Finite Group Databasebeta

← #28487834

Subgroups — #28487834

Order 15104.

650 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21424076C211yesmaximal, characteristic, complemented
2C2×#16837139C212yesmaximal, complemented
3C2×#16837136C212yesmaximal
4#21422323C211yesmaximal, characteristic, complemented
5C59×#2850C211yesmaximal, characteristic
6C2×C4⋊(C59⋊C8)—21no—
7C2×C22⋊(C59⋊C8)C2212yes—
8C2×C59⋊#95—21no—
9C2×C59⋊#95—21no—
10C2×C59⋊#95C2212yes—
11#16837157—21no—
12#16837139—24no—
13#16837136—24no—
14C2×#124×C59C411yescharacteristic, complemented
15C2×C22⋊Q8×C59C2211yescharacteristic
16C2×#121×C59C2211yescharacteristic
17C22×C42×C59C2211yescharacteristic
18C59×#391C411yescharacteristic, complemented
19C59×#390C411yescharacteristic, complemented
20C59×#388—21no—
21C59×#388—21no—
22C59×#386—21no—
23C59×#386—21no—
24C59×#382C411yescharacteristic, complemented
25C23×Dic59—22no—
26C22×C59⋊C8—22no—
27C4⋊(C59⋊C8)—42no—
28C22⋊(C59⋊C8)—22no—
29C22⋊(C59⋊C8)—22no—
30C59⋊#95—22no—
31C59⋊#95—42no—
32C59⋊#95—42no—
33C59⋊#95D414yes—
34C22×Q8×C59C2×C411yescharacteristic
35C22×D4×C59—21no—
36C22×D4×C59C2×C411yescharacteristic
37C23×C4×C59D411yescharacteristic
38C23×C4×C59D411yescharacteristic
39C23×C4×C59C2311yescharacteristic
40#124×C59—42no—
41C22⋊Q8×C59—22no—
42C22⋊Q8×C59—22no—
43#121×C59—22no—
44#121×C59—22no—
45C2×C4⋊C4×C59—21no—
46C2×C4⋊C4×C59C2×C411yescharacteristic
47C2×C4⋊C4×C59C2×C411yescharacteristic
48C2×C22⋊C4×C59—21no—
49C2×C22⋊C4×C59—21no—
50C2×C22⋊C4×C59—21no—
51C2×C22⋊C4×C59—21no—
52C2×C22⋊C4×C59—21no—
53C2×C22⋊C4×C59—21no—
54C2×C42×C59—42no—
55C2×C42×C59C2×C411yescharacteristic
56C2×C42×C59C2×C411yescharacteristic
57C2×C42×C59D411yescharacteristic
58C2×C42×C59D411yescharacteristic
59#95×C59—21no—
60#95×C59—21no—
61#95×C59—21no—
62#95×C59—21no—
63C24×C59—21no—
64C24×C59C2×D411yescharacteristic
65C2×Q8×C59—22no—
66C2×Q8×C59—41no—
67C2×Q8×C59—41no—
68C2×D4×C59—22no—
69C2×D4×C59—41no—
70C2×D4×C59—41no—
71C2×D4×C59—42no—
72C2×D4×C59—42no—
73C2×D4×C59—41no—
74C2×D4×C59—41no—
75C22×C4×C59—22no—
76C22×C4×C59—21no—
77C22×C4×C59—21no—
78C22×C4×C59—22no—
79C22×C4×C59—21no—
80C22×C4×C59—21no—
81C22×C4×C59—21no—
82C22×C4×C59—41no—
83C22×C4×C59—41no—
84C22×C4×C59—41no—
85C22×C4×C59—41no—
86C22×C4×C59—41no—
87C22×C4×C59C22⋊C411yescharacteristic
88C22×C4×C59C22⋊C411yescharacteristic
89C22×C4×C59C22⋊C411yescharacteristic
90C22×C4×C59C22⋊C411yescharacteristic
91C22×C4×C59C22×C411yescharacteristic
92C22×C4×C59C2×D411yescharacteristic
93C22×C4×C59C2×D412yes—
94C22×C4×C59#4112yes—
95C22×Dic59—24no—
96C22×Dic59—42no—
97C22×Dic59—42no—
98C22×Dic59—42no—
99C22×Dic59—42no—
100C22×Dic59—42no—
101C22×Dic59—42no—
102C4⋊C4×C59—22no—
103C4⋊C4×C59—22no—
104C4⋊C4×C59—41no—
105C4⋊C4×C59—41no—
106C22⋊C4×C59—42no—
107C22⋊C4×C59—42no—
108C22⋊C4×C59—42no—
109C22⋊C4×C59—42no—
110C22⋊C4×C59—42no—
111C22⋊C4×C59—41no—
112C22⋊C4×C59—41no—
113C42×C59—22no—
114C42×C59—22no—
115C42×C59—41no—
116C42×C59—41no—
117C2×C59⋊C8—42no—
118C2×C59⋊C8—42no—
119C2×C59⋊C8—42no—
120C23×C59—21no—
121C23×C59—21no—
122C23×C59—21no—
123C23×C59—21no—
124C23×C59—41no—
125C23×C59—41no—
126C23×C59—41no—
127C23×C59—41no—
128C23×C59—41no—
129C23×C59—41no—
130C23×C59—41no—
131C23×C59C2×C22⋊C411yescharacteristic
132C23×C59#12112yes—
133C23×C59#13512yes—
134C23×C59AGL(1,ℤ/8)11yescharacteristic
135C23×C59#13711yescharacteristic
136Q8×C59—82no—
137D4×C59—82no—
138D4×C59—82no—
139D4×C59—82no—
140C2×C4×C59—21no—
141C2×C4×C59—21no—
142C2×C4×C59—21no—
143C2×C4×C59—21no—
144C2×C4×C59—21no—
145C2×C4×C59—21no—
146C2×C4×C59—21no—
147C2×C4×C59—21no—
148C2×C4×C59—21no—
149C2×C4×C59—21no—
150C2×C4×C59—21no—
151C2×C4×C59—21no—
152C2×C4×C59—21no—
153C2×C4×C59—21no—
154C2×C4×C59—21no—
155C2×C4×C59—42no—
156C2×C4×C59—42no—
157C2×C4×C59—44no—
158C2×C4×C59—42no—
159C2×C4×C59—42no—
160C2×C4×C59—41no—
161C2×C4×C59—41no—
162C2×C4×C59—41no—
163C2×C4×C59—41no—
164C2×C4×C59—41no—
165C2×C4×C59—41no—
166C2×C4×C59—41no—
167C2×C4×C59—41no—
168C2×C4×C59—41no—
169C2×C4×C59#9912yes—
170C2×C4×C59C4×D412yes—
171C2×C4×C59#12411yescharacteristic
172C2×C4×C59#12711yescharacteristic
173C2×Dic59—44no—
174C2×Dic59—44no—
175C2×Dic59—44no—
176C2×Dic59—82no—
177C2×Dic59—82no—
178C2×Dic59—82no—
179C2×Dic59—82no—
180C59⋊C8—82no—
181#9716—591nomaximal, Sylow
182C22×C59—22no—
183C22×C59—22no—
184C22×C59—21no—
185C22×C59—21no—
186C22×C59—21no—
187C22×C59—21no—
188C22×C59—22no—
189C22×C59—22no—
190C22×C59—81no—
191C22×C59—81no—
192C22×C59—81no—
193C22×C59—81no—
194C22×C59—81no—
195C22×C59—81no—
196C22×C59—81no—
197C22×C59#39011yescharacteristic
198C22×C59C2×#9911yescharacteristic
199C22×C59#41011yescharacteristic
200C22×C59#41611yescharacteristic
201C22×C59#41711yescharacteristic
202C22×C59#42111yescharacteristic
203C22×C59#42111yescharacteristic
204C22×C59#46312yes—
205C22×C59#46412yes—
206C4×C59—21no—
207C4×C59—22no—
208C4×C59—21no—
209C4×C59—22no—
210C4×C59—41no—
211C4×C59—41no—
212C4×C59—41no—
213C4×C59—81no—
214C4×C59—81no—
215C4×C59—81no—
216Dic59—84no—
217#2850D5911yescharacteristic
218#1797—591no—
219C2×#328—592no—
220C2×#327—592no—
221#1434—591no—
222C2×C59—22no—
223C2×C59—22no—
224C2×C59—161no—
225C2×C59#151211yescharacteristic
226C2×C59#151311yescharacteristic
227C2×C59#151611yescharacteristic
228C2×C59#152111yescharacteristic
229C2×C59#196311yescharacteristic
230C2×C59#196411yescharacteristic
231C2×C59#198311yescharacteristic
232C2×#124Dic5911yescharacteristic, complemented
233C2×C22⋊Q8C2×D5911yescharacteristic
234C2×#121C2×D5911yescharacteristic
235C22×C42C2×D5911yescharacteristic
236C2×C4⋊C8—1181no—
237C2×C22⋊C8—592no—
238#391Dic5911yescharacteristic, complemented
239#390Dic5911yescharacteristic, complemented
240#388—21no—
241#388—21no—
242#386—21no—
243#386—21no—
244#382Dic5911yescharacteristic, complemented
245C2×#95—592no—
246C2×#95—1181no—
247C2×#95—1181no—
248#336—1181no—
249#328—1184no—
250#327—1184no—
251C59#971611yescharacteristic, complemented, Sylow
252C22×Q8C2×Dic5911yescharacteristic
253C22×D4—21no—
254C22×D4C2×Dic5911yescharacteristic
255C23×C4—1182no—
256C23×C4C22⋊D5911yescharacteristic
257C23×C4C22⋊D5911yescharacteristic
258C23×C4C22×D5911yescharacteristic
259C22×C8—1182no—
260#124—42no—
261C22⋊Q8—22no—
262C22⋊Q8—22no—
263#121—22no—
264#121—22no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4C2×Dic5911yescharacteristic
267C2×C4⋊C4C2×Dic5911yescharacteristic
268C2×C22⋊C4—21no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C42—42no—
275C2×C42C2×Dic5911yescharacteristic
276C2×C42C2×Dic5911yescharacteristic
277C2×C42C22⋊D5911yescharacteristic
278C2×C42C22⋊D5911yescharacteristic
279C4⋊C8—2362no—
280C22⋊C8—1182no—
281C22⋊C8—1182no—
282#95—21no—
283#95—21no—
284#95—21no—
285#95—21no—
286#95—594no—
287#95—1182no—
288#95—2362no—
289#95—2362no—
290C24—21no—
291C24C2×C22⋊D5911yescharacteristic
292C2×Q8—22no—
293C2×Q8—41no—
294C2×Q8—41no—
295C2×D4—22no—
296C2×D4—41no—
297C2×D4—41no—
298C2×D4—42no—
299C2×D4—42no—
300C2×D4—41no—
301C2×D4—41no—
302C22×C4—22no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—22no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—1184no—
315C22×C4—2362no—
316C22×C4—2362no—
317C22×C4—2362no—
318C22×C4—2362no—
319C22×C4—2362no—
320C22×C4—2362no—
321C22×C4C22⋊Dic5911yescharacteristic
322C22×C4C22⋊Dic5911yescharacteristic
323C22×C4C22⋊Dic5911yescharacteristic
324C22×C4C22⋊Dic5911yescharacteristic
325C22×C4D4×D5912yes—
326C22×C4#1174568712yes—
327C22×C4C22×Dic5911yescharacteristic
328C22×C4C2×C22⋊D5911yescharacteristic
329C2×C8—2362no—
330C2×C8—2362no—
331C2×C8—2362no—
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C22⋊C4—42no—
337C22⋊C4—42no—
338C22⋊C4—42no—
339C22⋊C4—42no—
340C22⋊C4—42no—
341C22⋊C4—41no—
342C22⋊C4—41no—
343C42—22no—
344C42—22no—
345C42—41no—
346C42—41no—
347C23—21no—
348C23—21no—
349C23—21no—
350C23—21no—
351C23—41no—
352C23—41no—
353C23—41no—
354C23—41no—
355C23—41no—
356C23—41no—
357C23—41no—
358C23C2×C22⋊Dic5911yescharacteristic
359C23#1427369812yes—
360C23#1427374712yes—
361C23#1427375211yescharacteristic
362C23#1427375711yescharacteristic
363Q8—82no—
364D4—82no—
365D4—82no—
366D4—82no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—44no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—4722no—
397C2×C4—4722no—
398C2×C4—4722no—
399C2×C4—4722no—
400C2×C4—2364no—
401C2×C4—2364no—
402C2×C4—2364no—
403C2×C4#1427364212yes—
404C2×C4D4×Dic5912yes—
405C2×C4#1427371411yescharacteristic
406C2×C4#1427372311yescharacteristic
407C8—4722no—
408C22—22no—
409C22—22no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—22no—
415C22—22no—
416C22—81no—
417C22—81no—
418C22—81no—
419C22—81no—
420C22—81no—
421C22—81no—
422C22—81no—
423C22#1683730511yescharacteristic
424C22C2×#1427364211yescharacteristic
425C22#1683737611yescharacteristic
426C22#1683739811yescharacteristic
427C22#1683740311yescharacteristic
428C22#1683741911yescharacteristic
429C22#1683741911yescharacteristic
430C22#1683761412yes—
431C22#1683762112yes—
432C4—21no—
433C4—22no—
434C4—21no—
435C4—22no—
436C4—41no—
437C4—41no—
438C4—41no—
439C4—81no—
440C4—81no—
441C4—81no—
442C4—4724no—
443C2—22no—
444C2—22no—
445C2—161no—
446C2#2142263411yescharacteristic
447C2#2142263911yescharacteristic
448C2#2142265211yescharacteristic
449C2#2142267311yescharacteristic
450C2#2142492911yescharacteristic
451C2#2142493311yescharacteristic
452C2#2142502911yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic59
commutatorC22×C4×C59C22×C4
FrattiniC23×C4C22×D59
FittingC59×#2850C2
socleC23×C59C2×C22⋊C4
prime radicalC2×#121×C59C22
cosocleC23×C4×C59C23
hypercenter#2850D59
hypocenterC59#9716
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2850

Complements

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

NH ≅ G/Ntype
1. #21424076C2semidirect
2. C2×#16837139C2semidirect
4. #21422323C2semidirect
14. C2×#124×C59C4semidirect
18. C59×#391C4semidirect
19. C59×#390C4semidirect
24. C59×#382C4semidirect
232. C2×#124Dic59semidirect
238. #391Dic59semidirect
239. #390Dic59semidirect
244. #382Dic59semidirect
251. C59#9716semidirect

Maximal subgroups

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

subgroupindex
#214240762
C2×#168371392
C2×#168371362
#214223232
C59×#28502
#971659