Finite Group Databasebeta

← #28567221

Subgroups — #28567221

Order 15104.

966 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21435044C211yesmaximal, characteristic, complemented
2#21434104C211yesmaximal, characteristic
3#21433901C211yesmaximal, characteristic, complemented
4#21423303C214yesmaximal, complemented
5#21423230C214yesmaximal, complemented
6#21422777C211yesmaximal, characteristic, complemented
7#21422521C211yesmaximal, characteristic
8#21422491C211yesmaximal, characteristic, complemented
9C2×C59×#554C211yesmaximal, characteristic
10#16837959C2211yescharacteristic
11#16837781C2212yescomplemented
12#16837752C2212yescomplemented
13#16837732—21no—
14#16837709—22no—
15#16837697C2212yescomplemented
16#16837692C2212yes—
17#16837687C2212yescomplemented
18#16837609—22no—
19#16837607C2212yes—
20#16837558—22no—
21#16837553—21no—
22C4×#11745670—22no—
23#16837455C2211yescharacteristic
24#16837455C2211yescharacteristic
25#16837442—21no—
26#16837426C2211yescharacteristic
27#16837411—21no—
28#16837151C2214yescomplemented
29Q8⋊(C59⋊C8)C2214yescomplemented
30D4⋊(C59⋊C8)C2214yes—
31C59×#554—22no—
32C59×#554—22no—
33C59×#554—22no—
34C59×#554C2214yes—
35C2×C4⋊Q8×C59C2211yescharacteristic
36C2×#125×C59—21no—
37C2×C22⋊Q8×C59—21no—
38C2×C22⋊Q8×C59—22no—
39C2×C4×Q8×C59C2211yescharacteristic
40C2×C4×D4×C59—21no—
41C2×C4×D4×C59C2211yescharacteristic
42C22×C4⋊C4×C59—21no—
43C2×#11745672—24no—
44C2×#11745670—22no—
45C2×#11745664—21no—
46#14273727C2311yescharacteristic
47#14273727C2311yescharacteristic
48#14273719—21no—
49#14273706—21no—
50#14273706—21no—
51C4×Dic118—21no—
52C4×Dic118—21no—
53#14273686—21no—
54#14273660—22no—
55#14273658—22no—
56#14273658—22no—
57C4⋊(C59⋊C8)—21no—
58C4⋊(C59⋊C8)—21no—
59C4⋊(C59⋊C8)C2312yes—
60C4⋊(C59⋊C8)C2312yes—
61#14273651—22no—
62#14273651—22no—
63#14273651—24no—
64#14273650—22no—
65#14273648—22no—
66C22⋊(C59⋊C8)—21no—
67C22⋊(C59⋊C8)—21no—
68C4×C59⋊C8—21no—
69C4×C59⋊C8C2312yes—
70C22×Q8×C59D411yescharacteristic
71C22×Q8×C59D412yes—
72C22×D4×C59D411yescharacteristic
73C23×C4×C59—22no—
74C23×C4×C59—21no—
75C4⋊Q8×C59—21no—
76C4⋊Q8×C59—21no—
77C4⋊Q8×C59D412yes—
78C4⋊Q8×C59C2312yes—
79#125×C59—24no—
80#125×C59—24no—
81C22⋊Q8×C59—24no—
82C22⋊Q8×C59—28no—
83C22⋊Q8×C59—24no—
84C22⋊Q8×C59—28no—
85C4×Q8×C59—21no—
86C4×Q8×C59—21no—
87C4×Q8×C59D412yes—
88C4×Q8×C59C2312yes—
89C4×D4×C59—24no—
90C4×D4×C59—24no—
91C4×D4×C59—21no—
92C4×D4×C59—21no—
93C4×D4×C59D412yes—
94C4×D4×C59C2312yes—
95C2×C4⋊C4×C59—22no—
96C2×C4⋊C4×C59—24no—
97C2×C4⋊C4×C59—22no—
98C2×C4⋊C4×C59—21no—
99C2×C4⋊C4×C59—21no—
100C2×C4⋊C4×C59—21no—
101C2×C4⋊C4×C59—24no—
102C2×C4⋊C4×C59D411yescharacteristic
103C2×C4⋊C4×C59D411yescharacteristic
104C2×C22⋊C4×C59—22no—
105C2×C22⋊C4×C59—21no—
106C2×C42×C59—21no—
107C2×C42×C59C2311yescharacteristic
108C24×C59—21no—
109C2×Q8×C59—21no—
110C2×Q8×C59—22no—
111C2×Q8×C59—22no—
112C2×Q8×C59—22no—
113C2×Q8×C59—82no—
114C2×Q8×C59C2×D412yes—
115C2×Q8×C59C2×D414yes—
116C2×D4×C59—24no—
117C2×D4×C59—21no—
118C2×D4×C59C2×D412yes—
119C22×C4×C59—21no—
120C22×C4×C59—21no—
121C22×C4×C59—21no—
122C22×C4×C59—22no—
123C22×C4×C59—28no—
124C22×C4×C59—22no—
125C22×C4×C59—22no—
126C22×C4×C59—42no—
127C22×C4×C59—42no—
128C22×C4×C59—84no—
129C22×C4×C59C2×D411yescharacteristic
130C22×C4×C59C2×D411yescharacteristic
131C22×C4×C59C2×D411yescharacteristic
132C2×Dic118—21no—
133C2×Dic118—21no—
134C4⋊C4×C59—28no—
135C4⋊C4×C59—28no—
136C4⋊C4×C59—21no—
137C4⋊C4×C59—22no—
138C4⋊C4×C59—22no—
139C4⋊C4×C59—22no—
140C4⋊C4×C59—21no—
141C4⋊C4×C59—22no—
142C4⋊C4×C59—22no—
143C4⋊C4×C59—22no—
144C4⋊C4×C59C2×D412yes—
145C4⋊C4×C59C2×D412yes—
146C22⋊C4×C59—28no—
147C22⋊C4×C59—22no—
148C22⋊C4×C59—22no—
149C42×C59—22no—
150C42×C59—22no—
151C42×C59C2×D411yescharacteristic
152C42×C59C2×D411yescharacteristic
153C42×C59C2×D411yescharacteristic
154C42×C59C2411yescharacteristic
155C22⋊Dic59—41no—
156#11745672—84no—
157#11745670—44no—
158C4⋊Dic59—21no—
159C4⋊Dic59—21no—
160C4⋊Dic59—21no—
161C4⋊Dic59—21no—
162Dic59⋊C4—41no—
163Dic59⋊C4—41no—
164C4×Dic59—41no—
165#11745664—42no—
166C2×C59⋊C8—22no—
167C2×C59⋊C8—22no—
168C2×C59⋊C8—22no—
169C23×C59—24no—
170C23×C59—22no—
171C23×C59—42no—
172C23×C59—42no—
173C23×C59#12011yescharacteristic
174Q8×C59—22no—
175Q8×C59—22no—
176Q8×C59—44no—
177Q8×C59—42no—
178Q8×C59—82no—
179D4×C59—24no—
180D4×C59—22no—
181D4×C59#13714yes—
182C2×C4×C59—22no—
183C2×C4×C59—22no—
184C2×C4×C59—22no—
185C2×C4×C59—22no—
186C2×C4×C59—24no—
187C2×C4×C59—41no—
188C2×C4×C59—42no—
189C2×C4×C59—44no—
190C2×C4×C59—42no—
191C2×C4×C59—42no—
192C2×C4×C59—42no—
193C2×C4×C59—42no—
194C2×C4×C59—44no—
195C2×C4×C59—88no—
196C2×C4×C59#12011yescharacteristic
197C2×C4×C59#12011yescharacteristic
198C2×C4×C59#12011yescharacteristic
199C2×C4×C59#13512yes—
200C2×C4×C59#13712yes—
201C2×C4×C59C22×D411yescharacteristic
202C2×C4×C59C22×D411yescharacteristic
203C2×C4×C59C22×D411yescharacteristic
204C2×Dic59—41no—
205C2×Dic59—41no—
206Dic118—42no—
207Dic118—42no—
208C59⋊C8—42no—
209C59⋊C8—42no—
210C59⋊C8—42no—
211#18753—591nomaximal, Sylow
212C22×C59—21no—
213C22×C59—21no—
214C22×C59—24no—
215C22×C59—48no—
216C22×C59—42no—
217C22×C59—42no—
218C22×C59#45212yes—
219C22×C59C2×#12011yescharacteristic
220C4×C59—21no—
221C4×C59—21no—
222C4×C59—21no—
223C4×C59—42no—
224C4×C59—42no—
225C4×C59—42no—
226C4×C59—44no—
227C4×C59#45212yes—
228C4×C59C2×#13511yescharacteristic
229C4×C59C2×#13711yescharacteristic
230C4×C59C2×#13712yes—
231Dic59—81no—
232Dic59—81no—
233C2×#554D5911yescharacteristic
234#3209—591no—
235#3101—591no—
236#3078—591no—
237#1643—594no—
238#1632—594no—
239#1548—591no—
240#1486—591no—
241#1479—591no—
242C2×C59—22no—
243C2×C59—44no—
244C2×C59C2×#45211yescharacteristic
245C2×C59#300411yescharacteristic
246C2×C59#301911yescharacteristic
247#554—22no—
248#554—22no—
249#554—22no—
250#554C2×D5914yes—
251#533—591no—
252C2×C4⋊Q8C2×D5911yescharacteristic
253C2×#125—21no—
254C2×C22⋊Q8—21no—
255C2×C22⋊Q8—22no—
256C2×C4×Q8C2×D5911yescharacteristic
257C2×C4×D4—21no—
258C2×C4×D4C2×D5911yescharacteristic
259C22×C4⋊C4—21no—
260#494—592no—
261#487—592no—
262#483—1181no—
263#479—1182no—
264#477—592no—
265#476—592no—
266#475—592no—
267#462—1182no—
268#461—592no—
269#451—1182no—
270#450—1181no—
271C4×SD16—1182no—
272#431—591no—
273#431—591no—
274#428—1181no—
275#423—591no—
276#419—1181no—
277#333—594no—
278Q8⋊C8—594no—
279D4⋊C8—594no—
280C59#1875311yescharacteristic, complemented, Sylow
281C22×Q8C22⋊D5911yescharacteristic
282C22×Q8C22⋊D5912yes—
283C22×D4C22⋊D5911yescharacteristic
284C23×C4—22no—
285C23×C4—21no—
286C2×Q16—1184no—
287C2×SD16—1182no—
288C2×M16—1181no—
289C4⋊Q8—21no—
290C4⋊Q8—21no—
291C4⋊Q8—591no—
292C4⋊Q8—591no—
293C4⋊Q8C22⋊D5912yes—
294C4⋊Q8C22×D5912yes—
295#125—24no—
296#125—24no—
297#125—1181no—
298C22⋊Q8—24no—
299C22⋊Q8—28no—
300C22⋊Q8—24no—
301C22⋊Q8—28no—
302C22⋊Q8—1181no—
303C22⋊Q8—1181no—
304C4×Q8—21no—
305C4×Q8—21no—
306C4×Q8—1181no—
307C4×Q8—1181no—
308C4×Q8C22⋊D5912yes—
309C4×Q8C22×D5912yes—
310C4×D4—24no—
311C4×D4—24no—
312C4×D4—21no—
313C4×D4—21no—
314C4×D4C22⋊D5912yes—
315C4×D4C22×D5912yes—
316#117—1181no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—24no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—24no—
324C2×C4⋊C4C22⋊D5911yescharacteristic
325C2×C4⋊C4C22⋊D5911yescharacteristic
326C2×C22⋊C4—22no—
327C2×C22⋊C4—21no—
328C2×C42—21no—
329C2×C42C22×D5911yescharacteristic
330#107—1182no—
331#106—1182no—
332#106—1182no—
333C4⋊C8—592no—
334C4⋊C8—592no—
335C4⋊C8—1181no—
336C4⋊C8—1181no—
337#103—1182no—
338#103—1182no—
339#103—1184no—
340#103—1182no—
341#102—1182no—
342C22⋊C8—1181no—
343C22⋊C8—1181no—
344C4×C8—592no—
345C4×C8—1181no—
346C24—21no—
347C2×Q8—21no—
348C2×Q8—22no—
349C2×Q8—22no—
350C2×Q8—22no—
351C2×Q8—82no—
352C2×Q8—1181no—
353C2×Q8—1181no—
354C2×Q8C2×C22⋊D5912yes—
355C2×Q8C2×C22⋊D5914yes—
356C2×D4—24no—
357C2×D4—21no—
358C2×D4C2×C22⋊D5912yes—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—22no—
363C22×C4—28no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—84no—
369C22×C4C2×C22⋊D5911yescharacteristic
370C22×C4C2×C22⋊D5911yescharacteristic
371C22×C4C2×C22⋊D5911yescharacteristic
372Q16—4724no—
373SD16—2364no—
374M16—2362no—
375C2×C8—1182no—
376C2×C8—1182no—
377C2×C8—1182no—
378C4⋊C4—28no—
379C4⋊C4—28no—
380C4⋊C4—21no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—21no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—1181no—
389C4⋊C4—1181no—
390C4⋊C4—1181no—
391C4⋊C4—1181no—
392C4⋊C4—2361no—
393C4⋊C4—2361no—
394C4⋊C4C2×C22⋊D5912yes—
395C4⋊C4C2×C22⋊D5912yes—
396C22⋊C4—28no—
397C22⋊C4—22no—
398C22⋊C4—22no—
399C22⋊C4—2361no—
400C42—22no—
401C42—22no—
402C42—2361no—
403C42C2×C22⋊D5911yescharacteristic
404C42C2×C22⋊D5911yescharacteristic
405C42C2×C22⋊D5911yescharacteristic
406C42C23×D5911yescharacteristic
407C23—24no—
408C23—22no—
409C23—42no—
410C23—42no—
411C23#1427369611yescharacteristic
412Q8—22no—
413Q8—22no—
414Q8—44no—
415Q8—42no—
416Q8—82no—
417Q8—2362no—
418Q8—2362no—
419D4—24no—
420D4—22no—
421D4#1427375814yes—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—24no—
427C2×C4—41no—
428C2×C4—42no—
429C2×C4—44no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—44no—
435C2×C4—88no—
436C2×C4—2361no—
437C2×C4—2361no—
438C2×C4#1427369611yescharacteristic
439C2×C4#1427369611yescharacteristic
440C2×C4#1427369611yescharacteristic
441C2×C4#1427374712yes—
442C2×C4#1427375712yes—
443C2×C4C22×C22⋊D5911yescharacteristic
444C2×C4C22×C22⋊D5911yescharacteristic
445C2×C4C22×C22⋊D5911yescharacteristic
446C8—2362no—
447C8—2362no—
448C8—2362no—
449C22—21no—
450C22—21no—
451C22—24no—
452C22—48no—
453C22—42no—
454C22—42no—
455C22#1683756412yes—
456C22C2×#1427369611yescharacteristic
457C4—21no—
458C4—21no—
459C4—21no—
460C4—42no—
461C4—44no—
462C4—42no—
463C4—42no—
464C4—4721no—
465C4—4721no—
466C4#1683756412yes—
467C4C2×#1427374711yescharacteristic
468C4C2×#1427375711yescharacteristic
469C4C2×#1427375812yes—
470C2—22no—
471C2—44no—
472C2C2×#1683756411yescharacteristic
473C2#2143333011yescharacteristic
474C2#2143345311yescharacteristic
subgroupG/H
centerC22C2×#14273696
commutator, cosocleC42×C59C24
FrattiniC42C23×D59
FittingC2×C59×#554C2
socleC22×C59C2×#120
prime radicalC22×D4×C59D4
hypercenterC2×#554D59
hypocenterC59#18753
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#554

Complements

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

NH ≅ G/Ntype
1. #21435044C2semidirect
3. #21433901C2semidirect
4. #21423303C2semidirect
5. #21423230C2semidirect
6. #21422777C2semidirect
8. #21422491C2semidirect
11. #16837781C22semidirect
12. #16837752C22semidirect
15. #16837697C22semidirect
17. #16837687C22semidirect
28. #16837151C22semidirect
29. Q8⋊(C59⋊C8)C22semidirect
280. C59#18753semidirect

Maximal subgroups

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

subgroupindex
#214350442
#214341042
#214339012
#214233032
#214232302
#214227772
#214225212
#214224912
C2×C59×#5542
#1875359