Finite Group Databasebeta

← C59×#12250 #28424359

Subgroups — C59×#12250 #28424359

Order 15104.

454 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 74 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
1C59×#2819C211yesmaximal, characteristic
2C59×#2061C211yesmaximal, characteristic, complemented
3C59×#1599C212yesmaximal, complemented
4C59×#1345C212yesmaximal, complemented
5C59×#1324C211yesmaximal, characteristic, complemented
6C59×#544C2211yescharacteristic
7C2×#124×C59—21no—
8C59×#450—22no—
9C59×#428—21no—
10C59×#394—21no—
11C59×#394C2212yes—
12C59×#393—22no—
13C59×#392—22no—
14C59×#382C2211yescharacteristic
15C59×#343C2211yescharacteristic
16C59×#339—21no—
17C59×#328—22no—
18C59×#324C2212yes—
19C22×Q8×C59—21no—
20C22×D4×C59—21no—
21C22×D4×C59—21no—
22C22×D4×C59D412yes—
23C2×SD16×C59—42no—
24C2×M16×C59—21no—
25C4⋊Q8×C59—41no—
26#125×C59—41no—
27#124×C59—42no—
28#124×C59—42no—
29C22⋊Q8×C59—22no—
30#117×C59—21no—
31C2×C4⋊C4×C59D411yescharacteristic
32C2×C4⋊C4×C59C2311yescharacteristic
33C2×C22⋊C4×C59—22no—
34C2×C22⋊C4×C59—22no—
35C2×C22⋊C4×C59—22no—
36C2×C22⋊C4×C59—22no—
37C2×C42×C59D411yescharacteristic
38#107×C59—41no—
39#106×C59—41no—
40#102×C59—42no—
41C22⋊C8×C59—22no—
42#95×C59—21no—
43#95×C59—22no—
44#95×C59D412yes—
45C24×C59—22no—
46C2×Q8×C59—22no—
47C2×Q8×C59—42no—
48C2×Q8×C59—81no—
49C2×Q8×C59—81no—
50C2×D4×C59—42no—
51C2×D4×C59—42no—
52C2×D4×C59—82no—
53C2×D4×C59—81no—
54C2×D4×C59—81no—
55C2×D4×C59—82no—
56C2×D4×C59—81no—
57C22×C4×C59—21no—
58C22×C4×C59—21no—
59C22×C4×C59—22no—
60C22×C4×C59C2×D411yescharacteristic
61C22×C4×C59C2×D412yes—
62SD16×C59—84no—
63M16×C59—22no—
64M16×C59—41no—
65C2×C8×C59—41no—
66C2×C8×C59—42no—
67C4⋊C4×C59—22no—
68C4⋊C4×C59—21no—
69C4⋊C4×C59—21no—
70C4⋊C4×C59—41no—
71C4⋊C4×C59—42no—
72C4⋊C4×C59—41no—
73C22⋊C4×C59—44no—
74C22⋊C4×C59—41no—
75C22⋊C4×C59—42no—
76C22⋊C4×C59—42no—
77C22⋊C4×C59—82no—
78C22⋊C4×C59—82no—
79C42×C59—21no—
80C42×C59—41no—
81C42×C59SD1612yescomplemented
82C23×C59—42no—
83C23×C59—42no—
84C23×C59—42no—
85C23×C59—42no—
86C23×C59—42no—
87C23×C59—42no—
88C23×C59—42no—
89C23×C59#12011yescharacteristic
90Q8×C59—82no—
91Q8×C59—84no—
92D4×C59—44no—
93D4×C59—82no—
94D4×C59—84no—
95D4×C59—84no—
96C2×C4×C59—21no—
97C2×C4×C59—21no—
98C2×C4×C59—22no—
99C2×C4×C59—22no—
100C2×C4×C59—41no—
101C2×C4×C59—41no—
102C2×C4×C59—42no—
103C2×C4×C59—42no—
104C2×C4×C59—42no—
105C2×C4×C59—42no—
106C2×C4×C59—42no—
107C2×C4×C59—41no—
108C2×C4×C59—41no—
109C2×C4×C59—41no—
110C2×C4×C59C2×SD1611yescharacteristic
111C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
112C8×C59—42no—
113C8×C59—82no—
114#12250C5911yesmaximal, characteristic, direct product factor, Sylow
115C22×C59—21no—
116C22×C59—21no—
117C22×C59—82no—
118C22×C59—82no—
119C22×C59—82no—
120C22×C59—82no—
121C22×C59—82no—
122C22×C59—82no—
123C22×C59—82no—
124C22×C59#45011yescharacteristic
125C22×C59#45611yescharacteristic
126C22×C59B(4,𝔽2)11yescharacteristic
127C4×C59—21no—
128C4×C59—21no—
129C4×C59—42no—
130C4×C59—42no—
131C4×C59—82no—
132C4×C59—81no—
133C4×C59—81no—
134C4×C59—81no—
135#2819C2×C5911yescharacteristic
136#2061C2×C5911yescharacteristic, complemented
137#1599C2×C5912yescomplemented
138#1345C2×C5912yescomplemented
139#1324C2×C5911yescharacteristic, complemented
140C2×C59—21no—
141C2×C59—21no—
142C2×C59—162no—
143C2×C59#160411yescharacteristic
144C2×C59#219911yescharacteristic
145C2×C59#220211yescharacteristic
146#544C22×C5911yescharacteristic
147C2×#124—21no—
148#450—22no—
149#428—21no—
150#394—21no—
151#394C22×C5912yes—
152#393—22no—
153#392—22no—
154#382C22×C5911yescharacteristic
155#343C22×C5911yescharacteristic
156#339—21no—
157#328—22no—
158#324C22×C5912yes—
159C59#1225011yescharacteristic, direct product factor, Sylow
160C22×Q8—21no—
161C22×D4—21no—
162C22×D4—21no—
163C22×D4D4×C5912yes—
164C2×SD16—42no—
165C2×M16—21no—
166C4⋊Q8—41no—
167#125—41no—
168#124—42no—
169#124—42no—
170C22⋊Q8—22no—
171#117—21no—
172C2×C4⋊C4D4×C5911yescharacteristic
173C2×C4⋊C4C23×C5911yescharacteristic
174C2×C22⋊C4—22no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—22no—
178C2×C42D4×C5911yescharacteristic
179#107—41no—
180#106—41no—
181#102—42no—
182C22⋊C8—22no—
183#95—21no—
184#95—22no—
185#95D4×C5912yes—
186C24—22no—
187C2×Q8—22no—
188C2×Q8—42no—
189C2×Q8—81no—
190C2×Q8—81no—
191C2×D4—42no—
192C2×D4—42no—
193C2×D4—82no—
194C2×D4—81no—
195C2×D4—81no—
196C2×D4—82no—
197C2×D4—81no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—22no—
201C22×C4C2×D4×C5911yescharacteristic
202C22×C4C2×D4×C5912yes—
203SD16—84no—
204M16—22no—
205M16—41no—
206C2×C8—41no—
207C2×C8—42no—
208C4⋊C4—22no—
209C4⋊C4—21no—
210C4⋊C4—21no—
211C4⋊C4—41no—
212C4⋊C4—41no—
213C4⋊C4—42no—
214C22⋊C4—44no—
215C22⋊C4—41no—
216C22⋊C4—42no—
217C22⋊C4—42no—
218C22⋊C4—82no—
219C22⋊C4—82no—
220C42—21no—
221C42—41no—
222C42SD16×C5912yescomplemented
223C23—42no—
224C23—42no—
225C23—42no—
226C23—42no—
227C23—42no—
228C23—42no—
229C23—42no—
230C23#120×C5911yescharacteristic
231Q8—82no—
232Q8—84no—
233D4—44no—
234D4—82no—
235D4—84no—
236D4—84no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4C2×SD16×C5911yescharacteristic
252C2×C4AGL(1,ℤ/8)×C5911yescharacteristic
253C8—42no—
254C8—82no—
255C22—21no—
256C22—21no—
257C22—82no—
258C22—82no—
259C22—82no—
260C22—82no—
261C22—82no—
262C22—82no—
263C22—82no—
264C22C59×#45011yescharacteristic
265C22C59×#45611yescharacteristic
266C22C59×B(4,𝔽2)11yescharacteristic
267C4—21no—
268C4—21no—
269C4—42no—
270C4—42no—
271C4—82no—
272C4—81no—
273C4—81no—
274C4—81no—
275C2—21no—
276C2—21no—
277C2—162no—
278C2C59×#160411yescharacteristic
279C2C59×#219911yescharacteristic
280C2C59×#220211yescharacteristic
subgroupG/H
center, socleC22×C59B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23×C59
prime radicalC59×#2819C2
LCSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C59 ◁ C23×C59 ◁ C2×C4⋊C4×C59 ◁ G

Complements

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

NH ≅ G/Ntype
2. C59×#2061C2semidirect
3. C59×#1599C2semidirect
4. C59×#1345C2semidirect
5. C59×#1324C2semidirect
81. C42×C59SD16semidirect
114. #12250C59direct (N × H)
136. #2061C2×C59semidirect
137. #1599C2×C59semidirect
138. #1345C2×C59semidirect
139. #1324C2×C59semidirect
159. C59#12250direct (N × H)
222. C42SD16×C59semidirect

Maximal subgroups

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

subgroupindex
C59×#28192
C59×#20612
C59×#15992
C59×#13452
C59×#13242
#1225059