Finite Group Databasebeta

← #28480265

Subgroups — #28480265

Order 15104.

794 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21424891C211yesmaximal, characteristic, complemented
2#21424088C211yesmaximal, characteristic, complemented
3#21422786C211yesmaximal, characteristic, complemented
4#21422773C211yesmaximal, characteristic, complemented
5#21422551C211yesmaximal, characteristic, complemented
6#21422547C211yesmaximal, characteristic, complemented
7C59×#3506C211yesmaximal, characteristic
8#16837426C2211yescharacteristic
9#16837426C2211yescharacteristic
10C2×C4⋊(C59⋊C8)C2211yescharacteristic
11C2×#14273644—21no—
12C2×C22⋊(C59⋊C8)—22no—
13C2×C4×C59⋊C8C2211yescharacteristic
14#16837171—21no—
15#16837151—21no—
16#16837148—21no—
17#16837148—21no—
18#16837146—21no—
19Q8⋊(C59⋊C8)—22no—
20D4⋊(C59⋊C8)—22no—
21C59×#559—22no—
22C59×#556—21no—
23C59×#555—21no—
24C59×#553—22no—
25C59×#550—21no—
26C59×#547—21no—
27C59×#538C414yes—
28C59×#535—21no—
29C59×#533C2211yescharacteristic
30C59×#532C2211yescharacteristic
31C59×#525—22no—
32C59×#520—21no—
33C2×C4×D4×C59C2211yescharacteristic
34C2×#11745664—21no—
35C2×#11745664—21no—
36C22×C59⋊C8—21no—
37C22×C59⋊C8—21no—
38C4⋊(C59⋊C8)—21no—
39C4⋊(C59⋊C8)—21no—
40C4⋊(C59⋊C8)—21no—
41C4⋊(C59⋊C8)—21no—
42C4⋊(C59⋊C8)—21no—
43C4⋊(C59⋊C8)—21no—
44#14273644—42no—
45C22⋊(C59⋊C8)—44no—
46C4×C59⋊C8—21no—
47C4×C59⋊C8—21no—
48C2×#41×C59—22no—
49C2×#41×C59—21no—
50C22×D4×C59D411yescharacteristic
51C23×C4×C59D412yes—
52C4⋊Q8×C59D411yescharacteristic
53C4⋊Q8×C59D411yescharacteristic
54C4⋊Q8×C59D411yescharacteristic
55#127×C59—22no—
56#127×C59D411yescharacteristic
57#126×C59—24no—
58#125×C59—21no—
59#124×C59—22no—
60#124×C59—24no—
61#124×C59—21no—
62#123×C59—24no—
63#123×C59—24no—
64C22⋊Q8×C59—22no—
65C22⋊Q8×C59—21no—
66C22⋊Q8×C59C2×C412yes—
67#121×C59—22no—
68#121×C59—22no—
69#121×C59—22no—
70#121×C59—24no—
71#121×C59—21no—
72#121×C59C2×C412yes—
73#120×C59—24no—
74C4×Q8×C59—22no—
75C4×Q8×C59—21no—
76C4×Q8×C59D412yes—
77C4×D4×C59—22no—
78C4×D4×C59—24no—
79C4×D4×C59—24no—
80C4×D4×C59—21no—
81C4×D4×C59—21no—
82C4×D4×C59—21no—
83C4×D4×C59—21no—
84C4×D4×C59—21no—
85C4×D4×C59D412yes—
86#117×C59—22no—
87#117×C59—22no—
88#117×C59—21no—
89C2×C4⋊C4×C59—22no—
90C2×C4⋊C4×C59D411yescharacteristic
91C2×C22⋊C4×C59C2×C412yes—
92C2×C42×C59C2311yescharacteristic
93C24×C59C22⋊C412yes—
94#41×C59—41no—
95#41×C59—41no—
96#41×C59—41no—
97#41×C59—41no—
98#41×C59—82no—
99#41×C59—82no—
100C2×Q8×C59—21no—
101C2×Q8×C59—21no—
102C2×Q8×C59C22⋊C412yes—
103C2×D4×C59—22no—
104C2×D4×C59—24no—
105C2×D4×C59—24no—
106C2×D4×C59—21no—
107C2×D4×C59—21no—
108C2×D4×C59—21no—
109C2×D4×C59—21no—
110C2×D4×C59—21no—
111C2×D4×C59—21no—
112C2×D4×C59—21no—
113C2×D4×C59—21no—
114C2×D4×C59C22⋊C412yes—
115C22×C4×C59—24no—
116C22×C4×C59—22no—
117C22×C4×C59—22no—
118C22×C4×C59—22no—
119C22×C4×C59—42no—
120C22×C4×C59—42no—
121C22×C4×C59C22⋊C412yes—
122C22×C4×C59C22×C411yescharacteristic
123C22×C4×C59C2×D411yescharacteristic
124C22×C4×C59C2×D411yescharacteristic
125C4⋊C4×C59—24no—
126C4⋊C4×C59—22no—
127C4⋊C4×C59—21no—
128C4⋊C4×C59—21no—
129C4⋊C4×C59—21no—
130C4⋊C4×C59—21no—
131C4⋊C4×C59—21no—
132C4⋊C4×C59—21no—
133C4⋊C4×C59—22no—
134C4⋊C4×C59C22⋊C412yes—
135C4⋊C4×C59C22⋊C412yes—
136C22⋊C4×C59—24no—
137C22⋊C4×C59—24no—
138C22⋊C4×C59—24no—
139C22⋊C4×C59—22no—
140C22⋊C4×C59—22no—
141C22⋊C4×C59—22no—
142C22⋊C4×C59—22no—
143C22⋊C4×C59—22no—
144C42×C59—24no—
145C42×C59—22no—
146C42×C59C2×D411yescharacteristic
147C42×C59C2×D411yescharacteristic
148C42×C59C2×D411yescharacteristic
149C42×C59C2×D411yescharacteristic
150#11745664—41no—
151#11745664—41no—
152#11745664—41no—
153#11745664—41no—
154C2×C59⋊C8—22no—
155C2×C59⋊C8—41no—
156C2×C59⋊C8—41no—
157C2×C59⋊C8—42no—
158C2×C59⋊C8—41no—
159C2×C59⋊C8—41no—
160C2×C59⋊C8—41no—
161C23×C59—22no—
162C23×C59—22no—
163C23×C59—22no—
164C23×C59—22no—
165C23×C59—42no—
166C23×C59—42no—
167C23×C59C2×C22⋊C411yescharacteristic
168Q8×C59—42no—
169Q8×C59—41no—
170Q8×C59—41no—
171Q8×C59—41no—
172Q8×C59—41no—
173D4×C59—42no—
174D4×C59—42no—
175D4×C59—42no—
176D4×C59—41no—
177D4×C59—41no—
178D4×C59—41no—
179D4×C59—41no—
180D4×C59—41no—
181D4×C59—41no—
182D4×C59—41no—
183D4×C59—41no—
184D4×C59—84no—
185D4×C59—84no—
186C2×C4×C59—21no—
187C2×C4×C59—21no—
188C2×C4×C59—22no—
189C2×C4×C59—22no—
190C2×C4×C59—22no—
191C2×C4×C59—22no—
192C2×C4×C59—41no—
193C2×C4×C59—42no—
194C2×C4×C59—42no—
195C2×C4×C59—42no—
196C2×C4×C59—42no—
197C2×C4×C59—42no—
198C2×C4×C59—82no—
199C2×C4×C59—84no—
200C2×C4×C59C2×C22⋊C411yescharacteristic
201C2×C4×C59C2×C22⋊C411yescharacteristic
202C2×C4×C59#12011yescharacteristic
203C2×C4×C59#12011yescharacteristic
204C2×C4×C59#12011yescharacteristic
205C2×C4×C59#12011yescharacteristic
206C2×C4×C59#13511yescharacteristic
207C2×C4×C59#13511yescharacteristic
208C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
209C2×C4×C59#13711yescharacteristic
210C59⋊C8—42no—
211C59⋊C8—81no—
212C59⋊C8—81no—
213C59⋊C8—81no—
214#8358—591nomaximal, Sylow
215C22×C59—21no—
216C22×C59—21no—
217C22×C59—44no—
218C22×C59—42no—
219C22×C59—42no—
220C22×C59—42no—
221C22×C59—42no—
222C22×C59—42no—
223C22×C59—42no—
224C22×C59—82no—
225C22×C59#37911yescharacteristic
226C22×C59#41611yescharacteristic
227C22×C59#41711yescharacteristic
228C4×C59—21no—
229C4×C59—21no—
230C4×C59—21no—
231C4×C59—42no—
232C4×C59—42no—
233C4×C59—42no—
234C4×C59—42no—
235C4×C59#44911yescharacteristic
236C4×C59#44911yescharacteristic
237C4×C59#45211yescharacteristic
238C4×C59#45211yescharacteristic
239C4×C59#45311yescharacteristic
240C4×C59#45511yescharacteristic
241#3506D5911yescharacteristic
242#1954—591no—
243#1800—591no—
244#1550—591no—
245#1547—591no—
246#1493—591no—
247#1492—591no—
248C2×C59—21no—
249C2×C59—21no—
250C2×C59—82no—
251C2×C59—82no—
252C2×C59#178811yescharacteristic
253C2×C59#179111yescharacteristic
254C2×C59#179211yescharacteristic
255#559—22no—
256#556—21no—
257#555—21no—
258#553—22no—
259#550—21no—
260#547—21no—
261#538Dic5914yes—
262#535—21no—
263#533C2×D5911yescharacteristic
264#532C2×D5911yescharacteristic
265#525—22no—
266#520—21no—
267C2×C4×D4C2×D5911yescharacteristic
268#423—591no—
269#423—591no—
270C2×C4⋊C8—591no—
271C2×#100—1181no—
272C2×C22⋊C8—1182no—
273C2×C4×C8—591no—
274#343—1181no—
275#333—1181no—
276#332—1181no—
277#332—1181no—
278#331—1181no—
279Q8⋊C8—1182no—
280D4⋊C8—1182no—
281C59#835811yescharacteristic, complemented, Sylow
282C2×#41—22no—
283C2×#41—21no—
284C22×D4C22⋊D5911yescharacteristic
285C23×C4C22⋊D5912yes—
286C2×M16—1181no—
287C2×M16—1181no—
288C22×C8—1181no—
289C22×C8—1181no—
290C4⋊Q8C22⋊D5911yescharacteristic
291C4⋊Q8C22⋊D5911yescharacteristic
292C4⋊Q8C22⋊D5911yescharacteristic
293#127—22no—
294#127C22⋊D5911yescharacteristic
295#126—24no—
296#125—21no—
297#124—22no—
298#124—24no—
299#124—21no—
300#123—24no—
301#123—24no—
302C22⋊Q8—22no—
303C22⋊Q8—21no—
304C22⋊Q8C2×Dic5912yes—
305#121—22no—
306#121—22no—
307#121—22no—
308#121—24no—
309#121—21no—
310#121C2×Dic5912yes—
311#120—24no—
312C4×Q8—22no—
313C4×Q8—21no—
314C4×Q8C22⋊D5912yes—
315C4×D4—22no—
316C4×D4—24no—
317C4×D4—24no—
318C4×D4—21no—
319C4×D4—21no—
320C4×D4—21no—
321C4×D4—21no—
322C4×D4—21no—
323C4×D4C22⋊D5912yes—
324#117—22no—
325#117—22no—
326#117—21no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4C22⋊D5911yescharacteristic
329C2×C22⋊C4C2×Dic5912yes—
330C2×C42C22×D5911yescharacteristic
331C4⋊C8—1181no—
332C4⋊C8—1181no—
333C4⋊C8—1181no—
334C4⋊C8—1181no—
335C4⋊C8—1181no—
336C4⋊C8—1181no—
337#100—2362no—
338C22⋊C8—2364no—
339C4×C8—1181no—
340C4×C8—1181no—
341C24C22⋊Dic5912yes—
342#41—41no—
343#41—41no—
344#41—41no—
345#41—41no—
346#41—82no—
347#41—82no—
348C2×Q8—21no—
349C2×Q8—21no—
350C2×Q8C22⋊Dic5912yes—
351C2×D4—22no—
352C2×D4—24no—
353C2×D4—24no—
354C2×D4—21no—
355C2×D4—21no—
356C2×D4—21no—
357C2×D4—21no—
358C2×D4—21no—
359C2×D4—21no—
360C2×D4—21no—
361C2×D4—21no—
362C2×D4C22⋊Dic5912yes—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4C22⋊Dic5912yes—
370C22×C4C22×Dic5911yescharacteristic
371C22×C4C2×C22⋊D5911yescharacteristic
372C22×C4C2×C22⋊D5911yescharacteristic
373M16—2361no—
374M16—2361no—
375M16—2361no—
376M16—2361no—
377C2×C8—1182no—
378C2×C8—2362no—
379C2×C8—2361no—
380C2×C8—2361no—
381C2×C8—2361no—
382C2×C8—2361no—
383C2×C8—2361no—
384C4⋊C4—24no—
385C4⋊C4—22no—
386C4⋊C4—21no—
387C4⋊C4—21no—
388C4⋊C4—21no—
389C4⋊C4—21no—
390C4⋊C4—21no—
391C4⋊C4—21no—
392C4⋊C4—22no—
393C4⋊C4C22⋊Dic5912yes—
394C4⋊C4C22⋊Dic5912yes—
395C22⋊C4—24no—
396C22⋊C4—24no—
397C22⋊C4—24no—
398C22⋊C4—22no—
399C22⋊C4—22no—
400C22⋊C4—22no—
401C22⋊C4—22no—
402C22⋊C4—22no—
403C42—24no—
404C42—22no—
405C42C2×C22⋊D5911yescharacteristic
406C42C2×C22⋊D5911yescharacteristic
407C42C2×C22⋊D5911yescharacteristic
408C42C2×C22⋊D5911yescharacteristic
409C23—22no—
410C23—22no—
411C23—22no—
412C23—22no—
413C23—42no—
414C23—42no—
415C23C2×C22⋊Dic5911yescharacteristic
416Q8—42no—
417Q8—41no—
418Q8—41no—
419Q8—41no—
420Q8—41no—
421D4—42no—
422D4—42no—
423D4—42no—
424D4—41no—
425D4—41no—
426D4—41no—
427D4—41no—
428D4—41no—
429D4—41no—
430D4—41no—
431D4—41no—
432D4—84no—
433D4—84no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—41no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—82no—
447C2×C4—84no—
448C2×C4C2×C22⋊Dic5911yescharacteristic
449C2×C4C2×C22⋊Dic5911yescharacteristic
450C2×C4#1427369611yescharacteristic
451C2×C4#1427369611yescharacteristic
452C2×C4#1427369611yescharacteristic
453C2×C4#1427369611yescharacteristic
454C2×C4#1427374711yescharacteristic
455C2×C4#1427374711yescharacteristic
456C2×C4#1427375211yescharacteristic
457C2×C4#1427375711yescharacteristic
458C8—4721no—
459C8—4721no—
460C8—4721no—
461C8—2362no—
462C22—21no—
463C22—21no—
464C22—44no—
465C22—42no—
466C22—42no—
467C22—42no—
468C22—42no—
469C22—42no—
470C22—42no—
471C22—82no—
472C22#1683725811yescharacteristic
473C22#1683739811yescharacteristic
474C22#1683740311yescharacteristic
475C4—21no—
476C4—21no—
477C4—21no—
478C4—42no—
479C4—42no—
480C4—42no—
481C4—42no—
482C4#1683754711yescharacteristic
483C4#1683754711yescharacteristic
484C4#1683756411yescharacteristic
485C4#1683756411yescharacteristic
486C4#1683757011yescharacteristic
487C4#1683758011yescharacteristic
488C2—21no—
489C2—21no—
490C2—82no—
491C2—82no—
492C2#2142403611yescharacteristic
493C2#2142404711yescharacteristic
494C2#2142405011yescharacteristic
subgroupG/H
centerC22#16837258
commutatorC22×C4×C59C22×C4
FrattiniC2×C42C22×D59
Fitting, prime radicalC59×#3506C2
socleC22×C59#379
cosocleC2×C42×C59C23
hypercenter#3506D59
hypocenterC59#8358
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3506

Complements

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

NH ≅ G/Ntype
1. #21424891C2semidirect
2. #21424088C2semidirect
3. #21422786C2semidirect
4. #21422773C2semidirect
5. #21422551C2semidirect
6. #21422547C2semidirect
281. C59#8358semidirect

Maximal subgroups

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

subgroupindex
#214248912
#214240882
#214227862
#214227732
#214225512
#214225472
C59×#35062
#835859