Finite Group Databasebeta

← #27287062

Subgroups — #27287062

Order 13568.

900 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 900 conjugacy classes of subgroups — too many to draw. Only its 115 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#20280829C211yesmaximal, characteristic, complemented
2#20274646C212yesmaximal
3#20272594C212yesmaximal, complemented
4#20272558C211yesmaximal, characteristic, complemented
5C53×#1438C211yesmaximal, characteristic
6C2×#13133196—21no—
7C2×#13133138—22no—
8C2×#13133138—22no—
9C2×C22⋊#88322C2212yes—
10#16810555—22no—
11#16810544—22no—
12#16810483—21no—
13#16810483C2212yes—
14C23×D4×C53C2211yescharacteristic
15C22×C22⋊C4×C53—21no—
16C22×C22⋊C4×C53—21no—
17C2×#95×C53C2211yescharacteristic
18C2×#95×C53C2211yescharacteristic
19C22×Dic106—21no—
20C22×#88322—22no—
21#13133196—42no—
22#13133196—42no—
23C2×C4⋊Dic53—21no—
24C2×C4⋊Dic53—21no—
25C2×C4⋊Dic53—21no—
26C2×C4⋊Dic53D412yes—
27C2×C4⋊Dic53D412yes—
28C2×#11719919—22no—
29C2×C22⋊Dic53—22no—
30#13133138—44no—
31#13133138—44no—
32C22⋊#88322—22no—
33C22⋊#88322—22no—
34#13133124—22no—
35#13133124—22no—
36C25×C53—21no—
37C22×D4×C53—22no—
38C22×D4×C53—22no—
39C22×D4×C53—21no—
40C22×D4×C53—42no—
41C22×D4×C53—42no—
42C22×D4×C53Q812yescomplemented
43C23×C4×C53—21no—
44C23×C4×C53D411yescharacteristic
45C23×C4×C53D411yescharacteristic
46C23×C4×C53C2311yescharacteristic
47C2×C22⋊C4×C53—42no—
48C2×C22⋊C4×C53—42no—
49C2×C22⋊C4×C53—42no—
50C2×C22⋊C4×C53—42no—
51C2×C22⋊C4×C53—42no—
52C2×C22⋊C4×C53—42no—
53#95×C53—22no—
54#95×C53—22no—
55C24×C53—22no—
56C24×C53—22no—
57C24×C53—22no—
58C24×C53—41no—
59C24×C53—41no—
60C24×C53—41no—
61C24×C53—41no—
62C24×C53—41no—
63C24×C53—41no—
64C24×C53—41no—
65C24×C53—41no—
66C24×C53—41no—
67C24×C53—41no—
68C24×C53—41no—
69C24×C53—41no—
70C24×C53C2×D411yescharacteristic
71C2×D4×C53—24no—
72C2×D4×C53—24no—
73C2×D4×C53—24no—
74C2×D4×C53—42no—
75C2×D4×C53—42no—
76C2×D4×C53—44no—
77C2×D4×C53—48no—
78C2×D4×C53—44no—
79C2×D4×C53—42no—
80C22×C4×C53—21no—
81C22×C4×C53—21no—
82C22×C4×C53—21no—
83C22×C4×C53—21no—
84C22×C4×C53—22no—
85C22×C4×C53—41no—
86C22×C4×C53—41no—
87C22×C4×C53—41no—
88C22×C4×C53—41no—
89C22×C4×C53—41no—
90C22×C4×C53—41no—
91C22×C4×C53—41no—
92C22×C4×C53—41no—
93C22×C4×C53—41no—
94C22×C4×C53—41no—
95C22×C4×C53—41no—
96C22×C4×C53—41no—
97C22×C4×C53D812yes—
98C22×C4×C53SD1612yes—
99C22×C4×C53C2×D412yes—
100C22×C4×C53C2×Q811yescharacteristic
101C22×C4×C53#4112yes—
102C22×C4×C53#4111yescharacteristic
103C22×Dic53—22no—
104C22×Dic53—22no—
105C2×Dic106—42no—
106C2×Dic106—81no—
107C2×Dic106—81no—
108C22⋊C4×C53—48no—
109C22⋊C4×C53—48no—
110C22⋊Dic53—42no—
111C22⋊Dic53—42no—
112C4⋊Dic53—42no—
113C4⋊Dic53—42no—
114C4⋊Dic53—42no—
115C4⋊Dic53—81no—
116C4⋊Dic53—81no—
117#11719919—44no—
118C2×#88322—42no—
119C2×#88322—42no—
120C2×#88322—42no—
121C23×C53—21no—
122C23×C53—21no—
123C23×C53—21no—
124C23×C53—21no—
125C23×C53—24no—
126C23×C53—42no—
127C23×C53—42no—
128C23×C53—42no—
129C23×C53—42no—
130C23×C53—42no—
131C23×C53—42no—
132C23×C53—42no—
133C23×C53—42no—
134C23×C53—42no—
135C23×C53—42no—
136C23×C53—42no—
137C23×C53—42no—
138C23×C53—42no—
139C23×C53—42no—
140C23×C53—42no—
141C23×C53—42no—
142C23×C53—42no—
143C23×C53—42no—
144C23×C53—81no—
145C23×C53—81no—
146C23×C53—81no—
147C23×C53—81no—
148C23×C53—81no—
149C23×C53—81no—
150C23×C53—81no—
151C23×C53—81no—
152C23×C53—81no—
153C23×C53—81no—
154C23×C53—81no—
155C23×C53—81no—
156C23×C53—81no—
157C23×C53—81no—
158C23×C53—81no—
159C23×C53—81no—
160C23×C53#12011yescharacteristic
161C23×C53C22⋊Q811yescharacteristic
162C23×C53#12411yescharacteristic
163C23×C53C2×D811yescharacteristic
164C23×C53C2×SD1611yescharacteristic
165C23×C53AGL(1,ℤ/8)11yescharacteristic
166C23×C53#13711yescharacteristic
167D4×C53—28no—
168D4×C53—44no—
169D4×C53—44no—
170D4×C53—44no—
171C2×C4×C53—22no—
172C2×C4×C53—22no—
173C2×C4×C53—22no—
174C2×C4×C53—22no—
175C2×C4×C53—41no—
176C2×C4×C53—41no—
177C2×C4×C53—42no—
178C2×C4×C53—41no—
179C2×C4×C53—41no—
180C2×C4×C53—41no—
181C2×C4×C53—41no—
182C2×C4×C53—42no—
183C2×C4×C53—42no—
184C2×C4×C53—42no—
185C2×C4×C53—42no—
186C2×C4×C53—42no—
187C2×C4×C53—81no—
188C2×C4×C53—81no—
189C2×C4×C53—81no—
190C2×C4×C53—81no—
191C2×C4×C53—81no—
192C2×C4×C53—81no—
193C2×C4×C53C22⋊Q812yes—
194C2×C4×C53#12412yes—
195C2×Dic53—42no—
196C2×Dic53—42no—
197C2×Dic53—42no—
198C2×Dic53—42no—
199C2×Dic53—42no—
200C2×Dic53—42no—
201Dic106—84no—
202#88322—82no—
203#10340—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—21no—
213C22×C53—21no—
214C22×C53—21no—
215C22×C53—21no—
216C22×C53—44no—
217C22×C53—44no—
218C22×C53—44no—
219C22×C53—82no—
220C22×C53—82no—
221C22×C53—82no—
222C22×C53—82no—
223C22×C53—82no—
224C22×C53—82no—
225C22×C53—82no—
226C22×C53—82no—
227C22×C53—82no—
228C22×C53—82no—
229C22×C53—82no—
230C22×C53—82no—
231C22×C53#39311yescharacteristic
232C22×C53#44711yescharacteristic
233C22×C53#44811yescharacteristic
234C22×C53#45311yescharacteristic
235C22×C53#45411yescharacteristic
236C22×C53B(4,𝔽2)11yescharacteristic
237C22×C53B(4,𝔽2)11yescharacteristic
238C22×C53#47411yescharacteristic
239C22×C53#47611yescharacteristic
240C22×C53#48611yescharacteristic
241C22×C53#48911yescharacteristic
242C4×C53—22no—
243C4×C53—22no—
244C4×C53—42no—
245C4×C53—42no—
246C4×C53—82no—
247Dic53—82no—
248Dic53—82no—
249#2659—531no—
250#1786—532no—
251#1438D5311yescharacteristic
252#1294—532no—
253#1280—531no—
254C2×C53—21no—
255C2×C53—21no—
256C2×C53—21no—
257C2×C53—21no—
258C2×C53—84no—
259C2×C53#159511yescharacteristic
260C2×C53#159611yescharacteristic
261C2×C53#159811yescharacteristic
262C2×C53#159911yescharacteristic
263C2×C53#202311yescharacteristic
264C2×C53#202711yescharacteristic
265C2×C53#202911yescharacteristic
266C23×D4C2×D5311yescharacteristic
267C2×C22⋊Q8—1061no—
268C22×C22⋊C4—21no—
269C22×C22⋊C4—21no—
270C2×#102—1062no—
271C2×#102—1062no—
272C2×C22⋊C8—532no—
273#398—1062no—
274#395—1062no—
275#380—532no—
276#380—1061no—
277C2×#95C2×D5311yescharacteristic
278C2×#95C2×D5311yescharacteristic
279C53#1034011yescharacteristic, complemented, Sylow
280C25—21no—
281C22×Q8—1061no—
282C22×D4—21no—
283C22×D4—22no—
284C22×D4—22no—
285C22×D4—42no—
286C22×D4—42no—
287C22×D4Dic10612yescomplemented
288C23×C4—21no—
289C23×C4C22⋊D5311yescharacteristic
290C23×C4C22⋊D5311yescharacteristic
291C23×C4C22×D5311yescharacteristic
292C22×C8—1062no—
293C22⋊Q8—2122no—
294C22⋊Q8—2122no—
295C2×C4⋊C4—532no—
296C2×C4⋊C4—532no—
297C2×C4⋊C4—1061no—
298C2×C4⋊C4—1061no—
299C2×C4⋊C4—1062no—
300C2×C4⋊C4—1061no—
301C2×C22⋊C4—42no—
302C2×C22⋊C4—42no—
303C2×C22⋊C4—42no—
304C2×C22⋊C4—42no—
305C2×C22⋊C4—42no—
306C2×C22⋊C4—42no—
307C2×C22⋊C4—1062no—
308#102—2124no—
309#102—2124no—
310C22⋊C8—1062no—
311C22⋊C8—1062no—
312#95—22no—
313#95—22no—
314#95—1062no—
315#95—1062no—
316C24—22no—
317C24—22no—
318C24—22no—
319C24—41no—
320C24—41no—
321C24—41no—
322C24—41no—
323C24—41no—
324C24—41no—
325C24—41no—
326C24—41no—
327C24—41no—
328C24—41no—
329C24—41no—
330C24—41no—
331C24C2×C22⋊D5311yescharacteristic
332C2×Q8—4241no—
333C2×Q8—4241no—
334C2×Q8—2122no—
335C2×D4—24no—
336C2×D4—24no—
337C2×D4—24no—
338C2×D4—42no—
339C2×D4—42no—
340C2×D4—44no—
341C2×D4—48no—
342C2×D4—44no—
343C2×D4—42no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—22no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—1062no—
362C22×C4—1062no—
363C22×C4#1171992212yes—
364C22×C4#1171992312yes—
365C22×C4C2×Dic10611yescharacteristic
366C22×C4#1171994511yescharacteristic
367C22×C4D4×D5312yes—
368C22×C4#1171994712yes—
369C2×C8—2122no—
370C2×C8—2122no—
371C2×C8—2122no—
372C4⋊C4—4241no—
373C4⋊C4—4241no—
374C4⋊C4—2122no—
375C4⋊C4—2122no—
376C4⋊C4—2124no—
377C4⋊C4—2122no—
378C22⋊C4—48no—
379C22⋊C4—48no—
380C22⋊C4—2122no—
381C22⋊C4—2122no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—21no—
386C23—24no—
387C23—42no—
388C23—42no—
389C23—42no—
390C23—42no—
391C23—42no—
392C23—42no—
393C23—42no—
394C23—42no—
395C23—42no—
396C23—42no—
397C23—42no—
398C23—42no—
399C23—42no—
400C23—42no—
401C23—42no—
402C23—42no—
403C23—42no—
404C23—42no—
405C23—81no—
406C23—81no—
407C23—81no—
408C23—81no—
409C23—81no—
410C23—81no—
411C23—81no—
412C23—81no—
413C23—81no—
414C23—81no—
415C23—81no—
416C23—81no—
417C23—81no—
418C23—81no—
419C23—81no—
420C23—81no—
421C23#1313318511yescharacteristic
422C23#1313319611yescharacteristic
423C23#1313320411yescharacteristic
424C23C2×#1171992211yescharacteristic
425C23C2×#1171992311yescharacteristic
426C23#1313324211yescharacteristic
427C23#1313324711yescharacteristic
428Q8—4244no—
429D4—28no—
430D4—44no—
431D4—44no—
432D4—44no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—42no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—81no—
450C2×C4—81no—
451C2×C4—81no—
452C2×C4—81no—
453C2×C4—81no—
454C2×C4—81no—
455C2×C4—2122no—
456C2×C4—2122no—
457C2×C4—2122no—
458C2×C4—2122no—
459C2×C4—2122no—
460C2×C4—2122no—
461C2×C4#1313319412yes—
462C2×C4#1313320312yes—
463C8—4242no—
464C22—21no—
465C22—21no—
466C22—21no—
467C22—21no—
468C22—21no—
469C22—21no—
470C22—21no—
471C22—21no—
472C22—21no—
473C22—21no—
474C22—21no—
475C22—21no—
476C22—44no—
477C22—44no—
478C22—44no—
479C22—82no—
480C22—82no—
481C22—82no—
482C22—82no—
483C22—82no—
484C22—82no—
485C22—82no—
486C22—82no—
487C22—82no—
488C22—82no—
489C22—82no—
490C22—82no—
491C22#1681053511yescharacteristic
492C22#1681075611yescharacteristic
493C22#1681076111yescharacteristic
494C22#1681079011yescharacteristic
495C22#1681079511yescharacteristic
496C22#1681080911yescharacteristic
497C22#1681080911yescharacteristic
498C22#1681090311yescharacteristic
499C22#1681091311yescharacteristic
500C22#1681096911yescharacteristic
501C22#1681098211yescharacteristic
502C4—22no—
503C4—22no—
504C4—42no—
505C4—42no—
506C4—82no—
507C4—4242no—
508C4—4242no—
509C2—21no—
510C2—21no—
511C2—21no—
512C2—21no—
513C2—84no—
514C2#2027363211yescharacteristic
515C2#2027363711yescharacteristic
516C2#2027364911yescharacteristic
517C2#2027365411yescharacteristic
518C2#2027585111yescharacteristic
519C2#2027587111yescharacteristic
520C2#2027587911yescharacteristic
subgroupG/H
centerC23#13133185
commutator, cosocleC23×C4×C53C23
FrattiniC23×C4C22×D53
FittingC53×#1438C2
socleC23×C53#120
prime radicalC23×D4×C53C22
hypercenter#1438D53
hypocenterC53#10340
LCSG ▷ C23×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1438

Complements

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

NH ≅ G/Ntype
1. #20280829C2semidirect
3. #20272594C2semidirect
4. #20272558C2semidirect
42. C22×D4×C53Q8semidirect
279. C53#10340semidirect
287. C22×D4Dic106semidirect

Maximal subgroups

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

subgroupindex
#202808292
#202746462
#202725942
#202725582
C53×#14382
#1034053