Finite Group Databasebeta

← C59×#23905 #28436014

Subgroups — C59×#23905 #28436014

Order 15104.

1744 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C59×#3485C211yesmaximal, characteristic
2C2×C59×#536C211yesmaximal, characteristic, complemented
3C2×C59×#458C212yesmaximal, complemented
4C59×#2614C212yesmaximal, complemented
5C59×#2022C214yesmaximal, complemented
6C59×#1899C212yesmaximal, complemented
7C59×#1890C212yesmaximal, complemented
8C59×#1782C211yesmaximal, characteristic, complemented
9C2×H(5,𝔽2)×C59C2211yescharacteristic
10C22×#41×C59C2211yescharacteristic
11C23×Q8×C59C2211yescharacteristic, complemented
12C59×#547—24no—
13C59×#546—24no—
14C59×#536—21no—
15C59×#536—28no—
16C59×#534—21no—
17C59×#534C2212yes—
18C2×C4⋊Q8×C59—21no—
19C2×#124×C59—21no—
20C59×#525—21no—
21C2×#123×C59—21no—
22C2×#123×C59C2214yes—
23C2×#123×C59C2212yes—
24C2×#123×C59C2212yes—
25C2×#123×C59C2212yescomplemented
26C2×C22⋊Q8×C59—21no—
27C2×C22⋊Q8×C59—21no—
28C59×#518C2211yescharacteristic
29C2×#117×C59C2211yescharacteristic, complemented
30C59×#458—28no—
31C2×#99×C59C2214yes—
32C2×#99×C59C2212yes—
33C2×#99×C59C2212yescomplemented
34C59×#395—24no—
35C59×#393—24no—
36C59×#381C2214yescomplemented
37C59×#381C2212yescomplemented
38C59×#342C2214yescomplemented
39H(5,𝔽2)×C59—24no—
40H(5,𝔽2)×C59—42no—
41C2×#41×C59—24no—
42C2×#41×C59—21no—
43C2×#41×C59—24no—
44C2×#41×C59—22no—
45C2×#41×C59—42no—
46C22×Q8×C59—21no—
47C22×Q8×C59—21no—
48C22×Q8×C59—24no—
49C22×Q8×C59—24no—
50C22×Q8×C59—42no—
51C22×Q8×C59D411yescharacteristic
52C22×D4×C59—21no—
53C22×D4×C59—22no—
54C22×D4×C59D411yescharacteristic
55C22×D4×C59C2312yes—
56C22×D4×C59C2311yescharacteristic
57C22×D4×C59C2311yescharacteristic
58C23×C4×C59D411yescharacteristic
59C23×C4×C59D411yescharacteristic
60C23×C4×C59C2312yes—
61C23×C4×C59C2311yescharacteristic
62C4⋊Q8×C59—28no—
63C4⋊Q8×C59—22no—
64#124×C59—28no—
65#124×C59—22no—
66#123×C59—24no—
67#123×C59—22no—
68#123×C59—24no—
69#123×C59—22no—
70#123×C59—22no—
71#123×C59—22no—
72#123×C59—24no—
73#123×C59—24no—
74#123×C59—22no—
75#123×C59—24no—
76#123×C59—28no—
77#123×C59—44no—
78#123×C59D418yescomplemented
79C22⋊Q8×C59—28no—
80C22⋊Q8×C59—28no—
81C22⋊Q8×C59—22no—
82C22⋊Q8×C59—24no—
83#121×C59—22no—
84#121×C59—24no—
85#121×C59—24no—
86#120×C59—22no—
87#120×C59—22no—
88#120×C59—22no—
89C4×D4×C59—24no—
90C4×D4×C59—24no—
91#117×C59—21no—
92#117×C59—24no—
93C2×C4⋊C4×C59—22no—
94C2×C4⋊C4×C59—21no—
95C2×C4⋊C4×C59—22no—
96C2×C4⋊C4×C59—22no—
97C2×C4⋊C4×C59—24no—
98C2×C4⋊C4×C59D414yes—
99C2×C22⋊C4×C59—22no—
100C2×C22⋊C4×C59—21no—
101C2×C22⋊C4×C59—22no—
102C2×C22⋊C4×C59—22no—
103C2×C22⋊C4×C59—24no—
104C2×C22⋊C4×C59D414yes—
105C2×C22⋊C4×C59C2314yes—
106C2×C22⋊C4×C59C2312yes—
107C2×C22⋊C4×C59C2312yes—
108C2×C42×C59—21no—
109#99×C59—24no—
110#99×C59—28no—
111#99×C59—42no—
112#95×C59—28no—
113#95×C59—24no—
114C24×C59—21no—
115C24×C59C2×D412yes—
116C24×C59C2×D411yescharacteristic
117C24×C59C2×D411yescharacteristic
118C24×C59C2411yescharacteristic
119#41×C59—416no—
120#41×C59—48no—
121#41×C59—42no—
122#41×C59—42no—
123C2×Q8×C59—24no—
124C2×Q8×C59—21no—
125C2×Q8×C59—24no—
126C2×Q8×C59—22no—
127C2×Q8×C59—22no—
128C2×Q8×C59—41no—
129C2×Q8×C59—416no—
130C2×Q8×C59—42no—
131C2×Q8×C59—42no—
132C2×Q8×C59—42no—
133C2×Q8×C59—42no—
134C2×D4×C59—22no—
135C2×D4×C59—22no—
136C2×D4×C59—22no—
137C2×D4×C59—21no—
138C2×D4×C59—22no—
139C2×D4×C59—24no—
140C2×D4×C59—22no—
141C2×D4×C59—22no—
142C2×D4×C59—22no—
143C2×D4×C59—22no—
144C2×D4×C59—22no—
145C2×D4×C59—24no—
146C2×D4×C59—24no—
147C2×D4×C59—22no—
148C2×D4×C59—41no—
149C2×D4×C59—48no—
150C2×D4×C59—42no—
151C2×D4×C59—42no—
152C2×D4×C59C2×D414yes—
153C22×C4×C59—22no—
154C22×C4×C59—22no—
155C22×C4×C59—21no—
156C22×C4×C59—22no—
157C22×C4×C59—22no—
158C22×C4×C59—24no—
159C22×C4×C59—22no—
160C22×C4×C59—22no—
161C22×C4×C59—22no—
162C22×C4×C59—24no—
163C22×C4×C59—22no—
164C22×C4×C59—22no—
165C22×C4×C59—22no—
166C22×C4×C59—22no—
167C22×C4×C59—24no—
168C22×C4×C59—41no—
169C22×C4×C59—41no—
170C22×C4×C59C2×D412yes—
171C22×C4×C59C2×D411yescharacteristic
172C22×C4×C59C2×D414yes—
173C22×C4×C59C2×D414yes—
174C22×C4×C59C2×D411yescharacteristic
175C22×C4×C59C2×D411yescharacteristic
176C22×C4×C59C2×D411yescharacteristic
177C22×C4×C59C2×D414yes—
178C4⋊C4×C59—24no—
179C4⋊C4×C59—24no—
180C4⋊C4×C59—28no—
181C4⋊C4×C59—22no—
182C4⋊C4×C59—24no—
183C4⋊C4×C59—28no—
184C4⋊C4×C59—24no—
185C4⋊C4×C59—48no—
186C22⋊C4×C59—22no—
187C22⋊C4×C59—24no—
188C22⋊C4×C59—28no—
189C22⋊C4×C59—24no—
190C22⋊C4×C59—28no—
191C22⋊C4×C59—22no—
192C22⋊C4×C59—22no—
193C22⋊C4×C59—22no—
194C22⋊C4×C59—24no—
195C22⋊C4×C59—24no—
196C22⋊C4×C59—24no—
197C22⋊C4×C59—24no—
198C22⋊C4×C59—48no—
199C22⋊C4×C59C2×D414yes—
200C42×C59—24no—
201C42×C59—22no—
202C23×C59—21no—
203C23×C59—21no—
204C23×C59—21no—
205C23×C59—22no—
206C23×C59—21no—
207C23×C59—21no—
208C23×C59—24no—
209C23×C59—42no—
210C23×C59—42no—
211C23×C59—42no—
212C23×C59—42no—
213C23×C59—44no—
214C23×C59—82no—
215C23×C59#12012yes—
216C23×C59C22×D411yescharacteristic
217C23×C59C22×D412yes—
218C23×C59C22×D412yes—
219C23×C59H(5,𝔽2)12yes—
220Q8×C59—44no—
221Q8×C59—48no—
222Q8×C59—48no—
223Q8×C59—42no—
224D4×C59—28no—
225D4×C59—44no—
226D4×C59—42no—
227D4×C59—44no—
228D4×C59—42no—
229D4×C59—42no—
230D4×C59—44no—
231D4×C59—44no—
232D4×C59—84no—
233C2×C4×C59—21no—
234C2×C4×C59—21no—
235C2×C4×C59—21no—
236C2×C4×C59—21no—
237C2×C4×C59—21no—
238C2×C4×C59—21no—
239C2×C4×C59—28no—
240C2×C4×C59—24no—
241C2×C4×C59—24no—
242C2×C4×C59—21no—
243C2×C4×C59—24no—
244C2×C4×C59—21no—
245C2×C4×C59—22no—
246C2×C4×C59—21no—
247C2×C4×C59—28no—
248C2×C4×C59—24no—
249C2×C4×C59—44no—
250C2×C4×C59—44no—
251C2×C4×C59—44no—
252C2×C4×C59—44no—
253C2×C4×C59—42no—
254C2×C4×C59—42no—
255C2×C4×C59—42no—
256C2×C4×C59—44no—
257C2×C4×C59—44no—
258C2×C4×C59—48no—
259C2×C4×C59—44no—
260C2×C4×C59—82no—
261C2×C4×C59#12012yes—
262C2×C4×C59#12012yes—
263C2×C4×C59#12012yes—
264#23905C5911yesmaximal, characteristic, direct product factor, Sylow
265C22×C59—24no—
266C22×C59—22no—
267C22×C59—22no—
268C22×C59—42no—
269C22×C59—42no—
270C22×C59—42no—
271C22×C59—42no—
272C22×C59—42no—
273C22×C59—41no—
274C22×C59—42no—
275C22×C59—42no—
276C22×C59—44no—
277C22×C59—81no—
278C22×C59—84no—
279C22×C59—81no—
280C22×C59—162no—
281C22×C59C2×#12011yescharacteristic
282C22×C59C2×#12011yescharacteristic
283C22×C59#53411yescharacteristic
284C22×C59D4214yes—
285C4×C59—22no—
286C4×C59—22no—
287C4×C59—22no—
288C4×C59—44no—
289C4×C59—42no—
290C4×C59—41no—
291C4×C59—44no—
292C4×C59—44no—
293C4×C59—84no—
294#3485C2×C5911yescharacteristic
295C2×#536C2×C5911yescharacteristic, complemented
296C2×#458C2×C5912yescomplemented
297#2614C2×C5912yescomplemented
298#2022C2×C5914yescomplemented
299#1899C2×C5912yescomplemented
300#1890C2×C5912yescomplemented
301#1782C2×C5911yescharacteristic, complemented
302C2×C59—21no—
303C2×C59—24no—
304C2×C59—21no—
305C2×C59—42no—
306C2×C59—81no—
307C2×C59—82no—
308C2×C59#240311yescharacteristic
309C2×C59#302812yes—
310C2×H(5,𝔽2)C22×C5911yescharacteristic
311C22×#41C22×C5911yescharacteristic
312C23×Q8C22×C5911yescharacteristic, complemented
313#547—24no—
314#546—24no—
315#536—21no—
316#536—28no—
317#534—21no—
318#534C22×C5912yes—
319C2×C4⋊Q8—21no—
320C2×#124—21no—
321#525—21no—
322C2×#123—21no—
323C2×#123C22×C5914yes—
324C2×#123C22×C5912yes—
325C2×#123C22×C5912yes—
326C2×#123C22×C5912yescomplemented
327C2×C22⋊Q8—21no—
328C2×C22⋊Q8—21no—
329#518C22×C5911yescharacteristic
330C2×#117C22×C5911yescharacteristic, complemented
331#458—28no—
332C2×#99C22×C5912yes—
333C2×#99C22×C5914yes—
334C2×#99C22×C5912yescomplemented
335#395—24no—
336#393—24no—
337#381C22×C5914yescomplemented
338#381C22×C5912yescomplemented
339#342C22×C5914yescomplemented
340C59#2390511yescharacteristic, direct product factor, Sylow
341H(5,𝔽2)—24no—
342H(5,𝔽2)—42no—
343C2×#41—24no—
344C2×#41—21no—
345C2×#41—24no—
346C2×#41—22no—
347C2×#41—42no—
348C22×Q8—21no—
349C22×Q8—21no—
350C22×Q8—24no—
351C22×Q8—24no—
352C22×Q8—42no—
353C22×Q8D4×C5911yescharacteristic
354C22×D4—21no—
355C22×D4—22no—
356C22×D4D4×C5911yescharacteristic
357C22×D4C23×C5912yes—
358C22×D4C23×C5911yescharacteristic
359C22×D4C23×C5911yescharacteristic
360C23×C4D4×C5911yescharacteristic
361C23×C4D4×C5911yescharacteristic
362C23×C4C23×C5912yes—
363C23×C4C23×C5911yescharacteristic
364C4⋊Q8—28no—
365C4⋊Q8—22no—
366#124—28no—
367#124—22no—
368#123—24no—
369#123—22no—
370#123—24no—
371#123—22no—
372#123—22no—
373#123—22no—
374#123—24no—
375#123—24no—
376#123—22no—
377#123—24no—
378#123—28no—
379#123—44no—
380#123D4×C5918yescomplemented
381C22⋊Q8—28no—
382C22⋊Q8—28no—
383C22⋊Q8—22no—
384C22⋊Q8—24no—
385#121—22no—
386#121—24no—
387#121—24no—
388#120—22no—
389#120—22no—
390#120—22no—
391C4×D4—24no—
392C4×D4—24no—
393#117—21no—
394#117—24no—
395C2×C4⋊C4—22no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—22no—
398C2×C4⋊C4—22no—
399C2×C4⋊C4—24no—
400C2×C4⋊C4D4×C5914yes—
401C2×C22⋊C4—22no—
402C2×C22⋊C4—21no—
403C2×C22⋊C4—22no—
404C2×C22⋊C4—22no—
405C2×C22⋊C4—24no—
406C2×C22⋊C4D4×C5914yes—
407C2×C22⋊C4C23×C5914yes—
408C2×C22⋊C4C23×C5912yes—
409C2×C22⋊C4C23×C5912yes—
410C2×C42—21no—
411#99—24no—
412#99—28no—
413#99—42no—
414#95—28no—
415#95—24no—
416C24—21no—
417C24C2×D4×C5912yes—
418C24C2×D4×C5911yescharacteristic
419C24C2×D4×C5911yescharacteristic
420C24C24×C5911yescharacteristic
421#41—416no—
422#41—48no—
423#41—42no—
424#41—42no—
425C2×Q8—24no—
426C2×Q8—21no—
427C2×Q8—24no—
428C2×Q8—22no—
429C2×Q8—22no—
430C2×Q8—41no—
431C2×Q8—416no—
432C2×Q8—42no—
433C2×Q8—42no—
434C2×Q8—42no—
435C2×Q8—42no—
436C2×D4—22no—
437C2×D4—22no—
438C2×D4—22no—
439C2×D4—21no—
440C2×D4—22no—
441C2×D4—24no—
442C2×D4—22no—
443C2×D4—22no—
444C2×D4—22no—
445C2×D4—22no—
446C2×D4—22no—
447C2×D4—24no—
448C2×D4—24no—
449C2×D4—22no—
450C2×D4—41no—
451C2×D4—48no—
452C2×D4—42no—
453C2×D4—42no—
454C2×D4C2×D4×C5914yes—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—21no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—24no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—24no—
470C22×C4—41no—
471C22×C4—41no—
472C22×C4C2×D4×C5912yes—
473C22×C4C2×D4×C5911yescharacteristic
474C22×C4C2×D4×C5914yes—
475C22×C4C2×D4×C5914yes—
476C22×C4C2×D4×C5911yescharacteristic
477C22×C4C2×D4×C5911yescharacteristic
478C22×C4C2×D4×C5911yescharacteristic
479C22×C4C2×D4×C5914yes—
480C4⋊C4—24no—
481C4⋊C4—24no—
482C4⋊C4—28no—
483C4⋊C4—22no—
484C4⋊C4—24no—
485C4⋊C4—28no—
486C4⋊C4—24no—
487C4⋊C4—48no—
488C22⋊C4—22no—
489C22⋊C4—24no—
490C22⋊C4—28no—
491C22⋊C4—24no—
492C22⋊C4—28no—
493C22⋊C4—22no—
494C22⋊C4—22no—
495C22⋊C4—22no—
496C22⋊C4—24no—
497C22⋊C4—24no—
498C22⋊C4—24no—
499C22⋊C4—24no—
500C22⋊C4—48no—
501C22⋊C4C2×D4×C5914yes—
502C42—24no—
503C42—22no—
504C23—21no—
505C23—21no—
506C23—21no—
507C23—22no—
508C23—21no—
509C23—21no—
510C23—24no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—44no—
516C23—82no—
517C23#120×C5912yes—
518C23C22×D4×C5911yescharacteristic
519C23C22×D4×C5912yes—
520C23C22×D4×C5912yes—
521C23H(5,𝔽2)×C5912yes—
522Q8—44no—
523Q8—48no—
524Q8—48no—
525Q8—42no—
526D4—28no—
527D4—44no—
528D4—42no—
529D4—44no—
530D4—42no—
531D4—42no—
532D4—44no—
533D4—44no—
534D4—84no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—28no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—21no—
545C2×C4—24no—
546C2×C4—22no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—28no—
550C2×C4—24no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—48no—
561C2×C4—44no—
562C2×C4—82no—
563C2×C4#120×C5912yes—
564C2×C4#120×C5912yes—
565C2×C4#120×C5912yes—
566C22—24no—
567C22—22no—
568C22—22no—
569C22—42no—
570C22—42no—
571C22—42no—
572C22—42no—
573C22—42no—
574C22—42no—
575C22—44no—
576C22—41no—
577C22—42no—
578C22—81no—
579C22—84no—
580C22—81no—
581C22—162no—
582C22C2×#120×C5911yescharacteristic
583C22C2×#120×C5911yescharacteristic
584C22C59×#53411yescharacteristic
585C22D42×C5914yes—
586C4—22no—
587C4—22no—
588C4—22no—
589C4—44no—
590C4—44no—
591C4—42no—
592C4—41no—
593C4—44no—
594C4—84no—
595C2—21no—
596C2—24no—
597C2—21no—
598C2—42no—
599C2—81no—
600C2—82no—
601C2C59×#240311yescharacteristic
602C2C59×#302812yes—
subgroupG/H
center, socleC22×C59C2×#120
commutator, Frattini, cosocleC24C24×C59
prime radicalC59×#3485C2
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22×C59 ◁ C23×C4×C59 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C59×#536C2semidirect
3. C2×C59×#458C2semidirect
4. C59×#2614C2semidirect
5. C59×#2022C2semidirect
6. C59×#1899C2semidirect
7. C59×#1890C2semidirect
8. C59×#1782C2semidirect
11. C23×Q8×C59C22semidirect
25. C2×#123×C59C22semidirect
29. C2×#117×C59C22semidirect
33. C2×#99×C59C22semidirect
36. C59×#381C22semidirect
37. C59×#381C22semidirect
38. C59×#342C22semidirect
78. #123×C59D4semidirect
264. #23905C59direct (N × H)
295. C2×#536C2×C59semidirect
296. C2×#458C2×C59semidirect
297. #2614C2×C59semidirect
298. #2022C2×C59semidirect
299. #1899C2×C59semidirect
300. #1890C2×C59semidirect
301. #1782C2×C59semidirect
312. C23×Q8C22×C59semidirect
326. C2×#123C22×C59semidirect
330. C2×#117C22×C59semidirect
334. C2×#99C22×C59semidirect
337. #381C22×C59semidirect
338. #381C22×C59semidirect
339. #342C22×C59semidirect
340. C59#23905direct (N × H)
380. #123D4×C59semidirect

Maximal subgroups

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

subgroupindex
C59×#34852
C2×C59×#5362
C2×C59×#4582
C59×#26142
C59×#20222
C59×#18992
C59×#18902
C59×#17822
#2390559