Finite Group Databasebeta

← #27302764

Subgroups — #27302764

Order 13568.

500 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 93 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#20279886C211yesmaximal, characteristic
2#20274701C211yesmaximal, characteristic
3#99×Dic53C211yesmaximal, characteristic
4#20272840C211yesmaximal, characteristic
5#20272839C211yesmaximal, characteristic
6#20272839C211yesmaximal, characteristic
7C53×#1393C211yesmaximal, characteristic
8C2×#13133216C411yescharacteristic, complemented
9C2×D4×Dic53C2211yescharacteristic
10C2×#13133132C2211yescharacteristic
11C2×#13133132C2211yescharacteristic
12#16810535—21no—
13#16810535—21no—
14#16810526C2211yescharacteristic
15#16810489C411yescharacteristic, complemented
16#16810488C411yescharacteristic, complemented
17#16810481—21no—
18#16810481—21no—
19#16810481—21no—
20C4×#11719919C411yescharacteristic, complemented
21C22⋊C4×Dic53—21no—
22#16810389—21no—
23#16810389—21no—
24C2×#99×C53C2211yescharacteristic
25C2×#99×C53C2211yescharacteristic
26C53×#390C2211yescharacteristic
27C22×Dic106—21no—
28C23×Dic53D411yescharacteristic
29C23×Dic53D411yescharacteristic
30#13133216—41no—
31#13133216—41no—
32D4×Dic53—41no—
33D4×Dic53—41no—
34C2×C4⋊Dic53C2×C411yescharacteristic
35C2×C4⋊Dic53C2×C411yescharacteristic
36C2×#11719919—21no—
37C2×#11719919—21no—
38C2×#11719919—41no—
39C2×C22⋊Dic53—21no—
40C2×C22⋊Dic53—21no—
41C2×C22⋊Dic53—21no—
42C2×C22⋊Dic53—21no—
43C2×C22⋊Dic53D411yescharacteristic
44C2×C22⋊Dic53D411yescharacteristic
45C2×C4×Dic53—41no—
46C2×C4×Dic53C2×C411yescharacteristic
47C2×C4×Dic53C2×C411yescharacteristic
48#13133132—21no—
49#13133132—21no—
50#13133132—21no—
51#13133132Q812yescomplemented
52#13133124—21no—
53#13133124—21no—
54#13133124—21no—
55#13133124—21no—
56#13133124—21no—
57#13133124—41no—
58#13133124—41no—
59C22×D4×C53C2311yescharacteristic
60C2×C4⋊C4×C53C2×C411yescharacteristic
61C2×C22⋊C4×C53—21no—
62C2×C22⋊C4×C53—21no—
63C2×C22⋊C4×C53—21no—
64C2×C22⋊C4×C53—21no—
65C2×C42×C53C2×C411yescharacteristic
66#99×C53—21no—
67#99×C53—21no—
68#99×C53—21no—
69#99×C53Q812yescomplemented
70C24×C53C2×D411yescharacteristic
71C24×C53C2×D411yescharacteristic
72C2×D4×C53—41no—
73C2×D4×C53—41no—
74C2×D4×C53C2×Q811yescharacteristic
75C2×D4×C53C2×Q811yescharacteristic
76C2×D4×C53#4111yescharacteristic
77C2×D4×C53#4111yescharacteristic
78C22×C4×C53—21no—
79C22×C4×C53—21no—
80C22×C4×C53—41no—
81C22×C4×C53—41no—
82C22×C4×C53C22×C411yescharacteristic
83C22×Dic53—21no—
84C22×Dic53—21no—
85C22×Dic53—21no—
86C22×Dic53—21no—
87C22×Dic53—21no—
88C22×Dic53—21no—
89C22×Dic53—42no—
90C22×Dic53—42no—
91C22×Dic53—41no—
92C22×Dic53—41no—
93C22×Dic53C22⋊C411yescharacteristic
94C22×Dic53C22⋊C411yescharacteristic
95C22×Dic53C22⋊C411yescharacteristic
96C22×Dic53C22⋊C411yescharacteristic
97C2×Dic106—42no—
98C2×Dic106—81no—
99C2×Dic106—81no—
100C4⋊C4×C53—41no—
101C22⋊C4×C53—22no—
102C22⋊C4×C53—41no—
103C22⋊C4×C53—41no—
104C22⋊C4×C53—41no—
105C22⋊C4×C53—81no—
106C22⋊C4×C53—81no—
107C42×C53—41no—
108C22⋊Dic53—22no—
109C22⋊Dic53—41no—
110C22⋊Dic53—41no—
111C22⋊Dic53—41no—
112C22⋊Dic53—41no—
113C22⋊Dic53—41no—
114C22⋊Dic53—81no—
115C22⋊Dic53—81no—
116C4⋊Dic53—41no—
117C4⋊Dic53—41no—
118#11719919—42no—
119#11719919—41no—
120#11719919—41no—
121#11719919—82no—
122C4×Dic53—21no—
123C4×Dic53—21no—
124C4×Dic53—41no—
125C4×Dic53—82no—
126C23×C53—21no—
127C23×C53—21no—
128C23×C53—21no—
129C23×C53—21no—
130C23×C53—41no—
131C23×C53—41no—
132C23×C53—41no—
133C23×C53—41no—
134C23×C53C2×C22⋊C411yescharacteristic
135C23×C53C22⋊Q811yescharacteristic
136C23×C53C22⋊Q811yescharacteristic
137C23×C53#12411yescharacteristic
138C23×C53C4⋊Q811yescharacteristic
139D4×C53—41no—
140D4×C53—41no—
141D4×C53—41no—
142D4×C53—41no—
143C2×C4×C53—21no—
144C2×C4×C53—21no—
145C2×C4×C53—41no—
146C2×C4×C53—41no—
147C2×C4×C53—41no—
148C2×C4×C53—41no—
149C2×C4×C53—42no—
150C2×C4×C53—41no—
151C2×C4×C53—41no—
152C2×C4×C53—81no—
153C2×C4×C53—81no—
154C2×C4×C53—81no—
155C2×C4×C53—81no—
156C2×C4×C53—81no—
157C2×C4×C53C4×Q811yescharacteristic
158C2×C4×C53C4×Q811yescharacteristic
159C2×Dic53—22no—
160C2×Dic53—21no—
161C2×Dic53—21no—
162C2×Dic53—21no—
163C2×Dic53—42no—
164C2×Dic53—42no—
165C2×Dic53—41no—
166C2×Dic53—41no—
167C2×Dic53—41no—
168C2×Dic53—41no—
169C2×Dic53—41no—
170C2×Dic53—41no—
171C2×Dic53—41no—
172C2×Dic53—42no—
173C2×Dic53—41no—
174C2×Dic53—41no—
175C2×Dic53—81no—
176C2×Dic53—81no—
177C2×Dic53—81no—
178C2×Dic53—81no—
179C2×Dic53—81no—
180C2×Dic53—81no—
181C2×Dic53—81no—
182C2×Dic53#9911yescharacteristic
183C2×Dic53#9911yescharacteristic
184Dic106—84no—
185#13299—531nomaximal, Sylow
186C22×C53—21no—
187C22×C53—21no—
188C22×C53—21no—
189C22×C53—21no—
190C22×C53—21no—
191C22×C53—21no—
192C22×C53—41no—
193C22×C53—41no—
194C22×C53—41no—
195C22×C53—41no—
196C22×C53—41no—
197C22×C53—41no—
198C22×C53—41no—
199C22×C53—41no—
200C22×C53—81no—
201C22×C53—81no—
202C22×C53#39111yescharacteristic
203C22×C53C2×#9911yescharacteristic
204C22×C53#41011yescharacteristic
205C4×C53—21no—
206C4×C53—21no—
207C4×C53—81no—
208C4×C53—82no—
209C4×C53—81no—
210C4×C53—161no—
211Dic53—21no—
212Dic53—41no—
213Dic53—41no—
214Dic53—41no—
215Dic53—81no—
216Dic53—82no—
217Dic53—81no—
218Dic53—161no—
219Dic53#35412yes—
220#2563—531no—
221#1799—531no—
222C4×#99—531no—
223#1393—531no—
224#1393—531no—
225#1393—531no—
226#1393D5311yescharacteristic
227C2×C53—21no—
228C2×C53—21no—
229C2×C53—41no—
230C2×C53—41no—
231C2×C53—41no—
232C2×C53—41no—
233C2×C53#199611yescharacteristic
234C2×C53C2×#35411yescharacteristic
235C2×C53#212811yescharacteristic
236C2×C4⋊Q8—531no—
237C2×C4×D4—531no—
238C2×#99—531no—
239C2×#99—531no—
240C2×#99C2×D5311yescharacteristic
241C2×#99C2×D5311yescharacteristic
242#393—1061no—
243#393—1061no—
244#390—531no—
245#390C2×D5311yescharacteristic
246#382—531no—
247#382—531no—
248#380—1061no—
249#380—1061no—
250#380—1061no—
251C4×C4⋊C4—531no—
252C4×C22⋊C4—1061no—
253#342—1061no—
254#342—1061no—
255C53#1329911yescharacteristic, complemented, Sylow
256C22×Q8—1061no—
257C22×D4C22×D5311yescharacteristic
258C23×C4—531no—
259C23×C4—531no—
260C4⋊Q8—2121no—
261C4⋊Q8—2121no—
262C4×D4—2121no—
263C4×D4—2121no—
264C2×C4⋊C4—531no—
265C2×C4⋊C4—531no—
266C2×C4⋊C4—1061no—
267C2×C4⋊C4—1061no—
268C2×C4⋊C4—2121no—
269C2×C4⋊C4C4×D5311yescharacteristic
270C2×C22⋊C4—21no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—531no—
275C2×C22⋊C4—531no—
276C2×C22⋊C4—1061no—
277C2×C22⋊C4—1061no—
278C2×C22⋊C4—1061no—
279C2×C22⋊C4—1061no—
280C2×C42—531no—
281C2×C42—531no—
282C2×C42—2121no—
283C2×C42C4×D5311yescharacteristic
284#99—21no—
285#99—21no—
286#99—21no—
287#99—532no—
288#99—1061no—
289#99—1061no—
290#99—1061no—
291#99Dic10612yescomplemented
292#95—1061no—
293#95—1061no—
294#95—1061no—
295#95—1061no—
296#95—1061no—
297#95—2121no—
298#95—2121no—
299C24D4×D5311yescharacteristic
300C24D4×D5311yescharacteristic
301C2×Q8—4241no—
302C2×Q8—4241no—
303C2×Q8—2122no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4C2×Dic10611yescharacteristic
307C2×D4#1171994511yescharacteristic
308C2×D4#1171994711yescharacteristic
309C2×D4Q8×D5311yescharacteristic
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—531no—
315C22×C4—531no—
316C22×C4—531no—
317C22×C4—531no—
318C22×C4—1061no—
319C22×C4—1061no—
320C22×C4—1061no—
321C22×C4—1061no—
322C22×C4—1061no—
323C22×C4—1061no—
324C22×C4—2122no—
325C22×C4—2122no—
326C22×C4—2121no—
327C22×C4—2121no—
328C22×C4C2×C4×D5311yescharacteristic
329C4⋊C4—41no—
330C4⋊C4—4242no—
331C4⋊C4—2121no—
332C4⋊C4—2121no—
333C4⋊C4—2121no—
334C4⋊C4—2121no—
335C4⋊C4—2122no—
336C22⋊C4—22no—
337C22⋊C4—41no—
338C22⋊C4—41no—
339C22⋊C4—41no—
340C22⋊C4—81no—
341C22⋊C4—81no—
342C22⋊C4—1062no—
343C22⋊C4—4241no—
344C22⋊C4—4241no—
345C22⋊C4—2121no—
346C22⋊C4—2121no—
347C22⋊C4—2121no—
348C22⋊C4—2121no—
349C22⋊C4—2121no—
350C42—41no—
351C42—1061no—
352C42—1061no—
353C42—4242no—
354C42—2121no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—41no—
360C23—41no—
361C23—41no—
362C23—41no—
363C23C22⋊C4×D5311yescharacteristic
364C23#1313319411yescharacteristic
365C23#1313319511yescharacteristic
366C23#1313320311yescharacteristic
367C23#1313321611yescharacteristic
368Q8—4244no—
369D4—41no—
370D4—41no—
371D4—41no—
372D4—41no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—42no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—531no—
388C2×C4—531no—
389C2×C4—1062no—
390C2×C4—1061no—
391C2×C4—1061no—
392C2×C4—1061no—
393C2×C4—4241no—
394C2×C4—4241no—
395C2×C4—4241no—
396C2×C4—4241no—
397C2×C4—4241no—
398C2×C4—4241no—
399C2×C4—4241no—
400C2×C4—2122no—
401C2×C4—2122no—
402C2×C4—2122no—
403C2×C4—2121no—
404C2×C4—2121no—
405C2×C4—2121no—
406C2×C4—2121no—
407C2×C4—2121no—
408C2×C4—2121no—
409C2×C4—2121no—
410C2×C4—2121no—
411C2×C4—2121no—
412C2×C4#1313318211yescharacteristic
413C2×C4C4×Dic10611yescharacteristic
414C22—21no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—41no—
421C22—41no—
422C22—41no—
423C22—41no—
424C22—41no—
425C22—41no—
426C22—41no—
427C22—41no—
428C22—81no—
429C22—81no—
430C22#1681052711yescharacteristic
431C22#99×D5311yescharacteristic
432C22#1681059511yescharacteristic
433C4—21no—
434C4—21no—
435C4—81no—
436C4—82no—
437C4—81no—
438C4—161no—
439C4—532no—
440C4—8481no—
441C4—1061no—
442C4—4241no—
443C4—4242no—
444C4—4241no—
445C4—2121no—
446C4—2121no—
447C4—2121no—
448C2—21no—
449C2—21no—
450C2—41no—
451C2—41no—
452C2—41no—
453C2—41no—
454C2#2027569811yescharacteristic
455C2#354×D5311yescharacteristic
456C2#2027638511yescharacteristic
subgroupG/H
centerC22#99×D53
commutatorC22×C4×C53C22×C4
FrattiniC22×D4C22×D53
FittingC53×#1393C2
socleC22×C53C2×#99
prime radical, cosocleC22×D4×C53C23
hypercenter#1393D53
hypocenterC53#13299
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ #1393

Complements

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

NH ≅ G/Ntype
8. C2×#13133216C4semidirect
15. #16810489C4semidirect
16. #16810488C4semidirect
20. C4×#11719919C4semidirect
51. #13133132Q8semidirect
69. #99×C53Q8semidirect
255. C53#13299semidirect
291. #99Dic106semidirect

Maximal subgroups

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

subgroupindex
#202798862
#202747012
#99×Dic532
#202728402
#202728392
C53×#13932
#1329953