Finite Group Databasebeta

← #28573912

Subgroups — #28573912

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
1C2×#16837689C211yesmaximal, characteristic, complemented
2#21433556C211yesmaximal, characteristic
3#21433119C211yesmaximal, characteristic, complemented
4#21423502C214yesmaximal, complemented
5#21423468C214yesmaximal, complemented
6#21422867C211yesmaximal, characteristic, complemented
7#21422777C211yesmaximal, characteristic, complemented
8#21422745C211yesmaximal, characteristic, complemented
9C59×#3227C211yesmaximal, characteristic, complemented
10C2×#14273752—21no—
11C2×C4×D236C2211yescharacteristic
12#16837689—22no—
13#16837689C2214yes—
14#16837614—22no—
15#16837604C2212yes—
16#16837594C2212yes—
17C8×D236C2214yes—
18#16837467—22no—
19#16837455C2211yescharacteristic
20C2×#14273658—21no—
21#16837426C2211yescharacteristic
22C2×C4⋊(C59⋊C8)C2211yescharacteristic
23#16837403—21no—
24C2×#14273647—21no—
25C22⋊C8×D59—22no—
26#16837153C2214yescomplemented
27#16837151C2214yescomplemented
28#16837148C2212yescomplemented
29#16837146C2212yescomplemented
30C59×#532C2211yescharacteristic
31C2×C4⋊Q8×C59C2211yescharacteristic
32C59×#498C2212yescomplemented
33C59×#486C2212yescomplemented
34C59×#481—22no—
35C59×#476—22no—
36C59×#431C2211yescharacteristic
37C22×D236D411yescharacteristic
38C22×C4×D59—21no—
39#14273752—44no—
40C2×#11745671—22no—
41C2×#11745669—22no—
42C2×#11745664—21no—
43C22×C59⋊C8—21no—
44C2×C8×D59—24no—
45C4×D236—21no—
46C4×D236—21no—
47C4×D236D412yes—
48C4×D236C2312yes—
49C2×C4⋊Dic59D411yescharacteristic
50C2×(C2×D59)⋊C4—21no—
51#14273658—24no—
52C4⋊(C59⋊C8)—21no—
53C4⋊(C59⋊C8)—21no—
54C4⋊(C59⋊C8)C2312yes—
55C4⋊(C59⋊C8)C2312yes—
56C4⋊(C59⋊C8)C2312yes—
57#14273651—22no—
58#14273648—22no—
59#14273647—24no—
60C22⋊(C59⋊C8)—21no—
61(C2×D59)⋊C8—24no—
62C4×C59⋊C8—21no—
63C2×#41×C59—21no—
64C22×Q8×C59—21no—
65C4⋊Q8×C59—21no—
66C4⋊Q8×C59—21no—
67C4⋊Q8×C59—21no—
68C4⋊Q8×C59C2311yescharacteristic
69C4⋊Q8×C59C2312yes—
70#127×C59C2311yescharacteristic
71#124×C59—21no—
72#121×C59—21no—
73#121×C59—21no—
74C4×D4×C59—21no—
75C4×D4×C59—21no—
76C2×C4⋊C4×C59—21no—
77C2×C4⋊C4×C59—21no—
78C2×C42×C59C2311yescharacteristic
79#106×C59—24no—
80C4⋊C8×C59D412yes—
81#102×C59—24no—
82C22⋊C8×C59D412yes—
83C4×C8×C59C2312yes—
84C23×D59—21no—
85#41×C59—42no—
86#41×C59—42no—
87C2×Q8×C59—21no—
88C2×Q8×C59—22no—
89C2×Q8×C59—41no—
90C2×Q8×C59—82no—
91C2×D4×C59—21no—
92C2×D4×C59—21no—
93C2×D4×C59—21no—
94C2×D4×C59—41no—
95C22×C4×C59—41no—
96C22×C4×C59—41no—
97C22×C4×C59C2×D411yescharacteristic
98C22×C4×C59C2×D411yescharacteristic
99C22×C4×C59#4111yescharacteristic
100C22×Dic59—21no—
101C2×D236—21no—
102C2×D236—22no—
103C2×D236SD1614yes—
104C2×D236C2×D412yes—
105C2×C4×D59—22no—
106C2×C4×D59—22no—
107C2×C4×D59—82no—
108C2×C8×C59C2×D414yes—
109C4⋊C4×C59—22no—
110C4⋊C4×C59—21no—
111C4⋊C4×C59—22no—
112C4⋊C4×C59—21no—
113C4⋊C4×C59—41no—
114C4⋊C4×C59—41no—
115C22⋊C4×C59—41no—
116C22⋊C4×C59—41no—
117C42×C59C2×D411yescharacteristic
118C42×C59C2×D411yescharacteristic
119C42×C59#4111yescharacteristic
120C42×C59C2411yescharacteristic
121#11745671—44no—
122#11745669—44no—
123(C2×D59)⋊C4—22no—
124(C2×D59)⋊C4—22no—
125C4⋊Dic59—21no—
126C4⋊Dic59C2×D412yes—
127#11745664—42no—
128C2×C59⋊C8—22no—
129C2×C59⋊C8—22no—
130C2×C59⋊C8—22no—
131C2×C59⋊C8—42no—
132C8×D59—84no—
133C23×C59—41no—
134C23×C59#12111yescharacteristic
135C22×D59—22no—
136C22×D59—24no—
137C22×D59—81no—
138C22×D59—81no—
139Q8×C59—41no—
140Q8×C59—41no—
141Q8×C59—82no—
142Q8×C59—82no—
143D4×C59—41no—
144D4×C59—41no—
145D4×C59—42no—
146D4×C59—81no—
147D4×C59—82no—
148C2×C4×C59—22no—
149C2×C4×C59—41no—
150C2×C4×C59—41no—
151C2×C4×C59—42no—
152C2×C4×C59—82no—
153C2×C4×C59—82no—
154C2×C4×C59#12111yescharacteristic
155C2×C4×C59#12111yescharacteristic
156C2×C4×C59#12111yescharacteristic
157C2×C4×C59C2×SD1612yes—
158C2×C4×C59#13512yes—
159C2×C4×C59C22×D411yescharacteristic
160C2×C4×C59C22×D411yescharacteristic
161C2×C4×C59C2×#4111yescharacteristic
162C2×Dic59—22no—
163C2×Dic59—81no—
164D236—22no—
165D236—24no—
166D236C2×SD1614yes—
167C4×D59—44no—
168C4×D59—82no—
169C8×C59—24no—
170C59⋊C8—42no—
171C59⋊C8—42no—
172C59⋊C8—42no—
173#19391—591nomaximal, Sylow
174C22×C59—21no—
175C22×C59—21no—
176C22×C59—81no—
177C22×C59—81no—
178C22×C59—81no—
179C22×C59#46512yes—
180C22×C59C2×#12111yescharacteristic
181C2×D59—24no—
182C2×D59—42no—
183C2×D59—42no—
184C2×D59—84no—
185C4×C59—21no—
186C4×C59—21no—
187C4×C59—21no—
188C4×C59—81no—
189C4×C59—82no—
190C4×C59#46512yes—
191C4×C59C22×SD1611yescharacteristic
192C4×C59C2×#13511yescharacteristic
193C4×C59#57712yes—
194Dic59—42no—
195#3227D5911yescharacteristic, complemented
196C2×#476—591no—
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—161no—
206C2×C59#304211yescharacteristic
207C2×C59C2×#46511yescharacteristic
208C2×C59#306811yescharacteristic
209D59—44no—
210C2×AGL(1,ℤ/8)—1181no—
211#532C2×D5911yescharacteristic
212C2×C4⋊Q8C2×D5911yescharacteristic
213C2×C4×D4—591no—
214#498C2×D5912yescomplemented
215#486C2×D5912yescomplemented
216#481—22no—
217#476—22no—
218#476—594no—
219#476—1182no—
220#463—1182no—
221#461—592no—
222#459—592no—
223C8×D4—594no—
224C8×D4—1182no—
225#431—591no—
226#431C2×D5911yescharacteristic
227C2×#106—1181no—
228#423—591no—
229C2×C4⋊C8—591no—
230#417—1181no—
231C2×#102—1181no—
232C2×C22⋊C8—1182no—
233#334—594no—
234#333—594no—
235#332—592no—
236#331—592no—
237C59#1939111yescharacteristic, complemented, Sylow
238C2×#41—21no—
239C22×Q8—21no—
240C22×D4—591no—
241C23×C4—1181no—
242AGL(1,ℤ/8)—2364no—
243C2×SD16—1182no—
244C2×D8—1182no—
245C2×M16—1181no—
246C22×C8—1181no—
247C22×C8—1184no—
248C4⋊Q8—21no—
249C4⋊Q8—21no—
250C4⋊Q8—21no—
251C4⋊Q8C22×D5912yes—
252C4⋊Q8C22×D5911yescharacteristic
253#127C22×D5911yescharacteristic
254#124—21no—
255#121—21no—
256#121—21no—
257C4×D4—21no—
258C4×D4—21no—
259C4×D4—592no—
260C4×D4—592no—
261C4×D4—1181no—
262C4×D4—1181no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—591no—
266C2×C22⋊C4—1181no—
267C2×C42C22×D5911yescharacteristic
268#106—24no—
269#106—1184no—
270C4⋊C8—592no—
271C4⋊C8—592no—
272C4⋊C8—592no—
273C4⋊C8—1181no—
274C4⋊C8—1181no—
275C4⋊C8C22⋊D5912yes—
276#103—1182no—
277#102—24no—
278#102—1184no—
279#102—1182no—
280C22⋊C8—1181no—
281C22⋊C8—1184no—
282C22⋊C8C22⋊D5912yes—
283C4×C8—1181no—
284C4×C8C22×D5912yes—
285C24—1181no—
286#41—42no—
287#41—42no—
288C2×Q8—21no—
289C2×Q8—22no—
290C2×Q8—41no—
291C2×Q8—82no—
292C2×D4—21no—
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—41no—
296C2×D4—594no—
297C2×D4—592no—
298C2×D4—1181no—
299C2×D4—1182no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—1181no—
303C22×C4—1182no—
304C22×C4—1182no—
305C22×C4—4722no—
306C22×C4D4×D5911yescharacteristic
307C22×C4#1174568711yescharacteristic
308C22×C4C2×C22⋊D5911yescharacteristic
309SD16—2364no—
310D8—2364no—
311M16—2362no—
312C2×C8—1182no—
313C2×C8—1182no—
314C2×C8—1182no—
315C2×C8—4724no—
316C2×C8—2362no—
317C2×C8C2×C22⋊D5914yes—
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—21no—
321C4⋊C4—21no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—592no—
325C4⋊C4—1181no—
326C22⋊C4—41no—
327C22⋊C4—41no—
328C22⋊C4—1182no—
329C22⋊C4—1182no—
330C42D4×D5911yescharacteristic
331C42#1174568711yescharacteristic
332C42C2×C22⋊D5911yescharacteristic
333C42C23×D5911yescharacteristic
334C23—41no—
335C23—1184no—
336C23—1182no—
337C23—4721no—
338C23—4721no—
339C23#1427369811yescharacteristic
340Q8—41no—
341Q8—41no—
342Q8—82no—
343Q8—82no—
344D4—41no—
345D4—41no—
346D4—42no—
347D4—81no—
348D4—82no—
349D4—594no—
350D4—1182no—
351D4—1184no—
352C2×C4—22no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—42no—
356C2×C4—82no—
357C2×C4—82no—
358C2×C4—1182no—
359C2×C4—4722no—
360C2×C4—4721no—
361C2×C4—2364no—
362C2×C4#1427369811yescharacteristic
363C2×C4#1427369811yescharacteristic
364C2×C4#1427369811yescharacteristic
365C2×C4SD16×D5912yes—
366C2×C4#1427374812yes—
367C2×C4C2×D4×D5911yescharacteristic
368C2×C4C22×C22⋊D5911yescharacteristic
369C2×C4C2×#1174568711yescharacteristic
370C8—24no—
371C8—2362no—
372C8—2362no—
373C8—2362no—
374C22—21no—
375C22—21no—
376C22—81no—
377C22—81no—
378C22—81no—
379C22—1184no—
380C22—4724no—
381C22—2362no—
382C22—2362no—
383C22#1683762812yes—
384C22C2×#1427369811yescharacteristic
385C4—21no—
386C4—21no—
387C4—21no—
388C4—81no—
389C4—82no—
390C4—2362no—
391C4#1683762812yes—
392C4C2×SD16×D5911yescharacteristic
393C4C2×#1427374811yescharacteristic
394C4#1683821012yes—
395C2—22no—
396C2—161no—
397C2—2364no—
398C2#2143360411yescharacteristic
399C2C2×#1683762811yescharacteristic
400C2#2143382011yescharacteristic
subgroupG/H
centerC22C2×#14273698
commutator, cosocleC42×C59C24
FrattiniC42C23×D59
FittingC59×#3227C2
socleC22×C59C2×#121
prime radical#21433556C2
hypercenter#3227D59
hypocenterC59#19391
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3227

Complements

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

NH ≅ G/Ntype
1. C2×#16837689C2semidirect
3. #21433119C2semidirect
4. #21423502C2semidirect
5. #21423468C2semidirect
6. #21422867C2semidirect
7. #21422777C2semidirect
8. #21422745C2semidirect
9. C59×#3227C2semidirect
26. #16837153C22semidirect
27. #16837151C22semidirect
28. #16837148C22semidirect
29. #16837146C22semidirect
32. C59×#498C22semidirect
33. C59×#486C22semidirect
195. #3227D59semidirect
214. #498C2×D59semidirect
215. #486C2×D59semidirect
237. C59#19391semidirect

Maximal subgroups

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

subgroupindex
C2×#168376892
#214335562
#214331192
#214235022
#214234682
#214228672
#214227772
#214227452
C59×#32272
#1939159