Finite Group Databasebeta

← #28492204

Subgroups — #28492204

Order 15104.

630 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21429009C211yesmaximal, characteristic
2#21424253C211yesmaximal, characteristic, complemented
3#21424097C211yesmaximal, characteristic, complemented
4#21424096C211yesmaximal, characteristic, complemented
5C2×#16837131C211yesmaximal, characteristic, complemented
6#21421980C211yesmaximal, characteristic, complemented
7C2×C59×D4⋊C8C211yesmaximal, characteristic, complemented
8C2×#14273701—21no—
9C2×#14273698—21no—
10C2×C4×D236C2211yescharacteristic
11C2×#14273659—22no—
12C2×C4⋊(C59⋊C8)C2211yescharacteristic
13C2×#14273649—21no—
14C2×#14273648—21no—
15#16837318—22no—
16#16837272C2211yescharacteristic
17#16837272C2211yescharacteristic
18#16837262—21no—
19#16837259—21no—
20#16837165—21no—
21#16837164—21no—
22#16837131—22no—
23C2×C4×D4×C59C2211yescharacteristic
24C2×C4⋊C8×C59C2211yescharacteristic
25C2×C4×C8×C59C2211yescharacteristic
26C59×D4⋊C8—22no—
27C22×D236D411yescharacteristic
28C22×C22⋊D59—22no—
29C22×C4×D59—21no—
30C22×C59⋊C8—21no—
31#14273701—42no—
32#14273701—42no—
33#14273698—42no—
34#14273698—42no—
35C4×D236—21no—
36C4×D236—21no—
37C4×D236—21no—
38C4×D236—21no—
39C2×C4⋊Dic59—21no—
40C2×C4⋊Dic59—21no—
41C2×C4⋊Dic59—21no—
42C2×C4⋊Dic59—21no—
43C2×C4⋊Dic59—21no—
44C2×C4⋊Dic59D411yescharacteristic
45C2×C4⋊Dic59D412yes—
46C2×C22⋊Dic59—22no—
47C2×(C2×D59)⋊C4—22no—
48C2×(C2×D59)⋊C4—21no—
49#14273659—44no—
50C4⋊(C59⋊C8)—21no—
51C4⋊(C59⋊C8)—21no—
52#14273649—42no—
53#14273648—42no—
54C59⋊#95—22no—
55C22×D4×C59D411yescharacteristic
56C23×C4×C59—21no—
57C22×C8×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×C59D411yescharacteristic
64C2×C22⋊C4×C59—21no—
65C2×C42×C59C2311yescharacteristic
66C4⋊C8×C59—21no—
67C4⋊C8×C59—21no—
68C4×C8×C59—21no—
69C4×C8×C59D412yescomplemented
70C24×C59—21no—
71C23×D59—21no—
72C2×D4×C59—21no—
73C2×D4×C59—21no—
74C2×D4×C59—21no—
75C2×D4×C59—21no—
76C2×D4×C59—21no—
77C2×D4×C59SD1612yes—
78C22×C4×C59—21no—
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—
93C22×Dic59—41no—
94C2×D236—21no—
95C2×D236—21no—
96C2×D236—21no—
97C2×D236—21no—
98C2×D236—21no—
99C2×D236SD1612yes—
100C2×C4×D59—41no—
101C2×C4×D59—41no—
102C2×C4×D59—41no—
103C2×C4×D59—41no—
104C2×C4×D59—41no—
105C2×C4×D59—41no—
106C2×C8×C59—22no—
107C2×C8×C59—22no—
108C2×C8×C59—22no—
109C2×C8×C59—41no—
110C2×C8×C59—41no—
111C2×C8×C59—41no—
112C4⋊C4×C59—21no—
113C4⋊C4×C59—21no—
114C22⋊C4×C59—41no—
115C22⋊C4×C59—41no—
116C42×C59C2×D411yescharacteristic
117C42×C59#4111yescharacteristic
118C42×C59#4111yescharacteristic
119C42×C59#4111yescharacteristic
120C22⋊Dic59—44no—
121(C2×D59)⋊C4—44no—
122(C2×D59)⋊C4—41no—
123(C2×D59)⋊C4—41no—
124C4⋊Dic59—21no—
125C4⋊Dic59—21no—
126C4⋊Dic59—41no—
127C4⋊Dic59—41no—
128C4⋊Dic59—41no—
129C4⋊Dic59—41no—
130C4⋊Dic59—42no—
131C4⋊Dic59—42no—
132C4⋊Dic59—41no—
133C4⋊Dic59—41no—
134C4⋊Dic59—41no—
135C4⋊Dic59—41no—
136C2×C59⋊C8—41no—
137C2×C59⋊C8—41no—
138C2×C59⋊C8—41no—
139C23×C59—22no—
140C23×C59—41no—
141C23×C59—41no—
142C23×C59—41no—
143C23×C59—41no—
144C23×C59—41no—
145C23×C59—41no—
146C23×C59#12011yescharacteristic
147C22×D59—22no—
148C22×D59—41no—
149C22×D59—41no—
150C22×D59—41no—
151C22×D59—41no—
152C22×D59—41no—
153C22×D59—41no—
154D4×C59—22no—
155D4×C59—22no—
156D4×C59—41no—
157D4×C59—41no—
158C2×C4×C59—21no—
159C2×C4×C59—21no—
160C2×C4×C59—41no—
161C2×C4×C59—41no—
162C2×C4×C59—41no—
163C2×C4×C59—81no—
164C2×C4×C59—81no—
165C2×C4×C59—81no—
166C2×C4×C59—81no—
167C2×C4×C59#12111yescharacteristic
168C2×C4×C59#12111yescharacteristic
169C2×C4×C59#12111yescharacteristic
170C2×C4×C59#12311yescharacteristic
171C2×C4×C59#12311yescharacteristic
172C2×C4×C59#12411yescharacteristic
173C2×C4×C59C2×SD1611yescharacteristic
174C2×C4×C59C2×SD1611yescharacteristic
175C2×C4×C59#13511yescharacteristic
176C2×C4×C59#13511yescharacteristic
177C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
178C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
179C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
180C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
181C22⋊D59—88no—
182C2×Dic59—41no—
183C2×Dic59—41no—
184C2×Dic59—42no—
185C2×Dic59—42no—
186C2×Dic59—41no—
187C2×Dic59—42no—
188C2×Dic59—81no—
189C2×Dic59—81no—
190C2×Dic59—81no—
191D236—22no—
192D236—22no—
193D236—41no—
194D236—41no—
195C4×D59—81no—
196C4×D59—81no—
197C4×D59—81no—
198C4×D59—81no—
199C8×C59—24no—
200C8×C59—81no—
201C59⋊C8—81no—
202#10432—591nomaximal, Sylow
203C22×C59—42no—
204C22×C59—42no—
205C22×C59—42no—
206C22×C59—81no—
207C22×C59—81no—
208C22×C59—81no—
209C22×C59—81no—
210C22×C59#39411yescharacteristic
211C22×C59#44911yescharacteristic
212C22×C59#44911yescharacteristic
213C22×C59#45011yescharacteristic
214C22×C59#45011yescharacteristic
215C22×C59#45311yescharacteristic
216C22×C59#45611yescharacteristic
217C2×D59—42no—
218C2×D59—42no—
219C2×D59—42no—
220C2×D59—81no—
221C2×D59—81no—
222C2×D59—81no—
223C2×D59—81no—
224C4×C59—21no—
225C4×C59—21no—
226C4×C59—81no—
227C4×C59#46511yescharacteristic
228C4×C59#46511yescharacteristic
229C4×C59#46911yescharacteristic
230C4×C59#46911yescharacteristic
231C4×C59#48111yescharacteristic
232C4×C59#48111yescharacteristic
233C4×C59#48211yescharacteristic
234C4×C59#48211yescharacteristic
235Dic59—82no—
236Dic59—81no—
237Dic59—161no—
238#2537—591no—
239#1844—591no—
240#1802—591no—
241#1802—591no—
242C2×D4⋊C8—591no—
243C2×D4⋊C8D5911yescharacteristic, complemented
244#1296—591no—
245C2×C59—82no—
246C2×C59#162411yescharacteristic
247C2×C59#162911yescharacteristic
248C2×C59#165411yescharacteristic
249C2×C59#165411yescharacteristic
250C2×C59#203411yescharacteristic
251C2×C59#203411yescharacteristic
252C2×C59#203911yescharacteristic
253D59—82no—
254C2×#121—1181no—
255C2×#121—1181no—
256C2×C4×D4—591no—
257C2×C4×D4C2×D5911yescharacteristic
258C2×#106—1182no—
259C2×C4⋊C8—591no—
260C2×C4⋊C8C2×D5911yescharacteristic
261C2×#102—1181no—
262C2×#102—1181no—
263C2×C4×C8C2×D5911yescharacteristic
264#394—1182no—
265#384—591no—
266#384—591no—
267#380—1181no—
268#380—1181no—
269#340—1181no—
270#340—1181no—
271D4⋊C8—22no—
272D4⋊C8—1182no—
273C59#1043211yescharacteristic, complemented, Sylow
274C22×D4—591no—
275C22×D4—1182no—
276C22×D4D23611yescharacteristic
277C23×C4—21no—
278C23×C4—1181no—
279C22×C8—21no—
280C22×C8—21no—
281C22×C8—1181no—
282#121—2362no—
283#121—2362no—
284#121—2362no—
285#121—2362no—
286C4×D4—21no—
287C4×D4—21no—
288C4×D4—21no—
289C4×D4—21no—
290C4×D4—1181no—
291C4×D4—1181no—
292C4×D4—1181no—
293C4×D4—1181no—
294C2×C4⋊C4—591no—
295C2×C4⋊C4—592no—
296C2×C4⋊C4—1181no—
297C2×C4⋊C4—1181no—
298C2×C4⋊C4—1181no—
299C2×C4⋊C4—1181no—
300C2×C4⋊C4—1181no—
301C2×C4⋊C4D23611yescharacteristic
302C2×C22⋊C4—21no—
303C2×C22⋊C4—1182no—
304C2×C22⋊C4—1182no—
305C2×C22⋊C4—1181no—
306C2×C42C22×D5911yescharacteristic
307#106—2364no—
308C4⋊C8—21no—
309C4⋊C8—21no—
310C4⋊C8—1181no—
311C4⋊C8—1181no—
312#102—2362no—
313#102—2362no—
314C4×C8—21no—
315C4×C8C22⋊D5912yescomplemented
316#95—1182no—
317C24—21no—
318C24—1181no—
319C2×D4—21no—
320C2×D4—21no—
321C2×D4—21no—
322C2×D4—21no—
323C2×D4—21no—
324C2×D4—592no—
325C2×D4—1181no—
326C2×D4—1181no—
327C2×D4—1181no—
328C2×D4—1181no—
329C2×D4—1181no—
330C2×D4—4722no—
331C2×D4—2364no—
332C2×D4—2364no—
333C2×D4#1174566012yes—
334C22×C4—21no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—1181no—
342C22×C4—1182no—
343C22×C4—2361no—
344C22×C4—2361no—
345C22×C4—2361no—
346C22×C4—2361no—
347C22×C4—2361no—
348C22×C4—2361no—
349C22×C4—2361no—
350C22×C4C2×D23611yescharacteristic
351C22×C4D4×D5911yescharacteristic
352C22×C4D4×D5911yescharacteristic
353C2×C8—22no—
354C2×C8—22no—
355C2×C8—22no—
356C2×C8—41no—
357C2×C8—41no—
358C2×C8—41no—
359C2×C8—2361no—
360C2×C8—2361no—
361C2×C8—2361no—
362C4⋊C4—21no—
363C4⋊C4—21no—
364C4⋊C4—1181no—
365C4⋊C4—1181no—
366C4⋊C4—2362no—
367C4⋊C4—2362no—
368C4⋊C4—2361no—
369C4⋊C4—2361no—
370C4⋊C4—2361no—
371C4⋊C4—2361no—
372C4⋊C4—2361no—
373C4⋊C4—2361no—
374C4⋊C4—2361no—
375C4⋊C4—2361no—
376C22⋊C4—41no—
377C22⋊C4—41no—
378C22⋊C4—2364no—
379C22⋊C4—2364no—
380C22⋊C4—2361no—
381C22⋊C4—2361no—
382C42#1174568511yescharacteristic
383C42#1174568711yescharacteristic
384C42#1174568711yescharacteristic
385C42C2×C22⋊D5911yescharacteristic
386C23—22no—
387C23—41no—
388C23—41no—
389C23—41no—
390C23—41no—
391C23—41no—
392C23—41no—
393C23—1182no—
394C23—2361no—
395C23—2361no—
396C23—2361no—
397C23—2361no—
398C23—2361no—
399C23—2361no—
400C23#1427369711yescharacteristic
401D4—22no—
402D4—22no—
403D4—41no—
404D4—41no—
405D4—1182no—
406D4—1182no—
407D4—4728no—
408D4—2361no—
409D4—2361no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—81no—
416C2×C4—81no—
417C2×C4—81no—
418C2×C4—81no—
419C2×C4—4721no—
420C2×C4—4721no—
421C2×C4—4721no—
422C2×C4—4721no—
423C2×C4—4721no—
424C2×C4—4721no—
425C2×C4—4721no—
426C2×C4—2361no—
427C2×C4—2362no—
428C2×C4—2361no—
429C2×C4—2362no—
430C2×C4—2362no—
431C2×C4—2361no—
432C2×C4#1427369811yescharacteristic
433C2×C4#1427369911yescharacteristic
434C2×C4#1427370111yescharacteristic
435C2×C4#1427370911yescharacteristic
436C2×C4#1427371111yescharacteristic
437C2×C4#1427371311yescharacteristic
438C2×C4SD16×D5911yescharacteristic
439C2×C4C2×#1174566011yescharacteristic
440C2×C4#1427374811yescharacteristic
441C2×C4#1427374911yescharacteristic
442C2×C4#1427375111yescharacteristic
443C2×C4#1427375111yescharacteristic
444C2×C4#1427375511yescharacteristic
445C2×C4#1427375511yescharacteristic
446C8—24no—
447C8—81no—
448C8—4721no—
449C22—42no—
450C22—42no—
451C22—42no—
452C22—81no—
453C22—81no—
454C22—81no—
455C22—81no—
456C22—4721no—
457C22—4721no—
458C22—4721no—
459C22—4721no—
460C22—2362no—
461C22—2362no—
462C22—2362no—
463C22#1683731811yescharacteristic
464C22#1683754811yescharacteristic
465C22#1683754911yescharacteristic
466C22#1683755411yescharacteristic
467C22#1683755511yescharacteristic
468C22#1683757111yescharacteristic
469C22#1683758611yescharacteristic
470C4—21no—
471C4—21no—
472C4—81no—
473C4—9441no—
474C4—4722no—
475C4—4721no—
476C4#1683762811yescharacteristic
477C4#1683763111yescharacteristic
478C4#1683765211yescharacteristic
479C4#1683765511yescharacteristic
480C4#1683771911yescharacteristic
481C4#1683772111yescharacteristic
482C4#1683772511yescharacteristic
483C4#1683772711yescharacteristic
484C2—82no—
485C2—4722no—
486C2#2142318111yescharacteristic
487C2#2142321411yescharacteristic
488C2#2142338211yescharacteristic
489C2#2142338311yescharacteristic
490C2#2142528411yescharacteristic
491C2#2142528611yescharacteristic
492C2#2142531911yescharacteristic
subgroupG/H
centerC23#14273697
commutator, cosocleC2×C42×C59C23
FrattiniC2×C42C22×D59
FittingC2×C59×D4⋊C8C2
socleC23×C59#120
prime radical#21429009C2
hypercenterC2×D4⋊C8D59
hypocenterC59#10432
LCSG ▷ C2×C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×D4⋊C8

Complements

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

NH ≅ G/Ntype
2. #21424253C2semidirect
3. #21424097C2semidirect
4. #21424096C2semidirect
5. C2×#16837131C2semidirect
6. #21421980C2semidirect
7. C2×C59×D4⋊C8C2semidirect
69. C4×C8×C59D4semidirect
243. C2×D4⋊C8D59semidirect
273. C59#10432semidirect
315. C4×C8C22⋊D59semidirect

Maximal subgroups

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

subgroupindex
#214290092
#214242532
#214240972
#214240962
C2×#168371312
#214219802
C2×C59×D4⋊C82
#1043259