Finite Group Databasebeta

← #27281742

Subgroups — #27281742

Order 13568.

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

normal subgroups only

This group has 770 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, complemented
2#20274633C211yesmaximal, characteristic, complemented
3#20273244C212yesmaximal, complemented
4#20272568C212yesmaximal, complemented
5C53×#2431C211yesmaximal, characteristic
6C22×#11719917—21no—
7C2×#13133132—21no—
8C2×#13133132—21no—
9#16810590—24no—
10#16810587—21no—
11#16810587C2212yes—
12#16810486C2212yes—
13#16810357—24no—
14C22×#41×C53C2211yescharacteristic
15C2×#127×C53C411yescharacteristic, complemented
16C2×#124×C53C411yescharacteristic, complemented
17C2×#121×C53C2211yescharacteristic
18C53×#392—21no—
19C53×#392—21no—
20C53×#391C411yescharacteristic, complemented
21C53×#390C411yescharacteristic, complemented
22C53×#386—21no—
23C53×#386—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⋊#88322D414yescomplemented
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×D4×C53—21no—
44C22×D4×C53—21no—
45C22×D4×C53—21no—
46C22×D4×C53C2×C411yescharacteristic
47C22×D4×C53C2×C411yescharacteristic
48C22×D4×C53D411yescharacteristic
49C22×D4×C53D411yescharacteristic
50C23×C4×C53D411yescharacteristic
51C23×C4×C53D411yescharacteristic
52C23×C4×C53C2311yescharacteristic
53#127×C53—42no—
54#124×C53—42no—
55#121×C53—22no—
56#121×C53—22no—
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×C53—21no—
70C24×C53C22⋊C411yescharacteristic
71C24×C53C22⋊C411yescharacteristic
72C24×C53C2×D411yescharacteristic
73#41×C53—48no—
74#41×C53—44no—
75#41×C53—44no—
76C2×Q8×C53—22no—
77C2×Q8×C53—22no—
78C2×Q8×C53—41no—
79C2×D4×C53—22no—
80C2×D4×C53—22no—
81C2×D4×C53—22no—
82C2×D4×C53—22no—
83C2×D4×C53—22no—
84C2×D4×C53—22no—
85C2×D4×C53—22no—
86C2×D4×C53—41no—
87C2×D4×C53—41no—
88C2×D4×C53—41no—
89C2×D4×C53—42no—
90C2×D4×C53—42no—
91C2×D4×C53—41no—
92C2×D4×C53—41no—
93C2×D4×C53—42no—
94C2×D4×C53—42no—
95C2×D4×C53—42no—
96C2×D4×C53—81no—
97C2×D4×C53—81no—
98C2×D4×C53—81no—
99C2×D4×C53—81no—
100C22×C4×C53—22no—
101C22×C4×C53—22no—
102C22×C4×C53—21no—
103C22×C4×C53—21no—
104C22×C4×C53—21no—
105C22×C4×C53—21no—
106C22×C4×C53—21no—
107C22×C4×C53—21no—
108C22×C4×C53—21no—
109C22×C4×C53—41no—
110C22×C4×C53—41no—
111C22×C4×C53—41no—
112C22×C4×C53—41no—
113C22×C4×C53C22⋊C411yescharacteristic
114C22×C4×C53C22⋊C411yescharacteristic
115C22×C4×C53C22×C411yescharacteristic
116C22×C4×C53C2×D411yescharacteristic
117C22×C4×C53C2×D412yes—
118C22×C4×C53#4112yes—
119C22×Dic53—42no—
120C4⋊C4×C53—22no—
121C22⋊C4×C53—42no—
122C22⋊C4×C53—42no—
123C22⋊C4×C53—42no—
124C22⋊C4×C53—41no—
125C22⋊C4×C53—41no—
126C22⋊C4×C53—81no—
127C22⋊C4×C53—81no—
128C42×C53—41no—
129C42×C53—41no—
130C22⋊Dic53—42no—
131C22⋊Dic53—42no—
132#11719917—44no—
133#11719917—42no—
134#11719917—42no—
135C2×#88322—24no—
136C2×#88322—44no—
137C23×C53—21no—
138C23×C53—21no—
139C23×C53—21no—
140C23×C53—21no—
141C23×C53—21no—
142C23×C53—21no—
143C23×C53—21no—
144C23×C53—41no—
145C23×C53—41no—
146C23×C53—41no—
147C23×C53—41no—
148C23×C53—41no—
149C23×C53—41no—
150C23×C53—41no—
151C23×C53—41no—
152C23×C53—41no—
153C23×C53—41no—
154C23×C53—41no—
155C23×C53—41no—
156C23×C53—41no—
157C23×C53#9912yes—
158C23×C53C2×C22⋊C411yescharacteristic
159C23×C53#12112yes—
160Q8×C53—44no—
161D4×C53—44no—
162D4×C53—44no—
163D4×C53—44no—
164D4×C53—82no—
165D4×C53—84no—
166D4×C53—84no—
167D4×C53—84no—
168C2×C4×C53—24no—
169C2×C4×C53—22no—
170C2×C4×C53—22no—
171C2×C4×C53—21no—
172C2×C4×C53—21no—
173C2×C4×C53—21no—
174C2×C4×C53—21no—
175C2×C4×C53—21no—
176C2×C4×C53—21no—
177C2×C4×C53—22no—
178C2×C4×C53—21no—
179C2×C4×C53—21no—
180C2×C4×C53—21no—
181C2×C4×C53—42no—
182C2×C4×C53—42no—
183C2×C4×C53—42no—
184C2×C4×C53—42no—
185C2×C4×C53—41no—
186C2×C4×C53—41no—
187C2×C4×C53—41no—
188C2×C4×C53—41no—
189C2×C4×C53—41no—
190C2×C4×C53—41no—
191C2×C4×C53—41no—
192C2×C4×C53—41no—
193C2×C4×C53#9912yes—
194C2×C4×C53C4×D412yes—
195C2×C4×C53#12011yescharacteristic
196C2×C4×C53#12311yescharacteristic
197C2×Dic53—82no—
198C2×Dic53—82no—
199C2×Dic53—82no—
200#88322—44no—
201#9445—531nomaximal, Sylow
202C22×C53—21no—
203C22×C53—21no—
204C22×C53—22no—
205C22×C53—22no—
206C22×C53—22no—
207C22×C53—22no—
208C22×C53—42no—
209C22×C53—42no—
210C22×C53—41no—
211C22×C53—41no—
212C22×C53—41no—
213C22×C53—41no—
214C22×C53—41no—
215C22×C53—81no—
216C22×C53—81no—
217C22×C53—81no—
218C22×C53—81no—
219C22×C53—81no—
220C22×C53—81no—
221C22×C53—81no—
222C22×C53—81no—
223C22×C53—81no—
224C22×C53—81no—
225C22×C53—81no—
226C22×C53#38611yescharacteristic
227C22×C53C2×#9911yescharacteristic
228C22×C53C2×#9911yescharacteristic
229C4×C53—21no—
230C4×C53—21no—
231C4×C53—22no—
232C4×C53—22no—
233C4×C53—42no—
234C4×C53—41no—
235C4×C53—81no—
236C4×C53—81no—
237Dic53—162no—
238#2879—531no—
239#2431D5311yescharacteristic
240#1782—531no—
241#1510—532no—
242#1284—532no—
243C2×C53—21no—
244C2×C53—21no—
245C2×C53—42no—
246C2×C53—42no—
247C2×C53—81no—
248C2×C53—161no—
249C2×C53#188411yescharacteristic
250C2×C53#189811yescharacteristic
251C2×C53#191011yescharacteristic
252C22×#41C2×D5311yescharacteristic
253C22×M16—1061no—
254C2×#127Dic5311yescharacteristic, complemented
255C2×#124Dic5311yescharacteristic, complemented
256C2×#121C2×D5311yescharacteristic
257C2×#99—1061no—
258C2×#99—1061no—
259#408—1064no—
260#407—532no—
261#407—1061no—
262#392—21no—
263#392—21no—
264#391Dic5311yescharacteristic, complemented
265#390Dic5311yescharacteristic, complemented
266#386—21no—
267#386—21no—
268#381—532no—
269C4×C22⋊C4C2×D5311yescharacteristic
270#327—1064no—
271C53#944511yescharacteristic, complemented, Sylow
272C2×#41—22no—
273C2×#41—22no—
274C2×#41—42no—
275C2×#41—42no—
276C22×Q8C2×Dic5311yescharacteristic
277C22×D4—21no—
278C22×D4—21no—
279C22×D4—21no—
280C22×D4C2×Dic5311yescharacteristic
281C22×D4C2×Dic5311yescharacteristic
282C22×D4C22⋊D5311yescharacteristic
283C22×D4C22⋊D5311yescharacteristic
284C23×C4C22⋊D5311yescharacteristic
285C23×C4C22⋊D5311yescharacteristic
286C23×C4C22×D5311yescharacteristic
287C2×M16—1062no—
288C2×M16—1062no—
289C2×M16—2122no—
290C2×M16—2122no—
291C22×C8—1062no—
292#127—42no—
293#124—42no—
294#121—22no—
295#121—22no—
296C2×C4⋊C4C2×Dic5311yescharacteristic
297C2×C22⋊C4—21no—
298C2×C22⋊C4—21no—
299C2×C22⋊C4—21no—
300C2×C22⋊C4—21no—
301C2×C22⋊C4—21no—
302C2×C22⋊C4—21no—
303C2×C22⋊C4—1062no—
304C2×C42C2×Dic5311yescharacteristic
305C2×C42C2×Dic5311yescharacteristic
306#99—2122no—
307#99—2122no—
308C22⋊C8—534no—
309C22⋊C8—1064no—
310C22⋊C8—1062no—
311#95—21no—
312#95—21no—
313#95—21no—
314#95—1062no—
315#95—1062no—
316C24—21no—
317C24C22⋊Dic5311yescharacteristic
318C24C22⋊Dic5311yescharacteristic
319C24C2×C22⋊D5311yescharacteristic
320#41—44no—
321#41—44no—
322#41—48no—
323C2×Q8—22no—
324C2×Q8—22no—
325C2×Q8—41no—
326C2×D4—22no—
327C2×D4—22no—
328C2×D4—22no—
329C2×D4—22no—
330C2×D4—22no—
331C2×D4—22no—
332C2×D4—22no—
333C2×D4—41no—
334C2×D4—41no—
335C2×D4—41no—
336C2×D4—41no—
337C2×D4—41no—
338C2×D4—42no—
339C2×D4—42no—
340C2×D4—42no—
341C2×D4—42no—
342C2×D4—42no—
343C2×D4—81no—
344C2×D4—81no—
345C2×D4—81no—
346C2×D4—81no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—2122no—
361C22×C4C22⋊Dic5311yescharacteristic
362C22×C4C22⋊Dic5311yescharacteristic
363C22×C4D4×D5312yes—
364C22×C4#1171994712yes—
365C22×C4C22×Dic5311yescharacteristic
366C22×C4C2×C22⋊D5311yescharacteristic
367M16—2124no—
368M16—2122no—
369M16—2122no—
370C2×C8—1064no—
371C2×C8—2124no—
372C4⋊C4—22no—
373C22⋊C4—42no—
374C22⋊C4—42no—
375C22⋊C4—42no—
376C22⋊C4—41no—
377C22⋊C4—41no—
378C22⋊C4—81no—
379C22⋊C4—81no—
380C22⋊C4—2122no—
381C22⋊C4—2122no—
382C42—41no—
383C42—41no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23—21no—
389C23—21no—
390C23—21no—
391C23—41no—
392C23—41no—
393C23—41no—
394C23—41no—
395C23—41no—
396C23—41no—
397C23—41no—
398C23—41no—
399C23—41no—
400C23—41no—
401C23—41no—
402C23—41no—
403C23—41no—
404C23#1313313212yes—
405C23C2×C22⋊Dic5311yescharacteristic
406C23#1313318912yes—
407Q8—44no—
408D4—44no—
409D4—44no—
410D4—44no—
411D4—82no—
412D4—84no—
413D4—84no—
414D4—84no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—24no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—22no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—4242no—
441C2×C4—4242no—
442C2×C4—4242no—
443C2×C4#1313313212yes—
444C2×C4D4×Dic5312yes—
445C2×C4#1313318511yescharacteristic
446C2×C4#1313319811yescharacteristic
447C8—2124no—
448C22—21no—
449C22—21no—
450C22—22no—
451C22—22no—
452C22—22no—
453C22—22no—
454C22—42no—
455C22—42no—
456C22—41no—
457C22—41no—
458C22—41no—
459C22—41no—
460C22—41no—
461C22—81no—
462C22—81no—
463C22—81no—
464C22—81no—
465C22—81no—
466C22—81no—
467C22—81no—
468C22—81no—
469C22—81no—
470C22—81no—
471C22—81no—
472C22#1681050311yescharacteristic
473C22C2×#1313313211yescharacteristic
474C22C2×#1313313211yescharacteristic
475C4—21no—
476C4—21no—
477C4—22no—
478C4—22no—
479C4—42no—
480C4—41no—
481C4—81no—
482C4—81no—
483C4—8482no—
484C2—21no—
485C2—21no—
486C2—42no—
487C2—42no—
488C2—81no—
489C2—161no—
490C2#2027509911yescharacteristic
491C2#2027516911yescharacteristic
492C2#2027524311yescharacteristic
subgroupG/H
centerC22#16810503
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
Fitting, prime radicalC53×#2431C2
socleC22×C53#386
cosocleC23×C4×C53C23
hypercenter#2431D53
hypocenterC53#9445
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2431

Complements

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

NH ≅ G/Ntype
1. #20283160C2semidirect
2. #20274633C2semidirect
3. #20273244C2semidirect
4. #20272568C2semidirect
15. C2×#127×C53C4semidirect
16. C2×#124×C53C4semidirect
20. C53×#391C4semidirect
21. C53×#390C4semidirect
35. C22⋊#88322D4semidirect
254. C2×#127Dic53semidirect
255. C2×#124Dic53semidirect
264. #391Dic53semidirect
265. #390Dic53semidirect
271. C53#9445semidirect

Maximal subgroups

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

subgroupindex
#202831602
#202746332
#202732442
#202725682
C53×#24312
#944553