Finite Group Databasebeta

← #27302747

Subgroups — #27302747

Order 13568.

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

normal subgroups only

This group has 488 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#20279029C211yesmaximal, characteristic
2#20274705C211yesmaximal, characteristic
3#99×Dic53C211yesmaximal, characteristic
4#20272834C211yesmaximal, characteristic
5#20272833C211yesmaximal, characteristic
6#20272832C211yesmaximal, characteristic
7C53×#1390C211yesmaximal, characteristic
8C2×D4×Dic53C2211yescharacteristic
9C22×#11719919C411yescharacteristic, complemented
10#16810646—21no—
11C2×#13133134C2211yescharacteristic
12C2×#13133132C2211yescharacteristic
13#16810565—21no—
14#16810559—21no—
15#16810520—21no—
16#16810503C2211yescharacteristic
17#16810498C411yescharacteristic, complemented
18#16810496C411yescharacteristic, complemented
19#16810481—21no—
20#16810481C411yescharacteristic, complemented
21C22⋊C4×Dic53—21no—
22#16810391—21no—
23#16810389—21no—
24C2×#100×C53C2211yescharacteristic
25C2×#99×C53C2211yescharacteristic
26C53×#386C2211yescharacteristic
27C23×Dic53C2×C411yescharacteristic
28C23×Dic53C2×C411yescharacteristic
29C23×Dic53D411yescharacteristic
30C2×#11719917—21no—
31D4×Dic53—41no—
32D4×Dic53—41no—
33C2×C4⋊Dic53—21no—
34C2×C4⋊Dic53D411yescharacteristic
35C2×#11719919—21no—
36C2×#11719919—21no—
37C2×#11719919—21no—
38C2×#11719919—21no—
39C2×#11719919—41no—
40C2×#11719919—41no—
41C2×C22⋊Dic53—21no—
42C2×C22⋊Dic53—21no—
43C2×C22⋊Dic53C2×C411yescharacteristic
44C2×C22⋊Dic53C2×C411yescharacteristic
45C2×C22⋊Dic53D411yescharacteristic
46C2×C4×Dic53—41no—
47C2×C4×Dic53D411yescharacteristic
48#13133146—41no—
49#13133146—41no—
50#13133134—21no—
51#13133134—21no—
52#13133132—21no—
53#13133132Q812yes—
54#13133124—21no—
55#13133124—21no—
56#13133124—21no—
57#13133124—21no—
58#13133124—41no—
59C22×D4×C53C2311yescharacteristic
60C23×C4×C53C2×C411yescharacteristic
61C2×M16×C53—21no—
62C2×C22⋊C4×C53—21no—
63C2×C22⋊C4×C53—21no—
64C2×C22⋊C4×C53C2×C411yescharacteristic
65#100×C53—21no—
66#100×C53—21no—
67#99×C53—21no—
68#99×C53Q812yes—
69#95×C53—21no—
70C24×C53C22×C411yescharacteristic
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×C53—81no—
83C22×C4×C53C2×D411yescharacteristic
84C22×Dic53—21no—
85C22×Dic53—21no—
86C22×Dic53—21no—
87C22×Dic53—21no—
88C22×Dic53—21no—
89C22×Dic53—21no—
90C22×Dic53—42no—
91C22×Dic53—42no—
92C22×Dic53—41no—
93C22×Dic53—41no—
94C22×Dic53—81no—
95C22×Dic53C22⋊C411yescharacteristic
96C22×Dic53C22⋊C411yescharacteristic
97C22×Dic53C22⋊C411yescharacteristic
98C22×Dic53C22⋊C411yescharacteristic
99M16×C53—41no—
100M16×C53—41no—
101C2×C8×C53—41no—
102C22⋊C4×C53—22no—
103C22⋊C4×C53—41no—
104C22⋊C4×C53—41no—
105C22⋊C4×C53—81no—
106C22⋊Dic53—22no—
107C22⋊Dic53—41no—
108C22⋊Dic53—41no—
109C22⋊Dic53—41no—
110C22⋊Dic53—41no—
111C22⋊Dic53—81no—
112C4⋊Dic53—41no—
113C4⋊Dic53—81no—
114#11719919—42no—
115#11719919—42no—
116#11719919—41no—
117#11719919—41no—
118#11719919—81no—
119C4×Dic53—21no—
120C4×Dic53—21no—
121C4×Dic53—82no—
122#11719917—41no—
123#11719917—41no—
124C2×#88322—41no—
125C23×C53—21no—
126C23×C53—21no—
127C23×C53—21no—
128C23×C53—21no—
129C23×C53—41no—
130C23×C53—41no—
131C23×C53—41no—
132C23×C53—41no—
133C23×C53C2×C22⋊C411yescharacteristic
134C23×C53C4×Q811yescharacteristic
135C23×C53C4×Q811yescharacteristic
136C23×C53C22⋊Q811yescharacteristic
137C23×C53C22⋊Q811yescharacteristic
138D4×C53—41no—
139D4×C53—41no—
140D4×C53—41no—
141D4×C53—41no—
142C2×C4×C53—21no—
143C2×C4×C53—21no—
144C2×C4×C53—41no—
145C2×C4×C53—41no—
146C2×C4×C53—41no—
147C2×C4×C53—41no—
148C2×C4×C53—42no—
149C2×C4×C53—41no—
150C2×C4×C53—41no—
151C2×C4×C53—81no—
152C2×C4×C53—81no—
153C2×C4×C53—82no—
154C2×C4×C53#12411yescharacteristic
155C2×C4×C53C4⋊Q811yescharacteristic
156C2×Dic53—22no—
157C2×Dic53—21no—
158C2×Dic53—21no—
159C2×Dic53—21no—
160C2×Dic53—42no—
161C2×Dic53—42no—
162C2×Dic53—42no—
163C2×Dic53—41no—
164C2×Dic53—41no—
165C2×Dic53—41no—
166C2×Dic53—41no—
167C2×Dic53—41no—
168C2×Dic53—41no—
169C2×Dic53—41no—
170C2×Dic53—42no—
171C2×Dic53—41no—
172C2×Dic53—41no—
173C2×Dic53—81no—
174C2×Dic53—81no—
175C2×Dic53—81no—
176C2×Dic53—82no—
177C2×Dic53#9911yescharacteristic
178C2×Dic53#9911yescharacteristic
179C8×C53—81no—
180#88322—81no—
181#13295—531nomaximal, Sylow
182C22×C53—21no—
183C22×C53—21no—
184C22×C53—21no—
185C22×C53—21no—
186C22×C53—21no—
187C22×C53—21no—
188C22×C53—41no—
189C22×C53—41no—
190C22×C53—41no—
191C22×C53—41no—
192C22×C53—42no—
193C22×C53—41no—
194C22×C53—41no—
195C22×C53—41no—
196C22×C53—41no—
197C22×C53—81no—
198C22×C53#39111yescharacteristic
199C22×C53C2×#9911yescharacteristic
200C22×C53#41011yescharacteristic
201C4×C53—21no—
202C4×C53—21no—
203C4×C53—81no—
204C4×C53—82no—
205C4×C53—81no—
206Dic53—21no—
207Dic53—41no—
208Dic53—41no—
209Dic53—41no—
210Dic53—81no—
211Dic53—82no—
212Dic53—81no—
213Dic53#35212yes—
214#2487—531no—
215#1800—531no—
216C4×#99—531no—
217#1390—531no—
218#1390—531no—
219#1390—531no—
220#1390D5311yescharacteristic
221C2×C53—21no—
222C2×C53—21no—
223C2×C53—41no—
224C2×C53—41no—
225C2×C53—41no—
226C2×C53—41no—
227C2×C53#199611yescharacteristic
228C2×C53C2×#35211yescharacteristic
229C2×C53#212211yescharacteristic
230C2×C4×D4—531no—
231C22×C4⋊C4—531no—
232#423—1061no—
233C2×#100—531no—
234C2×#100C2×D5311yescharacteristic
235C2×#99—531no—
236C2×#99C2×D5311yescharacteristic
237#400—1061no—
238#399—1061no—
239#389—1061no—
240#386—531no—
241#386C2×D5311yescharacteristic
242#385—531no—
243#385—531no—
244#380—531no—
245#380—1061no—
246C4×C22⋊C4—1061no—
247#343—1061no—
248#342—1061no—
249C53#1329511yescharacteristic, complemented, Sylow
250C22×D4C22×D5311yescharacteristic
251C23×C4—531no—
252C23×C4—531no—
253C23×C4—531no—
254C23×C4C4×D5311yescharacteristic
255C2×M16—21no—
256C2×M16—1061no—
257C4×D4—2121no—
258C4×D4—2121no—
259C2×C4⋊C4—531no—
260C2×C4⋊C4—1061no—
261C2×C4⋊C4—1061no—
262C2×C4⋊C4—1061no—
263C2×C4⋊C4—1061no—
264C2×C4⋊C4—1061no—
265C2×C4⋊C4—2121no—
266C2×C4⋊C4—2121no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—531no—
270C2×C22⋊C4—531no—
271C2×C22⋊C4—531no—
272C2×C22⋊C4—1061no—
273C2×C22⋊C4—1061no—
274C2×C22⋊C4C4×D5311yescharacteristic
275C2×C42—531no—
276C2×C42—2121no—
277C4⋊C8—2121no—
278C4⋊C8—2121no—
279#100—21no—
280#100—21no—
281#100—1061no—
282#100—1061no—
283#99—21no—
284#99—532no—
285#99—1061no—
286#99Dic10612yes—
287#95—21no—
288#95—1061no—
289#95—1061no—
290#95—1061no—
291#95—1061no—
292#95—2121no—
293C24C2×C4×D5311yescharacteristic
294C24D4×D5311yescharacteristic
295C2×D4—41no—
296C2×D4—41no—
297C2×D4C2×Dic10611yescharacteristic
298C2×D4#1171994511yescharacteristic
299C2×D4#1171994711yescharacteristic
300C2×D4Q8×D5311yescharacteristic
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—81no—
306C22×C4—531no—
307C22×C4—531no—
308C22×C4—531no—
309C22×C4—531no—
310C22×C4—1061no—
311C22×C4—1061no—
312C22×C4—1061no—
313C22×C4—1061no—
314C22×C4—1061no—
315C22×C4—1061no—
316C22×C4—4241no—
317C22×C4—2122no—
318C22×C4—2122no—
319C22×C4—2121no—
320C22×C4—2121no—
321C22×C4D4×D5311yescharacteristic
322M16—41no—
323M16—41no—
324M16—2121no—
325M16—2121no—
326C2×C8—41no—
327C2×C8—2121no—
328C4⋊C4—4241no—
329C4⋊C4—4241no—
330C4⋊C4—2121no—
331C4⋊C4—2121no—
332C4⋊C4—2121no—
333C4⋊C4—2122no—
334C4⋊C4—2122no—
335C22⋊C4—22no—
336C22⋊C4—41no—
337C22⋊C4—41no—
338C22⋊C4—81no—
339C22⋊C4—1062no—
340C22⋊C4—4241no—
341C22⋊C4—2121no—
342C22⋊C4—2121no—
343C22⋊C4—2121no—
344C22⋊C4—2121no—
345C42—1061no—
346C42—1061no—
347C42—4242no—
348C23—21no—
349C23—21no—
350C23—21no—
351C23—21no—
352C23—41no—
353C23—41no—
354C23—41no—
355C23—41no—
356C23C22⋊C4×D5311yescharacteristic
357C23#1313318211yescharacteristic
358C23C4×Dic10611yescharacteristic
359C23#1313319411yescharacteristic
360C23#1313319511yescharacteristic
361D4—41no—
362D4—41no—
363D4—41no—
364D4—41no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—42no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—82no—
377C2×C4—531no—
378C2×C4—531no—
379C2×C4—1062no—
380C2×C4—1061no—
381C2×C4—1061no—
382C2×C4—1061no—
383C2×C4—4241no—
384C2×C4—4241no—
385C2×C4—4241no—
386C2×C4—4242no—
387C2×C4—2122no—
388C2×C4—2122no—
389C2×C4—2122no—
390C2×C4—2122no—
391C2×C4—2121no—
392C2×C4—2121no—
393C2×C4—2121no—
394C2×C4—2121no—
395C2×C4—2121no—
396C2×C4—2121no—
397C2×C4—2121no—
398C2×C4—2121no—
399C2×C4—2121no—
400C2×C4#1313320311yescharacteristic
401C2×C4#1313321611yescharacteristic
402C8—81no—
403C8—4241no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—42no—
415C22—41no—
416C22—41no—
417C22—41no—
418C22—41no—
419C22—81no—
420C22#1681052711yescharacteristic
421C22#99×D5311yescharacteristic
422C22#1681059511yescharacteristic
423C4—21no—
424C4—21no—
425C4—81no—
426C4—82no—
427C4—81no—
428C4—532no—
429C4—1061no—
430C4—4241no—
431C4—4242no—
432C4—4241no—
433C4—2121no—
434C4—2121no—
435C4—2121no—
436C2—21no—
437C2—21no—
438C2—41no—
439C2—41no—
440C2—41no—
441C2—41no—
442C2#2027569811yescharacteristic
443C2#352×D5311yescharacteristic
444C2#2027636311yescharacteristic
subgroupG/H
centerC22#99×D53
commutatorC24×C53C22×C4
FrattiniC22×D4C22×D53
FittingC53×#1390C2
socleC22×C53C2×#99
prime radical, cosocleC22×D4×C53C23
hypercenter#1390D53
hypocenterC53#13295
LCSG ▷ C24×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ #1390

Complements

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

NH ≅ G/Ntype
9. C22×#11719919C4semidirect
17. #16810498C4semidirect
18. #16810496C4semidirect
20. #16810481C4semidirect
249. C53#13295semidirect

Maximal subgroups

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

subgroupindex
#202790292
#202747052
#99×Dic532
#202728342
#202728332
#202728322
C53×#13902
#1329553