Finite Group Databasebeta

← C59×#11781 #28423890

Subgroups — C59×#11781 #28423890

Order 15104.

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

normal subgroups only

This group has 482 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
1C59×#2548C211yesmaximal, characteristic
2C4×#101×C59C211yesmaximal, characteristic, complemented
3C59×#1467C211yesmaximal, characteristic, complemented
4C59×#1345C214yesmaximal, complemented
5C2×#127×C59C411yescharacteristic
6C2×#124×C59—21no—
7C2×C4×Q8×C59C2211yescharacteristic
8C59×#432—21no—
9C2×#101×C59—21no—
10C4×M16×C59—21no—
11C59×#394C412yes—
12C59×#394C2212yes—
13C59×#390—21no—
14C59×#390—21no—
15C59×#390—21no—
16C4×C4⋊C4×C59C411yescharacteristic
17C59×#343C2212yes—
18C59×#324—21no—
19C59×#324C2212yes—
20C22×Q8×C59D411yescharacteristic
21C22×D4×C59—21no—
22C22×D4×C59—21no—
23C22×D4×C59C2×C412yes—
24C2×M16×C59—22no—
25#127×C59—41no—
26#127×C59—41no—
27#124×C59—44no—
28C4×Q8×C59—41no—
29C4×Q8×C59—41no—
30C2×C4⋊C4×C59—21no—
31C2×C4⋊C4×C59C2×C411yescharacteristic
32C2×C4⋊C4×C59D411yescharacteristic
33C2×C4⋊C4×C59C2311yescharacteristic
34C2×C22⋊C4×C59—22no—
35C2×C22⋊C4×C59—22no—
36C2×C22⋊C4×C59—22no—
37C2×C22⋊C4×C59—22no—
38C2×C42×C59—21no—
39C2×C42×C59C2×C411yescharacteristic
40C2×C42×C59D411yescharacteristic
41C2×C42×C59D411yescharacteristic
42C4⋊C8×C59—41no—
43#101×C59—24no—
44C22⋊C8×C59—22no—
45#97×C59—41no—
46#97×C59—41no—
47C4×C8×C59—41no—
48#95×C59C2×C412yes—
49C24×C59—22no—
50C2×Q8×C59—22no—
51C2×Q8×C59—22no—
52C2×Q8×C59—41no—
53C2×D4×C59—42no—
54C2×D4×C59—42no—
55C2×D4×C59—42no—
56C2×D4×C59—42no—
57C2×D4×C59—42no—
58C2×D4×C59—41no—
59C2×D4×C59—41no—
60C2×D4×C59—42no—
61C2×D4×C59—82no—
62C22×C4×C59—22no—
63C22×C4×C59C22⋊C411yescharacteristic
64C22×C4×C59C22⋊C411yescharacteristic
65C22×C4×C59C22⋊C411yescharacteristic
66C22×C4×C59C22⋊C411yescharacteristic
67C22×C4×C59C22×C411yescharacteristic
68C22×C4×C59C2×D411yescharacteristic
69C22×C4×C59C2×D411yescharacteristic
70M16×C59—24no—
71M16×C59—42no—
72C2×C8×C59—42no—
73C2×C8×C59—42no—
74C4⋊C4×C59—41no—
75C4⋊C4×C59—41no—
76C4⋊C4×C59—41no—
77C4⋊C4×C59—42no—
78C22⋊C4×C59—44no—
79C22⋊C4×C59—44no—
80C22⋊C4×C59—82no—
81C22⋊C4×C59—82no—
82C42×C59—21no—
83C42×C59—21no—
84C42×C59—21no—
85C42×C59—21no—
86C42×C59—21no—
87C42×C59—21no—
88C42×C59—42no—
89C23×C59—42no—
90C23×C59—42no—
91C23×C59—42no—
92C23×C59—42no—
93C23×C59—42no—
94C23×C59—42no—
95C23×C59—42no—
96C23×C59C2×C22⋊C411yescharacteristic
97Q8×C59—44no—
98D4×C59—82no—
99D4×C59—84no—
100D4×C59—84no—
101D4×C59—82no—
102C2×C4×C59—21no—
103C2×C4×C59—21no—
104C2×C4×C59—21no—
105C2×C4×C59—21no—
106C2×C4×C59—22no—
107C2×C4×C59—21no—
108C2×C4×C59—21no—
109C2×C4×C59—21no—
110C2×C4×C59—21no—
111C2×C4×C59—21no—
112C2×C4×C59—21no—
113C2×C4×C59—41no—
114C2×C4×C59—41no—
115C2×C4×C59—41no—
116C2×C4×C59—42no—
117C2×C4×C59—42no—
118C2×C4×C59—42no—
119C2×C4×C59#9911yescharacteristic
120C2×C4×C59#9911yescharacteristic
121C2×C4×C59#10412yes—
122C2×C4×C59AGL(1,ℤ/8)12yes—
123C8×C59—44no—
124C8×C59—82no—
125#11781C5911yesmaximal, characteristic, direct product factor, Sylow
126C22×C59—21no—
127C22×C59—21no—
128C22×C59—82no—
129C22×C59—82no—
130C22×C59—82no—
131C22×C59—82no—
132C22×C59—82no—
133C22×C59—82no—
134C22×C59—82no—
135C22×C59C2×#9911yescharacteristic
136C22×C59#41811yescharacteristic
137C22×C59C2×#10411yescharacteristic
138C4×C59—21no—
139C4×C59—22no—
140C4×C59—22no—
141C4×C59—41no—
142C4×C59—41no—
143C4×C59—41no—
144C4×C59—41no—
145C4×C59—82no—
146C4×C59#35512yes—
147#2548C2×C5911yescharacteristic
148C4×#101C2×C5911yescharacteristic, complemented
149#1467C2×C5911yescharacteristic, complemented
150#1345C2×C5914yescomplemented
151C2×C59—21no—
152C2×C59—21no—
153C2×C59—162no—
154C2×C59#150711yescharacteristic
155C2×C59#212211yescharacteristic
156C2×C59C2×#35511yescharacteristic
157C2×#127C4×C5911yescharacteristic
158C2×#124—21no—
159C2×C4×Q8C22×C5911yescharacteristic
160#432—21no—
161C2×#101—21no—
162C4×M16—21no—
163#394C4×C5912yes—
164#394C22×C5912yes—
165#390—21no—
166#390—21no—
167#390—21no—
168C4×C4⋊C4C4×C5911yescharacteristic
169#343C22×C5912yes—
170#324—21no—
171#324C22×C5912yes—
172C59#1178111yescharacteristic, direct product factor, Sylow
173C22×Q8D4×C5911yescharacteristic
174C22×D4—21no—
175C22×D4—21no—
176C22×D4C2×C4×C5912yes—
177C2×M16—22no—
178#127—41no—
179#127—41no—
180#124—44no—
181C4×Q8—41no—
182C4×Q8—41no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4C2×C4×C5911yescharacteristic
185C2×C4⋊C4D4×C5911yescharacteristic
186C2×C4⋊C4C23×C5911yescharacteristic
187C2×C22⋊C4—22no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—22no—
191C2×C42—21no—
192C2×C42C2×C4×C5911yescharacteristic
193C2×C42D4×C5911yescharacteristic
194C2×C42D4×C5911yescharacteristic
195C4⋊C8—41no—
196#101—24no—
197C22⋊C8—22no—
198#97—41no—
199#97—41no—
200C4×C8—41no—
201#95C2×C4×C5912yes—
202C24—22no—
203C2×Q8—22no—
204C2×Q8—22no—
205C2×Q8—41no—
206C2×D4—42no—
207C2×D4—42no—
208C2×D4—42no—
209C2×D4—42no—
210C2×D4—42no—
211C2×D4—41no—
212C2×D4—41no—
213C2×D4—42no—
214C2×D4—82no—
215C22×C4—22no—
216C22×C4C22⋊C4×C5911yescharacteristic
217C22×C4C22⋊C4×C5911yescharacteristic
218C22×C4C22⋊C4×C5911yescharacteristic
219C22×C4C22⋊C4×C5911yescharacteristic
220C22×C4C22×C4×C5911yescharacteristic
221C22×C4C2×D4×C5911yescharacteristic
222C22×C4C2×D4×C5911yescharacteristic
223M16—24no—
224M16—42no—
225C2×C8—42no—
226C2×C8—42no—
227C4⋊C4—41no—
228C4⋊C4—41no—
229C4⋊C4—41no—
230C4⋊C4—42no—
231C22⋊C4—44no—
232C22⋊C4—44no—
233C22⋊C4—82no—
234C22⋊C4—82no—
235C42—21no—
236C42—21no—
237C42—21no—
238C42—21no—
239C42—21no—
240C42—21no—
241C42—42no—
242C23—42no—
243C23—42no—
244C23—42no—
245C23—42no—
246C23—42no—
247C23—42no—
248C23—42no—
249C23C2×C22⋊C4×C5911yescharacteristic
250Q8—44no—
251D4—84no—
252D4—82no—
253D4—84no—
254D4—82no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4#99×C5911yescharacteristic
273C2×C4#99×C5911yescharacteristic
274C2×C4#104×C5912yes—
275C2×C4AGL(1,ℤ/8)×C5912yes—
276C8—44no—
277C8—82no—
278C22—21no—
279C22—21no—
280C22—82no—
281C22—82no—
282C22—82no—
283C22—82no—
284C22—82no—
285C22—82no—
286C22—82no—
287C22C2×#99×C5911yescharacteristic
288C22C59×#41811yescharacteristic
289C22C2×#104×C5911yescharacteristic
290C4—22no—
291C4—22no—
292C4—21no—
293C4—41no—
294C4—41no—
295C4—41no—
296C4—41no—
297C4—82no—
298C4C59×#35512yes—
299C2—21no—
300C2—21no—
301C2—162no—
302C2C59×#150711yescharacteristic
303C2C59×#212211yescharacteristic
304C2C2×C59×#35511yescharacteristic
subgroupG/H
center, socleC22×C59C2×#99
commutatorC22×C4C22×C4×C59
Frattini, cosocleC2×C4⋊C4C23×C59
prime radicalC2×#127×C59C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C59 ◁ C22×C4×C59 ◁ C2×C4×Q8×C59 ◁ G

Complements

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

NH ≅ G/Ntype
2. C4×#101×C59C2semidirect
3. C59×#1467C2semidirect
4. C59×#1345C2semidirect
125. #11781C59direct (N × H)
148. C4×#101C2×C59semidirect
149. #1467C2×C59semidirect
150. #1345C2×C59semidirect
172. C59#11781direct (N × H)

Maximal subgroups

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

subgroupindex
C59×#25482
C4×#101×C592
C59×#14672
C59×#13452
#1178159