Finite Group Databasebeta

← C59×#8997 #28421106

Subgroups — C59×#8997 #28421106

Order 15104.

452 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C59×#3171C211yesmaximal, characteristic
2C59×#2995C211yesmaximal, characteristic, complemented
3C59×#1963C211yesmaximal, characteristic, complemented
4C59×#1500C211yesmaximal, characteristic, complemented
5C59×#1496C211yesmaximal, characteristic, complemented
6C59×#1300C211yesmaximal, characteristic, complemented
7C59×#1299C211yesmaximal, characteristic, complemented
8C59×#532C2211yescharacteristic
9C59×#529C2211yescharacteristic
10C59×#501C411yescharacteristic, complemented
11C59×#496C411yescharacteristic, complemented
12C59×#490C411yescharacteristic, complemented
13C59×#489C411yescharacteristic, complemented
14C59×#469—21no—
15C59×#468—21no—
16C59×#446—21no—
17C4×SD16×C59—21no—
18C4×D8×C59—21no—
19C59×#436—21no—
20C59×#435—21no—
21C59×#435—21no—
22C59×#433—21no—
23C59×#432C2211yescharacteristic
24C59×#431C2211yescharacteristic
25C59×#424—21no—
26C59×#423C2211yescharacteristic
27C2×#104×C59—21no—
28C2×#104×C59—21no—
29C4×M16×C59C2211yescharacteristic
30C59×#376C2211yescharacteristic
31C59×#328—21no—
32C59×#327—21no—
33C59×D4⋊C8—21no—
34C59×D4⋊C8—21no—
35C2×#41×C59—21no—
36C2×SD16×C59—21no—
37C2×D8×C59—21no—
38C2×M16×C59—21no—
39C2×M16×C59—21no—
40C2×M16×C59D411yescharacteristic
41C2×M16×C59D411yescharacteristic
42C22×C8×C59—21no—
43C4⋊Q8×C59C2×C411yescharacteristic
44#127×C59C2×C411yescharacteristic
45#126×C59—21no—
46#125×C59C2×C411yescharacteristic
47#124×C59—21no—
48#124×C59C2×C411yescharacteristic
49#123×C59—21no—
50C22⋊Q8×C59D411yescharacteristic
51#121×C59—21no—
52#121×C59—21no—
53#121×C59D411yescharacteristic
54C4×Q8×C59D411yescharacteristic
55C4×D4×C59—21no—
56C4×D4×C59—21no—
57C4×D4×C59—21no—
58C4×D4×C59D411yescharacteristic
59#117×C59—21no—
60C2×C42×C59—21no—
61C2×C42×C59C2311yescharacteristic
62#107×C59—21no—
63#106×C59—21no—
64C4⋊C8×C59—21no—
65C4⋊C8×C59—21no—
66C4⋊C8×C59—21no—
67C4⋊C8×C59—21no—
68C4⋊C8×C59—21no—
69C4⋊C8×C59D411yescharacteristic
70C4⋊C8×C59D411yescharacteristic
71#104×C59—42no—
72#104×C59—42no—
73#103×C59—21no—
74#102×C59—21no—
75#102×C59—21no—
76#102×C59—21no—
77C22⋊C8×C59—21no—
78C22⋊C8×C59—21no—
79C22⋊C8×C59—21no—
80C22⋊C8×C59—21no—
81#97×C59—21no—
82#97×C59—21no—
83#97×C59C2×C411yescharacteristic
84#97×C59C2×C411yescharacteristic
85C4×C8×C59—21no—
86C4×C8×C59—21no—
87C4×C8×C59—21no—
88#95×C59—21no—
89#95×C59—21no—
90#41×C59—41no—
91#41×C59—41no—
92#41×C59—41no—
93#41×C59—41no—
94C2×Q8×C59—21no—
95C2×Q8×C59C22⋊C411yescharacteristic
96C2×D4×C59—21no—
97C2×D4×C59—21no—
98C2×D4×C59—21no—
99C2×D4×C59—21no—
100C2×D4×C59—41no—
101C2×D4×C59C22⋊C411yescharacteristic
102C22×C4×C59—21no—
103C22×C4×C59—41no—
104C22×C4×C59—41no—
105C22×C4×C59C2×D411yescharacteristic
106C22×C4×C59C2×D411yescharacteristic
107C22×C4×C59#4111yescharacteristic
108SD16×C59—42no—
109D8×C59—42no—
110M16×C59—42no—
111M16×C59—41no—
112M16×C59—41no—
113M16×C59—81no—
114C2×C8×C59—21no—
115C2×C8×C59—21no—
116C2×C8×C59—21no—
117C2×C8×C59—21no—
118C2×C8×C59—21no—
119C2×C8×C59—21no—
120C2×C8×C59—41no—
121C2×C8×C59—81no—
122C4⋊C4×C59—21no—
123C4⋊C4×C59—21no—
124C4⋊C4×C59—21no—
125C4⋊C4×C59—21no—
126C4⋊C4×C59—21no—
127C4⋊C4×C59—21no—
128C4⋊C4×C59C22⋊C411yescharacteristic
129C4⋊C4×C59C22⋊C411yescharacteristic
130C22⋊C4×C59—21no—
131C22⋊C4×C59—21no—
132C22⋊C4×C59—21no—
133C22⋊C4×C59—21no—
134C22⋊C4×C59—21no—
135C22⋊C4×C59—41no—
136C22⋊C4×C59—41no—
137C42×C59—21no—
138C42×C59—22no—
139C42×C59—41no—
140C42×C59C22×C411yescharacteristic
141C42×C59C2×D411yescharacteristic
142C42×C59C2×D411yescharacteristic
143C42×C59#4111yescharacteristic
144C23×C59—21no—
145C23×C59—41no—
146C23×C59#12111yescharacteristic
147Q8×C59—41no—
148Q8×C59—41no—
149Q8×C59—41no—
150D4×C59—41no—
151D4×C59—41no—
152D4×C59—41no—
153D4×C59—42no—
154D4×C59—41no—
155D4×C59—41no—
156D4×C59—81no—
157D4×C59—81no—
158D4×C59—81no—
159C2×C4×C59—21no—
160C2×C4×C59—21no—
161C2×C4×C59—21no—
162C2×C4×C59—21no—
163C2×C4×C59—21no—
164C2×C4×C59—41no—
165C2×C4×C59—41no—
166C2×C4×C59—41no—
167C2×C4×C59—41no—
168C2×C4×C59—42no—
169C2×C4×C59—41no—
170C2×C4×C59—81no—
171C2×C4×C59—81no—
172C2×C4×C59—81no—
173C2×C4×C59—81no—
174C2×C4×C59C2×C22⋊C411yescharacteristic
175C2×C4×C59C4×D411yescharacteristic
176C2×C4×C59C4×D411yescharacteristic
177C2×C4×C59#12011yescharacteristic
178C2×C4×C59#12111yescharacteristic
179C2×C4×C59#12311yescharacteristic
180C8×C59—22no—
181C8×C59—41no—
182C8×C59—41no—
183C8×C59—41no—
184C8×C59—41no—
185C8×C59—41no—
186C8×C59—81no—
187#8997C5911yesmaximal, characteristic, direct product factor, Sylow
188C22×C59—21no—
189C22×C59—21no—
190C22×C59—21no—
191C22×C59—41no—
192C22×C59—41no—
193C22×C59—41no—
194C22×C59—81no—
195C22×C59—81no—
196C22×C59—81no—
197C22×C59#38611yescharacteristic
198C4×C59—21no—
199C4×C59—21no—
200C4×C59—21no—
201C4×C59—21no—
202C4×C59—21no—
203C4×C59—41no—
204C4×C59—41no—
205C4×C59—41no—
206C4×C59—82no—
207C4×C59—81no—
208C4×C59#45311yescharacteristic
209C4×C59#45411yescharacteristic
210#3171C2×C5911yescharacteristic
211#2995C2×C5911yescharacteristic, complemented
212#1963C2×C5911yescharacteristic, complemented
213#1500C2×C5911yescharacteristic, complemented
214#1496C2×C5911yescharacteristic, complemented
215#1300C2×C5911yescharacteristic, complemented
216#1299C2×C5911yescharacteristic, complemented
217C2×C59—41no—
218C2×C59—81no—
219C2×C59—161no—
220C2×C59#188111yescharacteristic
221C2×C59#188311yescharacteristic
222C2×C59#188411yescharacteristic
223#532C22×C5911yescharacteristic
224#529C22×C5911yescharacteristic
225#501C4×C5911yescharacteristic, complemented
226#496C4×C5911yescharacteristic, complemented
227#490C4×C5911yescharacteristic, complemented
228#489C4×C5911yescharacteristic, complemented
229#469—21no—
230#468—21no—
231#446—21no—
232C4×SD16—21no—
233C4×D8—21no—
234#436—21no—
235#435—21no—
236#435—21no—
237#433—21no—
238#432C22×C5911yescharacteristic
239#431C22×C5911yescharacteristic
240#424—21no—
241#423C22×C5911yescharacteristic
242C2×#104—21no—
243C2×#104—21no—
244C4×M16C22×C5911yescharacteristic
245#376C22×C5911yescharacteristic
246#328—21no—
247#327—21no—
248D4⋊C8—21no—
249D4⋊C8—21no—
250C59#899711yescharacteristic, direct product factor, Sylow
251C2×#41—21no—
252C2×SD16—21no—
253C2×D8—21no—
254C2×M16—21no—
255C2×M16—21no—
256C2×M16D4×C5911yescharacteristic
257C2×M16D4×C5911yescharacteristic
258C22×C8—21no—
259C4⋊Q8C2×C4×C5911yescharacteristic
260#127C2×C4×C5911yescharacteristic
261#126—21no—
262#125C2×C4×C5911yescharacteristic
263#124—21no—
264#124C2×C4×C5911yescharacteristic
265#123—21no—
266C22⋊Q8D4×C5911yescharacteristic
267#121—21no—
268#121—21no—
269#121D4×C5911yescharacteristic
270C4×Q8D4×C5911yescharacteristic
271C4×D4—21no—
272C4×D4—21no—
273C4×D4—21no—
274C4×D4D4×C5911yescharacteristic
275#117—21no—
276C2×C42—21no—
277C2×C42C23×C5911yescharacteristic
278#107—21no—
279#106—21no—
280C4⋊C8—21no—
281C4⋊C8—21no—
282C4⋊C8—21no—
283C4⋊C8—21no—
284C4⋊C8—21no—
285C4⋊C8D4×C5911yescharacteristic
286C4⋊C8D4×C5911yescharacteristic
287#104—42no—
288#104—42no—
289#103—21no—
290#102—21no—
291#102—21no—
292#102—21no—
293C22⋊C8—21no—
294C22⋊C8—21no—
295C22⋊C8—21no—
296C22⋊C8—21no—
297#97—21no—
298#97—21no—
299#97C2×C4×C5911yescharacteristic
300#97C2×C4×C5911yescharacteristic
301C4×C8—21no—
302C4×C8—21no—
303C4×C8—21no—
304#95—21no—
305#95—21no—
306#41—41no—
307#41—41no—
308#41—41no—
309#41—41no—
310C2×Q8—21no—
311C2×Q8C22⋊C4×C5911yescharacteristic
312C2×D4—21no—
313C2×D4—21no—
314C2×D4—21no—
315C2×D4—21no—
316C2×D4—41no—
317C2×D4C22⋊C4×C5911yescharacteristic
318C22×C4—21no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4C2×D4×C5911yescharacteristic
322C22×C4C2×D4×C5911yescharacteristic
323C22×C4#41×C5911yescharacteristic
324SD16—42no—
325D8—42no—
326M16—42no—
327M16—41no—
328M16—41no—
329M16—81no—
330C2×C8—21no—
331C2×C8—21no—
332C2×C8—21no—
333C2×C8—21no—
334C2×C8—21no—
335C2×C8—21no—
336C2×C8—41no—
337C2×C8—81no—
338C4⋊C4—21no—
339C4⋊C4—21no—
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4C22⋊C4×C5911yescharacteristic
345C4⋊C4C22⋊C4×C5911yescharacteristic
346C22⋊C4—21no—
347C22⋊C4—21no—
348C22⋊C4—21no—
349C22⋊C4—21no—
350C22⋊C4—21no—
351C22⋊C4—41no—
352C22⋊C4—41no—
353C42—21no—
354C42—22no—
355C42—41no—
356C42C22×C4×C5911yescharacteristic
357C42C2×D4×C5911yescharacteristic
358C42C2×D4×C5911yescharacteristic
359C42#41×C5911yescharacteristic
360C23—21no—
361C23—41no—
362C23#121×C5911yescharacteristic
363Q8—41no—
364Q8—41no—
365Q8—41no—
366D4—41no—
367D4—41no—
368D4—41no—
369D4—42no—
370D4—41no—
371D4—41no—
372D4—81no—
373D4—81no—
374D4—81no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—42no—
385C2×C4—41no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4C2×C22⋊C4×C5911yescharacteristic
391C2×C4C4×D4×C5911yescharacteristic
392C2×C4C4×D4×C5911yescharacteristic
393C2×C4#120×C5911yescharacteristic
394C2×C4#121×C5911yescharacteristic
395C2×C4#123×C5911yescharacteristic
396C8—22no—
397C8—41no—
398C8—41no—
399C8—41no—
400C8—41no—
401C8—41no—
402C8—81no—
403C22—21no—
404C22—21no—
405C22—21no—
406C22—41no—
407C22—41no—
408C22—41no—
409C22—81no—
410C22—81no—
411C22—81no—
412C22C59×#38611yescharacteristic
413C4—21no—
414C4—21no—
415C4—21no—
416C4—21no—
417C4—21no—
418C4—41no—
419C4—41no—
420C4—41no—
421C4—82no—
422C4—81no—
423C4C59×#45311yescharacteristic
424C4C59×#45411yescharacteristic
425C2—41no—
426C2—81no—
427C2—161no—
428C2C59×#188111yescharacteristic
429C2C59×#188311yescharacteristic
430C2C59×#188411yescharacteristic
subgroupG/H
centerC2×C4×C59#120
commutatorC42C22×C4×C59
Frattini, cosocleC2×C42C23×C59
socleC22×C59#386
prime radicalC59×#3171C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C59 ◁ C2×C42×C59 ◁ G

Complements

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

NH ≅ G/Ntype
2. C59×#2995C2semidirect
3. C59×#1963C2semidirect
4. C59×#1500C2semidirect
5. C59×#1496C2semidirect
6. C59×#1300C2semidirect
7. C59×#1299C2semidirect
10. C59×#501C4semidirect
11. C59×#496C4semidirect
12. C59×#490C4semidirect
13. C59×#489C4semidirect
187. #8997C59direct (N × H)
211. #2995C2×C59semidirect
212. #1963C2×C59semidirect
213. #1500C2×C59semidirect
214. #1496C2×C59semidirect
215. #1300C2×C59semidirect
216. #1299C2×C59semidirect
225. #501C4×C59semidirect
226. #496C4×C59semidirect
227. #490C4×C59semidirect
228. #489C4×C59semidirect
250. C59#8997direct (N × H)

Maximal subgroups

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

subgroupindex
C59×#31712
C59×#29952
C59×#19632
C59×#15002
C59×#14962
C59×#13002
C59×#12992
#899759