Finite Group Databasebeta

← #28492198

Subgroups — #28492198

Order 15104.

614 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21429009C211yesmaximal, characteristic
2#21424223C211yesmaximal, characteristic, complemented
3#21424093C211yesmaximal, characteristic, complemented
4C2×D4⋊(C59⋊C8)C211yesmaximal, characteristic, complemented
5C2×#16837129C211yesmaximal, characteristic, complemented
6#21421980C211yesmaximal, characteristic, complemented
7C59×#1802C211yesmaximal, characteristic, complemented
8C2×#14273701—21no—
9C2×#14273698—21no—
10C2×C4×D236C2211yescharacteristic
11C2×C4⋊(C59⋊C8)C2211yescharacteristic
12C2×#14273647—21no—
13C2×C4×C59⋊C8C2211yescharacteristic
14#16837318—22no—
15#16837272C2211yescharacteristic
16#16837262—21no—
17#16837259—21no—
18#16837164—22no—
19#16837164—21no—
20D4⋊(C59⋊C8)—22no—
21#16837129—22no—
22C2×C4×D4×C59C2211yescharacteristic
23C2×C4⋊C8×C59C2211yescharacteristic
24C2×#102×C59—21no—
25C59×#384C2211yescharacteristic
26C59×#340—21no—
27C22×D236D411yescharacteristic
28C22×C22⋊D59—22no—
29C22×C4×D59—21no—
30C22×C59⋊C8—21no—
31C22×C59⋊C8—21no—
32#14273701—42no—
33#14273701—42no—
34#14273698—42no—
35#14273698—42no—
36C4×D236—21no—
37C4×D236—21no—
38C4×D236—21no—
39C4×D236—21no—
40C2×C4⋊Dic59—21no—
41C2×C4⋊Dic59—21no—
42C2×C4⋊Dic59D411yescharacteristic
43C2×C4⋊Dic59D412yes—
44C2×C22⋊Dic59—22no—
45C2×(C2×D59)⋊C4—22no—
46C2×(C2×D59)⋊C4—21no—
47C4⋊(C59⋊C8)—21no—
48C4⋊(C59⋊C8)—21no—
49#14273647—42no—
50C4×C59⋊C8—21no—
51C4×C59⋊C8D412yescomplemented
52C59⋊#95—22no—
53C22×D4×C59D411yescharacteristic
54C23×C4×C59—21no—
55C22×C8×C59—21no—
56C4×D4×C59—21no—
57C4×D4×C59—21no—
58C4×D4×C59—21no—
59C4×D4×C59—21no—
60C2×C4⋊C4×C59—21no—
61C2×C4⋊C4×C59—21no—
62C2×C4⋊C4×C59—21no—
63C2×C4⋊C4×C59D411yescharacteristic
64C2×C22⋊C4×C59—21no—
65C2×C42×C59C2311yescharacteristic
66C4⋊C8×C59—21no—
67C4⋊C8×C59—21no—
68#102×C59—42no—
69C24×C59—21no—
70C23×D59—21no—
71C2×D4×C59—21no—
72C2×D4×C59—21no—
73C2×D4×C59—21no—
74C2×D4×C59—21no—
75C2×D4×C59—21no—
76C2×D4×C59SD1612yes—
77C22×C4×C59—21no—
78C22×C4×C59—41no—
79C22×C4×C59—41no—
80C22×C4×C59—41no—
81C22×C4×C59—41no—
82C22×C4×C59—41no—
83C22×C4×C59—41no—
84C22×C4×C59—41no—
85C22×C4×C59C2×D411yescharacteristic
86C22×C4×C59C2×D411yescharacteristic
87C22×C4×C59C2×D411yescharacteristic
88C2×C22⋊D59—44no—
89C2×C22⋊D59—44no—
90C2×C22⋊D59—82no—
91C22×Dic59—21no—
92C22×Dic59—22no—
93C2×D236—21no—
94C2×D236—21no—
95C2×D236—21no—
96C2×D236—21no—
97C2×D236—21no—
98C2×D236D812yes—
99C2×C4×D59—41no—
100C2×C4×D59—41no—
101C2×C4×D59—41no—
102C2×C4×D59—41no—
103C2×C4×D59—41no—
104C2×C4×D59—41no—
105C2×C8×C59—41no—
106C2×C8×C59—41no—
107C2×C8×C59—41no—
108C4⋊C4×C59—21no—
109C4⋊C4×C59—21no—
110C4⋊C4×C59—42no—
111C4⋊C4×C59—41no—
112C4⋊C4×C59—41no—
113C4⋊C4×C59—41no—
114C4⋊C4×C59—41no—
115C22⋊C4×C59—41no—
116C22⋊C4×C59—41no—
117C42×C59C2×D411yescharacteristic
118C42×C59#4111yescharacteristic
119C42×C59#4111yescharacteristic
120C42×C59#4111yescharacteristic
121C22⋊Dic59—44no—
122(C2×D59)⋊C4—44no—
123(C2×D59)⋊C4—41no—
124(C2×D59)⋊C4—41no—
125C4⋊Dic59—21no—
126C4⋊Dic59—21no—
127C4⋊Dic59—41no—
128C4⋊Dic59—41no—
129C4⋊Dic59—41no—
130C4⋊Dic59—41no—
131C4⋊Dic59—42no—
132C2×C59⋊C8—22no—
133C2×C59⋊C8—41no—
134C2×C59⋊C8—41no—
135C2×C59⋊C8—41no—
136C2×C59⋊C8—41no—
137C2×C59⋊C8—41no—
138C23×C59—22no—
139C23×C59—41no—
140C23×C59—41no—
141C23×C59—41no—
142C23×C59—41no—
143C23×C59—41no—
144C23×C59—41no—
145C23×C59#12011yescharacteristic
146C22×D59—22no—
147C22×D59—41no—
148C22×D59—41no—
149C22×D59—41no—
150C22×D59—41no—
151C22×D59—41no—
152C22×D59—41no—
153D4×C59—22no—
154D4×C59—22no—
155D4×C59—41no—
156D4×C59—41no—
157C2×C4×C59—21no—
158C2×C4×C59—21no—
159C2×C4×C59—41no—
160C2×C4×C59—41no—
161C2×C4×C59—41no—
162C2×C4×C59—81no—
163C2×C4×C59—81no—
164C2×C4×C59—81no—
165C2×C4×C59—81no—
166C2×C4×C59—81no—
167C2×C4×C59—81no—
168C2×C4×C59—81no—
169C2×C4×C59#12111yescharacteristic
170C2×C4×C59#12111yescharacteristic
171C2×C4×C59#12111yescharacteristic
172C2×C4×C59#12311yescharacteristic
173C2×C4×C59#12311yescharacteristic
174C2×C4×C59#12411yescharacteristic
175C2×C4×C59C2×D811yescharacteristic
176C2×C4×C59C2×SD1611yescharacteristic
177C2×C4×C59#13511yescharacteristic
178C2×C4×C59#13511yescharacteristic
179C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
180C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
181C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
182C2×C4×C59#13711yescharacteristic
183C22⋊D59—88no—
184C2×Dic59—41no—
185C2×Dic59—42no—
186C2×Dic59—41no—
187C2×Dic59—42no—
188C2×Dic59—41no—
189C2×Dic59—42no—
190D236—22no—
191D236—22no—
192D236—41no—
193D236—41no—
194C4×D59—81no—
195C4×D59—81no—
196C4×D59—81no—
197C4×D59—81no—
198C8×C59—81no—
199C59⋊C8—42no—
200C59⋊C8—81no—
201#10431—591nomaximal, Sylow
202C22×C59—42no—
203C22×C59—42no—
204C22×C59—42no—
205C22×C59—81no—
206C22×C59—81no—
207C22×C59—81no—
208C22×C59—81no—
209C22×C59#39411yescharacteristic
210C22×C59#44711yescharacteristic
211C22×C59#44911yescharacteristic
212C22×C59#45011yescharacteristic
213C22×C59#45211yescharacteristic
214C22×C59#45411yescharacteristic
215C22×C59#45511yescharacteristic
216C2×D59—42no—
217C2×D59—42no—
218C2×D59—42no—
219C2×D59—81no—
220C2×D59—81no—
221C2×D59—81no—
222C2×D59—81no—
223C4×C59—21no—
224C4×C59—21no—
225C4×C59—81no—
226C4×C59—161no—
227C4×C59#46511yescharacteristic
228C4×C59#46611yescharacteristic
229C4×C59#46811yescharacteristic
230C4×C59#46911yescharacteristic
231C4×C59#48011yescharacteristic
232C4×C59#48111yescharacteristic
233C4×C59#48211yescharacteristic
234C4×C59#48211yescharacteristic
235Dic59—82no—
236Dic59—81no—
237#2537—591no—
238#1834—591no—
239#1802—591no—
240#1802D5911yescharacteristic, complemented
241C2×D4⋊C8—591no—
242C2×D4⋊C8—591no—
243#1296—591no—
244C2×C59—82no—
245C2×C59#162511yescharacteristic
246C2×C59#163511yescharacteristic
247C2×C59#163911yescharacteristic
248C2×C59#164611yescharacteristic
249C2×C59#203311yescharacteristic
250C2×C59#203411yescharacteristic
251C2×C59#204011yescharacteristic
252D59—82no—
253C2×#121—1181no—
254C2×#121—1181no—
255C2×C4×D4—591no—
256C2×C4×D4C2×D5911yescharacteristic
257C2×C4⋊C8—591no—
258C2×C4⋊C8C2×D5911yescharacteristic
259C2×#102—21no—
260C2×#102—1181no—
261C2×C4×C8—591no—
262#394—1182no—
263#384—591no—
264#384C2×D5911yescharacteristic
265#380—1181no—
266#380—1181no—
267#340—21no—
268#340—1181no—
269#340—1182no—
270D4⋊C8—1182no—
271D4⋊C8—1182no—
272C59#1043111yescharacteristic, complemented, Sylow
273C22×D4—591no—
274C22×D4—1182no—
275C22×D4C22⋊D5911yescharacteristic
276C23×C4—21no—
277C23×C4—1181no—
278C22×C8—21no—
279C22×C8—1181no—
280C22×C8—1181no—
281#121—2362no—
282#121—2362no—
283#121—2362no—
284#121—2362no—
285C4×D4—21no—
286C4×D4—21no—
287C4×D4—21no—
288C4×D4—21no—
289C4×D4—1181no—
290C4×D4—1181no—
291C4×D4—1181no—
292C4×D4—1181no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—591no—
297C2×C4⋊C4—592no—
298C2×C4⋊C4—1181no—
299C2×C4⋊C4—1181no—
300C2×C4⋊C4C22⋊D5911yescharacteristic
301C2×C22⋊C4—21no—
302C2×C22⋊C4—1182no—
303C2×C22⋊C4—1182no—
304C2×C22⋊C4—1181no—
305C2×C42C22×D5911yescharacteristic
306C4⋊C8—21no—
307C4⋊C8—21no—
308C4⋊C8—1181no—
309C4⋊C8—1181no—
310#102—42no—
311#102—2362no—
312C4×C8—592no—
313C4×C8—1181no—
314#95—1182no—
315C24—21no—
316C24—1181no—
317C2×D4—21no—
318C2×D4—21no—
319C2×D4—21no—
320C2×D4—21no—
321C2×D4—21no—
322C2×D4—592no—
323C2×D4—1181no—
324C2×D4—1181no—
325C2×D4—1181no—
326C2×D4—1181no—
327C2×D4—1181no—
328C2×D4—4722no—
329C2×D4—2364no—
330C2×D4—2364no—
331C2×D4#1174567112yes—
332C22×C4—21no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—1181no—
341C22×C4—1182no—
342C22×C4—2361no—
343C22×C4—2361no—
344C22×C4—2361no—
345C22×C4—2361no—
346C22×C4—2361no—
347C22×C4—2361no—
348C22×C4D4×D5911yescharacteristic
349C22×C4D4×D5911yescharacteristic
350C22×C4C2×C22⋊D5911yescharacteristic
351C2×C8—41no—
352C2×C8—41no—
353C2×C8—41no—
354C2×C8—1182no—
355C2×C8—2361no—
356C2×C8—2361no—
357C2×C8—2361no—
358C2×C8—2361no—
359C2×C8—2361no—
360C4⋊C4—21no—
361C4⋊C4—21no—
362C4⋊C4—42no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—1181no—
368C4⋊C4—1181no—
369C4⋊C4—2362no—
370C4⋊C4—2361no—
371C4⋊C4—2361no—
372C4⋊C4—2361no—
373C4⋊C4—2361no—
374C22⋊C4—41no—
375C22⋊C4—41no—
376C22⋊C4—2364no—
377C22⋊C4—2364no—
378C22⋊C4—2361no—
379C22⋊C4—2361no—
380C42#1174568511yescharacteristic
381C42#1174568511yescharacteristic
382C42D4×D5911yescharacteristic
383C42#1174568711yescharacteristic
384C23—22no—
385C23—41no—
386C23—41no—
387C23—41no—
388C23—41no—
389C23—41no—
390C23—41no—
391C23—1182no—
392C23—2361no—
393C23—2361no—
394C23—2361no—
395C23—2361no—
396C23—2361no—
397C23—2361no—
398C23#1427369511yescharacteristic
399D4—22no—
400D4—22no—
401D4—41no—
402D4—41no—
403D4—1182no—
404D4—1182no—
405D4—4728no—
406D4—2361no—
407D4—2361no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—81no—
414C2×C4—81no—
415C2×C4—81no—
416C2×C4—81no—
417C2×C4—81no—
418C2×C4—81no—
419C2×C4—81no—
420C2×C4—4721no—
421C2×C4—4721no—
422C2×C4—4721no—
423C2×C4—4721no—
424C2×C4—2362no—
425C2×C4—2361no—
426C2×C4—2361no—
427C2×C4—2362no—
428C2×C4—2361no—
429C2×C4—2362no—
430C2×C4#1427369911yescharacteristic
431C2×C4#1427370011yescharacteristic
432C2×C4#1427370011yescharacteristic
433C2×C4#1427370911yescharacteristic
434C2×C4#1427371011yescharacteristic
435C2×C4#1427371311yescharacteristic
436C2×C4D8×D5911yescharacteristic
437C2×C4C2×#1174567111yescharacteristic
438C2×C4#1427374711yescharacteristic
439C2×C4#1427374811yescharacteristic
440C2×C4#1427375111yescharacteristic
441C2×C4#1427375211yescharacteristic
442C2×C4#1427375311yescharacteristic
443C2×C4#1427375611yescharacteristic
444C8—81no—
445C8—4721no—
446C8—2362no—
447C22—42no—
448C22—42no—
449C22—42no—
450C22—81no—
451C22—81no—
452C22—81no—
453C22—81no—
454C22—4721no—
455C22—4721no—
456C22—4721no—
457C22—4721no—
458C22—2362no—
459C22—2362no—
460C22—2362no—
461C22#1683731711yescharacteristic
462C22#1683753411yescharacteristic
463C22#1683754511yescharacteristic
464C22#1683755211yescharacteristic
465C22#1683756111yescharacteristic
466C22#1683757311yescharacteristic
467C22#1683757811yescharacteristic
468C4—21no—
469C4—21no—
470C4—81no—
471C4—161no—
472C4—4721no—
473C4—4722no—
474C4#1683763311yescharacteristic
475C4#1683763711yescharacteristic
476C4#1683764711yescharacteristic
477C4#1683765311yescharacteristic
478C4#1683771411yescharacteristic
479C4#1683772011yescharacteristic
480C4#1683772511yescharacteristic
481C4#1683772611yescharacteristic
482C2—82no—
483C2—4722no—
484C2#2142318411yescharacteristic
485C2#2142324511yescharacteristic
486C2#2142327411yescharacteristic
487C2#2142332211yescharacteristic
488C2#2142527611yescharacteristic
489C2#2142528511yescharacteristic
490C2#2142532311yescharacteristic
subgroupG/H
centerC23#14273695
commutator, cosocleC2×C42×C59C23
FrattiniC2×C42C22×D59
FittingC59×#1802C2
socleC23×C59#120
prime radical#21429009C2
hypercenter#1802D59
hypocenterC59#10431
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. #21424223C2semidirect
3. #21424093C2semidirect
4. C2×D4⋊(C59⋊C8)C2semidirect
5. C2×#16837129C2semidirect
6. #21421980C2semidirect
7. C59×#1802C2semidirect
51. C4×C59⋊C8D4semidirect
240. #1802D59semidirect
272. C59#10431semidirect

Maximal subgroups

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

subgroupindex
#214290092
#214242232
#214240932
C2×D4⋊(C59⋊C8)2
C2×#168371292
#214219802
C59×#18022
#1043159