Finite Group Databasebeta

← #28559361

Subgroups — #28559361

Order 15104.

600 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21434651C211yesmaximal, characteristic
2#21433193C211yesmaximal, characteristic, complemented
3#21433172C211yesmaximal, characteristic
4#21422995C212yesmaximal, complemented
5#21422989C212yesmaximal, complemented
6#21422905C212yesmaximal, complemented
7#21422899C212yesmaximal, complemented
8#21422857C211yesmaximal, characteristic, complemented
9#21422570C211yesmaximal, characteristic, complemented
10#21422380C211yesmaximal, characteristic, complemented
11C59×#1496C211yesmaximal, characteristic
12#16837959C2211yescharacteristic
13#16837798C411yescharacteristic
14#16837774C411yescharacteristic
15#16837754C411yescharacteristic
16#16837748C411yescharacteristic
17#16837629C412yes—
18#16837629C412yes—
19#16837530—22no—
20#16837529C2212yes—
21#16837526C2212yes—
22C4×#11745670C2212yes—
23C4×#11745670C2212yes—
24#16837480C2212yes—
25#16837474—22no—
26D4×C59⋊C8C2212yes—
27#16837460—22no—
28#16837459C2211yescharacteristic
29#16837455C2211yescharacteristic
30#16837369—22no—
31C2×C4×C59⋊C8C2211yescharacteristic
32#16837154C2212yes—
33#16837152C2212yes—
34#16837134C2212yes—
35#16837134C2212yes—
36#16837124C2211yescharacteristic
37#16837124C2211yescharacteristic
38C8×C59⋊C8C2211yescharacteristic
39C8×C59⋊C8C2211yescharacteristic
40C59×#532C2211yescharacteristic
41C59×#432C2211yescharacteristic
42C59×#431C2211yescharacteristic
43C59×D4⋊C8C2212yes—
44C59×D4⋊C8C2212yes—
45C2×#11745670C2×C412yes—
46C2×#11745670C2×C412yes—
47C2×#11745664—22no—
48C22×C59⋊C8—22no—
49C22×C59⋊C8C2×C411yescharacteristic
50C22×C59⋊C8C2×C411yescharacteristic
51#14273727C2×C411yescharacteristic
52#14273727C2×C411yescharacteristic
53#14273719—21no—
54#14273706—21no—
55#14273706C2×C412yes—
56C4×Dic118—21no—
57C4×Dic118C2312yes—
58#14273686—21no—
59#14273658C2×C412yes—
60#14273658C2×C412yes—
61C4⋊(C59⋊C8)D412yes—
62C4⋊(C59⋊C8)C2312yes—
63Dic59⋊C8—22no—
64Dic59⋊C8—22no—
65#14273650C2×C412yes—
66#14273650C2×C412yes—
67#14273648C2×C412yes—
68#14273648C2×C412yes—
69C22⋊(C59⋊C8)—21no—
70C22⋊(C59⋊C8)—22no—
71C22⋊(C59⋊C8)—22no—
72C22⋊(C59⋊C8)D412yes—
73#14273638—21no—
74#14273637—22no—
75C4×C59⋊C8C2×C411yescharacteristic
76C4×C59⋊C8C2×C411yescharacteristic
77C4×C59⋊C8C2311yescharacteristic
78C4×C59⋊C8C2311yescharacteristic
79C4×C59⋊C8C2311yescharacteristic
80C4×C59⋊C8C2311yescharacteristic
81C8×Dic59—22no—
82C2×#41×C59—21no—
83C4⋊Q8×C59C2×C411yescharacteristic
84#127×C59C2×C411yescharacteristic
85#124×C59—21no—
86#121×C59—21no—
87#121×C59C2×C412yes—
88C4×D4×C59—21no—
89C4×D4×C59C2312yes—
90C2×C42×C59C2311yescharacteristic
91C4⋊C8×C59—21no—
92C4⋊C8×C59C2312yes—
93C22⋊C8×C59—21no—
94C22⋊C8×C59—21no—
95#97×C59—21no—
96C4×C8×C59C2311yescharacteristic
97C4×C8×C59C2311yescharacteristic
98#41×C59—82no—
99C2×Q8×C59—21no—
100C2×D4×C59—22no—
101C2×D4×C59—21no—
102C2×D4×C59—21no—
103C2×D4×C59C22×C412yes—
104C22×C4×C59—22no—
105C22×C4×C59C22×C411yescharacteristic
106C22×C4×C59C2×D411yescharacteristic
107C22×C4×C59#4111yescharacteristic
108C2×Dic118—21no—
109C2×Dic118C22×C412yes—
110C2×C8×C59—22no—
111C2×C8×C59—21no—
112C2×C8×C59—21no—
113C4⋊C4×C59—21no—
114C4⋊C4×C59C22×C412yes—
115C22⋊C4×C59—22no—
116C22⋊C4×C59—22no—
117C42×C59C22×C411yescharacteristic
118C42×C59C2×D411yescharacteristic
119C42×C59#4111yescharacteristic
120C42×C59C2411yescharacteristic
121C22⋊Dic59—22no—
122#11745670—24no—
123#11745670—24no—
124C4⋊Dic59—21no—
125C4⋊Dic59—21no—
126C4⋊Dic59—21no—
127C4⋊Dic59C22×C412yes—
128Dic59⋊C4—22no—
129Dic59⋊C4—22no—
130C4×Dic59—22no—
131#11745664—82no—
132C2×C59⋊C8—22no—
133C2×C59⋊C8—22no—
134C2×C59⋊C8—22no—
135C2×C59⋊C8—82no—
136C2×C59⋊C8C22×C411yescharacteristic
137C2×C59⋊C8C22×C411yescharacteristic
138C2×C59⋊C8C22×C411yescharacteristic
139C2×C59⋊C8C22×C411yescharacteristic
140C2×C59⋊C8C2×D412yes—
141C2×C59⋊C8C2×D412yes—
142C23×C59—22no—
143C23×C59C4×D411yescharacteristic
144Q8×C59—81no—
145D4×C59—22no—
146D4×C59—22no—
147D4×C59—81no—
148D4×C59—81no—
149D4×C59—82no—
150C2×C4×C59—21no—
151C2×C4×C59—21no—
152C2×C4×C59—22no—
153C2×C4×C59—22no—
154C2×C4×C59—41no—
155C2×C4×C59—42no—
156C2×C4×C59—42no—
157C2×C4×C59C4×D411yescharacteristic
158C2×C4×C59C4×D411yescharacteristic
159C2×C4×C59C4×D411yescharacteristic
160C2×C4×C59C23×C411yescharacteristic
161C2×C4×C59C22×D411yescharacteristic
162C2×C4×C59C2×#4111yescharacteristic
163C2×Dic59—22no—
164C2×Dic59—22no—
165Dic118—22no—
166Dic118—22no—
167Dic118—81no—
168C8×C59—41no—
169C8×C59—41no—
170C8×C59—42no—
171C59⋊C8—21no—
172C59⋊C8—21no—
173C59⋊C8—22no—
174C59⋊C8—22no—
175C59⋊C8—42no—
176C59⋊C8#13112yes—
177C59⋊C8#13112yes—
178#17971—591nomaximal, Sylow
179C22×C59—21no—
180C22×C59—21no—
181C22×C59—21no—
182C22×C59—42no—
183C22×C59—42no—
184C22×C59—42no—
185C22×C59C2×C4×D411yescharacteristic
186C4×C59—21no—
187C4×C59—21no—
188C4×C59—21no—
189C4×C59—21no—
190C4×C59—42no—
191C4×C59C2×#13111yescharacteristic
192C4×C59C2×#13111yescharacteristic
193C4×C59#57712yes—
194Dic59—42no—
195Dic59—42no—
196#3167—591no—
197#2985—591no—
198#2981—591no—
199#1592—592no—
200#1590—592no—
201SD16⋊C8—592no—
202C8×SD16—592no—
203#1568—591no—
204#1497—591no—
205#1496D5911yescharacteristic
206#1450—591no—
207C2×C59—41no—
208C2×C59—82no—
209C2×C59#293111yescharacteristic
210C2×C59#294911yescharacteristic
211C2×C59#295711yescharacteristic
212#533—591no—
213#532C2×D5911yescharacteristic
214#498—591no—
215#492—591no—
216#487—591no—
217#486—591no—
218#465—592no—
219#465—592no—
220#446—592no—
221#446—1182no—
222C8×Q8—592no—
223C4×SD16—592no—
224C4×SD16—592no—
225#436—592no—
226#435—1182no—
227C8×D4—592no—
228#433—1182no—
229#432—591no—
230#432C2×D5911yescharacteristic
231#431—591no—
232#431C2×D5911yescharacteristic
233#408—1182no—
234C2×C4×C8—591no—
235#335—592no—
236#334—592no—
237Q8⋊C8—592no—
238Q8⋊C8—592no—
239D4⋊C8C2×D5912yes—
240D4⋊C8C2×D5912yes—
241#322—591no—
242#322—591no—
243C82—591no—
244C82—591no—
245C59#1797111yescharacteristic, complemented, Sylow
246C2×#41—21no—
247C2×SD16—592no—
248C2×SD16—592no—
249C2×M16—1182no—
250C22×C8—591no—
251C22×C8—591no—
252C22×C8—1182no—
253C4⋊Q8—591no—
254C4⋊Q8—591no—
255C4⋊Q8C4×D5911yescharacteristic
256#127C4×D5911yescharacteristic
257#125—1181no—
258#124—21no—
259C22⋊Q8—592no—
260C22⋊Q8—1181no—
261#121—21no—
262#121C4×D5912yes—
263C4×Q8—592no—
264C4×Q8—1181no—
265C4×D4—21no—
266C4×D4C22×D5912yes—
267#117—1181no—
268C2×C42C22×D5911yescharacteristic
269#106—592no—
270#106—592no—
271C4⋊C8—21no—
272C4⋊C8—592no—
273C4⋊C8—592no—
274C4⋊C8—1182no—
275C4⋊C8—1182no—
276C4⋊C8C22×D5912yes—
277#103—592no—
278#103—592no—
279#102—592no—
280#102—592no—
281C22⋊C8—21no—
282C22⋊C8—21no—
283C22⋊C8—592no—
284C22⋊C8—1182no—
285C22⋊C8—1182no—
286C22⋊C8—1181no—
287#97—21no—
288#97—1182no—
289#97—1181no—
290C4×C8—591no—
291C4×C8—591no—
292C4×C8—591no—
293C4×C8—591no—
294C4×C8—591no—
295C4×C8—591no—
296C4×C8—1182no—
297C4×C8C22×D5911yescharacteristic
298C4×C8C22×D5911yescharacteristic
299#41—82no—
300C2×Q8—21no—
301C2×Q8—592no—
302C2×Q8—1181no—
303C2×D4—22no—
304C2×D4—21no—
305C2×D4—21no—
306C2×D4C2×C4×D5912yes—
307C22×C4—22no—
308C22×C4C2×C4×D5911yescharacteristic
309C22×C4D4×D5911yescharacteristic
310C22×C4#1174568711yescharacteristic
311SD16—1184no—
312SD16—1184no—
313M16—4722no—
314C2×C8—22no—
315C2×C8—21no—
316C2×C8—21no—
317C2×C8—591no—
318C2×C8—591no—
319C2×C8—591no—
320C2×C8—591no—
321C2×C8—592no—
322C2×C8—592no—
323C2×C8—1182no—
324C2×C8—1182no—
325C2×C8—1182no—
326C2×C8—4722no—
327C4⋊C4—21no—
328C4⋊C4—592no—
329C4⋊C4—1181no—
330C4⋊C4—1181no—
331C4⋊C4—1182no—
332C4⋊C4—1181no—
333C4⋊C4—1182no—
334C4⋊C4C2×C4×D5912yes—
335C22⋊C4—22no—
336C22⋊C4—22no—
337C22⋊C4—1182no—
338C42—1182no—
339C42C2×C4×D5911yescharacteristic
340C42D4×D5911yescharacteristic
341C42#1174568711yescharacteristic
342C42C23×D5911yescharacteristic
343C23—22no—
344C23(C22⋊D59)⋊C411yescharacteristic
345Q8—81no—
346Q8—1182no—
347Q8—1182no—
348Q8—4721no—
349D4—22no—
350D4—22no—
351D4—81no—
352D4—81no—
353D4—82no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—41no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—1182no—
362C2×C4—1182no—
363C2×C4(C22⋊D59)⋊C411yescharacteristic
364C2×C4(C22⋊D59)⋊C411yescharacteristic
365C2×C4(C22⋊D59)⋊C411yescharacteristic
366C2×C4C22×C4×D5911yescharacteristic
367C2×C4C2×D4×D5911yescharacteristic
368C2×C4C2×#1174568711yescharacteristic
369C8—41no—
370C8—41no—
371C8—42no—
372C8—592no—
373C8—592no—
374C8—1181no—
375C8—1181no—
376C8—1182no—
377C8—1182no—
378C8—2362no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—42no—
383C22—42no—
384C22—42no—
385C22C2×(C22⋊D59)⋊C411yescharacteristic
386C4—21no—
387C4—21no—
388C4—21no—
389C4—21no—
390C4—42no—
391C4—2362no—
392C4—2362no—
393C4#131×D5911yescharacteristic
394C4#131×D5911yescharacteristic
395C4#1683821212yes—
396C2—41no—
397C2—82no—
398C2#2143280411yescharacteristic
399C2#2143295811yescharacteristic
400C2#2143300911yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC42C23×D59
FittingC59×#1496C2
socleC22×C59C2×C4×D4
prime radicalC59×#532C22
cosocleC42×C59C24
hypercenter#1496D59
hypocenterC59#17971
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1496

Complements

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

NH ≅ G/Ntype
2. #21433193C2semidirect
4. #21422995C2semidirect
5. #21422989C2semidirect
6. #21422905C2semidirect
7. #21422899C2semidirect
8. #21422857C2semidirect
9. #21422570C2semidirect
10. #21422380C2semidirect
245. C59#17971semidirect

Maximal subgroups

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

subgroupindex
#214346512
#214331932
#214331722
#214229952
#214229892
#214229052
#214228992
#214228572
#214225702
#214223802
C59×#14962
#1797159