Finite Group Databasebeta

← #27281732

Subgroups — #27281732

Order 13568.

712 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 712 conjugacy classes of subgroups — too many to draw. Only its 111 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#20283160C211yesmaximal, characteristic
2#20274633C211yesmaximal, characteristic
3#20273248C212yesmaximal, complemented
4#20272568C212yesmaximal, complemented
5C53×#2432C211yesmaximal, characteristic
6C22×#11719917—21no—
7C2×#13133132—21no—
8C2×#13133132—21no—
9#16810590—24no—
10#16810587—21no—
11#16810587C2212yes—
12#16810486C2212yes—
13#16810360—24no—
14C22×#41×C53C2211yescharacteristic
15C2×C4⋊Q8×C53C411yescharacteristic, complemented
16C2×#124×C53C411yescharacteristic, complemented
17C2×C22⋊Q8×C53C2211yescharacteristic
18C53×#394—21no—
19C53×#394—21no—
20C53×#391C411yescharacteristic, complemented
21C53×#390C411yescharacteristic, complemented
22C53×#385—21no—
23C53×#385—21no—
24C4×C22⋊C4×C53C2211yescharacteristic
25C2×#11719917—22no—
26C2×#11719917—22no—
27C2×#11719917—42no—
28C2×#11719917—42no—
29C22×#88322—22no—
30C2×C22⋊Dic53—22no—
31#13133132—42no—
32#13133132—42no—
33C22⋊#88322—24no—
34C22⋊#88322—22no—
35C22⋊#88322D414yes—
36#13133124—22no—
37#13133124—22no—
38C2×#41×C53—22no—
39C2×#41×C53—22no—
40C2×#41×C53—42no—
41C2×#41×C53—42no—
42C22×Q8×C53C2×C411yescharacteristic
43C22×Q8×C53C2×C411yescharacteristic
44C22×D4×C53C2×C411yescharacteristic
45C22×D4×C53D411yescharacteristic
46C22×D4×C53D411yescharacteristic
47C23×C4×C53D411yescharacteristic
48C23×C4×C53D411yescharacteristic
49C23×C4×C53C2311yescharacteristic
50C4⋊Q8×C53—42no—
51#124×C53—42no—
52C22⋊Q8×C53—22no—
53C22⋊Q8×C53—22no—
54C2×C4⋊C4×C53—21no—
55C2×C4⋊C4×C53—21no—
56C2×C4⋊C4×C53—21no—
57C2×C4⋊C4×C53C2×C411yescharacteristic
58C2×C22⋊C4×C53—21no—
59C2×C22⋊C4×C53—21no—
60C2×C22⋊C4×C53—21no—
61C2×C22⋊C4×C53—21no—
62C2×C22⋊C4×C53—21no—
63C2×C22⋊C4×C53—21no—
64C2×C42×C53C2×C411yescharacteristic
65C2×C42×C53C2×C411yescharacteristic
66#95×C53—21no—
67#95×C53—21no—
68#95×C53—21no—
69C24×C53C22⋊C411yescharacteristic
70C24×C53C22⋊C411yescharacteristic
71C24×C53C2×D411yescharacteristic
72#41×C53—48no—
73#41×C53—44no—
74#41×C53—44no—
75C2×Q8×C53—22no—
76C2×Q8×C53—22no—
77C2×Q8×C53—22no—
78C2×Q8×C53—41no—
79C2×Q8×C53—41no—
80C2×Q8×C53—41no—
81C2×D4×C53—22no—
82C2×D4×C53—22no—
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—
90C22×C4×C53—22no—
91C22×C4×C53—22no—
92C22×C4×C53—21no—
93C22×C4×C53—21no—
94C22×C4×C53—21no—
95C22×C4×C53—21no—
96C22×C4×C53—21no—
97C22×C4×C53—21no—
98C22×C4×C53—21no—
99C22×C4×C53—21no—
100C22×C4×C53—41no—
101C22×C4×C53—41no—
102C22×C4×C53—41no—
103C22×C4×C53—41no—
104C22×C4×C53C22⋊C411yescharacteristic
105C22×C4×C53C22⋊C411yescharacteristic
106C22×C4×C53C22×C411yescharacteristic
107C22×C4×C53C2×D411yescharacteristic
108C22×C4×C53C2×D412yes—
109C22×C4×C53#4112yes—
110C22×Dic53—42no—
111C4⋊C4×C53—22no—
112C4⋊C4×C53—42no—
113C4⋊C4×C53—42no—
114C4⋊C4×C53—41no—
115C4⋊C4×C53—41no—
116C22⋊C4×C53—42no—
117C22⋊C4×C53—42no—
118C22⋊C4×C53—42no—
119C22⋊C4×C53—42no—
120C22⋊C4×C53—42no—
121C22⋊C4×C53—42no—
122C42×C53—41no—
123C42×C53—41no—
124C22⋊Dic53—42no—
125C22⋊Dic53—42no—
126#11719917—44no—
127#11719917—42no—
128#11719917—42no—
129C2×#88322—24no—
130C2×#88322—44no—
131C23×C53—21no—
132C23×C53—21no—
133C23×C53—21no—
134C23×C53—21no—
135C23×C53—21no—
136C23×C53—21no—
137C23×C53—21no—
138C23×C53—41no—
139C23×C53—41no—
140C23×C53—41no—
141C23×C53—41no—
142C23×C53—41no—
143C23×C53—41no—
144C23×C53#9912yes—
145C23×C53C2×C22⋊C411yescharacteristic
146C23×C53#12112yes—
147Q8×C53—44no—
148Q8×C53—82no—
149D4×C53—44no—
150D4×C53—44no—
151D4×C53—44no—
152C2×C4×C53—24no—
153C2×C4×C53—22no—
154C2×C4×C53—22no—
155C2×C4×C53—21no—
156C2×C4×C53—21no—
157C2×C4×C53—21no—
158C2×C4×C53—21no—
159C2×C4×C53—21no—
160C2×C4×C53—21no—
161C2×C4×C53—22no—
162C2×C4×C53—21no—
163C2×C4×C53—21no—
164C2×C4×C53—21no—
165C2×C4×C53—42no—
166C2×C4×C53—42no—
167C2×C4×C53—42no—
168C2×C4×C53—42no—
169C2×C4×C53—41no—
170C2×C4×C53—41no—
171C2×C4×C53—41no—
172C2×C4×C53—41no—
173C2×C4×C53—41no—
174C2×C4×C53—41no—
175C2×C4×C53—41no—
176C2×C4×C53—41no—
177C2×C4×C53—41no—
178C2×C4×C53—41no—
179C2×C4×C53—41no—
180C2×C4×C53#9912yes—
181C2×C4×C53C4×D412yes—
182C2×C4×C53#12011yescharacteristic
183C2×C4×C53#12311yescharacteristic
184C2×Dic53—82no—
185C2×Dic53—82no—
186C2×Dic53—82no—
187#88322—44no—
188#9443—531nomaximal, Sylow
189C22×C53—21no—
190C22×C53—21no—
191C22×C53—22no—
192C22×C53—22no—
193C22×C53—22no—
194C22×C53—22no—
195C22×C53—42no—
196C22×C53—42no—
197C22×C53—41no—
198C22×C53—41no—
199C22×C53—41no—
200C22×C53—41no—
201C22×C53—41no—
202C22×C53—81no—
203C22×C53—81no—
204C22×C53—81no—
205C22×C53—81no—
206C22×C53#38611yescharacteristic
207C22×C53C2×#9911yescharacteristic
208C22×C53C2×#9911yescharacteristic
209C4×C53—21no—
210C4×C53—21no—
211C4×C53—22no—
212C4×C53—22no—
213C4×C53—42no—
214C4×C53—41no—
215C4×C53—81no—
216C4×C53—81no—
217C4×C53—81no—
218Dic53—162no—
219#2879—531no—
220#2432D5311yescharacteristic
221#1782—531no—
222#1511—532no—
223#1284—532no—
224C2×C53—21no—
225C2×C53—21no—
226C2×C53—42no—
227C2×C53—42no—
228C2×C53—81no—
229C2×C53#188411yescharacteristic
230C2×C53#189811yescharacteristic
231C2×C53#190911yescharacteristic
232C22×#41C2×D5311yescharacteristic
233C22×M16—1061no—
234C2×C4⋊Q8Dic5311yescharacteristic, complemented
235C2×#124Dic5311yescharacteristic, complemented
236C2×C22⋊Q8C2×D5311yescharacteristic
237C2×#99—1061no—
238C2×#99—1061no—
239#408—1064no—
240#407—532no—
241#407—1061no—
242#394—21no—
243#394—21no—
244#391Dic5311yescharacteristic, complemented
245#390Dic5311yescharacteristic, complemented
246#385—21no—
247#385—21no—
248#381—532no—
249C4×C22⋊C4C2×D5311yescharacteristic
250#328—1064no—
251C53#944311yescharacteristic, complemented, Sylow
252C2×#41—22no—
253C2×#41—22no—
254C2×#41—42no—
255C2×#41—42no—
256C22×Q8C2×Dic5311yescharacteristic
257C22×Q8C2×Dic5311yescharacteristic
258C22×D4C2×Dic5311yescharacteristic
259C22×D4C22⋊D5311yescharacteristic
260C22×D4C22⋊D5311yescharacteristic
261C23×C4C22⋊D5311yescharacteristic
262C23×C4C22⋊D5311yescharacteristic
263C23×C4C22×D5311yescharacteristic
264C2×M16—1062no—
265C2×M16—1062no—
266C2×M16—2122no—
267C2×M16—2122no—
268C22×C8—1062no—
269C4⋊Q8—42no—
270#124—42no—
271C22⋊Q8—22no—
272C22⋊Q8—22no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4C2×Dic5311yescharacteristic
277C2×C22⋊C4—21no—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—21no—
280C2×C22⋊C4—21no—
281C2×C22⋊C4—21no—
282C2×C22⋊C4—21no—
283C2×C22⋊C4—1062no—
284C2×C42C2×Dic5311yescharacteristic
285C2×C42C2×Dic5311yescharacteristic
286#99—2122no—
287#99—2122no—
288C22⋊C8—534no—
289C22⋊C8—1064no—
290C22⋊C8—1062no—
291#95—21no—
292#95—21no—
293#95—21no—
294#95—1062no—
295#95—1062no—
296C24C22⋊Dic5311yescharacteristic
297C24C22⋊Dic5311yescharacteristic
298C24C2×C22⋊D5311yescharacteristic
299#41—44no—
300#41—44no—
301#41—48no—
302C2×Q8—22no—
303C2×Q8—22no—
304C2×Q8—22no—
305C2×Q8—41no—
306C2×Q8—41no—
307C2×Q8—41no—
308C2×D4—22no—
309C2×D4—22no—
310C2×D4—22no—
311C2×D4—22no—
312C2×D4—22no—
313C2×D4—22no—
314C2×D4—41no—
315C2×D4—41no—
316C2×D4—41no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—2122no—
332C22×C4C22⋊Dic5311yescharacteristic
333C22×C4C22⋊Dic5311yescharacteristic
334C22×C4D4×D5312yes—
335C22×C4#1171994712yes—
336C22×C4C22×Dic5311yescharacteristic
337C22×C4C2×C22⋊D5311yescharacteristic
338M16—2124no—
339M16—2122no—
340M16—2122no—
341C2×C8—1064no—
342C2×C8—2124no—
343C4⋊C4—22no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C22⋊C4—42no—
349C22⋊C4—42no—
350C22⋊C4—42no—
351C22⋊C4—42no—
352C22⋊C4—42no—
353C22⋊C4—42no—
354C22⋊C4—2122no—
355C22⋊C4—2122no—
356C42—41no—
357C42—41no—
358C23—21no—
359C23—21no—
360C23—21no—
361C23—21no—
362C23—21no—
363C23—21no—
364C23—21no—
365C23—41no—
366C23—41no—
367C23—41no—
368C23—41no—
369C23—41no—
370C23—41no—
371C23#1313313212yes—
372C23C2×C22⋊Dic5311yescharacteristic
373C23#1313318912yes—
374Q8—44no—
375Q8—82no—
376D4—44no—
377D4—44no—
378D4—44no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—24no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—22no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—4242no—
408C2×C4—4242no—
409C2×C4—4242no—
410C2×C4#1313313212yes—
411C2×C4D4×Dic5312yes—
412C2×C4#1313318511yescharacteristic
413C2×C4#1313319811yescharacteristic
414C8—2124no—
415C22—21no—
416C22—21no—
417C22—22no—
418C22—22no—
419C22—22no—
420C22—22no—
421C22—42no—
422C22—42no—
423C22—41no—
424C22—41no—
425C22—41no—
426C22—41no—
427C22—41no—
428C22—81no—
429C22—81no—
430C22—81no—
431C22—81no—
432C22#1681050311yescharacteristic
433C22C2×#1313313211yescharacteristic
434C22C2×#1313313211yescharacteristic
435C4—21no—
436C4—21no—
437C4—22no—
438C4—22no—
439C4—42no—
440C4—41no—
441C4—81no—
442C4—81no—
443C4—81no—
444C4—8482no—
445C2—21no—
446C2—21no—
447C2—42no—
448C2—42no—
449C2—81no—
450C2#2027509911yescharacteristic
451C2#2027516911yescharacteristic
452C2#2027523811yescharacteristic
subgroupG/H
centerC22#16810503
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#2432C2
socleC22×C53#386
prime radicalC22×#41×C53C22
cosocleC23×C4×C53C23
hypercenter#2432D53
hypocenterC53#9443
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2432

Complements

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

NH ≅ G/Ntype
3. #20273248C2semidirect
4. #20272568C2semidirect
15. C2×C4⋊Q8×C53C4semidirect
16. C2×#124×C53C4semidirect
20. C53×#391C4semidirect
21. C53×#390C4semidirect
234. C2×C4⋊Q8Dic53semidirect
235. C2×#124Dic53semidirect
244. #391Dic53semidirect
245. #390Dic53semidirect
251. C53#9443semidirect

Maximal subgroups

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

subgroupindex
#202831602
#202746332
#202732482
#202725682
C53×#24322
#944353