Finite Group Databasebeta

← #28628681

Subgroups — #28628681

Order 15104.

922 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837721C211yesmaximal, characteristic
2#21432427C211yesmaximal, characteristic
3C2×#16837300C211yesmaximal, characteristic, complemented
4#21425594C212yesmaximal, complemented
5#21425483C212yesmaximal, complemented
6#21424834C212yesmaximal, complemented
7#21424793C212yesmaximal, complemented
8#21424411C211yesmaximal, characteristic
9#21424286C211yesmaximal, characteristic, complemented
10#21424187C211yesmaximal, characteristic, complemented
11C2×C59×#432C211yesmaximal, characteristic, complemented
12C2×#14273710—21no—
13C2×#14273701C2211yescharacteristic
14C22×C4⋊Dic59C2211yescharacteristic
15C22×C4⋊Dic59C2211yescharacteristic
16C22×Dic59⋊C4—21no—
17#16837721—24no—
18C2×#14273661C2212yes—
19C2×#14273659C2212yes—
20C2×#14273649C2212yes—
21C2×#14273649C2212yes—
22#16837304C2211yescharacteristic
23#16837304C2211yescharacteristic
24#16837300—21no—
25#16837300—21no—
26#16837300C2212yes—
27#16837300C2211yescharacteristic
28#16837300C2211yescharacteristic
29#16837300C2212yes—
30#16837293—21no—
31#16837293—21no—
32#16837281—21no—
33#16837262C2211yescharacteristic
34C2×C59⋊#95—21no—
35#16837165C2212yes—
36#16837165C2212yes—
37#16837165C2212yes—
38#16837165C2212yes—
39C22×C42×C59C2211yescharacteristic
40C59×#432—24no—
41C2×C4⋊C8×C59C2212yes—
42C2×C22⋊C8×C59C2211yescharacteristic
43C2×C22⋊C8×C59C2211yescharacteristic
44C2×#97×C59C2212yes—
45C22×D236C2311yescharacteristic
46C22×C22⋊D59—21no—
47C23×Dic59—21no—
48C23×Dic59—21no—
49#14273710—44no—
50#14273701—22no—
51#14273701—22no—
52C2×C4⋊Dic59—21no—
53C2×C4⋊Dic59—24no—
54C2×C4⋊Dic59—22no—
55C2×C4⋊Dic59—22no—
56C2×C4⋊Dic59—21no—
57C2×C4⋊Dic59C2311yescharacteristic
58C2×C4⋊Dic59C2312yes—
59C2×C4⋊Dic59C2311yescharacteristic
60C2×C4⋊Dic59C2311yescharacteristic
61C2×C4⋊Dic59C2312yes—
62C2×Dic59⋊C4—21no—
63C2×Dic59⋊C4—21no—
64C2×Dic59⋊C4—22no—
65C2×Dic59⋊C4—22no—
66C2×Dic59⋊C4—44no—
67C2×C22⋊Dic59—21no—
68C2×(C2×D59)⋊C4—21no—
69C2×(C2×D59)⋊C4—21no—
70C2×(C2×D59)⋊C4—21no—
71C2×(C2×D59)⋊C4—21no—
72C2×(C2×D59)⋊C4—21no—
73#14273661—24no—
74#14273659—24no—
75#14273649—24no—
76#14273649—24no—
77C59⋊#95—22no—
78C59⋊#95—22no—
79C59⋊#95—21no—
80C59⋊#95—21no—
81C59⋊#95—21no—
82C23×C4×C59D411yescharacteristic
83C23×C4×C59D411yescharacteristic
84C23×C4×C59C2311yescharacteristic
85C22×C8×C59C2312yes—
86C22×C8×C59C2312yes—
87C2×C42×C59—22no—
88C2×C42×C59—22no—
89C2×C42×C59D411yescharacteristic
90C2×C42×C59D411yescharacteristic
91C2×C42×C59C2311yescharacteristic
92C2×C42×C59C2311yescharacteristic
93C4⋊C8×C59—24no—
94C22⋊C8×C59—22no—
95C22⋊C8×C59D414yes—
96#97×C59—24no—
97C24×C59C2×D411yescharacteristic
98C23×D59—21no—
99C22×C4×C59—22no—
100C22×C4×C59—22no—
101C22×C4×C59—22no—
102C22×C4×C59—22no—
103C22×C4×C59—22no—
104C22×C4×C59—22no—
105C22×C4×C59—22no—
106C22×C4×C59—41no—
107C22×C4×C59C2×D411yescharacteristic
108C22×C4×C59C2×D411yescharacteristic
109C22×C4×C59C2×D411yescharacteristic
110C22×C4×C59C2×D411yescharacteristic
111C22×C4×C59C2×D412yes—
112C22×C4×C59C2×D411yescharacteristic
113C22×C4×C59#4112yes—
114C22×C4×C59C2411yescharacteristic
115C2×C22⋊D59—42no—
116C2×C22⋊D59—42no—
117C2×C22⋊D59—42no—
118C22×Dic59—24no—
119C22×Dic59—22no—
120C22×Dic59—21no—
121C22×Dic59—21no—
122C22×Dic59—21no—
123C22×Dic59—22no—
124C22×Dic59—42no—
125C22×Dic59—42no—
126C22×Dic59—42no—
127C22×Dic59—42no—
128C22×Dic59—42no—
129C2×D236—21no—
130C2×D236—22no—
131C2×D236—21no—
132C2×D236—22no—
133C2×C8×C59—22no—
134C2×C8×C59—22no—
135C2×C8×C59—22no—
136C2×C8×C59—22no—
137C2×C8×C59—22no—
138C2×C8×C59C2×D414yes—
139C42×C59—22no—
140C42×C59—22no—
141C42×C59—21no—
142C42×C59—21no—
143C42×C59—21no—
144C42×C59—21no—
145C22⋊Dic59—42no—
146(C2×D59)⋊C4—42no—
147(C2×D59)⋊C4—42no—
148(C2×D59)⋊C4—42no—
149(C2×D59)⋊C4—42no—
150(C2×D59)⋊C4—42no—
151C4⋊Dic59—24no—
152C4⋊Dic59—22no—
153C4⋊Dic59—22no—
154C4⋊Dic59—21no—
155C4⋊Dic59—22no—
156C4⋊Dic59—22no—
157C4⋊Dic59—21no—
158C4⋊Dic59—22no—
159C4⋊Dic59—42no—
160C4⋊Dic59#4114yes—
161Dic59⋊C4—44no—
162Dic59⋊C4—42no—
163Dic59⋊C4—42no—
164Dic59⋊C4—44no—
165C23×C59—21no—
166C23×C59—21no—
167C23×C59—21no—
168C23×C59—21no—
169C23×C59#12112yes—
170C23×C59AGL(1,ℤ/8)12yes—
171C23×C59#13712yes—
172C23×C59C22×D411yescharacteristic
173C22×D59—41no—
174C22×D59—41no—
175C22×D59—41no—
176C22×D59—41no—
177C22×D59—41no—
178C22×D59—41no—
179C22×D59—41no—
180C2×C4×C59—21no—
181C2×C4×C59—21no—
182C2×C4×C59—21no—
183C2×C4×C59—21no—
184C2×C4×C59—24no—
185C2×C4×C59—24no—
186C2×C4×C59—21no—
187C2×C4×C59—21no—
188C2×C4×C59—21no—
189C2×C4×C59—21no—
190C2×C4×C59—21no—
191C2×C4×C59—21no—
192C2×C4×C59—21no—
193C2×C4×C59—21no—
194C2×C4×C59—21no—
195C2×C4×C59—44no—
196C2×C4×C59—44no—
197C2×C4×C59#12112yes—
198C2×C4×C59AGL(1,ℤ/8)12yes—
199C2×C4×C59C22×D411yescharacteristic
200C2×C4×C59C2×#4112yes—
201C2×C4×C59C2×#4111yescharacteristic
202C2×C4×C592−1+412yes—
203C22⋊D59—84no—
204C2×Dic59—24no—
205C2×Dic59—48no—
206C2×Dic59—42no—
207C2×Dic59—42no—
208C2×Dic59—42no—
209C2×Dic59—41no—
210C2×Dic59—41no—
211C2×Dic59—41no—
212C2×Dic59—42no—
213C2×Dic59—41no—
214C2×Dic59—41no—
215C2×Dic59—41no—
216C2×Dic59—41no—
217C2×Dic59—41no—
218C2×Dic59—41no—
219C2×Dic59—42no—
220C2×Dic59—84no—
221D236—42no—
222D236—42no—
223C8×C59—24no—
224C8×C59—42no—
225#24607—591nomaximal, Sylow
226C22×C59—22no—
227C22×C59—22no—
228C22×C59—22no—
229C22×C59—22no—
230C22×C59—22no—
231C22×C59—22no—
232C22×C59#46814yes—
233C22×C59C2×#12111yescharacteristic
234C22×C59#52511yescharacteristic
235C22×C59#53611yescharacteristic
236C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
237C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
238C22×C59C2×#13711yescharacteristic
239C22×C59#57511yescharacteristic
240C2×D59—81no—
241C2×D59—81no—
242C2×D59—81no—
243C2×D59—81no—
244C2×D59—81no—
245C2×D59—81no—
246C2×D59—81no—
247C4×C59—22no—
248C4×C59—22no—
249C4×C59—41no—
250C4×C59—41no—
251C4×C59—41no—
252C4×C59#54712yes—
253C4×C59#55212yes—
254Dic59—44no—
255Dic59—81no—
256Dic59—81no—
257Dic59—81no—
258Dic59—82no—
259C2×#481—591no—
260C2×#432D5911yescharacteristic, complemented
261#2854—591no—
262C2×#389—591no—
263#2088—592no—
264#2067—592no—
265#1943—592no—
266#1936—592no—
267#1874—591no—
268#1852—591no—
269#1824—591no—
270C2×C59—24no—
271C2×C59#241311yescharacteristic
272C2×C59C2×#46811yescharacteristic
273C2×C59#305511yescharacteristic
274C2×C59#310911yescharacteristic
275C2×C59#311711yescharacteristic
276C2×C59#323411yescharacteristic
277C2×C59#323911yescharacteristic
278D59—161no—
279C2×#123—1181no—
280C2×#121—591no—
281C22×C4⋊C4—591no—
282C22×C4⋊C4—591no—
283C22×C4⋊C4—1181no—
284C22×C42C2×D5911yescharacteristic
285#481—1184no—
286#432—24no—
287C2×#107—592no—
288C2×#106—592no—
289C2×C4⋊C8C2×D5912yes—
290C2×#102—592no—
291C2×#102—592no—
292C2×C22⋊C8C2×D5911yescharacteristic
293C2×C22⋊C8C2×D5911yescharacteristic
294C2×#97C2×D5912yes—
295#390—591no—
296#390—591no—
297#389—592no—
298#389—591no—
299#389—591no—
300#389—592no—
301#389—1181no—
302#389—1181no—
303#388—1181no—
304#388—1181no—
305#386—1181no—
306#380—591no—
307C2×#95—1181no—
308#340—592no—
309#340—592no—
310#340—592no—
311#340—592no—
312C59#2460711yescharacteristic, complemented, Sylow
313C22×D4—591no—
314C22×D4—1181no—
315C23×C4—1181no—
316C23×C4—1181no—
317C23×C4D23611yescharacteristic
318C23×C4D23611yescharacteristic
319C23×C4C22×D5911yescharacteristic
320C22×C8C22×D5912yes—
321C22×C8C22×D5912yes—
322#123—2364no—
323#121—1182no—
324#121—1182no—
325C2×C4⋊C4—592no—
326C2×C4⋊C4—592no—
327C2×C4⋊C4—591no—
328C2×C4⋊C4—591no—
329C2×C4⋊C4—591no—
330C2×C4⋊C4—1184no—
331C2×C4⋊C4—1181no—
332C2×C4⋊C4—1182no—
333C2×C4⋊C4—1182no—
334C2×C4⋊C4—1181no—
335C2×C4⋊C4—1182no—
336C2×C4⋊C4—1182no—
337C2×C4⋊C4—1181no—
338C2×C4⋊C4—1181no—
339C2×C4⋊C4—2364no—
340C2×C22⋊C4—1181no—
341C2×C22⋊C4—1181no—
342C2×C22⋊C4—1181no—
343C2×C22⋊C4—1181no—
344C2×C22⋊C4—1181no—
345C2×C22⋊C4—1181no—
346C2×C42—22no—
347C2×C42—22no—
348C2×C42D23611yescharacteristic
349C2×C42D23611yescharacteristic
350C2×C42C22×D5911yescharacteristic
351C2×C42C22×D5911yescharacteristic
352#107—1184no—
353#106—1184no—
354C4⋊C8—24no—
355#102—1184no—
356#102—1184no—
357C22⋊C8—22no—
358C22⋊C8C22⋊D5914yes—
359#97—24no—
360#95—1182no—
361#95—1182no—
362#95—1181no—
363#95—1181no—
364#95—1181no—
365C24—1181no—
366C24C2×D23611yescharacteristic
367C2×D4—1181no—
368C2×D4—1182no—
369C2×D4—1181no—
370C2×D4—1182no—
371C2×D4—2362no—
372C2×D4—2362no—
373C2×D4—2362no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—41no—
382C22×C4—1184no—
383C22×C4—1182no—
384C22×C4—1181no—
385C22×C4—1181no—
386C22×C4—1181no—
387C22×C4—1182no—
388C22×C4—2362no—
389C22×C4—2362no—
390C22×C4—2362no—
391C22×C4—2362no—
392C22×C4—2362no—
393C22×C4C2×D23611yescharacteristic
394C22×C4C2×D23611yescharacteristic
395C22×C4C2×D23611yescharacteristic
396C22×C4C2×D23611yescharacteristic
397C22×C4C2×D23611yescharacteristic
398C22×C4#1174568512yes—
399C22×C4C2×C22⋊D5912yes—
400C22×C4C23×D5911yescharacteristic
401C2×C8—22no—
402C2×C8—22no—
403C2×C8—22no—
404C2×C8—22no—
405C2×C8—22no—
406C2×C8C2×C22⋊D5914yes—
407C4⋊C4—594no—
408C4⋊C4—1184no—
409C4⋊C4—1182no—
410C4⋊C4—1182no—
411C4⋊C4—1181no—
412C4⋊C4—1182no—
413C4⋊C4—1182no—
414C4⋊C4—1181no—
415C4⋊C4—1182no—
416C4⋊C4—2364no—
417C4⋊C4—2362no—
418C4⋊C4—2362no—
419C4⋊C4—2362no—
420C4⋊C4—2364no—
421C22⋊C4—2362no—
422C22⋊C4—2362no—
423C22⋊C4—2362no—
424C22⋊C4—2362no—
425C22⋊C4—2362no—
426C22⋊C4—2362no—
427C42—22no—
428C42—22no—
429C42—21no—
430C42—21no—
431C42—21no—
432C42—21no—
433C23—21no—
434C23—21no—
435C23—21no—
436C23—21no—
437C23—2361no—
438C23—2361no—
439C23—2361no—
440C23—2361no—
441C23—2361no—
442C23—2361no—
443C23—2361no—
444C23#1427370112yes—
445C23#1427375512yes—
446C23#1427376012yes—
447C23C22×D23611yescharacteristic
448D4—4724no—
449D4—2362no—
450D4—2362no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—1184no—
469C2×C4—4724no—
470C2×C4—2361no—
471C2×C4—2362no—
472C2×C4—2361no—
473C2×C4—2361no—
474C2×C4—2361no—
475C2×C4—2361no—
476C2×C4—2362no—
477C2×C4—2361no—
478C2×C4—2361no—
479C2×C4—2361no—
480C2×C4—2362no—
481C2×C4—2361no—
482C2×C4—2368no—
483C2×C4—2362no—
484C2×C4—2362no—
485C2×C4#1427370112yes—
486C2×C4#1427375512yes—
487C2×C4C22×C22⋊D5911yescharacteristic
488C2×C4#41×D5912yes—
489C2×C4C2×#1174568511yescharacteristic
490C2×C4#1427377412yes—
491C8—24no—
492C8—42no—
493C22—22no—
494C22—22no—
495C22—22no—
496C22—22no—
497C22—22no—
498C22—22no—
499C22—4721no—
500C22—4721no—
501C22—4721no—
502C22—4721no—
503C22—4721no—
504C22—4721no—
505C22—4721no—
506C22#1683764814yes—
507C22C2×#1427370111yescharacteristic
508C22#1683791411yescharacteristic
509C22#1683797811yescharacteristic
510C22C2×#1427375511yescharacteristic
511C22C2×#1427375511yescharacteristic
512C22C2×#1427376011yescharacteristic
513C22#1683820211yescharacteristic
514C4—22no—
515C4—22no—
516C4—41no—
517C4—41no—
518C4—41no—
519C4—4721no—
520C4—4721no—
521C4—4722no—
522C4—4721no—
523C4—2364no—
524C4#1683805112yes—
525C4#1683808512yes—
526C2—24no—
527C2—9441no—
528C2#2142749711yescharacteristic
529C2C2×#1683764811yescharacteristic
530C2#2143371511yescharacteristic
531C2#2143418811yescharacteristic
532C2#2143426911yescharacteristic
533C2#2143528811yescharacteristic
534C2#2143532811yescharacteristic
subgroupG/H
centerC23C22×D236
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC2×C59×#432C2
socleC23×C59C22×D4
prime radicalC2×#14273701C22
hypercenterC2×#432D59
hypocenterC59#24607
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#432

Complements

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

NH ≅ G/Ntype
3. C2×#16837300C2semidirect
4. #21425594C2semidirect
5. #21425483C2semidirect
6. #21424834C2semidirect
7. #21424793C2semidirect
9. #21424286C2semidirect
10. #21424187C2semidirect
11. C2×C59×#432C2semidirect
260. C2×#432D59semidirect
312. C59#24607semidirect

Maximal subgroups

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

subgroupindex
C2×#168377212
#214324272
C2×#168373002
#214255942
#214254832
#214248342
#214247932
#214244112
#214242862
#214241872
C2×C59×#4322
#2460759