Finite Group Databasebeta

← #27302794

Subgroups — #27302794

Order 13568.

618 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133161C211yesmaximal, characteristic, complemented
2C2×#16810864C211yesmaximal, characteristic, complemented
3C2×#16810426C211yesmaximal, characteristic, complemented
4C2×#16810424C211yesmaximal, characteristic, complemented
5C2×C22⋊#11719908C211yesmaximal, characteristic, complemented
6#20272806C211yesmaximal, characteristic, complemented
7C2×C53×#466C211yesmaximal, characteristic
8C22×Dic212C2211yescharacteristic
9C2×#13133196—21no—
10C2×#13133161—22no—
11C2×#13133161—22no—
12C2×#13133161—22no—
13C22×#11719908C2211yescharacteristic
14C22×#11719908C2211yescharacteristic
15#16810864—22no—
16#16810864—22no—
17C2×#13133151C2211yescharacteristic
18C2×#13133142—21no—
19#16810426—22no—
20#16810424—22no—
21C22⋊#11719908—22no—
22C22×D8×C53C2211yescharacteristic
23C23×C8×C53C2211yescharacteristic
24C2×#121×C53—21no—
25C53×#466—22no—
26C53×#466—22no—
27C2×#107×C53C2211yescharacteristic
28C2×#102×C53—21no—
29C22×Dic106—21no—
30C2×Dic212—21no—
31C2×Dic212—21no—
32C2×Dic212—21no—
33C2×Dic212—21no—
34C2×Dic212—21no—
35C2×Dic212D412yescomplemented
36#13133196—24no—
37#13133196—42no—
38C2×C4⋊Dic53—21no—
39C2×#11719919—41no—
40C2×C22⋊Dic53—41no—
41#13133161—28no—
42C2×#11719908—21no—
43C2×#11719908—21no—
44C2×#11719908—21no—
45C2×#11719908—21no—
46C2×#11719908—21no—
47C2×#11719908D412yescomplemented
48#13133151—21no—
49#13133151D412yescomplemented
50#13133142—24no—
51C22×D4×C53—21no—
52C22×D4×C53—41no—
53C23×C4×C53D411yescharacteristic
54C2×D8×C53—21no—
55C2×D8×C53—21no—
56C2×D8×C53—21no—
57C2×D8×C53—21no—
58C2×D8×C53—21no—
59C2×D8×C53D412yes—
60C22×C8×C53—21no—
61C22×C8×C53—21no—
62C22×C8×C53—21no—
63C22×C8×C53—21no—
64C22×C8×C53—21no—
65C22×C8×C53—21no—
66C22×C8×C53D411yescharacteristic
67C22×C8×C53C2311yescharacteristic
68#121×C53—24no—
69#121×C53—42no—
70C2×C4⋊C4×C53—21no—
71C2×C22⋊C4×C53—41no—
72#107×C53—21no—
73#107×C53D412yes—
74#102×C53—24no—
75C24×C53—41no—
76C24×C53D811yescharacteristic
77C2×D4×C53—22no—
78C2×D4×C53—44no—
79C2×D4×C53—41no—
80C2×D4×C53—41no—
81C2×D4×C53—41no—
82C2×D4×C53—41no—
83C2×D4×C53—41no—
84C2×D4×C53—82no—
85C2×D4×C53—82no—
86C22×C4×C53—21no—
87C22×C4×C53—21no—
88C22×C4×C53—21no—
89C22×C4×C53—21no—
90C22×C4×C53—21no—
91C22×C4×C53—21no—
92C22×C4×C53—41no—
93C22×C4×C53D811yescharacteristic
94C22×C4×C53C2×D411yescharacteristic
95C22×Dic53—41no—
96C22×Dic53—41no—
97C2×Dic106—22no—
98C2×Dic106—41no—
99C2×Dic106—41no—
100C2×Dic106—41no—
101C2×Dic106—41no—
102C2×Dic106—41no—
103D8×C53—22no—
104D8×C53—22no—
105D8×C53—41no—
106D8×C53—41no—
107C2×C8×C53—21no—
108C2×C8×C53—21no—
109C2×C8×C53—21no—
110C2×C8×C53—41no—
111C2×C8×C53—41no—
112C2×C8×C53—41no—
113C2×C8×C53—41no—
114C2×C8×C53C2×D411yescharacteristic
115C2×C8×C53C2×D411yescharacteristic
116C2×C8×C53C2×D411yescharacteristic
117C2×C8×C53C2×D411yescharacteristic
118C2×C8×C53C2×D411yescharacteristic
119C2×C8×C53#4111yescharacteristic
120C4⋊C4×C53—22no—
121C4⋊C4×C53—41no—
122C22⋊C4×C53—44no—
123C22⋊Dic53—44no—
124C4⋊Dic53—22no—
125C4⋊Dic53—41no—
126#11719919—44no—
127Dic212—22no—
128Dic212—22no—
129Dic212—41no—
130Dic212—41no—
131#11719908—22no—
132#11719908—41no—
133C23×C53—21no—
134C23×C53—21no—
135C23×C53—21no—
136C23×C53—21no—
137C23×C53—21no—
138C23×C53—21no—
139C23×C53—42no—
140C23×C53—81no—
141C23×C53—81no—
142C23×C53—81no—
143C23×C53—81no—
144C23×C53—81no—
145C23×C53—81no—
146C23×C53SD3212yescomplemented
147C23×C53C2×D811yescharacteristic
148D4×C53—42no—
149D4×C53—42no—
150D4×C53—88no—
151D4×C53—81no—
152D4×C53—81no—
153C2×C4×C53—21no—
154C2×C4×C53—21no—
155C2×C4×C53—21no—
156C2×C4×C53—41no—
157C2×C4×C53—41no—
158C2×C4×C53—41no—
159C2×C4×C53—41no—
160C2×C4×C53—42no—
161C2×C4×C53—81no—
162C2×C4×C53—81no—
163C2×C4×C53#12011yescharacteristic
164C2×C4×C53#12011yescharacteristic
165C2×C4×C53#12111yescharacteristic
166C2×C4×C53#13511yescharacteristic
167C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
168C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
169C2×Dic53—42no—
170C2×Dic53—42no—
171C2×Dic53—81no—
172C2×Dic53—81no—
173C2×Dic53—81no—
174C2×Dic53—81no—
175Dic106—42no—
176Dic106—42no—
177Dic106—81no—
178Dic106—81no—
179C8×C53—41no—
180C8×C53#12011yescharacteristic
181C8×C53#12111yescharacteristic
182C8×C53#12111yescharacteristic
183C8×C53#12711yescharacteristic
184#13305—531nomaximal, Sylow
185C22×C53—22no—
186C22×C53—22no—
187C22×C53—22no—
188C22×C53—41no—
189C22×C53—41no—
190C22×C53—41no—
191C22×C53—41no—
192C22×C53—82no—
193C22×C53—82no—
194C22×C53—82no—
195C22×C53—161no—
196C22×C53—161no—
197C22×C53—161no—
198C22×C53—161no—
199C22×C53#44711yescharacteristic
200C22×C53#44711yescharacteristic
201C22×C53#46611yescharacteristic
202C22×C53C2×SD3211yescharacteristic
203C22×C53#50811yescharacteristic
204C22×C53#50911yescharacteristic
205C22×C53#51011yescharacteristic
206C4×C53—41no—
207C4×C53—82no—
208C4×C53#39211yescharacteristic
209C4×C53#44911yescharacteristic
210C4×C53#44911yescharacteristic
211C4×C53#46911yescharacteristic
212Dic53—82no—
213Dic53—82no—
214C22×SD32—531no—
215C2×#467—531no—
216C2×#466D5311yescharacteristic
217C2×#358—531no—
218C2×#357—531no—
219C2×C22⋊C16—531no—
220#1380—531no—
221C2×C53—42no—
222C2×C53—162no—
223C2×C53#199911yescharacteristic
224C2×C53#218511yescharacteristic
225C2×C53#218611yescharacteristic
226C2×C53#218711yescharacteristic
227C2×C53#218911yescharacteristic
228C2×C53#221711yescharacteristic
229C2×C53#222011yescharacteristic
230C22×Q16—531no—
231C22×D8C2×D5311yescharacteristic
232C23×C8C2×D5311yescharacteristic
233C2×C22⋊Q8—1061no—
234C2×#121—21no—
235C2×SD32—1062no—
236C2×SD32—1062no—
237C2×SD32—1062no—
238C22×C16—531no—
239C22×C16—531no—
240#467—1062no—
241#467—1062no—
242#466—22no—
243#466—22no—
244C2×#107—531no—
245C2×#107C2×D5311yescharacteristic
246C2×#103—1061no—
247C2×#102—21no—
248#358—1062no—
249#357—1062no—
250C22⋊C16—1062no—
251C53#1330511yescharacteristic, complemented, Sylow
252C22×Q8—1061no—
253C22×D4—21no—
254C22×D4—41no—
255C23×C4C22⋊D5311yescharacteristic
256C2×Q16—532no—
257C2×Q16—1061no—
258C2×Q16—1061no—
259C2×Q16—1061no—
260C2×Q16—1061no—
261C2×Q16—1061no—
262C2×D8—21no—
263C2×D8—21no—
264C2×D8—21no—
265C2×D8—21no—
266C2×D8—21no—
267C2×D8C22⋊D5312yes—
268C22×C8—21no—
269C22×C8—21no—
270C22×C8—21no—
271C22×C8—21no—
272C22×C8—21no—
273C22×C8—21no—
274C22×C8C22⋊D5311yescharacteristic
275C22×C8C22×D5311yescharacteristic
276C22⋊Q8—1064no—
277C22⋊Q8—2122no—
278#121—24no—
279#121—42no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—1061no—
282C2×C4⋊C4—2121no—
283C2×C22⋊C4—41no—
284C2×C22⋊C4—2121no—
285SD32—1068no—
286C2×C16—532no—
287C2×C16—1061no—
288C2×C16—1061no—
289C2×C16—1061no—
290C2×C16—1061no—
291C2×C16—1061no—
292#107—21no—
293#107—532no—
294#107—1061no—
295#107C22⋊D5312yes—
296#103—1064no—
297#102—24no—
298C24—41no—
299C24#1171992211yescharacteristic
300C2×Q8—1062no—
301C2×Q8—2121no—
302C2×Q8—2121no—
303C2×Q8—2121no—
304C2×Q8—2121no—
305C2×Q8—2121no—
306C2×D4—22no—
307C2×D4—44no—
308C2×D4—41no—
309C2×D4—41no—
310C2×D4—41no—
311C2×D4—41no—
312C2×D4—41no—
313C2×D4—82no—
314C2×D4—82no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—41no—
322C22×C4—2121no—
323C22×C4—2121no—
324C22×C4#1171992211yescharacteristic
325C22×C4C2×C22⋊D5311yescharacteristic
326Q16—1062no—
327Q16—1062no—
328Q16—2121no—
329Q16—2121no—
330D8—22no—
331D8—22no—
332D8—41no—
333D8—41no—
334C2×C8—21no—
335C2×C8—21no—
336C2×C8—21no—
337C2×C8—41no—
338C2×C8—41no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8D4×D5311yescharacteristic
342C2×C8D4×D5311yescharacteristic
343C2×C8D4×D5311yescharacteristic
344C2×C8#1171994711yescharacteristic
345C2×C8C2×C22⋊D5311yescharacteristic
346C2×C8C2×C22⋊D5311yescharacteristic
347C4⋊C4—22no—
348C4⋊C4—41no—
349C4⋊C4—1062no—
350C4⋊C4—2124no—
351C4⋊C4—2121no—
352C22⋊C4—44no—
353C22⋊C4—2124no—
354C16—1062no—
355C16—2121no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—21no—
361C23—21no—
362C23—42no—
363C23—81no—
364C23—81no—
365C23—81no—
366C23—81no—
367C23—81no—
368C23—81no—
369C23#1313316112yescomplemented
370C23C2×#1171992211yescharacteristic
371Q8—4241no—
372Q8—4241no—
373Q8—2122no—
374Q8—2122no—
375D4—42no—
376D4—42no—
377D4—88no—
378D4—81no—
379D4—81no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—42no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—4241no—
391C2×C4—4241no—
392C2×C4—4241no—
393C2×C4—4241no—
394C2×C4—2122no—
395C2×C4—2122no—
396C2×C4#1313318511yescharacteristic
397C2×C4#1313318611yescharacteristic
398C2×C4#1313318911yescharacteristic
399C2×C4#1313323711yescharacteristic
400C2×C4#1313324211yescharacteristic
401C2×C4#1313324311yescharacteristic
402C8—41no—
403C8#1313318511yescharacteristic
404C8#1313318811yescharacteristic
405C8#1313318911yescharacteristic
406C8#1313321311yescharacteristic
407C22—22no—
408C22—22no—
409C22—22no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—82no—
415C22—82no—
416C22—82no—
417C22—161no—
418C22—161no—
419C22—161no—
420C22—161no—
421C22#1681075611yescharacteristic
422C22#1681075711yescharacteristic
423C22#1681085711yescharacteristic
424C22C2×#1313316111yescharacteristic
425C22#1681105311yescharacteristic
426C22#1681105911yescharacteristic
427C22#1681106311yescharacteristic
428C4—41no—
429C4—82no—
430C4—4242no—
431C4—4242no—
432C4#1681053111yescharacteristic
433C4#1681076611yescharacteristic
434C4#1681076811yescharacteristic
435C4#1681087411yescharacteristic
436C2—42no—
437C2—162no—
438C2#2027571111yescharacteristic
439C2#2027663111yescharacteristic
440C2#2027663611yescharacteristic
441C2#2027664411yescharacteristic
442C2#2027665511yescharacteristic
443C2#2027680911yescharacteristic
444C2#2027682611yescharacteristic
subgroupG/H
centerC23C2×#11719922
commutator, cosocleC22×C8×C53C23
FrattiniC22×C8C22×D53
Fitting, prime radicalC2×C53×#466C2
socleC23×C53C2×D8
hypercenterC2×#466D53
hypocenterC53#13305
LCSG ▷ C22×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C23×C8 ◁ C2×#466

Complements

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

NH ≅ G/Ntype
1. C22×#13133161C2semidirect
2. C2×#16810864C2semidirect
3. C2×#16810426C2semidirect
4. C2×#16810424C2semidirect
5. C2×C22⋊#11719908C2semidirect
6. #20272806C2semidirect
35. C2×Dic212D4semidirect
47. C2×#11719908D4semidirect
49. #13133151D4semidirect
146. C23×C53SD32semidirect
251. C53#13305semidirect
369. C23#13133161semidirect

Maximal subgroups

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

subgroupindex
C22×#131331612
C2×#168108642
C2×#168104262
C2×#168104242
C2×C22⋊#117199082
#202728062
C2×C53×#4662
#1330553