Finite Group Databasebeta

← #27299823

Subgroups — #27299823

Order 13568.

882 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810439C211yesmaximal, characteristic, complemented
2C2×#16810430C211yesmaximal, characteristic, complemented
3#20276427C211yesmaximal, characteristic, complemented
4C2×#16810427C211yesmaximal, characteristic, complemented
5C2×#16810407C211yesmaximal, characteristic, complemented
6#20274779C211yesmaximal, characteristic, complemented
7C2×C53×#578C211yesmaximal, characteristic
8C2×#13133157C2211yescharacteristic
9C22×#11719908C2211yescharacteristic
10C2×#13133153C2211yescharacteristic
11C2×#13133151C2211yescharacteristic
12#16810624—21no—
13#16810439—22no—
14#16810430—22no—
15#16810427—22no—
16#16810427—21no—
17#16810424—21no—
18#16810407—22no—
19#16810391—21no—
20C2×2−1+4×C53—21no—
21C53×#578—22no—
22C53×#578—22no—
23C53×#578—22no—
24C53×#578C414yescomplemented
25C2×#137×C53—22no—
26C2×#137×C53—21no—
27C2×#135×C53—21no—
28C2×#135×C53C2211yescharacteristic
29C22×Q16×C53—21no—
30C22×Q16×C53C2211yescharacteristic
31C2×#131×C53C2211yescharacteristic
32C2×#11719917—21no—
33C2×C4⋊Dic53—21no—
34#13133157—21no—
35#13133157—21no—
36C2×#11719908—21no—
37C2×#11719908—21no—
38C2×#11719908—21no—
39C2×#11719908—21no—
40#13133153—21no—
41#13133153—21no—
42#13133151—21no—
43#13133151—21no—
44#13133141—41no—
45#13133138—41no—
462−1+4×C53—24no—
472−1+4×C53—24no—
482−1+4×C53—42no—
492−1+4×C53—42no—
50C2×#41×C53—22no—
51C2×#41×C53—21no—
52C2×#41×C53—42no—
53C2×#41×C53—41no—
54C2×#41×C53D411yescharacteristic
55C22×Q8×C53—22no—
56C22×Q8×C53—21no—
57C22×Q8×C53—41no—
58#137×C53—28no—
59#137×C53—24no—
60#137×C53—28no—
61#137×C53—24no—
62#135×C53—24no—
63#135×C53—24no—
64#135×C53—21no—
65#135×C53—21no—
66#135×C53C2×C412yes—
67#135×C53D412yes—
68C2×Q16×C53—22no—
69C2×Q16×C53—24no—
70C2×Q16×C53—24no—
71C2×Q16×C53—21no—
72C2×Q16×C53—21no—
73C2×Q16×C53—21no—
74C2×Q16×C53C2×C412yes—
75C2×Q16×C53D411yescharacteristic
76C2×Q16×C53D412yes—
77C2×Q16×C53D412yes—
78C2×SD16×C53—22no—
79C2×SD16×C53—21no—
80C2×D8×C53D411yescharacteristic
81#131×C53—21no—
82#131×C53—21no—
83#131×C53C2×C412yes—
84#131×C53D412yes—
85C2×M16×C53—21no—
86C2×M16×C53D411yescharacteristic
87C22×C8×C53—21no—
88C22×C8×C53C2311yescharacteristic
89#41×C53—21no—
90#41×C53—21no—
91#41×C53—28no—
92#41×C53—22no—
93#41×C53—22no—
94#41×C53—44no—
95#41×C53—48no—
96#41×C53—48no—
97#41×C53—44no—
98#41×C53—44no—
99#41×C53—41no—
100#41×C53—41no—
101#41×C53C22⋊C412yes—
102#41×C53SD1612yes—
103C2×Q8×C53—28no—
104C2×Q8×C53—22no—
105C2×Q8×C53—22no—
106C2×Q8×C53—21no—
107C2×Q8×C53—22no—
108C2×Q8×C53—42no—
109C2×Q8×C53—44no—
110C2×Q8×C53—44no—
111C2×Q8×C53—44no—
112C2×Q8×C53—41no—
113C2×Q8×C53—41no—
114C2×Q8×C53—41no—
115C2×Q8×C53D811yescharacteristic
116C2×D4×C53—21no—
117C2×D4×C53—21no—
118C2×D4×C53—42no—
119C2×D4×C53—41no—
120C2×D4×C53D811yescharacteristic
121C22×C4×C53—21no—
122C22×C4×C53—42no—
123C22×C4×C53—42no—
124C22×C4×C53—41no—
125C22×C4×C53—41no—
126C22×C4×C53C2×D411yescharacteristic
127C22×Dic53—41no—
128Q16×C53—28no—
129Q16×C53—22no—
130Q16×C53—21no—
131Q16×C53—22no—
132Q16×C53—22no—
133Q16×C53C22⋊C412yes—
134Q16×C53C22⋊C414yes—
135SD16×C53—28no—
136SD16×C53—22no—
137SD16×C53—22no—
138D8×C53—21no—
139D8×C53C22⋊C412yes—
140M16×C53—21no—
141M16×C53—22no—
142M16×C53—22no—
143M16×C53C22⋊C412yes—
144C2×C8×C53—21no—
145C2×C8×C53—22no—
146C2×C8×C53—22no—
147C2×C8×C53C22×C411yescharacteristic
148C2×C8×C53C2×D411yescharacteristic
149C2×C8×C53C2×D411yescharacteristic
150C2×C8×C53C2×D411yescharacteristic
151C2×C8×C53C2×D411yescharacteristic
152C2×C8×C53C2×D411yescharacteristic
153C4⋊Dic53—41no—
154C4⋊Dic53—41no—
155#11719917—41no—
156#11719917—41no—
157C2×#88322—41no—
158#11719908—41no—
159#11719908—41no—
160C23×C53—21no—
161C23×C53—41no—
162C23×C53AGL(1,ℤ/8)11yescharacteristic
163Q8×C53—21no—
164Q8×C53—22no—
165Q8×C53—24no—
166Q8×C53—48no—
167Q8×C53—42no—
168Q8×C53—42no—
169Q8×C53—41no—
170Q8×C53—42no—
171Q8×C53#10212yes—
172D4×C53—22no—
173D4×C53—22no—
174D4×C53—22no—
175D4×C53—21no—
176D4×C53—48no—
177D4×C53—42no—
178D4×C53—42no—
179D4×C53—41no—
180D4×C53#10212yes—
181C2×C4×C53—21no—
182C2×C4×C53—22no—
183C2×C4×C53—22no—
184C2×C4×C53—48no—
185C2×C4×C53—48no—
186C2×C4×C53—42no—
187C2×C4×C53—42no—
188C2×C4×C53—41no—
189C2×C4×C53—42no—
190C2×C4×C53—42no—
191C2×C4×C53—42no—
192C2×C4×C53C2×C22⋊C411yescharacteristic
193C2×C4×C53#12011yescharacteristic
194C2×C4×C53#12011yescharacteristic
195C2×C4×C53C2×D811yescharacteristic
196C2×C4×C53C2×SD1611yescharacteristic
197C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
198C2×Dic53—81no—
199C2×Dic53—161no—
200C8×C53—22no—
201C8×C53C2×C22⋊C411yescharacteristic
202C8×C53C2×C22⋊C411yescharacteristic
203C8×C53#12011yescharacteristic
204C8×C53#12011yescharacteristic
205#88322—81no—
206#12749—531nomaximal, Sylow
207C22×C53—21no—
208C22×C53—21no—
209C22×C53—22no—
210C22×C53—41no—
211C22×C53—41no—
212C22×C53—42no—
213C22×C53—81no—
214C22×C53—81no—
215C22×C53#41811yescharacteristic
216C22×C53#44711yescharacteristic
217C22×C53#45011yescharacteristic
218C4×C53—22no—
219C4×C53—44no—
220C4×C53—42no—
221C4×C53#37911yescharacteristic
222C4×C53C2×#10211yescharacteristic
223C4×C53#44711yescharacteristic
224C4×C53#45011yescharacteristic
225Dic53—161no—
226C2×#578D5311yescharacteristic
227C2×#362—531no—
228C2×#359—531no—
229#2139—531no—
230C2×#358—531no—
231C2×#350—531no—
232#1816—531no—
233C2×C53—21no—
234C2×C53—21no—
235C2×C53—42no—
236C2×C53—82no—
237C2×C53#178611yescharacteristic
238C2×C53#219311yescharacteristic
239C2×C53#219511yescharacteristic
240C2×2−1+4—21no—
241#578—22no—
242#578—22no—
243#578—22no—
244#578Dic5314yescomplemented
245C2×#137—22no—
246C2×#137—21no—
247C2×#135—21no—
248C2×#135C2×D5311yescharacteristic
249C22×Q16—21no—
250C22×Q16C2×D5311yescharacteristic
251C2×#131C2×D5311yescharacteristic
252C2×M32—531no—
253C22×C16—531no—
254C2×#108—531no—
255C2×#107—531no—
256#417—1061no—
257#362—1062no—
258#359—1062no—
259#358—1062no—
260#358—1061no—
261#357—1061no—
262#350—1062no—
263#343—1061no—
264C53#1274911yescharacteristic, complemented, Sylow
2652−1+4—24no—
2662−1+4—24no—
2672−1+4—42no—
2682−1+4—42no—
269C2×#41—22no—
270C2×#41—21no—
271C2×#41—42no—
272C2×#41—41no—
273C2×#41C22⋊D5311yescharacteristic
274C22×Q8—22no—
275C22×Q8—21no—
276C22×Q8—41no—
277#137—24no—
278#137—28no—
279#137—28no—
280#137—24no—
281#135—24no—
282#135—24no—
283#135—21no—
284#135—21no—
285#135C2×Dic5312yes—
286#135C22⋊D5312yes—
287C2×Q16—22no—
288C2×Q16—24no—
289C2×Q16—24no—
290C2×Q16—21no—
291C2×Q16—21no—
292C2×Q16—21no—
293C2×Q16C2×Dic5312yes—
294C2×Q16C22⋊D5311yescharacteristic
295C2×Q16C22⋊D5312yes—
296C2×Q16C22⋊D5312yes—
297C2×SD16—22no—
298C2×SD16—21no—
299C2×D8C22⋊D5311yescharacteristic
300#131—21no—
301#131—21no—
302#131C2×Dic5312yes—
303#131C22⋊D5312yes—
304C2×M16—21no—
305C2×M16—1061no—
306C2×M16C22⋊D5311yescharacteristic
307C22×C8—21no—
308C22×C8C22×D5311yescharacteristic
309C2×C4⋊C4—1061no—
310M32—1061no—
311M32—1061no—
312C2×C16—1061no—
313C2×C16—1061no—
314C2×C16—1061no—
315C2×C16—1061no—
316#108—1061no—
317#108—1061no—
318#107—1061no—
319#107—1061no—
320#103—2121no—
321#102—2121no—
322#41—21no—
323#41—21no—
324#41—28no—
325#41—22no—
326#41—22no—
327#41—44no—
328#41—48no—
329#41—48no—
330#41—44no—
331#41—44no—
332#41—41no—
333#41—41no—
334#41#1171992312yes—
335#41C22⋊Dic5312yes—
336C2×Q8—28no—
337C2×Q8—22no—
338C2×Q8—22no—
339C2×Q8—21no—
340C2×Q8—22no—
341C2×Q8—42no—
342C2×Q8—44no—
343C2×Q8—44no—
344C2×Q8—44no—
345C2×Q8—41no—
346C2×Q8—41no—
347C2×Q8—41no—
348C2×Q8#1171992211yescharacteristic
349C2×D4—21no—
350C2×D4—21no—
351C2×D4—42no—
352C2×D4—41no—
353C2×D4#1171992211yescharacteristic
354C22×C4—21no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—2121no—
360C22×C4C2×C22⋊D5311yescharacteristic
361Q16—28no—
362Q16—22no—
363Q16—21no—
364Q16—22no—
365Q16—22no—
366Q16C22⋊Dic5312yes—
367Q16C22⋊Dic5314yes—
368SD16—28no—
369SD16—22no—
370SD16—22no—
371D8—21no—
372D8C22⋊Dic5312yes—
373M16—21no—
374M16—22no—
375M16—22no—
376M16—2121no—
377M16—2121no—
378M16C22⋊Dic5312yes—
379C2×C8—21no—
380C2×C8—22no—
381C2×C8—22no—
382C2×C8—2121no—
383C2×C8C22×Dic5311yescharacteristic
384C2×C8C2×C22⋊D5311yescharacteristic
385C2×C8C2×C22⋊D5311yescharacteristic
386C2×C8C2×C22⋊D5311yescharacteristic
387C2×C8C2×C22⋊D5311yescharacteristic
388C2×C8C2×C22⋊D5311yescharacteristic
389C4⋊C4—2121no—
390C4⋊C4—2121no—
391C16—2121no—
392C16—2121no—
393C23—21no—
394C23—41no—
395C23#1313324311yescharacteristic
396Q8—21no—
397Q8—24no—
398Q8—22no—
399Q8—48no—
400Q8—42no—
401Q8—41no—
402Q8—42no—
403Q8—42no—
404Q8#1313313812yes—
405D4—22no—
406D4—22no—
407D4—21no—
408D4—22no—
409D4—48no—
410D4—42no—
411D4—42no—
412D4—41no—
413D4#1313313812yes—
414C2×C4—21no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—48no—
418C2×C4—48no—
419C2×C4—42no—
420C2×C4—41no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—8481no—
426C2×C4—4241no—
427C2×C4C2×C22⋊Dic5311yescharacteristic
428C2×C4#1313318611yescharacteristic
429C2×C4#1313318611yescharacteristic
430C2×C4C2×#1171992211yescharacteristic
431C2×C4C2×#1171992311yescharacteristic
432C2×C4#1313324311yescharacteristic
433C8—22no—
434C8—4241no—
435C8C2×C22⋊Dic5311yescharacteristic
436C8C2×C22⋊Dic5311yescharacteristic
437C8#1313318611yescharacteristic
438C8#1313318611yescharacteristic
439C22—21no—
440C22—21no—
441C22—22no—
442C22—41no—
443C22—41no—
444C22—42no—
445C22—81no—
446C22—81no—
447C22#1681062811yescharacteristic
448C22#1681075711yescharacteristic
449C22#1681077411yescharacteristic
450C4—22no—
451C4—44no—
452C4—42no—
453C4—8481no—
454C4#1681047911yescharacteristic
455C4C2×#1313313811yescharacteristic
456C4#1681075711yescharacteristic
457C4#1681077411yescharacteristic
458C2—21no—
459C2—21no—
460C2—42no—
461C2—82no—
462C2#2027464611yescharacteristic
463C2#2027667911yescharacteristic
464C2#2027669311yescharacteristic
subgroupG/H
centerC22#16810757
commutatorC2×C8×C53C22×C4
FrattiniC22×C8C22×D53
Fitting, prime radicalC2×C53×#578C2
socleC22×C53#447
cosocleC22×C8×C53C23
hypercenterC2×#578D53
hypocenterC53#12749
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#131 ◁ C2×#578

Complements

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

NH ≅ G/Ntype
1. C2×#16810439C2semidirect
2. C2×#16810430C2semidirect
3. #20276427C2semidirect
4. C2×#16810427C2semidirect
5. C2×#16810407C2semidirect
6. #20274779C2semidirect
24. C53×#578C4semidirect
244. #578Dic53semidirect
264. C53#12749semidirect

Maximal subgroups

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

subgroupindex
C2×#168104392
C2×#168104302
#202764272
C2×#168104272
C2×#168104072
#202747792
C2×C53×#5782
#1274953