Finite Group Databasebeta

← #28600344

Subgroups — #28600344

Order 15104.

968 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837715C211yesmaximal, characteristic
2#21426602C211yesmaximal, characteristic
3C2×#16837301C211yesmaximal, characteristic, complemented
4#21424833C212yesmaximal, complemented
5#21424794C212yesmaximal, complemented
6#21424691C212yesmaximal, complemented
7#21424411C211yesmaximal, characteristic
8#21424358C212yesmaximal, complemented
9#21424288C211yesmaximal, characteristic, complemented
10#21423860C211yesmaximal, characteristic, complemented
11C59×#1823C211yesmaximal, characteristic, complemented
12C2×#14273701C2211yescharacteristic
13C2×#14273689C2211yescharacteristic
14C2×(C22⋊D59)⋊C4—21no—
15C22×C4⋊Dic59—21no—
16C22×C4⋊Dic59C2211yescharacteristic
17C22×C4×Dic59C2211yescharacteristic
18#16837715C418yescomplemented
19C2×#14273660C2212yes—
20C2×#14273658C2212yes—
21C2×#14273649C2212yes—
22C2×#14273647C2212yes—
23C2×C22⋊(C59⋊C8)C2211yescharacteristic
24C2×#14273637C2212yes—
25#16837303C2211yescharacteristic
26#16837301—21no—
27#16837301—21no—
28#16837301C2212yes—
29#16837301C2212yes—
30#16837298C2211yescharacteristic
31#16837291—21no—
32#16837279—21no—
33C4⋊C4×Dic59C2211yescharacteristic
34#16837249—21no—
35C2×C59⋊#95—21no—
36#16837165C2212yes—
37#16837164C2212yes—
38#16837156C2212yes—
39C22×C4⋊C4×C59C2211yescharacteristic
40C2×C22⋊C8×C59C2211yescharacteristic
41C59×#380C2211yescharacteristic
42C59×#340C2212yes—
43C59×#340C2212yes—
44C22×D236C2311yescharacteristic
45C22×C22⋊D59—21no—
46C23×Dic59—21no—
47C23×Dic59D411yescharacteristic
48C23×Dic59D411yescharacteristic
49C22×C4×D59—21no—
50C22×C59⋊C8C2312yes—
51#14273701—22no—
52#14273701C2×C414yes—
53#14273689—42no—
54(C22⋊D59)⋊C4—44no—
55C2×C4⋊Dic59—21no—
56C2×C4⋊Dic59—22no—
57C2×C4⋊Dic59—22no—
58C2×C4⋊Dic59—42no—
59C2×C4⋊Dic59—44no—
60C2×C4⋊Dic59C2312yes—
61C2×C4⋊Dic59C2311yescharacteristic
62C2×Dic59⋊C4—21no—
63C2×(C2×D59)⋊C4—21no—
64C2×(C2×D59)⋊C4—21no—
65C2×(C2×D59)⋊C4—21no—
66C2×(C2×D59)⋊C4—21no—
67C2×C4×Dic59—21no—
68C2×C4×Dic59—42no—
69C2×C4×Dic59D411yescharacteristic
70C2×C4×Dic59D411yescharacteristic
71C2×C4×Dic59C2311yescharacteristic
72C2×C4×Dic59C2311yescharacteristic
73#14273660C2×C418yescomplemented
74#14273658—42no—
75#14273649—42no—
76#14273647C2×C418yescomplemented
77C22⋊(C59⋊C8)C2×C414yescomplemented
78#14273637—42no—
79C59⋊#95—22no—
80C59⋊#95—22no—
81C59⋊#95—21no—
82C59⋊#95—21no—
83C23×C4×C59—21no—
84C23×C4×C59C2311yescharacteristic
85C22×C8×C59C2312yes—
86C2×C4⋊C4×C59—21no—
87C2×C4⋊C4×C59—22no—
88C2×C4⋊C4×C59C2×C414yescomplemented
89C2×C4⋊C4×C59C2311yescharacteristic
90C2×C4⋊C4×C59C2311yescharacteristic
91C2×C4⋊C4×C59C2312yes—
92C2×C22⋊C4×C59—21no—
93C22⋊C8×C59—22no—
94#95×C59—21no—
95C24×C59C2×D411yescharacteristic
96C23×D59—21no—
97C22×C4×C59—22no—
98C22×C4×C59—22no—
99C22×C4×C59—24no—
100C22×C4×C59—22no—
101C22×C4×C59—21no—
102C22×C4×C59—21no—
103C22×C4×C59—42no—
104C22×C4×C59—42no—
105C22×C4×C59C22×C412yes—
106C22×C4×C59C2×D411yescharacteristic
107C22×C4×C59#4112yes—
108C22×C4×C59C2411yescharacteristic
109C2×C22⋊D59—24no—
110C2×C22⋊D59—42no—
111C2×C22⋊D59—42no—
112C22×Dic59—22no—
113C22×Dic59—22no—
114C22×Dic59—21no—
115C22×Dic59—22no—
116C22×Dic59—22no—
117C22×Dic59—41no—
118C22×Dic59—41no—
119C22×Dic59—41no—
120C22×Dic59—42no—
121C22×Dic59—42no—
122C22×Dic59—42no—
123C22×Dic59C2×D411yescharacteristic
124C22×Dic59C2×D411yescharacteristic
125C22×Dic59C2×D411yescharacteristic
126C22×Dic59C2×D411yescharacteristic
127C2×D236—22no—
128C2×D236—22no—
129C2×D236—21no—
130C2×D236C22×C412yes—
131C2×C4×D59—42no—
132C2×C4×D59—42no—
133C2×C4×D59—42no—
134C2×C8×C59—22no—
135C2×C8×C59—22no—
136C2×C8×C59—22no—
137C4⋊C4×C59—24no—
138C4⋊C4×C59—22no—
139C4⋊C4×C59—22no—
140C4⋊C4×C59—22no—
141C4⋊C4×C59C22×C414yes—
142C22⋊C4×C59—42no—
143(C2×D59)⋊C4—24no—
144(C2×D59)⋊C4—42no—
145(C2×D59)⋊C4—42no—
146(C2×D59)⋊C4—42no—
147C4⋊Dic59—21no—
148C4⋊Dic59—22no—
149C4⋊Dic59—22no—
150C4⋊Dic59—44no—
151C4⋊Dic59—44no—
152C4⋊Dic59—42no—
153C4⋊Dic59C22×C412yes—
154Dic59⋊C4—42no—
155C4×Dic59—22no—
156C4×Dic59—22no—
157C4×Dic59—41no—
158C4×Dic59—41no—
159C4×Dic59—42no—
160C2×C59⋊C8—22no—
161C2×C59⋊C8—22no—
162C2×C59⋊C8C22×C414yes—
163C23×C59—21no—
164C23×C59—21no—
165C23×C59—21no—
166C23×C59—21no—
167C23×C59C4×D412yes—
168C23×C59AGL(1,ℤ/8)12yes—
169C23×C59AGL(1,ℤ/8)12yes—
170C23×C59C22×D411yescharacteristic
171C22×D59—22no—
172C22×D59—41no—
173C22×D59—41no—
174C22×D59—41no—
175C22×D59—41no—
176C22×D59—41no—
177C22×D59—41no—
178C2×C4×C59—21no—
179C2×C4×C59—21no—
180C2×C4×C59—24no—
181C2×C4×C59—24no—
182C2×C4×C59—24no—
183C2×C4×C59—42no—
184C2×C4×C59—42no—
185C2×C4×C59—48no—
186C2×C4×C59—41no—
187C2×C4×C59—41no—
188C2×C4×C59—41no—
189C2×C4×C59—41no—
190C2×C4×C59—41no—
191C2×C4×C59—41no—
192C2×C4×C59C4×D412yes—
193C2×C4×C59C23×C411yescharacteristic
194C2×C4×C59C2×#4111yescharacteristic
195C2×C4×C59H(5,𝔽2)12yes—
196C2×C4×C592−1+412yes—
197C22⋊D59—48no—
198C2×Dic59—21no—
199C2×Dic59—21no—
200C2×Dic59—22no—
201C2×Dic59—21no—
202C2×Dic59—21no—
203C2×Dic59—21no—
204C2×Dic59—21no—
205C2×Dic59—21no—
206C2×Dic59—21no—
207C2×Dic59—21no—
208C2×Dic59—21no—
209C2×Dic59—21no—
210C2×Dic59—44no—
211C2×Dic59—44no—
212C2×Dic59—41no—
213C2×Dic59—41no—
214C2×Dic59—42no—
215C2×Dic59—42no—
216C2×Dic59—42no—
217C2×Dic59—84no—
218C2×Dic59#13712yes—
219D236—24no—
220D236—42no—
221C4×D59—84no—
222C8×C59—42no—
223C59⋊C8—24no—
224#22081—591nomaximal, Sylow
225C22×C59—22no—
226C22×C59—22no—
227C22×C59—22no—
228C22×C59—22no—
229C22×C59—22no—
230C22×C59—22no—
231C22×C59#44214yes—
232C22×C59C2×C4×D411yescharacteristic
233C22×C59#53411yescharacteristic
234C22×C59#53611yescharacteristic
235C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
236C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
237C22×C59C2×#13711yescharacteristic
238C22×C59#57511yescharacteristic
239C2×D59—42no—
240C2×D59—42no—
241C2×D59—42no—
242C2×D59—81no—
243C2×D59—81no—
244C2×D59—81no—
245C2×D59—81no—
246C4×C59—22no—
247C4×C59—44no—
248C4×C59—81no—
249C4×C59—81no—
250C4×C59#52012yes—
251C4×C59#53912yes—
252Dic59—22no—
253Dic59—41no—
254Dic59—41no—
255Dic59—41no—
256Dic59—42no—
257Dic59—82no—
258C2×#480—591no—
259C4×#121—591no—
260C2×#389—591no—
261#1943—592no—
262#1936—592no—
263#1921—592no—
264#1874—591no—
265#1864—592no—
266#1853—591no—
267#1823D5911yescharacteristic, complemented
268#1749—591no—
269C2×C59—24no—
270C2×C59#232211yescharacteristic
271C2×C59C2×#44211yescharacteristic
272C2×C59C4×#13711yescharacteristic
273C2×C59#319311yescharacteristic
274C2×C59#319411yescharacteristic
275C2×C59#323411yescharacteristic
276C2×C59#323811yescharacteristic
277D59—82no—
278C2×#121—591no—
279C2×C4×D4—591no—
280C2×C4×D4—1181no—
281C22×C4⋊C4—591no—
282C22×C4⋊C4—1181no—
283C22×C4⋊C4C2×D5911yescharacteristic
284C22×C42—591no—
285#480—598no—
286C2×#107—592no—
287C2×#106—592no—
288C2×#102—592no—
289C2×#102—592no—
290C2×C22⋊C8—591no—
291C2×C22⋊C8C2×D5911yescharacteristic
292C2×#97—592no—
293#390—591no—
294#389—592no—
295#389—592no—
296#389—591no—
297#389—1181no—
298#389—1181no—
299#388—1181no—
300#386—1181no—
301#380C2×D5911yescharacteristic
302C4×C4⋊C4—591no—
303C4×C22⋊C4—1181no—
304C2×#95—1181no—
305#340—592no—
306#340—592no—
307#340C2×D5912yes—
308#340C2×D5912yes—
309#336—592no—
310C59#2208111yescharacteristic, complemented, Sylow
311C22×D4—591no—
312C22×D4—1181no—
313C23×C4—21no—
314C23×C4—591no—
315C23×C4—591no—
316C23×C4—1181no—
317C23×C4—1181no—
318C23×C4C22×D5911yescharacteristic
319C22×C8—592no—
320C22×C8C22×D5912yes—
321#121—594no—
322#121—1182no—
323C4×D4—2362no—
324C4×D4—2364no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—591no—
328C2×C4⋊C4—592no—
329C2×C4⋊C4—1182no—
330C2×C4⋊C4—1182no—
331C2×C4⋊C4—1181no—
332C2×C4⋊C4—1181no—
333C2×C4⋊C4—2362no—
334C2×C4⋊C4—2364no—
335C2×C4⋊C4C4×D5914yescomplemented
336C2×C4⋊C4C22×D5911yescharacteristic
337C2×C4⋊C4C22×D5911yescharacteristic
338C2×C4⋊C4C22×D5912yes—
339C2×C22⋊C4—21no—
340C2×C22⋊C4—1181no—
341C2×C22⋊C4—1181no—
342C2×C22⋊C4—1181no—
343C2×C22⋊C4—1181no—
344C2×C42—591no—
345C2×C42—591no—
346C2×C42—591no—
347C2×C42—591no—
348C2×C42—1181no—
349C2×C42—2362no—
350#107—598no—
351#106—2362no—
352#102—598no—
353#102—2362no—
354C22⋊C8—22no—
355C22⋊C8—594no—
356#97—2362no—
357#95—21no—
358#95—1182no—
359#95—1182no—
360#95—1181no—
361#95—1181no—
362C24—1181no—
363C24D4×D5911yescharacteristic
364C2×D4—592no—
365C2×D4—1184no—
366C2×D4—1181no—
367C2×D4—1182no—
368C2×D4—1182no—
369C2×D4—2362no—
370C2×D4—2362no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—591no—
380C22×C4—591no—
381C22×C4—591no—
382C22×C4—591no—
383C22×C4—1182no—
384C22×C4—1182no—
385C22×C4—1182no—
386C22×C4—1181no—
387C22×C4—1182no—
388C22×C4—2362no—
389C22×C4—2362no—
390C22×C4—2362no—
391C22×C4—2361no—
392C22×C4—2361no—
393C22×C4—2361no—
394C22×C4—2362no—
395C22×C4—2362no—
396C22×C4—2362no—
397C22×C4C2×C4×D5912yes—
398C22×C4D4×D5911yescharacteristic
399C22×C4#1174568712yes—
400C22×C4C23×D5911yescharacteristic
401C2×C8—22no—
402C2×C8—22no—
403C2×C8—22no—
404C2×C8—594no—
405C2×C8—1182no—
406C2×C8—1182no—
407C4⋊C4—24no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—592no—
412C4⋊C4—1181no—
413C4⋊C4—1182no—
414C4⋊C4—1182no—
415C4⋊C4—2362no—
416C4⋊C4—2364no—
417C4⋊C4—2364no—
418C4⋊C4—2362no—
419C4⋊C4C2×C4×D5914yes—
420C22⋊C4—42no—
421C22⋊C4—1184no—
422C22⋊C4—2362no—
423C22⋊C4—2362no—
424C22⋊C4—2362no—
425C42—1182no—
426C42—1182no—
427C42—2361no—
428C42—2361no—
429C42—2362no—
430C23—21no—
431C23—21no—
432C23—21no—
433C23—21no—
434C23—1182no—
435C23—2361no—
436C23—2361no—
437C23—2361no—
438C23—2361no—
439C23—2361no—
440C23—2361no—
441C23(C22⋊D59)⋊C412yes—
442C23#1427375112yes—
443C23#1427375412yes—
444C23C2×D4×D5911yescharacteristic
445D4—1184no—
446D4—2368no—
447D4—2362no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—48no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—592no—
463C2×C4—1181no—
464C2×C4—1181no—
465C2×C4—1181no—
466C2×C4—1181no—
467C2×C4—1181no—
468C2×C4—1181no—
469C2×C4—1181no—
470C2×C4—1181no—
471C2×C4—1181no—
472C2×C4—1181no—
473C2×C4—1181no—
474C2×C4—1182no—
475C2×C4—4724no—
476C2×C4—4724no—
477C2×C4—2364no—
478C2×C4—2362no—
479C2×C4—2362no—
480C2×C4—2362no—
481C2×C4—2364no—
482C2×C4—2361no—
483C2×C4—2361no—
484C2×C4(C22⋊D59)⋊C412yes—
485C2×C4C22×C4×D5911yescharacteristic
486C2×C4C2×#1174568711yescharacteristic
487C2×C4#1427377312yes—
488C2×C4#1427377412yes—
489C8—42no—
490C8—1184no—
491C22—22no—
492C22—22no—
493C22—22no—
494C22—22no—
495C22—22no—
496C22—22no—
497C22—4721no—
498C22—4721no—
499C22—4721no—
500C22—4721no—
501C22—2362no—
502C22—2362no—
503C22—2362no—
504C22#1683751114yes—
505C22C2×(C22⋊D59)⋊C411yescharacteristic
506C22#1683796311yescharacteristic
507C22#1683797411yescharacteristic
508C22C2×#1427375111yescharacteristic
509C22C2×#1427375411yescharacteristic
510C22#137×D5911yescharacteristic
511C22#1683820111yescharacteristic
512C4—22no—
513C4—44no—
514C4—81no—
515C4—81no—
516C4—1182no—
517C4—4722no—
518C4—2361no—
519C4—2361no—
520C4—2361no—
521C4—2362no—
522C4#1683788212yes—
523C4#1683799612yes—
524C2—24no—
525C2—4722no—
526C2#2142676211yescharacteristic
527C2C2×#1683751111yescharacteristic
528C2#2143291111yescharacteristic
529C2#2143490411yescharacteristic
530C2#2143491511yescharacteristic
531C2#2143528311yescharacteristic
532C2#2143532011yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#1823C2
socleC23×C59C22×D4
prime radicalC2×#14273701C22
cosocleC22×C4×C59C24
hypercenter#1823D59
hypocenterC59#22081
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1823

Complements

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

NH ≅ G/Ntype
3. C2×#16837301C2semidirect
4. #21424833C2semidirect
5. #21424794C2semidirect
6. #21424691C2semidirect
8. #21424358C2semidirect
9. #21424288C2semidirect
10. #21423860C2semidirect
11. C59×#1823C2semidirect
18. #16837715C4semidirect
73. #14273660C2×C4semidirect
76. #14273647C2×C4semidirect
77. C22⋊(C59⋊C8)C2×C4semidirect
88. C2×C4⋊C4×C59C2×C4semidirect
267. #1823D59semidirect
310. C59#22081semidirect
335. C2×C4⋊C4C4×D59semidirect

Maximal subgroups

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

subgroupindex
C2×#168377152
#214266022
C2×#168373012
#214248332
#214247942
#214246912
#214244112
#214243582
#214242882
#214238602
C59×#18232
#2208159