Finite Group Databasebeta

← #28567296

Subgroups — #28567296

Order 15104.

932 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21435217C211yesmaximal, characteristic
2#21434127C211yesmaximal, characteristic
3#21433568C211yesmaximal, characteristic
4#21423338C214yesmaximal, complemented
5#21423192C214yesmaximal, complemented
6#21422768C211yesmaximal, characteristic, complemented
7#21422525C211yesmaximal, characteristic, complemented
8#21422491C211yesmaximal, characteristic, complemented
9C2×C59×#557C211yesmaximal, characteristic, complemented
10C2×#14273758—21no—
11#16837952C2211yescharacteristic
12#16837805C2212yes—
13#16837750C2212yes—
14#16837720—21no—
15#16837715—22no—
16#16837687C2212yes—
17#16837679—22no—
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#14273660—24no—
57#14273658—22no—
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×C59D412yes—
104C2×C4⋊C4×C59D411yescharacteristic
105C2×C42×C59—22no—
106C2×C42×C59—21no—
107C2×C42×C59—21no—
108C2×C42×C59C2311yescharacteristic
109C2×Q8×C59—24no—
110C2×Q8×C59—21no—
111C2×Q8×C59—21no—
112C2×Q8×C59—22no—
113C2×Q8×C59—22no—
114C2×Q8×C59—22no—
115C2×Q8×C59—22no—
116C2×Q8×C59—42no—
117C2×Q8×C59—42no—
118C2×Q8×C59C2×D412yes—
119C2×Q8×C59C2×D412yes—
120C22×C4×C59—21no—
121C22×C4×C59—21no—
122C22×C4×C59—21no—
123C22×C4×C59—21no—
124C22×C4×C59—22no—
125C22×C4×C59C2×D411yescharacteristic
126C22×C4×C59C2×D411yescharacteristic
127C22×C4×C59C2×D411yescharacteristic
128C2×C22⋊D59—41no—
129#11745685—42no—
130#11745685—42no—
131C2×D236—21no—
132C2×D236—21no—
133C2×D236—21no—
134C2×C4×D59—41no—
135C2×Dic118—21no—
136C4⋊C4×C59—28no—
137C4⋊C4×C59—28no—
138C4⋊C4×C59—28no—
139C4⋊C4×C59—22no—
140C4⋊C4×C59—22no—
141C4⋊C4×C59—22no—
142C4⋊C4×C59—21no—
143C4⋊C4×C59—21no—
144C4⋊C4×C59—22no—
145C4⋊C4×C59—22no—
146C4⋊C4×C59—22no—
147C4⋊C4×C59—22no—
148C4⋊C4×C59C2×D412yes—
149C4⋊C4×C59C2×D412yes—
150C4⋊C4×C59C2×D414yes—
151C42×C59—28no—
152C42×C59—22no—
153C42×C59—22no—
154C42×C59—22no—
155C42×C59—22no—
156C42×C59C2×D411yescharacteristic
157C42×C59C2×D411yescharacteristic
158C42×C59C2×D411yescharacteristic
159C42×C59C2411yescharacteristic
160#11745672—44no—
161#11745671—44no—
162#11745671—44no—
163(C2×D59)⋊C4—41no—
164(C2×D59)⋊C4—41no—
165C4⋊Dic59—21no—
166C4⋊Dic59—21no—
167#11745664—42no—
168C2×C59⋊C8—22no—
169C2×C59⋊C8—22no—
170C2×C59⋊C8—22no—
171C23×C59#12011yescharacteristic
172C22×D59—41no—
173Q8×C59—24no—
174Q8×C59—22no—
175Q8×C59—22no—
176Q8×C59—22no—
177Q8×C59—42no—
178Q8×C59—42no—
179Q8×C59—42no—
180Q8×C59—44no—
181Q8×C59#13714yescomplemented
182C2×C4×C59—24no—
183C2×C4×C59—22no—
184C2×C4×C59—22no—
185C2×C4×C59—22no—
186C2×C4×C59—22no—
187C2×C4×C59—22no—
188C2×C4×C59—24no—
189C2×C4×C59—41no—
190C2×C4×C59—42no—
191C2×C4×C59—42no—
192C2×C4×C59—42no—
193C2×C4×C59—44no—
194C2×C4×C59#12011yescharacteristic
195C2×C4×C59#12011yescharacteristic
196C2×C4×C59#12011yescharacteristic
197C2×C4×C59#13512yes—
198C2×C4×C59AGL(1,ℤ/8)12yes—
199C2×C4×C59C22×D411yescharacteristic
200C2×C4×C59C22×D411yescharacteristic
201C2×C4×C59C22×D411yescharacteristic
202C22⋊D59—82no—
203C2×Dic59—41no—
204D236—42no—
205D236—41no—
206D236—41no—
207D236—42no—
208C4×D59—82no—
209Dic118—41no—
210Dic118—41no—
211C59⋊C8—42no—
212C59⋊C8—42no—
213C59⋊C8—42no—
214#18760—591nomaximal, Sylow
215C22×C59—21no—
216C22×C59—21no—
217C22×C59#44912yes—
218C22×C59C2×#12011yescharacteristic
219C2×D59—81no—
220C2×D59—81no—
221C2×D59—81no—
222C4×C59—24no—
223C4×C59—21no—
224C4×C59—21no—
225C4×C59—21no—
226C4×C59—42no—
227C4×C59—44no—
228C4×C59—42no—
229C4×C59—42no—
230C4×C59#44912yes—
231C4×C59C2×#13511yescharacteristic
232C4×C59C2×AGL(1,ℤ/8)11yescharacteristic
233C4×C59C2×#13712yes—
234Dic59—81no—
235C2×#557D5911yescharacteristic, complemented
236#3226—591no—
237#3103—591no—
238#3037—591no—
239#1648—594no—
240#1626—594no—
241#1546—591no—
242#1487—591no—
243#1479—591no—
244C2×C59—22no—
245C2×C59C2×#44911yescharacteristic
246C2×C59#300411yescharacteristic
247C2×C59#302011yescharacteristic
248D59—161no—
249C2×#137—1181no—
250#557—22no—
251#557—22no—
252#557—22no—
253#557C2×D5914yes—
254#532—591no—
255C2×C4⋊Q8—21no—
256C2×C4⋊Q8C2×D5911yescharacteristic
257C2×#125—22no—
258C2×#125—21no—
259C2×C4×Q8—21no—
260C2×C4×Q8—21no—
261C2×C4×Q8C2×D5911yescharacteristic
262C2×C4×Q8C2×D5911yescharacteristic
263#500—592no—
264#486—592no—
265#481—1181no—
266#480—1182no—
267#475—592no—
268#474—1182no—
269#464—1182no—
270#462—592no—
271#460—592no—
272#460—592no—
273#448—1181no—
274C4×SD16—1182no—
275#431—591no—
276#431—591no—
277#423—591no—
278#417—1181no—
279#332—594no—
280Q8⋊C8—594no—
281Q8⋊C8—594no—
282C59#1876011yescharacteristic, complemented, Sylow
283C2×#41—1181no—
284C22×Q8—21no—
285C22×Q8C22⋊D5911yescharacteristic
286C22×Q8C22⋊D5911yescharacteristic
287#137—2364no—
288C2×Q16—1182no—
289C2×SD16—1182no—
290C2×SD16—1182no—
291C2×M16—1181no—
292C4⋊Q8—24no—
293C4⋊Q8—24no—
294C4⋊Q8—21no—
295C4⋊Q8—21no—
296C4⋊Q8—591no—
297C4⋊Q8C22⋊D5912yes—
298C4⋊Q8C22×D5912yes—
299#127—591no—
300#125—28no—
301#125—24no—
302#125—28no—
303#125—24no—
304#124—1181no—
305#121—1181no—
306#121—1181no—
307C4×Q8—24no—
308C4×Q8—24no—
309C4×Q8—24no—
310C4×Q8—24no—
311C4×Q8—21no—
312C4×Q8—21no—
313C4×Q8—21no—
314C4×Q8—21no—
315C4×Q8C22⋊D5912yes—
316C4×Q8C22⋊D5912yes—
317C4×Q8C22×D5912yes—
318C4×Q8C22×D5912yes—
319C4×D4—1181no—
320C4×D4—1181no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4C22⋊D5911yescharacteristic
328C2×C4⋊C4C22⋊D5911yescharacteristic
329C2×C4⋊C4C22⋊D5912yes—
330C2×C42—22no—
331C2×C42—21no—
332C2×C42—21no—
333C2×C42C22×D5911yescharacteristic
334#107—1184no—
335#106—1182no—
336C4⋊C8—592no—
337C4⋊C8—592no—
338C4⋊C8—1181no—
339C4⋊C8—1181no—
340#103—1182no—
341#103—1182no—
342#102—1182no—
343#102—1182no—
344#102—1184no—
345C22⋊C8—1181no—
346C22⋊C8—1181no—
347C4×C8—592no—
348C4×C8—1181no—
349#41—2362no—
350#41—2362no—
351C2×Q8—24no—
352C2×Q8—21no—
353C2×Q8—21no—
354C2×Q8—22no—
355C2×Q8—22no—
356C2×Q8—22no—
357C2×Q8—22no—
358C2×Q8—42no—
359C2×Q8—42no—
360C2×Q8—1181no—
361C2×Q8C2×C22⋊D5912yes—
362C2×Q8C2×C22⋊D5912yes—
363C2×D4—1181no—
364C2×D4—1181no—
365C2×D4—1181no—
366C2×D4—2361no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—22no—
372C22×C4—2361no—
373C22×C4C2×C22⋊D5911yescharacteristic
374C22×C4C2×C22⋊D5911yescharacteristic
375C22×C4C2×C22⋊D5911yescharacteristic
376Q16—2364no—
377SD16—2364no—
378SD16—2364no—
379M16—2362no—
380C2×C8—1182no—
381C2×C8—1182no—
382C2×C8—1182no—
383C4⋊C4—28no—
384C4⋊C4—28no—
385C4⋊C4—28no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—21no—
390C4⋊C4—21no—
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—1181no—
396C4⋊C4—1181no—
397C4⋊C4C2×C22⋊D5912yes—
398C4⋊C4C2×C22⋊D5912yes—
399C4⋊C4C2×C22⋊D5914yes—
400C22⋊C4—2361no—
401C22⋊C4—2361no—
402C42—28no—
403C42—22no—
404C42—22no—
405C42—22no—
406C42—22no—
407C42C2×C22⋊D5911yescharacteristic
408C42C2×C22⋊D5911yescharacteristic
409C42C2×C22⋊D5911yescharacteristic
410C42C23×D5911yescharacteristic
411C23—2361no—
412C23#1427369611yescharacteristic
413Q8—24no—
414Q8—22no—
415Q8—22no—
416Q8—22no—
417Q8—42no—
418Q8—42no—
419Q8—42no—
420Q8—44no—
421Q8—2361no—
422Q8—2361no—
423Q8#1427375814yescomplemented
424D4—4722no—
425D4—2362no—
426D4—2361no—
427D4—2361no—
428D4—2362no—
429C2×C4—24no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—24no—
436C2×C4—41no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—44no—
441C2×C4—4722no—
442C2×C4—2361no—
443C2×C4#1427369611yescharacteristic
444C2×C4#1427369611yescharacteristic
445C2×C4#1427369611yescharacteristic
446C2×C4#1427374712yes—
447C2×C4#1427375212yes—
448C2×C4C22×C22⋊D5911yescharacteristic
449C2×C4C22×C22⋊D5911yescharacteristic
450C2×C4C22×C22⋊D5911yescharacteristic
451C8—2362no—
452C8—2362no—
453C8—2362no—
454C22—21no—
455C22—21no—
456C22—4721no—
457C22—4721no—
458C22—4721no—
459C22#1683754712yes—
460C22C2×#1427369611yescharacteristic
461C4—21no—
462C4—21no—
463C4—24no—
464C4—21no—
465C4—42no—
466C4—42no—
467C4—42no—
468C4—44no—
469C4—4721no—
470C4#1683754712yes—
471C4C2×#1427374711yescharacteristic
472C4C2×#1427375211yescharacteristic
473C4C2×#1427375812yes—
474C2—22no—
475C2—9441no—
476C2C2×#1683754711yescharacteristic
477C2#2143333011yescharacteristic
478C2#2143346011yescharacteristic
subgroupG/H
centerC22C2×#14273696
commutator, cosocleC42×C59C24
FrattiniC42C23×D59
FittingC2×C59×#557C2
socleC22×C59C2×#120
prime radical#16837952C22
hypercenterC2×#557D59
hypocenterC59#18760
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. #21423338C2semidirect
5. #21423192C2semidirect
6. #21422768C2semidirect
7. #21422525C2semidirect
8. #21422491C2semidirect
9. C2×C59×#557C2semidirect
181. Q8×C59#137semidirect
235. C2×#557D59semidirect
282. C59#18760semidirect
423. Q8#14273758semidirect

Maximal subgroups

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

subgroupindex
#214352172
#214341272
#214335682
#214233382
#214231922
#214227682
#214225252
#214224912
C2×C59×#5572
#1876059