Finite Group Databasebeta

← #28574235

Subgroups — #28574235

Order 15104.

684 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21435021C211yesmaximal, characteristic, complemented
2#21433560C211yesmaximal, characteristic
3#21433137C211yesmaximal, characteristic, complemented
4#21423641C214yesmaximal, complemented
5#21423563C214yesmaximal, complemented
6#21422870C211yesmaximal, characteristic, complemented
7#21422781C211yesmaximal, characteristic, complemented
8#21422746C211yesmaximal, characteristic, complemented
9C2×C59×#474C211yesmaximal, characteristic, complemented
10C2×#14273753—21no—
11#16837952C2211yescharacteristic
12#16837776C2212yescomplemented
13#16837752C2212yescomplemented
14#16837617—22no—
15#16837607C2212yes—
16#16837604—22no—
17#16837597C2212yes—
18#16837541—22no—
19#16837480—22no—
20#16837480C2214yes—
21#16837455C2211yescharacteristic
22#16837455C2211yescharacteristic
23#16837426C2211yescharacteristic
24#16837401—21no—
25#16837366—22no—
26#16837154C2214yescomplemented
27#16837149C2212yescomplemented
28#16837148C2214yescomplemented
29#16837145C2212yescomplemented
30C2×C4⋊Q8×C59C2211yescharacteristic
31C2×C4×D4×C59C2211yescharacteristic
32C59×#474—22no—
33C59×#474C2214yes—
34C2×#107×C59—21no—
35C2×C4⋊C8×C59C2211yescharacteristic
36C2×#102×C59—21no—
37C2×#11745685—21no—
38#14273753—44no—
39C2×#11745670—22no—
40C2×#11745671—24no—
41C2×#11745669—22no—
42C2×#11745664—21no—
43C2×#11745664—24no—
44#14273727C2311yescharacteristic
45#14273724C2311yescharacteristic
46#14273715—21no—
47#14273701—21no—
48#14273701—21no—
49C4×D236—21no—
50C4×D236—21no—
51C4⋊(C59⋊C8)—21no—
52C4⋊(C59⋊C8)D412yes—
53C4⋊(C59⋊C8)C2312yes—
54C4⋊(C59⋊C8)C2312yes—
55#14273651—24no—
56#14273650—22no—
57#14273647—22no—
58C22⋊(C59⋊C8)—21no—
59C22⋊(C59⋊C8)—24no—
60C22⋊(C59⋊C8)D412yes—
61C4×C59⋊C8—21no—
62C4×C59⋊C8C2312yes—
63C22×Q8×C59—21no—
64C22×D4×C59D411yescharacteristic
65C23×C4×C59—21no—
66C22×C8×C59—21no—
67C4⋊Q8×C59—21no—
68C4⋊Q8×C59—21no—
69C4⋊Q8×C59—21no—
70C4⋊Q8×C59C2312yes—
71C4×D4×C59—21no—
72C4×D4×C59—21no—
73C4×D4×C59D412yes—
74C4×D4×C59C2312yes—
75C2×C4⋊C4×C59—21no—
76C2×C4⋊C4×C59—21no—
77C2×C4⋊C4×C59D411yescharacteristic
78C2×C22⋊C4×C59—21no—
79C2×C42×C59C2311yescharacteristic
80#107×C59—24no—
81C4⋊C8×C59—21no—
82C4⋊C8×C59C2312yes—
83#102×C59—24no—
84C24×C59—21no—
85C2×Q8×C59—22no—
86C2×Q8×C59—41no—
87C2×Q8×C59—82no—
88C2×D4×C59—21no—
89C2×D4×C59—22no—
90C2×D4×C59—22no—
91C2×D4×C59C2×D412yes—
92C22×C4×C59—21no—
93C22×C4×C59—22no—
94C22×C4×C59—22no—
95C22×C4×C59—41no—
96C22×C4×C59—82no—
97C22×C4×C59C2×D411yescharacteristic
98C22×C4×C59C2×D411yescharacteristic
99C22×C4×C59#4111yescharacteristic
100C2×C22⋊D59—41no—
101#11745685—42no—
102#11745685—42no—
103C2×D236—21no—
104C2×D236—21no—
105C2×D236—21no—
106C2×C4×D59—41no—
107C2×Dic118—21no—
108C2×C8×C59—22no—
109C2×C8×C59—42no—
110C4⋊C4×C59—21no—
111C4⋊C4×C59—22no—
112C4⋊C4×C59—22no—
113C4⋊C4×C59—41no—
114C4⋊C4×C59—41no—
115C4⋊C4×C59C2×D412yes—
116C22⋊C4×C59—22no—
117C22⋊C4×C59—22no—
118C42×C59C2×D411yescharacteristic
119C42×C59C2×D411yescharacteristic
120C42×C59#4111yescharacteristic
121C42×C59C2411yescharacteristic
122#11745671—84no—
123#11745670—44no—
124#11745669—44no—
125(C2×D59)⋊C4—41no—
126(C2×D59)⋊C4—41no—
127C4⋊Dic59—21no—
128C4⋊Dic59—21no—
129#11745664—42no—
130#11745664—84no—
131C2×C59⋊C8—22no—
132C2×C59⋊C8—22no—
133C2×C59⋊C8C2×D414yes—
134C23×C59—22no—
135C23×C59—42no—
136C23×C59—81no—
137C23×C59—81no—
138C23×C59#12111yescharacteristic
139C22×D59—41no—
140Q8×C59—44no—
141Q8×C59—82no—
142D4×C59—22no—
143D4×C59—22no—
144D4×C59—42no—
145C2×C4×C59—22no—
146C2×C4×C59—22no—
147C2×C4×C59—41no—
148C2×C4×C59—44no—
149C2×C4×C59—42no—
150C2×C4×C59—82no—
151C2×C4×C59—81no—
152C2×C4×C59—82no—
153C2×C4×C59#12111yescharacteristic
154C2×C4×C59#12111yescharacteristic
155C2×C4×C59#12111yescharacteristic
156C2×C4×C59AGL(1,ℤ/8)12yes—
157C2×C4×C59#13712yes—
158C2×C4×C59C22×D411yescharacteristic
159C2×C4×C59C22×D411yescharacteristic
160C2×C4×C59C2×#4111yescharacteristic
161C22⋊D59—82no—
162C2×Dic59—41no—
163D236—41no—
164D236—41no—
165D236—42no—
166D236—81no—
167C4×D59—82no—
168Dic118—41no—
169Dic118—41no—
170C8×C59—42no—
171C59⋊C8—24no—
172C59⋊C8—42no—
173C59⋊C8—42no—
174#19421—591nomaximal, Sylow
175C22×C59—21no—
176C22×C59—21no—
177C22×C59—42no—
178C22×C59—42no—
179C22×C59—42no—
180C22×C59—84no—
181C22×C59#46812yes—
182C22×C59C2×#12111yescharacteristic
183C2×D59—81no—
184C2×D59—81no—
185C2×D59—81no—
186C4×C59—21no—
187C4×C59—21no—
188C4×C59—21no—
189C4×C59—42no—
190C4×C59—82no—
191C4×C59#46812yes—
192C4×C59C2×AGL(1,ℤ/8)11yescharacteristic
193C4×C59C2×#13711yescharacteristic
194C4×C59#57612yes—
195Dic59—81no—
196#3206—591no—
197C2×#474D5911yescharacteristic, complemented
198#3036—591no—
199#2975—591no—
200#1699—594no—
201#1685—594no—
202#1571—591no—
203#1549—591no—
204#1540—591no—
205C2×C59—22no—
206C2×C59—82no—
207C2×C59#304111yescharacteristic
208C2×C59C2×#46811yescharacteristic
209C2×C59#306911yescharacteristic
210D59—161no—
211C2×AGL(1,ℤ/8)—1181no—
212#532—591no—
213C2×C4⋊Q8C2×D5911yescharacteristic
214C2×C4×D4C2×D5911yescharacteristic
215#492—592no—
216#487—592no—
217#474—22no—
218#474C2×D5914yes—
219#463—1182no—
220#461—592no—
221#461—1182no—
222#459—592no—
223#448—1182no—
224#436—594no—
225#436—1182no—
226#431—591no—
227#431—591no—
228C2×#107—21no—
229#423—591no—
230C2×C4⋊C8C2×D5911yescharacteristic
231#417—1181no—
232C2×#102—21no—
233#407—1182no—
234#335—594no—
235#332—594no—
236#332—592no—
237#331—592no—
238C59#1942111yescharacteristic, complemented, Sylow
239C2×#41—1181no—
240C22×Q8—21no—
241C22×D4C22⋊D5911yescharacteristic
242C23×C4—21no—
243AGL(1,ℤ/8)—2364no—
244C2×SD16—1184no—
245C2×SD16—1182no—
246C2×D8—1182no—
247C2×M16—1184no—
248C2×M16—1181no—
249C22×C8—21no—
250C4⋊Q8—21no—
251C4⋊Q8—21no—
252C4⋊Q8—21no—
253C4⋊Q8—591no—
254C4⋊Q8C22×D5912yes—
255#127—591no—
256#124—1181no—
257#121—1181no—
258#121—1181no—
259C4×D4—21no—
260C4×D4—21no—
261C4×D4—1181no—
262C4×D4—1181no—
263C4×D4C22⋊D5912yes—
264C4×D4C22×D5912yes—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4C22⋊D5911yescharacteristic
268C2×C22⋊C4—21no—
269C2×C42C22×D5911yescharacteristic
270#107—24no—
271C4⋊C8—21no—
272C4⋊C8—592no—
273C4⋊C8—592no—
274C4⋊C8—592no—
275C4⋊C8—1181no—
276C4⋊C8C22×D5912yes—
277#103—1184no—
278#103—1182no—
279#102—24no—
280#102—1182no—
281C22⋊C8—592no—
282C22⋊C8—1184no—
283C22⋊C8—1181no—
284C4×C8—592no—
285C4×C8—1181no—
286C24—21no—
287#41—2362no—
288#41—2362no—
289C2×Q8—22no—
290C2×Q8—41no—
291C2×Q8—82no—
292C2×Q8—1181no—
293C2×D4—21no—
294C2×D4—22no—
295C2×D4—22no—
296C2×D4—1181no—
297C2×D4—1181no—
298C2×D4—1181no—
299C2×D4—2361no—
300C2×D4C2×C22⋊D5912yes—
301C22×C4—21no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—41no—
305C22×C4—82no—
306C22×C4—2361no—
307C22×C4D4×D5911yescharacteristic
308C22×C4#1174568711yescharacteristic
309C22×C4C2×C22⋊D5911yescharacteristic
310SD16—4724no—
311SD16—2364no—
312D8—2364no—
313M16—4724no—
314M16—2362no—
315C2×C8—22no—
316C2×C8—42no—
317C2×C8—594no—
318C2×C8—1182no—
319C2×C8—1182no—
320C4⋊C4—21no—
321C4⋊C4—22no—
322C4⋊C4—22no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—1181no—
326C4⋊C4—1181no—
327C4⋊C4C2×C22⋊D5912yes—
328C22⋊C4—22no—
329C22⋊C4—22no—
330C22⋊C4—2361no—
331C22⋊C4—2361no—
332C42D4×D5911yescharacteristic
333C42#1174568711yescharacteristic
334C42C2×C22⋊D5911yescharacteristic
335C42C23×D5911yescharacteristic
336C23—22no—
337C23—42no—
338C23—81no—
339C23—81no—
340C23—2361no—
341C23#1427369911yescharacteristic
342Q8—44no—
343Q8—82no—
344Q8—2361no—
345Q8—2361no—
346D4—22no—
347D4—22no—
348D4—42no—
349D4—4721no—
350D4—4722no—
351D4—2361no—
352D4—2361no—
353D4—2362no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—41no—
357C2×C4—44no—
358C2×C4—42no—
359C2×C4—82no—
360C2×C4—81no—
361C2×C4—82no—
362C2×C4—4722no—
363C2×C4—2361no—
364C2×C4#1427369911yescharacteristic
365C2×C4#1427369911yescharacteristic
366C2×C4#1427369911yescharacteristic
367C2×C4#1427375212yes—
368C2×C4#1427375812yes—
369C2×C4C2×D4×D5911yescharacteristic
370C2×C4C22×C22⋊D5911yescharacteristic
371C2×C4C2×#1174568711yescharacteristic
372C8—42no—
373C8—1184no—
374C8—2362no—
375C8—2362no—
376C22—21no—
377C22—21no—
378C22—42no—
379C22—42no—
380C22—42no—
381C22—84no—
382C22—4721no—
383C22—4721no—
384C22—4721no—
385C22#1683765012yes—
386C22C2×#1427369911yescharacteristic
387C4—21no—
388C4—21no—
389C4—21no—
390C4—42no—
391C4—82no—
392C4—4721no—
393C4#1683765012yes—
394C4C2×#1427375211yescharacteristic
395C4C2×#1427375811yescharacteristic
396C4#1683820312yes—
397C2—22no—
398C2—82no—
399C2—9441no—
400C2#2143359611yescharacteristic
401C2C2×#1683765011yescharacteristic
402C2#2143383011yescharacteristic
subgroupG/H
centerC22C2×#14273699
commutator, cosocleC42×C59C24
FrattiniC42C23×D59
FittingC2×C59×#474C2
socleC22×C59C2×#121
prime radical#21433560C2
hypercenterC2×#474D59
hypocenterC59#19421
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#474

Complements

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

NH ≅ G/Ntype
1. #21435021C2semidirect
3. #21433137C2semidirect
4. #21423641C2semidirect
5. #21423563C2semidirect
6. #21422870C2semidirect
7. #21422781C2semidirect
8. #21422746C2semidirect
9. C2×C59×#474C2semidirect
12. #16837776C22semidirect
13. #16837752C22semidirect
26. #16837154C22semidirect
27. #16837149C22semidirect
28. #16837148C22semidirect
29. #16837145C22semidirect
197. C2×#474D59semidirect
238. C59#19421semidirect

Maximal subgroups

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

subgroupindex
#214350212
#214335602
#214331372
#214236412
#214235632
#214228702
#214227812
#214227462
C2×C59×#4742
#1942159