Finite Group Databasebeta

← #28559155

Subgroups — #28559155

Order 15104.

606 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21434104C211yesmaximal, characteristic
2#21433190C211yesmaximal, characteristic, complemented
3#21422998C212yesmaximal, complemented
4#21422984C212yesmaximal, complemented
5#21422972C212yesmaximal, complemented
6#21422937C212yesmaximal, complemented
7#21422874C211yesmaximal, characteristic, complemented
8#21422539C211yesmaximal, characteristic
9#21422516C211yesmaximal, characteristic, complemented
10#21422391C211yesmaximal, characteristic, complemented
11C59×#2975C211yesmaximal, characteristic
12#16837959C2211yescharacteristic
13#16837732C412yes—
14#16837692C412yes—
15#16837607C412yes—
16#16837553C412yes—
17#16837532C2212yescomplemented
18#16837531—22no—
19C8×Dic118C2212yescomplemented
20C4×#11745670C2212yes—
21C4×#11745670C2212yes—
22#16837460—22no—
23#16837459C2211yescharacteristic
24#16837455C2211yescharacteristic
25C4×#11745664C2211yescharacteristic
26#16837154C2212yescomplemented
27#16837152C2212yescomplemented
28#16837132C2212yescomplemented
29#16837132C2212yescomplemented
30D4⋊(C59⋊C8)C2212yes—
31D4⋊(C59⋊C8)C2212yes—
32#16837124C2211yescharacteristic, complemented
33#16837124C2211yescharacteristic, complemented
34#16837123C2211yescharacteristic, complemented
35C8×C59⋊C8C2211yescharacteristic, complemented
36C2×C4×D4×C59C2211yescharacteristic
37C59×#436—22no—
38C59×#436C2212yes—
39C59×#436C2212yes—
40C59×#431C2211yescharacteristic
41C59×#431C2211yescharacteristic
42C59×#407—22no—
43C2×#11745670C2×C412yes—
44C2×#11745670C2×C412yes—
45C2×#11745664—21no—
46#14273727C2×C411yescharacteristic
47#14273727C2×C411yescharacteristic
48#14273719—21no—
49#14273706—21no—
50#14273706C2×C412yes—
51C4×Dic118—21no—
52C4×Dic118C2312yes—
53#14273686—21no—
54#14273658C2×C412yes—
55#14273658C2×C412yes—
56C4⋊(C59⋊C8)C2×C412yes—
57C4⋊(C59⋊C8)C2312yes—
58Dic59⋊C8—22no—
59Dic59⋊C8—22no—
60#14273650C2×C412yes—
61#14273650C2×C412yes—
62#14273648C2×C412yes—
63#14273648C2×C412yes—
64C22⋊(C59⋊C8)—21no—
65C22⋊(C59⋊C8)C2×C412yes—
66#14273638—21no—
67#14273638—21no—
68#14273637—22no—
69C4×C59⋊C8C2311yescharacteristic
70C4×C59⋊C8C2311yescharacteristic
71C4×C59⋊C8C2311yescharacteristic
72C4×C59⋊C8C2311yescharacteristic
73C8×Dic59—22no—
74C22×D4×C59C2×C411yescharacteristic
75C23×C4×C59—21no—
76C2×M16×C59—22no—
77C2×M16×C59—22no—
78C4×D4×C59—21no—
79C4×D4×C59—21no—
80C4×D4×C59C2×C412yes—
81C4×D4×C59C2312yes—
82C2×C4⋊C4×C59C2×C411yescharacteristic
83C2×C22⋊C4×C59—21no—
84C2×C42×C59C2311yescharacteristic
85C4⋊C8×C59D412yes—
86C4⋊C8×C59C2312yes—
87C22⋊C8×C59—21no—
88C22⋊C8×C59—22no—
89C22⋊C8×C59—22no—
90C22⋊C8×C59D412yes—
91C4×C8×C59—21no—
92C4×C8×C59C2311yescharacteristic
93C4×C8×C59C2311yescharacteristic
94C24×C59—21no—
95C2×D4×C59—21no—
96C2×D4×C59—42no—
97C2×D4×C59C22×C412yes—
98C22×C4×C59—21no—
99C22×C4×C59—22no—
100C22×C4×C59—22no—
101C22×C4×C59—82no—
102C22×C4×C59C22×C411yescharacteristic
103C22×C4×C59C2×D411yescharacteristic
104C22×C4×C59#4111yescharacteristic
105C2×Dic118—21no—
106C2×Dic118C22×C412yes—
107M16×C59—82no—
108M16×C59—82no—
109C2×C8×C59—22no—
110C2×C8×C59C2×D412yes—
111C2×C8×C59C2×D412yes—
112C4⋊C4×C59—21no—
113C4⋊C4×C59C22×C412yes—
114C22⋊C4×C59—22no—
115C22⋊C4×C59—22no—
116C42×C59C22×C411yescharacteristic
117C42×C59C2×D411yescharacteristic
118C42×C59#4111yescharacteristic
119C42×C59C2411yescharacteristic
120C22⋊Dic59—22no—
121#11745670—42no—
122#11745670—42no—
123C4⋊Dic59—21no—
124C4⋊Dic59—21no—
125C4⋊Dic59—21no—
126C4⋊Dic59C22×C412yes—
127Dic59⋊C4—22no—
128Dic59⋊C4—22no—
129C4×Dic59—22no—
130#11745664—42no—
131C2×C59⋊C8—21no—
132C2×C59⋊C8—22no—
133C2×C59⋊C8—21no—
134C2×C59⋊C8C22×C412yes—
135C2×C59⋊C8C22×C412yes—
136C23×C59—22no—
137C23×C59—42no—
138C23×C59—81no—
139C23×C59—81no—
140C23×C59C4×D411yescharacteristic
141D4×C59—22no—
142D4×C59—22no—
143D4×C59—42no—
144C2×C4×C59—22no—
145C2×C4×C59—22no—
146C2×C4×C59—22no—
147C2×C4×C59—41no—
148C2×C4×C59—42no—
149C2×C4×C59—42no—
150C2×C4×C59—82no—
151C2×C4×C59—81no—
152C2×C4×C59C4×D411yescharacteristic
153C2×C4×C59C4×D411yescharacteristic
154C2×C4×C59C4×D411yescharacteristic
155C2×C4×C59C23×C411yescharacteristic
156C2×C4×C59C22×D411yescharacteristic
157C2×C4×C59C2×#4111yescharacteristic
158C2×Dic59—22no—
159C2×Dic59—22no—
160Dic118—22no—
161Dic118—22no—
162Dic118—81no—
163C8×C59—22no—
164C8×C59—22no—
165C8×C59—42no—
166C59⋊C8—22no—
167C59⋊C8—22no—
168C59⋊C8—22no—
169C59⋊C8—41no—
170C59⋊C8—42no—
171#17952—591nomaximal, Sylow
172C22×C59—21no—
173C22×C59—21no—
174C22×C59—21no—
175C22×C59—42no—
176C22×C59—42no—
177C22×C59—42no—
178C22×C59—84no—
179C22×C59C2×C4×D411yescharacteristic
180C4×C59—21no—
181C4×C59—21no—
182C4×C59—21no—
183C4×C59—21no—
184C4×C59—42no—
185C4×C59#56812yes—
186C4×C59#57712yes—
187Dic59—42no—
188Dic59—42no—
189#3101—591no—
190#2985—591no—
191#2975D5911yescharacteristic
192#1592—592no—
193#1590—592no—
194#1588—592no—
195#1583—592no—
196#1572—591no—
197#1490—591no—
198#1485—591no—
199#1453—591no—
200C2×C59—41no—
201C2×C59—82no—
202C2×C59#293011yescharacteristic
203C2×C59#294911yescharacteristic
204C2×C59#295611yescharacteristic
205#533—591no—
206C2×C4×D4C2×D5911yescharacteristic
207#483—592no—
208#476—592no—
209#461—592no—
210#450—592no—
211#446—592no—
212#446—1182no—
213C8×Q8—592no—
214C4×SD16—592no—
215C4×SD16—592no—
216#436—22no—
217#436C2×D5912yes—
218#436C2×D5912yes—
219#433—1182no—
220#432—591no—
221#431—591no—
222#431C2×D5911yescharacteristic
223#431C2×D5911yescharacteristic
224#407—22no—
225C4×M16—591no—
226#335—592no—
227#334—592no—
228Q8⋊C8—592no—
229Q8⋊C8—592no—
230D4⋊C8—592no—
231D4⋊C8—592no—
232#322—591no—
233#322—591no—
234#322—591no—
235C82—591no—
236C59#1795211yescharacteristic, complemented, Sylow
237C22×D4C4×D5911yescharacteristic
238C23×C4—21no—
239C2×SD16—592no—
240C2×SD16—592no—
241C2×M16—22no—
242C2×M16—22no—
243C2×M16—1181no—
244C4⋊Q8—591no—
245C4⋊Q8—591no—
246#125—1181no—
247C22⋊Q8—592no—
248C22⋊Q8—1181no—
249C4×Q8—592no—
250C4×Q8—1181no—
251C4×D4—21no—
252C4×D4—21no—
253C4×D4C4×D5912yes—
254C4×D4C22×D5912yes—
255#117—1181no—
256C2×C4⋊C4C4×D5911yescharacteristic
257C2×C22⋊C4—21no—
258C2×C42C22×D5911yescharacteristic
259#106—592no—
260#106—592no—
261C4⋊C8—592no—
262C4⋊C8—592no—
263C4⋊C8—1182no—
264C4⋊C8—1182no—
265C4⋊C8C22⋊D5912yes—
266C4⋊C8C22×D5912yes—
267#103—592no—
268#103—592no—
269#102—592no—
270#102—592no—
271C22⋊C8—21no—
272C22⋊C8—22no—
273C22⋊C8—22no—
274C22⋊C8—592no—
275C22⋊C8—1181no—
276C22⋊C8C22⋊D5912yes—
277#97—1182no—
278#97—1181no—
279#97—1181no—
280C4×C8—21no—
281C4×C8—591no—
282C4×C8—591no—
283C4×C8—591no—
284C4×C8—591no—
285C4×C8—1182no—
286C4×C8C22×D5911yescharacteristic
287C4×C8C22×D5911yescharacteristic
288C24—21no—
289C2×Q8—592no—
290C2×Q8—1181no—
291C2×D4—21no—
292C2×D4—42no—
293C2×D4C2×C4×D5912yes—
294C22×C4—21no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—82no—
298C22×C4C2×C4×D5911yescharacteristic
299C22×C4#1174568511yescharacteristic
300C22×C4C2×C22⋊D5911yescharacteristic
301SD16—2362no—
302SD16—2362no—
303M16—82no—
304M16—82no—
305M16—2362no—
306C2×C8—22no—
307C2×C8—592no—
308C2×C8—592no—
309C2×C8—1181no—
310C2×C8—1182no—
311C2×C8—1181no—
312C2×C8C2×C22⋊D5912yes—
313C2×C8C2×C22⋊D5912yes—
314C4⋊C4—21no—
315C4⋊C4—592no—
316C4⋊C4—1181no—
317C4⋊C4—1181no—
318C4⋊C4—1182no—
319C4⋊C4—1181no—
320C4⋊C4—1182no—
321C4⋊C4C2×C4×D5912yes—
322C22⋊C4—22no—
323C22⋊C4—22no—
324C22⋊C4—1182no—
325C42—1182no—
326C42C2×C4×D5911yescharacteristic
327C42#1174568511yescharacteristic
328C42C2×C22⋊D5911yescharacteristic
329C42C23×D5911yescharacteristic
330C23—22no—
331C23—42no—
332C23—81no—
333C23—81no—
334C23C4×C22⋊D5911yescharacteristic
335Q8—1182no—
336Q8—1182no—
337Q8—4721no—
338D4—22no—
339D4—22no—
340D4—42no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—41no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—82no—
348C2×C4—81no—
349C2×C4—1182no—
350C2×C4—1182no—
351C2×C4C4×C22⋊D5911yescharacteristic
352C2×C4C4×C22⋊D5911yescharacteristic
353C2×C4C4×C22⋊D5911yescharacteristic
354C2×C4C22×C4×D5911yescharacteristic
355C2×C4C22×C22⋊D5911yescharacteristic
356C2×C4C2×#1174568511yescharacteristic
357C8—22no—
358C8—22no—
359C8—42no—
360C8—1182no—
361C8—1182no—
362C8—1182no—
363C8—2361no—
364C8—2362no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—42no—
369C22—42no—
370C22—42no—
371C22—84no—
372C22C2×C4×C22⋊D5911yescharacteristic
373C4—21no—
374C4—21no—
375C4—21no—
376C4—21no—
377C4—42no—
378C4—2362no—
379C4—2362no—
380C4#1683816812yes—
381C4#1683821312yes—
382C2—41no—
383C2—82no—
384C2#2143279911yescharacteristic
385C2#2143295311yescharacteristic
386C2#2143300311yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D59
commutatorC2×C4×C59C23×C4
FrattiniC42C23×D59
FittingC59×#2975C2
socleC22×C59C2×C4×D4
prime radicalC22×D4×C59C2×C4
cosocleC42×C59C24
hypercenter#2975D59
hypocenterC59#17952
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ #431 ◁ #2975

Complements

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

NH ≅ G/Ntype
2. #21433190C2semidirect
3. #21422998C2semidirect
4. #21422984C2semidirect
5. #21422972C2semidirect
6. #21422937C2semidirect
7. #21422874C2semidirect
9. #21422516C2semidirect
10. #21422391C2semidirect
17. #16837532C22semidirect
19. C8×Dic118C22semidirect
26. #16837154C22semidirect
27. #16837152C22semidirect
28. #16837132C22semidirect
29. #16837132C22semidirect
32. #16837124C22semidirect
33. #16837124C22semidirect
34. #16837123C22semidirect
35. C8×C59⋊C8C22semidirect
236. C59#17952semidirect

Maximal subgroups

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

subgroupindex
#214341042
#214331902
#214229982
#214229842
#214229722
#214229372
#214228742
#214225392
#214225162
#214223912
C59×#29752
#1795259