Finite Group Databasebeta

← C59×#10239 #28422348

Subgroups — C59×#10239 #28422348

Order 15104.

610 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C59×#3226C211yesmaximal, characteristic, complemented
2C59×#3036C211yesmaximal, characteristic
3C59×#2547C211yesmaximal, characteristic, complemented
4C59×#1546C211yesmaximal, characteristic, complemented
5C59×#1502C211yesmaximal, characteristic, complemented
6C59×#1478C211yesmaximal, characteristic, complemented
7C59×#1303C211yesmaximal, characteristic, complemented
8C2×AGL(1,ℤ/8)×C59—21no—
9C59×#532C2211yescharacteristic
10C2×C4⋊Q8×C59C2211yescharacteristic
11C2×#124×C59—21no—
12C2×C4×D4×C59C2211yescharacteristic
13C59×#500—21no—
14C59×#486—21no—
15C59×#480—22no—
16C59×#474—22no—
17C59×#463—22no—
18C59×#461—21no—
19C59×#459—21no—
20C59×#431C2211yescharacteristic
21C59×#431C2211yescharacteristic
22C59×#423C2211yescharacteristic
23C59×#417—21no—
24C59×#397—22no—
25C59×#391—21no—
26C59×#380—22no—
27C4×C4⋊C4×C59C2211yescharacteristic
28C59×#332—22no—
29C59×#327—22no—
30C59×D4⋊C8—22no—
31C2×#41×C59—21no—
32C22×Q8×C59—21no—
33C22×D4×C59D411yescharacteristic
34C23×C4×C59—21no—
35AGL(1,ℤ/8)×C59—42no—
36AGL(1,ℤ/8)×C59—42no—
37C2×SD16×C59—22no—
38C2×D8×C59—22no—
39C2×M16×C59—21no—
40C4⋊Q8×C59—21no—
41C4⋊Q8×C59—21no—
42C4⋊Q8×C59—21no—
43C4⋊Q8×C59—21no—
44C4⋊Q8×C59D411yescharacteristic
45#127×C59D411yescharacteristic
46#124×C59—21no—
47#124×C59—42no—
48#124×C59—42no—
49#121×C59—21no—
50#121×C59—21no—
51C4×D4×C59—21no—
52C4×D4×C59—21no—
53C4×D4×C59—21no—
54C4×D4×C59—21no—
55C4×D4×C59—21no—
56C4×D4×C59—21no—
57C2×C4⋊C4×C59—21no—
58C2×C4⋊C4×C59—21no—
59C2×C4⋊C4×C59D412yes—
60C2×C4⋊C4×C59D411yescharacteristic
61C2×C22⋊C4×C59—24no—
62C2×C22⋊C4×C59—21no—
63C2×C42×C59—21no—
64C2×C42×C59C2311yescharacteristic
65#107×C59—24no—
66C4⋊C8×C59—21no—
67C4⋊C8×C59—21no—
68C4⋊C8×C59D412yes—
69C4⋊C8×C59D412yes—
70#103×C59—22no—
71#102×C59—22no—
72#102×C59—24no—
73C22⋊C8×C59—21no—
74C22⋊C8×C59—21no—
75C4×C8×C59—21no—
76C4×C8×C59—21no—
77#95×C59—24no—
78#95×C59D412yes—
79C24×C59—21no—
80#41×C59—41no—
81#41×C59—41no—
82#41×C59—41no—
83#41×C59—41no—
84C2×Q8×C59—21no—
85C2×Q8×C59—41no—
86C2×Q8×C59—41no—
87C2×Q8×C59—41no—
88C2×Q8×C59—41no—
89C2×Q8×C59—41no—
90C2×Q8×C59—41no—
91C2×D4×C59—21no—
92C2×D4×C59—21no—
93C2×D4×C59—21no—
94C2×D4×C59—21no—
95C2×D4×C59—21no—
96C2×D4×C59—21no—
97C2×D4×C59—21no—
98C2×D4×C59—21no—
99C2×D4×C59—41no—
100C2×D4×C59D812yes—
101C22×C4×C59—21no—
102C22×C4×C59—22no—
103C22×C4×C59—22no—
104C22×C4×C59—44no—
105C22×C4×C59—41no—
106C22×C4×C59—41no—
107C22×C4×C59—41no—
108C22×C4×C59C2×D411yescharacteristic
109C22×C4×C59C2×D411yescharacteristic
110C22×C4×C59C2×D411yescharacteristic
111SD16×C59—44no—
112D8×C59—44no—
113M16×C59—41no—
114M16×C59—41no—
115C2×C8×C59—22no—
116C2×C8×C59—22no—
117C2×C8×C59—41no—
118C4⋊C4×C59—21no—
119C4⋊C4×C59—22no—
120C4⋊C4×C59—21no—
121C4⋊C4×C59—21no—
122C4⋊C4×C59—22no—
123C4⋊C4×C59—21no—
124C4⋊C4×C59—42no—
125C4⋊C4×C59—41no—
126C4⋊C4×C59—41no—
127C22⋊C4×C59—48no—
128C22⋊C4×C59—41no—
129C22⋊C4×C59—41no—
130C22⋊C4×C59—41no—
131C22⋊C4×C59—41no—
132C42×C59—41no—
133C42×C59—41no—
134C42×C59C2×D411yescharacteristic
135C42×C59C2×D411yescharacteristic
136C42×C59C2×D411yescharacteristic
137C42×C59#4111yescharacteristic
138C23×C59—22no—
139C23×C59—42no—
140C23×C59—42no—
141C23×C59—41no—
142C23×C59—41no—
143C23×C59—41no—
144C23×C59#12011yescharacteristic
145Q8×C59—41no—
146Q8×C59—41no—
147Q8×C59—81no—
148Q8×C59—81no—
149Q8×C59—81no—
150Q8×C59—81no—
151D4×C59—22no—
152D4×C59—22no—
153D4×C59—41no—
154D4×C59—41no—
155D4×C59—41no—
156D4×C59—42no—
157D4×C59—41no—
158D4×C59—81no—
159D4×C59—81no—
160D4×C59—81no—
161C2×C4×C59—22no—
162C2×C4×C59—41no—
163C2×C4×C59—42no—
164C2×C4×C59—42no—
165C2×C4×C59—42no—
166C2×C4×C59—41no—
167C2×C4×C59—41no—
168C2×C4×C59—42no—
169C2×C4×C59—42no—
170C2×C4×C59—42no—
171C2×C4×C59—42no—
172C2×C4×C59—82no—
173C2×C4×C59—82no—
174C2×C4×C59—81no—
175C2×C4×C59—81no—
176C2×C4×C59#12011yescharacteristic
177C2×C4×C59#12011yescharacteristic
178C2×C4×C59#12111yescharacteristic
179C2×C4×C59#12111yescharacteristic
180C2×C4×C59#12111yescharacteristic
181C2×C4×C59#12711yescharacteristic
182C2×C4×C59C2×D811yescharacteristic
183C2×C4×C59#13511yescharacteristic
184C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
185C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
186C8×C59—42no—
187C8×C59—42no—
188C8×C59—81no—
189#10239C5911yesmaximal, characteristic, direct product factor, Sylow
190C22×C59—21no—
191C22×C59—21no—
192C22×C59—44no—
193C22×C59—42no—
194C22×C59—82no—
195C22×C59—81no—
196C22×C59—81no—
197C22×C59—81no—
198C22×C59—81no—
199C22×C59—81no—
200C22×C59#39211yescharacteristic
201C22×C59#44711yescharacteristic
202C22×C59#44911yescharacteristic
203C4×C59—21no—
204C4×C59—21no—
205C4×C59—21no—
206C4×C59—82no—
207C4×C59—82no—
208C4×C59—81no—
209C4×C59—81no—
210C4×C59#44711yescharacteristic
211C4×C59#44911yescharacteristic
212C4×C59#45412yes—
213C4×C59#46611yescharacteristic
214C4×C59#46911yescharacteristic
215#3226C2×C5911yescharacteristic, complemented
216#3036C2×C5911yescharacteristic
217#2547C2×C5911yescharacteristic, complemented
218#1546C2×C5911yescharacteristic, complemented
219#1502C2×C5911yescharacteristic, complemented
220#1478C2×C5911yescharacteristic, complemented
221#1303C2×C5911yescharacteristic, complemented
222C2×C59—21no—
223C2×C59—21no—
224C2×C59—82no—
225C2×C59—161no—
226C2×C59#199911yescharacteristic
227C2×C59#200311yescharacteristic
228C2×C59#200911yescharacteristic
229C2×AGL(1,ℤ/8)—21no—
230#532C22×C5911yescharacteristic
231C2×C4⋊Q8C22×C5911yescharacteristic
232C2×#124—21no—
233C2×C4×D4C22×C5911yescharacteristic
234#500—21no—
235#486—21no—
236#480—22no—
237#474—22no—
238#463—22no—
239#461—21no—
240#459—21no—
241#431C22×C5911yescharacteristic
242#431C22×C5911yescharacteristic
243#423C22×C5911yescharacteristic
244#417—21no—
245#397—22no—
246#391—21no—
247#380—22no—
248C4×C4⋊C4C22×C5911yescharacteristic
249#332—22no—
250#327—22no—
251D4⋊C8—22no—
252C59#1023911yescharacteristic, direct product factor, Sylow
253C2×#41—21no—
254C22×Q8—21no—
255C22×D4D4×C5911yescharacteristic
256C23×C4—21no—
257AGL(1,ℤ/8)—42no—
258AGL(1,ℤ/8)—42no—
259C2×SD16—22no—
260C2×D8—22no—
261C2×M16—21no—
262C4⋊Q8—21no—
263C4⋊Q8—21no—
264C4⋊Q8—21no—
265C4⋊Q8—21no—
266C4⋊Q8D4×C5911yescharacteristic
267#127D4×C5911yescharacteristic
268#124—21no—
269#124—42no—
270#124—42no—
271#121—21no—
272#121—21no—
273C4×D4—21no—
274C4×D4—21no—
275C4×D4—21no—
276C4×D4—21no—
277C4×D4—21no—
278C4×D4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4D4×C5911yescharacteristic
282C2×C4⋊C4D4×C5912yes—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—21no—
285C2×C42—21no—
286C2×C42C23×C5911yescharacteristic
287#107—24no—
288C4⋊C8—21no—
289C4⋊C8—21no—
290C4⋊C8D4×C5912yes—
291C4⋊C8D4×C5912yes—
292#103—22no—
293#102—22no—
294#102—24no—
295C22⋊C8—21no—
296C22⋊C8—21no—
297C4×C8—21no—
298C4×C8—21no—
299#95—24no—
300#95D4×C5912yes—
301C24—21no—
302#41—41no—
303#41—41no—
304#41—41no—
305#41—41no—
306C2×Q8—21no—
307C2×Q8—41no—
308C2×Q8—41no—
309C2×Q8—41no—
310C2×Q8—41no—
311C2×Q8—41no—
312C2×Q8—41no—
313C2×D4—21no—
314C2×D4—21no—
315C2×D4—21no—
316C2×D4—21no—
317C2×D4—21no—
318C2×D4—21no—
319C2×D4—21no—
320C2×D4—21no—
321C2×D4—41no—
322C2×D4D8×C5912yes—
323C22×C4—21no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—44no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4C2×D4×C5911yescharacteristic
331C22×C4C2×D4×C5911yescharacteristic
332C22×C4C2×D4×C5911yescharacteristic
333SD16—44no—
334D8—44no—
335M16—41no—
336M16—41no—
337C2×C8—22no—
338C2×C8—22no—
339C2×C8—41no—
340C4⋊C4—21no—
341C4⋊C4—22no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—22no—
345C4⋊C4—21no—
346C4⋊C4—42no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C22⋊C4—48no—
350C22⋊C4—41no—
351C22⋊C4—41no—
352C22⋊C4—41no—
353C22⋊C4—41no—
354C42—41no—
355C42—41no—
356C42C2×D4×C5911yescharacteristic
357C42C2×D4×C5911yescharacteristic
358C42C2×D4×C5911yescharacteristic
359C42#41×C5911yescharacteristic
360C23—22no—
361C23—42no—
362C23—42no—
363C23—41no—
364C23—41no—
365C23—41no—
366C23#120×C5911yescharacteristic
367Q8—41no—
368Q8—41no—
369Q8—81no—
370Q8—81no—
371Q8—81no—
372Q8—81no—
373D4—22no—
374D4—22no—
375D4—41no—
376D4—41no—
377D4—41no—
378D4—42no—
379D4—41no—
380D4—81no—
381D4—81no—
382D4—81no—
383C2×C4—22no—
384C2×C4—41no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—82no—
395C2×C4—82no—
396C2×C4—81no—
397C2×C4—81no—
398C2×C4#120×C5911yescharacteristic
399C2×C4#120×C5911yescharacteristic
400C2×C4#121×C5911yescharacteristic
401C2×C4#121×C5911yescharacteristic
402C2×C4#121×C5911yescharacteristic
403C2×C4#127×C5911yescharacteristic
404C2×C4C2×D8×C5911yescharacteristic
405C2×C4#135×C5911yescharacteristic
406C2×C4AGL(1,ℤ/8)×C5911yescharacteristic
407C2×C4AGL(1,ℤ/8)×C5911yescharacteristic
408C8—42no—
409C8—42no—
410C8—81no—
411C22—21no—
412C22—21no—
413C22—44no—
414C22—42no—
415C22—82no—
416C22—81no—
417C22—81no—
418C22—81no—
419C22—81no—
420C22—81no—
421C22C59×#39211yescharacteristic
422C22C59×#44711yescharacteristic
423C22C59×#44911yescharacteristic
424C4—21no—
425C4—21no—
426C4—21no—
427C4—82no—
428C4—82no—
429C4—81no—
430C4—81no—
431C4C59×#44711yescharacteristic
432C4C59×#44911yescharacteristic
433C4C59×#45412yes—
434C4C59×#46611yescharacteristic
435C4C59×#46911yescharacteristic
436C2—21no—
437C2—21no—
438C2—82no—
439C2—161no—
440C2C59×#199911yescharacteristic
441C2C59×#200311yescharacteristic
442C2C59×#200911yescharacteristic
subgroupG/H
center, socleC22×C59#392
commutator, Frattini, cosocleC2×C42C23×C59
prime radicalC59×#3036C2
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C59 ◁ C2×C42×C59 ◁ G

Complements

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

NH ≅ G/Ntype
1. C59×#3226C2semidirect
3. C59×#2547C2semidirect
4. C59×#1546C2semidirect
5. C59×#1502C2semidirect
6. C59×#1478C2semidirect
7. C59×#1303C2semidirect
189. #10239C59direct (N × H)
215. #3226C2×C59semidirect
217. #2547C2×C59semidirect
218. #1546C2×C59semidirect
219. #1502C2×C59semidirect
220. #1478C2×C59semidirect
221. #1303C2×C59semidirect
252. C59#10239direct (N × H)

Maximal subgroups

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

subgroupindex
C59×#32262
C59×#30362
C59×#25472
C59×#15462
C59×#15022
C59×#14782
C59×#13032
#1023959