Finite Group Databasebeta

← #27357994

Subgroups — #27357994

Order 13568.

1282 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285463C212yesmaximal, complemented
2#20280457C211yesmaximal, characteristic, complemented
3#20273718C214yesmaximal, complemented
4#20273650C214yesmaximal, complemented
5#20273283C212yesmaximal, complemented
6C2×#16810349C211yesmaximal, characteristic, complemented
7C2×C53×#537C211yesmaximal, characteristic, complemented
8C2×#13133200—21no—
9C2×#13133191C2211yescharacteristic
10#16810936—22no—
11#16810934C2214yescomplemented
12#16810857—24no—
13#16810757—22no—
14#16810756C2214yescomplemented
15C2×C22⋊#88322C2212yes—
16#16810553C2212yescomplemented
17#16810545—22no—
18#16810540C2212yescomplemented
19#16810502—21no—
20#16810483C2211yescharacteristic
21#16810357C2218yescomplemented
22#16810349C2214yescomplemented
23C22×#41×C53—21no—
24C53×#537—24no—
25C53×#537—22no—
26C53×#537C2214yes—
27C2×C22⋊Q8×C53—22no—
28C2×#121×C53—22no—
29C2×#121×C53—21no—
30C2×#121×C53C2212yes—
31C22×C4⋊C4×C53C2211yescharacteristic
32C22×D212C2311yescharacteristic
33C22×C22⋊D53—21no—
34C2×#11719922—24no—
35C22×#88322—22no—
36#13133200—44no—
37#13133191—22no—
38#13133191—22no—
39C2×C4⋊Dic53—21no—
40C2×C4⋊Dic53C2311yescharacteristic
41C2×#11719919—22no—
42C2×C22⋊Dic53—21no—
43C2×#11719921—22no—
44C2×#11719921—21no—
45C2×#11719921—22no—
46#13133150—24no—
47#13133138—24no—
48#13133137—24no—
49C22⋊#88322—22no—
50C22⋊#88322C2314yes—
51#13133124—22no—
52#13133124C2312yes—
53C2×#41×C53—24no—
54C2×#41×C53—24no—
55C2×#41×C53—44no—
56C2×#41×C53—44no—
57C22×Q8×C53—21no—
58C22×D4×C53—21no—
59C22×D4×C53—22no—
60C22×D4×C53—21no—
61C22×D4×C53D412yes—
62C23×C4×C53—22no—
63C23×C4×C53D411yescharacteristic
64C23×C4×C53D411yescharacteristic
65C23×C4×C53C2311yescharacteristic
66C22⋊Q8×C53—28no—
67C22⋊Q8×C53—28no—
68#121×C53—28no—
69#121×C53—24no—
70#121×C53—28no—
71#121×C53—24no—
72#121×C53—22no—
73#121×C53—22no—
74#121×C53D414yes—
75#121×C53C2314yes—
76C2×C4⋊C4×C53—21no—
77C2×C4⋊C4×C53—22no—
78C2×C4⋊C4×C53—22no—
79C2×C4⋊C4×C53D412yes—
80C2×C4⋊C4×C53D412yes—
81C2×C4⋊C4×C53C2312yes—
82C2×C22⋊C4×C53—22no—
83C2×C22⋊C4×C53—22no—
84C24×C53—22no—
85C24×C53C2×D411yescharacteristic
86C23×D53—21no—
87#41×C53—48no—
88#41×C53—416no—
89#41×C53—48no—
90C2×Q8×C53—24no—
91C2×Q8×C53—44no—
92C2×D4×C53—24no—
93C2×D4×C53—24no—
94C2×D4×C53—24no—
95C2×D4×C53—24no—
96C2×D4×C53—22no—
97C2×D4×C53—44no—
98C2×D4×C53—44no—
99C2×D4×C53—44no—
100C2×D4×C53—82no—
101C2×D4×C53—82no—
102C2×D4×C53C2×D414yes—
103C22×C4×C53—21no—
104C22×C4×C53—21no—
105C22×C4×C53—22no—
106C22×C4×C53—28no—
107C22×C4×C53—21no—
108C22×C4×C53—22no—
109C22×C4×C53—21no—
110C22×C4×C53—48no—
111C22×C4×C53—81no—
112C22×C4×C53—81no—
113C22×C4×C53D814yescomplemented
114C22×C4×C53C2×D412yes—
115C22×C4×C53C2×D412yes—
116C22×C4×C53C2×D411yescharacteristic
117C22×C4×C53C2×D412yes—
118C22×C4×C53C2×D412yes—
119C22×C4×C53C2411yescharacteristic
120C2×C22⋊D53—41no—
121C2×C22⋊D53—41no—
122C2×C22⋊D53—42no—
123C2×C22⋊D53—42no—
124C22×Dic53—21no—
125C22×Dic53—22no—
126C2×D212—22no—
127C2×D212—81no—
128C4⋊C4×C53—24no—
129C4⋊C4×C53—24no—
130C4⋊C4×C53—22no—
131C4⋊C4×C53—24no—
132C4⋊C4×C53C2×D414yes—
133C22⋊C4×C53—28no—
134C22⋊C4×C53—24no—
135C22⋊C4×C53—24no—
136C22⋊Dic53—42no—
137#11719922—84no—
138#11719921—42no—
139#11719921—44no—
140#11719921—82no—
141C4⋊Dic53—22no—
142C4⋊Dic53—81no—
143#11719919—44no—
144C2×#88322—24no—
145C2×#88322—82no—
146C23×C53—24no—
147C23×C53—41no—
148C23×C53—41no—
149C23×C53—44no—
150C23×C53—44no—
151C23×C53—44no—
152C23×C53#12011yescharacteristic
153C23×C53#12011yescharacteristic
154C23×C53C2×D812yes—
155C23×C53AGL(1,ℤ/8)12yes—
156C23×C53C22×D411yescharacteristic
157C22×D53—42no—
158C22×D53—42no—
159C22×D53—41no—
160C22×D53—41no—
161C22×D53—41no—
162Q8×C53—48no—
163D4×C53—44no—
164D4×C53—48no—
165D4×C53—44no—
166D4×C53—44no—
167D4×C53—48no—
168D4×C53—82no—
169D4×C53—84no—
170C2×C4×C53—22no—
171C2×C4×C53—22no—
172C2×C4×C53—24no—
173C2×C4×C53—24no—
174C2×C4×C53—24no—
175C2×C4×C53—22no—
176C2×C4×C53—41no—
177C2×C4×C53—42no—
178C2×C4×C53—42no—
179C2×C4×C53—41no—
180C2×C4×C53—416no—
181C2×C4×C53—42no—
182C2×C4×C53—44no—
183C2×C4×C53—42no—
184C2×C4×C53—81no—
185C2×C4×C53#12012yes—
186C2×C4×C53C2×D814yes—
187C2×C4×C53C22×D412yes—
188C22⋊D53—82no—
189C22⋊D53—82no—
190C2×Dic53—42no—
191C2×Dic53—42no—
192C2×Dic53—41no—
193C2×Dic53—41no—
194C2×Dic53—42no—
195C2×Dic53—41no—
196D212—44no—
197#88322—44no—
198#18292—531nomaximal, Sylow
199C22×C53—21no—
200C22×C53—21no—
201C22×C53—21no—
202C22×C53—21no—
203C22×C53—21no—
204C22×C53—21no—
205C22×C53—42no—
206C22×C53—42no—
207C22×C53—44no—
208C22×C53—44no—
209C22×C53—44no—
210C22×C53—88no—
211C22×C53#44712yes—
212C22×C53#44712yes—
213C22×C53C2×#12011yescharacteristic
214C22×C53C22×D811yescharacteristic
215C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
216C2×D53—82no—
217C2×D53—82no—
218C2×D53—81no—
219C2×D53—81no—
220C2×D53—81no—
221C4×C53—24no—
222C4×C53—22no—
223C4×C53—22no—
224C4×C53—44no—
225C4×C53—42no—
226Dic53—82no—
227Dic53—81no—
228C2×#537D5311yescharacteristic, complemented
229#3186—532no—
230#2618—531no—
231#1610—534no—
232#1598—534no—
233#1519—532no—
234C2×#324—531no—
235C2×C53—21no—
236C2×C53—21no—
237C2×C53—22no—
238C2×C53—22no—
239C2×C53—84no—
240C2×C53C2×#44711yescharacteristic
241C2×C53#302011yescharacteristic
242C2×C53#302611yescharacteristic
243D53—161no—
244C22×#41—21no—
245#537—24no—
246#537—22no—
247#537C2×D5314yes—
248C2×#123—1061no—
249C2×C22⋊Q8—22no—
250C2×#121—22no—
251C2×#121—21no—
252C2×#121—531no—
253C2×#121C2×D5312yes—
254C22×C4⋊C4C2×D5311yescharacteristic
255#480—534no—
256#480—1062no—
257#466—1064no—
258#447—534no—
259#447—1062no—
260C2×C22⋊C8—532no—
261#398—532no—
262#396—1062no—
263#394—532no—
264#386—1061no—
265#380—531no—
266#327—538no—
267#324—534no—
268C53#1829211yescharacteristic, complemented, Sylow
269C2×#41—24no—
270C2×#41—24no—
271C2×#41—44no—
272C2×#41—44no—
273C22×Q8—21no—
274C22×D4—21no—
275C22×D4—22no—
276C22×D4—21no—
277C22×D4—531no—
278C22×D4—1061no—
279C22×D4C22⋊D5312yes—
280C23×C4—22no—
281C23×C4C22⋊D5311yescharacteristic
282C23×C4C22⋊D5311yescharacteristic
283C23×C4C22×D5311yescharacteristic
284C2×D8—1064no—
285C22×C8—1062no—
286#123—2124no—
287C22⋊Q8—28no—
288C22⋊Q8—28no—
289#121—24no—
290#121—28no—
291#121—28no—
292#121—24no—
293#121—22no—
294#121—22no—
295#121—1062no—
296#121—1062no—
297#121C22⋊D5314yes—
298#121C22×D5314yes—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—531no—
303C2×C4⋊C4—1061no—
304C2×C4⋊C4—1062no—
305C2×C4⋊C4C22⋊D5312yes—
306C2×C4⋊C4C22⋊D5312yes—
307C2×C4⋊C4C22×D5312yes—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—1062no—
311C2×C22⋊C4—1061no—
312C2×C22⋊C4—1061no—
313C2×C22⋊C4—1062no—
314#107—1064no—
315#102—1064no—
316#102—1064no—
317C22⋊C8—534no—
318C22⋊C8—1062no—
319#95—532no—
320#95—1062no—
321C24—22no—
322C24—1061no—
323C24C2×C22⋊D5311yescharacteristic
324#41—416no—
325#41—48no—
326#41—48no—
327C2×Q8—24no—
328C2×Q8—44no—
329C2×D4—24no—
330C2×D4—24no—
331C2×D4—24no—
332C2×D4—24no—
333C2×D4—22no—
334C2×D4—44no—
335C2×D4—44no—
336C2×D4—44no—
337C2×D4—82no—
338C2×D4—82no—
339C2×D4—1062no—
340C2×D4—4241no—
341C2×D4—2121no—
342C2×D4—2121no—
343C2×D4—2122no—
344C2×D4—2122no—
345C2×D4C2×C22⋊D5314yes—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—22no—
349C22×C4—28no—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—21no—
353C22×C4—48no—
354C22×C4—81no—
355C22×C4—81no—
356C22×C4—1062no—
357C22×C4—1061no—
358C22×C4#1171992214yescomplemented
359C22×C4C2×C22⋊D5312yes—
360C22×C4C2×C22⋊D5312yes—
361C22×C4C2×C22⋊D5311yescharacteristic
362C22×C4C2×C22⋊D5312yes—
363C22×C4C2×C22⋊D5312yes—
364C22×C4C23×D5311yescharacteristic
365D8—4244no—
366C2×C8—1064no—
367C2×C8—4242no—
368C4⋊C4—24no—
369C4⋊C4—24no—
370C4⋊C4—22no—
371C4⋊C4—24no—
372C4⋊C4—1062no—
373C4⋊C4—4241no—
374C4⋊C4—2124no—
375C4⋊C4C2×C22⋊D5314yes—
376C22⋊C4—28no—
377C22⋊C4—24no—
378C22⋊C4—24no—
379C22⋊C4—4242no—
380C22⋊C4—2122no—
381C22⋊C4—2122no—
382C22⋊C4—2124no—
383C23—24no—
384C23—41no—
385C23—41no—
386C23—44no—
387C23—44no—
388C23—44no—
389C23—2122no—
390C23—2122no—
391C23—2121no—
392C23—2121no—
393C23—2121no—
394C23#1313318611yescharacteristic
395C23#1313318611yescharacteristic
396C23C2×#1171992212yes—
397C23#1313324312yes—
398C23C22×C22⋊D5311yescharacteristic
399Q8—48no—
400D4—44no—
401D4—48no—
402D4—48no—
403D4—44no—
404D4—44no—
405D4—82no—
406D4—84no—
407D4—4242no—
408D4—4242no—
409D4—2124no—
410C2×C4—24no—
411C2×C4—24no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—24no—
416C2×C4—41no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—41no—
420C2×C4—416no—
421C2×C4—42no—
422C2×C4—44no—
423C2×C4—42no—
424C2×C4—81no—
425C2×C4—2122no—
426C2×C4—2122no—
427C2×C4—2121no—
428C2×C4—2122no—
429C2×C4—2121no—
430C2×C4—2121no—
431C2×C4#1313318612yes—
432C2×C4C2×#1171992214yes—
433C2×C4C22×C22⋊D5312yes—
434C8—2124no—
435C22—21no—
436C22—21no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—21no—
441C22—42no—
442C22—42no—
443C22—44no—
444C22—44no—
445C22—44no—
446C22—88no—
447C22—4242no—
448C22—4242no—
449C22—4241no—
450C22—4241no—
451C22—4241no—
452C22#1681075712yes—
453C22#1681075712yes—
454C22C2×#1313318611yescharacteristic
455C22C22×#1171992211yescharacteristic
456C22C2×#1313324311yescharacteristic
457C4—22no—
458C4—22no—
459C4—24no—
460C4—42no—
461C4—44no—
462C4—4242no—
463C4—4241no—
464C2—21no—
465C2—22no—
466C2—21no—
467C2—22no—
468C2—84no—
469C2—8481no—
470C2C2×#1681075711yescharacteristic
471C2#2028407911yescharacteristic
472C2#2028411211yescharacteristic
subgroupG/H
centerC22C2×#13133186
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#537C2
socleC22×C53C2×#120
hypercenterC2×#537D53
hypocenterC53#18292
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#537

Complements

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

NH ≅ G/Ntype
1. #20285463C2semidirect
2. #20280457C2semidirect
3. #20273718C2semidirect
4. #20273650C2semidirect
5. #20273283C2semidirect
6. C2×#16810349C2semidirect
7. C2×C53×#537C2semidirect
11. #16810934C22semidirect
14. #16810756C22semidirect
16. #16810553C22semidirect
18. #16810540C22semidirect
21. #16810357C22semidirect
22. #16810349C22semidirect
113. C22×C4×C53D8semidirect
228. C2×#537D53semidirect
268. C53#18292semidirect
358. C22×C4#11719922semidirect

Maximal subgroups

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

subgroupindex
#202854632
#202804572
#202737182
#202736502
#202732832
C2×#168103492
C2×C53×#5372
#1829253