Finite Group Databasebeta

← #27281127

Subgroups — #27281127

Order 13568.

576 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 576 conjugacy classes of subgroups — too many to draw. Only its 99 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#20285512C211yesmaximal, characteristic
2#20275550C211yesmaximal, characteristic, complemented
3#20273243C211yesmaximal, characteristic, complemented
4#20273231C211yesmaximal, characteristic, complemented
5#20272621C211yesmaximal, characteristic, complemented
6#20272567C211yesmaximal, characteristic, complemented
7C53×#2377C211yesmaximal, characteristic, complemented
8C2×#13133191C2211yescharacteristic
9#16810941—21no—
10#16810939C412yescomplemented
11#16810874—22no—
12#16810757—21no—
13#16810756C412yescomplemented
14C2×#13133131—21no—
15C2×#13133131—21no—
16#16810587C2211yescharacteristic
17#16810587C2211yescharacteristic
18C4×C22⋊Dic53C2211yescharacteristic
19#16810357—22no—
20#16810356—21no—
21#16810356—21no—
22C2×#121×C53C2211yescharacteristic
23C53×#390—21no—
24C53×#388—21no—
25C53×#387—21no—
26C53×#386—21no—
27C53×#386—21no—
28C53×#385—21no—
29C53×#381C2211yescharacteristic
30C53×#380C2211yescharacteristic
31C22×D212C2×C411yescharacteristic, complemented
32C22×C22⋊D53—21no—
33C2×#11719922—22no—
34C2×#11719917—21no—
35C2×#11719917—21no—
36#13133191—21no—
37#13133191—21no—
38#13133191—21no—
39#13133191—21no—
40C2×C4⋊Dic53C2×C411yescharacteristic, complemented
41C2×C22⋊Dic53—21no—
42C2×#11719921—21no—
43C2×C4×Dic53—21no—
44#13133148—22no—
45#13133138—22no—
46#13133137—22no—
47#13133131—42no—
48#13133131—42no—
49C22⋊#88322—21no—
50C22⋊#88322—21no—
51C22⋊#88322—21no—
52C22⋊#88322C2×C412yescomplemented
53#13133124—21no—
54C22×D4×C53—21no—
55C22×D4×C53D411yescharacteristic
56C23×C4×C53—21no—
57C23×C4×C53C2311yescharacteristic
58#121×C53—21no—
59#121×C53—21no—
60#121×C53C2×C412yes—
61#121×C53D412yes—
62C2×C4⋊C4×C53—21no—
63C2×C4⋊C4×C53—21no—
64C2×C4⋊C4×C53D411yescharacteristic
65C2×C22⋊C4×C53—21no—
66C2×C22⋊C4×C53—21no—
67C2×C22⋊C4×C53—21no—
68C2×C22⋊C4×C53—21no—
69C2×C22⋊C4×C53—21no—
70C2×C22⋊C4×C53D411yescharacteristic
71C2×C22⋊C4×C53D411yescharacteristic
72C2×C42×C53—21no—
73#95×C53—21no—
74#95×C53—21no—
75#95×C53—21no—
76#95×C53—21no—
77#95×C53D411yescharacteristic
78#95×C53D411yescharacteristic
79C24×C53—21no—
80C24×C53C2×D411yescharacteristic
81C23×D53—21no—
82C2×D4×C53—22no—
83C2×D4×C53—22no—
84C2×D4×C53—21no—
85C2×D4×C53—81no—
86C2×D4×C53—81no—
87C2×D4×C53—81no—
88C2×D4×C53C22⋊C412yes—
89C22×C4×C53—21no—
90C22×C4×C53—21no—
91C22×C4×C53—21no—
92C22×C4×C53—21no—
93C22×C4×C53—21no—
94C22×C4×C53—81no—
95C22×C4×C53—81no—
96C22×C4×C53—81no—
97C22×C4×C53—81no—
98C22×C4×C53C22×C411yescharacteristic
99C22×C4×C53C2×D411yescharacteristic
100C22×C4×C53C2×D411yescharacteristic
101C22×C4×C53C2×D411yescharacteristic
102C22×C4×C53#4111yescharacteristic
103C22×C4×C53#4111yescharacteristic
104C2×C22⋊D53—41no—
105C2×C22⋊D53—41no—
106C2×C22⋊D53—41no—
107C2×C22⋊D53—41no—
108C2×C22⋊D53—41no—
109C2×C22⋊D53—41no—
110C22×Dic53—21no—
111C22×Dic53—41no—
112C2×D212—21no—
113C2×D212—21no—
114C2×D212—81no—
115C4⋊C4×C53—21no—
116C4⋊C4×C53—81no—
117C4⋊C4×C53—81no—
118C4⋊C4×C53C22⋊C412yes—
119C22⋊C4×C53—22no—
120C22⋊C4×C53—22no—
121C22⋊C4×C53—41no—
122C22⋊C4×C53—41no—
123C22⋊C4×C53—81no—
124C22⋊C4×C53—81no—
125C22⋊C4×C53—81no—
126C22⋊C4×C53—81no—
127C42×C53—81no—
128C22⋊Dic53—22no—
129C22⋊Dic53—41no—
130#11719922—82no—
131#11719921—41no—
132#11719921—41no—
133C4⋊Dic53—21no—
134C4⋊Dic53—21no—
135C4×Dic53—81no—
136#11719917—81no—
137#11719917—81no—
138C2×#88322—22no—
139C2×#88322—41no—
140C23×C53—21no—
141C23×C53—21no—
142C23×C53—22no—
143C23×C53—41no—
144C23×C53—41no—
145C23×C53—41no—
146C23×C53—41no—
147C23×C53—41no—
148C23×C53—41no—
149C23×C53—41no—
150C23×C53—41no—
151C23×C53C4×D411yescharacteristic
152C23×C53#12011yescharacteristic
153C23×C53#12111yescharacteristic
154C22×D53—41no—
155C22×D53—41no—
156C22×D53—41no—
157C22×D53—81no—
158C22×D53—81no—
159D4×C53—42no—
160D4×C53—81no—
161D4×C53—82no—
162D4×C53—82no—
163C2×C4×C53—22no—
164C2×C4×C53—41no—
165C2×C4×C53—41no—
166C2×C4×C53—41no—
167C2×C4×C53—41no—
168C2×C4×C53—41no—
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—41no—
181C2×C4×C53—41no—
182C2×C4×C53—41no—
183C2×C4×C53—41no—
184C2×C4×C53—84no—
185C2×C4×C53C2×C22⋊C411yescharacteristic
186C2×C4×C53C4×D411yescharacteristic
187C2×C4×C53#12111yescharacteristic
188C2×C4×C53#12311yescharacteristic
189C22⋊D53—81no—
190C22⋊D53—81no—
191C22⋊D53—81no—
192C22⋊D53—81no—
193C2×Dic53—41no—
194C2×Dic53—41no—
195C2×Dic53—42no—
196C2×Dic53—41no—
197C2×Dic53—81no—
198C2×Dic53—81no—
199D212—42no—
200D212—81no—
201#88322—42no—
202#88322—81no—
203#9343—531nomaximal, Sylow
204C22×C53—21no—
205C22×C53—21no—
206C22×C53—21no—
207C22×C53—21no—
208C22×C53—21no—
209C22×C53—21no—
210C22×C53—21no—
211C22×C53—21no—
212C22×C53—42no—
213C22×C53—42no—
214C22×C53—42no—
215C22×C53—81no—
216C22×C53—81no—
217C22×C53—81no—
218C22×C53—81no—
219C22×C53—81no—
220C22×C53—81no—
221C22×C53#38611yescharacteristic
222C22×C53#45311yescharacteristic
223C22×C53#45411yescharacteristic
224C2×D53—81no—
225C2×D53—81no—
226C2×D53—81no—
227C2×D53—161no—
228C2×D53—161no—
229C4×C53—21no—
230C4×C53—21no—
231C4×C53—21no—
232C4×C53—21no—
233C4×C53—42no—
234C4×C53—81no—
235C4×C53—81no—
236C4×C53—81no—
237C4×C53—81no—
238Dic53—82no—
239Dic53—81no—
240#3192—531no—
241#2377D5311yescharacteristic, complemented
242#1964—531no—
243#1510—531no—
244#1507—531no—
245#1305—531no—
246#1284—531no—
247C2×C53—21no—
248C2×C53—21no—
249C2×C53—41no—
250C2×C53—41no—
251C2×C53—82no—
252C2×C53#188311yescharacteristic
253C2×C53#190411yescharacteristic
254C2×C53#191311yescharacteristic
255D53—161no—
256C2×#121—531no—
257C2×#121C2×D5311yescharacteristic
258#481—532no—
259#481—1061no—
260#469—1062no—
261#447—532no—
262#447—1061no—
263C2×#99—1061no—
264C2×#99—1061no—
265#407—531no—
266#407—531no—
267#390—21no—
268#388—21no—
269#387—21no—
270#386—21no—
271#386—21no—
272#385—21no—
273#381C2×D5311yescharacteristic
274#380C2×D5311yescharacteristic
275C4×C22⋊C4—531no—
276#327—1062no—
277#327—1061no—
278#327—1061no—
279C53#934311yescharacteristic, complemented, Sylow
280C22×D4—21no—
281C22×D4—531no—
282C22×D4—1061no—
283C22×D4C22⋊D5311yescharacteristic
284C23×C4—21no—
285C23×C4C22×D5311yescharacteristic
286C2×D8—1062no—
287C2×M16—1061no—
288C2×M16—1061no—
289#121—21no—
290#121—21no—
291#121—1061no—
292#121—1061no—
293#121—1061no—
294#121—1061no—
295#121C4×D5312yes—
296#121D21212yes—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—531no—
300C2×C4⋊C4C22⋊D5311yescharacteristic
301C2×C22⋊C4—21no—
302C2×C22⋊C4—21no—
303C2×C22⋊C4—21no—
304C2×C22⋊C4—21no—
305C2×C22⋊C4—21no—
306C2×C22⋊C4—1061no—
307C2×C22⋊C4—1061no—
308C2×C22⋊C4C22⋊D5311yescharacteristic
309C2×C22⋊C4C22⋊D5311yescharacteristic
310C2×C42—21no—
311C2×C42—1061no—
312#106—1062no—
313#102—1062no—
314#102—1062no—
315#99—2122no—
316#99—2122no—
317C22⋊C8—532no—
318C22⋊C8—1061no—
319C22⋊C8—1061no—
320C22⋊C8—1061no—
321#95—21no—
322#95—21no—
323#95—21no—
324#95—21no—
325#95—1061no—
326#95C22⋊D5311yescharacteristic
327#95C22⋊D5311yescharacteristic
328C24—21no—
329C24—1061no—
330C24C2×C22⋊D5311yescharacteristic
331C2×D4—22no—
332C2×D4—22no—
333C2×D4—21no—
334C2×D4—81no—
335C2×D4—81no—
336C2×D4—81no—
337C2×D4—1061no—
338C2×D4—1061no—
339C2×D4—4241no—
340C2×D4—2121no—
341C2×D4—2121no—
342C2×D4—2121no—
343C2×D4—2121no—
344C2×D4—2121no—
345C2×D4—2121no—
346C2×D4#1171992112yes—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—81no—
353C22×C4—81no—
354C22×C4—81no—
355C22×C4—81no—
356C22×C4—1061no—
357C22×C4—2121no—
358C22×C4C2×C4×D5311yescharacteristic
359C22×C4C2×D21211yescharacteristic
360C22×C4#1171994511yescharacteristic
361C22×C4#1171994511yescharacteristic
362C22×C4C2×C22⋊D5311yescharacteristic
363C22×C4C2×C22⋊D5311yescharacteristic
364D8—4242no—
365M16—4241no—
366M16—4241no—
367C2×C8—1062no—
368C2×C8—2121no—
369C4⋊C4—21no—
370C4⋊C4—81no—
371C4⋊C4—81no—
372C4⋊C4—1061no—
373C4⋊C4—1061no—
374C4⋊C4#1171992112yes—
375C22⋊C4—22no—
376C22⋊C4—22no—
377C22⋊C4—41no—
378C22⋊C4—41no—
379C22⋊C4—81no—
380C22⋊C4—81no—
381C22⋊C4—81no—
382C22⋊C4—81no—
383C22⋊C4—1062no—
384C22⋊C4—2121no—
385C22⋊C4—2121no—
386C22⋊C4—2121no—
387C42—81no—
388C42—4241no—
389C23—21no—
390C23—21no—
391C23—22no—
392C23—41no—
393C23—41no—
394C23—41no—
395C23—41no—
396C23—41no—
397C23—41no—
398C23—41no—
399C23—41no—
400C23—4241no—
401C23—4241no—
402C23—2121no—
403C23—2121no—
404C23—2121no—
405C23C4×C22⋊D5311yescharacteristic
406C23#1313318611yescharacteristic
407C23#1313319111yescharacteristic
408D4—42no—
409D4—81no—
410D4—82no—
411D4—82no—
412D4—4241no—
413D4—4241no—
414D4—4241no—
415D4—4241no—
416D4—4241no—
417D4—2122no—
418C2×C4—22no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—84no—
440C2×C4—4241no—
441C2×C4—4241no—
442C2×C4—2122no—
443C2×C4—2121no—
444C2×C4—2121no—
445C2×C4—2121no—
446C2×C4C2×#1171992111yescharacteristic
447C2×C4C4×C22⋊D5311yescharacteristic
448C2×C4#1313319111yescharacteristic
449C2×C4#1313320011yescharacteristic
450C8—4241no—
451C8—2122no—
452C22—21no—
453C22—21no—
454C22—21no—
455C22—21no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—42no—
461C22—42no—
462C22—42no—
463C22—81no—
464C22—81no—
465C22—81no—
466C22—81no—
467C22—81no—
468C22—81no—
469C22—8481no—
470C22—8481no—
471C22—4241no—
472C22—4241no—
473C22—4241no—
474C22#1681050211yescharacteristic
475C22#1681079111yescharacteristic
476C22#1681079611yescharacteristic
477C4—21no—
478C4—21no—
479C4—21no—
480C4—21no—
481C4—42no—
482C4—81no—
483C4—81no—
484C4—81no—
485C4—81no—
486C4—4242no—
487C4—4241no—
488C2—21no—
489C2—21no—
490C2—41no—
491C2—41no—
492C2—82no—
493C2—8481no—
494C2#2027509111yescharacteristic
495C2#2027520211yescharacteristic
496C2#2027525711yescharacteristic
subgroupG/H
centerC22#16810502
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#2377C2
socleC22×C53#386
prime radical#20285512C2
cosocleC23×C4×C53C23
hypercenter#2377D53
hypocenterC53#9343
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2377

Complements

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

NH ≅ G/Ntype
2. #20275550C2semidirect
3. #20273243C2semidirect
4. #20273231C2semidirect
5. #20272621C2semidirect
6. #20272567C2semidirect
7. C53×#2377C2semidirect
10. #16810939C4semidirect
13. #16810756C4semidirect
31. C22×D212C2×C4semidirect
40. C2×C4⋊Dic53C2×C4semidirect
52. C22⋊#88322C2×C4semidirect
241. #2377D53semidirect
279. C53#9343semidirect

Maximal subgroups

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

subgroupindex
#202855122
#202755502
#202732432
#202732312
#202726212
#202725672
C53×#23772
#934353