Finite Group Databasebeta

← C59×#19006 #28431115

Subgroups — C59×#19006 #28431115

Order 15104.

938 conjugacy classes of subgroups; only the 222 normal subgroups are shown.

normal subgroups only

This group has 938 conjugacy classes of subgroups — too many to draw. Only its 222 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×#562C211yesmaximal, characteristic
2C59×#3215C212yesmaximal, complemented
3C59×#3166C211yesmaximal, characteristic, complemented
4C59×#1661C218yesmaximal, complemented
5C59×#1534C211yesmaximal, characteristic, complemented
6C59×#1527C212yesmaximal, complemented
7C59×#562—22no—
8C59×#562—24no—
9C59×#562C2214yes—
10C59×#533C2211yescharacteristic
11C2×#125×C59C2211yescharacteristic
12C2×#124×C59C2212yes—
13C2×#123×C59—22no—
14C2×C22⋊Q8×C59—22no—
15C2×#121×C59—22no—
16C59×#501C2214yescomplemented
17C59×#499—22no—
18C59×#497C2214yescomplemented
19C59×#489C2214yescomplemented
20C59×#464—24no—
21C59×#451—22no—
22C59×#450—22no—
23C59×#432C2212yes—
24C59×#428—22no—
25C4×M16×C59C2211yescharacteristic
26C59×#330C2214yescomplemented
27C59×#329C2218yescomplemented
28C22×Q8×C59D412yes—
29C22×D4×C59—22no—
30C22×D4×C59D412yes—
31C23×C4×C59—22no—
32C2×Q16×C59—24no—
33C2×SD16×C59—24no—
34C2×M16×C59—21no—
35C4⋊Q8×C59C2312yes—
36#125×C59—22no—
37#125×C59—21no—
38#125×C59D412yes—
39#125×C59C2312yes—
40#124×C59—22no—
41#124×C59—22no—
42#124×C59D414yes—
43#124×C59C2314yes—
44#123×C59—28no—
45#123×C59—28no—
46C22⋊Q8×C59—28no—
47C22⋊Q8×C59—28no—
48C22⋊Q8×C59—22no—
49#121×C59—28no—
50#121×C59—28no—
51C4×Q8×C59—22no—
52#117×C59—21no—
53C2×C4⋊C4×C59—22no—
54C2×C4⋊C4×C59—22no—
55C2×C4⋊C4×C59D412yes—
56C2×C22⋊C4×C59—22no—
57C2×C22⋊C4×C59—22no—
58C2×C22⋊C4×C59—22no—
59C2×C42×C59C2311yescharacteristic
60#107×C59—24no—
61#106×C59—24no—
62C4⋊C8×C59—22no—
63#103×C59—24no—
64#102×C59—24no—
65C22⋊C8×C59—22no—
66#97×C59C2314yes—
67#97×C59C2312yes—
68C4×C8×C59—21no—
69C24×C59—22no—
70C2×Q8×C59—22no—
71C2×Q8×C59—22no—
72C2×Q8×C59—82no—
73C2×Q8×C59C2×D414yes—
74C2×D4×C59—28no—
75C2×D4×C59—22no—
76C2×D4×C59—82no—
77C2×D4×C59—84no—
78C2×D4×C59C2×D414yes—
79C22×C4×C59—22no—
80C22×C4×C59—22no—
81C22×C4×C59—28no—
82C22×C4×C59—84no—
83C22×C4×C59C2×D411yescharacteristic
84C22×C4×C59C2×D412yes—
85Q16×C59—84no—
86SD16×C59—84no—
87M16×C59—24no—
88C2×C8×C59—22no—
89C2×C8×C59—24no—
90C4⋊C4×C59—28no—
91C4⋊C4×C59—24no—
92C4⋊C4×C59—22no—
93C4⋊C4×C59—24no—
94C4⋊C4×C59—22no—
95C4⋊C4×C59—22no—
96C4⋊C4×C59—42no—
97C4⋊C4×C59C2×D414yes—
98C22⋊C4×C59—28no—
99C22⋊C4×C59—24no—
100C22⋊C4×C59—24no—
101C22⋊C4×C59—24no—
102C22⋊C4×C59—24no—
103C22⋊C4×C59—41no—
104C42×C59—41no—
105C42×C59C2×D411yescharacteristic
106C42×C59C2×D412yes—
107C42×C59C2411yescharacteristic
108C23×C59—24no—
109C23×C59—44no—
110C23×C59—44no—
111C23×C59—44no—
112C23×C59#12011yescharacteristic
113Q8×C59—44no—
114Q8×C59—44no—
115Q8×C59—82no—
116D4×C59—44no—
117D4×C59—82no—
118D4×C59—88no—
119C2×C4×C59—22no—
120C2×C4×C59—24no—
121C2×C4×C59—24no—
122C2×C4×C59—42no—
123C2×C4×C59—44no—
124C2×C4×C59—44no—
125C2×C4×C59—42no—
126C2×C4×C59—88no—
127C2×C4×C59#12011yescharacteristic
128C2×C4×C59#12012yes—
129C2×C4×C59C22×D411yescharacteristic
130C2×C4×C59C22×D412yes—
131C8×C59—24no—
132C8×C59—44no—
133#19006C5911yesmaximal, characteristic, direct product factor, Sylow
134C22×C59—22no—
135C22×C59—44no—
136C22×C59—44no—
137C22×C59—44no—
138C22×C59—88no—
139C22×C59#45612yes—
140C22×C59C2×#12011yescharacteristic
141C4×C59—21no—
142C4×C59—21no—
143C4×C59—22no—
144C4×C59—22no—
145C4×C59—44no—
146C4×C59—44no—
147C4×C59—82no—
148C2×#562C2×C5911yescharacteristic
149#3215C2×C5912yescomplemented
150#3166C2×C5911yescharacteristic, complemented
151#1661C2×C5918yescomplemented
152#1534C2×C5911yescharacteristic, complemented
153#1527C2×C5912yescomplemented
154C2×C59—22no—
155C2×C59—84no—
156C2×C59C2×#45611yescharacteristic
157C2×C59#301912yes—
158#562—22no—
159#562—24no—
160#562C22×C5914yes—
161#533C22×C5911yescharacteristic
162C2×#125C22×C5911yescharacteristic
163C2×#124C22×C5912yes—
164C2×#123—22no—
165C2×C22⋊Q8—22no—
166C2×#121—22no—
167#501C22×C5914yescomplemented
168#499—22no—
169#497C22×C5914yescomplemented
170#489C22×C5914yescomplemented
171#464—24no—
172#451—22no—
173#450—22no—
174#432C22×C5912yes—
175#428—22no—
176C4×M16C22×C5911yescharacteristic
177#330C22×C5914yescomplemented
178#329C22×C5918yescomplemented
179C59#1900611yescharacteristic, direct product factor, Sylow
180C22×Q8D4×C5912yes—
181C22×D4—22no—
182C22×D4D4×C5912yes—
183C23×C4—22no—
184C2×Q16—24no—
185C2×SD16—24no—
186C2×M16—21no—
187C4⋊Q8C23×C5912yes—
188#125—22no—
189#125—21no—
190#125D4×C5912yes—
191#125C23×C5912yes—
192#124—22no—
193#124—22no—
194#124D4×C5914yes—
195#124C23×C5914yes—
196#123—28no—
197#123—28no—
198C22⋊Q8—28no—
199C22⋊Q8—28no—
200C22⋊Q8—22no—
201#121—28no—
202#121—28no—
203C4×Q8—22no—
204#117—21no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4D4×C5912yes—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C42C23×C5911yescharacteristic
212#107—24no—
213#106—24no—
214C4⋊C8—22no—
215#103—24no—
216#102—24no—
217C22⋊C8—22no—
218#97C23×C5912yes—
219#97C23×C5914yes—
220C4×C8—21no—
221C24—22no—
222C2×Q8—22no—
223C2×Q8—22no—
224C2×Q8—82no—
225C2×Q8C2×D4×C5914yes—
226C2×D4—28no—
227C2×D4—22no—
228C2×D4—82no—
229C2×D4—84no—
230C2×D4C2×D4×C5914yes—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—28no—
234C22×C4—84no—
235C22×C4C2×D4×C5911yescharacteristic
236C22×C4C2×D4×C5912yes—
237Q16—84no—
238SD16—84no—
239M16—24no—
240C2×C8—22no—
241C2×C8—24no—
242C4⋊C4—28no—
243C4⋊C4—24no—
244C4⋊C4—22no—
245C4⋊C4—24no—
246C4⋊C4—22no—
247C4⋊C4—22no—
248C4⋊C4—42no—
249C4⋊C4C2×D4×C5914yes—
250C22⋊C4—28no—
251C22⋊C4—24no—
252C22⋊C4—24no—
253C22⋊C4—24no—
254C22⋊C4—24no—
255C22⋊C4—41no—
256C42—41no—
257C42C2×D4×C5911yescharacteristic
258C42C2×D4×C5912yes—
259C42C24×C5911yescharacteristic
260C23—24no—
261C23—44no—
262C23—44no—
263C23—44no—
264C23#120×C5911yescharacteristic
265Q8—44no—
266Q8—44no—
267Q8—82no—
268D4—44no—
269D4—82no—
270D4—88no—
271C2×C4—22no—
272C2×C4—24no—
273C2×C4—24no—
274C2×C4—42no—
275C2×C4—44no—
276C2×C4—44no—
277C2×C4—42no—
278C2×C4—88no—
279C2×C4#120×C5911yescharacteristic
280C2×C4#120×C5912yes—
281C2×C4C22×D4×C5911yescharacteristic
282C2×C4C22×D4×C5912yes—
283C8—24no—
284C8—44no—
285C22—22no—
286C22—44no—
287C22—44no—
288C22—44no—
289C22—88no—
290C22C59×#45612yes—
291C22C2×#120×C5911yescharacteristic
292C4—21no—
293C4—21no—
294C4—22no—
295C4—22no—
296C4—44no—
297C4—44no—
298C4—82no—
299C2—22no—
300C2—84no—
301C2C2×C59×#45611yescharacteristic
302C2C59×#301912yes—
subgroupG/H
center, socleC22×C59C2×#120
commutator, Frattini, cosocleC42C24×C59
prime radicalC2×C59×#562C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C59 ◁ C2×C42×C59 ◁ G

Complements

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

NH ≅ G/Ntype
2. C59×#3215C2semidirect
3. C59×#3166C2semidirect
4. C59×#1661C2semidirect
5. C59×#1534C2semidirect
6. C59×#1527C2semidirect
16. C59×#501C22semidirect
18. C59×#497C22semidirect
19. C59×#489C22semidirect
26. C59×#330C22semidirect
27. C59×#329C22semidirect
133. #19006C59direct (N × H)
149. #3215C2×C59semidirect
150. #3166C2×C59semidirect
151. #1661C2×C59semidirect
152. #1534C2×C59semidirect
153. #1527C2×C59semidirect
167. #501C22×C59semidirect
169. #497C22×C59semidirect
170. #489C22×C59semidirect
177. #330C22×C59semidirect
178. #329C22×C59semidirect
179. C59#19006direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C59×#5622
C59×#32152
C59×#31662
C59×#16612
C59×#15342
C59×#15272
#1900659