Finite Group Databasebeta

← #28492054

Subgroups — #28492054

Order 15104.

708 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21429042C211yesmaximal, characteristic
2#21424901C211yesmaximal, characteristic, complemented
3#21424857C211yesmaximal, characteristic, complemented
4C2×#16837145C211yesmaximal, characteristic, complemented
5C2×D4⋊(C59⋊C8)C211yesmaximal, characteristic, complemented
6#21421935C211yesmaximal, characteristic, complemented
7C59×#1802C211yesmaximal, characteristic, complemented
8C2×#14273724C2211yescharacteristic
9C2×#14273723—21no—
10C2×#14273701—21no—
11C2×#14273701—21no—
12C2×#14273658—21no—
13C2×C4⋊(C59⋊C8)C2211yescharacteristic
14C2×#14273647—22no—
15C2×C4×C59⋊C8C2211yescharacteristic
16#16837311—22no—
17#16837303—21no—
18C4×C4⋊Dic59C2211yescharacteristic
19#16837164—21no—
20#16837145—22no—
21D4⋊(C59⋊C8)—22no—
22C2×C4×D4×C59C2211yescharacteristic
23C2×C4⋊C8×C59C2211yescharacteristic
24C2×#102×C59—21no—
25C59×#384C2211yescharacteristic
26C59×#340—21no—
27C22×D236—21no—
28C22×D236—21no—
29C22×D236D412yes—
30C22×C22⋊D59—22no—
31C22×C22⋊D59—22no—
32C22×C59⋊C8—21no—
33C22×C59⋊C8—21no—
34#14273724—21no—
35#14273724—21no—
36#14273724—21no—
37#14273724—21no—
38#14273723—42no—
39#14273723—42no—
40#14273701—42no—
41#14273701—42no—
42#14273701—42no—
43#14273701—42no—
44C2×C4⋊Dic59D411yescharacteristic
45C2×C4⋊Dic59D411yescharacteristic
46C2×(C2×D59)⋊C4—22no—
47C2×(C2×D59)⋊C4—22no—
48C2×C4×Dic59—21no—
49#14273658—42no—
50C4⋊(C59⋊C8)—21no—
51C4⋊(C59⋊C8)D412yescomplemented
52#14273647—44no—
53C4×C59⋊C8—21no—
54C4×C59⋊C8—21no—
55C59⋊#95—21no—
56C22×D4×C59D411yescharacteristic
57C23×C4×C59—21no—
58C22×C8×C59—21no—
59C4×D4×C59—21no—
60C4×D4×C59—21no—
61C4×D4×C59—21no—
62C4×D4×C59—21no—
63C2×C4⋊C4×C59—21no—
64C2×C4⋊C4×C59—21no—
65C2×C4⋊C4×C59—21no—
66C2×C4⋊C4×C59D411yescharacteristic
67C2×C22⋊C4×C59—21no—
68C2×C42×C59C2311yescharacteristic
69C4⋊C8×C59—21no—
70C4⋊C8×C59—21no—
71#102×C59—42no—
72C24×C59—21no—
73C23×D59—22no—
74C2×D4×C59—21no—
75C2×D4×C59—21no—
76C2×D4×C59—21no—
77C2×D4×C59—21no—
78C2×D4×C59—21no—
79C2×D4×C59SD1612yes—
80C22×C4×C59—21no—
81C22×C4×C59—41no—
82C22×C4×C59—41no—
83C22×C4×C59—41no—
84C22×C4×C59—41no—
85C22×C4×C59—41no—
86C22×C4×C59—41no—
87C22×C4×C59—41no—
88C22×C4×C59C2×D411yescharacteristic
89C22×C4×C59C2×D411yescharacteristic
90C22×C4×C59C2×D411yescharacteristic
91C2×C22⋊D59—44no—
92C2×C22⋊D59—44no—
93C2×C22⋊D59—44no—
94C2×C22⋊D59—44no—
95C2×C22⋊D59—82no—
96C2×C22⋊D59—82no—
97C22×Dic59—21no—
98C22×Dic59—21no—
99C2×D236—22no—
100C2×D236—22no—
101C2×D236—42no—
102C2×D236—42no—
103C2×D236—41no—
104C2×D236—41no—
105C2×D236—41no—
106C2×D236—41no—
107C2×D236—41no—
108C2×D236—41no—
109C2×D236—41no—
110C2×D236—41no—
111C2×D236—41no—
112C2×D236—41no—
113C2×D236—41no—
114C2×D236—41no—
115C2×C8×C59—41no—
116C2×C8×C59—41no—
117C2×C8×C59—41no—
118C4⋊C4×C59—21no—
119C4⋊C4×C59—21no—
120C4⋊C4×C59—42no—
121C4⋊C4×C59—41no—
122C4⋊C4×C59—41no—
123C4⋊C4×C59—41no—
124C4⋊C4×C59—41no—
125C22⋊C4×C59—41no—
126C22⋊C4×C59—41no—
127C42×C59C2×D411yescharacteristic
128C42×C59#4111yescharacteristic
129C42×C59#4111yescharacteristic
130C42×C59#4111yescharacteristic
131(C2×D59)⋊C4—44no—
132(C2×D59)⋊C4—44no—
133C4⋊Dic59—21no—
134C4⋊Dic59—21no—
135C4⋊Dic59D812yes—
136C4⋊Dic59SD1612yes—
137C4×Dic59—41no—
138C4×Dic59—41no—
139C2×C59⋊C8—22no—
140C2×C59⋊C8—22no—
141C2×C59⋊C8—41no—
142C2×C59⋊C8—41no—
143C2×C59⋊C8—41no—
144C2×C59⋊C8—41no—
145C23×C59—22no—
146C23×C59—41no—
147C23×C59—41no—
148C23×C59—41no—
149C23×C59—41no—
150C23×C59—41no—
151C23×C59—41no—
152C23×C59#12011yescharacteristic
153C22×D59—42no—
154C22×D59—42no—
155C22×D59—42no—
156C22×D59—42no—
157C22×D59—42no—
158C22×D59—42no—
159C22×D59—42no—
160D4×C59—22no—
161D4×C59—22no—
162D4×C59—41no—
163D4×C59—41no—
164C2×C4×C59—21no—
165C2×C4×C59—21no—
166C2×C4×C59—41no—
167C2×C4×C59—41no—
168C2×C4×C59—41no—
169C2×C4×C59—81no—
170C2×C4×C59—81no—
171C2×C4×C59—81no—
172C2×C4×C59—81no—
173C2×C4×C59—81no—
174C2×C4×C59—81no—
175C2×C4×C59—81no—
176C2×C4×C59#12111yescharacteristic
177C2×C4×C59#12111yescharacteristic
178C2×C4×C59#12111yescharacteristic
179C2×C4×C59#12311yescharacteristic
180C2×C4×C59#12311yescharacteristic
181C2×C4×C59#12411yescharacteristic
182C2×C4×C59C2×D811yescharacteristic
183C2×C4×C59C2×SD1611yescharacteristic
184C2×C4×C59C2×SD1611yescharacteristic
185C2×C4×C59#13511yescharacteristic
186C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
187C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
188C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
189C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
190C22⋊D59—88no—
191C22⋊D59—88no—
192C2×Dic59—22no—
193C2×Dic59—22no—
194C2×Dic59—41no—
195C2×Dic59—41no—
196C2×Dic59—41no—
197C2×Dic59—41no—
198D236—44no—
199D236—82no—
200D236—81no—
201D236—81no—
202D236—81no—
203D236—81no—
204D236—81no—
205D236—81no—
206D236—81no—
207D236—81no—
208C8×C59—81no—
209C59⋊C8—42no—
210C59⋊C8—42no—
211#10409—591nomaximal, Sylow
212C22×C59—42no—
213C22×C59—42no—
214C22×C59—42no—
215C22×C59—81no—
216C22×C59—81no—
217C22×C59—81no—
218C22×C59—81no—
219C22×C59#39411yescharacteristic
220C22×C59#44711yescharacteristic
221C22×C59#44911yescharacteristic
222C22×C59#45011yescharacteristic
223C22×C59#45011yescharacteristic
224C22×C59#45311yescharacteristic
225C22×C59#45511yescharacteristic
226C2×D59—82no—
227C2×D59—82no—
228C2×D59—82no—
229C2×D59—82no—
230C2×D59—82no—
231C2×D59—82no—
232C2×D59—82no—
233C4×C59—21no—
234C4×C59—21no—
235C4×C59—81no—
236C4×C59—161no—
237C4×C59#45911yescharacteristic
238C4×C59#46011yescharacteristic
239C4×C59#46511yescharacteristic
240C4×C59#46911yescharacteristic
241C4×C59#48111yescharacteristic
242C4×C59#48211yescharacteristic
243C4×C59#48611yescharacteristic
244C4×C59#49011yescharacteristic
245Dic59—42no—
246Dic59—42no—
247#2541—591no—
248#1957—591no—
249#1948—591no—
250#1802D5911yescharacteristic, complemented
251C2×#331—591no—
252C2×D4⋊C8—591no—
253#1279—591no—
254C2×C59—82no—
255C2×C59#162111yescharacteristic
256C2×C59#162911yescharacteristic
257C2×C59#163911yescharacteristic
258C2×C59#164211yescharacteristic
259C2×C59#203411yescharacteristic
260C2×C59#203711yescharacteristic
261C2×C59#204111yescharacteristic
262D59—162no—
263C2×#127—591no—
264C2×#127—1181no—
265C2×#121—1181no—
266C2×#121—1181no—
267C2×C4×D4C2×D5911yescharacteristic
268C2×#106—1181no—
269C2×C4⋊C8—591no—
270C2×C4⋊C8C2×D5911yescharacteristic
271C2×#102—21no—
272C2×#102—1182no—
273C2×C4×C8—591no—
274#392—1182no—
275#390—1181no—
276#384C2×D5911yescharacteristic
277C4×C4⋊C4—591no—
278#340—21no—
279#340—1181no—
280#331—1182no—
281D4⋊C8—1182no—
282C59#1040911yescharacteristic, complemented, Sylow
283C22×D4—592no—
284C22×D4—1182no—
285C22×D4—1182no—
286C22×D4—1181no—
287C22×D4—1181no—
288C22×D4C22⋊D5911yescharacteristic
289C23×C4—21no—
290C22×C8—21no—
291C22×C8—1181no—
292C22×C8—1181no—
293#127—1181no—
294#127—1181no—
295#127—1181no—
296#127—1181no—
297#127—2362no—
298#127—2362no—
299#121—2362no—
300#121—2362no—
301#121—2362no—
302#121—2362no—
303C4×D4—21no—
304C4×D4—21no—
305C4×D4—21no—
306C4×D4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—591no—
311C2×C4⋊C4—591no—
312C2×C4⋊C4C22⋊D5911yescharacteristic
313C2×C22⋊C4—21no—
314C2×C22⋊C4—1182no—
315C2×C22⋊C4—1182no—
316C2×C42—1181no—
317C2×C42C22×D5911yescharacteristic
318#106—2362no—
319C4⋊C8—21no—
320C4⋊C8—21no—
321C4⋊C8—592no—
322C4⋊C8—1181no—
323#102—42no—
324#102—2364no—
325C4×C8—1181no—
326C4×C8—1181no—
327#95—1181no—
328C24—21no—
329C24—1182no—
330C2×D4—21no—
331C2×D4—21no—
332C2×D4—21no—
333C2×D4—21no—
334C2×D4—21no—
335C2×D4—1182no—
336C2×D4—1182no—
337C2×D4—4722no—
338C2×D4—4722no—
339C2×D4—2362no—
340C2×D4—2362no—
341C2×D4—2364no—
342C2×D4—2364no—
343C2×D4—2364no—
344C2×D4—2364no—
345C2×D4—2361no—
346C2×D4—2361no—
347C2×D4—2361no—
348C2×D4—2361no—
349C2×D4—2361no—
350C2×D4—2361no—
351C2×D4—2361no—
352C2×D4—2361no—
353C2×D4—2361no—
354C2×D4—2361no—
355C2×D4—2361no—
356C2×D4—2361no—
357C2×D4#1174567112yes—
358C22×C4—21no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—1181no—
367C22×C4—1181no—
368C22×C4D4×D5911yescharacteristic
369C22×C4D4×D5911yescharacteristic
370C22×C4C2×C22⋊D5911yescharacteristic
371C2×C8—41no—
372C2×C8—41no—
373C2×C8—41no—
374C2×C8—1182no—
375C2×C8—1182no—
376C2×C8—2361no—
377C2×C8—2361no—
378C2×C8—2361no—
379C2×C8—2361no—
380C4⋊C4—21no—
381C4⋊C4—21no—
382C4⋊C4—42no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C4⋊C4—592no—
388C4⋊C4—592no—
389C4⋊C4—1181no—
390C4⋊C4—1181no—
391C22⋊C4—41no—
392C22⋊C4—41no—
393C22⋊C4—2364no—
394C22⋊C4—2364no—
395C42—2361no—
396C42—2361no—
397C42#1174568511yescharacteristic
398C42#1174568511yescharacteristic
399C42D4×D5911yescharacteristic
400C42#1174568711yescharacteristic
401C23—22no—
402C23—41no—
403C23—41no—
404C23—41no—
405C23—41no—
406C23—41no—
407C23—41no—
408C23—2362no—
409C23—2362no—
410C23—2362no—
411C23—2362no—
412C23—2362no—
413C23—2362no—
414C23—2362no—
415C23#1427369511yescharacteristic
416D4—22no—
417D4—22no—
418D4—41no—
419D4—41no—
420D4—4722no—
421D4—4728no—
422D4—4728no—
423D4—4721no—
424D4—4721no—
425D4—4721no—
426D4—4721no—
427D4—4721no—
428D4—4721no—
429D4—4721no—
430D4—4721no—
431D4—2364no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—81no—
438C2×C4—81no—
439C2×C4—81no—
440C2×C4—81no—
441C2×C4—81no—
442C2×C4—81no—
443C2×C4—81no—
444C2×C4—1182no—
445C2×C4—1182no—
446C2×C4—2361no—
447C2×C4—2361no—
448C2×C4—2361no—
449C2×C4—2361no—
450C2×C4#1427369911yescharacteristic
451C2×C4#1427370011yescharacteristic
452C2×C4#1427370011yescharacteristic
453C2×C4#1427370911yescharacteristic
454C2×C4#1427371011yescharacteristic
455C2×C4#1427371311yescharacteristic
456C2×C4D8×D5911yescharacteristic
457C2×C4SD16×D5911yescharacteristic
458C2×C4C2×#1174567111yescharacteristic
459C2×C4#1427374711yescharacteristic
460C2×C4#1427375111yescharacteristic
461C2×C4#1427375211yescharacteristic
462C2×C4#1427375311yescharacteristic
463C2×C4#1427375411yescharacteristic
464C8—81no—
465C8—2362no—
466C8—2362no—
467C22—42no—
468C22—42no—
469C22—42no—
470C22—81no—
471C22—81no—
472C22—81no—
473C22—81no—
474C22—4722no—
475C22—4722no—
476C22—4722no—
477C22—4722no—
478C22—4722no—
479C22—4722no—
480C22—4722no—
481C22#1683731711yescharacteristic
482C22#1683753411yescharacteristic
483C22#1683754511yescharacteristic
484C22#1683755111yescharacteristic
485C22#1683755211yescharacteristic
486C22#1683756811yescharacteristic
487C22#1683757811yescharacteristic
488C4—21no—
489C4—21no—
490C4—81no—
491C4—161no—
492C4—2362no—
493C4—2362no—
494C4#1683759611yescharacteristic
495C4#1683760111yescharacteristic
496C4#1683763311yescharacteristic
497C4#1683765311yescharacteristic
498C4#1683772011yescharacteristic
499C4#1683772611yescharacteristic
500C4#1683774711yescharacteristic
501C4#1683776511yescharacteristic
502C2—82no—
503C2—9442no—
504C2#2142316011yescharacteristic
505C2#2142321111yescharacteristic
506C2#2142327411yescharacteristic
507C2#2142329411yescharacteristic
508C2#2142528511yescharacteristic
509C2#2142530411yescharacteristic
510C2#2142532811yescharacteristic
subgroupG/H
centerC23#14273695
commutator, cosocleC2×C42×C59C23
FrattiniC2×C42C22×D59
FittingC59×#1802C2
socleC23×C59#120
prime radical#21429042C2
hypercenter#1802D59
hypocenterC59#10409
LCSG ▷ C2×C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1802

Complements

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

NH ≅ G/Ntype
2. #21424901C2semidirect
3. #21424857C2semidirect
4. C2×#16837145C2semidirect
5. C2×D4⋊(C59⋊C8)C2semidirect
6. #21421935C2semidirect
7. C59×#1802C2semidirect
51. C4⋊(C59⋊C8)D4semidirect
250. #1802D59semidirect
282. C59#10409semidirect

Maximal subgroups

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

subgroupindex
#214290422
#214249012
#214248572
C2×#168371452
C2×D4⋊(C59⋊C8)2
#214219352
C59×#18022
#1040959