Finite Group Databasebeta

← #28743933

Subgroups — #28743933

Order 15104.

526 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21433007C211yesmaximal, characteristic, complemented
2#21426313C211yesmaximal, characteristic, complemented
3#21426308C211yesmaximal, characteristic, complemented
4#21426269C212yesmaximal, complemented
5#21426162C212yesmaximal, complemented
6#21426046C211yesmaximal, characteristic, complemented
7#21426041C211yesmaximal, characteristic
8#21425992C212yesmaximal, complemented
9#21425985C212yesmaximal, complemented
10#21425908C211yesmaximal, characteristic, complemented
11C59×#3183C211yesmaximal, characteristic, complemented
12#16838202C2211yescharacteristic
13#16838168—21no—
14#16837843C2212yes—
15#16837838C2212yes—
16#16837525—21no—
17#16837525C2212yes—
18#16837520—21no—
19#16837520C2212yes—
20#16837451C2211yescharacteristic
21C2×#14273655—21no—
22#16837224C2212yescomplemented
23#16837224C2212yescomplemented
24#16837223C2212yescomplemented
25#16837218C2212yescomplemented
26#16837215C2212yescomplemented
27#16837211C2212yes—
28#16837211C2212yes—
29#16837184C2211yescharacteristic
30#16837184C2211yescharacteristic
31#16837180C2212yescomplemented
32C59×#529—21no—
33C59×#499C2211yescharacteristic, complemented
34C59×#495C2211yescharacteristic, complemented
35C59×#488—21no—
36C59×#470C2211yescharacteristic
37C59×#469C2211yescharacteristic
38C59×#468—21no—
39C59×#442C2211yescharacteristic, complemented
40C59×#441C2211yescharacteristic, complemented
41C59×#440—21no—
42C4×M16×C59C2211yescharacteristic
43C2×#11745685—21no—
44#14273760—22no—
45#14273755—22no—
46#14273749—21no—
47#14273749C2312yes—
48#14273735—42no—
49#14273733—22no—
50#14273733—22no—
51C2×#11745664—21no—
52M16×D59—42no—
53#14273673—22no—
54#14273671—22no—
55#14273670—22no—
56#14273668—22no—
57#14273667C2312yes—
58#14273667C2312yes—
59#14273663—21no—
60#14273663C2312yes—
61#14273655—22no—
62#14273655—22no—
63C2×Q16×C59C2311yescharacteristic
64C2×SD16×C59—21no—
65C2×D8×C59C2311yescharacteristic
66C2×M16×C59D411yescharacteristic
67C2×M16×C59C2311yescharacteristic
68#126×C59—41no—
69#125×C59—21no—
70#124×C59—21no—
71#123×C59—41no—
72C22⋊Q8×C59—21no—
73#121×C59—21no—
74C4×Q8×C59—21no—
75C4×D4×C59—21no—
76C4×D4×C59—41no—
77#117×C59—41no—
78C2×C42×C59D411yescharacteristic
79#107×C59—21no—
80#106×C59C2311yescharacteristic
81#106×C59C2311yescharacteristic
82#103×C59—21no—
83#103×C59—21no—
84#102×C59—21no—
85#102×C59—21no—
86#97×C59D411yescharacteristic
87#97×C59C2311yescharacteristic
88C4×C8×C59D411yescharacteristic
89C4×C8×C59C2311yescharacteristic
90C2×Q8×C59—21no—
91C2×D4×C59—21no—
92C2×D4×C59—41no—
93C22×C4×C59—21no—
94C22×C4×C59—41no—
95C22×C4×C59C2×D411yescharacteristic
96C2×C22⋊D59—41no—
97#11745685—22no—
98#11745685—22no—
99#11745685—41no—
100#11745685—41no—
101C2×D236—21no—
102C2×C4×D59—41no—
103C2×Dic118—21no—
104Q16×C59—22no—
105SD16×C59—42no—
106D8×C59—22no—
107M16×C59—22no—
108M16×C59—22no—
109C2×C8×C59C2×D411yescharacteristic
110C2×C8×C59C2×D411yescharacteristic
111C2×C8×C59C2×D411yescharacteristic
112C2×C8×C59C2411yescharacteristic
113C4⋊C4×C59—21no—
114C4⋊C4×C59—21no—
115C4⋊C4×C59—41no—
116C4⋊C4×C59—41no—
117C4⋊C4×C59—41no—
118C4⋊C4×C59—41no—
119C22⋊C4×C59—41no—
120C22⋊C4×C59—41no—
121C22⋊C4×C59—41no—
122C22⋊C4×C59—41no—
123C22⋊C4×C59—41no—
124C42×C59—21no—
125C42×C59—41no—
126C42×C59C2×D411yescharacteristic
127C42×C59C2×D411yescharacteristic
128#11745664—22no—
129#11745664—41no—
130C2×C59⋊C8—41no—
131Dic236—21no—
132Dic236#4112yes—
133D472—21no—
134D472#4112yes—
135#11745660—22no—
136#11745660—22no—
137#11745659—42no—
138#11745659—42no—
139C8×D59—42no—
140C8×D59—42no—
141C59⋊C16—22no—
142C59⋊C16—22no—
143C23×C59—41no—
144C23×C59AGL(1,ℤ/8)11yescharacteristic
145C22×D59—41no—
146Q8×C59—41no—
147Q8×C59—41no—
148D4×C59—41no—
149D4×C59—41no—
150D4×C59—82no—
151C2×C4×C59—21no—
152C2×C4×C59—21no—
153C2×C4×C59—21no—
154C2×C4×C59—21no—
155C2×C4×C59—41no—
156C2×C4×C59—41no—
157C2×C4×C59—41no—
158C2×C4×C59—81no—
159C2×C4×C59—82no—
160C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
161C2×C4×C59C22×D411yescharacteristic
162C22⋊D59—42no—
163C22⋊D59—42no—
164C2×Dic59—41no—
165D236—22no—
166D236—41no—
167C4×D59—42no—
168C4×D59—42no—
169Dic118—22no—
170Dic118—41no—
171C8×C59—21no—
172C8×C59—21no—
173C8×C59AGL(1,ℤ/8)12yes—
174C8×C59C2×#4112yes—
175C59⋊C8—42no—
176#33690—591nomaximal, Sylow
177C22×C59—21no—
178C22×C59—41no—
179C22×C59—81no—
180C22×C59—81no—
181C22×C59—81no—
182C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
183C2×D59—42no—
184C2×D59—161no—
185C4×C59—21no—
186C4×C59—41no—
187C4×C59—41no—
188C4×C59—81no—
189C4×C59—81no—
190C4×C59—81no—
191C4×C59#52511yescharacteristic
192C4×C59C2×AGL(1,ℤ/8)11yescharacteristic
193Dic59—42no—
194#3183D5911yescharacteristic, complemented
195#2956—591no—
196#2239—591no—
197#2238—591no—
198#2231—592no—
199#2212—592no—
200#2191—591no—
201#2190—591no—
202AGL(1,ℤ/16)—592no—
203#2180—592no—
204#2165—591no—
205C2×C59—21no—
206C2×C59—41no—
207C2×C59—161no—
208C2×C59#310911yescharacteristic
209D59—82no—
210#575—591no—
211#568—1181no—
212#529—21no—
213#510—592no—
214#509—592no—
215#499C2×D5911yescharacteristic, complemented
216#495C2×D5911yescharacteristic, complemented
217#488—21no—
218#470C2×D5911yescharacteristic
219#469C2×D5911yescharacteristic
220#468—21no—
221#444—592no—
222#444—1181no—
223#443—592no—
224#443—1181no—
225#442C2×D5911yescharacteristic, complemented
226#441C2×D5911yescharacteristic, complemented
227#440—21no—
228#430—591no—
229C2×#104—1181no—
230C4×M16C2×D5911yescharacteristic
231#365—592no—
232#365—592no—
233#364—592no—
234#362—592no—
235#361—592no—
236#360—592no—
237#360—592no—
238#349—591no—
239#349—591no—
240#347—592no—
241C59#3369011yescharacteristic, complemented, Sylow
242C2×#41—1181no—
243#137—1182no—
244AGL(1,ℤ/8)—1182no—
245#135—592no—
246#135—1181no—
247C2×Q16C22×D5911yescharacteristic
248C2×SD16—21no—
249C2×D8C22×D5911yescharacteristic
250#131—1182no—
251#131—1182no—
252#131—2362no—
253C2×M16—1181no—
254C2×M16—2362no—
255C2×M16C22⋊D5911yescharacteristic
256C2×M16C22×D5911yescharacteristic
257#126—41no—
258#125—21no—
259#124—21no—
260#123—41no—
261C22⋊Q8—21no—
262#121—21no—
263C4×Q8—21no—
264C4×D4—21no—
265C4×D4—41no—
266#117—41no—
267C2×C42C22⋊D5911yescharacteristic
268Q32—1182no—
269SD32—1182no—
270SD32—1182no—
271D16—1182no—
272M32—592no—
273M32—592no—
274#108—592no—
275#108—1181no—
276#107—21no—
277#106C22×D5911yescharacteristic
278#106C22×D5911yescharacteristic
279#104—1182no—
280#104—1182no—
281#103—21no—
282#103—21no—
283#102—21no—
284#102—21no—
285#97C22⋊D5911yescharacteristic
286#97C22×D5911yescharacteristic
287C4×C8C22⋊D5911yescharacteristic
288C4×C8C22×D5911yescharacteristic
289#41—1182no—
290#41—1182no—
291#41—2361no—
292#41—2361no—
293C2×Q8—21no—
294C2×Q8—1181no—
295C2×D4—21no—
296C2×D4—41no—
297C2×D4—1181no—
298C2×D4—2361no—
299C22×C4—21no—
300C22×C4—41no—
301C22×C4—2361no—
302C22×C4C2×C22⋊D5911yescharacteristic
303Q16—22no—
304Q16—592no—
305Q16—1181no—
306SD16—42no—
307SD16—1182no—
308SD16—1182no—
309D8—22no—
310D8—592no—
311D8—1181no—
312M16—22no—
313M16—22no—
314M16—1182no—
315M16—2362no—
316M16—2361no—
317M16—2362no—
318C2×C8—2362no—
319C2×C8—2361no—
320C2×C8—2362no—
321C2×C8C2×C22⋊D5911yescharacteristic
322C2×C8C2×C22⋊D5911yescharacteristic
323C2×C8C2×C22⋊D5911yescharacteristic
324C2×C8C23×D5911yescharacteristic
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C22⋊C4—41no—
332C22⋊C4—41no—
333C22⋊C4—41no—
334C22⋊C4—41no—
335C22⋊C4—41no—
336C42—21no—
337C42—41no—
338C42C2×C22⋊D5911yescharacteristic
339C42C2×C22⋊D5911yescharacteristic
340C16—1182no—
341C16—1182no—
342C23—41no—
343C23—2361no—
344C23#1427375311yescharacteristic
345Q8—41no—
346Q8—41no—
347Q8—1182no—
348Q8—2361no—
349D4—41no—
350D4—41no—
351D4—82no—
352D4—1182no—
353D4—2361no—
354D4—2362no—
355D4—2362no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—81no—
364C2×C4—82no—
365C2×C4—2362no—
366C2×C4—2361no—
367C2×C4—2362no—
368C2×C4#1427375311yescharacteristic
369C2×C4C22×C22⋊D5911yescharacteristic
370C8—21no—
371C8—21no—
372C8—2362no—
373C8#1427375212yes—
374C8#41×D5912yes—
375C22—21no—
376C22—41no—
377C22—81no—
378C22—81no—
379C22—81no—
380C22—9441no—
381C22—2362no—
382C22C2×#1427375311yescharacteristic
383C4—21no—
384C4—41no—
385C4—41no—
386C4—81no—
387C4—81no—
388C4—81no—
389C4—2362no—
390C4#1683790911yescharacteristic
391C4C2×#1427375211yescharacteristic
392C2—21no—
393C2—41no—
394C2—161no—
395C2—4722no—
396C2#2143418311yescharacteristic
subgroupG/H
centerC2#21434183
commutator, cosocleC2×C8×C59C24
FrattiniC2×C8C23×D59
FittingC59×#3183C2
socleC2×C59#3109
prime radical#21426041C2
hypercenter#3183D59
hypocenterC59#33690
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×M16 ◁ #3183

Complements

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

NH ≅ G/Ntype
1. #21433007C2semidirect
2. #21426313C2semidirect
3. #21426308C2semidirect
4. #21426269C2semidirect
5. #21426162C2semidirect
6. #21426046C2semidirect
8. #21425992C2semidirect
9. #21425985C2semidirect
10. #21425908C2semidirect
11. C59×#3183C2semidirect
22. #16837224C22semidirect
23. #16837224C22semidirect
24. #16837223C22semidirect
25. #16837218C22semidirect
26. #16837215C22semidirect
31. #16837180C22semidirect
33. C59×#499C22semidirect
34. C59×#495C22semidirect
39. C59×#442C22semidirect
40. C59×#441C22semidirect
194. #3183D59semidirect
215. #499C2×D59semidirect
216. #495C2×D59semidirect
225. #442C2×D59semidirect
226. #441C2×D59semidirect
241. C59#33690semidirect

Maximal subgroups

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

subgroupindex
#214330072
#214263132
#214263082
#214262692
#214261622
#214260462
#214260412
#214259922
#214259852
#214259082
C59×#31832
#3369059