Finite Group Databasebeta

← #28484280

Subgroups — #28484280

Order 15104.

674 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837459C211yesmaximal, characteristic, complemented
2C2×#16837136C212yesmaximal
3#21422323C211yesmaximal, characteristic, complemented
4#21421974C212yesmaximal, complemented
5C59×#2854C211yesmaximal, characteristic
6#16837459—24no—
7C2×C4⋊(C59⋊C8)—21no—
8C2×C22⋊(C59⋊C8)C2212yes—
9C2×#14273638—21no—
10C2×C59⋊#95—21no—
11C2×C59⋊#95—21no—
12C2×C59⋊#95C2212yes—
13#16837136—24no—
14C2×#123×C59—21no—
15C2×#121×C59C2211yescharacteristic
16C22×C42×C59C2211yescharacteristic
17C59×#390C411yescharacteristic, complemented
18C59×#390C411yescharacteristic, complemented
19C59×#389C411yescharacteristic, complemented
20C59×#389C411yescharacteristic, complemented
21C59×#388—21no—
22C59×#388—21no—
23C59×#386—21no—
24C59×#380C2211yescharacteristic
25C23×Dic59—22no—
26C22×C59⋊C8—22no—
27C4⋊(C59⋊C8)—24no—
28C22⋊(C59⋊C8)—22no—
29C22⋊(C59⋊C8)D414yes—
30#14273638—24no—
31C59⋊#95—24no—
32C59⋊#95—22no—
33C59⋊#95—22no—
34C59⋊#95—42no—
35C22×D4×C59—21no—
36C22×D4×C59C2×C411yescharacteristic
37C23×C4×C59D411yescharacteristic
38C23×C4×C59D411yescharacteristic
39C23×C4×C59C2311yescharacteristic
40#123×C59—44no—
41#121×C59—22no—
42#121×C59—22no—
43C2×C4⋊C4×C59—21no—
44C2×C4⋊C4×C59—21no—
45C2×C4⋊C4×C59C2×C411yescharacteristic
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—22no—
55C2×C42×C59—22no—
56C2×C42×C59C2×C411yescharacteristic
57C2×C42×C59C2×C411yescharacteristic
58C2×C42×C59D411yescharacteristic
59C2×C42×C59D411yescharacteristic
60#95×C59—21no—
61#95×C59—21no—
62#95×C59—21no—
63C24×C59—21no—
64C24×C59C2×D411yescharacteristic
65C2×D4×C59—22no—
66C2×D4×C59—41no—
67C2×D4×C59—41no—
68C2×D4×C59—42no—
69C2×D4×C59—42no—
70C2×D4×C59—41no—
71C2×D4×C59—41no—
72C22×C4×C59—22no—
73C22×C4×C59—22no—
74C22×C4×C59—22no—
75C22×C4×C59—21no—
76C22×C4×C59—21no—
77C22×C4×C59—22no—
78C22×C4×C59—21no—
79C22×C4×C59—21no—
80C22×C4×C59—21no—
81C22×C4×C59—41no—
82C22×C4×C59—41no—
83C22×C4×C59—41no—
84C22×C4×C59C22⋊C411yescharacteristic
85C22×C4×C59C22⋊C411yescharacteristic
86C22×C4×C59C22⋊C411yescharacteristic
87C22×C4×C59C22⋊C411yescharacteristic
88C22×C4×C59C22×C411yescharacteristic
89C22×C4×C59C2×D411yescharacteristic
90C22×C4×C59C2×D412yes—
91C22×C4×C59#4112yes—
92C22×Dic59—24no—
93C22×Dic59—42no—
94C22×Dic59—42no—
95C22×Dic59—42no—
96C22×Dic59—42no—
97C22×Dic59—42no—
98C22×Dic59—42no—
99C4⋊C4×C59—22no—
100C4⋊C4×C59—42no—
101C4⋊C4×C59—42no—
102C4⋊C4×C59—41no—
103C4⋊C4×C59—41no—
104C22⋊C4×C59—42no—
105C22⋊C4×C59—42no—
106C22⋊C4×C59—42no—
107C22⋊C4×C59—42no—
108C22⋊C4×C59—42no—
109C22⋊C4×C59—41no—
110C22⋊C4×C59—41no—
111C42×C59—22no—
112C42×C59—22no—
113C42×C59—21no—
114C42×C59—21no—
115C42×C59—21no—
116C42×C59—21no—
117C2×C59⋊C8—24no—
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×C59AGL(1,ℤ/8)11yescharacteristic
134C23×C59AGL(1,ℤ/8)12yes—
135C23×C59#13711yescharacteristic
136D4×C59—82no—
137D4×C59—82no—
138D4×C59—82no—
139C2×C4×C59—21no—
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—42no—
155C2×C4×C59—42no—
156C2×C4×C59—44no—
157C2×C4×C59—42no—
158C2×C4×C59—42no—
159C2×C4×C59—41no—
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#10412yes—
169C2×C4×C59C4×D412yes—
170C2×C4×C59#12011yescharacteristic
171C2×C4×C59#12311yescharacteristic
172C2×Dic59—44no—
173C2×Dic59—44no—
174C2×Dic59—44no—
175C2×Dic59—82no—
176C2×Dic59—82no—
177C2×Dic59—82no—
178C2×Dic59—82no—
179C59⋊C8—44no—
180#9118—591nomaximal, Sylow
181C22×C59—22no—
182C22×C59—22no—
183C22×C59—22no—
184C22×C59—21no—
185C22×C59—21no—
186C22×C59—21no—
187C22×C59—21no—
188C22×C59—22no—
189C22×C59—81no—
190C22×C59—81no—
191C22×C59—81no—
192C22×C59—81no—
193C22×C59—81no—
194C22×C59—81no—
195C22×C59—81no—
196C22×C59#38611yescharacteristic
197C22×C59#41011yescharacteristic
198C22×C59#41011yescharacteristic
199C22×C59#41711yescharacteristic
200C22×C59#41811yescharacteristic
201C22×C59C2×#10411yescharacteristic
202C22×C59#42111yescharacteristic
203C22×C59#46812yes—
204C22×C59#46912yes—
205C4×C59—22no—
206C4×C59—21no—
207C4×C59—22no—
208C4×C59—21no—
209C4×C59—41no—
210C4×C59—41no—
211C4×C59—41no—
212C4×C59—81no—
213C4×C59—81no—
214C4×C59—81no—
215Dic59—84no—
216C2×#432—591no—
217#2854D5911yescharacteristic
218C2×#327—592no—
219#1434—591no—
220#1294—592no—
221C2×C59—22no—
222C2×C59—22no—
223C2×C59—161no—
224C2×C59#150511yescharacteristic
225C2×C59#150911yescharacteristic
226C2×C59#151211yescharacteristic
227C2×C59#151411yescharacteristic
228C2×C59#187111yescharacteristic
229C2×C59#189511yescharacteristic
230C2×C59#189911yescharacteristic
231C2×#123—21no—
232C2×#121C2×D5911yescharacteristic
233C22×C42C2×D5911yescharacteristic
234#432—1184no—
235C2×C4⋊C8—1181no—
236C2×C22⋊C8—592no—
237C2×#97—1181no—
238#390Dic5911yescharacteristic, complemented
239#390Dic5911yescharacteristic, complemented
240#389Dic5911yescharacteristic, complemented
241#389Dic5911yescharacteristic, complemented
242#388—21no—
243#388—21no—
244#386—21no—
245#380C2×D5911yescharacteristic
246C2×#95—592no—
247C2×#95—1181no—
248C2×#95—1181no—
249#327—1184no—
250C59#911811yescharacteristic, complemented, Sylow
251C22×D4—21no—
252C22×D4C2×Dic5911yescharacteristic
253C23×C4—1182no—
254C23×C4C22⋊D5911yescharacteristic
255C23×C4C22⋊D5911yescharacteristic
256C23×C4C22×D5911yescharacteristic
257C22×C8—1182no—
258#123—44no—
259#121—22no—
260#121—22no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4C2×Dic5911yescharacteristic
264C2×C4⋊C4C2×Dic5911yescharacteristic
265C2×C4⋊C4C2×Dic5911yescharacteristic
266C2×C22⋊C4—21no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—21no—
272C2×C42—22no—
273C2×C42—22no—
274C2×C42C2×Dic5911yescharacteristic
275C2×C42C2×Dic5911yescharacteristic
276C2×C42C22⋊D5911yescharacteristic
277C2×C42C22⋊D5911yescharacteristic
278C4⋊C8—1184no—
279C22⋊C8—594no—
280C22⋊C8—1182no—
281#97—1184no—
282#95—21no—
283#95—21no—
284#95—21no—
285#95—1184no—
286#95—1182no—
287#95—1182no—
288#95—2362no—
289C24—21no—
290C24C2×C22⋊D5911yescharacteristic
291C2×D4—22no—
292C2×D4—41no—
293C2×D4—41no—
294C2×D4—42no—
295C2×D4—41no—
296C2×D4—42no—
297C2×D4—41no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—21no—
301C22×C4—22no—
302C22×C4—21no—
303C22×C4—22no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—1184no—
311C22×C4—2362no—
312C22×C4—2362no—
313C22×C4—2362no—
314C22×C4—2362no—
315C22×C4—2362no—
316C22×C4—2362no—
317C22×C4C22⋊Dic5911yescharacteristic
318C22×C4C22⋊Dic5911yescharacteristic
319C22×C4C22⋊Dic5911yescharacteristic
320C22×C4C22⋊Dic5911yescharacteristic
321C22×C4D4×D5912yes—
322C22×C4#1174568712yes—
323C22×C4C22×Dic5911yescharacteristic
324C22×C4C2×C22⋊D5911yescharacteristic
325C2×C8—1184no—
326C2×C8—2362no—
327C2×C8—2362no—
328C4⋊C4—22no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C22⋊C4—42no—
334C22⋊C4—42no—
335C22⋊C4—42no—
336C22⋊C4—42no—
337C22⋊C4—42no—
338C22⋊C4—41no—
339C22⋊C4—41no—
340C42—22no—
341C42—22no—
342C42—21no—
343C42—21no—
344C42—21no—
345C42—21no—
346C23—21no—
347C23—21no—
348C23—21no—
349C23—21no—
350C23—41no—
351C23—41no—
352C23—41no—
353C23—41no—
354C23—41no—
355C23—41no—
356C23—41no—
357C23C2×C22⋊Dic5911yescharacteristic
358C23#1427369912yes—
359C23#1427375211yescharacteristic
360C23#1427375312yes—
361C23#1427375711yescharacteristic
362D4—82no—
363D4—82no—
364D4—82no—
365C2×C4—21no—
366C2×C4—21no—
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—42no—
381C2×C4—42no—
382C2×C4—44no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—4722no—
395C2×C4—4722no—
396C2×C4—4722no—
397C2×C4—4722no—
398C2×C4—2364no—
399C2×C4—2364no—
400C2×C4—2364no—
401C2×C4#1427365412yes—
402C2×C4D4×Dic5912yes—
403C2×C4#1427369511yescharacteristic
404C2×C4#1427370811yescharacteristic
405C8—2364no—
406C22—22no—
407C22—22no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—22no—
413C22—22no—
414C22—81no—
415C22—81no—
416C22—81no—
417C22—81no—
418C22—81no—
419C22—81no—
420C22—81no—
421C22#1683728211yescharacteristic
422C22#1683737611yescharacteristic
423C22#1683737611yescharacteristic
424C22#1683740311yescharacteristic
425C22#1683740711yescharacteristic
426C22C2×#1427365411yescharacteristic
427C22#1683741911yescharacteristic
428C22#1683764612yes—
429C22#1683765312yes—
430C4—22no—
431C4—21no—
432C4—22no—
433C4—21no—
434C4—41no—
435C4—41no—
436C4—41no—
437C4—81no—
438C4—81no—
439C4—81no—
440C4—4724no—
441C2—22no—
442C2—22no—
443C2—161no—
444C2#2142260411yescharacteristic
445C2#2142262111yescharacteristic
446C2#2142263411yescharacteristic
447C2#2142264311yescharacteristic
448C2#2142439911yescharacteristic
449C2#2142453511yescharacteristic
450C2#2142455511yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic59
commutatorC22×C4×C59C22×C4
FrattiniC23×C4C22×D59
FittingC59×#2854C2
socleC23×C59C2×C22⋊C4
prime radicalC2×#121×C59C22
cosocleC23×C4×C59C23
hypercenter#2854D59
hypocenterC59#9118
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2854

Complements

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

NH ≅ G/Ntype
1. C2×#16837459C2semidirect
3. #21422323C2semidirect
4. #21421974C2semidirect
17. C59×#390C4semidirect
18. C59×#390C4semidirect
19. C59×#389C4semidirect
20. C59×#389C4semidirect
238. #390Dic59semidirect
239. #390Dic59semidirect
240. #389Dic59semidirect
241. #389Dic59semidirect
250. C59#9118semidirect

Maximal subgroups

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

subgroupindex
C2×#168374592
C2×#168371362
#214223232
#214219742
C59×#28542
#911859