Finite Group Databasebeta

← #28558604

Subgroups — #28558604

Order 15104.

698 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21434104C211yesmaximal, characteristic
2#21433026C211yesmaximal, characteristic, complemented
3C2×D4×C59⋊C8C211yesmaximal, characteristic
4#21422950C214yesmaximal, complemented
5#21422905C214yesmaximal, complemented
6#21422872C211yesmaximal, characteristic, complemented
7#21422500C211yesmaximal, characteristic, complemented
8#21422389C211yesmaximal, characteristic, complemented
9C59×#1486C211yesmaximal, characteristic
10C23×C59⋊C8—21no—
11#16837959C2211yescharacteristic
12#16837732C412yes—
13#16837692C412yes—
14#16837607C412yes—
15#16837553C412yes—
16#16837529—21no—
17#16837529C2212yescomplemented
18#16837526—21no—
19#16837526C2212yescomplemented
20C4×#11745670C2214yes—
21D4×C59⋊C8—21no—
22D4×C59⋊C8—21no—
23D4×C59⋊C8C2212yes—
24D4×C59⋊C8C2212yes—
25#16837455C2211yescharacteristic
26#16837430—21no—
27C2×C4⋊(C59⋊C8)C2211yescharacteristic
28C2×C22⋊(C59⋊C8)—21no—
29#16837360—21no—
30C2×C4×C59⋊C8C2211yescharacteristic
31#16837154C2214yescomplemented
32#16837134C2214yescomplemented
33#16837124C2212yescomplemented
34#16837123C2212yescomplemented
35C2×C4×D4×C59C2211yescharacteristic
36C59×#431C2211yescharacteristic
37C59×#423C2211yescharacteristic
38C59×D4⋊C8C2214yes—
39C2×#11745670C2×C414yes—
40C22×C59⋊C8—21no—
41C22×C59⋊C8—22no—
42C22×C59⋊C8—22no—
43C22×C59⋊C8—44no—
44C22×C59⋊C8D411yescharacteristic
45C22×C59⋊C8D411yescharacteristic
46#14273727C2×C411yescharacteristic
47#14273727C2×C411yescharacteristic
48#14273719—21no—
49#14273706—21no—
50#14273706C2×C412yes—
51C4×Dic118—21no—
52C4×Dic118C2312yes—
53#14273686—21no—
54#14273658C2×C414yes—
55C4⋊(C59⋊C8)—21no—
56C4⋊(C59⋊C8)C2×C412yes—
57C4⋊(C59⋊C8)C2312yes—
58Dic59⋊C8—22no—
59Dic59⋊C8—22no—
60#14273650C2×C414yes—
61#14273648C2×C414yes—
62C22⋊(C59⋊C8)—22no—
63C22⋊(C59⋊C8)—22no—
64C22⋊(C59⋊C8)C2×C412yes—
65#14273637—22no—
66C4×C59⋊C8D411yescharacteristic
67C4×C59⋊C8D411yescharacteristic
68C4×C59⋊C8C2312yes—
69C4×C59⋊C8C2311yescharacteristic
70C4×C59⋊C8C2311yescharacteristic
71C8×Dic59—22no—
72C22×D4×C59C2×C411yescharacteristic
73C23×C4×C59—21no—
74C2×M16×C59—21no—
75C4×D4×C59—21no—
76C4×D4×C59—21no—
77C4×D4×C59C2×C412yes—
78C4×D4×C59C2312yes—
79C2×C4⋊C4×C59C2×C411yescharacteristic
80C2×C22⋊C4×C59—21no—
81C2×C42×C59C2311yescharacteristic
82C4⋊C8×C59—21no—
83C4⋊C8×C59—21no—
84C4⋊C8×C59C2312yes—
85C22⋊C8×C59—21no—
86C4×C8×C59C2312yes—
87C24×C59—21no—
88C2×D4×C59—24no—
89C2×D4×C59—21no—
90C2×D4×C59C22×C412yes—
91C22×C4×C59—21no—
92C22×C4×C59—22no—
93C22×C4×C59—22no—
94C22×C4×C59—42no—
95C22×C4×C59—42no—
96C22×C4×C59C22×C411yescharacteristic
97C22×C4×C59C2×D411yescharacteristic
98C22×C4×C59#4111yescharacteristic
99C2×Dic118—21no—
100C2×Dic118C22×C412yes—
101M16×C59—42no—
102C2×C8×C59—22no—
103C2×C8×C59—22no—
104C4⋊C4×C59—21no—
105C4⋊C4×C59C22×C412yes—
106C22⋊C4×C59—22no—
107C22⋊C4×C59—22no—
108C42×C59C22×C411yescharacteristic
109C42×C59C2×D411yescharacteristic
110C42×C59#4111yescharacteristic
111C42×C59C2411yescharacteristic
112C22⋊Dic59—22no—
113#11745670—28no—
114C4⋊Dic59—21no—
115C4⋊Dic59—21no—
116C4⋊Dic59—21no—
117C4⋊Dic59C22×C412yes—
118Dic59⋊C4—22no—
119Dic59⋊C4—22no—
120C4×Dic59—22no—
121C2×C59⋊C8—22no—
122C2×C59⋊C8—22no—
123C2×C59⋊C8—22no—
124C2×C59⋊C8—44no—
125C2×C59⋊C8—44no—
126C2×C59⋊C8—42no—
127C2×C59⋊C8C22×C414yes—
128C2×C59⋊C8C2×D411yescharacteristic
129C2×C59⋊C8C2×D411yescharacteristic
130C2×C59⋊C8C2×D411yescharacteristic
131C2×C59⋊C8C2×D411yescharacteristic
132C23×C59—24no—
133C23×C59—22no—
134C23×C59—42no—
135C23×C59—42no—
136C23×C59C4×D411yescharacteristic
137D4×C59—24no—
138D4×C59—22no—
139D4×C59#13114yes—
140C2×C4×C59—22no—
141C2×C4×C59—22no—
142C2×C4×C59—41no—
143C2×C4×C59—42no—
144C2×C4×C59—42no—
145C2×C4×C59—42no—
146C2×C4×C59—44no—
147C2×C4×C59C4×D411yescharacteristic
148C2×C4×C59C4×D411yescharacteristic
149C2×C4×C59C4×D411yescharacteristic
150C2×C4×C59AGL(1,ℤ/8)12yes—
151C2×C4×C59#13712yes—
152C2×C4×C59C23×C411yescharacteristic
153C2×C4×C59C22×D411yescharacteristic
154C2×C4×C59C2×#4111yescharacteristic
155C2×Dic59—22no—
156C2×Dic59—22no—
157Dic118—24no—
158Dic118—42no—
159C8×C59—42no—
160C8×C59—42no—
161C59⋊C8—24no—
162C59⋊C8—21no—
163C59⋊C8—21no—
164C59⋊C8—21no—
165C59⋊C8—21no—
166C59⋊C8—42no—
167#17905—591nomaximal, Sylow
168C22×C59—21no—
169C22×C59—21no—
170C22×C59—24no—
171C22×C59—48no—
172C22×C59—42no—
173C22×C59—42no—
174C22×C59#44012yes—
175C22×C59C2×C4×D411yescharacteristic
176C4×C59—21no—
177C4×C59—21no—
178C4×C59—21no—
179C4×C59—42no—
180C4×C59#44012yes—
181C4×C59C2×#13112yes—
182C4×C59C2×AGL(1,ℤ/8)11yescharacteristic
183C4×C59C2×#13711yescharacteristic
184Dic59—42no—
185Dic59—42no—
186#3101—591no—
187#2960—591no—
188C2×C8×D4—591no—
189#1585—594no—
190SD16⋊C8—594no—
191#1571—591no—
192#1486D5911yescharacteristic
193#1481—591no—
194#1452—591no—
195C2×C59—22no—
196C2×C59—44no—
197C2×C59#292911yescharacteristic
198C2×C59C2×#44011yescharacteristic
199C2×C59#295511yescharacteristic
200C23×C8—1181no—
201#533—591no—
202C2×C4×D4C2×D5911yescharacteristic
203#483—592no—
204#476—592no—
205#461—592no—
206#450—592no—
207#446—592no—
208#446—1181no—
209C8×Q8—592no—
210C8×Q8—1181no—
211C4×SD16—594no—
212C8×D4—592no—
213C8×D4—592no—
214C8×D4—1181no—
215C8×D4—1181no—
216#431—591no—
217#431C2×D5911yescharacteristic
218#424—1181no—
219#423C2×D5911yescharacteristic
220C2×C4⋊C8—591no—
221C2×C22⋊C8—1181no—
222#405—1181no—
223C2×C4×C8—591no—
224#335—594no—
225Q8⋊C8—594no—
226D4⋊C8C2×D5914yes—
227#322—592no—
228#322—592no—
229C59#1790511yescharacteristic, complemented, Sylow
230C22×D4C4×D5911yescharacteristic
231C23×C4—21no—
232C2×SD16—594no—
233C2×M16—21no—
234C22×C8—591no—
235C22×C8—591no—
236C22×C8—1182no—
237C22×C8—1182no—
238C22×C8—1181no—
239C22×C8—2364no—
240C4⋊Q8—591no—
241C4⋊Q8—591no—
242#125—1181no—
243C22⋊Q8—592no—
244C22⋊Q8—1181no—
245C4×Q8—592no—
246C4×Q8—1181no—
247C4×D4—21no—
248C4×D4—21no—
249C4×D4C4×D5912yes—
250C4×D4C22×D5912yes—
251#117—1181no—
252C2×C4⋊C4C4×D5911yescharacteristic
253C2×C22⋊C4—21no—
254C2×C42C22×D5911yescharacteristic
255#106—594no—
256C4⋊C8—21no—
257C4⋊C8—21no—
258C4⋊C8—592no—
259C4⋊C8—592no—
260C4⋊C8—1182no—
261C4⋊C8—1182no—
262C4⋊C8—1181no—
263C4⋊C8C22×D5912yes—
264#103—594no—
265#102—594no—
266C22⋊C8—21no—
267C22⋊C8—592no—
268C22⋊C8—1182no—
269C22⋊C8—1182no—
270#97—1182no—
271C4×C8—591no—
272C4×C8—591no—
273C4×C8—592no—
274C4×C8—591no—
275C4×C8—591no—
276C4×C8—1182no—
277C4×C8C22×D5912yes—
278C24—21no—
279C2×Q8—592no—
280C2×Q8—1181no—
281C2×D4—24no—
282C2×D4—21no—
283C2×D4C2×C4×D5912yes—
284C22×C4—21no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4C2×C4×D5911yescharacteristic
290C22×C4D4×D5911yescharacteristic
291C22×C4#1174568711yescharacteristic
292SD16—1188no—
293M16—42no—
294C2×C8—22no—
295C2×C8—22no—
296C2×C8—594no—
297C2×C8—591no—
298C2×C8—591no—
299C2×C8—591no—
300C2×C8—591no—
301C2×C8—1182no—
302C2×C8—1182no—
303C2×C8—1182no—
304C2×C8—2364no—
305C2×C8—2364no—
306C2×C8—2362no—
307C4⋊C4—21no—
308C4⋊C4—592no—
309C4⋊C4—1181no—
310C4⋊C4—1181no—
311C4⋊C4—1182no—
312C4⋊C4—1181no—
313C4⋊C4—1182no—
314C4⋊C4C2×C4×D5912yes—
315C22⋊C4—22no—
316C22⋊C4—22no—
317C22⋊C4—1182no—
318C42—1182no—
319C42C2×C4×D5911yescharacteristic
320C42D4×D5911yescharacteristic
321C42#1174568711yescharacteristic
322C42C23×D5911yescharacteristic
323C23—24no—
324C23—22no—
325C23—42no—
326C23—42no—
327C23(C22⋊D59)⋊C411yescharacteristic
328Q8—1184no—
329Q8—2362no—
330D4—24no—
331D4—22no—
332D4#1427373514yes—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—41no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—44no—
340C2×C4—1182no—
341C2×C4—1182no—
342C2×C4(C22⋊D59)⋊C411yescharacteristic
343C2×C4(C22⋊D59)⋊C411yescharacteristic
344C2×C4(C22⋊D59)⋊C411yescharacteristic
345C2×C4#1427375112yes—
346C2×C4#1427375612yes—
347C2×C4C22×C4×D5911yescharacteristic
348C2×C4C2×D4×D5911yescharacteristic
349C2×C4C2×#1174568711yescharacteristic
350C8—42no—
351C8—42no—
352C8—1184no—
353C8—1181no—
354C8—1181no—
355C8—1181no—
356C8—1181no—
357C8—2362no—
358C22—21no—
359C22—21no—
360C22—24no—
361C22—48no—
362C22—42no—
363C22—42no—
364C22#1683749912yes—
365C22C2×(C22⋊D59)⋊C411yescharacteristic
366C4—21no—
367C4—21no—
368C4—21no—
369C4—42no—
370C4—2362no—
371C4—2362no—
372C4#1683749912yes—
373C4C2×#1427373512yes—
374C4C2×#1427375111yescharacteristic
375C4C2×#1427375611yescharacteristic
376C2—22no—
377C2—44no—
378C2#2143278611yescharacteristic
379C2C2×#1683749911yescharacteristic
380C2#2143299511yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC42C23×D59
FittingC59×#1486C2
socleC22×C59C2×C4×D4
prime radicalC22×D4×C59C2×C4
cosocleC42×C59C24
hypercenter#1486D59
hypocenterC59#17905
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1486

Complements

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

NH ≅ G/Ntype
2. #21433026C2semidirect
4. #21422950C2semidirect
5. #21422905C2semidirect
6. #21422872C2semidirect
7. #21422500C2semidirect
8. #21422389C2semidirect
17. #16837529C22semidirect
19. #16837526C22semidirect
31. #16837154C22semidirect
32. #16837134C22semidirect
33. #16837124C22semidirect
34. #16837123C22semidirect
229. C59#17905semidirect

Maximal subgroups

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

subgroupindex
#214341042
#214330262
C2×D4×C59⋊C82
#214229502
#214229052
#214228722
#214225002
#214223892
C59×#14862
#1790559