Finite Group Databasebeta

← #28488969

Subgroups — #28488969

Order 15104.

734 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837732C211yesmaximal, characteristic, complemented
2#21429901C211yesmaximal, characteristic
3#21424182C211yesmaximal, characteristic, complemented
4C2×#16837138C211yesmaximal, characteristic, complemented
5C2×#16837126C211yesmaximal, characteristic
6#21421975C211yesmaximal, characteristic, complemented
7C2×C59×#397C211yesmaximal, characteristic
8C2×#14273706C2211yescharacteristic
9#16837732—22no—
10#16837732—22no—
11C2×#14273658—21no—
12C2×#14273650—21no—
13C2×C22⋊(C59⋊C8)C2211yescharacteristic
14C2×C22⋊(C59⋊C8)C2211yescharacteristic
15#16837338—21no—
16#16837325—22no—
17#16837314—21no—
18#16837275—21no—
19#16837262C2211yescharacteristic
20#16837258—21no—
21#16837164—22no—
22#16837138—22no—
23#16837126—22no—
24C22×C4⋊C4×C59C2211yescharacteristic
25C22×C22⋊C4×C59—21no—
26C22×C22⋊C4×C59C2211yescharacteristic
27C59×#397—22no—
28C59×#397—22no—
29C2×#95×C59—21no—
30C2×#95×C59C2211yescharacteristic
31C22×Dic118D411yescharacteristic
32C22×C59⋊C8—21no—
33C22×C59⋊C8—21no—
34#14273706—21no—
35#14273706—21no—
36#14273706—21no—
37#14273706—21no—
38C2×C4⋊Dic59D411yescharacteristic
39C2×C4⋊Dic59D412yes—
40C2×Dic59⋊C4—21no—
41C2×Dic59⋊C4—22no—
42C2×C22⋊Dic59—21no—
43C2×C22⋊Dic59—21no—
44C2×C22⋊Dic59—22no—
45C2×C22⋊Dic59—22no—
46#14273658—24no—
47#14273650—24no—
48C22⋊(C59⋊C8)—21no—
49C22⋊(C59⋊C8)—21no—
50C22⋊(C59⋊C8)—21no—
51C22⋊(C59⋊C8)D412yes—
52C59⋊#95—22no—
53C59⋊#95—21no—
54C25×C59D411yescharacteristic
55C23×C4×C59—21no—
56C23×C4×C59—21no—
57C23×C4×C59D411yescharacteristic
58C23×C4×C59C2311yescharacteristic
59C2×C4⋊C4×C59—21no—
60C2×C4⋊C4×C59—21no—
61C2×C4⋊C4×C59—21no—
62C2×C4⋊C4×C59—21no—
63C2×C4⋊C4×C59—21no—
64C2×C4⋊C4×C59D412yes—
65C2×C22⋊C4×C59—24no—
66C2×C22⋊C4×C59—21no—
67C2×C22⋊C4×C59—21no—
68C2×C22⋊C4×C59—21no—
69C2×C22⋊C4×C59—21no—
70C2×C22⋊C4×C59—21no—
71C2×C22⋊C4×C59—21no—
72C2×C22⋊C4×C59—42no—
73C2×C22⋊C4×C59—42no—
74#95×C59—24no—
75#95×C59—21no—
76#95×C59D412yes—
77C24×C59—21no—
78C24×C59—21no—
79C24×C59—21no—
80C24×C59—21no—
81C24×C59—21no—
82C24×C59—21no—
83C24×C59—42no—
84C24×C59—42no—
85C24×C59SD1612yes—
86C24×C59C2×D411yescharacteristic
87C22×C4×C59—21no—
88C22×C4×C59—22no—
89C22×C4×C59—22no—
90C22×C4×C59—21no—
91C22×C4×C59—21no—
92C22×C4×C59—42no—
93C22×C4×C59—42no—
94C22×C4×C59—42no—
95C22×C4×C59—42no—
96C22×C4×C59—41no—
97C22×C4×C59—41no—
98C22×C4×C59—41no—
99C22×C4×C59—41no—
100C22×C4×C59—41no—
101C22×C4×C59—41no—
102C22×C4×C59—41no—
103C22×C4×C59—41no—
104C22×C4×C59C2×D411yescharacteristic
105C22×C4×C59C2×D411yescharacteristic
106C22×C4×C59C2×D411yescharacteristic
107C22×C4×C59C2×D411yescharacteristic
108C22×C4×C59C2×D411yescharacteristic
109C22×C4×C59#4111yescharacteristic
110C22×Dic59—22no—
111C22×Dic59—21no—
112C22×Dic59—21no—
113C2×Dic118—21no—
114C2×Dic118—21no—
115C2×Dic118—42no—
116C4⋊C4×C59—22no—
117C4⋊C4×C59—22no—
118C4⋊C4×C59—41no—
119C4⋊C4×C59—41no—
120C22⋊C4×C59—48no—
121C22⋊C4×C59—41no—
122C22⋊C4×C59—41no—
123C22⋊C4×C59—41no—
124C22⋊C4×C59—41no—
125C22⋊Dic59—41no—
126C22⋊Dic59—41no—
127C22⋊Dic59—41no—
128C22⋊Dic59—41no—
129C22⋊Dic59—82no—
130C22⋊Dic59—82no—
131C4⋊Dic59—21no—
132C4⋊Dic59—21no—
133C4⋊Dic59—42no—
134Dic59⋊C4—41no—
135Dic59⋊C4—41no—
136Dic59⋊C4—82no—
137C2×C59⋊C8—22no—
138C2×C59⋊C8—42no—
139C2×C59⋊C8—42no—
140C2×C59⋊C8—41no—
141C23×C59—22no—
142C23×C59—22no—
143C23×C59—22no—
144C23×C59—22no—
145C23×C59—22no—
146C23×C59—44no—
147C23×C59—44no—
148C23×C59—42no—
149C23×C59—42no—
150C23×C59—41no—
151C23×C59—41no—
152C23×C59—41no—
153C23×C59—42no—
154C23×C59—42no—
155C23×C59—41no—
156C23×C59—82no—
157C23×C59—82no—
158C23×C59—81no—
159C23×C59—81no—
160C23×C59#12011yescharacteristic
161C23×C59#12011yescharacteristic
162C23×C59#12111yescharacteristic
163C23×C59C2×SD1611yescharacteristic
164C23×C59#13511yescharacteristic
165C23×C59AGL(1,ℤ/8)11yescharacteristic
166C23×C59#13711yescharacteristic
167C2×C4×C59—22no—
168C2×C4×C59—22no—
169C2×C4×C59—41no—
170C2×C4×C59—41no—
171C2×C4×C59—41no—
172C2×C4×C59—41no—
173C2×C4×C59—42no—
174C2×C4×C59—42no—
175C2×C4×C59—42no—
176C2×C4×C59—41no—
177C2×C4×C59—42no—
178C2×C4×C59—42no—
179C2×C4×C59—42no—
180C2×C4×C59—42no—
181C2×C4×C59—42no—
182C2×C4×C59—82no—
183C2×C4×C59—82no—
184C2×C4×C59—81no—
185C2×C4×C59—81no—
186C2×C4×C59—81no—
187C2×C4×C59—81no—
188C2×C4×C59—81no—
189C2×C4×C59#12011yescharacteristic
190C2×C4×C59#12111yescharacteristic
191C2×C4×C59#12111yescharacteristic
192C2×C4×C59#12711yescharacteristic
193C2×Dic59—41no—
194C2×Dic59—41no—
195C2×Dic59—42no—
196C2×Dic59—42no—
197C2×Dic59—42no—
198C2×Dic59—42no—
199C2×Dic59—42no—
200Dic118—42no—
201Dic118—81no—
202C59⋊C8—42no—
203C59⋊C8—81no—
204#9926—591nomaximal, Sylow
205C22×C59—22no—
206C22×C59—22no—
207C22×C59—22no—
208C22×C59—22no—
209C22×C59—21no—
210C22×C59—21no—
211C22×C59—21no—
212C22×C59—21no—
213C22×C59—44no—
214C22×C59—44no—
215C22×C59—42no—
216C22×C59—82no—
217C22×C59—82no—
218C22×C59—82no—
219C22×C59—82no—
220C22×C59—81no—
221C22×C59—81no—
222C22×C59#39211yescharacteristic
223C22×C59#44811yescharacteristic
224C22×C59#44911yescharacteristic
225C22×C59#45011yescharacteristic
226C22×C59#45211yescharacteristic
227C22×C59#45512yes—
228C22×C59#45812yes—
229C22×C59#46511yescharacteristic
230C22×C59#46811yescharacteristic
231C4×C59—21no—
232C4×C59—21no—
233C4×C59—21no—
234C4×C59—21no—
235C4×C59—82no—
236C4×C59—82no—
237C4×C59—81no—
238Dic59—82no—
239Dic59—81no—
240Dic59—81no—
241C2×#483—591no—
242#2626—591no—
243C2×#397D5911yescharacteristic
244#1823—591no—
245C2×#328—591no—
246C2×#323—591no—
247#1294—591no—
248C2×C59—21no—
249C2×C59—21no—
250C2×C59—21no—
251C2×C59—21no—
252C2×C59—82no—
253C2×C59#160812yes—
254C2×C59#161212yes—
255C2×C59#200011yescharacteristic
256C2×C59#200411yescharacteristic
257C2×C59#202211yescharacteristic
258C2×C22⋊Q8—591no—
259C22×C4⋊C4C2×D5911yescharacteristic
260C22×C22⋊C4—21no—
261C22×C22⋊C4C2×D5911yescharacteristic
262#483—1182no—
263#483—1182no—
264C2×#106—1181no—
265C2×#103—1181no—
266C2×C22⋊C8—591no—
267C2×C22⋊C8—591no—
268#399—1181no—
269#397—22no—
270#397—22no—
271#396—1182no—
272#393—1181no—
273#385—1181no—
274#380—591no—
275#379—1181no—
276C2×#95—21no—
277C2×#95C2×D5911yescharacteristic
278#340—1182no—
279#328—1182no—
280#323—1182no—
281C59#992611yescharacteristic, complemented, Sylow
282C25C22⋊D5911yescharacteristic
283C22×Q8—591no—
284C23×C4—21no—
285C23×C4—21no—
286C23×C4C22⋊D5911yescharacteristic
287C23×C4C22×D5911yescharacteristic
288C22×C8—1181no—
289C22×C8—1181no—
290C22⋊Q8—1181no—
291C22⋊Q8—1181no—
292C22⋊Q8—1181no—
293C22⋊Q8—1181no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—591no—
300C2×C4⋊C4—592no—
301C2×C4⋊C4—1182no—
302C2×C4⋊C4—1181no—
303C2×C4⋊C4C22⋊D5912yes—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—21no—
306C2×C22⋊C4—21no—
307C2×C22⋊C4—21no—
308C2×C22⋊C4—21no—
309C2×C22⋊C4—21no—
310C2×C22⋊C4—21no—
311C2×C22⋊C4—42no—
312C2×C22⋊C4—42no—
313C2×C22⋊C4—1182no—
314C2×C22⋊C4—1182no—
315C2×C22⋊C4—1181no—
316C2×C22⋊C4—1181no—
317#106—1184no—
318#103—1184no—
319C22⋊C8—592no—
320C22⋊C8—1181no—
321C22⋊C8—1181no—
322C22⋊C8—1181no—
323#95—24no—
324#95—21no—
325#95—1182no—
326#95—1181no—
327#95C22⋊D5912yes—
328C24—21no—
329C24—21no—
330C24—21no—
331C24—21no—
332C24—21no—
333C24—21no—
334C24—42no—
335C24—42no—
336C24#1174567012yes—
337C24C2×C22⋊D5911yescharacteristic
338C2×Q8—1181no—
339C2×Q8—1181no—
340C2×Q8—2362no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—21no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—1182no—
359C22×C4—1181no—
360C22×C4—1181no—
361C22×C4D4×D5911yescharacteristic
362C22×C4D4×D5911yescharacteristic
363C22×C4D4×D5911yescharacteristic
364C22×C4#1174568711yescharacteristic
365C22×C4C2×C22⋊D5911yescharacteristic
366C22×C4C2×C22⋊D5911yescharacteristic
367C2×C8—1182no—
368C2×C8—2362no—
369C2×C8—2361no—
370C2×C8—2362no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—1181no—
376C4⋊C4—1181no—
377C4⋊C4—4722no—
378C4⋊C4—2362no—
379C4⋊C4—2361no—
380C4⋊C4—2361no—
381C22⋊C4—48no—
382C22⋊C4—41no—
383C22⋊C4—41no—
384C22⋊C4—41no—
385C22⋊C4—41no—
386C22⋊C4—4722no—
387C22⋊C4—4722no—
388C22⋊C4—2361no—
389C22⋊C4—2361no—
390C22⋊C4—2361no—
391C22⋊C4—2361no—
392C23—22no—
393C23—22no—
394C23—22no—
395C23—22no—
396C23—22no—
397C23—44no—
398C23—44no—
399C23—42no—
400C23—42no—
401C23—41no—
402C23—41no—
403C23—41no—
404C23—41no—
405C23—42no—
406C23—42no—
407C23—82no—
408C23—82no—
409C23—81no—
410C23—81no—
411C23#1427369511yescharacteristic
412C23#1427369611yescharacteristic
413C23#1427369911yescharacteristic
414C23C2×#1174567011yescharacteristic
415C23#1427374711yescharacteristic
416C23#1427375311yescharacteristic
417C23#1427375711yescharacteristic
418Q8—4721no—
419Q8—2362no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—41no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4—81no—
438C2×C4—81no—
439C2×C4—81no—
440C2×C4—81no—
441C2×C4—81no—
442C2×C4—2362no—
443C2×C4—2361no—
444C2×C4—2361no—
445C2×C4—2362no—
446C2×C4—2362no—
447C2×C4—2362no—
448C2×C4—2362no—
449C2×C4#1427369511yescharacteristic
450C2×C4#1427369811yescharacteristic
451C2×C4#1427369911yescharacteristic
452C2×C4#1427372311yescharacteristic
453C8—4721no—
454C8—2362no—
455C22—22no—
456C22—22no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—21no—
461C22—22no—
462C22—22no—
463C22—44no—
464C22—42no—
465C22—44no—
466C22—82no—
467C22—82no—
468C22—82no—
469C22—82no—
470C22—81no—
471C22—81no—
472C22#1683731011yescharacteristic
473C22#1683754011yescharacteristic
474C22#1683754511yescharacteristic
475C22#1683755311yescharacteristic
476C22#1683756411yescharacteristic
477C22#1683757912yes—
478C22#1683759112yes—
479C22#1683762911yescharacteristic
480C22#1683764611yescharacteristic
481C4—21no—
482C4—21no—
483C4—21no—
484C4—21no—
485C4—82no—
486C4—82no—
487C4—81no—
488C4—4722no—
489C4—4721no—
490C4—4721no—
491C2—21no—
492C2—21no—
493C2—21no—
494C2—21no—
495C2—82no—
496C2#2142308612yes—
497C2#2142310812yes—
498C2#2142509911yescharacteristic
499C2#2142512511yescharacteristic
500C2#2142522411yescharacteristic
subgroupG/H
centerC23#14273695
commutator, cosocleC23×C4×C59C23
FrattiniC23×C4C22×D59
FittingC2×C59×#397C2
socleC23×C59#120
prime radicalC25×C59D4
hypercenterC2×#397D59
hypocenterC59#9926
LCSG ▷ C23×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#397

Complements

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

NH ≅ G/Ntype
1. C2×#16837732C2semidirect
3. #21424182C2semidirect
4. C2×#16837138C2semidirect
6. #21421975C2semidirect
281. C59#9926semidirect

Maximal subgroups

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

subgroupindex
C2×#168377322
#214299012
#214241822
C2×#168371382
C2×#168371262
#214219752
C2×C59×#3972
#992659