Finite Group Databasebeta

← #27280553

Subgroups — #27280553

Order 13568.

566 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810941C211yesmaximal, characteristic
2#20275551C211yesmaximal, characteristic
3#20274805C211yesmaximal, characteristic, complemented
4#20273282C211yesmaximal, characteristic
5#20272582C211yesmaximal, characteristic, complemented
6#20272563C211yesmaximal, characteristic, complemented
7C53×#2345C211yesmaximal, characteristic, complemented
8C2×#13133191C2211yescharacteristic
9#16810941—22no—
10#16810941—21no—
11#16810941—21no—
12C2×#13133148C412yescomplemented
13C2×#13133137C412yescomplemented
14C2×#13133131—21no—
15C2×#13133131—21no—
16C2×C22⋊#88322C2211yescharacteristic
17C2×C22⋊#88322C2211yescharacteristic
18#16810483C2211yescharacteristic
19#16810385—22no—
20#16810356—22no—
21C22×C4⋊C4×C53C2211yescharacteristic
22C53×#389—21no—
23C53×#387—22no—
24C53×#380—21no—
25C53×#380C2211yescharacteristic
26C4×C4⋊C4×C53—21no—
27C4×C22⋊C4×C53—21no—
28C4×C22⋊C4×C53C2211yescharacteristic
29C22×D212C2×C411yescharacteristic
30C22×C22⋊D53—21no—
31C22×#88322—21no—
32C22×#88322C2×C412yes—
33#13133191—21no—
34#13133191—21no—
35#13133191—21no—
36#13133191—21no—
37C2×C4⋊Dic53—21no—
38C2×C4⋊Dic53C2×C411yescharacteristic
39C2×C22⋊Dic53—21no—
40C2×#11719921—21no—
41#13133148—22no—
42#13133148—22no—
43#13133137—22no—
44#13133137—22no—
45#13133131—42no—
46#13133131—42no—
47C22⋊#88322—21no—
48C22⋊#88322—21no—
49C22⋊#88322—21no—
50C22⋊#88322—21no—
51#13133124—21no—
52C23×C4×C53—21no—
53C23×C4×C53C2311yescharacteristic
54C2×C4⋊C4×C53—22no—
55C2×C4⋊C4×C53—21no—
56C2×C4⋊C4×C53—21no—
57C2×C4⋊C4×C53—21no—
58C2×C4⋊C4×C53—21no—
59C2×C4⋊C4×C53C2×C412yes—
60C2×C4⋊C4×C53D412yes—
61C2×C4⋊C4×C53D411yescharacteristic
62C2×C4⋊C4×C53D411yescharacteristic
63C2×C22⋊C4×C53—21no—
64C2×C22⋊C4×C53D411yescharacteristic
65C2×C22⋊C4×C53D411yescharacteristic
66C2×C42×C53—22no—
67C2×C42×C53—21no—
68#95×C53—22no—
69#95×C53—22no—
70#95×C53—21no—
71#95×C53D412yes—
72C24×C53C2×D411yescharacteristic
73C23×D53—21no—
74C22×C4×C53—21no—
75C22×C4×C53—21no—
76C22×C4×C53—21no—
77C22×C4×C53—22no—
78C22×C4×C53—22no—
79C22×C4×C53—21no—
80C22×C4×C53—22no—
81C22×C4×C53—42no—
82C22×C4×C53—82no—
83C22×C4×C53C22×C411yescharacteristic
84C22×C4×C53C2×D411yescharacteristic
85C22×C4×C53C2×D411yescharacteristic
86C22×C4×C53C2×D411yescharacteristic
87C22×C4×C53#4111yescharacteristic
88C22×C4×C53#4111yescharacteristic
89C2×C22⋊D53—41no—
90C2×C22⋊D53—41no—
91C2×C22⋊D53—41no—
92C2×C22⋊D53—41no—
93C2×C22⋊D53—41no—
94C2×C22⋊D53—41no—
95C22×Dic53—21no—
96C22×Dic53—41no—
97C2×D212—21no—
98C2×D212—21no—
99C2×D212—42no—
100C4⋊C4×C53—22no—
101C4⋊C4×C53—22no—
102C4⋊C4×C53—21no—
103C4⋊C4×C53—21no—
104C4⋊C4×C53—42no—
105C4⋊C4×C53—82no—
106C4⋊C4×C53C22⋊C412yes—
107C4⋊C4×C53C22⋊C412yes—
108C22⋊C4×C53—21no—
109C22⋊C4×C53—21no—
110C22⋊C4×C53—21no—
111C22⋊C4×C53—41no—
112C22⋊C4×C53—41no—
113C22⋊C4×C53SD1612yes—
114C42×C53—42no—
115C42×C53—82no—
116C22⋊Dic53—41no—
117C22⋊Dic53—41no—
118#11719921—41no—
119#11719921—41no—
120C4⋊Dic53—21no—
121C4⋊Dic53—21no—
122C4⋊Dic53—42no—
123C2×#88322—22no—
124C2×#88322—42no—
125C2×#88322—42no—
126C2×#88322—41no—
127C23×C53—41no—
128C23×C53—41no—
129C23×C53C4×D411yescharacteristic
130C23×C53#12011yescharacteristic
131C23×C53#12111yescharacteristic
132C23×C53C2×SD1611yescharacteristic
133C23×C53#13511yescharacteristic
134C23×C53AGL(1,ℤ/8)11yescharacteristic
135C23×C53#13711yescharacteristic
136C22×D53—41no—
137C22×D53—41no—
138C22×D53—41no—
139C22×D53—81no—
140C22×D53—81no—
141C2×C4×C53—22no—
142C2×C4×C53—22no—
143C2×C4×C53—22no—
144C2×C4×C53—22no—
145C2×C4×C53—22no—
146C2×C4×C53—22no—
147C2×C4×C53—41no—
148C2×C4×C53—41no—
149C2×C4×C53—41no—
150C2×C4×C53—42no—
151C2×C4×C53—42no—
152C2×C4×C53—41no—
153C2×C4×C53—41no—
154C2×C4×C53—41no—
155C2×C4×C53—42no—
156C2×C4×C53—42no—
157C2×C4×C53—41no—
158C2×C4×C53—41no—
159C2×C4×C53—42no—
160C2×C4×C53—82no—
161C2×C4×C53—82no—
162C2×C4×C53C2×C22⋊C411yescharacteristic
163C2×C4×C53C4×D411yescharacteristic
164C2×C4×C53#12111yescharacteristic
165C2×C4×C53#12311yescharacteristic
166C22⋊D53—81no—
167C22⋊D53—81no—
168C22⋊D53—81no—
169C22⋊D53—81no—
170C2×Dic53—41no—
171C2×Dic53—41no—
172C2×Dic53—41no—
173C2×Dic53—81no—
174C2×Dic53—81no—
175C2×Dic53—81no—
176D212—42no—
177D212—81no—
178#88322—42no—
179#88322—81no—
180#9261—531nomaximal, Sylow
181C22×C53—21no—
182C22×C53—21no—
183C22×C53—21no—
184C22×C53—21no—
185C22×C53—21no—
186C22×C53—21no—
187C22×C53—41no—
188C22×C53—41no—
189C22×C53—41no—
190C22×C53—41no—
191C22×C53#38611yescharacteristic
192C22×C53C4×SD1611yescharacteristic
193C22×C53#44011yescharacteristic
194C22×C53#44811yescharacteristic
195C22×C53#44911yescharacteristic
196C22×C53#46011yescharacteristic
197C22×C53#46411yescharacteristic
198C2×D53—81no—
199C2×D53—81no—
200C2×D53—81no—
201C2×D53—161no—
202C2×D53—161no—
203C4×C53—21no—
204C4×C53—21no—
205C4×C53—21no—
206C4×C53—42no—
207C4×C53—42no—
208C4×C53—42no—
209C4×C53—81no—
210C4×C53—82no—
211C4×C53#41012yes—
212Dic53—81no—
213Dic53—161no—
214C2×#481—531no—
215#2345D5311yescharacteristic, complemented
216#1964—531no—
217#1824—531no—
218#1519—531no—
219#1289—531no—
220#1282—531no—
221C2×C53—21no—
222C2×C53—21no—
223C2×C53—21no—
224C2×C53—21no—
225C2×C53—21no—
226C2×C53—21no—
227C2×C53#188011yescharacteristic
228C2×C53#189911yescharacteristic
229C2×C53#190411yescharacteristic
230D53—161no—
231C2×#121—531no—
232C22×C4⋊C4C2×D5311yescharacteristic
233#481—1062no—
234#481—1061no—
235#481—1061no—
236C2×#106—532no—
237C2×#102—532no—
238C2×#99—1061no—
239C2×#99—1061no—
240C2×C22⋊C8—531no—
241C2×C22⋊C8—531no—
242#389—21no—
243#387—22no—
244#380—21no—
245#380—531no—
246#380C2×D5311yescharacteristic
247C4×C4⋊C4—21no—
248C4×C22⋊C4—21no—
249C4×C22⋊C4C2×D5311yescharacteristic
250#340—1062no—
251#327—1062no—
252C53#926111yescharacteristic, complemented, Sylow
253C22×D4—531no—
254C22×D4—1061no—
255C23×C4—21no—
256C23×C4C22×D5311yescharacteristic
257C22×C8—532no—
258C22×C8—1061no—
259#121—1061no—
260#121—1061no—
261#121—1061no—
262#121—1061no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—531no—
269C2×C4⋊C4—1061no—
270C2×C4⋊C4C4×D5312yes—
271C2×C4⋊C4D21212yes—
272C2×C4⋊C4C22⋊D5311yescharacteristic
273C2×C4⋊C4C22⋊D5311yescharacteristic
274C2×C22⋊C4—21no—
275C2×C22⋊C4—1061no—
276C2×C22⋊C4—1061no—
277C2×C22⋊C4C22⋊D5311yescharacteristic
278C2×C22⋊C4C22⋊D5311yescharacteristic
279C2×C42—21no—
280C2×C42—22no—
281#106—1062no—
282#106—1062no—
283#102—1062no—
284#102—1062no—
285#99—2122no—
286#99—2122no—
287C22⋊C8—1061no—
288C22⋊C8—1061no—
289C22⋊C8—1061no—
290C22⋊C8—1061no—
291#95—22no—
292#95—22no—
293#95—21no—
294#95—1061no—
295#95C22⋊D5312yes—
296C24—1061no—
297C24C2×C22⋊D5311yescharacteristic
298C2×D4—1061no—
299C2×D4—1061no—
300C2×D4—2122no—
301C2×D4—2121no—
302C2×D4—2121no—
303C2×D4—2121no—
304C2×D4—2121no—
305C2×D4—2121no—
306C2×D4—2121no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—21no—
313C22×C4—22no—
314C22×C4—42no—
315C22×C4—82no—
316C22×C4—1061no—
317C22×C4—2121no—
318C22×C4C2×C4×D5311yescharacteristic
319C22×C4C2×D21211yescharacteristic
320C22×C4#1171994511yescharacteristic
321C22×C4#1171994511yescharacteristic
322C22×C4C2×C22⋊D5311yescharacteristic
323C22×C4C2×C22⋊D5311yescharacteristic
324C2×C8—1062no—
325C2×C8—2122no—
326C2×C8—2121no—
327C2×C8—2122no—
328C4⋊C4—22no—
329C4⋊C4—22no—
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4—42no—
333C4⋊C4—82no—
334C4⋊C4—1061no—
335C4⋊C4—1061no—
336C4⋊C4—2122no—
337C4⋊C4#1171992112yes—
338C4⋊C4#1171992112yes—
339C22⋊C4—21no—
340C22⋊C4—21no—
341C22⋊C4—21no—
342C22⋊C4—41no—
343C22⋊C4—41no—
344C22⋊C4—2121no—
345C22⋊C4—2121no—
346C22⋊C4—2121no—
347C22⋊C4—2121no—
348C22⋊C4#1171992412yes—
349C42—42no—
350C42—82no—
351C23—41no—
352C23—41no—
353C23—4241no—
354C23—4241no—
355C23—2121no—
356C23—2121no—
357C23—2121no—
358C23C4×C22⋊D5311yescharacteristic
359C23#1313318611yescharacteristic
360C23#1313319111yescharacteristic
361C23C2×#1171992411yescharacteristic
362C23#1313323711yescharacteristic
363C23#1313324211yescharacteristic
364C23#1313324811yescharacteristic
365D4—4241no—
366D4—4241no—
367D4—4241no—
368D4—4241no—
369D4—4241no—
370D4—2122no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—41no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—42no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4—4241no—
393C2×C4—4241no—
394C2×C4—4241no—
395C2×C4—2121no—
396C2×C4—2121no—
397C2×C4—2121no—
398C2×C4C2×#1171992111yescharacteristic
399C2×C4C4×C22⋊D5311yescharacteristic
400C2×C4#1313319111yescharacteristic
401C2×C4#1313320011yescharacteristic
402C8—4241no—
403C8—2122no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—8481no—
415C22—8481no—
416C22—4241no—
417C22—4241no—
418C22—4241no—
419C22#1681050211yescharacteristic
420C22C4×#1171992411yescharacteristic
421C22#1681072111yescharacteristic
422C22#1681076211yescharacteristic
423C22#1681076811yescharacteristic
424C22#1681082311yescharacteristic
425C22#1681084511yescharacteristic
426C4—21no—
427C4—21no—
428C4—21no—
429C4—42no—
430C4—42no—
431C4—42no—
432C4—81no—
433C4—82no—
434C4—8481no—
435C4—4241no—
436C4#1681059612yes—
437C2—21no—
438C2—21no—
439C2—21no—
440C2—21no—
441C2—21no—
442C2—21no—
443C2—8481no—
444C2#2027507011yescharacteristic
445C2#2027517311yescharacteristic
446C2#2027520211yescharacteristic
subgroupG/H
centerC22#16810502
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#2345C2
socleC22×C53#386
prime radicalC2×#13133191C22
cosocleC23×C4×C53C23
hypercenter#2345D53
hypocenterC53#9261
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2345

Complements

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

NH ≅ G/Ntype
3. #20274805C2semidirect
5. #20272582C2semidirect
6. #20272563C2semidirect
7. C53×#2345C2semidirect
12. C2×#13133148C4semidirect
13. C2×#13133137C4semidirect
215. #2345D53semidirect
252. C53#9261semidirect

Maximal subgroups

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

subgroupindex
C2×#168109412
#202755512
#202748052
#202732822
#202725822
#202725632
C53×#23452
#926153