Finite Group Databasebeta

← #27278728

Subgroups — #27278728

Order 13568.

540 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
1#20283832C211yesmaximal, characteristic, complemented
2C4×#13133242C211yesmaximal, characteristic
3#20275509C211yesmaximal, characteristic, complemented
4#20273107C211yesmaximal, characteristic, complemented
5#20272564C211yesmaximal, characteristic, complemented
6#20272552C211yesmaximal, characteristic, complemented
7C53×#1476C211yesmaximal, characteristic, complemented
8C2×#13133242C411yescharacteristic
9C2×C4×D212C2211yescharacteristic
10#16810754—21no—
11#16810733—21no—
12#16810721—21no—
13C4×#11719924—21no—
14C4×#11719922—21no—
15#16810701—21no—
16#16810695—22no—
17#16810680—21no—
18#16810676C2211yescharacteristic
19#16810666C411yescharacteristic
20#16810647—21no—
21C2×C4⋊#88322C2211yescharacteristic
22#16810627C411yescharacteristic
23#16810624C411yescharacteristic
24C2×#13133129—22no—
25C4×#11719917C2211yescharacteristic
26#16810352—21no—
27#16810352—21no—
28C4×#41×C53C2211yescharacteristic
29C53×#431C2211yescharacteristic
30C2×C4×C8×C53C2211yescharacteristic
31C53×Q8⋊C8—21no—
32C53×D4⋊C8—21no—
33C22×D212C2×C411yescharacteristic
34C22×C4×D53—21no—
35#13133242—22no—
36#13133242—22no—
37C2×#11719924—21no—
38C2×#11719922—21no—
39C2×#11719917—22no—
40C2×#11719917C2×C411yescharacteristic
41C2×#11719917C2×C411yescharacteristic
42C22×#88322D412yes—
43C4×D212—21no—
44C4×D212—21no—
45C4×D212—21no—
46C4×D212—21no—
47C2×C4⋊Dic53C2×C411yescharacteristic
48C2×#11719921—21no—
49#13133150—21no—
50#13133148—21no—
51C4⋊#88322—22no—
52C4⋊#88322—21no—
53C4⋊#88322—21no—
54C4⋊#88322D411yescharacteristic
55C4⋊#88322D411yescharacteristic
56#13133141—21no—
57#13133138—21no—
58#13133137—21no—
59#13133137—21no—
60C22⋊#88322—22no—
61C22⋊#88322—21no—
62#13133129—44no—
63#13133128—22no—
64#13133128—21no—
65C4×#88322—21no—
66C4×#88322—21no—
67C2×#41×C53C2×C411yescharacteristic
68C22×C8×C53—21no—
69C4×Q8×C53D411yescharacteristic
70C4×Q8×C53D411yescharacteristic
71C4×D4×C53—21no—
72C4×D4×C53—21no—
73C4×D4×C53D411yescharacteristic
74C4×D4×C53D411yescharacteristic
75#117×C53—21no—
76#117×C53C2×C411yescharacteristic
77C2×C42×C53—21no—
78C2×C42×C53C2311yescharacteristic
79C4⋊C8×C53—21no—
80C22⋊C8×C53—21no—
81C4×C8×C53—21no—
82C4×C8×C53—21no—
83C4×C8×C53—21no—
84C23×D53—21no—
85#41×C53—21no—
86#41×C53—21no—
87#41×C53—21no—
88#41×C53—21no—
89C2×Q8×C53C22⋊C411yescharacteristic
90C2×D4×C53—21no—
91C2×D4×C53C22⋊C411yescharacteristic
92C22×C4×C53—21no—
93C22×C4×C53—21no—
94C22×C4×C53—21no—
95C22×C4×C53C22×C411yescharacteristic
96C22×C4×C53C2×D411yescharacteristic
97C22×C4×C53#4111yescharacteristic
98C22×Dic53—21no—
99C2×D212—21no—
100C2×D212—21no—
101C2×D212—21no—
102C2×D212—21no—
103C2×D212—21no—
104C2×D212M1612yescomplemented
105C2×C4×D53—42no—
106C2×C4×D53—42no—
107C2×C4×D53—41no—
108C2×C4×D53—41no—
109C2×C8×C53—22no—
110C2×C8×C53—41no—
111C2×C8×C53—41no—
112C2×C8×C53—41no—
113C4⋊C4×C53—21no—
114C4⋊C4×C53—21no—
115C4⋊C4×C53C22⋊C411yescharacteristic
116C4⋊C4×C53C22⋊C411yescharacteristic
117C22⋊C4×C53—21no—
118C22⋊C4×C53—21no—
119C22⋊C4×C53—21no—
120C42×C53—21no—
121C42×C53—21no—
122C42×C53—21no—
123C42×C53C2×D411yescharacteristic
124C42×C53C2×D411yescharacteristic
125C42×C53C2×D411yescharacteristic
126C42×C53#4111yescharacteristic
127#11719924—24no—
128#11719922—24no—
129#11719921—41no—
130#11719921—41no—
131C4⋊Dic53—21no—
132C4⋊Dic53—21no—
133#11719917—21no—
134#11719917—21no—
135#11719917—21no—
136#11719917—82no—
137#11719917M1612yescomplemented
138C2×#88322—21no—
139C2×#88322—21no—
140C2×#88322—21no—
141C2×#88322—21no—
142C2×#88322—22no—
143C2×#88322—42no—
144C23×C53—21no—
145C23×C53C4×D411yescharacteristic
146C22×D53—22no—
147C22×D53—42no—
148C22×D53—42no—
149C22×D53—41no—
150C22×D53—41no—
151Q8×C53—21no—
152Q8×C53—21no—
153D4×C53—21no—
154D4×C53—21no—
155D4×C53—41no—
156D4×C53—41no—
157C2×C4×C53—21no—
158C2×C4×C53—21no—
159C2×C4×C53—21no—
160C2×C4×C53—21no—
161C2×C4×C53—21no—
162C2×C4×C53—41no—
163C2×C4×C53—41no—
164C2×C4×C53—41no—
165C2×C4×C53—41no—
166C2×C4×C53—41no—
167C2×C4×C53—81no—
168C2×C4×C53C2×C22⋊C411yescharacteristic
169C2×C4×C53C4×D411yescharacteristic
170C2×C4×C53#12011yescharacteristic
171C2×C4×C53#12111yescharacteristic
172C2×C4×C53#12111yescharacteristic
173C2×C4×C53#12311yescharacteristic
174C2×C4×C53C2×M1611yescharacteristic
175C2×C4×C53C2×M1611yescharacteristic
176C2×C4×C53#13111yescharacteristic
177C2×C4×C53#13111yescharacteristic
178C2×Dic53—42no—
179C2×Dic53—41no—
180D212—22no—
181D212—22no—
182D212—41no—
183D212—41no—
184C4×D53—82no—
185C4×D53—81no—
186C4×D53—81no—
187C8×C53—42no—
188C8×C53—81no—
189#88322—22no—
190#88322—41no—
191#88322—42no—
192#88322—41no—
193#88322—41no—
194#8956—531nomaximal, Sylow
195C22×C53—21no—
196C22×C53—21no—
197C22×C53—41no—
198C22×C53—41no—
199C22×C53—41no—
200C22×C53#38611yescharacteristic
201C22×C53#43511yescharacteristic
202C22×C53#43611yescharacteristic
203C2×D53—44no—
204C2×D53—42no—
205C2×D53—82no—
206C2×D53—81no—
207C2×D53—81no—
208C4×C53—21no—
209C4×C53—21no—
210C4×C53—21no—
211C4×C53—41no—
212C4×C53—41no—
213C4×C53—41no—
214C4×C53#40711yescharacteristic
215C4×C53#40811yescharacteristic
216C4×C53#43511yescharacteristic
217C4×C53#43611yescharacteristic
218C4×C53#45311yescharacteristic
219C4×C53#45411yescharacteristic
220Dic53—81no—
221#2989—531no—
222C4×AGL(1,ℤ/8)—531no—
223#1954—531no—
224#1478—531no—
225#1476D5311yescharacteristic, complemented
226#1283—531no—
227#1278—531no—
228C2×C53—21no—
229C2×C53—21no—
230C2×C53—81no—
231C2×C53#187011yescharacteristic
232C2×C53#188111yescharacteristic
233C2×C53#188311yescharacteristic
234D53—82no—
235C2×AGL(1,ℤ/8)—531no—
236C4×#41C2×D5311yescharacteristic
237C2×C4×D4—531no—
238#446—1061no—
239#442—1061no—
240#440—1061no—
241C4×SD16—1061no—
242C4×D8—1061no—
243#436—1061no—
244#435—1062no—
245#432—1061no—
246#431—531no—
247#431C2×D5311yescharacteristic
248#428—531no—
249#423—1061no—
250C2×C4⋊C8—531no—
251#418—531no—
252#417—531no—
253C2×C22⋊C8—1062no—
254C4×M16—531no—
255C2×C4×C8C2×D5311yescharacteristic
256Q8⋊C8—21no—
257D4⋊C8—21no—
258D4⋊C8—1061no—
259D4⋊C8—1061no—
260C53#895611yescharacteristic, complemented, Sylow
261C2×#41C4×D5311yescharacteristic
262C22×D4—531no—
263C23×C4—1061no—
264AGL(1,ℤ/8)—1062no—
265AGL(1,ℤ/8)—1062no—
266C2×SD16—1061no—
267C2×D8—1061no—
268C2×M16—531no—
269C2×M16—531no—
270C2×M16—1062no—
271C22×C8—21no—
272C22×C8—532no—
273C4×Q8D21211yescharacteristic
274C4×Q8C22⋊D5311yescharacteristic
275C4×D4—21no—
276C4×D4—21no—
277C4×D4—1061no—
278C4×D4—1061no—
279C4×D4—1061no—
280C4×D4—1061no—
281C4×D4D21211yescharacteristic
282C4×D4C22⋊D5311yescharacteristic
283#117—21no—
284#117C4×D5311yescharacteristic
285C2×C4⋊C4—531no—
286C2×C22⋊C4—1061no—
287C2×C42—21no—
288C2×C42C22×D5311yescharacteristic
289#107—1061no—
290#106—1061no—
291C4⋊C8—21no—
292C4⋊C8—531no—
293C4⋊C8—531no—
294C4⋊C8—1062no—
295C4⋊C8—1061no—
296C4⋊C8—1061no—
297#103—1061no—
298#102—1061no—
299#102—1061no—
300#102—1061no—
301C22⋊C8—21no—
302C22⋊C8—1062no—
303C22⋊C8—1061no—
304C22⋊C8—2124no—
305#97—1062no—
306#97—1061no—
307C4×C8—21no—
308C4×C8—21no—
309C4×C8—21no—
310C4×C8—1061no—
311C4×C8—1061no—
312C24—1061no—
313#41—21no—
314#41—21no—
315#41—21no—
316#41—21no—
317C2×Q8#1171992111yescharacteristic
318C2×D4—21no—
319C2×D4—532no—
320C2×D4—1061no—
321C2×D4—1061no—
322C2×D4—1061no—
323C2×D4—1061no—
324C2×D4—1061no—
325C2×D4#1171992111yescharacteristic
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—1061no—
330C22×C4—2122no—
331C22×C4—2122no—
332C22×C4—2121no—
333C22×C4—2121no—
334C22×C4C2×C4×D5311yescharacteristic
335C22×C4D4×D5311yescharacteristic
336C22×C4#1171994711yescharacteristic
337SD16—1064no—
338D8—1064no—
339M16—532no—
340M16—1061no—
341M16—1061no—
342M16—1061no—
343M16—4242no—
344C2×C8—22no—
345C2×C8—41no—
346C2×C8—41no—
347C2×C8—41no—
348C2×C8—1061no—
349C2×C8—1061no—
350C2×C8—1061no—
351C2×C8—1061no—
352C2×C8—1062no—
353C2×C8—2122no—
354C4⋊C4—21no—
355C4⋊C4—21no—
356C4⋊C4—1061no—
357C4⋊C4—1061no—
358C4⋊C4#1171992111yescharacteristic
359C4⋊C4#1171992111yescharacteristic
360C22⋊C4—21no—
361C22⋊C4—21no—
362C22⋊C4—21no—
363C22⋊C4—2121no—
364C22⋊C4—2121no—
365C42—21no—
366C42—21no—
367C42—21no—
368C42C2×D21211yescharacteristic
369C42D4×D5311yescharacteristic
370C42#1171994711yescharacteristic
371C42C2×C22⋊D5311yescharacteristic
372C23—21no—
373C23—1062no—
374C23—2122no—
375C23—2122no—
376C23—2121no—
377C23—2121no—
378C23#1313317711yescharacteristic
379Q8—21no—
380Q8—21no—
381D4—21no—
382D4—21no—
383D4—41no—
384D4—41no—
385D4—1062no—
386D4—1062no—
387D4—2121no—
388D4—2121no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—81no—
400C2×C4—4242no—
401C2×C4—4241no—
402C2×C4—4241no—
403C2×C4—2122no—
404C2×C4—2121no—
405C2×C4C2×#1171992111yescharacteristic
406C2×C4#1313317711yescharacteristic
407C2×C4#1313318711yescharacteristic
408C2×C4#1313318911yescharacteristic
409C2×C4#1313318911yescharacteristic
410C2×C4#1313320111yescharacteristic
411C2×C4M16×D5311yescharacteristic
412C2×C4M16×D5311yescharacteristic
413C2×C4#1313322311yescharacteristic
414C2×C4#1313322311yescharacteristic
415C8—42no—
416C8—81no—
417C8—1062no—
418C8—2122no—
419C8—2121no—
420C8—2121no—
421C8—2121no—
422C22—21no—
423C22—21no—
424C22—41no—
425C22—41no—
426C22—41no—
427C22—4242no—
428C22—4241no—
429C22—4241no—
430C22—2124no—
431C22—2122no—
432C22#1681050011yescharacteristic
433C22#1681069711yescharacteristic
434C22#1681069811yescharacteristic
435C4—21no—
436C4—21no—
437C4—21no—
438C4—41no—
439C4—41no—
440C4—41no—
441C4—4241no—
442C4#1681058611yescharacteristic
443C4#1681058911yescharacteristic
444C4#1681069711yescharacteristic
445C4#1681069811yescharacteristic
446C4#1681079211yescharacteristic
447C4#1681079711yescharacteristic
448C2—21no—
449C2—21no—
450C2—81no—
451C2—4242no—
452C2#2027501011yescharacteristic
453C2#2027507511yescharacteristic
454C2#2027508911yescharacteristic
subgroupG/H
centerC2×C4#13133187
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1476C2
socleC22×C53#386
prime radicalC2×#13133242C4
cosocleC2×C42×C53C23
hypercenter#1476D53
hypocenterC53#8956
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1476

Complements

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

NH ≅ G/Ntype
1. #20283832C2semidirect
3. #20275509C2semidirect
4. #20273107C2semidirect
5. #20272564C2semidirect
6. #20272552C2semidirect
7. C53×#1476C2semidirect
104. C2×D212M16semidirect
137. #11719917M16semidirect
225. #1476D53semidirect
260. C53#8956semidirect

Maximal subgroups

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

subgroupindex
#202838322
C4×#131332422
#202755092
#202731072
#202725642
#202725522
C53×#14762
#895653