Finite Group Databasebeta

← #27301596

Subgroups — #27301596

Order 13568.

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

normal subgroups only

This group has 724 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×#16810412C212yesmaximal, complemented
3#20274643C211yesmaximal, characteristic
4#20272832C212yesmaximal, complemented
5C2×C53×#410C211yesmaximal, characteristic
6C2×#13133198C411yescharacteristic, complemented
7C2×#13133198C2211yescharacteristic
8C22×#11719919C411yescharacteristic, complemented
9C2×#13133136—21no—
10C2×#13133134C2212yes—
11#16810587—21no—
12#16810542—21no—
13#16810537—21no—
14#16810503C411yescharacteristic, complemented
15#16810503C2211yescharacteristic
16#16810496—21no—
17#16810496—21no—
18#16810486C411yescharacteristic, complemented
19#16810412—24no—
20C22×#41×C53C2211yescharacteristic
21C2×#117×C53—21no—
22C53×#410—24no—
23C2×#99×C53—21no—
24C2×#99×C53C2212yes—
25C23×Dic53C2×C411yescharacteristic
26C23×Dic53C2×C411yescharacteristic
27C2×#11719917—22no—
28#13133198—22no—
29#13133198—22no—
30#13133198—42no—
31C2×#11719919—21no—
32C2×#11719919—21no—
33C2×#11719919—21no—
34C2×#11719919—21no—
35C2×#11719919—41no—
36C2×#11719919—41no—
37C2×C22⋊Dic53—21no—
38C2×C22⋊Dic53—21no—
39C2×C22⋊Dic53—21no—
40C2×C22⋊Dic53—21no—
41C2×C22⋊Dic53C2×C411yescharacteristic
42C2×C22⋊Dic53C2×C411yescharacteristic
43#13133136—42no—
44#13133134—22no—
45#13133134—22no—
46C22⋊#88322—41no—
47C22⋊#88322—41no—
48#13133124—21no—
49#13133124—21no—
50#13133124—21no—
51#13133124—21no—
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
64#117×C53—24no—
65#117×C53—42no—
66C2×C4⋊C4×C53—41no—
67C2×C22⋊C4×C53—22no—
68C2×C42×C53—41no—
69#99×C53—24no—
70#99×C53—22no—
71#99×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—22no—
80C2×Q8×C53—22no—
81C2×Q8×C53—41no—
82C2×D4×C53—22no—
83C2×D4×C53—21no—
84C2×D4×C53—21no—
85C2×D4×C53—22no—
86C2×D4×C53—41no—
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—21no—
94C22×C4×C53—21no—
95C22×C4×C53—21no—
96C22×C4×C53—21no—
97C22×C4×C53—22no—
98C22×C4×C53—22no—
99C22×C4×C53—21no—
100C22×C4×C53—21no—
101C22×C4×C53—21no—
102C22×C4×C53—21no—
103C22×C4×C53—22no—
104C22×C4×C53—41no—
105C22×C4×C53—41no—
106C22×C4×C53—42no—
107C22×C4×C53C22⋊C411yescharacteristic
108C22×C4×C53C22⋊C411yescharacteristic
109C22×C4×C53C22⋊C411yescharacteristic
110C22×C4×C53C2×D411yescharacteristic
111C22×Dic53—21no—
112C22×Dic53—21no—
113C22×Dic53—21no—
114C22×Dic53—21no—
115C22×Dic53—22no—
116C22×Dic53—42no—
117C22×Dic53—41no—
118C22×Dic53—41no—
119C22×Dic53—41no—
120C4⋊C4×C53—44no—
121C22⋊C4×C53—24no—
122C22⋊C4×C53—42no—
123C42×C53—44no—
124C22⋊Dic53—22no—
125C22⋊Dic53—42no—
126C22⋊Dic53—42no—
127C22⋊Dic53—42no—
128C22⋊Dic53—41no—
129C22⋊Dic53—41no—
130C22⋊Dic53—41no—
131#11719919—42no—
132#11719919—41no—
133#11719919—41no—
134#11719919—42no—
135#11719919—42no—
136#11719917—42no—
137#11719917—42no—
138C2×#88322—42no—
139C23×C53—21no—
140C23×C53—21no—
141C23×C53—21no—
142C23×C53—21no—
143C23×C53—21no—
144C23×C53—21no—
145C23×C53—21no—
146C23×C53—41no—
147C23×C53—41no—
148C23×C53—41no—
149C23×C53—41no—
150C23×C53—41no—
151C23×C53—41no—
152C23×C53C2×C22⋊C411yescharacteristic
153C23×C53C4×D412yes—
154C23×C53#12011yescharacteristic
155C23×C53#12311yescharacteristic
156Q8×C53—44no—
157D4×C53—42no—
158D4×C53—42no—
159D4×C53—44no—
160D4×C53—42no—
161D4×C53—42no—
162C2×C4×C53—22no—
163C2×C4×C53—22no—
164C2×C4×C53—22no—
165C2×C4×C53—22no—
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—44no—
180C2×C4×C53—82no—
181C2×C4×C53—82no—
182C2×C4×C53#9911yescharacteristic
183C2×C4×C53#9911yescharacteristic
184C2×C4×C53#9912yes—
185C2×C4×C53#12112yes—
186C2×Dic53—22no—
187C2×Dic53—41no—
188C2×Dic53—42no—
189C2×Dic53—41no—
190C2×Dic53—42no—
191C2×Dic53—41no—
192C2×Dic53—41no—
193C2×Dic53—41no—
194C2×Dic53—41no—
195C2×Dic53—41no—
196C2×Dic53—41no—
197C2×Dic53—41no—
198C2×Dic53—41no—
199C2×Dic53—41no—
200C2×Dic53—82no—
201#88322—82no—
202#13085—531nomaximal, Sylow
203C22×C53—21no—
204C22×C53—21no—
205C22×C53—22no—
206C22×C53—21no—
207C22×C53—21no—
208C22×C53—42no—
209C22×C53—41no—
210C22×C53—41no—
211C22×C53—42no—
212C22×C53—42no—
213C22×C53—41no—
214C22×C53—41no—
215C22×C53—41no—
216C22×C53—41no—
217C22×C53—41no—
218C22×C53—41no—
219C22×C53—41no—
220C22×C53—81no—
221C22×C53—81no—
222C22×C53#38611yescharacteristic
223C22×C53C2×#9911yescharacteristic
224C22×C53C2×#9911yescharacteristic
225C4×C53—22no—
226C4×C53—21no—
227C4×C53—21no—
228C4×C53—22no—
229C4×C53—41no—
230C4×C53—84no—
231Dic53—42no—
232Dic53—81no—
233Dic53—81no—
234Dic53—81no—
235C2×#410D5311yescharacteristic
236#2406—531no—
237C2×#352—532no—
238#1785—531no—
239#1390—532no—
240C2×C53—21no—
241C2×C53—21no—
242C2×C53—41no—
243C2×C53—42no—
244C2×C53—41no—
245C2×C53—81no—
246C2×C53#189811yescharacteristic
247C2×C53#212011yescharacteristic
248C2×C53#212311yescharacteristic
249C22×#41C2×D5311yescharacteristic
250C2×#123—531no—
251C2×#123—531no—
252C2×#117—21no—
253C22×C4⋊C4—531no—
254C2×#101—1061no—
255C2×#100—532no—
256#410—24no—
257C2×#99—21no—
258C2×#99C2×D5312yes—
259#407—1061no—
260#395—1061no—
261#394—1061no—
262#386—531no—
263#386—531no—
264#385—1061no—
265#385—1061no—
266#381—531no—
267#352—1064no—
268C53#1308511yescharacteristic, complemented, Sylow
269C2×#41—22no—
270C2×#41—22no—
271C2×#41—41no—
272C2×#41—41no—
273C2×#41—42no—
274C22×Q8D21211yescharacteristic
275C22×D4C4×D5311yescharacteristic
276C22×D4C22⋊D5311yescharacteristic
277C22×D4C22×D5311yescharacteristic
278C23×C4—531no—
279C23×C4—531no—
280C23×C4C4×D5311yescharacteristic
281C23×C4D21211yescharacteristic
282C23×C4C22⋊D5311yescharacteristic
283C2×M16—1062no—
284#123—1062no—
285#123—1062no—
286#123—2122no—
287#117—24no—
288#117—42no—
289C2×C4⋊C4—41no—
290C2×C4⋊C4—1061no—
291C2×C4⋊C4—1061no—
292C2×C4⋊C4—1061no—
293C2×C4⋊C4—1061no—
294C2×C4⋊C4—2121no—
295C2×C4⋊C4—2121no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—531no—
298C2×C22⋊C4—531no—
299C2×C22⋊C4—1061no—
300C2×C22⋊C4—1061no—
301C2×C22⋊C4—1061no—
302C2×C22⋊C4—1061no—
303C2×C42—41no—
304#101—2122no—
305#100—1062no—
306#100—1062no—
307#99—24no—
308#99—22no—
309#99C22⋊D5314yes—
310C22⋊C8—2121no—
311C22⋊C8—2121no—
312#95—1061no—
313#95—1061no—
314#95—1061no—
315#95—1061no—
316C24#1171992111yescharacteristic
317C24C2×C4×D5311yescharacteristic
318C24C2×C22⋊D5311yescharacteristic
319#41—44no—
320#41—44no—
321#41—44no—
322#41—44no—
323C2×Q8—22no—
324C2×Q8—22no—
325C2×Q8—41no—
326C2×D4—22no—
327C2×D4—21no—
328C2×D4—21no—
329C2×D4—22no—
330C2×D4—41no—
331C2×D4—41no—
332C2×D4—41no—
333C2×D4—41no—
334C2×D4—41no—
335C2×D4#1171994512yes—
336C2×D4C2×C22⋊D5312yes—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—22no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—42no—
351C22×C4—1061no—
352C22×C4—1062no—
353C22×C4—1061no—
354C22×C4—1061no—
355C22×C4—1061no—
356C22×C4—2122no—
357C22×C4—2121no—
358C22×C4—2121no—
359C22×C4—2121no—
360C22×C4#1171992111yescharacteristic
361C22×C4#1171992111yescharacteristic
362C22×C4#1171992111yescharacteristic
363C22×C4C2×D21211yescharacteristic
364M16—2122no—
365M16—2122no—
366C2×C8—2122no—
367C4⋊C4—44no—
368C4⋊C4—2122no—
369C4⋊C4—2121no—
370C4⋊C4—2121no—
371C4⋊C4—2122no—
372C4⋊C4—2122no—
373C22⋊C4—24no—
374C22⋊C4—42no—
375C22⋊C4—1062no—
376C22⋊C4—2122no—
377C22⋊C4—2122no—
378C22⋊C4—2122no—
379C22⋊C4—2121no—
380C22⋊C4—2121no—
381C22⋊C4—2121no—
382C42—44no—
383C23—21no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23—21no—
389C23—21no—
390C23—41no—
391C23—41no—
392C23—41no—
393C23—41no—
394C23—41no—
395C23—41no—
396C23C2×#1171992111yescharacteristic
397C23C4×C22⋊D5312yes—
398C23#1313318611yescharacteristic
399C23#1313320011yescharacteristic
400Q8—44no—
401D4—42no—
402D4—42no—
403D4—44no—
404D4—42no—
405D4—42no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—44no—
424C2×C4—82no—
425C2×C4—82no—
426C2×C4—1062no—
427C2×C4—4242no—
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—2121no—
437C2×C4—2121no—
438C2×C4—2122no—
439C2×C4—2121no—
440C2×C4—2122no—
441C2×C4#1313313112yes—
442C2×C4#1313313111yescharacteristic
443C2×C4#1313313111yescharacteristic
444C2×C4#1313319112yes—
445C8—4242no—
446C22—21no—
447C22—21no—
448C22—21no—
449C22—21no—
450C22—22no—
451C22—42no—
452C22—41no—
453C22—41no—
454C22—42no—
455C22—42no—
456C22—41no—
457C22—41no—
458C22—41no—
459C22—41no—
460C22—41no—
461C22—41no—
462C22—41no—
463C22—81no—
464C22—81no—
465C22#1681050211yescharacteristic
466C22C2×#1313313111yescharacteristic
467C22C2×#1313313111yescharacteristic
468C4—22no—
469C4—21no—
470C4—21no—
471C4—22no—
472C4—41no—
473C4—84no—
474C4—4241no—
475C4—4241no—
476C4—4241no—
477C4—2122no—
478C2—21no—
479C2—21no—
480C2—41no—
481C2—42no—
482C2—41no—
483C2—81no—
484C2#2027516811yescharacteristic
485C2#2027635411yescharacteristic
486C2#2027636811yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC24×C53C22×C4
FrattiniC22×D4C22×D53
FittingC2×C53×#410C2
socleC22×C53C2×#99
prime radicalC22×#41×C53C22
cosocleC22×D4×C53C23
hypercenterC2×#410D53
hypocenterC53#13085
LCSG ▷ C24×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ C2×#410

Complements

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

NH ≅ G/Ntype
2. C2×#16810412C2semidirect
4. #20272832C2semidirect
6. C2×#13133198C4semidirect
8. C22×#11719919C4semidirect
14. #16810503C4semidirect
18. #16810486C4semidirect
268. C53#13085semidirect

Maximal subgroups

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

subgroupindex
#202780462
C2×#168104122
#202746432
#202728322
C2×C53×#4102
#1308553