Finite Group Databasebeta

← C59×#32898 #28445007

Subgroups — C59×#32898 #28445007

Order 15104.

4360 conjugacy classes of subgroups; only the 370 normal subgroups are shown.

normal subgroups only

This group has 4360 conjugacy classes of subgroups — too many to draw. Only its 370 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
1C22×H(5,𝔽2)×C59C211yesmaximal, characteristic, complemented
2C2×C59×B(4,𝔽2)C214yesmaximal, complemented
3C59×#2403C213yesmaximal, complemented
4C59×#1790C214yesmaximal, complemented
5C59×#1781C213yesmaximal, complemented
6C2×H(5,𝔽2)×C59—216no—
7C2×H(5,𝔽2)×C59—26no—
8C2×H(5,𝔽2)×C59C2214yes—
9C22×#41×C59—23no—
10C23×D4×C59—23no—
11C23×D4×C59C2213yes—
12C23×D4×C59C2211yescharacteristic, complemented
13C2×#120×C59—23no—
14C2×#120×C59C2216yescomplemented
15C59×B(4,𝔽2)—216no—
16C2×#99×C59C22112yescomplemented
17C59×#392—26no—
18C59×#386—23no—
19C59×#386C2216yescomplemented
20C59×#379C2213yescomplemented
21C25×C59—21no—
22C25×C59D413yescomplemented
23C25×C59C2311yescharacteristic
24C25×C59C2311yescharacteristic, complemented
25H(5,𝔽2)×C59—248no—
26H(5,𝔽2)×C59—248no—
27H(5,𝔽2)×C59—224no—
28H(5,𝔽2)×C59D4116yescomplemented
29C2×#41×C59—212no—
30C2×#41×C59—248no—
31C2×#41×C59—212no—
32C22×Q8×C59—21no—
33C22×D4×C59—23no—
34C22×D4×C59—248no—
35C22×D4×C59—212no—
36C22×D4×C59—26no—
37C22×D4×C59—212no—
38C22×D4×C59—26no—
39C22×D4×C59—224no—
40C22×D4×C59—23no—
41C22×D4×C59—23no—
42C22×D4×C59—26no—
43C22×D4×C59—44no—
44C22×D4×C59D416yescomplemented
45C22×D4×C59C2316yes—
46C23×C4×C59—23no—
47C23×C4×C59D413yescomplemented
48C23×C4×C59C2311yescharacteristic, complemented
49#120×C59—224no—
50#120×C59—412no—
51C2×C22⋊C4×C59—23no—
52C2×C22⋊C4×C59—26no—
53C2×C22⋊C4×C59—26no—
54C2×C22⋊C4×C59—23no—
55C2×C22⋊C4×C59C2316yescomplemented
56#99×C59—224no—
57#95×C59—26no—
58C24×C59—28no—
59C24×C59—24no—
60C24×C59—23no—
61C24×C59—23no—
62C24×C59—212no—
63C24×C59—24no—
64C24×C59—23no—
65C24×C59—24no—
66C24×C59—42no—
67C24×C59—42no—
68C24×C59—46no—
69C24×C59—81no—
70C24×C59—81no—
71C24×C59—83no—
72C24×C59—82no—
73C24×C59—82no—
74C24×C59C2×D413yes—
75C24×C59C2×D413yes—
76C24×C59C2×D416yes—
77C24×C59C2411yescharacteristic
78#41×C59—296no—
79#41×C59—248no—
80#41×C59—248no—
81C2×Q8×C59—28no—
82C2×Q8×C59—24no—
83C2×D4×C59—224no—
84C2×D4×C59—248no—
85C2×D4×C59—212no—
86C2×D4×C59—296no—
87C2×D4×C59—248no—
88C2×D4×C59—248no—
89C2×D4×C59—212no—
90C2×D4×C59—248no—
91C2×D4×C59—212no—
92C2×D4×C59—212no—
93C2×D4×C59—212no—
94C2×D4×C59—212no—
95C2×D4×C59—212no—
96C2×D4×C59—46no—
97C2×D4×C59—46no—
98C2×D4×C59—46no—
99C2×D4×C59—43no—
100C2×D4×C59—41no—
101C2×D4×C59—412no—
102C2×D4×C59—412no—
103C2×D4×C59—86no—
104C2×D4×C59C2×D4124yes—
105C22×C4×C59—224no—
106C22×C4×C59—212no—
107C22×C4×C59—23no—
108C22×C4×C59—23no—
109C22×C4×C59—212no—
110C22×C4×C59—23no—
111C22×C4×C59—26no—
112C22×C4×C59—42no—
113C22×C4×C59C2×D416yes—
114C22⋊C4×C59—212no—
115C22⋊C4×C59—412no—
116C22⋊C4×C59—46no—
117C22⋊C4×C59—412no—
118C22⋊C4×C59—83no—
119C23×C59—212no—
120C23×C59—212no—
121C23×C59—216no—
122C23×C59—26no—
123C23×C59—26no—
124C23×C59—26no—
125C23×C59—26no—
126C23×C59—26no—
127C23×C59—26no—
128C23×C59—41no—
129C23×C59—43no—
130C23×C59—41no—
131C23×C59—43no—
132C23×C59—43no—
133C23×C59—46no—
134C23×C59—46no—
135C23×C59—46no—
136C23×C59—43no—
137C23×C59—46no—
138C23×C59—46no—
139C23×C59—46no—
140C23×C59—41no—
141C23×C59—83no—
142C23×C59—83no—
143C23×C59—88no—
144C23×C59—88no—
145C23×C59—824no—
146C23×C59—816no—
147C23×C59—84no—
148C23×C59—84no—
149C23×C59#12013yes—
150C23×C59#12011yescharacteristic
151C23×C59#120112yescomplemented
152C23×C59C22×D413yes—
153C23×C59C22×D414yes—
154Q8×C59—216no—
155D4×C59—248no—
156D4×C59—248no—
157D4×C59—248no—
158D4×C59—412no—
159D4×C59—44no—
160D4×C59—824no—
161C2×C4×C59—212no—
162C2×C4×C59—212no—
163C2×C4×C59—248no—
164C2×C4×C59—26no—
165C2×C4×C59—26no—
166C2×C4×C59—26no—
167C2×C4×C59—43no—
168C2×C4×C59—41no—
169C2×C4×C59—46no—
170C2×C4×C59—46no—
171C2×C4×C59—46no—
172C2×C4×C59#120112yescomplemented
173#32898C5911yesmaximal, characteristic, direct product factor, Sylow
174C22×C59—21no—
175C22×C59—23no—
176C22×C59—23no—
177C22×C59—21no—
178C22×C59—23no—
179C22×C59—212no—
180C22×C59—23no—
181C22×C59—23no—
182C22×C59—44no—
183C22×C59—412no—
184C22×C59—412no—
185C22×C59—424no—
186C22×C59—412no—
187C22×C59—46no—
188C22×C59—46no—
189C22×C59—42no—
190C22×C59—86no—
191C22×C59—86no—
192C22×C59—86no—
193C22×C59—812no—
194C22×C59—816no—
195C22×C59—81no—
196C22×C59—81no—
197C22×C59B(4,𝔽2)12yescomplemented
198C22×C59C2×#12016yes—
199C22×C59C2×#12011yescharacteristic
200C4×C59—212no—
201C4×C59—42no—
202C4×C59—86no—
203C22×H(5,𝔽2)C2×C5911yescharacteristic, complemented
204C2×B(4,𝔽2)C2×C5914yescomplemented
205#2403C2×C5913yescomplemented
206#1790C2×C5914yescomplemented
207#1781C2×C5913yescomplemented
208C2×C59—26no—
209C2×C59—22no—
210C2×C59—43no—
211C2×C59—412no—
212C2×C59—82no—
213C2×C59#284611yescharacteristic
214C2×C59C2×B(4,𝔽2)11yescharacteristic
215C2×C59#302611yescharacteristic
216C2×H(5,𝔽2)—216no—
217C2×H(5,𝔽2)—26no—
218C2×H(5,𝔽2)C22×C5914yes—
219C22×#41—23no—
220C23×D4—23no—
221C23×D4C22×C5913yes—
222C23×D4C22×C5911yescharacteristic, complemented
223C2×#120—23no—
224C2×#120C22×C5916yescomplemented
225B(4,𝔽2)—216no—
226C2×#99C22×C59112yescomplemented
227#392—26no—
228#386—23no—
229#386C22×C5916yescomplemented
230#379C22×C5913yescomplemented
231C59#3289811yescharacteristic, direct product factor, Sylow
232C25—21no—
233C25D4×C5913yescomplemented
234C25C23×C5911yescharacteristic
235C25C23×C5911yescharacteristic, complemented
236H(5,𝔽2)—248no—
237H(5,𝔽2)—248no—
238H(5,𝔽2)—224no—
239H(5,𝔽2)D4×C59116yescomplemented
240C2×#41—212no—
241C2×#41—248no—
242C2×#41—212no—
243C22×Q8—21no—
244C22×D4—23no—
245C22×D4—248no—
246C22×D4—212no—
247C22×D4—26no—
248C22×D4—212no—
249C22×D4—26no—
250C22×D4—224no—
251C22×D4—23no—
252C22×D4—23no—
253C22×D4—26no—
254C22×D4—44no—
255C22×D4D4×C5916yescomplemented
256C22×D4C23×C5916yes—
257C23×C4—23no—
258C23×C4D4×C5913yescomplemented
259C23×C4C23×C5911yescharacteristic, complemented
260#120—224no—
261#120—412no—
262C2×C22⋊C4—23no—
263C2×C22⋊C4—26no—
264C2×C22⋊C4—26no—
265C2×C22⋊C4—23no—
266C2×C22⋊C4C23×C5916yescomplemented
267#99—224no—
268#95—26no—
269C24—28no—
270C24—24no—
271C24—23no—
272C24—23no—
273C24—212no—
274C24—24no—
275C24—23no—
276C24—24no—
277C24—42no—
278C24—42no—
279C24—46no—
280C24—81no—
281C24—81no—
282C24—83no—
283C24—82no—
284C24—82no—
285C24C2×D4×C5913yes—
286C24C2×D4×C5913yes—
287C24C2×D4×C5916yes—
288C24C24×C5911yescharacteristic
289#41—296no—
290#41—248no—
291#41—248no—
292C2×Q8—28no—
293C2×Q8—24no—
294C2×D4—224no—
295C2×D4—248no—
296C2×D4—212no—
297C2×D4—296no—
298C2×D4—248no—
299C2×D4—248no—
300C2×D4—212no—
301C2×D4—248no—
302C2×D4—212no—
303C2×D4—212no—
304C2×D4—212no—
305C2×D4—212no—
306C2×D4—212no—
307C2×D4—46no—
308C2×D4—46no—
309C2×D4—46no—
310C2×D4—43no—
311C2×D4—41no—
312C2×D4—412no—
313C2×D4—412no—
314C2×D4—86no—
315C2×D4C2×D4×C59124yes—
316C22×C4—224no—
317C22×C4—212no—
318C22×C4—23no—
319C22×C4—23no—
320C22×C4—212no—
321C22×C4—23no—
322C22×C4—26no—
323C22×C4—42no—
324C22×C4C2×D4×C5916yes—
325C22⋊C4—212no—
326C22⋊C4—412no—
327C22⋊C4—46no—
328C22⋊C4—412no—
329C22⋊C4—83no—
330C23—212no—
331C23—212no—
332C23—216no—
333C23—26no—
334C23—26no—
335C23—26no—
336C23—26no—
337C23—26no—
338C23—26no—
339C23—41no—
340C23—43no—
341C23—41no—
342C23—43no—
343C23—43no—
344C23—46no—
345C23—46no—
346C23—46no—
347C23—43no—
348C23—46no—
349C23—46no—
350C23—46no—
351C23—41no—
352C23—83no—
353C23—83no—
354C23—88no—
355C23—88no—
356C23—824no—
357C23—816no—
358C23—84no—
359C23—84no—
360C23#120×C5913yes—
361C23#120×C5911yescharacteristic
362C23#120×C59112yescomplemented
363C23C22×D4×C5913yes—
364C23C22×D4×C5914yes—
365Q8—216no—
366D4—248no—
367D4—248no—
368D4—248no—
369D4—412no—
370D4—44no—
371D4—824no—
372C2×C4—212no—
373C2×C4—212no—
374C2×C4—248no—
375C2×C4—26no—
376C2×C4—26no—
377C2×C4—26no—
378C2×C4—43no—
379C2×C4—41no—
380C2×C4—46no—
381C2×C4—46no—
382C2×C4—46no—
383C2×C4#120×C59112yescomplemented
384C22—23no—
385C22—23no—
386C22—21no—
387C22—21no—
388C22—23no—
389C22—23no—
390C22—23no—
391C22—212no—
392C22—44no—
393C22—412no—
394C22—412no—
395C22—424no—
396C22—412no—
397C22—46no—
398C22—46no—
399C22—42no—
400C22—86no—
401C22—86no—
402C22—86no—
403C22—812no—
404C22—816no—
405C22—81no—
406C22—81no—
407C22C59×B(4,𝔽2)12yescomplemented
408C22C2×#120×C5916yes—
409C22C2×#120×C5911yescharacteristic
410C4—212no—
411C4—42no—
412C4—86no—
413C2—26no—
414C2—22no—
415C2—43no—
416C2—412no—
417C2—82no—
418C2C59×#284611yescharacteristic
419C2C2×C59×B(4,𝔽2)11yescharacteristic
420C2C59×#302611yescharacteristic
subgroupG/H
center, socleC22×C59C2×#120
commutator, Frattini, cosocleC24C24×C59
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22×C59 ◁ C25×C59 ◁ G

Complements

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

NH ≅ G/Ntype
1. C22×H(5,𝔽2)×C59C2semidirect
2. C2×C59×B(4,𝔽2)C2semidirect
3. C59×#2403C2semidirect
4. C59×#1790C2semidirect
5. C59×#1781C2semidirect
12. C23×D4×C59C22semidirect
14. C2×#120×C59C22semidirect
16. C2×#99×C59C22semidirect
19. C59×#386C22semidirect
20. C59×#379C22semidirect
22. C25×C59D4semidirect
24. C25×C59C23semidirect
28. H(5,𝔽2)×C59D4semidirect
44. C22×D4×C59D4semidirect
47. C23×C4×C59D4semidirect
48. C23×C4×C59C23semidirect
55. C2×C22⋊C4×C59C23semidirect
151. C23×C59#120semidirect
172. C2×C4×C59#120semidirect
173. #32898C59direct (N × H)
197. C22×C59B(4,𝔽2)semidirect
203. C22×H(5,𝔽2)C2×C59semidirect
204. C2×B(4,𝔽2)C2×C59semidirect
205. #2403C2×C59semidirect
206. #1790C2×C59semidirect
207. #1781C2×C59semidirect
222. C23×D4C22×C59semidirect
224. C2×#120C22×C59semidirect
226. C2×#99C22×C59semidirect
229. #386C22×C59semidirect
230. #379C22×C59semidirect
231. C59#32898direct (N × H)
233. C25D4×C59semidirect
235. C25C23×C59semidirect
239. H(5,𝔽2)D4×C59semidirect
255. C22×D4D4×C59semidirect
258. C23×C4D4×C59semidirect
259. C23×C4C23×C59semidirect
266. C2×C22⋊C4C23×C59semidirect
362. C23#120×C59semidirect
383. C2×C4#120×C59semidirect
407. C22C59×B(4,𝔽2)semidirect

Maximal subgroups

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

subgroupindex
C22×H(5,𝔽2)×C592
C2×C59×B(4,𝔽2)2
C59×#24032
C59×#17902
C59×#17812
#3289859