Finite Group Databasebeta

← C59×#11784 #28423893

Subgroups — C59×#11784 #28423893

Order 15104.

460 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 102 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×#2580C211yesmaximal, characteristic
2C4×#99×C59C211yesmaximal, characteristic
3C59×#1465C211yesmaximal, characteristic
4C59×#1352C212yesmaximal, complemented
5C59×#1351C212yesmaximal, complemented
6C2×C4×Q8×C59C411yescharacteristic, complemented
7C2×C4×D4×C59C2211yescharacteristic
8C2×#99×C59—21no—
9C59×#407—21no—
10C59×#399C411yescharacteristic, complemented
11C59×#399C2211yescharacteristic
12C59×#393C411yescharacteristic, complemented
13C59×#393C2211yescharacteristic
14C59×#390—21no—
15C59×#389—21no—
16C59×#385—21no—
17C59×#385—21no—
18C59×#382C411yescharacteristic, complemented
19C4×C22⋊C4×C59—21no—
20C59×#342C2212yes—
21C59×#324—21no—
22C59×#323C2212yes—
23C22×Q8×C59C2×C411yescharacteristic
24C22×D4×C59D411yescharacteristic
25C23×C4×C59D411yescharacteristic
26C23×C4×C59D411yescharacteristic
27C2×M16×C59—41no—
28C4×Q8×C59—41no—
29C4×Q8×C59—41no—
30C4×D4×C59—41no—
31C4×D4×C59—41no—
32C2×C4⋊C4×C59—21no—
33C2×C4⋊C4×C59—21no—
34C2×C4⋊C4×C59—21no—
35C2×C4⋊C4×C59—21no—
36C2×C4⋊C4×C59C2×C411yescharacteristic
37C2×C4⋊C4×C59D411yescharacteristic
38C2×C22⋊C4×C59—22no—
39C2×C22⋊C4×C59—21no—
40C2×C22⋊C4×C59—21no—
41C2×C22⋊C4×C59—21no—
42C2×C22⋊C4×C59—21no—
43C2×C22⋊C4×C59C2×C411yescharacteristic
44C2×C22⋊C4×C59C2311yescharacteristic
45C2×C42×C59—21no—
46C2×C42×C59—41no—
47C2×C42×C59C2×C411yescharacteristic
48#99×C59—24no—
49C22⋊C8×C59—22no—
50C22⋊C8×C59—41no—
51#95×C59—21no—
52#95×C59—21no—
53#95×C59—21no—
54#95×C59—41no—
55#95×C59C2×C411yescharacteristic
56#95×C59C2×C411yescharacteristic
57C24×C59C22⋊C411yescharacteristic
58C24×C59C2×D411yescharacteristic
59C2×Q8×C59—42no—
60C2×Q8×C59—41no—
61C2×D4×C59—22no—
62C2×D4×C59—22no—
63C2×D4×C59—41no—
64C22×C4×C59—21no—
65C22×C4×C59—21no—
66C22×C4×C59—21no—
67C22×C4×C59—21no—
68C22×C4×C59—21no—
69C22×C4×C59—21no—
70C22×C4×C59—21no—
71C22×C4×C59—21no—
72C22×C4×C59—42no—
73C22×C4×C59—81no—
74C22×C4×C59—81no—
75C22×C4×C59C22⋊C411yescharacteristic
76C22×C4×C59C22⋊C411yescharacteristic
77C22×C4×C59C22⋊C411yescharacteristic
78C22×C4×C59C22×C411yescharacteristic
79C22×C4×C59C2×D411yescharacteristic
80M16×C59—82no—
81C2×C8×C59—42no—
82C4⋊C4×C59—41no—
83C4⋊C4×C59—41no—
84C4⋊C4×C59—41no—
85C4⋊C4×C59—41no—
86C4⋊C4×C59—42no—
87C4⋊C4×C59—81no—
88C4⋊C4×C59—81no—
89C22⋊C4×C59—24no—
90C22⋊C4×C59—41no—
91C22⋊C4×C59—41no—
92C22⋊C4×C59—42no—
93C22⋊C4×C59—42no—
94C22⋊C4×C59—42no—
95C22⋊C4×C59—81no—
96C22⋊C4×C59—81no—
97C42×C59—21no—
98C42×C59—21no—
99C42×C59—42no—
100C42×C59—82no—
101C23×C59—21no—
102C23×C59—21no—
103C23×C59—21no—
104C23×C59—21no—
105C23×C59—21no—
106C23×C59—41no—
107C23×C59—41no—
108C23×C59—41no—
109C23×C59—41no—
110C23×C59C2×C22⋊C411yescharacteristic
111C23×C59#13712yes—
112Q8×C59—42no—
113Q8×C59—42no—
114D4×C59—44no—
115C2×C4×C59—21no—
116C2×C4×C59—21no—
117C2×C4×C59—22no—
118C2×C4×C59—21no—
119C2×C4×C59—21no—
120C2×C4×C59—21no—
121C2×C4×C59—41no—
122C2×C4×C59—42no—
123C2×C4×C59—42no—
124C2×C4×C59—41no—
125C2×C4×C59—41no—
126C2×C4×C59—41no—
127C2×C4×C59—41no—
128C2×C4×C59—41no—
129C2×C4×C59—44no—
130C2×C4×C59—41no—
131C2×C4×C59—41no—
132C2×C4×C59—41no—
133C2×C4×C59—41no—
134C2×C4×C59—41no—
135C2×C4×C59—41no—
136C2×C4×C59—41no—
137C2×C4×C59—41no—
138C2×C4×C59—41no—
139C2×C4×C59—81no—
140C2×C4×C59—81no—
141C2×C4×C59—82no—
142C2×C4×C59—82no—
143C2×C4×C59#9911yescharacteristic
144C2×C4×C59#9911yescharacteristic
145C2×C4×C59#10412yes—
146C8×C59—82no—
147#11784C5911yesmaximal, characteristic, direct product factor, Sylow
148C22×C59—21no—
149C22×C59—21no—
150C22×C59—22no—
151C22×C59—42no—
152C22×C59—42no—
153C22×C59—41no—
154C22×C59—41no—
155C22×C59—41no—
156C22×C59—41no—
157C22×C59—41no—
158C22×C59—81no—
159C22×C59—81no—
160C22×C59C2×#9911yescharacteristic
161C22×C59#41911yescharacteristic
162C22×C59C2×#10411yescharacteristic
163C4×C59—21no—
164C4×C59—21no—
165C4×C59—22no—
166C4×C59—42no—
167C4×C59—41no—
168C4×C59—41no—
169C4×C59—41no—
170C4×C59—81no—
171C4×C59—81no—
172C4×C59—84no—
173C4×C59—81no—
174#2580C2×C5911yescharacteristic
175C4×#99C2×C5911yescharacteristic
176#1465C2×C5911yescharacteristic
177#1352C2×C5912yescomplemented
178#1351C2×C5912yescomplemented
179C2×C59—21no—
180C2×C59—21no—
181C2×C59—42no—
182C2×C59—81no—
183C2×C59#150811yescharacteristic
184C2×C59#212011yescharacteristic
185C2×C59#212711yescharacteristic
186C2×C4×Q8C4×C5911yescharacteristic, complemented
187C2×C4×D4C22×C5911yescharacteristic
188C2×#99—21no—
189#407—21no—
190#399C4×C5911yescharacteristic, complemented
191#399C22×C5911yescharacteristic
192#393C4×C5911yescharacteristic, complemented
193#393C22×C5911yescharacteristic
194#390—21no—
195#389—21no—
196#385—21no—
197#385—21no—
198#382C4×C5911yescharacteristic, complemented
199C4×C22⋊C4—21no—
200#342C22×C5912yes—
201#324—21no—
202#323C22×C5912yes—
203C59#1178411yescharacteristic, direct product factor, Sylow
204C22×Q8C2×C4×C5911yescharacteristic
205C22×D4D4×C5911yescharacteristic
206C23×C4D4×C5911yescharacteristic
207C23×C4D4×C5911yescharacteristic
208C2×M16—41no—
209C4×Q8—41no—
210C4×Q8—41no—
211C4×D4—41no—
212C4×D4—41no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4C2×C4×C5911yescharacteristic
218C2×C4⋊C4D4×C5911yescharacteristic
219C2×C22⋊C4—22no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4C2×C4×C5911yescharacteristic
225C2×C22⋊C4C23×C5911yescharacteristic
226C2×C42—21no—
227C2×C42—41no—
228C2×C42C2×C4×C5911yescharacteristic
229#99—24no—
230C22⋊C8—22no—
231C22⋊C8—41no—
232#95—21no—
233#95—21no—
234#95—21no—
235#95—41no—
236#95C2×C4×C5911yescharacteristic
237#95C2×C4×C5911yescharacteristic
238C24C22⋊C4×C5911yescharacteristic
239C24C2×D4×C5911yescharacteristic
240C2×Q8—42no—
241C2×Q8—41no—
242C2×D4—22no—
243C2×D4—22no—
244C2×D4—41no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—42no—
254C22×C4—81no—
255C22×C4—81no—
256C22×C4C22⋊C4×C5911yescharacteristic
257C22×C4C22⋊C4×C5911yescharacteristic
258C22×C4C22⋊C4×C5911yescharacteristic
259C22×C4C22×C4×C5911yescharacteristic
260C22×C4C2×D4×C5911yescharacteristic
261M16—82no—
262C2×C8—42no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—42no—
268C4⋊C4—81no—
269C4⋊C4—81no—
270C22⋊C4—24no—
271C22⋊C4—41no—
272C22⋊C4—41no—
273C22⋊C4—42no—
274C22⋊C4—42no—
275C22⋊C4—42no—
276C22⋊C4—81no—
277C22⋊C4—81no—
278C42—21no—
279C42—21no—
280C42—42no—
281C42—82no—
282C23—21no—
283C23—21no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23—41no—
288C23—41no—
289C23—41no—
290C23—41no—
291C23C2×C22⋊C4×C5911yescharacteristic
292C23#137×C5912yes—
293Q8—42no—
294Q8—42no—
295D4—44no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—22no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—41no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—44no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—81no—
321C2×C4—81no—
322C2×C4—82no—
323C2×C4—82no—
324C2×C4#99×C5911yescharacteristic
325C2×C4#99×C5911yescharacteristic
326C2×C4#104×C5912yes—
327C8—82no—
328C22—21no—
329C22—21no—
330C22—22no—
331C22—42no—
332C22—42no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—81no—
339C22—81no—
340C22C2×#99×C5911yescharacteristic
341C22C59×#41911yescharacteristic
342C22C2×#104×C5911yescharacteristic
343C4—22no—
344C4—21no—
345C4—21no—
346C4—42no—
347C4—41no—
348C4—41no—
349C4—41no—
350C4—81no—
351C4—81no—
352C4—84no—
353C4—81no—
354C2—21no—
355C2—21no—
356C2—42no—
357C2—81no—
358C2C59×#150811yescharacteristic
359C2C59×#212011yescharacteristic
360C2C59×#212711yescharacteristic
subgroupG/H
center, socleC22×C59C2×#99
commutatorC22×C4C22×C4×C59
Frattini, cosocleC2×C22⋊C4C23×C59
prime radicalC22×D4×C59D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C59 ◁ C22×C4×C59 ◁ C2×C4×D4×C59 ◁ G

Complements

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

NH ≅ G/Ntype
4. C59×#1352C2semidirect
5. C59×#1351C2semidirect
6. C2×C4×Q8×C59C4semidirect
10. C59×#399C4semidirect
12. C59×#393C4semidirect
18. C59×#382C4semidirect
147. #11784C59direct (N × H)
177. #1352C2×C59semidirect
178. #1351C2×C59semidirect
186. C2×C4×Q8C4×C59semidirect
190. #399C4×C59semidirect
192. #393C4×C59semidirect
198. #382C4×C59semidirect
203. C59#11784direct (N × H)

Maximal subgroups

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

subgroupindex
C59×#25802
C4×#99×C592
C59×#14652
C59×#13522
C59×#13512
#1178459