Finite Group Databasebeta

← #28606394

Subgroups — #28606394

Order 15104.

810 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#107×D59C211yesmaximal, characteristic
2#21426972C212yesmaximal
3#21424835C211yesmaximal, characteristic, complemented
4#21424670C212yesmaximal, complemented
5#21424664C212yesmaximal, complemented
6#21424185C212yesmaximal
7#21424118C211yesmaximal, characteristic
8#21423996C211yesmaximal, characteristic, complemented
9#21423993C211yesmaximal, characteristic, complemented
10#21423873C211yesmaximal, characteristic
11C59×#1943C211yesmaximal, characteristic, complemented
12C22×C8×D59C2211yescharacteristic
13C2×C4⋊C4×D59C2212yes—
14C2×#14273661C2211yescharacteristic
15C2×#14273660C2212yes—
16#107×D59C418yescomplemented
17C2×#14273659C2211yescharacteristic
18C2×(C2×D59)⋊C8C2211yescharacteristic
19C2×#14273637C2211yescharacteristic
20#16837301C2212yes—
21#16837289—22no—
22#16837283—22no—
23#16837259—22no—
24#16837259C2212yes—
25C4⋊C4×Dic59C2212yes—
26C4×C4⋊Dic59C2212yes—
27C4×(C2×D59)⋊C4C2211yescharacteristic
28#16837249—22no—
29#16837165C2212yes—
30#16837164C2212yes—
31#16837164C2212yes—
32#16837164C2212yes—
33#16837157C2211yescharacteristic
34#16837156C2211yescharacteristic
35C2×#107×C59C2211yescharacteristic
36C2×#106×C59C2211yescharacteristic
37C2×#97×C59C2211yescharacteristic
38C59×#389C2212yes—
39C59×#340C2212yes—
40C22×C4×D59—22no—
41C22×C4×D59C2311yescharacteristic
42C22×C59⋊C8C2311yescharacteristic
43C22×C59⋊C8C2311yescharacteristic
44C2×C8×D59—22no—
45C2×C8×D59—22no—
46C2×C8×D59C2×C414yes—
47C2×C4⋊Dic59—22no—
48C2×C4⋊Dic59C2312yes—
49C2×C4⋊Dic59C2312yes—
50C2×Dic59⋊C4—22no—
51C4⋊C4×D59—24no—
52C4⋊C4×D59C2×C418yes—
53C2×(C2×D59)⋊C4—22no—
54C2×(C2×D59)⋊C4D412yes—
55C2×C4×Dic59—22no—
56C2×C4×Dic59—22no—
57C2×C4×Dic59C2311yescharacteristic
58#14273661C2×C414yes—
59#14273660C2×C418yes—
60#14273659—22no—
61(C2×D59)⋊C8—22no—
62#14273637—22no—
63C59⋊#95—22no—
64C59⋊#95—22no—
65C59⋊#95—22no—
66C59⋊#95—22no—
67C59⋊#95D412yes—
68C22×C8×C59C2311yescharacteristic
69C22×C8×C59C2311yescharacteristic
70C2×C4⋊C4×C59—22no—
71C2×C4⋊C4×C59C2312yes—
72C2×C4⋊C4×C59C2312yes—
73C2×C42×C59C2311yescharacteristic
74#107×C59C2×C414yes—
75#106×C59—22no—
76#97×C59—22no—
77#95×C59—22no—
78C23×D59C2×D411yescharacteristic
79C22×C4×C59—22no—
80C22×C4×C59—22no—
81C22×C4×C59C2×D412yes—
82C22×C4×C59C2411yescharacteristic
83C22×Dic59—22no—
84C22×Dic59—22no—
85C22×Dic59C2×D412yes—
86C22×Dic59C2×D411yescharacteristic
87C2×C4×D59—21no—
88C2×C4×D59—21no—
89C2×C4×D59—21no—
90C2×C4×D59—21no—
91C2×C4×D59—28no—
92C2×C4×D59—82no—
93C2×C4×D59—82no—
94C2×C4×D59C22×C412yes—
95C2×C4×D59#4112yes—
96C2×C8×C59—21no—
97C2×C8×C59—21no—
98C2×C8×C59—21no—
99C2×C8×C59C22×C412yes—
100C2×C8×C59#4112yes—
101C2×C8×C59#4112yes—
102C4⋊C4×C59—22no—
103C4⋊C4×C59—22no—
104C4⋊C4×C59—22no—
105C4⋊C4×C59—82no—
106C4⋊C4×C59C22×C414yes—
107C42×C59—21no—
108C42×C59—21no—
109(C2×D59)⋊C4—24no—
110(C2×D59)⋊C4—44no—
111C4⋊Dic59—22no—
112C4⋊Dic59—22no—
113C4⋊Dic59—22no—
114C4⋊Dic59—82no—
115C4⋊Dic59C22×C414yes—
116Dic59⋊C4—28no—
117C4×Dic59—21no—
118C4×Dic59—21no—
119C4×Dic59—82no—
120C4×Dic59—82no—
121C2×C59⋊C8—21no—
122C2×C59⋊C8—21no—
123C2×C59⋊C8—21no—
124C2×C59⋊C8—21no—
125C2×C59⋊C8—21no—
126C2×C59⋊C8C22×C412yes—
127C8×D59—28no—
128C23×C59C22×D411yescharacteristic
129C22×D59—21no—
130C22×D59—21no—
131C22×D59—22no—
132C22×D59C4×D412yes—
133C22×D59#13714yes—
134C2×C4×C59—24no—
135C2×C4×C59—22no—
136C2×C4×C59—22no—
137C2×C4×C59—22no—
138C2×C4×C59—42no—
139C2×C4×C59—42no—
140C2×C4×C59—42no—
141C2×C4×C59—42no—
142C2×C4×C59—42no—
143C2×C4×C59C23×C411yescharacteristic
144C2×C4×C59C2×#4111yescharacteristic
145C2×C4×C59C2×#4111yescharacteristic
146C2×C4×C59C2×#4111yescharacteristic
147C2×C4×C592−1+411yescharacteristic
148C2×C4×C592−1+411yescharacteristic
149C2×Dic59—21no—
150C2×Dic59—21no—
151C2×Dic59—24no—
152C2×Dic59—22no—
153C2×Dic59—22no—
154C2×Dic59—22no—
155C2×Dic59—42no—
156C2×Dic59—42no—
157C2×Dic59—42no—
158C2×Dic59—42no—
159C2×Dic59—42no—
160C2×Dic59C4×D412yes—
161C4×D59—22no—
162C4×D59—22no—
163C4×D59—22no—
164C4×D59—22no—
165C4×D59—88no—
166C8×C59—41no—
167C8×C59#11714yes—
168C59⋊C8—22no—
169C59⋊C8—41no—
170#22618—591nomaximal, Sylow
171C22×C59C2×C4×D411yescharacteristic
172C22×C59#54711yescharacteristic
173C22×C59#54711yescharacteristic
174C22×C59C2×#13712yes—
175C22×C59#57612yes—
176C2×D59—24no—
177C2×D59—24no—
178C2×D59—22no—
179C2×D59—22no—
180C2×D59#44114yes—
181C4×C59—42no—
182C4×C59—44no—
183C4×C59—82no—
184C4×C59C2×#11711yescharacteristic
185C4×C59#51911yescharacteristic
186C4×C59#55211yescharacteristic
187C4×C59#55211yescharacteristic
188Dic59—22no—
189Dic59—42no—
190Dic59—44no—
191Dic59—82no—
192C22×#107—591no—
193#2345—592no—
194#1943—591no—
195#1943D5911yescharacteristic, complemented
196#1918—592no—
197#1918—592no—
198#1824—592no—
199#1807—591no—
200#1776—591no—
201#1776—591no—
202#1752—591no—
203C2×C59#234411yescharacteristic
204C2×C59C2×#44111yescharacteristic
205C2×C59#294811yescharacteristic
206C2×C59#334612yes—
207C2×C59#334612yes—
208D59—24no—
209C23×C8—591no—
210C22×C4⋊C4—592no—
211C2×#107—591no—
212C2×#107—592no—
213C2×#107—598no—
214C2×#107C2×D5911yescharacteristic
215C2×#106—591no—
216C2×#106C2×D5911yescharacteristic
217C2×C22⋊C8—591no—
218C2×#97—591no—
219C2×#97C2×D5911yescharacteristic
220#389—592no—
221#389C2×D5912yes—
222#387—1182no—
223#387—1182no—
224#380—592no—
225#380—1182no—
226C4×C4⋊C4—592no—
227C4×C4⋊C4—592no—
228C4×C22⋊C4—591no—
229C4×C22⋊C4—1182no—
230#340—592no—
231#340—592no—
232#340—592no—
233#340—592no—
234#340C2×D5912yes—
235#336—591no—
236#336—591no—
237C59#2261811yescharacteristic, complemented, Sylow
238C23×C4—591no—
239C23×C4—1182no—
240C22×C8—591no—
241C22×C8—594no—
242C22×C8—591no—
243C22×C8—1182no—
244C22×C8—1182no—
245C22×C8C22×D5911yescharacteristic
246C22×C8C22×D5911yescharacteristic
247C2×C4⋊C4—22no—
248C2×C4⋊C4—592no—
249C2×C4⋊C4—598no—
250C2×C4⋊C4—592no—
251C2×C4⋊C4—1184no—
252C2×C4⋊C4—1182no—
253C2×C4⋊C4—1182no—
254C2×C4⋊C4C22×D5912yes—
255C2×C4⋊C4C22×D5912yes—
256C2×C22⋊C4—592no—
257C2×C22⋊C4—1182no—
258C2×C42—591no—
259C2×C42—1182no—
260C2×C42—1182no—
261C2×C42C22×D5911yescharacteristic
262#107—594no—
263#107—598no—
264#107C4×D5914yes—
265#106—22no—
266#106—1182no—
267C22⋊C8—1182no—
268#97—22no—
269#97—1182no—
270#95—22no—
271#95—592no—
272#95—1182no—
273#95—1182no—
274#95—1182no—
275#95—1182no—
276C24—591no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—591no—
280C22×C4—592no—
281C22×C4—592no—
282C22×C4—592no—
283C22×C4—1188no—
284C22×C4—1181no—
285C22×C4—1181no—
286C22×C4—1181no—
287C22×C4—1181no—
288C22×C4—1182no—
289C22×C4—1182no—
290C22×C4—4722no—
291C22×C4—4722no—
292C22×C4D4×D5912yes—
293C22×C4C23×D5911yescharacteristic
294C2×C8—21no—
295C2×C8—21no—
296C2×C8—21no—
297C2×C8—592no—
298C2×C8—1181no—
299C2×C8—1181no—
300C2×C8—1188no—
301C2×C8—1181no—
302C2×C8—1181no—
303C2×C8—1181no—
304C2×C8C2×C4×D5912yes—
305C2×C8#1174568712yes—
306C2×C8#1174568912yes—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—22no—
310C4⋊C4—82no—
311C4⋊C4—594no—
312C4⋊C4—1188no—
313C4⋊C4—1182no—
314C4⋊C4—1182no—
315C4⋊C4—1182no—
316C4⋊C4—4722no—
317C4⋊C4C2×C4×D5914yes—
318C22⋊C4—1184no—
319C22⋊C4—2364no—
320C42—21no—
321C42—21no—
322C42—1181no—
323C42—1181no—
324C42—4722no—
325C42—4722no—
326C23—594no—
327C23—592no—
328C23—1182no—
329C23—1181no—
330C23—1181no—
331C23C2×D4×D5911yescharacteristic
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—24no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—592no—
342C2×C4—1181no—
343C2×C4—1181no—
344C2×C4—1182no—
345C2×C4—1182no—
346C2×C4—1182no—
347C2×C4—1182no—
348C2×C4—1182no—
349C2×C4—1182no—
350C2×C4—1182no—
351C2×C4—1184no—
352C2×C4—4728no—
353C2×C4—2362no—
354C2×C4—2362no—
355C2×C4—2362no—
356C2×C4—2362no—
357C2×C4—2362no—
358C2×C4C22×C4×D5911yescharacteristic
359C2×C4#41×D5911yescharacteristic
360C2×C4C2×#1174568711yescharacteristic
361C2×C4C2×#1174568911yescharacteristic
362C2×C4#1427377411yescharacteristic
363C2×C4#1427377511yescharacteristic
364C8—41no—
365C8—1182no—
366C8—2361no—
367C8#1427368514yes—
368C22—594no—
369C22—1182no—
370C22—1182no—
371C22—1184no—
372C22—1184no—
373C22C4×D4×D5911yescharacteristic
374C22#1683805211yescharacteristic
375C22#1683805511yescharacteristic
376C22#137×D5912yes—
377C22#1683820612yes—
378C4—44no—
379C4—42no—
380C4—82no—
381C4—1182no—
382C4—4722no—
383C4—2364no—
384C4—2362no—
385C4C2×#1427368511yescharacteristic
386C4#1683787811yescharacteristic
387C4#1683808611yescharacteristic
388C4#1683809011yescharacteristic
389C2—1184no—
390C2#2142696411yescharacteristic
391C2#441×D5911yescharacteristic
392C2#2143294511yescharacteristic
393C2#2143657612yes—
394C2#2143658012yes—
subgroupG/H
centerC23C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#1943C2
socleC23×C59C22×D4
prime radicalC23×D59C2×D4
cosocleC22×C4×C59C24
hypercenter#1943D59
hypocenterC59#22618
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1943

Complements

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

NH ≅ G/Ntype
3. #21424835C2semidirect
4. #21424670C2semidirect
5. #21424664C2semidirect
8. #21423996C2semidirect
9. #21423993C2semidirect
11. C59×#1943C2semidirect
16. #107×D59C4semidirect
195. #1943D59semidirect
237. C59#22618semidirect

Maximal subgroups

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

subgroupindex
C2×#107×D592
#214269722
#214248352
#214246702
#214246642
#214241852
#214241182
#214239962
#214239932
#214238732
C59×#19432
#2261859