Finite Group Databasebeta

← C59×#9453 #28421562

Subgroups — C59×#9453 #28421562

Order 15104.

498 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 498 conjugacy classes of subgroups — too many to draw. Only its 126 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×#2973C211yesmaximal, characteristic
2C59×#2563C211yesmaximal, characteristic
3C59×#1556C212yesmaximal, complemented
4C59×#1466C211yesmaximal, characteristic
5C59×#1285C212yesmaximal, complemented
6C2×C4⋊Q8×C59C2211yescharacteristic
7C2×C4×D4×C59C2211yescharacteristic
8C59×#435—24no—
9C59×#432C2212yes—
10C59×#432C2212yes—
11C59×#407—21no—
12C2×C22⋊C8×C59—21no—
13C59×#393—21no—
14C59×#393—21no—
15C59×#390C412yes—
16C59×#382C412yes—
17C59×#380—21no—
18C59×#380—21no—
19C4×C4⋊C4×C59C2211yescharacteristic
20C59×#332—24no—
21C59×#323—21no—
22C59×#323—21no—
23C22×Q8×C59—21no—
24C22×D4×C59C2×C411yescharacteristic
25C23×C4×C59D411yescharacteristic
26C23×C4×C59D411yescharacteristic
27C2×M16×C59—22no—
28C22×C8×C59—22no—
29C4⋊Q8×C59—22no—
30C4⋊Q8×C59—22no—
31C4×D4×C59—22no—
32C4×D4×C59—22no—
33C2×C4⋊C4×C59—21no—
34C2×C4⋊C4×C59—21no—
35C2×C4⋊C4×C59C2×C412yes—
36C2×C4⋊C4×C59C2×C411yescharacteristic
37C2×C22⋊C4×C59—22no—
38C2×C22⋊C4×C59—22no—
39C2×C22⋊C4×C59D411yescharacteristic
40C2×C22⋊C4×C59D411yescharacteristic
41C2×C42×C59C2×C412yes—
42C2×C42×C59C2311yescharacteristic
43C4⋊C8×C59—22no—
44C4⋊C8×C59D414yes—
45C22⋊C8×C59—22no—
46C22⋊C8×C59—22no—
47C22⋊C8×C59—22no—
48C22⋊C8×C59—22no—
49#97×C59—22no—
50#97×C59—22no—
51#95×C59—22no—
52#95×C59—22no—
53#95×C59—21no—
54C24×C59C22⋊C411yescharacteristic
55C24×C59C22⋊C411yescharacteristic
56C2×Q8×C59—42no—
57C2×Q8×C59—82no—
58C2×D4×C59—22no—
59C2×D4×C59—22no—
60C2×D4×C59—41no—
61C22×C4×C59—21no—
62C22×C4×C59—21no—
63C22×C4×C59—21no—
64C22×C4×C59—21no—
65C22×C4×C59—22no—
66C22×C4×C59—22no—
67C22×C4×C59—42no—
68C22×C4×C59—42no—
69C22×C4×C59C22⋊C411yescharacteristic
70C22×C4×C59C22⋊C411yescharacteristic
71C22×C4×C59C22×C411yescharacteristic
72C22×C4×C59C2×D411yescharacteristic
73C22×C4×C59C2×D411yescharacteristic
74M16×C59—82no—
75C2×C8×C59—24no—
76C2×C8×C59—42no—
77C2×C8×C59—82no—
78C4⋊C4×C59—22no—
79C4⋊C4×C59—24no—
80C4⋊C4×C59—42no—
81C4⋊C4×C59—41no—
82C4⋊C4×C59—41no—
83C22⋊C4×C59—22no—
84C22⋊C4×C59—22no—
85C22⋊C4×C59—82no—
86C22⋊C4×C59—82no—
87C42×C59—42no—
88C42×C59C2×D412yes—
89C42×C59#4112yes—
90C23×C59—21no—
91C23×C59—21no—
92C23×C59—21no—
93C23×C59—21no—
94C23×C59—21no—
95C23×C59—41no—
96C23×C59—41no—
97C23×C59—41no—
98C23×C59—41no—
99C23×C59#10112yes—
100C23×C59C2×C22⋊C411yescharacteristic
101Q8×C59—84no—
102D4×C59—44no—
103C2×C4×C59—21no—
104C2×C4×C59—21no—
105C2×C4×C59—22no—
106C2×C4×C59—21no—
107C2×C4×C59—21no—
108C2×C4×C59—41no—
109C2×C4×C59—42no—
110C2×C4×C59—42no—
111C2×C4×C59—41no—
112C2×C4×C59—41no—
113C2×C4×C59—42no—
114C2×C4×C59—42no—
115C2×C4×C59—42no—
116C2×C4×C59—42no—
117C2×C4×C59—42no—
118C2×C4×C59—42no—
119C2×C4×C59—81no—
120C2×C4×C59—81no—
121C2×C4×C59C4×D412yes—
122C2×C4×C59#12011yescharacteristic
123C2×C4×C59#12112yes—
124C2×C4×C59#12311yescharacteristic
125C8×C59—44no—
126C8×C59—82no—
127#9453C5911yesmaximal, characteristic, direct product factor, Sylow
128C22×C59—21no—
129C22×C59—21no—
130C22×C59—22no—
131C22×C59—42no—
132C22×C59—42no—
133C22×C59—41no—
134C22×C59—41no—
135C22×C59—41no—
136C22×C59—41no—
137C22×C59—41no—
138C22×C59—81no—
139C22×C59—81no—
140C22×C59#38611yescharacteristic
141C22×C59C2×#10111yescharacteristic
142C22×C59#41311yescharacteristic
143C4×C59—21no—
144C4×C59—22no—
145C4×C59—22no—
146C4×C59—41no—
147C4×C59—41no—
148C4×C59—82no—
149C4×C59—82no—
150C4×C59#45512yes—
151#2973C2×C5911yescharacteristic
152#2563C2×C5911yescharacteristic
153#1556C2×C5912yescomplemented
154#1466C2×C5911yescharacteristic
155#1285C2×C5912yescomplemented
156C2×C59—21no—
157C2×C59—21no—
158C2×C59—42no—
159C2×C59—81no—
160C2×C59#188511yescharacteristic
161C2×C59#190111yescharacteristic
162C2×C59#191011yescharacteristic
163C2×C4⋊Q8C22×C5911yescharacteristic
164C2×C4×D4C22×C5911yescharacteristic
165#435—24no—
166#432C22×C5912yes—
167#432C22×C5912yes—
168#407—21no—
169C2×C22⋊C8—21no—
170#393—21no—
171#393—21no—
172#390C4×C5912yes—
173#382C4×C5912yes—
174#380—21no—
175#380—21no—
176C4×C4⋊C4C22×C5911yescharacteristic
177#332—24no—
178#323—21no—
179#323—21no—
180C59#945311yescharacteristic, direct product factor, Sylow
181C22×Q8—21no—
182C22×D4C2×C4×C5911yescharacteristic
183C23×C4D4×C5911yescharacteristic
184C23×C4D4×C5911yescharacteristic
185C2×M16—22no—
186C22×C8—22no—
187C4⋊Q8—22no—
188C4⋊Q8—22no—
189C4×D4—22no—
190C4×D4—22no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4C2×C4×C5912yes—
194C2×C4⋊C4C2×C4×C5911yescharacteristic
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4D4×C5911yescharacteristic
198C2×C22⋊C4D4×C5911yescharacteristic
199C2×C42C2×C4×C5912yes—
200C2×C42C23×C5911yescharacteristic
201C4⋊C8—22no—
202C4⋊C8D4×C5914yes—
203C22⋊C8—22no—
204C22⋊C8—22no—
205C22⋊C8—22no—
206C22⋊C8—22no—
207#97—22no—
208#97—22no—
209#95—22no—
210#95—22no—
211#95—21no—
212C24C22⋊C4×C5911yescharacteristic
213C24C22⋊C4×C5911yescharacteristic
214C2×Q8—42no—
215C2×Q8—82no—
216C2×D4—22no—
217C2×D4—22no—
218C2×D4—41no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4C22⋊C4×C5911yescharacteristic
228C22×C4C22⋊C4×C5911yescharacteristic
229C22×C4C22×C4×C5911yescharacteristic
230C22×C4C2×D4×C5911yescharacteristic
231C22×C4C2×D4×C5911yescharacteristic
232M16—82no—
233C2×C8—24no—
234C2×C8—42no—
235C2×C8—82no—
236C4⋊C4—22no—
237C4⋊C4—24no—
238C4⋊C4—42no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C22⋊C4—22no—
242C22⋊C4—22no—
243C22⋊C4—82no—
244C22⋊C4—82no—
245C42—42no—
246C42C2×D4×C5912yes—
247C42#41×C5912yes—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—21no—
252C23—21no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23#101×C5912yes—
258C23C2×C22⋊C4×C5911yescharacteristic
259Q8—84no—
260D4—44no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—41no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—81no—
278C2×C4—81no—
279C2×C4C4×D4×C5912yes—
280C2×C4#120×C5911yescharacteristic
281C2×C4#121×C5912yes—
282C2×C4#123×C5911yescharacteristic
283C8—44no—
284C8—82no—
285C22—21no—
286C22—21no—
287C22—22no—
288C22—42no—
289C22—42no—
290C22—41no—
291C22—41no—
292C22—41no—
293C22—41no—
294C22—41no—
295C22—81no—
296C22—81no—
297C22C59×#38611yescharacteristic
298C22C2×#101×C5911yescharacteristic
299C22C59×#41311yescharacteristic
300C4—21no—
301C4—22no—
302C4—22no—
303C4—41no—
304C4—41no—
305C4—82no—
306C4—82no—
307C4C59×#45512yes—
308C2—21no—
309C2—21no—
310C2—42no—
311C2—81no—
312C2C59×#188511yescharacteristic
313C2C59×#190111yescharacteristic
314C2C59×#191011yescharacteristic
subgroupG/H
center, socleC22×C59#386
commutatorC22×C4C22×C4×C59
Frattini, cosocleC2×C42C23×C59
prime radicalC22×D4×C59C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C59 ◁ C2×C42×C59 ◁ G

Complements

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

NH ≅ G/Ntype
3. C59×#1556C2semidirect
5. C59×#1285C2semidirect
127. #9453C59direct (N × H)
153. #1556C2×C59semidirect
155. #1285C2×C59semidirect
180. C59#9453direct (N × H)

Maximal subgroups

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

subgroupindex
C59×#29732
C59×#25632
C59×#15562
C59×#14662
C59×#12852
#945359