Finite Group Databasebeta

← C59×#10256 #28422365

Subgroups — C59×#10256 #28422365

Order 15104.

580 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C59×#3246C211yesmaximal, characteristic, complemented
2C59×#3077C211yesmaximal, characteristic, complemented
3C59×#3036C211yesmaximal, characteristic
4C59×#1954C211yesmaximal, characteristic, complemented
5C59×#1493C211yesmaximal, characteristic, complemented
6C59×#1492C211yesmaximal, characteristic, complemented
7C59×#1292C211yesmaximal, characteristic, complemented
8C2×AGL(1,ℤ/8)×C59—21no—
9C59×#533C2211yescharacteristic
10C59×#532C2211yescharacteristic
11C2×C4×D4×C59C2211yescharacteristic
12C59×#478—22no—
13C59×#476—21no—
14C59×#476—21no—
15C59×#475—21no—
16C59×#474—21no—
17C59×#465—22no—
18C59×#463—22no—
19C59×#461—21no—
20C59×#461—21no—
21C59×#460—21no—
22C59×#459—21no—
23C59×#428—21no—
24C59×#423C2211yescharacteristic
25C59×#423C2211yescharacteristic
26C2×C4⋊C8×C59C2211yescharacteristic
27C59×#418—21no—
28C59×#417—21no—
29C2×C22⋊C8×C59—22no—
30C2×C4×C8×C59C2211yescharacteristic
31C59×Q8⋊C8—22no—
32C59×D4⋊C8—22no—
33C2×#41×C59—21no—
34C22×D4×C59D411yescharacteristic
35C23×C4×C59—21no—
36AGL(1,ℤ/8)×C59—42no—
37AGL(1,ℤ/8)×C59—42no—
38C2×SD16×C59—22no—
39C2×SD16×C59—22no—
40C2×D8×C59—22no—
41C2×M16×C59—21no—
42C2×M16×C59—21no—
43C22×C8×C59—21no—
44C22×C8×C59D412yes—
45C4⋊Q8×C59D411yescharacteristic
46C4⋊Q8×C59D411yescharacteristic
47C4⋊Q8×C59D411yescharacteristic
48#127×C59D411yescharacteristic
49#125×C59—21no—
50#124×C59—21no—
51C22⋊Q8×C59—21no—
52C22⋊Q8×C59—21no—
53#121×C59—21no—
54#121×C59—21no—
55C4×Q8×C59—21no—
56C4×Q8×C59—21no—
57C4×D4×C59—21no—
58C4×D4×C59—21no—
59C4×D4×C59—21no—
60C4×D4×C59—21no—
61C4×D4×C59—21no—
62C4×D4×C59—21no—
63#117×C59—21no—
64C2×C4⋊C4×C59D411yescharacteristic
65C2×C22⋊C4×C59—21no—
66C2×C42×C59C2311yescharacteristic
67#107×C59—22no—
68#106×C59—22no—
69#106×C59—22no—
70C4⋊C8×C59—21no—
71C4⋊C8×C59—21no—
72C4⋊C8×C59—21no—
73C4⋊C8×C59—21no—
74C4⋊C8×C59D412yes—
75C4⋊C8×C59D412yes—
76#103×C59—22no—
77#103×C59—22no—
78#102×C59—22no—
79#102×C59—22no—
80#102×C59—22no—
81#102×C59—22no—
82C22⋊C8×C59—44no—
83C4×C8×C59—21no—
84C4×C8×C59—21no—
85C24×C59—21no—
86#41×C59—41no—
87#41×C59—41no—
88#41×C59—41no—
89#41×C59—41no—
90C2×Q8×C59—21no—
91C2×Q8×C59—21no—
92C2×Q8×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—21no—
100C2×D4×C59—21no—
101C2×D4×C59—41no—
102C2×D4×C59SD1612yescomplemented
103C22×C4×C59—21no—
104C22×C4×C59—42no—
105C22×C4×C59—42no—
106C22×C4×C59—41no—
107C22×C4×C59—41no—
108C22×C4×C59—41no—
109C22×C4×C59C2×D411yescharacteristic
110C22×C4×C59C2×D411yescharacteristic
111C22×C4×C59#4111yescharacteristic
112SD16×C59—44no—
113SD16×C59—82no—
114D8×C59—44no—
115M16×C59—41no—
116M16×C59—41no—
117M16×C59—41no—
118M16×C59—41no—
119C2×C8×C59—22no—
120C2×C8×C59—22no—
121C2×C8×C59—22no—
122C2×C8×C59—22no—
123C2×C8×C59—41no—
124C2×C8×C59—41no—
125C2×C8×C59—42no—
126C4⋊C4×C59—21no—
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—21no—
134C4⋊C4×C59—41no—
135C4⋊C4×C59—41no—
136C22⋊C4×C59—41no—
137C22⋊C4×C59—41no—
138C22⋊C4×C59—41no—
139C22⋊C4×C59—41no—
140C22⋊C4×C59—41no—
141C42×C59—41no—
142C42×C59C2×D411yescharacteristic
143C42×C59C2×D411yescharacteristic
144C42×C59C2×D411yescharacteristic
145C42×C59C2×D411yescharacteristic
146C23×C59—22no—
147C23×C59—42no—
148C23×C59—42no—
149C23×C59—41no—
150C23×C59—41no—
151C23×C59—41no—
152C23×C59#12111yescharacteristic
153Q8×C59—42no—
154Q8×C59—41no—
155Q8×C59—41no—
156Q8×C59—41no—
157Q8×C59—41no—
158D4×C59—22no—
159D4×C59—22no—
160D4×C59—42no—
161D4×C59—41no—
162D4×C59—41no—
163D4×C59—41no—
164D4×C59—41no—
165D4×C59—41no—
166D4×C59—41no—
167D4×C59—81no—
168D4×C59—81no—
169C2×C4×C59—21no—
170C2×C4×C59—21no—
171C2×C4×C59—41no—
172C2×C4×C59—42no—
173C2×C4×C59—41no—
174C2×C4×C59—41no—
175C2×C4×C59—41no—
176C2×C4×C59—41no—
177C2×C4×C59—82no—
178C2×C4×C59—81no—
179C2×C4×C59—81no—
180C2×C4×C59—81no—
181C2×C4×C59—81no—
182C2×C4×C59#12011yescharacteristic
183C2×C4×C59#12011yescharacteristic
184C2×C4×C59#12011yescharacteristic
185C2×C4×C59#12111yescharacteristic
186C2×C4×C59#12111yescharacteristic
187C2×C4×C59#12711yescharacteristic
188C2×C4×C59C2×SD1611yescharacteristic
189C2×C4×C59#13511yescharacteristic
190C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
191C2×C4×C59#13711yescharacteristic
192C8×C59—42no—
193C8×C59—42no—
194C8×C59—42no—
195C8×C59—42no—
196#10256C5911yesmaximal, characteristic, direct product factor, Sylow
197C22×C59—21no—
198C22×C59—21no—
199C22×C59—44no—
200C22×C59—42no—
201C22×C59—82no—
202C22×C59—81no—
203C22×C59—81no—
204C22×C59—81no—
205C22×C59—81no—
206C22×C59—81no—
207C22×C59#39211yescharacteristic
208C22×C59#46511yescharacteristic
209C22×C59#46811yescharacteristic
210C4×C59—21no—
211C4×C59—21no—
212C4×C59—21no—
213C4×C59—81no—
214C4×C59—81no—
215C4×C59—81no—
216C4×C59—81no—
217C4×C59#44811yescharacteristic
218C4×C59#44911yescharacteristic
219C4×C59#45011yescharacteristic
220C4×C59#45211yescharacteristic
221C4×C59#45311yescharacteristic
222C4×C59#45411yescharacteristic
223#3246C2×C5911yescharacteristic, complemented
224#3077C2×C5911yescharacteristic, complemented
225#3036C2×C5911yescharacteristic
226#1954C2×C5911yescharacteristic, complemented
227#1493C2×C5911yescharacteristic, complemented
228#1492C2×C5911yescharacteristic, complemented
229#1292C2×C5911yescharacteristic, complemented
230C2×C59—21no—
231C2×C59—21no—
232C2×C59—82no—
233C2×C59—161no—
234C2×C59#200011yescharacteristic
235C2×C59#200611yescharacteristic
236C2×C59#201711yescharacteristic
237C2×AGL(1,ℤ/8)—21no—
238#533C22×C5911yescharacteristic
239#532C22×C5911yescharacteristic
240C2×C4×D4C22×C5911yescharacteristic
241#478—22no—
242#476—21no—
243#476—21no—
244#475—21no—
245#474—21no—
246#465—22no—
247#463—22no—
248#461—21no—
249#461—21no—
250#460—21no—
251#459—21no—
252#428—21no—
253#423C22×C5911yescharacteristic
254#423C22×C5911yescharacteristic
255C2×C4⋊C8C22×C5911yescharacteristic
256#418—21no—
257#417—21no—
258C2×C22⋊C8—22no—
259C2×C4×C8C22×C5911yescharacteristic
260Q8⋊C8—22no—
261D4⋊C8—22no—
262C59#1025611yescharacteristic, direct product factor, Sylow
263C2×#41—21no—
264C22×D4D4×C5911yescharacteristic
265C23×C4—21no—
266AGL(1,ℤ/8)—42no—
267AGL(1,ℤ/8)—42no—
268C2×SD16—22no—
269C2×SD16—22no—
270C2×D8—22no—
271C2×M16—21no—
272C2×M16—21no—
273C22×C8—21no—
274C22×C8D4×C5912yes—
275C4⋊Q8D4×C5911yescharacteristic
276C4⋊Q8D4×C5911yescharacteristic
277C4⋊Q8D4×C5911yescharacteristic
278#127D4×C5911yescharacteristic
279#125—21no—
280#124—21no—
281C22⋊Q8—21no—
282C22⋊Q8—21no—
283#121—21no—
284#121—21no—
285C4×Q8—21no—
286C4×Q8—21no—
287C4×D4—21no—
288C4×D4—21no—
289C4×D4—21no—
290C4×D4—21no—
291C4×D4—21no—
292C4×D4—21no—
293#117—21no—
294C2×C4⋊C4D4×C5911yescharacteristic
295C2×C22⋊C4—21no—
296C2×C42C23×C5911yescharacteristic
297#107—22no—
298#106—22no—
299#106—22no—
300C4⋊C8—21no—
301C4⋊C8—21no—
302C4⋊C8—21no—
303C4⋊C8—21no—
304C4⋊C8D4×C5912yes—
305C4⋊C8D4×C5912yes—
306#103—22no—
307#103—22no—
308#102—22no—
309#102—22no—
310#102—22no—
311#102—22no—
312C22⋊C8—44no—
313C4×C8—21no—
314C4×C8—21no—
315C24—21no—
316#41—41no—
317#41—41no—
318#41—41no—
319#41—41no—
320C2×Q8—21no—
321C2×Q8—21no—
322C2×Q8—21no—
323C2×D4—21no—
324C2×D4—21no—
325C2×D4—21no—
326C2×D4—21no—
327C2×D4—21no—
328C2×D4—21no—
329C2×D4—21no—
330C2×D4—21no—
331C2×D4—41no—
332C2×D4SD16×C5912yescomplemented
333C22×C4—21no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4C2×D4×C5911yescharacteristic
340C22×C4C2×D4×C5911yescharacteristic
341C22×C4#41×C5911yescharacteristic
342SD16—44no—
343SD16—82no—
344D8—44no—
345M16—41no—
346M16—41no—
347M16—41no—
348M16—41no—
349C2×C8—22no—
350C2×C8—22no—
351C2×C8—22no—
352C2×C8—22no—
353C2×C8—41no—
354C2×C8—41no—
355C2×C8—42no—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—21no—
359C4⋊C4—21no—
360C4⋊C4—21no—
361C4⋊C4—21no—
362C4⋊C4—21no—
363C4⋊C4—21no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C22⋊C4—41no—
367C22⋊C4—41no—
368C22⋊C4—41no—
369C22⋊C4—41no—
370C22⋊C4—41no—
371C42—41no—
372C42C2×D4×C5911yescharacteristic
373C42C2×D4×C5911yescharacteristic
374C42C2×D4×C5911yescharacteristic
375C42C2×D4×C5911yescharacteristic
376C23—22no—
377C23—42no—
378C23—42no—
379C23—41no—
380C23—41no—
381C23—41no—
382C23#121×C5911yescharacteristic
383Q8—42no—
384Q8—41no—
385Q8—41no—
386Q8—41no—
387Q8—41no—
388D4—22no—
389D4—22no—
390D4—42no—
391D4—41no—
392D4—41no—
393D4—41no—
394D4—41no—
395D4—41no—
396D4—41no—
397D4—81no—
398D4—81no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—41no—
402C2×C4—42no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—82no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4—81no—
411C2×C4—81no—
412C2×C4#120×C5911yescharacteristic
413C2×C4#120×C5911yescharacteristic
414C2×C4#120×C5911yescharacteristic
415C2×C4#121×C5911yescharacteristic
416C2×C4#121×C5911yescharacteristic
417C2×C4#127×C5911yescharacteristic
418C2×C4C2×SD16×C5911yescharacteristic
419C2×C4#135×C5911yescharacteristic
420C2×C4AGL(1,ℤ/8)×C5911yescharacteristic
421C2×C4#137×C5911yescharacteristic
422C8—42no—
423C8—42no—
424C8—42no—
425C8—42no—
426C22—21no—
427C22—21no—
428C22—44no—
429C22—42no—
430C22—82no—
431C22—81no—
432C22—81no—
433C22—81no—
434C22—81no—
435C22—81no—
436C22C59×#39211yescharacteristic
437C22C59×#46511yescharacteristic
438C22C59×#46811yescharacteristic
439C4—21no—
440C4—21no—
441C4—21no—
442C4—81no—
443C4—81no—
444C4—81no—
445C4—81no—
446C4C59×#44811yescharacteristic
447C4C59×#44911yescharacteristic
448C4C59×#45011yescharacteristic
449C4C59×#45211yescharacteristic
450C4C59×#45311yescharacteristic
451C4C59×#45411yescharacteristic
452C2—21no—
453C2—21no—
454C2—82no—
455C2—161no—
456C2C59×#200011yescharacteristic
457C2C59×#200611yescharacteristic
458C2C59×#201711yescharacteristic
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×#3246C2semidirect
2. C59×#3077C2semidirect
4. C59×#1954C2semidirect
5. C59×#1493C2semidirect
6. C59×#1492C2semidirect
7. C59×#1292C2semidirect
102. C2×D4×C59SD16semidirect
196. #10256C59direct (N × H)
223. #3246C2×C59semidirect
224. #3077C2×C59semidirect
226. #1954C2×C59semidirect
227. #1493C2×C59semidirect
228. #1492C2×C59semidirect
229. #1292C2×C59semidirect
262. C59#10256direct (N × H)
332. C2×D4SD16×C59semidirect

Maximal subgroups

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

subgroupindex
C59×#32462
C59×#30772
C59×#30362
C59×#19542
C59×#14932
C59×#14922
C59×#12922
#1025659