Subgroups — #28328390

Order 13568.

642 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20291492	C₂	1	2	yes	maximal, complemented
2	#20291117	C₂	1	4	yes	maximal, complemented
3	#20291116	C₂	1	2	yes	maximal, complemented
4	#20291114	C₂	1	2	yes	maximal, complemented
5	C₂×#16811551	C₂	1	1	yes	maximal, characteristic, complemented
6	#20291029	C₂	1	2	yes	maximal, complemented
7	#20290923	C₂	1	1	yes	maximal, characteristic, complemented
8	#508×D₅₃	C₂	1	1	yes	maximal, characteristic
9	#16811702	—	2	2	no	—
10	D₅₃⋊M₃₂	C₂²	1	1	yes	characteristic
11	#16811668	—	2	4	no	—
12	#16811667	C₂²	1	4	yes	complemented
13	#16811665	C₂²	1	4	yes	complemented
14	C₂×#13133288	C₂²	1	2	yes	—
15	#16811620	—	2	2	no	—
16	#16811586	C₂²	1	1	yes	characteristic
17	#16811551	C₂²	1	2	yes	complemented
18	#16811551	C₂²	1	2	yes	complemented
19	#16811538	C₂²	1	4	yes	complemented
20	#16811537	C₂²	1	4	yes	complemented
21	#16811510	C₂²	1	2	yes	complemented
22	C₁₆×#6157	C₂²	1	2	yes	complemented
23	#135×D₅₃	C₂²	1	2	yes	—
24	#16811056	C₂²	1	1	yes	characteristic
25	#16811055	C₄	1	1	yes	characteristic, complemented
26	#16811054	C₂²	1	2	yes	—
27	#16811053	C₄	1	2	yes	complemented
28	#16811052	C₂²	1	1	yes	characteristic
29	Q₃₂×D₅₃	C₄	1	1	yes	characteristic, complemented
30	SD₃₂×D₅₃	C₄	1	2	yes	complemented
31	D₁₆×D₅₃	C₄	1	1	yes	characteristic, complemented
32	C₂×C₁₆×D₅₃	C₂²	1	1	yes	characteristic
33	C₅₃×#508	C₄	1	1	yes	characteristic, complemented
34	#13133305	—	2	4	no	—
35	#13133304	—	2	4	no	—
36	#13133304	—	4	4	no	—
37	C₂×#11719936	—	2	2	no	—
38	C₂×#11719936	—	4	2	no	—
39	D₅₃⋊M₁₆	—	4	2	no	—
40	D₅₃⋊M₁₆	D₄	1	1	yes	characteristic
41	#13133296	D₄	1	1	yes	characteristic
42	#13133292	—	2	1	no	—
43	#13133291	—	2	1	no	—
44	D₅₃⋊C₁₆	C₂³	1	2	yes	—
45	#13133288	—	2	2	no	—
46	#13133288	C₂³	1	4	yes	—
47	#13133282	—	2	4	no	—
48	#13133281	—	2	4	no	—
49	#13133271	D₄	1	2	yes	—
50	C₈×#6157	C₂³	1	2	yes	—
51	#41×D₅₃	—	2	2	no	—
52	#13133240	C₂³	1	2	yes	—
53	#13133239	C₂×C₄	1	2	yes	—
54	#13133238	—	2	2	no	—
55	#13133237	—	2	2	no	—
56	#13133236	C₂³	1	2	yes	—
57	Q₁₆×D₅₃	C₂×C₄	1	2	yes	—
58	SD₁₆×D₅₃	—	2	2	no	—
59	D₈×D₅₃	C₂×C₄	1	2	yes	—
60	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
61	Dic₄₂₄	C₂×C₄	1	1	yes	characteristic
62	#13133163	C₂×C₄	1	2	yes	—
63	#13133162	C₂×C₄	1	2	yes	—
64	#13133161	C₂×C₄	1	2	yes	—
65	#13133160	C₂×C₄	1	2	yes	—
66	D₈₄₈	C₂×C₄	1	1	yes	characteristic
67	#13133158	C₂×C₄	1	2	yes	—
68	C₂×#11719908	C₂×C₄	1	1	yes	characteristic
69	C₁₆×D₅₃	C₂×C₄	1	1	yes	characteristic
70	C₁₆×D₅₃	C₂×C₄	1	1	yes	characteristic
71	C₁₆×D₅₃	C₂³	1	1	yes	characteristic
72	C₁₆×D₅₃	C₂³	1	1	yes	characteristic
73	#135×C₅₃	C₂×C₄	1	2	yes	—
74	Q₃₂×C₅₃	C₂×C₄	1	1	yes	characteristic
75	SD₃₂×C₅₃	C₂×C₄	1	2	yes	—
76	D₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
77	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
78	#41×C₅₃	—	2	2	no	—
79	#11719949	—	2	2	no	—
80	Q₈×D₅₃	—	2	2	no	—
81	#11719947	—	2	2	no	—
82	#11719947	—	4	2	no	—
83	D₄×D₅₃	—	2	2	no	—
84	D₄×D₅₃	—	4	2	no	—
85	#11719945	—	2	2	no	—
86	#11719945	—	4	2	no	—
87	C₂×C₄×D₅₃	—	4	2	no	—
88	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
89	C₂²⋊#6157	—	2	1	no	—
90	#11719940	—	4	4	no	—
91	#11719940	D₈	1	2	yes	—
92	C₂×#88324	—	4	2	no	—
93	C₂×#88324	—	4	4	no	—
94	C₄⋊#6157	—	2	1	no	—
95	C₄×#6157	C₂×D₄	1	2	yes	—
96	#11719936	—	2	4	no	—
97	#11719936	—	4	2	no	—
98	#11719936	—	4	4	no	—
99	#11719936	D₈	1	1	yes	characteristic
100	#11719936	D₈	1	1	yes	characteristic
101	D₅₃⋊C₈	—	4	4	no	—
102	D₅₃⋊C₈	C₂×D₄	1	2	yes	—
103	Q₁₆×C₅₃	C₂²×C₄	1	2	yes	—
104	SD₁₆×C₅₃	—	2	2	no	—
105	D₈×C₅₃	C₂²×C₄	1	2	yes	—
106	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
107	#11719925	—	2	2	no	—
108	#11719924	—	2	2	no	—
109	#11719923	—	2	2	no	—
110	#11719922	—	2	2	no	—
111	C₂×#88322	#41	1	1	yes	characteristic
112	Dic₂₁₂	C₂²×C₄	1	2	yes	—
113	D₄₂₄	C₂²×C₄	1	2	yes	—
114	#11719913	—	2	2	no	—
115	C₈×D₅₃	C₂²×C₄	1	1	yes	characteristic
116	C₈×D₅₃	C₂×D₄	1	1	yes	characteristic
117	C₈×D₅₃	#41	1	1	yes	characteristic
118	C₈×D₅₃	C₂⁴	1	1	yes	characteristic
119	#11719910	—	2	2	no	—
120	C₁₆×C₅₃	C₂²×C₄	1	1	yes	characteristic
121	C₁₆×C₅₃	C₂²×C₄	1	1	yes	characteristic
122	#11719908	C₂²×C₄	1	1	yes	characteristic
123	#11719908	C₂²×C₄	1	1	yes	characteristic
124	C₂²×D₅₃	—	4	2	no	—
125	C₂²×D₅₃	#135	1	1	yes	characteristic
126	C₂×#6157	—	2	2	no	—
127	Q₈×C₅₃	—	2	2	no	—
128	D₄×C₅₃	—	2	2	no	—
129	D₄×C₅₃	—	4	2	no	—
130	C₂×C₄×C₅₃	—	4	2	no	—
131	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
132	C₂²⋊D₅₃	—	4	2	no	—
133	C₂²⋊D₅₃	—	4	2	no	—
134	C₂²⋊D₅₃	—	4	2	no	—
135	C₂×Dic₅₃	—	4	2	no	—
136	C₂×Dic₅₃	C₂×D₈	1	1	yes	characteristic
137	D₂₁₂	—	2	2	no	—
138	D₂₁₂	—	4	2	no	—
139	C₄×D₅₃	—	4	2	no	—
140	C₄×D₅₃	—	4	2	no	—
141	C₄×D₅₃	—	4	2	no	—
142	C₄×D₅₃	C₄×D₄	1	1	yes	characteristic
143	C₄×D₅₃	C₂×D₈	1	1	yes	characteristic
144	C₄×D₅₃	#135	1	1	yes	characteristic
145	C₄×D₅₃	C₂²×D₄	1	1	yes	characteristic
146	Dic₁₀₆	—	2	2	no	—
147	Dic₁₀₆	—	4	2	no	—
148	#88324	—	4	4	no	—
149	#88324	C₂×D₈	1	2	yes	—
150	#88324	C₂×D₈	1	2	yes	—
151	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
152	C₈×C₅₃	C₂³×C₄	1	1	yes	characteristic
153	#88322	C₄×D₄	1	1	yes	characteristic
154	#88322	C₂×#41	1	1	yes	characteristic
155	#33465	—	53	1	no	maximal, Sylow
156	C₂²×C₅₃	—	4	2	no	—
157	C₂²×C₅₃	C₄×D₈	1	1	yes	characteristic
158	C₂×D₅₃	—	4	1	no	—
159	C₂×D₅₃	—	4	2	no	—
160	C₂×D₅₃	—	16	2	no	—
161	C₂×D₅₃	C₄×D₈	1	1	yes	characteristic
162	C₂×D₅₃	C₂×#135	1	1	yes	characteristic
163	#6157	—	4	2	no	—
164	C₄×C₅₃	—	4	2	no	—
165	C₄×C₅₃	C₄×D₈	1	1	yes	characteristic
166	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
167	Dic₅₃	—	4	2	no	—
168	Dic₅₃	C₄×D₈	1	1	yes	characteristic
169	Dic₅₃	C₂²×D₈	1	1	yes	characteristic
170	C₂×#508	—	53	1	no	—
171	#2957	—	53	2	no	—
172	#2179	—	53	4	no	—
173	#2178	—	53	2	no	—
174	#2178	—	53	2	no	—
175	C₂×#368	—	53	1	no	—
176	#2143	—	53	2	no	—
177	#2108	—	53	1	no	—
178	C₂×C₅₃	—	2	1	no	—
179	C₂×C₅₃	—	8	2	no	—
180	C₂×C₅₃	C₂×C₄×D₈	1	1	yes	characteristic
181	D₅₃	—	2	1	no	—
182	D₅₃	—	2	1	no	—
183	D₅₃	—	8	2	no	—
184	C₂×#135	—	53	2	no	—
185	#568	—	106	2	no	—
186	#508	—	53	2	no	—
187	#508	—	53	1	no	—
188	#508	—	53	2	no	—
189	#508	—	53	1	no	—
190	#508	—	53	1	no	—
191	#508	#6157	1	1	yes	characteristic, complemented
192	C₂×Q₃₂	—	53	1	no	—
193	C₂×SD₃₂	—	53	2	no	—
194	C₂×D₁₆	—	53	1	no	—
195	C₂×M₃₂	—	53	1	no	—
196	C₂²×C₁₆	—	53	1	no	—
197	#444	—	106	4	no	—
198	#443	—	53	4	no	—
199	#443	—	53	4	no	—
200	C₂×#108	—	53	2	no	—
201	#421	—	106	2	no	—
202	#405	—	53	1	no	—
203	#368	—	53	2	no	—
204	#368	—	53	2	no	—
205	#359	—	53	4	no	—
206	#359	—	53	4	no	—
207	#346	—	53	2	no	—
208	C₄×C₁₆	—	53	2	no	—
209	C₅₃	#33465	1	1	yes	characteristic, complemented, Sylow
210	C₂×#41	—	106	2	no	—
211	#135	—	53	2	no	—
212	#135	—	53	2	no	—
213	#135	—	53	2	no	—
214	#135	—	106	2	no	—
215	#135	—	106	2	no	—
216	#135	C₂×#6157	1	2	yes	—
217	C₂×Q₁₆	—	53	2	no	—
218	C₂×SD₁₆	—	106	2	no	—
219	C₂×D₈	—	53	2	no	—
220	#131	—	106	4	no	—
221	#131	—	106	4	no	—
222	#131	—	212	4	no	—
223	C₂×M₁₆	—	53	1	no	—
224	C₂×M₁₆	—	106	2	no	—
225	C₂×M₁₆	—	212	2	no	—
226	C₂×M₁₆	—	212	2	no	—
227	C₂²×C₈	—	53	1	no	—
228	#117	—	53	1	no	—
229	Q₃₂	—	53	2	no	—
230	Q₃₂	—	53	1	no	—
231	Q₃₂	C₂×#6157	1	1	yes	characteristic
232	SD₃₂	—	53	2	no	—
233	SD₃₂	—	53	2	no	—
234	SD₃₂	—	53	2	no	—
235	SD₃₂	C₂×#6157	1	2	yes	—
236	D₁₆	—	53	2	no	—
237	D₁₆	—	53	1	no	—
238	D₁₆	C₂×#6157	1	1	yes	characteristic
239	M₃₂	—	106	1	no	—
240	M₃₂	—	106	1	no	—
241	C₂×C₁₆	—	53	1	no	—
242	C₂×C₁₆	—	53	1	no	—
243	C₂×C₁₆	—	53	1	no	—
244	C₂×C₁₆	—	53	1	no	—
245	C₂×C₁₆	—	53	1	no	—
246	C₂×C₁₆	—	53	2	no	—
247	C₂×C₁₆	C₂×#6157	1	1	yes	characteristic
248	#108	—	53	4	no	—
249	#108	—	106	2	no	—
250	#104	—	106	4	no	—
251	#104	—	106	4	no	—
252	#97	—	53	2	no	—
253	C₄×C₈	—	53	2	no	—
254	#41	—	2	2	no	—
255	#41	—	106	2	no	—
256	#41	—	106	2	no	—
257	#41	—	106	2	no	—
258	#41	—	212	2	no	—
259	#41	—	212	2	no	—
260	C₂×Q₈	—	106	2	no	—
261	C₂×D₄	—	106	2	no	—
262	C₂×D₄	—	212	2	no	—
263	C₂²×C₄	—	53	1	no	—
264	C₂²×C₄	—	212	2	no	—
265	Q₁₆	—	53	2	no	—
266	Q₁₆	—	106	2	no	—
267	Q₁₆	C₂²×#6157	1	2	yes	—
268	SD₁₆	—	2	2	no	—
269	SD₁₆	—	106	2	no	—
270	SD₁₆	—	106	2	no	—
271	SD₁₆	—	106	2	no	—
272	D₈	—	53	2	no	—
273	D₈	—	106	2	no	—
274	D₈	C₂²×#6157	1	2	yes	—
275	M₁₆	—	53	1	no	—
276	M₁₆	—	53	1	no	—
277	M₁₆	—	53	2	no	—
278	M₁₆	—	106	4	no	—
279	M₁₆	—	212	4	no	—
280	M₁₆	—	212	4	no	—
281	M₁₆	—	212	2	no	—
282	C₂×C₈	—	53	2	no	—
283	C₂×C₈	—	53	1	no	—
284	C₂×C₈	—	53	1	no	—
285	C₂×C₈	—	53	1	no	—
286	C₂×C₈	—	53	1	no	—
287	C₂×C₈	—	53	1	no	—
288	C₂×C₈	—	212	4	no	—
289	C₂×C₈	—	212	4	no	—
290	C₂×C₈	—	212	2	no	—
291	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
292	C₄⋊C₄	—	106	1	no	—
293	C₂²⋊C₄	—	106	1	no	—
294	C₄²	—	53	2	no	—
295	C₁₆	—	53	1	no	—
296	C₁₆	—	53	1	no	—
297	C₁₆	—	106	2	no	—
298	C₁₆	C₂²×#6157	1	1	yes	characteristic
299	C₁₆	C₂²×#6157	1	1	yes	characteristic
300	C₂³	—	53	1	no	—
301	C₂³	—	212	2	no	—
302	Q₈	—	2	2	no	—
303	Q₈	—	106	2	no	—
304	Q₈	—	212	2	no	—
305	D₄	—	2	2	no	—
306	D₄	—	4	2	no	—
307	D₄	—	106	2	no	—
308	D₄	—	212	2	no	—
309	D₄	—	212	2	no	—
310	D₄	—	212	2	no	—
311	D₄	—	212	2	no	—
312	C₂×C₄	—	4	2	no	—
313	C₂×C₄	—	53	1	no	—
314	C₂×C₄	—	53	1	no	—
315	C₂×C₄	—	53	1	no	—
316	C₂×C₄	—	53	1	no	—
317	C₂×C₄	—	53	1	no	—
318	C₂×C₄	—	106	2	no	—
319	C₂×C₄	—	212	2	no	—
320	C₂×C₄	—	212	2	no	—
321	C₂×C₄	—	212	2	no	—
322	C₂×C₄	—	212	2	no	—
323	C₂×C₄	D₄×#6157	1	1	yes	characteristic
324	C₈	—	53	2	no	—
325	C₈	—	53	2	no	—
326	C₈	—	53	1	no	—
327	C₈	—	53	1	no	—
328	C₈	—	212	4	no	—
329	C₈	D₄×#6157	1	1	yes	characteristic
330	C₈	C₂³×#6157	1	1	yes	characteristic
331	C₂²	—	4	2	no	—
332	C₂²	—	53	1	no	—
333	C₂²	—	53	1	no	—
334	C₂²	—	848	2	no	—
335	C₂²	—	212	1	no	—
336	C₂²	—	212	2	no	—
337	C₂²	D₈×#6157	1	1	yes	characteristic
338	C₄	—	4	2	no	—
339	C₄	—	53	1	no	—
340	C₄	—	53	1	no	—
341	C₄	—	212	2	no	—
342	C₄	—	212	2	no	—
343	C₄	D₈×#6157	1	1	yes	characteristic
344	C₄	C₂×D₄×#6157	1	1	yes	characteristic
345	C₂	—	2	1	no	—
346	C₂	—	8	2	no	—
347	C₂	—	106	1	no	—
348	C₂	—	106	1	no	—
349	C₂	—	424	2	no	—
350	C₂	C₂×D₈×#6157	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₈×#6157
commutator	C₈×C₅₃	C₂³×C₄
Frattini	C₈	C₂³×#6157
Fitting	C₅₃×#508	C₄
socle	C₂×C₅₃	C₂×C₄×D₈
prime radical	#508×D₅₃	C₂
cosocle	C₈×D₅₃	C₂⁴
hypercenter	#508	#6157
hypocenter	C₅₃	#33465
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #508

Complements

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

N	H ≅ G/N	type
1. #20291492	C₂	semidirect
2. #20291117	C₂	semidirect
3. #20291116	C₂	semidirect
4. #20291114	C₂	semidirect
5. C₂×#16811551	C₂	semidirect
6. #20291029	C₂	semidirect
7. #20290923	C₂	semidirect
12. #16811667	C₂²	semidirect
13. #16811665	C₂²	semidirect
17. #16811551	C₂²	semidirect
18. #16811551	C₂²	semidirect
19. #16811538	C₂²	semidirect
20. #16811537	C₂²	semidirect
21. #16811510	C₂²	semidirect
22. C₁₆×#6157	C₂²	semidirect
25. #16811055	C₄	semidirect
27. #16811053	C₄	semidirect
29. Q₃₂×D₅₃	C₄	semidirect
30. SD₃₂×D₅₃	C₄	semidirect
31. D₁₆×D₅₃	C₄	semidirect
33. C₅₃×#508	C₄	semidirect
191. #508	#6157	semidirect
209. C₅₃	#33465	semidirect

Maximal subgroups

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

subgroup	index
#20291492	2
#20291117	2
#20291116	2
#20291114	2
C₂×#16811551	2
#20291029	2
#20290923	2
#508×D₅₃	2
#33465	53