Finite Group Databasebeta

← #28479212

Subgroups — #28479212

Order 15104.

710 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#21423858C211yesmaximal, characteristic, complemented
2C4×C22⋊(C59⋊C8)C211yesmaximal, characteristic, complemented
3C2×#16837126C212yesmaximal
4#21421939C212yesmaximal, complemented
5C2×C59×#388C211yesmaximal, characteristic
6C2×C22⋊(C59⋊C8)C2212yes—
7C2×C22⋊(C59⋊C8)C2212yes—
8C2×#14273638—21no—
9C2×C4×C59⋊C8—21no—
10#16837157—21no—
11#16837157—21no—
12#16837126C418yescomplemented
13C22×C4⋊C4×C59C411yescharacteristic
14C22×C22⋊C4×C59C411yescharacteristic
15C22×C22⋊C4×C59C411yescharacteristic
16C22×C22⋊C4×C59C2211yescharacteristic
17C22×C42×C59C2211yescharacteristic
18C59×#388—21no—
19C59×#388—21no—
20C59×#388—21no—
21C59×#388—21no—
22C2×#95×C59C411yescharacteristic
23C2×#95×C59C2211yescharacteristic
24C22×C59⋊C8—22no—
25C22×C59⋊C8—22no—
26C22⋊(C59⋊C8)—22no—
27C22⋊(C59⋊C8)—22no—
28C22⋊(C59⋊C8)C2×C414yes—
29C22⋊(C59⋊C8)C2×C414yes—
30#14273638—42no—
31C4×C59⋊C8—42no—
32C25×C59C2×C411yescharacteristic
33C23×C4×C59C2×C411yescharacteristic
34C23×C4×C59C2×C411yescharacteristic
35C23×C4×C59C2×C411yescharacteristic
36C23×C4×C59C2×C411yescharacteristic
37C23×C4×C59C2×C411yescharacteristic
38C23×C4×C59C2311yescharacteristic
39C2×C4⋊C4×C59—21no—
40C2×C4⋊C4×C59—21no—
41C2×C4⋊C4×C59—42no—
42C2×C22⋊C4×C59—21no—
43C2×C22⋊C4×C59—21no—
44C2×C22⋊C4×C59—22no—
45C2×C22⋊C4×C59—21no—
46C2×C22⋊C4×C59—21no—
47C2×C22⋊C4×C59—21no—
48C2×C22⋊C4×C59—21no—
49C2×C22⋊C4×C59—42no—
50C2×C22⋊C4×C59—42no—
51C2×C22⋊C4×C59C2×C414yes—
52C2×C42×C59—42no—
53C2×C42×C59D411yescharacteristic
54C2×C42×C59D411yescharacteristic
55C2×C42×C59D411yescharacteristic
56C2×C42×C59D411yescharacteristic
57#95×C59—21no—
58#95×C59—21no—
59#95×C59—21no—
60#95×C59—21no—
61C24×C59—21no—
62C24×C59—21no—
63C24×C59—22no—
64C24×C59—22no—
65C24×C59—41no—
66C24×C59—41no—
67C24×C59—41no—
68C24×C59—41no—
69C24×C59C4212yes—
70C24×C59C22×C411yescharacteristic
71C22×C4×C59—21no—
72C22×C4×C59—21no—
73C22×C4×C59—22no—
74C22×C4×C59—22no—
75C22×C4×C59—22no—
76C22×C4×C59—22no—
77C22×C4×C59—21no—
78C22×C4×C59—21no—
79C22×C4×C59—22no—
80C22×C4×C59—22no—
81C22×C4×C59—41no—
82C22×C4×C59—41no—
83C22×C4×C59—41no—
84C22×C4×C59—41no—
85C22×C4×C59—41no—
86C22×C4×C59—41no—
87C22×C4×C59—41no—
88C22×C4×C59—41no—
89C22×C4×C59—41no—
90C22×C4×C59—41no—
91C22×C4×C59C4212yes—
92C22×C4×C59C22⋊C411yescharacteristic
93C22×C4×C59C22⋊C411yescharacteristic
94C22×C4×C59C22⋊C411yescharacteristic
95C22×C4×C59C22⋊C411yescharacteristic
96C22×C4×C59C22×C412yes—
97C22×C4×C59C2×D411yescharacteristic
98C22×C4×C59C2×D411yescharacteristic
99C22×C4×C59#4112yes—
100C4⋊C4×C59—42no—
101C4⋊C4×C59—42no—
102C22⋊C4×C59—24no—
103C22⋊C4×C59—24no—
104C22⋊C4×C59—42no—
105C22⋊C4×C59—42no—
106C22⋊C4×C59—42no—
107C22⋊C4×C59—42no—
108C42×C59—22no—
109C42×C59—22no—
110C42×C59—41no—
111C42×C59—41no—
112C2×C59⋊C8—24no—
113C2×C59⋊C8—24no—
114C2×C59⋊C8—42no—
115C2×C59⋊C8—42no—
116C2×C59⋊C8—42no—
117C2×C59⋊C8—42no—
118C23×C59—21no—
119C23×C59—21no—
120C23×C59—21no—
121C23×C59—21no—
122C23×C59—22no—
123C23×C59—24no—
124C23×C59—42no—
125C23×C59—42no—
126C23×C59—42no—
127C23×C59—42no—
128C23×C59—41no—
129C23×C59—41no—
130C23×C59—41no—
131C23×C59—41no—
132C23×C59—41no—
133C23×C59—41no—
134C23×C59—42no—
135C23×C59—82no—
136C23×C59—82no—
137C23×C59—82no—
138C23×C59—82no—
139C23×C59C2×C4211yescharacteristic
140C23×C59C2×C22⋊C411yescharacteristic
141C23×C59#11711yescharacteristic
142C23×C59#13112yes—
143C23×C59#13112yes—
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—22no—
153C2×C4×C59—21no—
154C2×C4×C59—21no—
155C2×C4×C59—21no—
156C2×C4×C59—21no—
157C2×C4×C59—21no—
158C2×C4×C59—21no—
159C2×C4×C59—42no—
160C2×C4×C59—42no—
161C2×C4×C59—44no—
162C2×C4×C59—44no—
163C2×C4×C59—42no—
164C2×C4×C59—41no—
165C2×C4×C59—41no—
166C2×C4×C59—42no—
167C2×C4×C59—41no—
168C2×C4×C59—41no—
169C2×C4×C59—41no—
170C2×C4×C59—41no—
171C2×C4×C59—41no—
172C2×C4×C59—41no—
173C2×C4×C59—82no—
174C2×C4×C59—82no—
175C2×C4×C59#9912yes—
176C2×C4×C59#10012yes—
177C2×C4×C59C4×D412yes—
178C2×C4×C59C4×D412yes—
179C59⋊C8—44no—
180C59⋊C8—44no—
181#8155—591nomaximal, Sylow
182C22×C59—22no—
183C22×C59—22no—
184C22×C59—22no—
185C22×C59—22no—
186C22×C59—21no—
187C22×C59—21no—
188C22×C59—21no—
189C22×C59—21no—
190C22×C59—42no—
191C22×C59—42no—
192C22×C59—42no—
193C22×C59—84no—
194C22×C59—84no—
195C22×C59—81no—
196C22×C59—81no—
197C22×C59—81no—
198C22×C59—81no—
199C22×C59C4×C22⋊C411yescharacteristic
200C22×C59#40512yes—
201C22×C59#40811yescharacteristic
202C22×C59#40811yescharacteristic
203C22×C59C2×#9911yescharacteristic
204C22×C59#41011yescharacteristic
205C22×C59C2×#10011yescharacteristic
206C22×C59#41311yescharacteristic
207C22×C59#43312yes—
208C4×C59—22no—
209C4×C59—22no—
210C4×C59—22no—
211C4×C59—22no—
212C4×C59—41no—
213C4×C59—41no—
214C4×C59—42no—
215C4×C59—81no—
216C4×C59—81no—
217C2×#388D5911yescharacteristic
218#1749—591no—
219C4×C22⋊C8—591no—
220C2×#323—592no—
221#1280—592no—
222C2×C59—22no—
223C2×C59—22no—
224C2×C59—82no—
225C2×C59#146811yescharacteristic
226C2×C59#146911yescharacteristic
227C2×C59#147111yescharacteristic
228C2×C59#147311yescharacteristic
229C2×C59#175011yescharacteristic
230C2×C59C4×#9911yescharacteristic
231C2×C59C4×#10011yescharacteristic
232C22×C4⋊C4Dic5911yescharacteristic
233C22×C22⋊C4Dic5911yescharacteristic
234C22×C22⋊C4Dic5911yescharacteristic
235C22×C22⋊C4C2×D5911yescharacteristic
236C22×C42C2×D5911yescharacteristic
237C2×C22⋊C8—592no—
238C2×C22⋊C8—592no—
239C2×#97—1181no—
240C2×C4×C8—1181no—
241#388—21no—
242#388—21no—
243#388—21no—
244#388—21no—
245C2×#95Dic5911yescharacteristic
246C2×#95C2×D5911yescharacteristic
247#336—1181no—
248#336—1181no—
249#323—598no—
250C59#815511yescharacteristic, complemented, Sylow
251C25C2×Dic5911yescharacteristic
252C23×C4C2×Dic5911yescharacteristic
253C23×C4C2×Dic5911yescharacteristic
254C23×C4C2×Dic5911yescharacteristic
255C23×C4C2×Dic5911yescharacteristic
256C23×C4C2×Dic5911yescharacteristic
257C23×C4C22×D5911yescharacteristic
258C22×C8—1182no—
259C22×C8—1182no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—42no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—42no—
271C2×C22⋊C4—42no—
272C2×C22⋊C4C4×D5914yes—
273C2×C42—42no—
274C2×C42C22⋊D5911yescharacteristic
275C2×C42C22⋊D5911yescharacteristic
276C2×C42C22⋊D5911yescharacteristic
277C2×C42C22⋊D5911yescharacteristic
278C22⋊C8—594no—
279C22⋊C8—594no—
280C22⋊C8—1182no—
281C22⋊C8—1182no—
282#97—2362no—
283C4×C8—2362no—
284#95—21no—
285#95—21no—
286#95—21no—
287#95—21no—
288C24—21no—
289C24—21no—
290C24—22no—
291C24—22no—
292C24—41no—
293C24—41no—
294C24—41no—
295C24—41no—
296C24C4×Dic5912yes—
297C24C22×Dic5911yescharacteristic
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4C4×Dic5912yes—
319C22×C4C22⋊Dic5911yescharacteristic
320C22×C4C22⋊Dic5911yescharacteristic
321C22×C4C22⋊Dic5911yescharacteristic
322C22×C4C22⋊Dic5911yescharacteristic
323C22×C4C2×C4×D5912yes—
324C22×C4#1174568512yes—
325C22×C4C2×C22⋊D5911yescharacteristic
326C22×C4C2×C22⋊D5911yescharacteristic
327C2×C8—1184no—
328C2×C8—1184no—
329C2×C8—2362no—
330C2×C8—2362no—
331C2×C8—2362no—
332C2×C8—2362no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C22⋊C4—24no—
336C22⋊C4—24no—
337C22⋊C4—42no—
338C22⋊C4—42no—
339C22⋊C4—42no—
340C22⋊C4—42no—
341C42—22no—
342C42—22no—
343C42—41no—
344C42—41no—
345C23—21no—
346C23—21no—
347C23—21no—
348C23—21no—
349C23—22no—
350C23—24no—
351C23—42no—
352C23—42no—
353C23—42no—
354C23—42no—
355C23—41no—
356C23—41no—
357C23—41no—
358C23—42no—
359C23—41no—
360C23—41no—
361C23—41no—
362C23—82no—
363C23—82no—
364C23—82no—
365C23—82no—
366C23C2×C4×Dic5911yescharacteristic
367C23C2×C22⋊Dic5911yescharacteristic
368C23#1427368611yescharacteristic
369C23#1427373412yes—
370C23#1427373412yes—
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—22no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—44no—
389C2×C4—44no—
390C2×C4—42no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—42no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—82no—
401C2×C4—82no—
402C2×C4#1427364212yes—
403C2×C4#1427364412yes—
404C2×C4C4×C22⋊D5912yes—
405C2×C4C4×C22⋊D5912yes—
406C8—2364no—
407C8—2364no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—22no—
413C22—22no—
414C22—22no—
415C22—22no—
416C22—42no—
417C22—42no—
418C22—42no—
419C22—84no—
420C22—84no—
421C22—81no—
422C22—81no—
423C22—81no—
424C22—81no—
425C22C4×C22⋊Dic5911yescharacteristic
426C22#1683735812yes—
427C22#1683736911yescharacteristic
428C22#1683736911yescharacteristic
429C22C2×#1427364211yescharacteristic
430C22#1683737611yescharacteristic
431C22C2×#1427364411yescharacteristic
432C22#1683738611yescharacteristic
433C22#1683746312yes—
434C4—22no—
435C4—22no—
436C4—22no—
437C4—22no—
438C4—42no—
439C4—41no—
440C4—41no—
441C4—81no—
442C4—81no—
443C2—22no—
444C2—22no—
445C2—82no—
446C2#2142244411yescharacteristic
447C2#2142244811yescharacteristic
448C2#2142245611yescharacteristic
449C2#2142246511yescharacteristic
450C2#2142386211yescharacteristic
451C2C4×#1427364211yescharacteristic
452C2C4×#1427364411yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic59
commutatorC23×C59C2×C42
FrattiniC23×C4C22×D59
FittingC2×C59×#388C2
socleC23×C59C2×C22⋊C4
prime radicalC25×C59C2×C4
cosocleC23×C4×C59C23
hypercenterC2×#388D59
hypocenterC59#8155
LCSG ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#388

Complements

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

NH ≅ G/Ntype
1. #21423858C2semidirect
2. C4×C22⋊(C59⋊C8)C2semidirect
4. #21421939C2semidirect
12. #16837126C4semidirect
250. C59#8155semidirect

Maximal subgroups

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

subgroupindex
#214238582
C4×C22⋊(C59⋊C8)2
C2×#168371262
#214219392
C2×C59×#3882
#815559