Finite Group Databasebeta

← #27280429

Subgroups — #27280429

Order 13568.

514 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283750C211yesmaximal, characteristic, complemented
2#20273313C211yesmaximal, characteristic, complemented
3#20273158C211yesmaximal, characteristic, complemented
4#20273067C211yesmaximal, characteristic, complemented
5#20272585C211yesmaximal, characteristic, complemented
6#20272565C211yesmaximal, characteristic, complemented
7C2×C53×#463C211yesmaximal, characteristic
8#16810701—21no—
9#16810695—22no—
10D4×#88322—21no—
11#16810680C2211yescharacteristic
12#16810680C2211yescharacteristic
13#16810676C2211yescharacteristic
14#16810587—21no—
15C2×C22⋊#88322—21no—
16C4×#11719917C2211yescharacteristic
17#16810363—22no—
18D4⋊#88322—21no—
19D4⋊#88322—21no—
20#16810347—21no—
21#16810347—21no—
22C22×SD16×C53C411yescharacteristic
23C22×D8×C53C411yescharacteristic
24C2×#124×C53C2211yescharacteristic
25C2×C4×D4×C53C2211yescharacteristic
26C53×#463—21no—
27C53×#463—21no—
28C53×#463—21no—
29C53×#463—21no—
30C2×C4⋊C8×C53C2211yescharacteristic
31C2×#103×C53C411yescharacteristic
32C2×#102×C53C411yescharacteristic
33C2×#11719917—22no—
34C2×#11719917—21no—
35C22×#88322—22no—
36C4⋊#88322—21no—
37C4⋊#88322—21no—
38C4⋊#88322—21no—
39C22⋊#88322—22no—
40C22⋊#88322—22no—
41C22⋊#88322—21no—
42C22⋊#88322—21no—
43C22⋊#88322D411yescharacteristic
44C22⋊#88322D411yescharacteristic
45#13133128—21no—
46#13133128—21no—
47#13133128D412yes—
48C4×#88322—21no—
49C4×#88322—21no—
50C22×Q8×C53C2×C411yescharacteristic
51C22×D4×C53C2×C411yescharacteristic
52C22×D4×C53C2×C411yescharacteristic
53C23×C4×C53—21no—
54C2×SD16×C53—21no—
55C2×SD16×C53—21no—
56C2×SD16×C53—42no—
57C2×D8×C53—21no—
58C2×D8×C53—21no—
59C2×D8×C53—42no—
60C22×C8×C53C2×C411yescharacteristic
61C22×C8×C53C2×C411yescharacteristic
62#124×C53—21no—
63#124×C53—21no—
64#124×C53—21no—
65#124×C53—21no—
66C4×D4×C53—21no—
67C4×D4×C53—21no—
68C4×D4×C53D411yescharacteristic
69C4×D4×C53D411yescharacteristic
70C4×D4×C53D411yescharacteristic
71C4×D4×C53D411yescharacteristic
72C2×C4⋊C4×C53C2×C411yescharacteristic
73C2×C22⋊C4×C53—21no—
74C2×C22⋊C4×C53—21no—
75C2×C22⋊C4×C53—21no—
76C2×C42×C53C2311yescharacteristic
77C4⋊C8×C53—21no—
78C4⋊C8×C53—21no—
79#103×C53—21no—
80#103×C53—21no—
81#102×C53—21no—
82#102×C53—21no—
83C24×C53—21no—
84C24×C53—21no—
85C2×Q8×C53—21no—
86C2×Q8×C53—21no—
87C2×Q8×C53—41no—
88C2×Q8×C53—41no—
89C2×D4×C53—22no—
90C2×D4×C53—21no—
91C2×D4×C53—21no—
92C2×D4×C53—21no—
93C2×D4×C53—41no—
94C2×D4×C53—41no—
95C2×D4×C53—41no—
96C2×D4×C53C22⋊C411yescharacteristic
97C2×D4×C53C22⋊C411yescharacteristic
98C22×C4×C53—21no—
99C22×C4×C53—21no—
100C22×C4×C53—21no—
101C22×C4×C53—21no—
102C22×C4×C53—21no—
103C22×C4×C53—21no—
104C22×C4×C53—82no—
105C22×C4×C53C22×C411yescharacteristic
106C22×C4×C53C2×D411yescharacteristic
107C22×C4×C53#4111yescharacteristic
108SD16×C53—42no—
109SD16×C53—42no—
110D8×C53—42no—
111D8×C53—42no—
112C2×C8×C53—21no—
113C2×C8×C53—21no—
114C2×C8×C53—41no—
115C2×C8×C53—41no—
116C4⋊C4×C53—21no—
117C4⋊C4×C53C22⋊C411yescharacteristic
118C4⋊C4×C53C22⋊C411yescharacteristic
119C22⋊C4×C53—21no—
120C22⋊C4×C53—21no—
121C22⋊C4×C53—21no—
122C22⋊C4×C53—21no—
123C22⋊C4×C53—41no—
124C22⋊C4×C53—41no—
125C22⋊C4×C53—41no—
126C22⋊C4×C53—41no—
127C42×C53C2×D411yescharacteristic
128C42×C53C2×D411yescharacteristic
129C42×C53C2×D411yescharacteristic
130C42×C53#4111yescharacteristic
131#11719917—22no—
132#11719917—41no—
133#11719917—82no—
134C2×#88322—21no—
135C2×#88322—21no—
136C2×#88322—22no—
137C2×#88322—41no—
138C2×#88322—41no—
139C2×#88322—82no—
140C23×C53—21no—
141C23×C53—21no—
142C23×C53—42no—
143C23×C53—41no—
144C23×C53—41no—
145C23×C53—41no—
146C23×C53—41no—
147C23×C53—41no—
148C23×C53—81no—
149C23×C53—81no—
150C23×C53—81no—
151C23×C53C4×D411yescharacteristic
152Q8×C53—41no—
153Q8×C53—41no—
154Q8×C53—81no—
155D4×C53—21no—
156D4×C53—21no—
157D4×C53—21no—
158D4×C53—42no—
159D4×C53—41no—
160D4×C53—41no—
161D4×C53—81no—
162D4×C53#10012yes—
163C2×C4×C53—21no—
164C2×C4×C53—21no—
165C2×C4×C53—21no—
166C2×C4×C53—21no—
167C2×C4×C53—21no—
168C2×C4×C53—21no—
169C2×C4×C53—41no—
170C2×C4×C53—42no—
171C2×C4×C53—42no—
172C2×C4×C53—41no—
173C2×C4×C53—81no—
174C2×C4×C53—81no—
175C2×C4×C53—81no—
176C2×C4×C53—81no—
177C2×C4×C53C2×C22⋊C411yescharacteristic
178C2×C4×C53C4×D411yescharacteristic
179C2×C4×C53#12011yescharacteristic
180C2×C4×C53#12111yescharacteristic
181C2×C4×C53#12111yescharacteristic
182C2×C4×C53#12311yescharacteristic
183C8×C53—41no—
184C8×C53—41no—
185#88322—41no—
186#88322—42no—
187#88322—41no—
188#88322—42no—
189#88322—81no—
190#9243—531nomaximal, Sylow
191C22×C53—21no—
192C22×C53—21no—
193C22×C53—22no—
194C22×C53—41no—
195C22×C53—41no—
196C22×C53—41no—
197C22×C53—41no—
198C22×C53—41no—
199C22×C53—82no—
200C22×C53—82no—
201C22×C53—81no—
202C22×C53—81no—
203C22×C53—81no—
204C22×C53—161no—
205C22×C53—161no—
206C22×C53#38611yescharacteristic
207C22×C53#44311yescharacteristic
208C22×C53#44411yescharacteristic
209C4×C53—21no—
210C4×C53—21no—
211C4×C53—21no—
212C4×C53—21no—
213C4×C53—21no—
214C4×C53—41no—
215C4×C53—41no—
216C4×C53—81no—
217C4×C53C2×#10011yescharacteristic
218C4×C53#41311yescharacteristic
219C2×#463D5311yescharacteristic
220#2974—531no—
221#1527—531no—
222#1490—531no—
223#1469—531no—
224#1290—531no—
225#1283—531no—
226C2×C53—21no—
227C2×C53—21no—
228C2×C53—42no—
229C2×C53—81no—
230C2×C53—161no—
231C2×C53#187611yescharacteristic
232C2×C53#190011yescharacteristic
233C2×C53#190411yescharacteristic
234C22×SD16Dic5311yescharacteristic
235C22×D8Dic5311yescharacteristic
236C2×#124C2×D5311yescharacteristic
237C2×C4×D4C2×D5311yescharacteristic
238#463—21no—
239#463—21no—
240#463—21no—
241#463—21no—
242#436—1061no—
243#435—1062no—
244C8×D4—1061no—
245#432—531no—
246#432—531no—
247#431—531no—
248C2×C4⋊C8C2×D5311yescharacteristic
249C2×#103Dic5311yescharacteristic
250C2×#102Dic5311yescharacteristic
251#407—1061no—
252C2×C22⋊C8—1061no—
253C4×M16—531no—
254#329—1062no—
255D4⋊C8—1061no—
256D4⋊C8—1061no—
257#323—1061no—
258#323—1061no—
259C53#924311yescharacteristic, complemented, Sylow
260C22×Q8C2×Dic5311yescharacteristic
261C22×D4C2×Dic5311yescharacteristic
262C22×D4C2×Dic5311yescharacteristic
263C23×C4—21no—
264C2×SD16—21no—
265C2×SD16—21no—
266C2×SD16—42no—
267C2×D8—21no—
268C2×D8—21no—
269C2×D8—42no—
270C2×M16—1062no—
271C2×M16—1061no—
272C22×C8—1062no—
273C22×C8C2×Dic5311yescharacteristic
274C22×C8C2×Dic5311yescharacteristic
275#124—21no—
276#124—21no—
277#124—21no—
278#124—21no—
279C4×D4—21no—
280C4×D4—21no—
281C4×D4C22⋊D5311yescharacteristic
282C4×D4C22⋊D5311yescharacteristic
283C4×D4C22⋊D5311yescharacteristic
284C4×D4C22⋊D5311yescharacteristic
285C2×C4⋊C4C2×Dic5311yescharacteristic
286C2×C22⋊C4—21no—
287C2×C22⋊C4—21no—
288C2×C22⋊C4—21no—
289C2×C42C22×D5311yescharacteristic
290C4⋊C8—21no—
291C4⋊C8—21no—
292C4⋊C8—1061no—
293C4⋊C8—1061no—
294C4⋊C8—1061no—
295#103—21no—
296#103—21no—
297#102—21no—
298#102—21no—
299C22⋊C8—531no—
300C22⋊C8—531no—
301C22⋊C8—1062no—
302C22⋊C8—1062no—
303C22⋊C8—1061no—
304C22⋊C8—1061no—
305#97—532no—
306#97—1061no—
307#97—1061no—
308C4×C8—1061no—
309C4×C8—1061no—
310C24—21no—
311C24—21no—
312C2×Q8—21no—
313C2×Q8—21no—
314C2×Q8—41no—
315C2×Q8—41no—
316C2×D4—22no—
317C2×D4—21no—
318C2×D4—21no—
319C2×D4—21no—
320C2×D4—41no—
321C2×D4—41no—
322C2×D4—41no—
323C2×D4C22⋊Dic5311yescharacteristic
324C2×D4C22⋊Dic5311yescharacteristic
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—82no—
332C22×C4D4×D5311yescharacteristic
333C22×C4#1171994711yescharacteristic
334C22×C4C22×Dic5311yescharacteristic
335SD16—42no—
336SD16—42no—
337D8—42no—
338D8—42no—
339M16—1062no—
340M16—4242no—
341M16—2121no—
342C2×C8—21no—
343C2×C8—21no—
344C2×C8—41no—
345C2×C8—41no—
346C2×C8—1061no—
347C2×C8—1061no—
348C2×C8—1062no—
349C2×C8—4242no—
350C2×C8—2121no—
351C2×C8—2121no—
352C4⋊C4—21no—
353C4⋊C4C22⋊Dic5311yescharacteristic
354C4⋊C4C22⋊Dic5311yescharacteristic
355C22⋊C4—21no—
356C22⋊C4—21no—
357C22⋊C4—21no—
358C22⋊C4—21no—
359C22⋊C4—41no—
360C22⋊C4—41no—
361C22⋊C4—41no—
362C22⋊C4—41no—
363C42D4×D5311yescharacteristic
364C42#1171994711yescharacteristic
365C42C2×C22⋊D5311yescharacteristic
366C42C2×C22⋊D5311yescharacteristic
367C23—21no—
368C23—21no—
369C23—42no—
370C23—41no—
371C23—41no—
372C23—41no—
373C23—41no—
374C23—41no—
375C23—81no—
376C23—81no—
377C23—81no—
378C23D4×Dic5311yescharacteristic
379Q8—41no—
380Q8—41no—
381Q8—81no—
382D4—21no—
383D4—21no—
384D4—21no—
385D4—42no—
386D4—41no—
387D4—41no—
388D4—81no—
389D4#1313313412yes—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—41no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—41no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—81no—
404C2×C4C2×C22⋊Dic5311yescharacteristic
405C2×C4D4×Dic5311yescharacteristic
406C2×C4#1313318511yescharacteristic
407C2×C4#1313318911yescharacteristic
408C2×C4#1313318911yescharacteristic
409C2×C4#1313319811yescharacteristic
410C8—41no—
411C8—41no—
412C8—4241no—
413C8—2121no—
414C8—2122no—
415C8—2121no—
416C8—2122no—
417C22—21no—
418C22—21no—
419C22—22no—
420C22—41no—
421C22—41no—
422C22—41no—
423C22—41no—
424C22—41no—
425C22—82no—
426C22—82no—
427C22—81no—
428C22—81no—
429C22—81no—
430C22—161no—
431C22—161no—
432C22#1681050311yescharacteristic
433C22#1681074011yescharacteristic
434C22#1681074511yescharacteristic
435C4—21no—
436C4—21no—
437C4—21no—
438C4—21no—
439C4—21no—
440C4—41no—
441C4—41no—
442C4—81no—
443C4C2×#1313313411yescharacteristic
444C4#1681060711yescharacteristic
445C2—21no—
446C2—21no—
447C2—42no—
448C2—81no—
449C2—161no—
450C2#2027504411yescharacteristic
451C2#2027517911yescharacteristic
452C2#2027520411yescharacteristic
subgroupG/H
centerC22#16810503
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC2×C53×#463C2
socleC22×C53#386
prime radicalC22×D8×C53C4
cosocleC2×C42×C53C23
hypercenterC2×#463D53
hypocenterC53#9243
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#463

Complements

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

NH ≅ G/Ntype
1. #20283750C2semidirect
2. #20273313C2semidirect
3. #20273158C2semidirect
4. #20273067C2semidirect
5. #20272585C2semidirect
6. #20272565C2semidirect
259. C53#9243semidirect

Maximal subgroups

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

subgroupindex
#202837502
#202733132
#202731582
#202730672
#202725852
#202725652
C2×C53×#4632
#924353