Finite Group Databasebeta

← #27284379

Subgroups — #27284379

Order 13568.

654 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810953C211yesmaximal, characteristic
2#20280528C211yesmaximal, characteristic
3#20274801C211yesmaximal, characteristic
4C2×#16810359C211yesmaximal, characteristic
5C2×#16810349C211yesmaximal, characteristic
6#20272594C211yesmaximal, characteristic
7C2×C53×#398C211yesmaximal, characteristic
8C2×#13133196C2211yescharacteristic
9#16810953—22no—
10#16810953—22no—
11C2×#13133148—21no—
12C2×#13133140—21no—
13C2×C22⋊#88322C2211yescharacteristic
14C2×C22⋊#88322C2211yescharacteristic
15#16810562—21no—
16#16810556—22no—
17#16810543—21no—
18#16810496—21no—
19#16810486—21no—
20#16810483C2211yescharacteristic
21#16810385—22no—
22#16810359—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×Dic106D411yescharacteristic
32C22×#88322—21no—
33C22×#88322—21no—
34#13133196—21no—
35#13133196—21no—
36#13133196—21no—
37#13133196—21no—
38C2×C4⋊Dic53D411yescharacteristic
39C2×C4⋊Dic53D412yes—
40C2×#11719919—22no—
41C2×#11719919—21no—
42C2×C22⋊Dic53—21no—
43C2×C22⋊Dic53—21no—
44#13133148—24no—
45#13133140—24no—
46C22⋊#88322—21no—
47C22⋊#88322—21no—
48C22⋊#88322—21no—
49C22⋊#88322D412yes—
50#13133124—22no—
51#13133124—22no—
52#13133124—22no—
53#13133124—21no—
54C23×C4×C53—21no—
55C23×C4×C53—21no—
56C23×C4×C53D411yescharacteristic
57C23×C4×C53D411yescharacteristic
58C23×C4×C53C2311yescharacteristic
59C2×C4⋊C4×C53—24no—
60C2×C4⋊C4×C53—21no—
61C2×C4⋊C4×C53—21no—
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—42no—
72C2×C4⋊C4×C53—42no—
73C2×C4⋊C4×C53D412yes—
74#95×C53—24no—
75#95×C53—21no—
76#95×C53D412yes—
77C24×C53C2×D411yescharacteristic
78C22×C4×C53—24no—
79C22×C4×C53—22no—
80C22×C4×C53—21no—
81C22×C4×C53—21no—
82C22×C4×C53—21no—
83C22×C4×C53—21no—
84C22×C4×C53—21no—
85C22×C4×C53—21no—
86C22×C4×C53—21no—
87C22×C4×C53—22no—
88C22×C4×C53—21no—
89C22×C4×C53—21no—
90C22×C4×C53—42no—
91C22×C4×C53—42no—
92C22×C4×C53—41no—
93C22×C4×C53—41no—
94C22×C4×C53—42no—
95C22×C4×C53—41no—
96C22×C4×C53—41no—
97C22×C4×C53—41no—
98C22×C4×C53—41no—
99C22×C4×C53—41no—
100C22×C4×C53—41no—
101C22×C4×C53SD1612yes—
102C22×C4×C53C2×D411yescharacteristic
103C22×C4×C53C2×D411yescharacteristic
104C22×C4×C53C2×D411yescharacteristic
105C22×C4×C53C2×D411yescharacteristic
106C22×C4×C53C2×D411yescharacteristic
107C22×C4×C53#4111yescharacteristic
108C22×Dic53—21no—
109C22×Dic53—22no—
110C22×Dic53—21no—
111C2×Dic106—21no—
112C2×Dic106—21no—
113C2×Dic106—42no—
114C4⋊C4×C53—22no—
115C4⋊C4×C53—22no—
116C4⋊C4×C53—48no—
117C4⋊C4×C53—41no—
118C4⋊C4×C53—41no—
119C4⋊C4×C53—41no—
120C4⋊C4×C53—41no—
121C4⋊C4×C53—41no—
122C4⋊C4×C53—41no—
123C22⋊Dic53—41no—
124C22⋊Dic53—41no—
125C22⋊Dic53—41no—
126C22⋊Dic53—41no—
127C4⋊Dic53—21no—
128C4⋊Dic53—21no—
129C4⋊Dic53—42no—
130#11719919—41no—
131#11719919—41no—
132#11719919—82no—
133C2×#88322—22no—
134C2×#88322—42no—
135C2×#88322—42no—
136C2×#88322—41no—
137C23×C53—22no—
138C23×C53—22no—
139C23×C53#12011yescharacteristic
140C23×C53#12011yescharacteristic
141C23×C53#12111yescharacteristic
142C23×C53C2×SD1611yescharacteristic
143C23×C53#13511yescharacteristic
144C23×C53AGL(1,ℤ/8)11yescharacteristic
145C23×C53#13711yescharacteristic
146C2×C4×C53—22no—
147C2×C4×C53—22no—
148C2×C4×C53—24no—
149C2×C4×C53—22no—
150C2×C4×C53—22no—
151C2×C4×C53—22no—
152C2×C4×C53—41no—
153C2×C4×C53—41no—
154C2×C4×C53—41no—
155C2×C4×C53—41no—
156C2×C4×C53—44no—
157C2×C4×C53—44no—
158C2×C4×C53—42no—
159C2×C4×C53—41no—
160C2×C4×C53—41no—
161C2×C4×C53—42no—
162C2×C4×C53—41no—
163C2×C4×C53—41no—
164C2×C4×C53—42no—
165C2×C4×C53—41no—
166C2×C4×C53—42no—
167C2×C4×C53—42no—
168C2×C4×C53—42no—
169C2×C4×C53—42no—
170C2×C4×C53—81no—
171C2×C4×C53—81no—
172C2×C4×C53—81no—
173C2×C4×C53—81no—
174C2×C4×C53—81no—
175C2×C4×C53#12011yescharacteristic
176C2×C4×C53#12111yescharacteristic
177C2×C4×C53#12111yescharacteristic
178C2×C4×C53#12711yescharacteristic
179C2×Dic53—41no—
180C2×Dic53—41no—
181C2×Dic53—42no—
182C2×Dic53—42no—
183C2×Dic53—42no—
184C2×Dic53—42no—
185C2×Dic53—42no—
186Dic106—42no—
187Dic106—81no—
188#88322—42no—
189#88322—81no—
190#9927—531nomaximal, Sylow
191C22×C53—22no—
192C22×C53—22no—
193C22×C53—22no—
194C22×C53—22no—
195C22×C53—21no—
196C22×C53—21no—
197C22×C53—21no—
198C22×C53—21no—
199C22×C53#39211yescharacteristic
200C22×C53#44811yescharacteristic
201C22×C53#44911yescharacteristic
202C22×C53#45011yescharacteristic
203C22×C53#45211yescharacteristic
204C22×C53#45612yes—
205C22×C53#45812yes—
206C22×C53#46511yescharacteristic
207C22×C53#46811yescharacteristic
208C4×C53—21no—
209C4×C53—21no—
210C4×C53—21no—
211C4×C53—21no—
212C4×C53—44no—
213C4×C53—42no—
214C4×C53—82no—
215C4×C53—81no—
216Dic53—82no—
217Dic53—81no—
218Dic53—81no—
219C2×#483—531no—
220#2627—531no—
221C2×#398D5311yescharacteristic
222#1823—531no—
223C2×#328—531no—
224C2×#324—531no—
225#1294—531no—
226C2×C53—21no—
227C2×C53—21no—
228C2×C53—21no—
229C2×C53—21no—
230C2×C53#160912yes—
231C2×C53#161112yes—
232C2×C53#200011yescharacteristic
233C2×C53#200511yescharacteristic
234C2×C53#202211yescharacteristic
235C2×C22⋊Q8—531no—
236C22×C4⋊C4—21no—
237C22×C4⋊C4C2×D5311yescharacteristic
238C22×C4⋊C4C2×D5311yescharacteristic
239#483—1062no—
240#483—1062no—
241C2×#106—1061no—
242C2×#103—1061no—
243C2×C22⋊C8—531no—
244C2×C22⋊C8—531no—
245#400—1061no—
246#398—22no—
247#398—22no—
248#398—1062no—
249#395—1061no—
250#385—1061no—
251#381—1061no—
252#380—531no—
253C2×#95—21no—
254C2×#95C2×D5311yescharacteristic
255#340—1062no—
256#328—1062no—
257#324—1062no—
258C53#992711yescharacteristic, complemented, Sylow
259C22×Q8—531no—
260C23×C4—21no—
261C23×C4—21no—
262C23×C4C22⋊D5311yescharacteristic
263C23×C4C22⋊D5311yescharacteristic
264C23×C4C22×D5311yescharacteristic
265C22×C8—1061no—
266C22×C8—1061no—
267C22⋊Q8—1061no—
268C22⋊Q8—1061no—
269C22⋊Q8—1061no—
270C22⋊Q8—1061no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—42no—
284C2×C4⋊C4—42no—
285C2×C4⋊C4—531no—
286C2×C4⋊C4—532no—
287C2×C4⋊C4—1062no—
288C2×C4⋊C4—1061no—
289C2×C4⋊C4C22⋊D5312yes—
290C2×C22⋊C4—1061no—
291C2×C22⋊C4—1061no—
292#106—1064no—
293#103—1064no—
294C22⋊C8—532no—
295C22⋊C8—1061no—
296C22⋊C8—1061no—
297C22⋊C8—1061no—
298#95—24no—
299#95—21no—
300#95—1062no—
301#95—1062no—
302#95—1062no—
303#95—1061no—
304#95C22⋊D5312yes—
305C24C2×C22⋊D5311yescharacteristic
306C2×Q8—1061no—
307C2×Q8—1061no—
308C2×Q8—2122no—
309C22×C4—24no—
310C22×C4—21no—
311C22×C4—22no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—22no—
320C22×C4—21no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—42no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—1062no—
333C22×C4—1061no—
334C22×C4—1061no—
335C22×C4#1171992312yes—
336C22×C4D4×D5311yescharacteristic
337C22×C4D4×D5311yescharacteristic
338C22×C4D4×D5311yescharacteristic
339C22×C4#1171994711yescharacteristic
340C22×C4C2×C22⋊D5311yescharacteristic
341C22×C4C2×C22⋊D5311yescharacteristic
342C2×C8—1062no—
343C2×C8—2122no—
344C2×C8—2121no—
345C2×C8—2122no—
346C4⋊C4—22no—
347C4⋊C4—22no—
348C4⋊C4—48no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—41no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C4⋊C4—41no—
355C4⋊C4—1061no—
356C4⋊C4—1061no—
357C4⋊C4—4242no—
358C4⋊C4—2122no—
359C4⋊C4—2121no—
360C4⋊C4—2121no—
361C22⋊C4—2121no—
362C22⋊C4—2121no—
363C22⋊C4—2121no—
364C22⋊C4—2121no—
365C23—22no—
366C23—22no—
367C23#1313318511yescharacteristic
368C23#1313318611yescharacteristic
369C23#1313318911yescharacteristic
370C23C2×#1171992311yescharacteristic
371C23#1313323711yescharacteristic
372C23#1313324311yescharacteristic
373C23#1313324711yescharacteristic
374Q8—4241no—
375Q8—2122no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—24no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—42no—
389C2×C4—41no—
390C2×C4—42no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—41no—
399C2×C4—42no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—81no—
404C2×C4—81no—
405C2×C4—2122no—
406C2×C4—2121no—
407C2×C4—2121no—
408C2×C4—2122no—
409C2×C4—2122no—
410C2×C4—2122no—
411C2×C4—2122no—
412C2×C4#1313318511yescharacteristic
413C2×C4#1313318811yescharacteristic
414C2×C4#1313318911yescharacteristic
415C2×C4#1313321311yescharacteristic
416C8—4241no—
417C8—2122no—
418C22—21no—
419C22—21no—
420C22—22no—
421C22—22no—
422C22—22no—
423C22—22no—
424C22—21no—
425C22—21no—
426C22#1681053111yescharacteristic
427C22#1681076111yescharacteristic
428C22#1681076611yescharacteristic
429C22#1681077411yescharacteristic
430C22#1681078511yescharacteristic
431C22#1681080512yes—
432C22#1681081212yes—
433C22#1681085011yescharacteristic
434C22#1681086711yescharacteristic
435C4—21no—
436C4—21no—
437C4—21no—
438C4—21no—
439C4—44no—
440C4—42no—
441C4—82no—
442C4—81no—
443C4—4241no—
444C4—4242no—
445C4—4241no—
446C2—21no—
447C2—21no—
448C2—21no—
449C2—21no—
450C2#2027371012yes—
451C2#2027372212yes—
452C2#2027571811yescharacteristic
453C2#2027575111yescharacteristic
454C2#2027584311yescharacteristic
subgroupG/H
centerC23#13133185
commutator, cosocleC23×C4×C53C23
FrattiniC23×C4C22×D53
FittingC2×C53×#398C2
socleC23×C53#120
prime radicalC24×C53C2×D4
hypercenterC2×#398D53
hypocenterC53#9927
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
258. C53#9927semidirect

Maximal subgroups

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

subgroupindex
C2×#168109532
#202805282
#202748012
C2×#168103592
C2×#168103492
#202725942
C2×C53×#3982
#992753