Finite Group Databasebeta

← C59×#13354 #28425463

Subgroups — C59×#13354 #28425463

Order 15104.

692 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 692 conjugacy classes of subgroups — too many to draw. Only its 94 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C59×#3200C211yesmaximal, characteristic, complemented
2C59×#3028C211yesmaximal, characteristic, complemented
3C59×#3021C211yesmaximal, characteristic
4C59×#2135C211yesmaximal, characteristic, complemented
5C59×#2122C211yesmaximal, characteristic, complemented
6C59×#1791C211yesmaximal, characteristic, complemented
7C59×#1395C211yesmaximal, characteristic, complemented
8C59×#585C2211yescharacteristic
9C2×AGL(1,ℤ/8)×C59—21no—
10C59×#536—21no—
11C59×#536C2211yescharacteristic
12C59×#535C2211yescharacteristic
13C59×#485—21no—
14C59×#482—21no—
15C59×#468—21no—
16C59×#465—21no—
17C59×#458—22no—
18C59×#458—22no—
19C59×#454—22no—
20C59×#453—22no—
21C59×#450—21no—
22C59×#448—21no—
23C59×#421—21no—
24C59×#413—21no—
25C59×#413C2211yescharacteristic
26C59×#413C2211yescharacteristic
27C59×#410—21no—
28C59×#410C2211yescharacteristic
29C59×#408—21no—
30C59×#408C2211yescharacteristic
31C59×#356—22no—
32C59×AΣL(1,𝔽16)—22no—
332−1+4×C59—21no—
342−1+4×C59—41no—
35H(5,𝔽2)×C59—21no—
36H(5,𝔽2)×C59—21no—
37H(5,𝔽2)×C59—41no—
38H(5,𝔽2)×C59D412yes—
39C2×#41×C59—21no—
40C2×#41×C59—21no—
41C2×#41×C59—21no—
42C2×#41×C59—42no—
43C2×#41×C59C2311yescharacteristic
44C22×Q8×C59D411yescharacteristic
45C22×D4×C59D411yescharacteristic
46AGL(1,ℤ/8)×C59—24no—
47AGL(1,ℤ/8)×C59—42no—
48C2×SD16×C59—21no—
49C2×SD16×C59—21no—
50C2×SD16×C59—41no—
51C2×D8×C59—41no—
52C2×M16×C59—21no—
53C2×M16×C59—21no—
54C2×M16×C59D411yescharacteristic
55C22×C8×C59D411yescharacteristic
56C4⋊Q8×C59—21no—
57C4⋊Q8×C59—41no—
58#127×C59—21no—
59#124×C59—21no—
60#124×C59—21no—
61#124×C59—41no—
62#123×C59—24no—
63#123×C59—21no—
64#123×C59—21no—
65C22⋊Q8×C59—21no—
66C22⋊Q8×C59—21no—
67C22⋊Q8×C59—42no—
68#121×C59—21no—
69#121×C59—21no—
70#120×C59—21no—
71#120×C59—21no—
72#117×C59—21no—
73#117×C59D411yescharacteristic
74#117×C59D411yescharacteristic
75#107×C59—21no—
76#106×C59—21no—
77#104×C59—24no—
78#103×C59—21no—
79#103×C59—21no—
80#102×C59—21no—
81#102×C59—21no—
82#101×C59—21no—
83#101×C59—41no—
84#101×C59D412yes—
85#100×C59—21no—
86#100×C59—21no—
87#100×C59—41no—
88#99×C59—24no—
89#99×C59—21no—
90#99×C59D412yes—
91C22⋊C8×C59—21no—
92C22⋊C8×C59—21no—
93C22⋊C8×C59—42no—
94C24×C59—21no—
95#41×C59—22no—
96#41×C59—22no—
97#41×C59—41no—
98#41×C59—41no—
99#41×C59—42no—
100#41×C59—41no—
101#41×C59—41no—
102#41×C59—41no—
103#41×C59—41no—
104#41×C59—41no—
105#41×C59—42no—
106#41×C59—42no—
107#41×C59—42no—
108#41×C59—41no—
109#41×C59—41no—
110#41×C59—41no—
111C2×Q8×C59—21no—
112C2×Q8×C59—21no—
113C2×Q8×C59—21no—
114C2×Q8×C59—21no—
115C2×Q8×C59—21no—
116C2×Q8×C59—41no—
117C2×Q8×C59—41no—
118C2×Q8×C59—42no—
119C2×Q8×C59C2×D411yescharacteristic
120C2×D4×C59—21no—
121C2×D4×C59—21no—
122C2×D4×C59—21no—
123C2×D4×C59—22no—
124C2×D4×C59—22no—
125C2×D4×C59—21no—
126C2×D4×C59—21no—
127C2×D4×C59—41no—
128C2×D4×C59—41no—
129C2×D4×C59—42no—
130C2×D4×C59—42no—
131C2×D4×C59—42no—
132C2×D4×C59—41no—
133C2×D4×C59—41no—
134C2×D4×C59—41no—
135C2×D4×C59C2×D411yescharacteristic
136C2×D4×C59C2×D411yescharacteristic
137C2×D4×C59#4111yescharacteristic
138C22×C4×C59—22no—
139C22×C4×C59—22no—
140C22×C4×C59—21no—
141C22×C4×C59—41no—
142C22×C4×C59C2×D411yescharacteristic
143C22×C4×C59C2×D411yescharacteristic
144C22×C4×C59C2×D411yescharacteristic
145SD16×C59—44no—
146SD16×C59—81no—
147SD16×C59—81no—
148D8×C59—44no—
149M16×C59—22no—
150M16×C59—41no—
151M16×C59—41no—
152M16×C59—41no—
153M16×C59—41no—
154C2×C8×C59—21no—
155C2×C8×C59—21no—
156C2×C8×C59—41no—
157C2×C8×C59—41no—
158C2×C8×C59—41no—
159C4⋊C4×C59—21no—
160C4⋊C4×C59—21no—
161C4⋊C4×C59—41no—
162C4⋊C4×C59—44no—
163C4⋊C4×C59—41no—
164C4⋊C4×C59—41no—
165C22⋊C4×C59—21no—
166C22⋊C4×C59—22no—
167C22⋊C4×C59—21no—
168C22⋊C4×C59—44no—
169C22⋊C4×C59—41no—
170C22⋊C4×C59—41no—
171C22⋊C4×C59—41no—
172C22⋊C4×C59—41no—
173C42×C59—21no—
174C42×C59—21no—
175C42×C59—41no—
176C23×C59—22no—
177C23×C59—22no—
178C23×C59—41no—
179C23×C59—41no—
180C23×C59—81no—
181C23×C59—161no—
182C23×C59#12011yescharacteristic
183C23×C59#12111yescharacteristic
184C23×C59#12111yescharacteristic
185Q8×C59—22no—
186Q8×C59—41no—
187Q8×C59—41no—
188Q8×C59—41no—
189Q8×C59—41no—
190Q8×C59—41no—
191Q8×C59—41no—
192Q8×C59—81no—
193D4×C59—22no—
194D4×C59—41no—
195D4×C59—41no—
196D4×C59—41no—
197D4×C59—41no—
198D4×C59—41no—
199D4×C59—42no—
200D4×C59—42no—
201D4×C59—42no—
202D4×C59—42no—
203D4×C59—42no—
204D4×C59—41no—
205D4×C59—41no—
206D4×C59—41no—
207D4×C59—161no—
208D4×C59—161no—
209C2×C4×C59—21no—
210C2×C4×C59—21no—
211C2×C4×C59—21no—
212C2×C4×C59—21no—
213C2×C4×C59—21no—
214C2×C4×C59—21no—
215C2×C4×C59—22no—
216C2×C4×C59—22no—
217C2×C4×C59—42no—
218C2×C4×C59—42no—
219C2×C4×C59—42no—
220C2×C4×C59—41no—
221C2×C4×C59—42no—
222C2×C4×C59—42no—
223C2×C4×C59—42no—
224C2×C4×C59—41no—
225C2×C4×C59—41no—
226C2×C4×C59—41no—
227C2×C4×C59#12011yescharacteristic
228C2×C4×C59#12011yescharacteristic
229C2×C4×C59#12111yescharacteristic
230C2×C4×C59#12711yescharacteristic
231C8×C59—42no—
232C8×C59—41no—
233C8×C59—41no—
234C8×C59—81no—
235#13354C5911yesmaximal, characteristic, direct product factor, Sylow
236C22×C59—21no—
237C22×C59—21no—
238C22×C59—21no—
239C22×C59—41no—
240C22×C59—41no—
241C22×C59—41no—
242C22×C59—42no—
243C22×C59—81no—
244C22×C59—81no—
245C22×C59—82no—
246C22×C59—82no—
247C22×C59—81no—
248C22×C59—161no—
249C22×C59—162no—
250C22×C59#39211yescharacteristic
251C4×C59—21no—
252C4×C59—21no—
253C4×C59—21no—
254C4×C59—42no—
255C4×C59—81no—
256C4×C59—81no—
257C4×C59—82no—
258C4×C59—81no—
259C4×C59B(4,𝔽2)11yescharacteristic
260C4×C59B(4,𝔽2)11yescharacteristic
261#3200C2×C5911yescharacteristic, complemented
262#3028C2×C5911yescharacteristic, complemented
263#3021C2×C5911yescharacteristic
264#2135C2×C5911yescharacteristic, complemented
265#2122C2×C5911yescharacteristic, complemented
266#1791C2×C5911yescharacteristic, complemented
267#1395C2×C5911yescharacteristic, complemented
268C2×C59—21no—
269C2×C59—41no—
270C2×C59—41no—
271C2×C59—41no—
272C2×C59—82no—
273C2×C59—161no—
274C2×C59#202111yescharacteristic
275#585C22×C5911yescharacteristic
276C2×AGL(1,ℤ/8)—21no—
277#536—21no—
278#536C22×C5911yescharacteristic
279#535C22×C5911yescharacteristic
280#485—21no—
281#482—21no—
282#468—21no—
283#465—21no—
284#458—22no—
285#458—22no—
286#454—22no—
287#453—22no—
288#450—21no—
289#448—21no—
290#421—21no—
291#413—21no—
292#413C22×C5911yescharacteristic
293#413C22×C5911yescharacteristic
294#410—21no—
295#410C22×C5911yescharacteristic
296#408—21no—
297#408C22×C5911yescharacteristic
298#356—22no—
299AΣL(1,𝔽16)—22no—
300C59#1335411yescharacteristic, direct product factor, Sylow
3012−1+4—21no—
3022−1+4—41no—
303H(5,𝔽2)—21no—
304H(5,𝔽2)—21no—
305H(5,𝔽2)—41no—
306H(5,𝔽2)D4×C5912yes—
307C2×#41—21no—
308C2×#41—21no—
309C2×#41—21no—
310C2×#41—42no—
311C2×#41C23×C5911yescharacteristic
312C22×Q8D4×C5911yescharacteristic
313C22×D4D4×C5911yescharacteristic
314AGL(1,ℤ/8)—24no—
315AGL(1,ℤ/8)—42no—
316C2×SD16—21no—
317C2×SD16—21no—
318C2×SD16—41no—
319C2×D8—41no—
320C2×M16—21no—
321C2×M16—21no—
322C2×M16D4×C5911yescharacteristic
323C22×C8D4×C5911yescharacteristic
324C4⋊Q8—21no—
325C4⋊Q8—41no—
326#127—21no—
327#124—21no—
328#124—21no—
329#124—41no—
330#123—24no—
331#123—21no—
332#123—21no—
333C22⋊Q8—21no—
334C22⋊Q8—21no—
335C22⋊Q8—42no—
336#121—21no—
337#121—21no—
338#120—21no—
339#120—21no—
340#117—21no—
341#117D4×C5911yescharacteristic
342#117D4×C5911yescharacteristic
343#107—21no—
344#106—21no—
345#104—24no—
346#103—21no—
347#103—21no—
348#102—21no—
349#102—21no—
350#101—21no—
351#101—41no—
352#101D4×C5912yes—
353#100—21no—
354#100—21no—
355#100—41no—
356#99—24no—
357#99—21no—
358#99D4×C5912yes—
359C22⋊C8—21no—
360C22⋊C8—21no—
361C22⋊C8—42no—
362C24—21no—
363#41—22no—
364#41—22no—
365#41—41no—
366#41—41no—
367#41—42no—
368#41—41no—
369#41—41no—
370#41—41no—
371#41—41no—
372#41—41no—
373#41—42no—
374#41—42no—
375#41—42no—
376#41—41no—
377#41—41no—
378#41—41no—
379C2×Q8—21no—
380C2×Q8—21no—
381C2×Q8—21no—
382C2×Q8—21no—
383C2×Q8—21no—
384C2×Q8—41no—
385C2×Q8—41no—
386C2×Q8—42no—
387C2×Q8C2×D4×C5911yescharacteristic
388C2×D4—21no—
389C2×D4—21no—
390C2×D4—21no—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—21no—
394C2×D4—21no—
395C2×D4—41no—
396C2×D4—41no—
397C2×D4—42no—
398C2×D4—42no—
399C2×D4—42no—
400C2×D4—41no—
401C2×D4—41no—
402C2×D4—41no—
403C2×D4C2×D4×C5911yescharacteristic
404C2×D4C2×D4×C5911yescharacteristic
405C2×D4#41×C5911yescharacteristic
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—41no—
410C22×C4C2×D4×C5911yescharacteristic
411C22×C4C2×D4×C5911yescharacteristic
412C22×C4C2×D4×C5911yescharacteristic
413SD16—44no—
414SD16—81no—
415SD16—81no—
416D8—44no—
417M16—22no—
418M16—41no—
419M16—41no—
420M16—41no—
421M16—41no—
422C2×C8—21no—
423C2×C8—21no—
424C2×C8—41no—
425C2×C8—41no—
426C2×C8—41no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4—41no—
430C4⋊C4—44no—
431C4⋊C4—41no—
432C4⋊C4—41no—
433C22⋊C4—21no—
434C22⋊C4—22no—
435C22⋊C4—21no—
436C22⋊C4—44no—
437C22⋊C4—41no—
438C22⋊C4—41no—
439C22⋊C4—41no—
440C22⋊C4—41no—
441C42—21no—
442C42—21no—
443C42—41no—
444C23—22no—
445C23—22no—
446C23—41no—
447C23—41no—
448C23—81no—
449C23—161no—
450C23#120×C5911yescharacteristic
451C23#121×C5911yescharacteristic
452C23#121×C5911yescharacteristic
453Q8—22no—
454Q8—41no—
455Q8—41no—
456Q8—41no—
457Q8—41no—
458Q8—41no—
459Q8—41no—
460Q8—81no—
461D4—22no—
462D4—41no—
463D4—41no—
464D4—41no—
465D4—41no—
466D4—41no—
467D4—42no—
468D4—42no—
469D4—42no—
470D4—42no—
471D4—42no—
472D4—41no—
473D4—41no—
474D4—41no—
475D4—161no—
476D4—161no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—41no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4#120×C5911yescharacteristic
496C2×C4#120×C5911yescharacteristic
497C2×C4#121×C5911yescharacteristic
498C2×C4#127×C5911yescharacteristic
499C8—42no—
500C8—41no—
501C8—41no—
502C8—81no—
503C22—21no—
504C22—21no—
505C22—21no—
506C22—41no—
507C22—41no—
508C22—41no—
509C22—42no—
510C22—81no—
511C22—81no—
512C22—82no—
513C22—82no—
514C22—81no—
515C22—161no—
516C22—162no—
517C22C59×#39211yescharacteristic
518C4—21no—
519C4—21no—
520C4—21no—
521C4—42no—
522C4—81no—
523C4—81no—
524C4—82no—
525C4—81no—
526C4C59×B(4,𝔽2)11yescharacteristic
527C4C59×B(4,𝔽2)11yescharacteristic
528C2—21no—
529C2—41no—
530C2—41no—
531C2—41no—
532C2—82no—
533C2—161no—
534C2C59×#202111yescharacteristic
subgroupG/H
center, socleC2×C59#2021
commutator, Frattini, cosocleC2×#41C23×C59
prime radicalC59×#3021C2
LCSG ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C59 ◁ C2×C4×C59 ◁ C2×#41×C59 ◁ G

Complements

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

NH ≅ G/Ntype
1. C59×#3200C2semidirect
2. C59×#3028C2semidirect
4. C59×#2135C2semidirect
5. C59×#2122C2semidirect
6. C59×#1791C2semidirect
7. C59×#1395C2semidirect
235. #13354C59direct (N × H)
261. #3200C2×C59semidirect
262. #3028C2×C59semidirect
264. #2135C2×C59semidirect
265. #2122C2×C59semidirect
266. #1791C2×C59semidirect
267. #1395C2×C59semidirect
300. C59#13354direct (N × H)

Maximal subgroups

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

subgroupindex
C59×#32002
C59×#30282
C59×#30212
C59×#21352
C59×#21222
C59×#17912
C59×#13952
#1335459