Finite Group Databasebeta

← #28567482

Subgroups — #28567482

Order 15104.

932 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#21435224C211yesmaximal, characteristic
2#21434127C211yesmaximal, characteristic
3#21433568C211yesmaximal, characteristic
4#21423373C214yesmaximal, complemented
5#21423192C214yesmaximal, complemented
6#21422781C211yesmaximal, characteristic, complemented
7#21422525C211yesmaximal, characteristic, complemented
8#21422491C211yesmaximal, characteristic, complemented
9C2×C59×#557C211yesmaximal, characteristic, complemented
10C2×#14273758—21no—
11#16837952C2211yescharacteristic
12#16837798C2212yes—
13#16837750C2212yes—
14#16837720—21no—
15#16837720—22no—
16#16837689—22no—
17#16837687C2212yes—
18#16837624—22no—
19#16837612C2212yes—
20#16837602C2212yes—
21#16837602C2212yes—
22#16837541—21no—
23C4×#11745671—22no—
24#16837455C2211yescharacteristic
25#16837455C2211yescharacteristic
26#16837426C2211yescharacteristic
27#16837401—21no—
28#16837148C2214yes—
29Q8⋊(C59⋊C8)C2214yes—
30Q8⋊(C59⋊C8)C2214yes—
31C59×#557—22no—
32C59×#557—22no—
33C59×#557—22no—
34C59×#557C2214yes—
35C2×C4⋊Q8×C59—21no—
36C2×C4⋊Q8×C59C2211yescharacteristic
37C2×#125×C59—22no—
38C2×#125×C59—21no—
39C2×C4×Q8×C59—21no—
40C2×C4×Q8×C59—21no—
41C2×C4×Q8×C59C2211yescharacteristic
42C2×C4×Q8×C59C2211yescharacteristic
43C2×#11745685—21no—
44#14273758—44no—
45C2×#11745672—22no—
46C2×#11745671—22no—
47C2×#11745671—22no—
48C2×#11745664—21no—
49#14273727C2311yescharacteristic
50#14273724C2311yescharacteristic
51#14273715—21no—
52#14273701—21no—
53#14273701—21no—
54C4×D236—21no—
55C4×D236—21no—
56#14273658—22no—
57#14273658—24no—
58C4⋊(C59⋊C8)—21no—
59C4⋊(C59⋊C8)—21no—
60C4⋊(C59⋊C8)C2312yes—
61C4⋊(C59⋊C8)C2312yes—
62#14273651—22no—
63#14273650—22no—
64#14273647—22no—
65#14273647—24no—
66#14273647—22no—
67C22⋊(C59⋊C8)—21no—
68C22⋊(C59⋊C8)—21no—
69C4×C59⋊C8—21no—
70C4×C59⋊C8C2312yes—
71C22×Q8×C59—21no—
72C22×Q8×C59D411yescharacteristic
73C22×Q8×C59D411yescharacteristic
74C4⋊Q8×C59—24no—
75C4⋊Q8×C59—24no—
76C4⋊Q8×C59—21no—
77C4⋊Q8×C59—21no—
78C4⋊Q8×C59D412yes—
79C4⋊Q8×C59C2312yes—
80#125×C59—28no—
81#125×C59—24no—
82#125×C59—28no—
83#125×C59—24no—
84C4×Q8×C59—24no—
85C4×Q8×C59—24no—
86C4×Q8×C59—24no—
87C4×Q8×C59—24no—
88C4×Q8×C59—21no—
89C4×Q8×C59—21no—
90C4×Q8×C59—21no—
91C4×Q8×C59—21no—
92C4×Q8×C59D412yes—
93C4×Q8×C59D412yes—
94C4×Q8×C59C2312yes—
95C4×Q8×C59C2312yes—
96C2×C4⋊C4×C59—22no—
97C2×C4⋊C4×C59—22no—
98C2×C4⋊C4×C59—22no—
99C2×C4⋊C4×C59—21no—
100C2×C4⋊C4×C59—21no—
101C2×C4⋊C4×C59—21no—
102C2×C4⋊C4×C59D411yescharacteristic
103C2×C4⋊C4×C59D411yescharacteristic
104C2×C4⋊C4×C59D412yes—
105C2×C42×C59—22no—
106C2×C42×C59—21no—
107C2×C42×C59—21no—
108C2×C42×C59C2311yescharacteristic
109C2×Q8×C59—21no—
110C2×Q8×C59—21no—
111C2×Q8×C59—22no—
112C2×Q8×C59—22no—
113C2×Q8×C59—22no—
114C2×Q8×C59—42no—
115C2×Q8×C59—44no—
116C2×Q8×C59SD1614yescomplemented
117C2×Q8×C59C2×D412yes—
118C2×Q8×C59C2×D412yes—
119C22×C4×C59—21no—
120C22×C4×C59—21no—
121C22×C4×C59—21no—
122C22×C4×C59—21no—
123C22×C4×C59—22no—
124C22×C4×C59C2×D411yescharacteristic
125C22×C4×C59C2×D411yescharacteristic
126C22×C4×C59C2×D411yescharacteristic
127C2×C22⋊D59—41no—
128#11745685—42no—
129#11745685—42no—
130C2×D236—21no—
131C2×D236—21no—
132C2×D236—21no—
133C2×C4×D59—41no—
134C2×Dic118—21no—
135C4⋊C4×C59—28no—
136C4⋊C4×C59—28no—
137C4⋊C4×C59—28no—
138C4⋊C4×C59—22no—
139C4⋊C4×C59—22no—
140C4⋊C4×C59—22no—
141C4⋊C4×C59—21no—
142C4⋊C4×C59—21no—
143C4⋊C4×C59—22no—
144C4⋊C4×C59—22no—
145C4⋊C4×C59—22no—
146C4⋊C4×C59—22no—
147C4⋊C4×C59C2×D412yes—
148C4⋊C4×C59C2×D412yes—
149C4⋊C4×C59C2×D414yes—
150C42×C59—28no—
151C42×C59—22no—
152C42×C59—22no—
153C42×C59—22no—
154C42×C59—22no—
155C42×C59C2×D411yescharacteristic
156C42×C59C2×D411yescharacteristic
157C42×C59C2×D411yescharacteristic
158C42×C59C2411yescharacteristic
159#11745672—44no—
160#11745671—44no—
161#11745671—44no—
162(C2×D59)⋊C4—41no—
163(C2×D59)⋊C4—41no—
164C4⋊Dic59—21no—
165C4⋊Dic59—21no—
166#11745664—42no—
167C2×C59⋊C8—22no—
168C2×C59⋊C8—22no—
169C2×C59⋊C8—22no—
170C23×C59#12011yescharacteristic
171C22×D59—41no—
172Q8×C59—22no—
173Q8×C59—24no—
174Q8×C59—22no—
175Q8×C59—22no—
176Q8×C59—42no—
177Q8×C59—44no—
178Q8×C59—44no—
179Q8×C59C2×SD1614yes—
180C2×C4×C59—22no—
181C2×C4×C59—24no—
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—42no—
190C2×C4×C59—42no—
191C2×C4×C59—44no—
192C2×C4×C59#12011yescharacteristic
193C2×C4×C59#12011yescharacteristic
194C2×C4×C59#12011yescharacteristic
195C2×C4×C59C2×SD1612yes—
196C2×C4×C59#13712yes—
197C2×C4×C59C22×D411yescharacteristic
198C2×C4×C59C22×D411yescharacteristic
199C2×C4×C59C22×D411yescharacteristic
200C22⋊D59—82no—
201C2×Dic59—41no—
202D236—42no—
203D236—41no—
204D236—41no—
205D236—42no—
206C4×D59—82no—
207Dic118—41no—
208Dic118—41no—
209C59⋊C8—42no—
210C59⋊C8—42no—
211C59⋊C8—42no—
212#18778—591nomaximal, Sylow
213C22×C59—21no—
214C22×C59—21no—
215C22×C59#44812yes—
216C22×C59C2×#12011yescharacteristic
217C2×D59—81no—
218C2×D59—81no—
219C2×D59—81no—
220C4×C59—24no—
221C4×C59—21no—
222C4×C59—21no—
223C4×C59—21no—
224C4×C59—42no—
225C4×C59—44no—
226C4×C59—42no—
227C4×C59—42no—
228C4×C59#44812yes—
229C4×C59C22×SD1611yescharacteristic
230C4×C59C2×#13711yescharacteristic
231C4×C59#57512yes—
232Dic59—81no—
233C2×#557D5911yescharacteristic, complemented
234#3227—591no—
235#3103—591no—
236#3037—591no—
237#1653—594no—
238#1626—594no—
239#1549—591no—
240#1487—591no—
241#1479—591no—
242C2×C59—22no—
243C2×C59C2×#44811yescharacteristic
244C2×C59#300511yescharacteristic
245C2×C59#302011yescharacteristic
246D59—161no—
247C2×#137—1181no—
248#557—22no—
249#557—22no—
250#557—22no—
251#557C2×D5914yes—
252#532—591no—
253C2×C4⋊Q8—21no—
254C2×C4⋊Q8C2×D5911yescharacteristic
255C2×#125—22no—
256C2×#125—21no—
257C2×C4×Q8—21no—
258C2×C4×Q8—21no—
259C2×C4×Q8C2×D5911yescharacteristic
260C2×C4×Q8C2×D5911yescharacteristic
261#498—592no—
262#486—592no—
263#481—1181no—
264#481—1182no—
265#476—1182no—
266#475—592no—
267#464—1182no—
268#462—592no—
269#460—592no—
270#460—592no—
271#448—1181no—
272C4×SD16—1182no—
273#431—591no—
274#431—591no—
275#423—591no—
276#417—1181no—
277#332—594no—
278Q8⋊C8—594no—
279Q8⋊C8—594no—
280C59#1877811yescharacteristic, complemented, Sylow
281C2×#41—1181no—
282C22×Q8—21no—
283C22×Q8C22⋊D5911yescharacteristic
284C22×Q8C22⋊D5911yescharacteristic
285#137—2364no—
286C2×Q16—1182no—
287C2×SD16—1182no—
288C2×SD16—1182no—
289C2×M16—1181no—
290C4⋊Q8—24no—
291C4⋊Q8—24no—
292C4⋊Q8—21no—
293C4⋊Q8—21no—
294C4⋊Q8—591no—
295C4⋊Q8C22⋊D5912yes—
296C4⋊Q8C22×D5912yes—
297#127—591no—
298#125—28no—
299#125—24no—
300#125—28no—
301#125—24no—
302#124—1181no—
303#121—1181no—
304#121—1181no—
305C4×Q8—24no—
306C4×Q8—24no—
307C4×Q8—24no—
308C4×Q8—24no—
309C4×Q8—21no—
310C4×Q8—21no—
311C4×Q8—21no—
312C4×Q8—21no—
313C4×Q8C22⋊D5912yes—
314C4×Q8C22⋊D5912yes—
315C4×Q8C22×D5912yes—
316C4×Q8C22×D5912yes—
317C4×D4—1181no—
318C4×D4—1181no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4C22⋊D5911yescharacteristic
326C2×C4⋊C4C22⋊D5911yescharacteristic
327C2×C4⋊C4C22⋊D5912yes—
328C2×C42—22no—
329C2×C42—21no—
330C2×C42—21no—
331C2×C42C22×D5911yescharacteristic
332#106—1182no—
333#106—1184no—
334C4⋊C8—592no—
335C4⋊C8—592no—
336C4⋊C8—1181no—
337C4⋊C8—1181no—
338#103—1182no—
339#103—1182no—
340#102—1182no—
341#102—1182no—
342#102—1184no—
343C22⋊C8—1181no—
344C22⋊C8—1181no—
345C4×C8—592no—
346C4×C8—1181no—
347#41—2362no—
348#41—2362no—
349C2×Q8—21no—
350C2×Q8—21no—
351C2×Q8—22no—
352C2×Q8—22no—
353C2×Q8—22no—
354C2×Q8—42no—
355C2×Q8—44no—
356C2×Q8—1181no—
357C2×Q8#1174567114yescomplemented
358C2×Q8C2×C22⋊D5912yes—
359C2×Q8C2×C22⋊D5912yes—
360C2×D4—1181no—
361C2×D4—1181no—
362C2×D4—1181no—
363C2×D4—2361no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—22no—
369C22×C4—2361no—
370C22×C4C2×C22⋊D5911yescharacteristic
371C22×C4C2×C22⋊D5911yescharacteristic
372C22×C4C2×C22⋊D5911yescharacteristic
373Q16—2364no—
374SD16—2364no—
375SD16—2364no—
376M16—2362no—
377C2×C8—1182no—
378C2×C8—1182no—
379C2×C8—1182no—
380C4⋊C4—28no—
381C4⋊C4—28no—
382C4⋊C4—28no—
383C4⋊C4—22no—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4—21no—
387C4⋊C4—21no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—1181no—
393C4⋊C4—1181no—
394C4⋊C4C2×C22⋊D5912yes—
395C4⋊C4C2×C22⋊D5912yes—
396C4⋊C4C2×C22⋊D5914yes—
397C22⋊C4—2361no—
398C22⋊C4—2361no—
399C42—28no—
400C42—22no—
401C42—22no—
402C42—22no—
403C42—22no—
404C42C2×C22⋊D5911yescharacteristic
405C42C2×C22⋊D5911yescharacteristic
406C42C2×C22⋊D5911yescharacteristic
407C42C23×D5911yescharacteristic
408C23—2361no—
409C23#1427369611yescharacteristic
410Q8—22no—
411Q8—24no—
412Q8—22no—
413Q8—22no—
414Q8—42no—
415Q8—44no—
416Q8—44no—
417Q8—2361no—
418Q8—2361no—
419Q8C2×#1174567114yes—
420D4—4722no—
421D4—2362no—
422D4—2361no—
423D4—2361no—
424D4—2362no—
425C2×C4—22no—
426C2×C4—24no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—24no—
432C2×C4—41no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—44no—
437C2×C4—4722no—
438C2×C4—2361no—
439C2×C4#1427369611yescharacteristic
440C2×C4#1427369611yescharacteristic
441C2×C4#1427369611yescharacteristic
442C2×C4C2×#1174567112yes—
443C2×C4#1427375812yes—
444C2×C4C22×C22⋊D5911yescharacteristic
445C2×C4C22×C22⋊D5911yescharacteristic
446C2×C4C22×C22⋊D5911yescharacteristic
447C8—2362no—
448C8—2362no—
449C8—2362no—
450C22—21no—
451C22—21no—
452C22—4721no—
453C22—4721no—
454C22—4721no—
455C22#1683754112yes—
456C22C2×#1427369611yescharacteristic
457C4—21no—
458C4—21no—
459C4—24no—
460C4—21no—
461C4—42no—
462C4—42no—
463C4—42no—
464C4—44no—
465C4—4721no—
466C4#1683754112yes—
467C4C22×#1174567111yescharacteristic
468C4C2×#1427375811yescharacteristic
469C4#1683820012yes—
470C2—22no—
471C2—9441no—
472C2C2×#1683754111yescharacteristic
473C2#2143333811yescharacteristic
474C2#2143346011yescharacteristic
subgroupG/H
centerC22C2×#14273696
commutator, cosocleC42×C59C24
FrattiniC42C23×D59
FittingC2×C59×#557C2
socleC22×C59C2×#120
prime radical#16837952C22
hypercenterC2×#557D59
hypocenterC59#18778
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#557

Complements

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

NH ≅ G/Ntype
4. #21423373C2semidirect
5. #21423192C2semidirect
6. #21422781C2semidirect
7. #21422525C2semidirect
8. #21422491C2semidirect
9. C2×C59×#557C2semidirect
116. C2×Q8×C59SD16semidirect
233. C2×#557D59semidirect
280. C59#18778semidirect
357. C2×Q8#11745671semidirect

Maximal subgroups

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

subgroupindex
#214352242
#214341272
#214335682
#214233732
#214231922
#214227812
#214225252
#214224912
C2×C59×#5572
#1877859