Finite Group Databasebeta

← D53×#2159 #27537122

Subgroups — D53×#2159 #27537122

Order 13568.

718 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×M32×D53C211yesmaximal, characteristic
2#427×D53C211yesmaximal, characteristic
3C2×#107×D53C211yesmaximal, characteristic
4#20276504C211yesmaximal, characteristic, complemented
5#20276503C211yesmaximal, characteristic, complemented
6#20276502C212yesmaximal, complemented
7#20276501C211yesmaximal, characteristic, complemented
8#20276500C212yesmaximal, complemented
9#367×D53C212yesmaximal, complemented
10#366×D53C212yesmaximal, complemented
11C53×#2159C211yesmaximal, characteristic, complemented
12C22×C8×D53C2211yescharacteristic
13#117×D53—21no—
14C2×C4⋊C4×D53—21no—
15C2×#13133157C2211yescharacteristic
16C2×#13133156C2212yes—
17M32×D53C418yescomplemented
18C2×C16×D53C2212yes—
19#16810661C2211yescharacteristic
20#16810660C2211yescharacteristic
21#16810660C2211yescharacteristic
22#16810659—21no—
23#16810658—21no—
24C2×#13133151C2211yescharacteristic
25C2×#13133150—21no—
26#107×D53—21no—
27#107×D53—21no—
28#107×D53C2211yescharacteristic
29#107×D53C2212yes—
30#107×D53C2211yescharacteristic
31#107×D53C2212yes—
32#106×D53—21no—
33#16810451C2212yescomplemented
34#16810450C2212yescomplemented
35#16810450C2212yescomplemented
36#16810449C2212yescomplemented
37#16810448C2212yescomplemented
38#16810448C2212yescomplemented
39C2×M32×C53C2211yescharacteristic
40C53×#427C2211yescharacteristic
41C2×#107×C53C2211yescharacteristic
42C53×#367C2212yescomplemented
43C53×#366C2212yescomplemented
44C22×C4×D53—41no—
45C22×C4×D53D411yescharacteristic
46C22×#88322D411yescharacteristic
47C2×C8×D53C2×C414yes—
48C2×C8×D53D411yescharacteristic
49C2×C8×D53D411yescharacteristic
50C2×C8×D53Q814yes—
51C2×C8×D53C2311yescharacteristic
52C2×C8×D53C2311yescharacteristic
53#13133176—21no—
54#13133175—21no—
55#13133175—21no—
56#13133174—41no—
57#13133173—41no—
58C2×C4⋊Dic53—21no—
59C2×#11719919—41no—
60C4⋊C4×D53—22no—
61C4⋊C4×D53—22no—
62C4⋊C4×D53—21no—
63C4⋊C4×D53—21no—
64C4⋊C4×D53—41no—
65C4⋊C4×D53—41no—
66C22⋊C4×D53—41no—
67C42×D53—41no—
68#13133157C2×C414yescomplemented
69#13133156C2×C418yescomplemented
70C2×#11719908C2312yes—
71C16×D53C2×C418yescomplemented
72#13133151—21no—
73#13133151C2312yes—
74#13133151C2311yescharacteristic
75#13133151C2311yescharacteristic
76#13133150—22no—
77#13133150—22no—
78#13133150—21no—
79#13133150—21no—
80#13133149—21no—
81#13133148—21no—
82#13133148—21no—
83C22×C8×C53C2311yescharacteristic
84#117×C53—21no—
85C2×C4⋊C4×C53—21no—
86M32×C53C2×C414yescomplemented
87C2×C16×C53C2312yes—
88#107×C53—21no—
89#107×C53C2311yescharacteristic
90#107×C53C2312yes—
91#107×C53C2311yescharacteristic
92#106×C53—21no—
93C23×D53D811yescharacteristic
94C22×C4×C53—41no—
95C22×C4×C53C2×D411yescharacteristic
96C22×Dic53—41no—
97C22×Dic53D811yescharacteristic
98C2×C4×D53—41no—
99C2×C4×D53—41no—
100C2×C4×D53—42no—
101C2×C4×D53—42no—
102C2×C4×D53—84no—
103C2×C4×D53C4⋊C414yes—
104C2×C4×D53D811yescharacteristic
105C2×C4×D53D811yescharacteristic
106C2×C4×D53Q1614yes—
107C2×C4×D53C2×D411yescharacteristic
108C2×C4×D53C2×D411yescharacteristic
109C2×C8×C53C22×C412yes—
110C2×C8×C53C2×D411yescharacteristic
111C2×C8×C53C2×Q812yes—
112C2×C8×C53C2411yescharacteristic
113C4⋊C4×C53—21no—
114C4⋊C4×C53—21no—
115C4⋊C4×C53—22no—
116C4⋊C4×C53—41no—
117C22⋊C4×C53—41no—
118C42×C53—41no—
119C22⋊Dic53—41no—
120#11719921—41no—
121#11719921—41no—
122C4⋊Dic53—21no—
123C4⋊Dic53—21no—
124C4⋊Dic53—22no—
125C4⋊Dic53—41no—
126#11719919—42no—
127#11719919—42no—
128#11719919—41no—
129#11719919—41no—
130C4×Dic53—41no—
131C4×Dic53—41no—
132C4×Dic53—41no—
133C2×#88322C4⋊C414yes—
134C2×#88322C2×D411yescharacteristic
135C2×#88322C2×D411yescharacteristic
136C8×D53C4⋊C414yes—
137C8×D53C4⋊C414yes—
138C8×D53C22×C412yes—
139C8×D53C22×C412yes—
140C8×D53C2×Q812yes—
141C8×D53C2×Q812yes—
142C16×C53C22×C414yescomplemented
143#11719908C22×C414yescomplemented
144C23×C53C2×D811yescharacteristic
145C22×D53—22no—
146C22×D53—22no—
147C22×D53#10714yes—
148C22×D53C2×D811yescharacteristic
149C22×D53C2×D811yescharacteristic
150C2×C4×C53—42no—
151C2×C4×C53—41no—
152C2×C4×C53—41no—
153C2×C4×C53—82no—
154C2×C4×C53C2×C4⋊C412yes—
155C2×C4×C53C2×D811yescharacteristic
156C2×C4×C53C2×Q1612yes—
157C2×C4×C53C22×D411yescharacteristic
158C2×Dic53—42no—
159C2×Dic53—41no—
160C2×Dic53—41no—
161C2×Dic53—82no—
162C2×Dic53#10714yes—
163C2×Dic53C2×D811yescharacteristic
164C2×Dic53C2×D811yescharacteristic
165C4×D53—84no—
166C4×D53—82no—
167C4×D53—84no—
168C4×D53—82no—
169C4×D53#10714yes—
170C4×D53#10714yes—
171C4×D53C2×C4⋊C412yes—
172C4×D53C2×C4⋊C412yes—
173C4×D53C2×Q1612yes—
174C4×D53C2×Q1612yes—
175C8×C53C2×C4⋊C412yes—
176C8×C53C23×C411yescharacteristic
177C8×C53C22×Q811yescharacteristic
178#88322C2×C4⋊C412yes—
179#88322C2×C4⋊C412yes—
180C2×#2159—531nomaximal, Sylow
181C22×C53—21no—
182C22×C53—21no—
183C22×C53C2×#10712yes—
184C22×C53C22×D811yescharacteristic
185C2×D53—21no—
186C2×D53—21no—
187C2×D53—28no—
188C2×D53C2×#10712yes—
189C2×D53C2×#10712yes—
190C2×D53#50912yes—
191C2×D53#51012yes—
192C4×C53—82no—
193C4×C53—81no—
194C4×C53—81no—
195C4×C53C2×#10712yes—
196C4×C53C22×C4⋊C411yescharacteristic
197C4×C53C22×Q1611yescharacteristic
198Dic53—82no—
199Dic53—81no—
200Dic53—81no—
201Dic53C2×#10712yes—
202Dic53C2×#10712yes—
203C22×M32—531no—
204C2×#427—531no—
205C22×#107—531no—
206#2159—531no—
207#2159—531no—
208#2159—531no—
209#2159—532no—
210#2159—532no—
211#2159D5311yescharacteristic, direct product factor
212C2×#367—532no—
213C2×#366—532no—
214C2×C53—22no—
215C2×C53C22×#10711yescharacteristic
216C2×C53C2×#50911yescharacteristic
217C2×C53C2×#51011yescharacteristic
218D53—22no—
219D53#215914yesdirect product factor
220C23×C8—531no—
221C2×#117—1061no—
222C22×C4⋊C4—1061no—
223C2×M32—531no—
224C2×M32—538no—
225C2×M32—532no—
226C2×M32C2×D5311yescharacteristic
227C22×C16—532no—
228#427—531no—
229#427—531no—
230#427—531no—
231#427—1061no—
232#427—1061no—
233#427C2×D5311yescharacteristic
234C2×#107—531no—
235C2×#107—532no—
236C2×#107—532no—
237C2×#107—531no—
238C2×#107—531no—
239C2×#107—1061no—
240C2×#107—1061no—
241C2×#107—1061no—
242C2×#107C2×D5311yescharacteristic
243C2×#106—1061no—
244#367—532no—
245#367—532no—
246#367—532no—
247#367C2×D5312yescomplemented
248#366—532no—
249#366—532no—
250#366—532no—
251#366C2×D5312yescomplemented
252C53C2×#215911yescharacteristic, complemented, Sylow
253C23×C4—531no—
254C23×C4—2121no—
255C22×C8—531no—
256C22×C8—534no—
257C22×C8—531no—
258C22×C8—531no—
259C22×C8—534no—
260C22×C8—531no—
261C22×C8—531no—
262C22×C8C22×D5311yescharacteristic
263#117—21no—
264#117—1061no—
265#117—1061no—
266#117—1061no—
267#117—2121no—
268#117—2121no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—1062no—
271C2×C4⋊C4—1062no—
272C2×C4⋊C4—1061no—
273C2×C4⋊C4—1061no—
274C2×C4⋊C4—1061no—
275C2×C4⋊C4—2121no—
276C2×C4⋊C4—2121no—
277C2×C4⋊C4—2121no—
278C2×C22⋊C4—2121no—
279C2×C42—2121no—
280M32—534no—
281M32—538no—
282M32C4×D5314yescomplemented
283C2×C16—532no—
284C2×C16—538no—
285C2×C16C22×D5312yes—
286#107—21no—
287#107—531no—
288#107—532no—
289#107—531no—
290#107—1062no—
291#107—1062no—
292#107—1061no—
293#107—1061no—
294#107—1061no—
295#107C22×D5311yescharacteristic
296#107C22×D5312yes—
297#107C22×D5311yescharacteristic
298#106—21no—
299#106—1061no—
300#106—1061no—
301#106—1061no—
302C24—531no—
303C22×C4—41no—
304C22×C4—531no—
305C22×C4—534no—
306C22×C4—531no—
307C22×C4—531no—
308C22×C4—534no—
309C22×C4—531no—
310C22×C4—531no—
311C22×C4—4244no—
312C22×C4—2121no—
313C22×C4—2121no—
314C22×C4—2122no—
315C22×C4—2122no—
316C22×C4—2121no—
317C22×C4D4×D5311yescharacteristic
318C2×C8—534no—
319C2×C8—531no—
320C2×C8—531no—
321C2×C8—534no—
322C2×C8—534no—
323C2×C8—532no—
324C2×C8—532no—
325C2×C8—532no—
326C2×C8—532no—
327C2×C8C2×C4×D5312yes—
328C2×C8D4×D5311yescharacteristic
329C2×C8Q8×D5312yes—
330C2×C8C23×D5311yescharacteristic
331C4⋊C4—21no—
332C4⋊C4—21no—
333C4⋊C4—22no—
334C4⋊C4—41no—
335C4⋊C4—1061no—
336C4⋊C4—1061no—
337C4⋊C4—1062no—
338C4⋊C4—2122no—
339C4⋊C4—2122no—
340C4⋊C4—2121no—
341C4⋊C4—2121no—
342C4⋊C4—2121no—
343C22⋊C4—41no—
344C22⋊C4—2121no—
345C22⋊C4—2121no—
346C22⋊C4—2121no—
347C42—41no—
348C42—2121no—
349C42—2121no—
350C42—2121no—
351C16—534no—
352C16C2×C4×D5314yescomplemented
353C23—531no—
354C23—531no—
355C23—534no—
356C23—1062no—
357C23—1062no—
358C23D8×D5311yescharacteristic
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—42no—
362C2×C4—82no—
363C2×C4—531no—
364C2×C4—531no—
365C2×C4—534no—
366C2×C4—532no—
367C2×C4—532no—
368C2×C4—532no—
369C2×C4—532no—
370C2×C4—534no—
371C2×C4—534no—
372C2×C4—4244no—
373C2×C4—4242no—
374C2×C4—4242no—
375C2×C4—4244no—
376C2×C4—4242no—
377C2×C4—2121no—
378C2×C4—2121no—
379C2×C4—2122no—
380C2×C4C4⋊C4×D5312yes—
381C2×C4D8×D5311yescharacteristic
382C2×C4Q16×D5312yes—
383C2×C4C2×D4×D5311yescharacteristic
384C8—532no—
385C8—532no—
386C8C4⋊C4×D5312yes—
387C8C22×C4×D5311yescharacteristic
388C8C2×Q8×D5311yescharacteristic
389C22—21no—
390C22—21no—
391C22—532no—
392C22—532no—
393C22—532no—
394C22—532no—
395C22—1068no—
396C22—1061no—
397C22—1061no—
398C22#107×D5312yes—
399C22C2×D8×D5311yescharacteristic
400C4—82no—
401C4—81no—
402C4—81no—
403C4—532no—
404C4—532no—
405C4—4242no—
406C4—4241no—
407C4—4241no—
408C4#107×D5312yes—
409C4C2×C4⋊C4×D5311yescharacteristic
410C4C2×Q16×D5311yescharacteristic
411C2—22no—
412C2—534no—
413C2—1062no—
414C2C2×#107×D5311yescharacteristic
415C2#509×D5311yescharacteristic
416C2#510×D5311yescharacteristic
subgroupG/H
centerC22C2×D8×D53
commutatorC8×C53C23×C4
FrattiniC2×C8C23×D53
FittingC53×#2159C2
socleC22×C53C22×D8
prime radicalC23×D53D8
cosocleC2×C8×C53C24
hypercenter#2159D53
hypocenterC53C2×#2159
LCSG ▷ C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2159

Complements

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

NH ≅ G/Ntype
4. #20276504C2semidirect
5. #20276503C2semidirect
6. #20276502C2semidirect
7. #20276501C2semidirect
8. #20276500C2semidirect
9. #367×D53C2semidirect
10. #366×D53C2semidirect
11. C53×#2159C2semidirect
17. M32×D53C4semidirect
33. #16810451C22semidirect
34. #16810450C22semidirect
35. #16810450C22semidirect
36. #16810449C22semidirect
37. #16810448C22semidirect
38. #16810448C22semidirect
42. C53×#367C22semidirect
43. C53×#366C22semidirect
68. #13133157C2×C4semidirect
69. #13133156C2×C4semidirect
71. C16×D53C2×C4semidirect
86. M32×C53C2×C4semidirect
142. C16×C53C22×C4semidirect
143. #11719908C22×C4semidirect
211. #2159D53direct (N × H)
219. D53#2159direct (N × H)
247. #367C2×D53semidirect
251. #366C2×D53semidirect
252. C53C2×#2159semidirect
282. M32C4×D53semidirect
352. C16C2×C4×D53semidirect

Maximal subgroups

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

subgroupindex
C2×M32×D532
#427×D532
C2×#107×D532
#202765042
#202765032
#202765022
#202765012
#202765002
#367×D532
#366×D532
C53×#21592
C2×#215953