Finite Group Databasebeta

← #28492053

Subgroups — #28492053

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#21429043C211yesmaximal, characteristic
2#21424903C211yesmaximal, characteristic, complemented
3#21424860C211yesmaximal, characteristic, complemented
4#21424097C211yesmaximal, characteristic, complemented
5C2×#16837129C211yesmaximal, characteristic, complemented
6#21421935C211yesmaximal, characteristic, complemented
7C2×C59×#331C211yesmaximal, characteristic, complemented
8C2×#14273723—21no—
9C2×#14273698—21no—
10C2×#14273698—21no—
11C2×C4×D236C2211yescharacteristic
12C2×#14273659—21no—
13C2×#14273649—21no—
14C2×#14273648—22no—
15C2×C4×C59⋊C8C2211yescharacteristic
16#16837310—22no—
17#16837305—21no—
18#16837272C2211yescharacteristic
19C4×C4⋊Dic59C2211yescharacteristic
20#16837165—21no—
21#16837165—21no—
22#16837129—22no—
23C2×#127×C59C2211yescharacteristic
24C2×C4⋊C8×C59C2211yescharacteristic
25C2×C4⋊C8×C59C2211yescharacteristic
26C59×#331—22no—
27C22×D236D411yescharacteristic
28C22×C22⋊D59—22no—
29C22×C22⋊D59—22no—
30C22×C4×D59—21no—
31C22×C59⋊C8—21no—
32#14273723—42no—
33#14273723—42no—
34#14273698—42no—
35#14273698—42no—
36#14273698—42no—
37#14273698—42no—
38C4×D236—21no—
39C4×D236—21no—
40C4×D236—21no—
41C4×D236—21no—
42C2×C4⋊Dic59—21no—
43C2×C4⋊Dic59—21no—
44C2×C4⋊Dic59—21no—
45C2×C4⋊Dic59D411yescharacteristic
46C2×C4⋊Dic59D411yescharacteristic
47C2×C4⋊Dic59D411yescharacteristic
48C2×C22⋊Dic59—22no—
49C2×C22⋊Dic59—22no—
50C2×(C2×D59)⋊C4—21no—
51C2×C4×Dic59—21no—
52#14273659—42no—
53#14273649—42no—
54#14273648—44no—
55C4×C59⋊C8—21no—
56C4×C59⋊C8—21no—
57C59⋊#95—21no—
58C22×D4×C59—21no—
59C22×D4×C59—21no—
60C22×D4×C59D412yes—
61C22×C8×C59—21no—
62C22×C8×C59—21no—
63#127×C59—21no—
64#127×C59—21no—
65#127×C59—21no—
66#127×C59—21no—
67C2×C42×C59C2311yescharacteristic
68C4⋊C8×C59—21no—
69C4⋊C8×C59—21no—
70C4⋊C8×C59—21no—
71C4⋊C8×C59D412yescomplemented
72C24×C59—22no—
73C23×D59—21no—
74C2×D4×C59—22no—
75C2×D4×C59—22no—
76C2×D4×C59—42no—
77C2×D4×C59—42no—
78C2×D4×C59—41no—
79C2×D4×C59—41no—
80C2×D4×C59—41no—
81C2×D4×C59—41no—
82C2×D4×C59—41no—
83C2×D4×C59—41no—
84C2×D4×C59—41no—
85C2×D4×C59—41no—
86C2×D4×C59—41no—
87C2×D4×C59—41no—
88C2×D4×C59—41no—
89C2×D4×C59—41no—
90C22×C4×C59C2×D411yescharacteristic
91C22×C4×C59C2×D411yescharacteristic
92C22×C4×C59C2×D411yescharacteristic
93C2×C22⋊D59—44no—
94C2×C22⋊D59—44no—
95C2×C22⋊D59—44no—
96C2×C22⋊D59—44no—
97C2×C22⋊D59—82no—
98C2×C22⋊D59—82no—
99C22×Dic59—21no—
100C22×Dic59—21no—
101C22×Dic59—21no—
102C22×Dic59—41no—
103C2×D236—21no—
104C2×D236—21no—
105C2×D236—21no—
106C2×D236—21no—
107C2×D236—21no—
108C2×D236SD1612yes—
109C2×C4×D59—41no—
110C2×C4×D59—41no—
111C2×C4×D59—41no—
112C2×C4×D59—41no—
113C2×C4×D59—41no—
114C2×C4×D59—41no—
115C2×C8×C59—22no—
116C2×C8×C59—41no—
117C2×C8×C59—41no—
118C2×C8×C59—41no—
119C2×C8×C59—41no—
120C2×C8×C59—41no—
121C42×C59C2×D411yescharacteristic
122C42×C59#4111yescharacteristic
123C42×C59#4111yescharacteristic
124C42×C59#4111yescharacteristic
125C22⋊Dic59—44no—
126C22⋊Dic59—44no—
127(C2×D59)⋊C4—41no—
128(C2×D59)⋊C4—41no—
129C4⋊Dic59—21no—
130C4⋊Dic59—21no—
131C4⋊Dic59—21no—
132C4⋊Dic59—21no—
133C4⋊Dic59—42no—
134C4⋊Dic59—41no—
135C4⋊Dic59—41no—
136C4⋊Dic59—41no—
137C4⋊Dic59—41no—
138C4⋊Dic59D812yes—
139C4⋊Dic59SD1612yes—
140C4×Dic59—41no—
141C4×Dic59—41no—
142C2×C59⋊C8—22no—
143C2×C59⋊C8—41no—
144C2×C59⋊C8—41no—
145C23×C59—42no—
146C23×C59—42no—
147C23×C59—42no—
148C23×C59—42no—
149C23×C59—42no—
150C23×C59—42no—
151C23×C59—42no—
152C23×C59#12011yescharacteristic
153C22×D59—22no—
154C22×D59—41no—
155C22×D59—41no—
156C22×D59—41no—
157C22×D59—41no—
158C22×D59—41no—
159C22×D59—41no—
160D4×C59—44no—
161D4×C59—82no—
162D4×C59—81no—
163D4×C59—81no—
164D4×C59—81no—
165D4×C59—81no—
166D4×C59—81no—
167D4×C59—81no—
168D4×C59—81no—
169D4×C59—81no—
170C2×C4×C59—21no—
171C2×C4×C59—21no—
172C2×C4×C59#12111yescharacteristic
173C2×C4×C59#12111yescharacteristic
174C2×C4×C59#12111yescharacteristic
175C2×C4×C59#12311yescharacteristic
176C2×C4×C59#12311yescharacteristic
177C2×C4×C59#12411yescharacteristic
178C2×C4×C59C2×D811yescharacteristic
179C2×C4×C59C2×SD1611yescharacteristic
180C2×C4×C59C2×SD1611yescharacteristic
181C2×C4×C59#13511yescharacteristic
182C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
183C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
184C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
185C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
186C22⋊D59—88no—
187C22⋊D59—88no—
188C2×Dic59—22no—
189C2×Dic59—22no—
190C2×Dic59—41no—
191C2×Dic59—41no—
192C2×Dic59—41no—
193C2×Dic59—41no—
194C2×Dic59—41no—
195C2×Dic59—41no—
196C2×Dic59—41no—
197C2×Dic59—81no—
198C2×Dic59—81no—
199C2×Dic59—81no—
200D236—22no—
201D236—22no—
202D236—41no—
203D236—41no—
204C4×D59—81no—
205C4×D59—81no—
206C4×D59—81no—
207C4×D59—81no—
208C8×C59—42no—
209C8×C59—81no—
210C59⋊C8—42no—
211#10409—591nomaximal, Sylow
212C22×C59—82no—
213C22×C59—82no—
214C22×C59—82no—
215C22×C59—82no—
216C22×C59—82no—
217C22×C59—82no—
218C22×C59—82no—
219C22×C59#39411yescharacteristic
220C22×C59#44711yescharacteristic
221C22×C59#44911yescharacteristic
222C22×C59#45011yescharacteristic
223C22×C59#45011yescharacteristic
224C22×C59#45311yescharacteristic
225C22×C59#45511yescharacteristic
226C2×D59—42no—
227C2×D59—42no—
228C2×D59—42no—
229C2×D59—81no—
230C2×D59—81no—
231C2×D59—81no—
232C2×D59—81no—
233C4×C59—21no—
234C4×C59—21no—
235C4×C59#45911yescharacteristic
236C4×C59#46011yescharacteristic
237C4×C59#46511yescharacteristic
238C4×C59#46911yescharacteristic
239C4×C59#48111yescharacteristic
240C4×C59#48211yescharacteristic
241C4×C59#48611yescharacteristic
242C4×C59#49011yescharacteristic
243Dic59—42no—
244Dic59—42no—
245Dic59—81no—
246Dic59—161no—
247#2541—591no—
248#1957—591no—
249#1948—591no—
250#1802—591no—
251C2×#331D5911yescharacteristic, complemented
252C2×D4⋊C8—591no—
253#1279—591no—
254C2×C59—162no—
255C2×C59#162111yescharacteristic
256C2×C59#162911yescharacteristic
257C2×C59#163911yescharacteristic
258C2×C59#164211yescharacteristic
259C2×C59#203411yescharacteristic
260C2×C59#203711yescharacteristic
261C2×C59#204111yescharacteristic
262D59—82no—
263C2×#127—1181no—
264C2×#127C2×D5911yescharacteristic
265C2×#121—1181no—
266C2×#121—1181no—
267C2×C4×D4—591no—
268C2×#106—1181no—
269C2×C4⋊C8C2×D5911yescharacteristic
270C2×C4⋊C8C2×D5911yescharacteristic
271C2×#102—1181no—
272C2×#102—1182no—
273C2×C4×C8—591no—
274#392—1182no—
275#390—1181no—
276#384—591no—
277C4×C4⋊C4—591no—
278#340—1181no—
279#340—1181no—
280#331—22no—
281D4⋊C8—1182no—
282C59#1040911yescharacteristic, complemented, Sylow
283C22×D4—21no—
284C22×D4—21no—
285C22×D4—591no—
286C22×D4—1182no—
287C22×D4—1182no—
288C22×D4D23612yes—
289C23×C4—1181no—
290C22×C8—21no—
291C22×C8—21no—
292C22×C8—1181no—
293#127—21no—
294#127—21no—
295#127—21no—
296#127—21no—
297#127—2362no—
298#127—2362no—
299#121—2362no—
300#121—2362no—
301#121—2362no—
302#121—2362no—
303C4×D4—1181no—
304C4×D4—1181no—
305C4×D4—1181no—
306C4×D4—1181no—
307C2×C4⋊C4—591no—
308C2×C4⋊C4—591no—
309C2×C4⋊C4—591no—
310C2×C4⋊C4—1181no—
311C2×C4⋊C4—1181no—
312C2×C4⋊C4—1181no—
313C2×C22⋊C4—1182no—
314C2×C22⋊C4—1182no—
315C2×C22⋊C4—1181no—
316C2×C42—1181no—
317C2×C42C22×D5911yescharacteristic
318#106—2362no—
319C4⋊C8—21no—
320C4⋊C8—21no—
321C4⋊C8—21no—
322C4⋊C8C22⋊D5912yescomplemented
323#102—2362no—
324#102—2364no—
325C4×C8—1181no—
326C4×C8—1181no—
327#95—1181no—
328C24—22no—
329C24—1181no—
330C2×D4—22no—
331C2×D4—22no—
332C2×D4—42no—
333C2×D4—42no—
334C2×D4—41no—
335C2×D4—41no—
336C2×D4—41no—
337C2×D4—41no—
338C2×D4—41no—
339C2×D4—41no—
340C2×D4—41no—
341C2×D4—41no—
342C2×D4—41no—
343C2×D4—41no—
344C2×D4—41no—
345C2×D4—41no—
346C2×D4—592no—
347C2×D4—1181no—
348C2×D4—1181no—
349C2×D4—1181no—
350C2×D4—1181no—
351C2×D4—1181no—
352C2×D4—4722no—
353C2×D4—4722no—
354C2×D4—2364no—
355C2×D4—2364no—
356C2×D4—2364no—
357C2×D4—2364no—
358C22×C4—1181no—
359C22×C4—1181no—
360C22×C4—1181no—
361C22×C4—2361no—
362C22×C4—2361no—
363C22×C4—2361no—
364C22×C4—2361no—
365C22×C4—2361no—
366C22×C4—2361no—
367C22×C4—2361no—
368C22×C4C2×D23611yescharacteristic
369C22×C4D4×D5911yescharacteristic
370C22×C4D4×D5911yescharacteristic
371C2×C8—22no—
372C2×C8—41no—
373C2×C8—41no—
374C2×C8—41no—
375C2×C8—41no—
376C2×C8—41no—
377C2×C8—1182no—
378C2×C8—2361no—
379C2×C8—2361no—
380C4⋊C4—592no—
381C4⋊C4—592no—
382C4⋊C4—1181no—
383C4⋊C4—1181no—
384C4⋊C4—1181no—
385C4⋊C4—1181no—
386C4⋊C4—2362no—
387C4⋊C4—2361no—
388C4⋊C4—2361no—
389C4⋊C4—2361no—
390C4⋊C4—2361no—
391C22⋊C4—2364no—
392C22⋊C4—2364no—
393C22⋊C4—2361no—
394C22⋊C4—2361no—
395C42—2361no—
396C42—2361no—
397C42#1174568511yescharacteristic
398C42#1174568711yescharacteristic
399C42#1174568711yescharacteristic
400C42C2×C22⋊D5911yescharacteristic
401C23—42no—
402C23—42no—
403C23—42no—
404C23—42no—
405C23—42no—
406C23—42no—
407C23—42no—
408C23—1182no—
409C23—2361no—
410C23—2361no—
411C23—2361no—
412C23—2361no—
413C23—2361no—
414C23—2361no—
415C23#1427369711yescharacteristic
416D4—44no—
417D4—82no—
418D4—81no—
419D4—81no—
420D4—81no—
421D4—81no—
422D4—81no—
423D4—81no—
424D4—81no—
425D4—81no—
426D4—1182no—
427D4—1182no—
428D4—4728no—
429D4—4728no—
430D4—2361no—
431D4—2361no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—1182no—
435C2×C4—1182no—
436C2×C4—4721no—
437C2×C4—4721no—
438C2×C4—4721no—
439C2×C4—4721no—
440C2×C4—4721no—
441C2×C4—4721no—
442C2×C4—4721no—
443C2×C4—2361no—
444C2×C4—2361no—
445C2×C4—2361no—
446C2×C4—2361no—
447C2×C4—2361no—
448C2×C4—2361no—
449C2×C4—2361no—
450C2×C4#1427369811yescharacteristic
451C2×C4#1427369911yescharacteristic
452C2×C4#1427370111yescharacteristic
453C2×C4#1427370911yescharacteristic
454C2×C4#1427371111yescharacteristic
455C2×C4#1427371311yescharacteristic
456C2×C4D8×D5911yescharacteristic
457C2×C4SD16×D5911yescharacteristic
458C2×C4SD16×D5911yescharacteristic
459C2×C4#1427374811yescharacteristic
460C2×C4#1427375111yescharacteristic
461C2×C4#1427375111yescharacteristic
462C2×C4#1427375111yescharacteristic
463C2×C4#1427375411yescharacteristic
464C8—42no—
465C8—81no—
466C8—2362no—
467C22—82no—
468C22—82no—
469C22—82no—
470C22—82no—
471C22—82no—
472C22—82no—
473C22—82no—
474C22—4721no—
475C22—4721no—
476C22—4721no—
477C22—4721no—
478C22—2362no—
479C22—2362no—
480C22—2362no—
481C22#1683731811yescharacteristic
482C22#1683753711yescharacteristic
483C22#1683754811yescharacteristic
484C22#1683755411yescharacteristic
485C22#1683755411yescharacteristic
486C22#1683757211yescharacteristic
487C22#1683758211yescharacteristic
488C4—21no—
489C4—21no—
490C4—9441no—
491C4—4721no—
492C4—2362no—
493C4—2362no—
494C4#1683759511yescharacteristic
495C4#1683760011yescharacteristic
496C4#1683762811yescharacteristic
497C4#1683765211yescharacteristic
498C4#1683771911yescharacteristic
499C4#1683772511yescharacteristic
500C4#1683774711yescharacteristic
501C4#1683776511yescharacteristic
502C2—162no—
503C2—4722no—
504C2#2142316211yescharacteristic
505C2#2142321311yescharacteristic
506C2#2142327611yescharacteristic
507C2#2142329711yescharacteristic
508C2#2142528411yescharacteristic
509C2#2142530711yescharacteristic
510C2#2142532911yescharacteristic
subgroupG/H
centerC23#14273697
commutator, cosocleC2×C42×C59C23
FrattiniC2×C42C22×D59
FittingC2×C59×#331C2
socleC23×C59#120
prime radical#21429043C2
hypercenterC2×#331D59
hypocenterC59#10409
LCSG ▷ C2×C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#331

Complements

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

NH ≅ G/Ntype
2. #21424903C2semidirect
3. #21424860C2semidirect
4. #21424097C2semidirect
5. C2×#16837129C2semidirect
6. #21421935C2semidirect
7. C2×C59×#331C2semidirect
71. C4⋊C8×C59D4semidirect
251. C2×#331D59semidirect
282. C59#10409semidirect
322. C4⋊C8C22⋊D59semidirect

Maximal subgroups

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

subgroupindex
#214290432
#214249032
#214248602
#214240972
C2×#168371292
#214219352
C2×C59×#3312
#1040959