Finite Group Databasebeta

← #27301627

Subgroups — #27301627

Order 13568.

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

normal subgroups only

This group has 724 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
1C2×#16810597C211yesmaximal, characteristic
2C2×#16810418C212yesmaximal, complemented
3#20274633C211yesmaximal, characteristic
4#20272840C212yesmaximal, complemented
5C53×#2432C211yesmaximal, characteristic
6C2×#13133176—21no—
7#16810597—24no—
8C2×#13133132—21no—
9C2×#13133132—21no—
10C2×#13133132C2212yes—
11C2×#13133132C2212yes—
12#16810486—21no—
13#16810418—24no—
14C22×#41×C53C2211yescharacteristic
15C2×C4⋊Q8×C53C411yescharacteristic, complemented
16C2×#124×C53C2211yescharacteristic
17C2×C22⋊Q8×C53C411yescharacteristic, complemented
18C53×#394—21no—
19C53×#394—21no—
20C53×#391C411yescharacteristic, complemented
21C53×#390C2211yescharacteristic
22C53×#385—21no—
23C53×#385—21no—
24C4×C22⋊C4×C53C411yescharacteristic, complemented
25#13133176—24no—
26#13133176—42no—
27C2×C4⋊Dic53—41no—
28C2×C22⋊Dic53—22no—
29C2×C22⋊Dic53—22no—
30C2×C4×Dic53—41no—
31#13133132—24no—
32#13133132—22no—
33#13133132—22no—
34#13133132—22no—
35#13133132—42no—
36#13133132D414yes—
37#13133124—41no—
38#13133124—41no—
39C2×#41×C53—22no—
40C2×#41×C53—22no—
41C2×#41×C53—41no—
42C2×#41×C53—41no—
43C2×#41×C53—42no—
44C22×Q8×C53C2×C411yescharacteristic
45C22×Q8×C53C2×C411yescharacteristic
46C22×D4×C53D411yescharacteristic
47C22×D4×C53D411yescharacteristic
48C22×D4×C53C2311yescharacteristic
49C23×C4×C53C2×C411yescharacteristic
50C23×C4×C53D411yescharacteristic
51C23×C4×C53D411yescharacteristic
52C4⋊Q8×C53—42no—
53#124×C53—22no—
54#124×C53—22no—
55C22⋊Q8×C53—41no—
56C22⋊Q8×C53—41no—
57C2×C4⋊C4×C53—21no—
58C2×C4⋊C4×C53—21no—
59C2×C4⋊C4×C53—21no—
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×C22⋊C4×C53—21no—
66C2×C22⋊C4×C53—21no—
67C2×C42×C53C2×C411yescharacteristic
68C2×C42×C53C2×C411yescharacteristic
69#95×C53—21no—
70#95×C53—21no—
71#95×C53—21no—
72C24×C53C22⋊C411yescharacteristic
73C24×C53C2×D411yescharacteristic
74C24×C53C2×D411yescharacteristic
75#41×C53—44no—
76#41×C53—44no—
77#41×C53—44no—
78#41×C53—44no—
79C2×Q8×C53—22no—
80C2×Q8×C53—22no—
81C2×Q8×C53—22no—
82C2×Q8×C53—41no—
83C2×Q8×C53—41no—
84C2×Q8×C53—41no—
85C2×D4×C53—21no—
86C2×D4×C53—22no—
87C2×D4×C53—21no—
88C2×D4×C53—22no—
89C2×D4×C53—41no—
90C2×D4×C53—41no—
91C2×D4×C53—41no—
92C2×D4×C53—41no—
93C2×D4×C53—41no—
94C2×D4×C53C2×D412yes—
95C2×D4×C53#4112yes—
96C22×C4×C53—21no—
97C22×C4×C53—21no—
98C22×C4×C53—21no—
99C22×C4×C53—21no—
100C22×C4×C53—22no—
101C22×C4×C53—21no—
102C22×C4×C53—22no—
103C22×C4×C53—21no—
104C22×C4×C53—21no—
105C22×C4×C53—21no—
106C22×C4×C53—21no—
107C22×C4×C53—22no—
108C22×C4×C53—21no—
109C22×C4×C53—21no—
110C22×C4×C53—21no—
111C22×C4×C53—41no—
112C22×C4×C53—41no—
113C22×C4×C53C22⋊C411yescharacteristic
114C22×C4×C53C22⋊C411yescharacteristic
115C22×C4×C53C22⋊C411yescharacteristic
116C22×C4×C53C22×C411yescharacteristic
117C22×Dic53—42no—
118C22×Dic53—42no—
119C4⋊C4×C53—42no—
120C4⋊C4×C53—41no—
121C4⋊C4×C53—41no—
122C4⋊C4×C53—41no—
123C4⋊C4×C53—42no—
124C22⋊C4×C53—42no—
125C22⋊C4×C53—42no—
126C22⋊C4×C53—42no—
127C22⋊C4×C53—42no—
128C22⋊C4×C53—41no—
129C22⋊C4×C53—42no—
130C22⋊C4×C53—41no—
131C42×C53—22no—
132C42×C53—41no—
133C22⋊Dic53—24no—
134C22⋊Dic53—42no—
135C22⋊Dic53—42no—
136C22⋊Dic53—42no—
137C4⋊Dic53—44no—
138C4×Dic53—44no—
139C23×C53—21no—
140C23×C53—21no—
141C23×C53—21no—
142C23×C53—21no—
143C23×C53—21no—
144C23×C53—21no—
145C23×C53—21no—
146C23×C53—41no—
147C23×C53—41no—
148C23×C53—41no—
149C23×C53—41no—
150C23×C53—41no—
151C23×C53—41no—
152C23×C53C2×C22⋊C411yescharacteristic
153C23×C53#12011yescharacteristic
154C23×C53#12112yes—
155C23×C53#12311yescharacteristic
156Q8×C53—44no—
157Q8×C53—82no—
158D4×C53—42no—
159D4×C53—42no—
160D4×C53—44no—
161D4×C53—42no—
162D4×C53—42no—
163C2×C4×C53—22no—
164C2×C4×C53—22no—
165C2×C4×C53—22no—
166C2×C4×C53—22no—
167C2×C4×C53—21no—
168C2×C4×C53—21no—
169C2×C4×C53—21no—
170C2×C4×C53—21no—
171C2×C4×C53—21no—
172C2×C4×C53—21no—
173C2×C4×C53—22no—
174C2×C4×C53—21no—
175C2×C4×C53—21no—
176C2×C4×C53—21no—
177C2×C4×C53—42no—
178C2×C4×C53—42no—
179C2×C4×C53—42no—
180C2×C4×C53—42no—
181C2×C4×C53—41no—
182C2×C4×C53—41no—
183C2×C4×C53—41no—
184C2×C4×C53—41no—
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#9911yescharacteristic
193C2×C4×C53#9911yescharacteristic
194C2×C4×C53#9912yes—
195C2×C4×C53C4×D412yes—
196C2×Dic53—44no—
197C2×Dic53—82no—
198C2×Dic53—82no—
199C2×Dic53—82no—
200C2×Dic53—82no—
201C2×Dic53—82no—
202#13091—531nomaximal, Sylow
203C22×C53—21no—
204C22×C53—21no—
205C22×C53—21no—
206C22×C53—22no—
207C22×C53—21no—
208C22×C53—41no—
209C22×C53—42no—
210C22×C53—41no—
211C22×C53—41no—
212C22×C53—41no—
213C22×C53—41no—
214C22×C53—41no—
215C22×C53—41no—
216C22×C53—41no—
217C22×C53—41no—
218C22×C53—81no—
219C22×C53—81no—
220C22×C53—81no—
221C22×C53—81no—
222C22×C53#38611yescharacteristic
223C22×C53C2×#9911yescharacteristic
224C22×C53C2×#9911yescharacteristic
225C4×C53—21no—
226C4×C53—21no—
227C4×C53—22no—
228C4×C53—22no—
229C4×C53—42no—
230C4×C53—41no—
231C4×C53—81no—
232C4×C53—81no—
233C4×C53—81no—
234Dic53—84no—
235Dic53—162no—
236C2×#410—531no—
237#2432D5311yescharacteristic
238C2×#354—532no—
239#1782—531no—
240#1393—532no—
241C2×C53—21no—
242C2×C53—21no—
243C2×C53—41no—
244C2×C53—42no—
245C2×C53—41no—
246C2×C53—81no—
247C2×C53#189811yescharacteristic
248C2×C53#212611yescharacteristic
249C2×C53#212911yescharacteristic
250C22×#41C2×D5311yescharacteristic
251C2×C4⋊Q8Dic5311yescharacteristic, complemented
252C2×#124C2×D5311yescharacteristic
253C2×C22⋊Q8Dic5311yescharacteristic, complemented
254C2×#117—1061no—
255#410—1064no—
256C2×#99—532no—
257C2×#99—532no—
258C2×#99—1061no—
259C2×#99—1061no—
260#394—21no—
261#394—21no—
262#391Dic5311yescharacteristic, complemented
263#390C2×D5311yescharacteristic
264#385—21no—
265#385—21no—
266#381—1061no—
267C4×C22⋊C4Dic5311yescharacteristic, complemented
268#354—1064no—
269C53#1309111yescharacteristic, complemented, Sylow
270C2×#41—22no—
271C2×#41—22no—
272C2×#41—41no—
273C2×#41—41no—
274C2×#41—42no—
275C22×Q8C2×Dic5311yescharacteristic
276C22×Q8C2×Dic5311yescharacteristic
277C22×D4C22⋊D5311yescharacteristic
278C22×D4C22⋊D5311yescharacteristic
279C22×D4C22×D5311yescharacteristic
280C23×C4C2×Dic5311yescharacteristic
281C23×C4C22⋊D5311yescharacteristic
282C23×C4C22⋊D5311yescharacteristic
283C4⋊Q8—42no—
284#124—22no—
285#124—22no—
286C22⋊Q8—41no—
287C22⋊Q8—41no—
288#117—1064no—
289#117—2122no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—2121no—
294C2×C4⋊C4C2×Dic5311yescharacteristic
295C2×C22⋊C4—21no—
296C2×C22⋊C4—21no—
297C2×C22⋊C4—21no—
298C2×C22⋊C4—21no—
299C2×C22⋊C4—21no—
300C2×C22⋊C4—21no—
301C2×C22⋊C4—1062no—
302C2×C22⋊C4—1062no—
303C2×C42—2121no—
304C2×C42C2×Dic5311yescharacteristic
305C2×C42C2×Dic5311yescharacteristic
306#99—534no—
307#99—1064no—
308#99—1062no—
309#99—1062no—
310#99—1062no—
311#99—2122no—
312#95—21no—
313#95—21no—
314#95—21no—
315#95—2121no—
316#95—2121no—
317C24C22⋊Dic5311yescharacteristic
318C24C2×C22⋊D5311yescharacteristic
319C24C2×C22⋊D5311yescharacteristic
320#41—44no—
321#41—44no—
322#41—44no—
323#41—44no—
324C2×Q8—22no—
325C2×Q8—22no—
326C2×Q8—22no—
327C2×Q8—41no—
328C2×Q8—41no—
329C2×Q8—41no—
330C2×D4—21no—
331C2×D4—22no—
332C2×D4—21no—
333C2×D4—22no—
334C2×D4—41no—
335C2×D4—41no—
336C2×D4—41no—
337C2×D4—41no—
338C2×D4—41no—
339C2×D4D4×D5312yes—
340C2×D4#1171994712yes—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—22no—
346C22×C4—21no—
347C22×C4—22no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—22no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—2122no—
359C22×C4—2122no—
360C22×C4C22⋊Dic5311yescharacteristic
361C22×C4C22⋊Dic5311yescharacteristic
362C22×C4C22⋊Dic5311yescharacteristic
363C22×C4C22×Dic5311yescharacteristic
364C4⋊C4—41no—
365C4⋊C4—42no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4—42no—
369C4⋊C4—2124no—
370C22⋊C4—42no—
371C22⋊C4—42no—
372C22⋊C4—42no—
373C22⋊C4—42no—
374C22⋊C4—41no—
375C22⋊C4—42no—
376C22⋊C4—41no—
377C22⋊C4—1064no—
378C22⋊C4—2122no—
379C22⋊C4—2122no—
380C22⋊C4—2122no—
381C42—22no—
382C42—41no—
383C42—2124no—
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—
397C23C2×C22⋊Dic5311yescharacteristic
398C23#1313318511yescharacteristic
399C23#1313318912yes—
400C23#1313319811yescharacteristic
401Q8—44no—
402Q8—82no—
403D4—42no—
404D4—42no—
405D4—44no—
406D4—42no—
407D4—42no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—22no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
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—4242no—
438C2×C4—4242no—
439C2×C4—4242no—
440C2×C4—4242no—
441C2×C4—4242no—
442C2×C4—2124no—
443C2×C4#1313313212yes—
444C2×C4#1313313211yescharacteristic
445C2×C4#1313313211yescharacteristic
446C2×C4D4×Dic5312yes—
447C22—21no—
448C22—21no—
449C22—21no—
450C22—21no—
451C22—22no—
452C22—41no—
453C22—42no—
454C22—41no—
455C22—41no—
456C22—41no—
457C22—41no—
458C22—41no—
459C22—41no—
460C22—41no—
461C22—41no—
462C22—81no—
463C22—81no—
464C22—81no—
465C22—81no—
466C22#1681050311yescharacteristic
467C22C2×#1313313211yescharacteristic
468C22C2×#1313313211yescharacteristic
469C4—21no—
470C4—21no—
471C4—22no—
472C4—22no—
473C4—42no—
474C4—41no—
475C4—81no—
476C4—81no—
477C4—81no—
478C4—8482no—
479C4—4244no—
480C2—21no—
481C2—21no—
482C2—42no—
483C2—41no—
484C2—41no—
485C2—81no—
486C2#2027516911yescharacteristic
487C2#2027638011yescharacteristic
488C2#2027639011yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutatorC22×C4×C53C22×C4
FrattiniC22×D4C22×D53
FittingC53×#2432C2
socleC22×C53C2×#99
prime radicalC22×#41×C53C22
cosocleC22×D4×C53C23
hypercenter#2432D53
hypocenterC53#13091
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ #2432

Complements

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

NH ≅ G/Ntype
2. C2×#16810418C2semidirect
4. #20272840C2semidirect
15. C2×C4⋊Q8×C53C4semidirect
17. C2×C22⋊Q8×C53C4semidirect
20. C53×#391C4semidirect
24. C4×C22⋊C4×C53C4semidirect
251. C2×C4⋊Q8Dic53semidirect
253. C2×C22⋊Q8Dic53semidirect
262. #391Dic53semidirect
267. C4×C22⋊C4Dic53semidirect
269. C53#13091semidirect

Maximal subgroups

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

subgroupindex
C2×#168105972
C2×#168104182
#202746332
#202728402
C53×#24322
#1309153