Finite Group Databasebeta

← #27538493

Subgroups — #27538493

Order 13568.

706 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#20287849C211yesmaximal, characteristic, complemented
2C2×#16811058C211yesmaximal, characteristic, complemented
3#20285201C211yesmaximal, characteristic, complemented
4#20276791C214yesmaximal, complemented
5#20276784C214yesmaximal, complemented
6#20276481C211yesmaximal, characteristic, complemented
7#20276462C211yesmaximal, characteristic, complemented
8C2×#16810435C211yesmaximal, characteristic, complemented
9C53×#2957C211yesmaximal, characteristic, complemented
10C2×#13133245—21no—
11C2×#13133239C2211yescharacteristic
12C22×D424C2211yescharacteristic
13#16811192—21no—
14#16811064—21no—
15#16811063C2212yescomplemented
16#16811059—21no—
17#16811058—22no—
18#16811058C2214yes—
19#16811058C2212yescomplemented
20#16811053—22no—
21C2×#13133160—21no—
22C2×#13133158—21no—
23C2×#13133157C2211yescharacteristic
24C2×#13133157C2211yescharacteristic
25#16811012—22no—
26#16811007C2212yescomplemented
27#16811000C2212yescomplemented
28#16810444C2214yescomplemented
29#16810438C2212yescomplemented
30#16810435C2214yescomplemented
31#16810435C2212yescomplemented
32C2×#135×C53C2211yescharacteristic
33C53×#568—21no—
34C53×#444—22no—
35C53×#443C2214yescomplemented
36C2×#108×C53C2211yescharacteristic
37C53×#421—21no—
38C53×#405C2211yescharacteristic
39C2×#11719945—21no—
40C22×D212—21no—
41#13133245—42no—
42#13133245—42no—
43#13133239—21no—
44#13133239—21no—
45#13133239—21no—
46#13133239—21no—
47C2×Dic212C2311yescharacteristic
48C2×#11719913—21no—
49C2×#11719913—22no—
50C2×D424—21no—
51C2×D424—22no—
52C2×D424—21no—
53C2×D424—21no—
54C2×D424—21no—
55C2×D424—21no—
56C2×D424C2312yes—
57C2×D424C2311yescharacteristic
58#13133214—22no—
59#13133205—22no—
60#13133192—42no—
61#13133187—42no—
62#13133163—22no—
63#13133161—22no—
64#13133160—24no—
65#13133160—22no—
66#13133158—24no—
67#13133158—22no—
68#13133157—21no—
69#13133157—21no—
70#13133157C2312yes—
71#13133157C2312yes—
72C2×#11719908—21no—
73C2×#11719908—21no—
74C2×#41×C53—21no—
75#135×C53—21no—
76#135×C53—21no—
77#135×C53D412yescomplemented
78#135×C53C2312yes—
79C2×Q16×C53D411yescharacteristic, complemented
80C2×SD16×C53—21no—
81C2×D8×C53D411yescharacteristic, complemented
82#131×C53—24no—
83#131×C53—42no—
84C2×M16×C53—21no—
85C2×M16×C53—42no—
86C2×M16×C53D411yescharacteristic
87C22×C8×C53C2311yescharacteristic
88#117×C53D411yescharacteristic
89#108×C53—21no—
90#108×C53C2312yes—
91#104×C53—24no—
92#97×C53D412yes—
93C4×C8×C53C2312yes—
94C23×D53—41no—
95#41×C53—22no—
96#41×C53—22no—
97#41×C53—41no—
98#41×C53—41no—
99C2×Q8×C53—21no—
100C2×D4×C53—21no—
101C2×D4×C53—41no—
102C22×C4×C53—41no—
103C22×C4×C53C2×D411yescharacteristic
104C2×C22⋊D53—41no—
105#11719945—41no—
106#11719945—41no—
107#11719945—41no—
108#11719945—41no—
109C2×D212—22no—
110C2×D212—21no—
111C2×D212—41no—
112C2×D212—41no—
113C2×D212—41no—
114C2×D212—41no—
115C2×D212—41no—
116C2×D212—44no—
117C2×C4×D53—41no—
118C2×Dic106—21no—
119Q16×C53—21no—
120Q16×C53C2×D412yes—
121SD16×C53—22no—
122SD16×C53—22no—
123D8×C53—21no—
124D8×C53C2×D412yes—
125M16×C53—22no—
126M16×C53—41no—
127M16×C53—44no—
128M16×C53D812yes—
129M16×C53D812yes—
130C2×C8×C53—41no—
131C2×C8×C53—44no—
132C2×C8×C53C2×D411yescharacteristic
133C2×C8×C53C2×D411yescharacteristic
134C2×C8×C53C2×D411yescharacteristic
135C2×C8×C53C2×D412yes—
136C2×C8×C53#4111yescharacteristic
137C2×C8×C53#4111yescharacteristic
138C2×C8×C53C2411yescharacteristic
139C4⋊C4×C53—21no—
140C22⋊C4×C53—21no—
141C42×C53C2×D412yes—
142#11719921—44no—
143Dic212—21no—
144Dic212—21no—
145D424—22no—
146D424—21no—
147D424—22no—
148D424—21no—
149D424—41no—
150D424—41no—
151D424—44no—
152#11719913—41no—
153#11719913—41no—
154#11719913—44no—
155#11719908—22no—
156#11719908—22no—
157C23×C53—41no—
158C23×C53AGL(1,ℤ/8)11yescharacteristic
159C22×D53—41no—
160C22×D53—42no—
161C22×D53—81no—
162C22×D53—81no—
163C22×D53—162no—
164Q8×C53—22no—
165Q8×C53—41no—
166D4×C53—22no—
167D4×C53—42no—
168D4×C53—42no—
169D4×C53—41no—
170C2×C4×C53—22no—
171C2×C4×C53—41no—
172C2×C4×C53—42no—
173C2×C4×C53—42no—
174C2×C4×C53#12111yescharacteristic
175C2×C4×C53#12111yescharacteristic
176C2×C4×C53C2×D811yescharacteristic
177C2×C4×C53C2×D811yescharacteristic
178C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
179C2×C4×C53C22×D411yescharacteristic
180C22⋊D53—81no—
181C22⋊D53—81no—
182C2×Dic53—41no—
183D212—42no—
184D212—41no—
185D212—42no—
186D212—41no—
187D212—81no—
188D212—81no—
189D212—164no—
190C4×D53—81no—
191C4×D53—81no—
192Dic106—41no—
193Dic106—41no—
194C8×C53—42no—
195C8×C53#12111yescharacteristic
196C8×C53#12111yescharacteristic
197C8×C53C2×D814yes—
198C8×C53C22×D411yescharacteristic
199C8×C53C2×#4111yescharacteristic
200#33595—531nomaximal, Sylow
201C22×C53—41no—
202C22×C53—42no—
203C22×C53—161no—
204C22×C53#45911yescharacteristic
205C22×C53#45911yescharacteristic
206C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
207C2×D53—82no—
208C2×D53—84no—
209C2×D53—81no—
210C2×D53—82no—
211C2×D53—162no—
212C2×D53—162no—
213C4×C53—42no—
214C4×C53—42no—
215C4×C53#45911yescharacteristic
216C4×C53#45911yescharacteristic
217C4×C53C2×#12111yescharacteristic
218C4×C53C22×D811yescharacteristic
219Dic53—81no—
220#3414—531no—
221C2×#509—531no—
222#3155—531no—
223#2957D5311yescharacteristic, complemented
224#2213—534no—
225#2212—534no—
226#2153—531no—
227#2148—531no—
228C2×#361—531no—
229C2×C53—21no—
230C2×C53—21no—
231C2×C53—21no—
232C2×C53—82no—
233C2×C53C2×#45911yescharacteristic
234D53—161no—
235D53—162no—
236C2×AGL(1,ℤ/8)—1061no—
237C2×#135—531no—
238C2×#135C2×D5311yescharacteristic
239C22×D8—531no—
240#568—21no—
241#535—1061no—
242#510—532no—
243#510—1061no—
244#509—534no—
245#509—532no—
246#509—1062no—
247#509—1061no—
248#508—1062no—
249C2×SD32—1061no—
250C2×D16—1061no—
251C2×M32—531no—
252C2×M32—531no—
253#496—1062no—
254#495—532no—
255#493—532no—
256#444—22no—
257#443C2×D5314yescomplemented
258C2×#108C2×D5311yescharacteristic
259#421—21no—
260#405C2×D5311yescharacteristic
261#364—534no—
262#362—532no—
263#361—534no—
264#361—532no—
265C53#3359511yescharacteristic, complemented, Sylow
266C2×#41—21no—
267C2×#41—1061no—
268C22×D4—1061no—
269AGL(1,ℤ/8)—2122no—
270AGL(1,ℤ/8)—2122no—
271#135—21no—
272#135—21no—
273#135—1061no—
274#135—1061no—
275#135—1061no—
276#135—1061no—
277#135D21212yescomplemented
278#135C22×D5312yes—
279C2×Q16—531no—
280C2×Q16D21211yescharacteristic, complemented
281C2×SD16—21no—
282C2×SD16—1061no—
283C2×SD16—1062no—
284C2×D8—532no—
285C2×D8—531no—
286C2×D8—1061no—
287C2×D8—1061no—
288C2×D8—1061no—
289C2×D8—1061no—
290C2×D8—1061no—
291C2×D8—1062no—
292C2×D8D21211yescharacteristic, complemented
293#131—24no—
294#131—42no—
295C2×M16—21no—
296C2×M16—42no—
297C2×M16C22⋊D5311yescharacteristic
298C22×C8C22×D5311yescharacteristic
299#127—1062no—
300#124—1062no—
301#121—2122no—
302#120—2122no—
303#117C22⋊D5311yescharacteristic
304Q32—1062no—
305SD32—1064no—
306SD32—1062no—
307SD32—1062no—
308D16—1064no—
309D16—1062no—
310M32—532no—
311M32—532no—
312M32—1061no—
313M32—1061no—
314C2×C16—1061no—
315C2×C16—1061no—
316#108—21no—
317#108C22×D5312yes—
318#104—24no—
319#97C22⋊D5312yes—
320C4×C8C22×D5312yes—
321C24—2121no—
322#41—22no—
323#41—22no—
324#41—41no—
325#41—41no—
326#41—2121no—
327#41—2121no—
328#41—2121no—
329#41—2121no—
330C2×Q8—21no—
331C2×Q8—1061no—
332C2×D4—21no—
333C2×D4—41no—
334C2×D4—1062no—
335C2×D4—1061no—
336C2×D4—2124no—
337C2×D4—2121no—
338C2×D4—2121no—
339C2×D4—2121no—
340C2×D4—2121no—
341C2×D4—2121no—
342C2×D4—2121no—
343C22×C4—41no—
344C22×C4—2121no—
345C22×C4C2×C22⋊D5311yescharacteristic
346Q16—21no—
347Q16—1061no—
348Q16—1061no—
349Q16C2×D21212yes—
350SD16—22no—
351SD16—22no—
352SD16—2121no—
353SD16—2121no—
354SD16—2124no—
355D8—21no—
356D8—1062no—
357D8—1061no—
358D8—1062no—
359D8—1061no—
360D8—2121no—
361D8—2121no—
362D8—2124no—
363D8C2×D21212yes—
364M16—22no—
365M16—41no—
366M16—44no—
367M16#1171992212yes—
368M16#1171992212yes—
369C2×C8—41no—
370C2×C8—44no—
371C2×C8C2×D21211yescharacteristic
372C2×C8C2×D21211yescharacteristic
373C2×C8#1171994511yescharacteristic
374C2×C8#1171994511yescharacteristic
375C2×C8C2×C22⋊D5311yescharacteristic
376C2×C8C2×C22⋊D5312yes—
377C2×C8C23×D5311yescharacteristic
378C4⋊C4—21no—
379C22⋊C4—21no—
380C22⋊C4—2124no—
381C42C2×C22⋊D5312yes—
382C16—1062no—
383C16—1062no—
384C23—41no—
385C23—8482no—
386C23—4241no—
387C23—4241no—
388C23—2122no—
389C23—2121no—
390C23#1313324211yescharacteristic
391Q8—22no—
392Q8—41no—
393Q8—2121no—
394Q8—2121no—
395D4—22no—
396D4—42no—
397D4—42no—
398D4—41no—
399D4—8484no—
400D4—4241no—
401D4—4241no—
402D4—4241no—
403D4—4241no—
404D4—2122no—
405D4—2121no—
406D4—2121no—
407D4—2122no—
408C2×C4—22no—
409C2×C4—41no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—4241no—
413C2×C4—4241no—
414C2×C4—2121no—
415C2×C4#1313319111yescharacteristic
416C2×C4#1313319111yescharacteristic
417C2×C4C2×#1171992211yescharacteristic
418C2×C4C2×#1171992211yescharacteristic
419C2×C4#1313324211yescharacteristic
420C2×C4C22×C22⋊D5311yescharacteristic
421C8—42no—
422C8#1313319111yescharacteristic
423C8#1313319111yescharacteristic
424C8C2×#1171992214yes—
425C8C22×D21211yescharacteristic
426C8C2×#1171994511yescharacteristic
427C22—41no—
428C22—42no—
429C22—161no—
430C22—8482no—
431C22—8482no—
432C22—4242no—
433C22—4244no—
434C22—4241no—
435C22—4242no—
436C22#1681081811yescharacteristic
437C22#1681081811yescharacteristic
438C22C2×#1313324211yescharacteristic
439C4—42no—
440C4—42no—
441C4—4241no—
442C4#1681081811yescharacteristic
443C4#1681081811yescharacteristic
444C4C2×#1313319111yescharacteristic
445C4C22×#1171992211yescharacteristic
446C2—21no—
447C2—21no—
448C2—21no—
449C2—82no—
450C2—8482no—
451C2—8481no—
452C2C2×#1681081811yescharacteristic
subgroupG/H
centerC2C2×#16810818
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC53×#2957C2
socleC2×C53C2×#459
hypercenter#2957D53
hypocenterC53#33595
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C22×C4 ◁ #405 ◁ #2957

Complements

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

NH ≅ G/Ntype
1. #20287849C2semidirect
2. C2×#16811058C2semidirect
3. #20285201C2semidirect
4. #20276791C2semidirect
5. #20276784C2semidirect
6. #20276481C2semidirect
7. #20276462C2semidirect
8. C2×#16810435C2semidirect
9. C53×#2957C2semidirect
15. #16811063C22semidirect
19. #16811058C22semidirect
26. #16811007C22semidirect
27. #16811000C22semidirect
28. #16810444C22semidirect
29. #16810438C22semidirect
30. #16810435C22semidirect
31. #16810435C22semidirect
35. C53×#443C22semidirect
77. #135×C53D4semidirect
79. C2×Q16×C53D4semidirect
81. C2×D8×C53D4semidirect
223. #2957D53semidirect
257. #443C2×D53semidirect
265. C53#33595semidirect
277. #135D212semidirect
280. C2×Q16D212semidirect
292. C2×D8D212semidirect

Maximal subgroups

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

subgroupindex
#202878492
C2×#168110582
#202852012
#202767912
#202767842
#202764812
#202764622
C2×#168104352
C53×#29572
#3359553