Finite Group Databasebeta

← #28599290

Subgroups — #28599290

Order 15104.

762 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837463C211yesmaximal, characteristic
2#21426705C211yesmaximal, characteristic
3#21424862C212yesmaximal, complemented
4#21424852C212yesmaximal, complemented
5#21424295C212yesmaximal
6#21424284C212yesmaximal
7#21424265C211yesmaximal, characteristic, complemented
8#21424255C211yesmaximal, characteristic, complemented
9#21424231C211yesmaximal, characteristic, complemented
10#21424226C211yesmaximal, characteristic, complemented
11C59×#2317C211yesmaximal, characteristic
12#16837463C418yescomplemented
13C2×#14273660C2212yes—
14C2×#14273660C2212yes—
15C2×#14273660C2212yes—
16C2×#14273658C2212yes—
17C2×C4⋊(C59⋊C8)C2212yes—
18C2×C22⋊(C59⋊C8)C2212yes—
19C2×#14273638C2211yescharacteristic
20C2×C4×C59⋊C8C2211yescharacteristic
21#16837295—22no—
22#16837288—22no—
23#16837272C2211yescharacteristic
24#16837268C2211yescharacteristic
25#16837262C2212yes—
26C4×C4⋊Dic59C2212yes—
27#16837164C2212yes—
28#16837164C2212yes—
29#16837164C2212yes—
30#16837164C2212yes—
31C2×#117×C59C2211yescharacteristic
32C2×#117×C59C2211yescharacteristic
33C22×C4⋊C4×C59C2211yescharacteristic
34C59×#389—22no—
35C59×#385—22no—
36C59×#384C2211yescharacteristic
37C59×#384C2211yescharacteristic
38C59×#382C2211yescharacteristic
39C59×#382C2211yescharacteristic
40C22×C59⋊C8C2312yes—
41C22×C59⋊C8C2312yes—
42C2×C4⋊Dic59—22no—
43C2×C4⋊Dic59C2312yes—
44C2×C4⋊Dic59C2312yes—
45C2×C22⋊Dic59—22no—
46C2×C4×Dic59—22no—
47#14273660—42no—
48#14273660—42no—
49#14273660—42no—
50#14273658—42no—
51C4⋊(C59⋊C8)C2×C418yescomplemented
52C22⋊(C59⋊C8)C2×C418yes—
53#14273638C2×C414yescomplemented
54C4×C59⋊C8C2×C414yescomplemented
55C59⋊#95—22no—
56C59⋊#95—22no—
57C59⋊#95—22no—
58C23×C4×C59—21no—
59C23×C4×C59C2311yescharacteristic
60#117×C59—42no—
61#117×C59C2×C414yes—
62#117×C59Q814yes—
63C2×C4⋊C4×C59—21no—
64C2×C4⋊C4×C59—22no—
65C2×C4⋊C4×C59—22no—
66C2×C4⋊C4×C59—42no—
67C2×C4⋊C4×C59D412yes—
68C2×C4⋊C4×C59C2311yescharacteristic
69C2×C4⋊C4×C59C2311yescharacteristic
70C2×C4⋊C4×C59C2311yescharacteristic
71C2×C4⋊C4×C59C2311yescharacteristic
72C2×C22⋊C4×C59—21no—
73C2×C22⋊C4×C59D412yes—
74C2×C42×C59—21no—
75C2×C42×C59C2311yescharacteristic
76C2×C42×C59C2311yescharacteristic
77#95×C59—22no—
78#95×C59—22no—
79C24×C59C2×D411yescharacteristic
80C22×C4×C59—22no—
81C22×C4×C59—22no—
82C22×C4×C59—21no—
83C22×C4×C59—21no—
84C22×C4×C59—21no—
85C22×C4×C59—21no—
86C22×C4×C59—42no—
87C22×C4×C59—82no—
88C22×C4×C59C22×C412yes—
89C22×C4×C59C2×D412yes—
90C22×C4×C59C2×D412yes—
91C22×C4×C59C2×D411yescharacteristic
92C22×C4×C59C2×Q812yes—
93C22×C4×C59C2411yescharacteristic
94C22×Dic59—22no—
95C22×Dic59—22no—
96C4⋊C4×C59—22no—
97C4⋊C4×C59—22no—
98C4⋊C4×C59—22no—
99C4⋊C4×C59—22no—
100C4⋊C4×C59—42no—
101C4⋊C4×C59—82no—
102C4⋊C4×C59—82no—
103C4⋊C4×C59C4⋊C418yes—
104C22⋊C4×C59—42no—
105C22⋊C4×C59C4⋊C418yes—
106C42×C59—42no—
107C42×C59C22×C412yes—
108C42×C59C22×C412yes—
109C42×C59C2×Q812yes—
110C42×C59C2×Q812yes—
111C22⋊Dic59—82no—
112C4⋊Dic59—22no—
113C4⋊Dic59—22no—
114C4⋊Dic59—24no—
115C4⋊Dic59—82no—
116C4×Dic59—82no—
117C2×C59⋊C8—22no—
118C2×C59⋊C8—22no—
119C2×C59⋊C8—22no—
120C2×C59⋊C8—22no—
121C2×C59⋊C8C22×C414yes—
122C2×C59⋊C8C22×C414yes—
123C23×C59—21no—
124C23×C59—21no—
125C23×C59—22no—
126C23×C59—22no—
127C23×C59C2×C4⋊C412yes—
128C23×C59C22×D411yescharacteristic
129C2×C4×C59—21no—
130C2×C4×C59—21no—
131C2×C4×C59—22no—
132C2×C4×C59—22no—
133C2×C4×C59—22no—
134C2×C4×C59—22no—
135C2×C4×C59—24no—
136C2×C4×C59—24no—
137C2×C4×C59—41no—
138C2×C4×C59—41no—
139C2×C4×C59—41no—
140C2×C4×C59—41no—
141C2×C4×C59—42no—
142C2×C4×C59—42no—
143C2×C4×C59—42no—
144C2×C4×C59—42no—
145C2×C4×C59—84no—
146C2×C4×C59C2×C4⋊C414yes—
147C2×C4×C59C2×C4⋊C414yes—
148C2×C4×C59C2×C4⋊C412yes—
149C2×C4×C59C23×C411yescharacteristic
150C2×C4×C59C22×Q811yescharacteristic
151C2×C4×C59H(5,𝔽2)12yes—
152C2×C4×C592−1+412yes—
153C2×Dic59—42no—
154C2×Dic59—42no—
155C2×Dic59—42no—
156C2×Dic59—42no—
157C2×Dic59—42no—
158C2×Dic59—42no—
159C59⋊C8—24no—
160C59⋊C8—24no—
161#21991—591nomaximal, Sylow
162C22×C59—22no—
163C22×C59—24no—
164C22×C59—41no—
165C22×C59—41no—
166C22×C59—42no—
167C22×C59C22×C4⋊C411yescharacteristic
168C22×C59#53511yescharacteristic
169C22×C59#53611yescharacteristic
170C22×C59#57612yes—
171C22×C59#57812yes—
172C4×C59—22no—
173C4×C59—24no—
174C4×C59—24no—
175C4×C59—81no—
176C4×C59—81no—
177C4×C59—81no—
178C4×C59—81no—
179C4×C59#52012yes—
180C4×C59#54412yes—
181Dic59—82no—
182Dic59—82no—
183C2×#433—591no—
184#2317D5911yescharacteristic
185#2314—591no—
186#1949—592no—
187#1947—592no—
188#1854—592no—
189#1852—592no—
190#1847—591no—
191#1845—591no—
192#1836—591no—
193#1835—591no—
194C2×C59—42no—
195C2×C59#231711yescharacteristic
196C2×C59#291312yes—
197C2×C59#322011yescharacteristic
198C2×C59#322311yescharacteristic
199C2×C59#324312yes—
200C2×#117C2×D5911yescharacteristic
201C2×#117C2×D5911yescharacteristic
202C22×C4⋊C4C2×D5911yescharacteristic
203#433—598no—
204C2×#107—592no—
205C2×#107—592no—
206C2×#107—592no—
207C2×#106—592no—
208C2×C4⋊C8—592no—
209C2×C22⋊C8—592no—
210C2×#97—591no—
211C2×C4×C8—591no—
212#389—22no—
213#388—1182no—
214#387—1182no—
215#385—22no—
216#384—591no—
217#384C2×D5911yescharacteristic
218#384C2×D5911yescharacteristic
219#382—591no—
220#382C2×D5911yescharacteristic
221#382C2×D5911yescharacteristic
222#380—592no—
223C4×C4⋊C4—592no—
224#340—592no—
225#340—592no—
226#340—592no—
227#340—592no—
228C59#2199111yescharacteristic, complemented, Sylow
229C23×C4—21no—
230C23×C4C22×D5911yescharacteristic
231C22×C8—592no—
232C22×C8—592no—
233#117—42no—
234#117Dic11814yes—
235#117C4×D5914yes—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—592no—
241C2×C4⋊C4—592no—
242C2×C4⋊C4—1182no—
243C2×C4⋊C4C22⋊D5912yes—
244C2×C4⋊C4C22×D5911yescharacteristic
245C2×C4⋊C4C22×D5911yescharacteristic
246C2×C4⋊C4C22×D5911yescharacteristic
247C2×C4⋊C4C22×D5911yescharacteristic
248C2×C22⋊C4—21no—
249C2×C22⋊C4—1182no—
250C2×C22⋊C4C22⋊D5912yes—
251C2×C42—21no—
252C2×C42—1182no—
253C2×C42C22×D5911yescharacteristic
254C2×C42C22×D5911yescharacteristic
255#107—2362no—
256#107—2362no—
257#107—2362no—
258#106—2362no—
259C4⋊C8—598no—
260C22⋊C8—598no—
261#97—594no—
262C4×C8—594no—
263#95—22no—
264#95—22no—
265#95—1182no—
266#95—1182no—
267#95—1182no—
268C24C2×C22⋊D5911yescharacteristic
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—42no—
276C22×C4—82no—
277C22×C4—1182no—
278C22×C4—1182no—
279C22×C4C2×Dic11812yes—
280C22×C4C2×C4×D5912yes—
281C22×C4C2×C22⋊D5912yes—
282C22×C4C2×C22⋊D5912yes—
283C22×C4C2×C22⋊D5911yescharacteristic
284C22×C4C23×D5911yescharacteristic
285C2×C8—594no—
286C2×C8—594no—
287C2×C8—1182no—
288C2×C8—1182no—
289C2×C8—1182no—
290C2×C8—1182no—
291C4⋊C4—22no—
292C4⋊C4—22no—
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—42no—
296C4⋊C4—82no—
297C4⋊C4—82no—
298C4⋊C4—1182no—
299C4⋊C4—1182no—
300C4⋊C4—1184no—
301C4⋊C4—4722no—
302C4⋊C4Dic59⋊C418yes—
303C22⋊C4—42no—
304C22⋊C4—4722no—
305C22⋊C4Dic59⋊C418yes—
306C42—42no—
307C42—4722no—
308C42C2×Dic11812yes—
309C42C2×Dic11812yes—
310C42C2×C4×D5912yes—
311C42C2×C4×D5912yes—
312C23—22no—
313C23—22no—
314C23—21no—
315C23—21no—
316C23C2×Dic59⋊C412yes—
317C23C22×C22⋊D5911yescharacteristic
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—24no—
325C2×C4—24no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—84no—
335C2×C4—2362no—
336C2×C4—2362no—
337C2×C4—2362no—
338C2×C4—2362no—
339C2×C4—2362no—
340C2×C4—2362no—
341C2×C4C2×Dic59⋊C414yes—
342C2×C4C2×Dic59⋊C414yes—
343C2×C4C2×Dic59⋊C412yes—
344C2×C4C22×C4×D5911yescharacteristic
345C2×C4C22×Dic11811yescharacteristic
346C2×C4#1427377212yes—
347C2×C4#1427377512yes—
348C8—1184no—
349C8—1184no—
350C22—22no—
351C22—24no—
352C22—41no—
353C22—41no—
354C22—42no—
355C22C22×Dic59⋊C411yescharacteristic
356C22#1683796911yescharacteristic
357C22#1683797611yescharacteristic
358C22#1683820512yes—
359C22#1683821712yes—
360C4—22no—
361C4—24no—
362C4—24no—
363C4—81no—
364C4—81no—
365C4—81no—
366C4—81no—
367C4—4722no—
368C4—4722no—
369C4#1683788312yes—
370C4#1683803312yes—
371C2—42no—
372C2#2142672011yescharacteristic
373C2#2143269412yes—
374C2#2143514611yescharacteristic
375C2#2143517911yescharacteristic
376C2#2143537912yes—
subgroupG/H
centerC23C22×C22⋊D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#2317C2
socleC23×C59C22×D4
prime radicalC24×C59C2×D4
cosocleC22×C4×C59C24
hypercenter#2317D59
hypocenterC59#21991
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2317

Complements

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

NH ≅ G/Ntype
3. #21424862C2semidirect
4. #21424852C2semidirect
7. #21424265C2semidirect
8. #21424255C2semidirect
9. #21424231C2semidirect
10. #21424226C2semidirect
12. #16837463C4semidirect
51. C4⋊(C59⋊C8)C2×C4semidirect
53. #14273638C2×C4semidirect
54. C4×C59⋊C8C2×C4semidirect
228. C59#21991semidirect

Maximal subgroups

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

subgroupindex
C2×#168374632
#214267052
#214248622
#214248522
#214242952
#214242842
#214242652
#214242552
#214242312
#214242262
C59×#23172
#2199159