Finite Group Databasebeta

← #27284323

Subgroups — #27284323

Order 13568.

716 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810936C211yesmaximal, characteristic
2#20280457C211yesmaximal, characteristic
3#20274801C211yesmaximal, characteristic, complemented
4C2×#16810356C211yesmaximal, characteristic
5C2×#16810349C211yesmaximal, characteristic, complemented
6#20272594C211yesmaximal, characteristic, complemented
7C2×C53×#398C211yesmaximal, characteristic, complemented
8C2×#13133200—21no—
9C2×#13133191C2211yescharacteristic
10#16810936—22no—
11#16810936—22no—
12C2×#13133150—21no—
13C2×#13133137—21no—
14C2×C22⋊#88322C2211yescharacteristic
15C2×C22⋊#88322C2211yescharacteristic
16#16810553—21no—
17#16810545—22no—
18#16810540—21no—
19#16810502—21no—
20#16810483C2211yescharacteristic
21#16810385—22no—
22#16810356—22no—
23#16810349—22no—
24C22×C4⋊C4×C53—21no—
25C22×C4⋊C4×C53C2211yescharacteristic
26C22×C4⋊C4×C53C2211yescharacteristic
27C53×#398—22no—
28C53×#398—22no—
29C2×#95×C53—21no—
30C2×#95×C53C2211yescharacteristic
31C22×D212D411yescharacteristic
32C22×C22⋊D53—21no—
33C22×#88322—21no—
34C22×#88322—21no—
35#13133200—42no—
36#13133200—42no—
37#13133191—21no—
38#13133191—21no—
39#13133191—21no—
40#13133191—21no—
41C2×C4⋊Dic53D411yescharacteristic
42C2×C4⋊Dic53D412yes—
43C2×#11719919—22no—
44C2×C22⋊Dic53—21no—
45C2×#11719921—22no—
46C2×#11719921—22no—
47C2×#11719921—21no—
48#13133150—24no—
49#13133137—24no—
50C22⋊#88322—21no—
51C22⋊#88322—21no—
52C22⋊#88322—21no—
53C22⋊#88322D412yes—
54#13133124—22no—
55#13133124—21no—
56C23×C4×C53—21no—
57C23×C4×C53—21no—
58C23×C4×C53D411yescharacteristic
59C23×C4×C53D411yescharacteristic
60C23×C4×C53C2311yescharacteristic
61C2×C4⋊C4×C53—24no—
62C2×C4⋊C4×C53—21no—
63C2×C4⋊C4×C53—21no—
64C2×C4⋊C4×C53—21no—
65C2×C4⋊C4×C53—21no—
66C2×C4⋊C4×C53—21no—
67C2×C4⋊C4×C53—21no—
68C2×C4⋊C4×C53—21no—
69C2×C4⋊C4×C53—21no—
70C2×C4⋊C4×C53—21no—
71C2×C4⋊C4×C53—21no—
72C2×C4⋊C4×C53—21no—
73C2×C4⋊C4×C53—42no—
74C2×C4⋊C4×C53—42no—
75C2×C4⋊C4×C53D412yes—
76#95×C53—24no—
77#95×C53—21no—
78#95×C53D412yes—
79C24×C53C2×D411yescharacteristic
80C23×D53—21no—
81C22×C4×C53—24no—
82C22×C4×C53—22no—
83C22×C4×C53—21no—
84C22×C4×C53—21no—
85C22×C4×C53—21no—
86C22×C4×C53—21no—
87C22×C4×C53—21no—
88C22×C4×C53—21no—
89C22×C4×C53—21no—
90C22×C4×C53—22no—
91C22×C4×C53—21no—
92C22×C4×C53—21no—
93C22×C4×C53—42no—
94C22×C4×C53—42no—
95C22×C4×C53—41no—
96C22×C4×C53—41no—
97C22×C4×C53—42no—
98C22×C4×C53—41no—
99C22×C4×C53—41no—
100C22×C4×C53—41no—
101C22×C4×C53—41no—
102C22×C4×C53—41no—
103C22×C4×C53—41no—
104C22×C4×C53D812yes—
105C22×C4×C53C2×D411yescharacteristic
106C22×C4×C53C2×D411yescharacteristic
107C22×C4×C53C2×D411yescharacteristic
108C22×C4×C53C2×D411yescharacteristic
109C22×C4×C53C2×D411yescharacteristic
110C22×C4×C53#4111yescharacteristic
111C2×C22⋊D53—41no—
112C2×C22⋊D53—41no—
113C2×C22⋊D53—41no—
114C2×C22⋊D53—41no—
115C2×C22⋊D53—41no—
116C2×C22⋊D53—41no—
117C22×Dic53—22no—
118C22×Dic53—21no—
119C2×D212—21no—
120C2×D212—21no—
121C2×D212—42no—
122C4⋊C4×C53—22no—
123C4⋊C4×C53—22no—
124C4⋊C4×C53—48no—
125C4⋊C4×C53—41no—
126C4⋊C4×C53—41no—
127C4⋊C4×C53—41no—
128C4⋊C4×C53—41no—
129C4⋊C4×C53—41no—
130C4⋊C4×C53—41no—
131C22⋊Dic53—41no—
132C22⋊Dic53—41no—
133#11719921—44no—
134#11719921—41no—
135#11719921—41no—
136#11719921—82no—
137C4⋊Dic53—21no—
138C4⋊Dic53—21no—
139C4⋊Dic53—42no—
140#11719919—44no—
141C2×#88322—22no—
142C2×#88322—42no—
143C2×#88322—42no—
144C2×#88322—41no—
145C23×C53—22no—
146C23×C53—22no—
147C23×C53#12011yescharacteristic
148C23×C53#12011yescharacteristic
149C23×C53#12111yescharacteristic
150C23×C53C2×D811yescharacteristic
151C23×C53#13511yescharacteristic
152C23×C53AGL(1,ℤ/8)11yescharacteristic
153C23×C53AGL(1,ℤ/8)11yescharacteristic
154C22×D53—41no—
155C22×D53—41no—
156C22×D53—41no—
157C22×D53—42no—
158C22×D53—42no—
159C2×C4×C53—22no—
160C2×C4×C53—22no—
161C2×C4×C53—24no—
162C2×C4×C53—22no—
163C2×C4×C53—22no—
164C2×C4×C53—22no—
165C2×C4×C53—41no—
166C2×C4×C53—41no—
167C2×C4×C53—41no—
168C2×C4×C53—41no—
169C2×C4×C53—44no—
170C2×C4×C53—44no—
171C2×C4×C53—42no—
172C2×C4×C53—41no—
173C2×C4×C53—42no—
174C2×C4×C53—41no—
175C2×C4×C53—41no—
176C2×C4×C53—41no—
177C2×C4×C53—42no—
178C2×C4×C53—41no—
179C2×C4×C53—42no—
180C2×C4×C53—42no—
181C2×C4×C53—42no—
182C2×C4×C53—42no—
183C2×C4×C53—81no—
184C2×C4×C53—81no—
185C2×C4×C53—81no—
186C2×C4×C53—81no—
187C2×C4×C53—81no—
188C2×C4×C53#12011yescharacteristic
189C2×C4×C53#12111yescharacteristic
190C2×C4×C53#12111yescharacteristic
191C2×C4×C53#12711yescharacteristic
192C22⋊D53—81no—
193C22⋊D53—81no—
194C22⋊D53—81no—
195C22⋊D53—81no—
196C2×Dic53—41no—
197C2×Dic53—41no—
198C2×Dic53—42no—
199C2×Dic53—41no—
200C2×Dic53—42no—
201C2×Dic53—42no—
202D212—42no—
203D212—81no—
204#88322—42no—
205#88322—81no—
206#9919—531nomaximal, Sylow
207C22×C53—22no—
208C22×C53—22no—
209C22×C53—22no—
210C22×C53—22no—
211C22×C53—21no—
212C22×C53—21no—
213C22×C53—21no—
214C22×C53—21no—
215C22×C53#39211yescharacteristic
216C22×C53#44711yescharacteristic
217C22×C53#44711yescharacteristic
218C22×C53#44911yescharacteristic
219C22×C53#44911yescharacteristic
220C22×C53#45612yes—
221C22×C53#45812yes—
222C22×C53#46611yescharacteristic
223C22×C53#46911yescharacteristic
224C2×D53—82no—
225C2×D53—82no—
226C2×D53—81no—
227C2×D53—81no—
228C2×D53—81no—
229C4×C53—21no—
230C4×C53—21no—
231C4×C53—21no—
232C4×C53—21no—
233C4×C53—44no—
234C4×C53—42no—
235C4×C53—82no—
236C4×C53—81no—
237Dic53—82no—
238Dic53—81no—
239C2×#480—531no—
240#2618—531no—
241C2×#398D5311yescharacteristic, complemented
242#1823—531no—
243C2×#327—531no—
244C2×#324—531no—
245#1294—531no—
246C2×C53—21no—
247C2×C53—21no—
248C2×C53—21no—
249C2×C53—21no—
250C2×C53#160912yes—
251C2×C53#161012yes—
252C2×C53#199911yescharacteristic
253C2×C53#200511yescharacteristic
254C2×C53#202211yescharacteristic
255D53—161no—
256C2×#123—1061no—
257C2×#121—531no—
258C22×C4⋊C4—21no—
259C22×C4⋊C4C2×D5311yescharacteristic
260C22×C4⋊C4C2×D5311yescharacteristic
261#480—1062no—
262#480—1062no—
263C2×#107—1061no—
264C2×#102—1061no—
265C2×C22⋊C8—531no—
266C2×C22⋊C8—531no—
267#398—22no—
268#398—22no—
269#398—1061no—
270#396—1062no—
271#394—1061no—
272#386—1061no—
273#380—531no—
274C2×#95—21no—
275C2×#95C2×D5311yescharacteristic
276#340—1062no—
277#327—1062no—
278#324—1062no—
279C53#991911yescharacteristic, complemented, Sylow
280C22×D4—531no—
281C22×D4—1061no—
282C23×C4—21no—
283C23×C4—21no—
284C23×C4C22⋊D5311yescharacteristic
285C23×C4C22⋊D5311yescharacteristic
286C23×C4C22×D5311yescharacteristic
287C22×C8—1061no—
288C22×C8—1061no—
289#123—2122no—
290#123—2122no—
291#121—1061no—
292#121—1061no—
293#121—1061no—
294#121—1061no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—42no—
308C2×C4⋊C4—42no—
309C2×C4⋊C4—531no—
310C2×C4⋊C4—532no—
311C2×C4⋊C4—1062no—
312C2×C4⋊C4C22⋊D5312yes—
313C2×C22⋊C4—1062no—
314C2×C22⋊C4—1062no—
315C2×C22⋊C4—1061no—
316C2×C22⋊C4—1061no—
317#107—1064no—
318#102—1064no—
319C22⋊C8—532no—
320C22⋊C8—1061no—
321C22⋊C8—1061no—
322C22⋊C8—1061no—
323#95—24no—
324#95—21no—
325#95—1062no—
326#95—1061no—
327#95C22⋊D5312yes—
328C24—1061no—
329C24C2×C22⋊D5311yescharacteristic
330C2×D4—1061no—
331C2×D4—1061no—
332C2×D4—2122no—
333C2×D4—2121no—
334C2×D4—2121no—
335C2×D4—2121no—
336C2×D4—2121no—
337C2×D4—2121no—
338C2×D4—2121no—
339C22×C4—24no—
340C22×C4—21no—
341C22×C4—22no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—22no—
350C22×C4—21no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—42no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—1062no—
363C22×C4—1061no—
364C22×C4#1171992212yes—
365C22×C4D4×D5311yescharacteristic
366C22×C4D4×D5311yescharacteristic
367C22×C4D4×D5311yescharacteristic
368C22×C4#1171994711yescharacteristic
369C22×C4C2×C22⋊D5311yescharacteristic
370C22×C4C2×C22⋊D5311yescharacteristic
371C2×C8—1062no—
372C2×C8—2122no—
373C2×C8—2121no—
374C2×C8—2122no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—48no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—41no—
382C4⋊C4—41no—
383C4⋊C4—41no—
384C4⋊C4—1061no—
385C4⋊C4—1061no—
386C4⋊C4—2122no—
387C4⋊C4—2124no—
388C22⋊C4—4242no—
389C22⋊C4—2124no—
390C22⋊C4—2121no—
391C22⋊C4—2121no—
392C22⋊C4—2121no—
393C22⋊C4—2121no—
394C23—22no—
395C23—22no—
396C23—2121no—
397C23—2121no—
398C23—2121no—
399C23—2122no—
400C23—2122no—
401C23#1313318511yescharacteristic
402C23#1313318611yescharacteristic
403C23#1313318911yescharacteristic
404C23C2×#1171992211yescharacteristic
405C23#1313323711yescharacteristic
406C23#1313324211yescharacteristic
407C23#1313324311yescharacteristic
408D4—4241no—
409D4—4241no—
410D4—4241no—
411D4—4241no—
412D4—4241no—
413D4—2122no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—24no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—41no—
437C2×C4—42no—
438C2×C4—81no—
439C2×C4—81no—
440C2×C4—81no—
441C2×C4—81no—
442C2×C4—81no—
443C2×C4—2122no—
444C2×C4—2121no—
445C2×C4—2121no—
446C2×C4—2121no—
447C2×C4—2122no—
448C2×C4—2122no—
449C2×C4#1313318511yescharacteristic
450C2×C4#1313318811yescharacteristic
451C2×C4#1313318911yescharacteristic
452C2×C4#1313321311yescharacteristic
453C8—4241no—
454C8—2122no—
455C22—21no—
456C22—21no—
457C22—22no—
458C22—22no—
459C22—22no—
460C22—22no—
461C22—21no—
462C22—21no—
463C22—4241no—
464C22—4242no—
465C22—4242no—
466C22—4241no—
467C22—4241no—
468C22#1681053111yescharacteristic
469C22#1681075611yescharacteristic
470C22#1681075711yescharacteristic
471C22#1681076611yescharacteristic
472C22#1681076811yescharacteristic
473C22#1681080512yes—
474C22#1681081212yes—
475C22#1681085711yescharacteristic
476C22#1681087411yescharacteristic
477C4—21no—
478C4—21no—
479C4—21no—
480C4—21no—
481C4—44no—
482C4—42no—
483C4—81no—
484C4—82no—
485C4—4242no—
486C4—4241no—
487C2—21no—
488C2—21no—
489C2—21no—
490C2—21no—
491C2—8481no—
492C2#2027371012yes—
493C2#2027371712yes—
494C2#2027571111yescharacteristic
495C2#2027575111yescharacteristic
496C2#2027584311yescharacteristic
subgroupG/H
centerC23#13133185
commutator, cosocleC23×C4×C53C23
FrattiniC23×C4C22×D53
FittingC2×C53×#398C2
socleC23×C53#120
prime radicalC2×#13133191C22
hypercenterC2×#398D53
hypocenterC53#9919
LCSG ▷ C23×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#398

Complements

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

NH ≅ G/Ntype
3. #20274801C2semidirect
5. C2×#16810349C2semidirect
6. #20272594C2semidirect
7. C2×C53×#398C2semidirect
241. C2×#398D53semidirect
279. C53#9919semidirect

Maximal subgroups

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

subgroupindex
C2×#168109362
#202804572
#202748012
C2×#168103562
C2×#168103492
#202725942
C2×C53×#3982
#991953