Finite Group Databasebeta

← C59×#11802 #28423911

Subgroups — C59×#11802 #28423911

Order 15104.

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

normal subgroups only

This group has 758 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×#394C211yesmaximal, characteristic
2C59×#1821C211yesmaximal, characteristic, complemented
3C59×#1463C211yesmaximal, characteristic, complemented
4C59×#1345C214yesmaximal, complemented
5C23×D4×C59C411yescharacteristic
6C22×C4⋊C4×C59C2211yescharacteristic
7C22×C22⋊C4×C59—21no—
8C22×C22⋊C4×C59—21no—
9C59×#407—21no—
10C59×#407—21no—
11C59×#394—21no—
12C59×#394—21no—
13C59×#394C412yes—
14C59×#394C2212yes—
15C2×#95×C59C411yescharacteristic
16C59×#343—21no—
17C59×#343C2212yescomplemented
18C59×#324—21no—
19C59×#324C2212yescomplemented
20C25×C59—21no—
21C22×D4×C59—21no—
22C22×D4×C59—41no—
23C22×D4×C59—41no—
24C22×D4×C59—44no—
25C22×D4×C59C2×C412yes—
26C23×C4×C59—21no—
27C23×C4×C59C2×C411yescharacteristic
28C23×C4×C59D411yescharacteristic
29C23×C4×C59D411yescharacteristic
30C2×M16×C59—22no—
31C2×M16×C59—41no—
32C2×C4⋊C4×C59—41no—
33C2×C4⋊C4×C59—41no—
34C2×C4⋊C4×C59C2×C411yescharacteristic
35C2×C4⋊C4×C59D411yescharacteristic
36C2×C4⋊C4×C59D411yescharacteristic
37C2×C4⋊C4×C59C2311yescharacteristic
38C2×C22⋊C4×C59—22no—
39C2×C22⋊C4×C59—22no—
40C2×C22⋊C4×C59—22no—
41C2×C22⋊C4×C59—22no—
42C2×C22⋊C4×C59—44no—
43C2×C22⋊C4×C59—44no—
44C22⋊C8×C59—22no—
45C22⋊C8×C59—41no—
46C22⋊C8×C59—41no—
47C22⋊C8×C59—41no—
48#95×C59—21no—
49#95×C59C2×C412yes—
50C24×C59—24no—
51C24×C59—22no—
52C24×C59—42no—
53C24×C59—42no—
54C24×C59—42no—
55C24×C59—42no—
56C24×C59—42no—
57C24×C59—41no—
58C24×C59—41no—
59C24×C59C22⋊C411yescharacteristic
60C2×D4×C59—42no—
61C2×D4×C59—42no—
62C2×D4×C59—44no—
63C2×D4×C59—48no—
64C2×D4×C59—44no—
65C2×D4×C59—44no—
66C2×D4×C59—44no—
67C22×C4×C59—21no—
68C22×C4×C59—21no—
69C22×C4×C59—21no—
70C22×C4×C59—21no—
71C22×C4×C59—21no—
72C22×C4×C59—21no—
73C22×C4×C59—22no—
74C22×C4×C59—22no—
75C22×C4×C59—41no—
76C22×C4×C59—42no—
77C22×C4×C59—42no—
78C22×C4×C59—42no—
79C22×C4×C59—42no—
80C22×C4×C59—42no—
81C22×C4×C59C22⋊C411yescharacteristic
82C22×C4×C59C22⋊C411yescharacteristic
83C22×C4×C59C22⋊C411yescharacteristic
84C22×C4×C59C22×C411yescharacteristic
85C22×C4×C59C2×D411yescharacteristic
86C22×C4×C59C2×D411yescharacteristic
87M16×C59—42no—
88M16×C59—42no—
89M16×C59—82no—
90C2×C8×C59—42no—
91C2×C8×C59—42no—
92C4⋊C4×C59—41no—
93C4⋊C4×C59—41no—
94C4⋊C4×C59—41no—
95C4⋊C4×C59—41no—
96C22⋊C4×C59—44no—
97C22⋊C4×C59—44no—
98C22⋊C4×C59—44no—
99C22⋊C4×C59—44no—
100C23×C59—22no—
101C23×C59—22no—
102C23×C59—21no—
103C23×C59—21no—
104C23×C59—24no—
105C23×C59—42no—
106C23×C59—44no—
107C23×C59—48no—
108C23×C59—48no—
109C23×C59—42no—
110C23×C59—42no—
111C23×C59—44no—
112C23×C59—42no—
113C23×C59—42no—
114C23×C59—42no—
115C23×C59—84no—
116C23×C59—84no—
117C23×C59—82no—
118C23×C59—82no—
119C23×C59—82no—
120C23×C59—82no—
121C23×C59#9911yescharacteristic
122C23×C59#9911yescharacteristic
123C23×C59C2×C22⋊C411yescharacteristic
124D4×C59—44no—
125D4×C59—48no—
126D4×C59—44no—
127C2×C4×C59—21no—
128C2×C4×C59—21no—
129C2×C4×C59—21no—
130C2×C4×C59—21no—
131C2×C4×C59—21no—
132C2×C4×C59—21no—
133C2×C4×C59—21no—
134C2×C4×C59—21no—
135C2×C4×C59—21no—
136C2×C4×C59—42no—
137C2×C4×C59—42no—
138C2×C4×C59—42no—
139C2×C4×C59—42no—
140C2×C4×C59—42no—
141C2×C4×C59—41no—
142C2×C4×C59—41no—
143C2×C4×C59—41no—
144C2×C4×C59—41no—
145C2×C4×C59—42no—
146C2×C4×C59—42no—
147C2×C4×C59—42no—
148C2×C4×C59—81no—
149C2×C4×C59—84no—
150C2×C4×C59AGL(1,ℤ/8)12yes—
151C8×C59—82no—
152C8×C59—82no—
153#11802C5911yesmaximal, characteristic, direct product factor, Sylow
154C22×C59—24no—
155C22×C59—24no—
156C22×C59—21no—
157C22×C59—21no—
158C22×C59—21no—
159C22×C59—21no—
160C22×C59—21no—
161C22×C59—21no—
162C22×C59—44no—
163C22×C59—44no—
164C22×C59—44no—
165C22×C59—82no—
166C22×C59—82no—
167C22×C59—88no—
168C22×C59—88no—
169C22×C59—82no—
170C22×C59—82no—
171C22×C59AΣL(1,𝔽16)12yes—
172C22×C59#35512yes—
173C22×C59C2×#9911yescharacteristic
174C22×C59#41811yescharacteristic
175C22×C59#42111yescharacteristic
176C4×C59—22no—
177C4×C59—41no—
178C4×C59—41no—
179C4×C59—41no—
180C4×C59—41no—
181C4×C59—41no—
182C4×C59—82no—
183C2×#394C2×C5911yescharacteristic
184#1821C2×C5911yescharacteristic, complemented
185#1463C2×C5911yescharacteristic, complemented
186#1345C2×C5914yescomplemented
187C2×C59—21no—
188C2×C59—21no—
189C2×C59—22no—
190C2×C59—22no—
191C2×C59—84no—
192C2×C59#151011yescharacteristic
193C2×C59C2×AΣL(1,𝔽16)11yescharacteristic
194C2×C59C2×#35511yescharacteristic
195C23×D4C4×C5911yescharacteristic
196C22×C4⋊C4C22×C5911yescharacteristic
197C22×C22⋊C4—21no—
198C22×C22⋊C4—21no—
199#407—21no—
200#407—21no—
201#394—21no—
202#394—21no—
203#394C4×C5912yes—
204#394C22×C5912yes—
205C2×#95C4×C5911yescharacteristic
206#343—21no—
207#343C22×C5912yescomplemented
208#324—21no—
209#324C22×C5912yescomplemented
210C59#1180211yescharacteristic, direct product factor, Sylow
211C25—21no—
212C22×D4—21no—
213C22×D4—41no—
214C22×D4—41no—
215C22×D4—44no—
216C22×D4C2×C4×C5912yes—
217C23×C4—21no—
218C23×C4C2×C4×C5911yescharacteristic
219C23×C4D4×C5911yescharacteristic
220C23×C4D4×C5911yescharacteristic
221C2×M16—22no—
222C2×M16—41no—
223C2×C4⋊C4—41no—
224C2×C4⋊C4—41no—
225C2×C4⋊C4C2×C4×C5911yescharacteristic
226C2×C4⋊C4D4×C5911yescharacteristic
227C2×C4⋊C4D4×C5911yescharacteristic
228C2×C4⋊C4C23×C5911yescharacteristic
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—44no—
234C2×C22⋊C4—44no—
235C22⋊C8—22no—
236C22⋊C8—41no—
237C22⋊C8—41no—
238C22⋊C8—41no—
239#95—21no—
240#95C2×C4×C5912yes—
241C24—24no—
242C24—22no—
243C24—42no—
244C24—42no—
245C24—42no—
246C24—42no—
247C24—42no—
248C24—41no—
249C24—41no—
250C24C22⋊C4×C5911yescharacteristic
251C2×D4—42no—
252C2×D4—42no—
253C2×D4—44no—
254C2×D4—44no—
255C2×D4—48no—
256C2×D4—44no—
257C2×D4—44no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—41no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4C22⋊C4×C5911yescharacteristic
273C22×C4C22⋊C4×C5911yescharacteristic
274C22×C4C22⋊C4×C5911yescharacteristic
275C22×C4C22×C4×C5911yescharacteristic
276C22×C4C2×D4×C5911yescharacteristic
277C22×C4C2×D4×C5911yescharacteristic
278M16—42no—
279M16—42no—
280M16—82no—
281C2×C8—42no—
282C2×C8—42no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C22⋊C4—44no—
288C22⋊C4—44no—
289C22⋊C4—44no—
290C22⋊C4—44no—
291C23—22no—
292C23—22no—
293C23—21no—
294C23—21no—
295C23—24no—
296C23—42no—
297C23—44no—
298C23—48no—
299C23—48no—
300C23—42no—
301C23—42no—
302C23—44no—
303C23—42no—
304C23—42no—
305C23—42no—
306C23—84no—
307C23—84no—
308C23—82no—
309C23—82no—
310C23—82no—
311C23—82no—
312C23#99×C5911yescharacteristic
313C23#99×C5911yescharacteristic
314C23C2×C22⋊C4×C5911yescharacteristic
315D4—44no—
316D4—44no—
317D4—48no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—81no—
340C2×C4—84no—
341C2×C4AGL(1,ℤ/8)×C5912yes—
342C8—82no—
343C8—82no—
344C22—24no—
345C22—24no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—44no—
353C22—44no—
354C22—44no—
355C22—82no—
356C22—82no—
357C22—88no—
358C22—88no—
359C22—82no—
360C22—82no—
361C22C59×AΣL(1,𝔽16)12yes—
362C22C59×#35512yes—
363C22C2×#99×C5911yescharacteristic
364C22C59×#41811yescharacteristic
365C22C59×#42111yescharacteristic
366C4—22no—
367C4—41no—
368C4—41no—
369C4—41no—
370C4—41no—
371C4—41no—
372C4—82no—
373C2—21no—
374C2—21no—
375C2—22no—
376C2—22no—
377C2—84no—
378C2C59×#151011yescharacteristic
379C2C2×C59×AΣL(1,𝔽16)11yescharacteristic
380C2C2×C59×#35511yescharacteristic
subgroupG/H
center, socleC22×C59C2×#99
commutatorC22×C4C22×C4×C59
Frattini, cosocleC2×C4⋊C4C23×C59
prime radicalC23×D4×C59C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C59 ◁ C24×C59 ◁ C22×C4⋊C4×C59 ◁ G

Complements

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

NH ≅ G/Ntype
2. C59×#1821C2semidirect
3. C59×#1463C2semidirect
4. C59×#1345C2semidirect
17. C59×#343C22semidirect
19. C59×#324C22semidirect
153. #11802C59direct (N × H)
184. #1821C2×C59semidirect
185. #1463C2×C59semidirect
186. #1345C2×C59semidirect
207. #343C22×C59semidirect
209. #324C22×C59semidirect
210. C59#11802direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C59×#3942
C59×#18212
C59×#14632
C59×#13452
#1180259