	2	no	—
285	C₂²⋊C₄	—	212	2	no	—
286	C₂²⋊C₄	—	212	2	no	—
287	C₂²⋊C₄	—	212	2	no	—
288	C₄²	—	212	2	no	—
289	C₄²	D₄×D₅₃	1	1	yes	characteristic
290	C₁₆	—	2	1	no	—
291	C₁₆	—	106	2	no	—
292	C₁₆	#11719945	1	2	yes	complemented
293	C₂³	—	4	1	no	—
294	C₂³	—	4	1	no	—
295	C₂³	—	53	2	no	—
296	Q₈	—	848	1	no	—
297	Q₈	—	424	2	no	—
298	D₄	—	4	1	no	—
299	D₄	—	4	1	no	—
300	D₄	—	4	1	no	—
301	D₄	—	4	1	no	—
302	D₄	—	848	2	no	—
303	D₄	—	106	2	no	—
304	D₄	—	424	4	no	—
305	C₂×C₄	—	4	1	no	—
306	C₂×C₄	—	4	1	no	—
307	C₂×C₄	—	8	2	no	—
308	C₂×C₄	—	8	2	no	—
309	C₂×C₄	—	53	2	no	—
310	C₂×C₄	—	106	2	no	—
311	C₂×C₄	—	106	2	no	—
312	C₂×C₄	—	424	4	no	—
313	C₂×C₄	—	212	2	no	—
314	C₂×C₄	—	212	2	no	—
315	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
316	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
317	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
318	C₈	—	2	1	no	—
319	C₈	—	106	2	no	—
320	C₈	C₂×#11719945	1	1	yes	characteristic
321	C₈	#13133264	1	1	yes	characteristic
322	C₂²	—	8	1	no	—
323	C₂²	—	8	1	no	—
324	C₂²	—	8	1	no	—
325	C₂²	—	8	1	no	—
326	C₂²	—	8	1	no	—
327	C₂²	—	8	1	no	—
328	C₂²	—	106	2	no	—
329	C₂²	—	106	2	no	—
330	C₂²	—	106	2	no	—
331	C₂²	C₂×D₈×D₅₃	1	1	yes	characteristic
332	C₄	—	2	1	no	—
333	C₄	—	8	1	no	—
334	C₄	—	8	1	no	—
335	C₄	—	106	2	no	—
336	C₄	—	424	2	no	—
337	C₄	—	424	2	no	—
338	C₄	#16811057	1	2	yes	—
339	C₄	#16811278	1	1	yes	characteristic
340	C₄	#16811437	1	1	yes	characteristic
341	C₂	—	16	1	no	—
342	C₂	—	16	1	no	—
343	C₂	—	212	2	no	—
344	C₂	#20287064	1	1	yes	characteristic
345	C₂	C₂×#16811057	1	1	yes	characteristic
346	C₂	#20287858	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₅₃
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₄×D₁₆×C₅₃	C₂
socle	C₂²×C₅₃	C₂²×D₈
hypercenter	C₄×D₁₆	D₅₃
hypocenter	C₅₃	#33921
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×D₁₆

Complements

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

N	H ≅ G/N	type
1. #20287139	C₂	semidirect
2. #20287064	C₂	semidirect
3. #20276997	C₂	semidirect
4. #20276905	C₂	semidirect
5. #20276899	C₂	semidirect
6. #20276862	C₂	semidirect
7. #20276852	C₂	semidirect
8. #20276815	C₂	semidirect
9. #20276548	C₂	semidirect
10. #20276500	C₂	semidirect
11. C₄×D₁₆×C₅₃	C₂	semidirect
117. C₁₆×C₅₃	#41	semidirect
176. C₄×D₁₆	D₅₃	semidirect
214. C₅₃	#33921	semidirect
292. C₁₆	#11719945	semidirect

Maximal subgroups

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

subgroup	index
#20287139	2
#20287064	2
#20276997	2
#20276905	2
#20276899	2
#20276862	2
#20276852	2
#20276815	2
#20276548	2
#20276500	2
C₄×D₁₆×C₅₃	2
#33921	53