Finite Group Databasebeta

← C59×#18204 #28430313

Subgroups — C59×#18204 #28430313

Order 15104.

1002 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×#3501C211yesmaximal, characteristic
2C2×C59×#483C212yesmaximal, complemented
3C59×#2486C211yesmaximal, characteristic
4C59×#1611C214yesmaximal, complemented
5C59×#1608C214yesmaximal, complemented
6C59×#1520C212yesmaximal, complemented
7C59×#1459C211yesmaximal, characteristic
8C59×#554—24no—
9C59×#554—22no—
10D4×Q8×C59—22no—
11C59×#544C2212yes—
12C59×#543—21no—
13C59×#543C2212yes—
14C59×#533—21no—
15C2×#123×C59—21no—
16C2×C22⋊Q8×C59—21no—
17C2×C22⋊Q8×C59C2212yes—
18C59×#518—21no—
19C2×C4×D4×C59C2211yescharacteristic
20C22×C4⋊C4×C59C2211yescharacteristic
21C59×#483—24no—
22C59×#483C2218yes—
23C2×#106×C59—22no—
24C2×#103×C59—22no—
25C2×C22⋊C8×C59C2212yes—
26C59×#398C2212yes—
27C59×#397C2212yes—
28C59×#389—21no—
29C59×#381—21no—
30C59×#380—21no—
31C59×#380C2211yescharacteristic
32C59×#328C2218yes—
33C59×#324C2212yes—
34C59×#323C2212yes—
35C22×Q8×C59—21no—
36C22×Q8×C59D412yes—
37C22×D4×C59D411yescharacteristic
38C23×C4×C59—21no—
39C23×C4×C59D411yescharacteristic
40C23×C4×C59C2311yescharacteristic
41C22×C8×C59—22no—
42C4⋊Q8×C59—21no—
43C4⋊Q8×C59—21no—
44C4⋊Q8×C59—24no—
45#125×C59—22no—
46#125×C59—24no—
47#123×C59—44no—
48C22⋊Q8×C59—22no—
49C22⋊Q8×C59—22no—
50C22⋊Q8×C59—24no—
51C22⋊Q8×C59—24no—
52C22⋊Q8×C59—24no—
53C22⋊Q8×C59—22no—
54C22⋊Q8×C59—22no—
55C22⋊Q8×C59—22no—
56C22⋊Q8×C59—22no—
57C22⋊Q8×C59—24no—
58C22⋊Q8×C59D414yes—
59C22⋊Q8×C59C2314yes—
60C4×Q8×C59—22no—
61C4×Q8×C59—22no—
62C4×D4×C59—24no—
63C4×D4×C59—24no—
64C4×D4×C59—22no—
65C4×D4×C59—22no—
66#117×C59—22no—
67C2×C4⋊C4×C59—21no—
68C2×C4⋊C4×C59—22no—
69C2×C4⋊C4×C59—22no—
70C2×C4⋊C4×C59—22no—
71C2×C4⋊C4×C59—22no—
72C2×C4⋊C4×C59—21no—
73C2×C4⋊C4×C59—21no—
74C2×C4⋊C4×C59—22no—
75C2×C4⋊C4×C59D412yes—
76C2×C4⋊C4×C59C2312yes—
77C2×C4⋊C4×C59C2311yescharacteristic
78C2×C22⋊C4×C59—22no—
79C2×C22⋊C4×C59—22no—
80C2×C22⋊C4×C59—22no—
81C2×C22⋊C4×C59—21no—
82C2×C22⋊C4×C59D411yescharacteristic
83C2×C22⋊C4×C59C2311yescharacteristic
84C2×C42×C59D411yescharacteristic
85#106×C59—28no—
86#103×C59—28no—
87C22⋊C8×C59—22no—
88C22⋊C8×C59C2314yes—
89#95×C59—22no—
90#95×C59—22no—
91#95×C59C2312yes—
92C24×C59C2×D411yescharacteristic
93C24×C59C2×D411yescharacteristic
94C2×Q8×C59—22no—
95C2×Q8×C59—22no—
96C2×Q8×C59—22no—
97C2×Q8×C59—44no—
98C2×Q8×C59—82no—
99C2×Q8×C59C2×D414yes—
100C2×D4×C59—22no—
101C2×D4×C59—22no—
102C2×D4×C59SD1614yes—
103C22×C4×C59—22no—
104C22×C4×C59—22no—
105C22×C4×C59—22no—
106C22×C4×C59—22no—
107C22×C4×C59—21no—
108C22×C4×C59—21no—
109C22×C4×C59—22no—
110C22×C4×C59—22no—
111C22×C4×C59—42no—
112C22×C4×C59—42no—
113C22×C4×C59—42no—
114C22×C4×C59—42no—
115C22×C4×C59—42no—
116C22×C4×C59C2×D411yescharacteristic
117C22×C4×C59C2×D412yes—
118C22×C4×C59C2×D411yescharacteristic
119C22×C4×C59C2×D411yescharacteristic
120C22×C4×C59C2×D411yescharacteristic
121C22×C4×C59C2×D412yes—
122C22×C4×C59C2411yescharacteristic
123C2×C8×C59—24no—
124C2×C8×C59—44no—
125C4⋊C4×C59—22no—
126C4⋊C4×C59—24no—
127C4⋊C4×C59—22no—
128C4⋊C4×C59—22no—
129C4⋊C4×C59—24no—
130C4⋊C4×C59—22no—
131C4⋊C4×C59—24no—
132C4⋊C4×C59—22no—
133C4⋊C4×C59—24no—
134C4⋊C4×C59—22no—
135C4⋊C4×C59—24no—
136C4⋊C4×C59—42no—
137C4⋊C4×C59—41no—
138C4⋊C4×C59—41no—
139C4⋊C4×C59—44no—
140C4⋊C4×C59C2×D414yes—
141C22⋊C4×C59—24no—
142C22⋊C4×C59—24no—
143C22⋊C4×C59—22no—
144C22⋊C4×C59—24no—
145C22⋊C4×C59—22no—
146C22⋊C4×C59—42no—
147C22⋊C4×C59—44no—
148C42×C59—24no—
149C42×C59—21no—
150C42×C59—21no—
151C23×C59—21no—
152C23×C59—21no—
153C23×C59—41no—
154C23×C59—41no—
155C23×C59—41no—
156C23×C59—41no—
157C23×C59#12012yes—
158C23×C59C2×SD1612yes—
159C23×C59C2×SD1612yes—
160C23×C59#13712yes—
161C23×C59C22×D411yescharacteristic
162Q8×C59—44no—
163Q8×C59—44no—
164Q8×C59—82no—
165Q8×C59—82no—
166D4×C59—28no—
167C2×C4×C59—21no—
168C2×C4×C59—21no—
169C2×C4×C59—22no—
170C2×C4×C59—24no—
171C2×C4×C59—21no—
172C2×C4×C59—24no—
173C2×C4×C59—24no—
174C2×C4×C59—21no—
175C2×C4×C59—42no—
176C2×C4×C59—41no—
177C2×C4×C59—42no—
178C2×C4×C59—41no—
179C2×C4×C59—41no—
180C2×C4×C59—42no—
181C2×C4×C59—42no—
182C2×C4×C59—42no—
183C2×C4×C59—42no—
184C2×C4×C59—42no—
185C2×C4×C59—42no—
186C2×C4×C59—81no—
187C2×C4×C59—82no—
188C2×C4×C59—82no—
189C2×C4×C59—82no—
190C2×C4×C59—82no—
191C2×C4×C59—84no—
192C2×C4×C59#12011yescharacteristic
193C2×C4×C59#12011yescharacteristic
194C2×C4×C59C2×SD1612yes—
195C2×C4×C59C22×D412yes—
196C8×C59—44no—
197#18204C5911yesmaximal, characteristic, direct product factor, Sylow
198C22×C59—21no—
199C22×C59—22no—
200C22×C59—21no—
201C22×C59—22no—
202C22×C59—22no—
203C22×C59—42no—
204C22×C59—42no—
205C22×C59—42no—
206C22×C59—41no—
207C22×C59—41no—
208C22×C59—81no—
209C22×C59#44814yes—
210C22×C59C2×#12011yescharacteristic
211C22×C59C22×SD1611yescharacteristic
212C22×C59C2×#13711yescharacteristic
213C4×C59—22no—
214C4×C59—22no—
215C4×C59—21no—
216C4×C59—41no—
217C4×C59—44no—
218C4×C59—41no—
219C4×C59—44no—
220C4×C59—82no—
221C4×C59—82no—
222C4×C59#45812yes—
223#3501C2×C5911yescharacteristic
224C2×#483C2×C5912yescomplemented
225#2486C2×C5911yescharacteristic
226#1611C2×C5914yescomplemented
227#1608C2×C5914yescomplemented
228#1520C2×C5912yescomplemented
229#1459C2×C5911yescharacteristic
230C2×C59—21no—
231C2×C59—21no—
232C2×C59—24no—
233C2×C59—42no—
234C2×C59C2×#44811yescharacteristic
235C2×C59#302211yescharacteristic
236C2×C59C2×#45811yescharacteristic
237#554—24no—
238#554—22no—
239D4×Q8—22no—
240#544C22×C5912yes—
241#543—21no—
242#543C22×C5912yes—
243#533—21no—
244C2×#123—21no—
245C2×C22⋊Q8—21no—
246C2×C22⋊Q8C22×C5912yes—
247#518—21no—
248C2×C4×D4C22×C5911yescharacteristic
249C22×C4⋊C4C22×C5911yescharacteristic
250#483—24no—
251#483C22×C5918yes—
252C2×#106—22no—
253C2×#103—22no—
254C2×C22⋊C8C22×C5912yes—
255#398C22×C5912yes—
256#397C22×C5912yes—
257#389—21no—
258#381—21no—
259#380—21no—
260#380C22×C5911yescharacteristic
261#328C22×C5918yes—
262#324C22×C5912yes—
263#323C22×C5912yes—
264C59#1820411yescharacteristic, direct product factor, Sylow
265C22×Q8—21no—
266C22×Q8D4×C5912yes—
267C22×D4D4×C5911yescharacteristic
268C23×C4—21no—
269C23×C4D4×C5911yescharacteristic
270C23×C4C23×C5911yescharacteristic
271C22×C8—22no—
272C4⋊Q8—21no—
273C4⋊Q8—21no—
274C4⋊Q8—24no—
275#125—22no—
276#125—24no—
277#123—44no—
278C22⋊Q8—22no—
279C22⋊Q8—22no—
280C22⋊Q8—24no—
281C22⋊Q8—24no—
282C22⋊Q8—24no—
283C22⋊Q8—22no—
284C22⋊Q8—22no—
285C22⋊Q8—22no—
286C22⋊Q8—22no—
287C22⋊Q8—24no—
288C22⋊Q8D4×C5914yes—
289C22⋊Q8C23×C5914yes—
290C4×Q8—22no—
291C4×Q8—22no—
292C4×D4—24no—
293C4×D4—24no—
294C4×D4—22no—
295C4×D4—22no—
296#117—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4D4×C5912yes—
306C2×C4⋊C4C23×C5912yes—
307C2×C4⋊C4C23×C5911yescharacteristic
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—21no—
312C2×C22⋊C4D4×C5911yescharacteristic
313C2×C22⋊C4C23×C5911yescharacteristic
314C2×C42D4×C5911yescharacteristic
315#106—28no—
316#103—28no—
317C22⋊C8—22no—
318C22⋊C8C23×C5914yes—
319#95—22no—
320#95—22no—
321#95C23×C5912yes—
322C24C2×D4×C5911yescharacteristic
323C24C2×D4×C5911yescharacteristic
324C2×Q8—22no—
325C2×Q8—22no—
326C2×Q8—22no—
327C2×Q8—44no—
328C2×Q8—82no—
329C2×Q8C2×D4×C5914yes—
330C2×D4—22no—
331C2×D4—22no—
332C2×D4SD16×C5914yes—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4C2×D4×C5911yescharacteristic
347C22×C4C2×D4×C5912yes—
348C22×C4C2×D4×C5911yescharacteristic
349C22×C4C2×D4×C5911yescharacteristic
350C22×C4C2×D4×C5911yescharacteristic
351C22×C4C2×D4×C5912yes—
352C22×C4C24×C5911yescharacteristic
353C2×C8—24no—
354C2×C8—44no—
355C4⋊C4—22no—
356C4⋊C4—24no—
357C4⋊C4—22no—
358C4⋊C4—22no—
359C4⋊C4—24no—
360C4⋊C4—22no—
361C4⋊C4—24no—
362C4⋊C4—22no—
363C4⋊C4—24no—
364C4⋊C4—22no—
365C4⋊C4—24no—
366C4⋊C4—42no—
367C4⋊C4—41no—
368C4⋊C4—41no—
369C4⋊C4—44no—
370C4⋊C4C2×D4×C5914yes—
371C22⋊C4—24no—
372C22⋊C4—24no—
373C22⋊C4—22no—
374C22⋊C4—24no—
375C22⋊C4—22no—
376C22⋊C4—42no—
377C22⋊C4—44no—
378C42—24no—
379C42—21no—
380C42—21no—
381C23—21no—
382C23—21no—
383C23—41no—
384C23—41no—
385C23—41no—
386C23—41no—
387C23#120×C5912yes—
388C23C2×SD16×C5912yes—
389C23C2×SD16×C5912yes—
390C23#137×C5912yes—
391C23C22×D4×C5911yescharacteristic
392Q8—44no—
393Q8—44no—
394Q8—82no—
395Q8—82no—
396D4—28no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—22no—
400C2×C4—24no—
401C2×C4—21no—
402C2×C4—24no—
403C2×C4—24no—
404C2×C4—21no—
405C2×C4—42no—
406C2×C4—41no—
407C2×C4—42no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—81no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4—82no—
421C2×C4—84no—
422C2×C4#120×C5911yescharacteristic
423C2×C4#120×C5911yescharacteristic
424C2×C4C2×SD16×C5912yes—
425C2×C4C22×D4×C5912yes—
426C8—44no—
427C22—21no—
428C22—22no—
429C22—21no—
430C22—22no—
431C22—22no—
432C22—42no—
433C22—42no—
434C22—42no—
435C22—41no—
436C22—41no—
437C22—81no—
438C22C59×#44814yes—
439C22C2×#120×C5911yescharacteristic
440C22C22×SD16×C5911yescharacteristic
441C22C2×#137×C5911yescharacteristic
442C4—22no—
443C4—22no—
444C4—21no—
445C4—41no—
446C4—44no—
447C4—41no—
448C4—44no—
449C4—82no—
450C4—82no—
451C4C59×#45812yes—
452C2—21no—
453C2—21no—
454C2—24no—
455C2—42no—
456C2C2×C59×#44811yescharacteristic
457C2C59×#302211yescharacteristic
458C2C2×C59×#45811yescharacteristic
subgroupG/H
center, socleC22×C59C2×#120
commutator, Frattini, cosocleC22×C4C24×C59
prime radicalC22×D4×C59D4
LCSG ▷ C22×C4 ▷ C22 ▷ 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×#483C2semidirect
4. C59×#1611C2semidirect
5. C59×#1608C2semidirect
6. C59×#1520C2semidirect
197. #18204C59direct (N × H)
224. C2×#483C2×C59semidirect
226. #1611C2×C59semidirect
227. #1608C2×C59semidirect
228. #1520C2×C59semidirect
264. C59#18204direct (N × H)

Maximal subgroups

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

subgroupindex
C59×#35012
C2×C59×#4832
C59×#24862
C59×#16112
C59×#16082
C59×#15202
C59×#14592
#1820459