Finite Group Databasebeta

← #27301576

Subgroups — #27301576

Order 13568.

722 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 722 conjugacy classes of subgroups — too many to draw. Only its 107 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20278046C211yesmaximal, characteristic
2C2×#16810414C212yesmaximal, complemented
3#20274633C211yesmaximal, characteristic
4#20272834C212yesmaximal, complemented
5C2×C53×#413C211yesmaximal, characteristic
6C2×#13133198C411yescharacteristic, complemented
7C2×#13133198C2211yescharacteristic
8C22×#11719919C411yescharacteristic, complemented
9C2×#13133132—21no—
10C2×#13133132C2212yes—
11#16810542—21no—
12#16810537—21no—
13#16810503C411yescharacteristic, complemented
14#16810503C2211yescharacteristic
15#16810496—21no—
16#16810496—21no—
17#16810486—21no—
18#16810486C411yescharacteristic, complemented
19#16810414—24no—
20C22×#41×C53C2211yescharacteristic
21C22×M16×C53—21no—
22C53×#413—24no—
23C2×#101×C53—21no—
24C2×#100×C53C2212yes—
25C23×Dic53C2×C411yescharacteristic
26C23×Dic53C2×C411yescharacteristic
27#13133198—22no—
28#13133198—22no—
29#13133198—42no—
30C2×#11719919—21no—
31C2×#11719919—21no—
32C2×#11719919—21no—
33C2×#11719919—21no—
34C2×#11719919—41no—
35C2×#11719919—41no—
36C2×C22⋊Dic53—21no—
37C2×C22⋊Dic53—21no—
38C2×C22⋊Dic53—21no—
39C2×C22⋊Dic53—21no—
40C2×C22⋊Dic53—22no—
41C2×C22⋊Dic53C2×C411yescharacteristic
42C2×C22⋊Dic53C2×C411yescharacteristic
43#13133132—22no—
44#13133132—22no—
45#13133132—42no—
46#13133124—21no—
47#13133124—21no—
48#13133124—21no—
49#13133124—21no—
50#13133124—41no—
51#13133124—41no—
52C2×#41×C53—22no—
53C2×#41×C53—22no—
54C2×#41×C53—41no—
55C2×#41×C53—41no—
56C2×#41×C53—42no—
57C22×Q8×C53D411yescharacteristic
58C22×D4×C53C2×C411yescharacteristic
59C22×D4×C53D411yescharacteristic
60C22×D4×C53C2311yescharacteristic
61C23×C4×C53C2×C411yescharacteristic
62C23×C4×C53D411yescharacteristic
63C23×C4×C53D411yescharacteristic
64C2×M16×C53—24no—
65C2×M16×C53—22no—
66C2×M16×C53—42no—
67C2×M16×C53—41no—
68C22×C8×C53—41no—
69#101×C53—24no—
70#100×C53—22no—
71#100×C53D414yes—
72C24×C53C22⋊C411yescharacteristic
73C24×C53C22×C411yescharacteristic
74C24×C53C2×D411yescharacteristic
75#41×C53—44no—
76#41×C53—44no—
77#41×C53—44no—
78#41×C53—44no—
79C2×Q8×C53—21no—
80C2×Q8×C53—21no—
81C2×Q8×C53—41no—
82C2×Q8×C53—41no—
83C2×D4×C53—22no—
84C2×D4×C53—22no—
85C2×D4×C53—22no—
86C2×D4×C53—22no—
87C2×D4×C53—41no—
88C2×D4×C53—41no—
89C2×D4×C53—41no—
90C2×D4×C53—41no—
91C2×D4×C53C2×D412yes—
92C2×D4×C53#4112yes—
93C22×C4×C53—22no—
94C22×C4×C53—21no—
95C22×C4×C53—21no—
96C22×C4×C53—21no—
97C22×C4×C53—21no—
98C22×C4×C53—22no—
99C22×C4×C53—22no—
100C22×C4×C53—21no—
101C22×C4×C53—21no—
102C22×C4×C53—21no—
103C22×C4×C53—21no—
104C22×C4×C53—41no—
105C22×C4×C53—41no—
106C22×C4×C53C22⋊C411yescharacteristic
107C22×C4×C53C22⋊C411yescharacteristic
108C22×C4×C53C22⋊C411yescharacteristic
109C22×C4×C53C2×D411yescharacteristic
110C22×Dic53—21no—
111C22×Dic53—21no—
112C22×Dic53—21no—
113C22×Dic53—21no—
114C22×Dic53—22no—
115C22×Dic53—42no—
116C22×Dic53—41no—
117C22×Dic53—41no—
118C22×Dic53—41no—
119C22×Dic53—42no—
120M16×C53—24no—
121M16×C53—44no—
122M16×C53—42no—
123C2×C8×C53—44no—
124C2×C8×C53—42no—
125C22⋊Dic53—22no—
126C22⋊Dic53—42no—
127C22⋊Dic53—42no—
128C22⋊Dic53—42no—
129C22⋊Dic53—42no—
130C22⋊Dic53—41no—
131C22⋊Dic53—42no—
132C22⋊Dic53—42no—
133#11719919—42no—
134#11719919—42no—
135#11719919—41no—
136#11719919—41no—
137#11719919—41no—
138#11719919—41no—
139C23×C53—21no—
140C23×C53—21no—
141C23×C53—21no—
142C23×C53—21no—
143C23×C53—21no—
144C23×C53—21no—
145C23×C53—41no—
146C23×C53—41no—
147C23×C53—41no—
148C23×C53—41no—
149C23×C53—41no—
150C23×C53—41no—
151C23×C53#9912yes—
152C23×C53C2×C22⋊C411yescharacteristic
153C23×C53C4×D412yes—
154C23×C53#12112yes—
155Q8×C53—42no—
156Q8×C53—42no—
157D4×C53—42no—
158D4×C53—42no—
159D4×C53—44no—
160D4×C53—44no—
161C2×C4×C53—22no—
162C2×C4×C53—22no—
163C2×C4×C53—22no—
164C2×C4×C53—22no—
165C2×C4×C53—21no—
166C2×C4×C53—21no—
167C2×C4×C53—21no—
168C2×C4×C53—21no—
169C2×C4×C53—21no—
170C2×C4×C53—21no—
171C2×C4×C53—21no—
172C2×C4×C53—21no—
173C2×C4×C53—21no—
174C2×C4×C53—21no—
175C2×C4×C53—42no—
176C2×C4×C53—42no—
177C2×C4×C53—42no—
178C2×C4×C53—42no—
179C2×C4×C53#9911yescharacteristic
180C2×C4×C53#9911yescharacteristic
181C2×C4×C53#12011yescharacteristic
182C2×C4×C53#12311yescharacteristic
183C2×Dic53—22no—
184C2×Dic53—41no—
185C2×Dic53—42no—
186C2×Dic53—41no—
187C2×Dic53—42no—
188C2×Dic53—41no—
189C2×Dic53—41no—
190C2×Dic53—41no—
191C2×Dic53—41no—
192C2×Dic53—41no—
193C2×Dic53—41no—
194C2×Dic53—41no—
195C2×Dic53—41no—
196C2×Dic53—41no—
197C2×Dic53—82no—
198C2×Dic53—82no—
199C2×Dic53—82no—
200C2×Dic53—82no—
201C8×C53—44no—
202#13081—531nomaximal, Sylow
203C22×C53—21no—
204C22×C53—21no—
205C22×C53—22no—
206C22×C53—22no—
207C22×C53—22no—
208C22×C53—42no—
209C22×C53—42no—
210C22×C53—42no—
211C22×C53—42no—
212C22×C53—41no—
213C22×C53—41no—
214C22×C53—41no—
215C22×C53—41no—
216C22×C53—41no—
217C22×C53—41no—
218C22×C53—81no—
219C22×C53—81no—
220C22×C53#38611yescharacteristic
221C22×C53C2×#9911yescharacteristic
222C22×C53C2×#9911yescharacteristic
223C4×C53—21no—
224C4×C53—21no—
225C4×C53—21no—
226C4×C53—21no—
227C4×C53—41no—
228C4×C53—41no—
229Dic53—42no—
230Dic53—81no—
231Dic53—81no—
232Dic53—81no—
233Dic53—162no—
234C2×#413D5311yescharacteristic
235#2406—531no—
236C2×#352—532no—
237#1782—531no—
238#1390—532no—
239C2×C53—21no—
240C2×C53—21no—
241C2×C53—42no—
242C2×C53—42no—
243C2×C53—42no—
244C2×C53#189811yescharacteristic
245C2×C53#212011yescharacteristic
246C2×C53#212311yescharacteristic
247C22×#41C2×D5311yescharacteristic
248C22×M16—21no—
249C2×#123—531no—
250C2×#123—531no—
251C22×C4⋊C4—531no—
252#413—24no—
253C2×#101—21no—
254C2×#100C2×D5312yes—
255C2×#99—532no—
256C2×#99—1061no—
257#395—1061no—
258#394—1061no—
259#386—531no—
260#386—531no—
261#385—1061no—
262#385—1061no—
263#381—531no—
264#381—1061no—
265#352—1064no—
266C53#1308111yescharacteristic, complemented, Sylow
267C2×#41—22no—
268C2×#41—22no—
269C2×#41—41no—
270C2×#41—41no—
271C2×#41—42no—
272C22×Q8C22⋊D5311yescharacteristic
273C22×D4C4×D5311yescharacteristic
274C22×D4D21211yescharacteristic
275C22×D4C22×D5311yescharacteristic
276C23×C4—531no—
277C23×C4—531no—
278C23×C4C4×D5311yescharacteristic
279C23×C4D21211yescharacteristic
280C23×C4C22⋊D5311yescharacteristic
281C2×M16—24no—
282C2×M16—22no—
283C2×M16—42no—
284C2×M16—41no—
285C22×C8—41no—
286#123—1062no—
287#123—1062no—
288#123—2122no—
289C2×C4⋊C4—1061no—
290C2×C4⋊C4—1061no—
291C2×C4⋊C4—1061no—
292C2×C4⋊C4—1061no—
293C2×C4⋊C4—2121no—
294C2×C4⋊C4—2121no—
295C2×C22⋊C4—531no—
296C2×C22⋊C4—531no—
297C2×C22⋊C4—1061no—
298C2×C22⋊C4—1061no—
299C2×C22⋊C4—1061no—
300C2×C22⋊C4—1061no—
301C2×C22⋊C4—1062no—
302#101—24no—
303#100—22no—
304#100C22⋊D5314yes—
305#99—1062no—
306#99—1062no—
307#99—2122no—
308#95—1061no—
309#95—1061no—
310#95—1061no—
311#95—1061no—
312#95—2121no—
313#95—2121no—
314C24#1171992111yescharacteristic
315C24C2×C4×D5311yescharacteristic
316C24C2×D21211yescharacteristic
317#41—44no—
318#41—44no—
319#41—44no—
320#41—44no—
321C2×Q8—21no—
322C2×Q8—21no—
323C2×Q8—41no—
324C2×Q8—41no—
325C2×D4—22no—
326C2×D4—22no—
327C2×D4—22no—
328C2×D4—22no—
329C2×D4—41no—
330C2×D4—41no—
331C2×D4—41no—
332C2×D4—41no—
333C2×D4#1171994512yes—
334C2×D4C2×C22⋊D5312yes—
335C22×C4—22no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—1061no—
349C22×C4—1062no—
350C22×C4—1061no—
351C22×C4—1061no—
352C22×C4—1061no—
353C22×C4—2122no—
354C22×C4—2122no—
355C22×C4—2121no—
356C22×C4—2121no—
357C22×C4—2121no—
358C22×C4#1171992111yescharacteristic
359C22×C4#1171992111yescharacteristic
360C22×C4#1171992111yescharacteristic
361C22×C4C2×C22⋊D5311yescharacteristic
362M16—24no—
363M16—44no—
364M16—42no—
365C2×C8—44no—
366C2×C8—42no—
367C4⋊C4—2122no—
368C4⋊C4—2122no—
369C4⋊C4—2121no—
370C4⋊C4—2121no—
371C4⋊C4—2121no—
372C4⋊C4—2121no—
373C22⋊C4—1062no—
374C22⋊C4—2122no—
375C22⋊C4—2122no—
376C22⋊C4—2122no—
377C22⋊C4—2122no—
378C22⋊C4—2122no—
379C22⋊C4—2122no—
380C22⋊C4—2121no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—41no—
388C23—41no—
389C23—41no—
390C23—41no—
391C23—41no—
392C23—41no—
393C23#1313313112yes—
394C23C2×#1171992111yescharacteristic
395C23C4×C22⋊D5312yes—
396C23#1313319112yes—
397Q8—42no—
398Q8—42no—
399D4—42no—
400D4—42no—
401D4—44no—
402D4—44no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—1062no—
422C2×C4—4242no—
423C2×C4—4242no—
424C2×C4—4242no—
425C2×C4—4242no—
426C2×C4—2121no—
427C2×C4—2121no—
428C2×C4—2121no—
429C2×C4—2121no—
430C2×C4—2121no—
431C2×C4—2121no—
432C2×C4—2121no—
433C2×C4—2121no—
434C2×C4—2121no—
435C2×C4—2121no—
436C2×C4—2122no—
437C2×C4—2121no—
438C2×C4—2122no—
439C2×C4#1313313111yescharacteristic
440C2×C4#1313313111yescharacteristic
441C2×C4#1313318611yescharacteristic
442C2×C4#1313320011yescharacteristic
443C8—44no—
444C22—21no—
445C22—21no—
446C22—22no—
447C22—22no—
448C22—22no—
449C22—42no—
450C22—42no—
451C22—42no—
452C22—42no—
453C22—41no—
454C22—41no—
455C22—41no—
456C22—41no—
457C22—41no—
458C22—41no—
459C22—81no—
460C22—81no—
461C22#1681050211yescharacteristic
462C22C2×#1313313111yescharacteristic
463C22C2×#1313313111yescharacteristic
464C4—21no—
465C4—21no—
466C4—21no—
467C4—21no—
468C4—41no—
469C4—41no—
470C4—8482no—
471C4—4241no—
472C4—4241no—
473C4—4241no—
474C4—2122no—
475C2—21no—
476C2—21no—
477C2—42no—
478C2—42no—
479C2—42no—
480C2#2027516811yescharacteristic
481C2#2027635711yescharacteristic
482C2#2027637011yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC24×C53C22×C4
FrattiniC22×D4C22×D53
FittingC2×C53×#413C2
socleC22×C53C2×#99
prime radicalC22×#41×C53C22
cosocleC22×D4×C53C23
hypercenterC2×#413D53
hypocenterC53#13081
LCSG ▷ C24×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ C2×#413

Complements

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

NH ≅ G/Ntype
2. C2×#16810414C2semidirect
4. #20272834C2semidirect
6. C2×#13133198C4semidirect
8. C22×#11719919C4semidirect
13. #16810503C4semidirect
18. #16810486C4semidirect
266. C53#13081semidirect

Maximal subgroups

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

subgroupindex
#202780462
C2×#168104142
#202746332
#202728342
C2×C53×#4132
#1308153