Finite Group Databasebeta

← #28504441

Subgroups — #28504441

Order 15104.

888 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21425845C211yesmaximal, characteristic, complemented
2C2×#16837212C211yesmaximal, characteristic, complemented
3C2×#16837212C211yesmaximal, characteristic, complemented
4#21425824C211yesmaximal, characteristic, complemented
5#21425724C211yesmaximal, characteristic, complemented
6#21424155C211yesmaximal, characteristic, complemented
7C59×#3584C211yesmaximal, characteristic
8C2×#14273667C2211yescharacteristic, complemented
9C2×#14273667C2211yescharacteristic, complemented
10C2×#14273663C2211yescharacteristic, complemented
11C2×#14273659C2211yescharacteristic, complemented
12#16837403—21no—
13#16837218—21no—
14#16837215—21no—
15#16837212—22no—
16#16837212—22no—
17#16837209—22no—
18#16837186—22no—
19#16837170—21no—
20C59×#585—21no—
21C59×#578—21no—
22C59×#578C412yescomplemented
23C59×#577—22no—
24C59×#577—22no—
25C59×#576—21no—
26C59×#576C412yescomplemented
27C59×#575—22no—
28C59×#575—21no—
29C2×#135×C59—22no—
30C2×#135×C59C2211yescharacteristic
31C2×#135×C59C2211yescharacteristic
32C2×#131×C59C2211yescharacteristic
33C2×#11745664—21no—
34C2×C4⋊Dic59—21no—
35#14273667—21no—
36#14273667—21no—
37#14273667—21no—
38#14273667—21no—
39C2×C59⋊C16—21no—
40C2×C59⋊C16—21no—
41#14273663—21no—
42#14273663—21no—
43#14273659—21no—
44#14273659—21no—
45#14273651—41no—
46#14273648—41no—
472−1+4×C59—22no—
482−1+4×C59—22no—
492−1+4×C59—41no—
50H(5,𝔽2)×C59—22no—
51H(5,𝔽2)×C59—22no—
52H(5,𝔽2)×C59—42no—
53H(5,𝔽2)×C59—41no—
54C2×#41×C59—24no—
55C2×#41×C59—21no—
56C2×#41×C59—21no—
57C2×#41×C59—42no—
58C2×#41×C59—42no—
59C2×#41×C59D411yescharacteristic
60#137×C59—22no—
61#137×C59—24no—
62#137×C59—24no—
63#137×C59—22no—
64AGL(1,ℤ/8)×C59—22no—
65AGL(1,ℤ/8)×C59—24no—
66AGL(1,ℤ/8)×C59—24no—
67AGL(1,ℤ/8)×C59—22no—
68#135×C59—24no—
69#135×C59—24no—
70#135×C59—22no—
71#135×C59—22no—
72#135×C59—21no—
73#135×C59—21no—
74#135×C59—21no—
75#135×C59—21no—
76#135×C59C2×C412yes—
77#135×C59D412yes—
78#135×C59D412yes—
79#135×C59D412yes—
80C2×Q16×C59—22no—
81C2×Q16×C59C2×C411yescharacteristic
82C2×Q16×C59D411yescharacteristic
83C2×SD16×C59—24no—
84C2×SD16×C59—21no—
85C2×SD16×C59—21no—
86C2×D8×C59—22no—
87C2×D8×C59C2×C411yescharacteristic
88C2×D8×C59D411yescharacteristic
89#131×C59—21no—
90#131×C59—21no—
91#131×C59C2×C412yes—
92#131×C59D412yes—
93C2×M16×C59—21no—
94C2×M16×C59D411yescharacteristic
95C22×C8×C59—21no—
96C22×C8×C59C2311yescharacteristic
97#41×C59—21no—
98#41×C59—21no—
99#41×C59—24no—
100#41×C59—24no—
101#41×C59—22no—
102#41×C59—22no—
103#41×C59—22no—
104#41×C59—22no—
105#41×C59—44no—
106#41×C59—44no—
107#41×C59—44no—
108#41×C59—44no—
109#41×C59—42no—
110#41×C59—42no—
111#41×C59—42no—
112#41×C59—42no—
113#41×C59—41no—
114#41×C59—41no—
115#41×C59—41no—
116#41×C59—41no—
117#41×C59C22⋊C412yes—
118#41×C59SD1612yes—
119C2×Q8×C59—24no—
120C2×Q8×C59—21no—
121C2×Q8×C59—21no—
122C2×Q8×C59—42no—
123C2×Q8×C59—42no—
124C2×Q8×C59D811yescharacteristic
125C2×D4×C59—21no—
126C2×D4×C59—24no—
127C2×D4×C59—21no—
128C2×D4×C59—21no—
129C2×D4×C59—44no—
130C2×D4×C59—44no—
131C2×D4×C59—42no—
132C2×D4×C59—44no—
133C2×D4×C59—41no—
134C2×D4×C59—41no—
135C2×D4×C59—42no—
136C2×D4×C59D811yescharacteristic
137C22×C4×C59—21no—
138C22×C4×C59—44no—
139C22×C4×C59—41no—
140C22×C4×C59—41no—
141C22×C4×C59C2×D411yescharacteristic
142C22×Dic59—41no—
143Q16×C59—24no—
144Q16×C59—21no—
145Q16×C59—21no—
146Q16×C59C22⋊C412yes—
147Q16×C59C22⋊C412yes—
148SD16×C59—24no—
149SD16×C59—24no—
150SD16×C59—22no—
151SD16×C59—22no—
152SD16×C59—22no—
153SD16×C59—22no—
154D8×C59—24no—
155D8×C59—21no—
156D8×C59—21no—
157D8×C59C22⋊C412yes—
158D8×C59C22⋊C412yes—
159M16×C59—21no—
160M16×C59—22no—
161M16×C59—22no—
162M16×C59C22⋊C412yes—
163C2×C8×C59—21no—
164C2×C8×C59—22no—
165C2×C8×C59—22no—
166C2×C8×C59C22×C411yescharacteristic
167C2×C8×C59C2×D411yescharacteristic
168C2×C8×C59C2×D411yescharacteristic
169C2×C8×C59C2×D411yescharacteristic
170C2×C8×C59C2×D411yescharacteristic
171C2×C8×C59C2×D411yescharacteristic
172C4⋊Dic59—41no—
173C4⋊Dic59—41no—
174#11745664—41no—
175#11745664—41no—
176C2×C59⋊C8—41no—
177C59⋊C16—41no—
178C59⋊C16—41no—
179C23×C59—21no—
180C23×C59—44no—
181C23×C59—41no—
182C23×C59—41no—
183C23×C59AGL(1,ℤ/8)11yescharacteristic
184Q8×C59—21no—
185Q8×C59—22no—
186Q8×C59—22no—
187Q8×C59—44no—
188Q8×C59—41no—
189Q8×C59—41no—
190Q8×C59#10212yes—
191D4×C59—22no—
192D4×C59—22no—
193D4×C59—22no—
194D4×C59—22no—
195D4×C59—21no—
196D4×C59—44no—
197D4×C59—44no—
198D4×C59—44no—
199D4×C59—42no—
200D4×C59—42no—
201D4×C59—42no—
202D4×C59—42no—
203D4×C59—41no—
204D4×C59—41no—
205D4×C59#10212yes—
206C2×C4×C59—21no—
207C2×C4×C59—22no—
208C2×C4×C59—22no—
209C2×C4×C59—44no—
210C2×C4×C59—44no—
211C2×C4×C59—44no—
212C2×C4×C59—41no—
213C2×C4×C59—41no—
214C2×C4×C59—42no—
215C2×C4×C59—42no—
216C2×C4×C59—42no—
217C2×C4×C59—42no—
218C2×C4×C59C2×C22⋊C411yescharacteristic
219C2×C4×C59#12011yescharacteristic
220C2×C4×C59#12011yescharacteristic
221C2×C4×C59C2×D811yescharacteristic
222C2×C4×C59C2×SD1611yescharacteristic
223C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
224C2×Dic59—81no—
225C2×Dic59—161no—
226C8×C59—22no—
227C8×C59C2×C22⋊C411yescharacteristic
228C8×C59C2×C22⋊C411yescharacteristic
229C8×C59#12011yescharacteristic
230C8×C59#12011yescharacteristic
231C59⋊C8—81no—
232#12752—591nomaximal, Sylow
233C22×C59—22no—
234C22×C59—41no—
235C22×C59—42no—
236C22×C59—42no—
237C22×C59—81no—
238C22×C59—161no—
239C22×C59—161no—
240C22×C59—164no—
241C22×C59#41811yescharacteristic
242C22×C59#44711yescharacteristic
243C22×C59#45011yescharacteristic
244C4×C59—22no—
245C4×C59—42no—
246C4×C59—42no—
247C4×C59#37911yescharacteristic
248C4×C59C2×#10211yescharacteristic
249C4×C59#44711yescharacteristic
250C4×C59#45011yescharacteristic
251Dic59—161no—
252#3584D5911yescharacteristic
253#2148—591no—
254C2×#360—591no—
255C2×#360—591no—
256#2143—591no—
257#2117—591no—
258#1815—591no—
259C2×C59—21no—
260C2×C59—21no—
261C2×C59—21no—
262C2×C59—42no—
263C2×C59—82no—
264C2×C59—82no—
265C2×C59#178611yescharacteristic
266#585—21no—
267#578—21no—
268#578Dic5912yescomplemented
269#577—22no—
270#577—22no—
271#576—21no—
272#576Dic5912yescomplemented
273#575—22no—
274#575—21no—
275C2×#135—22no—
276C2×#135C2×D5911yescharacteristic
277C2×#135C2×D5911yescharacteristic
278C2×#131C2×D5911yescharacteristic
279C2×M32—591no—
280C2×M32—591no—
281C2×#108—591no—
282C2×#106—591no—
283#417—1181no—
284#362—1181no—
285#361—1181no—
286#360—1182no—
287#360—1182no—
288#359—1182no—
289#350—1182no—
290#343—1181no—
291C59#1275211yescharacteristic, complemented, Sylow
2922−1+4—22no—
2932−1+4—22no—
2942−1+4—41no—
295H(5,𝔽2)—22no—
296H(5,𝔽2)—22no—
297H(5,𝔽2)—42no—
298H(5,𝔽2)—41no—
299C2×#41—24no—
300C2×#41—21no—
301C2×#41—21no—
302C2×#41—42no—
303C2×#41—42no—
304C2×#41C22⋊D5911yescharacteristic
305#137—22no—
306#137—24no—
307#137—24no—
308#137—22no—
309AGL(1,ℤ/8)—22no—
310AGL(1,ℤ/8)—24no—
311AGL(1,ℤ/8)—24no—
312AGL(1,ℤ/8)—22no—
313#135—24no—
314#135—24no—
315#135—22no—
316#135—22no—
317#135—21no—
318#135—21no—
319#135—21no—
320#135—21no—
321#135C2×Dic5912yes—
322#135C22⋊D5912yes—
323#135C22⋊D5912yes—
324#135C22⋊D5912yes—
325C2×Q16—22no—
326C2×Q16C2×Dic5911yescharacteristic
327C2×Q16C22⋊D5911yescharacteristic
328C2×SD16—24no—
329C2×SD16—21no—
330C2×SD16—21no—
331C2×D8—22no—
332C2×D8C2×Dic5911yescharacteristic
333C2×D8C22⋊D5911yescharacteristic
334#131—21no—
335#131—21no—
336#131C2×Dic5912yes—
337#131C22⋊D5912yes—
338C2×M16—21no—
339C2×M16—1181no—
340C2×M16C22⋊D5911yescharacteristic
341C22×C8—21no—
342C22×C8C22×D5911yescharacteristic
343C2×C4⋊C4—1181no—
344M32—1181no—
345M32—1181no—
346M32—1181no—
347M32—1181no—
348C2×C16—1181no—
349C2×C16—1181no—
350#108—1181no—
351#108—1181no—
352#106—1181no—
353#106—1181no—
354#103—2361no—
355#102—2361no—
356#41—21no—
357#41—21no—
358#41—24no—
359#41—24no—
360#41—22no—
361#41—22no—
362#41—22no—
363#41—22no—
364#41—44no—
365#41—44no—
366#41—44no—
367#41—44no—
368#41—42no—
369#41—42no—
370#41—42no—
371#41—42no—
372#41—41no—
373#41—41no—
374#41—41no—
375#41—41no—
376#41#1174567012yes—
377#41C22⋊Dic5912yes—
378C2×Q8—24no—
379C2×Q8—21no—
380C2×Q8—21no—
381C2×Q8—42no—
382C2×Q8—42no—
383C2×Q8#1174566911yescharacteristic
384C2×D4—21no—
385C2×D4—24no—
386C2×D4—21no—
387C2×D4—21no—
388C2×D4—44no—
389C2×D4—44no—
390C2×D4—42no—
391C2×D4—44no—
392C2×D4—41no—
393C2×D4—41no—
394C2×D4—42no—
395C2×D4#1174566911yescharacteristic
396C22×C4—21no—
397C22×C4—44no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—2361no—
401C22×C4C2×C22⋊D5911yescharacteristic
402Q16—24no—
403Q16—21no—
404Q16—21no—
405Q16C22⋊Dic5912yes—
406Q16C22⋊Dic5912yes—
407SD16—24no—
408SD16—24no—
409SD16—22no—
410SD16—22no—
411SD16—22no—
412SD16—22no—
413D8—24no—
414D8—21no—
415D8—21no—
416D8C22⋊Dic5912yes—
417D8C22⋊Dic5912yes—
418M16—21no—
419M16—22no—
420M16—22no—
421M16—2361no—
422M16—2361no—
423M16C22⋊Dic5912yes—
424C2×C8—21no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—2361no—
428C2×C8C22×Dic5911yescharacteristic
429C2×C8C2×C22⋊D5911yescharacteristic
430C2×C8C2×C22⋊D5911yescharacteristic
431C2×C8C2×C22⋊D5911yescharacteristic
432C2×C8C2×C22⋊D5911yescharacteristic
433C2×C8C2×C22⋊D5911yescharacteristic
434C4⋊C4—2361no—
435C4⋊C4—2361no—
436C16—2361no—
437C16—2361no—
438C23—21no—
439C23—44no—
440C23—41no—
441C23—41no—
442C23#1427375311yescharacteristic
443Q8—21no—
444Q8—22no—
445Q8—22no—
446Q8—44no—
447Q8—41no—
448Q8—41no—
449Q8#1427364812yes—
450D4—22no—
451D4—22no—
452D4—22no—
453D4—22no—
454D4—21no—
455D4—44no—
456D4—44no—
457D4—44no—
458D4—42no—
459D4—42no—
460D4—42no—
461D4—42no—
462D4—41no—
463D4—41no—
464D4#1427364812yes—
465C2×C4—21no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—9441no—
478C2×C4—4721no—
479C2×C4C2×C22⋊Dic5911yescharacteristic
480C2×C4#1427369611yescharacteristic
481C2×C4#1427369611yescharacteristic
482C2×C4C2×#1174566911yescharacteristic
483C2×C4C2×#1174567011yescharacteristic
484C2×C4#1427375311yescharacteristic
485C8—22no—
486C8—4721no—
487C8C2×C22⋊Dic5911yescharacteristic
488C8C2×C22⋊Dic5911yescharacteristic
489C8#1427369611yescharacteristic
490C8#1427369611yescharacteristic
491C22—22no—
492C22—41no—
493C22—42no—
494C22—42no—
495C22—81no—
496C22—161no—
497C22—161no—
498C22—164no—
499C22#1683740711yescharacteristic
500C22#1683753611yescharacteristic
501C22#1683755311yescharacteristic
502C4—22no—
503C4—42no—
504C4—42no—
505C4—9441no—
506C4#1683725811yescharacteristic
507C4C2×#1427364811yescharacteristic
508C4#1683753611yescharacteristic
509C4#1683755311yescharacteristic
510C2—21no—
511C2—21no—
512C2—21no—
513C2—42no—
514C2—82no—
515C2—82no—
516C2#2142402711yescharacteristic
subgroupG/H
centerC2#21424027
commutatorC2×C8×C59C22×C4
FrattiniC22×C8C22×D59
Fitting, prime radicalC59×#3584C2
socleC2×C59#1786
cosocleC22×C8×C59C23
hypercenter#3584D59
hypocenterC59#12752
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ #3584

Complements

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

NH ≅ G/Ntype
1. #21425845C2semidirect
2. C2×#16837212C2semidirect
3. C2×#16837212C2semidirect
4. #21425824C2semidirect
5. #21425724C2semidirect
6. #21424155C2semidirect
8. C2×#14273667C22semidirect
9. C2×#14273667C22semidirect
10. C2×#14273663C22semidirect
11. C2×#14273659C22semidirect
22. C59×#578C4semidirect
26. C59×#576C4semidirect
268. #578Dic59semidirect
272. #576Dic59semidirect
291. C59#12752semidirect

Maximal subgroups

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

subgroupindex
#214258452
C2×#168372122
#214258242
#214257242
#214241552
C59×#35842
#1275259