Finite Group Databasebeta

← #28485761

Subgroups — #28485761

Order 15104.

470 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21432754C211yesmaximal, characteristic, complemented
2#21424191C211yesmaximal, characteristic, complemented
3#21422664C211yesmaximal, characteristic
4#21422456C211yesmaximal, characteristic
5#21422001C211yesmaximal, characteristic, complemented
6#21421983C211yesmaximal, characteristic, complemented
7C59×#3094C211yesmaximal, characteristic
8#16837463—22no—
9#16837459—21no—
10#16837455—21no—
11#16837429—21no—
12#16837366C2211yescharacteristic
13C2×C22⋊(C59⋊C8)C2211yescharacteristic
14C2×C22⋊(C59⋊C8)C2211yescharacteristic
15#16837358—21no—
16#16837262C2211yescharacteristic
17#16837164—21no—
18#16837164—21no—
19#16837136—21no—
20#16837136—21no—
21#16837126—22no—
22C2×#121×C59C2211yescharacteristic
23C2×#117×C59C2211yescharacteristic
24C59×#482—21no—
25C59×#482—21no—
26C59×#481—21no—
27C59×#480—21no—
28C59×#428C411yescharacteristic, complemented
29C59×#428C411yescharacteristic, complemented
30C59×#418C411yescharacteristic, complemented
31C59×#418C411yescharacteristic, complemented
32C59×#407C2211yescharacteristic
33C2×#11745664—21no—
34C22×C59⋊C8—21no—
35C22×C59⋊C8—21no—
36C2×C4⋊Dic59—21no—
37C2×C22⋊Dic59—21no—
38C4⋊(C59⋊C8)—22no—
39C4⋊(C59⋊C8)—22no—
40C22⋊(C59⋊C8)—21no—
41C22⋊(C59⋊C8)—21no—
42C22⋊(C59⋊C8)—21no—
43C22⋊(C59⋊C8)—21no—
44C22⋊(C59⋊C8)D412yes—
45C22⋊(C59⋊C8)D411yescharacteristic
46C22⋊(C59⋊C8)D411yescharacteristic
47#14273638—22no—
48C4×C59⋊C8—22no—
49C59⋊#95—21no—
50C22×D4×C59—21no—
51C22×D4×C59C2×C411yescharacteristic
52C23×C4×C59C2311yescharacteristic
53C2×M16×C59C2×C411yescharacteristic
54C2×M16×C59C2×C411yescharacteristic
55#121×C59—21no—
56#121×C59—21no—
57#121×C59—21no—
58#121×C59—21no—
59#117×C59—21no—
60#117×C59—21no—
61#117×C59C2×C411yescharacteristic
62#117×C59C2×C411yescharacteristic
63#117×C59D411yescharacteristic
64#117×C59D411yescharacteristic
65C2×C4⋊C4×C59C2×C411yescharacteristic
66C2×C4⋊C4×C59D411yescharacteristic
67C2×C4⋊C4×C59D411yescharacteristic
68C2×C22⋊C4×C59—21no—
69C2×C22⋊C4×C59—21no—
70C2×C42×C59—21no—
71#107×C59—21no—
72#107×C59—21no—
73#106×C59—21no—
74#106×C59—21no—
75#102×C59—21no—
76#102×C59—21no—
77#102×C59—21no—
78#102×C59—21no—
79C22⋊C8×C59—21no—
80C22⋊C8×C59—21no—
81C24×C59—21no—
82C24×C59#4111yescharacteristic
83C2×D4×C59—21no—
84C2×D4×C59—21no—
85C2×D4×C59—41no—
86C2×D4×C59—41no—
87C2×D4×C59—41no—
88C2×D4×C59—41no—
89C2×D4×C59—81no—
90C2×D4×C59—81no—
91C22×C4×C59—21no—
92C22×C4×C59—21no—
93C22×C4×C59—21no—
94C22×C4×C59—41no—
95C22×C4×C59—41no—
96C22×C4×C59C22×C411yescharacteristic
97C22×C4×C59C2×D411yescharacteristic
98C22×C4×C59C2×D411yescharacteristic
99C22×C4×C59C2×D411yescharacteristic
100C22×C4×C59C2×D411yescharacteristic
101C22×C4×C59#4111yescharacteristic
102C22×Dic59—41no—
103M16×C59—41no—
104M16×C59—41no—
105C2×C8×C59—21no—
106C2×C8×C59—21no—
107C4⋊C4×C59—21no—
108C4⋊C4×C59—21no—
109C4⋊C4×C59—21no—
110C4⋊C4×C59—21no—
111C4⋊C4×C59C22⋊C411yescharacteristic
112C4⋊C4×C59C22⋊C411yescharacteristic
113C4⋊C4×C59C22⋊C411yescharacteristic
114C4⋊C4×C59C22⋊C411yescharacteristic
115C22⋊C4×C59—21no—
116C22⋊C4×C59—21no—
117C22⋊C4×C59—21no—
118C22⋊C4×C59—21no—
119C22⋊C4×C59—41no—
120C22⋊C4×C59—41no—
121C42×C59—21no—
122C42×C59—21no—
123C42×C59—21no—
124C42×C59—21no—
125C22⋊Dic59—81no—
126C4⋊Dic59—41no—
127C4⋊Dic59—41no—
128#11745664—41no—
129#11745664—41no—
130C2×C59⋊C8—21no—
131C2×C59⋊C8—22no—
132C2×C59⋊C8—21no—
133C2×C59⋊C8—41no—
134C2×C59⋊C8—41no—
135C2×C59⋊C8—41no—
136C2×C59⋊C8—41no—
137C2×C59⋊C8—41no—
138C23×C59—21no—
139C23×C59—21no—
140C23×C59—41no—
141C23×C59—41no—
142C23×C59—42no—
143C23×C59—41no—
144C23×C59—81no—
145C23×C59—81no—
146C23×C59C4×D411yescharacteristic
147C23×C59#12111yescharacteristic
148C23×C59#12311yescharacteristic
149D4×C59—41no—
150D4×C59—41no—
151D4×C59—81no—
152D4×C59—82no—
153D4×C59—81no—
154D4×C59—81no—
155C2×C4×C59—21no—
156C2×C4×C59—21no—
157C2×C4×C59—21no—
158C2×C4×C59—21no—
159C2×C4×C59—21no—
160C2×C4×C59—21no—
161C2×C4×C59—21no—
162C2×C4×C59—21no—
163C2×C4×C59—21no—
164C2×C4×C59—21no—
165C2×C4×C59—41no—
166C2×C4×C59—41no—
167C2×C4×C59—41no—
168C2×C4×C59—41no—
169C2×C4×C59—41no—
170C2×C4×C59—41no—
171C2×C4×C59—81no—
172C2×C4×C59—81no—
173C2×C4×C59C2×C22⋊C411yescharacteristic
174C2×C4×C59C4×D411yescharacteristic
175C2×C4×C59#12011yescharacteristic
176C2×C4×C59#12111yescharacteristic
177C2×Dic59—81no—
178C2×Dic59—81no—
179C2×Dic59—81no—
180C8×C59—41no—
181C8×C59—41no—
182C59⋊C8—41no—
183C59⋊C8—41no—
184C59⋊C8—42no—
185C59⋊C8—81no—
186#9348—591nomaximal, Sylow
187C22×C59—21no—
188C22×C59—21no—
189C22×C59—21no—
190C22×C59—21no—
191C22×C59—21no—
192C22×C59—21no—
193C22×C59—21no—
194C22×C59—21no—
195C22×C59—81no—
196C22×C59—81no—
197C22×C59—81no—
198C22×C59—81no—
199C22×C59—81no—
200C22×C59—162no—
201C22×C59#38611yescharacteristic
202C22×C59#47111yescharacteristic
203C22×C59#47211yescharacteristic
204C4×C59—21no—
205C4×C59—21no—
206C4×C59—21no—
207C4×C59—41no—
208C4×C59—41no—
209C4×C59—41no—
210C4×C59—41no—
211C4×C59—81no—
212C4×C59#41011yescharacteristic
213C4×C59#41011yescharacteristic
214Dic59—161no—
215#3094D5911yescharacteristic
216#2924—591no—
217#1825—591no—
218#1519—591no—
219#1471—591no—
220#1305—591no—
221#1298—591no—
222C2×C59—21no—
223C2×C59—21no—
224C2×C59—41no—
225C2×C59—41no—
226C2×C59—161no—
227C2×C59#188111yescharacteristic
228C2×C59#189911yescharacteristic
229C2×C59#190711yescharacteristic
230C2×#121C2×D5911yescharacteristic
231C2×#117C2×D5911yescharacteristic
232#482—21no—
233#482—21no—
234#481—21no—
235#480—21no—
236#433—1182no—
237#432—1181no—
238#431—1181no—
239#428Dic5911yescharacteristic, complemented
240#428Dic5911yescharacteristic, complemented
241#424—1181no—
242#418Dic5911yescharacteristic, complemented
243#418Dic5911yescharacteristic, complemented
244#407—591no—
245#407C2×D5911yescharacteristic
246C2×C22⋊C8—591no—
247C2×C22⋊C8—591no—
248#405—1181no—
249#380—591no—
250#340—1181no—
251#340—1181no—
252#327—1181no—
253#327—1181no—
254#323—1182no—
255C59#934811yescharacteristic, complemented, Sylow
256C22×D4—21no—
257C22×D4C2×Dic5911yescharacteristic
258C23×C4C22×D5911yescharacteristic
259C2×M16—1181no—
260C2×M16C2×Dic5911yescharacteristic
261C2×M16C2×Dic5911yescharacteristic
262C22×C8—1181no—
263C22×C8—1181no—
264#121—21no—
265#121—21no—
266#121—21no—
267#121—21no—
268#117—21no—
269#117—21no—
270#117C2×Dic5911yescharacteristic
271#117C2×Dic5911yescharacteristic
272#117C22⋊D5911yescharacteristic
273#117C22⋊D5911yescharacteristic
274C2×C4⋊C4—1181no—
275C2×C4⋊C4C2×Dic5911yescharacteristic
276C2×C4⋊C4C22⋊D5911yescharacteristic
277C2×C4⋊C4C22⋊D5911yescharacteristic
278C2×C22⋊C4—21no—
279C2×C22⋊C4—21no—
280C2×C22⋊C4—1181no—
281C2×C42—21no—
282#107—21no—
283#107—21no—
284#106—21no—
285#106—21no—
286C4⋊C8—1182no—
287C4⋊C8—1182no—
288#102—21no—
289#102—21no—
290#102—21no—
291#102—21no—
292C22⋊C8—21no—
293C22⋊C8—21no—
294C22⋊C8—592no—
295C22⋊C8—591no—
296C22⋊C8—591no—
297C22⋊C8—1181no—
298C22⋊C8—1181no—
299C22⋊C8—1181no—
300C22⋊C8—1181no—
301#97—1182no—
302C4×C8—1182no—
303#95—1181no—
304C24—21no—
305C24#1174568711yescharacteristic
306C2×D4—21no—
307C2×D4—21no—
308C2×D4—41no—
309C2×D4—41no—
310C2×D4—41no—
311C2×D4—41no—
312C2×D4—81no—
313C2×D4—81no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—2361no—
320C22×C4D4×D5911yescharacteristic
321C22×C4D4×D5911yescharacteristic
322C22×C4#1174568711yescharacteristic
323C22×C4C22×Dic5911yescharacteristic
324C22×C4C2×C22⋊D5911yescharacteristic
325C22×C4C2×C22⋊D5911yescharacteristic
326M16—41no—
327M16—41no—
328M16—2361no—
329M16—2361no—
330C2×C8—21no—
331C2×C8—21no—
332C2×C8—1181no—
333C2×C8—1182no—
334C2×C8—1181no—
335C2×C8—2361no—
336C2×C8—2361no—
337C2×C8—2361no—
338C2×C8—2361no—
339C2×C8—2361no—
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—2361no—
345C4⋊C4—2361no—
346C4⋊C4C22⋊Dic5911yescharacteristic
347C4⋊C4C22⋊Dic5911yescharacteristic
348C4⋊C4C22⋊Dic5911yescharacteristic
349C4⋊C4C22⋊Dic5911yescharacteristic
350C22⋊C4—21no—
351C22⋊C4—21no—
352C22⋊C4—21no—
353C22⋊C4—21no—
354C22⋊C4—41no—
355C22⋊C4—41no—
356C22⋊C4—4721no—
357C42—21no—
358C42—21no—
359C42—21no—
360C42—21no—
361C23—21no—
362C23—21no—
363C23—41no—
364C23—41no—
365C23—42no—
366C23—41no—
367C23—81no—
368C23—81no—
369C23D4×Dic5911yescharacteristic
370C23#1427369911yescharacteristic
371C23#1427370811yescharacteristic
372D4—41no—
373D4—41no—
374D4—81no—
375D4—82no—
376D4—81no—
377D4—81no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4—4721no—
397C2×C4—4721no—
398C2×C4—4721no—
399C2×C4C2×C22⋊Dic5911yescharacteristic
400C2×C4D4×Dic5911yescharacteristic
401C2×C4#1427369511yescharacteristic
402C2×C4#1427369911yescharacteristic
403C8—41no—
404C8—41no—
405C8—4721no—
406C8—2361no—
407C8—2361no—
408C8—2362no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—81no—
418C22—81no—
419C22—81no—
420C22—81no—
421C22—81no—
422C22—162no—
423C22#1683728211yescharacteristic
424C22#1683766311yescharacteristic
425C22#1683767011yescharacteristic
426C4—21no—
427C4—21no—
428C4—21no—
429C4—41no—
430C4—41no—
431C4—41no—
432C4—41no—
433C4—81no—
434C4—9441no—
435C4#1683737611yescharacteristic
436C4#1683737611yescharacteristic
437C2—21no—
438C2—21no—
439C2—41no—
440C2—41no—
441C2—161no—
442C2#2142446011yescharacteristic
443C2#2142455511yescharacteristic
444C2#2142460611yescharacteristic
subgroupG/H
centerC22#16837282
commutatorC22×C4×C59C22×C4
FrattiniC23×C4C22×D59
FittingC59×#3094C2
socleC22×C59#386
prime radicalC2×#121×C59C22
cosocleC23×C4×C59C23
hypercenter#3094D59
hypocenterC59#9348
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3094

Complements

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

NH ≅ G/Ntype
1. #21432754C2semidirect
2. #21424191C2semidirect
5. #21422001C2semidirect
6. #21421983C2semidirect
28. C59×#428C4semidirect
29. C59×#428C4semidirect
30. C59×#418C4semidirect
31. C59×#418C4semidirect
239. #428Dic59semidirect
240. #428Dic59semidirect
242. #418Dic59semidirect
243. #418Dic59semidirect
255. C59#9348semidirect

Maximal subgroups

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

subgroupindex
#214327542
#214241912
#214226642
#214224562
#214220012
#214219832
C59×#30942
#934859