Finite Group Databasebeta

← #28592363

Subgroups — #28592363

Order 15104.

1638 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837950C211yesmaximal, characteristic
2C2×#16837400C214yesmaximal, complemented
3C2×#16837354C211yesmaximal, characteristic, complemented
4#382×D59C211yesmaximal, characteristic, complemented
5#21424921C211yesmaximal, characteristic, complemented
6#21424916C211yesmaximal, characteristic, complemented
7#21424915C211yesmaximal, characteristic, complemented
8#21424906C214yesmaximal, complemented
9C59×#1961C211yesmaximal, characteristic, complemented
10C22×#11745687C2212yes—
11C22×#11745659C2212yes—
12#16837950—24no—
13#16837950C418yescomplemented
14C2×#14273727C2211yescharacteristic
15C2×#14273714C2212yes—
16C2×#14273698—22no—
17C2×D4×Dic59—22no—
18C2×C4⋊C4×D59C2212yes—
19C2×C42×D59C2211yescharacteristic
20#16837400—216no—
21C2×#14273652C2214yes—
22C2×#14273650C2214yes—
23C2×#14273648C2214yes—
24#16837354—24no—
25C2×#14273638C2211yescharacteristic
26C2×C8×Dic59C2212yes—
27#16837268C2211yescharacteristic
28#16837267C2212yes—
29#16837266—22no—
30#95×D59—22no—
31C2×#127×C59C2211yescharacteristic
32C2×#102×C59C2214yes—
33C2×#97×C59C2211yescharacteristic
34C59×#382C2211yescharacteristic
35C2×#11745687—28no—
36C2×#11745687—24no—
37C2×#11745687—28no—
38C2×#11745687C2×C418yes—
39C22×Dic118C2312yes—
40C22×C22⋊D59—22no—
41C23×Dic59—22no—
42C22×C4×D59—22no—
43C22×C4×D59D412yes—
44C22×C4×D59C2311yescharacteristic
45C2×#11745659—24no—
46C2×#11745659—24no—
47C2×#11745659D418yescomplemented
48C22×C59⋊C8C2312yes—
49#14273727—22no—
50#14273727C2×C414yescomplemented
51#14273714—24no—
52#14273714C2×C418yescomplemented
53#14273698—216no—
54D4×Dic59—216no—
55C2×C4⋊Dic59—22no—
56C2×C4⋊Dic59C2312yes—
57C2×Dic59⋊C4—22no—
58C4⋊C4×D59—28no—
59C2×C22⋊Dic59—24no—
60C2×C4×Dic59D412yes—
61C2×C4×Dic59C2311yescharacteristic
62C42×D59C2×C414yes—
63C42×D59D414yes—
64#14273652—28no—
65#14273650—28no—
66#14273648—28no—
67#14273638—22no—
68C8×Dic59—24no—
69C59⋊#95—22no—
70C59⋊#95—24no—
71C22×D4×C59—22no—
72C22×D4×C59C2312yes—
73C22×C8×C59C2312yes—
74#127×C59—22no—
75#127×C59C2×C414yescomplemented
76C2×C4⋊C4×C59C2312yes—
77C2×C42×C59C2311yescharacteristic
78#102×C59—28no—
79#97×C59—22no—
80#95×C59—22no—
81C24×C59—22no—
82C23×D59C2×D411yescharacteristic
83C2×D4×C59—28no—
84C2×D4×C59—24no—
85C2×D4×C59—22no—
86C2×D4×C59—24no—
87C2×D4×C59—48no—
88C2×D4×C59C22×C414yes—
89C22×C4×C59—22no—
90C22×C4×C59C2×D412yes—
91C22×C4×C59C2411yescharacteristic
92C2×C22⋊D59—28no—
93C2×C22⋊D59—44no—
94C2×C22⋊D59—44no—
95C22×Dic59—22no—
96C22×Dic59—28no—
97C22×Dic59—22no—
98C22×Dic59—44no—
99C22×Dic59—44no—
100C22×Dic59C2×D412yes—
101C22×Dic59C2×D411yescharacteristic
102#11745687—232no—
103#11745687—416no—
104C2×C4×D59—44no—
105C2×C4×D59—48no—
106C2×C4×D59—44no—
107C2×C4×D59C22⋊C418yes—
108C2×C4×D59C22×C412yes—
109C2×C4×D59C2×D414yes—
110C2×C4×D59C2×D412yes—
111C2×C4×D59#4114yes—
112C2×Dic118—24no—
113C2×Dic118—24no—
114C2×Dic118—22no—
115C2×Dic118C22×C414yes—
116C2×C8×C59—22no—
117C2×C8×C59—22no—
118C2×C8×C59C2×D414yes—
119C4⋊C4×C59—24no—
120C42×C59C22×C412yes—
121C42×C59C2×D412yes—
122C22⋊Dic59—216no—
123C4⋊Dic59—28no—
124C4⋊Dic59—24no—
125Dic59⋊C4—44no—
126C4×Dic59C22⋊C418yes—
127C4×Dic59C22×C412yes—
128C4×Dic59C2×D412yes—
129C2×C59⋊C8—22no—
130C2×C59⋊C8—22no—
131C2×C59⋊C8C2×D414yes—
132#11745659—216no—
133C23×C59—24no—
134C23×C59—42no—
135C23×C59—42no—
136C23×C59—48no—
137C23×C59C22×D411yescharacteristic
138C22×D59—24no—
139C22×D59C2×C22⋊C412yes—
140C22×D59#12412yes—
141C22×D59#12712yes—
142D4×C59—28no—
143D4×C59—416no—
144D4×C59—44no—
145C2×C4×C59—24no—
146C2×C4×C59—42no—
147C2×C4×C59—44no—
148C2×C4×C59C2×C22⋊C414yes—
149C2×C4×C59C23×C411yescharacteristic
150C2×C4×C59C22×D412yes—
151C2×C4×C59C22×D411yescharacteristic
152C2×C4×C59C2×#4112yes—
153C22⋊D59—416no—
154C2×Dic59—24no—
155C2×Dic59—24no—
156C2×Dic59—42no—
157C2×Dic59—42no—
158C2×Dic59—416no—
159C2×Dic59—42no—
160C2×Dic59—44no—
161C2×Dic59C2×C22⋊C414yes—
162C2×Dic59C2×C22⋊C412yes—
163C2×Dic59#12412yes—
164C2×Dic59#12712yes—
165C4×D59—48no—
166C4×D59—88no—
167C4×D59C4×D418yes—
168C4×D59#12118yes—
169Dic118—28no—
170Dic118—44no—
171C8×C59—24no—
172C59⋊C8—24no—
173#21201—591nomaximal, Sylow
174C22×C59—48no—
175C22×C59—44no—
176C22×C59—84no—
177C22×C59—84no—
178C22×C59C22×C22⋊C411yescharacteristic
179C22×C59C2×#12411yescharacteristic
180C22×C59C2×#12711yescharacteristic
181C22×C59#57514yes—
182C2×D59—24no—
183C2×D59—24no—
184C2×D59—24no—
185C2×D59#39014yes—
186C4×C59—24no—
187C4×C59—82no—
188C4×C59C2×C4×D412yes—
189C4×C59C2×#12112yes—
190Dic59—24no—
191Dic59—44no—
192Dic59—82no—
193Dic59#39014yes—
194C2×#532—591no—
195C2×#417—594no—
196C2×C4×M16—591no—
197C2×#382—591no—
198#1961—591no—
199#1961D5911yescharacteristic, complemented
200#1960—591no—
201#1960—591no—
202#1958—594no—
203C2×C59—84no—
204C2×C59C2×#39011yescharacteristic
205C2×C59#289612yes—
206C2×C59#314212yes—
207C2×C59#315312yes—
208D59—24no—
209C22×#41—592no—
210C22×M16—592no—
211#532—598no—
212#532—1184no—
213C2×C4⋊Q8—591no—
214C2×#127C2×D5911yescharacteristic
215C2×#124—592no—
216C2×#121—1182no—
217C2×C4×D4—1182no—
218C22×C4⋊C4—592no—
219C22×C42—591no—
220#417—11816no—
221C2×#103—594no—
222C2×#103—594no—
223C2×#102—594no—
224C2×#102C2×D5914yes—
225C4×M16—1184no—
226C2×#97—591no—
227C2×#97C2×D5911yescharacteristic
228C2×C4×C8—592no—
229#382—592no—
230#382—591no—
231#382—1182no—
232#382C2×D5911yescharacteristic
233C2×#95—1182no—
234C59#2120111yescharacteristic, complemented, Sylow
235C2×#41—598no—
236C2×#41—1184no—
237C2×#41—1188no—
238C2×#41—1188no—
239C22×Q8—592no—
240C22×D4—22no—
241C22×D4—1182no—
242C22×D4C22×D5912yes—
243C23×C4—591no—
244C23×C4—592no—
245C23×C4—1182no—
246C23×C4—1182no—
247C2×M16—598no—
248C2×M16—1184no—
249C2×M16—1184no—
250C22×C8—592no—
251C22×C8C22×D5912yes—
252C4⋊Q8—594no—
253C4⋊Q8—1182no—
254#127—22no—
255#127C4×D5914yescomplemented
256#124—598no—
257#124—1184no—
258#121—11816no—
259C4×D4—11816no—
260C2×C4⋊C4—592no—
261C2×C4⋊C4—1188no—
262C2×C4⋊C4—1182no—
263C2×C4⋊C4—1182no—
264C2×C4⋊C4C22×D5912yes—
265C2×C22⋊C4—1184no—
266C2×C42—592no—
267C2×C42—591no—
268C2×C42—594no—
269C2×C42—594no—
270C2×C42C22×D5911yescharacteristic
271#103—1188no—
272#103—1188no—
273#102—28no—
274#102—1188no—
275#97—22no—
276#97—1182no—
277C4×C8—1184no—
278#95—22no—
279#95—1184no—
280#95—1182no—
281C24—22no—
282C24—591no—
283#41—11832no—
284#41—23616no—
285C2×Q8—594no—
286C2×Q8—1182no—
287C2×Q8—1184no—
288C2×Q8—1184no—
289C2×D4—28no—
290C2×D4—24no—
291C2×D4—22no—
292C2×D4—24no—
293C2×D4—48no—
294C2×D4—1188no—
295C2×D4—2364no—
296C2×D4—2364no—
297C2×D4C2×C4×D5914yes—
298C22×C4—22no—
299C22×C4—591no—
300C22×C4—592no—
301C22×C4—592no—
302C22×C4—594no—
303C22×C4—594no—
304C22×C4—592no—
305C22×C4—598no—
306C22×C4—1188no—
307C22×C4—1182no—
308C22×C4—1182no—
309C22×C4—2364no—
310C22×C4—2364no—
311C22×C4—2368no—
312C22×C4—2364no—
313C22×C4—2364no—
314C22×C4D4×D5912yes—
315C22×C4C23×D5911yescharacteristic
316M16—11816no—
317C2×C8—22no—
318C2×C8—22no—
319C2×C8—594no—
320C2×C8—1182no—
321C2×C8—1182no—
322C2×C8D4×D5914yes—
323C4⋊C4—24no—
324C4⋊C4—1188no—
325C4⋊C4—1184no—
326C4⋊C4—2364no—
327C22⋊C4—11816no—
328C42—598no—
329C42—592no—
330C42—592no—
331C42C2×C4×D5912yes—
332C42D4×D5912yes—
333C23—24no—
334C23—42no—
335C23—42no—
336C23—48no—
337C23—592no—
338C23—592no—
339C23—592no—
340C23—1184no—
341C23C2×D4×D5911yescharacteristic
342Q8—1188no—
343Q8—2364no—
344D4—28no—
345D4—416no—
346D4—44no—
347D4—23616no—
348C2×C4—24no—
349C2×C4—42no—
350C2×C4—44no—
351C2×C4—592no—
352C2×C4—592no—
353C2×C4—592no—
354C2×C4—598no—
355C2×C4—598no—
356C2×C4—594no—
357C2×C4—1184no—
358C2×C4—1184no—
359C2×C4—4728no—
360C2×C4—2368no—
361C2×C4—23616no—
362C2×C4—2362no—
363C2×C4—2364no—
364C2×C4—2362no—
365C2×C4—2362no—
366C2×C4C22⋊C4×D5914yes—
367C2×C4C22×C4×D5911yescharacteristic
368C2×C4C2×D4×D5911yescharacteristic
369C2×C4C2×D4×D5912yes—
370C2×C4#41×D5912yes—
371C8—24no—
372C8—1184no—
373C22—48no—
374C22—44no—
375C22—84no—
376C22—84no—
377C22—594no—
378C22—1184no—
379C22—1184no—
380C22—1184no—
381C22C2×C22⋊C4×D5911yescharacteristic
382C22#124×D5911yescharacteristic
383C22#127×D5911yescharacteristic
384C22#1683819814yes—
385C4—24no—
386C4—82no—
387C4—594no—
388C4—1184no—
389C4—4722no—
390C4—2364no—
391C4C4×D4×D5912yes—
392C4#121×D5912yes—
393C2—84no—
394C2—1184no—
395C2#390×D5911yescharacteristic
396C2#2143261012yes—
397C2#2143449512yes—
398C2#2143456112yes—
subgroupG/H
centerC23C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#1961C2
socleC23×C59C22×D4
prime radicalC2×#16837950C2
cosocleC22×C4×C59C24
hypercenter#1961D59
hypocenterC59#21201
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1961

Complements

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

NH ≅ G/Ntype
2. C2×#16837400C2semidirect
3. C2×#16837354C2semidirect
4. #382×D59C2semidirect
5. #21424921C2semidirect
6. #21424916C2semidirect
7. #21424915C2semidirect
8. #21424906C2semidirect
9. C59×#1961C2semidirect
13. #16837950C4semidirect
47. C2×#11745659D4semidirect
50. #14273727C2×C4semidirect
52. #14273714C2×C4semidirect
75. #127×C59C2×C4semidirect
199. #1961D59semidirect
234. C59#21201semidirect
255. #127C4×D59semidirect

Maximal subgroups

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

subgroupindex
C2×#168379502
C2×#168374002
C2×#168373542
#382×D592
#214249212
#214249162
#214249152
#214249062
C59×#19612
#2120159