Finite Group Databasebeta

← #28573913

Subgroups — #28573913

Order 15104.

696 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21435224C211yesmaximal, characteristic, complemented
2#21433560C211yesmaximal, characteristic
3#21433120C211yesmaximal, characteristic, complemented
4#21423507C214yesmaximal, complemented
5#21423473C214yesmaximal, complemented
6#21422866C211yesmaximal, characteristic, complemented
7#21422777C211yesmaximal, characteristic, complemented
8#21422746C211yesmaximal, characteristic, complemented
9C2×C59×#476C211yesmaximal, characteristic, complemented
10C2×#14273753—21no—
11#16837952C2211yescharacteristic
12#16837798C2212yescomplemented
13#16837750C2212yescomplemented
14#16837720—22no—
15#16837689—22no—
16#16837617—22no—
17#16837607C2212yes—
18#16837597C2212yes—
19D4×C59⋊C8—22no—
20D4×C59⋊C8C2214yes—
21#16837455C2211yescharacteristic
22#16837455C2211yescharacteristic
23#16837426C2211yescharacteristic
24#16837401—21no—
25C2×C22⋊(C59⋊C8)—22no—
26#16837152C2214yescomplemented
27#16837151C2214yescomplemented
28#16837149C2212yescomplemented
29#16837145C2212yescomplemented
30C2×C4⋊Q8×C59C2211yescharacteristic
31C2×C4×D4×C59C2211yescharacteristic
32C59×#476—22no—
33C59×#476C2214yes—
34C2×#106×C59—21no—
35C2×C4⋊C8×C59C2211yescharacteristic
36C2×#102×C59—21no—
37C2×#11745685—21no—
38#14273753—44no—
39C2×#11745670—22no—
40C2×#11745669—22no—
41C2×#11745664—21no—
42C22×C59⋊C8—24no—
43#14273727C2311yescharacteristic
44#14273724C2311yescharacteristic
45#14273715—21no—
46#14273701—21no—
47#14273701—21no—
48C4×D236—21no—
49C4×D236—21no—
50#14273658—24no—
51C4⋊(C59⋊C8)—21no—
52C4⋊(C59⋊C8)D412yes—
53C4⋊(C59⋊C8)C2312yes—
54C4⋊(C59⋊C8)C2312yes—
55#14273650—22no—
56#14273647—22no—
57#14273647—24no—
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#106×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×C59SD1614yes—
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#11745670—44no—
123#11745669—44no—
124(C2×D59)⋊C4—41no—
125(C2×D59)⋊C4—41no—
126C4⋊Dic59—21no—
127C4⋊Dic59—21no—
128#11745664—42no—
129C2×C59⋊C8—22no—
130C2×C59⋊C8—22no—
131C2×C59⋊C8—84no—
132C2×C59⋊C8C2×D414yes—
133C23×C59—22no—
134C23×C59—24no—
135C23×C59—81no—
136C23×C59—81no—
137C23×C59#12111yescharacteristic
138C22×D59—41no—
139Q8×C59—82no—
140Q8×C59—82no—
141D4×C59—22no—
142D4×C59—24no—
143D4×C59C2×SD1614yes—
144C2×C4×C59—22no—
145C2×C4×C59—22no—
146C2×C4×C59—41no—
147C2×C4×C59—44no—
148C2×C4×C59—42no—
149C2×C4×C59—82no—
150C2×C4×C59—81no—
151C2×C4×C59—82no—
152C2×C4×C59#12111yescharacteristic
153C2×C4×C59#12111yescharacteristic
154C2×C4×C59#12111yescharacteristic
155C2×C4×C59C2×SD1612yes—
156C2×C4×C59#13512yes—
157C2×C4×C59C22×D411yescharacteristic
158C2×C4×C59C22×D411yescharacteristic
159C2×C4×C59C2×#4111yescharacteristic
160C22⋊D59—82no—
161C2×Dic59—41no—
162D236—41no—
163D236—41no—
164D236—42no—
165D236—81no—
166C4×D59—82no—
167Dic118—41no—
168Dic118—41no—
169C8×C59—42no—
170C59⋊C8—24no—
171C59⋊C8—42no—
172C59⋊C8—42no—
173#19391—591nomaximal, Sylow
174C22×C59—21no—
175C22×C59—21no—
176C22×C59—24no—
177C22×C59—42no—
178C22×C59—42no—
179C22×C59—84no—
180C22×C59#46512yes—
181C22×C59C2×#12111yescharacteristic
182C2×D59—81no—
183C2×D59—81no—
184C2×D59—81no—
185C4×C59—21no—
186C4×C59—21no—
187C4×C59—21no—
188C4×C59—42no—
189C4×C59—82no—
190C4×C59#46512yes—
191C4×C59C22×SD1611yescharacteristic
192C4×C59C2×#13511yescharacteristic
193C4×C59#57712yes—
194Dic59—81no—
195#3227—591no—
196C2×#476D5911yescharacteristic, complemented
197#3036—591no—
198#2972—591no—
199#1677—594no—
200#1671—594no—
201#1570—591no—
202#1548—591no—
203#1540—591no—
204C2×C59—22no—
205C2×C59—44no—
206C2×C59#304211yescharacteristic
207C2×C59C2×#46511yescharacteristic
208C2×C59#306811yescharacteristic
209D59—161no—
210C2×AGL(1,ℤ/8)—1181no—
211#532—591no—
212C2×C4⋊Q8C2×D5911yescharacteristic
213C2×C4×D4C2×D5911yescharacteristic
214#498—592no—
215#486—592no—
216#481—1182no—
217#476—22no—
218#476—1182no—
219#476C2×D5914yes—
220#463—1182no—
221#461—592no—
222#459—592no—
223C8×D4—594no—
224C8×D4—1182no—
225#431—591no—
226#431—591no—
227C2×#106—21no—
228#423—591no—
229C2×C4⋊C8C2×D5911yescharacteristic
230#417—1181no—
231C2×#102—21no—
232C2×C22⋊C8—1182no—
233#334—594no—
234#333—594no—
235#332—592no—
236#331—592no—
237C59#1939111yescharacteristic, complemented, Sylow
238C2×#41—1181no—
239C22×Q8—21no—
240C22×D4C22⋊D5911yescharacteristic
241C23×C4—21no—
242AGL(1,ℤ/8)—2364no—
243C2×SD16—1182no—
244C2×D8—1182no—
245C2×M16—1181no—
246C22×C8—21no—
247C22×C8—1184no—
248C4⋊Q8—21no—
249C4⋊Q8—21no—
250C4⋊Q8—21no—
251C4⋊Q8—591no—
252C4⋊Q8C22×D5912yes—
253#127—591no—
254#124—1181no—
255#121—1181no—
256#121—1181no—
257C4×D4—21no—
258C4×D4—21no—
259C4×D4—1181no—
260C4×D4—1181no—
261C4×D4C22⋊D5912yes—
262C4×D4C22×D5912yes—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4C22⋊D5911yescharacteristic
266C2×C22⋊C4—21no—
267C2×C42C22×D5911yescharacteristic
268#106—24no—
269#106—1184no—
270C4⋊C8—21no—
271C4⋊C8—592no—
272C4⋊C8—592no—
273C4⋊C8—592no—
274C4⋊C8—1181no—
275C4⋊C8C22×D5912yes—
276#103—1182no—
277#102—24no—
278#102—1184no—
279#102—1182no—
280C22⋊C8—592no—
281C22⋊C8—1184no—
282C22⋊C8—1181no—
283C4×C8—592no—
284C4×C8—1181no—
285C24—21no—
286#41—2362no—
287#41—2362no—
288C2×Q8—22no—
289C2×Q8—41no—
290C2×Q8—82no—
291C2×Q8—1181no—
292C2×D4—21no—
293C2×D4—22no—
294C2×D4—1181no—
295C2×D4—1181no—
296C2×D4—1181no—
297C2×D4—2361no—
298C2×D4#1174567114yes—
299C2×D4C2×C22⋊D5912yes—
300C22×C4—21no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—41no—
304C22×C4—82no—
305C22×C4—2361no—
306C22×C4D4×D5911yescharacteristic
307C22×C4#1174568711yescharacteristic
308C22×C4C2×C22⋊D5911yescharacteristic
309SD16—2364no—
310D8—2364no—
311M16—2362no—
312C2×C8—22no—
313C2×C8—42no—
314C2×C8—594no—
315C2×C8—1182no—
316C2×C8—1182no—
317C2×C8—4724no—
318C4⋊C4—21no—
319C4⋊C4—22no—
320C4⋊C4—22no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C4⋊C4—1181no—
324C4⋊C4—1181no—
325C4⋊C4C2×C22⋊D5912yes—
326C22⋊C4—22no—
327C22⋊C4—22no—
328C22⋊C4—2361no—
329C22⋊C4—2361no—
330C42D4×D5911yescharacteristic
331C42#1174568711yescharacteristic
332C42C2×C22⋊D5911yescharacteristic
333C42C23×D5911yescharacteristic
334C23—22no—
335C23—24no—
336C23—81no—
337C23—81no—
338C23—2361no—
339C23#1427369911yescharacteristic
340Q8—82no—
341Q8—82no—
342Q8—2361no—
343Q8—2361no—
344D4—24no—
345D4—22no—
346D4—4721no—
347D4—4722no—
348D4—2361no—
349D4—2361no—
350D4—2362no—
351D4C2×#1174567114yes—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—41no—
355C2×C4—44no—
356C2×C4—42no—
357C2×C4—82no—
358C2×C4—81no—
359C2×C4—82no—
360C2×C4—4722no—
361C2×C4—2361no—
362C2×C4#1427369911yescharacteristic
363C2×C4#1427369911yescharacteristic
364C2×C4#1427369911yescharacteristic
365C2×C4C2×#1174567112yes—
366C2×C4#1427374712yes—
367C2×C4C2×D4×D5911yescharacteristic
368C2×C4C22×C22⋊D5911yescharacteristic
369C2×C4C2×#1174568711yescharacteristic
370C8—42no—
371C8—1184no—
372C8—2362no—
373C8—2362no—
374C22—21no—
375C22—21no—
376C22—24no—
377C22—42no—
378C22—42no—
379C22—84no—
380C22—4721no—
381C22—4721no—
382C22—4721no—
383C22#1683763312yes—
384C22C2×#1427369911yescharacteristic
385C4—21no—
386C4—21no—
387C4—21no—
388C4—42no—
389C4—82no—
390C4—4721no—
391C4#1683763312yes—
392C4C22×#1174567111yescharacteristic
393C4C2×#1427374711yescharacteristic
394C4#1683820812yes—
395C2—22no—
396C2—44no—
397C2—9441no—
398C2#2143360511yescharacteristic
399C2C2×#1683763311yescharacteristic
400C2#2143382111yescharacteristic
subgroupG/H
centerC22C2×#14273699
commutator, cosocleC42×C59C24
FrattiniC42C23×D59
FittingC2×C59×#476C2
socleC22×C59C2×#121
prime radical#21433560C2
hypercenterC2×#476D59
hypocenterC59#19391
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#476

Complements

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

NH ≅ G/Ntype
1. #21435224C2semidirect
3. #21433120C2semidirect
4. #21423507C2semidirect
5. #21423473C2semidirect
6. #21422866C2semidirect
7. #21422777C2semidirect
8. #21422746C2semidirect
9. C2×C59×#476C2semidirect
12. #16837798C22semidirect
13. #16837750C22semidirect
26. #16837152C22semidirect
27. #16837151C22semidirect
28. #16837149C22semidirect
29. #16837145C22semidirect
196. C2×#476D59semidirect
237. C59#19391semidirect

Maximal subgroups

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

subgroupindex
#214352242
#214335602
#214331202
#214235072
#214234732
#214228662
#214227772
#214227462
C2×C59×#4762
#1939159