Finite Group Databasebeta

← C59×#11796 #28423905

Subgroups — C59×#11796 #28423905

Order 15104.

1068 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 1068 conjugacy classes of subgroups — too many to draw. Only its 106 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
1C2×C59×#392C211yesmaximal, characteristic
2C59×#1819C211yesmaximal, characteristic, complemented
3C59×#1462C211yesmaximal, characteristic, complemented
4C59×#1343C214yesmaximal, complemented
5C23×D4×C59—21no—
6C23×D4×C59C411yescharacteristic, complemented
7C22×C22⋊C4×C59—21no—
8C22×C22⋊C4×C59C2211yescharacteristic
9C59×#407—21no—
10C59×#392—21no—
11C59×#392—21no—
12C59×#392C412yescomplemented
13C59×#392C2212yes—
14C59×#379—21no—
15C2×#95×C59C411yescharacteristic, complemented
16C59×#342—21no—
17C59×#342C2212yescomplemented
18C59×#323—21no—
19C59×#323C2212yescomplemented
20C25×C59—21no—
21C25×C59D411yescharacteristic
22C22×D4×C59—22no—
23C22×D4×C59—22no—
24C22×D4×C59—21no—
25C22×D4×C59—41no—
26C22×D4×C59—41no—
27C22×D4×C59—44no—
28C22×D4×C59—44no—
29C22×D4×C59—44no—
30C22×D4×C59—44no—
31C22×D4×C59C2×C412yes—
32C23×C4×C59—21no—
33C23×C4×C59C2×C411yescharacteristic
34C23×C4×C59D411yescharacteristic
35C2×M16×C59—41no—
36C2×C22⋊C4×C59—22no—
37C2×C22⋊C4×C59—22no—
38C2×C22⋊C4×C59—22no—
39C2×C22⋊C4×C59—41no—
40C2×C22⋊C4×C59—41no—
41C2×C22⋊C4×C59—44no—
42C2×C22⋊C4×C59—41no—
43C2×C22⋊C4×C59—41no—
44C2×C22⋊C4×C59C2×C411yescharacteristic
45C2×C22⋊C4×C59D411yescharacteristic
46C2×C22⋊C4×C59D411yescharacteristic
47C2×C22⋊C4×C59C2311yescharacteristic
48C22⋊C8×C59—22no—
49C22⋊C8×C59—41no—
50#95×C59—21no—
51#95×C59C2×C412yes—
52C24×C59—24no—
53C24×C59—22no—
54C24×C59—22no—
55C24×C59—21no—
56C24×C59—21no—
57C24×C59—41no—
58C24×C59—42no—
59C24×C59—42no—
60C24×C59—42no—
61C24×C59—42no—
62C24×C59—42no—
63C24×C59—42no—
64C24×C59—42no—
65C24×C59—41no—
66C24×C59—41no—
67C24×C59C22⋊C411yescharacteristic
68C24×C59C22⋊C411yescharacteristic
69C24×C59C2×D411yescharacteristic
70C2×D4×C59—44no—
71C2×D4×C59—44no—
72C2×D4×C59—42no—
73C2×D4×C59—42no—
74C2×D4×C59—44no—
75C2×D4×C59—44no—
76C2×D4×C59—44no—
77C2×D4×C59—44no—
78C2×D4×C59—48no—
79C2×D4×C59—432no—
80C2×D4×C59—44no—
81C2×D4×C59—44no—
82C2×D4×C59—44no—
83C2×D4×C59—44no—
84C22×C4×C59—24no—
85C22×C4×C59—21no—
86C22×C4×C59—21no—
87C22×C4×C59—21no—
88C22×C4×C59—22no—
89C22×C4×C59—21no—
90C22×C4×C59—42no—
91C22×C4×C59—42no—
92C22×C4×C59—42no—
93C22×C4×C59—42no—
94C22×C4×C59—42no—
95C22×C4×C59C22⋊C411yescharacteristic
96C22×C4×C59C22⋊C411yescharacteristic
97C22×C4×C59C22×C411yescharacteristic
98C22×C4×C59C2×D411yescharacteristic
99M16×C59—82no—
100C2×C8×C59—42no—
101C22⋊C4×C59—41no—
102C22⋊C4×C59—41no—
103C22⋊C4×C59—41no—
104C22⋊C4×C59—41no—
105C22⋊C4×C59—44no—
106C22⋊C4×C59—44no—
107C22⋊C4×C59—42no—
108C22⋊C4×C59—42no—
109C22⋊C4×C59—82no—
110C22⋊C4×C59—82no—
111C23×C59—22no—
112C23×C59—22no—
113C23×C59—21no—
114C23×C59—21no—
115C23×C59—21no—
116C23×C59—21no—
117C23×C59—24no—
118C23×C59—21no—
119C23×C59—21no—
120C23×C59—21no—
121C23×C59—42no—
122C23×C59—42no—
123C23×C59—41no—
124C23×C59—41no—
125C23×C59—41no—
126C23×C59—41no—
127C23×C59—42no—
128C23×C59—48no—
129C23×C59—48no—
130C23×C59—42no—
131C23×C59—44no—
132C23×C59—42no—
133C23×C59—48no—
134C23×C59—42no—
135C23×C59—44no—
136C23×C59—42no—
137C23×C59—42no—
138C23×C59—42no—
139C23×C59—42no—
140C23×C59—81no—
141C23×C59—81no—
142C23×C59—84no—
143C23×C59—84no—
144C23×C59—84no—
145C23×C59—82no—
146C23×C59—82no—
147C23×C59—82no—
148C23×C59—82no—
149C23×C59#9911yescharacteristic
150C23×C59#9911yescharacteristic
151C23×C59C2×C22⋊C411yescharacteristic
152C23×C59AGL(1,ℤ/8)12yes—
153D4×C59—44no—
154D4×C59—416no—
155D4×C59—416no—
156D4×C59—48no—
157D4×C59—44no—
158C2×C4×C59—24no—
159C2×C4×C59—21no—
160C2×C4×C59—21no—
161C2×C4×C59—21no—
162C2×C4×C59—21no—
163C2×C4×C59—48no—
164C2×C4×C59—42no—
165C2×C4×C59—42no—
166C2×C4×C59—42no—
167C2×C4×C59—42no—
168C2×C4×C59—42no—
169C2×C4×C59—41no—
170C2×C4×C59—41no—
171C2×C4×C59—41no—
172C2×C4×C59—41no—
173C2×C4×C59—81no—
174C2×C4×C59—82no—
175C2×C4×C59—82no—
176C2×C4×C59—82no—
177C8×C59—82no—
178#11796C5911yesmaximal, characteristic, direct product factor, Sylow
179C22×C59—24no—
180C22×C59—24no—
181C22×C59—21no—
182C22×C59—21no—
183C22×C59—21no—
184C22×C59—21no—
185C22×C59—21no—
186C22×C59—21no—
187C22×C59—22no—
188C22×C59—42no—
189C22×C59—42no—
190C22×C59—44no—
191C22×C59—44no—
192C22×C59—41no—
193C22×C59—41no—
194C22×C59—44no—
195C22×C59—41no—
196C22×C59—41no—
197C22×C59—41no—
198C22×C59—81no—
199C22×C59—84no—
200C22×C59—84no—
201C22×C59—81no—
202C22×C59—82no—
203C22×C59—82no—
204C22×C59—88no—
205C22×C59—88no—
206C22×C59—82no—
207C22×C59—82no—
208C22×C59AΣL(1,𝔽16)12yes—
209C22×C59#35312yes—
210C22×C59C2×#9911yescharacteristic
211C22×C59#41811yescharacteristic
212C22×C59#42111yescharacteristic
213C4×C59—44no—
214C4×C59—41no—
215C4×C59—41no—
216C4×C59—41no—
217C4×C59—41no—
218C4×C59—162no—
219C2×#392C2×C5911yescharacteristic
220#1819C2×C5911yescharacteristic, complemented
221#1462C2×C5911yescharacteristic, complemented
222#1343C2×C5914yescomplemented
223C2×C59—21no—
224C2×C59—21no—
225C2×C59—22no—
226C2×C59—22no—
227C2×C59—42no—
228C2×C59—81no—
229C2×C59—84no—
230C2×C59#151011yescharacteristic
231C2×C59C2×AΣL(1,𝔽16)11yescharacteristic
232C2×C59C2×#35311yescharacteristic
233C23×D4—21no—
234C23×D4C4×C5911yescharacteristic, complemented
235C22×C22⋊C4—21no—
236C22×C22⋊C4C22×C5911yescharacteristic
237#407—21no—
238#392—21no—
239#392—21no—
240#392C4×C5912yescomplemented
241#392C22×C5912yes—
242#379—21no—
243C2×#95C4×C5911yescharacteristic, complemented
244#342—21no—
245#342C22×C5912yescomplemented
246#323—21no—
247#323C22×C5912yescomplemented
248C59#1179611yescharacteristic, direct product factor, Sylow
249C25—21no—
250C25D4×C5911yescharacteristic
251C22×D4—22no—
252C22×D4—22no—
253C22×D4—21no—
254C22×D4—41no—
255C22×D4—41no—
256C22×D4—44no—
257C22×D4—44no—
258C22×D4—44no—
259C22×D4—44no—
260C22×D4C2×C4×C5912yes—
261C23×C4—21no—
262C23×C4C2×C4×C5911yescharacteristic
263C23×C4D4×C5911yescharacteristic
264C2×M16—41no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—41no—
269C2×C22⋊C4—41no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—41no—
272C2×C22⋊C4—41no—
273C2×C22⋊C4C2×C4×C5911yescharacteristic
274C2×C22⋊C4D4×C5911yescharacteristic
275C2×C22⋊C4D4×C5911yescharacteristic
276C2×C22⋊C4C23×C5911yescharacteristic
277C22⋊C8—22no—
278C22⋊C8—41no—
279#95—21no—
280#95C2×C4×C5912yes—
281C24—24no—
282C24—22no—
283C24—22no—
284C24—21no—
285C24—21no—
286C24—41no—
287C24—42no—
288C24—42no—
289C24—42no—
290C24—42no—
291C24—42no—
292C24—42no—
293C24—42no—
294C24—41no—
295C24—41no—
296C24C22⋊C4×C5911yescharacteristic
297C24C22⋊C4×C5911yescharacteristic
298C24C2×D4×C5911yescharacteristic
299C2×D4—42no—
300C2×D4—44no—
301C2×D4—44no—
302C2×D4—42no—
303C2×D4—44no—
304C2×D4—44no—
305C2×D4—44no—
306C2×D4—44no—
307C2×D4—44no—
308C2×D4—48no—
309C2×D4—432no—
310C2×D4—44no—
311C2×D4—44no—
312C2×D4—44no—
313C22×C4—24no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—22no—
318C22×C4—21no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4C22⋊C4×C5911yescharacteristic
325C22×C4C22⋊C4×C5911yescharacteristic
326C22×C4C22×C4×C5911yescharacteristic
327C22×C4C2×D4×C5911yescharacteristic
328M16—82no—
329C2×C8—42no—
330C22⋊C4—41no—
331C22⋊C4—41no—
332C22⋊C4—41no—
333C22⋊C4—41no—
334C22⋊C4—44no—
335C22⋊C4—44no—
336C22⋊C4—42no—
337C22⋊C4—42no—
338C22⋊C4—82no—
339C22⋊C4—82no—
340C23—22no—
341C23—22no—
342C23—21no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—24no—
347C23—21no—
348C23—21no—
349C23—21no—
350C23—42no—
351C23—42no—
352C23—41no—
353C23—41no—
354C23—41no—
355C23—41no—
356C23—42no—
357C23—48no—
358C23—48no—
359C23—42no—
360C23—44no—
361C23—42no—
362C23—48no—
363C23—42no—
364C23—44no—
365C23—42no—
366C23—42no—
367C23—42no—
368C23—42no—
369C23—81no—
370C23—81no—
371C23—84no—
372C23—84no—
373C23—84no—
374C23—82no—
375C23—82no—
376C23—82no—
377C23—82no—
378C23#99×C5911yescharacteristic
379C23#99×C5911yescharacteristic
380C23C2×C22⋊C4×C5911yescharacteristic
381C23AGL(1,ℤ/8)×C5912yes—
382D4—44no—
383D4—416no—
384D4—416no—
385D4—44no—
386D4—48no—
387C2×C4—24no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—48no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—81no—
403C2×C4—82no—
404C2×C4—82no—
405C2×C4—82no—
406C8—82no—
407C22—24no—
408C22—24no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—22no—
416C22—42no—
417C22—42no—
418C22—44no—
419C22—44no—
420C22—41no—
421C22—41no—
422C22—41no—
423C22—44no—
424C22—41no—
425C22—41no—
426C22—81no—
427C22—84no—
428C22—84no—
429C22—81no—
430C22—82no—
431C22—82no—
432C22—88no—
433C22—88no—
434C22—82no—
435C22—82no—
436C22C59×AΣL(1,𝔽16)12yes—
437C22C59×#35312yes—
438C22C2×#99×C5911yescharacteristic
439C22C59×#41811yescharacteristic
440C22C59×#42111yescharacteristic
441C4—44no—
442C4—41no—
443C4—41no—
444C4—41no—
445C4—41no—
446C4—162no—
447C2—21no—
448C2—21no—
449C2—22no—
450C2—22no—
451C2—42no—
452C2—81no—
453C2—84no—
454C2C59×#151011yescharacteristic
455C2C2×C59×AΣL(1,𝔽16)11yescharacteristic
456C2C2×C59×#35311yescharacteristic
subgroupG/H
center, socleC22×C59C2×#99
commutatorC22×C4C22×C4×C59
Frattini, cosocleC2×C22⋊C4C23×C59
prime radicalC2×C59×#392C2
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C59 ◁ C24×C59 ◁ C22×C22⋊C4×C59 ◁ G

Complements

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

NH ≅ G/Ntype
2. C59×#1819C2semidirect
3. C59×#1462C2semidirect
4. C59×#1343C2semidirect
6. C23×D4×C59C4semidirect
12. C59×#392C4semidirect
15. C2×#95×C59C4semidirect
17. C59×#342C22semidirect
19. C59×#323C22semidirect
178. #11796C59direct (N × H)
220. #1819C2×C59semidirect
221. #1462C2×C59semidirect
222. #1343C2×C59semidirect
234. C23×D4C4×C59semidirect
240. #392C4×C59semidirect
243. C2×#95C4×C59semidirect
245. #342C22×C59semidirect
247. #323C22×C59semidirect
248. C59#11796direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C59×#3922
C59×#18192
C59×#14622
C59×#13432
#1179659